2016 硕士论文 Performance of risk-based asset allocation strategies

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2016 硕士论文 Performance of risk-based asset allocation strategies 156页

Contents
1 Introduction 1
1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theory 5
2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Sample covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Shrinkage estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Shrinkage targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Shrinkage intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Markowitz portfolio theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Justication for the mean-variance strategy . . . . . . . . . . . . . . . . . 23
2.4.2 The unconstrained global minimum variance portfolio . . . . . . . . . . . 24
2.4.3 The long-only global minimum variance portfolio . . . . . . . . . . . . . . 30
2.5 Maximum diversication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Diversication ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 The most diversied portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 The equally-weighted portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Risk parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7.1 Risk contribution and marginal risk contribution . . . . . . . . . . . . . . 36
2.7.2 The equally-weighted risk contributions portfolio . . . . . . . . . . . . . . 37
2.8 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.8.1 Equal volatility example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8.2 Constant correlation example . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9 Statistical measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9.1 Sharpe ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9.2 Sortino ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9.3 Maximum drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9.4 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.10 Turnover and transaction costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Methodology 51
3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Data source and data type . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 The investment universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Risk-free rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Calculation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Basic calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Covariance matrix estimation . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Global minimum variance portfolio . . . . . . . . . . . . . . . . . . . . . . 63
3.2.4 Most diversied portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.5 Equally-weighted risk contributions portfolio . . . . . . . . . . . . . . . . 65
3.2.6 Equally-weighted portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.7 Sharpe ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.8 Sortino ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.9 Maximum drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.10 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.11 Turnover and transaction costs . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.12 Constant risk targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Results 72
4.1 How well do these portfolios perform when re-investing returns? . . . . . . . . . . 73
4.2 What are the characteristics of the portfolios returns distribution? . . . . . . . . 79
4.3 How concentrated are the portfolio weights? . . . . . . . . . . . . . . . . . . . . . 81
4.4 What are the levels of turnover and transaction costs? . . . . . . . . . . . . . . . 84
4.5 To what degree can these portfolios be used to obtain a constant risk target? . . 87
4.6 How can our choice of covariance matrix aect results? . . . . . . . . . . . . . . . 88
5 Discussion 96
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Conclusion 100
Bibliography 103
Appendix A Results using constant correlation as target 110
A.1 Figures using the constant correlation model . . . . . . . . . . . . . . . . . . . . 110
A.2 Tables using the constant correlation model . . . . . . . . . . . . . . . . . . . . . 115
Appendix B R program 117
B.1 Main program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.2 Functions used in main program . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B.3 Graphing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

List of Figures
2.1 The ecient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Nominal/risk allocation in the 60/40 strategy . . . . . . . . . . . . . . . . . . . . 35
2.3 Equal risk contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Number of stocks fullling data requirement . . . . . . . . . . . . . . . . . . . . . 53
3.2 Net return EURO STOXX 50 in euro . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Risk-free rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Cumulative performance - single-index model . . . . . . . . . . . . . . . . . . . . 73
4.2 Return-standard deviation - single-index model . . . . . . . . . . . . . . . . . . . 75
4.3 Rolling 24-month volatility - single-index model . . . . . . . . . . . . . . . . . . . 77
4.4 Rolling 24-month Sharpe ratio - single-index model . . . . . . . . . . . . . . . . . 78
4.5 Rolling 24-month Sortino ratio - single-index model . . . . . . . . . . . . . . . . . 79
4.6 Density of portfolio returns - single-index model . . . . . . . . . . . . . . . . . . 80
4.7 Number of stocks invested in for each portfolio - single-index model . . . . . . . . 82
4.8 Maximum absolute values of portfolio weights - single-index model . . . . . . . . 83
4.9 Minimum absolute values of portfolio weights - single-index model . . . . . . . . 84
4.10 Annualized turnover - single-index model . . . . . . . . . . . . . . . . . . . . . . 85
4.11 Performance after transaction costs - single-index model . . . . . . . . . . . . . . 86
4.12 Rolling 24-month volatilities with volatility target 10% - single-index model . . . 87
4.13 Leverage constant over time - single-index model . . . . . . . . . . . . . . . . . . 88
4.14 Rolling 24-month volatilities with volatility target 10% - constant correlation model 89
4.15 Leverage constant over time - constant correlation model . . . . . . . . . . . . . 90
4.16 Shrinkage intensity - single-index model . . . . . . . . . . . . . . . . . . . . . . . 91
4.17 Shrinkage intensity - constant correlation model . . . . . . . . . . . . . . . . . . . 91
4.18 Cumulative performance - constant correlation model . . . . . . . . . . . . . . . 92
A.1 Rolling 24-month volatility - constant correlation model . . . . . . . . . . . . . . 110
A.2 Return-standard deviations - constant correlation model . . . . . . . . . . . . . . 111
A.3 Rolling 24-month Sharpe ratio - constant correlation model . . . . . . . . . . . . 111
A.4 Rolling 24-month Sortino ratio - constant correlation model . . . . . . . . . . . . 112
A.5 Density of portfolio returns - constant correlation model . . . . . . . . . . . . . . 112
A.6 Number of stocks invested in for each portfolio - constant correlation model . . . 113
iv
A.7 Maximum absolute values of portfolio weights - constant correlation model . . . 113
A.8 Minimum absolute values of portfolio weights - constant correlation model . . . . 114
A.9 Annualized turnover - constant correlation model . . . . . . . . . . . . . . . . . . 114
A.10 Performance after transaction costs - constant correlation model . . . . . . . . . 115
vList of Figures
2.1 The ecient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Nominal/risk allocation in the 60/40 strategy . . . . . . . . . . . . . . . . . . . . 35
2.3 Equal risk contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Number of stocks fullling data requirement . . . . . . . . . . . . . . . . . . . . . 53
3.2 Net return EURO STOXX 50 in euro . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Risk-free rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Cumulative performance - single-index model . . . . . . . . . . . . . . . . . . . . 73
4.2 Return-standard deviation - single-index model . . . . . . . . . . . . . . . . . . . 75
4.3 Rolling 24-month volatility - single-index model . . . . . . . . . . . . . . . . . . . 77
4.4 Rolling 24-month Sharpe ratio - single-index model . . . . . . . . . . . . . . . . . 78
4.5 Rolling 24-month Sortino ratio - single-index model . . . . . . . . . . . . . . . . . 79
4.6 Density of portfolio returns - single-index model . . . . . . . . . . . . . . . . . . 80
4.7 Number of stocks invested in for each portfolio - single-index model . . . . . . . . 82
4.8 Maximum absolute values of portfolio weights - single-index model . . . . . . . . 83
4.9 Minimum absolute values of portfolio weights - single-index model . . . . . . . . 84
4.10 Annualized turnover - single-index model . . . . . . . . . . . . . . . . . . . . . . 85
4.11 Performance after transaction costs - single-index model . . . . . . . . . . . . . . 86
4.12 Rolling 24-month volatilities with volatility target 10% - single-index model . . . 87
4.13 Leverage constant over time - single-index model . . . . . . . . . . . . . . . . . . 88
4.14 Rolling 24-month volatilities with volatility target 10% - constant correlation model 89
4.15 Leverage constant over time - constant correlation model . . . . . . . . . . . . . 90
4.16 Shrinkage intensity - single-index model . . . . . . . . . . . . . . . . . . . . . . . 91
4.17 Shrinkage intensity - constant correlation model . . . . . . . . . . . . . . . . . . . 91
4.18 Cumulative performance - constant correlation model . . . . . . . . . . . . . . . 92
A.1 Rolling 24-month volatility - constant correlation model . . . . . . . . . . . . . . 110
A.2 Return-standard deviations - constant correlation model . . . . . . . . . . . . . . 111
A.3 Rolling 24-month Sharpe ratio - constant correlation model . . . . . . . . . . . . 111
A.4 Rolling 24-month Sortino ratio - constant correlation model . . . . . . . . . . . . 112
A.5 Density of portfolio returns - constant correlation model . . . . . . . . . . . . . . 112
A.6 Number of stocks invested in for each portfolio - constant correlation model . . . 113
iv
A.7 Maximum absolute values of portfolio weights - constant correlation model . . . 113
A.8 Minimum absolute values of portfolio weights - constant correlation model . . . . 114
A.9 Annualized turnover - constant correlation model . . . . . . . . . . . . . . . . . . 114
A.10 Performance after transaction costs - constant correlation model . . . . . . . . . 115

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