Contents
List of Tables XIII
List of Figures XV
I Utility and risk analysis 1
1 Utility theory 5
1.1 Preferences under uncertainty . . . . . . . . . . . . . . . . . 7
1.1.1 Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Axioms on pref erences . . . . . . . . . . . . . . . . . . 8
1.2 Expected utility . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Arrow-Pratt measures ofri sk aversion . . . . . . . . . 13
1.3.2 Standard utility functions . . . . . . . . . . . . . . . . 15
1.3.3 Applications to portfolio allocation . . . . . . . . . . . 17
1.4 Stochastic dominance . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Alternative expected utility theory . . . . . . . . . . . . . . . 24
1.5.1 Weighted utility theory . . . . . . . . . . . . . . . . . 25
1.5.2 Rank dependent expected utility theory . . . . . . . . 27
1.5.3 Non-additive expected utility . . . . . . . . . . . . . . 32
1.5.4 Regret theory . . . . . . . . . . . . . . . . . . . . . . . 33
1.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Riskmeasures 37
2.1 Coherent and convex risk measures . . . . . . . . . . . . . . 37
2.1.1 Coherent riskmeasures . . . . . . . . . . . . . . . . . 38
2.1.2 Convex riskmeasures . . . . . . . . . . . . . . . . . . 39
2.1.3 Representation ofri sk measures . . . . . . . . . . . . . 40
2.1.4 Risk measures and utility . . . . . . . . . . . . . . . . 41
2.1.5 Dynamic riskmeasures . . . . . . . . . . . . . . . . . 43
2.2 Standard riskmeasures . . . . . . . . . . . . . . . . . . . . . 48
2.2.1 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.2 CVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.3 Spectral measures of risk . . . . . . . . . . . . . . . . 59
2.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 62
X Portfolio Optimization and Performance Analysis
II Standard portfolio optimization 65
3 Static optimization 67
3.1 Mean-variance analysis . . . . . . . . . . . . . . . . . . . . . 68
3.1.1 Diversification effect . . . . . . . . . . . . . . . . . . . 68
3.1.2 Optimal weights . . . . . . . . . . . . . . . . . . . . . 71
3.1.3 Additional constraints . . . . . . . . . . . . . . . . . . 78
3.1.4 Estimation problems . . . . . . . . . . . . . . . . . . . 82
3.2 Alternative criteria . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 Expected utility maximization . . . . . . . . . . . . . 85
3.2.2 Risk measure minimization . . . . . . . . . . . . . . . 93
3.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Indexed funds and benchmarking 103
4.1 Indexed funds . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.1 Tracking error . . . . . . . . . . . . . . . . . . . . . . 104
4.1.2 Simple index tracking methods . . . . . . . . . . . . . 105
4.1.3 The threshold accepting algorithm . . . . . . . . . . . 106
4.1.4 Cointegration tracking method . . . . . . . . . . . . . 112
4.2 Benchmark portf olio optimization . . . . . . . . . . . . . . . 117
4.2.1 Tracking-error definition . . . . . . . . . . . . . . . . . 118
4.2.2 Tracking-errorminimization . . . . . . . . . . . . . . . 119
4.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Portfolio performance 129
5.1 Standard performance measures . . . . . . . . . . . . . . . . 130
5.1.1 The Capital Asset Pricing Model . . . . . . . . . . . . 130
5.1.2 The three standard performance measures . . . . . . . 132
5.1.3 Other performance measures . . . . . . . . . . . . . . 140
5.1.4 Beyond the CAPM . . . . . . . . . . . . . . . . . . . . 145
5.2 Perf ormance decomposition . . . . . . . . . . . . . . . . . . . 151
5.2.1 The Fama decomposition . . . . . . . . . . . . . . . . 151
5.2.2 Other performance attributions . . . . . . . . . . . . . 153
5.2.3 The external attribution . . . . . . . . . . . . . . . . . 153
5.2.4 The internal attribution . . . . . . . . . . . . . . . . . 155
5.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 163
III Dynamic portfolio optimization 165
6 Dynamic programming optimization 169
6.1 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.1 Calculus ofv ariations . . . . . . . . . . . . . . . . . . 169
6.1.2 Pontryagin and Bellman principles . . . . . . . . . . . 175
6.1.3 Stochastic optimal control . . . . . . . . . . . . . . . . 182
6.2 Lifetime portfolio selection . . . . . . . . . . . . . . . . . . . 187
Contents XI
6.2.1 The optimization problem . . . . . . . . . . . . . . . . 187
6.2.2 The deterministic coefficients case . . . . . . . . . . . 188
6.2.3 The general case . . . . . . . . . . . . . . . . . . . . . 195
6.2.4 Recursive utility in continuous-time . . . . . . . . . . 203
6.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 205
7 Optimal payoff profiles and long-term management 207
7.1 Optimal payoffs as functions of a benchmark . . . . . . . . . 207
7.1.1 Linear versus option-based strategy . . . . . . . . . . 207
7.2 Application to long-term management . . . . . . . . . . . . . 214
7.2.1 Assets dynamics and optimal portfolios . . . . . . . . 214
7.2.2 Exponential utility . . . . . . . . . . . . . . . . . . . . 220
7.2.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . 223
7.2.4 Distribution ofthe optimal portfolio return . . . . . . 225
7.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 226
8 Optimization within specific markets 229
8.1 Optimization in incomplete markets . . . . . . . . . . . . . . 230
8.1.1 General result based on martingale method . . . . . . 230
8.1.2 Dynamic programming and viscosity solutions . . . . 238
8.2 Optimization with constraints . . . . . . . . . . . . . . . . . 242
8.2.1 General result . . . . . . . . . . . . . . . . . . . . . . . 242
8.2.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . 249
8.3 Optimization with transaction costs . . . . . . . . . . . . . . 256
8.3.1 The infinite-horizon case . . . . . . . . . . . . . . . . . 256
8.3.2 The finite-horizon case . . . . . . . . . . . . . . . . . . 260
8.4 Other f rameworks . . . . . . . . . . . . . . . . . . . . . . . . 263
8.4.1 Labor income . . . . . . . . . . . . . . . . . . . . . . . 263
8.4.2 Stochastic horizon . . . . . . . . . . . . . . . . . . . . 272
8.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 276
IV Structured portfolio management 279
9 Portfolio insurance 281
9.1 The Option Based Portfolio Insurance . . . . . . . . . . . . . 282
9.1.1 The standard OBPI method . . . . . . . . . . . . . . . 284
9.1.2 Extensions oft he OBPI method . . . . . . . . . . . . 286
9.2 The Constant Proportion Portfolio Insurance . . . . . . . . . 294
9.2.1 The standard CPPI method . . . . . . . . . . . . . . . 295
9.2.2 CPPI extensions . . . . . . . . . . . . . . . . . . . . . 303
9.3 Comparison between OBPI and CPPI . . . . . . . . . . . . . 305
9.3.1 Comparison at maturity . . . . . . . . . . . . . . . . . 305
9.3.2 The dynamic behavior ofOBPI and CPPI . . . . . . . 310
9.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 318
XII Portfolio Optimization and Performance Analysis
10 Optimal dynamic portfolio with risk limits 319
10.1 Optimal insured portfolio: discrete-time case . . . . . . . . . 321
10.1.1 Optimal insured portfolio with a fixed number of assets 321
10.1.2 Optimal insured payoffs as functions of a benchmark . 326
10.2 Optimal Insured Portfolio: the dynamically complete case . . 333
10.2.1 Guarantee atmaturity . . . . . . . . . . . . . . . . . . 333
10.2.2 Risk exposure and utility function . . . . . . . . . . . 335
10.2.3 Optimal portfolio with controlled drawdowns . . . . . 337
10.3 Value-at-Risk and expected shortfall based management . . . 340
10.3.1 Dynamic saf ety criteria . . . . . . . . . . . . . . . . . 340
10.3.2 Expected utility under VaR/CVaR constraints . . . . 347
10.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 350
11 Hedge funds 351
11.1 The hedge funds industry . . . . . . . . . . . . . . . . . . . . 351
11.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 351
11.1.2 Main strategies . . . . . . . . . . . . . . . . . . . . . . 352
11.2 Hedge f und perf ormance . . . . . . . . . . . . . . . . . . . . 354
11.2.1 Return distributions . . . . . . . . . . . . . . . . . . . 354
11.2.2 Sharpe ratio limits . . . . . . . . . . . . . . . . . . . . 355
11.2.3 Alternative performance measures . . . . . . . . . . . 362
11.2.4 Benchmarks for alternative investment . . . . . . . . . 368
11.2.5 Measure oft he performance persistence . . . . . . . . 369
11.3 Optimal allocation in hedge funds . . . . . . . . . . . . . . . 370
11.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 371
A Appendix A: Arch Models 373
B Appendix B: Stochastic Processes 381
References 397
Symbol Description 431
Index 433