Practical Risk-Adjusted Performance Measurement
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Content
Preface xv
Acknowledgements xvii
1 Introduction 1
Definition of risk 1
Risk types 1
Risk management v risk control 4
Risk aversion 4
Ex-post and ex-ante 4
Dispersion 5
2 Descriptive Statistics 7
Mean (or arithmetic mean) 7
Annualised return 8
Continuously compounded returns (or log returns) 8
Winsorised mean 9
Mean absolute deviation (or mean deviation) 9
Variance 10
Mean difference (absolute mean difference or Gini mean difference) 12
Relative mean difference 14
Bessel's correction (population or sample, n or n-1) 14
Sample variance 17
Standard deviation (variability or volatility) 17
Annualised risk (or time aggregation) 18
The Central Limit Theorem 19
Janssen annualisation 19
Frequency and number of data points 19
Normal (or Gaussian) distribution 21
Histograms 22
Skewness (Fisher's or moment skewness) 22
Sample skewness 24
Kurtosis (Pearson's kurtosis) 24
Excess kurtosis (or Fisher's kurtosis) 25
Sample kurtosis 25
Bera-Jarque statistic (or Jarque-Bera) 27
Covariance 28
Sample covariance 30
Correlation (¿) 30
Sample correlation 32
Up capture indicator 32
Down capture indicator 34
Up number ratio 34
Down number ratio 34
Up percentage ratio 35
Down percentage ratio 35
Percentage gain ratio 35
Hurst index (or Hurst exponent) 35
Bias ratio 37
3 Simple Risk Measures 43
Performance appraisal 43
Sharpe ratio (reward to variability, Sharpe index) 43
Roy ratio 46
Risk free rate 46
Alternative Sharpe ratio 47
Revised Sharpe ratio 48
Adjusted Sharpe ratio 48
Skewness-kurtosis ratio 49
MAD ratio 49
Gini ratio 52
Relative risk 53
Tracking error (or tracking risk, relative risk, active risk) 53
Relative skewness 54
Relative kurtosis 55
Information ratio 55
Geometric information ratio 56
Modified information ratio 57
Adjusted information ratio 61
Relative Hurst 61
4 Regression Analysis 69
Regression equation 69
Regression alpha (aR) 70
Regression beta (ßR) 70
Regression epsilon (eR) 70
Capital Asset Pricing Model (CAPM) 71
Beta (ß) (systematic risk or volatility) 71
Jensen's alpha (Jensen's measure or Jensen's differential return or ex-post alpha) 72
Annualised alpha 72
Bull beta (ß +) 73
Bear beta (ß -) 73
Beta timing ratio 73
Market timing 78
Systematic risk 81
R2 (or coefficient of determination) 83
Specific or residual risk 83
Treynor ratio (reward to volatility) 84
Modified Treynor ratio 86
Appraisal ratio (or Treynor-Black ratio) 86
Modified Jensen 87
Fama decomposition 88
Selectivity 88
Diversification 88
Net selectivity 89
Fama-French three factor model 89
Three factor alpha (or Fama-French alpha) 91
Carhart four factor model 91
Four factor alpha (or Carhart's alpha) 91
K ratio 91
5 Drawdown 97
Drawdown 97
Average drawdown 97
Maximum drawdown (or peak to valley drawdown) 98
Largest individual drawdown 98
Recovery time (or drawdown duration) 98
Drawdown deviation 98
Ulcer index 99
Pain index 100
Calmar ratio (or drawdown ratio) 100
MAR ratio 100
Sterling ratio 100
Sterling-Calmar ratio 101
Burke ratio 102
Modified Burke ratio 102
Martin ratio (or Ulcer performance index) 102
Pain ratio 103
Lake ratio 103
Peak ratio 106
6 Partial Moments 107
Downside risk (or semi-standard deviation) 107
Pure downside risk 108
Half variance (or semi-variance) 108
Upside risk (or upside uncertainty) 108
Mean absolute moment 109
Omega ratio () 110
Bernardo and Ledoit (or gain-loss) ratio 110
d ratio 110
Omega-Sharpe ratio 111
Sortino ratio 112
Reward to half-variance 112
Downside risk Sharpe ratio 113
Downside information ratio 113
Kappa (Kl) (or Sortino-Satchell ratio) 113
Upside potential ratio 114
Volatility skewness 114
Variability skewness 115
Farinelli-Tibiletti ratio 115
Prospect ratio 117
7 Extreme Risk 119
Extreme events 119
Extreme value theory 119
Value at risk (VaR) 119
Relative VaR 120
Ex-post VaR 120
Potential upside (gain at risk) 121
Percentile rank 121
VaR calculation methodology 122
Parametric VaR 124
Modified VaR 125
Historical simulation (or non-parametric) 125
Monte Carlo simulation 126
Which methodology for calculating VaR should be used? 126
Frequency and time aggregation 127
Time horizon 127
Window length 127
Reward to VaR 128
Reward to relative VaR 129
Double VaR ratio 129
Conditional VaR (expected shortfall, tail loss, tail VaR or average VaR) 130
Upper CVaR or CVaR+ 131
Lower CVaR or CVaR- 131
Tail gain (expected gain or expected upside) 132
Conditional Sharpe ratio (STARR ratio or reward to conditional VaR) 133
Modified Sharpe ratio (reward to modified VaR) 136
Tail risk 136
Tail ratio 137
Rachev ratio (or R ratio) 137
Generalised Rachev ratio 137
Drawdown at risk 138
Conditional drawdown at risk 138
Reward to conditional drawdown 138
Generalised Z ratio 138
8 Fixed Income Risk 141
Pricing fixed income instruments 141
Redemption yield (yield to maturity) 141
Weighted average cash flow 141
Duration (effective mean term, discounted mean term or volatility) 142
Macaulay duration 142
Macaulay-Weil duration 143
Modified duration 143
Portfolio duration 144
Effective duration (or option-adjusted duration) 145
Duration to worst 146
Convexity 147
Modified convexity 147
Effective convexity 148
Portfolio convexity 148
Bond returns 149
Duration beta 150
Reward to duration 151
9 Risk-adjusted Return 153
Risk-adjusted return 153
M2 153
M2 excess return 154
Differential return 155
GH1 (Graham & Harvey 1) 156
GH2 (Graham & Harvey 2) 156
Correlation and risk-adjusted return M3 157
Return adjusted for downside risk 158
Adjusted M2 160
Omega excess return 161
10 Which Risk Measure to Use? 163
Why measure ex-post risk? 163
Which risk measures to use? 164
Hedge funds 164
Smoothing 169
Outliers 171
Data mining 171
Risk measures and the Global Investment
Performance Standards (GIPS R ) 172
Fund rating systems 174
Risk efficiency ratio 175
Which measures are actually used? 176
Which risk measures should really be used? 178
11 Risk Control 181
Regulations in the investment risk area 181
Risk control structure 182
Risk management 183
Glossary of Key Terms 189
Appendix A - Composite Internal Risk Measures 193
Appendix B - Absolute Risk Dashboard 195
Appendix C - Relative Risk Dashboard 199
Bibliography 203
Index 209
