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.

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 () (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 (+) 106

Bear 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®) 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

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