Edward J. AndersonThe University of Sydney Business School, Australia

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• Preface xiii1 What is risk management? 11.1 Introduction 21.2 Identifying and documenting risk 51.3 Fallacies and traps in risk management 71.4 Why safety is different 91.5 The Basel framework 111.6 Hold or hedge? 121.7 Learning from a disaster 131.7.1 What went wrong? 15Notes 17References 18Exercises 192 The structure of risk 222.1 Introduction to probability and risk 232.2 The structure of risk 252.2.1 Intersection and union risk 252.2.2 Maximum of random variables 282.3 Portfolios and diversification 302.3.1 Adding random variables 302.3.2 Portfolios with minimum variance 332.3.3 Optimal portfolio theory 372.3.4 When risk follows a normal distribution 382.4 The impact of correlation 402.4.1 Using covariance in combining random variables 412.4.2 Minimum variance portfolio with covariance 432.4.3 The maximum of variables that are positively correlated 442.4.4 Multivariate normal 462.5 Using copulas to model multivariate distributions 492.5.1 *Details on copula modeling 52Notes 58References 59Exercises 603 Measuring risk 633.1 How can we measure risk? 643.2 Value at risk 673.3 Combining and comparing risks 733.4 VaR in practice 763.5 Criticisms of VaR 793.6 Beyond value at risk 823.6.1 *More details on expected shortfall 86Notes 88References 88Exercises 894 Understanding the tails 924.1 Heavy-tailed distributions 934.1.1 Defining the tail index 934.1.2 Estimating the tail index 954.1.3 *More details on the tail index 984.2 Limiting distributions for the maximum 1004.2.1 *More details on maximum distributions and Fisher–Tippett 1064.3 Excess distributions 1094.3.1 *More details on threshold exceedances 1144.4 Estimation using extreme value theory 1154.4.1 Step 1. Choose a threshold u 1164.4.2 Step 2. Estimate the parameters ξ and β 1184.4.3 Step 3. Estimate the risk measures of interest 119Notes 121References 122Exercises 1235 Making decisions under uncertainty 1255.1 Decisions, states and outcomes 1265.1.1 Decisions 1265.1.2 States 1275.1.3 Outcomes 1275.1.4 Probabilities 1285.1.5 Values 1295.2 Expected Utility Theory 1305.2.1 Maximizing expected profit 1305.2.2 Expected utility 1325.2.3 No alternative to Expected Utility Theory 1355.2.4 *A sketch proof of the theorem 1395.2.5 What shape is the utility function? 1425.2.6 *Expected utility when probabilities are subjective 1455.3 Stochastic dominance and risk profiles 1485.3.1 *More details on stochastic dominance 1525.4 Risk decisions for managers 1565.4.1 Managers and shareholders 1565.4.2 A single company-wide view of risk 1585.4.3 Risk of insolvency 158Notes 160References 161Exercises 1626 Understanding risk behavior 1646.1 Why decision theory fails 1656.1.1 The meaning of utility 1656.1.2 Bounded rationality 1676.1.3 Inconsistent choices under uncertainty 1686.1.4 Problems from scaling utility functions 1716.2 Prospect Theory 1726.2.1 Foundations for behavioral decision theory 1736.2.2 Decision weights and subjective values 1756.3 Cumulative Prospect Theory 1806.3.1 *More details on Prospect Theory 1836.3.2 Applying Prospect Theory 1856.3.3 Why Prospect Theory does not always predict well 1876.4 Decisions with ambiguity 1896.5 How managers treat risk 191Notes 194References 194Exercises 1957 Stochastic optimization 1987.1 Introduction to stochastic optimization 1997.1.1 A review of optimization 1997.1.2 Two-stage recourse problems 2037.1.3 Ordering with stochastic demand 2087.2 Choosing scenarios 2127.2.1 How to carry out Monte Carlo simulation 2137.2.2 Alternatives to Monte Carlo 2177.3 Multistage stochastic optimization 2187.3.1 Non-anticipatory constraints 2207.4 Value at risk constraints 224Notes 228References 228Exercises 2298 Robust optimization 2328.1 True uncertainty: Beyond probabilities 2338.2 Avoiding disaster when there is uncertainty 2348.2.1 *More details on constraint reformulation 2408.2.2 Budget of uncertainty 2438.2.3 *More details on budgets of uncertainty 2478.3 Robust optimization and the minimax approach 2508.3.1 *Distributionally robust optimization 254Notes 261References 262Exercises 2639 Real options 2659.1 Introduction to real options 2669.2 Calculating values with real options 2679.2.1 *Deriving the formula for the surplus with a normal distribution 2729.3 Combining real options and net present value 2739.4 The connection with financial options 2789.5 Using Monte Carlo simulation to value real options 2829.6 Some potential problems with the use of real options 285Notes 287References 287Exercises 28810 Credit risk 29110.1 Introduction to credit risk 29210.2 Using credit scores for credit risk 29410.2.1 A Markov chain analysis of defaults 29610.3 Consumer credit 30110.3.1 Probability, odds and log odds 30210.4 Logistic regression 30810.4.1 *More details on logistic regression 31310.4.2 Building a scorecard 31510.4.3 Other scoring applications 317Notes 317References 318Exercises 319Appendix A Tutorial on probability theory 323A. 1 Random events 323A. 2 Bayes’ rule and independence 326A. 3 Random variables 327A. 4 Means and variances 329A. 5 Combinations of random variables 332A. 6 The normal distribution and the Central Limit Theorem 336Appendix B Answers to even-numbered exercises 340Index 361

Edward J. AndersonThe University of Sydney Business School, Australia

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• Preface xiii1 What is risk management? 11.1 Introduction 21.2 Identifying and documenting risk 51.3 Fallacies and traps in risk management 71.4 Why safety is different 91.5 The Basel framework 111.6 Hold or hedge? 121.7 Learning from a disaster 131.7.1 What went wrong? 15Notes 17References 18Exercises 192 The structure of risk 222.1 Introduction to probability and risk 232.2 The structure of risk 252.2.1 Intersection and union risk 252.2.2 Maximum of random variables 282.3 Portfolios and diversification 302.3.1 Adding random variables 302.3.2 Portfolios with minimum variance 332.3.3 Optimal portfolio theory 372.3.4 When risk follows a normal distribution 382.4 The impact of correlation 402.4.1 Using covariance in combining random variables 412.4.2 Minimum variance portfolio with covariance 432.4.3 The maximum of variables that are positively correlated 442.4.4 Multivariate normal 462.5 Using copulas to model multivariate distributions 492.5.1 *Details on copula modeling 52Notes 58References 59Exercises 603 Measuring risk 633.1 How can we measure risk? 643.2 Value at risk 673.3 Combining and comparing risks 733.4 VaR in practice 763.5 Criticisms of VaR 793.6 Beyond value at risk 823.6.1 *More details on expected shortfall 86Notes 88References 88Exercises 894 Understanding the tails 924.1 Heavy-tailed distributions 934.1.1 Defining the tail index 934.1.2 Estimating the tail index 954.1.3 *More details on the tail index 984.2 Limiting distributions for the maximum 1004.2.1 *More details on maximum distributions and Fisher–Tippett 1064.3 Excess distributions 1094.3.1 *More details on threshold exceedances 1144.4 Estimation using extreme value theory 1154.4.1 Step 1. Choose a threshold u 1164.4.2 Step 2. Estimate the parameters ξ and β 1184.4.3 Step 3. Estimate the risk measures of interest 119Notes 121References 122Exercises 1235 Making decisions under uncertainty 1255.1 Decisions, states and outcomes 1265.1.1 Decisions 1265.1.2 States 1275.1.3 Outcomes 1275.1.4 Probabilities 1285.1.5 Values 1295.2 Expected Utility Theory 1305.2.1 Maximizing expected profit 1305.2.2 Expected utility 1325.2.3 No alternative to Expected Utility Theory 1355.2.4 *A sketch proof of the theorem 1395.2.5 What shape is the utility function? 1425.2.6 *Expected utility when probabilities are subjective 1455.3 Stochastic dominance and risk profiles 1485.3.1 *More details on stochastic dominance 1525.4 Risk decisions for managers 1565.4.1 Managers and shareholders 1565.4.2 A single company-wide view of risk 1585.4.3 Risk of insolvency 158Notes 160References 161Exercises 1626 Understanding risk behavior 1646.1 Why decision theory fails 1656.1.1 The meaning of utility 1656.1.2 Bounded rationality 1676.1.3 Inconsistent choices under uncertainty 1686.1.4 Problems from scaling utility functions 1716.2 Prospect Theory 1726.2.1 Foundations for behavioral decision theory 1736.2.2 Decision weights and subjective values 1756.3 Cumulative Prospect Theory 1806.3.1 *More details on Prospect Theory 1836.3.2 Applying Prospect Theory 1856.3.3 Why Prospect Theory does not always predict well 1876.4 Decisions with ambiguity 1896.5 How managers treat risk 191Notes 194References 194Exercises 1957 Stochastic optimization 1987.1 Introduction to stochastic optimization 1997.1.1 A review of optimization 1997.1.2 Two-stage recourse problems 2037.1.3 Ordering with stochastic demand 2087.2 Choosing scenarios 2127.2.1 How to carry out Monte Carlo simulation 2137.2.2 Alternatives to Monte Carlo 2177.3 Multistage stochastic optimization 2187.3.1 Non-anticipatory constraints 2207.4 Value at risk constraints 224Notes 228References 228Exercises 2298 Robust optimization 2328.1 True uncertainty: Beyond probabilities 2338.2 Avoiding disaster when there is uncertainty 2348.2.1 *More details on constraint reformulation 2408.2.2 Budget of uncertainty 2438.2.3 *More details on budgets of uncertainty 2478.3 Robust optimization and the minimax approach 2508.3.1 *Distributionally robust optimization 254Notes 261References 262Exercises 2639 Real options 2659.1 Introduction to real options 2669.2 Calculating values with real options 2679.2.1 *Deriving the formula for the surplus with a normal distribution 2729.3 Combining real options and net present value 2739.4 The connection with financial options 2789.5 Using Monte Carlo simulation to value real options 2829.6 Some potential problems with the use of real options 285Notes 287References 287Exercises 28810 Credit risk 29110.1 Introduction to credit risk 29210.2 Using credit scores for credit risk 29410.2.1 A Markov chain analysis of defaults 29610.3 Consumer credit 30110.3.1 Probability, odds and log odds 30210.4 Logistic regression 30810.4.1 *More details on logistic regression 31310.4.2 Building a scorecard 31510.4.3 Other scoring applications 317Notes 317References 318Exercises 319Appendix A Tutorial on probability theory 323A. 1 Random events 323A. 2 Bayes’ rule and independence 326A. 3 Random variables 327A. 4 Means and variances 329A. 5 Combinations of random variables 332A. 6 The normal distribution and the Central Limit Theorem 336Appendix B Answers to even-numbered exercises 340Index 361

Edward J. AndersonThe University of Sydney Business School, Australia

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• Preface xiii1 What is risk management? 11.1 Introduction 21.2 Identifying and documenting risk 51.3 Fallacies and traps in risk management 71.4 Why safety is different 91.5 The Basel framework 111.6 Hold or hedge? 121.7 Learning from a disaster 131.7.1 What went wrong? 15Notes 17References 18Exercises 192 The structure of risk 222.1 Introduction to probability and risk 232.2 The structure of risk 252.2.1 Intersection and union risk 252.2.2 Maximum of random variables 282.3 Portfolios and diversification 302.3.1 Adding random variables 302.3.2 Portfolios with minimum variance 332.3.3 Optimal portfolio theory 372.3.4 When risk follows a normal distribution 382.4 The impact of correlation 402.4.1 Using covariance in combining random variables 412.4.2 Minimum variance portfolio with covariance 432.4.3 The maximum of variables that are positively correlated 442.4.4 Multivariate normal 462.5 Using copulas to model multivariate distributions 492.5.1 *Details on copula modeling 52Notes 58References 59Exercises 603 Measuring risk 633.1 How can we measure risk? 643.2 Value at risk 673.3 Combining and comparing risks 733.4 VaR in practice 763.5 Criticisms of VaR 793.6 Beyond value at risk 823.6.1 *More details on expected shortfall 86Notes 88References 88Exercises 894 Understanding the tails 924.1 Heavy-tailed distributions 934.1.1 Defining the tail index 934.1.2 Estimating the tail index 954.1.3 *More details on the tail index 984.2 Limiting distributions for the maximum 1004.2.1 *More details on maximum distributions and Fisher–Tippett 1064.3 Excess distributions 1094.3.1 *More details on threshold exceedances 1144.4 Estimation using extreme value theory 1154.4.1 Step 1. Choose a threshold u 1164.4.2 Step 2. Estimate the parameters ξ and β 1184.4.3 Step 3. Estimate the risk measures of interest 119Notes 121References 122Exercises 1235 Making decisions under uncertainty 1255.1 Decisions, states and outcomes 1265.1.1 Decisions 1265.1.2 States 1275.1.3 Outcomes 1275.1.4 Probabilities 1285.1.5 Values 1295.2 Expected Utility Theory 1305.2.1 Maximizing expected profit 1305.2.2 Expected utility 1325.2.3 No alternative to Expected Utility Theory 1355.2.4 *A sketch proof of the theorem 1395.2.5 What shape is the utility function? 1425.2.6 *Expected utility when probabilities are subjective 1455.3 Stochastic dominance and risk profiles 1485.3.1 *More details on stochastic dominance 1525.4 Risk decisions for managers 1565.4.1 Managers and shareholders 1565.4.2 A single company-wide view of risk 1585.4.3 Risk of insolvency 158Notes 160References 161Exercises 1626 Understanding risk behavior 1646.1 Why decision theory fails 1656.1.1 The meaning of utility 1656.1.2 Bounded rationality 1676.1.3 Inconsistent choices under uncertainty 1686.1.4 Problems from scaling utility functions 1716.2 Prospect Theory 1726.2.1 Foundations for behavioral decision theory 1736.2.2 Decision weights and subjective values 1756.3 Cumulative Prospect Theory 1806.3.1 *More details on Prospect Theory 1836.3.2 Applying Prospect Theory 1856.3.3 Why Prospect Theory does not always predict well 1876.4 Decisions with ambiguity 1896.5 How managers treat risk 191Notes 194References 194Exercises 1957 Stochastic optimization 1987.1 Introduction to stochastic optimization 1997.1.1 A review of optimization 1997.1.2 Two-stage recourse problems 2037.1.3 Ordering with stochastic demand 2087.2 Choosing scenarios 2127.2.1 How to carry out Monte Carlo simulation 2137.2.2 Alternatives to Monte Carlo 2177.3 Multistage stochastic optimization 2187.3.1 Non-anticipatory constraints 2207.4 Value at risk constraints 224Notes 228References 228Exercises 2298 Robust optimization 2328.1 True uncertainty: Beyond probabilities 2338.2 Avoiding disaster when there is uncertainty 2348.2.1 *More details on constraint reformulation 2408.2.2 Budget of uncertainty 2438.2.3 *More details on budgets of uncertainty 2478.3 Robust optimization and the minimax approach 2508.3.1 *Distributionally robust optimization 254Notes 261References 262Exercises 2639 Real options 2659.1 Introduction to real options 2669.2 Calculating values with real options 2679.2.1 *Deriving the formula for the surplus with a normal distribution 2729.3 Combining real options and net present value 2739.4 The connection with financial options 2789.5 Using Monte Carlo simulation to value real options 2829.6 Some potential problems with the use of real options 285Notes 287References 287Exercises 28810 Credit risk 29110.1 Introduction to credit risk 29210.2 Using credit scores for credit risk 29410.2.1 A Markov chain analysis of defaults 29610.3 Consumer credit 30110.3.1 Probability, odds and log odds 30210.4 Logistic regression 30810.4.1 *More details on logistic regression 31310.4.2 Building a scorecard 31510.4.3 Other scoring applications 317Notes 317References 318Exercises 319Appendix A Tutorial on probability theory 323A. 1 Random events 323A. 2 Bayes’ rule and independence 326A. 3 Random variables 327A. 4 Means and variances 329A. 5 Combinations of random variables 332A. 6 The normal distribution and the Central Limit Theorem 336Appendix B Answers to even-numbered exercises 340Index 361