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 (ρ) 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 (αR) 70

Regression beta (βR) 70

Regression epsilon (εR) 70

Capital Asset Pricing Model (CAPM) 71

Beta (β) (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 (β +) 73

Bear beta (β −) 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