Practical Risk-Adjusted Performance Measurement

Chapter 1 Introduction 15 Definition of risk 15 Risk types 15 Risk management v Risk control 18 Risk aversion 19 Ex-post and ex-ante 19 Dispersion 20 Chapter 2 Descriptive statistics 21 Mean (or arithmetic mean) 21 Annualised return 22 Continuously compounded returns (or log returns) 22 Winsorised mean 23 Mean absolute deviation (or mean deviation) 24 Variance 25 Mean difference (absolute mean difference or Gini mean difference) 30 Relative mean difference 31 Bessel's correction (population or sample, n or n-1) 31 Sample variance 35 Standard deviation (variability or volatility) 36 Annualised risk (or time aggregation) 37 The Central Limit Theorem 38 Frequency and number of data points 38 Alternative risk annualisation methods 39 Normal (or Gaussian) distribution 40 Histograms 42 Skewness (Fisher's or moment skewness) 43 Sample skewness 44 Kurtosis (Pearson's kurtosis) 45 Excess kurtosis (or Fisher's kurtosis) 47 Sample kurtosis 47 Bera-Jarque statistic (or Jarque-Bera) 48 Covariance 53 Sample covariance 54 Correlation ( ) 54 Sample correlation 55 Autocovariance 55 Autocorrelation (or serial correlation) 57 Annualised variability if returns are autocorrelated 60 Chapter 3 APPRAISAL MEASURES 62 Performance appraisal 62 Sharpe ratio (reward to variability, Sharpe index) 63 Roy ratio 65 Risk-free rate 66 Alternative Sharpe ratio 66 Revised Sharpe ratio 67 Adjusted Sharpe Ratio 68 Skew-adjusted Sharpe Ratio 69 Skewness-Kurtosis ratio 74 Alternative adjusted Sharpe Ratios 74 Smoothing-adjusted Sharpe Ratio 75 MAD ratio 76 Gini ratio 76 Relative risk 77 Tracking error (or tracking risk, relative risk, active risk) 77 Relative skewness 78 Relative kurtosis 79 Information ratio 79 Geometric information ratio 80 Modified information ratio 87 Adjusted information ratio 88 Skew-adjusted information ratio 88 Chapter 4: Regression Analysis 94 Regression analysis 94 Regression equation 95 Regression alpha 95 Regression beta 95 Regression epsilon 95 Capital Asset Pricing Model (CAPM) 96 Beta (beta) (systematic risk or volatility) 97 Jensen's alpha (Jensen's measure or Jensen's differential return or ex-post alpha) 97 Annualised alpha 98 Bull beta (beta+) 106 Bear beta (beta-) 106 Beta timing ratio 106 Market timing 107 Systematic risk 115 Correlation 115 R2(or coefficient of determination) 116 Specific (or residual) risk 117 The Geometry of Risk 120 Treynor ratio (Reward to volatility) 124 Modified Treynor ratio 124 Appraisal ratio (or Treynor-Black ratio) 125 Modified Jensen 126 Fama decomposition 126 Selectivity 127 Diversification 127 Net selectivity 127 Fama-French three factor model 128 Three factor alpha (or Fama-French alpha) 129 Carhart four factor model 129 Four factor alpha (or Carhart's alpha) 130 Types of Alpha 130 Multi-factor Models 131 Chapter 5 Drawdown 132 Drawdown 132 Average drawdown 132 Maximum drawdown 133 Largest individual drawdown 133 Recovery time (or drawdown duration) 133 Drawdown deviation 134 Ulcer index 134 Pain index 135 Calmar ratio (or Drawdown ratio) 136 MAR ratio 136 Sterling ratio 136 Sterling-Calmar ratio 137 Burke ratio 138 Modified Burke ratio 138 Martin ratio (or Ulcer performance index) 138 Pain ratio 138 Active (or relative) Drawdown 143 Chapter 6 Partial Moments 148 Downside risk (or semi-standard deviation) 148 Downside potential 149 Pure downside risk 149 Half variance (or semi-variance) 149 Upside risk (or upside uncertainty) 150 Mean absolute moment 150 Omega ratio ( ) 151 Bernardo & Ledoit (or gain-loss) ratio 151 d ratio 151 Omega-Sharpe ratio 152 Sortino ratio 153 Reward to half-variance 153 Downside risk Sharpe ratio 154 Downside information ratio 154 Sortino-Satchell ratio 155 Kappa ratio 155 Upside potential ratio 156 Volatility skewness 156 Variability skewness 157 Farinelli- Tibiletti Ratio 160 Gain-loss skewness 160 Downside Skewness & Kurtosis 161 Sortino Ratio with higher order moments 161 Chapter 7 Prospect Theory 165 Prospect ratio 165 New Prospect ratio 166 Omega-Prospect ratio 166 Chapter 8 Extreme Risk 170 Extreme events 170 Extreme value theory 170 Value at Risk (VaR) 170 Relative VaR 171 Ex-post VaR 171 Potential upside (gain at risk) 172 Percentile rank 172 VaR calculation methodology 175 Parametric VaR 175 Modified VaR 176 Historical simulation (or non-parametric) 177 Monte Carlo simulation 177 Which methodology for calculating VaR should be used? 178 VaR Interpretation 178 Frequency and time aggregation 180 Time horizon 180 Window length 181 Reward to VaR 181 Reward to relative VaR 182 Double VaR ratio 183 Conditional VaR (expected shortfall, tail loss, tail VaR or average VaR) 183 Upper CVaR or CVaR+ 184 Lower CVaR or CVaR- 184 Tail gain (expected gain or expected upside) 186 Conditional Sharpe ratio (STARR ratio or reward to conditional VaR) 191 Modified Sharpe ratio (reward to modified VaR) 191 Tail risk 191 Tail ratio 192 Rachev ratio (or R ratio) 192 Generalised Rachev ratio 194 Drawdown at risk 194 Conditional drawdown at risk 194 Reward to conditional drawdown 195 Generalised Z ratio 195 Chapter 9 Fixed Income Risk 197 Pricing fixed income instruments 197 Redemption yield (yield to maturity) 197 Weighted average cash flow 197 Duration (effective mean term, discounted mean term or volatility) 198 Macaulay duration 198 Macaulay-Weil duration 199 Modified duration 199 Portfolio duration 200 Effective duration (or option-adjusted duration) 202 Duration to worst 204 Convexity 204 Modified convexity 205 Effective convexity 205 Portfolio convexity 207 Bond returns 207 Duration beta 209 Reward to duration 209 Chapter 10 miscellaneous Risk Measures 210 Upside Capture Ratio (or up capture indicator) 210 Downside capture ratio (or down capture indicator) 210 Up/down capture (or Capture ratio) 211 Up number ratio 216 Down number ratio 216 Up percentage ratio 217 Down percentage ratio 217 Percentage gain ratio 217 Batting Average (or Relative Batting Average) 217 Hurst index (or Hurst exponent) 218 Relative Hurst Index (or Active Hurst) 225 Bias ratio 231 Active Share 237 K ratio 239 Chapter 11 Risk-adjusted Return 248 Risk-adjusted return 248 M2 248 M2 excess return 250 Differential return 250 GH1 (Graham & Harvey 1) 252 GH2 (Graham & Harvey 2) 252 Correlation and risk-adjusted return M3 253 Return adjusted for downside risk 253 Adjusted M2 257 Skew-adjusted M2 257 Omega excess return 258 Chapter 12: A Periodic Table of Risk Measures 259 A Periodic Table of Risk Measures 259 Periodic Table Design 260 Filling the Periodic Table 261 Notation 264 Chapter 13: Risk-adjusted Performance Fees 269 Performance Fees 269 Asymmetric or Symmetric 269 Performance Fees in Practice 273 Chapter 14: Performance Dashboards 276 Effective dashboards 276 Data visualisation tools 277 Chapter 15: Manager Selection 279 Asset Manager Selection 279 Manager Evaluation 280 Portfolio Evaluation 281 Monitoring and Control 282 Chapter 16: The Four Dimensions of Performance 284 Ex-post Return (The traditional dimension) 285 Ex-post Risk (The neglected dimension) 285 Ex-ante Return (The unknown dimension) 285 Ex-ante Risk (The "sexy" dimension) 286 Risk efficiency ratio 286 Performance efficiency 287 Ex-ante Risk Standards 287 Consistency in calculations and comparison 288 Disclosure 288 Recognition of adherence to best practice 288 More robust internal process and control 288 Chapter 17: Which Risk Measure to Use? 291 Why measure ex-post risk? 291 Which risk measures to use? 291 Hedge funds 295 Smoothing 296 Outliers 299 Data mining 300 Risk measures and the Global Investment Performance Standards (GIPS(r)) 300 Fund rating systems 303 Which measures are actually used? 304 Which risk measures should really be used? 309 Common Errors to avoid 310 Chapter 18: Risk Control 311 Regulations in the investment risk area 311 Risk control structure 312 Risk management 313 Glossary of Key Terms 318 Appendix A - Composite Internal Risk Measures 321 Bibliography 323