CARL R. BACON, CIPM, is Chief Advisor to Confluence. He is a member of the Advisory Board of the Journal of Performance Measurement and Founder of The Freedom Index Company. He was formerly Chairman of StatPro Plc from 2000 to 2017.

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