Dynamic Asset Allocation Claus Munk Until August 2012: Aarhus University, e-mail: [email protected] From August 2012: Copenhagen Business School, e-mail: [email protected] this version: July 3, 2012 The document contains graphs in color, use color printer for best results. Contents Preface v 1 Introduction to asset allocation 1 1.1 Introduction . .1 1.2 Investor classes and motives for investments . .1 1.3 Typical investment advice . .3 1.4 How do individuals allocate their wealth? . .3 1.5 An overview of the theory of optimal investments . .3 1.6 The future of investment management and services . .3 1.7 Outline of the rest . .3 1.8 Notation . .3 2 Preferences 5 2.1 Introduction . .5 2.2 Consumption plans and preference relations . .6 2.3 Utility indices . .9 2.4 Expected utility representation of preferences . 10 2.5 Risk aversion . 16 2.6 Utility functions in models and in reality . 20 2.7 Preferences for multi-date consumption plans . 26 2.8 Exercises . 33 3 One-period models 37 3.1 Introduction . 37 3.2 The general one-period model . 37 3.3 Mean-variance analysis . 43 3.4 A numerical example . 49 3.5 Mean-variance analysis with constraints . 49 3.6 Estimation . 49 i ii Contents 3.7 Critique of the one-period framework . 49 3.8 Exercises . 50 4 Discrete-time multi-period models 51 4.1 Introduction . 51 4.2 A multi-period, discrete-time framework for asset allocation . 51 4.3 Dynamic programming in discrete-time models . 54 5 Introduction to continuous-time modelling 59 5.1 Introduction . 59 5.2 The basic continuous-time setting . 59 5.3 Dynamic programming in continuous-time models . 62 5.4 Loss from suboptimal strategies . 66 5.5 Exercises . 67 6 Asset allocation with constant investment opportunities 69 6.1 Introduction . 69 6.2 General utility function . 70 6.3 CRRA utility function . 72 6.4 Logarithmic utility . 75 6.5 Discussion of the optimal investment strategy for CRRA utility . 76 6.6 The life-cycle . 78 6.7 Loss due to suboptimal investments . 80 6.8 Infrequent rebalancing of the portfolio . 81 6.9 Exercises . 84 7 Stochastic investment opportunities: the general case 85 7.1 Introduction . 85 7.2 General utility functions . 86 7.3 CRRA utility . 93 7.4 Logarithmic utility . 105 7.5 How costly are deviations from the optimal investment strategy? . 105 7.6 Exercises . 107 8 The martingale approach 111 8.1 The martingale approach in complete markets . 111 8.2 Complete markets and constant investment opportunities . 115 8.3 Complete markets and stochastic investment opportunities . 119 8.4 The martingale approach with portfolio constraints . 120 8.5 Exercises . 127 9 Numerical methods for solving dynamic asset allocation problems 129 Contents iii 10 Asset allocation with stochastic interest rates 131 10.1 Introduction . 131 10.2 One-factor Vasicek interest rate dynamics . 132 10.3 One-factor CIR dynamics . 135 10.4 A numerical example . 137 10.5 Two-factor Vasicek model . 143 10.6 Other studies with stochastic interest rates . 146 10.7 Exercises . 149 11 Asset allocation with stochastic market prices of risk 153 11.1 Introduction . 153 11.2 Mean reversion in stock returns . 153 11.3 Stochastic volatility . 160 11.4 More . 164 11.5 Exercises . 164 12 Inflation risk and asset allocation with no risk-free asset 167 12.1 Introduction . 167 12.2 Real and nominal price dynamics . 167 12.3 Constant investment opportunities . 169 12.4 General stochastic investment opportunities . 172 12.5 Hedging real interest rate risk without real bonds . 172 13 Labor income 179 13.1 Introduction . 179 13.2 A motivating example . 179 13.3 Exogenous income in a complete market . 181 13.4 Exogenous income in incomplete markets . 189 13.5 Endogenous labor supply and income . 191 13.6 More . 194 14 Consumption and portfolio choice with housing 195 15 Other variations of the problem... 197 15.1 Multiple and/or durable consumption goods . 197 15.2 Uncertain time of death; insurance . 197 16 International asset allocation 199 17 Non-standard assumptions on investors 201 17.1 Preferences with habit formation . 201 17.2 Recursive utility . 203 17.3 Model/parameter uncertainty, incomplete information, learning . 210 17.4 Ambiguity aversion . 210 17.5 Other objective functions . 210 17.6 Consumption and portfolio choice for non-price takers . 210 iv Contents 17.7 Non-utility based portfolio choice . 210 17.8 Allowing for bankruptcy . 211 18 Trading and information imperfections 213 18.1 Trading constraints . 213 18.2 Transaction costs . 213 A Results on the lognormal distribution 219 B Stochastic processes and stochastic calculus 223 B.1 Introduction . 223 B.2 What is a stochastic process? . 224 B.3 Brownian motions . 231 B.4 Diffusion processes . 234 B.5 It^oprocesses . 237 B.6 Stochastic integrals . 237 B.7 It^o'sLemma . 241 B.8 Important diffusion processes . ..