prm

The Professional Risk Managers Handbook A Comprehensive Guide to Current Theory and Best Practices

___________________________________________________

Edited by Carol Alexander and Elizabeth Sheedy

Introduced by David R. Koenig

Volume I: Finance Theory, Financial Instruments and Markets

The Official Handbook for the PRM Certification

PDF created with pdfFactory Pro trial version www.pdffactory.com

The PRM Handbook Volume I

Copyright 2004 The Authors and The Professional Risk Managers International Association 2

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Published by PRMIA Publications, Wilmington, DE

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ISBN 0-9766097-0-3 (3 Volume Set)

ISBN 0-9766097-1-1 (Volume I)




Contents Introduction................................................................................................................................................11Preface to Volume I: Finance Theory, Financial Instruments and Markets..........................13

I.A.1 Risk and Risk Aversion.....................................................................................................................19

I.A.1.1 Introduction ..........................................................................................................................19

I.A.1.2 Mathematical Expectations: Prices or Utilities?...............................................................20

I.A.1.3 The Axiom of Independence of Choice ...........................................................................22

I.A.1.4 Maximising Expected Utility...............................................................................................24

I.A.1.5 Encoding a Utility Function ...............................................................................................28

I.A.1.6 The MeanVariance Criterion ............................................................................................34

I.A.1.7 Risk-Adjusted Performance Measures ..............................................................................37

I.A.1.8 Summary ................................................................................................................................49

References ................................................................................................................................................52

Appendix I.A.1.A: Terminology ...........................................................................................................54

Appendix I.A.1.B: Utility Functions ....................................................................................................55

I.A.2 Portfolio Mathematics ......................................................................................................................61

I.A.2.1 Means and Variances of Past Returns ....................................................................................61

I.A.2.2 Mean and Variance of Future Returns ...................................................................................68

I.A.2.3 MeanVariance Tradeoffs ........................................................................................................73

I.A.2.4 Multiple Assets ...........................................................................................................................78

I.A.2.5 A Hedging Example ..................................................................................................................81

I.A.2.6 Serial Correlation .......................................................................................................................87

I.A.2.7 Normally Distributed Returns .................................................................................................89

I.A.3 Capital Allocation ..............................................................................................................................95

I.A.3.1 An Overview .........................................................................................................................95

I.A.3.2 MeanVariance Criterion ................................................................................................. 100

I.A.3.3 Efficient Frontier: Two Risky Assets ............................................................................. 101

I.A.3.4 Asset Allocation ................................................................................................................. 106

I.A.3.5 Combining the Risk-Free Asset with Risky Assets ...................................................... 107

I.A.3.6 The Market Portfolio and the CML ............................................................................... 112

I.A.3.7 The Market Price of Risk and the Sharpe Ratio ........................................................... 113

I.A.3.8 Separation Principle........................................................................................................... 113

I.A.3.9 Summary ............................................................................................................................. 114

Appendix: Mathematics of the MeanVariance Model ................................................................. 116

I.A.4 The CAPM and Multifactor Models............................................................................................ 119

I.A.4.1 Overview............................................................................................................................. 120

I.A.4.2 Capital Asset Pricing Model............................................................................................. 121




I.A.4.3 Security Market Line ......................................................................................................... 123

I.A.4.4 Performance Measures...................................................................................................... 125

I.A.4.5 The Single-Index Model ................................................................................................... 128

I.A.4.6 Multifactor Models and the APT.................................................................................... 130

I.A.4.7 Summary ............................................................................................................................. 132

References ............................................................................................................................................. 133

I.A.5 Basics of Capital Structure ............................................................................................................ 135

I.A.5.1 Introduction ....................................................................................................................... 135

I.A.5.2 Maximising Shareholder Value, Incentives and Agency Costs................................... 139

I.A.5.3 Characteristics of Debt and Equity................................................................................. 143

I.A.5.4 Choice of Capital Structure.............................................................................................. 144

I.A.5.5 Making the Capital Structure Decision .......................................................................... 155

I.A.5.6 Conclusion .......................................................................................................................... 158

References ............................................................................................................................................. 158

I.A.6 The Term Structure of Interest Rates ......................................................................................... 161

I.A.6.1 Compounding Methods.................................................................................................... 161

I.A.6.2 Term Structure A Definition........................................................................................ 166

I.A.6.3 Shapes of the Yield Curve................................................................................................ 168

I.A.6.4 Spot and Forward Rates ................................................................................................... 171

I.A.6.5 Term Structure Theories .................................................................................................. 177

I.A.6.6 Summary ............................................................................................................................. 179

I.A.7 Valuing Forward Contracts........................................................................................................... 181

I.A.7.1 The Difference between Pricing and Valuation for Forward Contracts .................. 181

I.A.7.2 Principles of Pricing and Valuation for Forward Contracts on Assets..................... 182

I.A.7.3 Principles of Pricing and Valuation for Forward Contracts on Interest Rates ........ 192

I.A.7.4 The Relationship Between Forward and Futures Prices ............................................. 197

References ............................................................................................................................................. 197

I.A.8 Basic Principles of Option Pricing............................................................................................... 199

I.A.8.1 Factors Affecting Option Prices ..................................................................................... 199

I.A.8.2 PutCall Parity ................................................................................................................... 200

I.A.8.3 One-step Binomial Model and the Riskless Portfolio ................................................. 202

I.A.8.4 Delta Neutrality and Simple Delta Hedging.................................................................. 204

I.A.8.5 Risk-Neutral Valuation ..................................................................................................... 210

I.A.8.6 Real versus Risk-Neutral .................................................................................................. 211

I.A.8.7 The BlackScholesMerton Pricing Formula ............................................................... 212

I.A.8.8 The Greeks ......................................................................................................................... 216

I.A.8.9 Implied Volatility ............................................................................................................... 218




I.A.8.10 Intrinsic versus Time Value......................................................................................... 220

References ............................................................................................................................................. 220

I.B.1 General Characteristics of Bonds................................................................................................. 221

I.B.1.1 Definition of a Bullet Bond ............................................................................................. 221

I.B.1.2 Terminology and Convention.......................................................................................... 222

I.B.1.3 Market Quotes ................................................................................................................... 227

I.B.1.4 Non-bullet Bonds.............................................................................................................. 231

I.B.1.5 Summary ............................................................................................................................. 236

Reference............................................................................................................................................... 237

I.B.2 The Analysis of Bonds ................................................................................................................... 239

I.B.2.1 Features of Bonds.............................................................................................................. 240

I.B.2.2 Non-conventional Bonds ................................................................................................. 242

I.B.2.3 Pricing a Conventional Bond.......................................................................................... 244

I.B.2.4 Market Yield ....................................................................................................................... 252

I.B.2.5 Relationship between Bond Yield and Bond Price ...................................................... 256

I.B.2.6 Duration .............................................................................................................................. 259

I.B.2.7 Hedging Bond Positions................................................................................................... 264

I.B.2.8 Convexity ............................................................................................................................ 266

I.B.2.9 A Summary of Risks Associated with Bonds................................................................ 271

References ............................................................................................................................................. 273

I.B.3 Futures and Forwards .................................................................................................................... 275

I.B.3.1 Introduction ....................................................................................................................... 275

I.B.3.2 Stock Index Futures .......................................................................................................... 278

I.B.3.3 Currency Forwards and Futures...................................................................................... 286

I.B.3.4 Commodity Futures .......................................................................................................... 293

I.B.3.5 Forward Rate Agreements ............................................................................................... 294

I.B.3.6 Short-Term Interest-Rate Futures .................................................................................. 296

I.B.3.7 T-bond Futures .................................................................................................................. 305

I.B.3.8 Stack and Strip Hedges ..................................................................................................... 312

I.B.3.9 Concluding Remarks ......................................................................................................... 314

References ............................................................................................................................................. 314

I.B.4 Swaps ................................................................................................................................................ 317

I.B.4.1 What is a swap? ....................................................................................................................... 318

I.B.4.2 Types of Swaps........................................................................................................................ 320

I.B.4.3 Engineering Interest Rate Swaps.......................................................................................... 326

I.B.4.4 Risks of swaps ......................................................................................................................... 329

I.B.4.5 Other Swaps............................................................................................................................. 330




I.B.4.6 Uses of Swap............................................................................................................................ 330

I.B.4.7 Swap Conventions .................................................................................................................. 331

I.B.4.8 Conclusions.............................................................................................................................. 332

I.B.5 Vanilla Options ............................................................................................................................... 333

I.B.5.1 Stock Options Characteristics and Payoff Diagrams ............................................... 333

I.B.5.2 American versus European Options .............................................................................. 335

I.B.5.3 Strategies Involving a Single Option and a Stock......................................................... 336

I.B.5.4 Spread Strategies ................................................................................................................ 337

I.B.5.5 Other Strategies ................................................................................................................. 339

I.B.6 Credit Derivatives ........................................................................................................................... 345

I.B.6.1 Introduction ....................................................................................................................... 345

I.B.6.2 Credit Default Swaps ........................................................................................................ 350

I.B.6.3 Credit-Linked Notes ......................................................................................................... 353

I.B.6.4 Total Return Swaps ........................................................................................................... 356

I.B.6.5 Credit Options ................................................................................................................... 362

I.B.6.6 Synthetic Collateralised Debt Obligations ..................................................................... 363

I.B.6.7 General Applications of Credit Derivatives .................................................................. 372

I.B.6.8 Unintended Risks in Credit Derivatives......................................................................... 376

I.B.6.9 Summary ............................................................................................................................. 378

References ............................................................................................................................................. 378

I.B.7 Caps, Floors and Swaptions .......................................................................................................... 379

I.B.7.1 Caps, Floors and Collars: Definition and Terminology............................................... 379

I.B.7.2 Pricing Caps, Floors and Collars ..................................................................................... 381

I.B.7.3 Uses of Caps, Floors and Collars .................................................................................... 384

I.B.7.4 Swaptions: Definition and Terminology........................................................................ 389

I.B.7.5 Pricing Swaptions .............................................................................................................. 390

I.B.7.6 Uses of Swaptions ............................................................................................................. 392

I.B.7.7 Summary ............................................................................................................................. 393

References ............................................................................................................................................. 393

I.B.8 Convertible Bonds.......................................................................................................................... 395

I.B.8.1 Introduction ....................................................................................................................... 395

I.B.8.2 Characteristics of Convertibles........................................................................................ 398

I.B.8.3 Capital Structure Implications (for Banks) .................................................................... 408

I.B.8.4 Mandatory Convertibles ................................................................................................... 409

I.B.8.5 Valuation and Risk Assessment....................................................................................... 411

I.B.8.6 Summary ............................................................................................................................ 415

References ............................................................................................................................................. 415




I.B.9 Simple Exotics Catriona March.................................................................................................... 417

I.B.9.1 Introduction ....................................................................................................................... 417

I.B.9.2 A Short History.................................................................................................................. 418

I.B.9.3 Classifying Exotics............................................................................................................. 420

I.B.9.4 Notation .............................................................................................................................. 421

I.B.9.5 Digital Options .................................................................................................................. 422

I.B.9.6 Two Asset Options ........................................................................................................... 428

I.B.9.7 Quantos............................................................................................................................... 431

I.B.9.8 Second-Order Contracts................................................................................................... 434

I.B.9.9 Decision Options............................................................................................................... 437

I.B.9.10 Average Options ........................................................................................................... 438

I.B.9.11 Options on Baskets of Assets ..................................................................................... 441

I.B.9.12 Barrier and Related Options........................................................................................ 443

I.B.9.13 Other Path-Dependent Options................................................................................. 450

I.B.9.14 Resolution Methods...................................................................................................... 453

I.B.9.15 Summary......................................................................................................................... 455

References ............................................................................................................................................. 456

I.C.1 The Structure of Financial Markets ............................................................................................. 457

I.C.1.1 Introduction ....................................................................................................................... 457

I.C.1.2 Global Markets and Their Terminology ........................................................................ 458

I.C.1.3 Drivers of Liquidity........................................................................................................... 463

I.C.1.4 Liquidity and Financial Risk Management..................................................................... 467

I.C.1.5 Exchanges versus OTC Markets..................................................................................... 469

I.C.1.6 Technological Change....................................................................................................... 471

I.C.1.7 Post-trade Processing........................................................................................................ 475

I.C.1.8 Retail and Wholesale Brokerage ...................................................................................... 477

I.C.1.9 New Financial Markets ..................................................................................................... 478

I.C.1.10 Conclusion ..................................................................................................................... 480

References ............................................................................................................................................. 482

I.C.2 The Money Markets ....................................................................................................................... 483

I.C.2.1 Introduction ....................................................................................................................... 483

I.C.2.2 Characteristics of Money Market Instruments.............................................................. 483

I.C.2.3 Deposits and Loans........................................................................................................... 485

I.C.2.4 Money Market Securities .................................................................................................. 492

I.C.2.5 Summary ............................................................................................................................. 496

I.C.3 Bond Markets .................................................................................................................................. 499

I.C.3.1 Introduction ....................................................................................................................... 499




I.C.3.2 The Players ......................................................................................................................... 500

I.C.3.3 Bonds by Issuers................................................................................................................ 503

I.C.3.4 The Markets........................................................................................................................ 512

I.C.3.5 Credit Risk .......................................................................................................................... 519

I.C.3.6 Summary ............................................................................................................................. 521

References ............................................................................................................................................. 521

I.C.4 The Foreign Exchange Market ..................................................................................................... 523

I.C.4.1 Introduction ....................................................................................................................... 523

I.C.4.2 The Interbank Market....................................................................................................... 523

I.C.4.3 Exchange-Rate Quotations .............................................................................................. 525

I.C.4.4 Determinants of Foreign Exchange Rates..................................................................... 529

I.C.4.5 Spot and Forward Markets............................................................................................... 533

I.C.4.6 Structure of a Foreign Exchange Operation ................................................................. 538

I.C.4.7 Summary/Conclusion....................................................................................................... 540

I.C.5 The Stock Market ........................................................................................................................... 543

I.C.5.1 Introduction ....................................................................................................................... 543

I.C.5.2 The Characteristics of Common Stock .......................................................................... 544

I.C.5.3 Stock Markets and their Participants .............................................................................. 550

I.C.5.4 The Primary Market IPOs and Private Placements.................................................. 552

I.C.5.5 The Secondary Market the Exchange versus OTC Market..................................... 554

I.C.5.6 Trading Costs ..................................................................................................................... 556

I.C.5.7 Buying on Margin .............................................................................................................. 557

I.C.5.8 Short Sales and Stock Borrowing Costs......................................................................... 559

I.C.5.9 Exchange-Traded Derivatives on Stocks....................................................................... 561

I.C.5.10 Summary......................................................................................................................... 562

References ............................................................................................................................................. 563

I.C.6 The Futures Markets ...................................................................................................................... 565

I.C.6.1 Introduction ....................................................................................................................... 565

I.C.6.2 History of Forward-Based Derivatives and Futures Markets..................................... 565

I.C.6.3 Futures Contracts and Markets ....................................................................................... 568

I.C.6.4 Options on Futures ........................................................................................................... 580

I.C.6.5 Futures Exchanges and Clearing Houses ...................................................................... 585

I.C.6.6 Market Participants Hedgers ........................................................................................ 591

I.C.6.7 Market Participants Speculators................................................................................... 595

I.C.6.8 Market Participants Managed Futures Investors....................................................... 598

I.C.6.9 Summary and Conclusion ................................................................................................ 599

I.C.7 The Structure of Commodities Markets ..................................................................................... 601




I.C.7.1 Introduction ....................................................................................................................... 601

I.C.7.2 The Commodity Universe and Anatomy of Markets .................................................. 602

I.C.7.3 SpotForward Pricing Relationships .............................................................................. 610

I.C.7.4 Short Squeezes, Corners and Regulation ....................................................................... 616

I.C.7.5 Risk Management at the Commodity Trading Desk.................................................... 620

I.C.7.6 The Distribution of Commodity Returns...................................................................... 625

I.C.7.7 Conclusions ........................................................................................................................ 627

References ............................................................................................................................................. 628

I.C.8 The Energy Markets ....................................................................................................................... 629

I.C.8.1 Introduction ....................................................................................................................... 629

I.C.8.2 Market Overview ............................................................................................................... 629

I.C.8.3 Energy Futures Markets ................................................................................................... 634

I.C.8.4 OTC Energy Derivative Markets .................................................................................... 646

I.C.8.5 Emerging Energy Commodity Markets ......................................................................... 652

I.C.8.6 The Future of Energy Trading ........................................................................................ 657

I.C.8.7 Conclusion .......................................................................................................................... 660




Introduction

If you're reading this, you are seeking to attain a higher standard. Congratulations!

Those who have been a part of financial risk management for the past twenty years, have seen it

change from an on-the-fly profession, with improvisation as a rule, to one with substantially

higher standards, many of which are now documented and expected to be followed. Its no

longer enough to say you know. Now, you and your team need to prove it.

As its title implies, this book is the Handbook for the Professional Risk Manager. It is for those

professionals who seek to demonstrate their skills through certification as a Professional Risk

Manager (PRM) in the field of financial risk management. And it is for those looking simply to

develop their skills through an excellent reference source.

With contributions from nearly 40 leading authors, the Handbook is designed to provide you

with the materials needed to gain the knowledge and understanding of the building blocks of

professional financial risk management. Financial risk management is not about avoiding risk.

Rather, it is about understanding and communicating risk, so that risk can be taken more

confidently and in a better way. Whether your specialism is in insurance, banking, energy, asset

management, weather, or one of myriad other industries, this Handbook is your guide.

We encourage you to work through it sequentially. In Section I, we introduce the foundations of

finance theory, the financial instruments that provide tools for the mitigation or transfer of risk,

and the financial markets in which instruments are traded and capital is raised. After studying this

section, you will have read the materials necessary for passing Exam I of the PRM Certification

program.

Those preparing for the PRM certification will also be preparing for Exam II on the

Mathematical Foundations of Risk Measurement, covered in Volume II of the PRM Handbook,

Exam III on Risk Management Practices, covered in Volume III of the PRM Handbook and

Exam IV - Case Studies, Standards of Best Practice Conduct and Ethics and PRMIA

Governance. Exam IV is where we study some failed practices, standards for the performance of

the duties of a Professional Risk Manager, and the governance structure of our association, the

Professional Risk Managers International Association. The materials for this exam are freely

available on our web site (see http://www.prmia.org/pdf/Web_based_Resources.htm) and are

thus outside of the Handbook.

At the end of your progression through these materials, you will find that you have broadened

your knowledge and skills in ways that you might not have imagined. You will have challenged




yourself as well. And, you will be a better risk manager. It is for this reason that we have created

the Professional Risk Managers Handbook.

Our deepest appreciation is extended to Prof. Carol Alexander and Prof. Elizabeth Sheedy, both

of PRMIAs Academic Advisory Council, for their editorial work on this document. The

commitment they have shown to ensuring the highest level of quality and relevance is beyond

description. Our thanks also go to Laura Bianco, past President of PRMIA Publications, who has

tirelessly kept the work process moving forward and who has dedicated herself to demanding the

finest quality output. We also thank Richard Leigh, our London-based copyeditor, for his skilful

and timely work.

Finally, we express our thanks to the authors who have shared their insights with us. The

demands for sharing of their expertise are frequent. Yet, they have each taken special time for this

project and have dedicated themselves to making the Handbook and you a success. We are very

proud to bring you such a fine assembly.

Much like PRMIA, the Handbook is a place where the best ideas of the risk profession meet. We

hope that you will take these ideas, put them into practice and certify your knowledge by attaining

the PRM designation. Among our membership are several hundred Chief Risk Officers / Heads

of Risk and tens of thousands of other risk professionals who will note your achievements. They

too know the importance of setting high standards and the trust that capital providers and

stakeholders have put in them. Now they put their trust in you and you can prove your

commitment and distinction to them.

We wish you much success during your studies and for your performance in the PRM exams!

David R. Koenig, Executive Director, Chair, Board of Directors, PRMIA




Preface to Volume I: Finance Theory, Financial Instruments and Markets

Section I of this Handbook has been written by a group of leading scholars and practitioners and

represents a broad overview of the theory, instruments and markets of finance. This section

corresponds to Exam I in the Professional Risk Manager (PRM) certification programme.

The modern theory of finance is the solid basis of risk management and thus it naturally

represents the basis of the PRM certification programme. All major areas of finance are involved

in the process of risk management: from the expected utility approach and risk aversion, which

were the forerunners of the capital asset pricing model (CAPM), to portfolio theory and the risk-

neutral approach to pricing derivatives. All of these great financial theories and their interactions

are presented in Part I.A (Finance Theory). Many examples demonstrate how the concepts are

applied in practical situations.

Part I.B (Financial Instruments) describes a wide variety of financial products and connects them

to the theoretical development in Part I.A. The ability to value all the instruments/assets within a

trading or asset portfolio is fundamental to risk management. This part examines the valuation of

financial instruments and also explains how many of them can be used for risk management.

The designers of the PRM curriculum have correctly determined that financial risk managers

should have a sound knowledge of financial markets. Market liquidity, the role of intermediaries

and the role of exchanges are all features that vary considerably from one market to the next and

over time. It is crucial that professional risk managers understand how these features vary and

their consequences for the practice of risk management. Part I.C (Financial Markets) describes

where and how instruments are traded, the features of each type of financial asset or commodity

and the various conventions and rules governing their trade.

This background is absolutely necessary for professional risk management, and Exam I therefore

represents a significant portion of the whole PRM certification programme. For a practitioner

who left academic studies several years ago, this part of the Handbook will provide efficient

revision of finance theory, financial instruments and markets, with emphasis on practical

application to risk management. Such a person will find the chapters related to his/her work easy

reading and will have to study other topics more deeply.

The coverage of financial topics included in Section I of the Handbook is typically deeper and

broader than that of a standard MBA syllabus. But the concepts are well explained and




appropriately linked together. For example, Chapter I.B.6 on credit derivatives covers many

examples (such as credit-linked notes and credit default swaps) that are not always included in a

standard MBA-level elective course on fixed income. Chapter I.B.9 on simple exotics also

provides examples of path-dependent derivatives beyond the scope of a standard course on

options. All chapters are written for professionals and assume a basic understanding of markets

and their participants.

Finance Theory Chapter I.A.1 provides a general overview of risk and risk aversion, introduces the utility function

and meanvariance criteria. Various risk-adjusted performance measures are described. A

summary of several widely used utility functions is presented in the appendix.

Chapter I.A.2 provides an introduction to portfolio mathematics, from means and variances of

returns to correlation and portfolio variance. This leads the reader to the efficient frontier,

portfolio theory and the concept of portfolio diversification. Eventually this chapter discusses

normally distributed returns and basic applications for value-at-risk, as well as the probability of

reaching a target or beating a benchmark. This chapter is very useful for anybody with little

experience in applying basic mathematical models in finance.

The concept of capital allocation is another fundamental notion for risk managers. Chapter I.A.3

describes how capital is allocated between portfolios of risky and riskless assets, depending on

risk preference. Then the efficient frontier, the capital markets line, the Sharpe ratio and the

separation principle are introduced. These concepts lead naturally to a discussion of the CAPM

model and the idea that marginal risk (rather than absolute risk) is the key issue when pricing

risky assets. Chapter I.A.4 provides a rigorous description of the CAPM model, including betas,

systematic risk, alphas and performance measures. Arbitrage pricing theory and multifactor

models are also introduced in this chapter.

Capital structure is an important theoretical concept for risk managers since capital is viewed as

the last defence against extreme, unexpected outcomes. Chapter I.A.5 introduces capital

structure, advantages and costs related to debt financing, various agency costs, various types of

debt and equity, return on equity decomposition, examples of attractive and unattractive debt,

bankruptcy and financial distress costs.

Most valuation problems involve discounting future cash flows, a process that requires

knowledge of the term structure of interest rates. Chapter I.A.6 describes various types of




interest rates and discounting, defines the term structure of interest rates, introduces forward

rates and explains the three main economic term structure theories.

These days all risk managers must be well versed in the use and valuation of derivatives. The two

basic types of derivatives are forwards (having a linear payoff) and options (having a non-linear

payoff). All other derivatives can be decomposed to these underlying payoffs or alternatively they

are variations on these basic ideas. Chapter I.A.7 describes valuation methods used for forward

contracts. Discounting is used to value forward contracts with and without intermediate cash

flow. Chapter I.A.8 introduces the principles of option pricing. It starts with definitions of basic

put and call options, putcall parity, binomial models, risk-neutral methods and simple delta

hedging. Then the BlackScholesMerton formula is introduced. Finally, implied volatility and

smile effects are briefly described.

Financial Instruments Having firmly established the theoretical basis for valuation in Part I.A, Part I.B applies these

theories to the most commonly used financial instruments.

Chapter I.B.1 introduces bonds, defines the main types of bonds and describes the market

conventions for major types of treasuries, strips, floaters (floating-rate notes) and inflation-

protected bonds in different countries. Bloomberg screens are used to show how the market

information is presented. Chapter I.B.2 analyses the main types of bonds, describes typical cash

flows and other features of bonds and also gives a brief description of non-conventional

instruments. Examples of discounting, day conventions and accrued interest are provided, as

well as yield calculations. The connection between yield and price is described, thus naturally

leading the reader to duration, convexity and hedging interest-rate risk.

While Chapter I.A.7 explained the principles of forward valuation, Chapter I.B.3 examines and

compares futures and forward contracts. Usage of these contracts for hedging and speculation is

discussed. Examples of currency, commodity, bonds and interest-rate contracts are used to

explain the concept and its applications. Mark-to-market, quotation, settlements and other

specifications are described here as well. The principles of forward valuation are next applied to

swap contracts, which may be considered to be bundles of forward contracts. Chapter I.B.4

analyses some of the most popular swap varieties, explaining how they may be priced and used

for managing risk.

The remaining chapters in Part I.B all apply the principles of option valuation as introduced in

Chapter I.A.8. The power of the option concept is obvious when we see its applications to so




many instruments and risk management problems. Chapter I.B.5 begins with an analysis of

vanilla options. Chapter I.B.6 covers one of the newer applications of options: the use of credit

risk derivatives to manage credit risk. Chapter I.B.7 addresses caps, floors and swaptions, which

are the main option strategies used in interest-rate markets. Yet another application of the option

principle is found in Chapter I.B.8 convertible bonds. These give investors the right to convert

a debt security into equity. Finally, Chapter I.B.9 examines exotic option payoffs. In every case

the author defines the instrument, discusses its pricing and illustrates its use for risk management

purposes.

Financial Markets

Financial risk management takes place in the context of markets and varies depending on the

nature of the market. Chapter I.C.1 is a general introduction to world financial markets. They can

be variously classified geographically, by type of exchange, by issuers, liquidity and type of

instruments all are provided here. The importance of liquidity, the distinction between

exchange and over-the-counter markets and the role of intermediaries in their various forms are

explained in more detail.

Money markets are the subject of Chapter I.C.2. These markets are of vital importance to the

risk manager as the closest thing to a risk-free asset is found here. This chapter covers all short-

term debt securities, whether issued by governments or corporations. It also explains the repo

markets markets for borrowing/lending on a secured basis. The market for longer-term debt

securities is discussed in Chapter I.C.3, which classifies bonds by issuer: government, agencies,

corporate and municipal. There is a comparison of bond markets in major countries and a

description of the main intermediaries and their roles. International bond markets are introduced

as well.

Chapter I.C.4 turns to the foreign exchange market the market with the biggest volume of

trade. Various aspects of this market are explained, such as quotation conventions, types of

brokers, and examples of cross rates. Economic theories of exchange rates are briefly presented

here along with central banks policies. Forward rates are introduced together with currency

swaps. Interest-rate parity is explained with several useful examples.

Chapter I.C.5 provides a broad introduction to stock markets. This includes the description and

characteristics of several types of stocks, stock market indices and priorities in the case of

liquidation. Dividends and dividend-based stock valuation methods are described in this

chapter. Primary and secondary markets are distinguished. Market mechanics, including types of




orders, market participants, margin and short trades, are explained here with various examples

clarifying these transactions. Some exchange-traded options on stocks are introduced as well.

Chapter I.C.6 introduces the futures markets; this includes a comparison of the main exchange-

traded markets, options on futures, specifications of the most popular contracts, the use of

futures for hedging, trade orders for futures contracts, mark-to-market procedures, and various

expiration conventions. A very interesting description of the main market participants concludes

this chapter.

Chapter I.C.7 introduces the structure of the commodities market. It starts with the spot market

and then moves to commodity forwards and futures. Specific features, such as delivery and

settlement methods, are described. The spotforward pricing relationship is used to decompose

the forward price into spot and carry. Various types of price term structure (such as

backwardation and contango) are described, together with some economic theory. The chapter

also describes short squeezes and regulations. Risk management at the commodity trading desk is

given at a good intuitive level. The chapter concludes with some interesting facts on distribution

of commodity returns.

Finally, Chapter I.C.8 examines one of the most rapidly developing markets for risk the energy

markets. These markets allow participants to manage the price risks of oil and gas, electricity,

coal and so forth. Some other markets closely linked with energy are also briefly discussed here,

including markets for greenhouse gas emissions, weather derivatives and freight. Energy markets

create enormous challenges and opportunities for risk managers in part because of the extreme

volatility of prices that can occur.

As a whole, Section I gives an overview of the theoretical and practical aspects of finance that are

used in the management of financial risks. Many concepts, some quite complex, are explained in a

relatively simple language and are demonstrated with numerous examples. Studying this part of

the Handbook should refresh your knowledge of financial models, products and markets and

provide the background for risk management applications.

Zvi Wiener, Co-chair of PRMIAs Education and Standards Committee




I.A.1 Risk and Risk Aversion

Jacques Pier1

I.A.1.1 Introduction Risk management, in a wide sense, is the art of making decisions in an uncertain world. Such

decisions involve a weighting of risks and rewards, a choice between doing the safe thing and

taking a risk. For example, we may ponder whether to invest in a new venture, whether to

diversify or hedge a portfolio of assets, or at what price it would be worth insuring a person or a

system. Risk attitude determines such decisions. Utility theory offers a rational method for

expressing risk attitude and should therefore be regarded as a main pillar of risk management.

The other two pillars of risk management are the generation of good alternatives without which

there would be nothing to decide and the assessment of probabilities without which we could

not tell the likely consequences of our actions.

Rationality, in the context of utility theory, means simply that decisions should be logically

consistent with a set of preference axioms and in line with patterns of risk attitude expressed in

simple, easily understood circumstances. So, utility theory does not dictate what risk attitude

should be that remains a personal matter or a matter of company policy it merely provides a

logical framework to extend risk preferences from simple cases to complex situations.

But why should one seek an axiomatic framework to express risk preferences? Alas, experience

shows that unaided intuition is an unreliable guide. It is relatively easy to construct simple

decision problems where intuitive choices seem to contradict each other, that is, seem to violate

basic rules of behaviour that we hold as self-evident. It seems wise, therefore, to start by agreeing

a basic set of rules and then draw the logical consequences.

Thus, utility theory is neither purely descriptive nor purely normative. It brings about a more

disciplined, quantitative approach to the expression of risk attitude than is commonly found in

everyday life. In other words, where too often risk taking is seat of the pants or based on gut

feel or nose, it tries to bring the brain into play. By questioning instinctive reactions to risky

situations, it leads decision makers and firms to understand better what risk attitude they ought to

adopt, to express it formally as an element of corporate policy and to convey it through the

organisation so that decisions under uncertainty can be safely delegated.

1 Visiting Professor, ISMA Centre, University of Reading, UK.




This chapter introduces some concepts that are absolutely fundamental to the management of

financial risks. Section I.A.1.2 introduces the idea of utility maximisation following Bernoullis

original ideas. Section I.A.1.3 discusses the axiom of independence of choice, one of the basic

axioms that must be satisfied if preferences over risky outcomes are to be represented by a utility

function. Section I.A.1.4 introduces the principle of maximum expected utility and the concept of

risk aversion (and its inverse, risk tolerance). Section I.A.1.5 explains how to encode your

personal attitude to risk in your own utility function. Section I.A.1.6 shows under what

circumstances the principle of maximum expected utility reduces to a meanvariance criterion to

distinguish between different investments. A comprehensive treatment of risk-adjusted

performance measures is given in Section I.A.1.7. We pay particular attention to the

circumstances in which the risks to be compared are not normally distributed and investors are

mainly concerned with downside risks. Section I.A.1.8 summarises and indicates which types of

decision criteria and performance measures may be appropriate in which circumstances.

Much of the material that is introduced in this chapter will be more fully discussed in other parts

of the Handbook. Thus you will find many references to subsequent chapters in Part I.A, Part II

and Part III of the Handbook. A thorough treatment of utility theory, whilst fundamental to our

understanding of risk and risk aversion, is beyond the scope of the PRM exam. However, for

completeness, and for readers seeking to use this chapter as a resource that goes further than the

PRM syllabus, we have provided extensive footnotes of the mathematical derivations.

Furthermore, we have added an Appendix that describes the properties of standard utility

functions. However, it should be stressed that neither the mathematical derivations in the

footnotes nor the material in Appendix B are part of the PRM exam.

I.A.1.2 Mathematical Expectations: Prices or Utilities? It may seem curious nowadays that early probabilists, who liked to study games of chance, took it

for granted that the mathematical expectation of cash outcomes was the only rational criterion for

choosing between gambles. The expected value of a gamble is defined as the sum of its cash

outcomes weighted by their respective probabilities; the gamble with the highest expected value

was deemed to be the best. Fairness in gambling was the main argument in support of this

principle (among zero-sum games, where the gains of one player are the losses of the other, only

zero-expectation games are fair). Another argument drew on the weak law of large numbers, which

implies that, if the consequences of each gamble are small relative to the wealth of the players,

then, in the long run, after many independent gambles, only the average result would matter.

Daniel Bernoulli (1738) was the first mathematician to question the principle of maximising

expected value and to try to justify departures from it observed in daily life. He questioned




choices that fly in the face of the principle of maximising expected value. For example, he asked,

if a poor man were offered an equal chance to win a fortune or nothing, should he be regarded as

irrational if he tried to negotiate a sure reward of slightly less than half the potential fortune? Or

is it insane to insure a precious asset and thus knowingly contribute an expected profit to the

insurance company and therefore an equivalent expected decrease in ones wealth? To reconcile

common behaviour with a maximum-expectation principle, Bernoulli suggested applying the

principle not to cash outcomes but to utilities2 associated with cash outcomes. Bernoulli thus

pre-dates by half a century the core tenet of the Utilitarianism school of social philosophy, the

distinction between:

the utility, i.e. the personal value of an asset,

and

the price, i.e. the exchange value of an asset.

Bernoullis principle was that actions should be directed at maximising expected utility. The problem

that inspired Bernoulli and which has gained fame under the name of the St Petersburg paradox3

runs as follows: Peter tosses a coin and continues to do so until it should land heads. He agrees

to give Paul one ducat if he gets heads on the very first throw, two ducats if he gets it on the

second, four if on the third, eight if on the fourth, and so on, so that with each additional throw

the number of ducats he must pay is doubled. We seek to determine the value of Pauls

expectation.

Since, with a fairly tossed, symmetrical coin, the probability of landing heads for the first time on

the kth toss is 2k and the corresponding reward is 2k 1 ducats, the contribution to Pauls

monetary expectation of this outcome is half a ducat. And since there is an infinite number of

possible outcomes k = 1, k = 2, etc., Pauls monetary expectation is infinite. But, then as now,

gamblers are not willing to pay more than a few ducats for the right to play the game, hence the

paradox.

Bernoulli suggested that the utility of a cash reward depends on the existing wealth of the

recipient. He even made the far stronger assumption that utility is always inversely proportional to

existing wealth, in other words, that a gain of one ducat to someone worth a thousand ducats has the

same utility as a gain of a thousand ducats to someone worth a million ducats.

In this case a small change in utility, du, would be related to a small change in wealth, dx, by

du = dx/x.

2 In the Latin original, to calculate an emolumentum medium. 3 Simply because Bernoullis paper was published in the Commentaries from the Academy of Sciences of St Petersburg.




This leads, by integration (see Section II.C.6), to a logarithmic utility function,

u(x) = ln (x).

If we apply a logarithmic utility to the St Petersburg paradox, it no longer appears to be a

paradox. For instance, a gambler whose only wealth is the game itself would perceive an expected

utility of ln(2), which is the same as the utility of 2 ducats, and this is quite a small number far

short of infinity! In general, if the gambler has a logarithmic utility function, the larger the initial

wealth of the gambler, the larger his perceived utility of the game.

I.A.1.3 The Axiom of Independence of Choice Rarely is the power of a new idea fully understood on first encounter. Bernoullis introduction of

a utility function did influence the development of classical economics, where it was transposed

into a deterministic context. But it took more than two hundred years for the concept to be

revived in its original probabilistic context and to be re-erected on a firmer footing. In a seminal

book on games theory the mathematician J. von Neumann and the economist O. Morgenstern

(1947) postulated a basic set of rules from which it will follow that a utility function provides a

complete description of an individuals risk attitude.4

Bernoulli made a very strong assumption that the utility of a gain is inversely proportional to

existing wealth. By contrast, von Neumann and Morgenstern only assumed a minimal set of rules

that should appeal to all decision makers and which would result in the existence of utilities

without specifying what these utilities will be. These rules, or preference axioms, should seem

so fundamental that if, in some circumstances, a decision maker accidentally violates one of them,

she would re-examine her choice and correct it rather than knowingly abuse one of the rules.

To illustrate this point, consider the following preference axiom:

A choice between two gambles should not be influenced by the way the gambles are presented, provided that all

presentations contain the same relevant information.

This is an axiom because it cannot be derived from more fundamental principles. It is called the

axiom of independence of choice. One is free to accept or reject it, though most decision makers freely

accept it as self-evident. However, this axiom is easily violated by instinctive choices.

Daniel Kahneman, a Nobel prize winning expert in cognitive psychology, and his long time

colleague Amos Tversky designed the following simple, if somewhat dramatic test to show how a

4 The axiomatic approach pioneered by von Neumann, Morgenstern, Savage and others is often referred to as the American school of axiomatic utility theory.




change of presentation can affect our decisions. Their test consists of presenting two variants of a

choice between two public health programmes that address a threat to the lives of 600 people.

The first variant is:

With programme A we know that 200 lives will be saved, whereas with programme B there is a one-third chance

of saving all 600 lives and a two-thirds chance of saving none.

Kahneman and Tversky found that a clear majority of the people they presented with this choice

preferred A to B.5 The second variant is:

With programme C we know that 400 lives will be lost, whereas with programme D there is a one-third chance

that none will die and a two-thirds chance that all 600 people will die.

A majority of the people presented with this choice prefer D to C. Now, looking at the four

programmes, it becomes clear that, on the one hand, A and C are the same and, on the other

hand, C and D are also the same; the people saved in one presentation are the people not dying in

the other. So, whether one prefers A to B or the reverse, one ought express the same order of

preference between C and D, and that is not the case with many of the people interviewed; these

people are violating the axiom of independence of choice.

Kahneman and Tversky (1979) developed a new theory to explain their findings. They suggested

that people are generally risk averse when choosing between a sure gain and a chance of a larger

gain, but the same people may take a chance when forced to choose between a sure loss and only

a probability of a worse loss. The snag is that what appears as a sure gain or a sure loss is often a

question of perspective that can be easily manipulated by the way a problem is presented. Aware of

the importance attached to presentation, we provide in Appendix I.A.1.A a brief glossary of some

of the terms used in this chapter in order to dispel any unintended meaning.

More generally, cognitive psychologists have shown that we, as decision makers, may be swayed

by cognitive biases in the same way as untrained observers may be tricked by optical illusions.

We recognise the possibility of such biases when dealing with unusual events, for example, rare

events or extreme circumstances, or when our thoughts are too accustomed to a status quo, or

when they are blurred by emotions. But it may be unsafe to dismiss all instinctive reactions as

mere biases. After all, human instincts have evolved over millennia and must have some

survival value; important features of human risk behaviour could be overlooked by a nae

axiomatic approach.

5 Similar hypothetical questions were presented to numerous audiences of students and university faculty (the Hebrew University of Jerusalem, University of Stockholm, University of Michigan, among others) with similar results and repeated with business men in National Science Foundation sponsored studies.




I.A.1.4 Maximising Expected Utility There are a few variations of the axiomatic formulation of utility theory. We give here an

intuitive, if less than rigorous presentation.6 For students information we give in a footnote the

derivation of the principle of maximum expected utility, but the derivation is not examinable in

the PRM.

I.A.1.4.1 The Four Basic Axioms

(A1) Transitivity of Choice: All possible outcomes of the decision under consideration can

be ranked in order of preference; that is, if among three outcomes A, B and C, we strictly

prefer A to B and B to C then we ought to strictly prefer A to C.

(A2) Continuity of Choice: If among three outcomes A, B, C we strictly prefer A to B and

B to C, then B is the certain equivalent of some lottery between A and C, that is, there exists

a unique probability p for which we should be indifferent between receiving B or playing

a lottery offering A with probability p and C with probability 1 p.

(A3) Independence of Choice:7 Our preference order between two lotteries should not be

affected if these lotteries are part of the same wider range of possibilities.

(A4) Stochastic Dominance: Between two lotteries offering the same two possible

outcomes, we ought to prefer the lottery offering the larger probability of yielding the

preferred outcome.

Whether these axioms are nae or reasonable will remain an open debate; they are certainly not

always descriptive of intuitive human behaviour see Allais (1953) as well as Khaneman but

they may be useful guides as we try to improve on intuition. What is remarkable is that these four

axioms are sufficient to establish the concept of utility and lead to a unique decision criterion

known as the principle of maximum expected utility (maximum EU, for short), namely: the lottery with

the largest expected utility ought to be preferred over others.8

6 For the original presentation, see von Neumann and Morgenstern (1947). For alternative presentations, see Savage (1954), Fishburn (1970) or Kreps (1988). 7 The axiom of independence of choice has been formulated in many ways. In this form, it is also known as the axiom of substitution or simply of no fun in gambling.8 Suppose we face a choice between two lotteries A and B, each offering some of a finite number of outcomes {xi}, i= 1 to n. We associate probability pAi to outcome xi in lottery A and pBi in lottery B, respectively. We seek a criterion that will transform the choice between the two lotteries into determining which of two real numbers is the largest. Axiom (A1) requires that we should be able to rank all outcomes in a simple preference order and therefore that we should be able to identify at least one outcome that is not less desirable than any other, call it M, and at least one outcome that is not more desirable than any other, call it m. Axiom (A2) implies that to any outcome xi corresponds a probability ui such that xi can be regarded as the certain equivalent of a lottery offering M with probability ui and mwith probability 1 ui. Now, for each prize offered in lotteries A and B, substitute the equivalent lottery between Mand m. According to the axiom of independence of choice (A3), our preference between the new, compounded lotteries, call them A and B , should be the same as our preferences between A and B. But the compounded lotteries A and B offer the same two outcomes M and m. To make a choice, according to axiom (A4), we simply have to compare the probabilities of winning the preferred outcome M. These probabilities are EA[u] = pAiui and EB[u] = pAiui, that is, renaming as utilities the probabilities ui, they are the expected utilities of lotteries A and B. Therefore the preferred lottery ought to be the lottery with maximum expected utility.




Any decision criterion other than maximum EU that leads to a different choice would violate at

least one of the four basic axioms. It is therefore somewhat mystifying that the maximum EU

principle is not routinely used in risk management. We address this paradox in Section I.A.1.9.

I.A.1.4.2 Introducing the Utility Function Assigning utilities to possible outcomes is the key. We explain how this may be done in the next

section. But let us remark first that, for most financial risks, outcomes are already expressed on a

monetary scale, for instance, company profit or shareholder value. That is no mean feat and one

can only hope that there is no significant loss of information or distortion in the translation

process. Outcomes are generally complex, multi-faceted, and perceived differently by various

interested parties: shareholders, investors, clients, employees, management, etc. We must be

confident that between two outcomes A and B we prefer A to B simply because the cash value of

A is greater than the cash value of B.

Utility theory does not require the expression of all outcomes on a monetary scale and therefore

can address more general decision problems. However, when outcomes are already expressed in

terms of cash, utilities become a function of cash; we limit our discussion to this case.

The utility function u(x), where x is a cash amount expressing wealth and u(x) its utility to the

owner of the wealth, should be a continuous, non-decreasing function of x. It should be

continuous in as much as cash itself can be considered as continuous and a small increase in cash

should produce small increase in utility.9 It should be non-decreasing in as much as more cash is

preferred to less, a proposition that is not necessarily obvious and that is therefore put forward as

an additional axiom, the axiom of non-satiation.

On the other hand, we are free to choose the origin and the unit scale of utility without affecting

preferences. To simplify comparisons, we choose u(0) = 0 and a slope of 1 at the origin, that is

u(0) = 1.10

It is also common practice to choose the current level of wealth as the origin of the cash scale so

we have zero utility for our current level of wealth. In this case, future wealth is valued against the

current level of wealth rather than in absolute terms. We follow this practice here. But we should

remember that the level of wealth is unlikely to remain unchanged over time, and this may affect

9 We ignore pathological cases where, because of crude modelling of outcomes, an infinitesimal increase in cash could apparently lead to vastly different consequences such as having just enough money to get bail or to buy a new house. 10 . The expectation operator is linear, that is, E[(a.u(X) + b)] = a.E[u(X)] +b, with X a lottery and a and b two scalar constants. Therefore the order of preference set by the maximum EU principle remains unchanged under a positive linear transformation (a > 0) of the utility function. Without loss of generality, one may choose a utility scale as in




risk attitude. This is not a major drawback as risk attitude may evolve over time anyway and it is

therefore prudent to check regularly whether the utility function being used is still representative

of risk preferences.

I.A.1.4.3 Risk Aversion (and Risk Tolerance) It is the curvature of the utility function that captures the risk attitude of a decision maker or a

firm. A downward curvature (concave utility function) expresses risk aversion: the minimum selling

price of a risky opportunity is less than its expected value.

Figure I.A.1.1: Describing risk attitude with a utility function

Example I.A.1.1: Faced with the prospect of winning or losing 500m with equal probabilities, a firm using the

utility function plotted in Figure I.A.1.1 would perceive an expected utility of 270, the average

of the utilities of the two outcomes read of the curve: u(500) = 280 and u( 500) = 820. Now

reading back from the curve (black arrows) we find that 270 is the utility of a sure loss of

220m. In other words, the firm would be willing to pay up to 220m to have the risky prospect

taken away.

If the curvature were upwards (convex utility function), a risky opportunity would be perceived as

having greater expected utility than its expected value, which would reveal a risk-seeking attitude.

Figure I.A.1.1 where u(0) = 0 and the first derivative u (0) = 1, so that for infinitesimal variations around the origin utility and cash have the same unit.

-1500

-1000

-500

0

500

1000

-750 -500 -250 0 250 500 750 1000 1250

Million Euros

Equal chances of w inning or losing 500m w ould have a certain

equivalent of -220m




Finally, no curvature, that is, a straight-line utility function, would reflect a risk-neutral attitude

the expected value of outcomes is the choice criterion. Risk aversion is the norm, at least for

business decisions, whereas a risk-seeking attitude is usually regarded as pathological.11 We shall

argue later why a utility function should be very smooth (continuous first- and second-order

derivatives) and concave for business decisions.

Mathematically, the curvature of a twice differentiable function is defined as the ratio of its second-

order derivative to its first-order derivative. For a concave utility function such as that in Figure

I.A.1.1 the curvature is negative because u (x) < 0. We call minus the curvature the local coefficient

of risk aversion at x. That is:

Local Coefficient of Risk Aversion = u (x)/ u (x).

Its inverse is called, quite naturally, the local coefficient of risk tolerance at x; it is expressed in the same

monetary units as x and therefore may be easier to interpret.12 According to the age-old principle

of assigning a Greek letter to an unknown parameter, we shall call the local coefficient of risk

tolerance. Thus

= u (x)/ u (x). (I.A.1.1)

Stipulating the coefficient of risk tolerance (or the coefficient of risk aversion) over various levels

of wealth is equivalent to stipulating a utility function (see Pratt, 1964).

I.A.1.4.4 Certain Equivalence We call the certain equivalent (CE) of a gamble X the sure quantity that we would be willing to

exchange for the gamble, (i.e. u(CE(X)) = E[u(X)]). In the previous example, minus 220m is the

certain equivalent of the project. Clearly,choosing the alternative with the maximum EU is

equivalent to choosing the alternative with the maximum CE.

I.A.1.4.5 Summary Financial risks are gambles. For our purposes, a gamble is a set of cash-value outcomes, with some

probabilities attached to each outcome. Then, rational decisions between financial risks are

achieved by:

i. defining a utility function u(x), a monotonically increasing function of cash value x;

ii. calculating the expected utility E[u(X)] of each gamble X;

11 Gambling has always fascinated men. It is not only the subject of gripping stories (such as Dostoyevskys The Gambler) but it also arouses principled and even religious reactions, usually in the form of condemnations. But that is not to say that rational people should necessarily be risk-averse. 12 Mathematicians usually prefer to use the coefficient of risk aversion whereas practitioners usually prefer to use its inverse, the coefficient of risk tolerance; which coefficient is used does not really matter. We shall side here with the practitioners




iii. choosing the gamble that has the maximum expected utility or, equivalently,

choosing the gamble with maximum certain equivalent.

Although the current level of wealth is usually taken as the origin of the scale on which future

outcomes are valued, each new course of action should not be considered independently of the

status quo. The uncertainties we have in the future will depend on what we do today. Each

future choice should therefore be considered in the context of current uncertainties.

I.A.1.5 Encoding a Utility Function I.A.1.5.1 For an Individual

The first step in implementing utility theory is to draw a utility function over possible states of

wealth of an individual or a firm. It is a tricky exercise best conducted by an experienced and

independent experimentalist.

An individuals risk attitude can, in theory, be inferred from a series of decisions, provided the

other elements of the decisions (i.e. the outcomes, probabilities, alternatives) are clearly

understood by all. It is best, of course, if the problems submitted for decision are:

i. Realistic. One should avoid game playing with all the distortions it may create (e.g.

displays of bravado).

ii. Meaningful. The range of monetary outcomes should be on a scale of gains and

losses for which we can define a utility function.

iii. Clear and simple. One should avoid ambiguities, or inducements that could lead to

misinterpretations of the problem, or biases. In particular, probabilities should be

clearly stated and these probabilities should not be so extreme that they cannot be

comprehended.

We think of the decision maker as a bank executive or a successful trader. We start by defining a

monetary range of interest for our decision maker by choosing a minimum and a maximum cash

amount, say minus 3 million and plus 10 million. This range should cover the personal impact

of decisions she may have to face, for example, insuring her life, deciding whether to accept a

new incentive

prm

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