Project Economics And Decision Analys (volume 2).pdf 3i105h

Project Economics and

Decision Analysis

Volume 2 Probabilistic Models

Disclaimer: The recommendations, advice, descriptions, and the methods in this book are presented solely for educational purposes. The author and publisher assume no liability whatsoever for any loss or damage that results from the use of any of the material in this book. Use of the material in this book is solely at the risk of the .

Copyright© 2002 by PennWell Corporation 1421 South Sheridan Road Tulsa, Oklahoma 74112-6600 USA 800.752.9764 +1.918.831.9421 [email protected] www.pennwellbooks.com www.pennwell.com Marketing Manager: Julie Simmons National Executive: Barbara McGee Director: Mary McGee Managing Editor: Marla M. Patterson Production/Operations Manager: Traci Huntsman Cover Designer: Clark Bell Book Designer: Amy Spehar Library of Congress Catag-in-Publication Data Available on Request Mian, M. A. Project Economics and Decision Analysis, Volume II: Probabilistic Models ISBN 978-0-87814-855-4 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed in any form or by any means, electronic or mechanical, including photocopying and recording, without the prior written permission of the publisher. Printed in the United States of America 6 7 8 9 10

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PREFACE The international oil and gas industry remains one of the most important, highly capital-intensive and risky industries at global, regional, and local levels. Capital and exploration spending in this sector average in excess of some $40 billion/year (Oil & Gas Journal {OGJ}: June 10, 1996). Our profit margins are under real pressure from many factors, including the higher costs of developing new reserves, less oil found per foot drilled, rising inflationary costs of doing business, oversupply of crude, crude oil price volatility, competition for oil company investments, competition for acreage/concessions, competition for funds, and the overall business risk and uncertainty put our profit margins under real pressure. Therefore, it is very crucial to carry out prudent economic evaluations of any capital investment’s commitment before resources are committed. This, of course, requires a thorough understanding of the techniques available and their application by all those involved in decision-making. To assist with in achieving this goal, the industry deserves a comprehensive guide that will provide all the necessary concepts of capital investment evaluation, capital budgeting, and decision analysis. This two-book series, Project Economics and Decision Analysis, Volumes I and II, hopefully meets this requirement.

Objective One goal in writing this book has been to provide the students, practicing engineers, geologists, economists, planners, and managers with a solid foundation in the dynamic and growing field of capital investment evaluation with emphasis on the uncertainty aspect. It describes how investment decisions are currently made under different stages of uncertainty, and prescribes techniques that will assist in making

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rational decisions. Therefore, the book is a compilation of knowledge about the prescriptive styled application of the paradigm of decision analysis to capital investment problems. This two-volume set describes the philosophy, process, and methods of capital investment evaluation and decision analysis. In summary, the main objectives of these books are to: • • • • • •

Explain the ever-expanding role of economics in prudent capital investment decision-making. Assist readers in developing a knowledgeable vocabulary of the associated with economic analysis. Review the procedures used in preparing capital investment evaluations and decision analysis. Relate the new vocabulary and knowledge to some specific problems. Provide ways of interpreting estimates that include uncertainty (i.e., converting probabilistic description into a measure of profitability). Assist in achieving solid hands-on experience with capital investment evaluation and decision analysis.

Emphasis and Style The book presents a balanced blend of theoretical concepts and their practical utility. I prefer to focus less (compared to other books) on providing extensive theoretical discussions. Theory, I feel, distracts the reader from the most important concepts and their practical application. Moreover, theory will seem sterile and pointless unless its usefulness is made clear. Therefore, I have focused more on the practical application. The underlying concepts are stressed, and made concrete with liberal use of illustrations, many of them taken from actual real-life capital investment evaluations. Algebraic formulations and spreadsheets are used side-byside to help readers develop conceptual thinking skills. Emphasis is placed on model formulation and interpretation rather than algorithms.

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The technical materials have been developed with considerable patience—assuming very little foreknowledge and proceeding in a stepby-step manner in order to minimize the possibility of the reader getting lost along the way. Moreover, I have resorted to a greater degree of informality in the presentation in the belief that readability should be an overriding consideration. Toward the same goal, intuitive and economic explanations are frequently provided to clarify the why of a particular concept/technique. This book is primarily intended for use by economists, earth scientists, engineers, and students. It also is intended to serve as a refresher, and perhaps as a self-study textbook. The problem-solving approach is instructive in nature, but the foundational principles show the practical application of the material. Its chief purpose is two-fold: (1) to render a systematic exposition of certain basic deterministic investment evaluation methods, and (2) to relate these to the decision analysis so the mutual relevance of the two is clearly brought out. Therefore, the book is literally divided into two separate, yet complementary, volumes. This book, Volume II deals with the concepts of decision analysis (incorporating risk and uncertainty as applied to capital investments). Generally, each topic is introduced by a practical or conceptual overview, followed by a brief discussion related to its application in practice and a solved example. Volume I is essentially introductory and deals with the deterministic evaluation tools used for capital investment evaluations. These concepts are seldom covered, as broadly or from the same viewpoints in economics and other courses, yet they are fundamental to the proper understanding of all evaluation work. For optimum benefit, it is recommended readers explore both volumes and benefit from their integrated instruction. Additionally, a CD is included with this volume and provides helpful software, spreadsheets, and tables to enhance the practical application of this material.

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Examples and Assignment Problems Included in these two volumes are an abundance of solved real-life examples (some 100+) and end-of-chapter assignment material (some 200+ questions and problems) are included in the book. Examples are presented to reinforce the reader’s learning process. Each solved example is straightforward, fully explained, and avoids sudden leaps to conclusions that might confuse the reader. The assignment material is divided into questions and problems. The questions primarily address key concepts and in the chapter. The problems either consolidate a number of chapter topics or else focus on a comprehensive analysis of a single topic. As always, careful choices among the wide variety of assignment material will offer practical knowledge since the assignments include various combinations of breadth and depth, theory and procedures, simplicity and complexity. For maximum benefit, the reader should work out as many of these problems as possible, if indeed not all of them.

Spreadsheet Applications An additional unique feature of this book is embedded application of computers in solving investment evaluation and decision analysis problems. Rapid advancements in computer hardware and software are revolutionizing our working environment. Powerful computers are now available at affordable prices, and new state-of-the-art software makes it easy to do things much faster and more efficiently. Much of this was not feasible several years ago. Today, most of us are using spreadsheet programs to build models of the decision problems we face as a regular part of our daily work activities. Because of this, it makes sense to use spreadsheets as the vehicle for introducing undergraduate and graduate students, dealing with investment evaluation and decision analysis, to the concepts and tools covered in this area. Spreadsheets also capture s’ interest and add a new relevance to investment evaluation and decision analysis.

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Examples are provided to show how computers can be used to help make better evaluations and, hence, better decisions. The Microsoft Excel spreadsheet is making it increasingly easy and practical to do sensitivity and scenario analyses. Its use has gained acceptance in the industry and makes it feasible to do a variety of analysis of with a multitude of problems. The latest version of DecisionTools™ Suite (an Excel add-in) by Palisade Corp., including the award winning @RISK, PrecisionTree, BestFit, TopRank, and RISKView is used where applicable. Screen captures of the various menus of DecisionTools Suite are used. A copy of the software is included on the CD-ROM accompanying this volume. The CD-ROM also includes all the tables and spreadsheets incorporated in the book. This greatly enhances the utility of the book and permits the quantification of project evaluation and risk analysis in a practical manner.

Request for Suggestions Considerable amount of dedication and investment (time and capital) goes into writing and publishing such a book. PennWell Books and I have made every effort to introduce this two-volume book as the most comprehensive stand-alone desk reference. However, we sincerely welcome your thoughts as an end- to help us further improve the contents, presentation, and utility of this book so as to make it a standard for the new generation of petroleum industry personnel. The author and publisher believe this book to be substantially error-free. For any errors remaining, I will always be very grateful for your comments, suggestions, or corrections sent to me directly or through PennWell books.

Acknowledgements Many people are involved in the successful publication of a book. I wish to thank the following for making significant contributions to this book. Without their assistance, this project could not have been possible.

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All the people at PennWell who contributed their efforts to the production of this book, especially Editorial Manager Marla Patterson, and the production, and editing staff, as well as Sue Rhodes Dodd, my editor and president of Amethyst Enterprises in Tulsa, Oklahoma. Mr. Randy Heffernan of Palisade Corporation for sending me a copy of the latest version of their DecisionTools™ Suite and allowing me to include this tool on the CD-ROM accompanying this volume. This has significantly enhanced the utility of this book. I would like to thank James A. MacKay of Texaco, Houston, Texas, and Dr. Roger Eraj Ertefai (Petrochemical and Refining Group Leader) at Qatar Petroleum for reviewing some parts of the manuscript. Their comments were very valuable. I would like to thank my wife, three daughters, and son for their continued patience, , encouragement, love, and back rubs throughout this project. They contributed to this effort in ways that I probably will never know or understand. Special thanks to my daughters at the University of Colorado at Boulder for being on the Dean’s list; this has contributed a lot to my enthusiasm. Last but not least, I thank my parents for their continued and inspiration and for being together for more than 50 years.

M. A. Mian, P.E

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CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1.

INTRODUCTION Handling Uncertainty in Capital Investments . . . . . . . . . 4 Industry Practice Over the Years . . . . . . . . . . . . . . . . . . . 6 Classification of Decision Situations . . . . . . . . . . . . . . . . . 7 Terminology Used in Decision Analysis . . . . . . . . . . . . . . . 8 The Decision Analysis Cycle . . . . . . . . . . . . . . . . . . . . . . . 11 Advantages of Decision Analysis . . . . . . . . . . . . . . . . . . . 16 Misconceptions Regarding Decision Analysis . . . . . . . . . 17 Applications of Decision Analysis . . . . . . . . . . . . . . . . . . 19 Typical Industry Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.

STATISTICS AND PROBABILITY CONCEPTS Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . 30 Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . 31 Geometric, Harmonic, and Quadratic Mean . . . . . . 35 Weighted Average . . . . . . . . . . . . . . . . . . . . . . . . . 39 Choice of a Suitable Average . . . . . . . . . . . . . . . . . . . . 42 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . . . 44 Range, Variance, and Standard Deviation . . . . . . . . 45 Mean Absolute Deviation . . . . . . . . . . . . . . . . . . . . 46 Coefficient of Variation . . . . . . . . . . . . . . . . . . . . . 47

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Descriptive Statistics from Grouped Data . . . . . . . . . . . 50 Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . 52 Geometric and Harmonic Mean . . . . . . . . . . . . . . . 53 Variance and Standard Deviation . . . . . . . . . . . . . . 53 Graphical Presentation of Data . . . . . . . . . . . . . . . . . . . 56 Other Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Dociles and Percentiles . . . . . . . . . . . . . . . . . . . . . 64 Coefficient of Peakedness and Skewness . . . . . . . . 66 Spreadsheet Application . . . . . . . . . . . . . . . . . . . . . . . . 66 Understanding Probability Concepts . . . . . . . . . . . . . . . . 70 Objective Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . 72 Empirical Approach . . . . . . . . . . . . . . . . . . . . . . . . 75 Subjective Probability . . . . . . . . . . . . . . . . . . . . . . . . . 77 Modern Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Operations on Event Sets . . . . . . . . . . . . . . . . . . . . 81 Characteristics of Events . . . . . . . . . . . . . . . . . . . . 82 Rules of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Complementation Rule . . . . . . . . . . . . . . . . . . . . . 85 Addition Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Multiplication Rule . . . . . . . . . . . . . . . . . . . . . . . . 87 Probability Table and Probability Tree . . . . . . . . . . . . . 92 Baye’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Theoretical Probability Distributions . . . . . . . . . . . . . . . 99 Discrete Probability Distributions . . . . . . . . . . . . . . . . 101 Binomial Probability Distribution . . . . . . . . . . . . 101 Multinomial Probability Distribution . . . . . . . . . . 105 Hypergeometric Probability Distribution . . . . . . . 107 Poisson Probability Distribution . . . . . . . . . . . . . . 110 Continuous Probability Distributions . . . . . . . . . . . . . 113 Normal Probability Distribution . . . . . . . . . . . . . . 114 Lognormal Probability Distribution . . . . . . . . . . . 122 Uniform Probability Distribution . . . . . . . . . . . . . 126 Triangular Probability Distribution . . . . . . . . . . . 128 viii

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Spreadsheet Application . . . . . . . . . . . . . . . . . . . . . . . . . 132 Excel Built-in Functions . . . . . . . . . . . . . . . . . . . . . . 132 Introducing BestFit . . . . . . . . . . . . . . . . . . . . . . . 138 BestFit Add-in . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.

EXPECTED VALUE AND DECISION TREES Expected Value Concepts . . . . . . . . . . . . . . . . . . . . . . . . 153 Expected Value of Random Variable . . . . . . . . . . . . . . 154 Standard Deviation of Random Variable . . . . . . . . . . . 155 Expected Monetary Value (EMV) . . . . . . . . . . . . . . . 157 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Expected Profitability Index (EPI) . . . . . . . . . . . . . . . 166 Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Expected Opportunity Loss (EOL) . . . . . . . . . . . . . . . 172 Mean–Variance and Stochastic Dominance . . . . . . . . 174 Meaning and Interpretation of Expected Value . . . . . . 181 Characteristics of Expected Value . . . . . . . . . . . . . . . . 185 Value of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Expected Value of Perfect Information (EVPI) . . . . . . 187 Expected Value of Imperfect Information (EVII) . . . . 190 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Decision Tree Convention . . . . . . . . . . . . . . . . . . . . . 198 Guidelines for Deg Trees . . . . . . . . . . . . . . . . . 200 Solving a Decision Tree . . . . . . . . . . . . . . . . . . . . . . . 201 Constructing Risk Profiles . . . . . . . . . . . . . . . . . . . . . 207 Advantages of Decision Trees . . . . . . . . . . . . . . . . . . 208 Spreadsheet Applications . . . . . . . . . . . . . . . . . . . . . . . . 209 Excel Built-in Functions . . . . . . . . . . . . . . . . . . . . . . 209 Introducing PrecisionTree . . . . . . . . . . . . . . . . . . . . . 210 PrecisionTree Add-in . . . . . . . . . . . . . . . . . . . . . . 212 Constructing the Tree . . . . . . . . . . . . . . . . . . . . . . 213

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Risk Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 220 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . 227

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INCORPORATING ATTITUDES TOWARD RISK The Expected Utility Theory . . . . . . . . . . . . . . . . . . . . . 237 Typical Attitudes toward Risk . . . . . . . . . . . . . . . . . . . 237 The Axioms of Utility . . . . . . . . . . . . . . . . . . . . . . . . 241 Risk Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Certainty Equivalent and Risk . . . . . . . . . . . 246 Assessing the Utility Function . . . . . . . . . . . . . . . . . . 250 Mathematical Representation of Utility Functions . . . . . 255 Approximation to Certainty Equivalent . . . . . . . . . . . 262 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Expected Utility Decision Criteria . . . . . . . . . . . . . . . 268 Spreadsheet Applications . . . . . . . . . . . . . . . . . . . . . . . . 269 Fitting the Utility Curve . . . . . . . . . . . . . . . . . . . . . . . 269 Finding the Certainty Equivalent . . . . . . . . . . . . . . . . 273 Critical Risk Tolerance . . . . . . . . . . . . . . . . . . . . . . . . 274 PrecisionTree and Utility Functions . . . . . . . . . . . . . . 276 Questions and Problems . . . . . . . . . . . . . . . . . . . . . . . . . 280

5.

DETERMINING VENTURE PARTICIPATION Gambler’s Ruin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Working Interest and RAV . . . . . . . . . . . . . . . . . . . . . . . 289 Based on Exponential Risk Aversion . . . . . . . . . . . . . 289 Optimum Working Interest . . . . . . . . . . . . . . . . . . 290 Apparent Risk Tolerance . . . . . . . . . . . . . . . . . . . 290 Breakeven Working Interest . . . . . . . . . . . . . . . . . 291 Based on Hyperbolic Risk Aversion . . . . . . . . . . . . . . 293 Parabolic Approximation to Cozzolino’s RAV . . . . . . 298 Modified Risk-Adjusted Value . . . . . . . . . . . . . . . . . . 300

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Portfolio Balancing and RAV . . . . . . . . . . . . . . . . . . . . . 303 Portfolio Balancing—Cozzolino RAV Formula . . . . . 304 Portfolio Balancing—Parabolic RAV Formula . . . . . . 305 Spreadsheet Applications . . . . . . . . . . . . . . . . . . . . . . . . 307

6.

SIMULATION IN DECISION ANALYSIS Deg the Simulation Model . . . . . . . . . . . . . . . . . . 314 Real Applications of Simulation . . . . . . . . . . . . . . . . . 315 Steps in Simulation Modeling . . . . . . . . . . . . . . . . . . 316 Random Sampling Methods . . . . . . . . . . . . . . . . . . . . 322 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . 322 Monte Carlo Sampling . . . . . . . . . . . . . . . . . . . . . 324 Latin Hypercube Sampling . . . . . . . . . . . . . . . . . 326 Behind-The-Scene Calculations . . . . . . . . . . . . . . . . . . . 328 Recognizing Dependence on Input Variables . . . . . . . . . 333 Simulating Total Dependence . . . . . . . . . . . . . . . . . . . 338 Simulating Diffuse Dependence . . . . . . . . . . . . . . . . . 339 Spreadsheet Applications . . . . . . . . . . . . . . . . . . . . . . . . 347 Using Excel for Simulation . . . . . . . . . . . . . . . . . . . . 348 Generating Random Numbers in Excel . . . . . . . . 348 Probability Distributions in Excel . . . . . . . . . . . . 349 Introducing @RISK . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Loading @RISK . . . . . . . . . . . . . . . . . . . . . . . . . 354 Developing the Simulation Model . . . . . . . . . . . . 355 Running the Simulation . . . . . . . . . . . . . . . . . . . . 359 Analyzing the Output . . . . . . . . . . . . . . . . . . . . . . 361 Handling Dependency on @RISK . . . . . . . . . . . . 370 Combining PrecisionTree and @RISK . . . . . . . . . . . . . 373 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

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APPENDIX A Probability Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

APPENDIX B Contents of the Accompanying CD-ROM . . . . . . . . . . 395

INDEX

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chapter

ONE

Introduction

In Volume I of this series, it was assumed the results and implications of investment decision are known at the time of the decision, meaning the future is known and it is known with certainty. In reality, such conditions simply do not exist. In practice, different investment options have inherently different degrees of associated uncertainty and consequently different degrees of risk. The uncertainty and risk are formally defined in the following pages. However, the deterministic concepts presented in Volume I of this series (as will be shown in this chapter) form an integral part of decision analysis. The following main topics are presented in Volume I. 1. The time value of money concepts. 2. Before-tax and after-tax cash-flow models. 3. International petroleum economics (concessionary systems and production-sharing agreements). 4. Capital budgeting techniques. 5. Investment selection decision making. 1

Project Economics and Decision Analysis Volume II: Probabilistic Models

Uncertainty about the variables in the investment model complicates the decision-making process. Many important decisions have to be made without knowing exactly what will happen in the future or exactly what the ultimate outcome will be from a decision made today. This introductory chapter sets a stage, by presenting basic concepts and definitions, for decision analysis techniques presented in the following chapters. The purpose of decision analysis is to help a decision maker think systematically about complex problems and to improve the quality of the resulting decisions. In this regard, it is important to distinguish between a good decision and a lucky outcome. A good decision is one that is made based on a thorough understanding of the problem and careful thought regarding the important issues. Outcomes, on the other hand, may be unlucky, regardless of decision quality. Therefore, decision analysis allows making effective decisions more consistently.1 When faced with decision choices under uncertain conditions, informal analysis methods and quantitative techniques can be used to ascertain risk and uncertainty associated with the available investment options. Informal analysis involves a decision maker’s experience, intuition, judgment, and gut feeling. The quantitative analyses use a logical and consistent decisions strategy by incorporating the effects of risk and uncertainty into the analysis results. Several different approaches are available to quantitatively incorporate risk and uncertainty into investment analysis.2 The worldwide oil and gas industry applies advanced techniques of investment analysis. Because of the dynamics of high risk, highly capital-intensive investments, complexity of operations, and profit potential, companies are compelled to seek the most sophisticated investment evaluation methods. The term decision analysis and risk analysis are often used interchangeably. In preferred usage, decision analysis refers to the methodology of problem-solving approach, while risk analysis refers to the sub-process of assessing the probability distributions of a single parameter.3

2

Introduction

Today the oil industry is operating in an ever-changing technical, economical, and political environment. Factors affecting decision making include technology advancements, new products and techniques, uncertainty of oil and gas prices, operating costs, equipment costs, inflation, market conditions, political risks, environmental threats, reliability of the aging production facilities, capital availability constraints, scarcity of good prospects (more difficult E & P challenges), and governmental regulations. All these and other factors necessitate knowing your risks, thus posing new challenges to the decision analysts and the decision makers. Historically, decision analysis techniques have been used mainly in the oil and gas exploration evaluations. Newendorp and Megill wrote the classic books about the application of decision analysis techniques in petroleum exploration. However, the trend is now changing and many other decision problems are now subject to rigorous decision analysis. One reason for this change is that technically more advanced projects are now being planned in today’s volatile economic and political climates. A summary of the various petroleum industry’s applications of decision analysis are given in this chapter.4–6 Many oil and gas investments involve a relatively high risk that the investment may not achieve the desired results. On the other hand, some of these investments may possibly generate better than the desired results or even a bonanza. Therefore, the investment decisions have to be based on sound trade-offs between the risk of complete or partial loss and the potential of significant gains. To appropriately assist in the comparison of risk versus reward, a number of techniques have been developed. The product of these techniques incorporates probability distributions and possible measure of profitability relevant to the problem under evaluation. The decision analysis techniques presented in this book serve as valuable tools for making decisions in complex situations. As part of the business cycle, every decision maker needs a clear understanding of the risks. It is therefore not surprising that the technoeconomic feasibility analysis methods for project evaluations are becoming more elaborate. Methods properly ing for risk and

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uncertainties in all aspects of project evaluation are becoming more popular than ever before. Even if the future cash flow from a project and industry trends are calculated with a good deal of deliberation and accuracy, there is some inherent degree of uncertainty associated with the overall profitability of a project. Although the uncertainties are considered part of the business, they need not be vague (the risk-benefit tradeoffs shall be thoroughly evaluated). The techniques presented, if applied correctly, explicitly quantify those variables that can significantly change return from the investment.

HANDLING UNCERTAINTY IN CAPITAL INVESTMENTS A typical capital investment project evaluation requires input variables such as future product prices, production forecast over the economic life of the project, initial capital expenditure and ongoing operating expenditures, useful lifetime of facilities/equipment, salvage value at the end of the economic life of the project, and interest rates. The estimates of some of these variables have little associated uncertainty and can be determined with relative confidence. Other estimates used in the analysis may have higher uncertainty. The uncertainty of some of the variables may be very detrimental to the profitability of the investment as compared to the others. Many approaches, with varying degree of sophistication, are used in the industry for treating uncertainty in capital investment decisions. These are: 1. Sensitivity Analysis. Sensitivity analysis, also referred to as what if analysis, is a technique indicating exactly how much the profitability of a project will change in response to a given arbitrary change in an input variable, other variables held constant. The analysis begins with a base case situation using the most likely (expected) input values. Then each variable is changed at a time by a specific percentage above and below the expected value and

4

Introduction

profitability calculated. For example, to run sensitivity on oil price of $18/barrel, $20/barrel, and $22/barrel, $20/barrel may be used as the base case or expected value. The derived profitability measures are then plotted against the changed variable. The plot shows how sensitive the profitability of the investment is to changes in each of the input variables. 2. Scenario Analysis. Scenario analysis is a technique that considers the sensitivity of investment’s profitability to changes in key variables and the range of likely variable values. In scenario analysis, a bad set of circumstances (such as lower production, less ultimate recovery, lower price, high operating cost, higher capital cost, and so on), an average set, and a good or optimistic set are picked. The investment’s profitability at these three conditions are then calculated and compared. The bad set, average set, and good set are typically referred to as worst-case scenario, most likely scenario, and best-case scenario, respectively. 3. Probability Approach I. Probability approach utilizing the expected value of a decision criterion for each alternative and determining the preferred course of action based on expected value. 4. Probability Approach II. Probability approach where an explicit measure of risk is used in addition to the expected value. 5. Computer Simulation. Computer simulation where different combinations of uncertain variables are derived from probability distributions of each variable. The outcome for a measure of effectiveness on the profitability measure is determined for each combination. Thousands of randomly generated combinations for each variable are normally tried.

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Project Economics and Decision Analysis Volume II: Probabilistic Models

INDUSTRY PRACTICE OVER THE YEARS In an industry survey conducted by Boyle and Schenck (1985), participants were asked how they ed for risk in capital investment analysis. The responses indicated considerable sophistication in this area, 100% of the respondents reported they explicitly incorporate risk in investment analysis.7 Sensitivity analysis was reported as the most common method of ing for risk. The second method that was used by 38% of the respondents was to arbitrarily increase the discount rate to for risk. These methods were followed by the methods of subjective adjustments and probability factors. A majority of the firms, 59%, reported use of the most popular single method of expected value (probability weighted cash flows). The larger oil and gas corporations reported the common use of Monte Carlo computer simulation. Computer simulation was least popular with the smaller firms, because of the high degree of complexity required, needed resources, and the cost of conducting simulations. A similar survey conducted by Dougherty and Sarkar8 (1993) showed a decline in popularity of subjective judgments and an increase in popularity of probability factors to for risk. To for risk by raising discount rate still dominated, with 46.2% of the firms reported using this method. However, arbitrarily raising the discount rate to for risk is not correct. As Newendorp points out, the problem with an arbitrary minimum is “it does not explicitly consider the varying levels of risk between competing investments.”4–5 The later survey showed some shift over the years towards the use of more sophisticated quantitative techniques in ing for risk. However, most of the firms need to critically review their decision analysis techniques. The smaller companies need to be even more aware of risk and uncertainty as compared to the larger firms because, with fewer assets, financial ruin is more likely to occur due to bad luck.7 In other words, small firms have limited funds, and therefore the size of their investment portfolio is small. This does not provide them with enough room for averaging the return on bad projects with the return on good projects.

6

Introduction

Any individual who participates in the decision-making process is ethically and professionally bound to assure that only informed and adequately analyzed decisions are made. One of the key elements contributing to sound decision making is an assessment of uncertainty associated with the chances of economic failure and upside potential of the investment project being evaluated.

CLASSIFICATION OF DECISION SITUATIONS Depending on the type of available information on a project being evaluated to the decision maker, he or she may be faced with decision making under (1) certainty, (2) risk, or (3) uncertainty.9 When complete information is available for an outcome resulting from each decision alternative under consideration, the decision situation is referred to as decision under certainty. If each one of the decision alternatives under consideration is expected to result in several probable outcomes and if the probability of each outcome can be assessed, such decision situation is referred to as decision under risk. For example, drilling a well is expected to result in either a dry hole or a productive well. If the well is productive, it may have estimated recoverable reserves of 100,000 or 200,000 barrels. The probability of dry hole is 30% and that of the productive well is 70%. The probability of 100,000-barrel reserve is 60%, and that of 200,000 barrels is 40%. In other words, there is uncertainty that can be quantified in of probabilities. On the other hand, if this same situation occurs but the analyst has no idea of the probabilities to be assigned to each possible outcome the decision is made under uncertainty. Uncertainty is usually described by words such as unlikely, probably, impossible, expected, and most likely. Decision analysts have treated the risk and uncertainty for many decades as distinguishable entities. In decision analysis, however, it is assumed the probabilities of the chance events can be subjectively assessed; therefore, the distinction is unnecessary. Many times the two have been used interchangeably.10

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Project Economics and Decision Analysis Volume II: Probabilistic Models

TERMINOLOGY USED IN DECISION ANALYSIS In this section, some of the basic routinely used in decision analysis are defined. Other definitions specific to a certain concept will be given when these concepts are presented and discussed. •

•

•

8

Objective. Objective is defined as an indication of the preferred direction of movement. Typical objectives in decision analysis would be to minimize cost, maximize net worth, minimize the chance of the worst possible outcome or maximize the chance of the best possible outcome.11 Outcome Criteria. Variables used to rank or measure desirability of any outcome is called outcome criteria. Typical outcome criteria used in this book are net present value, rate of return, present value ratio, and present value of costs, profitability index, benefit-cost ratio, and utility. The outcome criteria may be deterministic if no probability is included and probabilistic if probability is included. If probabilities are included, it is called expected value and so on. Risk. Risk, in general , is defined as the probability of occurrence of unfavorable outcomes. In relation to project/investment evaluation, risk refers to the probabilities that the returns from a project may have alternative outcomes (multiple possible outcomes for each decision or action). ing for risk in petroleum investment analysis is important because the outcome of an investment decision depends on a wide range of uncertain variables. The presence and producibility of oil and gas are uncertain estimates. This uncertainty is further compounded by the uncertainty in product prices, expenses, future investments, and tax laws, etc. In addition, the mechanical risks associated with production facilities, human error, and new technologies add to the geologic and economic uncertainties. The role of a good investment plan is to accurately define the levels of risks assumed and the expected yield from the investment in light of the identified risks and uncertainties. An efficient plan

Introduction

•

of action has to be provided to justify the exposure of funds to risk and for managing risk. Risk Attitude. Most decision makers are not impartial to money. They have specific attitudes and feelings about money due to factors such as asset position, risk preferences, and any immediate and/or longer-term goals. Preference theory is used to arrive at the decision maker’s expected preference value (also referred to as expected utility). This quantitative parameter incorporates the decision maker’s risk attitudes towards money. The following three possible human attitudes towards risk are identified. 1. Risk-averse: Risk-averse attitude represents conservative attitude for risk taking. Individuals or companies who exhibit this attitude are called risk averters. Except for pathological gamblers, most of us are risk averse. Risk aversion safeguards against unwanted losses. However, in many situations it imposes a subtle yet heavy penalty by limiting the decision makers’ ability to realize optimal returns from investments. A decision maker with risk-averse behavior will prefer a given dollar payoff (for certain) to a lottery offering an equal expected dollar payoff. For instance, a sure gain of $5,000 to a 50:50 chance of obtaining $0 or $10,000 is preferred. 2. Risk-seeking: A person with attitude or preference towards high risk is the exact opposite of being conservative. The attitude towards risk in this situation is the opposite of risk aversion. For instance, a 50:50 chance of obtaining $0 or $10,000 is preferred to a sure gain of $5,000. 3. Risk-neutral: This third attitude is an indifference to risk or risk neutral attitude. A risk-neutral decision maker will be indifferent to choosing between a sure gain of $5,000 versus a 50:50 chance of obtaining $0 or $10,000.

•

Decision. Decision, in general, is defined as the act of making up one’s mind to take a certain action. In the context of capital budgeting, making a decision means making a choice between

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Project Economics and Decision Analysis Volume II: Probabilistic Models

•

10

alternative ways of allocating resources. This means there is more than one distinct alternative to choose from. Of course, if there is no choice then there is no problem. Distinct alternatives mean they result in different outcomes in of rewards or penalties for the decision maker. If no resources are committed, then no decision has been made. However, once the resources are allocated, then it may be impossible or extremely costly to revert to the situation that existed before the decision was made. Thus for our purpose, a decision is not merely a mental commitment to take a certain action but rather the actual implementation of that course of action. Events and Outcomes. Many computations in probability and statistics are based on the occurrence of equally likely events. An event is one possible outcome of an experiment. Equally likely events are those with the same chance of occurring under stated conditions. For example, drilling of a test well is a decision, result of the test well is an event, and whether the well is a dry hole or a commercial well is an outcome. If the occurrence of one event is dependent on the occurrence of a second event, the events are called conditional events. An event is called independent event if its occurrence is not dependent on the occurrence of any other event. In decision analysis, a chance event is used, referring to an experiment, process, or measurement where the outcome is not known beforehand. The result of this chance event is then called an outcome. The chance events could be discrete or continuous. If there are only two, three, or four (any finite number) of possible outcomes, and each outcome has a quantifiable probability of occurrence, the chance event is called discrete. On the other hand, continuous events refer to an infinite number of outcomes. Many companies represent possible outcomes when analyzing a drilling decision by four outcomes, such as dry hole, low reserves, most likely reserves, and high reserves that can reasonably be expected. The probabilities of each one of these outcomes are then determined.

Introduction

•

Analysis. Analysis is defined as the separation of anything into its constituent parts or elements to find out what it is made of. Philosophically, the breaking up of a concept or event into its constituent or into its causes to reveal concealed content or form is called analysis.

THE DECISION ANALYSIS CYCLE Decision analysis is a term used to describe a decision-making process that combines the methods of statistical decision theory, imaginative creativity, systems analysis, and operations research. In other words, it is a procedure of logically balancing the factors influencing a decision. Typically, these factors may be technical, economical, environmental, political, legal, etc. The procedure deals with uncertainties, values, and preferences. A decision analysis problem-solving process serves as a philosophy of the problem, as a decision framework, as a decision-making process, and as a methodology of decision making. The whole process is meant to represent the decision makers’ information and preferences concerning the uncertain, complex, and dynamic features of the problem under consideration.12 Statistical decision theory provides logical decisions in simple uncertain situations. It assists in describing the uncertainties, values, and preferences relevant to the decision problem and provides mathematical techniques for solving these uncertain problems. In decision analysis, the key word is analysis as defined in the previous section. In this context, it refers to the decomposition of decision into smaller, easily manageable problems. This allows detailed separate study of each constituent element of the problem. The reduced complexity and scope also lends the problem to effective use of intuition. After each constituent problem element is analyzed, then a subset of these elements is chosen for inclusion in the integrated problem to give a complete picture. This process has been referred to as the “divide and conquer orientation” of decision analysis.13

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Project Economics and Decision Analysis Volume II: Probabilistic Models

A decision analysis process, in general, consists of • • • •

problem identification and definition, assessment of the various factors impacting the problem, sensitivity analysis, and implementation.

The process is iterative, typically comprised of four phases referred to as the 1. basis development phase, 2. deterministic phase, 3. probabilistic phase, and 4. information phase. The four phases are schematically illustrated in Figure 1–1. The concepts in this book are also arranged according to these phases of decision analysis.

Fig. 1–1 The decision analysis cycle (after Howard)12

The basis development phase consists of the basic structuring of the decision problem and then using this structure to identify the crucial

12

Introduction

variables. In basis development, the task is mainly to identify the real decision, the scope and method of approach, requirements, and objectives of the analysis. The decision analyst’s first responsibility is to lucidly define the decision to be made. This by itself is an iterative process between the decision makers, relevant specialists from other disciplines (geophysics, geology, reservoir engineering, petrophysics, production engineering, drilling, and so on), and the analyst. The problem definition depends mainly on •

• •

The judgment about such things as uncertainty, the alternative courses of actions available, the possible outcomes of each course of action available, and the preferences of the decision maker, Previous experience with problems of similar nature, and Intuition of the experts involved in assisting the analyst in formulating the problem.

The decision analysis process provides a framework to clarify and articulate the judgments mentioned above. The problem definition phase is important, perhaps more important than subsequent computations because it provides indirect value of the analysis.14 Close group interaction is essential for good decision making. Routine discussions between the groups of specialists are healthy; they further lead to greater understanding of the problem and creative thinking (two minds are better than one). The structuring of a problem, at first , may look like pieces from many different jigsaw puzzles. The analyst has to identify the relevant pieces to the problem under consideration and put them together. Then the missing pieces and their source must be obtained to complete the picture. Morgan and Henrion emphasize iterative refinement of the problem and statement. They claim that, without exception, good analysts follow the approach as schematically shown in Figure 1–2. The iterative process helps in refining the analysis as more data becomes available and more is learned about the decision problem.15

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Fig. 1–2 Illustration of the “good” decision analysis process (adopted from Morgan and Henrion)15

A model without any probabilistic components is called deterministic model. In deterministic models, the output is determined once the set of input quantities and relationships in the model have been specified. The determin14

Introduction

istic phase treats the problem quantitatively by determining critical uncertainties. Financial cash-flow models are developed, and parameters such as net present value, profitability index, etc. are determined for each outcome. After this project-specific cash-flow model is developed, the sensitivity analysis is performed on variables. This highlights the variables that have greatest uncertainty and are critical for good decision making. Sensitivities may be performed on variables like discount rate, product prices, initial investment, operating costs, production volumes, and project timing, etc. The probabilistic phase involves asg probability distributions to the crucial uncertain events and variables, thus explicitly incorporating uncertainty. The probability distributions are obtained by eliciting the judgment of knowledgeable personnel involved in the analysis. Since uncertainty is the heart of most decision problems, decision analysis requires specifying the amount of uncertainty that exists given available information. Once the probabilities are assigned, the expected values of the chance events are calculated. In the information phase, the results of the first three phases are reviewed to determine if any additional information eliminates or reduces uncertainty in each of the important variables. Comparing the value of the additional information with the cost to achieve this information is evaluated. The information is obtained if it is determined that the additional information, at a certain cost, enhances the value of the variables and subsequently improves the expected payoff from the project under consideration. If the information can be obtained from more than one source, the preferred source has to be determined. This process is also called preposterior analysis. In general, the greater the uncertainty, the greater will be the expected value of additional information. In light of the new information, the original analyses are revisited, (the three phases are repeated). This iterative process continues until an optimum solution is achieved. Sensitivity analysis should be performed at each stage of the analysis. It is used to examine the robustness of an alternative to changes in key variables. These analyses enhance efficiency and effectiveness in the decision analysis process. Using the principle of dominance or process of elimination chooses the best economic alternative. When one alternative or chance event is superior to another, it is said the first alternative or event dominates the

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Project Economics and Decision Analysis Volume II: Probabilistic Models

second. The principle of dominance permits a major reduction in the number of alternatives, thus reaching a final decision about the preferred course of action. Decision analysis results are better represented in graphics form such as decision trees, probability distributions, influence diagrams, sensitivity analysis (spider diagram) plots, etc. Therefore, it should be kept in mind while presenting a problem.

ADVANTAGES OF DECISION ANALYSIS Several advantages of the decision analysis approach versus the commonly used deterministic tools of investment analysis are given in this section (adopted from Newendorp).4–5 •

•

•

•

• •

16

In order to answer complex problems, decision makers and the analysts must thoroughly and systematically contemplate the problem. The thinking process in decision analysis is more beneficial than the answers generated by the analysis. For example, creative thinking may result in newer and possibly superior courses of action. Decision analysis provides guidance on additional information needs and their worth. For example, is it worth acquiring 3D seismic survey at a cost of $500,000? Decision analysis provides a convincing rationale for choice and a language for communication among those involved in the process. This clarifies the basis for a decision rather than only the decision itself. Certain techniques of decision analysis provide excellent tools for evaluating sensitivity of various complex variables affecting the overall worth. For example, what if a well encounters blowout and it costs an additional $100,000 to cap it? Decision analysis provides a means for comparing riskiness of different projects or investment alternatives. Decision analysis forces a more explicit look at the possible outcomes that could occur if the decision maker accepts a given prospect. For example, the different levels of reserves found and

Introduction

•

•

the associated probabilities instead of saying the well will be a dry hole or productive. Decision analysis provides an unambiguous way for communicating judgments about risk. For example, it does not allow adjectives such as “it is likely,” “there is a good chance,” or “it is a risky prospect.” Such are subjective; what is “most likely” in the judgment of one person may not be “most likely” in the judgment of another person. On the other hand, saying there is a 70% chance of a productive well conveys a much clearer magnitude of the probability of occurrence to the decision maker. In decision analysis, the description of risk and uncertainty is given in the form of probabilities. Decision analyses techniques help analyze exceedingly complex investment alternatives.

MISCONCEPTIONS REGARDING DECISION ANALYSIS There are several critical misconceptions and misunderstandings regarding the use of decision analysis. Misconception #1—Decision analysis is considered as a technique to eliminate or reduce risk in decision making. The reality is decision analysis does not reduce or eliminate the degree of risk that a decision maker faces in this uncertain environment; rather it assists in taking the right risk (calculated risk). Therefore, the purpose of decision analysis is not to eliminate or reduce risk, but to provide a tool for properly evaluating, quantifying, and understanding risk. This insight helps the decision makers devise a decision strategy that hopefully minimizes the firm’s exposure to risk.4–5 A meaningful message is given by Peter Drucker: To try to eliminate risk in business enterprise is futile. Risk is inherent in the commitment of present resources to future expectations. Indeed, economic progress can be defined as the ability to take greater risks. The attempt to 17

Project Economics and Decision Analysis Volume II: Probabilistic Models

eliminate risks, even the attempt to minimize them, can only make them irrational and unbearable. It can only result in that greatest risk of all: rigidity. The main goal of management science must be to enable business to take the right risk. Indeed, it must be to enable business to take greater risks—by providing knowledge and understanding of alternative risks and alternative expectations: by identifying the resources and efforts needed for desired results; by mobilizing energies for expectations, thereby providing means for early correction of wrong or inadequate decisions.16 Misconception #2—Decision analysis is considered as a method of producing optimal solution to a decision problem under consideration. Keeney clarifies this misconception as follows. Decision analysis will not solve a decision problem, nor is it intended to. Its purpose is to produce insight and promote creativity to help decision makers make better decisions.13 Misconception #3—Decision analysis replaces professionals’ judgment or intuition. As it has been mentioned in the preceding pages, decision analysis is not meant to replace, rather supplement the judgment and intuition of the professionals involved in the decision-making process. Misconception #4—Decision makers tend to equate the quality of the decision with the quality of its outcome. They think a good decision ought to result in good outcome; that the quality of a decision is gauged by its outcome. However, a good decision may result in a poor outcome, and a poor decision may end up with a good outcome. In dealing with decision analysis, it is important to differentiate between the quality of the decision and the quality of the outcome. It just happens that if the outcome of a decision is good, many people may assume a good decision was made.

18

Introduction

In this context, it is necessary to define a good decision. A good decision is a logical decision—one based on the uncertainties, values, preferences, and good professional judgment of the experts involved, and s for all available information. The analysis ing for the above factors and conducted systematically and consistently on all projects minimizes the consequences of getting an unfavorable outcome. However, the decision analysis is not expected to shield us from “bad luck.”

APPLICATIONS OF DECISION ANALYSIS Decision analysis can be used in many complex problems. A brief list of the various types of problems encountered in the petroleum industry where decision analysis may be used is as follows. • • • • •

• • • • • • • • • • • •

Selecting projects to obtain a balanced research portfolio. Decision about research on parallel or competing technologies. Evaluating basic research strategies. Single and multi-prospect development farm-outs. Determining venture participation based on review of interrelation of venture profitability, total risk investment, aversion to risk, profitability of success, and available risk investment funds. How to quantify uncertainties and leveraging issues in oil and gas project economics. Making a choice between two or more wildcat ventures. Competitive bidding situations. Litigation planning. Single- and dual-string gas lift completions. Risk analysis of well completion systems. Economics of fishing operations. Development drilling in a reservoir with pay discontinuities. Operational decision making for stuck pipe incidents. Reservoir engineering forecasts. Reserves estimate of exploratory and producing fields. Evaluating acquisition of information, such as seismic surveys, well logs, additional data analysis, laboratory tests, reservoir simulation studies (sector models or full field models), and so on. 19

Project Economics and Decision Analysis Volume II: Probabilistic Models

• • •

Deciding on the type of well completion and stimulation methods. Investment decisions related to plants and surface facilities. Exploration ventures.

Decisions regarding a majority of the different nature of these problems are made using decision analysis. Examples are provided later in the book to show the application of decision analysis to the various problems listed.

TYPICAL INDUSTRY RISKS A list of some typical examples of risks and uncertainties affecting various types of decision problems is given in Table 1–1.

Product Market

Competition

•

Decreased oil, gas, and product prices

•

Lease acquisition

•

Speculation in the markets, causing

•

For drilling capital

volatility

•

For competent people

Decreased energy or product demand, i.e.

•

Preferred contractor committed to another

•

spot shortage •

Fuel switching, price sensitivity

•

Effects:

company •

availability

ƒ

Lower project revenue

•

ƒ

Lower production volume

customers

ƒ

Difficult to plan, higher cost of capital •

Effects:

to maintain liquidity

ƒ

Missed opportunities

ƒ

Higher costs

ƒ

Delays in commissioning projects

ƒ

Design compromises

Table 1–1 Typical industry risks (after Schuyler)17

20

Limited rig, service, or materials Competing for supply sources and

Introduction

Technology

Catastrophe

•

Competitive fuels (coal, biomas, natural

•

Weather damage

gas vehicles, alternate fuels, etc.)

•

Petroleum or chemical spill

More efficient processes, finding

•

Blowout, casing, or tubing collapse

advanced techniques of drilling and

•

Collision with production station,

•

completion (horizontal/multi-lateral

wellhead jackets, or pipelines

wells).

•

Accident or failure causing plant disaster

•

Obsolescence

•

Foreign country nationalization

•

Safer processes

•

Product tampering, sabotage

•

More environmentally acceptable

•

Civic unrest, riots, war, terrorism

processes

•

Effects:

•

Environmental protection groups

ƒ

Extreme costs or loss

•

Effects:

ƒ

Adverse public opinion

ƒ

Product market changes

ƒ

Adverse legal and regulatory action

ƒ

Product or distribution obsolescence

ƒ

Non-competitive costs

Environmental Risks

Uncertainties in Oil Price Forecast

•

Delay risks; government intervention

•

OPEC s strategy

•

Hazardous materials

•

World oil reserves

•

Permitting delays

•

Political stability in the Gulf (Ex: Gulf

•

Destruction of the environment

•

Atmosphere

•

Watersheds

•

Soil fertility

•

New technologies

•

Visual pollution

•

Price elasticity of demand

•

Effects:

•

Mideast production and capability

ƒ

Delay

•

Economic growth

ƒ

Unexpected costs

•

Alternate fuels availability

ƒ

Low probability, high impact

•

Seasonality and weather

contingencies

•

Effects: ƒ Uncertain revenue projection

War) •

Finding rate (unit technical cost to develop reserves)

Table 1–1 continued . . .

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Operations (Drilling & Production)

Business Interruption

•

Faulty design, materials, workmanship

•

•

Equipment breakdown, availability

•

Weather delays; site access delays

•

Litigation; injunction

•

Material delivery delay; poor quality

•

Loss of market share

•

Manpower, accidents

•

Partners or contractors fail to perform

•

Formation or well damage

•

Union strikes

•

Poor reservoir development plan

•

Key persons leave or get sick

•

Incompetent outside operator (cost

•

Takeover; reorganization

overruns; liability)

•

Effects:

Unavailable facility (unscheduled shutdowns)

•

Protest demonstrations

ƒ

Permanently lost market share

•

Unexpected corrosion, wear

ƒ

Lost transaction opportunities

•

Theft and vandalism

ƒ

Project delays

•

Unitization

ƒ

Company fails to perform to its

•

Effects: ƒ

Remedial costs

ƒ

Business interruption

contractual obligations ƒ

Goals and projections missed

Monetary Risks

Legislative and Regulatory

•

Unexpected inflation levels

•

Unexpected tax laws

•

Unexpected foreign currency exchange

•

FERC, clean air act

rate changes

•

Production allowable (curtailment)

•

Interest and capital rate risks

•

Contract abrogation

•

Attitude change of lenders, creditors,

•

Business liability

parent company

•

Corporate regulations

Effects:

•

International agreements

ƒ

Smaller real profit margins

•

Permitting

ƒ

Increased interest rate expense

•

Effects:

ƒ

Value lost through changes in

ƒ

Risk-sharing relationships changed

exchange rate

ƒ

Burdensome record––keeping and

•

ƒ

Difficulty in raising future required capital (equity or debt)

reporting requirements ƒ

Higher liability risk transactions avoided

Table 1–1 continued . . .

22

Introduction

This list should be kept in mind when structuring a decision analysis problem. Here is further explanation and examples of some of these risks. •

•

•

•

• •

Geological Risks. Dry hole, bottomhole location, field size, reservoir heterogeneity (pay thickness, permeability variation, porosity variation, faults, fracture, adequate reservoir pressure, etc.) Financial Risk. Price volatility, operating expenditure, capital expenditure overruns, inflation, currency devaluation, taxes, discount (interest) rates, etc. Drilling and Completion Risk. Blowout, casing collapse, extended fishing, plug back and sidetrack, poor cementing resulting in reservoir cross flow, etc. Production Risk. Reservoir management, lower than expected production profile, lower ultimate recovery, water coning, gas cusping, facility limitations, etc. Catastrophe. Fire of Piper Alpha (North Sea), Exxon Valdez oil spill, Union Carbide chemical leak in India, etc. Political Risk. Regulations, taxes, nationalization, etc.

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Project Economics and Decision Analysis Volume II: Probabilistic Models

QUESTIONS and PROBLEMS

24

1.1

What is the difference between decision making under (a) certainty, (b) risk, and (c) uncertainty?

1.2

Briefly define the outcome criteria, risk, risk attitude, decision, and events and outcomes.

1.3

What are the different phases of decision analysis? What is the difference between the deterministic phase and probabilistic phase?

1.4

How is it decided if obtaining additional information in decision analysis is worth acquiring?

1.5

Briefly describe three advantages of carrying out decision analysis.

1.6

The purpose of decision analysis is to eliminate risk. True or false? Explain.

1.7

List five problems where decision analysis can be applied.

1.8

What is the difference between decision analysis and risk analysis?

Introduction

REFERENCES 1

2

3

4

5

6

7

8

9

Clemen, R. T., Making Hard Decisions—An Introduction to Decision Analysis, 2nd Edition Duxbury Press, Wadsworth Publishing Co., 10 Davis Drive, Belmont, CA, USA, 1996. Stermole, F. J. and Stermole, J. M., Economic Evaluation and Investment Decision Methods, Tenth Edition, Investment Evaluation Corporation, 3070 South Newcombe Way, Lakewood, CO 80227, USA, 2000 Schuyler, J.R., “Credible Projections Now Require Decision Analysis,” Cost Engineering, vol. 34, no. 3, March 1992, pp.15–19. Newendorp, P.D., Decision Analysis for Petroleum Exploration, PennWell Publishing Co., Tulsa, Oklahoma, USA, 1975. Newendorp, P.D. and Schuyler, J. R., Decision Analysis for Petroleum Exploration, 2nd Edition, Planning Press, 15492 East Chenango Avenue, Aurora, CO 80015, USA, 2000. Megill, R.E., An Introduction to Exploration Economics, 2nd Edition, PennWell Publishing Co., Tulsa, Oklahoma, USA, 1979. Boyle, H.F. Jr. and Schenk, G.K., “Analysis: U.S. Oil and Gas Producers Score High in University Survey,” Journal of Petroleum Technology, April 1985, pp. 680–690. Dougherty, E.L. and Sarkar, J., “Current Investment Practices and Procedures: Results of a Survey of U.S. Oil and Gas Producers and Petroleum Consultants,” SPE Paper 25824, presented at the SPE Hydrocarbon Economics and Evaluation Symposium held in Dallas, Texas, USA, March 29–30, 1993. Ramalingam, P., System Analysis for Managerial Decisions—A Computer Approach, John Wiley and Sons, Inc., New York, USA, 1976.

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10

11

12

13

14

15

16

17

26

Swanson, D., Managerial Decision Analysis, Richard D. Irwin, Inc., Homewood, Illinois, USA, 1988. Keeney, R.L. and Rafia, H., Decisions with Multiple Objectives: Preferences and Value Tradeoffs, John Wiley and Sons, Inc., New York, USA, 1976. Howard, R.A., “The Evolution of Decision Analysis,” Readings on the Principles and Applications of Decision Analysis, Vol. 1, Strategic Decisions Group, Menlo Park, California, USA, 1984. Keeney, R.L., “Decision Analysis: An Overview,” Operations Research 30, No. 5, September 1982, pp. 803–838. Goodwin, P. and Wright, G., Decision Analysis for Management Judgment, John Wiley and Sons, Ltd., England (UK), 1991. Morgan, M.G. and Henrion, M., Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis, Cambridge University Press, New York, U.S.A., 1992. Drucker, P.F., Management: Tasks, Responsibilities, Practices, Harper and Row Publishers, Inc., New York, USA, 1974. Schuyler, J.R., Petroleum Risk and Decision Analysis, Oil & Gas Consultants International, Inc., Tulsa, Oklahoma, USA, 1992, pp. 060-9–060-18.

chapter

TWO

Statistics and Probability Concepts

This chapter provides some concepts of descriptive statistics and probability. These concepts form the basis for the more detailed decision analysis tools presented in the following chapters. For more details and formal treatment of the subject, the reader is referred to books and courses devoted entirely to the subject. Statistics and probability studies involve many subtle ideas. Care must be exercised at the beginning to understand the fundamental concepts. Modern statistics provides for the following: • • •

Estimation of population parameters based on sample statistics. Drawing inferences about sample statistics based on population parameters. Decision making under uncertainty where the outcome is not precisely known.

Therefore, statistics is concerned with methods of converting numerical data into information useful for scientific research, business decision making, and other similar purposes. Familiarity with basic statistical methods used for organization and presentation of data in

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Project Economics and Decision Analysis Volume II: Probabilistic Models

charts, graphs, tables, and summary measures is essential. These methods are called descriptive statistics. Other methods enable one to draw conclusions about entire populations of data, based upon information taken from a representative sample of this population. These methods are collectively referred to as inferential statistics or inductive processes of generalizing the population parameters from its sample statistics. These inferential methods require familiarity with the overall statistical theory if they are to be properly applied in decision making. Probability theory forms important and fundamental bases of inferential statistics. In this chapter, the following concepts are presented. • • •

Descriptive statistics Probability concepts Theoretical probability distributions

DESCRIPTIVE STATISTICS Descriptive statistics is concerned with methods of organizing, summarizing, and presenting data. In Volume I, single-point descriptions of cash inflows and cash outflows (such as the net present value, internal rate of return, payback period, and technical cost, etc.) were derived from cash flows. The objective of these single-point descriptions was to give a manager the gist of an investment’s profitability in just a few and easily interpretable numbers. On the other hand, presenting the management with an array of numbers arranged in several columns and rows of a spreadsheet would be, if not meaningless, hard to interpret into useful decisions. In the oil industry, technical data concerning porosity, permeability, reserves, pay thickness, bit records, bit penetration rate, water saturation, oil saturation, etc. are routinely analyzed. Other data include weights of packages filled by machines, employee attitudes toward a new policy, length of time a computer program spends in queue, costs of alternative production methods, daily demand for an inventory item, hourly wages, overhead costs, consumer reaction to a new product, gross national

28

Statistics and Probability Concepts

product, etc. There are many other examples related to other data in oil and gas and other professions. Whatever the type or source of data, the descriptive statistics techniques are the same. This involves (a) (b) (c) (d)

collection of data, arranging and/or grouping the acquired data to reduce volume, graphing data for display, and finally expressing data measures to appropriately describe the central tendencies and variability.

Since data can be very voluminous, these tasks are often a necessary first step in the decision-making process. In this chapter, descriptions of statistical properties of various types of data are presented. Figure 2–1 schematically shows different ways data can be handled.

Fig. 2–1 Series of quantitative variables

29

Project Economics and Decision Analysis Volume II: Probabilistic Models

Descriptive statistics dealing with • • •

measures of central tendency for individual data elements and grouped data, measures of variability for individual data elements and grouped data, and graphical presentation of individual data elements and grouped data

are presented in this section.

Measures of Central Tendency If a data set contains all possible items of interest, it is called a population. If a data set contains only a few random or otherwise representative elements of a population (larger data set), it is called a sample. Most of the collected data is normally too detailed, in its raw form, to be useful. It is necessary for this raw data to be organized into a useful form, so it can be easily and effectively interpreted by decision-makers. One method allows discovering the most likely or common outcome from the data. The most likely value to describe a data set is called its measure of central tendency. It is usually near the center of an array of data sets, and all other elements of the data set are inclined to behave as though each had a particular value. Measures of central tendency are most commonly referred to as averages. This term is commonly associated with one particular measure of central tendency, called the arithmetic mean. The term average is further classified into three sub categories as (a) mathematical average, (b) positional average, and (c) moving or progressive average. Figure 2–2 shows various measures of central tendency to be discussed in this section.

30

Statistics and Probability Concepts

Fig. 2–2 Various measures of central tendency

Measures of central tendency and other statistical characteristics that describe the population are called parameters. The corresponding measures and statistical characteristics of a sample are called statistics. Mean, Median and Mode. As described earlier, the arithmetic mean (symbol or X) is the figure most people think of as an average. Simply, adding the values of all items in a data set and dividing this total by the total number of items in the data set calculates the mean. The purpose of calculating an average value for a set of observations is to obtain a single value representative of all items that the mind can grasp simply and quickly. The single value is the point of location around which the individual items cluster. The mean has a number of the following interesting characteristics that should be kept in mind while describing a data set. • •

Every element in the data set contributes to the value of the mean. An average provides a common denominator for comparing one set of data with others, and conclusions can be drawn about the characteristics of the separate data sets. 31

Project Economics and Decision Analysis Volume II: Probabilistic Models

•

• •

The mean is influenced by the extreme values in the data set. In some cases, particularly when dealing with small data sets, the extreme values are discarded in order to avoid distortion of the mean. This, however, should be done with extreme care. Alternatively, other measures of central tendency may be used in such cases. These are discussed in the following pages. The mean may not be an actual element of the data set. The sum of deviations from the mean is zero, and the sum of squared deviations is minimized when those deviations are measured from the mean. Mathematically, these properties of the mean are expressed as: (2.1) N

∑( X i =1

i

− X) = 0 (2.2)

N

∑(X i =1

i

− X ) 2 = minimum

The formula for calculating the arithmetic mean is: (2.3) n

X =

∑X

i

i =1

n

where X = mean of a sample Xi = sum of all data elements (i = 1, 2, 3, … n) in a sample n = total number of data elements in a sample The sample mean is represented by X and the population mean is represented by . Similarly, the number of items in a sample and population are represented by n and N, respectively.

32

Statistics and Probability Concepts

The median is usually defined as the central value in an array of a data set. If there is an odd number of data elements in a data set, the actual data element in the middle of the data set is the median of this data set. For example, if there are 25 data elements then the median is the 13th data element, since there are 12 data elements higher and 12 lower. However, if the data set contains an even number of data elements, then the median is the average of the two data elements in the middle of the data array. For example, if there are 24 data elements, then the median is the average of the 12th and 13th data elements. To find the median of a data set, it is preferred to first arrange the data set in a descending or ascending order so the mid-point can be identified easily. For example, the median of the numbers 3, 6, 8, 10 and 12 would be 8. If a firm had production of 200, 220, 240 and 260 units, the median would be half way between 220 and 240. Following are the properties of the median. •

• • •

The median is not influenced by extreme values since only one or at the most two data elements determine the value of the median. This makes the median useful in describing the central tendency of data that are limited in one direction only. The median may not be an actual value of the data set, and it ignores other data elements of a data set. The median is not used in any common subsequent statistical methods. For a perfectly symmetrical data set, the mean is equal to the median. The median divides a frequency curve (to be discussed in the following pages) into equal areas.

The mode is the most frequently occurring data element in a data set. Following are the properties of the mode. •

A data set may have more than one mode. A data set is called bimodal when two data elements occur an equal number of times (more than once and more than any other data element). If a data set has more than two modes, the worth of the mode as a measure of central tendency becomes questionable.

33

Project Economics and Decision Analysis Volume II: Probabilistic Models

• •

•

• •

As with the median, the mode is unaffected by extreme data elements of a data set. The main problem with the mode value is it does not take into all data elements in a data set. Therefore, it might prove misleading when taken as a measure of central tendency. A data set may not have any mode. If all the data elements of a data set are different (no data element occurs twice), the data set has no mode. A mode is always a data element in the data set. For a perfectly symmetrical data set: the mean, median, and the mode are same. The highest point on a frequency curve of a noncontinuous data occurs at the mode.

Example 2–1 The performance of a certain type of rock bit in a given formation has to be analyzed. After searching the drilling records for the pertinent data, information given in Table 2–1 is gathered. Using this data, calculate the mean, median, and mode of the footage drilled by the bit under evaluation.1 Solution: The Column 2 of Table 2–1 lists the footage drilled by each bit. This data is sorted in an ascending order of footage drilled, Column 3. Using Equation (2.3), the mean is n

X =

∑X i =1

n

i

=

2,013 = 100.65 feet 20

Since n = 20 (even), the median is the average of the 10th and 11th record. Median =

102 + 105 = 103.5 feet 2

The mode is simply read from Table 2–1 as 102 feet (the number that appears twice)

34

Statistics and Probability Concepts

Bit

Footage

Footage

No.

Drilled, ft

Drilled, ft

1

69

53

2

72

69

3

123

72

4

135

76

5

95

80

6

89

89

7

139

90

8

109

95

9

125

102

10

80

102

11

104

105

12

53

108

13

76

109

14

102

110

15

90

115

16

116

116

17

105

123

18

108

125

19

115

135

20

110

139

n = 20

X i = 2,013

Table 2–1 Bit record for Example 2–1

Geometric, Harmonic, and Quadratic Mean. The geometric mean (symbol Gm) of a data set containing N data elements is the Nth root of the product of the individual data elements. If there are two data elements

35

Project Economics and Decision Analysis Volume II: Probabilistic Models

in a data set, the square root of the product of these two data elements is the geometric mean. Similarly, the geometric mean of three data elements is the cube root of the product of the three data elements, and so on. Mathematically, the geometric mean is given by the following equation. (2.4)

Gm =

n

X1 × X 2 × X 3 × ⋅ ⋅ ⋅ ⋅ ×X n

For a larger number of data elements, the computation of geometric mean may be simplified by using logarithms as shown below. (2.5) ⎛1 ⎜

n

⎞ ⎟

log X i ⎜ n∑ ⎟ ⎛1 n ⎞ ⎠ Gm = Anti log ⎜ ∑ log X i ⎟ = 10⎝ i =1 n ⎝ i =1 ⎠

OR in of natural logarithm, (2.5a) ⎛1 ⎞ ⎜ ln X i ⎟ ⎜n ⎟ ⎝ i =1 ⎠ n

Gm = e • • • • •

36

∑

The Gm is biased towards lower values. Therefore, it is appropriate for data sets that are skewed (asymmetrical distributions). Unlike X, the Gm is not affected much by the presence of extremely small or large data elements in a data set. Gm cannot be calculated for data sets containing any negative data elements or if any of the data elements are equal to zero. The bias for lower values may not suit in cases where disparities have to be brought out as in the case of incomes, living standard, etc. The Gm may not be the actual data element of the data set.

Statistics and Probability Concepts

The harmonic mean (symbol Hm) of a data set is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the data elements of a data set. In averaging rates and ratios involving speed, time and distance, or price, quantity and amount of money spent, etc., the following rule must apply. If the given ratios are stated as X units per Y, then use (i) Hm when Xs are given, (ii) X when Ys are given. For example, to calculate average speed in mph use Hm when miles (distance traveled) is given but use X when hours (times of journey) are given. Mathematically, Hm is calculated using the following equation. (2.6)

Hm =

n n

1

∑X i =1

• • •

i

Harmonic mean is suitable when it is desired to give greater weight to smaller data elements and less weight to the larger ones. It s for all the data elements in a data set. The Hm may not be the actual data element of its data set.

The quadratic mean (symbol Qm), also referred to as the root mean square, is calculated by dividing the sum of the squared data elements by the total number of data elements in the data set and then extracting the root of the resulting value. Mathematically, Qm is calculated using the following equation. (2.7) 1/ 2

⎞ ⎛ ⎜ ∑ X i2 ⎟ ⎟ Qm = ⎜ i =1 ⎜ n ⎟ ⎜ ⎟ ⎠ ⎝ n

37

Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 2–2 Calculate the geometric, harmonic, and quadratic means for the bit record in Table 2–1. Solution: The data of Table 2–1 is rearranged as shown in Table 2–2.

Bit

Footage, ft

Record

Drilled (X)

Log XI

1/Xi

1

69

1.8388

.0145

4,761

2

72

1.8573

.0139

5,184

3

123

2.0899

.0081

15,129

4

135

2.1303

.0074

18,225

5

95

1.9777

.0105

9,025

6

89

1.9494

.0112

7,921

7

139

2.1430

.0072

19,321

8

109

2.0374

.0092

11,881

9

125

2.0969

.0080

15,625

10

80

1.9031

.0125

6,400

11

102

2.0086

.0098

10,404

12

53

1.7243

.0189

2,809

13

76

1.8808

.0132

5,776

14

102

2.0086

.0098

10,404

15

90

1.9542

.0111

8,100

16

116

2.0645

.0086

13,456

17

105

2.0212

.0095

11,025

18

108

2.0334

.0093

11,664

19

115

2.0607

.0087

13,225

20

110

2.0414

.0091

12,100

39.8215

.2105

212,435

N = 20

Table 2–2 Data of Table 2–1 for calculating Gm, Hm, and Qm

38

Xi2

Statistics and Probability Concepts

Using Equation (2.5), the geometric mean is

Gm = 10

⎛1 n ⎞ ⎜ log X i ⎟ ⎜n ⎟ ⎝ i =1 ⎠

∑

⎛ 1 ⎞ ⎜ × 39.8217⎟ ⎠

= 10⎝ 20

= 97.97 feet

Using Equation (2.6), the harmonic mean is n 20 Hm = n = = 95.01 feet 1 0.2105 ∑ i =1 X i Using Equation (2.7), the quadratic mean is

⎛ n 2⎞ ⎜ ∑ Xi ⎟ ⎟ Qm = ⎜ i =1 ⎜ n ⎟ ⎟ ⎜ ⎠ ⎝

1/ 2

⎛ 212,435 ⎞ =⎜ ⎟ ⎝ 20 ⎠

1/ 2

= 103.06 feet

Weighted Average. The weighted average is commonly used in a number of situations, especially when the weighting of each data element in a data set is different. The term is very familiar since most of the reservoir properties (porosity, permeability, formation thickness, product price, etc.) are reported on weighted average basis. These weighted average properties are required if the intervals these properties are reported for are not of equal thickness and aerial extent. For example, if the porosity measurements are reported for each two-foot interval, then the weighted average porosity and the simple arithmetic mean of these porosity measurements is same. Similarly, the averages of many financial related data are calculated using the weighted average. For example, if 100 shares of a company are bought at $5 per share and another 150 shares of the same company are bought at $2.5 per share, the average price per share is not simply ($5 + $2.5)/2 = $3.75. In this case, the weighted average price per share is calculated since the number of shares purchased at each price is different.

39

Project Economics and Decision Analysis Volume II: Probabilistic Models

The weighted average is therefore, an average obtained by application of an appropriate weighting factor to each observation. Mathematically, the weighted average is. (2.8) n

XW =

∑w X i

i

i =1

n

∑w

i

i =1

where wi = weighting factor of the data element Xi Like arithmetic mean (X), weighted geometric mean (symbol Gwm), and weighted harmonic mean (symbol Hwm) can also be calculated using the following equations. (2.9)

⎛ ⎞ ⎜ ∑ wi log X i ⎟ ⎟ = anti log⎜ i =1 n ⎜ ⎟ ⎜ ∑ wi ⎟ i =1 ⎝ ⎠ n

Gwm

(2.10)

n

H wm =

∑w

i

i =1 n

wi

∑X i =1

i

Example 2–3 A porosity log recently run in a well has been analyzed. A total of 22foot pay has been identified with the porosity and pay thickness as shown in Table 2–3. Calculate the weighted average porosity of this formation.

40

Statistics and Probability Concepts

Porosity, %

Thickness, ft

φ

h

φxh

15.2

2

30.4

10.5

3

31.5

13.2

5

66.0

18.6

2

37.2

11.4

6

68.4

12.6

4

50.4

Σ h = 22

Σ φ h = 283.9

Table 2–3 Porosity Data for Example 2–3 Solution: The weighted porosity (h) is calculated as shown in Column 3 of Table 2–3. Using Equation (2.8), the weighted average porosity is n

w

=

∑

h

i i

i =1 n

∑h

=

283.9 ≅ 12.9 22

i

i =1

Example 2–4 A company is ready to invest in a new project that will cost $500,000. Out of the total capital of this $500,000, $150,000 will be equity at a cost of 8% and $350,000 will be a long-term debt at a cost of 18%. What is the weighted average cost of capital for this project? Solution: Using Equation (2.8), the weighted average cost of capital is, iw =

$150,000 × 0.08 + $350,000 × 0.18 $500,000

=

$12,000 + $63,000 × 100 = 15.0% $500,000 41

Project Economics and Decision Analysis Volume II: Probabilistic Models

Choice of a Suitable Average As presented in the preceding pages, there are many measures of central tendency (averages). Each one of these measures has its own characteristics and usefulness. The question now is, which measure to use in which situation. The answer to this question depends upon the consideration of the nature of data and the ultimate purpose of investigation. If proper choice is not made, then the comparison of these measures from one data set to another is misleading and the conclusions thus obtained are not trustworthy. The justification of employing one measure against another must be determined by an appeal to all the facts in light of the peculiar characteristics of the different types. In Example 2–1 and Example 2–2, the difference between various measures of central tendency is not significant (within 5% of the arithmetic mean). However, this may not be the case for other data sets. Following are some guidelines to follow when making a choice between particular measures of central tendency to use. If all the values in a data set are to be given equal importance, use arithmetic mean. To determine the most frequently occurring item, mode should be used. If the objective is to determine the position or ranking of a data element in relation to all other data elements, median should be used. If smaller data elements are to be given greater importance than the larger data elements, use geometric or harmonic mean. The measure chosen should represent the basic characteristics of the entire data set. For example, the geometric mean best represents the ratio or percentage changes while the arithmetic mean represents changes in absolute magnitudes. The choice depends upon the nature of the distribution of data elements. •

• •

42

If the data elements are distributed symmetrically or nearly symmetrical then mean, median or mode may be used almost interchangeably. If the distribution is skewed in one direction, use mode or median. When a data set contains open-end classes or when it has extreme values, both the mode and median are suitable. The positional measures are suitable for data sets that contain few extraordinary

Statistics and Probability Concepts

•

•

• •

data elements not representative of the rest of the data. For example, median is commonly used to report wages of workers since these may contain very high or low wage rates. If a frequency distribution (to be discussed in the following sections) involves varying class intervals, then median is the most suitable average because its computation is based on the size of the class interval of the median group. In this case, the arithmetic mean may not be very suitable as it may greatly differ from the mean of ungrouped data. The calculation of mode also requires the size of class interval should be the same for pre-modal class, modal class, and post-modal class. In case of J or reverse J-shaped curve, the median is more appropriate. For example, in case of price distribution it is preferred to use median. If there is no wide variation in the data elements of a data set, the arithmetic mean is more suitable. Geometric mean is recommended for data sets comprised of rates, percentages or ratios instead of actual observations or quantities.

If the measure of central tendency has to be used in further analysis, such as probability distributions, the arithmetic mean is commonly used. It possesses many mathematical properties due to what is included as a variable in many other statistical calculations (calculating standard deviation, etc.). When it is desired to give relative importance to the different data elements of a data set, weighted arithmetic mean is calculated. The arithmetic mean of permeability (weighted for formation thickness if necessary) represents the upper bound of performance, and the harmonic mean represents the lower limit. The arithmetic mean is equivalent to the effective permeability of a completely stratified reservoir composed of parallel layers, while harmonic mean is equivalent to the effective permeability of formation in which layers are arranged in series. Reservoirs composed of layers arranged randomly could be expected to have performance intermediate between these two extremes. The average permeability computed from pressure buildup tests usually falls somewhere between the arithmetic and harmonic means. 43

Project Economics and Decision Analysis Volume II: Probabilistic Models

Following are some general rules ascertaining the differences in various measures of central tendency. • •

•

•

For perfectly symmetrical distributions of data, mean = median = mode. If the values of all data elements in a distribution are not identical, the geometric mean is smaller than the arithmetic mean and the harmonic mean, i.e., X Gm Hm. In distributions with higher concentration of lower values and a long tail towards higher values: the X has the highest value, the mode the lowest, and the median is about one-third the distances from mean towards the mode. In distributions with higher concentration of larger values and a long tail towards lower values: the X has the lowest value, the mode the largest, and the median is about one-third the distances from mean towards the mode. If any two measures of a moderately skewed distribution are known, the third one can be approximated using the following empirical relationships.

The mode is unstable measure of central tendency; its value depends upon the size of the class intervals used in grouping data into a frequency distribution. A change in the size of the class interval changes the value of the mode.

Mean = Mode = 3 (Mean − Median ) 3 Mean = Mode + (Median − Mode ) = 2 Mode = 3 Median − 2 Mean 2 Median = Mode + (Mean − Mode ) = 3

1 3(Median − Mode ) 2 1 2(Mean + Mode ) 3

Measures of Variability The measures of central tendency of a set of data elements provide a single descriptive measure of the most likely value of the entire data set.

44

Statistics and Probability Concepts

However, they do not show how the independent data elements within the data set are clustered around this most likely value. Therefore, variability of the data is just as important as the measure of central tendency. In fact, the description of a data set is incomplete without this additional measure. To ascertain a complete picture of the clustering of the data elements around the mean, other measures such as (a) (b) (c) (d) (e)

range variance standard deviation mean absolute deviation and coefficient of variation are presented in this section.

Range, Variance, and Standard Deviation. The range (symbol R) of a data set is the difference between the highest and the lowest values in the data set. When data are arrayed in ascending order, the range is simply the value of the last data element minus the value of the first data element. The range of the footage drilled by the 20 bits presented in Table 2–1 is 86 (139 – 53 = 86). The range is given as a single value, 86 feet, although in practice it is common to say, “the range is 53 to 139.” Mathematically, the range is given by the following simple equation. (2.11)

R = X max − X min

The range is not a very useful measure of dispersion because (1) it uses only two data elements (the extreme values) of the entire data set while ignoring all others, and (2) it gives no idea of how the data is clustered. The commonly used measure of dispersion is the standard deviation (symbol s). It gives a numerical value used to describe the clustering tendency of the data. The numerical value is presented in the same units as the unit of the data elements. A large standard deviation is an indication of wider spread of the data elements around the mean. A small standard deviation indicates the data is more closely clustered around the mean. This measure brings out how two data sets with the same mean value may

45

Project Economics and Decision Analysis Volume II: Probabilistic Models

have wide differences in the spread of their individual data elements around the mean. Mathematically, the standard deviation is calculated by using the following equation. (2.12) 2⎤ ⎡ n ⎢ ∑(X i − X ) ⎥ ⎥ s = ⎢ i =1 n −1 ⎥ ⎢ ⎥⎦ ⎢⎣

1/ 2

OR (2.12a)

⎤ ⎡ n 2 ⎥ ⎢∑ Xi n s = ⎢ i =1 X 2⎥ − ⎢ n −1 n −1 ⎥ ⎥⎦ ⎢⎣

1/ 2

The mean (X) and standard deviation (s) together give more information than any other measure. They tell a great deal about the observed data. The deviation from the mean is typically reported as X s. The term (Xi - X )2 in Equation (2.12) is referred to as the mean squared deviation. The standard deviation is the average of the mean squared deviation. The n–1 is used to obtain an unbiased estimate of the standard deviation for a sample. The bias effect diminishes as the sample size n approaches the size of the population from which n is extracted. The standard deviation of a population is represented by the symbol while the n replaces (n–1) in Equation (2.12). Technically, the n–1 represents degrees of freedom and is reduced by 1 because the mean (X) represents one of the , which is an estimate by itself. The variance (symbol s2) is simply the square of the standard deviation. Physically the variance has no direct implementation in descriptive statistics. Mean Absolute Deviation. The mean absolute deviation (symbol dm) is defined as the average deviation of data from the mean over all observations of the data. Mathematically, the mean absolute deviation is given by the following equation. 46

Statistics and Probability Concepts

(2.13) n

dm = n

where

∑ (X i =1

i

∑ (X

i

i =1

−X)

n

− X ) is absolute.

In practice, the following equation can use the dm to approximate the more detailed computation of standard deviation (s). However, this approximation is limited to symmetric (bell shaped) frequency distributions. (2.14)

s = 1.25d m

Coefficient of Variation. The coefficient of variation (symbol ), also referred to as the coefficient of mean deviation, is defined as the ratio of the standard deviation to the mean of a data set. It is used to express the standard deviation as a percentage of the mean. Mathematically, the coefficient of variation is calculated using the following equation. (2.15)

s υ = × 100 X

For example, a certain data is collected one day. After one month, the same process is repeated to collect another set of data. By comparing the v values of the two different data sets, the stability of standard deviation with respect to the mean can be determined. The coefficient of variation is especially useful when comparing data sets, which have different units (porosity in percent and reservoir drainage in acres). It is also a useful parameter for representing model sensitivity among different units.

47

Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 2–5 Calculate (a) range, (b) standard deviation, (c) variance, (d) mean absolute deviation, and (e) coefficient of variation for the bit record in Table 2–1. Solution: The bit records of Table 2–1 are reiterated in Table 2–4, along with some additional data manipulation.

Footage

Absolute

Bit No.

Drilled, ft

N

X

X- X

(X - X )

X

2

X- X

1

69

(31.65)

1,001.72

4,761

31.65

2

72

(28.65)

820.82

5,184

28.65

3

123

22.35

499.52

15,129

22.35

4

135

34.35

1,179.92

18,225

34.35

5

95

(5.65)

31.92

9,025

5.65

6

89

(11.65)

135.72

7,921

11.65

7

139

38.35

1,470.72

19,321

38.35

8

109

8.35

69.72

11,881

8.35

9

125

24.35

592.92

15,625

24.35

10

80

(20.65)

426.42

6,400

20.65

11

102

1.35

1.82

10,404

1.35

12

53

(47.65)

2,270.52

2,809

47.65

13

76

(24.65)

607.62

5,776

24.65

14

102

1.35

1.82

10,404

1.35

15

90

(10.65)

113.42

8,100

10.65

16

116

15.35

235.62

13,456

15.35

17

105

4.35

18.92

11,025

4.35

18

108

7.35

54.02

11,664

7.35

19

115

14.35

205.92

13,225

14.35

20

110

9.35

87.42

12,100

9.35

2,013

0

9,826.55

212,435

362.4

Arithmetic Mean, X =

100.65

Feet

Table 2–4 Data of Table 2–1 used for Example 2–5 48

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Project Economics and Decision Analysis Volume II: Probabilistic Models

In this particular example, using Equation (2.14) and the dm calculated above gives the same standard deviation as calculated above, so s =1.25dm = 1.25 x 18.12 = 22.65 feet.

Descriptive Statistics from Grouped Data In the preceding section, raw (ungrouped) data was analyzed. In this section, descriptive statistics parameters are calculated for grouped data. Grouping is usually not required for smaller data sets. However, grouping of larger data sets greatly simplifies the calculations of various parameters defined above. The common method of condensing data is to group them into classes. As shown in Figure 2–1, the grouped observations can be arranged in discrete or continuous series of observations. In the discrete form of distribution, the frequency refers to a given discrete value and not to a range of values. Discrete variables are the ones where the individual values differ from each other by definite amounts. For example, houses in a city, children per family, students in a class can be given in whole numbers. In the continuous form of distribution, the frequencies refer to group of values. Continuous variables are the ones that can take any intermediate value between the smallest and the largest value in the distribution (height, temperature, pay thickness, porosity, permeability, water saturation, reserves, etc.). The grouping may be done directly from source data, an unordered data list, or an array. The important exercise in analyzing grouped data is to: 1. Define classes for the data set to be analyzed. 2. Determine the frequency (number of times) of data elements appearing in each class. 3. Calculate the absolute and relative frequency. The frequency distribution showing the actual number of data elements falling in each class is called absolute frequency distribution. The proportion or percentage of data elements falling in each class is referred to as relative frequency distribution. Both the absolute

50

Statistics and Probability Concepts

and relative frequency distributions may be either cumulative (showing number of data elements in and below each class or decumulative (number of data elements in or above each class). 4. Calculate mid point or class mark (CM) of each class. This is done by taking average of the lower and upper limits of each class. 5. Perform calculations. The appropriate number of classes for a data set can be determined by using some mathematical approximations. However, it is usually more convenient to apply subjective criteria. The following rules assist in defining the classes in a frequency distribution (number of times data appearing in each class). 1. The number of classes should be between 5 and 20, inclusive. The choice of the number of class intervals depends on: • • • •

Number of data items to be classified, Magnitude of the data items, Accuracy desired, and The ease of calculation for further processing of the data.

The number of classes and the size of the class interval are inversely related, (if one is increased, the other is automatically squeezed). Lesser number of classes would result in the size of a class interval being too big. This may contribute to the inaccuracy of the results and camouflage their characteristics. On the other hand, smaller size makes the data too fragmented to give only a broad picture of the distribution as a whole. Alternatively, the number of classes may be obtained using the following equation (N = number of total data elements in a data set). (2.16)

Number of Classes ≅ 1 + 3.322 log N

51

Project Economics and Decision Analysis Volume II: Probabilistic Models

2. The size of the class, or class interval (class width), should be the same for the entire data set. The following equation may be used to calculate the class width. (2.17)

X max − X min Class Width = Number of Classes 3. The classes must be defined so every data element in the data set falls into one and only one class. The endpoints, or boundaries, of the classes should avoid overlap (a situation in which one data element could be in either of two classes). 4. No class should be empty, (no data falling in it). This may be avoided by redefining the number and/or interval of classes or by using an open-end class (greater than or less than, such as less than 20 or greater than 50 etc.). Once the above exercise is complete, the following equations are used to calculate the desired measures of central tendency and measures of variability of grouped data. Mean, Median, and Mode. (2.18) n

X =

∑ f (CM ) i

i

i =1

n

∑f

i

i =1

where CM = class mark, mid value of individual class fi = the frequency of individual class (2.19)

⎛n −a⎞ ⎟ Median = L + CI ⎜ 2 ⎜ b ⎟ ⎝ ⎠ 52

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 2–6 Group the bit records given in Table 2–1. Use this grouped data to calculate the measures of central tendency and variability. Compare the parameters calculated from the grouped data with those calculated in Examples 2–1, 2–2, and 2–5. Solution: The first step in solving this problem is to group the data of Table 2–1. Using Equation (2.16), the number of classes for N=20 bit records is

Number of Classes ≅ 1 + 3.322 log N ≅ 1 + 3.322 log (20) = 5.322 ≅ 5 Using Equation (2.17), approximate class width is

Class Width =

X max − X min 139 − 53 = = 17.2 ≅ 17 Number of Classes 5

The data is grouped, using class width of 17, as shown in Table 2–5.

Class

Class

Mark

Frequency

CM

f

f(CM)2

f(CM)

f/CM

52 — 69

60.5

2

121.0

7,320.50

8.2053

0.0331

70 — 87

78.5

3

235.5

18,486.75

13.0893

0.0382

88 — 105

96.5

b=6

579.0

55,873.50

27.4173

0.0622

106 — 123

114.5

6

687.0

78,661.50

28.4434

0.0524

124 — 141

132.5

3

397.5

52,668.75

14.6597

0.0226

2,020.0

213,011.00

91.8150

0.2085

Σ f = 20

Table 2–5 Grouped data of Table 2–1 for Example 2–6

54

f ln(CM)

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Graphical Presentation of Data Data may be graphically presented in many different ways. In this section, three important graphs (histograms, ogives, and frequency curves) are discussed. These are generated from the frequency distribution of grouped data. A frequency distribution, also called a histogram, is an elementary graphical format used to present distribution of the frequencies over the observed values of the variable. The histogram may be used for ungrouped (usually used for discrete variables) or grouped data (for discrete or continuous variables). Histograms for grouped data is discussed here. Frequency distribution of ungrouped data do not give a good idea of how the data is distributed since most observed data elements are different. The three graphs presented in this section are extensively utilized in the following chapters on decision analysis where they represent continuous probability distributions. The horizontal or X-axis of a histogram displays boundaries of the classes. The vertical or Y-axis of a histogram represents frequency at which the data elements occur in each class. The frequencies may be plotted at the mid point (class mark) of the class boundaries while the cell boundaries are also indicated on the graph. The histogram consists of rectangles formed by the class boundaries (along the X-axis). The heights of these rectangles (along the Y-axis) represent the frequency of occurrence of the data being analyzed. Histograms are typically drawn without gaps between class rectangles. A frequency distribution showing the actual number of data elements in each class is called an absolute frequency distribution. If a frequency distribution shows the percentage or proportion of data items in each class, it is called relative frequency distribution. Both of these distributions can be shown as cumulative by including the number (or proportion) of data elements or frequency in and below each class or decumulative by including the number (or proportion) of data elements or frequency in and above each class. In these cases, only the Y-axis is changed so it represents the relative, cumulative or de-cumulative frequency rather than the absolute frequency. If the number of observations less than some class boundary is desired, use cumulative plot. If the number of observations greater than some class boundary is desired, use de-cumulative plot. A histogram with cumulative frequency distribution slopes upward while a histogram with 56

Statistics and Probability Concepts

de-cumulative frequency distribution slopes downward. In the cumulative form, the relative frequency of observations in each class is progressively accumulated, so the histogram bar at the extreme right equals 100% along the cumulative scale. The choice of the type of histogram depends on its use. To compare histograms, prepared using different number of observations, they must be plotted with relative frequencies rather than absolute frequencies. Therefore, the scale along the Y-axis is expressed as a percentage of the total number of data elements. Otherwise, the scales of the Y-axis of different histograms differ according to the number of observations in the data set. If obtaining values in number of items is desired, absolute frequency distribution is used. If the purpose is to determine percentages, it should be based on relative frequency. A frequency curve is a frequency polygon in which the data plots are fitted to a smooth curve. This curve is based on the data of the histogram but reflecting area under the curve rather than a length. The following equation is used to fit a normal curve to a histogram. (2.23)

f ( x) =

n × CI s 2π

⎛ x− X − 0.5 ⎜⎜ e ⎝ s

⎞ ⎟ ⎟ ⎠

2

where CI = length of class interval used to draw the histogram n = number of observations, n=1 if histogram is drawn on proportional frequency. Another very useful form of presenting the cumulative or decumulative relative frequency distribution is to translate the respective histogram into a corresponding ogive. The purpose of this graph is to determine percentages. The ogive may be constructed from an absolute frequency distribution to obtain values in numbers of items rather than percentages or proportions. For a cumulative (equal to or less than type) ogive, the cumulative frequencies (relative or absolute) are plotted at the upper boundary of each class. For a decumulative (equal to or greater than type) ogive, the decumulative

57

Project Economics and Decision Analysis Volume II: Probabilistic Models

frequencies (relative or absolute) are plotted at the lower boundary of each class. As a visual aid, the cumulative ogive faces right while the decumulative ogive faces left. For a perfectly symmetrical data, the ogive results in a lazy S-shaped curve. In the section on theoretical probability distributions, various useful forms of probability distributions are presented. These are xy plots relating probability to value but similar to the frequency curve and ogive. The ogive of a cumulative probability distribution is called a cumulative distribution function (CDF). The companion plot (frequency curve), called a probability density function (PDF) is simply the derivative of the CDF. The steps in graphing a frequency distribution are: 1. Group the data to be analyzed into appropriate classes, using the guidelines presented in the previous section. 2. Record the number of observations (frequency) falling in each class. 3. Calculate the cumulative frequency by progressively accumulating the frequencies. 4. Calculate the relative frequencies. 5. Plot the data as required. The following example assists in understanding the graphing of frequency distributions.

Example 2–7 Construct a histogram, frequency curve, and ogives (cumulative and decumulative) for the bit records given in Table 2–1. Solution: The grouped bit records of Example 2–6 are used to start with the problem. For this grouped data, the cumulative frequency, cumulative relative frequency, and decumulative relative frequency calculations are shown in Table 2–6.

58

Statistics and Probability Concepts

Class

Class 52 — 69

Cumulative

Cumulative

Decumulative

Mark

Frequency

Frequency

Relative

Relative

CM

f

CF

Frequency

Frequency

2/20 = 0.10

20/20 = 1.00

60.5

2

2

70 — 87

78.5

3

3+2=5

5/20 = 0.25

18/20 = 0.90

88 — 105

96.5

6

6+5=11

11/20 = 0.55

15/20 = 0.75

106 — 123

114.5

6

6+11=17

17/20 = 0.85

9/20 = 0.45

124 — 141

132.5

3

3+17=20

20/20 = 1.00

3/20 = 0.15

20

Table 2–6 Frequency table for the bit records of Table 2–1 Column 3 of Table 2–6 gives the frequency of occurrence of various footages drilled by bits in the assigned class interval, Column 4 shows the cumulative frequency, Column 5 shows the cumulative relative frequency as a proportion of the total frequency, and Column 6 shows the decumulative relative frequency. The histogram representing Column 1 and Column 3 is shown in Figure 2–3. Figure 2–4 shows the cumulative frequency distribution representing Column 1 and Column 5. Figure 2–5 shows the decumulative frequency distribution representing Column 1 and Column 6.

Fig. 2–3 Frequency distribution of footage drilled by bits in Table 2–1 59

Project Economics and Decision Analysis Volume II: Probabilistic Models

Fig. 2–4 Cumulative relative frequency curve for footage drilled by bits in Table 2–1

Fig. 2–5 Decumulative relative frequency curve for footage drilled by bits in Table 2–1 In some cases, the graphs generated from a set of data become more meaningful and useful by transforming the data into a different form. The most popular transformations include ln(x), x2, xn, ln(x+a), 1/(x+a), etc. For example, sandstone permeabilities can be represented by a normal distribution if the data is transformed into a form k' = lnk / ln(2). This transformed permeability plots as a straight line on probability graph paper. In addition, the logs of field sizes (transformation of field sizes into logarithmic values) often plot as a straight line on Cartesian probability paper (or field sizes plot as a straight line on log-probability paper). 60

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Fig. 2–6 Normal curve fitted to the histogram of Figure 2–3, using data of Table 2–1

Other Measures The mean, median, and mode were discussed in the section on measures of central tendency. To reiterate, median is a measure dividing a data set into two halves. A few more measures, referred to as fractiles, representing different parts of a data set are discussed in this section. The common fractiles are (a) quartiles, (b) dociles, and (c) percentiles. Quartiles. Quartiles, as the name implies, divide a data set into four equal parts. As shown in Figure 2–7, the first quartile (Q1) segregates the first and second parts; the second quartile (Q2) segregates the second and third parts; and the third quartile (Q3) segregates the third and fourth parts. The median and the second quartile are the same and can be found at the [(N + 1) / 2]th position in an array. Similarly, the first quartile can be treated as the median of the fist half of a data set while the third quartile as the median of the last half of a data set. The first quartile can be located at the [(N + 1) / 4]th position and the third quartile at the [3(N + 1) / 4]th position in an array.

62

Statistics and Probability Concepts

Fig. 2–7 Pictorial presentation of quartile and interquartile range For grouped data, quartiles can be calculated by using the following equations. These equations are similar to Equation (2.19) used for calculating the median from grouped data. (2.24)

⎛ n / 4 − a1 ⎞ ⎟⎟ Q1 = L1 + CI ⎜⎜ ⎝ b1 ⎠ (2.25)

⎛ 2n / 4 − a2 ⎞ ⎟⎟ Q2 = L 2 + CI ⎜⎜ b2 ⎝ ⎠ (2.26)

⎛ 3n / 4 − a3 ⎞ ⎟ Q3 = L3 + CI ⎜⎜ ⎟ b3 ⎝ ⎠

where L1 , L2 , L3 = lower limits of the classes containing Q1, Q2, and Q3, respectively a1, a2, a3 = are analogous to a in the equation for computational median b1, b2, b3 = are analogous to b in equation for computational median

63

Project Economics and Decision Analysis Volume II: Probabilistic Models

Dociles and Percentiles. Dociles (symbol D) divide a data set into 10 equal parts, and percentiles (symbol P) divide a set into 100 equal parts. The 5th docile (D5) and the 50th percentile (P50) are both equal to the median. The following steps are to be followed for calculating fractiles for ungrouped data. 1. Order the data in ascending order of value. 2. Determine the desired fractile and the associated k value (maximum number of divisions) where k=100 for percentiles, k=10 for docile and k=4 for quartile. 3. Calculate the position indicator (F) using Equation (2.27). 4. Count through the ungrouped, ordered data until the value in Step 3 is reached. This observed value is the desired fractile. If the value obtained in Step 3 is not an integer, use linear interpolation to approximate the desired fractile. (2.27)

i F = (n + 1) k

where F = fractile indicator (Pi, Di or QI) is the position at which the fractile is read from the ungrouped data. i = the magnitude of the fractile desired, i.e. i=7 if D7 is desired and i=50 if P50 is desired, and so on.

Example 2–9 For the ungrouped bit records in Table 2–1, determine the footage drilled for (a) 50th percentile, (b) 7th docile, (c) 3rd quartile, and (d) 5th docile. Solution: The data is arranged in ascending order as shown in the second column of Table 2–1.

64

Statistics and Probability Concepts

(a) For P50, k=100, i=50, and n=20: using Equation (6.27) F ( P50 ) =

50 (20 + 1)) = 10.5 100

Therefore, the P50 is the average of the 10th and 11th value, i.e. (102+105)/2 = 103.5 feet. This value is the same as the median calculated in Example 2–1. (b) For D7, k=10, i=7, and n=20: using Equation (2.27) F ( D7 ) =

7 (20 + 1) = 14.7 10

For linear interpolation, the bit records are read at the 14th and 15th positions. These are 110 and 115 at the 14th and the 15th positions, respectively. By linear interpolation, the D7 is D7 = 110 + 0.7(115 − 110 ) = 113.5 feet

(c ) For Q3, k=4, i=3, and n=20: using Equation (2.27) F (Q3 ) =

3 (20 + 1) = 15.75 4

The Q3 value is obtained by interpolation between the values at the 15th and 16th position. Therefore, Q3 = 115 + 0.75(116 − 115) = 115.75 feet

(d) for D5, k=10, i=5 and n=20: using Equation (2.27) F ( D5 ) =

5 (20 + 1) = 10.5 10

Note the position of the 5th docile is the same as the position of the 50th percentile calculated in (a). Therefore, the 5th docile is 103.5 feet. For grouped data, a graphical method can be used to approximate the desired fractile. The desired fractile can be read directly from cumulative relative frequency curves. 65

Project Economics and Decision Analysis Volume II: Probabilistic Models

Coefficient of Peakedness and Skewness. These additional equations may be used to determine the peakedness and skewness of the frequency distributions. The coefficient of peakedness a4 is dimensionless and computed as follows. (2.28)

fourth central moment m4 a4 = = 4 ( s)4 s

∑ f (x n

i

i

i =1

− X)

4

n s4

=

A dimensionless coefficient of skewness a3 may be computed as: (2.29)

third central moment m3 a3 = = 3 (s)3 s n

∑ f (x − X )

3

i

i

i =1

=

n s3

If a3 = 0, symmetric, the curve is bell shaped a3 > 0, the curve is skewed to the right a3 < 0, the curve is skewed to the left

Spreadsheet Application Most of the information presented in the preceding sections can be easily generated by the built-in capabilities of Microsoft Excel or Lotus 1-2-3. Table 2–8 shows an Excel spreadsheet for the descriptive statistics of Example 2–1 and Example 2–2 for the data in Table 2–1. When using this option of the spreadsheet, the data do not have to be arranged in ascending or descending order. 66

Statistics and Probability Concepts

A

B

3

C

D

E

Footage Drilled, feet

4

69

95

125

76

105

5

72

89

80

102

108

6

123

139

102

90

115

7

135

109

53

116

110

20

=COUNT(A4:E7)

8 9 Number of Data Elements 10 Arithmetic Mean, ft

100.65

=AVERAGE(A4:E7)

11 Median, ft

103.50

=MEDIAN(A4:E7)

12 Mode, ft

102.00

=MODE(A4:E7)

13 Geometric Mean, ft

97.97

=GEOMEAN(A4:E7)

14 Harmonic Mean, ft

95.02

=HARMEAN(A4:E7)

15 Standard Deviation, ft

22.74

=STDEV(A4:E7)

16 Variance, ft

517.19

=VAR(A4:E7)

Table 2–8 Descriptive statistics (spreadsheet application) Alternatively, the descriptive statistics option of the Data Analysis option of the Excel tools can be used (This requires use of the Analysis Tool Pak in the Excel Tool Add-ins). However, this option does not give the harmonic and geometric means, but it does give additional parameters of kurtosis (peakedness) and skewness of the data distribution. The following steps are followed for generating descriptive statistics using this option. 1. Arrange the raw data of Table 2–1 in a spreadsheet shown in Table 2–9. 2. Click on Tools in the tool bar menu at the top of the spreadsheet. 3. Click on Data Analysis in the drop down menu. 4. Select Descriptive Statistics from the Data Analysis dropdown menu and click on OK. The Descriptive Statistics dropdown menu appears. 5. Specify the input range by entering the first and the last cell positions of the column containing the data to be analyzed. The input range input is $A$5:$A$24. 67

Project Economics and Decision Analysis Volume II: Probabilistic Models

6. Select “Grouped by Columns,” since the data to be analyzed is in a column. 7. Specify the Output Range where the analysis is printed, in this case the Cell $C$4 is specified. 8. Select Summary Statistics and click on OK, the analysis is printed as shown in Table 2–9.

A

B

C

D

3

Footage

4

Drilled, ft

5 6

53 69

Mean

7

72

Standard Error

8

76

Median

9

80

Mode

10

89

Standard Deviation

11

90

Sample Variance

Column 1 100.65 5.09 103.5 102.00 22.74 517.19

12

95

Kurtosis

(0.38)

13

102

Skewness

(0.29)

14

102

Range

86.00

15

105

Minimum

53.00

16

108

Maximum

139.00

17

109

Sum

18

110

Count

19

115

20

116

21

123

22

125

23

135

24

139

2,013.00 20.00

Table 2–9 Descriptive statistics (alternative spreadsheet application)

The histogram and cumulative relative frequency distribution can also be generated using the built-in Data Analysis capabilities of Excel, the following steps achieve this. 68

Statistics and Probability Concepts

1. Arrange the raw data of Table 2–1 in a spreadsheet as shown in Table 2–10. 2. Click on Tools in the tool bar menu at the top of the spreadsheet. 3. Click on Data Analysis in the drop down menu. 4. Select Histogram from the Data Analysis dropdown menu and click on OK. The Histogram dropdown menu appears. 5. Specify the input range by entering the first and the last cell positions of the columns containing the data to be analyzed. The input range input is $A$4:$E$7. 6. Specify the Bin Range as $A$10:$A$13 (the upper boundary of each class interval). If the Bin Range is left blank, then Excel calculates its own class width and class intervals. 7. Specify the Output Range where the analysis is printed, in this case Cell $C$9 is specified. 8. Select Cumulative Percentage and Chart Output. Click on OK, the analysis is printed as shown in Table 2–10. Some formatting is required in order to appropriately place and size the graph as shown in Table 2–10.

A

B

3

C

D

E

Footage Drilled, feet

4

69

95

125

76

105

5

72

89

80

102

108

6

123

139

102

90

115

7

135

109

53

116

110

8 9

Bin

Bin

Frequency

Cumulative

10

69

69

2

10.00%

11

87

87

3

25.00%

12

105

105

6

55.00%

13

123

123

6

85.00%

3

100.00%

14

More

Table 2–10 Histogram and cumulative frequency curve for data of Table 2–1, using Excel Tools 69

Project Economics and Decision Analysis Volume II: Probabilistic Models

Table 2–10 continued . . .

UNDERSTANDING PROBABILITY CONCEPTS Decision theories are based on fundamental laws of probability and expected value derived from them. The empirical probability concept, based on experimental tests, provides scope for the application of probability in real-life situations. The use of probability theory is increasing in general economic decision making and the concept is very helpful in situations of risk and uncertainty. In order to make proper decisions in the face of uncertainty, the fundamental laws of probability are used. The theory of probability enables a person to make educated guesses on matters where either full fact are not known or there are uncertainties about the outcome. Probability employs a number of mathematical concepts to

70

Statistics and Probability Concepts

analyze past experience. It also employs personal beliefs after proper testing and verification. Probability is used informally in everyday life, such as: • • • •

Possibly, it will rain tonight. There is a high chance of you getting a job in October. This year’s demand for the product is likely to exceed last year’s demand. The odds are 2:1 in favor of Denver Broncos to win this year’s Super Bowl.

The above sentences, with the words possibly, high chance, likely, and odds are expressions indicating a degree of uncertainty associated with respective events. The probability concepts discussed in this chapter put such conjectures into formal quantitative measures. The appropriate use of probability can assess uncertainty, which is the central principle of decision analysis. In this section, some of the basic principles of working with probability and probability models are discussed. The objective is to be able to create and analyze a model that represents uncertainty. In the following chapters, a variety of ways to model uncertainty in decision problems by using probability are discussed. A broad classification of the concepts of probability is presented in Figure 2–8.

Fig. 2–8 Broad classification of the concepts used in probability 71

Project Economics and Decision Analysis Volume II: Probabilistic Models

Objective Probability The objective probability is based on certain undisputed laws of nature or on some experiments conducted for the purpose. It is not based on the impressions of the individuals, as is the case with subjective probability. These theories are, therefore, free from personal bias and ensure objectivity. There are two approaches to objective probability (1) classical approach and (2) empirical approach. Classical Approach. The classical approach is based on certain a priori laws of nature. It employs deductive logic to derive probability measures from certain undisputed laws of nature. Being based on abstract mathematical logic, it is also called as abstract or mathematical. This is why a considerable use is made of familiar objects like flip of coin, dice, cards, etc. The theory of chance requires identifying the total number of all possible outcomes of an event and the number of outcomes favorable to the event whose probability is sought. For example, the number of all possible outcomes on a flip of coin are two (head and tail) and the number of outcomes each time a coin is flipped is one (either head or tail). The ratio of the number of favorable outcomes and all the possible outcomes is the measure of probability. Therefore, it is a fraction, whose numerator is the number of favorable outcomes and whose denominator is the number of all possible outcomes. Therefore, the probability of occurrence of an event expressed as P(A) is given by: (2.30)

m P( A) = n

where m = the favorable cases n = the total number of all possible cases and the probability of non-happening of event A, also referred to as the complement of A, expressed as P(A) is given by: n−m P( A ) = n 72

(2.31)

Statistics and Probability Concepts

The previous concept can also be stated as the law of proportion or the law of chance. This is stated as follows: Wherever a particular action can have more than one outcome, then if all possible outcomes have an equal chance of occurring, the probability of any one of them occurring in a single test is the proportion that a particular outcome bears to all the possible outcomes.

Example 2–10 A box contains 56 black balls, 20 white balls, and 24 green balls. If one ball is randomly drawn from the box, what is the probability the ball drawn will be black, white, or green? The ball drawn each time is replaced in the box before drawing another ball. Solution: Total number of all cases possible is 100 (56+20+24=100) Black =

56 = 0.56 or 56% 100

White =

20 = 0.20 or 20% 100

Green =

24 = 0.24 or 24% 100

The three basic conditions implicit in the classical approach are that the events are (1) equally likely, (2) collectively exhausted, and (3) mutually exclusive. Equally Likely. According to this assumption, there is symmetry and homogeneity in the occurrence of events. Such an assumption is necessary for determining the chance of occurrence of an event. This principle is also called the principle of insufficient reason to believe

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otherwise. For example, the assumption of fair dice, fair coin, fair deck of cards is very basic to predict the chance of occurrence. If the coin is not fair, how can the chance of getting a head or a tail be equal or 50:50? Collectively Exhaustive. The sum of all favorable and unfavorable cases should be equal to the total number of cases (the number of favorable cases cannot be greater than the total number of cases). It also implies all possible events have been known intuitively if not through experiments. For example, the outcome in a toss of a fair coin can be either a head or a tail; there is no third possibility. The outcome in a throw of a die can be any one of six possibilities: 1, 2, 3, 4, 5 or 6 dots. Similarly, black and red cards are examples of collectively exhaustive events in a draw from a deck of cards. Mutually Exclusive. The events are said to be mutually exclusive (dist) if they cannot occur simultaneously in a single draw. Therefore, the occurrence of one automatically excludes the occurrence of other. For example, in a single throw of a die, either an odd number or an even number will come up.

Example 2–11 What is the chance of getting (a) queen, (b) a club, and (c) no club in a draw of a card from a pack of 52 cards? Each card is replaced in the pack before the successive card is drawn. Solution: (a) The number of cases that can happen = 52 Number of favorable cases (total number of queens) = 4

∴ Probability of drawing a queen =

74

4 1 = = 0.0769 or 7.69% 52 13

Statistics and Probability Concepts

(b) The probability of drawing a club in a single draw is P (Club) =

13 1 = = 0.25 or 25% 52 4

(c) The probability no club will be drawn is

⎛ 52 − 13 ⎞ 3 P (No Club) = ⎜ ⎟ = = 0.75 or 75% ⎝ 52 ⎠ 4

Example 2–12 A ball is drawn at random from an urn containing 12 white, 10 red, and 5 green balls. Determine the probability the ball drawn is (a) white, (b) red, (c) green, (d) not red, and (d) red or green. Solution: The total number of equally likely and exhaustive cases is n = 12 + 10 + 5 = 27 (a) (b)

12 = 0.4444 or 44.44% 27 10 = 0.3704 or 37.04% Probability of drawing a red ball = 27

Probability of drawing a white ball =

5 = 0.1852 or 18.52% 27

(c)

Probability of drawing a green ball =

(d)

Probability the ball is not red =

(e)

Probability the ball drawn is red or green =

27 − 10 = 0.6296 or 62.96% 27 10 + 5 = 0.5556 or 55.56% 27

Empirical Approach. According to this approach, the probability of an event ordinarily represents the proportion of times (under identical

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Project Economics and Decision Analysis Volume II: Probabilistic Models

circumstances) the outcome can be expected to occur. This value refers to the event’s long-run frequency of occurrence. The main assumptions under this approach are as follows. • •

The experiments or observations are random. As there is no bias in favor of any outcome, all elements enjoy equal chance of selection. There are a large number of observations.

It is only when these two assumptions are satisfied that the relative frequency becomes stable since it is subject to the law of statistical regularity. This aspect is clearly brought out in the following definition. If an experiment is repeated a large number of times, under essentially identical conditions, then the limiting value of the ratio of the number of times the event A happens to the total number of trials of the experiment as the number of trials increases indefinitely, is called the probability of the occurrence of A. Assuming that the limit is finite and unique, if m is the number of times when event A occurs in a series of n trials of experiment, then the probability of occurrence of A denoted by P(A) is (2.32)

m P( A) = lim n n→∞

provided the limit exists. The only objection to the previous definition is that it is difficult to prove the existence of a limit of relative frequency. The earlier classical concept is based on a priori or abstract reasoning. The above empirical concept is based on experiments and observations, therefore, called empirical. An experiment is unique and non-repeating only in the case of subjective probability. In other cases, there are a large number of experiments or trials to establish the chance of occurrence of an event. This is particularly true in the case of empirical probability. In classical

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Statistics and Probability Concepts

probability also, repeated experiments may be made to whether a deduction based on certain axioms or undisputed laws is justified. Only after repeated trials, can it be established that the chance of a head in a toss of a fair coin is half (0.5). In practice, the classical probability concept based on cards, coin, etc., is not very helpful. In such situations, the prediction of occurrence has to be based on actual experiments.

Subjective Probability In the subjective or personal interpretation of probability, a probability is interpreted as a measure of degree of belief, or as the quantified judgment of a particular individual based on his experience and intuitive. Thus, the personal or subjective concept of probability measures the confidence an individual has in the truth of a particular proposition. It is bound to vary from individual to individual, therefore, called a subjective measure of probability. It is a calculated guess of an experienced geologist, economist, or engineer and is based only on qualitative factors, on his/her belief, experience in similar situations, and even intuition (gut feeling). This concept plays a considerable role, particularly when there is neither any priori laws of nature to guide nor the experiments can be repeatedly performed to establish the chance of occurrence. Statisticians of the objectivist or classical school reject the use of subjective probability while those of the Bayesian School accept it.

Modern Approach The modern concept to probability combines both the objective and subjective concepts of probability. This new approach is introduced using the theory of sets. All basic ideas like equally likely events, favorable outcomes, character of collectively exhaustive, mutually exclusive events, dependence and independence of events, etc., are explained through the theory and operations of sets. The use of Venn and tree diagrams and graphs has been made freely, which has rendered the modern concept very useful and interesting. Some of the basic components of this approach are given here.

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Random Experiment or Trial. An action, event or an operation that can produce any result or outcome is called a random experiment or a trial. Following are some examples. a. Tossing a fair coin is an experiment, whether the coin will throw a head or tail is unpredictable. b. Rolling an unbiased die is an experiment, how many dots it will actually throw is unpredictable and is subject to chance. c. Drawing a card from a well-shuffled deck of playing cards is an experiment. As there are 52 cards in the deck and any of these may be drawn in a specific trial, which card it will turn out to be is unpredictable. d. When a coin is tossed 100 times or 100 coins are tossed together, there are 100 experiments. e. Experiments in real world situations can be (i) observation of a number of defective items produced by a machine; or (ii) drilling of a well might result in a commercial discovery or a dry hole, and so on. In each of the above examples, certain conditions are set such as tossing a coin or drawing a card from a deck of cards that can have two or more unpredictable results. The establishment of such conditions with the intention of noting the result is known as trial or a random experiment. The word random may be taken as one depending on chance without any bias. Elementary Event. Each one of the possible results in a single trial or experiment is called an elementary event. The total number of possible outcomes in any trial or experiment depends on the number and type of elementary events and the number of trials performed. The outcomes, in modern terminology, are described by sample space and sample points. The former describes all possible events, the latter, each elementary event comprised in the sample space. a. In an experiment of tossing a coin, there are two possible elementary events, the head or the tail. b. In an experiment of throwing a six-faced die, the possible elementary events are 1, 2, 3, 4, 5, or 6.

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Statistics and Probability Concepts

c. In an experiment of drawing a card from a deck of cards, there are four possible outcomes corresponding to four suits designated as heart, spade, diamond, or club. d. In an experiment of drilling a well, it may be (i) a dry hole or (ii) a commercial discovery. If it is a commercial discovery, it may be worth 50,000 barrels, 100,000 barrels or more. In this case, there are two elementary events (a) dry hole versus commercial producer and (b) reserves size if it is a producer. Sample Space. Experiments or trials conducted on a given phenomenon yield certain outcomes or elementary events. The possible outcome depends upon the elementary events composing the phenomenon on which the trial is performed as well as the number of experiments performed, simultaneously or consecutively as the case may be. A set representing all possible outcomes from an experiment is called a sample space, denoted by S. Each possible outcome or element in a sample space is called a sample point. The number of sample points in S may be denoted by N(S). a. If one item is drawn from a manufactured product, the item may be either defective D, or non-defective N. Thus the sample space is: S = {D, N}. b. If two items, one at a time, are picked up at random from a box containing defective and non-defective items; then the sample space for this experiment is: S = {NN, ND, DN, DD}. c. When a coin and a die are tossed together, there are 12 sample points in the sample space: S = {(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6), (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)}. d. Now if a coin is thrown three times in succession or three coins at a time, the sample space S is given by: S = {H, T} {H, T} {H, T} = {HH, HT, TH, TT} {H, T} = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Event Sets. An event set is a subset of the sample space. Thus, a sample space may consist of two or more event sets consisting of a group of elementary events (sample points).

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Project Economics and Decision Analysis Volume II: Probabilistic Models

For example, in the experiment of picking two items, one at a time, at random, from a box containing defective and non-defective items, “both items may be defective {DD}” is one event, “both items are non-defective {NN}” is another event. Properties (or Axioms). See the following characteristics. a. With each event set a number between 0 and 1, called probability of that event, is associated. b. The sum of probabilities of all simple events (or sample points) constituting the sample space is equal to 1. c. The probability of a compound event is the sum of the probabilities of the simple events comprising the compound event. In particular, if there are n possible outcomes of a random experiment and these are considered equally likely then the sample space S contains n sample points and the probability associated with each sample point would be 1/n (due to second axiom). Now, if an event A consists of m sample points, then the probability of A is: P( A) = =

1 1 m + + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( m times) = n n n Number of Sample Points in A Total Number of Sample Points in S

Simple Probability. A simple probability, also referred to as independent probability, is the probability of occurrence that is independent of the occurrence of another event (unconditional). As shown above, this type of probability is denoted as P(A) or P(B), where A and B are events. Conditional Probability. As the name implies, this type of probability is conditional (dependent) on the occurrence of another event. For example, hydrocarbon accumulation is conditional upon the presence of a reservoir rock, porosity, permeability, and a trap. Symbolically, this type

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Statistics and Probability Concepts

of probability is denoted by P(A |B) and is read as the probability of A, given B has occurred. t Probability. This third type of probability is the probability of more than one event occurring simultaneously or in succession. Symbolically, the t probability is denoted by P(AB), which means the probability of both event A and event B occurs. For example, if probability of drilling a producer (event A) is 0.6 (60%) and the probability the producer will have reserves of 30,000 barrels (event B) is 0.8 (80%) then the t probability of getting the 30,000 barrels P(AB) = 0.48=0.6 0.8 or 48%. Marginal Probability. Marginal probabilities are the sum of t probabilities. Operations on Event Sets. The individual event sets can be subjected to some logical operation of union, intersection, and complementation as noted in the following list. •

•

•

Union of Two Events. The union of two event sets A and B, denoted by A傼B(read as A union B), is the event set that consists of all the outcomes (sample points) that belong either to A or B or both. Thus if A is an event that a person will take Math course and B is an event that he will take Biology course, then A傼B is the event that this person will take either Math or Biology or both. Intersection of Two Events. The intersection of two event sets A and B, denoted by A傽B(read as A intersection B), is an event set that consists of all the outcomes (sample points) the two event sets A and B have in common. With reference to the previous example, A傽B is an event the person will take both Math and Biology courses? For example, event A={1,2,3,4} and event B={2,4,6,8} then A傽B ={2,4} and A傼B ={1,2,3,4,6,8}. If the intersection contains no elements then it is empty (symbol ) set and thus called mutually exclusive. Complement of an Event Set. The complement of an event set A is the set of all sample points in the sample space S not contained in A. We denote the compliment of A by A.

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Characteristics of Events. Venn diagrams are utilized to solve probability problems. A Venn diagram gives a pictorial representation of sets and how they are related. A set is a collection of objects. The concepts presented in this section are described in reference to the Venn diagrams. 1. Mutually Exclusive Events. Two or more events are considered mutually exclusive if the events cannot occur together, i.e., the occurrence of any one of them precludes the occurrence of the other. For example, Producer and Dry Hole are mutually exclusive or dist events as shown in the Venn diagram of Figure 2–9.

Fig. 2–9 Venn diagram showing two mutually exclusive events

2. Partially Overlapping Events. Partially overlapping events are also called non-mutually exclusive or intersection events. Two or more events are considered partially overlapping if part of one event and part of another event can occur together. The partially overlapping events are shown in the Venn diagram (Fig. 2–10). For example, the events comprising spades and face cards are partially overlapping. Venn diagram in Figure 2–11 shows the union of two events.

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Statistics and Probability Concepts

Fig. 2–10 Venn diagram of partially overlapping events

Fig. 2–11 Venn diagram showing union of two events

3. Dependent or Independent Events. Two or more events are considered independent if the occurrence of one event in no way affects the occurrence of the other. The question of dependence or independence of events is relevant when experiments are consecutive and not simultaneous. For example, in the tossing of a coin a trial, is not affected by the result of the previous trial. The events, therefore, are independent. However, out of 52 cards if

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Project Economics and Decision Analysis Volume II: Probabilistic Models

one is drawn, then only 51 are left. Unless the card is put back or replaced, the composition stands changed and the probability of the second card is affected. In a second case, the events are dependent, as there has been no replacement of the card drawn in the first place. It is necessary to understand the distinction between overlapping and dependent events. The first expression describes the composition of the sets of events, which may have some elements in common. In dependence, it is examined whether the first experiment or trial will affect the result of the second one. Thus, while the first is associated with the character of events, the second is related to the occurrence of events in repeated experiments or trials when one follows another. The first has relevance for the addition rule and the second has relevance for the multiplication rule. They relate to simultaneous and consecutive experiments, respectively.

Rules of Probability Earlier in this section, marginal probability of event A was discussed, showing that it can be calculated by simply dividing the number of times A has occurred by the total number of trials. However, most decision problems require complicated calculations. Often, conditional and t probabilities for both dependent and independent events occur. In each case, there may be several events rather than just A and B events. For handling such complex situations, appropriate rules for dealing with probabilities need to be ed. These rules of probability are pictorially shown in Figure 2–12.

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Statistics and Probability Concepts

Fig. 2–12 Pictorial presentation of various probability rules

Complementation Rule. Since an event either occurs or does not occur, and since the sum of collectively exhaustive events must equal one, then (2.33)

P( A) + P( A ) = 1

where P(A) = marginal probability of event A P(A) = 1 – P(A) = marginal probability of no event A Addition Rule. If events are mutually exclusive, the probability of outcomes can be calculated by adding the probabilities of the possible ways the outcome can occur.

P (A or B ) = P( A) + P( B)

(2.34)

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Project Economics and Decision Analysis Volume II: Probabilistic Models

For N mutually exclusive events, this equation becomes

P( A or B or C

K or N ) = P( A) + P( B) + P(C ) + K + P( N )

(2.34a)

The statement “A or B” means either event A occurs or event B occurs. This is not to be confused with “A and B,” which means both A and B occur. If events are not mutually exclusive, then the probability of events A or B can be computed using the following equation. (2.35)

P( A or B) = P( A) + P( B) − P( AB ) Note P(AB) is a subset of P(A or B), so P(A or B) includes the occurrence of both A and B as well as A alone and B alone. In set’s notation the P(A or B) = P(A傼B) and P(AB) = P(A傽B). It is clear from Figure 2–11 that (A傼B) can be expressed as the union of three dist sets • • •

(A傽B) = Event A only and no Event B (A傽B) = Event A and Event B (A傽B) = Event B only and no Event A

Since event A can be expressed as the union of dist event sets (a) (A傽B) and (b) (A傽B); it follows that P (A) = P (A 傽 B )+ P (A 傽 B )

(2.35a)

Similarly, P (B ) = P( A 傽 B)+ P(A 傽 B)

86

(2.35b)

Statistics and Probability Concepts

Multiplication Rule. For calculating t probability that two events will occur while the events are independent (the outcome of any previous trial has no effect on the following trial), the following equation, also referred to as the equation of t probability, can be used.

P( AB) = P( A and B) = P( A) × P( B)

(2.36)

For more than two independent events, the above equation becomes

P( ABC

K N ) = P( A and B and C K and N ) = P( A) × P( B) × P(C ) × K × P( N )

(2.36a)

P(AB) is the probability of events A and B occurring together or in succession. For example, the multiplication rule is used to compute the probability of the occurrence of tail in three consecutive flips of a fair coin. The marginal probability is 1/2 for each time a coin is flipped. Therefore, the t probability for this example is

P(3 tails ) = P(tail , flip 1) × P(tail , flip 2) × P(tail , flip 3) = 1 / 2 × 1 / 2 × 1 / 2 = 1 / 8 or 12.5% If the probability of event A occurring, given event B has occurred, is different from marginal probability of event B, then the events are dependent and Equation (2.36) becomes (2.37)

P( AB) = P( A and B) = P( B) × P(A B )or P(A ) × P( B A)

The only difference between the computation of probability using Equation (2.36) and (2.37) is that (a) when events are dependent, the t probability equals the probability of one event times the conditional probability of the other event, and (b) when events are independent, the t probability results from straight multiplication of the marginal probabilities of event A and B. 87

Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 2–13 Assume 50 wells have been drilled in an area with blanket sands. The drilling resulted in (a) 8 productive wells in Zone A, (b) 11 productive wells in Zone B, and (c) 4 productive wells in both Zones A and B. With the help of Venn diagram and probability rules, calculate the following. 1. Number of wells productive in Zone A only, 2. Number of wells productive in Zone B only, 3. Number of wells discovered, 4. Number of wells productive in either Zone A or Zone B, and 5. Number of dry holes. Solution: The given information can be expressed in of set notations as n(S) = 50

n(A) = 8

n(B) = 11

n(A傽B) = 4

1. Wells productive in Zone A but not in Zone B are given by n(A 1 B ) = n (A) − n (A 1 B ) = 8 − 4 = 4

2. Wells productive in Zone B but not in Zone A are given by

(

)

n A 1 B = n(B )− n(A 1 B ) = 11 − 4 = 7

3. Number of wells discovered is given by

(

)

n(A 1 B ) + n A 1 B + n(A 1 B ) = 4 + 7 + 4 = 15

4. In addition, since number of wells discovered means each well has encountered either Zone A or Zone B or both, we use Equation (2.35) to get the number of wells discovered.

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Statistics and Probability Concepts

n (A 7 B ) = n(A)+ n(B ) − n (A 1 B ) = 8 + 11 − 4 = 15

5. The number of dry holes is simply

n (S ) − n(A 7 B ) = 50 − 15 = 35 The Venn diagram for this problem is shown in Figure 2–13.

Fig. 2–13 Venn diagram for Example 2–13

Example 2–14 A crude oil sample possibly contains sulphur and mercaptans. In 100 randomly picked crude oil samples, 45 contained sulphur, 40 contained mercaptans, and 35 contained both. Calculate the probability that the sample contained (a) both sulphur and mercaptans, (b) sulphur only, and (c) neither sulphur nor mercaptans. Solution: The given information is translated into set notations as n(U) = 100

n(S) = 45

n(M) = 40

n(S傽M) = 35

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Project Economics and Decision Analysis Volume II: Probabilistic Models

(a) Using Equation (2.35), probability that the sample contained both sulphur and mercaptans is p (S 7 M ) = =

[n(S )+ n(M )− n(S 7 M )] 100 45 + 40 − 35 = 0.5 or 50% 100

(b) the sample contained sulphur only p (S 7 M ) = =

n(S )− n(S 1 M ) 100 45 − 35 = 0.1 or 10% 100

(c ) the sample contained neither sulphur nor mercaptans

( 1 M ) = n(U )− n(S 7 M ) = 100 − 50 = 0.5 or 50% 100 100

p S

Example 2–15 A bag contains 17 counters, which are like the small numbered balls used in Bingo. The balls are marked with the numbers 1 to 17. A counter is drawn and replaced; a second counter is drawn. What is the probability that: 1. The first number drawn is even and the second odd? 2. The first number is odd and the second even. 3. How will the results in (1) and (2) be affected if the first counter drawn is not replaced? Solution: Let A denote drawing an even numbered counter and B denote drawing an odd numbered counter. Now from 1 to 17, there are 8 odd numbers and 9 even numbers.

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Statistics and Probability Concepts

1. Using Equation (2.36), probability of first even numbered counter and second odd numbered counter is P (A and B ) = P(A) × P (B ) =

8 9 × = 0.2491 or 24.91% 17 17

2. Similarly, the probability of first odd numbered counter and second even numbered counter is also 24.91%. 3. However, if the first counter drawn is not replaced before the second is drawn, then the two events are dependent. In this case, using Equation (2.37)

P (A and B ) =

P(A) 8 9 × P(B A)= × = .2647 or 26.47% P(B ) 17 16

Example 2–16 Ten prospective leases have been acquired. Seismic surveys conducted on the leases show three of the leases are expected to result in commercial discoveries. The leases have equal chances of success. If drilling of one well is planned for each lease, calculate the probability of drilling the first two wells as successive discoveries.2 Solution: Let W1 be the first well and W2 be the second well. 3 P W1 ) = 10

)

Three discoveries are expected out of the 10 prospects. Since the first well drilled is a discovery, there remains 2 additional expected discoveries out of the 9 remaining wells to be drilled. P (W2 W1 ) =

2 9

∴ P(W1W2 ) = P(W1 ) × P(W2 W1) =

3 2 6 1 × = = = 6.67% 10 9 90 15

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Probability Table and Probability Tree The use of probability table is another useful technique to solve problems similar to the ones presented above. In the probability table, number of elements and probabilities are presented in a table form rather than a diagram. Therefore, both the probability table and the Venn diagram represent all possible outcomes. A skeleton probability table for the problem in Example 2–13 is shown in Table 2–11.

Zone B Yes Zone A

Yes

TOTAL No

4

8

11

50

No TOTAL

Table 2–11 Probability table for the data of Example 2–13

In Table 2–11, the probabilities listed along the bottom row and the right-hand column (the margins) are marginal probabilities. These should sum to one in either direction. They represent simple or unconditional probabilities of four different events. To solve Table 2–11, each row across and each column down are added in such a way that they add to the pivot column (50 in this case). The pivot column represents a probability of 100% or total elements in the analysis (50 wells in this case). The completed Table is shown as Table 2–11a.

Zone B Yes Zone A TOTAL

TOTAL No

Yes

4

4

8

No

7

35

42

11

39

50

Table 2–11a Continued probability table for the data of Example 2–13 92

Statistics and Probability Concepts

The concluding table shows the number of dry holes is 35 (intersection of no Zone A and no Zone B). Four wells have encountered both Zones A and B (intersection of yes Zone A and yes Zone B), 7 wells have encountered Zone B, and 4 wells have encountered Zone A. Since the dry holes are 35, the producers are 50 – 35=15. These numbers can be easily converted into probability by dividing all the numbers in the table by 50. The four numbers in the body of the table (for example, 7) represent t probabilities since they denote more than one event. The marginal probability of yes Zone B is 11/50, no Zone B is 39/50, yes Zone A is 8/50, and no Zone A is 42/50. Probability trees are diagrams representing sequences of lines depicting probabilistic events branching to all possible sequences that can occur in any situation. The probability tree is very useful when probability computations of several options and stages are required. It is a pictorial presentation of conditional probabilities, which shows all possible branches for each event. Each branch of the tree is labeled with the event taking place at this branch and its corresponding probability of occurrence given the prior sequence necessary to reach that particular point in the tree. At the source of branches, called nodes, the branches must be mutually exclusive and collectively exhaustive. The probabilities at every node (called chance node) must sum to one. The probabilities on the first branch of the tree are simple or unconditional probabilities. All branches after the first branch are conditional probabilities because they assume branches to their left have taken place. If all the possible events are evaluated on the tree, the final right-hand values should add to one. When the probabilities from the origin of the tree to any terminal point of the tree are multiplied, t probability of that particular sequence of events is obtained. A typical probability tree is shown in Figure 2–14.

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Fig. 2–14 A typical probability tree

Baye’s Rule In the preceding sections, conditional probability of occurrence of an event, given the occurrence of an earlier event is discussed. In this section, the problem is reversed by revising the probability of an earlier event conditioned on the occurrence of a later event. The principle of revising probabilities based on test results has application in many functional areas. The revision of subjective probabilities based on empirical (sample) data is a particularly significant aspect of Bayesian decision analysis. It is a systematic approach for establishing an optimal strategy. This decision-making tool makes use of the decision maker’s judgment and the economic consequences of the status of nature and the possible actions.3 The following Baye’s formula is used to revise the probability of an earlier event given a later event.

P (Ai B) =

P (B Ai )× P(Ai ) k

∑ P(B A ) × P(A ) i

i

i =1

where P(Ai | B) = the posterior probabilities P(Ai ) = the prior or source event probabilities 94

(2.38)

Statistics and Probability Concepts

The Baye’s formula can be interpreted in of a probability tree as follows.

P(Ai B) =

Product of Branch Probabilities Leading to B throuh Ai Sum of all Branch Pr oducts Leading to B

The Baye’s theorem is used when solving problems concerning the assessment of value of additional information that will result in revised probabilities. The method involves inverting the order of chance nodes on probability tree and revising the probabilities accordingly.

Example 2–17 One box contains 3 green and 2 red pencils. A second box contains 1 green and 3 red pencils. A single fair die is rolled and if 1 or 2 comes up, a pencil is drawn from the first box; if 3, 4, 5, or 6 come up, then a pencil is drawn from the second. If the pencil drawn is green, then what is the probability it has been drawn from (a) the first box and (b) the second box? Solution: A probability tree as shown in Figure 2–15 is drawn to let B1 represent box 1, B2 box 2, G as green pencil, and R as a red pencil. Appropriate probabilities are assigned to the various outcome branches as:

Fig. 2–15 Probability tree for Example 2–17

95

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Statistics and Probability Concepts

acquisition of seismic will indicate structure when structure actually exist, and there is 10% chance it will indicate structure when structure actually does not exist. This additional information leads to additional branches of probability tree. c. Obtain the t probabilities by multiplying each prior probability by the conditional probability that follows it on the tree. d. Sum the t probabilities. e. Divide each t probability by the sum of the t probabilities to obtain the required posterior probability. Using Table Format: The table format is often more convenient than using the probability tree or the formula because the table format requires the information to be systematically displayed and manipulated. Secondly, any spreadsheet program can be used to apply Bayes rule using the table format. Table 2–12 and Equation (2.38) relate directly to each other. The following steps summarize the Bayesian application via the table format.

Prior

Conditional

t

Posterior

Event

Probability

Probability

Probability

1

2

3

4=2x3

Probability 5 = 4/Σ (4)

A

P(A)

P(X/A)

P(A) x P(X/A)

P(A) x P(X/A)/Σ (4)

B

P(B)

P(Y/B)

P(B) x P(Y/B)

P(B) x P(Y/B)/Σ (4)

:

:

:

:

:

:

:

:

:

:

N

P(N)

P(Z/N)

P(N) x P(Z/N)

P(N) x P(Z/N)/Σ (4)

Σ(2) = 1.0

Σ(4)

Σ (5) = 1.0

Table 2–12 A Bayesian revision calculation procedure

a. Input all the possible events in Column 1 of Table 2–12. b. Assess the prior probability associated with each event and input these in Column 2 (corresponding to the respective events). The sum of all probabilities in Column 2 must equal 1.0.

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Project Economics and Decision Analysis Volume II: Probabilistic Models

c. Input the likelihood probabilities into Column 3. These are conditional probabilities associated with the occurrence of each event in Column 1. The likelihood probabilities need not add to 1.0. d. Calculate the t probabilities in Column 4 by multiplying, rowby-row, Column 1 by Column 3. Sum the t probabilities in Column 4. e. Divide each row of Column 4 by the sum of Column 4. This gives the posterior probabilities in Column 5.

Example 2–18 A geologist has assigned a 50–50 chance for a prospect to be gas bearing (50% chance that gas exists and 50% chance that gas does not exist). A well is drilled and tested. The test confirms the existence of gas. However, based on experience, it is believed the test is reliable only 90% of the time. This means 10% of the time the test shows existence of gas while the gas does not actually exist. Based on this information, revise the prior probabilities. Solution: The calculations are performed using Bayes rule, as shown in Table 2–13.

Prior

Test Indicates

t

Posterior

Probability

Gas Exists

Probability

Probability

Gas Exists

0.50

0.90

0.45

0.90

Gas Does not Exist

0.50

0.10

0.05

0.10

1.00

1.00

0.50

1.00

Event

Table 2–13 Bayesian revision of prior probabilities for Example 2–18

A more general view of the relationship between estimated prior probabilities and the reliability of additional or new information can be seen in Figure 2–16 (plot of posterior probability on y-axis and prior probability on x-axis). Figure 2–16 shows if the test is reliable (giving

98

Statistics and Probability Concepts

correct result) only 50% of the time, then its result will not be of any interest and the posterior probability will be equal to the prior probability. In Figure 2–16, the distance of the curved lines from the diagonal line indicates that as the reliability of the new information increases, the modification of the prior probabilities also increases. If the prior probability of an event is zero, its posterior probability will also be zero, regardless of how reliable the new information is. On the other hand, the more unreliable the new information is, the greater is its effect on the prior probability.

Fig. 2–16 Effect of reliability of new information on the modification of prior probability of an event

THEORETICAL PROBABILITY DISTRIBUTIONS In the preceding section, probabilities were expressed only in of one specified event, other possible events being considered only to the extent to which they determine the denominator in the probability

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equation. When additional probabilities are determined, a probability distribution of all possible events in an experiment can be formed. A set of outcomes and their associated probability of occurrence can best represent variables faced in many uncertain situations. A probability distribution therefore, represents the pattern of the distribution of probabilities over all possible events. The distribution can be employed to determine the likelihood of the occurrence of all possible outcomes. The variable, whose pattern of variation is described by the probability distribution, is called a random variable. The probability distributions are classified into discrete probability distributions and continuous probability distributions. The discrete probability distribution is associated with a random variable that can take only a finite number of values. A continuous probability distribution is associated with a random variable that can take on infinite values. The breakdown of some of the probability distributions under each of the above categories is illustrated in Figure 2–17. In this section, some of the widely used probability distributions applicable to the problems in the oil and gas industry are presented.

Fig. 2–17 Classification of widely used probability distributions

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Statistics and Probability Concepts

Discrete Probability Distributions As shown in Figure 2–17, under the discrete probability distributions, four types of probability distributions (1) binomial, (2) multinomial, (3) hypergeometric, and (4) poison are presented. Binomial Probability Distribution. In many situations, there are only two possible outcomes, such as success or failure, yes or no, dry hole or producer, acceptable or unacceptable, and accept or reject. The binomial probability distribution can be used in situations when the probability of either outcome is constant from trial to trial, and the occurrence of outcomes does not adhere to any fixed pattern. The binomial distribution is more suitable for use in early exploratory efforts in a newly recognized geologic area. According to the binomial distribution, the probability of obtaining exactly x successes in n trials is given by the following equation. (2.39)

P(x ) = C × p q n x

C xn =

x

n−x

n! x!(n − x )! (2.40)

Mean = np (2.41)

Variance = npq where P(x) p q n

= = = =

probability of obtaining exactly x successes in n trials, probability of success, probability of failure = (1 – p), number of trials considered, the n and p are called parameters of distribution, x = number of successes, x = 0, 1, 2, . . . . , n.

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The term Cxn, called the binomial coefficient, is the number of possible combinations of n things taken X at a time. The following four critical assumptions define a binomial situation. •

•

•

•

The random variable may assume only one of two possible values in any single trial. The name bi-nomial implies the outcome has only two names (accept or reject, defective or acceptable, dry or producer, and so on). The probability of the two possible outcomes remains constant from one trial to the next, so the probability of getting a head on the flip of a coin is 1/2. In a sequence of trials, the outcome on any trial has no effect on the outcome of any other trial. This assumes the trials are independent events. There must be a discrete number of trials.

Tables of binomial probabilities (Appendix A) are available that give the binomial probabilities of x for various combinations of n and p. For hand calculations, using these tables greatly facilitates and expedites problem solving. In the following section, the use of Excel for solving problems following binomial distributions is presented.

Example 2–19 A company is planning to drill six exploratory wells with an estimated chance (probability) of success (COS) of 15%. What is the probability that (a) the drilling will result in exactly two discoveries, (b) there will be less than three successful wells, and (c) there will be more than three successful wells? Solution: n = 6, p = 0.15 and q = 1 – p = 0.85

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Statistics and Probability Concepts

Using Equation (2.39), the probability of exactly two successes is

P(2) = C xn × p x q n − x =

P(2) =

=

6! 6−2 ( 0.15 2)(0.85) 2 ! × (6 − 2 )!

6 × 5 × 4 × 3 × 2 ×1 × 0.0225 × 0.522 (2 × 1)(4 × 3 × 2 × 1) 720 × 0.0225 × 0.522 = 0.1762 or 17.62% 2 × 24

The probability of success of less than any x is the sum of the individual probabilities before x. Similarly, the probability of success greater than any x is the sum of individual probabilities after x and up to the total number of x. The probability of success of less than three wells is P(0) + P(1) + P(2),

P(0) =

P(1) =

6 × 5 × 4 × 3 × 2 ×1 6−0 × (0.150)(0.85) = 0.3771 (0)(6 × 5 × 4 × 3 × 2 × 1) 6 × 5 × 4 × 3 × 2 ×1 6 −1 × (0.151)(0.85) = 0.3993 (1)(5 × 4 × 3 × 2 × 1)

∴ P(x less than 3) = P(0)+ P(1)+ (2) = 0.3771 + 0.1762 + 0.3993 = 0.9526 or 95.26%

Similarly, P(x greater than 3) = P(4 )+ P(5)+ P(6) = 0.0055 + 0.0004 = 0.0059 or 0.59%

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Similar to the binomial distribution, is a negative binomial distribution. This distribution is concerned with the number of failures that will occur before any success is attained; assuming the probability of success is constant for each trial. The probability of exactly x failures is given by the following equation. (2.42)

⎛ r + x − 1⎞ r x ⎟p q P (x ) = ⎜ ⎝ x ⎠

where x = number of failures before achieving the rth success, r = required number of successes, p = probability of success on each trial, q = 1 – p = probability of failure on each trial. ⎛ r + x − 1⎞ (r + x − 1) ! ⎟⎟ = ⎜⎜ x! ⎝ x ⎠

Example 2–20 If the probability of success for drilling a successful well is p = 0.15 and the required number of success is r = 2 then what is the probability of (a) zero failures and (b) exactly one failure?4 Solution: Using Equation (2.42), the probability of zero failures is ⎛1 ⎞ ⎛ 2 + 0 − 1⎞ 2 0 2 0 ⎟ × (0.15) × (0.85) = ⎜ ⎟ × (0.15) × (0.85) P(0 ) = ⎜ 0 0 ⎝ ⎠ ⎠ ⎝

= 1 × 0.0225 × 1 = 0.0225 or 2.25%

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Statistics and Probability Concepts

The probability of exactly one failure is ⎛ 2⎞ ⎛ 2 + 1 − 1⎞ 2 1 2 1 ⎟ × (0.15) × (0.85) = ⎜ ⎟ × (0.15) × (0.85) P(1) = ⎜ 1 1 ⎝ ⎠ ⎠ ⎝

= 2 × 0.0225 × 0.85 = 0.0383 or 3.83% Since the cumulative probability, P(0) + P(1) = 6.08%, as it is way below the certainty level of unity, it shows the chance of drilling more than one dry hole is 93.92% (100 – 6.08 = 93.92). Multinomial Probability Distribution. When the outcomes have to be classified into more than two types, then the binomial distribution is not applicable. In this case, the multinomial distribution presented in this section can be used. The types of outcomes are classified into • • •

dry hole versus a producer, if producer then the expected reserves is 10 million barrels, or the expected reserves are 15 million barrels.

Therefore in this situation, there are three types of outcomes: zero reserves, 10 MMBls and 15 MMBls. Solution of this type of problem involves listing all the possible outcomes and calculating their probability. The multinomial probability distribution is given by the following equation.

N! P(S ) = P1k1 P2k 2 k1 !k 2 ! k m !

L

KP

(2.43)

km m

where P(S) = probability of the particular sample P1P2 . . . Pm = probabilities of drawing types 1, 2, . . . m from population N = k1 + k2 + . . . + km= size of the sample k1 + k2 + . . . + km = total number of outcomes of types 1, 2,. . . ., m, respectively, in the sample m = total number of different types (or classifications or outcomes) within the total sample. 105

Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 2–21 In a certain prospect, the company has grouped the possible outcomes of an exploratory well into three general classes as (a) dry hole (zero reserves), (b) discovery with 12 MMBbls reserves, and (c) discovery with 18 MMBbls reserves. Each of these reserves categories was assigned probabilities of 0.5, 0.35, and 0.15, respectively. If the company plans to drill three additional wells, what will be the probabilities of discovering various total reserves with these three additional wells?4 Solution: The following data is given m=3 N=3 P1 = 0.5 P2 = 0.35

P3 = 0.15

Let k1, k2, and k3 be the number of wells giving reserves of zero, 12 MMBbls, and 18 MMBbls, respectively. One possible sample combination out of the three wells drilled is k1 = 2, k2 = 1, and k3 = 0. The probability of this particular combination is calculated as follows:

P(S ) = =

3! (0.5)2 × (0.35)1 × (0.15)0 2!1!0! 3 × 2 ×1 × 0.25 × 0.35 × 1 = 0.263 2 × 1×1 ×1

Corresponding reserves = (2)(0) + (1)(12) + (0)(18) = 12 MMBls The expected reserves for this combination are thus equal to 3.15 (0.263 12 = 3.15) MMBbls. Calculations for all of the possible three-well combinations are shown in Table 2–14. The probability of reserves greater than for each combination is also calculated as shown in Column 6 of Table 2–14. This cumulative probability plotted versus the reserves gives further insight.

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Statistics and Probability Concepts

N=

3.00

P1 =

0.50

P2 =

0.35

P3 =

0.15

k1 =

- MMBbls

k2 =

12.00 MMBbls

k3 =

18.00 MMBbls Probability Probability Reserves P(S)

of Reserves > Than

(MMBbls)

Expected Reserves

k1

k2

k3

3

0

0

0.125

-

1.000

-

2

1

0

0.263

12.00

0.875

3.150

2

0

1

0.113

18.00

0.613

2.025

1

2

0

0.184

24.00

0.500

4.410

1

1

1

0.158

30.00

0.316

4.725

1

0

2

0.034

36.00

0.159

1.215

0

3

0

0.043

36.00

0.125

1.544

0

2

1

0.055

42.00

0.082

2.315

0

1

2

0.024

48.00

0.027

1.134

0

0

3

0.003

54.00

0.003

0.182

1.000

(MMBbls)

20.700

Table 2–14 Illustration of multinomial distribution's calculation for Example 2–21

Hypergeometric Probability Distribution. The hypergeometric probability distribution is a discrete distribution that finds application in statistical sampling. The hypergeometric distribution is applicable in cases when the trials are dependent and in cases of sampling without replacement. If X is a discrete random variable representing the number of successes observed when n items are selected without replacement from a finite population of N total items and the number of successes in the population is less than N, then X is termed a hypergeometric random

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variable. The hypergeometric random variable X has possible outcomes 0,1, . . . , up to n or C, whichever is smaller (the smallest possible outcome is n – N + C when n > N – C). The hypergeometric probability function is given by the following equation. (2.44)

⎛ C⎞ ⎛ N − C ⎞ ⎜⎜ ⎟ ⎜⎜ ⎟⎟ ⎝ x⎠ ⎝ n − x ⎠ P(x ) = ⎛N⎞ ⎜ ⎟ ⎝n⎠

where N C n x min{n,C}

= number of items in the population = number of total successes in the population = number of trials (size of the samples) = number of successes observed in the sample = the minimum of n and C

The mean and standard deviation of a hypergeometric probability distribution are given by the following equations. (2.45)

⎛C ⎞ E (X ) = n⎜ ⎟ ⎝N⎠ (2.46)

C⎞ ⎛ ⎛ N −n⎞ σ (X ) = ⎜ ⎟ × E (X )× ⎜1 − ⎟ N⎠ ⎝ N −1 ⎠ ⎝

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Statistics and Probability Concepts

Example 2–22 A box contains 20 pieces of chocolate. Eight pieces are caramels, and the remaining 12 are nuts. If a person selects 4 pieces at random, what is the probability that exactly one piece is a caramel? Solution: N = 20

x=1

C=8

n=4

⎛ 8 ⎞ ⎛ 20 − 8 ⎞ ⎛ 8 ⎞ ⎛12 ⎞ 8! 12! ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ × − 1 4 1 1 3 ⎠ = ⎝ ⎠ ⎝ ⎠ = 1!(8 − 1)! 3!(12 − 3)! P(x ) = ⎝ ⎠ ⎝ 20 ! ⎛ 20 ⎞ ⎛ 20 ⎞ ⎜ ⎟ ⎜ ⎟ 4 !(20 − 4 ) ! ⎝4⎠ ⎝4⎠

=

8 × 220 1,760 = = 0.3633 or 36.33% 4,845 4,845

Example 2–23 A company has 10 exploration prospects, 4 of which are expected to be productive. What is the probability 1 well will be productive if 3 wells are drilled? Solution: N = 10

C=4

x=1

n=3

⎛ 4 ⎞ ⎛10 − 4 ⎞ ⎛ 4 ⎞ ⎛ 6 ⎞ 4! 6! ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ × − 1 3 1 1 2 ( ) ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1! 4 − 1 ! 2!(6 − 2)! P(x ) = = = ⎛10 ⎞ ⎛10 ⎞ 10! ⎜ ⎟ ⎜ ⎟ 3!(10 − 3) ! ⎝3⎠ ⎝3⎠ =

4 × 15 60 = = 0.50 or 50.00% 120 120

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Poisson Probability Distribution. While the binomial distribution is particularly useful for representing successes in several trials, the Poisson distribution is a useful family of theoretical discrete probability distributions—good for representing occurrences of a particular event over time or space. This distribution is particularly useful in describing the distribution of number of occurrences per unit of volume, area, or time in cases where the average number of occurrences is small. For example, • • • • • •

Number of customers arriving at a bank in a certain time. Number of machine breakdowns in a factory over some period. Number of chocolate chips in a chocolate chip cookie. Number of telephone calls received at a switchboard in a 5-minute period. Frequency of pump breakdowns on a certain lease, and Well workover requirements.

The Poisson distribution is given by the following equation. (2.47) x

P (x) = λ e −λ x!

where

= average number of occurrences per interval of time or space x = number of occurrences per basic unit of measure P(x) = probability of exactly x occurrences Given a mean number of occurrences, the actual number of occurrences is a random variable designated by X. The Poisson standard deviation is given by = √ . Various characteristics of the Poisson distribution are as follows. 1. Discrete distribution. Like binomial distribution, Poisson distribution is also a discrete probability distribution, which means it is concerned with occurrences that can be described by a discrete random variable.

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Statistics and Probability Concepts

2. Values of p and q. It is applied in situations where the probability of success (p) of an event is very small as compared to that of failure (q), which is almost equal to 1 and n is also very large. 3. Main parameter. The main parameter of the Poisson distribution is the mean ( ). If the value of ( ) is known then the values of the other parameters can be ascertained easily. 4. Form of distribution. The Poisson distribution is a positively skewed distribution, (to the left). With increase in the value of ( ), the distribution shifts to the right and the skewness is reduced. 5. Assumptions. The Poisson distribution is based on the following assumptions: a. The average number of events over a unit of measure (time or space) under consideration must remain constant from one trial to another, no matter how far or how long the process has gone on. b. Events happen independently of other events, so from one trial to another, the number of occurrences of the random variable must be independent (of each other). c. The random variable must be an integer per unit of measurement. d. The event must be a relatively rare or uncommon occurrence within the unit of measurement (a judgment factor), so the probability of two or more occurrences in any time arrival is negligibly small, relative to the probability of one occurrence in the interval. 6. Approximation to Binomial distribution. The binomial distribution can be approximated with the Poisson distribution when n is large (beyond the binomial tables for n) and p is small such that np ≤ 5. On the other hand, the binomial distribution can be approximated with the normal distribution if n ≥ 30 and np and n(1 - p) ≥ 5. The normal approximation to the binomial distribution can be extended to the Poisson distribution when the mean of a Poisson process is greater than 10, i.e., ≥ 10.

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Example 2–24 Solve the following problems using the Poisson distribution. 1. If a pipeline averages 3 leaks per year, what is the probability of having exactly 4 leaks next year? 2. If a pipeline averages 5 leaks per 1000 miles, what is the probability of having no leaks in the first 100 miles? 3. The well workovers average 2 per day on a certain lease; on how many days per month will no breakdowns be reported? 4. A company discovers, on the average, 1 giant oil field every 2 years, how would the discoveries be distributed over a period of 10 years?4 Solution: 1. Using Equation (2.47), the probability of exactly 4 leaks next year is λ x −λ 34 −3 81 × 0.0498 P(x ) = e = e = = 0.1681 or 16.81% x! 4! 24 2. Using Equation (2.47), the probability of having no leak in the first 100 miles is λ x −λ 0.5 0 −0.5 1 × 0.6065 P(x ) = e = e = = 0.6065 or 60.65% x! 0! 1 3. Using Equation (2.47), the number of days when no breakdowns will occur is λ x −λ 0.5 0 −0.5 1 × 0.6065 P(x ) = e = e = = 0.6065 or 60.65% x! 0! 1

∴ days per month = 0.1353 ×

365 = 4.115 or 4 12

4. For this case, = 0.5 occurrences per year. Table 2–15 gives the expected distribution per year.

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Statistics and Probability Concepts

Fields Discovered During 1-Year

Number of

Fields

Period, x

P(x)

P(x) x 10 Years

Years

Discovered

0

0.6065

6.065

6

0

1

0.3033

3.033

3

3

2

0.0758

0.758

1

2

3

0.0126

0.126

0

0

4

0.0016

0.016

0

0

5

0.0002

0.002

0

0

1.0000

10.000

10

5

Table 2–15 Distribution of discoveries per year for Example 2–24

Continuous Probability Distributions Continuous probability distributions consist of random variables that assume any value within a given interval or intervals. A random variable that may take on any value on the continuum is called a continuous random variable. A random variable is a variable whose numerical value is determined by the outcome of a random trial. It is not meaningful to speak of a probability value being associated with each possible point on the continuum. Therefore, probability that a continuous random variable X takes on a value in a specified interval is found by determining the corresponding area under its probability density function f(x), commonly abbreviated as PDF. The probability density function of a continuous random variable X is a mathematical function for which the area under the curve corresponding to any interval is equal to the probability that X will take on a value in the interval. The value f(x) is called the probability density at x. The f(x) is always nonnegative and the total area under a probability density function must be 1. The probability that a continuous random variable will take on any particular value on the continuum is zero.

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A corresponding curve to the PDF is the cumulative distribution function, abbreviated as CDF. This curve is the same as the cumulative relative frequency curve discussed under descriptive statistics, with the exception that the cumulative relative frequency on the y-axis is replaced by the cumulative probability. This function is normally denoted by F(x). The CDF indicates the probability that the outcome of X in a random trial will be less than or equal to any specified value of x. The CDF corresponds to the area under the probability density function to the left of x. In this section, several important families of continuous probability distributions particularly applicable to solving problems of oil and gas industry are presented. The continuous probability distributions discussed include (1) normal, (2) lognormal, (3) uniform, and (4) triangular. Normal Probability Distribution. The family of normal probability distributions is one of the most important in decision analysis. Its graph, called the normal curve, is a bell-shaped curve extending indefinitely on both sides of the mean. The normal curve is also referred to as Gaussian distribution, named after the mathematician Karl Friedrich Gauss who derived the equation of the normal curve. Many types of random trials in practice involve the normal random variable. The application of normal probability distribution enables us to determine the expected range of outcomes from a particular population or sample. The normal probability density function is as follows (based on Equation 2.23): (2.23a)

f (x) =

⎡ 1 ⎛ x − X ⎞2 ⎤ 1 ⎟ ⎥ exp ⎢− ⎜ 2πs ⎢⎣ 2 ⎝ s ⎠ ⎥⎦

A typical normal probability distribution curve (PDF), bell shaped, is shown in Figure 2–18. The distribution is characterized by two parameters: (1) the mean (X or μ) and (2) the standard deviation (s or ). Each different pair of (X, s) values corresponds to a different probability distribution of the family of normal distribution. The spread of the data determines the curves’ steepness or shallowness. The spread is measured using standard deviation and the location of the mean. Some normal probability curves with variable mean and standard deviation are shown in Figure 2–19. Regardless of the spread or mean of the normal distribution curves, they have the following particular features. 114

Statistics and Probability Concepts

Fig. 2–18 A typical normal probability distribution curve

Fig. 2–19 Normal probability curves with variable mean and standard deviation

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• • • •

The curve is symmetrical about the mean, bell-shaped. The mean, mode and median of the distribution are equal. Fifty percent of all values lay either side of the mean value, so the mean divides the PDF into two halves. The theoretical limits of a normal distribution are – ∞ and + ∞. In practice, however, the curve can be divided into three standard deviations on either side of the mean as shown in Figure 2–20.

Fig. 2–20 A normal curve showing distribution of area under the curve

a. 68% of the population will lie between 1s (one standard deviation), b. 96% of the population will lie between 2s (two standard deviations), c. 99.8% of the population will lie between 3s (three standard deviations), d. A small portion (0.2%) will lie outside this range, but it is very small and we will seldom fall in this range in practice.

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• •

•

The curve is uniquely defined by two single value parameters (mean and standard deviation), usually reported as N(X, s). The mean and standard deviation of a normal distribution are given by mean = np and s = np(1 - p) (where n = the sample size and p = the probability of an event occurring, respectively. The cumulative frequency of a normally distributed data plots as an s-shaped curve on coordinate graph paper as shown in Figure 2–4. However, the cumulative frequency plots as a straight line when plotted on a special graph paper (normal probability graph paper) with a standard coordinate scale on the x-axis and probability scale on the y-axis or vice versa. The mean and standard deviation can be easily read from this straight-line plot; the mean is read at the 50 percentile. The standard deviation is the difference between the 84.1 percentile and the 50 percentiles (s = x84.1 – x50). Figure 2–21 shows a typical normal probability plot of porosity data. The graph is provided on the accompanied CD. It accommodates up to 32 data pairs of cumulative probability versus random variable. It performs regression on specified data pairs and draws a straight line through the data points. Using this line, it calculates the x84.1 and x50.

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Fig. 2–21 Cumulative relative frequency plotted on normal probability graph paper

Statistics and Probability Concepts

Any normal distribution, regardless of the values of mean and standard deviation, can be described in a single table called a normal probability table as shown in Appendix A. This table gives the normal probabilities in of the number of standard deviations a particular value of X is away from the mean. The random variable described by this distribution is termed Z and is referred to as a standard normal deviate. Any Z value may be computed using the following equation. (2.48)

Z=

X−X X −μ or = s σ

In Equation (2.48), the mean of X = 0 and its standard deviation s = 1. The value of Z represents the number of standard deviations a particular value of X is away from the mean. The following example clarifies the use of standard normal probability table.

Example 2–25 Porosities calculated from porosity logs of a certain formation show a mean porosity of 12% with standard deviation of 2.5%. What is the probability the formation’s porosity will be (a) between 12% and 15%, (b) less than 15%, (c) greater than 16%, and (d) between 10% and 16%. Solution: Equation (2.48) and Table in Appendix A are used to solve this problem. It is always helpful to sketch the desired area under the normal curve when solving a problem like this. Figure 2–22 shows the areas representing the probability of each problem.

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Fig. 2–22 Areas under the normal probability curves for Example 2–25

(a) For a porosity to be between 12% and 15%, the Z is Z=

15 − 12 = 1.2 2.5

The area under the normal curve is read from the normal probability table at Z = 1.2 as 0.3849. Thus, the probability that the porosity will be between 12% and 15% is 38.49%. (b) For porosity less than 15%, Z is the same as in (a) above. However, as shown in Figure 2–22(b), the area bounded by porosity less than 15% is the total area to the left of Z = 1.2. Since the total area on one side of the mean is 0.5, the probability that porosity will be less than 15% is 0.8849 = 0.5 + 0.3849 or 88.49%.

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(c) For porosity greater than 16% Z=

16 − 12 = 1. 6 2.5

The probability of porosity greater than the mean = 12% is 50% (half area of the normal curve) out of which the area from the mean up to Z = 1.6 is 0.4452. The probability of porosity greater than 12% is 0.5 – 0.4452 = 0.0548 or 5.48%. (d) For porosity between 10% and 16% Z 10% =

10 − 12 = − 0.80 2.5

and

Z16%

16 − 12 = 1.6 2.5

The solution of this problem requires two steps. First, the area between the mean and X = 10% must be found and then the area between the mean and X = 16% as shown in Figure 2–22(d). The Z values for these two steps are calculated above. Since the normal distribution is symmetrical, the area between the mean and Z = –0.80 is the same as that between the mean and Z = 0.80, that is 0.2881. The area between the mean and Z = 1.6 is 0.4552. Therefore, the probability that the porosity will be between 10% and 16% is 0.7433 (= 0.4552+0.2881) or 74.33%.

Example 2–26 A chance of discovering oil in a certain prospect is 15%. What is the probability of going broke before a discovery is made if there are enough funds to drill 10 wells?

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Solution: The example involves calculating the mean and standard deviation first.

mean = Np = 10 × 0.15 = 1.5 Standard Deviation = Np (1 − p ) = 10 × 0.15 × 0.85 = 1.275

To avoid being broke, there has to be at least one discovery. To calculate area under the normal probability curve between one discovery and the mean, Z=

1 − 1.5 = − 0.3922 ≅ −.39 1.275

The area under the curve at Z = -.39 is 0.1517 or 15.17%, which is the probability of going broke before an initial discovery is made. In other words, the chance of success is 84.83%. Lognormal Probability Distribution. Another useful distribution, based on the normal distribution, is the lognormal distribution. This type of distribution is widely used in environmental engineering, economics to represent the distribution of returns on investment, insurance claims, and many oil and gas related problems. The occurrence of oil and gas reserves has shown lognormal distributions. Other examples of possible random variables represented by the lognormal distribution include core permeability, and formation thickness, etc. As shown in Figure 2–23, the probability density function of the lognormal distribution is similar to the normal distribution except it is skewed to one side. The distribution can be skewed in either direction. When the curve is skewed to the right, as shown in Figure 2–23, the median lies to the right of the mode, and the mean lies to the right of the median. Further relationship between the mean, median, and mode of the skewed distributions were given in the section on descriptive statistics.

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Fig. 2–23 Lognormal probability distribution function

The lognormal distribution is the transformed version of the normal distribution, (the logarithms of the random variables are normally distributed). Finding probabilities from a lognormal distribution involves taking natural logarithm of the random variable X. For example, if X is lognormally distributed with parameters s and X ( and μ), the Y = ln(X)is normally distributed with these parameters. Therefore, when working with the lognormal distributions, the random variable is first converted into its natural logarithm. This is then used to find area under the curve from the table of normal probability distribution. The mean and standard deviation of X are given by the following equations. (2.49)

Mean, X = exp(α + 0.5β ) 2

(2.50) 2 Mode = exp(α − β )

(2.51)

Standard Deviation s = exp(2α + 2 β 2) − exp(2α + β 2)

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The cumulative frequency graph of a lognormal distribution, when plotted on a special graph paper called lognormal probability graph paper, plots as a straight line. This type of graph consists of probability on the y-axis and logarithmic scale rather than coordinate scale on the xaxis. A cumulative frequency graph of a lognormal distribution is shown in Figure 2–24. The mean and standard deviation of the lognormally distributed random variables can be calculated by using the conventional calculations performed on the tabulated data as done in the section on descriptive statistics. These parameters can also be calculated from the lognormal probability graph and the following equations. By algebraic manipulation, the following can be obtained (2.52)

(

2

)

X 2 eβ −1

s=

(2.53)

mode of X =

X e

1.5 β 2

where α = natural logarith of the variable read at the 50th percentile

β=

ln(X 2% ) − ln (X 98% ) z, between two tails

The lognormal probability graph paper (shown in Figure 2–24) is given on the accompanying CD. It can be used to plot up to 32 pairs of data points and perform regression analysis on specified data pairs. A straight line is drawn through the data points.

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125

Fig. 2–24 Log-probability plot for lognormally distributed data of Figure 2–23

Project Economics and Decision Analysis Volume II: Probabilistic Models

Uniform Probability Distribution. A random variable that can take on integer values within a given interval (maximum and minimum) with equal probabilities is useful for statistical sampling (all values of x between xmin and xmax are equally likely to occur). This random variable is called a discrete uniform random variable. The discrete uniform probability distribution has a continuous analog known as the continuous uniform or rectangular probability distribution. The cumulative relative frequency curve of the uniform distribution plots as a straight line on Cartesian coordinate graph paper. The uniform distribution and the corresponding cumulative relative frequency curves are shown in Figure 2–25. The continuous uniform probability density function is given by (2.54)

1 f ( x) = xmax − xmin

Fig. 2–25 Uniform distribution of variable x and the corresponding cumulative relative frequency distribution 126

Statistics and Probability Concepts

The mean and standard deviation of a uniform probability distribution are (2.55)

x + xmin X = max 2

(2.56)

s=

(xmax − xmin )

2

12

Any desired probability or percentile for a uniform probability distribution may be found by using its cumulative probability function. (2.57)

x − xmin P(X ≤ x ) = xmax − xmin Replacing P(X) with RN, the rectangular distribution random number, and solving for random variable X, the following equation is obtained. X = RN (Xmax − Xmin ) + X min

(2.58)

Example 2–27 An economist estimates the lease operating costs could vary between $2,500 and $5,000 per month, while any value within the range is equally likely. Calculate the mean, standard deviation, continuous uniform probability density function, and the probability the operating cost will be $3,500 or less.

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Solution: Using Equation (2.55), the mean is X =

xmax + xmin $5,000 + $2,500 = = $3,750 2 2

Using Equation (2.56), the standard deviation is s=

(xmax − xmin ) 2 = ($5,000 − $2,500 )2 = 12

12

$6,250,000 12

= $520,833.33 = $721.69

The density function is given by Equation (2.54)

f ( x) =

1 1 1 = = xmax − xmin $5,000 − $2,500 $2,500

Using Equation (2.57), the probability the operating cost will be $3,500 or less is

P(X ≤ x ) =

x − xmin $3,500 − $2,500 = = 0.40 or 40.00% xmax − xmin $5,000 − $2,500

Triangular Probability Distribution. The triangular distribution is a continuous probability distribution, which has the shape of a triangle. The triangle could be symmetrical or skewed in either direction. Use of the triangular distribution is indicated when the upper limit, lower limit, and the most likely value of the random variables can be specified. Figure 2–26 shows a triangular distribution and the corresponding cumulative probability distribution. The cumulative probability function of random variables exhibiting triangular distribution plots as an s-shaped curve on Cartesian coordinate graph paper.

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There may be limited number of random variables showing triangular distribution. The use of this distribution is more useful in simulation, which is discussed in the following chapters.

Fig. 2–26 Triangular distribution of a random variable x and its corresponding cumulative frequency distribution

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The mean and standard deviation of a triangular distribution are calculated using the following equations. (2.59)

X + XH + XM X = L 3

(2.60)

s=

(XH

− XL )(X − XL XH ) − XM XH (XH − XM ) − XL XM (XM − XL ) 18 (XH − XL ) 2 H

where XL = the minimum value of the random variable XM = the most likely value (mode) of the random variable XH = the maximum value of the random variable The following equations are used to convert a triangular distribution to its equivalent cumulative probability distribution. When XL ≤ X ≤ XM, the cumulative probability of X is given by (2.61)

⎛ X − XL f (x) = ⎜⎜ ⎝ XM − XL

⎞ ⎟⎟ ⎠

2

⎛ XM − XL ⎜⎜ ⎝ XH − XL

⎞ ⎟⎟ ⎠

When XM ≤ X ≤ XH, the cumulative probability of X is given by (2.62)

⎛ X −X ⎞ ⎟⎟ f (x ) = 1 − ⎜⎜ H ⎝ XH − XM ⎠

130

2

⎛ XH − XM ⎜⎜ ⎝ XH − XL

⎞ ⎟⎟ ⎠

Statistics and Probability Concepts

Replacing f(x) with RN, the uniformly distributed random number, and solving for random number X, the following equations result. If RN ≤ (X M − X L )/ (X H − X L ) (2.63)

X = XL +

(X M

− X L )(X H − X L ) RN

If RN ≥ (X M − XL ) / (XH − XL ) (2.64)

X = XH −

(XH

− XM )(XH − XL )(1 − R N )

Example 2–28 A bit record in a certain area shows the minimum and maximum footage drilled by the bit to be 100 and 200 feet, respectively. The drilling engineers have estimated the most probable value of the footage drilled by a bit will be 130 feet and the footage drilled follows triangular distribution as shown in Figure 2–26. Develop cumulative probability distribution for this data. Solution: XL = 100

XM = 130

XH = 200

Using Equation (2.62) with various values of random variable X generate the cumulative probability less than or equal to. For X = 110 feet [since 110 is less than the mode of 130, use Equation (2.61)]

⎛ X − XL f (x ) = ⎜⎜ ⎝ XM − XL

⎞ ⎟⎟ ⎠

2

2

⎛ X M − X L ⎞ ⎛ 110 − 100 ⎞ ⎛ 130 − 100 ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ X H − X L ⎠ ⎝ 130 − 100 ⎠ ⎝ 200 − 100 ⎠

= (0.3333) (0.3) = 0.033 or 3.33% 2

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Similarly, for X = 160 feet [since 160 is greater than the mode of 130, use Equation (2.62)]

⎛ X −X f (x ) = 1 − ⎜⎜ H ⎝ XH − XM

⎞ ⎟⎟ ⎠

2

⎛ XH − XM ⎜⎜ ⎝ XH − XL

2

⎞ 200 − 160 ⎞ ⎛ 200 − 130 ⎞ ⎟⎟ = 1 − ⎛⎜ ⎟ ⎜ ⎟ ⎝ 200 − 130 ⎠ ⎝ 200 − 100 ⎠ ⎠

= 1 − (0.5714) (0.7 ) = 1 − 0.2285 = 0.7715 or 77.15% 2

In the same way, the f(x) is calculated for few other values and then plotted as a cumulative probability curve shown in Figure 2–26.

SPREADSHEET APPLICATION In this section, the use of Microsoft Excel’s utility functions for solving various problems related to the probability calculations and probability distributions is presented. The Palisade Corporation’s BestFit software is also introduced. This is used to fit appropriate probability distributions to historical data.

Excel Built-in Functions In one of the preceding sections in this chapter, the use of Excel for calculating parameters required for descriptive statistics was presented. In this section, other statistical functions of Excel that can be used for solving problems involving probability distributions are presented. The examples presented on probability distributions in this chapter are solved using these capabilities of Excel. BINOMDIST. The BINOMDIST (number of successes in trials, number of independent trials, probability of success on each trial, cumulative) function returns the individual term binomial distribution probability. This function is used for problems with a fixed number of tests or trials, when the outcomes of any trial are only success or failure. The function represents Equation (2.39) and is able to return cumulative distribution function as well as the probability density function. Example

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2–19 is solved using the BINOMDIST function as shown in Table 2–16. The following steps are used to construct Table 2–16. a. Input the probability of success, total number of wells drilled, and the number of discoveries for which the probability is to be calculated in Cells C4, C5, and C6, respectively. b. Input the formula =BINOMDIST(C6,C5,C4,FALSE) in Cell C8 to calculate the probability of exactly two discoveries. The FALSE implies density function and not cumulative distribution function. c. Enter the formula =BINOMDIST(A12,C$5,C$4,FALSE) in Cell B12 and copy it to Cells B13:B18. Enter the number of discoveries from 0 to 6 in Cells A12 to A18. This generates a profile of the probabilities of discovering different number of wells.

A

B

C

4 Probability of Success

0.15

5 Number of Wells Drilled

6

6 Probability of Discovering Wells

2

7 8 Probability of 2 Discoveries

17.62%

9 10

Probability of Discovering

11

Wells

12

0

37.71%

13

1

39.93%

14

2

17.62%

15

3

4.15%

16

4

0.55%

17

5

0.04%

18

6

0.00%

Probability

Table 2–16 Using Excel’s binomial distribution function

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NEGBINOMDIST. The NEGBINOMDIST(number of failures, threshold number of successes, probability of success) function returns the negative binomial distribution. The function returns the probability there will be x number of failures before the yth number of success, when the constant probability of success is given. The function represents Equation (2.42). Example 2–20 is solved using the NEGBINOMDIST function as shown in Table 2–17. The following steps are used to construct Table 2–17. a. Input the probability of success for drilling a successful well in Cell C4 and the required number of successful wells in Cell C5. b. Enter the formula =NEGBINOMDIST(A9,C$5,C$4) in Cell B9 and copy it to Cells B10 to B12. c. Input the desired number of failures in Cells A9 to A12.

A

B

C

4 Probability of Success

0.15

5 Required Number of Success

2

6 7

Probability of Failures

8

Wells

9

0

2.25%

10

1

3.83%

11

2

4.88%

12

3

5.53%

Probability

Table 2–17 Using Excel’s negative binomial distribution function

HYPGEOMDIST. The HYPGEOMDIST (number of successes in the sample, size of the sample, number of successes in the population, population size) function returns the hypergeometric distribution. The function returns the probability of a given number of sample successes, given the sample size, population successes, and population size. The function

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represents Equation (2.44). Example 2–23 is solved using the HYPGEOMDIST function as shown in Table 2–18. The following steps are used to construct Table 2–18. a. Input the population size (number of available prospects) in Cell C4, input the number of expected discoveries out of the population (available prospects) in Cell C5, and input the number of wells to be drilled in Cell C6. b. Enter the formula =HYPGEOMDIST(A10,C$6,C$5,C$4) in Cell B10 and copy it to Cells B12 to B13. c. Input the desired number of successes in the sample in Cells A10 to A13.

A

B

C

4 Number of Total Prospects

10

5 Number of Expected Discoveries

4

6 Number of Wells to be Drilled

3

7 8

Probability of Failures

9

Wells

10

0

16.67%

11

1

50.00%

12

2

30.00%

13

3

3.33%

Probability

Table 2–18 Using Excel’s hypergeometric distribution function

POISSON. The POISSON (X, mean, cumulative) function returns the poisson distribution. The function represents Equation (2.47). Example 2–24 (Part 1) is solved using the POISSON function as shown in Table 2–19. The following steps are used to construct Table 2–19.

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a. Input the average leaks per year in Cell C4. b. Enter the formula =POISSON(A8,C$4,FALSE) in Cell B8 and copy it to Cells B9 to B12. c. Input the number of leaks for which probability is to be calculated Cells A8 to A12.

A

B

C

4 Average Number of Leaks per year

3

5 6 Probability of Leaks Next Year 7

Leaks

8

0

4.98%

9

1

14.94%

10

2

22.40%

11

3

22.40%

12

4

16.80%

Probability

Table 2–19 Using Excel’s Poisson distribution function

NORMDIST. The NORMDIST (X, mean, standard deviation, cumulative) function returns the cumulative distribution function or probability density function for a random variable X when the mean and standard deviation of the sample are given. The function represents Equation (2.23). The function can be used to generate PDFs and CDFs for random variables when mean and standards deviations are given. Alternatively, NORMINV(probability,mean,standard deviation) function is used to calculate the inverse of NORMDIST, i.e. to calculate the value of X when probability, mean, and standard deviation are given. NORMSDIST. The NORMSDIST(z) function returns the standard normal cumulative distribution function. The distribution has a mean of

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zero and a standard deviation of one. The term z, refereed to as standard normal deviate, is calculated using the STANDARDIZE(X, mean, standard deviation) function. Equation (2.48) is represented by the STANDARDIZE function. Example 2–25 is solved using the two functions as shown in Table 2–20. The following steps are used to construct Table 2–20.

A

B

C

4 Upper Porosity Limit, %

15.0

5 Lower Porosity Limit, %

12.0

6 Average Porosity, %

12.0

7 Standard Deviation of Porosity, %

2.5

8 9 Z at Upper Porosity Limit

1.2

10 Z at Lower Porosity Limit

0.0

11 12

Probability of Porosity

13 Between Mean and Upper Limit

38.49%

14 Greater than Upper Limit

11.51%

15 Less than Upper Limit

88.49%

16 Between Upper and Lower Limit

38.49%

17 Between Mean and Lower Limit

0.00%

18 Greater than Lower Limit

50.00%

19 Less than Lower Limit

50.00%

Table 2–20 Using Excel’s normal distribution and STANDARDIZE functions

a. Input the upper limit of porosity in Cell C4, the lower limit of porosity in Cell C5, the average porosity in Cell C6, and the standard deviation of porosity in Cell C7.

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b. Enter the formula =STANDARDIZE(C4,C$6,C$7) in Cell C9 and copy it to Cell C10. The standard normal deviate (z) is calculated for the upper and lower limits of porosities. c. Enter the function =NORMSDIST(C9)–0.50 in Cell C13 to calculate the probability of porosity between the lower and upper limits. For the probability of porosity greater than the upper limit enter =0.50–C13 in Cell C14. Similarly, enter =0.50+C13 in Cell C15, enter =0.50–NORMSDIST(C10)+C13 in Cell C16, enter =0.50–NORMSDIST(C10) in Cell C17, enter =1–NORMSDIST(C10) in Cell C18, and enter =0.50–C17 in Cell C19. LOGNORMDIST. The LOGNORMDIST(X, mean, standard deviation) function returns the cumulative lognormal distribution of random variable X, where ln(X) is normally distributed with parameters mean and standard deviation.

Introducing BestFit In the preceding pages of this chapter, Excel and its built-in statistical function was used to fit a normal curve to data (see Example 2–8, Table 2–7, and Figure 2–6). A spreadsheet add-in called BestFit can be used to fit several curves to historical data and suggest the curve that best fits the data. The bit record data of Table 2–1 is used to show the use of BestFit. BestFit is included in the DecisionTools suite (on the CD included) provided by Palisade Corporation. BestFit links directly to Excel using Excel style menus, toolbars, and functions. Figure 2–27 shows Excel with BestFit add-in, except for the change in the toolbar menu at the top. The @RISK add-in, presented in Chapter 6, also has built-in capability of curve fitting. Either one of these can be used.

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Fig. 2–27 The Excel spreadsheet with BestFit add-in, data of Table 2–1 is entered

BestFit Add-in. If BestFit is not already installed then the first thing to do is to install the DecisionTools suite with the setup program on the CDROM accompanying this book. The steps for installing the DecisionTools are given in Appendix B. The installation, of course, is done only once. The BestFit can be executed in one of the following three ways. •

•

•

If Excel is not currently running, both the Excel and BestFit can be launched together by double clicking on the BestFit icon already created when the DecisionTools suite was installed. If for some reason the icon has not been created, BestFit can be accessed by (a) click on Windows Start button, (b) click on Palisade DecisionTools, and then (c) click on BestFit. If Excel is already open and the BestFit needs to be loaded, then follow the previous instructions.

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When BestFit is executed, the menu as shown in Figure 2–27 appears. The data to be fitted with a curve is entered as shown in Figure 2–27. The BestFit icons in the toolbar at the top are described, from left to right, below. The same functions can also be accessed by clicking on the Fitting menu, fifth from left, in the top toolbar. 1. Fit Distribution to Data. Starts fitting specified distributions to the data. 2. Select Distribution to Fit. By clicking on this icon, a menu appears listing all the distributions available in BestFit. Select the desired distributions BestFit will use to fit the data. 3. Define Chi-Square Test Binning. Here limits for the bins can be specified or left at automatic so BestFit selects the bins it considers appropriate. 4. Input Data Options. BestFit allows fitting distributions to various types of data, (sampled values, density curve, or cumulative curve). 5. Sort Data. By clicking on this icon, the input data is automatically sorted in ascending order. 6. Generate Random Data. By clicking on this icon, a menu appears where the type of distributions and its parameters are specified. Based on these selections BestFit generates random input data. 7. Transform Data. This function can be used to transform input data into linear, power, logarithmic, exponential, or round type. Once the Fit Distribution to Data icon is clicked, BestFit shows the selected distributions to the data on the screen in order of best fit (see Figure 2–28.) At this time, the other five icons are activated. The manual of BestFit should be used to study other capabilities of BestFit.

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Fig. 2–28 Results of fitting the bit record data of Table 2–1 with probability distribution

The normal curve fitted to the data gives a mean of 100.75 and standard deviation of 22.752, which is the same as manually calculated in Example 2–5.

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QUESTIONS and PROBLEMS 2.1

What is meant by measure of central tendency? What are the various measures of central tendency?

2.2

What is the difference between mean, median, and mode? In which situation are the three measures the same. When would mode be chosen as a measure of central tendency?

2.3

Order mean, mode, and median for the data skews to the left.

2.4

List the situations where geometric and harmonic mean give realistic measures of central tendency versus the mathematical mean.

2.5

Define the weighted average of a set of numbers.

2.6

What purpose does a measure of variability serve? Name the various measures of variability.

2.7

Briefly, define standard deviation, variance, percentile, docile, and quartile. In which situation the percentile is equal to the docile?

2.8

What is coefficient of variation? When is it useful to use coefficient of variation?

2.9

Calculate the coefficient of variation for a set of data with standard deviation of 4.2, number of items 12, and the sum of the items is 138. What will be the coefficient of variation if the variance is 138 and the mean is 38?

2.10 Coefficient of variation of two series is 58% and 72%. Their standard deviations are 20 and 18. What are their means? 2.11 In a moderately skewed distribution, the arithmetic mean is 10 units and the mode is 7 units. Find the median.

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2.12 What is the difference between grouped data and ungrouped data? In which situation should grouped data be used for analysis? 2.13 Compare the relative advantages of presenting statistical data in the form of (a) tables, (b) diagrams, and (c) graphs. Illustrate your answer with suitable examples. 2.14 Name various ways frequency distribution can be graphically presented. 2.15 Graphically illustrate the difference between a histogram and an ogive. 2.16 What is meant by a frequency curve? State important characteristics of a frequency curve. 2.17 Explain with illustration the following: a. b. c. d. e. f.

Frequency distribution Relative frequency Cumulative frequency Class boundaries Class mark Width of class interval

2.18 Discuss the different steps required for the construction of a frequency distribution from raw data. 2.19 Discuss the problems one might face in constructing a frequency distribution, with particular emphasis on the choice of class interval and number of classes. What is the effect of too many or too few number of classes? 2.20 The class midpoints in a frequency distribution of water saturations are 0.25, 0.32, 0.39, 0.46, 0.53, and 0.60. Find (a) the size of the class interval, (b) the class boundaries, and (c) the class limits.

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2.21 A frequency distribution table has six class intervals with 3, 9, 15, 30, 18, and 5 as frequencies in order. The left end-point of the third class interval and the midpoint of the fifth class interval are 20 and 45, respectively. Set up the frequency table. 2.22 Define probability and briefly explain its importance in statistics. 2.23 What are the different Schools of thought on the interpretation of probability? 2.24 What is meant be subjective and objective probability? A tail on a flip of coin gives objective or subjective probability? 2.25 What is the difference between classical and modern concepts of probability? 2.26 In of probability concepts, what do we mean by (a) equally likely, (b) collectively exhaustive, and (c) mutually exclusive? 2.27 What do (a) simple probability, (b) conditional probability, and (c) t probability mean? 2.28 What do we mean by (a) prior probability and (b) posterior probability? 2.29 What is the difference between (a) mutually exclusive events and overlapping events, (b) mutually exclusive events and independent events, and (c) simple and composite events? 2.30 What is a probability distribution? How is it useful for decision making? 2.31 What is the difference between discrete and continuous probability distributions? 2.32 Name some of the discrete and continuous probability distributions.

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2.33 What is the difference between binomial distribution and hypergeometric probability distribution? 2.34 The following table lists recoverable reserves of 60 wells on a prospect. 12.65 171.35 55.20 57.50 58.65 47.15 94.30 139.15 85.10 159.85

115.00 63.25 138.00 185.15 94.30 127.65 50.60 173.65 220.80 230.00

23.00 36.80 14.95 88.55 135.70 174.80 180.55 44.85 109.25 55.20

29.90 133.40 126.50 46.00 127.65 126.50 331.20 280.34 95.45 161.04

109.35 112.72 48.34 58.67 79.43 27.62 146.12 33.37 36.82 72.47

241.65 70.15 247.25 98.96 19.55 25.31 65.57 36.83 265.65 96.48

a. Using the equation 1 + 3.322logN, classify the data in equal class intervals. b. What is the frequency of reserves in each interval? c. Generate the cumulative frequency distribution d. Generate the cumulative relative frequency and the decumulative relative frequency distribution. e. Plot the frequency distribution as a histogram and the cumulative relative frequency as an ogive. f. Using the information obtained above; what is the (a) probability of reserves greater than 48,000 Stb, (b) probability of reserves less than 100,000 Stb, and (c) probability of reserves between 160,000 Stb and 43,000 Stb. 2.35 Using the ungrouped data of Problem 2.34, calculate the following. a. Mean, mode, and median. b. Standard deviation, variance, and coefficient of variation. c. Geometric, harmonic, and quadratic mean

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2.36 Group the data of Problem 2.34, and calculate the following from the grouped data. a. b. c. d.

Mean, median, and mode Standard deviation, variance, and co-efficient of variation Geometric and harmonic mean Draw histogram and cumulative relative frequency curve for the grouped data.

2.37 Rework Problem 2.35 using the built-in functions of Excel. 2.38 Rework Problem 2.36 using the built-in functions of Excel. 2.39 Rework Problem 2.34 using BestFit. Based on the distribution fitted by BestFit, which distribution more appropriately represents the data? 2.40 Formation thickness and the corresponding porosity data are given in the following table. Calculate the weighted average porosity for the data. Porosity

Thickness

Porosity

Thickness

(%)

(feet)

(%)

(feet)

24.2

2.5

13.5

1.5

20.7

1.5

12.2

4.0

18.9

2.0

11.8

3.5

18.0

3.2

16.5

5.5

15.5

4.0

25.2

3.0

2.41 Use the normal-probability and log-normal probability graphs to plot the results arrived at in Problem 2.34. From these plots, which types of probability distribution you think more appropriately represent the data? Are the findings in line with what is determined by BestFit in Problem 2.39?

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2.42 Using the Venn diagram below, assign values to the following. A, B, U, A, and B A艛B, A艚B, A艚B, A艚B, (A艚B) and A艚B

75

35

90

2.43 If p(A) = 70, p(B) = 90, p(A艚B) = 30, and p(艛) = 200 then fill following table. A

Totals

B

A ?

?

?

B

?

?

?

Totals

?

?

?

2.44 An oil company is evaluating a new prospect. Based on current information, the company’s geologists estimate the probability of finding a commercial discovery is 65%. Further studies are then carried out that suggest a commercial discovery is certain. However, the probability of the studies giving a correct result is 80%. Calculate the posterior probability based on this additional information. 2.45 What is the probability of rolling a five on a roll of two dice? 2.46 There are 500 coins in a bag, 200 are dimes, 175 are quarters, and 125 are nickels. What is the probability a coin drawn from the box will be a (a) quarter, (b) dime, and (c) nickel.

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2.47 If the probability of drilling the next well as producer is 30%, two additional wells will be drilled. The probability of success of the second and third wells is 45% and 35%, respectively. If the first well is dry then no other wells will be drilled. a. What is the probability all three wells will be successful? b. What is the probability at least two wells will be successful? c. What is the probability none of the three wells will be successful? 2.48 You have 10 exploratory prospects to be drilled and the chance of success of each well is 35%. The management has decided that if the first three prospects drilled are not successful then the other seven prospects will not be drilled. However, if one of the first three prospects drilled is successful then the rest of the prospects will be drilled. a. What is the probability of drilling three or more successful wells? b. What is the probability of drilling more than four successful wells? c. What is the probability of drilling two or less successful wells? d. What is the probability of drilling at least six successful wells? 2.49 You have determined the reserve data of a certain prospect is normally distributed with a mean of 250,000 barrels and a standard deviation of 100,000 barrels. What is the probability of reserves (a) more than 300,000 barrels, (b) less than 100,000 barrels, (c) less than 250,000 barrels, and (d) between 100,000 and 280,000 barrels? 2.50 For the data in Problem 2.49, generate a probability density function and cumulative distribution function of the reserves. Use the appropriate statistical functions of Excel to solve this problem.

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REFERENCES 1

2

3

4

Stanley, L.T., Practical Statistics for Petroleum Engineers, Petroleum Publishing Co., Tulsa, Oklahoma, USA, 1973. Schuyler, J. R., Petroleum Risks and Decision Analysis, course manual, Oil and Gas Consultants International, Inc., 4554 South Harvard Avenue, Tulsa, OK 74135, USA, 1993. Lerche, I., and MacKay, J. A., Economic Risk in Hydrocarbon Exploration, Academic Press, 525B Street, Suite 1900, San Diego, CA 92101, USA, 1999. McCray, A. W., Petroleum Evaluation and Economic Decisions, Prentice-Hall, Inc., Englewood Cliffs, NJ, USA, 1975.

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THREE

Expected Value and Decision Trees

Models are constructed and used to predict consequences of various possible events and decisions. It is not rare to identify uncertainties in the inputs to these models. The objective here is to show how best these uncertainties can be propagated through the model in order to discover uncertainty in the predicted consequences. For example, how the uncertainty of porosity, water saturation, formation thickness, formation volume factor, recovery factor, and drainage area can be propagated through the reserves estimates. Similarly, how the uncertainty of oil price, oil production forecast, and operating expenses can be propagated through the cash flow and the resulting profitability indicators. Therefore, useful insights about the relative importance of forecasts of various assumptions, decisions, uncertainties, and disagreements in the inputs shall be obtained. Depending on the information available to a decision maker, the decision making might be classified into one of three situations.

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Decision making under certainty, deterministic models: As previously discussed, certainty refers to situations where the variables are treated as if they will result in only one specified outcome. This may never be true in reality but at least the decision maker assumes it so. Decision making under uncertainty, non-probabilistic models: This type of uncertainty is characterized by situations where each action may lead to any one of a set of possible outcomes, the probabilities of which are not known or cannot be estimated. Decision making under uncertainty, probabilistic models: This type of uncertainty is characterized by situations where each action may lead to any one of several outcomes, the probabilities of which either are known or can be estimated. The following methods are generally used to select a course of action under uncertain environments when probabilities are not known or cannot be estimated (they do not for event likelihood). • • • • •

Maximax decision rule Maximin decision rule Minimax regret criterion Hurwicz criterion Laplace criterion

However, these methods are not discussed in this book due to their limited use in oil and gas business and since decision analysis philosophy generally aims towards explicitly recognizing and ing for uncertainty. Any good management science books may be consulted for coverage of these topics. Other approaches of dealing with the decisions concerning the uncertain (non-probabilistic) situations are conducting scenario analysis or sensitivity analysis. A scenario is a particular situation, specified by a single value for each input variable. For example, a “base case” scenario consists of a single nominal value for each input. These are normally the initial “best guess” values for the inputs. Each of these may be the mean,

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median, or most likely values. Then there is a worst-case (pessimistic or low case) scenario and a best-case (optimistic or high case) scenario. Since the analysis of uncertainty involves, in part, measuring the degree to which each input contributes to uncertainty in the output, the simplest measure to quantify this degree can be termed as sensitivity. Sensitivity is the rate of change of the output with respect to variation in input of a certain variable. The method, in general, involves computing the effect on the output from its low to high value, while keeping the other inputs at their nominal or most likely values.1 In Chapters 3 through 7 of Volume I of this series, different aspects of project economics (based on deterministic models) are discussed. Chapter 2 of this volume discussed decision making under uncertainty (probabilistic rules). In this chapter, some basic probabilistic methods used for selecting a course of action under uncertain environments (when probability of each outcome is either known or can be estimated with reasonable certainty) are presented. The two subjects discussed in this chapter are (a) expected value concepts and (b) decision trees.

EXPECTED VALUE CONCEPTS This section presents the most commonly used concepts that quantify probabilistic treatment of uncertainty in decision making in of expected value relative to decision methods that ignore uncertainty. It introduces the notion of the expected value of including uncertainty, as a measure of the effect of considering uncertainty in the context of a specific decision problem. Two types of decision elements are normally considered when dealing with situations involving uncertainty. The first is the value associated with various alternatives and their outcomes. The value may be measured in monetary or any other dimensions (such as porosity, pay thickness, water saturation, and reserves, etc.). The second is the likelihood of occurrence of this value associated with its respective outcome. This is measured in of probability. The probabilistic rules for both these elements associated with various alternative actions.2

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This section presents (a) (b) (c) (d) (e) (f) (g)

expected value and variance of random variables, expected monetary value, expected opportunity loss, limitations of expected value, expected value rules in multistage decisions, value of information, and stochastic dominance of expected value rules.

Expected Value of Random Variable When working with situations involving uncertainty, the decision maker is interested in the average value of the occurrence of random variables in many trials. This average value is the weighted average of the possible outcomes, with the probability values used as weights. This weighted average of the random variable is referred to as the expected value of the random variable and it (expected value of X) is mathematically denoted by the following equation. (3.1) n

E{X }= ∑ xi P(xi ) i =1

where E{X} = the expectation operator, read as “expectation of ” P(xi) = Denotes P(X = xi), the unconditional probability associated with variable x The expected value of a random variable is like a measure of position for the probability distribution. Therefore, the E{X} is often referred to as the mean of the probability distribution of X or simply the mean of X. The E{X}, like any mean, will be a number that may or may not correspond to any of the actual possible outcomes of X.

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Standard Deviation of Random Variable Since the outcomes of a random variable are probabilistic, it is useful to have a measure of the dispersion or variability of the outcomes. This single parameter is important since the mean by itself conveys only the measure of position and not its variability. The mean and the standard deviation or variance play an important role in decision making. The standard deviation of a discrete random variable is given by the following equation. (3.2) n

2

s 2 {X }= ∑ (xi − E{X }) P(xi ) i =1

OR (3.2a)

s 2 {X }= E{X 2}− (E{X })

2

where s2{X} = Variance of X, referred to as variance operator. 2 The standard deviation of X is then s{X} = s {X} .

The variance is a weighted average of squared deviations, the deviations being the outcomes of X around their expected value, and the weights being the respective probabilities of occurrence. Therefore, the variance measures the extent to which the outcomes of X depart from their expected value. It is always measured in square units, such as dollars squared. To convert this to a more meaningful number, its square root is calculated and referred to as a standard deviation. In practice, there are situations when expected values or variances of different alternatives have to be added together. So if T = X1 + X2 + . . . Xn represents the sum of n independent variables then the expected value and

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variance of T(a sum of independent random variables) is given by the following equations.

n

E{T}= ∑ E{X i} = E{X 1} + E{X 2}+ i =1

n

K + E{X }

s 2 {T} = ∑ s 2 {X i} = s 2 {X 1}+ s 2 {X2} + i =1

(3.3)

n

K + s {X }

(3.4)

2

n

When the expectations and variance of each random variable are already known, then the expected value and variance of the sum can be directly obtained with the following equations. (3.5)

E{X + Y }= E{X }+ E{Y }

(3.6)

s 2 {X + Y} = s 2 {X }+ s 2 {Y}

Similarly, the expected value of the difference of two independent random variables is the difference of expected values of the two random variables. However, the variance of the difference is the same as the variance of the sum [Equation (3.6)]. For these simplifying equations to work, the variables must be both independent and random. When errors occur, it is because the variables are not both independent and random.

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Example 3–1 Drilling of a certain prospect is expected to result in an oil producer with a 30% chance of 20 MStb reserves, 50% chance of 60 MStb reserves, and a 20% chance of 95 MStb reserves. Calculate the mean, variance and standard deviation of the expected reserves encountered by the well. Solution: The calculations for this example are summarized in Table 3–1.

Probability

Reserves, Xi Expectancy E{X}=piXi

(Xi-E{X})2

Variance 2

pi(Xi-E{X})

pi

MStb

0.30

20

6.0

1,225.0

367.5

0.50

60

30.0

25.0

12.5

0.20

95

19.0

1,600.0

320.0

1.00

55.0

700.0

Table 3–1 Calculation of mean and variance for Example 3–1

The expected value of reserves is 55 MStb and the standard deviation is, therefore, s = 700.0 = 26.5 MStb. This means that on the average it is expected to discover 55 MStb over a large number of similar trials with about 67% confidence that the actual result will lie between 28.5 (55 – 26.5) and 81.5 (55 + 26.5) MStb.

Expected Monetary Value (EMV) If the random variable in the situation discussed previously is in monetary , then the calculated expected value is termed as expected monetary value (abbreviated as EMV). The EMV is a weighted average of the possible monetary values (usually net present value, NPV), weighted by their respective probabilities. The monetary values can be undiscounted or discounted. The expected value of the discounted net

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cash flow (NPV) is also referred to as the expected present value profit. The net present value (NPV) for EMV calculations is based on deterministic models. The calculations of NPV are shown in Chapter 6 of Volume I. Throughout this book EMV means E{NPV}, i.e., expected value of net present value. Alternatively, it is defined as the mean of the probability distribution of all possible monetary outcomes. The calculations involve generating event probabilities, multiplying the payoff of each event by the probability of occurrence for that event, and summing up the products. This sum is termed as the expected monetary value. The expected value of a decision alternative can be zero, positive, or negative. Mathematically, Equation (3.1) is modified to give the EMV as: (3.7) n

EMV = E{NPV } = ∑ NPVi × P(NPVi ) i =1

To make a choice between mutually exclusive investment alternatives, the expected monetary value of each alternative is calculated, and the alternative with the largest EMV is selected. For screening alternatives, all investments with EMV greater then zero (EMV > 0) are acceptable. If all the conditional values are expressed as costs, then the alternative with the lowest expected value of costs is selected. In some cases, it is useful to give the values in of opportunity losses, in which case the alternative with the lowest expected opportunity loss would be selected. The decision problems incorporating probabilities have the following common structural elements.3 Acts or strategies, Aj. These are the various available alternatives (drill a well, don’t drill a well, farmout, etc.). The decision problem involves deciding on which alternative to choose out of the available alternatives.

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Outcome states, Si. The different situations that may prevail and affect the consequences of Aj may include dry hole, 20 MStb reserves, etc. Drilling a well is in our control but whether it will be a producer or a dry hole is an uncontrollable variable/outcome. Consequences or payoffs, Cij. These are the gains, rewards, losses, etc. associated with the jth act that results in the ith outcome state. Outcome state probabilities, P(Si). These are the probabilities assigned to the outcome states. Criterion. This is the basis used by decision makers to decide on the appropriate course of action (such as choosing the act Aj ) that is best among the available courses of action. The previously mentioned structural elements can be displayed and then solved to arrive at the expected value in two convenient ways. These two devices are (1) the payoff table (tabular) and (2) the decision tree (graphical). Either one of these methods can be used depending on whether a tabular or graphic display is more effective to communicate with the decision maker. The payoff tables are used in this section. In the next section, decision trees are used to solve the same problems. A typical structure of a generalized payoff table is shown in Table 3–2.

Outcome

Act A1

Act A2

State

Probability

Value of A1

E{A1}

Value of A2

E{A2}

S1

P(S1)

C11

P(S1)C11

C12

P(S1)C12

S2

P(S2)

C21

P(S2)C21

C22

P(S2)C22

: Sx

: P(Sx)

: Cx1

: P(Sx)Cx1

: Cx2

: P(Sx)Cx2

EV=

ΣE{A } 1

EV=

ΣE{A } 2

Table 3–2 Generalized expected value computation

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Example 3–2 A drilling prospect is evaluated with an estimate that the probability of a successful well is 35% and the probability of a dry hole is 65%. Drilling of a dry hole will result in net loss of $250,000. If the well is successful, then the net present value of the future streams of net revenues will be $500,000. Instead of drilling the well the prospect can be farmed out (i.e., no exposure to drilling expenditure) and retain an overriding royalty interest in the well. The net present value of farmout will be $50,000. Is it economically better to drill the well or farm it out? Solution: The problem is solved in the payoff table of Table 3–3.

Outcome

Drill

Farmout

State

Probability

NPV

EMV

NPV

EMV

Dry Hole

0.65

-$250,000

-$162,500

$0

$0

Producer

0.35

+$500,000

+175,000

$50,000

$17,500

1.00

$12,500

$17,500

Table 3–3 Payoff table for the data in Example 3–2

In Example 3–2, the EMV = $17,500 of farmout is higher than the EMV = $12,500 of drilling ourselves. In order to maximize the expected net present value, it is prudent to farmout in this situation. The farmout is giving a net advantage of $5000 in expected present value profit over the drill option. However, would the decision hold if the probability of a producer were increased to 36% versus the 35%? Increasing the probability of the producer to 36% switches the decision against the farmout option. This shows that if one is not certain about the assigned probabilities, performing sensitivity analysis will give a range of probabilities in which each option will be feasible. Furthermore, variance of the drill option is

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significantly higher than the variance of the farmout option (the drilling option is much more riskier).

Example 3–3 A proposed drilling prospect has been evaluated. A well will be drilled on 160-acre prospect area. The company does not yet lease additional acreage of 60 acres, adjacent to the prospect. Evaluation is based on the condition that the company will be able to lease the remaining 60 acres. Basic data and economic analysis (net present value at 10% that would be realized from the prospect for each option, net of taxes, royalties, lease operating expense and costs of the well) of each option for the prospect are given in Table 3–4. The company has identified the following three options for participation in the deal, given that the company has acquired the remaining 60 acres. (after Newendorp).4

Gross Well Cost (including Lease Equipment)= Gross Dry Hole Cost =

$110,000 $80,000

Net Present Value Drill with

Farmout

37.5%

Retain ORI

Back-in

Outcomes

Probability

37.5% WI

Dry Hole

0.25

-$30,000

$0

$0

20 MStb

0.30

$4,357

$8,733

$750

35 MStb

0.25

$45,448

$14,646

$34,142

50 MStb

0.15

$87,411

$20,693

$73,712

65 MStb

0.05

$125,863

$26,401

$111,141

Table 3–4 Basic data for Example 3–3

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1. Participate in the drilling with 37.5% (60 acres/160 acres x 100 = 37.5%) non-operating WI. 2. Farmout acreage while retaining 1/8th of 7/8th royalty interest on 60 net acres. 3. Carry with a back-in privilege (37.5% WI) after 150% of the investment by the participating parties is recovered. Based on the above options and the data in Table 3–4, calculate the following. 1. Should the adjacent lease be acquired? If yes, how much (maximum) should be paid for it? 2. If the adjacent lease is acquired, which of the three options will be more valuable? Solution: The example is solved by first calculating the expected value of each available option. The expected value calculations for this example, using the data in Table 3–4, are shown in Table 3–5. The detailed cash flows for each option are shown in Table 3–5 on the CD.

Outcome State

Drill with 37.5%

Farmout with ORI NPV

Probabilit y

NPV

EMV

Dry Hole

0.25

($30,000)

($7,500)

$0

$0

$0

$0

20 MStb

0.30

$4,357

$1,307

$8,733

$2,620

$750

$225

35 MStb

0.25

$45,448

$11,362

$14,646

$3,662

$34,142

$8,536

50 MStb

0.15

$87,411

$13,112

$20,693

$3,104

$73,712

$11,057

65 MStb

0.05

$125,863

$6,293

$26,401

$1,320 $111,141

$5,557

NPV

EMV

EMV

$24,574

$10,706

$25,375

Standard deviation

$45,622

$7,809

$32,869

Table 3–5 Expected value computation for Example 3–3

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Expected Value and Decision Trees

From the expected value calculations in Table 3–5, the back-in option with the highest EMV of $25,375, is economically the most viable option. Therefore, the remaining acreage is worth acquiring. Now the question is, how much, maximum should be paid for the additional 60 acres. Since the expected net present value of the preferred option is $25,375, the maximum value of this additional acreage should not be more than $423/acre ($25,375/60 acres ≈ $423/acre). If an amount exactly equal to the expected NPV is paid, the rate of return will be equal to 10% (equivalent to the discount rate used for NPV calculations). If it is managed to acquire the acreage for less than $423/acre, this will generate a positive E{NPV} and a return higher than 10%.

Sensitivity Analysis Sensitivity analysis is generally used to examine the robustness of an alternative to changes in the variables used in arriving at the value of the alternative under consideration. The usual approach to any sensitivity analysis, in general, is to hold all aspects of the model constant and vary each parameter while observing the influence of the changes upon the optimal decision. In this section, sensitivity analysis on probabilities, used to calculate EMV of each alternative, is performed. These are normally the most uncertain parameters. The data of Example 3–2 and Table 3–3 are used to show how the sensitivity analysis can be performed. This is a simple case, i.e., two acts (drill or farmout) and two events (dry hole or producer). The objective is to determine the dry hole probability when the decision maker will be indifferent to the alternatives. If p is the probability of dry hole and (1 – p) is the probability of producer, then the decision problem becomes EV{Drill} = p(–$250,000) + (1 – p)($500,000) = –$250,000p + $500,000 – $500,000p = –$750,000p + $500,000 EV{Farmout} = p(0) + (1 – p)($50,000) = –$50,000p + $50,000

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The decision maker would be indifferent between the two alternatives if the two expected values are equal. The question is, at what dry hole probability the expected value of the two alternatives will be equal. This probability is computed as shown following. EV{Drill} = EV{Farmout} –$750,000p + $500,000 = –$50,000p +$50,000 –$700,000p = –$450,000

–$450,000 p = ––––––––– = 0.6429 or 64.29% –$750,000

This means that at the point of indifference, both expected values are equal. If the probability of a dry hole were higher than 64.29%, then the optimal act would be to farmout; and if the probability of a dry hole is less than 64.29%, then the optimal act would be to drill. The same result can be obtained graphically. Figure 3–1 is a graph of expected monetary value (EMV) as a function of the probability of dry hole. The correlation in this figure assists in estimating a general range of dry hole probability on which a decision can be made. In this example, if the analyst is not certain of the probability of encountering a producer but the analyst felt it was probably below 35.71% (dry hole probability above 64.29%), then the preferred choice will be to farmout. It is evident from Figure 3–1 that as long as the probability of dry hole is higher than 64.29%, the farmout option is the optimal choice. However, the drill option is highly sensitive to change in probability. A small improvement in the dry hole probability makes the drill option dominant over the farmout option.

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Fig. 3–1 Sensitivity analysis on the probability of dry hole in Example 3–2

Sensitivity analyses become more complex when there are more than two alternatives. For example, in Example 3–3 there are three alternatives (drill, farmout, and back in). This leads to solving three simultaneous equations, and graphical method is recommended in such situations. For Example 3–3, sensitivity is performed on the probability a well drilled will be a producer. A table used to perform sensitivity is included in the tables (Table 3–5) on the CD. For an important parameter such as this, it is important to evaluate its effect on the expected monetary value. Figure 3–1 for Example 3–2 and Figure 3–2 for Example 3–3 serve this purpose. As previously mentioned, such a correlation assists the analyst in estimating a general range of the probabilities of encountering a productive well.

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Fig. 3–2 Graph of EMV versus the probability to encounter producer

In Example 3–3 (Fig. 3–2), the graph shows if the probability of encountering a productive well is greater than 78% (crossover of drill and back-in option) the drill option is the preferred option. If the probability of encountering a productive well is less than 78%, then the back-in option is preferred. The farmout option is not the preferred choice in either case.

Expected Profitability Index (EPI) In the previous section, E{NPV} was used for choosing the most economically viable option out of a set of mutually exclusive investment alternatives. In this section, another profitability indicator that assists in choosing the most economically viable option(s) out of a set of nonmutually exclusive investment alternatives is introduced. This yields the highest expected value gain per unit of money invested under imposed capital constraint. The method presented here can also be used in choosing the most economically viable option out of a set of mutually exclusive investment alternatives.

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When working under limited capital constraints and with investments that are non-mutually exclusive, it is recommended to use the EPI criterion to arrive at the highest expected net gain per unit of expected investment. The description of the profitability index (PI) has been previously presented in Chapter 6 Volume I. Here the probabilistic (expected) version of that is presented. The EPI is basically the ratio of E{PV of Net Operating Revenue}/E{PV Capital Investment}. Alternatively, the E{PI} is given by 1 + E{NPV} /E{PV Capital Investment}. The objective of a decision maker would be to maximize this ratio if he/she does not have enough funds to invest in all of the available economically viable options. The numerator in the ratio is the same as that calculated in the previous section. The denominator is the expected value of investment or capital expenditure. If the investments or capital expenditures are not at time zero, then expected present value of these investments should be used. The present value of the investment is calculated at the same discount rate as that used for calculating E{NPV}. The ratio provides a measure of the expected gain per unit of expected investment. For example, an investment with EPI of 1.5 means it will generate a net expected profit, in present value , of $0.5 per every dollar of capital invested in the project. This means the project will recover the initial capital investment, in addition it will generate $0.5 for every dollar initially invested. The weakness of EMV is it does not give any idea of the magnitude of initial investment. For example, one project generates EMV of $1,500 for an investment of $1000, and another project generates an EMV of $500 for an investment of $200. Based on the EMV decision rule a decision maker will select the option that generates an EMV of $1,500 (EMV=$1,500 vs. EMV=$500). However, based on the option with EMV=$500 is better because its EPI=3.5 is much higher than the EPI=2.5 of the other option. For screening investments, all investments with EPI > 1 are economically viable projects. For deciding among mutually exclusive investments, choose the investment that yields the highest EPI is selected. For deciding among non-mutually exclusive investments, if there is no budget constraint, all investments with EPI > 1 are selected. Under budget constraints, all investments with EPI > 1 are arranged in descending order of EPI. Then starting at the top, investments are selected and their investment accumulated until the budgeted level of investment funds is achieved. 167

Project Economics and Decision Analysis Volume II: Probabilistic Models

Another profitability indicator giving the same result as the E{PI} is the E{PVR}, referred to as the expected present value ratio or expected discounted return on investment. The E{PVR} = EMV/E{PV of Capital Investment} or simply E{PI} – 1. Projects with E{PVR} > 0 are accepted, and projects with E{PVR} < 0 are rejected. Similarly, when choosing among mutually exclusive investment alternatives, the alternative with the highest E{PVR} is selected. The E{PI} and E{PVR} do not work for farmout or royalty type of investments because in these cases investment is zero. Therefore, with zero in the denominator the E{PI} is infinity.

Example 3–4 This example will clarify the use of EPI criterion in decision analysis. Three different drilling prospects are under evaluation. Detailed deterministic economic analysis for each prospect was conducted and probability assessment for each outcome was made. The economic analysis for each prospect included taxes, royalties, lease operating expenses, and costs of the well. A discount rate of 10% was used in each case to arrive at their NPV. Based on the data presented in Table 3–6, calculate 1. The EMV and EPI for each prospect. 2. Based on the decision rule of maximizing the EMV, which of these prospects will be economically most viable (assume the prospects are mutually exclusive)? 3. If the decision-maker has limited funds of $150,000 and he/she wants to maximize the expected worth per expected investment costs, which of these prospects would be the preferred choice? Solution: The calculations are shown in Table 3–6.

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Expected Value and Decision Trees

PROSPECT A Completed Well Cost

$120,000

Dry Hole Cost

$80,000

Outcome State

Probability

NPV

EMV

Dry Hole

0.40

-$80,000

-$32,000

100 MMScf

0.25

$90,000

$22,500

200 MMScf

0.20

$180,000

$36,000

300 MMScf

0.10

$270,000

$27,000

400 MMScf

0.05

$360,000

$18,000

1.00

$71,500

E{Drilling}

$104,000

EPI

1.688 PROSPECT B

Completed Well Cost

$225,000

Dry Hole Cost

$150,000

Outcome State

Probability

NPV

EMV

Dry Hole

0.35

-$150,000

-$52,500

100 MMScf

0.25

-$85,000

-$21,250

250 MMScf

0.20

$325,000

$65,000

400 MMScf

0.15

$520,000

$78,000

550 MMScf

0.05

$715,000

$35,750

1.00 E{Drilling} EPI

$105,000 $198,750 1.528

Table 3–6 Data and calculations for Example 3–4

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Project Economics and Decision Analysis Volume II: Probabilistic Models

PROSPECT C Completed Well Cost

$45,000

Dry Hole Cost

$30,000

Outcome State

Probability

NPV

EMV

Dry Hole

0.25

-$30,000

-$7,500

100 MMScf

0.30

$25,000

$7,500

250 MMScf

0.25

$45,000

$11,250

400 MMScf

0.15

$62,000

$9,300

550 MMScf

0.05

$90,000

$4,500

1.00

$25,050

E{Drilling}

$41,250 1.607

EPI

Table 3–6 continued . . .

The expected value of capital drilling cost (CAPEX) for Prospect A is calculated as E{CAPEX }A = P (Dry Hole )A × CAPEX (Dry Hole )A + P(Producer)A × CAPEX (Producer)A

= 0.40 × $80,000 + (1 − 0.40) × $120,000

= $32,000 + $72,000 = $104,000 The EPI for Prospect A is now calculated as

EPI A = 1 +

170

EMV A $71,500 =1+ = 1.688 $104,000 E{CAPEX}A

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Performance Index This measure of risk incorporates both the expected returns and the risk. It serves as a complete measure of the feasibility of an investment subject to risk constraint. The risk constraint means the investor’s decision is constrained by choosing the less risky investment. Mathematically, the performance index is given as, (3.8)

I=

EMV Expected Monetary Value = s Standard Deviation

Since the objective of any investment is to maximize the economic returns at the lowest risk, maximizing the value of the performance index achieves this objective. Typically, an investor sets a minimum performance index value as a threshold for screening investments. This minimum is based on the investors’ risk aversion and investment’s objective. Once this minimum desirable performance index threshold is set, then projects with performance index value greater than this minimum desirable value would be considered as economically feasible. For example, the performance indexes for the three investment alternatives in Table 3–5 are (a) 0.539 for the drill option, (b) 1.371 for the farmout option, and (c) 0.772 for the back-in option. The higher the performance index, the lesser is the associated risk. The performance index clearly ranks the riskiness of each alternative. For example, the farmout option carries minimum risk (therefore highest I of 1.371), followed by the back-in option (with I = 0.772), and then the drill option (with I = 0.539). However, as will be discussed in Chapter 4, maximizing the EMV criteria does not for the risk preference of the investor, meaning it assumes the investor is risk neutral (indifferent to risk).

Expected Opportunity Loss (EOL) Expected opportunity loss (EOL) is another probabilistic value criterion used to arrive at a selection of the most economically viable option out of several alternatives available. For any alternative, this is the expected value of regret associated with that action. 172

Expected Value and Decision Trees

By definition, EOL is the difference between an actual profit or loss and the profit or loss that would have resulted if the decision maker had had perfect information (as if he knew the true state of nature) at the time he made the decision. For example, based on the EMV calculations, the drill option in Table 3–4 had been selected. After drilling the well, it comes out to be a dry hole. This would result in a loss of $30,000 (dry hole cost). On the other hand, if it were known the well would be a dry hole, then the farmout or back-in options with zero loss would have been selected. Therefore, the EOL would be $30,000 (30,000 – 0 = $30,000). The EOL and the EMV rules give exactly the same preference rankings of alternatives. They are essentially the same rule, simply incorporating a linear transformation between them. In addition, the difference between EOL and EMV values is identical. The only difference between the two is that EMV is a maximization rule while the EOL is a minimization rule. Therefore, the EOL rule states that the most economically viable alternative out of the available alternatives is the one with the most preferred expected regret (EOL value). If thinking in of values of regret as against pure monetary values such as cash flows, it may be handy to calculate EOL. However, in large multistage decisions where EOL can become complex to interpret, it is recommended to use the EMV rule. The EOL concept is particularly useful in situations where the value of obtaining perfect information has to be determined. The following example clarifies the use of EOL decision rule.

Example 3–5 Rework the data of Example 3–3 (Table 3–4) using the EOL criterion. If a decision maker wants to run detailed seismic survey on the prospect before choosing one of the options, how much (maximum) should he pay for the seismic (the cost of this additional information)? Solution: First construct an opportunity loss table as shown in Table 3–7. The following steps are followed.

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Outcome State

Drill with 37.5% Probabilit y

OL

EOL

Farmout with ORI OL

EOL

Back-in with 37.5% OL

EOL

Dry Hole

0.25

$30,000

$7,500

$0

$0

$0

$0

20 MStb

0.30

$4,376

$1,312

$0

$0

$7,983

$2,395

35 MStb

0.25

$0

$0

$30,802

$7,701

$11,301

$2,825

50 MStb

0.15

$0

$0

$66,718

$10,008

$13,699

$2,055

65 MStb

0.05

$0

$0

$99,462

$4,973

$14,722

$736

1.00

$8,812

$22,682

$8,011

Table 3–7 Expected opportunity loss computation for Example 3–5

1. Identify the maximum value entry in each row of Table 3-4 and subtract each entry in the same row from this maximum value. For example, in Row 2 (20 MMStb) the maximum is $8,733 so the same row in the EOL table will be 4,376 ($8,733 – $4,357 = $4,376), $0 ($8,733 – $8,733 = $0), and $7,983 ($8,733 – $750 = $7,983). Similarly, the other rows are transformed. 2. The expected value computations are performed in the same way as they were done in the EMV computation, except in this case the probabilities are multiplied by conditional opportunity losses, rather than NPVs. From Table 3–7, the minimum EOL is for the back-in option. Therefore, this option is selected because it minimizes the expected opportunity loss. The selection is consistent with the option selected by using the EMV decision rule. The EOL values in Table 3–7 also represent the cost of uncertainty for each alternative.

Mean–Variance and Stochastic Dominance In this chapter so far, the concepts of EMV, EPI, and EOL are discussed. These criteria are used for choosing the most economically viable alternative among mutually exclusive and non-mutually exclusive decision

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Expected Value and Decision Trees

alternatives. The decision criteria used in the particular case are summarized as follows. •

•

•

If profit is the payoff variable and the available alternatives are mutually exclusive, then choose the alternative that yields the highest EMV. If payoffs are measured in monetary but represent costs rather than profits and at the same time the available alternatives are mutually exclusive, then choose the alternative that yields the lowest EOL. If profit is the payoff variable and the available alternatives are non-mutually exclusive, then rank the alternatives from highest EPI to the lowest EPI. Under budget constraint, rank the alternatives in descending order of EPI and select the alternatives from the top until the expected capital expenditure is equivalent to the allocated budget.

However, at times it is essential to use probabilistic rules involving more than just the expected value of an uncertain quantity. The decision criteria noted previously do not for the magnitude of associated risk in each alternative. Therefore, a decision maker who is sensitive to the degree of risk associated with various alternatives may use the EMV and standard deviation to screen alternatives or as a basis of ultimate choice. Based on the mean–variance approach, the investor who is a riskaverse wealth maximizer will try to choose the alternative that will yield the highest expected return with lowest variance. This approach is useful in situations where a risk-averse decision maker has to choose between a large numbers of possible investment portfolios. For example, if alternatives A and B have a mean–variance of A(8,15) and B(8,12), respectively, then alternative A is mean–variance dominated by B and can be eliminated. Similarly, if alternatives C and D have a mean–variance of C(12,30) and D(9,30), respectively, then D is mean–variance dominated by C and can be eliminated. For a large number of available options, it is best to plot the standard deviation of expected profit for each investment versus their corresponding expected profit (EMV) as shown in Figure 3–3. For the hypothetical investments shown in Figure 3–3, the decision maker would like to maximize the EMV, and being risk averse, he/she would also like to minimize the risk or uncertainty. 175

Project Economics and Decision Analysis Volume II: Probabilistic Models

Fig. 3–3 The mean–standard deviation screening method

In Figure 3–3, it is evident investments 1, 7, 8, 9, and 10 are said to lie on the efficient frontier. Only these investments will survive the screening process and be considered further. Investment 9 mean–variance dominates investment 7 (the same mean but investment 7 has higher standard deviation). Investment 10 shows the highest EMV but at the same time much higher risk of all the investments. The choice between the investments on the efficient frontier will depend on how risk averse the decision maker is. This approach is more appropriate in situations where the probability distribution of each alternative can be represented by its mean and standard deviation. The normal probability distribution is an example where this holds true. If the mean and standard deviation of a distribution do not completely describe it, then it is best to directly compare the distributions themselves rather than comparing their measures of central tendency and dispersion. In situations where complete probability density functions (PDF) and cumulative probability distribution functions (CDF) of alternatives can be compared, then the following dominance rules apply (Figure 3–4 is used to clarify the dominance rules).

176

(b)

Fig. 3–4 Probability and cumulative probability functions representing different investment alternatives

(c)

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Expected Value and Decision Trees

(a)

Project Economics and Decision Analysis Volume II: Probabilistic Models

1. Figure 3–4a shows the PDF and the corresponding CDF for the two investment alternatives A and B do not intersect, and alternative B has a higher expected mean and lower expected standard deviation. Therefore, alternative A is clearly dominated by alternative B with the EMV of B being certainly higher than A. The situation is referred to as deterministic dominance. In this case, alternatives are ranked based on maximized EMV or minimized EOL as the case may be, without knowing the risk preference of the decision maker. In addition, the CDF of investment B is always to the right of the CDF of investment A. This means that for any level of EMV, investment B offers the smallest probability of falling below that of EMV.5 2. In cases when the PDF of two investments do intersect, as shown in Figure 3–4b, it cannot be said with certainty one alternative will produce higher EMV than the other. However, in this case it can be said that for all EMVs alternative D is more likely to exceed the EMV of C. Furthermore, when PDFs intersect and the corresponding CDFs do not intersect, then a condition of firstdegree stochastic dominance exists. As in the case of deterministic dominance, alternatives in this case may be ranked based on maximized EMV or minimized EOL, without knowing the risk preference of the decision maker. 3. Look at the PDFs and CDFs of alternatives in Figure 3–4c. Both the PDFs and their corresponding CDFs intersect. In this particular case, the alternatives have the same expected value, but the standard deviation of alternative F is more than the standard deviation of alternative E. Therefore, alternative F is more risky. In such cases, the rules applied to deterministic dominance or first-order stochastic dominance cannot be applied. The CDF of Figure 3–4c shows up to the point of crossover, investment E dominates investment F while after the crossover point investment F dominates investment E. Comparison of the areas between the F dominated part and the E dominated part will give the overall extent of dominance. The part with the larger area is the dominating invest-

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Expected Value and Decision Trees

ment. In situations where the CDFs intersect each other more than once, the areas have to be added together in order to establish the extent to which one investment dominates the other.5 The dominance rules described previously are used on the data of Example 3–3. The standard deviations for the three alternatives are calculated as shown in Table 3–8. The EMV and the standard deviation are then used to generate their cumulative probability distribution function (assuming they are normally distributed) as shown in Figure 3–5.

Outcome State

Drill with

Farmout

Back-in with

37.5% WI

with ORI

37.5% WI

Probability (NPV-E{NPV})2 (NPV-E{NPV})2 (NPV-E{NPV})2

Dry Hole

0.25

$744,577,640

$28,651,397

$160,980,269

20 MStb

0.30

$122,616,914

$1,167,109

$181,926,053

35 MStb

0.25

$108,932,013

$3,882,082

$19,234,364

50 MStb

0.15

$592,275,170

$14,962,823

$350,461,135

65 MStb

0.05

$512,974,089

$12,317,593

$367,785,192

Total

1.00

$2,081,375,826

$60,981,004

$1,080,387,013

EMV from Table 3-5

$24,574

$10,706

$25,375

Standard Deviation

$45,622

$7,809

$32,869

Table 3–8 Standard deviation for data in Table 3–5

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Fig. 3–5 Cumulative probability distribution function (CDF) for the data in Table 3–8

Figure 3–5 shows how for EMVs between $0 and $27,000, the backin option dominates the drill option. For the range between $27,000 and $50,000, the drill option dominates the back-in option. This shows the back-in option is dominant over a larger range than the drill option. By comparing areas before and after the crossover, the area dominated by the back-in option is slightly larger than the area dominated by the drill option. Since the area of the back-in option is larger, this option has second-degree stochastic dominance over the drill option. The mean–variance analysis and stochastic dominance rules are useful as screening devices. Screening alternatives based on dominance is an important decision analysis tool. If alternatives can be eliminated early in the selection process on this basis, considerable cost can be saved in large-scale problems. Screening alternatives based on dominance begins implicitly in the structuring phase of decision analysis, i.e., when alternatives are considered, they are usually compared to other available alternatives. These concepts are further discussed when the concepts of preference theory are discussed in Chapter 4.

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Expected Value and Decision Trees

Meaning and Interpretation of Expected Value The EMV is thought of as the most probable outcome of selecting an alternative. It is also thought of as a 50% chance of achieving a profit equal to or greater than the EMV. Both these interpretations are incorrect. The correct interpretation is as follows. The expected value of a decision alternative is interpreted to mean the average value (monetary profit, cost, or any other variable) per decision that would be realized if the decision maker accepted the alternative over a series of repeated trials. The key words in this interpretation are “per decision” and “repeated trials.”4 The repeated trial in the previous definition does not mean the same prospect is drilled repeatedly for averaging. It means the company has an inventory of investment opportunities out of which the economically viable investments are selected for implementation by consistently using the expected value criteria. Some of these investments will result in returns below the expected value and some above it. If the expected value concept is used on rare occasions, it becomes the same as a one-time bet in the casino—a game of chance. Many lawyers use the expected value concepts in arriving at expected value of lawsuit settlements. They will develop a complex decision analysis model, incorporating the probabilities and the corresponding monetary values of offers, counteroffers, and acceptance between the parties. This expected value is then used for negotiations. Such lawsuits are unique and not repeated, but the expected value analysis gives insight into evaluating various courses of uncertain events and their outcomes. Schuyler states the benefits of applying the EMV concepts are more elusive than one might think. He shows that many prospects must be drilled in order to establish confidence in realizing the benefits predicted by expected monetary value theory.6

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Bear in mind that the concept of expected value is merely an abstract one. It represents a play-the-average strategy, and hence its role should be limited to that of a decision criterion. It is used only to model and provide inference about real choice processes and generally not to replace them. To summarize, consistently applying the probability of success (or failure) in investment selection decision making will, for many projects, maximize the expected value over longer-term. By definition, the expected value concept requires it be consistently applied to all project evaluations over a long period. The concept works only when applied over a long term. To elaborate on the definition of the expected value, a dice was rolled, and the sequence of even and odd numbers was observed each time the die was rolled. Since there are three even numbers (2, 4, and 6) and three odd numbers (1, 3, and 5) on a die, the probability of getting an even number or an odd number facing up each time the die is rolled is 50:50. This means there is 50% probability a die will roll with an even number facing up and 50% probability the number will be odd. To assign a monetary value to the outcomes, it was assumed an even number would win $5 and an odd number will loose $2. The trial process is repeated 30 times and the outcomes noted as shown in Table 3–9. Theoretically, the expected monetary value for this experiment would be EMV = 0.5(+$5) + 0.5(–$2) = $2.5 – 1 = +$1.5

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Expected Value and Decision Trees

Cumulative Trial

Outcome

Gain/Loss

Cumulative Gain

Gain/Loss

Per Trial

1 2

5 5

– $2 – $2

– $2 – $4

– $2/1 = – $2 – $4/2 = – $2

3

1

– $2

– $6

– $6/3 = – $2

4

4

+$5

– $1

– $1/4 = – $0.25

5

2

+$5

+$4

+$4/5 = +$.80

6

4

+$5

+$9

+$9/6 = +$1.50

7

5

– $2

+$7

+$7/7 = +$1.00

8

4

+$5

+$12

+$12/8 = +$1.50

9

2

+$5

+$17

+$17/9 = +$1.89

10

2

+$5

+$22

+$22/10 = +$2.20

11

3

– $2

+$20

+$20/11 = +$1.82

12

4

+$25/12 = +$2.08

5

+$5 – $2

+$25

13

+$23

+$23/13 = +$1.77

14

4

+$5

+$28

+$28/14 = +$2.00

15

2

+$5

+$33

+$33/15 = +$2.20

16

3

– $2

+$31

+$31/16 = +$1.94

17

2

+$5

+$36

+$36/17 = +$2.12

18

2

+$5

+$41

+$41/18 = +$2.28

19

5

– $2

+$39

+$39/19 = +$2.05

20

1

– $2

+$37

+$37/20 = +$1.85

21

2

+$5

+$42

+$42/21 = +$2.00

22

2

+$47

+$47/22 = +$2.14

23

3

+$5 – $2

+$45

+$45/23 = +$1.96

24

4

+$5

+$50

+$50/24 = +$2.08

25

4

+$55

+$55/25 = +$2.20

26

1

+$5 – $2

+$53

+$53/26 = +$2.04

27

1

– $2

+$51

+$51/27 = +$1.89

28

3

– $2

+$49

+$49/28 = +$1.75

29

3

– $2

+$47

+$47/29 = +$1.62

30

4

+$5

+$52

+$52/30 = +$1.73

Table 3–9 Illustration of EMV with a dice

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The cumulative gain per trial from Table 3–9 is plotted against the number of trials as shown in Figure 3–6. From this graph, the cumulative gain per trial from the die rolling experiment is beginning to converge to the theoretical value of EMV. This observation is in line with the definition of expected value, i.e., the expected value of a decision alternative represents the average value per decision realized if the alternatives were accepted over a series of repeated trials.

Fig. 3–6 Graph of cumulative gain per trial for a die-rolling experiment

The following important observations can be deduced from the data in Table 3–9 and Figure 3–6. •

184

Newendorp states “the gamble is said to have a certain cash equivalent value equal to its EMV,” which means expected value is the average per trial. By looking at the data in Table 3–9, the EMV is close to $1.73, and the actual cumulative payoff at the end of 30th trial is $52, i.e., EMV = 52/30 = 1.73.4

Expected Value and Decision Trees

• • •

The probability concept is correct, i.e., the even numbers are 16 and the odd numbers are 14 (almost 50:50). Table 3–9 shows that right after the 4th trial, the cumulative gain (payoff) was positive and increased thereafter. The repeated trials must come from acceptance of many investment projects over a period in order to offset the losses on some investments with the gains from the others.

Characteristics of Expected Value The concept of expected value is very general. Therefore, it can be applied in any uncertain situation where the probabilities can be estimated with reasonable certainty. The following characteristics and rules provide basis for use of the concept in more complex situations. 1. Unlimited number of events and outcomes (as shown in Table 3–2) can be considered as long as the probabilities of outcome state for each event add up to one. 2. Any number of decision alternatives can be considered. 3. The outcomes can be expressed in any form of random variables, not necessarily monetary values. Furthermore, the monetary values can be discounted net cash flow, undiscounted net revenue, after tax, before tax, costs, opportunity costs, and so on. As previously mentioned, if discounted net present value profits are used, the expected value of the decision alternative is called expected net present value profit. 4. The expected monetary value is not necessarily the same as the actual payoff realized. For example, in the dice rolling experiment, the expected value is $1.5 but the payoff is $1.73 per trial. Similarly, in Example 3–3, the maximum EMV is $17,500 but the only possible payoffs that could be received (if a well is drilled) would be a loss of $0 (dry hole) or a gain of $50,000 (producer).

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5. The concept is based on the assumption the investor is impartial to money and the magnitude of money involved. This means, among other things, the investor has a large amount of money and can afford any potential losses in the process of repeated trials. 6. The expected value rule is appropriate only when the decision maker wishes to gain insight about how best to play the averages and is not concerned about the spread of real outcomes about those averages. This means the rule does not for the decision makers’ attitude (aversion) towards risk. 7. The expected values of individual accepted alternatives can be added together in order to arrive at the composite expected value for all the selected alternatives. 8. If all the conditional values received are multiplied by constant value c, then the resulting EV will be equal to the original EV times the constant. For example, if EV for an investment is calculated for 100% working interest and it is desired to determine the EV at 50% working interest, the EV in this case can be arrived at by directly multiplying the original EV by half (1/2).

VALUE OF INFORMATION New or additional information, on an investment or problem subject to expected value evaluation, can remove or reduce the uncertainty involved in decision. Reducing or eliminating uncertainty in return is expected to increase the expected payoff and reducing variance. However, acquiring new or additional information costs money. For example, acquiring seismic surveys; carrying laboratory analysis; engaging services of a consultant or the need to carry out a market research survey if launching a new product. Before deciding on acquiring additional information, the following questions should be answered.

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Is it worth obtaining the additional information? Sometimes the process of determining whether it is worth acquiring additional information is referred to as preposterior analysis. If there are several potential sources of information to improve the decision process and ultimately the payoff, which one is to be preferred? The main purpose of gathering information is to improve the decision makers’ overall ability to improve the risk profile of the decision status, through reducing the uncertainty before committing to a decision strategy. Information about an uncertain outcome of an event can be perfect (leading to complete resolution of uncertainty) or imperfect (leading to partial resolution of uncertainty). The information has value only if it affects a decision strategy, especially improving it. Information can have value irrespective of the decision maker’s risk preference (risk neutral, risk seeking, or risk averse). In either case, useful additional information assists in making better decisions if uncertainty is reduced or resolved before commitment to an action. In this section, it is first assumed that if additional or new information is acquired, it will be perfectly reliable, thus reducing uncertainty and increasing payoff. Secondly, a much more common situation will be considered where the reliability of the additional information is not as perfect.

Expected Value of Perfect Information (EVPI) Perfect information is seldom, if ever, available. Additional information, even though it may not be perfect, often has some value. Decision analysis is useful in determining whether to buy additional information if there is any available. Such additional information may reduce the uncertainty but at cost. The cost may be in monetary or in lost time or both. A quantitative analysis must be applied to show if the cost of obtaining additional information is justified because it will increase expected financial gain.

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The most common information that could add value to analysis in oil and gas business activities is: • • • • • •

Running seismic prior to drilling a wildcat Acquiring 2D seismic versus 3D seismic Drilling additional appraisal wells prior to committing to full field development and infrastructure Doing market survey before developing a new tool Running drill stem test before setting casing Hiring a consultant to evaluate the integrity of an offshore pipeline (whether to replace part of it or total replacement)

The expected value of perfect information (EVPI) is the expected payoff with perfect information (EPPI) minus the expected payoff under uncertainty. For this, the maximum payoff of each event is taken and then the expected payoff is calculated by multiplying the maximum of each event by the respective event probabilities. This gives the expected value of perfect information. The difference between this amount and the expected payoff under uncertainty gives the value of perfect information. It is the difference between the most likely payoff and the best payoff that could have been achieved had the occurrence of event been precisely known. The EVPI is the incremental gain due to perfect information about the outcome state. The EVPI is the amount a person can spend on acquiring perfect information. On the other hand, the minimum expected opportunity loss is the value of perfect information. The calculation of EVPI gives an upper-bound value for imperfect information.

Example 3–6 For the decision problem in Example 3–3, the geologists feel that acquisition of additional seismic data will significantly reduce uncertainty. Calculate the maximum value that can be paid to acquire seismic data.

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Solution: In Example 3–3 and Table 3–5, it was determined that the backin option with the EMV (under uncertainty) of $25,375 is economically the most viable option out of the three available options. The expected payoff is now calculated as if perfect information was available. The following steps are used. 1. Choose the maximum value in each row of Table 3–5 and put them as shown in Column 3 of Table 3–10. Since information is perfect, the dry hole probability is zero and the probability of producer is 1.0.

Outcome

Perfect Information

State

Probabilit y

NPV

EPPI

Dry Hole

0.25

0

0

20 MStb

0.30

$8,733

$2,620

35 MStb

0.25

$45,448

$11,362

50 MStb

0.15

$87,411

$13,112

65 MStb

0.05

$125,863

$6,293

1.00

$33,387

Table 3–10 EPPI computation for Example 3–6

2. Multiply the values in Column 3 by their corresponding probabilities in Column 2 to arrive at the expected values in Column 4. 3. Add the expected values of Column 4 to get the expected value as if the information was perfect. 4. Subtract the EMV (under uncertainty) of $25,375 from the EPPI arrived at in Column 4 of Table 3–10.

EVPI = EPPI – EMV = $33,387 – $25,375 = $8,012

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The EPPI is the expected payoff when the act is chosen based on information that indicates the exact outcome state. Thus, the EVPI represents the maximum amount one would afford to spend on acquiring perfect information. Therefore, the seismic survey should be acquired to reduce uncertainty only if it costs less than $8,012. The same value was arrived at from EOL (Example 3–5) calculations, where $8,012 represents the cost of uncertainty. Therefore, to eliminate the uncertainty, up to $8,012 can be spent. If the EMV and EOL were added (EMV + EOL = k), an expected profit the decision maker would achieve if he chooses his option after learning the true state of nature will result. In this example, it would be $33,387, which is the same as that calculated in Table 3–10. If there are multiple sources of uncertainties in a decision problem, calculate the EVPI for each source of uncertainty separately. In such a situation, it is important to determine which of the uncertain information to obtain. This requires a sequential decision strategy.

Expected Value of Imperfect Information (EVII) The expected value of imperfect information (EVII) is the expected payoff with imperfect information minus the expected payoff under uncertainty. The imperfect here refers to the degree of reliability of the information obtained. The perfect information is considered 100% reliable, whereas the imperfect information may have a reliability of anything between but not equal to zero and 100%. For example, there is a 90% probability a seismic survey will indicate a structure when a structure is present, and there is 10% probability it will indicate a structure when the structure is actually not present. The seismic survey information thus obtained is considered imperfect information. The EVII is more complicated to calculate than EVPI. Perfect information is considered to provide complete certainty about an uncertain quantity. Imperfect information serves to change the degree and nature of uncertainty without eliminating it. For example, in the case of seismic

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survey, if the 90% and 10% probabilities are changed into 100% and 0%, respectively, then the information is considered perfect information. The EVII calculations require altering the prior probabilities into posterior probabilities based on the information obtained from experiments or tests. The expected net gain (ENG) is the expected value of information (perfect or imperfect) less the cost of obtaining the information. If expected value of information for an event is very low, then it may not be worth the effort and money to reduce uncertainty by acquiring additional information. On the other hand, if the expected value of information for an event is relatively high, then it may be worthwhile to acquire this information. Such information may have a relatively large payoff when it reduces uncertainty, thus improving the decision maker’s EMV. The Bayesian theory is used to revise prior probabilities and derive new posterior probabilities, calculated based on new information available through experiments or tests. Under the Bayesian decision theory, these posterior probabilities are substituted in place of prior probabilities in the outcome state. The expected payoff is thus calculated by taking into the posterior probabilities in regard to the outcome state in place of prior probabilities. The following steps assist in using the Bayes theorem. 1. Determine the course of action that would be chosen using only the prior probabilities and record the expected payoff (EMV) of this course of action. Prior probabilities are the original marginal probabilities giving the initial estimate of the outcome state. These are first in a decision tree diagram. 2. Identify the possible indications the new information can give. 3. Assign probabilities to the new information in Step 2. The additional information constitutes the conditional probability. 4. Calculate t probabilities. The product of prior probabilities in Step 1 and the conditional probabilities in Step 3 gives the t probabilities.

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5. Calculate marginal probabilities. Summing up the appropriate t probabilities given in Step 4 derives the new marginal probabilities. 6. Calculate posterior probabilities. The t probabilities of Step 4 are divided by the marginal probabilities of Step 5 to give posterior probabilities. 7. Revise prior probabilities. The posterior probabilities, in the form of conditional probabilities, derived in Step 6 show the altered outcome state in light of information available from experiments or tests.

Example 3–7 Exploration drilling on one of the offshore prospects has resulted in a commercial oil discovery. Facilities for this prospect need to be selected. The studies have indicated reserves in the range of 5 MMStb to 25 MMStb. The geologists and engineers have estimated that the there is 30% probability of a large field with 25 MMStb reserves, 45% probability of a medium field with 15 MMStb reserves, and 25% probability of a small field with 5 MMStb reserves. There are two options: (a) to design the facility based on the information available, or (b) drill delineation wells to further improve the probability and reservoir size. Based on the possibility of three different field sizes, three different sizes of facilities can be installed. Economics of the different combination of facilities are shown in Table 3–11.7

NPV of Each Field Size and Facility, MM$ Field Size

Probability

Size A

Size B

Size C

Large

0.30

290

350

450

Medium

0.45

90

210

160

Small

0.25

60

35

50

Table 3–11 Economics of various facilities for Example 3–7 192

Expected Value and Decision Trees

a. Using the EMV criterion, select the most economical facility size without obtaining any additional information. b. Calculate the expected value of perfect information using EMV and EOL. Based on the EVPI, how much maximum can be paid to acquire additional information? c. Calculate the expected value of imperfect information if it is decided to drill delineation wells at a cost of 15 MM$ before deciding on the size of facilities. The geologists consider that (i) there is a 90% chance that the delineation wells will show large reservoir, (ii) 60% chance that it will show medium reservoir, and (iii) 30% chance that it will show small reservoir. Solution: (a) the EMV calculations are shown in Table 3–12.

NPV of Each Field Size and Facility, MM$ Field Size

Probability

Size A

Size B

Size C

Large

0.30

290.0

350.0

450.0

Medium

0.45

90.0

210.0

160.0

Small

0.25

60.0

35.0

50.0

142.5

208.3

219.5

EMV

Table 3–12 EMV calculations for data in Table 3–11 (Example 3–7)

Based on the EMV calculations, Size C facility is the economic option. The EVPP calculations are shown in Table 3–13. that EVPI = EVPP – EMV.

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NPV of Each Field Size and Facility, MM$ Field Size

Probability

Large

0.30

Medium

0.45

Small

0.25

EVPP

244.50ƒÎ

Size A

Size B

Size C 450.0

210.0 60.0 15

94.5

135.0

Table 3–13 EVPP calculations for data in Table 3–11 (Example 3–7)

Therefore EVPI = 244.50 – 219.5 = 25 MM$ The EOL calculations are shown in Table 3–14.

NPV of Each Field Size and Facility, MM$ Field Size

Probability

Size A

Size B

Large

0.30

160

100

0

Medium

0.45

120

0

50

Small

0.25

0

25

10

102

36.25

25

EOL

Size C

Table 3–14 EOL calculations for data in Table 3–11 (Example 3–7)

The EOL = 25 MM$ confirms the selection of Size C facility. Also, the EVPI = 25 MM$ is the same as calculated by the EMV method. To calculate the EVII, consider two alternatives. (a) install platform without acquiring additional information (these calculations are already performed above), and (b) drill delineation wells and based on their result decide on the platform size. The decision tree for this option is shown in Figure 3–7. The outcome of delineation wells is considered Favorable and Unfavorable.

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Expected Value and Decision Trees

Fig. 3–7 Decision tree for Example 3–7

The probabilities of different field sizes expected to be assessed from the delineation wells are shown as a partial tree in Figure 3–8(a). As discussed in Chapter 2, the tree in Figure 3–8 has to be inverted (applying Bayes’ rule) in order to make use of the outcomes of delineation well results. In the inverted tree, the node of Delineation Well Results will come ahead of the Field Size nodes as shown in Figure 3–8(b).

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Fig. 3–8 Partial tree for (a) the result of delineation wells, and (b) its inverse

Once the posterior probabilities, based on the favorable and unfavorable outcome of delineation wells, are obtained, the expected values as shown in Table 3–15 are calculated.

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Expected Value and Decision Trees

Delineation Wells Favorable Posterior

NPV of Each Field Size and Facility, MM$

Field Size

Probability

Size A

Size B

Size C

Large

0.27/0.615

290.0

350.0

450.0

Medium

0.27/0.615

90.0

210.0

160.0

Small

0.075/0.615

60.0

35.0

50.0

174.15

250.12

273.90

EMV ƒÎ

Delineation Wells Unfavorable NPV of Each Field Size and Facility, MM$ Field Size

Probability

Size A

Size B

Size C

Large

0.03/0.385

290.0

350.0

450.0

Medium

0.18/0.385

90.0

210.0

160.0

Small

0.175/0.385

60.0

35.0

50.0

91.95

141.36

132.60

EMV ƒÎ

Table 3–15 EMV calculations for data in Table 3–11 (Example 3–7)

From Table 3–15, facility of Size C will be selected if the results of the delineation wells are favorable and facility Size B will be selected if the results are unfavorable. The EPII is 222.87 MM$ (Figure 3–8) and the EVII is 3.37 MM$ (222.87 – 219.50 = 3.37). Based on these calculations, it would not be worth paying more than 3.37 MM$ to drill the delineation wells. The EVPI of 25 MM$ (calculated in Table 3–14) is higher than 3.37 MM$, i.e. the value of the information obtained from delineation wells is less than perfectly reliable results.

DECISION TREES A decision problem with a set of alternative courses of action and a set of outcome states could be presented in the form of a matrix (table) or a tree. Since many decision problems involve complex sequence of decisions and outcome states, decision trees are more appropriate to use.

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Most decision analysis problems, when they are first encountered, appear to be overwhelmingly complicated. This is because a large number of interrelated elements are normally associated with the problem that, at best, provides the decision maker with only a hazy perception of the issues involved. In these circumstances, decision trees (diagrammatic representation of a decision situation) can be extremely useful in helping the decision maker to gain a visual understanding of the structure of the problem on hand. The decision trees can help a decision maker to develop a lucid view of the structure of a problem and make it easier to determine the possible scenarios that can result if a particular course of action is chosen. The decision trees also assist in judging the nature of information needed for tackling a certain problem. Decision trees depict pictorial views of the available courses of action, events, their outcomes, and probabilities associated with the outcomes. Therefore, they also serve as an excellent communication medium when a problem is discussed or presented to management or other individuals. Decision trees can be used to assist in reaching a decision to maximize the EMV. It is a method of tracking alternative outcomes of any decision and comparing the likely returns of those alternatives while minimizing the risks involved. The best method of learning the technique of decision tree construction and analysis is to practice it by solving many problems. The construction of a tree requires clear thinking about the problem. For this purpose, a number of solved problems are presented to assist in reinforcing the concept.

Decision Tree Convention Decision trees have a number of features. Figure 3–9 shows the decision tree for a company that has to decide whether to drill or not drill a certain well. Beginning on the left side of the tree, the first thing to happen is typically a decision, followed by either other decisions or chance events in chronological order. The following conventions are adopted in structuring a decision tree.

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Expected Value and Decision Trees

Fig. 3–9 A typical two-activity decision tree

Decision nodes: Points where decisions have to be made in a decision tree are represented by squares () and are called decision nodes. The decision maker has to choose the optimum alternative between the available courses of action. On the other hand, these are points at which the decision maker has control and must exercise choice. In this particular example, the decision is to drill or don’t drill the proposed well. Since there is only one decision node, it is denoted by D1. Similarly, any subsequent decision nodes are assigned sequential numbers for ease of reference. A decision node may be followed by another decision node or by a chance node. The branches emanating from a square, called decision forks, correspond to the choices available to the decision maker. There are no probabilities or uncertainties attached to the decision forks. Chance node: Circles () represent points where there are different possible outcomes in a decision tree and are called chance nodes. These are points where the decision maker does not have control and where chance or nature determines an outcome. A chance event is probabilistic. One of the major difficulties of decision analysis is to identify the events that may occur and their associated probabilities. Since there is only one chance node in Figure 3–9, it is denoted by C1. Similarly, any subsequent chance nodes are assigned sequential numbers for ease of reference. A chance node may be followed by a series of chance nodes or by decision nodes. The branches from a circle, called chance forks, represent the possible outcomes of a

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Expected Value and Decision Trees

1. The process of constructing a decision tree is iterative. The original structure changes as the decision maker’s understanding of the problem develops. 2. The decision trees should be kept as simple as possible. They represent a model of the actual problem and as such should be a simplification of the real problem. Very large and complex trees, representing every possible detail/scenario that may occur, can be counterproductive and thus obscuring the objectivity of ease of understanding the problem. 3. Define decision nodes so only one option may be chosen at each node and every possible option is described. 4. Define chance nodes so they are mutually exclusive and collectively exhaustive. A node where only one outcome is possible out of a set of multiple available outcomes is mutually exclusive. A node where all possibilities are described is collectively exhaustive. 5. The tree should proceed chronologically from left to right. 6. The sum of probabilities at each chance node must equal 1. 7. In many situations, it is possible to draw a single tree in a number of different ways that may look different but are structurally equivalent. When chance nodes and decision nodes are directly connected, they generally cannot be combined or reordered without altering the meaning of the tree. When nodes of the same type, either chance nodes or decision nodes, are directly connected, they can be combined or reordered.

Solving a Decision Tree Once the model has been defined and the decision tree constructed, the tree is ready to be solved. A decision analysis on a decision tree produces expected value of the model, standard deviation, and risk profile of the optimum strategy. The method of calculating the optimum path in decision trees is typically called folding back or rolling back a tree. To apply this method, the decision trees are analyzed from right to left by

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considering the later decisions first. A brief outline of the folding back procedure is described as follows. Chance node reduction. Calculate the expected value of the rightmost chance nodes and reduce to a single event. For example, the expected value of chance node C1 in Figure 3–9 is given by (3.9) n

E{C1} = ∑ Pi ( NPVi ) i =1

where pi is the conditional probability of following each branch emanating from the chance node. For a probability node, all branches are included in the summation, and the sum of the conditional probabilities is equal to unity. The NPVi is the payoff corresponding the branch with pi. Decision node reduction. Choose the optimum path of the rightmost decision nodes and reduce to a single event, (choose the maximum E{Ci} at the decision node). Repeat: return to Step 1 if there are nodes that have not been analyzed yet. Repeat this procedure until arriving at the final, leftmost, decision node.

Example 3–8 A company has to decide whether to drill a certain prospect. If it drills the prospect, geologists and engineers expect (a) probability of zero reserves (dry hole) to be 60%, (b) probability of 60,000 barrels to be 30%, and (c) probability of 90,000 barrels to be 10%. The net present value for each option is calculated as shown in the decision tree in Figure 3–9. Based on this information, determine the optimum course of action the company should take in order to maximize its EMV. Solution: There is only one chance node and one decision node in this problem. Using Equation (3.9), the expected value of chance node C1 is 202

Expected Value and Decision Trees

n

E{C1} = ∑ Pi ( NPVi ) = 0.60(−$65,000) + 0.30($120,000) + 0.10($180,000) i =1

= −$39,000 + $36,000 + $18,000 = $15,000

At the decision node D1, the expected value of don’t drill is zero and the expected value of the drill option is $15,000. Since the expected value of the drill option is greater than the expected value of don’t drill option, it is recommended to drill the well. The calculations shown above are shown in Figure 3–10.

Fig. 3–10 Illustration of decision tree calculation for Example 3–8 203

Project Economics and Decision Analysis Volume II: Probabilistic Models

Example 3–9 A company is planning to drill a well. Geologists and engineers estimate that there is a 65% chance that the well will be a producer and a 35% chance that it will be a dry hole. If the well is successful, it is estimated that there is a 60% chance that it will have reserves of 30,000 barrels, a 30% chance of 60,000 barrels, and a 10% chance of 90,000 barrels. The dry hole cost is $65,000 and the NPV corresponding each reserve value is $60,000, $120,000 or $180,000, respectively. Using decision tree, calculate the expected monetary value (EMV) for this proposal. Solution: A decision tree for the problem is first constructed as shown in Figure 3–11.

Fig. 3–11 Decision tree for Example 3–9

Using Equation (3.9), the expected value at each chance node, starting from the right of the tree, is calculated.

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Expected Value and Decision Trees

EMV (C 2 ) = 0.60($60,000)+ 0.30($120,000 )+ 0.10($180,000)

= $36,000 + $36,000 + $18,000 = $90,000 EMV (C1 )producer = p(C1 )× EMV (C 2 ) = 0.65($90,000)

= $58,500 Similarly, the EMV at chance node C1 (for the chance fork of dry hole) is given by EMV (C1 )dry hole = 0.35(− $65,000)

= −$22,750 The EMV of the Drill option is obtained by adding the EMVs of the chance forks emanating from C1. EMV = EMV (C1 )producer + EMV (C1 )dry hole

= $58,500 + (− $22,750 ) = $35,750 Alternatively, combining the chance nodes C1 and C2 can collapse the tree. Note the probabilities of the chance node C2 are conditional probabilities, conditional upon encountering a producer with a probability of 65%. By multiplying the simple probability of chance node C1 by the conditional probabilities of each branch of the chance node, C2 will give t probabilities. These t probabilities are then used to calculate the EMV at chance node C1. The calculations are shown in Figure 3–12.

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Fig. 3–12 EMV calculation for Example 3–9, using the t probabilities

Similarly, the decision tree of Figure 3–12 can be further collapsed and EMV calculated as shown in Figure 3–13. The decision trees in Figure 3–11, Figure 3–12, and Figure 3–13 look different, but they are structurally equivalent. However, significant collapsing of the trees is not recommended.

Fig. 3–13 Alternative view of the decision trees in Figure 3–11 and Figure 3–12

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Expected Value and Decision Trees

Constructing Risk Profiles The methods described previously dealt with determining the optimum path in a decision tree. In addition to determining the optimum path, maximum EMV, analysts may need to know the consequences of implementing the derived optimum path. Therefore, risk profile of the optimum has to be generated. A risk profile is a distribution function describing the chance associated with every possible outcome of the decision model. It graphically shows the uncertainty of the decision. The decision tree of Example 3–8 is used to show the steps involved in arriving at the risk profile for this particular example. The following steps are performed to generate a risk profile from a decision tree.7 1. Reduce chance nodes (tree collapsing). As shown in Figure 3–12 and Figure 3–13, multiplying probabilities on sequential chance nodes collapses the tree. The value of each path in the tree is calculated by summing the value for each branch in the path. Using this path value, the expected value is calculated for the remaining chance node. 2. Reduce decision nodes. Considering only the optimal branches reduces the decision nodes. 3. Repeat Steps 1 and 2. These steps are repeated until the tree is completely reduced to a single chance node with a set of values and corresponding probabilities. If any two outcomes have the same payoff value, they are combined into one chance event and their probabilities are added together. Similarly, if any two outcomes have the same probability value, they are combined into one chance event and their payoffs added together. 4. Generate risk profile. The final set of payoff and probability pairs defines a discrete probability distribution used to generate the risk profile. The risk profile can be graphed as discrete cumulative density distribution or a scatter diagram. 5. Calculate expected value, variance, and standard deviation.Once the expected value is calculated, the variance or standard deviation

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about the mean can be calculated by using the methods shown in the preceding section. The standard deviation, about the expected value, calculation for the decision tree in Figure 3–13 is shown in Table 3–16.

Unconditional

NPV

EMV

Variance (MM$)

Probability, p i

X i ($)

X i ($)

p i (X i – EMV) 2

0.350

-65,000

-22,750

3,552.697

0.065

180,000

11,700

1,352.524

0.195

120,000

23,400

1,384.122

0.390

60,000

23,400

229.344

35,750

6,518.687

s= 6,518.687=$80,738.39

Table 3–16 Calculation of EMV and variance for Example 3–9

Advantages of Decision Trees Decision trees can be applied to much more complicated problems. They have some of the following major advantages. • • •

•

•

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Decision trees portray the problems in a logical, straightforward, and easily reconciled manner. They enable a decision maker to think in a clear and logical manner. They require the decision maker to think about all the possible future contingencies, options, and “what if ” situations in a consistent manner. They have the ability to break a complex decision problem into a series of small parts. Once each part is solved, they are reassembled piece by piece to provide a rational basis for the initial decision. Once a detailed decision tree is ready, its course of events can be followed during decision implementation and at any decision

Expected Value and Decision Trees

• • •

node. If conditions change, the remaining alternatives can be reanalyzed to develop a revised strategy from that point forward. Constructing the tree diagram may show possible courses of action not previously considered. They involve placing numerical values on decisions. This tends to improve results and decision making. They force management to take of the risks involved in decisions and help to separate important from unimportant risks.

The technique also has some limitations. • •

The information the technique gives is not exact. Much of it is based on probabilities that are often estimated. Decisions are not only concerned with quantities and probabilities, they often involve people and are influenced by legal constraints or people’s preferences. These factors cannot always be shown by numerical values.

SPREADSHEET APPLICATIONS In this section, (a) some basic tips are presented to assist with the calculation of expected value, variance, and standard deviation in Excel; and (b) another package of the Palisade DecisionTools suite (PrecisionTree) is introduced to assist with generating and solving decision trees. Both the built-in Excel functions and the PrecisionTree are powerful tools to use when dealing with decision analyses problems.

Excel Built-in Functions The use of SUMPRODUCT built-in function of Excel greatly simplifies the calculations of expected value, variance, and standard deviation. Table 3–17 shows the EMV, variance, and standard deviation calculated using SUMPRODUCT.

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B

C

3

Unconditional

NPV

4

Probability, pi

Xi ($)

5

0.350

-$65,000

6

0.065

$180,000

7

0.195

$120,000

8

0.390

$60,000

9

1.000

10 EMV 11 Variance, s

$35,750 2

$6,518,687,500

12 Standard Deviation

$80,738

C10=SUMPRODUCT(B5:B8,C5:C8) C11=SUMPRODUCT(B5:B8,C5:C8^2)-C10^2 C12=SQRT(C11)

Table 3–17 Use of SUMPRODUCT function

Introducing PrecisionTree A spreadsheet add-in, called PrecisionTree, helps to create and analyze decision trees in Excel. PrecisionTree is used to implement the decision trees used in the preceding discussions. PrecisionTree is an add-in to Microsoft Excel that helps to structure complex decisions quickly and easily. When decision trees are used to model the sequence of events faced by decision makers, the PrecisionTree’s advanced decision analysis features identifies the best decision to make as well as the risks involved. PrecisionTree links directly to Excel using Excel style menus, toolbars, and functions. Figure 3–14 shows the Excel with PrecisionTree add-in. Except for the change in the toolbar menu at the top, there is hardly any difference. These provide all the necessary tools to set up and analyze the decision trees. This

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• • • •

clarifies options and rewards, describes uncertainty quantitatively, weighs multiple objectives simultaneously, and defines risk preferences—all in an Excel spreadsheet.

In addition, it enables us to tie decision analysis models directly to the existing spreadsheet models. For example, the NPV for each alternative can be read directly from the respective cash flow into the decision tree.

Fig. 3–14 The Excel spreadsheet with PrecisionTree add-in8

PrecisionTree has the capability to perform sensitivity analysis in order to identify critical factors influencing the decision. The results of the sensitivity analyses are displayed in spider and tornado graphs. Risk profile graphs are generated to show all the possible outcomes and the probability of their occurrence.

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PrecisionTree Add-in. If PrecisionTree is not already installed, then the first thing to do is to install the DecisionTools suite with the setup program on the CD-ROM accompanying this book. The steps for installing the DecisionTools are given in Appendix B. The installation, of course, is done only once. The PrecisionTree can be run in one of the following three ways. •

• •

If Excel is not currently running, both the Excel and PrecisionTree can be launched together by double clicking on the PrecisionTree icon already created when the Decision suite was installed. If for some reason the icon has not been created, PrecisionTree can be executed by (a) click on Windows Start button, (b) click on Palisade Decision Tools, and then (c) click on PrecisionTree. If you are already in Excel and want to load the PrecisionTree, just follow the second part of Step 1. If Excel is already open and the DecisionTools suite is already loaded, which is shown by the toolbar in Figure 3–14, the PrecisionTree can start by clicking on its icon (the third from left).

The only difference in the outlook of Excel, as shown in Figure 3–14, when any of the DecisionTools is ready for use, is the toolbar shown in Figure 3–14 and a PrecisionTree menu to the left of the Help menu of Excel menu bar. The DecisionTools’ icons in Figure 3–14 are described as follows. • •

212

Decision Tools toolbar— The icons on the top toolbar in Figure 3–14 are used to activate the other DecisionTools by Palisade. PrecisionTree Toolbar—These icons are used to quickly and easily perform tasks necessary to set up and run decision analysis. The functions done by these icons are the same as the commands in the PrecisionTree menu on the Excel menu bar (at the top, just before the Excel Help). There are eight PrecisionTree icons as shown in Figure 3–14.

Expected Value and Decision Trees

Constructing the Tree. In this section, the steps required to build a decision tree (using the features of PrecisionTree) from scratch, are presented. The basic steps are presented by constructing a tree for the data in Example 3–2 (Table 3–3). By practicing with the software, its advanced features can be easily learned. The end-of-chapter problems provide opportunity to further build on the skills of using the software. 1. While in windows, double click on the PrecisionTree icon. This executes the software along with Excel, and the screen looks like the one shown in Figure 3–14. 2. Click on the New Tree icon (first icon on the PrecisionTree toolbar) and then click in Cell A4 or any other Cell in the spreadsheet. By doing this, the first branch of the tree appears as shown in Figure 3–15.

Fig. 3–15 Beginning of a new tree (the starting branch of a tree) 3. Click on the name box of this tree, (where it says “tree #1”) to open the Tree Settings dialog box as shown in Figure 3–16. Type the desired descriptive tree name, (“Oil Well Drilling” in this case). If NPVs are used, then make sure the Optimum Path window shows “Maximum,” (to choose the alternative with the maximum EMV). If costs are used, then change this to “Minimum,” (to choose the alternative with the minimum expected cost). For now, ignore the Utility Function window. This will be used in the next chapter. The Delete Tree button on this menu can be used to delete any branch of the tree.

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Fig. 3–16 Tree Settings menu of Precision Tree 4. Add more nodes and branches to the tree by clicking on the Triangle (the end node of the first branch in Figure 3–15) to open the Node Settings dialog box as shown in Figure 3–17. In the Node Settings dialog box, click on the desired node type to insert. First, start with a decision node by clicking on the decision node button (the green square). Since the decision is between drill or don’t drill (two decision alternatives), set the number of branches to 2. The Decision in the Name window can be changed to EMV in this case or as the case may be. Click on OK, the tree with initial decision node and two branches appear as shown in Figure 3–18.

Fig. 3–17 Dialog box for adding a new decision or chance node and corresponding branches 214

Expected Value and Decision Trees

Fig. 3–18 Tree with initial decision node and branches 5. Click on either of the branches to open the Branch Settings dialog box. The boxes, as shown in Figure 3–18, saying “Branch” show the default labels for these branches. Change the title to a more descriptive name of the branch. This is done to label the two branches as “Drill” and “Don’t Drill” as shown in Figure 3–19. The Branch Settings dialog box also allows one to delete a branch and to move a branch upward or downward. Click on the upper branch, the Branch Settings dialog box appears, click on the Delete button. This deletes the branch. To add the same branch again, click on the decision node so the Node Settings dialog box appears. Now change the # of Branches to 2, this adds the second branch again.

Fig. 3–19 Tree with all the required decision nodes, chance nodes, and branches

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6. Add the probability node and branches to the Drill option (the top branch) by clicking on the triangle of this branch, the Node Settings dialog box appears, click on the chance node (red circle), specify three branches (dry hole, 60 MStb, and 90 MStb), and click on OK. The tree now appears as shown in Figure 3–19 with default probabilities on each branch. 7. Change the default branch labels as done in Step 5. Now start replacing the default probabilities and payoffs. The probabilities of 60%, 30%, and 10% are added in Cells C4, C8, and C10, respectively. Similarly, input the payoffs of $65,000, $120,000, and $180,000 in Cells C5, C9, and C11, respectively. Any values or formulas can be entered in these cells, exactly as it is done in Excel worksheet. Format the numbers as desired, using the formatting capabilities of Excel. The final completed decision tree with the best strategy and its associated EMV is shown in Figure 3–20. At any time, if any of the probabilities or the payoff values change, the tree is automatically updated. The TRUE on the branch means the Drill option is selected while the alternative with FALSE is rejected.

Fig. 3–20 The final Oil Well Drilling decision tree

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Expected Value and Decision Trees

8. Use PrecisionTree’s copy and paste feature for adding another probability node, which is structurally equivalent to the one already in the tree. Click on the probability node to be copied to open a dialog box and click on Copy. Then click on the desired end node, where the information needs to be pasted, to bring up the same dialog box and click on Paste.

Example 3–10 Using PrecisionTree, construct decision tree for the data of Example 3–3 (Table 3–4). Use the copy and paste commands of PrecisionTree to build the Farm-out and Back-in chance nodes after completing the Drill chance node. Solution: The tree is completed as shown in Figure 3–21.

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Fig. 3–21 The decision tree for Example 3–10 (for the data of Table 3–4)

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Expected Value and Decision Trees

Risk Profile. Once the decision tree is completed, PrecisionTree can be used to gain more information about the decision analysis. It can generate a statistics report, policy suggestion, risk profile, cumulative profile, and scatter profile of the optimal choice only or of all the available alternatives. To start the decision analysis, open the file that needs to be further analyzed and click on the Decision Analysis icon (staircase, fourth from the left) in the PrecisionTree toolbar. A Decision Analysis dialog box shown in Figure 3–22 appears with the name of the model (name on the first branch of the decision tree), in this case it is the “Drilling Prospect” of Figure 3–21. In the Decision Analysis dialog box (Fig. 3–22), the Statistics Report and the Analyze All Choices options are selected. By clicking on OK, the statistics report shown in Table 3–18 is created in a worksheet. Reports can be generated on the optimal choice, all available alternatives or any particular alternative. The Policy Suggestion option allows seeing only that part of the tree corresponding to the optimum decision (maximum or minimum EMV as the case may be). The Risk Profile option generates a graphical risk profile of the selected alternatives. The Cumulative Profile option generates a cumulative probability plot for the selected alternatives.

Fig. 3–22 Decision Analysis dialog box of PrecisionTree for the model in Figure 3–21

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EMV

Drill with

STATISTICS

37.5% WI

Mean

Farmout

37.5% Back-in

24,574

10,705

Minimum

-30,000

0

0

Maximum

125,863

26,401

111,141

4,357

8,733

750

Mode Std Dev

25,374

45,622

7,809

32,869

Skewness

1.0

0.0

1.0

Kurtosis

2.0

2.0

3.0

PROFILE: # 1

X -30,000

P 0.25

X

P 0

0.25

X

P 0

0.25

2

4,357

0.30

8,733

0.30

750

0.30

4

45,448

0.25

14,646

0.25

34,142

0.25

4

87,411

0.15

20,693

0.15

73,712

0.15

5

125,863

0.05

26,401

0.05

111,141

0.05

Table 3–18 PrecisionTree statistics report for drilling prospect of Figure 3–21.xls Sensitivity Analysis. PrecisionTree makes the task of performing sensitivity analysis relatively easy. To start with, first open the decision tree model file on which sensitivity analysis need to be performed. Build a decision tree for the problem in Example 3–2 (Table 3–3) as shown in Figure 3–23 (FIGURE 3–23.xls on CD. Enter probability of dry hole in Cell C3. Enter a formula =1-C3 in Cell C7. The other two probability cells (Cells C11 and C15) are referenced to these cells by inputting =C3 in Cell C11 and =C7 in Cell C15. Now the tree is ready for sensitivity analysis.

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Expected Value and Decision Trees

Fig. 3–23 Decision tree for the data in Table 3–3 (Example 3–2) Sensitivity analysis can be performed on any variables in the tree (i.e., probability, dry hole cost, or NPV of producer). The effect of changes in these variables can be seen in individual chance nodes as well as the decision nodes. Clicking at the Sensitivity Analysis icon, the fifth from the left on the toolbar (it looks like a tornado), in the PrecisionTree toolbar can perform the sensitivity analysis. This brings the Sensitivity Analysis dialog box shown in Figure 3–24.

Fig. 3–24 The Sensitivity Analysis dialog box of PrecisionTree 221

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If sensitivity of the EMV to the changes in probability of dry hole is required, the following steps are followed to perform sensitivity analysis. 1. Select the cell containing the tree’s EMV (Cell B10) as the Cell to Analyze. 2. Input the parameters to vary in the Input Editor, one by one. From the Input Editor, the cells to vary (for this problem C3 is entered as shown in Figure 3–24) are selected. 3. Specify the cell’s minimum value, maximum value, a base value from which to measure percentage changes in EMV, and the number of equally spaced steps between the minimum and maximum value at which the cell is evaluated. These can be entered manually or let PrecisionTree suggest values. If it is desired for PrecisionTree to suggest values, click on the Suggest Values icon. The min., max, base, and steps appear as shown in Figure 3–24. For example, for Cell C3 (probability of dry hole) PrecisionTree has been told to vary the base probability of 0.65 by 10%, i.e., from 0.50 to 0.72, and divide this range into 10 steps. Now click on Add icon, the parameters appear in the Cells to Vary as shown in Figure 3–24. 4. Repeat Steps 2 and 3 if more cells need to be varied. Otherwise, click on Run Analysis. The analysis is performed and the generated graphs appear in an Excel worksheet. The following three types of charts are generated. a. Tornado Graph. This graph measures the percentage change in EMV from the base value (calculated with all inputs at their base levels) as each input parameter is varied from its lowest to highest levels. b. Sensitivity Graph. A set of sensitivity graphs is generated to show how the optimal decision and EMV for the cell to analyze change as each input parameter varies between its minimum and maximum values. The sensitivity graph generated for the decision tree of Figure 3–23 is shown in Figure 3–25. This figure is similar to the one developed in Figure 3–1. c. Spider Graph. This graph shows how the EMV associated with the Cell to Analyze varies with changes in the selected input parameters. 222

Expected Value and Decision Trees

Fig. 3–25 Sensitivity graph for the decision tree in Figure 3–23

Example 3–11 Rework Example 7–10 (Chapter 7 of Volume I) with the following additional options. There is 24 km portion of the existing pipeline that can be replaced at a cost of 12.5 MM$. The cost will be allocated 70% in 2002 and 30% in 2003. The replacement will take approximately 18 months (starting 01/01/2002). It is assumed that once this portion of the pipeline is replaced, it will be in operating condition for a limited period before another portion becomes defective. However, the production will be switched to the spiking option when the pipeline is not operable. The following timing and probabilities are assumed. a. There is 60% chance that the pipeline will be good for two years. The NPV for this option is 68.03 MM$. b. There is 30% chance that the pipeline will be good for four years. The NPV for this option is 78.40 MM$. c. There is 10% chance that the pipeline will be good for six years. The NPV for this option is 85.42 MM$. Based on the information above, determine the economical option. Calculations for each option are given in Table 7–5 on the CD. 223

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Solution: The decision tree for Example 3–11 is shown in Figure 3–26. The best option is to replace the damaged portion of pipeline with EMV = 72.88 MM$.

Fig. 3-26 NGL pipeline replacement economics for Example 3–11

Example 3–12 A drilling company is considering bidding on a $150 million turnkey contract for drilling offshore oil wells. The company estimates that it has a 65% chance of winning the contract at its bid price. If the company is awarded the contract, it has three options. (a) use the existing rig to drill the wells, (b) buy a new rig, or (c) subcontract the drilling to another drilling company. The contract allows for sub-contracting the activities. The probabilities and payoffs for each option are given in Table 3–19.

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Expected Value and Decision Trees

Probability

NPV (MM$)

High Profit

0.35

60

Medium Profit

0.45

30

Loss

0.20

-20

High Profit

0.55

35

Medium Profit

0.35

25

Loss

0.10

-10

1.00

30

Using Existing Rig

Buying New Rig

Subcontract Medium Profit

Table 3–19 Probability and payoffs for Example 3–12

The cost of preparing the contract proposal is $1.5 million. If the company does not bid on this tender, it has an opportunity to make a guaranteed profit of $10 million elsewhere. Construct a decision tree for this situation and advise the contractor if he should bid. Solution: The decision tree for Example 3–12 is shown in Figure 3–27. The company should bid because it will result in an expected value of $19.3 million, and it should use the existing rig.

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226

Fig. 3–27 A sequential decision tree for the drilling contract example of Example 3–12

Expected Value and Decision Trees

QUESTIONS and PROBLEMS 3.1

What is meant by expected monetary value and expected profitability index? How would the two parameters be used in decision making?

3.2

What is meant by the stochastic dominance of one alternative by another?

3.3

What is meant by deterministic dominance? Why is deterministic dominance a special case of stochastic dominance?

3.4

Which factors would be considered when deciding on whether to obtain additional information?

3.5

What do expected value of perfect information (EVPI), expected value of perfect payoff (EVPP), and expected value of imperfect information (EVII) mean?

3.6

Give a brief description of the various branches and nodes of a decision tree.

3.7

Additional information can be acquired for $6500 to improve project risk. If the EVPI of the project without the additional information is $5000, should the information be acquired?

3.8

If the EVPI for a project is $6000; should the EVII be higher or lower than $6000. Why the EVPI is different from the EVII?

3.9

The probability of success of drilling a certain well is 45%. If the dry hole cost is $350,000, the completed well cost is $450,000; and if the well is successful it will have reserves of 250,000 barrels. What is the expected value of reserves? What is the risked investment (expected cost)?

3.10 There are three field development options (1) continue with primary depletion, (2) implement waterflood, or (3) implement gas injection.

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The net present value at the corporate discount rate and corresponding probabilities of realizing these NPVs are given in the table that follows. Using this data, perform expected value analysis to arrive at the most economically viable option. Base decision on EMV and EPI. Outcome

Probability

NPV, M$

Primary Depletion

Good

100%

345

Implement Waterflood

Good

45%

805

Moderate

30%

690

Poor

25%

460

Good

40%

920

Moderate

30%

805

Poor

30%

575

Implement Gas Injection

3.11 Rework the data of Problem 3.10 to calculate the standard deviation and performance index of each option. What would be deduced from the resulting calculations? 3.12 Use PrecisionTree to solve Problem 3.10. Perform sensitivity analysis on probability of each option. 3.13 Use the data of Problem 3.10 and the method of Expected Opportunity Loss (EOL) to arrive at the most economically viable option. Compare and comment on the results obtained in Problem 3.10 and using the EOL approach. 3.14 While drilling a wildcat well, a tool is stuck. The engineers have two options: (1) to fish for the tool or (2) sidetrack the well above the stuck position. The cost for each option and the associated probabilities are given in the table that follows. Based on expected value analysis, select the most economically viable option.

228

Outcome

Probability

Cost, M$

Fishing

Routine Troublesome

35% 65%

126 380

Sidetrack Immediately

Routine Troublesome

85% 15%

245 436

Expected Value and Decision Trees

3.15 Rework Problem 3.14 using Precision Tree 3.16 Perform sensitivity analysis on the probability of each option in Problem 3.14 and comment on the findings. 3.17 A company is planning to drill a wildcat well. Based on the past exploration experience and analyses of all geological and geophysical aspects of the area, the following data is available for evaluation. Calculate the expected monetary value (EMV), EPI, and standard deviation for the data. Reserves, MMStb Dry Hole

Probability

Cost, MM$

0

75%

2.6

10%

-4.2

7.9

20%

-6.3

13.0

30%

39.5

18.4

20%

110.5

23.6

15%

189.0

28.9

5%

288.5

Producer

-2.5

25%

3.18 Rework Problem 3.17 using PrecisionTree and perform sensitivity analysis on probability of dry hole in the range of 35% to 75% and comment on the results obtained. 3.19 A company manufacturing sucker rod pumping units has found the demand for its pumps has been increasing rapidly over the last two years. A decision has to be made to expand the production capacity in order to meet future demand. Three alternatives are available: (1) expand the existing facility, (2) build a new plant, and (3) subcontract work to another manufacturer. For each course of action there are three scenarios: (1) demand rising faster than expected, (2) demand continues to raise at the current rate, and (3) demand increasing at a slower rate. The net present values for each alternative and probabilities for each scenario are given in the following table.

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a. Based on expected value analysis, determine the most economically viable course of action the company should take. b. What will be the expected value of perfect information? Demand is

Demand rising

Demand increasing

Rising Faster

at Current rate

Slowly

Expand Plant

$525

$435

$175

Build New Plant

$735

$265

$285

Sub-Contract

$250

$190

$75

Probability

60%

35%

5%

3.20 There are two field development options for waterflood implementation. These are (1) to develop the field right away without a pilot and (2) to do a pilot first and if the pilot is successful then go for the full field implementation. Without the pilot, there is a 60% chance the project will generate NPV of 110 MM$, 40% chance the waterflood is not feasible, and the project loses NPV of –3.5 MM$. If the pilot option is opted for, it is estimated the probability of pilot’s success is 70%. If the pilot is successful, there is 90% probability of generating NPV of 80 MM$ and 10% chance of loosing 17.5 MM$. If the pilot fails, the resulting NPV will be –43 MM$. a. Using the EMV calculations, determine the most economically viable option. b. Use PrecisionTree to solve this problem. 3.21 The following data is given on a new wildcat prospect to be drilled. Based on this information calculate (a) the EMV, (b) EPI, and (c) E(Drilling). At what dry hole probability will the EMV be zero? Outcome

230

Probability

Investment, MM$

Net Cash Flow

Dry Hole

0.60

4.0

-2.8

5 MMStb

0.15

48.0

20.0

12 MMStb

0.10

55.0

106.0

23 MMStb

0.05

85.0

365.0

Expected Value and Decision Trees

3.22 The profits associated with combinations of acts and states of nature are as follows: State of Nature

Probability

Act A

Act B

Act C

I

0.30

60

50

30

II

0.30

40

50

80

III

0.40

70

60

60

a. Calculate the expected value of each Act. b. Which Act should be chosen based on the expected value profit calculated in Step a? c. Assuming the values in the table are costs rather than profits, which Act should be chosen in this case? d. What will be the expected value of perfect information (EVPI)? Base your calculations on EOL and EMV. 3.23 You have started drilling an oil well. It is estimated the well will produce at 300 barrels/day and has a 30% probability of success. If the well is successful, a second well will be drilled. This well is also expected to produce 300 barrels/day but its probability of success is 65%. If the first well is dry, then the second well will not be drilled. Both wells, if successful, will require pumping units from day one. Two secondhand pumping units are available for $65,000 each. However, the pumping units will not be available if you do not buy them immediately. If you buy these secondhand units and the first well is dry, you will be able to resell them at a cost of $65,000 each with probability of 20% or at a cost of $40,000 each with a probability of 80%. Buying new pumping units will cost you $90,000 each. Should you buy the secondhand pumping units now or wait until the first well is drilled to buy new ones at that time?

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3.24 You are planning to drill a well in an area where there is 35% probability of encountering an over-pressured formation at around 4000 feet. You can either a. Drill straight through the high pressure formation and face the risks and additional costs involved if over-pressured formation is encountered, b. Drill to the top of expected over-pressured formation and set intermediate casing before drilling further through the highpressured formation, or c. Drill to the top of expected over-pressured formation and run a density log to evaluate if over-pressured formation is expected, if the over-pressured formation is confirmed then set intermediate casing and drill ahead, if the over-pressured formation is not detected by the log then drill ahead without setting the intermediate casing. Other information required for this problem is given as follows. a. Drilling cost without intermediate casing set and no overpressured formation encountered is $450,000. b. Drilling cost without intermediate casing and over-pressured formation encountered is $900,000. c. Drilling cost with intermediate casing as a safety measure without running a log is $600,000. d. Drilling cost with intermediate casing set after running the log is $665,000. e. The probability of the log giving reliable information is 85%. f. Based on the previous information, should the log be run? What will be the EVII?

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REFERENCES 1

2

3

4

5

6

7

8

Ramalingam, P., Systems Analysis for Managerial Decisions—A Computer Approach, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158, USA, 1976. Samson, Danny, Managerial Decision Analysis, Irwin, Homewood, Illinois 60430 USA, 1988. Neter, J., Wasserman, W., and Whitmore, G.A., Applied Statistics, Second Edition, Allyn and Bacon, Inc., 470 Atlantic Avenue, Boston, Massachusetts 02210, USA, 1982. Newendorp, P. D., Decision Analysis for Petroleum Exploration, PennWell Books, PennWell Publishing Company, Tulsa, Oklahoma, USA, 1975. ——- Newendorp, P. D. and Schuyler, J., Decision Analysis for Petroleum Exploration, 2nd Edition, Planning Press, 15492 East Chenango Av., Aurora, CO 80015, U.S.A., 2000. Goodwin, P. and Wright, G., Decision Analysis for Management Judgment, John Wiley & Sons Ltd., Baffins Lane, Chichester, West Sussex PO19 1UD, England, 1991. Schuyler, J. R., “Applying the Expected Monetary Value concept: How many prospects is enough?” Oil & Gas Journal, Dec. 11, 1989, pp. 87–90. Schuyler, J. R., Petroleum Risks and Decision Analysis, short course manual, Oil & Gas Consultants International, P.O. Box 35448, Tulsa, OK 74153, USA, 1993. Precision Tree–’s Guide, Decision Analysis Add-in for Microsoft Excel, Palisade Corporation, 31 Decker Road, Newfield, NY 14867, USA, 2000.

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chapter

FOUR

Incorporating Attitudes Toward Risk The capital budgeting problem holds a critical place in both theory and practice of corporate finance. In certain situations, it is a widely accepted fact that choosing among independent and mutually exclusive investments based on net present value (NPV) is a consistent approach with owner wealth maximization. However, an important attribute of real-world decision making is the risk and uncertainty associated with future outcomes. In Chapter 3, the concept of expected value, as applied to the evaluation of alternative investments, was discussed. The expected value concept weighs financial consequences by their probabilities, and the resulting criterion implies the decision maker is impartial to money and the magnitudes of potential profit or losses. Many analysts consider this an adequate measure of considering risk. However, risk is not just a function of the probability distribution of outcomes (reserves or financial payoffs) but also the magnitude of capital being exposed to the chance of loss and whether this loss is sustainable by the decision maker. For example, a person with a total worth of $100,000 may be able to sustain

Project Economics and Decision Analysis Volume II: Probabilistic Models

a loss of $10,000 or even higher, but a person with a total worth of $10,000 cannot afford a loss of $10,000. While evaluating competing investment alternatives with widely varied risk characteristics, it is important for investors to use a formal and consistent means of risk policy. In view of this, setting a corporate risk policy is an important component in managing the evaluation of available investments.1 When the estimated possible loss is high, to the point where a significant fraction of the capital asset value of the investor is at risk, then most investors either • •

Downgrade the investment because of the total risk threshold that can be sustained, or Share risk by taking less than 100% stake in the investment. This is done in order to limit potential losses to values below what the investor considers harmful to its fiscal health.2

Efforts to avoid some of the pitfalls associated with the expected value concepts lead to the discussion of a fundamental decision science model known as preference theory, also referred to as expected utility theory. The theory encomes the decision makers’ attitude toward risk. The concepts presented in this chapter are extensions of the expected value concept, in that the investors’ attitudes about money are incorporated into a quantitative decision model. The result is a more realistic measure of value among competing investments characterized by risk and uncertainty. The preference theory concepts are based on fundamental and reasonable concepts about rational decision making. This powerful decision rule can encom multiple dimensions of value and nonlinear attitudes toward risk. It can be applied to decision trees using the same procedure as discussed in the previous chapter. In this chapter, first the basic theoretical aspects of the expected utility theory are presented. These are followed by the use of utility functions and application of the expected utility rule to decision trees. Once again, PrecisionTree is used to solve decision trees using the expected utility concepts. In Chapter 5, ways of incorporating risk threshold factors in estimating risk-adjusted value of a project and optimum participation level of an investor are discussed.

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THE EXPECTED UTILITY THEORY The expected value concepts are applicable in many routine, repetitive decisions where the investor is not exposed to gains or losses that are not large relative to the investor’s total assets or total worth. In such situations, it is usually valid to assume a linear preference (riskneutral) function. In cases where the stakes are high, relative to the size of the total portfolio of the investor, real preference functions are often considered nonlinear. Therefore, it is considered prudent to use the expected utility (EU) approach rather than the expected monetary value (EMV) approach discussed in the previous chapter. In this section, typical risk attitudes of a decision maker, axioms of utility theory, risk tolerance, risk , assessment of utility function, and mathematical representation of risk preferences are presented.

Typical Attitudes toward Risk The value of risk is fundamentally a subjective concept. The pleasure (utility) associated with winning $1000 is generally less than the displeasure and disappointment of losing the same amount or even less. Similarly, one derives more pleasure by winning $10 from $30 versus winning $15 from $1000. It depends upon such things as the total worth of the decision maker (individual, department, or an entire company) making the decision. It determines its security against bankruptcy and the budget level representing the liquid assets that are at risk of being lost. However, there is no set rule to measure how risk averse a decision maker ought to be. Table 4–1 shows outcomes of two projects with their corresponding probabilities of occurrence. The expected value and standard deviation of each project is calculated. As evident from Table 4–1, the EMV of Project A is much higher than the EMV of Project B. Based on the EMV criterion; investors should select Project A if they are mutually exclusive. However, the EMV criterion fails to give adequate weight to the decision maker’s exposure to the chance of a very large financial loss in Project A

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(–40 M$) versus Project B (–5 M$). The decision maker knows a loss of 40 M$ with Project A would be much more detrimental to his financial position. A risk-averse decision maker will obviously choose Project B with a chance of lower loss. In this situation, the payoffs in monetary units do not fully reflect the decision maker’s attitude toward risk. Hence, the comparative payoffs may not be in accord with the decision maker’s actual risk preferences.

Project A

Project B

Probability

NPV (M$)

Probability

NPV (M$)

0.80

80

0.80

30

0.20

-40

0.20

-5

EMV

56

23

Std. Dev. (s)

48

14

Table 4–1 Expected monetary value of two projects

A theoretical utility curve is shown in Figure 4–1. Theoretically, by analyzing past decisions, a utility curve can be constructed for an individual, a department, or an entire company (let’s call it decision maker). The problem is no one correct utility function can be determined for a decision maker. It is the one that reflects risk attitudes of previous decisions. Furthermore, a company’s risk response depends on factors such as the overall economic climate, changing budgets, management philosophy, and its fortune over time.

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Fig. 4–1 Theoretical utility/preference curves

Individuals who avoid risk or are sensitive to risk are called riskaverse (risk avoiders, risk averters, conservative). Attitude toward risk is when a decision maker is less likely to choose an alternative with a higher EMV if it includes proportionately higher risk. The risk-averse behavior is represented by the utility curve as shown in Figure 4–1, i.e. curved and opening downward (concave). A decision maker with risk-averse behavior will prefer to invest in a venture having a perceived high chance of success to a second venture having a low chance of success, even if the expected value of the second venture is clearly superior. However, not everyone displays risk-averse behavior all times, and so utility curves need not be concave for him or her. A convex (opening upward) utility curve represents risk-seeking (aggressive) behavior. This attitude is opposite of the attitude of risk-averse decision makers. Finally, an individual can be risk-neutral. A decision maker who always selects the alternative with the highest EMV, regardless of the associated risk, is considered risk-neutral. As shown in Figure 4–1, the diagonal line in all utility curves represents the risk neutral attitude. Therefore, it is represented by the EMV and is referred to as the EMV line. The linear relationship means the act that maximizes expected payoffs also maximizes EU.

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Therefore, there is no need to use utility numbers for risk-neutral persons. A risk-neutral decision maker will therefore use EMV for choosing among alternatives. Arps and Arps, in an important paper on “prudent risk-taking,” considered the following two concepts.3 •

•

In successive ventures, the amount of money at risk in each venture should be less than the amount that would create a breakeven situation in the longer run. In other words, one should pay less for a venture than its expected value. The amount of money risked on any one venture should not exceed the amount that would increase the risk of gambler’s ruin beyond acceptable limits. Arps and Arps defined gambler’s ruin as the situation where a risk-taker with limited funds goes broke through a continuous string of failures that exhaust his available funds.

For example, if a decision maker with only $40 million in total resources invests in Project A of Table 4–1 and the project goes bad, he will be bankrupt. On the other hand, he can invest the available $40 million in eight projects, each with $5 million cost. This enhances his chances of meeting the EMV, thus avoiding bankruptcy. Alternatively, the decision maker may elect to take partial interest in Project A (for example, 12.5% working interest). By taking partial interest in a venture, he can reduce his exposure to risk and possible bankruptcy. Chapter 5 presents application of utility theory in arriving at the optimum venture participation based on the decision maker’s risk tolerance. Following are some of the important properties of the utility/preference curves.4 •

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The vertical scale is dimensionless, representing the relative desirability of an amount of money. For example, in Figure 4–1, receiving $1 million is more desirable than receiving $0.5 million because the utility (0.79) of $1 million is greater than the utility (0.54) of $0.5 million. The magnitude of the scale is arbitrary and normally ranges from zero (0) to one (1) as shown in Figure 4–1.

Incorporating Attitudes Toward Risk

•

•

•

However, some curves may represent negative desirability. The point zero is generally interpreted as the point of indifference. The positive values show increasing desirability, and negative values show increasing dislike for the corresponding amounts of money. The horizontal axis represents monetary values corresponding to the various levels of desirability. The monetary values could be NPV, costs, incremental cash flows, current asset position, CEs, etc. The curve is a monotonically increasing function indicating that as the amount of money increases, the vertical parameter (utility/preference) increases in numerical value. It denotes preference of a decision maker changes as his total wealth or monetary position changes (getting more is always better). The preference values can be multiplied by their probability of occurrence to arrive at expected preference value or EU for a decision alternative in the same way as the EMV calculated in the previous chapter. Mathematically, the EU is given by (4.1) N

EU = ∑ pi × U (xi ) i =1

The Axioms of Utility The mathematical preference theory is based on certain assumptions, commonly referred to as EU axioms. Mathematical proofs are presented in literature ing the following EU axioms. If a decision maker accepts these axioms as the basis of rational decision, then it is possible his or her attitudes toward money can be described by a preference/utility curve. The Transitivity Axiom: According to this axiom, if the decision maker prefers alternative A to alternative B and alternative B to alternative C, then he also must prefer alternative A to alternative C (his preference must be

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transitive). For example, if a person prefers a Cadillac to Mercedes and Mercedes to Honda, then he prefers Cadillac to Honda. The Complete Ordering Axiom: A decision maker is able to order her preferences or indifference to any two alternatives. For example, if a decision maker has two alternatives A and B — she either prefers A to B or she prefers B to A, or she is indifferent between A and B. The Continuity Axiom: Suppose a decision maker is offered a choice between two lotteries shown in Figure 4–2a. Lottery 1 offers a no-risk reward of A, while Lottery 2 offers a reward of B with probability p and a reward of C with probability of 1 – p. Reward B is preferable over reward A, and A in turn is preferable over reward C. The continuity axiom states there must be some value of probability p at which the decision maker is indifferent between the two lotteries.

Fig. 4–2 Illustration of utility axioms: (a) illustrates the continuity axiom and (b) illustrates the substitution axiom (after Goodwin and Wright)5

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The Substitution Axiom: If a decision maker indicates indifference between the lotteries shown in Figure 4–2a, then according to the substitution axiom, if reward A appears as a reward in another lottery, it can always be substituted by Lottery 2 because the decision maker is indifferent to A and Lottery 2 (meaning both are equally preferable). For example, a decision maker is indifferent to Lottery 1 and Lottery 2 of Figure 4–2b. Therefore, according to the substitution axiom, the decision maker will also be indifferent to Lotteries 3 and 4. These two lotteries are identical except that in Lottery 4, the $15 of Lottery 3 is replaced by Lottery 2. Unequal Probability Axiom: If a decision maker prefers reward B to reward C. Then, according to this axiom, if he is offered two lotteries with only two outcomes B and C, he will prefer the lottery offering the highest probability of reward B. For example, in Figure 4–3a the Lottery 1 will be preferred to Lottery 2.

Fig. 4–3 Illustration of utility axioms: (a) illustrates unequal probability axiom and (b) illustrates compound lottery axiom (after Goodwin and Wright)5

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Compound Lottery Axiom: According to this axiom, a decision maker will be indifferent to a compound lottery and a simple lottery that offers the same outcomes with the same probabilities. For example, a decision maker will be indifferent between the two lotteries shown in Figure 4–3b. Invariance: All that is needed to determine a decision maker’s preferences among uncertain events are the payoffs (or consequences) and the associated probabilities. Finiteness: No consequences are considered infinitely bad or infinitely good.

Risk Tolerance In the preference theory approach, the risk tolerance value has a considerable effect on the valuation of a risky investment. Risk tolerance, R, is a measure of how much risk a decision maker will tolerate. By definition, the R-value represents the sum of money at which the decision makers will be indifferent between a 50:50 chance of winning that sum and losing half of that sum. The larger the value of R, the less risk averse the decision maker is. Therefore, a person or company with a large value of R is more willing to take risks than those with a smaller value of R. The R is reported in the same units as the monetary value. For example, if revenues and costs are reported in millions, then R has to be in millions. The reason for assessing the corporate risk tolerance is one of assessing tradeoffs between potential upside gains versus downside losses. The decision maker’s attitude about the magnitude of capital being exposed to the chance of loss is an important component of this analysis.1 A variety of techniques exists for determining R. As defined previously, R has an intuitive interpretation that makes its assessment relatively easy. Figure 4–4 provides some insight into the assessment of R in of decision about risky choices. Consider the following gamble Win $Y with probability of 0.5 Lose $Y/2 with probability of 0.5

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Fig. 4–4 Assessing risk tolerance (find the largest value of Y at which alternative A will be preferred)

Would a person be able to take this gamble if Y were 10, 100, 200, or 500? Thinking in of investment, how much would a person be willing to risk ($Y/2) in order to have a 50% chance of tripling the money (winning $Y and keeping $Y/2)? The decision maker is asked for the assessment of the value at which her risk becomes intolerable. The maximum value of Y at which the decision maker would accept the gamble gives a reasonable estimate of his R. A decision maker, willing to accept this gamble at only a small value of Y is risk averse, whereas a decision maker willing to play for larger value of Y is less risk averse. A decision maker is posed with lotteries of various monetary values ($Y) in order to assess the value to which he will be indifferent. For example, a decision maker is indifferent at a value of $30 million. This becomes his risk tolerance level. Any investments costing less than $30 million are accepted, and those higher than $30 million are rejected. Cozzolino (1977) and Howard (1988) suggest a relationship exists between certain financial measures (shareholder equity, net income, and capital budget size, etc.) and the firm’s risk tolerance. Howard suggests financial statements might be used to develop guidelines for establishing acceptable R levels, at least in certain industries. Howard suggests certain guidelines for determining a corporation’s risk tolerance in of total sales, net income, or equity. Reasonable values of R appear to be approximately 6.4% of total sales, 1.24 times net income, or 15.7% of equity. These

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figures are based on observations Howard made in the course of consulting with various companies.6–7 As a rule of thumb, some companies will take 25% of their total annual exploration budget as their risk tolerance. For example, if the annual exploration budget of a firm is $40 million, then it will consider its risk tolerance level to be approximately $10 million. Walls and Dyer performed an empirical study to measure the implied risk tolerance values of the top 50 independent and integrated oil companies from 1981 to 1990. They reconstructed each firm’s annual exploration budget allocations across a set of risky ventures. Based on the amount each firm was willing to pay for participating in these ventures, an implied R-value was estimated for each firm in each year. They found a significant positive relationship between firm size and corporate R, i.e. the larger the firm the greater was the R value.8

Certainty Equivalent and Risk The term certainty equivalent (CE) refers to the amount of money equivalent in a decision maker’s mind to a situation involving uncertainty. A closely related term, risk (RP), refers to the EMV a decision maker is willing to give up (or pay) in order to avoid a risky decision. These two concepts are closely linked to the idea of EU. Mathematically, the RP, EMV, and CE are related by the following equation. (4.2)

Risk = EMV – CE The following lottery explains these two concepts. Win $3000 with probability of 0.50 Lose $200 with probability of 0.50 The EMV of the gamble is $1400. However, if someone wants to buy the lottery ticket, the owner is willing to sell it for $500. This is a sure thing and no risk is involved in it. Therefore, the CE of the seller for this gamble is $500. In the seller’s mind, the lottery is equivalent to a sure

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amount of $500. If she is unable to sell the lottery at this price then she would prefer to keep it. According to Equation (4.2), the RP for this lottery is then equivalent to $900. This means the owner is willing to give up $900 in expected value in order to avoid the inherent risk in the lottery. The RP is, therefore, the (lost opportunity) one pays to avoid risk. In any given situation, the CE, EMV, and RP all depend on the decision maker’s utility function and the probability distribution of the payoffs. The EU and the CE can be used for ranking investments. If the CEs of two investments are same, their EU will also be same and the decision maker would be indifferent to a choice between the two alternatives. The alternative with a higher CE is preferred over the alternative with a lower CE and vice versa. Similarly, the alternative with a higher EU is preferred over the alternative with a lower EU and vice versa. For a risk-averse decision maker (concave utility curve) the RP is positive while for a risk-seeker (convex utility curve) it is negative. The CE is less than EMV for positive RP, and CE is greater than EMV for negative RP. The negative RP implies a decision maker would have to be paid to give up an opportunity to invest. The following steps are used to calculate the RP of an investment alternative. 1. Assess the utility function of the decision maker (to be discussed in the next section). 2. Find the EU of the investment. 3. Find the CE, or the certain amount with the utility value equal to the EU determined in Step 2. 4. Calculate the EMV of the investment. 5. Subtract the CE from EMV to find the RP. The higher the RP for an investment, the more risk averse the decision maker is. A negative shows risk-seeker attitude, and a RP of zero shows risk-neutral attitude.

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The CE valuation highlights tradeoffs between potential and uncertain upside gains versus downside losses with respect to the decision maker’s risk preference. It also provides the decision makers with a measure of the amount of expectation they give up by making certain participation choices. The following example is used to clarify the calculations of RP.

Example 4–1 Use the data of Table 4–1 to calculate the EU, CE, and risk of each project. The utility curve for the decision maker is given in Figure 4–5. Based on the decision maker’s utility function, which project will be the preferred choice? Use EMV, CE, and EU. Solution: The utility curve of Figure 4–5, with the payoffs and probabilities given in Table 4–1, are used to solve this example.

Fig. 4–5 Decision maker’s utility curve for the data in Table 4–1 and Example 4–1

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The following steps are followed to solve the problem. 1. This first step involves converting the dollar payoffs into their corresponding utility values. The utility values for the corresponding dollar payoffs are read from the utility curve of Figure 4–5 as, U($80) = 0.8983 U($30) = 0.5756 U(–$5) = –0.1536 U(–$40) = –2.1357 2. The expected utility (EU) for each project is calculated as EUA = 0.80 0.8983 + 0.20 –2.1357 = 0.2915 EUB = 0.80 0.5756 + 0.20 –0.1536 = 0.4298 3. For EUA = 0.2915 and EUB = 0.4298, the corresponding CE is read from Figure 4–5. This is done by starting at the vertical axis with the utility value of 0.2915, reading across to the utility curve, and then dropping down to the horizontal axis to read the CE. The CEA = 12.06 is shown by the dotted line in Figure 4–5. Similarly, the CEB is read as 19.66. 4. The EMV of each project is shown in Table 4–1. The expected value of Project A (56 M$) is greater than the expected value of Project B (23 M$). Based on EMV criteria, the decision maker will select Project A. However, based on the EU and CE, Project B is preferred over project A since EUB > EUA and CEB > CEA. Unlike the expected value analysis, the EU and certainty equivalent valuations make a clear distinction between the projects, based on the risk preference of the decision maker. 5. The risk (RP) for each of these projects is calculated as RPA = EMVA – CEA = $56 – $12.06 = 43.94 M$ RPB = EMVB – CEB = $23 – $19.66 = 3.34 M$

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Assessing the Utility Function A major problem in utilizing preference/utility theory is the assessment of preference/utility function of the decision maker. Once the utility function U(x) has been ascertained, finding the optimum investment alternative proceeds as before, except utility numbers rather than monetary payoffs are used. When the utility numbers rather than the monetary payoffs are used, the preferred act will be the one with the largest expected value of the outcomes expressed in utility numbers, i.e. the rational decision maker should select the alternative that will maximize his EU. The assessment of utility function is a matter of subjective judgment, just like the assessment of subjective probabilities. The assessment of utility function involves constructing a mathematical model to represent the risk preferences (risk attitudes) of the decision maker. This model is then included in the overall decision analysis process and used to analyze the situation at hand. Two approaches, (a) assessment using CEs and (b) assessment using probabilities, are in common use for the assessment of utility functions. A disadvantage of the probability approach is that thinking in of probability may be difficult. The most widely used is the certainty equivalence approach, which only requires the decision maker to think in of 50:50 gambles.5 The certainty equivalence method requires the analyst to assess several CEs. The CEs of the drill versus farm out problem addressed in Table 3–3 are assessed here. In this problem, a decision maker faces an uncertain situation. If she chooses to drill the well, the worst case is a dry hole with –$250,000, and the best case is to have a discovery yielding an NPV of $500,000. On the other hand, if she farms out the option and the farmee encounters a producer, her NPV will be $50,000 or she looses nothing if it is a dry hole. Therefore, there are a variety of options, each of which leads to some uncertain payoff ranging from a loss of $250,000 to a gain of $500,000. To evaluate the alternatives, the decision maker must assess her utility for payoffs in this range. The technique presented here involves eliciting five CE points, thus it is known as the five-point method of assessing utility function. The following steps are followed.

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Eliciting Point 1: To start with, identify the worst possible outcome. In this case, the negative payoff of –$250,000 for the dry hole (worst case) is taken, and a utility of zero is assigned to it. By asg zero to the negative payoff shows negative attitude toward losing money or preference not to lose money. Therefore, U(–$250,000) = 0 Eliciting Point 2: Identify a best possible outcome ($500,000 to encounter a producer). A utility value of one is assigned to it. Therefore, U(+$500,000) = 1 Eliciting Point 3: Now imagine the decision maker has an opportunity to play the following lottery, referred to as the reference lottery or reference gamble. Win $500,000 with probability 0.5 Lose $250,000 with probability 0.5 The expected value of the previous gamble is $125,000 (0.5 x $500,000 – 0.5 x $250,000). What is the minimum amount, CE, for which the decision maker would be willing to sell her opportunity to play this game? Note a risk-averse decision maker generally trades a gamble for a sure amount that is less than the expected value of the gamble. Suppose for this reference gamble, CE of $50,000 is elicited. This means the decision maker is truly indifferent between $50,000 and the risky gamble. The utility of this amount must equal the EU of the gamble, which is calculated as: U($50,000) = 0.5 U($500,000) + 0.5 U($250,000) = 0.5 1 + 0.5 0 = 0.5

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Eliciting Point 4: Now the CE for a new gamble is elicited, this time between the U($50,000) calculated in Eliciting Point 3 and the U($500,000) in Eliciting Point 2. Win $50,000 with probability of 0.5 Win $500,000 with probability of 0.5 The expected value of the previous gamble is $275,000. Suppose that for this reference gamble, the elicited CE is $225,000. In this case the utility of $225,000 is elicited because the decision maker knows that from Eliciting Point 3 U($50,000) = 0.5 and Eliciting Point 2 U($500,000) = 1.0. U($225,000) = 0.5 U($50,000) + 0.5 U($500,000) = 0.5 0.5 + 0.5 1.0 = 0.75 Eliciting Point 5: Now consider the following reference gamble. Win $50,000 with probability of 0.5 Lose $250,000 with probability of 0.5 The expected value of the previous gamble is –$100,000. For this reference gamble, the elicited CE is –$100,000. The utility value of this CE is calculated by using U($50,000) = 0.5 from Eliciting Point 3 and U(–$250,000) = 0 from Eliciting Point 1. U(–$100,000)= 0.5 U($50,000) + 0.5 U(–$250,000) = 0.5 0.5 + 0.5 0 = 0.25 The elicited utility values obtained are now plotted against their corresponding CEs. A curve is drawn through the five points.

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The following five eliciting points are used to construct a utility curve. The effect of this procedure is to elicit the monetary values in the range of $0 to $500,000 that have utilities of 0, 0.25, 0.50, 0.75, and 1.0. Monetary Utility

| –$250,000 | –$100,000 | $50,000 | 0.00 | 0.25 | 0.50

| $225,000 | $500,000 | 0.75 | 1.00

The CE values and their corresponding utility functions are plotted as shown in Figure 4–6. The shape of the curve in Figure 4–6 shows the decision maker is risk-averse for the range of monetary values in which the utility is assessed.

Fig. 4–6 Graph of the utility function assessed using the CE approach

The assessments and the graph are checked for consistency. If the graph is not reasonably smooth, then the assessments are checked, and some more assessments are made by deg additional gambles. Note the decision maker’s first response of $50,000 is used in subsequent lotteries, as both a best and worst outcome. This process is called chaining, and it propagates the very first judgmental error, if any, throughout the rest of the assessment.

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Once the utility functions for the complete range of the monetary values are assessed and graphed, these are then used in making decisions. Decision trees can be utilized to solve problems involving utilities. The first step is to replace the monetary values in a decision tree by their corresponding utility values. If a utility value of a particular monetary value is not elicited from the five elicited points, it can be read from the utility curve. The EU for each chance node is then calculated. The alternative with the highest EU is then selected as a viable option to satisfy the decision maker’s preference toward risk. The following example clarifies the calculations.

Example 4–2 Rework Example 3–2 (Chapter 3) using the EU criterion. Use the utility values assessed in the previous section. Based on the EU maximization, which alternative should be selected? Solution: The decision tree for the problem is shown in Figure 4–7.

Fig. 4–7 Decision tree for Example 4–2

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EU (Drill) = 0.65 × U (− $250,000 )+ 0.35 × U ($500,000 ) = 0.65 × 0 + 0.35 × 1 = 0.35

EU (Farm out) = 1.0 × U ($50,000) = 0.50

Since the EU of the farmout option is higher than the EU of the drill option [EU(Farmout) > EU(Drill)], the farmout option is selected. The decision tree calculation procedure is the same as used in Chapter 3. The only exception is the monetary values are replaced by their corresponding utility values.

Mathematical Representation of Utility Functions Utility function is an expression that measures risk by converting monetary payoffs related to an outcome against their corresponding utility units. The EU (probability weighted utilities) of one alternative is then compared to that of another alternative in order to select the alternative that will maximize the decision maker’s EU. In a complicated decision problem with numerous possible payoffs, it might be difficult and time consuming for a decision maker to determine the EU values corresponding to the possible payoffs from graphical presentations only. However, if the decision maker is risk-averse or riskseeker, then some typical mathematical utility functions can be used as an approximation of the decision maker’s actual utility function. Mathematical formulas representing the utility functions are as many and as varied as the individual corporations involved. However, some of the most common ones are presented in this section. The utility function curves, such as the one presented in Figure 4–1, are normally represented by mathematical relationships. Since mathematical functions give exact forms of expressions, it is usually more accurate and convenient to fit mathematical functions to utility curves as compared to

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drawing the curves by hand. The risk preferences may be represented by a variety of curves such as exponential, logarithmic, quadratic, linear, power, and hyperbolic. In this section, some of the most commonly used mathematical relationships representing the utility curves are examined. Exponential Utility Function: Many variations of the exponential equations appear in literature, although some of them are mathematically the same. These various forms of the exponential utility curves are defined by the following equations. (4.3)

U (x) = a + be U (x) = 1 − e

− cx

or (4.3a) −x / R

U (x) = a + b(1− e

−x /R

)

or (4.3b)

or (4.3c)

U (x) = a − be

−x / R

or (4.3d)

U (x) = ae

bx

or (4.3e)

U (x ) =

1 a

(1− e ) −x / R

where x = the dimension of monetary value in currency units a, b, and c = constants R = risk tolerance, a constant specified by the decision maker e = exponential constant (e = 2.71828)

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The utility curves developed by using Equation (4.3a) are shown in Figure 4–8. The curve with R = 100 represents a utility function with a higher degree of risk aversion than the lines represented by R = 200 and R = 300. This implies the farther from the EMV line the higher the degree of risk aversion. As R increases, the decision maker becomes more risk tolerant. At higher values of R, the curve becomes flat. If a constant is added to a utility curve or if the utility curve is multiplied by a constant, its properties remain the same.

Fig. 4–8 Representation of exponential utility curves, generated by Equation (4.3a)

As previously mentioned, constructing the utility curve is a subjective affair. Therefore, use exponential utility when the risk aversion level bounds the true utility function. Since a majority of the companies are not aware of their utility functions, bounding or estimating their risk aversion levels may be a practical approach for deriving the benefits of utility theory. Not all corporations choose exponential risk aversion, some have hyperbolic tangent type of risk weighting, others have empirical models based on prior evaluations of projects and the anticipated value to the corporate assets, and so on.

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Logarithmic Utility Function: The logarithmic utility function is given by the following equations. (4.4)

x⎞ ⎛ U (x ) = ln⎜1 + ⎟ ⎝ R⎠ (4.4a)

U (x) = a log(b + x ) + c (4.4b)

1 U (x) = ln(b + x) − c a

Quadratic Utility Function: The quadratic utility function is given by the following equation. (4.5)

U (x) = a + bx − cx

2

Linear Plus Exponential Function: This type of utility curve is represented by the following equation. (4.6)

U (x) = ax − be

−x /R

Power Utility Function: This type of utility curve is represented by the following equation. U (x) = a + bx

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Hyperbolic Tangent Utility Function: This type of utility function is given by the following equation. The idea behind the hyperbolic equation is that there is greater stability in the management of high loss scenarios than there is with the exponential rule. (4.8)

⎛x⎞ U (x ) = 1 − tanh ⎜ ⎟ ⎝R⎠

Some additional utility curves, representing Equations (4.3d), (4.3e), (4.4c), and (4.5) are shown in Figure 4–9.

Fig. 4–9 Illustration of utility curves representing Equations (4.3d), (4.3e), (4.4c), and (4.5)

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Since the decision maker’s risk preference or utility can be represented by a mathematical equation, the same equation can be used to calculate the CE. CE is the inverse of any equation representing a certain utility curve. For example the inverse of Equation (4.3 a) and Equation (4.4a) are given by (4.9)

CE = − R ln[1− U (x)]

⎛ U (x )− c ⎞ CE = Anti log⎜ ⎟−b a ⎝ ⎠

(4.10)

The utility curve presented in Figure 4–5 is generated using Equation (4.3 a) for R = 35. In addition, the CEA and CEB corresponding to the EUA and EUB were read from the curve in Figure 4–5 for use in Example 4–1. The same information can be obtained with more accuracy by using Equation (4.9) and the EUA and EUB from Example 4–1 shown as follows. Since EUA = 0.2915 and EUB = 0.4298, then the CEA and CEB using Equation (4.9) will be CEA = –Rln[1 – EUA] = –35ln[1 – 0.2915] = 12.0612 12.06 M$ CEB = –Rln[1 – EUB] = –35ln[1 – 0.4298] = 19.6619 19.66 M$ The same values were read from the utility curve in Figure 4–5. The other equations can be arranged in a similar way in order to calculate the CE from them.

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Example 4–3 Rework the data of Table 4–1, assuming the risk tolerance level of the decision maker is R = 100. Calculate the EU, CE, and risk for each project. Based on these parameters, select the most viable project. The decision maker’s risk preference can be modeled by exponential equation of the type 1 – e –x / R. Solution: First the payoffs of Table 4–1 are converted to their respective utilities, using the decision maker’s utility curve represented by the equation 1 – e –x / R. U ($80 ) = 1 − e −80 / 100 = 1 − e −0.80 = 1 − 0.4493 = 0.5507 U (− $40) = 1 − e 40 / 100 = 1 − e 0.40 = 1 − 1.4918 = −0.4918 U ($30 ) = 1 − e −30 / 100 = 1 − e −0.30 = 1 − 0.7408 = 0.2592 U (− $5) = 1 − e 5 / 100 = 1 − e 0.05 = 1 − 1.0513 = −0.0513

The next step is to calculate the EU of each project.

EU A = 0.80 × 0.5507 + 0.20 × −0.4918 = 0.4406 − 0.0984 = 0.3422 EU B = 0.80 × 0.2592 + 0.20 × −0.0513 = 0.2074 − 0.0103 = 0.1971 Due to the R = 100 versus the R = 35 (used in Example 4–1), the project preference is changed. Since EUA = 0.3422 > EUB = 0.1971, Project A is preferred over Project B. The calculations are also shown in Table 4–2.

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Project A

Project B

NPV

Utility

Probability

(M$)

1-e-x/100

0.8

80

0.2 EMV & EU

NPV

Utility

Probability

(M$)

1-e-x/100

0.5507

0.8

30

0.2592

– 40

– 0.4918

0.2

–5

– 0.0513

56

0.3422

23

0.1971

Table 4–2 EMV and EU calculations of two projects, Example 4–3

The next step is to calculate the CE of each project, using Equation (4.9) as CE A = − R ln(1 − EU A ) = −100 × ln (1 − 0.3422) = −100 × −0.4189 = 41.8854 CE B = − R ln(1 − EU B ) = −100 × ln(1 − 0.1971) = −100 × −0.2195 = 21.9525

The certainty equivalent calculations confirm the selection of Project A over Project B. The last step is now to calculate the risk of each project, using Equation (4.2) as RPA = EMV A − CE A = 56 − 41.8854 = 14.1146 M$ RPB = EMVB − CE B = 23 − 21.9525 = 1.0475 M$

Approximation to Certainty Equivalent The examples presented in the preceding sections used exponential utility function to translate the monetary values of outcomes into their corresponding utility values. These utility values were then used to calculate the EU. This EU was then fed back into the exponential utility function (or utility curve) to come up with the certainty equivalent (CE) in of monetary value. This exercise may become tedious when there are too many outcomes under evaluation.

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An approximate mathematical correlation is available that can be used to arrive at the CE. The approximation works well when the outcome’s probability distribution is represented by normal probability distribution, i.e. a symmetrical, bell-shaped curve. The following relationship can be used.9–11 (4.11)

CE ≈ EMV −

0.5s R

2

where s2 is the variance (discussed in Chapter 3) and R is the risk tolerance. When this is applied to the outcomes of the two investment alternatives given in Table 4–2, CE is calculated as

CEA ≈ 56 −

0.5[0.80 80 2 + 0.20 (–40) 2 – 56 2 ] 100

≈ 56 −

0.5 (5,120 + 320 – 3136) 100

≈ 56 −

0.5 2,304 = 44.48 100

Similarly, CEB = 22.02. The calculated CEA = 41.8854 and CEB = 21.9525 were calculated in Example 4–3. The approximate CE calculated for Project B is in close agreement with the one calculated in Example 4–3 (22.02 vs. 21.9525). The approximate CE calculated for Project A is not as close to the one calculated in Example 4–3 (44.48 vs. 41.8854).

Risk Aversion In the section on Mathematical Representation of Utility Functions, six different types of mathematical equations and their variations were presented. Although the shapes of the utility curves generated from these

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equations may appear to be quite similar, they signify quite different risk aversion levels. To show the important difference between these functions, a term called risk aversion (RA) is introduced.12 (4.12)

− U ′′(x) RA = U ′(x)

where U(x) and U(x) is the second and first derivative of U(x), respectively. The RA function measures the degree of aversion to uncertainty in a utility function. The numerator, second derivative of U(x), measures the curvature (rate of change of slope) of the utility curve. More curvature means more RA. The EMV line, being a straight linear line, shows an important concept of the proportionality of decision maker’s risk preference or utility to the amount of wealth. As previously mentioned, this means that as the total wealth level of the decision maker increases, he becomes more risk tolerant. However, this concept may not be generalized. The RA parameters for some of the equations are derived. In each case, the expected RA behavior of the decision maker is analyzed. For the exponential utility function of the form 1 – e–Rx, the RA is

U ′(x) = –(–R)e–Rx = Re–Rx U ′′(x) = –R 2 e–Rx RA =

264

– ( –R2e–Rx) Re–Rx

=

R2e–Rx Re–Rx

=R

Incorporating Attitudes Toward Risk

Similarly, for the exponential utility function of the form 1/R(1 – e–Rx), the RA is

1 U ′(x) = –(–R)e–Rx = e–Rx R U ′′(x) = –Re–Rx RA =

– ( –Re–Rx) e–Rx

=R

Similarly, the RA for Equation (4.4a) is

RA =

1 b+x

and for Equation (4.5), it is

RA =

2c b + 2cx

The relationships derived previously show RA is constant for the exponential utility function, regardless of the total wealth level of the decision maker. This risk attitude is called constant risk aversion. The constant risk aversion means that no matter how much wealth a decision maker has, she will view a particular investment in the same way. The constant risk aversion may be reasonable for some decision makers, but most of them might be less risk averse if they had more wealth (the risk tolerance level increases with increase in wealth). On the other hand, for the logarithmic and quadratic utility functions the RA decreases as the level of wealth, x, increases. The RP profile can also be used to illustrate the constant and decreasing risk aversion properties of exponential and logarithmic utility functions. It can be shown that the RP for exponential type utility function stays constant regardless of the increase in wealth while for the logarithmic type utility the RP decreases as the level of wealth increases.9

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If a constant amount is added to all payoffs of an investment and the decision maker’s preferences show decreasing risk aversion, then the calculated RP will decrease. On the other hand, if the decision maker’s preferences show constant risk aversion then the calculated RP will be constant. Therefore, in the constant risk aversion case the decision maker will feel the same regardless of increase in his wealth. In the decreasing risk aversion situation, an increase in the level of wealth will make the decision maker less risk-averse toward an investment thus resulting in lower RP. The decreasing risk aversion sounds more logical because, by nature, a wealthier decision maker can afford to take larger risks. Example 4–4 shows this concept.

Example 4–4 Consider the payoffs of Project B as shown in Table 4–3. If the decision maker’s risk preference can be modeled by an exponential utility function of the form 1 – e–x/R (R = 100), what will be the risk if the decision maker’s wealth is 100 M$, 150 M$, and 200 M$. Repeat the same problem assuming that the decision maker’s risk preference can be modeled by a logarithmic utility function of the form 1.73log(74.97 + x) – 3.23.

Project A Probability

Project B

NPV (M$)

Probability

NPV (M$)

0.20

95

0.5

48

0.80

-5

0.5

-18

Table 4–3 Expected monetary value of two projects Solution: To solve this problem, the existing wealth is added to each payoff, and the risk for the investment is calculated as shown in Table 4–4. The calculations for the first part are as follows.

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Wealth Level Outcome A Outcome A (M$)

(M$)

(M$)

100

148

82

150

198

132

200

248

182

Probability

0.5

0.5

Risk Tolerance

100

U(x)=1-e-x/100

-100In(1-EU)

Utility, A

Utility, B

EMV, M$

EU

CE, M$

RP, M$

0.7724

0.5596

115

0.6660

109.65

5.3490

0.8619

0.7329

165

0.7974

159.65

5.3490

0.9163

0.8380

215

0.8771

209.65

5.3490

(

1.73 log (74.97+x)-3.23

10

)-74.97

EU+3.23 1.73

Utility, A

Utility, B

EMV, M$

EU

CE, M$

RP, M$

0.8325

0.5688

115

0.7006

112.11

2.8882

0.9845

0.7765

165

0.8805

162.72

2.2799

1.1109

0.9391

215

1.0250

213.12

1.8839

Table 4–4 Risk from exponential & logarithmic functions

The monetary payoffs are first converted into their corresponding utility values U (A) = 1 − e − (100 + 48 ) / 100 = 0.7724 U (B ) = 1 − e − (100−18 )/ 100 = 0.5596

The EMV and EU for the investment are calculated

EMV = 0.50 × (100 + 48)+ 0.50(100 − 18) = 115 M$ EU = 0.50 × 0.7724 + 0.50 × 0.5596 = 0.6660

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Now using the inverse of 1 – e–x/100 , the CE is calculated

CE = −100 ln(1 − EU ) = −100 ln(1 − 0.6660 ) = 109.65 The RP is now calculated as RP = EMV − CE = 115 − 109.65 = 5.35

The same calculations are repeated, each time adding 150 M$ and 200 M$ to the payoffs. The calculations are shown in Table 4–4, showing constant risk of 5.349 M$. Similarly, Table 4–4 shows decreasing risk when the logarithmic utility function is used. The detailed calculations are shown in Excel, Table 4–4.xls on the CD.

Expected Utility Decision Criteria For all problems, the EMV for each alternative is first calculated. The alternative with the highest EMV is selected if the decision maker is riskneutral. If the decision maker is risk-averse or risk-seeker, then the investments should be checked to see if the alternatives are sensitive to changes in risk attitude. If the decision changes due to risk aversion, i.e., if the decision is sensitive to risk attitudes (risk-averse or risk-seeker), then risk attitude should be carefully modeled in the calculations. The incorporation of risk attitudes involves determining the utility function for the particular risk attitude and replacing all monetary payoffs in a decision tree or table by their corresponding utility values. The alternative with the highest EU or highest expected CE is selected. Both the highest EU and the highest CE would result in the same decision. However, since the CE is calculated in monetary values, it is easy to understand and explain.

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SPREADSHEET APPLICATIONS In this section, Excel is used to fit the mathematical equations (presented earlier in this chapter) to an elicited utility curve and calculate critical risk tolerance level. PrecisionTree is used to solve decision problems while incorporating risk attitudes.

Fitting the Utility Curve In this section, the mathematical equations presented earlier are fitted to the five points elicited in Figure 4–6 to determine the equation that best represents the utility curve. The SOLVER option of Excel is used to perform this task. This should be the standard way of selecting the particular mathematical equation rather than arbitrarily choosing the exponential equation as advocated in most literature on the subject. The following five utility points were generated in the section on Assessing the Utility Function. Monetary Utility

| –$250,000 | –$100,000 | $50,000 | 0.00 | 0.25 | 0.50

| $225,000 | $500,000 | 0.75 | 1.00

The following systematic procedure is used to fit the quadratic equation, U(x) = a + bx – cx2, to the data. All other equations are fitted, not shown here, to the same data as shown in Table 4–5 on the accompanied CD. Look at Table 4–5 on the CD to find out why the quadratic equation has been chosen and how the other equations represent the same utility curve.

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A

B

C

2

CE

Elicited

Predicted

3

(M$)

Utility

U(x) = a+bx-cx2

4

– 250

0.00

– 30.7500

945.5625

5

– 50

0.25

– 10.5000

115.5625

6

100

0.50

5.2500

22.5625

7

225

0.75

17.9375

295.4102

8

500

1.00

25.5000

600.2500

9

D

SSE

1,979.3477

10

0.5000

11

0.1000

12

0.0001

Table 4–5 Fitting a utility curve with mathematical equation

Step 1: Enter the elicited CEs in Cells A4:A8 with the corresponding utilities in Cells B4:B8. The data is arranged as shown in Table 4–5. Step 2: Enter arbitrary values of a, b, and c in Cells C10, C11, and C12, respectively. These (a, b, and c) are the three constants of the quadratic equation. Step 3: Input the quadratic formula: =C$10 + C$11*A4 – C$12*A4^2 in Cell C4, to calculate the predicted utility. Copying this formula to the range C5 to C8 generates predicted utility for the remaining four points. Step 4: Enter the formula =(B4 – C4)^2 in Cell D4 to calculate the squared error for the first point. Copying this formula to Cells D5 to D8 computes the squared error for the other four points on the utility curve. In Column D, the squared errors for each point on the utility curve are calculated by squaring the difference between the actual and predicted utility.

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Incorporating Attitudes Toward Risk

Step 5: Enter the formula =SUM(D4:D8) in Cell D9 to calculate the sum of squared errors of all five points. Step 6: Use SOLVER to find the constants a, b, and c that would best fit the curve, i.e. choose the values of a, b, and c that will minimize the sum of squared errors in Cell D9. The following steps are followed for using SOLVER. a. Click on the Tools in the Excel toolbar and then click on Solver in the dropdown menu. If the Solver command is not on the Tools menu, install the Solver add-in. b. Enter the cell reference of the target cell $D$9 in the Set Target Cell box, c. Click on Min, to have the value of the target cell as small ( the objective is to minimize the sum of squared error) as possible. d. Enter the reference for each adjustable cell: $C$10:$C$12 in the By Changing Cells box. e. Click on Solve. As soon as this is done, iterations start and Cell D9, Cells C4 to C8, and Cells C10 to C12 will be revised with the final numbers accordingly. The final equation fitted to the utility curve is now U(x) = 0.4278 + 0.0015x – 0x2 The final table will look like the one shown in Table 4–6. The fitted data is plotted on the actual data as shown in Figure 4–10, which shows a perfect fit. In order to select the equation that best represents the elicited utility curve, the previous procedure is repeated for all the equations, and the equation that gives the lowest SSE is chosen for further calculations.

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A

B

C

2

CE

Elicited

Predicted

3

(M$)

Utility

U(x) = a+bx-cx2

4

– 250

0.00

– 0.0063

0.0000

5

– 50

0.25

0.2658

0.0003

6

100

0.50

0.5031

0.0000

7

225

0.75

0.7359

0.0002

8

500

1.00

1.0060

0.0000

9

SSE

0.0005

10

0.4279

11

0.0015

12

0.0000

Table 4–6 Fitting a utility curve with mathematical equation

Fig. 4–10 Illustration of the actual and fitted utility curves

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Incorporating Attitudes Toward Risk

Finding the Certainty Equivalent As presented in the preceding pages, certainty equivalent (CE) can either be approximated by using Equation (4.11), or it can be back calculated from the inverse of the utility equation representing the decision maker’s risk preference. The inverse of one of the exponential utility equations and one of the logarithmic equations was derived and presented in Equation (4.9) and Equation (4.10). However, the inverse of some equations may not be easily derived, or it may require iterative solution. In addition, once the EU of the desired alternative (maximum EU) is arrived at, it will be desirable to determine the corresponding monetary value of the EU. The Goal Seek option of Excel can be used to perform this function. The following steps are followed to calculate the CE of EU = 0.6660 in Table 4–4 using exponential utility equation of the type 1 – e–x/100 . Step 1: Enter the EU for which the CE be calculated in Cell A2. Step 2: Input a trial value of the trial CE is in Cell B2. Step 3: Enter the utility equation used to calculate the EU in Cell C2, while replacing the x in the equation by the Cell reference B2. In this case, it is = 1 – exp ( –B2 / 100). The cells will look like this.

A

B

C

1

EU

CE

U(x)

2

0.6660

50.0

0.3935

Step 4: Click on the Tools on the Excel toolbar and then click on Goal Seek in the dropdown menu.

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Step 5: Specify (a) the Set cell as C2, (b) input 0.6660 in the To value window, and (c) specify the By changing cell as $B$2 in the Goal Seek dropdown menu (following Step 4). Step 6: Click on OK. This will revise Cells B2 and C2 by calculating the CE and U(x), respectively. The final output will look like this.

A

B

C

1

EU

CE

U(x)

2

0.6660

109.661

0.6660

In the same way, CE for Project B (Table 4–2) with EU of 0.1971 is calculated to be 21.877 M$ (R = 100). This payoff is less than the EMV of 23 M$ (Table 4–2). Since the utility function exhibits risk aversion, it is natural the project is valued at an amount less than its expected NPV (CEB = 21.877 < EMVB = 23).

Critical Risk Tolerance Critical risk tolerance is the risk tolerance at which the EU of the alternatives under consideration are equal. For example, in Example 4–3 (Table 4–2) the EU of Project A and Project B was calculated. Project B was selected because EUA = 0.3422 > EUB = 0.1971 at R = 100 and using exponential utility equation of the type 1 – e–x/R . The same problem was solved in Example 4–1, using R = 35. In Example 4–1, Project B was preferred over Project A (EUB = 0.4298 > EUA = 0.2915). This shows the risk tolerance at which the decision maker will be indifferent to the two projects lies between R = 35 and R = 100. Therefore, the critical risk tolerance for this example is the R at which the EUA = EUB. The critical risk tolerance can be used for sensitivity analysis (whether the decision maker’s risk tolerance is above or below the critical risk tolerance level). If the decision maker’s risk tolerance is above the

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Incorporating Attitudes Toward Risk

critical value then she should go for the riskier project; if it is less, then she should select the less risky project. The critical risk tolerance for the two projects is calculated as 43.78. Therefore, Project A is selected if the decision maker’s R > 43.78 and Project B is selected if R < 43.78. The simplest way to determine the critical risk tolerance is to use computer in searching for the R-value that will make the EU of the two projects equal. Once again, the SOLVER or GOAL SEEK options of Excel can be used to achieve this objective. The following steps and SOLVER are used here. Step 1: Input the outcome probability of Project A in Cells A5 and A6 and the corresponding payoffs in Cells B5 and B6. Similarly, input the outcome probability of Project B in Cells D5 and D6 with the corresponding payoffs in Cells E5 and E6. The data is arranged as shown in Table 4–7. Step 2: Enter a trial value of critical risk tolerance in Cell B9.

A 2

B

C

D

Project A

3

NPV

4 Probability

(M$)

E

F

Project B Utility -x/R

U(x) = 1-e

Probability

NPV

Utility

(M$)

U(x) = 1-e

-x/R

5

0.80

80

0.8329

0.80

30

0.4960

6

0.20

– 40

– 1.4935

0.20

–5

– 0.1210

56

0.3726

23

0.3726

7 EMV & EU 8 9 Critical RT 10 SSE

43.78 1.722E – 19

Table 4–7 Calculation of critical risk tolerance

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Step 3: Input equation = 1 – exp(–B5 / B$9) in Cell C5. Copy Cell C5 to Cell C6. Similarly, input equation: = 1 – exp(–E5 / B$9) in Cell F5 and copy it to Cell F6. Step 4: Input equation: =SUMPRODUCT(A5:A6,B5:B6) in Cell B7 and equation: =SUMPRODUCT(A5:A6,C5:C6) in Cell C7. Note the EMV and EU of Project A are calculated in Cells B7 and C7, respectively. Similarly, input equation: =SUMPRODUCT(D5:D6,E5:E6) in Cell E7 and equation: =SUMPRODUCT(D5:D6,F5,F6) in Cell F7. Step 5: Enter equation =(C7 – F7)^2 in Cell B10. This calculates the critical risk tolerance by minimizing the sum of squared errors between the EUA and EUB. Step 6: Use SOLVER to find the R that would make EUA = EUB. The following steps are followed for using the SOLVER. a. Click on the Tools in the Excel toolbar and then click on Solver in the dropdown menu. If the Solver command is not on the Tools menu, the Solver add-in should be installed. b. Enter the cell reference of the target cell $B$10 in the Set Target Cell box. c. Click on Min to have the value of the target cell as small as possible. d. Enter $B$9 in the By Changing Cells box. e. Click on Solve. As soon as this is done, iterations start and Cell C7, Cell F7, Cell B9, and Cell B10 will be revised with the final numbers accordingly. As shown in Table 4–7, the EU of Project A and B are equal at R = 43.78.

PrecisionTree and Utility Functions PrecisionTree was introduced in Chapter 3 for solving decision trees. However, in Chapter 3 it was used for decision analysis based on expected value concepts. In this chapter, PrecisionTree is used again to solve problems based on the concepts of CE and EU theory. PrecisionTree has

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Incorporating Attitudes Toward Risk

made it simple to incorporate CE or utility in decision trees. The decision trees developed in Chapter 3 can be easily converted to expected value, CE or EU basis (desired type of analysis can be easily switched from one version to the other and vice versa). PrecisionTree includes built-in exponential (1 – e–x/R ) and logarithmic (ln + R) utility functions. -defined utility functions can be added using Excel’s built-in programming language, VBA (Visual Basic for Applications). Once a utility function is selected, the monetary payoffs of the decision tree are automatically converted to the desired option (utility or CE). Optimum paths in a decision tree are selected based on whether EU or CE is opted for. The decision tree is first developed in exactly the same way as shown in Chapter 3. To apply a utility function to a decision tree’s calculations, the following steps are followed.

1. Click on the Branch name to open the Tree Settings dialog box as shown in Figure 4–11.

Fig. 4–11 PrecisionTree’s Tree Settings dialog box

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Project Economics and Decision Analysis Volume II: Probabilistic Models

2. Change the settings as desired, for example a. Click the check box Use Utility Function. Leaving this box blank will make the PrecisionTree perform expected value calculations. When we click on it, a tick mark will appear in the box. b. Select Function (exponential or logarithmic). c. Input the risk tolerance R. d. Select Display (EU or CE). e. Click on OK to start calculations. PrecisionTree allows defining a different utility function for every chance node in the tree. The utility function of any node can be changed at any time during the modeling process. By specifying R = 0 or by setting decision model to Expected Value will make PrecisionTree assume risk neutral decision behavior, thus it bases its decisions accordingly.

Example 4–5 Using PrecisionTree, construct a decision tree for the data in Example 4–3 (Table 4–3). As a first , calculations are based on expected value criteria. The same tree is now used with certainty equivalent criteria. Assume exponential utility function and R = 100. Solution: The decision tree with monetary payoffs is shown in Figure 4–12. EMV of $15 is calculated for both projects. The decision tree with certainty equivalents is shown in Figure 4–13. Based on certainty equivalent criteria, Project B is selected. The same tree can be used with EU criteria.

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Incorporating Attitudes Toward Risk

Fig. 4–12 Decision tree for Example 4–5, showing expected value calculations

Fig. 4–13 Decision tree for Example 4–5, showing certainty equivalent calculations

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Project Economics and Decision Analysis Volume II: Probabilistic Models

QUESTIONS and PROBLEMS 4.1

Briefly describe various risk attitudes. Why is it important to incorporate risk attitudes in investment decision making?

4.2

What is the difference between expected value and EU?

4.3

What is meant by (a) CE, (b) RP, and (c) risk tolerance?

4.4

Do you agree with the statement “the more wealth one has, the easier it is to take larger risks”?

4.5

Explain the concept of RA, its measure, and how it relates to utility function.

4.6

Is assessing a utility function of an investor a subjective or objective process?

4.7

A decision maker’s risk tolerance is $1300 and his risk preference can be modeled with an exponential utility function of the type U(x) = 1 – e–x/R . a. What will be U($1050), U($,750), U(0), and U(–$1200). b. Calculate the EU for the payoffs in 4.7a if their probability distribution is 0.35, 0.30, 0.20, and 0.15. c. Calculate the exact and approximate CEs of the investment. For each of these, calculate the RP. d. What will be the approximate CE for investment if its expected value is $2575 and its standard deviation is $250?

4.8

280

The best to worst monetary returns for a project are $50,000, $35,000, $15,000, and –$12,000. If we assign a utility of 1 to $50,000 and a utility of 0 to –$12,000, assess the utilities of the remaining two monetary returns.

Incorporating Attitudes Toward Risk

4.9

A utility function for an investor is given as follows. Payoff, MM$

Utility

12.0

1.00

5.8

0.75

3.5

0.60

2.2

0.45

0.0

0.00

a. Graph the utility function. Based on this graph, how would you classify the investor’s risk preference? b. Fit an exponential utility function of the type 1 – e–x/R to this risk attitude. What is the value of risk tolerance? c. Also fit logarithmic and quadratic utility functions to this risk attitude. d. For each of these utility functions as fitted, which alternative is preferable and why? e. Many analysts claim there is insignificant difference between the exponential utility function and other utility functions; therefore, exponential utility function can be used in all cases. Do you agree with this claim? 4.10 Rework Problem 3.10 (Chapter 3) if the investor’s risk preference can be modeled by an exponential utility function 1 – e–x/R when R = 50 M$. Calculate the EU, RP and CE for each investment. 4.11 Rework Problem 4.10 to calculate the critical risk tolerance 4.12 Rework Problem 4.10 using PrecisionTree. 4.13 Rework Problem 3.14 (Chapter 3) if the investor’s risk preference can be modeled by an exponential utility function 1 – e–x/R when R = 50 M$. Does your answer change if the R value is increased to R = 75 M$?

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REFERENCES 1

2

3

4

5

6

7

8

282

Walls, Michael, R., “Corporate Risk Tolerance and Capital Allocation: A Practical Approach to Implementing an Exploration Risk Policy,” Journal of Petroleum Technology, SPE, April 1995, p. 307–311. Lerche, I. and MacKay, J. A., “Energy Exploration & Exploitation,” Multi-Science Publishing Co. Ltd., 107 High Street, Brentwood, Essex CM14 4RX, England, Volume 14, No.2 1996. Arps, J. J. and Arps, J. L, “Prudent Risk-Taking,” Journal of Petroleum Technology, SPE, v. 26, 1974, p. 711–715. Newendorp, P. D., Decision Analysis for Petroleum Exploration, PennWell Books, 1421 South Sheridan Road, Tulsa, OK 74101, USA, 1975. —— Newendorp, P. D. and Schuyler, J., Decision Analysis for Petroleum Exploration, 2nd Edition, Planning Press, 15492 East Chenango Avenue, Aurora, CO 80015, USA, 2000. Goodwin, P. and Wright, G., Decision Analysis for Management Judgment, John Wiley & Sons Ltd., Baffins Lane, Chichester, West Sussex, PO 19 1UD, England, 1991. Cozzolino, J., “A Simplified Utility Framework for the Analysis of Financial Risk,” SPE paper 6359 presented at the SPE Hydrocarbon Economics and Evaluation Symposium, Dallas, Texas, Feb. 20–22, 1977. Howard, R. A., “Decision Analysis: Practice and Promise,” Management Science, 34, No. 6, 679, 1988. Walls, M. R. and Dyer, J., “Risk Propensity and Firm Performance: A Study of the Petroleum Exploration Industry,” paper 92S–113 presented at the Operations Research Society of American Institute of Management Science t National Meeting, San Francisco, Nov. 2, 1992.

Incorporating Attitudes Toward Risk

9

10

11

12

Clemen, R. T., Making Hard Decisions: Introduction to Decision Analysis, 2nd Edition, Wadsworth Publishing Co., 10 Davis Drive, Belmont, CA 94002, USA, 1996. Pratt, J., “Risk Aversion in the Small and in the Large.” Econometrica, 32, 1964, pp. 122–136. McNamee, P. and Celona, J., Decision Analysis for the Professional with Supertree, Scientific Press, Redwood City, CA, USA, 1987. Samson, D., Managerial Decision Analysis, Richard D. Irwin, Inc., 1988.

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FIVE

Determining Venture Participation

This chapter presents further application of the concepts presented in Chapters 3 and 4. In Chapter 4, the investor’s risk attitude was incorporated in investment decision making. The following two points were highlighted. •

•

When the estimated possible loss is high, to the point where a significant fraction of the capital asset value of the investor is at risk, then most investors either a. Downgrade the investment because of the total risk threshold that can be sustained, or b. Share risk by taking less than 100% stake in the investment. This is done to limit potential losses to values below what the investor considers harmful to its fiscal health. The amount of money risked on any one venture should not exceed the amount that would increase the risk of gambler’s ruin beyond acceptable limits. Gambler’s ruin is a situation in which a risk-taker with limited funds goes broke through a continuous string of failures that exhaust her available funds.

Project Economics and Decision Analysis Volume II: Probabilistic Models

Based on the these, it makes good sense for investors to also evaluate the decisions associated with what fraction of their budget should be committed to a given investment opportunity when many opportunities are available. This leads to deciding on the optimum participation factor in each opportunity. Because of high risk involved in petroleum exploration, these sorts of decisions are of paramount importance if the investor or corporation would desire to stay in business and be profitable. The participation factor depends upon the interrelation of the following: • • • • •

Venture profitability, the financial losses of each failure (dry hole) and the gains of each success, Total risk investment, the amount of capital that can be spent in the game, Aversion to risk (risk preference of the investor), Probability of success or failure on the successive trials, and Available risk investment funds or venture capital required.

Many concepts in this chapter are adopted from the excellent work of Cozzolino, Lerche, and MacKay. The book by Lerche and MacKay, Economic Risk in Hydrocarbon Exploration, is a valuable reference on the subject. It is highly recommended for further insight into the subject. Additional recommended sources are the articles by Greenwalt, Quick and Buck, and The Business of Petroleum Exploration by AAPG.1–5 This chapter includes (a) formal presentation of the concept of gambler’s ruin, (b) introduction of risk-adjusted value (RAV, same as CE discussed in Chapter 4) as a tool for determining venture participation, and (c) portfolio balancing. Two approaches (1) Cozzolino’s approach, based on exponential risk aversion and (2) Lerche and MacKay’s approach, based on hyperbolic risk aversion are presented for RAV calculations. Spreadsheets are provided on the accompanying CD-ROM to assist in applying the concepts presented in this chapter.

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GAMBLER’S RUIN The law of gambler’s ruin states there is a chance of going broke merely by normal runs of bad luck, regardless of the longer-term expectations. Gambler’s ruin is avoided by having sufficient capital to continue to participate in numerous ventures and ride out the run of bad lucks.6 Hedging against gambler’s ruin is to participate in as many ventures as practically and financially possible. If there is capital constraint, taking a proportionate interest in many ventures rather than full ownership in few ventures is a prudent course of action. In simple , don’t put all your eggs in one basket. Arps and Arps formulated the gambler’s ruin as (5.1)

v = (1− p)

C / Wx

where v = probability of gambler’s ruin (probability of exhausting the total capital by drilling C/x consecutive dry holes, the C/x must be the largest whole integer) p = probability of venture success (such as exploratory well) C = the total capital available x = total cost of each venture (drilling cost of each exploratory well) W = working interest in the venture, fraction

Example 5–1 Suppose $10 million is available for investment in oil and gas exploration. The probability (chance of success) of prospects to contain enough oil or gas to be considered as a commercial discovery is 15%. The cost of drilling each exploratory well is $1 million. What is the probability that the initial capital will be consumed without a single success (gambler’s ruin)? Assume 100% working interest.

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Solution: Since the total available capital is 10 MM$ and each exploratory well costs one million dollars, it follows that the maximum number of consecutive failures an investor could withstand before reaching gambler’s ruin is $10/$1 = 10 wells. Using Equation (5.1) with C = 10, x = 1 and p = 0.15, the probability of gambler’s ruin is v = (1 – 0.15) 10 / 1x1 100 = 19.69% Therefore, the probability that the initial capital will be consumed without a single success is about 19.7%. However, if participation (working interest) in the game is reduced from 100% to 50% (i.e. drilling costs of $500,000 each well), it will allow participation in 20 exploratory prospects instead of 10. As shown below, this will significantly reduce the probability of gambler’s ruin. v = (1 – 0.15) 10 / 0.5x1 100 = 3.88% By rearranging Equation (5.1) in of working interest, we get (5.2)

W =

C log (1− p) x log v

Example 5–2 Suppose $10 million is available for investment in oil and gas exploration. The probability (chance of success) of prospects to contain enough oil or gas to be considered as a commercial discovery is 15%. The cost of drilling each exploratory well is $1 million. Determine the maximum working interest that would allow for less than 5% chance of program failure.

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Solution: Using Equation (5.2), the working interest is

W =

10 log(1 − 0.15) × 100 = 54.25% 1.0 log(0.05)

Substituting the value of 0.5425 in Equation (5.1) gives the number of wells as 18.45. Since only a whole number of wells can be drilled, the 18.45 is rounded off to C/Wx = 19. This gives W = 10/19 or 52.63%. Substituting this value back in Equation (5.1) will give v = 0.0456.

WORKING INTEREST and RAV In this section, maximum working interest in a venture is derived by incorporating investors’ risk attitude into the computation. Cozzolino, Lerche, and others have made excellent contributions in presenting this concept. The only difference between the two approaches is the way each uses the dependence of risk aversion. Cozzolino’s approach is based on exponential dependence of risk aversion while Lerche and MacKay have incorporated hyperbolic dependence of risk aversion into the concept.

Based on Exponential Risk Aversion Cozzolino’s concept incorporates risk threshold, also referred to as risk tolerance, factors in estimating a RAV and fractional working interest in a project. His analysis incorporated exponential utility function of the type U(x) = e–x/R and the derived expected utility. Therefore, the requirement of such a model forces an exponential aversion utility function to be employed by any investor wishing to use this concept. The following equations are used to assess the relevant working interest (W) and RAV of a project. In Chapter 4, Equation (4.1) was given to calculate the expected utility of an investment. The same equation is used as follows but incorporates working interest in it. (5.3)

EU = pe

−WV / R

+ (1− p)e

WC / R

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where p W V C

= = = =

probability of success, fraction working interest or participation factor, fraction net present value (NPV) of the project, currency units investment (Cost) in currency units, equation is adjusted for negative cost (enter +ve C)

Based on Equation (5.3), Cozzolino proposed the following equation to calculate RAV.

[

RAV = − R ln (EU ) = − R ln pe−WV / R + (1− p)e WC / R

]

(5.4)

RAV is the same as the certainty equivalent (CE), Equation (4.9), discussed in Chapter 4. Equation (4.9) was used for exponential risk aversion equation of the type 1 – e –x / R. The only exception is the incorporation of working interest (W) in the equation. Any venture with positive RAV is accepted. Optimum Working Interest. The optimum working interest can be calculated for any given R, p, C, and V by using the following equation. (5.5)

Wopt =

⎡ pV ⎤ R ln ⎢ C + V ⎣ (1 − p )C ⎥⎦

Thus, replacing W by Wopt in Equation (5.4) gives the optimum riskadjusted value, i.e. RAVopt . Apparent Risk Tolerance. If a working interest in a venture is already committed, then to see if the corporate mandate of risk tolerance has overrisked or under-risked this particular venture, apparent risk tolerance is calculated. The apparent risk tolerance also enables one to assess the extent

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to which the particular working interest permits the apparent risk tolerance to be in reasonable accord with the corporate risk tolerance value. The apparent risk tolerance is given by the following equation. (5.6)

RA =

W (V + C ) ⎡ pV ⎤ ln ⎢ ⎥ ⎣ (1 − p )C⎦

Breakeven Working Interest. The breakeven working interest is calculated by changing the W in Equation (5.4) until a RAV of zero is obtained. The following equation is used to calculate the breakeven working interest. (5.7)

p − eWo V/ R + (1 − p) eWo (V + C) / R = 0

The equation can be easily solved using the Solver option of Excel. This can be done by following the steps. 1. Input p, V, C, and R in Cells C3, C4, C5, and C6, respectively. 2. Enter an estimated value of W in Cell B16. Solver uses this in the first iteration. 3. Enter Equation (5.7), =C3-EXP(B16*C4/C6)+(1-C3)*EXP (B16*(C4+C5)/C6), in Cell C16. 4. Click on Tools ➱ Solver. A drop-down Solver Parameters menu appears. Enter C16 in Set Target Cell, Click on Value of = 0, and enter B16 in By Changing Cells. Now Click on Solve. Solver performs iterations until the value in Cell C16 is zero while at the same time, final breakeven value of working interest is shown in Cell B16.

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Example 5–3 Using the concept of Cozzolino’s exponential risk aversion concept, calculate the (a) RAV, (b) optimum working interest and the corresponding RAV, (c) apparent risk tolerance for working interest of 20%, and (d) the breakeven working interest for the following data. W = 0.20,

p = 0.35,

V = 320 M$,

C = 50 M$,

and

R = 25 M$

Solution: (a) Using Equation (5.4), the RAV is ⎣⎡

RAV = −25 ln ⎡⎣0.35e −0.2×320 / 25 + (1 − 0.35)e 0.20×50 / 25

= −25 ln(0.35 × 0.077 + 0.65 × 1.492) = −25 × −0.00326 = 0.082 M$

(b) Using Equation (5.5), the optimum working interest is Wopt =

⎡ 0.35 × 320 ⎤ 25 ln ⎢ 320 + 50 ⎣ (1 − 0.35)50⎥⎦

= 0.0676 ln(3.446) = 0.0836 or 8.36% ⎣⎡

RAVopt = −25 ln ⎡⎣0.35e −0.0836×320 / 25 + (1 − 0.35)e 0.0836×50 / 25

= −25 ln(0.35 × 0.343 + 0.65 × 1.182) = −25 × −0.118 = 2.960 M$

(c) Using Equation (5.6), the apparent risk tolerance at 20% working interest is

RA =

0.20(320 + 50 ) 74 74 = = ⎡ 0.35 × 320 ⎤ ln (112 / 32.50) 1.237 ln ⎢ ⎥ ⎣ (1 − 0.35)× 50 ⎦

= 59.82 M$

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Determining Venture Participation

Figure 5–1 shows a plot of estimated RAV values (in MM$) versus a fixed working interest and at various risk tolerance levels from R = 10 to R =100. It can be seen from Figure 5–1 that as R increases, a greater degree of risk is considered acceptable. Therefore, the range of working interest over which a positive RAV can be obtained increases as the risk tolerance increases.

Fig. 5–1 Cozzolino’s RAV in MM$ versus working interest for various risk tolerance (R = 10 to R = 100) values

Based on Hyperbolic Risk Aversion Lerche and MacKay investigated the RAV rule introduced by Cozzolino in of hyperbolic utility function of the form U(x) = 1 – tanh(x / R). The reason for investigating this utility function versus the Cozzolino’s exponential utility version is there is greater stability in the management of high-loss scenarios than there is with the exponential utility function. They derived the following equations, equivalent to the equations by Cozzolino.

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The expected utility for hyperbolic utility function is given by the following equation. (5.8)

⎛ WC ⎞ ⎛WV ⎞ U ( x) = 1 + (1 − p) tanh ⎜ ⎟ − p tanh ⎜ ⎟ ⎝ R ⎠ ⎝ R ⎠ The RAV based on hyperbolic utility function is then given by (5.9)

⎧ ⎛ WV ⎞ ⎫ ⎛ WC ⎞ RAV = − R ln ⎨1 + (1 − p) tanh ⎜ ⎟⎬ ⎟ − p tanh ⎜ R ⎠ ⎝ R ⎠⎭ ⎝ ⎩

The optimum working interest, based on hyperbolic utility function, is iteratively obtained from the following equation. Again, the Solver function of Excel can be used to solve this type of equation. (5.10)

⎛ Wopt C ⎞ ⎛ (1 − p )C ⎞ ⎟⎟ − ⎜ ⎟ cosh ⎜⎜ ⎝ R ⎠ ⎝ pV ⎠

1/ 2

⎛ Wopt V cosh ⎜⎜ ⎝ R

⎞ ⎟⎟ = 0 ⎠

Once the Wopt is calculated from Equation (5.10), the RAVopt can be calculated using Equation (5.9) while replacing W by Wopt . The breakeven working interest is iteratively calculated by the following equation. (5.11)

Wo C ⎞ ⎛W V ⎞ ⎟ − p tanh ⎜ o ⎟ = 0 ⎝ R ⎠ ⎝ R ⎠

(1 − p ) tanh ⎛⎜

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Determining Venture Participation

Figure 5–2 shows a plot of estimated RAV values (in MM$) versus a fixed working interest and at various risk tolerance levels from R = 10 to R =100. It can be seen from Figure 5–2 that a similar change in RAV occurs with change in R and working interest, as displayed in Figure 5–1 for the exponential case. At a fixed working interest, the exponential risk aversion formula gives a conservative value of RAV as the risk tolerance increases. Therefore, a slightly higher R is required by the exponential formula to give the same RAV as obtained by the hyperbolic formula.

Fig. 5–2 Hyperbolic RAV in MM$ versus working interest for various risk tolerance (R = 10 to R = 100) values

The calculations of Wopt, RAVopt, and apparent risk tolerance based on the exponential and hyperbolic utility functions are shown in Table 5–1. Table 5–2 shows the calculation of RAV at various risk tolerance levels and fixed working interest. A plot of the RAV calculated by using the Cozzolino and Hyperbolic risk aversion formulae at variable working interests, for the data in Table 5–1, is shown in Figure 5–3.

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3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

A Probability of Success Net present Value, MM$ Cost, MM$ Risk Tolerance, MM$ Expected Value, MM$

Working Interest 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

B

C 0.20 90.00 10.00 20.00 10.00 @100% WI

Exponential Expected Cozzolino RAV, MM$ Utility 1.0000 0.9685 0.9655 0.9813 1.0102 1.0483 1.0933 1.1438 1.1989 1.2581 1.3212

0.0000 0.6393 0.7032 0.3772 –0.2026 –0.9434 –1.7845 –2.6875 –3.6285 –4.5926 –5.5708

E

Hyperbolic Expected Utility RAV, MM$ 1.0000 0.9556 0.9365 0.9443 0.9685 1.0003 1.0348 1.0698 1.1043 1.1376 1.1697

Selected Working Expected Value, MM$ Expected Utility RAV, MM$ Apparent R, MM$

50.0% 5.00 1.0483 –0.943 61.66

5.00 1.0003 –0.0066

Optimum Working Expected value, MM$ Expected Utility RAV Optimum, MM$

16.2% 1.622 0.964 0.734

21.6% 2.156 0.936 1.320

Table 5–1 Risk-adjusted value (RAV) and working interest

296

D

0.0000 0.9086 1.3127 1.1463 0.6393 –0.0066 –0.6851 –1.3501 –1.9835 –2.5791 –3.1357

0.0000

Determining Venture Participation

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

A Probability of Success Net Present Value, MM$ Cost, MM$ Working Interest Expected Value, MM$

Risk Tolerance, MM$ 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00

B 0.20 90.00 10.00 100.0% 10.00

C

D

E

@ 100%

Exponential Hyperbolic Expected Cozzolino Expected RAV, MM$ Utility RAV, MM$ Utility 2.1747 -7.7687 1.4093 -3.4308 1.3212 -5.5708 1.1697 -3.1357 1.1264 -3.5721 1.0582 -1.6971 1.0483 -1.8868 1.0003 -0.0132 1.0102 -0.5065 0.9685 1.5983 0.9897 0.6203 0.9511 3.0093 0.9781 1.5470 0.9419 4.1888 0.9714 2.3173 0.9376 5.1527 0.9676 2.9651 0.9362 5.9328 0.9655 3.5160 0.9365 6.5633 0.9644 3.9894 0.9377 7.0744

Selected Risk Tolerance Expected Value, MM$ Expected Utility RAV, MM$

20.0 20.00 1.321 -5.571

20.0 1.170 -3.136

Optimum Working Interest Expected Value, MM$ Expected Utility RAV Optimum, MM$

16.2% 1.622 0.964 0.734

21.6% 2.156 0.936 1.320

0.0000

Table 5–2 Risk-adjusted value at various risk tolerances

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Fig. 5–3 Comparison of Cozzolino and Hyperbolic RAVs at varied WI

Parabolic Approximation to Cozzolino’s RAV A useful approximation to the Cozzolino RAV formula is given by the parabolic RAV formula. The parabolic RAV formula is simpler to use and yields essentially identical RAVopt and Wopt at low working interests as does the more complex exponential formulae presented in the previous sections. At higher working interest values, the parabolic RAV formula shows more risk aversion than that shown by the exponential formulae. The following equations can be used for parabolic approximation to Cozzolino’s RAV formulae. (5.12)

Wopt =

R Eυ 2 (5.13)

⎡ 0.5Wopt υ E ⎤ 0.5 R RAVopt = Wopt × E ⎢1 − ⎥= 2 R υ ⎣ ⎦ 2

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Determining Venture Participation

where the expected value E = pV – (1 – p)C (C is taken as positive), and the volatility is given by (the cost C is taken as positive) (5.14)

υ=

pV 2 + (1 − p )C 2 s −1 = 2 E E

Equation (5.14) is the same as the coefficient of variation given by Equation (2.15) in Chapter 2, and it is the inverse of Performance Index given by Equation (3.8) in Chapter 3. The volatility quantifies the stability of the estimated expected monetary value (EMV) in this context, relative to the fluctuations about the mean. A small volatility ( < 1) implies there is little uncertainty in the expected value. A large volatility ( > 1) implies a considerable uncertainty in the expected value.2

Example 5–4 Using the data of Example 5–3 and the parabolic approximation to the Cozzolino’s exponential RAV formula, calculate the optimum working interest and the corresponding risk-adjusted value. Solution: The expected monetary value for the data is

E = 0.35(320 )− (1 − 0.35)(50) = 79.5 M$ Using Equation (5.14), the volatility about the expected value is 0.35(320) + (1 − 0.35)(50) −1 79.5 2 2

υ=

=

2

35,840 + 1,625 −1 6320.25

= 2.22 299

Project Economics and Decision Analysis Volume II: Probabilistic Models

Using Equation (5.12), the optimum working interest is Wopt =

25 = 0.0638 or 6.38% 79.5 × 2.22 2

Using Equation (5.13), the optimum RAV is ⎛ 0.5 × 0.0638 × 2.222 × 79.5 ⎞ ⎟⎟ RAVopt = 0.0638 × 79.5 ⎜⎜1 − 25 ⎠ ⎝

= 0.0638 × 79.5(1 − 0.50 ) = 2.536 M$

Modified Risk-Adjusted Value All equations (Cozzolino, hyperbolic, and parabolic) presented in the preceding sections fail to adequately reflect high gain situations if these situations do exist in reality. As shown in Figure 5–4, as the NPV of the project increases (keeping all other values constant) the optimum working interest decreases. This, however, is contrary to the concept that if gains are higher (relative to the downside risk) then one should go for higher working interest. On the other hand, the formula is saying take less working interest. However, in you heart you know that you would like more. This concept, however, is adequately represented in the normal gain situations. In normal gain situations, the optimum working interest should increase with an increase in gain (all other variables held constant), which implies that more profit is better so take a higher working interest.

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Determining Venture Participation

Fig. 5–4 Optimum working interest versus NPV (using Cozzolino’s formula)

The reason for this discrepancy is that risk aversion, in part, is trying to minimize uncertainty. Uncertainty is volatility (variance) and large ranges between gain (revenue) and/or cost increases this volatility. Volatility is not considered good by the risk aversion formulae. However, the risk aversion formulae do not recognize if the increase in volatility comes from more gain or more cost. If the volatility is due to more gain rather than more cost, then it is good and not bad. However, since the risk aversion formulae fail to recognize this factor, it reduces the investor’s participation even if the volatility is due to more gain. In such situations, the optimum working interest should be adjusted to the high gain situation. Usually when this situation occurs, it is more likely from an unrealistic assumption (projects are just not that good!). Therefore, before the adjustment is made, it should be confirmed the gain is realistic. The following steps should be followed in order to determine if the adjustment is required. 1. Input the probability of success, NPV, cost, and risk tolerance in Cells B3, B4, B5, and B6, respectively (as shown in Table 5–3).

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3 4 5 6 7 8

A Success Probability Net Present Value, MM$ Cost, MM$ Risk Tolerance Optimum Working Interest RAV at Wopt, MM$

B 20.0% 50 1 10 49.52% 1.54

C 20.0% 11.83 1 10 84.52% 0.57

Table 5–3 Modified RAV calculation

2. Enter =B3 in Cell C3. 3. Input a guessed value, say 10, for the NPV in Cell C4. 4. Enter =B5 in Cell C5 and =B6 in Cell C6. 5. Enter Equation (5.5) ➝ =B6/(B4+B5)*LN(B3*B4/((1-B3)*B5)) in Cell B7. Copy Cell B7 to Cell C7. 6. Enter Equation (5.4) ➝ =-B6*LN(B3*EXP(-B7*B4/B6)+(1B3)*EXP(B7*B5/B6)) in Cell B8. Copy Cell B8 to Cell C8. 7. Click on Tools ➯ Solver. A drop-down Solver Parameters menu appears. Enter the parameters as shown in Figure 5–5 and then Click on Solve. The Wopt is now changed to 84.52% as compared to the 49.52% and the NPV is changed from 50 MM$ to NPVmax of 11.83 MM$.

Fig. 5–5 Setting Solver parameters to solve for modified Wopt and RAV for data in Table 5–3 302

Determining Venture Participation

In Table 5–3, if the cost is changed to 4.3 MM$ and Solver is run again, the same Wopt and RAV for both the modified and non-modified cases are obtained. This means no adjustment would have been required if the cost had been equal to or greater than 4.3 MM$.

PORTFOLIO BALANCING and RAV The preceding sections of this chapter presented calculation techniques for determining venture participation using gambler’s ruin, Cozzolino’s exponential utility approach, and hyperbolic utility approach introduced by Lerche and MacKay. These methods allow calculating the optimum working interest in a venture based on probability of success, NPV, cost, and risk tolerance level. If the decision maker has a portfolio of N projects, Wopt for each project will be calculated in isolation. The cost exposure of the corporation taking the Wopt in each of these projects will then be Ctotal = Ni = 1 Ci (Wopt)i ; where Ci is the cost of each project, and (Wopt)i is the optimum working interest in each project. This situation, however, is applicable when there is no limit on the availability of funds. Since most corporations work under a certain budget allocation B (budget constrain), the Wi has to be determined so the total cost does not exceed the allocated budget. At the same time, an optimum RAV is achieved within the constrained budget value. In such cases, one cannot take Wopt in each opportunity. In this section, techniques used to optimize the total RAV from N opportunities while staying within the mandated corporate budget B (to balance the portfolio of opportunities) are presented. The classic method of resolving various optimum working interests is with a linear optimization scheme. The techniques presented here are based on the Cozzolino RAV formula and parabolic RAV formula.



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Portfolio Balancing—Cozzolino RAV Formula The following two equations are used to arrive at the working interest of each opportunity so a balanced portfolio within a constrained budget is achieved. (5.15)

⎡ p j ⎛ Vj ⎞⎤ ⎜ − H ⎟⎥ ⎢ ⎟⎥ N Cj ⎢ (1 − p j )⎜⎝ C j ⎠ B = R∑ ln ⎢ ⎥ ( ) + V C 1 H + j =1 j j ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ The H in Equation (5.15) is first determined using Solver. The value of H is then used in the following equation to calculate the working interest for each opportunity. (5.16)

Vj ⎡ pj × ⎢ (1 − p j ) C j − H R ln ⎢ Wj = (1 + H ) Vj + C j ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

The programming of Equations (5.15) and (5.16) are shown in the following section on Spreadsheet Applications. The application of this method is shown in Table 5–4.

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Determining Venture Participation

A Project

B NPV MM$

C C MM$

D R MM$

ps

A B

110 200

10 100

30 30

0.50 0.50

7

C

300

120

30

0.40

8

D

120

50

30

0.40

9

E

180

50

30

0.30

910

330

3 4 5 6

10 11

Wi x C

RAV

Wi

MM$

MM$

0.1836 0.1383

0.5509 0.0415

5.5090 4.1500

12.14 1.43

0.0365

0.0725

0.0181

2.1757

0.64

0.0829

0.0616

0.0370

1.8486

0.51

0.0566

0.0439

0.0263

1.3167

0.38

15.0000

15.11

12 13

Wopt

14 15

0.5995 0.0693

16 17 18 19 20 21

E

15.0000 H=

0.1931

Total Cost at Wopt =

24.28

Table 5–4 Portfolio balancing, Cozzolino formula

Portfolio Balancing—Parabolic RAV Formula The following equations can be used for obtaining the best working interests that will balance the portfolio of opportunities, using parabolic RAV formula. (5.17) N

∑CW i

H =

opt

−B

i =1

N

Ci2Wopt

i =1

Ei

∑

≥0

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The best working interest in each opportunity is then given by (5.18)

⎛ CH⎞ Wi = Wopt ⎜⎜1 − i ⎟⎟ Ei ⎠ ⎝

The Wopt for the parabolic approximation to the Cozzolino formula is given by Wopt = R / (2E) and the optimum RAV is given by Equation (5.13). The application of this method is shown in Table 5–5.

A

B

C

D

E

F

3 4

NPV

Cost

Project

MM$

MM$

6 7

A B

110.0

10.0

50.0%

30

50.00

200.0

100.0

50.0%

30

50.00

9.00

8

C

300.0

120.0

40.0%

30

48.00

18.38

5

p

s

R

Value

Volatility

MM$

MM$

v2 1.44

9

D

120.0

50.0

40.0%

30

18.00

21.41

10

E

180.0

50.0

30.0%

30

19.00

30.77

11 12

Total

910.0

330.0

13 14

Wopt

CWopt E

CWopt

185.00

Wi

Cost

RAV

MM$

MM$

15 16

0.4167 0.0667

0.833 13.333

4.167 6.667

0.4040 0.0464

4.0400 4.6403

10.407 1.513

17

0.0340

10.204

4.082

0.0211

2.5308

0.698

18

0.0779

10.813

3.893

0.0450

2.2494

0.576

19

0.0513

6.751

2.565

0.0308

1.5394

0.409

41.935

21.373

15.00

MM$

20 21 22 Budget 23 H =

13.604 Total Cost at Wopt =

0.152

24

Table 5–5 Portfolio balancing, parabolic RAV formula

306

G

Expected

21.37

MM$

Determining Venture Participation

SPREADSHEET APPLICATIONS In this section, the numerical algorithms are presented in the spreadsheet form to enable the interested s to quickly generate versions of the procedures for use in their own applications. Most of the equations presented in this section are programmed to generate the data given in the tables in this section. The details of each cell in the tables are given on the CD. SOLVER is used to perform the calculations requiring iterations. The Cells used with SOLVER are presented here for each table. Table 5–1. The calculations of optimum working interest (Wopt) and the risk-adjusted value (RAV), using the Cozzolino and Hyperbolic utility functions, are shown in this table. a. Click on Tools ➯ Solver b. Set Solver Parameters as (i) Set Target Cell to E30, (ii) Click on Value of: button and input value of 0, (iii) Input D30 in By Changing Cells, and (iv) Click on Solve to perform iterations and get the final value in Cell D30. Tables 5–2 and 5–3. In Table 5–1, the RAV profile at various working interests and a fixed value of risk tolerance was generated. In Tables 5–2 and 5–3, the RAV profile at various risk tolerances and a fixed value of working interest was generated. In addition to the RAV profile, other parameters are calculated. a. Click on Tools ➯ Solver b. Set Solver Parameters as (i) Set Target Cell to E29, (ii) Click on Value of: button and input value of 0, (iii) Input D29 in By Changing Cells, and (iv) Click on Solve to perform iterations and get the final value in Cell D29. Table 5–4. This table uses the Cozzolino RAV formulae to optimize venture participation in five investment opportunities under a constrained budget of 15 MM$. Equations (5.15) and (5.16) are used in Cells B14:B18 and Cells C14:C18, respectively. Equation (5.5) is used in Cells A14:A18. a. Cell B21 ➯ Input initial guess for the value of H.

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b. Click on Tools ➯ Solver . c. Set Solver Parameters as (i) Set Target Cell to B19, (ii) Click on Value of: button and input value of 15 (this is the budget constraint of 15 MM$), (iii) Input B21 in By Changing Cells, and (iv) Click on Solve to perform iterations and get the final value in Cell B21. The values in Cells B14 to B18 will change. d. Cell E21 ➯ In this cell the total cost is calculated based on the Optimum Working Interest participation. If there is no budget constraint then the total required cost would be this. Table 5–5. This table uses the parabolic approximation to the Cozzolino RAV formulae to optimize venture participation in five investment opportunities under a constrained budget of 15 MM$. Equation (5.14) is used in Cells G6:G10, Equation (5.12) is used in Cells A15:A19, Equation (5.17) is used in Cells B15:C19 and Cell B23, Equation (5.18) is used in Cells D15:D19, and Equation (5.13) is used in Cells F15:F19. Table 5–6. This table uses the Cozzolino RAV formulae to optimize venture participation in five investment opportunities under a constrained budget of 15 MM$, using linear optimization rather than using Equations (5.15) and (5.16). Equations (5.3) and (5.4) are used in Cells B10:F10.

A 3 Projects =>

B

C

D

E

F

G

E

TOTAL

A

B

C

D

4 Success Probability

50.0%

50.0%

40.0%

40.0%

30.0%

5 NPV at WI = 100%, MM$

110.0

200.0

300.0

120.0

180.0

6 Cost at WI = 100%, MM$

10.0

100.0

120.0

50.0

50.0

7 Risk Tolerance, MM$

30.0

30.0

30.0

30.0

30.0

4.15% 1.81%

8 9 Working Interest

3.70%

2.63%

12.14

1.43

0.64

0.51

0.38

15.11

12 EMV at WI, MM$

27.55

2.08

0.87

0.67

0.50

31.66

13 Cost at WI, MM$

5.51

4.15

2.17

1.85

1.31

15.00

10 RAV, Equation (5.4), MM$

55.11%

11

Table 5–6 Portfolio balancing using Cozzolino formulae and linear optimization 308

Determining Venture Participation

e. The objective here is to determine venture participation in each opportunity to optimize the RAV. Solver is used to do this. Click on Tools ➯ Solver . f. Set Solver Parameters as shown in Figure 5–6. Now Click on Solve to perform iterations and get the final values in Cells B9:F9, Cells B10:F10, Cells B12:F12, and Cells B13:F13. The result obtained, using linear optimization, is the same as shown in Table 5–4.

Fig. 5–6 Solver parameters for Table 5–6

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REFERENCES 1

2

3

4

5

6

310

Cozzolino, J. M., “A New Method for Measurement and Control of Exploration Risk,” SPE Paper 6632, March 1978. Lerche, Ian and MacKay, J. A., Economic Risk in Hydrocarbon Exploration, Academic Press, 525 B. Street, Suite 1900, San Diego, CA 92101, USA, 1999. Greenwalt, W. A., “Determining Venture Participation,” Journal of Petroleum Technology, Society of Petroleum Engineers, November 1981, pp. 2189–2195. Quick, A. N. and Buck, N. A., “Viability and Portfolio Analysis as Tools for Exploration Strategy Development,” Journal of Petroleum Technology, Society of Petroleum Engineers, April 1984, pp. 619–627. Steinmetz, R., The Business of Petroleum Exploration, American Association of Petroleum Geologists, Tulsa, Oklahoma, U.S.A. 74101, 1992. Arps, J. J. and Arps, J. L., “Prudent Risk-Taking,” Journal of Petroleum Technology, Society of Petroleum Engineers, July 1974, pp. 711–716.

chapter

SIX

Simulation in Decision Analysis

The simulation approach to project evaluation, which is known today as risk analysis, was originally developed by Hertz in 1964 in an article that has since become a Harvard Business Review reprint classic. The opening paragraph of this article conveys the importance of risk analysis as: Of all the decisions that business executives must make, none is more challenging—and none has received more attention—than choosing among alternative capital investment opportunities. What makes this kind of decision so demanding, of course, is not the problem of projecting return on investment under any given set of assumptions. The difficulty is in the assumptions and in their impact. Each assumption involves its own degree—often a high degree—of uncertainty; and, taken together, these combined uncertainties can multiply into a total uncertainty of critical proportions. This is where the element of risk enters, and it is in the evaluation of risk that the executive has been able to get little help from currently available tools and techniques.1

Project Economics and Decision Analysis Volume II: Probabilistic Models

Most Fortune 500 companies are adopting simulation as the method for doing capital budgeting and analyzing the introduction of new products etc. Wall Street uses simulation daily to price complex and exotic derivatives. Risk analysis is any form of analysis that studies and attempts to quantify risks associated with a particular uncertain problem under consideration, investments in particular. Other words and phrases associated with risk analysis are decision analysis, risk assessment, and risk management, etc. Decision analysis, in its broadest form, includes problem identification, specification of objectives and constraints, modeling, uncertainty analysis, and rules that lead to decisions. In the preceding chapters, expected value analysis using decision trees were presented. There, single value random variables with corresponding single value probability of occurrence were assumed. On the other hand, the expected value was also calculated into a single value as the probability weighted average outcome of the individual random variables. In simulation instead, the random variables are replaced by appropriate probability distributions rather than the single values. Any variable whose value cannot be predicted or set with certainty is called a random variable. In addition, the resulting expected value is a probability distribution rather than a single value. Therefore, simulation is another version of a complex decision tree analysis where the discrete outcomes are replaced by a quantum of uncertain outcomes and their associated probabilities. For example, the decision tree shown in Figure 6–1 replaces the decision tree of Figure 3–8 (Chapter 3).

312

Dry Hole

Probability

Simulation in Decision Analysis

50

60

70

80

Dry Hole Cost, M$

C1 Normal(150 50)

Producer

Probability

Drill

0

50

100

150

200

250

300

Probability

Net Present Value, M$

D1 0

50

100

150

200

250

Expected Monetary Value (EMV), M$

$0 Don’t Drill

Fig. 6–1 Chance nodes of decision tree represented by hypothetical probability distributions

Simulation is a quantitative technique used for evaluating alternative courses of action, based upon facts and assumptions, with a computerized mathematical model in order to represent actual decision making under conditions of uncertainty. In most instances, a simulation model is built to imitate a real-life situation. Often the simulation model can provide a decision maker with important information. The gist of the method involves the following. • • •

Constructing a model of complex real systems. Using the model via random sampling processes to obtain data or information. Using a deterministic model to combine the variables of the model at each sampling stage.

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• •

Repeating the process several times, and Analyzing the information output in order to make it possible to evaluate alternative courses of action.

Therefore, the main contribution of simulation is in carefully modeling and analyzing the combined effects of multiple sources of uncertainty to produce a composite of total uncertainty. The specific kind of simulation methodology addressed in this chapter is called the Monte Carlo simulation. The method offers a flexible and powerful tool to study complex problems. Monte Carlo simulation dwells on probabilistic processes and employs random variables to investigate the relationship between the models’ variables. It is easy to comprehend that if independent variables of a model are random variables, then the dependent variable derived from these independent variables also represent a random variable. The random sampling and probability distributions form the building blocks of simulation. Some of the commonly used probability distributions were discussed in Chapter 2. In this chapter, these probability distributions are used in simulation. A brief description of the overall simulation methodology is first discussed, followed by the application of spreadsheets in modeling simulation problems. Simulation software, called @RISK, by Palisade Corporation is introduced and extensively used for modeling various real-life problems.

DEG THE SIMULATION MODEL In this section, theoretical background to Monte Carlo simulation is discussed. Topics include: • • • • • • •

314

applications of simulation steps involved in building the model selecting probability distributions and parameters random sampling Latin hypercube sampling dependence of variables in simulations benefits of simulation

Simulation in Decision Analysis

Since simulations cannot be done without a computer, a section on “behind-the-scene calculations” is presented to show how computers perform the calculations to arrive at the final output(s). In the following section, Excel and @RISK are used to model various problems on computer.

Real Applications of Simulation There are many published applications of simulation in a wide variety of contexts. Some examples are listed as follows to illustrate the versatility of this technique. • •

• • • • • • • •

Simulating maintenance operation to determine the optimal size of repair crew. Simulating plant production operations to evaluate changes in operating practices and the capacity and configuration of the facilities and to determine the amount of in-process storage space that should be provided. Simulating large-scale distribution and inventory control systems to improve the design of these systems. Simulating the overall operation of an entire business firm to evaluate broad changes in the policies and operation of the firm. Using simulation to determine optimal maintenance policies. Using simulation to model equipment replacement decisions. Estimating probability distribution of project completion time and overall project management parameters. Modeling market share of new product and advertising effectiveness. Simulating stock prices, options, and hedging with futures. Choosing among several available investments. If the future cash flows are not known with certainty, then it is not clear how to choose between competing projects. When several investments are compared using simulation, it may become apparent some are riskier than others. The simulation enables decision makers to obtain the probability distributions and other statistics of the

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projects’ cash flows. These distributions and statistical data then assist in answering questions such as: a) Which of the available projects is the most risky? b) What is the probability a particular investment will yield at least a 20% return? c) What is the probability the NPV of an investment will be less than, say $10 million or between $5 and $10 million? •

Competitive bidding for contracts. Simulation is frequently used in determining appropriate bid value. Firms must often bid against competitors, whose bid values cannot be determined with certainty before the bids are opened to win a contract. Bidding too low may win the firm a contract but it may generate little profit. On the other hand, if the firm bids too high, it may not get the contract. Simulation assists in choosing the bid amount to maximize its expected profit.

In addition to many applications listed previously, typical oil industry specific applications include2–7 a) Field development program: estimate reserves, initial production rates, production decline, operating expenses, product prices, oil field equipment and infrastructure, and associated cash flows. b) Authority for Expenditure (AFE): preparation, estimate total drilling cost and times. c) Strategic plan: estimate reserves increases, NPV, and capital exposure. d) Assessment: of risk in workover evaluations. e) Bidding: on oil and gas lease acquisition.

Steps in Simulation Modeling The most arduous job in simulation modeling is to define the problem to be simulated, along with the inherent uncertainties, and translating it into a workable model on computer. In this section, the steps required to design a Monte Carlo simulation are discussed. The required steps correspond to the general steps required in any decision analysis problem as discussed in Chapter 1. 316

Simulation in Decision Analysis

1. Define the Problem: This step involves defining possible actions, uncertainties, and time horizon. Properly defining a problem and viewing it from the correct perspective is the most important step in any decision analysis problem. The correct perspective means appropriate breadth or scope of the problem should be identified in relation to the specific questions the model is supposed to answer. If the problem statement is broad, it is difficult to focus on relevant details. On the other hand, if the problem statement is too narrow in scope, it may result in a nice and easy solution to a portion of the broader problem. It may fail to address the broader (more critical) issue (resulting in suboptimal solution). For example, a company needs to increase its reserves base. From this broad definition, there follows a number of possibilities. Should the reserves be increased through exploration efforts or purchase of reserves? If it is through exploration, should it be in the United States or overseas? If it is overseas, which country will be a viable target? Should the company take a 100% stake in exploration or partner with another outfit? Similar questions can be asked about the purchase option. 2. Construct the Model: This step involves relating input variables to output variable. Many important choices have to be made at this stage regarding the input and output variables. The output variable serves as an objective function for the decision maker. For example, the objective function is to determine the cash flow of a certain project. To do this, an exponential decline curve are modeled with an initial rate, decline rate, economic limit, lease operating expenses, product price, and initial capital expenditure while incorporating their probability distributions to arrive at an expected production and cash-flow forecast of a project. However, this model addresses uncertainties only in the parameters mentioned. The question that must be asked is whether there is a need to include uncertainties of the parameters leading to the initial reserves calculations (such as porosity, water saturation, formation thickness, drainage area, and recovery factor, etc.). A flowchart of a typical cash-flow model is shown in Figure 6–2.

317

Probability

Probability

Probability

Project Economics and Decision Analysis Volume II: Probabilistic Models

10

12

14

16

18

20

7

9

11

13

15

17

19

1.2

1.3

1.4

1.5

0

1.6

26

32

38

44

Water Saturation (S w), percent

Probability

Probability 1.1

20

21

Porosity, percent

Probability

Formation Thickness, feet

50

100

150

200

250

20

25

Drainage Area, Acres

Oil Formation Volume Factor

30

35

40

45

50

Recovery Factor, percent

Probability

Probability

⎡ 7,758φ (1 − S w )⎤ NR = ⎢ ⎥ Ad hFR βo ⎣ ⎦

1500 2000 2500 3000 3500 4000 4500 5000

10

20

Initial Production Rate, Stb/month

30

40

50

Exponential Decline, percent

10

15

20

25

30

35

40

1050

Oil Price, $/Stb

Probability

Probability

Probability

4.00

1100

1150

1200

1250

1300

1350

3.00 2.00 1.00 0.00 4000

Initial Capital Investment, M$

Probability

Normal(485,60)

300

400

500

600

700

Net Present Value, M$

Fig. 6–2 Probability distributions for each variable feeding the NPV calculations 318

5000

6000

7000

Lease Operating Cost, $/month

8000

Simulation in Decision Analysis

3. Assess Input Variables: This step is normally broken down in two stages, namely: (a) preliminary sensitivity analysis of input variables and (b) assessment of probability distributions for the input variables. There may be some uncertain variables in the model, the uncertainty of which may not be detrimental to the output in comparison to all other factors. For example, we might discover the change in net cash flow will be insignificant if the fixed costs are changed from their lowest to the highest possible values. In such cases, it would be a waste of time and effort to elicit probability distribution for the fixed cost. Using the most likely value of this variable in the model will suffice. Therefore, it is recommended that sensitivity analyses be conducted at this stage. These analyses identify the critical input variables and those variables not requiring a probability distribution. Sensitivity analysis involves calculating the output variable by varying the value of each input variable. The following steps are required. a. Assume the values for each input variable, including the lowest, highest, and most likely. b. Calculate the output variable while keeping the first variable at its lowest value and the remaining variables at their most likely values. c. Repeat (b), this time keeping the first variable at its highest value and the remaining variables at their most likely values. d. Repeat (b) and (c) by varying, in turn, each of the other variables between their lowest and highest possible values while keeping the remaining variables at their most likely values. TopRank (another Palisade decision tool) can be used to perform sensitivity analysis. Once the critical variables are identified from sensitivity analysis, probability distributions are then elicited for these variables.

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4. Evaluate Probability Distributions: The probability distributions must describe the range of likely values for each parameter that the analyst believes to be its good representation. The probability distributions may be of standard form (such as normal or lognormal distributions), or they may have empirical forms such as rectangular, triangular, or more complicated shapes. This assessment can be done based on a. Fitting a theoretical distribution to historical data of the parameters from the same situation (say, reservoir) or similar situation the analyst believes is analogous to the one under evaluation. b. Experience of the analyst. c. Heuristic approach—a heuristic is a rule-of-thumb for making decisions the analyst believes might work well in some instances (for example, most analysts expect reserves to follow a lognormal distribution). The distributions, depending on the parameters, could be discrete distributions or continuous distributions. The parameters of the distribution (mean and standard deviation of normal distribution) may be chosen based on studies of the geological properties in the area or analogous areas or on personal and collective experience of other colleagues. Regardless of their source, the selected distributions and their respective parameters represent a codification of the expectations of the analyst. The probabilities distinguish between the likely and unlikely values. In the absence of specific knowledge about the form of distribution, it seems reasonable, in most cases, to assume normal or lognormal distribution especially when dealing with various geological distributions. Some properties must be bounded; for example, it is not possible to have negative reservoir thickness or percentage variables such as porosities or oil saturations outside a specific range (never outside 0% to 100%).8

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5. Perform Calculations: The Monte Carlo simulation involves repeated random sampling from the input distributions and subsequent calculation of a set of sample values for the output distribution. This process is repeated over several iterations. However, the statistics obtained from any simulation are estimates of the population parameters. Regardless of the number of iterations performed, the exact value of the population parameters will never be known. However, as the number of iterations increases, the probability an estimate of a population parameter is within a desired amount of the actual population value also increases. The analyst has to use his professional judgment in order to select adequate number of iterations. The number of iterations depends upon the accuracy required, complexity of the problem, and required computer time, etc. In most cases, a good idea of the output distribution can be achieved in hundreds or thousands of trials. In simulation analysis, the roll-forward of the decision tree is repetitively simulated in contrast to rolling it back. A data bank of the simulation trial outcomes is generated during simulation that provides a useful probabilistic description of the consequences of a decision strategy.9 6. Interpret the Output: Simulation runs generate a considerable amount of useful output information. This information has to be critically examined by the analyst, i.e., the assumptions made about model structure and input variables are critically analyzed in light of the output distribution achieved. For example, the analyst may want to assess the sensitivity of the output distribution to changes in any of the input distributions or the form of the model. Such sensitivity analysis requires rerunning the model with different assumptions and assessments, and then comparing the outputs with those of the base case. If such changes have minor effects on the former simulation runs, then the original model can be assumed adequate.

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7. Analyze the Simulated Alternatives: Most decision analysis problems involve making choices between alternatives. Therefore, all the alternatives under consideration should be simulated in a consistent manner to generate individual distributions and statistical information. The output probability distributions are then used to provide insights about the alternative and to guide choice. The techniques such as stochastic dominance, presented in Chapter 3, are used to rank alternatives.

Random Sampling Methods A simulation of any system or process, composed of inherent random components, requires a method of generating or obtaining numbers that are random in some sense. These random numbers are then transformed to obtain corresponding random variables from other distributions (such as normal, lognormal, triangular, exponential, etc.). The transformed random numbers are commonly referred to as random variates. A simulation with any random aspects at all must involve sampling or generating random variates from probability distributions. Generating a random variate refers to the activity of obtaining an observation on a random variable from the desired distribution. Thus, the random numbers and probability distributions form the building blocks of simulation.10 Random numbers and different methods of generating random variates are discussed in this section. Random Numbers. Since random numbers form the building blocks of any simulation model, they are required to obtain random observations (random variates) from probability distributions. Spinning disks, electronic randomizers, and computers are used to generate random numbers. Several tables have been generated in this way, and most books on statistics contain a table of random numbers as shown in Table 6–1. On the other hand, most spreadsheet and simulation softwares have a built-in capability of generating random numbers.

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Simulation in Decision Analysis

0.03134 0.39800 0.40564 0.55144

0.02472 0.76542 0.53449 0.95666

0.09009 0.30939 0.67990 0.06117

0.33071 0.92171 0.12977 0.66010

0.21537 0.53607 0.10273 0.97143

0.92353 0.05013 0.93055 0.30229

0.86184 0.12585 0.70737 0.58097

0.84805 0.05650 0.67381 0.27832

0.35013 0.03471 0.16286 0.67560

0.59619 0.02157 0.51379 0.88957

0.89659 0.73789 0.30397 0.00526

0.18518 0.24870 0.96631 0.13265

0.12432 0.07570 0.87368 0.11773

0.87536 0.35596 0.86953 0.81101

0.48047 0.60270 0.76924 0.93245

0.24868 0.91441 0.09078 0.78502

0.84931 0.27316 0.90977 0.04384

0.61568 0.59190 0.39769 0.43446

0.13443 0.19681 0.23976 0.60011

0.67273 0.13410 0.57713 0.23851

0.65334 0.09143 0.78631 0.03595

0.37068 0.70480 0.45439 0.00446

0.01132 0.74590 0.31639 0.39770

0.05395 0.89538 0.90827 0.43562

0.37911 0.03720 0.65572 0.01683

0.76155 0.90763 0.69865 0.03898

0.12245 0.27469 0.33260 0.07615

0.40015 0.00143 0.95684 0.33142

0.91815 0.48567 0.91302 0.41718

0.53332 0.02090 0.48251 0.07435

0.95050 0.18716 0.97003 0.49629

0.00944 0.82710 0.35187 0.25173

0.79012 0.72260 0.34112 0.67509

0.60203 0.20421 0.18044 0.05758

0.05900 0.16923 0.06201 0.44831

0.34995 0.86915 0.23153 0.91205

0.03103 0.75133 0.06152 0.98794

0.39139 0.02358 0.14712 0.49891

0.27117 0.57222 0.85887 0.62064

0.82049 0.20378 0.61731 0.98406

0.92390 0.38917 0.60548 0.97554

0.02563 0.36118 0.75241 0.19873

0.18735 0.06747 0.37942 0.80804

0.05743 0.68036 0.12261 0.22580

0.79512 0.27059 0.07646 0.56848

0.15900 0.09083 0.74563 0.13434

0.71377 0.69116 0.12631 0.51776

0.84752 0.51680 0.92943 0.91918

Table 6–1 Random numbers generated by Microsoft Excel

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Each successive number in a sequence of random numbers must have an equal probability of taking on any one of the possible values (for example, every number between 0 and 1 has the same chance of occurring). Additionally, each number must be statistically independent of the other numbers in the sequence. For example, if a computer generates one random number, it tells nothing about the values of other random numbers generated. The numbers need to be random observations from a uniform distribution between 0 and 1. Such random numbers are referred to as U(0,1) random numbers. Uniformly distributed random numbers between two fixed limits, say lower limit of A and higher limit of B, can be generated by using the formula A + x(B – A), where x is uniformly distributed between 0 and 1. For example, to obtain uniformly distributed numbers between 10 and 60, each value of x is replaced between 0 and 1 with the value 10 + x(60 – 10). Thus, when x = 0, we get 10 and when x = 1, we get 60 and so on. Monte Carlo Sampling. The second and most important requirement of simulation calculations is to generate random observations from probability distributions assigned to each input variable in the model. This is done by transforming the random numbers generated into their respective random variates. Computer softwares use complex algorithms to achieve this. An example of manual transformation is done here to show how computer softwares perform the sampling behind the scene. For manual calculations, the transformation can be done in the following two ways. 1. Graphical Approach: Using a simple graphical interpretation, random samples can be generated from distributions other than the uniform distribution. The cumulative distribution function (CDF) of the distribution of interest and a sampling procedure for the uniform distribution on (0,1) are required. Figure 6–3 shows the CDF of a triangular distribution of oil price, with parameters (10,18,26). The following steps are used to transform the random numbers 0.92353, 0.67381, and 0.27832 into random variates. a. Start with a uniformly distributed random number between 0 and 1, such as 0.92353. b. Enter the vertical axis of Figure 6–3 at RN = 0.92353. 324

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Fig. 6–3 Graphical procedure for obtaining a random observation from a CDF

c. Move horizontally to the CDF. d. Move vertically down to the horizontal axis, and e. Read the value of random variate as 22.87. However, since simulation normally requires a large number of iterations, this method may be cumbersome to use. 2. Using Equation: This method requires finding an equation for the CDF of the distribution of interest in the form RN = f(X) and solving for X as a function of RN, (find the inverse function of the CDF). This approach can be used for some simple distributions (such as uniform, triangular, and exponential) but not for the more complex distributions (such as normal, lognormal, and others). The method essentially requires integrating the probability density function (PDF) to get the CDF and then finding the inverse of CDF. However, only simple distributions (uniform, triangular, and exponential) have CDFs with simple analytical integrals. Other common distributions (including the normal and lognormal distributions) have no simple formula for CDF. The inverse functions for triangular, uniform, and exponential CDFs (in of RN, uniform random number between 0 and 1) follow.

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Sampling from triangular distributions T(XL , XM , XH) If RN ≤ ( XM – XL) / ( XH – XL) (6.1)

X = XL +

(X M

− X L )(X H − X L )R N

If RN ≥ ( XM – XL) / ( XH – XL)

X = XH −

(X H − XM )(X H − X L )(1 − RN )

(6.2)

Sampling from uniform distributions U(xmin , xmax) (6.3)

X = RN (xmax − xmin )+ xmin

Sampling from exponential distributions E(x, ) The PDF and CDF of exponential distributions are given by (6.4)

f (x ) = λe

− λx

(6.5)

P{X ≤ x} = 1 − e − λx

where 1 / = the mean of the distribution The inverse function of exponential CDF is (6.6)

X =

ln(1 − RN ) −λ

Latin Hypercube Sampling. In Monte Carlo sampling (also referred to as full distribution sampling), each random variable remains as an element of the distribution, thus leaving the entire statistical range available for sampling in subsequent iterations. In most cases, this results

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in clustering of sampling in some parts of the distribution while other parts are not sampled at all. Figure 6–4(a) shows the CDF of Figure 6–3 and the 10 full distribution samples marked by arrows. Note there are three areas of the CDF not sampled at all, and the interval from 0.9 to 1 is sampled three times. The interval with a probability of 90% to 100% is weighted more heavily. In Latin Hypercube Sampling (designated by LHS), the cumulative distribution function is first partitioned into nonoverlapping intervals of equal probability, in line with the number of required iterations. For example, if the assigned iterations are 10 then the distribution is divided into 10 equal parts. Random samples (one in this case) are then picked from each interval and transformed into random variates as shown in Figure 6–4(b). Therefore, the main distinction between the full distribution sampling and the LHS is the method of random variable selection. Full distribution technique employs sampling with replacement while LHS uses stratified sampling with replacement.

1.00

Probability

0.80 0.60 0.40 0.20 0.00 10

15

20

25

30

25

30

Oil Price, $/Stb

(a) 1.00

Probability

0.80 0.60 0.40 0.20 0.00 10

15

20

Oil Price, $/Stb

(b)

Fig. 6–4 Monte Carlo and Latin Hypercube Sampling 327

Project Economics and Decision Analysis Volume II: Probabilistic Models

LHS is a significant improvement over the conventional method because it guarantees all probabilities are represented as intended, thus giving equal weight to all probabilities on the CDF. If for instance, an event occurs with a 5% probability, then the event will, indeed, occur exactly five times in every 100 iterations. From a practical point of view, the LHS can reduce the necessary number of iterations by 30% (necessary means further extending the number of iterations would not show an appreciable change in the results). The more input random variables in a simulation model, the more iterations are necessary to accurately represent the output variable. In addition, if the probability distributions for the input variables are highly skewed, the number of required iterations further increase. In such situations, using the LHS greatly reduces the number of required iterations, thus saving on computer processing time.

BEHIND-THE-SCENE CALCULATIONS Simulations are carried out on computers. However, it is of interest to have at least a general idea of how the computers arrive at the simulated outputs. Therefore, a simple problem is manually solved in this section to show how simulation software arrives at the output distributions and other statistics. Assume a simple case of three variables as (a) oil reserves in thousand stock tank barrels, (b) oil price in $/barrel (net of operating cost and severance taxes, etc.), and (c) tax rate of 40%. It is further assumed that •

•

328

oil reserves are represented by uniform distribution with parameters U(100,250), i.e., minimum reserves of 100,000 barrels and maximum reserves of 250,000 barrels, and oil price is represented by a triangular distribution with parameters T(10,18,26), i.e. minimum price of $10/Stb, most likely price of $18/Stb, and the maximum price of $26/Stb.

Simulation in Decision Analysis

The tax rate is a certain parameter and is assumed constant in all cases. Based on these assumptions, the expected reserves need to be simulated. Systematic calculations for the first iteration are shown as follows, and calculations for a total of 10 iterations are shown in Table 6–2 (also see Table 6–2.xls on the accompanying CD).

Parameters Distribution

Low

High

Most Likely

Reserves

M Stb

Uniform

100

250

Oil Price

$/Stb

Triangular

10

26

Tax Rate

40.0% Random

Reserves

Random

Oil Price

Iteration

Number

(M Stb)

Number

($/Stb)

(M$)

1 2 3 4 5 6 7 8 9 10

0.03134 0.39800 0.40564 0.55144 0.86184 0.12585 0.70737 0.58097 0.12432 0.07570

104.701 159.700 160.846 182.716 229.276 118.878 206.106 187.146 118.648 111.355

0.92353 0.05013 0.93055 0.30229 0.18518 0.24870 0.96631 0.13265 0.61568 0.59190

22.871 12.533 23.018 16.220 14.869 15.642 23.923 14.121 18.986 18.772 E{NCF} =

1,436.79 1,200.92 2,221.46 1,778.23 2,045.40 1,115.70 2,958.45 1,585.56 1,351.61 1,254.25 1,694.84

18

Net Cash Flow

Table 6–2 Manual simulation of net cash flow (Monte Carlo Sampling)

Step 1: Choose two random numbers for reserves and oil price from Table 6–1 as 0.03134 (Column 1) and 0.92353 (Column 6), respectively.

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Step 2: Using the cumulative distribution function (CDF) of uniform distribution, transform the random number 0.03134 into its corresponding random variate from uniform distribution. To do this, use Equation (2.57) while replacing P(x) by random number RN = 0.03134 and solving for x. (2.57)

P (X ≤ x) =

x − xmin xmax − x min

X = RN (xmax − xmin )+ xmin

= 0.03134(250 − 100)+ 100 = 4.7010 + 100 = 104.7010 M Stb

Step 3: Using the cumulative distribution function (CDF) of triangular distribution, transform the random number 0.92353 into its corresponding random variate from triangular distribution. To do this, use Equation (6.1) or Equation (6.2), depending on the value of random number, with random number RN = 0.92353 and solving for X.

X M − X L 18 − 10 = = 0.5 X H − X L 26 − 10 Since RN > 0.5, we use Equation (6.2) as

X = XH − = 26 −

(XH − X M )(X H − X L )(1 − RN ) (26 − 18)(26 − 10)(1 − 0.92353)

= 26 − 8 × 16 × 0.0765 = 22.8714 $/Stb

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Step 4: Now calculate net cash flow with reserves and oil price from Step 2 and Step 3, respectively, and the tax rate of 40%. Net Cash Flow = [(Reserves x Price) – Investment] x (1 – T)

= [(104.7010 × 22.8714)− 0](1 − 0.40)

= 1,436.7951 M $ or 1.437 MM $

Step 5: Repeat Step 2 to Step 4 several times as shown in Table 6–2. The results obtained must now be organized and plotted just as though they were raw data, collected from repeated tests or experiments. The data and presentation of most interest are the cumulative probability distribution, expected value of the output variable, standard deviation of the output variable, and statistical information of interest that may include the 90% probability reserves are greater than a certain value, and so on. The expected net cash flow from the 10 iterations is simply the average of the net cash flows as $1,694,000. By using the expected values of the individual distributions, the expected value of the net cash flow is calculated as: Using Equation (2.55), the expected value of uniform distribution is XU =

xmin + xmax 100 + 250 = = 175 2 2

Using Equation (2.58), the expected value of triangular distribution is XT =

X L + X M + X H 10 + 18 + 26 = = 18 3 3

Therefore, the net expected cash flow is

E{NCF }= (175 × 18)(1 − 0.4 ) = 1,890 M$ Note that from the first 10 iterations in Table 6–2, the E{NCF} is only 1,694.84 M$.

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Project Economics and Decision Analysis Volume II: Probabilistic Models

The same problem is reworked, this time using LHS as shown in Table 6–3. The expected value from the first 10 iterations, using LHS, calculates to 1,855.90 M$ (after 30 iterations, the expected value is 1,873.57 M$ as shown in Table 6–3 on the CD). The running average net cash flow versus number of iterations (for data in Table 6–3) is plotted as shown in Figure 6–5 to show the convergence of the expected value.

Parameters Distribution

Low

High

Most Likely

Reserves

M Stb

Uniform

100

250

Oil Price

$/Stb

Triangular

10

26

Tax Rate

40.0% Random

Reserves

Random

Oil Price

Number

(M Stb)

Number

($/Stb)

(M$)

(M$)

0.03134 0.12585 0.23976 0.39800 0.40564 0.55144 0.65572 0.70737 0.86184 0.95050

104.701 118.878 135.964 159.700 160.846 182.716 198.358 206.106 229.276 242.575

0.89538 0.43446 0.18518 0.24870 0.61568 0.59190 0.75241 0.92353 0.30229 0.05013

22.341 17.457 14.869 15.642 18.986 18.772 20.370 22.871 16.220 12.533

1,403.45 1,245.17 1,212.95 1,498.83 1,832.32 2,058.02 2,424.39 2,828.35 2,231.37 1,824.13

1,403.45 1,324.31 1,287.19 1,340.10 1,438.54 1,541.79 1,667.87 1,812.93 1,859.43 1,855.90

Iteration 1 2 3 4 5 6 7 8 9 10

18

Net Cash

NCF

Flow

Average

Table 6–3 Manual simulation of net cash flow (Latin Hypercube Sampling)

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Simulation in Decision Analysis

Fig. 6–5 Plot of expected net cash flow versus number of iterations (Table 6–3)

The expected value of 1,855.90 M$ calculated using the LHS differs only slightly from the expected value of 1,890 M$ calculated using the expected value of the input distributions. There is 10.3% difference in expected value using Monte Carlo Sampling versus less than 1.8% difference using LHS.

RECOGNIZING DEPENDENCE ON INPUT VARIABLES In the preceding sections, the input variables were assumed probabilistically independent. This may not always be the case, and assuming the independence of input variables while some of them may actually be dependent will result in nonsensical results. It is important, therefore, that when building a simulation model, the probabilistic dependence of the input variables must first be determined and for the correlation, if there is any, between them. To for dependencies in Monte Carlo techniques, conditional probability distributions need to be specified.

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When the samplings of two or more input distributions are related (in some way) to one another, then they are correlated. For example, when interest rates increase, the number of houses sold decreases. However, the simulation model would not know this relationship unless it is specified. As a result, when the sampling of interest rate’s distribution returns low interest rate, the sampling of the housing sale may also return lower sales and vice versa. Therefore, the sampling of these two distributions should be related to avoid nonsensical results. Similarly, it is possible maintenance cost on a machine is related to the volume of product it will produce, since higher volume of production means more wear and tear on the machine. In addition, sales revenue from a product may depend on the advertising expenditure for that product (the higher the advertising expenditure, the higher the revenue from the product, and vice versa). In practical life, many such situations are faced where the inputs are correlated. In the volumetric reserve model developed in the preceding pages, distributions were specified for the geological variables such as porosity, water saturation, formation thickness, and drainage area, etc. Values of these geological variables are randomly sampled and multiplied together to arrive at the reserve estimate (the output distribution of volume of recoverable oil). There it was assumed that when a value of one variable is drawn, this value would have no effect on the variable drawn for another variable. This shows it was assumed the input variables are completely independent of one another. In order to be certain, one must make sure her assumptions are correct, as some of the geological properties may be dependent on others. For example, •

• •

334

There may be a positive relationship between field size and reservoir thickness (larger fields may have thicker hydrocarbon columns and larger drainage area). Formations with higher porosity may have lower water saturation (higher hydrocarbon saturation). Productivity may be correlated with geological characteristics of the reservoir (recovery may be low for tight formations).

Simulation in Decision Analysis

• •

The production from a waterflood can be correlated with the amount of water injection, and so on. Thicker hydrocarbon formations may have higher IP (initial production rate) and vice versa.

From courses on statistics, one may be already familiar with the techniques of determining correlation between variables. However, before discussing methods for simulating dependence relationship between variables in a simulation model, the various forms of these relationships should be known. One way of displaying the relationship between two variables is to draw a scatter diagram. A scatter diagram involves plotting one variable against the other; usually the dependent variable is plotted on the y-axis (vertical axis) and the independent on the x-axis (horizontal axis). If it is not known which of the two variables is the dependent variable, then just plotting one variable against the other will suffice. Once the variables are plotted as described, the patterns revealed by the scatter diagrams would typically have, at least, four basic characteristics as shown in Figure 6–6. Figure 6–6(a) shows total linear positive dependence between the variables, i.e., increase or decrease in independent variable will have a respective increase or decrease in dependent variable. Figure 6–6(b) shows a total nonlinear negative dependence between the variables. Figure 6–6(c) shows diffused positive dependence. The diffused dependence, negative or positive, means the variables are correlated but the correlation is not as perfect as it is in Figure 6–6(a) or Figure 6–6(b). Figure 6–6(d) shows there is no correlation of any type between the variables, meaning the variables are completely independent of each other. For the purpose of simulation, these relationships are divided into two categories. These are (1) total dependence and (2) diffuse dependence. ing for these two types of dependence in simulation is discussed in this section.

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Dependent Variable

Dependent Variable

Project Economics and Decision Analysis Volume II: Probabilistic Models

(b)

(a)

Independent Variable

Dependent Variable

Dependent Variabl e

Independent Variable

(d)

(c)

Independent Variable

Independent Variable

Fig. 6–6 Scatter diagrams showing typical correlations between two variables The description of dependence in the preceding paragraphs gives a visual and qualitative assessment between the data of two variables. We further need to determine how strong or weak is the dependence, if there is any, in the variables. Calculating the correlation coefficient does this. The correlation coefficient (designated by symbol r or rXY) is a numerical summary measure indicating the strength of relationship between a pair of variables. The correlation coefficient is given by the following equation. (6.7)

∑ (X − X )(Y − Y ) n

i

i =1

rXY =

where sX sY X Y 336

i

(n − 1) s X sY

= standard deviation of the X = standard deviation of the Y = average (mean) of the independent variables (X) = average (mean) of the dependent variables (Y)

Simulation in Decision Analysis

The built-in functions of Excel can be used to determine the correlation coefficient. There are two possibilities in Excel. To calculate a single correlation between variables X and Y, use Excel’s CORREL function in the form =CORREL(X-range,Y-range) Alternatively, use the Tools ➯ Data Analysis ➯ Regression to obtain a whole table of correlations and other important parameters of a set of variables. The correlations calculated using Equation (6.7) are between –1 and +1, inclusive. The sign of a correlation, plus or minus, determines whether the relationship between the two variables is positive (upward trend) or negative (downward trend). However, the strength of the relationship between the variables is measured by the absolute value, or magnitude, of the correlation. The closer this magnitude is to 1, the stronger the relationship. A correlation coefficient equal to 0 or near 0 practically indicates no correlation. Therefore, the correlation coefficient measures the strength of linear relationship. Looking back at the scatter plots in Figure 6–6, the correlation coefficient of the relationship in Figure 6–6(a) is 0.99934 (strong positive or perfect linear correlation). The rXY for data in Figure 6–6(c) is 0.9568 (good correlation) and for Figure 6–6(d) is 0.09399 (poor correlation). The relationship between the independent and dependent variables, when the scatter plot shows linear trend, can be quantified by fitting a straight line through the data of the dependent variable Y versus the independent variable X. The line fitting is done using the technique of least squares estimation (simple linear regression) using the following equation. The equation of a straight line is given by (6.8)

Y = a + bX

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Project Economics and Decision Analysis Volume II: Probabilistic Models

Here, a is the Y-intercept of the line, the value of Y when X = 0, and b is the slope of the line, the change in Y when X increases by one unit. Therefore, to specify the least squares line, all that is needed is to specify the slope and intercept. These are given by (6.9)

∑ (X − X)(Y − Y ) n

b=

i

i

i =1

∑ (X n

i =1

− X)

2

i

and (6.10)

a = Y − bX

Simulating Total Dependence In total dependence, all of the points in a scatter diagram fall on or are extremely close to a line (in linear relationship) or a curve (nonlinear relationship). The correlation coefficient r for this type of dependency is close to 1.0. Asg the same random number for both the independent and dependent variables can simulate the positive relationship (with a positive slope). For a negatively sloping relationship and if random numbers between 0 and 1 are used, the original random number is used to generate a value for the independent variable and 1.0 minus this random number is used to generate a value for the dependent variable. In simulation, the independent variable is sampled first. Its value then influences the value of the dependent variable by restricting its range. Alternatively, the distribution of one of the variables (preferably the independent variable) is specified. The distribution is then sampled at each iteration to determine its numerical value. Equation (6.8) is used to correlate the two variables in order to establish the value of the other variable (dependent variable or vice versa).

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Simulating Diffuse Dependence The diffused relationship is the most commonly occurring dependence situation, and there is relationship between the variables but it is not perfect as shown in Figure 6–6(d) and Figure 6–7. The correlation coefficient for this type of relationship is much less than 1.0. For example, the data in Figure 6–7 (although scattered) shows a downward sloping trend with a correlation coefficient of –0.5344.

Fig. 6–7 Selecting two correlated variables using the box method

The situation in Figure 6–7 represents a “gray” area between complete dependence and complete independence. Yet, this situation is often encountered. For this type of dependence, the sampling scheme is modified so the distributions of both the independent and dependent variables can be sampled in each iteration, while honoring the partial dependence at the same time. Newendorp presents the best explanation of handling partial dependence in simulation. The following steps are adopted from Newendorp.11

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1. Prepare a cross plot of random variables X and Y in which there is partial/diffuse dependence, as shown in Figure 6–7. 2. Draw a box (envelope) around the data points so a majority of the data points is bounded as shown in Figure 6–7. This is also called bounding, the boundary defines the maximum and minimum limits within which X and Y can vary. Any data points outside the box are considered rare occurrences, meaning the probability of their occurrence is zero. 3. Identify the type of variation of dependent variable Y within the box as a function of independent variable X. Note that since the dependence is not perfect, there exists a unique distribution, random occurrence, of Y for each value of X within the limits Xmin and Xmax (in Figure 6–7, Xmin = 0.08 and Xmax = 0.22). This step enables one to ascertain the possible distribution of Y along the vertical axis for a given value of X. For example, a. If the variation of Y as a function of X is random (as in Figure 6–7), the variability can be represented by a series of uniform, rectangular distributions. b. If the variation of Y as a function of X is not random but it is clustered midway between the upper and lower bound, then the variability of Y within the upper and lower limits may be represented by a series of normal or symmetrical triangular distributions. c. If the variation of Y as a function of X is clustered along the upper or lower boundary of the envelope, then the variability may be represented by a series of triangular distributions skewed toward the maximum (upper boundary) or minimum (lower boundary) values (as the case may be) of Y to reflect the respective clustering. 4. Generate normalized Y distribution. Step 3 noted that for each value of X (between Xmin and Xmax) there exists a unique distribution of Y since the Ymin and Ymax vary as X varies. For example, in Figure 6–7, if a uniform rectangular distribution is

340

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assumed for Y, then at X = 0.15 then Ymin = 0.39 and Ymax = 0.75. This means for a 100-iteration simulation, 100 values of X and 100 distributions of Y are picked. The value of X is established for each iteration. The value of Y for that iteration is sampled from the CDF of Y corresponding to that value of X. This, however, is tedious and cumbersome. In order to make the job more manageable, a normalized Y distribution is developed to represent so many distinct Y distributions in some form. The normalized probability distribution is obtained by converting the random variable scale to an equivalent dimensionless scale ranging from zero to one. For each iteration, this single representative normalized distribution enable one to sample a value of YNORM and give the fractional distance along the line from Ymin to Ymax for locating the desired sample value. The following equation is used to normalize the variability of Y. (6.11)

YNORM

Y − Ymin = Ymax − Ymin

where Ymin = minimum value of Y Ymax = maximum value of Y YNORM = normalized, dimensionless random variable 5. Develop cumulative probability distributions for the independent variable X and the normalized variable YNORM obtained in Step 4. 6. Generate random numbers and sample the distributions. At this stage, there are two cumulative distribution functions. The CDF for variable X and the CDF for YNORM. For the first iteration, two random numbers as RN1 and RN2 are generated. The RN1 is used to sample the X distribution and RN2 to sample the YNORM distribution. This gives X1 and (YNORM)1.

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7. Obtain Ymin and Ymax, corresponding to X1 in Step 6, from the envelop in Figure 6–7. The envelope of Figure 6–7 is entered with X1 to read Ymin and Ymax. For example, if X1 = 0.15 then Ymin = 0.39 and Ymax = 0.75. Instead of obtaining the Ymin and Ymax graphically, equations are developed to represent the two lines (as shown in Figure 6–7) in of X. These equations are then used to obtain Ymin and Ymax at any given X. 8. Calculate Y1 corresponding to X1. The Y1 is calculated from the parameters obtained in Step 6 and Step 7 by using the following equation.

YX = (Ymin )X + [(Ymax )X − (Ymin )X ](YNORM )X

(6.12)

where (Ymin)X = minimum possible value of Y, given X (Ymax)X = maximum possible value of Y, given X (YNORM)X = sampled value from the dimensionless normalized distribution Steps 1 through 5 are used to prepare the necessary data for simulation of partially dependent variables. Steps 6 to 8 are repeated for each iteration. The following example further clarifies handling of partial dependence in simulation.

Example 6–1 A petroleum engineer with Advanced Oil Company is asked to estimate reserves and oil production forecast for a newly drilled wildcat well. Being an expert in simulation, he decides to use simulation to achieve this objective. The performance and data of some 30 wells already developed in the same reservoir in the nearby field are used as analogy. While organizing

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the data, the engineer notices that there is partial dependency between the initial production rates and pay thickness of the 30 wells drilled. The data is given in Table 6–4. Using the methods discussed in this chapter, manually perform an iteration of simulation on the data given in Table 6–4.

Net Pay Initial Well Well Thickness Potential No.

(feet)

(Stb/day)

YNORM Y -80 = _______ 280 - 80

Net Pay Initial Well Well Thickness Potential No.

(feet)

(Stb/day)

YNORM Y -80 = _______ 280 - 80

BA-1

6

80

0.000

BA-16

31

205

0.625

BA-2

46

190

0.550

BA-17

33

145

0.325

BA-3

8

100

0.100

BA-18

34

110

0.150

BA-4

11

115

0.175

BA-19

40

185

0.525

BA-5

13

190

0.550

BA-20

41

145

0.325

BA-6

55

200

0.600

BA-21

43

221

0.705

BA-7

15

130

0.250

BA-22

48

235

0.775

BA-8

16

160

0.400

BA-23

50

186

0.530

BA-9

13

85

0.025

BA-24

57

235

0.775

BA-10

18

103

0.115

BA-25

62

178

0.490

BA-11

20

190

0.550

BA-26

7

160

0.400

BA-12

22

160

0.400

BA-27

69

225

0.725

BA-13

23

90

0.050

BA-28

75

230

0.750

BA-14

26

130

0.250

BA-29

8

140

0.300

BA-15

29

165

0.425

BA-30

83

280

1.000

Table 6–4 Net pay thickness and initial potential data for 30 wells

Solution: The series of steps representing data preparation and a simulation iteration are as follows. Step 1: Plot the data as shown in Figure 6–8 with net pay thickness as the independent variable and the initial well potential as the dependent variable. Enclose the plotted data in an envelop/box.

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Fig. 6–8 Cross-plot of net pay thickness and initial potential of Table 6–4

Step 2: Calculate the cumulative frequency distribution for the independent variable (net pay thickness, variable X). The calculations are shown in Table 6–5 and the data is plotted in Figure 6–9(a).

Independent Variable X Net Pay Thickness

No. of Wells with Thickness

Cumulative Frequency

(feet)

Frequency

(fraction)

0 –15 15 –30

8 7

0.2667 0.5000

30 –45

6

45 –60

5

60 –75 75 –90

Normalized Variable No. of Wells With YNORM

Cumulative Frequency

Frequency

(fraction)

0 –0.2

1 6

0.0333 0.2333

0.7000

0.2 –0.4

8

0.5000

0.8667

0.4 –0.6

8

0.7667

3

0.9667

0.6 –0.8

6

0.9667

1

1.000

0.8 –1.0

1

1.0000

YNORM 0

30

Table 6–5 Calculating cumulative probability of data in Table 6–4

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Simulation in Decision Analysis

Cumulative Probability

1.00 0.80 0.60 RN = 0.37942

0.40 0.20

h = 22

0.00 0

25

50

75

100

Net Pay Thickness, feet (a)

Cumulative Probability

1.00 0.80 RN = 0.5168

0.60 0.40 0.20

YNORM = 0.41

0.00 0.00

0.20

0.40

0.60

0.80

1.00

Normalized Initial Well Potential, Stb/day (b)

Fig. 6–9 Cumulative frequency graph of (a) net pay thickness data (independent random variable X) from Table 6–5 and (b) normalized variable YNORM representing variability of initial potential as a function of pay thickness in Table 6–4

Step 3: Calculate the cumulative frequency distribution for the normalized variable YNORM representing the variability of initial potential as a function of net pay thickness. The calculations are shown in Table 6–5 and the data is plotted in Figure 6–9(b). Step 4: Obtain equations for the upper and lower boundaries of the envelop in Figure 6–8 as a function of X. Since the data points are bounded by two straight lines with the coordinates as shown in Figure 6–8, it is easier to obtain their equations as follows. Equation of straight line (with intercept = a and slope = b) is (6.13)

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Project Economics and Decision Analysis Volume II: Probabilistic Models

The equation of a straight line can be computed using the x–y coordinates of two points on the line with the following equation: (6.14)

y − y1 y2 − y1 = x − x1 x2 − x1 Equation (6.14) is used to find equation of the upper boundary line of Figure 6–8. The coordinates of this line, as shown in Figure 6–8, are (x1 = 6, y1 = 185) and (x2 = 83, y2 = 280). y − 185 280 − 185 = 83 − 6 x−6

After simplification, the equation of the upper boundary reduces to yupper = 177.5974 + 1.2338 x

Similarly, equation of the lower boundary is given by ylower = 42.4545 + 2.0909 x

Step 5: Randomly pick two random numbers from Table 6–1 as 0.37942 and 0.5168. Using the random number 0.37942, sample the cumulative frequency distribution of random variable X from Figure 6–9(a) as h = 22. Similarly, using the random number 0.5168, sample the cumulative frequency distribution of YNORM from Figure 6–9(b) as YNORM = 0.41. Step 6: Using the straight line equations developed in Step 4 and the value of X = 22 sampled in Step 5, calculate Ymin and Ymax as

Ymin = 42.4545 + 2.0909(22 ) = 88.4543 Ymax = 177.5974 + 1.2338(22) = 204.741

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Step 7: Now using Equation (6.12), we get Y for the first iteration is. YX = (Ymin )X + [(Ymax )X − (Ymin )X ](YNORM )X

Y22 = 88.4543 + (204.741 − 88.4543)× 0.41 = 136.13 Stb/day

Therefore, the values used for the first iteration would be X = 22 feet and Y = 136 Stb/day. Repeat Steps 5 to 7 for any subsequent iterations.

SPREADSHEET APPLICATIONS Simulation modeling using the spreadsheet environment has become popular in recent years. The main reason for this is the ease with which the spreadsheets such as Excel and Lotus can be used to achieve this task. Additionally, the popularity is due to the availability of simulation add-ins such as Crystal Ball and @RISK to Excel or Lotus. Following are two inherent advantages for using the add-ins. •

•

The add-ins give easy excess to many probability distributions that form the basis of any simulation model. The Excel and Lotus are limited to only the few basic probability distributions. The add-ins provide other facilities to replicate a simulation such as (a) fitting a probability distribution, (b) tying the simulation models to decision trees, (c) calculating summary statistics (such as averages, standard deviations, and percentiles etc.), and (d) illustrating the outcomes through graphs, the most important part of simulation (presentation of results). Although Excel and Lotus can be tailored to achieve these tasks, it is cumbersome and time consuming.

This section provides examples showing the use of Excel for some basic simulation models. However, the use of @RISK add-in from Palisade Corporation is recommended. 347

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Using Excel for Simulation Excel can be tailored to perform simulation modeling by developing fit-for-purpose applications with Excel’s built-in macro language— Visual Basic for Applications (VBA). Only the built-in capability of Excel, used for simple simulations, is presented in this section. Generating Random Numbers in Excel. Both Excel and Lotus 1-2-3 are capable of generating random numbers between 0 and 1. In Excel, uniform random numbers between 0 and 1 are generated by entering the formula =RAND() in any cells. The random numbers generated by this function have the following properties. •

•

Each time the =RAND() function is used, the numbers generated between 0 and 1 have the same chance of occurring. For example, 10% of the numbers generated by this function will be between 0 and 0.1; 10% of the numbers will be between 0.65 and 0.75; 60% of the numbers will be between 0.2 and 0.8; and so on. This implies the generated random numbers are uniformly distributed between 0 and 1. The random numbers generated by the =RAND() function are probabilistically independent. This means when one random number is generated, it does not tell anything about the subsequent number(s) that will be generated.

One of the characteristics of the random generators is that every time different numbers are generated. In Excel, if the F9 (recalculate) key is pressed or if anything is done to the spreadsheet to activate the recalculation, all the cells containing the =RAND() function change. However, it may be desirable to reproduce a given stream of random numbers exactly, for at least two reasons. First, using exactly the same random numbers makes it easier for debugging or ing a model. Second, the use of identical random numbers in simulating different systems makes it easier to obtain a comparison that is more precise. Due to these reasons, it may be required, at times, to freeze the current random numbers that are generated. The following steps can be used to do this. 1. Select the range of cells that contain the =RAND() function. 348

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2. Use the Copy command to copy this range. 3. Click on Edit ➯ Paste Special ➯ Values ➯ OK. This pastes a copy of the random numbers as values only rather than the =RAND() function. These numbers will not change whenever the spreadsheet recalculates. Probability Distributions in Excel. In addition to the random numbers, the second most important input to simulation models is the probability distributions. Excel has a limited number of built-in probability distributions, including: • • • • •

•

• •

=BETAINV(probability, , ,A,B)—returns the inverse of the cumulative beta probability distribution. =CHIINV(probability,degrees_freedom)—returns the inverse of the one-tailed probability of the chi-squared distribution. =FINV(probability,degrees_freedom1,degrees_freedom2)— returns the inverse of the F probability distribution. =GAMMAINV(probability, , )—returns the inverse of the gamma probability distribution. =V(probability,,)—returns the inverse of the lognormal probability distribution, where and are the mean and standard deviation, respectively. =NORMINV(probability,,)—returns the inverse of the normal cumulative probability distribution, where and are the mean and standard deviation, respectively. =NORMSINV(probability)—returns the inverse of the standard normal cumulative probability distribution. =TINV(probability,degrees_freedom)—returns the inverse of the student’s t-distribution.

The Excel’s built-in statistical functions can be used directly by replacing the probability in the argument (parenthesis) to random number, in simulation. For example, =NORMINV(RAND(),,) is entered to calculate the random variate of normal distribution, and so on. In addition to these inverses of CDFs, Excel has other statistical functions to generate PDF the CDF values associated with a specified input value of the random variable but not the inverse. These are listed as follows; the functions that generate 349

Project Economics and Decision Analysis Volume II: Probabilistic Models

inverse functions also have similar counterparts to generate only PDF and CDF. For example, the counterpart of =NORMINV() is =NORMDIST(). • • • • • •

=BINOMDIST()—individual term binomial distribution. =EXPONDIST()—returns the exponential distribution. =HYPGEOMDIST()—returns the hypergeometric distribution. =NEGBINOMDIST()—returns the negative binomial distribution. =POISSON()—returns the Poisson distribution. =WEIBULL()—returns the Weibull distribution.

The statistical functions (CDFs) of Excel that do not have the inverse functions can be used with =VLOOKUP (a special spreadsheet table lookup function of Excel). Alternatively, if the inverse equation of the desired probability distribution can be easily derived [such as Equations (6.1) to (6.6)], then these should be used. The use of Excel’s =VLOOKUP function and exponential probability function is shown in Table 6–6. The simulation of Table 6–2 should be tried by inputting the =RAND() function in Columns B and D. The descriptive statistics information for this simulation can be obtained in the usual manner (discussed in Chapter 2).

2

A 0.0000

B 0.00

Cell A2: =EXPONDIST(B2,0.85,TRUE)

3

0.3462

0.50

Copy A2 from A3:A13

4

0.5726

1.00

Enter random variables in B2:B13

5

0.7206

1.50

6

0.8173

2.00

7

0.8806

2.50

8

0.9219

3.00

Press F9 and see how the value in Cell A15

9

0.9490

3.50

changes each time F9 is pressed.

10

0.9666

4.00

11

0.9782

4.50

12

0.9857

5.00

13

0.9907

5.50

Cell A15: =VLOOKUP(RAND(),A2:B13,2)

14 15

3.5

Table 6–6 Use of Excel’s =VLOOKUP function

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Example 6–2 Use Excel to set up a simulation model for calculating oil reserves using volumetrics. It is assumed that 1. Porosity is normally distributed with mean = 0.14 and standard deviation = 0.02. 2. Water saturation is represented by triangular distribution with the minimum, most likely, and maximum values of 0.2, 0.3, and 0.44, respectively. 3. Formation thickness is normally distributed with a mean of 15 feet and standard deviation of 1.5 feet. 4. Drainage area is normally distributed with mean of 77 acres and standard deviation of 63 acres. 5. Recovery factor is normally distributed with a mean of 0.34 and standard deviation of 0.05, and 6. The oil formation volume factor is represented by rectangular distribution with parameters 1.15 and 1.5. Solution: The model is set up as shown in Table 6–7. The following steps are followed in developing the simulation model in Table 6–7.

A

B

C

D

E

Distribution

3

F

4 Porosity 5 Water Saturation

Fraction

Normal

0.14

Fraction

Triangular

0.20

0.30

6 Formation Thickness 7 Drainage Area

Feet

Normal

15

1.50

Acres

Normal

77

63

8 Recovery Factor 9 Oil FVF

Fraction

Normal

0.34

0.05

1.15

1.50

Area

RF

RB/Stb

Uniform

10 Average Reserves Porosity 11

Stb

215,673 Sw

Thickness

12 No.

(fraction)

(feet)

(fraction)

G

H

Parameters 0.02 0.44

Oil FVF Reserves (Stb)

(acres) (fraction) (RB/Stb)

13

1

0.0895

0.2922

15.735

14

2

0.1478

0.2921

15.062

15

3

0.1480

0.2786

19.155

1.16E+02 0.2993 1.3736 196,013 6.98E+01 0.2999 1.4256 179,541 7.55E+01 0.3131 1.3892 270,083

16

4

0.1575

0.2841

16.462

4.37E+01 0.2924 1.1959 153,856

Table 6–7 Oil volumetric reserves simulation using Excel 351

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1. Enter the inputs for probability distributions in Cells E4 to G9. 2. Cell B13 ➯ =NORMINV(RAND(),E$4,F$4). 3. Cell C13 ➯ =IF(J13<=((F$5-E$5)/(G$5-E$5)),E$5+SQRT ((F$5-E$5)*(G$5-E$5)*J13),G$5-SQRT((G$5-F$5)*(G$5E$5)*(1-J13))). The formula in Cell C13 represent Equations (6.1) and (6.2). The Cell refers to random numbers in Cell J13 (not shown in Table 6–7), therefore, enter =RAND() in Cell J13 and copy it to Cell J550. 4. Cell D13 ➯ =NORMINV(RAND(),E$6,F$6). 5. Cell E13 ➯ =NORMINV(RAND(),E$7,F$7). 6. Cell F13 ➯ =NORMINV(RAND(),E$8,F$8). 7. Cell G13 ➯ =RAND()*(F$9-E$9)+E$9, this is Equation (6.3). 8. Cell H13 ➯ =((7758*B13*(1-C13)*D13*E13)/G13)*F13 ➯ volumetric reserve equation. 9. Copy Cells B13:H13 to Cells B550 to H550, this means 538 iterations of simulation. 10. Cell D10 ➯ =AVERAGE(H13:H550) ➯ calculates the average of reserves calculated for 538 iterations. When any value in the spreadsheet is changed, the average will be recalculated again. The number of iterations may be increased to 1000 to observe change in the average reserves each time the spreadsheet is recalculated.

Introducing @RISK @RISK is a revolutionary software system for the analysis of business and technical situations with risk exposure. It brings the decision analysis techniques to Microsoft Excel as an add-in. The combination of these two packages enables the decision maker to model any risky situation—from business to science and engineering. @RISK can be used to improve comprehension about the bottomline decision parameters subject to uncertainty.

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@RISK uses the technique of Monte Carlo simulation for risk analysis. The uncertain input variables in the spreadsheet are represented by corresponding probability distributions. The package provides more than 35 probability distributions for the decision maker to choose from. The distribution functions are identified by their name, shape, and range of the distribution. For example, a normal probability distribution is entered in the spreadsheet as =RISKNORM(100,20). The RISKNORM refers to the function name—a normal probability distribution. The shape and range of the distribution are given by the mean of 100 and standard deviation of 20. Similarly, the triangular distribution is input as =RISKTRIANG(10,20,30) where 10 is the minimum value, 20 is the most likely value, and 30 is the maximum value. The probability distributions are graphically displayed in @RISK’s “Define Distribution” window, which can be substituted for values in the spreadsheet formula. A change in the range of possible values for a certain input in the model can be seen be changing the displayed distribution. The package also provides built-in capability through which a probability distribution can be fitted to actual historical or experimental data. This is the same as the BestFit discussed in Chapter 2. In this section, the steps required to set up and analyze a model for the recoverable oil calculations (using the volumetric reserves equation) are presented. Additional examples are given to show the flexibility and utility of the software in different areas of analysis, such as production decline, optimum maintenance policy, cash budget, and complete cash flow. Any cell or range of cells in the simulation model can be specified as output cells. While running the simulation, the @Risk automatically keeps track of the statistics (such as averages, standard deviations, percentiles, and others) of the values generated in these output cells. It also generates useful graphical presentations of the values. The RISKSIMTABLE option of @Risk allows running the simulation several times, each time using a different variable for some key input of the model. @RISK links directly to Excel using Excel style menus, toolbars, and functions. Figure 6–10 shows Excel with @RISK add-in, except for the change in the toolbar menu at the top; there is hardly any difference. These provide all the necessary tools to set up and analyze a simulation model.

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Fig. 6–10 The Excel spreadsheet with DecisionTools add-in

Loading @RISK. If @RISK is not already installed, then the first thing to do is to install the DecisionTools suite with the setup program on the CD-ROM accompanying this book. The steps for installing the DecisionTools are given in Appendix B. The installation, of course, is done only once. The @RISK program can be run in one of the following three ways. 1. If Excel is not currently open, both the Excel and @RISK can be launched together by double clicking on the @RISK icon already created when the DecisionTools suite was installed. If for some reason the icon has not been created, @RISK can be accessed by (a) clicking on Windows Start button, (b) clicking on Palisade Decision Tools, and then (c) clicking on @RISK. 2. If Excel is already open, @RISK can be executed by following the second part of Step 1 (the lettered steps). 354

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3. If Excel and the DecisionTools suite are already open, shown by the toolbar displayed in Figure 6–10, @RISK can be started by clicking on its icon (the third from left). The DecisionTools’ icons in Figure 6–10 are described as follows. • •

DecisionTools Toolbar: The icons on the top toolbar in Figure 6–10 are used to activate the other DecisionTools by Palisade. @RISK Toolbar: These icons are used to quickly and easily perform tasks necessary to set up and run simulation. The functions performed by these icons are the same as the commands in @RISK menu on the Excel menu bar (at the top just before Excel Help). There are 11 @RISK icons as shown in Figure 6–10.

Developing the Simulation Model. The steps required to build a simulation model from scratch, using the features of @RISK, are given in this section. The basic steps are presented by constructing a model for calculating recoverable oil reserves using the volumetric equation, Equation (6.15). The probability distributions shown in Figure 6–2 are assumed for each variable. By practicing with the software, its advanced features can be learned.

NR =

7,758φ (1 − S w)hAd FR βo

(6.15)

where = porosity, fraction [assume normal distribution, (14%, 2%)] Sw = water saturation, fraction [assume triangular distribution, Sw (20,30,44)] h = formation thickness, feet [assume normal distribution, h(15, 1.5)] Ad = drainage area, acres [assume lognormal distribution, Ad(77, 63)] FR = recovery factor, fraction [assume normal distribution, FR(34%, 5%) ]

o = oil formation volume factor, RB/Stb [assume uniform distribution, o(1.15, 1.5)] 355

Project Economics and Decision Analysis Volume II: Probabilistic Models

1. Load @RISK: While in windows, double click on the @RISK icon. This executes the software with Excel, and the screen will look like the one shown in Figure 6–10. 2. Input distribution: Start developing the model as shown in Table 6–8. The probability distributions for each variable are input in various cells as follows.

A

B

C

D

E

F

2

G

H

Formation

3

Water

Formation Drainage Recovery

4

Porosity Saturation Thickness

5

(fraction) (fraction)

6

0.150

0.319

Area

(feet)

(acres)

14.38

123.05

Factor

Volume Factor

(fraction) (RB/STB) 0.339

1.430

Reserves (STB) 97,441

7 8

Summary measures of Inputs and Outputs

9 Minimum

0.085

0.211

11.07

5.65

0.209

1.153

17,111

10 Maximum

0.187

0.437

19.03

316.51

0.485

1.500

1,036,034

11 Mean

0.140

0.313

15.01

76.26

0.340

1.325

215,389

12 St. Dev.

0.020

0.050

1.51

58.18

0.050

0.102

173,457

13 Mode

0.140

0.287

15.56

39.51

0.332

1.282

82,919

14

Table 6–8 Simulation model for volumetric reserves calculations

a. b. c. d. e. f.

Cell B6 Cell C6 Cell D6 Cell E6 Cell F6 Cell G6

➯ = RISKNORMAL(0.14,0.02) ➯ = RISKTRIANG(0.20,0.30,0.40) ➯ = RISKNORMAL(15,1.5) ➯ = RISKLOGNORMAL(77,63) ➯ = RISKNORMAL(0.34,0.05) ➯ = RISKUNIFORM(1.15,1.5)

The lower and upper limits within any of these input distributions can be specified. For example, any negative, abnormally small, or abnormally high porosity values can be avoided by using =RISKNORMAL(0.14,0.02,RISKTRUNCATE(0.05,0.25)). This

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generates porosity distribution that follows a truncated normal distribution. This means @RISK will generate porosity from a normal distribution with mean of 14% and standard deviation of 2% and ignore it if the porosity is less than 5% or more than 25%. 3. Designate Output Cell: When the simulation is run, @RISK keeps track of the desired parameter to be calculated using the distributions given in Step 2. Cell H6 is designated as an output cell. The output cell may be designated in two ways. a. Input the formula, Equation (6.1), in Cell H6 preceded by the RISKOUTPUT()+ function as ➯ =RISKOUTPUT()+[7,758* B6* (1-C6)*D6*E6*F6]/G6. The plus sign following RISK OUTPUT() does not indicate addition. It is used by @RISK to keep track of the value in this cell (for reporting reasons) as the simulation progresses. Any number of cells can be designated as output cells in this way. They are typically the bottomline values of the desired parameters, generated after simulation. b. The second way is to highlight Cell H6 and then click on the Add Output Cell icon (fourth icon from left; the icon with one arrow) on the @RISK toolbar. The =RISKOUTPUT() function is automatically inserted. Now add the formula to this function in the formula bar. If a range of cells has to be tracked as output cells, then highlight the desired range of cells and then click on the output icon. The names for the input and output cells are chosen by @RISK based on the headings in the initial model. To change the name of an output cell/range of cells, or to delete an output cell/range of cells; click on the Display List of Outputs and Inputs icon (fifth from left; the icon with two arrows). This displays the @RISK Model window where the list of all the probability distributions for the input and output cells can be seen. The names can be edited here by simply typing over the existing name(s). 4. View Inputs and Outputs: @RISK keeps a list of all input cells (cells with the input probability distributions, @RISK’s random functions) and output cells. The list can be checked at any time by clicking on the Display List of Outputs and Inputs icon (red

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and blue arrows, fifth icon from the left) on the @RISK toolbar. This provides a list (as shown in Figure 6–11) that is like Windows Explorer. Next to this is a table showing the listing of the input and output cells and the assigned probability distributions for each.

Fig. 6–11 List of inputs and outputs

5. Input Summary Functions: @RISK provides several functions for summarizing output values once the simulation is run. The values in these cells are not of any use until the simulation is run. Once the simulation is run, these cells capture the desired summary statistics. Although the same summary, with additional details, also appears in @RISK’s output reports, some s prefer them to be in the same worksheet as the model. These functions are illustrated in Cells B9:H13 of Table 6–8. They contain the formulas a. Cell B9 ➯ =RISKMIN(B$6), for reporting the minimum value of porosity generated during simulation. b. Cell B10 ➯ =RISKMAX(B$6), for reporting the maximum value of porosity.

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c. Cell B11 ➯ =RISKMEAN(B$6), for reporting the average value of porosity. d. Cell B12 ➯ =RISKSTDDEV(B$6), for reporting the standard deviation of porosity. e. Cell B13 ➯ =RISKMODE(B$6), for reporting the mode of porosity. Once all the desired summary functions are input in Cells B9 to B13, then copy Cells B9:B13 ➯ Cells C9:H13. This generates the summary statistics for each variable in the model. Running the Simulation. The model for the volumetric reserves calculations is now ready for simulation. Before running the simulation, the simulation and report settings are specified, including • • • •

the desired type of sampling (Monte Carlo versus Hypercube), the number of desired iterations for simulation, the type of desired calculations (Monte Carlo or expected value) etc., and the details of desired outputs (graphs, sensitivities, detailed statistics, scenarios, etc.).

1. Select Simulation Settings: Here, @RISK is told how to run the simulation. To do so, click on the Simulation Settings icon on the @RISK toolbar. Click on the Iteration tab and fill out the desired number of iterations (we specify 100) as shown in the dialog box in Figure 6–12. This says the distributions will be sampled 100 times, each with a new random variable. If the number of iterations is not large enough, then each time the simulation is run significantly different results will generate. If the Update Display option is ticked, then when the simulation is running you will be able to see the calculations as each iteration is performed.

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Fig. 6–12 @RISK’s simulation settings dialog box (showing Iterations tab)

2. Select Type of Sampling: Next click on the Sampling tab and fill out the dialog box as shown in Figure 6–13. The Standard Recalc option in Figure 6–13 has three sub-options available. The selection of one of these options determines whether the random number generator (RNG) return random sample or expected values. The Expected Value option (default) rounds the expected value of a discrete RNG to its nearest integer value, whereas the True EV does not. These two options will generate the expected value of the output cell based on the assumption each input cell in the model assumed their expected values. However, in order for the RNGs to return random sample values, select the Monte Carlo option.

Fig. 6–13 @RISK’s simulation settings dialog box (showing Sampling tab) 360

Simulation in Decision Analysis

3. Select Report Settings: @RISK has many options for displaying outputs from a simulation. The outputs can be placed in an @RISK Results window or on new sheets in the existing Excel workbook, or both. Click on the Report Settings icon in the @RISK toolbar to select some of the options as shown in the dialog box of Figure 6–14. Once the simulation is run, @RISK automatically inserts additional worksheets in the desired workbook (see file TABLE 6–8.xls on the CD).

Fig. 6–14 @RISK’s report settings dialog box 4. Run the Simulation: Click on the Start Simulation icon (9th from left, looks like a normal distribution) on the @RISK toolbar. As soon as the Start Simulation icon is clicked, @RISK repeatedly generates a random number for each random input cell, recalculates the worksheet, and keeps track of all output cell values. The iteration in progress can be watched at the bottom left of the screen if the Update Display in the Simulation Settings dialog box has been ticked. Analyzing the Output. Analyzing the output of simulation is an important part of the whole process. @RISK generates a variety of reports 361

Project Economics and Decision Analysis Volume II: Probabilistic Models

as shown in Figure 6–14. Some of the most important and basic reports generated by @RISK are discussed in this section. When a simulation is completed, a screen as shown in Figure 6–15 appears along with additional icons in the @RISK toolbar used for viewing various reports. At the same time, the selected reports (the ones selected in Step 3, Select Report Settings, Figure 6–14) are automatically added to the active Excel workbook. Simulation reports and graphs generated in the Excel window are less interactive than those displayed in the @RISK – Results window. However, the Excel’s capabilities enable formatting the graphs and reports. In addition, @RISK reports generated in Excel can use pre-built template sheets that may contain custom formatting, titles, logos, and more.

Fig. 6–15 Summary of results provided by @RISK for the reserves calculation model

•

362

Summary Report: As shown in the summary of results, the average (or mean) value of reserves is approximately 215,389 barrels. The best-case and worst-case scenarios are also given in order to get an idea of the possible outcomes. This information is shown in Figure 6–15, with a minimum of 17,111 barrels and a maximum of 1,036,034 barrels. The summary also shows that for the 100 reserves values calculated, 5% were below 46,433.91 barrels and 95% were below 455,346 barrels. The summary measures reported here match those from =RISKMIN, =RISKMAX, and =RISK

Simulation in Decision Analysis

•

•

• •

MEAN functions used in Table 6–7. @RISK not only summarizes the output cell, but also the random input cells (porosity, water saturation, formation thickness, recovery factor, drainage area, and formation volume factor). Detailed Statistics: The detailed statistics of the input and output can be viewed by clicking on the Detailed Statistics icon. This report gives all of the information shown in Figure 6–15. In addition, it gives standard deviation, variance, skewness, kurtosis, and various percentiles. Target Values: By scrolling down the Detailed Statistics report to the bottom, one can enter any target value or target percentage. A target shows the probability of achieving a specific outcome or the value associated with any probability level. By entering a target value or percentage, @RISK calculates the corresponding percentage, and vice versa. This function enables answering questions such as “what is the chance of a result greater than 500,000 barrels?” Simulation Data: By clicking on the Data icon, the detailed calculations for input and output for each iteration can be viewed. Sensitivity Analysis Results: The sensitivity analysis results, as shown in Figure 6–16, are displayed by clicking on the Sensitivity Window icon. This report shows the sensitivity of each output variable to the input distribution in the model. This identifies the most critical inputs in the model. One should concentrate on these inputs while implementing the model. @RISK performs sensitivity analysis using either multivariate stepwise regression or a rank order correlation. As shown in Figure 6–16, the type of analysis desired is set using the Display Significant Inputs Using entry in the Sensitivity Window. The overall fit of the regression analysis is measured by the reported beta coefficient of the model. The lower the fit, the less stable the reported sensitivity statistics. The most influential variables are the ones with the largest coefficients (in absolute value).

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Fig. 6–16 The @RISK sensitivity analysis report

A beta coefficient for an independent variable indicates the number of standard deviations by which the dependent variable will increase/decrease if the independent variable increases by one standard deviation (assuming all other independent variables are held constant). Thus, if the drainage area is increased by one standard deviation, reserves will increase by 0.959 standard deviation. A negative beta coefficient means if the independent variable is increased by one standard deviation, then the dependent variable will decrease by the standard deviation equivalent to the beta coefficient. For example, if water saturation (with negative beta coefficient of 0.047) increases by one standard deviation, the reserves will decrease by 0.047 standard deviation. Sensitivity results are also shown graphically by a tornado graph as shown in Figure 6–17. A tornado graph is generated by right clicking on the desired output in the Explorer list (as shown in Figure 6–8) and selecting the Tornado Graph from the dropdown menu.

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Fig. 6–17 Tornado graph for sensitivity analysis

•

Scenario Analysis Results: The scenario analysis results can be viewed by clicking on the Scenario Window icon. Up to three scenario targets may be entered for each output variable. The @RISK Scenario option can also be used to do sensitivity analysis. To perform a sensitivity analysis with Scenarios, click on the Scenario icon (the icon with the % sign). @RISK is set at default scenarios of 75th, 25th, and 90th percentiles for the output cell. For each scenario, @RISK looks at the iterations that meet the scenario and identifies any random input variable whose median value for iterations meeting the scenario differs by 0.5 standard deviation or more from its overall median. By looking at the Scenario report (see worksheet in Table 6–8.xls on the CD), for the reserves >90% scenario, @RISK identified porosity (Cell B6) as having a median value of 0.58 standard deviation below the overall median for iterations in which the reserves were in the top 10% of all iterations.

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•

Viewing the Distributions: The probability distributions of the input and output cells can be easily viewed in @RISK. In addition, by specifying the Input Graphs and Output Graphs in the Report Setting dialog box (Figure 6–6), the graphs are displayed in the Excel workbook. For example, to view the graph of total reserves (output cell), right click on the Reserves in the @RISK Results window. Then click on the desired graph type from the drop-down menu. The resulting probability distribution is as shown in Figure 6–18. The graph is interactive; it can be easily changed into histogram or cumulative probability distribution by clicking on the respective graph icon in the @RISK toolbar.

Fig. 6–18 Distribution for reserves generated by @RISK

Example 6–3 In this example, a simulation model for a cash flow is developed. A typical deterministic cash flow model is set up in Excel as shown in Table 6–9. The deterministic cash flow model is first developed and then uncertainty is added to it by using the @RISK functions. Note that any existing cash flow models like the ones shown in Table 3–18 (Chapter 3 of Volume I) can be used for simulation.

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A

B

C

D

E

F

G

3 Input Parameters 4 Initial Production Rate

Stb/yr

25,000

5 Decline Rate

percent

12.0%

6 Gas—Oil Ratio

Scf/Stb

750

7 Oil Price

$/Stb

20

8 Gas Price

$/Mscf

1.2

9 Discount Rate

percent

12.0%

10 Gas Price Escalation

percent

3.0%

11 Oil Price Escalation

percent

4.0%

Production Forecast

Revenue

Costs

NCF

Disc. NCF

$

$

$

$

12 13 14

Year

Oil, Stb

Gas, MScf

15

2002

25,000

18,750

522,500

16

2003

22,173

16,630

481,753

20,000

461,753

368,107

17

2004

19,666

14,749

444,185

20,000

424,185

301,927

18

2005

17,442

13,081

409,549

20,000

389,549

247,565

19

2006

15,470

11,602

377,615

20,000

357,615

202,920

20

2007

13,720

10,290

348,172

20,000

328,172

166,262

21

2008

12,169

9,127

321,026

20,000

301,026

136,169

22

2009

10,793

8,095

295,997

20,000

275,997

111,471

23

2010

9,572

7,179

272,921

20,000

252,921

91,206

24

2011

8,490

6,367

251,645

20,000

231,645

74,583

25 TOTAL

154,494

115,871 3,725,362

1,980,000

1,745,362

559,584

1,800,000 (1,277,500) (1,140,625)

26 27 Net Present Value, $ 28 Rate of Return

559,584 25.8%

Table 6–9 Simulation of cash flow model

Solution: The deterministic cash flow model is developed as shown in Table 6–9. To add uncertainty to the uncertain variables, execute @RISK,

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open the Excel file TABLE 6–9.xls, and save it as TABLE 6–9a.xls. The following steps are required to add uncertainty to the model. 1. Assign normal probability distribution to the initial production rate with mean of 25,000 Stb/year and standard deviation of 6000 Stb/year. Cell D4➡ =RiskNormal(25000,6000) 2. Assign normal probability distribution to the decline rate with mean of 12% and standard deviation of 5%. Cell D5➡ =RiskNormal(0.12,0.05) 3. Assign triangular probability distribution to gas–oil ratio with minimum, most likely, and maximum values of 500, 750, and 1200 Scf/Stb, respectively. Cell D6➡ =RiskTriang(500,750,1200) 4. Add uncertainty to the oil price by asg triangular probability distribution with values of $12/Stb, $20/Stb, and $28/Stb, respectively. Cell D7➡ =RiskTriang(12,20,28) 5. Add uncertainty to the capital expenditure in the first year by asg triangular probability distribution with values of $1,600,000, $1,800,000, and $2,000,000. Cell E15➡ =RiskTriang(1600000,1800000,2000000) 6. If simulation at different discount rates is desired, the =RISKSIMTABLE is used. @RISK allows the to run several cases back-to-back. Each case is called a simulation and the number of simulations is set using the @RISK simulation settings. In this model, @RISK is asked to run three simulations at three different discount rates as specified in Cell D9. If the simulation is run for 200 iterations, the first 200 iterations will use 8% for a discount rate. This will be followed by another 200 iterations at 12% discount rate, and the third one at 16% discount rate, and so on. Cell D9➡ =RiskSimtable(0.08,0.12,0.16)

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7. The output cells are now specified. Three output cells (a) net present value (Cell C27), (b) the rate of return (Cell C28), and (c) the net cash flow (Cells F15 to F24) are specified. By asg multiple cells to the output, highlight the cells (for example, highlight Cells F15 to F24) and then click on Add Output icon in the @RISK toolbar (single down arrow, fourth icon from the left). Cell C27➡ =RiskOutput() + NPV(D9,F15:F24) Cell C28➡ =RiskOutput() + IRR(F15:F24) Cell F15➡ =RiskOutput(,”Net Cash Flow”,1) + D15-E15 As in the previous simulation example, the data and graphs generated by the simulation are stored in both the Excel workbook (TABLE 6–9a.xls) and the @RISK file (TABLE 6–9a.rsk). By clicking on the Open @RISK File icon (first icon on the @RISK toolbar), the @RISK file can be retrieved. All the simulation data generated during simulation can be viewed in these files. Figure 6–19 shows the band of uncertainty around the net cash flow generated by the simulation.

Fig. 6–19 The band of uncertainty around the net cash flow of Table 6–9a

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Handling Dependency on @RISK. In @RISK, dependency can be modeled by either inputting an equation to describe the nature of the relationship or a correlation coefficient. The correlation coefficient measures the strength of the association between the two variables. The @RISK functions used to for dependency are based on the Spearman rank order correlation coefficient. The rank order correlation between two variables is simply the ordinary correlations coefficient when the ranks are used in place of the raw data. This means if there are 30 porosity values ranging from 0.12 to 0.25, then the value 0.12 will be assigned a number 1, and the 0.25 will be assigned a number 30. Once these ranks are assigned to each variable, the correlation coefficient of these ranked variables is then calculated. The following @RISK functions are available when modeling dependency in @RISK. •

370

RISKINDEPC and RISKDEPC Functions. These two functions link two variables together. For example, porosity as independent variable and water saturation as dependent variable are represented by normal probability distributions. The parameters will be correlated using the RISKINDEPC and RISKDEPC functions as follows. a. For the independent porosity enter =RISKNORMAL( , s, RISK INDEPC(“PHI”)). The and s are the mean and standard deviation of porosity and “PHI” is simply an identifier for this pair of variables. There may be several pairs of related variables in the same worksheet, each pair with its unique identifier. Any other probability distribution can be used in the same way. b. For the dependent water saturation enter =RISKNORMAL ( Sw, s, RISKDEPC(“PHI”,Cc)). The Sw and s are the mean and standard deviation of water saturation. The Cc is the correlation coefficient that correlates the two variables (it could be any number from –1 to +1). To see how these functions work, enter the following functions in Excel worksheet and then run @RISK. The data (100 iterations) generated by the simulation are plotted as shown in Figure 6–20.

Simulation in Decision Analysis

A

B

1 Porosity

0.15

2 Water Saturation

0.35

Cell B1➡ =RiskNormal(0.15,0.05,RiskIndepC(“PHI”)) Cell B2➡ =RiskNormal(0.35,0.05,RiskDepC(“PHI”,-0.7))

Fig. 6–20 Correlation between and Sw, with correlation coefficient of – 0.70

•

RISKCORRMAT Function. The RISKCORRMAT is another @RISK function used to generate a group of random variables with a given correlation matrix. Suppose there are three normally distributed random variables each having a mean of 20 and a standard deviation of 5, which are correlated as follows. a. Variable 1 and Variable 2 are correlated with a correlation coefficient of 0.70, b. Variable 1 and Variable 3 are correlated with a correlation coefficient of 0.80, and

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c. Variable 2 and Variable 3 are correlated with a correlation coefficient of 0.65. To model the correlation of these variables, use =RiskCorrmat function of @RISK by simply entering the correlation matrix somewhere in the worksheet as shown in Table 6–10. The matrix in Table 6–10 represents the relationship between the three variables described here.

A 1

B

C

D

Variable 1

Variable 2

Variable 3

2

Variable 1

1.00

0.70

0.80

3

Variable 2

0.70

1.00

0.65

4

Variable 3

0.80

0.65

1.00

6

Variable 1

20

7

Variable 2

20

8

Variable 3

20

5

Table 6–10 Correlation matrix for three variables

In Cells B6, B7 and B8, enter the correlation functions with RiskNormal(20,5) probability distribution. Cell B6➡ =RiskNormal(20,5,RiskCorrmat(B2:D4,1)) ➡ 1 refers to Variable 1 Cell B7➡ =RiskNormal(20,5,RiskCormat(B2:D4,2)) ➡ 2 refers to Variable 2 Cell B8➡ =RiskNormal(20,5,RiskCorrmat(B2:D4,3)) ➡3 refers to Variable 3

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Simulation in Decision Analysis

Alternatively, correlations can be specified in a matrix by using the @RISK Model Window. Click on @RISK Model Window icon ➡ Click on Insert in the toolbar menu ➡ Click on Correlation Window in the drop down menu. A 2 x 2 matrix appears. Right Click on the NewMatrix (2x2) and click on Insert Row/Column in the drop down menu, add another row and column. Now the correlation coefficients are entered as shown in Figure 6–21. Click on Apply. The matrix is now ready in the worksheet and can be used for simulation modeling.

Fig. 6–21 Correlation window of @RISK

COMBINING PRECISIONTREE AND @RISK In Chapter 3, decision trees were solved where chance events were represented in of discrete outcomes (a chance node with finite number of outcome branches). In real life, many of these uncertain events may not be discrete. They will be represented by many continuous (infinite) uncertain

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events. For example, in the decision tree of Figure 3–9 (Chapter 3), the chance nodes were • • •

A dry hole probability of 60% with an estimated dry hole cost of $65,000, A producer with a 30% chance of encountering 60,000 Stbs of oil reserves with NPV of $120,000, and A producer with a 10% chance of encountering 90,000 Stbs of oil reserves with NPV of $180,000.

The same decision tree can be represented by probability distributions as (a) a dry hole cost represented by a triangular distribution, and (b) the reserves represented by a normal distribution. Using @RISK with PrecisionTree makes the modeling of such continuous events easier. During a normal decision analysis, the expected value of the distribution function will be used as the value for the branch. The expected value for a path in the tree will be calculated using this value. The following steps are required to use @RISK and PrecisionTree in combination. 1. Open PrecisionTree and set up the desired tree following the procedure presented in Chapter 3. The tree will look like the one shown in Figure 6–22.

Fig. 6–22 Decision tree used with @RISK 374

Simulation in Decision Analysis

2. Add probability distributions to the chance nodes as Cell C3 ➯ =–RiskTriang(55,65,80) Cell C7 ➯ =RiskNormal(150,50) 3. Click on the @RISK icon in the DecisionTools’ toolbar to load @RISK. Once the @RISK is loaded, follow these steps. a. Specify the output cell by clicking on the Add Output icon (fourth icon with single red arrow) on the @RISK toolbar, move the cursor to Cell B9 and press Enter. b. Click on Simulation Settings icon (seventh from left) to select the number of iterations, preferred sampling (LHS), and simulation (Monte Carlo). c. Click on the Report Settings icon (eighth from left) to select reports, etc. d. Click Decisions Follow Current Optimal Path option in the @RISK dialog box before running simulation. e. Click on Start Simulation icon to start simulation. The final tree with EMV of the Drill option is shown in Figure 6–22. The selected simulation reports are generated and saved in the files FIGURE 6–22.xls and FIGURE 6–22.rsk.

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REFERENCES 1

2

3

4

5

6

7

8

9

10

376

Hertz, D. B., “Risk Analysis in Capital Investment,” Harvard Business Review, Volume 57, Number 5, September–October, 1979, pp. 169–181. Murtha, J.A., “Monte Carlo Simulation: Its Status and Future,” Journal of Petroleum Technology, April 1997, pp. 361–373. Murtha, J.A., Corrections to “Monte Carlo Simulation: Its Status and Future,” Journal of Petroleum Technology, July 1997, pp. 756. Peterson, S. K., Murtha, J. A., and Schneider, F. F., “Brief: Risk Analysis and Monte Carlo Simulation Applied to the Generation of Drilling AFE Estimates,” Journal of Petroleum Technology, June 1995, pp. 504–506. Dhir, R., Dern, R. R., Jr., and Mavor, M. J., “Economic and Reserve Evaluation of Coalbed Methane Reservoirs,” Journal of Petroleum Technology, December 1991, pp. 1424–1431. Sears, G. F. and Phillips, N. V., “Fractured Reservoir Evaluation Using Monte Carlo Techniques,” Journal of Petroleum Technology, January 1987, pp. 71–76. Wiggins, M. L. and Zhang, Xu, “Using PCs and Monte Carlo Simulation to Assess Risk in Workover Evaluations,” Society of Petroleum Engineers, 1994. Sampson. D., Managerial Decision Analysis, Richard D. Irwin, Inc., Homewood, Illinois 60430, USA, 1988. Harbaugh, J. W., Davis, J. C., and Wendeboutg, J., Computing Risk for oil Prospects: Principles and Programs, Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK, 1995. Law, A. M. and Kelton, W. D., Simulation Modeling & Analysis, 2nd Edition, McGraw-Hill, Inc., USA, 1991.

Simulation in Decision Analysis

11

Newendorp, P. D., Decision Analysis for Petroleum Exploration, PennWell Publishing Co., Tulsa, Oklahoma, USA, 1975, pp. 409–430. —-Newendorp, P. D. and Schuyler, J., Decision Analysis for Petroleum Exploration, 2nd Edition, Planning Press, 15492 East Chenango Avenue, Aurora, CO 80015, USA, 2000.

377

Afterword

AFTERWORD The Monte Carlo simulation is a converging process. However, there is no hard and fast rule for selecting the number of iterations/replications required for a certain simulation model. The required number of iterations will vary from model to model and depends on the complexity of the model. With complex simulation models, running thousands of iterations may be time consuming. Therefore, there has to be a trade-off between the numbers of iterations and the time required to run these iterations. The following steps are recommended to determine the adequate number of iterations. 1. Identify an important output cell, preferably a single-cell output (for example, the NPV in a cash-flow model or reserves in volumetrics model). 2. Run the simulation, starting with 200 iterations. 3. Generate the CDF for the single-cell output selected in Step 1. Copy the CDF to the clipboard. 4. Run the simulation again, this time using 400 iterations. 5. Overlay the CDF for this output with the CDF generated in previous number of trials. This process is repeated until a satisfactory match between the current and the previous CDF is obtained, i.e. further increases in iterations would not yield significant changes in the CDF.

SUMMARIZING THE DECISION ANALYSIS PROCESS The 13 chapters in these 2 companion volumes present the processes and techniques used for developing an appropriate framework to perform decision analysis. Volume I dealt with the deterministic phase and Volume II

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Project Economics and Decision Analysis Volume II: Probabilistic Models

deals with the probabilistic phase of the decision analysis. All these chapters form the basis for understanding and developing good evaluations. The objective of decision analysis is to systematically lead the decision maker through the issues that make the decisions complicated. The requisite decision model should therefore capture all the essential elements of the problem. The model, whether it is a complex simulation model or a deterministic cash-flow model, should be valid and credible. Validation is concerned with determining whether the conceptual model is the actual representation of the real-life system. The model is called a credible model when its results are accepted by the decision maker as being valid, and are used as an aid in making decisions. The decision analysis cycle, on the macro level, was presented in Chapter 1. To provide the decision maker with a clear and compelling course of action, this section summarizes the recommended logical sequence for detailed analysis.1 •

380

Structuring Phase: Develop a shared understanding of the decision problem and generate a credible set of alternatives to evaluate. This phase requires the following steps. a. Identify the Problem or Opportunity. Understand the decision problem from the perspective of the decision maker. b. Generate Creative Alternatives. Develop a small but distinctly different list of alternatives. Some analysts will generate a long list of possible alternatives but only two or three of these will be truly different. Do not spend time on developing all the possible alternatives with only minor variations. The alternatives should be realistic (technically and financially viable). There is no sense in considering alternatives that are not technically viable even if they are commercially viable. c. Model the Situation. This step involves building a decision diagram and a quantitative model of the problem or opportunity. The decision diagram (such as decision trees) should convey the essence of the problem in a logical and understandable manner. The quantitative decision model (such as the spreadsheet cash flow model) should be designed to achieve clarity of action.

Afterword

•

Evaluation Phase: This phase involves analysis of the alternatives generated in the structuring phase. The objective is to learn where and why value is created. a. Discover What is Important. This step involves what-if analysis. Deterministic sensitivity analyses are performed to identify the key (critical) variables in the model. Use these key variables to further simplify the model and clarify the course of action. Use a tornado diagram to graphically display the dominating variables. Any inferior courses of action can be dropped at this stage. The most sensitive variables are then included in the probabilistic analysis. Palisade’s TopRank tool can be used to perform the sensitivity analysis. b. Quantify Risk and Return. This is the probabilistic phase of evaluation. This step involves modeling and analyzing uncertainties surrounding the critical uncertain variables identified in the previous step. Probabilities are assigned to these variables. Decision trees and/or a combination of decision trees and Monte Carlo simulation are used at this stage to deal with the risk of the project. Risk profiles should be used to quickly ascertain (i) the upside and downside potential of each alternative, along with the risk/reward trade-off and (ii) the relative risk of one strategy to another. Perform sensitivity analysis on the probabilities to identify changes in optimal policy with changes in the state of probabilities. The overall analysis must conform to the decision maker’s risk tolerance. Therefore, the risk attitudes of the decision maker must be included in the analysis when required. c. Determine the Value of Information. Acquiring new information about key uncertain variables may change the decision. Determine the value of information (value of perfect information and value of imperfect information). Assess the economic viability of acquiring such information. Is it worthwhile to spend money on acquiring additional information?

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Agreement Phase: This phase involves bringing together all parts of the analysis in a coherent and compelling form so that the decision maker can take an immediate action on the appropriate course of action. a. Decide on the Course of Action. This is the decision making stage. At this stage, any refinements that may change the recommended course of action should be focused on. b. Allocate the Appropriate Resources. At this point, the proposed course of action is chosen. A list of human and financial resources, necessary for the implementation of the decision, should be developed. c. Implement the Course of Action. This step involves the actual execution of the decision.

•

REFERENCES 1

382

Skinner, D. C., Introduction to Decision Analysis, Second Edition, Probabilistic Publishing, 5715 NW 67 Court, Gainesville, FL 32653, USA, 1999.

This book is dedicated to my brother, the late M. Rashid Mian, who ed away in a tragic car accident, and the memories that we shared together for more than 40 years. It was his idea to write this book with me. I have incorporated some of his handwritten notes on production-sharing economics in Chapter 5 of Volume I. He was a warm, dynamic, vivacious, and fun-loving person. He was truly one of the most beloved and ired of our family. We loved him deeply, and his loss leaves an enormous void in all our lives. May God bless his soul in peace.

Appendix A Probability Tables

z=

X − μ 15 − 12 = = 1.2 2.5 σ

f (x )

N(12, 2.5)

Z=1.2

0.3849

Mean 2

4

6

8

10

12

14

16

18

20

22

Porosity, Percent z 0.0 0.1 0.2 0.3 0.4

0.00 0.0398 0.0793 0.1179 0.1554

0.01 0.0040 0.0438 0.0832 0.1217 0.1591

0.02 0.0080 0.0478 0.0871 0.1255 0.1628

0.03 0.0120 0.0517 0.0910 0.1293 0.1664

0.04 0.0160 0.0557 0.0948 0.1331 0.1700

0.05 0.0199 0.0596 0.0987 0.1368 0.1736

0.06 0.0239 0.0636 0.1026 0.1406 0.1772

0.07 0.0279 0.0675 0.1064 0.1443 0.1808

0.08 0.0319 0.0714 0.1103 0.1480 0.1844

0.09 0.0359 0.0753 0.1141 0.1517 0.1879

0.5 0.6 0.7 0.8 0.9

0.1915 0.2257 0.2580 0.2881 0.3159

0.1950 0.2291 0.2611 0.2910 0.3186

0.1985 0.2324 0.2642 0.2939 0.3212

0.2019 0.2357 0.2673 0.2967 0.3238

0.2054 0.2389 0.2704 0.2995 0.3264

0.2088 0.2422 0.2734 0.3023 0.3289

0.2123 0.2454 0.2764 0.3051 0.3315

0.2157 0.2486 0.2794 0.3078 0.3340

0.2190 0.2517 0.2823 0.3106 0.3365

0.2224 0.2549 0.2852 0.3133 0.3389

Table A–1 Standard Normal Probability Distribution

Project Economics and Decision Analysis Volume II: Probabilistic Models

z 1.0 1.1 1.2 1.3 1.4

0.00 0.3413 0.3643 0.3849 0.4032 0.4192

0.01 0.3438 0.3665 0.3869 0.4049 0.4207

0.02 0.3461 0.3686 0.3888 0.4066 0.4222

0.03 0.3485 0.3708 0.3907 0.4082 0.4236

0.04 0.3508 0.3729 0.3925 0.4099 0.4251

0.05 0.3531 0.3749 0.3944 0.4115 0.4265

0.06 0.3554 0.3770 0.3962 0.4131 0.4279

0.07 0.3577 0.3790 0.3980 0.4147 0.4292

0.08 0.3599 0.3810 0.3997 0.4162 0.4306

0.09 0.3621 0.3830 0.4015 0.4177 0.4319

1.5 1.6 1.7 1.8 1.9

0.4332 0.4452 0.4554 0.4641 0.4713

0.4345 0.4463 0.4564 0.4649 0.4719

0.4357 0.4474 0.4573 0.4656 0.4726

0.4370 0.4484 0.4582 0.4664 0.4732

0.4382 0.4495 0.4591 0.4671 0.4738

0.4394 0.4505 0.4599 0.4678 0.4744

0.4406 0.4515 0.4608 0.4686 0.4750

0.4418 0.4525 0.4616 0.4693 0.4756

0.4429 0.4535 0.4625 0.4699 0.4761

0.4441 0.4545 0.4633 0.4706 0.4767

2.0 2.1 2.2 2.3 2.4

0.4772 0.4821 0.4861 0.4893 0.4918

0.4778 0.4826 0.4864 0.4896 0.4920

0.4783 0.4830 0.4868 0.4898 0.4922

0.4788 0.4834 0.4871 0.4901 0.4925

0.4793 0.4838 0.4875 0.4904 0.4927

0.4798 0.4842 0.4878 0.4906 0.4929

0.4803 0.4846 0.4881 0.4909 0.4931

0.4808 0.4850 0.4884 0.4911 0.4932

0.4812 0.4854 0.4887 0.4913 0.4934

0.4817 0.4857 0.4890 0.4916 0.4936

2.5 2.6 2.7 2.8 2.9

0.4938 0.4953 0.4965 0.4974 0.4981

0.4940 0.4955 0.4966 0.4975 0.4982

0.4941 0.4956 0.4967 0.4976 0.4982

0.4943 0.4957 0.4968 0.4977 0.4983

0.4945 0.4959 0.4969 0.4977 0.4984

0.4946 0.4960 0.4970 0.4978 0.4984

0.4948 0.4961 0.4971 0.4979 0.4985

0.4949 0.4962 0.4972 0.4979 0.4985

0.4951 0.4963 0.4973 0.4980 0.4986

0.4952 0.4964 0.4974 0.4981 0.4986

3.0 3.1 3.2 3.3 3.4

0.4987 0.4990 0.4993 0.4995 0.4997

0.4987 0.4991 0.4993 0.4995 0.4997

0.4987 0.4991 0.4994 0.4995 0.4997

0.4988 0.4991 0.4994 0.4996 0.4997

0.4988 0.4992 0.4994 0.4996 0.4997

0.4989 0.4992 0.4994 0.4996 0.4997

0.4989 0.4992 0.4994 0.4996 0.4997

0.4989 0.4992 0.4995 0.4996 0.4997

0.4990 0.4993 0.4995 0.4996 0.4997

0.4990 0.4993 0.4995 0.4997 0.4998

3.5 3.6 3.7 3.8 3.9 4.0

0.4998 0.4998 0.4999 0.4999 0.5000 0.5000

0.4998 0.4998 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

0.4998 0.4999 0.4999 0.4999 0.5000 0.5000

Table A–1 continued . . .

384

Appendix A: Probability Tables

Entry is Probability

x

P (X = x ) =

λx exp(− λ ) x!

λ

0 1 2 3 4

0.1 0.9048 0.0905 0.0045 0.0002 0.0000

0.2 0.8187 0.1637 0.0164 0.0011 0.0001

0.3 0.7408 0.2222 0.0333 0.0033 0.0003

0.4 0.6703 0.2681 0.0536 0.0072 0.0007

0.5 0.6065 0.3033 0.0758 0.0126 0.0016

0.6 0.5488 0.3293 0.0988 0.0198 0.0030

0.7 0.4966 0.3476 0.1217 0.0284 0.0050

0.8 0.4493 0.3595 0.1438 0.0383 0.0077

0.9 0.4066 0.3659 0.1647 0.0494 0.0111

1.0 0.3679 0.3679 0.1839 0.0613 0.0153

5 6 7

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0001 0.0000 0.0000

0.0002 0.0000 0.0000

0.0004 0.0000 0.0000

0.0007 0.0001 0.0000

0.0012 0.0002 0.0000

0.0020 0.0003 0.0000

0.0031 0.0005 0.0001

0 1 2 3 4

1.1 0.3329 0.3662 0.2014 0.0738 0.0203

1.2 0.3012 0.3614 0.2169 0.0867 0.0260

1.3 0.2725 0.3543 0.2303 0.0998 0.0324

1.4 0.2466 0.3452 0.2417 0.1128 0.0395

1.5 0.2231 0.3347 0.2510 0.1255 0.0471

1.6 0.2019 0.3230 0.2584 0.1378 0.0551

1.7 0.1827 0.3106 0.2640 0.1496 0.0636

1.8 0.1653 0.2975 0.2678 0.1607 0.0723

1.9 0.1496 0.2842 0.2700 0.1710 0.0812

2.0 0.1353 0.2707 0.2707 0.1804 0.0902

5 6 7 8 9

0.0045 0.0008 0.0001 0.0000 0.0000

0.0062 0.0012 0.0002 0.0000 0.0000

0.0084 0.0018 0.0003 0.0001 0.0000

0.0111 0.0026 0.0005 0.0001 0.0000

0.0141 0.0035 0.0008 0.0001 0.0000

0.0176 0.0047 0.0011 0.0002 0.0000

0.0216 0.0061 0.0015 0.0003 0.0001

0.0260 0.0078 0.0020 0.0005 0.0001

0.0309 0.0098 0.0027 0.0006 0.0001

0.0361 0.0120 0.0034 0.0009 0.0002

0 1 2 3 4

2.1 0.1225 0.2572 0.2700 0.1890 0.0992

2.2 0.1108 0.2438 0.2681 0.1966 0.1082

2.3 0.1003 0.2306 0.2652 0.2033 0.1169

2.4 0.0907 0.2177 0.2613 0.2090 0.1254

2.5 0.0821 0.2052 0.2565 0.2138 0.1336

2.6 0.0743 0.1931 0.2510 0.2176 0.1414

2.7 0.0672 0.1815 0.2450 0.2205 0.1488

2.8 0.0608 0.1703 0.2384 0.2225 0.1557

2.9 0.0550 0.1596 0.2314 0.2237 0.1622

3.0 0.0498 0.1494 0.2240 0.2240 0.1680

5 6 7 8 9

0.0417 0.0146 0.0044 0.0011 0.0003

0.0476 0.0174 0.0055 0.0015 0.0004

0.0538 0.0206 0.0068 0.0019 0.0005

0.0602 0.0241 0.0083 0.0025 0.0007

0.0668 0.0278 0.0099 0.0031 0.0009

0.0735 0.0319 0.0118 0.0038 0.0011

0.0804 0.0362 0.0139 0.0047 0.0014

0.0872 0.0407 0.0163 0.0057 0.0018

0.0940 0.0455 0.0188 0.0068 0.0022

0.1008 0.0504 0.0216 0.0081 0.0027

10 11 12

0.0001 0.0000 0.0000

0.0001 0.0000 0.0000

0.0001 0.0000 0.0000

0.0002 0.0000 0.0000

0.0002 0.0000 0.0000

0.0003 0.0001 0.0000

0.0004 0.0001 0.0000

0.0005 0.0001 0.0000

0.0006 0.0002 0.0000

0.0008 0.0002 0.0001

x

x

λ

λ

Example: For = 14 and x = 8, P(X = 8) = 0.0304

Table A–2 Poisson Probabilities

385

Project Economics and Decision Analysis Volume II: Probabilistic Models

x

λ

0 1 2 3 4

3.1 0.0450 0.1397 0.2165 0.2237 0.1733

3.2 0.0408 0.1304 0.2087 0.2226 0.1781

3.3 0.0369 0.1217 0.2008 0.2209 0.1823

3.4 0.0334 0.1135 0.1929 0.2186 0.1858

3.5 0.0302 0.1057 0.1850 0.2158 0.1888

3.6 0.0273 0.0984 0.1771 0.2125 0.1912

3.7 0.0247 0.0915 0.1692 0.2087 0.1931

3.8 0.0224 0.0850 0.1615 0.2046 0.1944

3.9 0.0202 0.0789 0.1539 0.2001 0.1951

4.0 0.0183 0.0733 0.1465 0.1954 0.1954

5 6 7 8 9

0.1075 0.0555 0.0246 0.0095 0.0033

0.1140 0.0608 0.0278 0.0111 0.0040

0.1203 0.0662 0.0312 0.0129 0.0047

0.1264 0.0716 0.0348 0.0148 0.0056

0.1322 0.0771 0.0385 0.0169 0.0066

0.1377 0.0826 0.0425 0.0191 0.0076

0.1429 0.0881 0.0466 0.0215 0.0089

0.1477 0.0936 0.0508 0.0241 0.0102

0.1522 0.0989 0.0551 0.0269 0.0116

0.1563 0.1042 0.0595 0.0298 0.0132

10 11 12 13 14

0.0010 0.0003 0.0001 0.0000 0.0000

0.0013 0.0004 0.0001 0.0000 0.0000

0.0016 0.0005 0.0001 0.0000 0.0000

0.0019 0.0006 0.0002 0.0000 0.0000

0.0023 0.0007 0.0002 0.0001 0.0000

0.0028 0.0009 0.0003 0.0001 0.0000

0.0033 0.0011 0.0003 0.0001 0.0000

0.0039 0.0013 0.0004 0.0001 0.0000

0.0045 0.0016 0.0005 0.0002 0.0000

0.0053 0.0019 0.0006 0.0002 0.0001

0 1 2 3 4

4.1 0.0166 0.0679 0.1393 0.1904 0.1951

4.2 0.0150 0.0630 0.1323 0.1852 0.1944

4.3 0.0136 0.0583 0.1254 0.1798 0.1933

4.4 0.0123 0.0540 0.1188 0.1743 0.1917

4.5 0.0111 0.0500 0.1125 0.1687 0.1898

4.6 0.0101 0.0462 0.1063 0.1631 0.1875

4.7 0.0091 0.0427 0.1005 0.1574 0.1849

4.8 0.0082 0.0395 0.0948 0.1517 0.1820

4.9 0.0074 0.0365 0.0894 0.1460 0.1789

5.0 0.0067 0.0337 0.0842 0.1404 0.1755

5 6 7 8 9

0.1600 0.1093 0.0640 0.0328 0.0150

0.1633 0.1143 0.0686 0.0360 0.0168

0.1662 0.1191 0.0732 0.0393 0.0188

0.1687 0.1237 0.0778 0.0428 0.0209

0.1708 0.1281 0.0824 0.0463 0.0232

0.1725 0.1323 0.0869 0.0500 0.0255

0.1738 0.1362 0.0914 0.0537 0.0281

0.1747 0.1398 0.0959 0.0575 0.0307

0.1753 0.1432 0.1002 0.0614 0.0334

0.1755 0.1462 0.1044 0.0653 0.0363

10 11 12 13 14

0.0061 0.0023 0.0008 0.0002 0.0001

0.0071 0.0027 0.0009 0.0003 0.0001

0.0081 0.0032 0.0011 0.0004 0.0001

0.0092 0.0037 0.0013 0.0005 0.0001

0.0104 0.0043 0.0016 0.0006 0.0002

0.0118 0.0049 0.0019 0.0007 0.0002

0.0132 0.0056 0.0022 0.0008 0.0003

0.0147 0.0064 0.0026 0.0009 0.0003

0.0164 0.0073 0.0030 0.0011 0.0004

0.0181 0.0082 0.0034 0.0013 0.0005

15

0.0000

0.0000

0.0000

0.0000

0.0001

0.0001

0.0001

0.0001

0.0001

0.0002

x

λ

Table A–2 continued . . .

386

Appendix A: Probability Tables

x

λ

0 1 2 3 4

5.1 0.0061 0.0311 0.0793 0.1348 0.1719

5.2 0.0055 0.0287 0.0746 0.1293 0.1681

5.3 0.0050 0.0265 0.0701 0.1239 0.1641

5.4 0.0045 0.0244 0.0659 0.1185 0.1600

5.5 0.0041 0.0225 0.0618 0.1133 0.1558

5.6 0.0037 0.0207 0.0580 0.1082 0.1515

5.7 0.0033 0.0191 0.0544 0.1033 0.1472

5.8 0.0030 0.0176 0.0509 0.0985 0.1428

5.9 0.0027 0.0162 0.0477 0.0938 0.1383

6.0 0.0025 0.0149 0.0446 0.0892 0.1339

5 6 7 8 9

0.1753 0.1490 0.1086 0.0692 0.0392

0.1748 0.1515 0.1125 0.0731 0.0423

0.1740 0.1537 0.1163 0.0771 0.0454

0.1728 0.1555 0.1200 0.0810 0.0486

0.1714 0.1571 0.1234 0.0849 0.0519

0.1697 0.1584 0.1267 0.0887 0.0552

0.1678 0.1594 0.1298 0.0925 0.0586

0.1656 0.1601 0.1326 0.0962 0.0620

0.1632 0.1605 0.1353 0.0998 0.0654

0.1606 0.1606 0.1377 0.1033 0.0688

10 11 12 13 14

0.0200 0.0093 0.0039 0.0015 0.0006

0.0220 0.0104 0.0045 0.0018 0.0007

0.0241 0.0116 0.0051 0.0021 0.0008

0.0262 0.0129 0.0058 0.0024 0.0009

0.0285 0.0143 0.0065 0.0028 0.0011

0.0309 0.0157 0.0073 0.0032 0.0013

0.0334 0.0173 0.0082 0.0036 0.0015

0.0359 0.0190 0.0092 0.0041 0.0017

0.0386 0.0207 0.0102 0.0046 0.0019

0.0413 0.0225 0.0113 0.0052 0.0022

15 16 17

0.0002 0.0001 0.0000

0.0002 0.0001 0.0000

0.0003 0.0001 0.0000

0.0003 0.0001 0.0000

0.0004 0.0001 0.0000

0.0005 0.0002 0.0001

0.0006 0.0002 0.0001

0.0007 0.0002 0.0001

0.0008 0.0003 0.0001

0.0009 0.0003 0.0001

0 1 2 3 4

6.1 0.0022 0.0137 0.0417 0.0848 0.1294

6.2 0.0020 0.0126 0.0390 0.0806 0.1249

6.3 0.0018 0.0116 0.0364 0.0765 0.1205

6.4 0.0017 0.0106 0.0340 0.0726 0.1162

6.5 0.0015 0.0098 0.0318 0.0688 0.1118

6.6 0.0014 0.0090 0.0296 0.0652 0.1076

6.7 0.0012 0.0082 0.0276 0.0617 0.1034

6.8 0.0011 0.0076 0.0258 0.0584 0.0992

6.9 0.0010 0.0070 0.0240 0.0552 0.0952

7.0 0.0009 0.0064 0.0223 0.0521 0.0912

5 6 7 8 9

0.1579 0.1605 0.1399 0.1066 0.0723

0.1549 0.1601 0.1418 0.1099 0.0757

0.1519 0.1595 0.1435 0.1130 0.0791

0.1487 0.1586 0.1450 0.1160 0.0825

0.1454 0.1575 0.1462 0.1188 0.0858

0.1420 0.1562 0.1472 0.1215 0.0891

0.1385 0.1546 0.1480 0.1240 0.0923

0.1349 0.1529 0.1486 0.1263 0.0954

0.1314 0.1511 0.1489 0.1284 0.0985

0.1277 0.1490 0.1490 0.1304 0.1014

10 11 12 13 14

0.0441 0.0244 0.0124 0.0058 0.0025

0.0469 0.0265 0.0137 0.0065 0.0029

0.0498 0.0285 0.0150 0.0073 0.0033

0.0528 0.0307 0.0164 0.0081 0.0037

0.0558 0.0330 0.0179 0.0089 0.0041

0.0588 0.0353 0.0194 0.0099 0.0046

0.0618 0.0377 0.0210 0.0108 0.0052

0.0649 0.0401 0.0227 0.0119 0.0058

0.0679 0.0426 0.0245 0.0130 0.0064

0.0710 0.0452 0.0263 0.0142 0.0071

15 16 17 18 19

0.0010 0.0004 0.0001 0.0000 0.0000

0.0012 0.0005 0.0002 0.0001 0.0000

0.0014 0.0005 0.0002 0.0001 0.0000

0.0016 0.0006 0.0002 0.0001 0.0000

0.0018 0.0007 0.0003 0.0001 0.0000

0.0020 0.0008 0.0003 0.0001 0.0000

0.0023 0.0010 0.0004 0.0001 0.0001

0.0026 0.0011 0.0004 0.0002 0.0001

0.0029 0.0013 0.0005 0.0002 0.0001

0.0033 0.0014 0.0006 0.0002 0.0001

x

λ

Table A–2 continued . . .

387

Project Economics and Decision Analysis Volume II: Probabilistic Models

x

λ

0 1 2 3 4

7.1 0.0008 0.0059 0.0208 0.0492 0.0874

7.2 0.0007 0.0054 0.0194 0.0464 0.0836

7.3 0.0007 0.0049 0.0180 0.0438 0.0799

7.4 0.0006 0.0045 0.0167 0.0413 0.0764

7.5 0.0006 0.0041 0.0156 0.0389 0.0729

7.6 0.0005 0.0038 0.0145 0.0366 0.0696

7.7 0.0005 0.0035 0.0134 0.0345 0.0663

7.8 0.0004 0.0032 0.0125 0.0324 0.0632

7.9 0.0004 0.0029 0.0116 0.0305 0.0602

8.0 0.0003 0.0027 0.0107 0.0286 0.0573

5 6 7 8 9

0.1241 0.1468 0.1489 0.1321 0.1042

0.1204 0.1445 0.1486 0.1337 0.1070

0.1167 0.1420 0.1481 0.1351 0.1096

0.1130 0.1394 0.1474 0.1363 0.1121

0.1094 0.1367 0.1465 0.1373 0.1144

0.1057 0.1339 0.1454 0.1381 0.1167

0.1021 0.1311 0.1442 0.1388 0.1187

0.0986 0.1282 0.1428 0.1392 0.1207

0.0951 0.1252 0.1413 0.1395 0.1224

0.0916 0.1221 0.1396 0.1396 0.1241

10 11 12 13 14

0.0740 0.0478 0.0283 0.0154 0.0078

0.0770 0.0504 0.0303 0.0168 0.0086

0.0800 0.0531 0.0323 0.0181 0.0095

0.0829 0.0558 0.0344 0.0196 0.0104

0.0858 0.0585 0.0366 0.0211 0.0113

0.0887 0.0613 0.0388 0.0227 0.0123

0.0914 0.0640 0.0411 0.0243 0.0134

0.0941 0.0667 0.0434 0.0260 0.0145

0.0967 0.0695 0.0457 0.0278 0.0157

0.0993 0.0722 0.0481 0.0296 0.0169

15 16 17 18 19

0.0037 0.0016 0.0007 0.0003 0.0001

0.0041 0.0019 0.0008 0.0003 0.0001

0.0046 0.0021 0.0009 0.0004 0.0001

0.0051 0.0024 0.0010 0.0004 0.0002

0.0057 0.0026 0.0012 0.0005 0.0002

0.0062 0.0030 0.0013 0.0006 0.0002

0.0069 0.0033 0.0015 0.0006 0.0003

0.0075 0.0037 0.0017 0.0007 0.0003

0.0083 0.0041 0.0019 0.0008 0.0003

0.0090 0.0045 0.0021 0.0009 0.0004

0 1 2 3 4

8.1 0.0003 0.0025 0.0100 0.0269 0.0544

8.2 0.0003 0.0023 0.0092 0.0252 0.0517

8.3 0.0002 0.0021 0.0086 0.0237 0.0491

8.4 0.0002 0.0019 0.0079 0.0222 0.0466

8.5 0.0002 0.0017 0.0074 0.0208 0.0443

8.6 0.0002 0.0016 0.0068 0.0195 0.0420

8.7 0.0002 0.0014 0.0063 0.0183 0.0398

8.8 0.0002 0.0013 0.0058 0.0171 0.0377

8.9 0.0001 0.0012 0.0054 0.0160 0.0357

9.0 0.0001 0.0011 0.0050 0.0150 0.0337

5 6 7 8 9

0.0882 0.1191 0.1378 0.1395 0.1256

0.0849 0.1160 0.1358 0.1392 0.1269

0.0816 0.1128 0.1338 0.1388 0.1280

0.0784 0.1097 0.1317 0.1382 0.1290

0.0752 0.1066 0.1294 0.1375 0.1299

0.0722 0.1034 0.1271 0.1366 0.1306

0.0692 0.1003 0.1247 0.1356 0.1311

0.0663 0.0972 0.1222 0.1344 0.1315

0.0635 0.0941 0.1197 0.1332 0.1317

0.0607 0.0911 0.1171 0.1318 0.1318

10 11 12 13 14

0.1017 0.0749 0.0505 0.0315 0.0182

0.1040 0.0776 0.0530 0.0334 0.0196

0.1063 0.0802 0.0555 0.0354 0.0210

0.1084 0.0828 0.0579 0.0374 0.0225

0.1104 0.0853 0.0604 0.0395 0.0240

0.1123 0.0878 0.0629 0.0416 0.0256

0.1140 0.0902 0.0654 0.0438 0.0272

0.1157 0.0925 0.0679 0.0459 0.0289

0.1172 0.0948 0.0703 0.0481 0.0306

0.1186 0.0970 0.0728 0.0504 0.0324

15 16 17 18 19

0.0098 0.0050 0.0024 0.0011 0.0005

0.0107 0.0055 0.0026 0.0012 0.0005

0.0116 0.0060 0.0029 0.0014 0.0006

0.0126 0.0066 0.0033 0.0015 0.0007

0.0136 0.0072 0.0036 0.0017 0.0008

0.0147 0.0079 0.0040 0.0019 0.0009

0.0158 0.0086 0.0044 0.0021 0.0010

0.0169 0.0093 0.0048 0.0024 0.0011

0.0182 0.0101 0.0053 0.0026 0.0012

0.0194 0.0109 0.0058 0.0029 0.0014

x

λ

Table A–2 continued . . .

388

Appendix A: Probability Tables

x

λ

0 1 2 3 4

9.1 0.0001 0.0010 0.0046 0.0140 0.0319

9.2 0.0001 0.0009 0.0043 0.0131 0.0302

9.3 0.0001 0.0009 0.0040 0.0123 0.0285

9.4 0.0001 0.0008 0.0037 0.0115 0.0269

9.5 0.0001 0.0007 0.0034 0.0107 0.0254

9.6 0.0001 0.0007 0.0031 0.0100 0.0240

9.7 0.0001 0.0006 0.0029 0.0093 0.0226

9.8 0.0001 0.0005 0.0027 0.0087 0.0213

9.9 0.0001 0.0005 0.0025 0.0081 0.0201

10.0 0.0000 0.0005 0.0023 0.0076 0.0189

5 6 7 8 9

0.0581 0.0881 0.1145 0.1302 0.1317

0.0555 0.0851 0.1118 0.1286 0.1315

0.0530 0.0822 0.1091 0.1269 0.1311

0.0506 0.0793 0.1064 0.1251 0.1306

0.0483 0.0764 0.1037 0.1232 0.1300

0.0460 0.0736 0.1010 0.1212 0.1293

0.0439 0.0709 0.0982 0.1191 0.1284

0.0418 0.0682 0.0955 0.1170 0.1274

0.0398 0.0656 0.0928 0.1148 0.1263

0.0378 0.0631 0.0901 0.1126 0.1251

10 11 12 13 14

0.1198 0.0991 0.0752 0.0526 0.0342

0.1210 0.1012 0.0776 0.0549 0.0361

0.1219 0.1031 0.0799 0.0572 0.0380

0.1228 0.1049 0.0822 0.0594 0.0399

0.1235 0.1067 0.0844 0.0617 0.0419

0.1241 0.1083 0.0866 0.0640 0.0439

0.1245 0.1098 0.0888 0.0662 0.0459

0.1249 0.1112 0.0908 0.0685 0.0479

0.1250 0.1125 0.0928 0.0707 0.0500

0.1251 0.1137 0.0948 0.0729 0.0521

15 16 17 18 19

0.0208 0.0118 0.0063 0.0032 0.0015

0.0221 0.0127 0.0069 0.0035 0.0017

0.0235 0.0137 0.0075 0.0039 0.0019

0.0250 0.0147 0.0081 0.0042 0.0021

0.0265 0.0157 0.0088 0.0046 0.0023

0.0281 0.0168 0.0095 0.0051 0.0026

0.0297 0.0180 0.0103 0.0055 0.0028

0.0313 0.0192 0.0111 0.0060 0.0031

0.0330 0.0204 0.0119 0.0065 0.0034

0.0347 0.0217 0.0128 0.0071 0.0037

20 21 22 23 24

0.0007 0.0003 0.0001 0.0000 0.0000

0.0008 0.0003 0.0001 0.0001 0.0000

0.0009 0.0004 0.0002 0.0001 0.0000

0.0010 0.0004 0.0002 0.0001 0.0000

0.0011 0.0005 0.0002 0.0001 0.0000

0.0012 0.0006 0.0002 0.0001 0.0000

0.0014 0.0006 0.0003 0.0001 0.0000

0.0015 0.0007 0.0003 0.0001 0.0001

0.0017 0.0008 0.0004 0.0002 0.0001

0.0019 0.0009 0.0004 0.0002 0.0001

Table A–2 continued . . .

389

Project Economics and Decision Analysis Volume II: Probabilistic Models

⎛n⎞ ⎝ x⎠

Entry is Probability P(X = x ) = ⎜ ⎟ p x (1 − p )n − x

n

x

0.01

0.02

0.03

0.04

p 0.05

0.06

0.07

0.08

0.09

x

n

2

0 1 2

0.9801 0.0198 0.0001

0.9604 0.0392 0.0004

0.9409 0.0582 0.0009

0.9216 0.0768 0.0016

0.9025 0.0950 0.0025

0.8836 0.1128 0.0036

0.8649 0.1302 0.0049

0.8464 0.1472 0.0064

0.8281 0.1638 0.0081

2 1 0

2

3

0 1 2 3

0.9703 0.0294 0.0003 0.0000

0.9412 0.0576 0.0012 0.0000

0.9127 0.0847 0.0026 0.0000

0.8847 0.1106 0.0046 0.0001

0.8574 0.1354 0.0071 0.0001

0.8306 0.1590 0.0102 0.0002

0.8044 0.1816 0.0137 0.0003

0.7787 0.2031 0.0177 0.0005

0.7536 0.2236 0.0221 0.0007

3 2 1 0

3

4

0 1 2 3 4

0.9606 0.0388 0.0006 0.0000 0.0000

0.9224 0.0753 0.0023 0.0000 0.0000

0.8853 0.1095 0.0051 0.0001 0.0000

0.8493 0.1416 0.0088 0.0002 0.0000

0.8145 0.1715 0.0135 0.0005 0.0000

0.7807 0.1993 0.0191 0.0008 0.0000

0.7481 0.2252 0.0254 0.0013 0.0000

0.7164 0.2492 0.0325 0.0019 0.0000

0.6857 0.2713 0.0402 0.0027 0.0001

4 3 2 1 0

4

5

0 1 2 3 4 5

0.9510 0.0480 0.0010 0.0000 0.0000

0.9039 0.0922 0.0038 0.0001 0.0000

0.8587 0.1328 0.0082 0.0003 0.0000

0.8154 0.1699 0.0142 0.0006 0.0000

0.7738 0.2036 0.0214 0.0011 0.0000

0.7339 0.2342 0.0299 0.0019 0.0001

0.6957 0.2618 0.0394 0.0030 0.0001

0.6591 0.2866 0.0498 0.0043 0.0002

0.6240 0.3086 0.0610 0.0060 0.0003

5 4 3 2 1 0

5

6

0 1 2 3 4 5 6

0.9415 0.0571 0.0014 0.0000 0.0000

0.8858 0.1085 0.0055 0.0002 0.0000

0.8330 0.1546 0.0120 0.0005 0.0000

0.7828 0.1957 0.0204 0.0011 0.0000

0.7351 0.2321 0.0305 0.0021 0.0001

0.6899 0.2642 0.0422 0.0036 0.0002

0.6470 0.2922 0.0550 0.0055 0.0003

0.6064 0.3164 0.0688 0.0080 0.0005

0.5679 0.3370 0.0833 0.0110 0.0008

6 5 4 3 2 1 0

6

7

0 1 2 3 4 5 6 7

0.9321 0.0659 0.0020 0.0000 0.0000 0.0000

0.8681 0.1240 0.0076 0.0003 0.0000 0.0000

0.8080 0.1749 0.0162 0.0008 0.0000 0.0000

0.7514 0.2192 0.0274 0.0019 0.0001 0.0000

0.6983 0.2573 0.0406 0.0036 0.0002 0.0000

0.6485 0.2897 0.0555 0.0059 0.0004 0.0000

0.6017 0.3170 0.0716 0.0090 0.0007 0.0000

0.5578 0.3396 0.0886 0.0128 0.0011 0.0001

0.5168 0.3578 0.1061 0.0175 0.0017 0.0001

7 6 5 4 3 2 1 0

7

8

0 1 2 3 4 5 6 7 8

0.9227 0.0746 0.0026 0.0001 0.0000 0.0000

0.8508 0.1389 0.0099 0.0004 0.0000 0.0000

0.7837 0.1939 0.0210 0.0013 0.0001 0.0000

0.7214 0.2405 0.0351 0.0029 0.0002 0.0000

0.6634 0.2793 0.0515 0.0054 0.0004 0.0000

0.6096 0.3113 0.0695 0.0089 0.0007 0.0000

0.5596 0.3370 0.0888 0.0134 0.0013 0.0001

0.5132 0.3570 0.1087 0.0189 0.0021 0.0001

0.4703 0.3721 0.1288 0.0255 0.0031 0.0002

8 7 6 5 4 3 2 1 0

8

9

0 1 2 3 4 5 6 7 8 9

0.9135 0.0830 0.0034 0.0001 0.0000 0.0000

0.8337 0.1531 0.0125 0.0006 0.0000 0.0000

0.7602 0.2116 0.0262 0.0019 0.0001 0.0000

0.6925 0.2597 0.0433 0.0042 0.0003 0.0000

0.6302 0.2985 0.0629 0.0077 0.0006 0.0000

0.5730 0.3292 0.0840 0.0125 0.0012 0.0001

0.5204 0.3525 0.1061 0.0186 0.0021 0.0002

0.4722 0.3695 0.1285 0.0261 0.0034 0.0003

0.4279 0.3809 0.1507 0.0348 0.0052 0.0005

0.99

0.98

0.97

0.96

0.95 p

0.94

0.93

0.92

0.91

9 8 7 6 5 4 3 2 1 0 x

9 n

Table A–3 Binomial Probabilities

390

Appendix A: Probability Tables

p 0.01 0.9044 0.0914 0.0042 0.0001 0.0000 0.0000 0.0000

0.02 0.8171 0.1667 0.0153 0.0008 0.0000 0.0000 0.0000

0.03 0.7374 0.2281 0.0317 0.0026 0.0001 0.0000 0.0000

0.04 0.6648 0.2770 0.0519 0.0058 0.0004 0.0000 0.0000

0.05 0.5987 0.3151 0.0746 0.0105 0.0010 0.0001 0.0000

0.06 0.5386 0.3438 0.0988 0.0168 0.0019 0.0001 0.0000

0.07 0.4840 0.3643 0.1234 0.0248 0.0033 0.0003 0.0000

0.08 0.4344 0.3777 0.1478 0.0343 0.0052 0.0005 0.0000

0.09 0.3894 0.3851 0.1714 0.0452 0.0078 0.0009 0.0001

10

0 1 2 3 4 5 6 7 8 9 10

10 9 8 7 6 5 4 3 2 1 0

12

0 1 2 3 4 5 6 7 8 9 10 11 12

0.8864 0.1074 0.0060 0.0002 0.0000 0.0000 0.0000

0.7847 0.1922 0.0216 0.0015 0.0001 0.0000 0.0000

0.6938 0.2575 0.0438 0.0045 0.0003 0.0000 0.0000

0.6127 0.3064 0.0702 0.0098 0.0009 0.0001 0.0000

0.5404 0.3413 0.0988 0.0173 0.0021 0.0002 0.0000

0.4759 0.3645 0.1280 0.0272 0.0039 0.0004 0.0000

0.4186 0.3781 0.1565 0.0393 0.0067 0.0008 0.0001

0.3677 0.3837 0.1835 0.0532 0.0104 0.0014 0.0001

10

0.3225 0.3827 0.2082 0.0686 0.0153 0.0024 0.0003

12 11 10 9 8 7 6 5 4 3 2 1 0

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.8601 0.1303 0.0092 0.0004 0.0000 0.0000 0.0000 0.0000

0.7386 0.2261 0.0323 0.0029 0.0002 0.0000 0.0000 0.0000

0.6333 0.2938 0.0636 0.0085 0.0008 0.0001 0.0000 0.0000

0.5421 0.3388 0.0988 0.0178 0.0022 0.0002 0.0000 0.0000

0.4633 0.3658 0.1348 0.0307 0.0049 0.0006 0.0000 0.0000

0.3953 0.3785 0.1691 0.0468 0.0090 0.0013 0.0001 0.0000

0.3367 0.3801 0.2003 0.0653 0.0148 0.0024 0.0003 0.0000

12

0.2863 0.3734 0.2273 0.0857 0.0223 0.0043 0.0006 0.0001

0.2430 0.3605 0.2496 0.1070 0.0317 0.0069 0.0011 0.0001

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.8179 0.1652 0.0159 0.0010 0.0000 0.0000 0.0000 0.0000 0.0000

0.6676 0.2725 0.0528 0.0065 0.0006 0.0000 0.0000 0.0000 0.0000

0.5438 0.3364 0.0988 0.0183 0.0024 0.0002 0.0000 0.0000 0.0000

0.4420 0.3683 0.1458 0.0364 0.0065 0.0009 0.0001 0.0000 0.0000

0.3585 0.3774 0.1887 0.0596 0.0133 0.0022 0.0003 0.0000 0.0000

0.2901 0.3703 0.2246 0.0860 0.0233 0.0048 0.0008 0.0001 0.0000

15

0.2342 0.3526 0.2521 0.1139 0.0364 0.0088 0.0017 0.0002 0.0000

0.1887 0.3282 0.2711 0.1414 0.0523 0.0145 0.0032 0.0005 0.0001

0.1516 0.3000 0.2818 0.1672 0.0703 0.0222 0.0055 0.0011 0.0002

0.93

0.92

0.91

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 x

0.99

0.98

0.97

0.96

0.95

0.94

20 n

p

Table A–3 continued . . .

391

Project Economics and Decision Analysis Volume II: Probabilistic Models

p n

x

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

2

0 1 2

0.8100 0.1800 0.0100

0.7225 0.2550 0.0225

0.6400 0.3200 0.0400

0.5625 0.3750 0.0625

0.4900 0.4200 0.0900

0.4225 0.4550 0.1225

0.3600 0.4800 0.1600

0.3025 0.4950 0.2025

0.2500 0.5000 0.2500

2 1 0

2

3

0 1 2 3

0.7290 0.2430 0.0270 0.0010

0.6141 0.3251 0.0574 0.0034

0.5120 0.3840 0.0960 0.0080

0.4219 0.4219 0.1406 0.0156

0.3430 0.4410 0.1890 0.0270

0.2746 0.4436 0.2389 0.0429

0.2160 0.4320 0.2880 0.0640

0.1664 0.4084 0.3341 0.0911

0.1250 0.3750 0.3750 0.1250

3 2 1 0

3

4

0 1 2 3 4

0.6561 0.2916 0.0486 0.0036 0.0001

0.5220 0.3685 0.0975 0.0115 0.0005

0.4096 0.4096 0.1536 0.0256 0.0016

0.3164 0.4219 0.2109 0.0469 0.0039

0.2401 0.4116 0.2646 0.0756 0.0081

0.1785 0.3845 0.3105 0.1115 0.0150

0.1296 0.3456 0.3456 0.1536 0.0256

0.0915 0.2995 0.3675 0.2005 0.0410

0.0625 0.2500 0.3750 0.2500 0.0625

4 3 2 1 0

4

5

0 1 2 3 4 5

0.5905 0.3281 0.0729 0.0081 0.0005 0.0000

0.4437 0.3915 0.1382 0.0244 0.0022 0.0001

0.3277 0.4096 0.2048 0.0512 0.0064 0.0003

0.2373 0.3955 0.2637 0.0879 0.0146 0.0010

0.1681 0.3602 0.3087 0.1323 0.0284 0.0024

0.1160 0.3124 0.3364 0.1811 0.0488 0.0053

0.0778 0.2592 0.3456 0.2304 0.0768 0.0102

0.0503 0.2059 0.3369 0.2757 0.1128 0.0185

0.0313 0.1563 0.3125 0.3125 0.1563 0.0313

5 4 3 2 1 0

5

6

0 1 2 3 4 5 6

0.5314 0.3543 0.0984 0.0146 0.0012 0.0001 0.0000

0.3771 0.3993 0.1762 0.0415 0.0055 0.0004 0.0000

0.2621 0.3932 0.2458 0.0819 0.0154 0.0015 0.0001

0.1780 0.3560 0.2966 0.1318 0.0330 0.0044 0.0002

0.1176 0.3025 0.3241 0.1852 0.0595 0.0102 0.0007

0.0754 0.2437 0.3280 0.2355 0.0951 0.0205 0.0018

0.0467 0.1866 0.3110 0.2765 0.1382 0.0369 0.0041

0.0277 0.1359 0.2780 0.3032 0.1861 0.0609 0.0083

0.0156 0.0938 0.2344 0.3125 0.2344 0.0938 0.0156

6 5 4 3 2 1 0

6

7

0 1 2 3 4 5 6 7

0.4783 0.3720 0.1240 0.0230 0.0026 0.0002 0.0000 0.0000

0.3206 0.3960 0.2097 0.0617 0.0109 0.0012 0.0001 0.0000

0.2097 0.3670 0.2753 0.1147 0.0287 0.0043 0.0004 0.0000

0.1335 0.3115 0.3115 0.1730 0.0577 0.0115 0.0013 0.0001

0.0824 0.2471 0.3177 0.2269 0.0972 0.0250 0.0036 0.0002

0.0490 0.1848 0.2985 0.2679 0.1442 0.0466 0.0084 0.0006

0.0280 0.1306 0.2613 0.2903 0.1935 0.0774 0.0172 0.0016

0.0152 0.0872 0.2140 0.2918 0.2388 0.1172 0.0320 0.0037

0.0078 0.0547 0.1641 0.2734 0.2734 0.1641 0.0547 0.0078

7 6 5 4 3 2 1 0

7

8

0 1 2 3 4 5 6 7 8

0.4305 0.3826 0.1488 0.0331 0.0046 0.0004 0.0000 0.0000 0.0000

0.2725 0.3847 0.2376 0.0839 0.0185 0.0026 0.0002 0.0000 0.0000

0.1678 0.3355 0.2936 0.1468 0.0459 0.0092 0.0011 0.0001 0.0000

0.1001 0.2670 0.3115 0.2076 0.0865 0.0231 0.0038 0.0004 0.0000

0.0576 0.1977 0.2965 0.2541 0.1361 0.0467 0.0100 0.0012 0.0001

0.0319 0.1373 0.2587 0.2786 0.1875 0.0808 0.0217 0.0033 0.0002

0.0168 0.0896 0.2090 0.2787 0.2322 0.1239 0.0413 0.0079 0.0007

0.0084 0.0548 0.1569 0.2568 0.2627 0.1719 0.0703 0.0164 0.0017

0.0039 0.0313 0.1094 0.2188 0.2734 0.2188 0.1094 0.0313 0.0039

8 7 6 5 4 3 2 1 0

8

9

0 1 2 3 4 5 6 7 8 9

0.3874 0.3874 0.1722 0.0446 0.0074 0.0008 0.0001 0.0000 0.0000 0.0000 0.90

0.2316 0.3679 0.2597 0.1069 0.0283 0.0050 0.0006 0.0000 0.0000 0.0000 0.85

0.1342 0.3020 0.3020 0.1762 0.0661 0.0165 0.0028 0.0003 0.0000 0.0000 0.80

0.0751 0.2253 0.3003 0.2336 0.1168 0.0389 0.0087 0.0012 0.0001 0.0000 0.75

0.0404 0.1556 0.2668 0.2668 0.1715 0.0735 0.0210 0.0039 0.0004 0.0000 0.70

0.0207 0.1004 0.2162 0.2716 0.2194 0.1181 0.0424 0.0098 0.0013 0.0001 0.65

0.0101 0.0605 0.1612 0.2508 0.2508 0.1672 0.0743 0.0212 0.0035 0.0003 0.60

0.0046 0.0339 0.1110 0.2119 0.2600 0.2128 0.1160 0.0407 0.0083 0.0008 0.55

0.0020 0.0176 0.0703 0.1641 0.2461 0.2461 0.1641 0.0703 0.0176 0.0020 0.50

9 8 7 6 5 4 3 2 1 0

9

x

n

p

Example: For n = 12, p = 0.25, and x = 3, P(X = 3) = 0.2581. In the context of oil exploration: n = number wells drilled, x = number of successful wells drilled, and p = regional success ratio.

Table A–3 continued . . .

392

Appendix A: Probability Tables

0.15 0.1969 0.3474 0.2759 0.1298 0.0401 0.0085 0.0012 0.0001 0.0000 0.0000 0.0000

0.20 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0000

0.25 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0000

p 0.30 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000

0.35 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000

0.40 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001

0.45 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003

0.50 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010

10 9 8 7 6 5 4 3 2 1 0

10

10

0 1 2 3 4 5 6 7 8 9 10

0.10 0.3487 0.3874 0.1937 0.0574 0.0112 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000

12

0 1 2 3 4 5 6 7 8 9 10 11 12

0.2824 0.3766 0.2301 0.0852 0.0213 0.0038 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.1422 0.3012 0.2924 0.1720 0.0683 0.0193 0.0040 0.0006 0.0001 0.0000 0.0000 0.0000 0.0000

0.0687 0.2062 0.2835 0.2362 0.1329 0.0532 0.0155 0.0033 0.0005 0.0001 0.0000 0.0000 0.0000

0.0317 0.1267 0.2323 0.2581 0.1936 0.1032 0.0401 0.0115 0.0024 0.0004 0.0000 0.0000 0.0000

0.0138 0.0712 0.1678 0.2397 0.2311 0.1585 0.0792 0.0291 0.0078 0.0015 0.0002 0.0000 0.0000

0.0057 0.0368 0.1088 0.1954 0.2367 0.2039 0.1281 0.0591 0.0199 0.0048 0.0008 0.0001 0.0000

0.0022 0.0174 0.0639 0.1419 0.2128 0.2270 0.1766 0.1009 0.0420 0.0125 0.0025 0.0003 0.0000

0.0008 0.0075 0.0339 0.0923 0.1700 0.2225 0.2124 0.1489 0.0762 0.0277 0.0068 0.0010 0.0001

0.0002 0.0029 0.0161 0.0537 0.1208 0.1934 0.2256 0.1934 0.1208 0.0537 0.0161 0.0029 0.0002

12 11 10 9 8 7 6 5 4 3 2 1 0

12

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.2059 0.3432 0.2669 0.1285 0.0428 0.0105 0.0019 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0874 0.2312 0.2856 0.2184 0.1156 0.0449 0.0132 0.0030 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000

0.0352 0.1319 0.2309 0.2501 0.1876 0.1032 0.0430 0.0138 0.0035 0.0007 0.0001 0.0000 0.0000 0.0000 0.0000

0.0134 0.0668 0.1559 0.2252 0.2252 0.1651 0.0917 0.0393 0.0131 0.0034 0.0007 0.0001 0.0000 0.0000 0.0000

0.0047 0.0305 0.0916 0.1700 0.2186 0.2061 0.1472 0.0811 0.0348 0.0116 0.0030 0.0006 0.0001 0.0000 0.0000

0.0016 0.0126 0.0476 0.1110 0.1792 0.2123 0.1906 0.1319 0.0710 0.0298 0.0096 0.0024 0.0004 0.0001 0.0000

0.0005 0.0047 0.0219 0.0634 0.1268 0.1859 0.2066 0.1771 0.1181 0.0612 0.0245 0.0074 0.0016 0.0003 0.0000

0.0001 0.0016 0.0090 0.0318 0.0780 0.1404 0.1914 0.2013 0.1647 0.1048 0.0515 0.0191 0.0052 0.0010 0.0001

0.0000 0.0005 0.0032 0.0139 0.0417 0.0916 0.1527 0.1964 0.1964 0.1527 0.0916 0.0417 0.0139 0.0032 0.0005

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.1216 0.2702 0.2852 0.1901 0.0898 0.0319 0.0089 0.0020 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0388 0.1368 0.2293 0.2428 0.1821 0.1028 0.0454 0.0160 0.0046 0.0011 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0115 0.0576 0.1369 0.2054 0.2182 0.1746 0.1091 0.0545 0.0222 0.0074 0.0020 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0032 0.0211 0.0669 0.1339 0.1897 0.2023 0.1686 0.1124 0.0609 0.0271 0.0099 0.0030 0.0008 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000

0.0008 0.0068 0.0278 0.0716 0.1304 0.1789 0.1916 0.1643 0.1144 0.0654 0.0308 0.0120 0.0039 0.0010 0.0002 0.0000 0.0000 0.0000 0.0000

0.0002 0.0020 0.0100 0.0323 0.0738 0.1272 0.1712 0.1844 0.1614 0.1158 0.0686 0.0336 0.0136 0.0045 0.0012 0.0003 0.0000 0.0000 0.0000

0.0000 0.0005 0.0031 0.0123 0.0350 0.0746 0.1244 0.1659 0.1797 0.1597 0.1171 0.0710 0.0355 0.0146 0.0049 0.0013 0.0003 0.0000 0.0000

0.0000 0.0001 0.0008 0.0040 0.0139 0.0365 0.0746 0.1221 0.1623 0.1771 0.1593 0.1185 0.0727 0.0366 0.0150 0.0049 0.0013 0.0002 0.0000

0.0000 0.0000 0.0002 0.0011 0.0046 0.0148 0.0370 0.0739 0.1201 0.1602 0.1762 0.1602 0.1201 0.0739 0.0370 0.0148 0.0046 0.0011 0.0002

0.90

0.85

0.80

0.75

0.70 p

0.65

0.60

0.55

0.50

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 x

20 n

20

Table A–3 continued . . .

393

Appendix B Contents of the Accompanying CD-ROM We are excited about bundling the most comprehensive decision analysis software on a CD-ROM that accompanies this book. Palisade’s DecisionTools™ Suite (an Excel add-in), including the award-winning @RISK, PrecisionTree, BestFit, TopRank, and RISKview. The addition of this software makes the chapters on decision trees and simulation very useful. The BestFit is used in Chapter 2, PrecisionTree is used extensively for decision making under uncertainty in Chapters 3 and 4, and Chapter 6 illustrates the use of @RISK to create and run simulation models. For more information about the Palisade Corporation and the DecisionTools Suite, visit Palisade’s Web site at www.palisade.com. The CD-ROM also contains the Excel workbooks that are used in the examples, the data files required for a number of example problems, and the solutions. In addition, comprehensive cash-flow spreadsheets are included that can be used for day-to-day economic evaluations. The functionality of these cash-flow spreadsheets is described in Appendix B of Volume I.

Project Economics and Decision Analysis Volume II: Probabilistic Models

The contents of the CD-ROM can be installed with the Setup program on the CD-ROM accompanying this book. For the Serial Number, input “Trial.” The Setup program creates four folders (i.e. DTOOLEG, DTOOLS, ECONPAC, and TABLES) on the root (C:\) directory and copies files from the respective folders on the CD-ROM to these folders. Brief description of the contents in each folder is given below. DTOOLEG: This directory contains all the example files of decision trees and simulation. The following file can be retrieved while BestFit is loaded. •

FIGURE 2–28.bfp — This file contains the results of fitting the bit record data of Table 2-1 with probability distributions, using BestFit.

The following files can be retrieved while PrecisionTree is loaded. • •

•

•

• • •

396

FIGURE 3–20.xls — The decision tree of the Final Oil Well Drilling example (Example 3–8, Figure 3–10). FIGURE 3–21.xls — Decision tree for Example 3–10. Includes a “Statistics” worksheet that shows the overall statistics of each option as shown in Table 3–18. FIGURE 3–23.xls — Decision tree for the data in Table 3–3 (Example 3–2). The workbook also includes “Statistics” and “Policy Suggestion” worksheets. FIGURE 3–26.xls — Decision tree for NGL pipeline replacement economics of Example 3–11 for the data in Example 7–10 (Volume I). FIGURE 3–27.xls — Decision tree of a turnkey bid for a drilling contract in Example 3–12. FIGURE 4–12.xls — Decision tree for Example 4–5, showing expected value calculations for two projects. FIGURE 4–13.xls — Decision tree for Example 4–5, showing Certainty Equivalent (CE) calculations for the same projects as shown in Figure 4–12.

Appendix B: Contents of the Accompanying CD-ROM

•

FIGURE 6–22.xls — Decision tree used in combination with @RISK (i.e. the discrete probabilities on chance nodes are replaced with probability distributions). The workbook includes worksheets for (a) (b) (c) (d) (e)

summary report, input statistics report, output statistics report, output graphs, and input graphs.

The following files are used for simulation using @RISK. •

•

TABLE 6–8.xls — Simulation model, using @RISK, for volumetric reserves calculations. The workbook includes other worksheets that are generated by @RISK. TABLE 6–9a.xls — Simulation of cash-flow model using @RISK. The workbook includes other worksheets that are generated by @RISK.

DTOOLS: This folder contains the Palisade’s DecisionTools suite. The Setup program installs @RISK, PrecisionTree, BestFit, RISKview, and TopRank and creates desktop icons for the individual programs. Each one of these programs can be retrieved by double clicking on the icon. This will load the respective program and Excel simultaneously. You will know a program that is retrieved is ready for use when you see its toolbar as shown in Figure 3–14 for PrecisionTree and Figure 6–10 for @RISK. A DecisionTools toolbar will also appear, which can be used to activate the programs.

397

Project Economics and Decision Analysis Volume II: Probabilistic Models

ECONPAC: This folder contains cash-flow spreadsheets. It contains four spreadsheets (a) (b) (c) (d)

ECONOIL, ECONGAS, ECONSUM, and ECONPSA.

A detailed description of these spreadsheets is given in Appendix B of Volume I. Each of these files can be retrieved while in Excel. TABLES: This folder contains spreadsheets for all the solved examples in Volume I and II. All the spreadsheets in this folder are linked through a MAIN MENU as shown in Appendix B (Volume I). The MAIN MENU directs the to the sub-menu for each chapter in Volume I or Volume II, as desired. This sub-menu gives a brief description of each table that is linked through this sub-menu. Once the MAIN MENU is retrieved, the can easily go back and forth from one spreadsheet to another. The can also go back and forth between the individual tables and the submenu by clicking on the BACK HOME icon or on the title of the table if the BACK HOME icon is not visible. This folder also includes the (a) interest tables of Appendix A (Volume I), (b) probability tables of Appendix A (Volume II), and (c) probability graph papers. These can also be accessed through the MAIN MENU. To get a full view of the table, it is recommended to maximize the view by clicking on View ➡ Full Screen.

398

INDEX

Index

Links

A @RISK for simulation

138

loading @RISK

354

developing the simulation model

355

running the simulation

359

analyzing output

361

handling dependency

370

with PrecisionTree

373

@RISK with PrecisionTree (simulation)

72

Absolute frequency distribution

50

Abstract probability

72

56

Addition rule

85

Advantages

16

decision trees

352

56

158

Actual number

decision analysis

347

373

A priori principles

Acts/strategies (alternatives)

314

208

16 208

Agreement phase (decision analysis)

382

Alternatives analysis

322

Analysis (definition)

11

Analyzing output (@RISK)

361

Apparent risk tolerance

290

This page has been reformatted by Knovel to provide easier navigation

Index Appendices

Links 383

probability tables

383

CD-ROM contents

395

Applications (decision analysis)

19

Approximation to binomial distribution

111

Approximation to certainty equivalent

262

Arithmetic mean

30

Assessing the utility function

250

Assumptions (Poisson distribution)

111

Asymmetrical distribution Attitudes toward risk

395

315

32

42

36 235

expected utility theory

237

spreadsheet applications

269

questions and problems

280

references

282

Authority for expenditure

316

Aversion (risk)

263

Axioms of expected utility

241

transivity

241

complete ordering

242

continuity

242

substitution

243

unequal probability

243

compound lottery

244

invariance

244

finiteness

244

B Basis development phase

12

Baye’s rule

94

191

This page has been reformatted by Knovel to provide easier navigation

Index Bayesian probability Baye’s rule

Links 77

328

Best payoff

188

BestFit

138

Excel add-in

139

Bidding (acquisition)

316 33

BINOMDIST (Excel)

132

Binomial coefficient

102

Binomial probability distribution

101

Breakeven working interest

291

Budget for investment

xiii

Business interruption risk

191

94

Behind-the-scene calculations

Bimodal data set

94

111

286

22

C Calculations (simulation)

321

328

Capital budgeting

xiii

286

Capital investment

xiii

4

problems

167

xiv

analysis

6

Catastrophe risk

21

23

CD-ROM

xv

xvii

395

44

54

contents Central tendency (grouped data) measures Central tendency measures

395 30 30 30

mean

31

median

33

mode

33

34

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Central tendency measures (Cont.) weighted average Certainty decision making

39 7

Certainty equivalence method

250

Certainty equivalent

246

method

250

approx imation

262

Certainty/deterministic model

10

Chance event

10

discrete

10

continuous

10

Chance node reduction

202

Chance nodes

199

definition of

201

reduction

202

52

Class empty

52

Class mark

51

Class number

51

Class size

52

Classical approach (probability)

72

Coefficient of mean deviation

47

Coefficient of variation (measure)

47 205

Collectively exhaustive principle

74

Competition risk

20

Competitive bidding Complement of an event set

200

207

313

185

Class definition

Collapse (decision tree)

273

152

Chance (definition)

Characteristics (expected value)

262

77

207

316 81

This page has been reformatted by Knovel to provide easier navigation

Index Complementation rule

Links 85

Complete ordering axiom

242

Compound lottery axiom

244

Conditional event Conditional probability distributions Conditional probability distributions

10 333 80 333

Consequences/payoffs

159

Constant risk aversion

265

Constant risk

268

Constructing risk profiles

207

Construction of decision tree

213

Continuity axiom

242

Continuous event

10

Continuous probability distribution

100

normal

114

lognormal

122

uniform

126

triangular

128

Continuous variables

333

113

50

Convention (decision trees)

198

Correlation coefficient

336

CORRELL (Excel)

337

Cozzolino risk-adjusted value formula

298

304

Cozzolino’s approach

286

289

risk-adjusted value formula

298

304

optimum working interest

290

apparent risk tolerance

290

breakeven working interest

291

Lerche and MacKay’s approach

293

304

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Cozzolino’s approach (Cont.) parabolic approximation

298

portfolio balancing

304

spreadsheet application

307

Criterion/basis

159

Critical risk calculation

269

Critical risk tolerance

274

305

Cumulative distribution function

58

114

330

Cumulative frequency distribution

51

56

58

Cumulative probability distribution

341

Curve fitting

269

Cycle (decision analysis)

11

basis development phase

12

deterministic phase

12

14

probabilistic phase

12

15

information phase

12

16

Data elements distribution

42

Data handling

28

Data set characteristics

42

Data variability

45

D

Decision (definition) Decision analysis cycle

9 11

basis development phase

12

deterministic phase

12

14

probabilistic phase

12

15

information phase

12

16

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Decision analysis process

379

structuring phase

380

evaluation phase

381

agreement phase

382

references

382

Decision analysis

xiii

uncertainty

4

industry practice

6

decision situation

7

terminology

8

cycle

11

advantages

16

misconceptions

17

applications

19

industry risks

20

questions and problems

24

references

25

process

268

Decision making

151

Decision node reduction

202

Decision nodes

199

defining

201

reduction

202

Decision situation classification

198

202

207

212

379

379

Decision criteria (expected utility)

Decision quality/outcome misconception

1

18 7

Decision Tools™ Suite

xvii

139

Decision tree collapse

205

207

Decision tree construction

213

Decision tree solution

201

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Decision trees

151

expected value

151

convention

198

guide-lines for design

200

solving a decision tree

201

collapse

205

constructing risk profiles

207

advantages

208

construction

213

De-cumulative frequency distribution

51

Degrees of freedom

46

Dependence on input variables

207

56

58

333

total

338

diffuse

339

Dependent events

83

Descriptive statistics

27

measures of central tendency

30

suitable average choice

42

variability measures

44

grouped data

50

graphical presentation of data

56

other measures

62

spreadsheet application

66

Design (simulation model)

314

real application of simulation

315

steps in simulation modeling

316

random sampling methods

322

Design guidelines (decision trees)

200

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Determining venture participation

285

gambler’s ruin

287

working interest and risk-adjusted value

289

portfolio balancing and risk adjusted value

303

spreadsheet applications

307

references

310

Deterministic criteria Deterministic dominance

8 178

Deterministic model

14

313

Deterministic phase

12

14

Developing simulation model (@RISK)

355

Diffuse dependence (simulation)

335

Discrete attribution

110

Discrete event Discrete probability distribution

10 100

binomial

101

multinomial

105

hypergeometric

107

Poisson

110

Discrete uniform random variable Discrete variables Distribution/inventory control systems

126 50

126

315

Dociles

64

Dominance rules

15

elimination rule

339

176

15

E Economic evaluation Educated guess

xiii 70

This page has been reformatted by Knovel to provide easier navigation

Index Elementary event Elicited utility values

Links 78 251

Elimination rule

15

Empirical approach (probability)

75

End/terminal/payoff node

200

Environmental risk

21

Equally likely

73

Equation approach (sampling)

325

Equipment replacement decisions

315

Evaluation phase (decision analysis)

381

Event sets

80

79

operations Events and outcomes chance

81 10 10

Events characteristics

82

Examples/assignment problems

xvi

34

38

40

48

54

58

61

64

73

88

95

98

102

104

106

109

112

119

121

127

131

157

160

168

173

192

202

204

217

223

248

254

261

266

278

287

292

299

342

351

366

SEE ALSO Questions and problems

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Excel add-in

138

BestFit

138

PrecisionTree

210

Excel built-in functions

xvii

212

132

209

269

307 337 BINOMDIST

132

NEGBINOMDIST

134

HYPGEOMDIST

134

POISSON

135

NORMDIST

136

STANDARDIZE

137

LOGNORMDIST

138

SUMPRODUCT

209

SOLVER

269

CORRELL

377

Excel for simulation

348

random numbers

348

probability distributions

349

307

Expected cash flow

331

Expected investment

171

Expected monetary value

157

Expected net gain

191

Expected net present value profit

185

Expected opportunity loss

172

Expected payoff

185

188

188

190

with perfect information Expected preference value

241

Expected profitability index

166

Expected random variable

154

185

190

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Expected utility axioms

241

transitivity

241

complete ordering

242

continuity

242

substitution

243

unequal probability

243

compound lottery

244

invariance

244

finiteness

244

Expected utility decision criteria

268

Expected utility theory

236

typical attitudes toward risk

237

axioms of utility

241

risk tolerance

244

certainty equivalent

246

risk

246

assessing the utility function

250

262

mathematical representation of utility functions

255

approximation to certainty equivalent

262

risk aversion

263

expected utility decision criteria

268

Expected value and decision trees

151

expected value concepts

153

value of information

186

decision trees

197

spreadsheet applications

209

Expected value characteristics

185

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Expected value concepts

153

expected random variable

154

standard deviation of random variable

155

expected monetary value

157

sensitivity analysis

163

expected profitability index

166

performance index

172

expected opportunity loss

172

mean-variance and stochastic dominance

174

expected value meaning/interpretation

181

expected value characteristics

185

Expected value meaning/interpretation

181

Exponential risk aversion (interest and value)

289

optimum working interest

290

apparent risk tolerance

290

breakeven working interest

291

Exponential utility function

256

F Favorable outcome

194

Field development program

316

Financial gain of success

286

Financial loss of failure

286

Financial risk

23

Finding the certainty equivalent

273

Finiteness axiom

244

First-degree stochastic dominance

178

Fitting (utility curve)

269

Five-point method

250

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Folding back/rolling back (decision tree)

201

Form of distribution

111

Fractiles

62

Frequency curve

56

Frequency distribution (data)

43

Full distribution sampling

50

56

38

43

53

37

44

53

326

G Gambler’s ruin

285

Gaussian distribution

114

Geological risk

23

Geometric mean

35

Good decision

19

Graphical approach (sampling)

324

Graphical presentation (data)

56

Grouped data

50

mean

52

median

52

mode

52

variance

53

standard deviation

53

central tendency

54

variability

54

Guidelines (decision tree design)

200

H Handling dependency (@RISK)

370

Harmonic mean

35

Histogram (data presentation)

56

Hurwicz criterion

152

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Hyperbolic risk aversion (interest and value)

293

Hyperbolic tangent utility function

259

Hypergeometric probability distribution

107

Hypergeometric random variable

107

HYPGEOMDIST (Excel)

134

I Imperfect information (expected value)

187

190

Independent event

10

83

Independent probability

80

Indifference to risk (neutral)

9

Indirect value

13

Inductive processes

28

Industry practice (evaluation)

6

Industry risks (decision analysis)

20

product market

20

competition

20

technology

21

catastrophe

21

environmental

21

oil price forecast uncertainties

21

operations (drilling/completion/production)

22

business interruption

22

monetary

22

legislative and regulatory

22

geological

23

financial

23

political

23

Inferential statistics

23

28

This page has been reformatted by Knovel to provide easier navigation

Index Information (value)

Links 186

expected value of perfect information

187

expected value of imperfect information

190

Information cost Information phase

186 12

Information sources

187

Input variables (dependence on)

319

assessment

319

total

338

diffuse

339

Input variables assessment

73

Interpretation of estimates

xiv

Intersection of two events

81

Invariance axiom

244

Investment analysis

316 xv

uncertainty

4

industry practice

6

decision situation classification

7

terminology (decision analysis)

8

decision analysis cycle

11

decision analysis advantages

16

misconceptions of decision analysis

17

applications of decision analysis

19

industry risks

20

questions and problems

24

references

25

Iteration (decision tree construction)

333

319

Insufficient reason principle

Investment evaluation

16

1

316

201

This page has been reformatted by Knovel to provide easier navigation

Index

Links

J t probability

81

93

L Laplace criterion

152

Latin hypercube sampling

326

Law of chance

73

Law of proportion

73

Least squares estimation

337

Legislative and regulatory risk

22

Lerche and MacKay’s approach

286

Likelihood of occurrence

153

Likely/common outcome

30

Linear plus exponential function

258

Loading @RISK

354

Logarithmic utility function

258

Lognormal probability distribution

122

LOGNORMDIST (Excel)

138

Lower boundary (class) Lower limit value

332

289

293

58 128

M Main parameter

111

Maintenance operations

315

Marginal probability

81

Market share modeling

315

Mathematical average

30

Mathematical probability

72

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Mathematical representation (utility functions)

255

Maximax decision rule

152

Maximin decision rule

152

Maximization rule

173

Mean (grouped data)

43

52

Mean (measure)

31

34

Mean absolute deviation (measure)

46

Mean squared deviation

46

Meaning/interpretation (expected value)

181

Mean-variance and stochastic dominance

174

mean-variance approach

175

Mean-variance approach

175

Median (grouped data)

52

Median (measure)

33

Minimax regret criterion

152

Minimization rule

173

Misconceptions (decision analysis)

17

Mode (grouped data)

44

Mode (measure)

33

Model construction

317

Model input (data)

151

Modeling/simulation

xiv

input data

151

real application of simulation

315

steps

316

construction

317

random sampling methods

322

52

151

314

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Modern probability approach

77

event sets operations

81

events characteristics

82

Modified risk-adjusted value

300

Monetary risk

22

Monetary value (expected)

157

Monte Carlo sampling

314

379

simulation

314

379

graphical approach

324

equation approach

325

Latin hypercube

326

full distribution

328

Monte Carlo simulation

314

Most likely payoff

188

Most likely value

128

Most probable outcome

181

Moving/progressive average Multinomial probability distribution

379

30 105

Multiplication rule

87

Mutually exclusive events

81

Mutually exclusive principle

74

N Negative desirability

241

NEGBINOMDIST (Excel)

134

Net cash flow

331

Net operating revenue

167

Net present value

157

Normal curve

114

Normal probability distribution

114

235

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Normalized distribution

340

NORMDIST (Excel)

136

Number of classes

51

O Objective (definition)

8

Objective probability

72

classical approach

72

empirical approach

75

Ogive (data presentation)

56

Oil price forecast uncertainties

21

Operations evaluation

315

Operations risk (drilling/ completion/production)

22

Opportunity loss (expected)

158

Optimal maintenance policies

315

Optimal solution misconception Optimum working interest Outcome criteria

172

18 290 8

Outcome state probabilities

159

185

Output analysis (simulation)

321

361

summary report

362

detailed statistics

363

target values

363

simulation data

363

sensitivity analysis

363

scenario analysis

365

viewing distributions

366

This page has been reformatted by Knovel to provide easier navigation

Index

Links

P Parabolic approximation to Cozzolino’s formula Parabolic risk-adjusted value formula

298 305

Parameter (definition)

31

Parameter (population)

27

Partially overlapping events

82

Payoff value

207

Peakedness coefficient

66

Percentiles

64

Perfect information (expected value)

187

Performance index

172

Petroleum industry practice

6

Plant production operations

315

POISSON (Excel)

135

Poisson probability distribution

110

Political risk

23

Population (data set)

30

Population parameters

27

Portfolio balancing and risk-adjusted value

303

Cozzolino’s formula

304

parabolic risk-adjusted value formula

305

Portfolio balancing

286

303

and risk-adjusted value Positional average

30

Power utility function

258

PrecisionTree with @RISK (simulation)

373

PrecisionTree

210

Excel add-in

276

373

212

This page has been reformatted by Knovel to provide easier navigation

Index

Links

construction of tree

213

risk profile

219

sensitivity analysis

220

with @RISK (simulation)

373

Preference theory

236

(risk)

246

Preposterior analysis

15

Price forecast uncertainties

21

Probabilistic criteria

8

Probabilistic description

xiv

Probabilistic phase

12

15

5

70

Probability concepts objective probability

72

subjective probability

77

modern probability approach

77

rules of probability

84

probability table

92

probability tree

92

Baye’s rule

94

Probability density function Probability distributions

58

113

100

312

evaluation

320

Probability rules

84

complementation

85

addition

85

multiplication

87

example problems

88

Probability table

92

Appendix A

383

Probability tables (Appendix)

200

286

320

349

383

383

This page has been reformatted by Knovel to provide easier navigation

Index

Links

standard normal probability distribution

383

Poisson probabilities

385

binomial probabilities

390

Probability theory

28

Probability tree

92

procedure

96

Probability value

207

Probability-weighted utilities

255

Probable outcomes

7

Problem definition

316

Product market risk

20

Professional judgment misconception

18

Profitability index (expected)

166

Profitability

xiii

profit margin

xiii

profitability index (expected)

166

Project completion/management

315

Properties or axioms

80

Proportion/percentage (data items)

56

96

166

Q Quadratic mean Quadratic utility function

35

37

258

Quantitative variables

29

Quartiles

62

Questions and problems (decision analysis)

24

Questions and problems (expected value and decision trees)

227

Questions and problems (risk)

280

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Questions and problems (statistics and probability)

142

R Random experiment/trial

78

Random numbers

322

generating

348

Random occurrence

340

Random sampling methods

313

random numbers

322

Monte Carlo sampling

324

Latin hypercube sampling

326

Random variable

100

expected

154

standard deviation

155

Random variates

322

Range (measure)

45

Realistic gain

301

Recognizing dependence (input variables)

333

simulating total dependence

338

simulating diffuse dependence

339

Rectangular probability distribution References

Relative desirability Relative frequency distribution Risk (definition) Risk analysis

329

348

322

154

312

48

126 25

149

233

310

376

382

282

240 50

56

8 xiii

1

7

311

316

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Risk analysis (Cont.) decision making assessment Risk attitudes

7 316 9

risk-averse

9

risk-seeking

9

risk-neutral

9

235

expected utility theory

237

spreadsheet applications

269

questions and problems

280

references

282

Risk aversion

9

175

286

289

exponential

289

hyperbolic

293

Risk investment

286

Risk neutral

239

Risk

246

Risk profile

207

construction Risk reduction misconception

219

17 285

Risk threshold

285

Risk tolerance

244

definition

244

apparent

290

274

290

303

Cozzolino’s RAV formula

304

parabolic RAV formula

305

Risk-adjusted value and working interest

263

207

Risk sharing

Risk-adjusted value and portfolio balancing

239

289

This page has been reformatted by Knovel to provide easier navigation

Index

Links

based on exponential risk aversion

289

based on hyperbolic risk aversion

293

parabolic approximation to Cozzolino’s RAV modified risk-adjusted value

298 300

Risk-adjusted value

286

working interest

289

modified

300

portfolio balancing

303

Risk seeking

239

Root mean square

37

Rules of probability

84

complementation

85

addition

85

multiplication

87

example problems

88

Running simulation (@RISK)

289

359

S Sample (data set)

30

Sample space

79

Sample statistics

27

Scatter diagram (definition/construction)

335

Scenario analysis

5

Screening criteria

166

Second-degree stochastic dominance

180

Selection without replacement

107

Sensitivity analysis

Sensitivity graph

152

4

15

152

211

220

274

163

222

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Set theory

77

Simple probability

80

Simulating diffuse dependence

339

Simulating total dependence

338

Simulation application

315

Simulation in decision analysis

5

deg the simulation model

314

application

315

behind the-scene calculations

328

311

377

recognizing dependence on input variables spreadsheet applications

333 347

combining PrecisionTree and @RISK references Simulation model design

373 376 314

real application of simulation

315

steps in simulation modeling

316

random sampling methods

322

Simulation modeling steps

316

Skewness coefficient

66

Software

xiv SEE ALSO Spread sheet applications

SOLVER (Excel)

269

Solving a decision tree

201

Spider graph

222

307

Spreadsheet applications (decision analysis simulation)

347

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Spreadsheet applications (decision analysis simulation) (Cont.) Excel for simulation

348

@RISK for simulation

352

Spreadsheet applications (descriptive statistics)

66

Spreadsheet applications (expected value/decision trees)

209

Excel built-in functions

209

PrecisionTree

210

questions and problems

227

references

233

Spreadsheet applications (risk)

269

fitting the utility curve

269

finding the certainty equivalent

273

critical risk tolerance

274

PrecisionTree and utility functions

276

Spreadsheet applications (statistics/probability)

132

Excel built-in functions

132

BestFit

138

Spreadsheet applications (venture participation)

307

Standard deviation (grouped data)

53

Standard deviation (measure)

45

Standard deviation (random variable)

155

STANDARDIZE (Excel)

137

Statistics (definition)

31

Statistics and probability concepts

27

descriptive statistics

48

28

This page has been reformatted by Knovel to provide easier navigation

Index

Links

probability concepts

70

theoretical probability distributions

99

spreadsheet applications

132

questions/problems

142

references

149

Steps in simulation/modeling

316

Stochastic dominance (mean-variance)

174

Stock prices/options/hedging

315

Strategic plan

316

Structuring phase (decision analysis)

380

Subjective probability Substitution axiom

77 243

Suitable average choice

42

SUMPRODUCT (Excel)

209

Sustainable loss

235

T Table format (procedure)

96

Technology risk

21

Terminology

xiv

decision analysis Theoretical probability distributions

8

8 99

discrete

101

continuous

113

Theory of sets

77

Tolerance (risk)

244

Tornado graph

222

Total dependence (simulation)

338

Total dependence

335

simulation

338

274

338

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Total risk investment

286

Transitivity axiom

241

Triangular probability distribution

128

330

xiii

1

71

151

U Uncertainty concept

capital investment model Uncertainty in capital investment

7

27

4 152 4

sensitivity analysis

4

scenario analysis

5

probability approaches

5

computer simulation

5

Uncertainty of outcome

27

Uncertainty/non-probabilistic model

152

Uncertainty/probabilistic model

152

Unequal probability axiom

243

Unfavorable outcome

194

Uniform probability distribution

126

Union of two events

81

Upper boundary (class)

57

Upper limit value

128

Useful information

187

Utility (expected)

237

criteria

268

Utility axioms

241

Utility curve fitting

269

Utility function assessment

250

Utility function mathematics

255

330

This page has been reformatted by Knovel to provide easier navigation

Index

Links

Utility functions

276

V Value measurement

153

Value of information

186

expected value of perfect information

187

expected value of imperfect information

190

Variability (data)

45

Variability (grouped data)

54

Variability measures

44

range

45

48

variance

45

48

standard deviation

45

48

mean absolute deviation

46

coefficient of variation

47

Variance (grouped data)

53

Variance (measure)

45

Venn diagram (definition)

82

Venture profitability

286

Volatility increase

301

48

W Weighted average (measure)

39

Working interest and risk-adjusted value

289

based on exponential risk aversion

289

based on hyperbolic risk aversion

293

157

parabolic approximation to Cozzolino’s formula modified risk-adjusted value

298 300

This page has been reformatted by Knovel to provide easier navigation