Practical Risk-Adjusted Performance Measurement

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Carl R. Bacon. Practical Risk-Adjusted Performance Measurement

Table of Contents

List of Tables

List of Illustrations

Guide

Pages

Practical Risk-Adjusted Performance Measurement

Preface

Note

Acknowledgements

About the Companion Website

CHAPTER 1 Introduction

DEFINITION OF RISK

RISK TYPES

RISK MANAGEMENT VERSUS RISK CONTROL

RISK AVERSION

EX POST AND EX ANTE

DISPERSION

NOTES

CHAPTER 2 Descriptive Statistics

MEAN (OR ARITHMETIC MEAN)

ANNUALISED RETURN

Caution

CONTINUOUSLY COMPOUNDED RETURNS (OR LOG RETURNS)

Note

WINSORISED MEAN

Note

MEAN ABSOLUTE DEVIATION (OR MEAN DEVIATION)

VARIANCE

EXHIBIT 2.1 Portfolio mean and variance

EXHIBIT 2.2 Benchmark mean and variance

MEAN DIFFERENCE (ABSOLUTE MEAN DIFFERENCE OR GINI MEAN DIFFERENCE)

RELATIVE MEAN DIFFERENCE

BESSEL'S CORRECTION (POPULATION OR SAMPLE, N OR N − 1)

EXHIBIT 2.3 Bessel's correction. FIRST MONTH IN EACH QUARTER

SECOND MONTH IN EACH QUARTER

THIRD MONTH IN EACH QUARTER

SAMPLE VARIANCE

STANDARD DEVIATION (VARIABILITY OR VOLATILITY)

Interpretation

Note

ANNUALISED RISK (OR TIME AGGREGATION)

Note

Note

THE CENTRAL LIMIT THEOREM

FREQUENCY AND NUMBER OF DATA POINTS

Caution

Caution

EXHIBIT 2.4 Standard deviation

ALTERNATIVE RISK ANNUALISATION METHODS

EXHIBIT 2.5 Alternative annualised standard deviation

NORMAL (OR GAUSSIAN) DISTRIBUTION

Caution

HISTOGRAMS

SKEWNESS (FISHER'S OR MOMENT SKEWNESS)

SAMPLE SKEWNESS

Note

KURTOSIS (PEARSON'S KURTOSIS)

Note

EXCESS KURTOSIS (OR FISHER'S KURTOSIS)

Caution

SAMPLE KURTOSIS

Caution

Note

BERA–JARQUE STATISTIC (OR JARQUE–BERA)

Note

COVARIANCE

EXHIBIT 2.6 Skewness, kurtosis and the Bera–Jarque hypothesis test

EXHIBIT 2.7 Benchmark skewness and kurtosis

SAMPLE COVARIANCE

CORRELATION (ρ)

Caution

SAMPLE CORRELATION

EXHIBIT 2.8 Covariance and correlation

AUTOCOVARIANCE

AUTOCORRELATION (OR SERIAL CORRELATION)

Caution

EXHIBIT 2.9 Autocovariance and autocorrelation

ANNUALISED VARIABILITY IF RETURNS ARE AUTOCORRELATED

Caution

Caution

EXHIBIT 2.10 Annualised variability adjusted for autocorrelation

NOTES

CHAPTER 3 Performance Appraisal Measures

PERFORMANCE APPRAISAL

SHARPE RATIO (REWARD TO VARIABILITY, SHARPE INDEX)

Caution

Interpretation

Interpretation

ROY RATIO

RISK-FREE RATE

Caution

Caution

ALTERNATIVE SHARPE RATIO

REVISED SHARPE RATIO

ADJUSTED SHARPE RATIO

Interpretation

Note

Interpretation

SKEW-ADJUSTED SHARPE RATIO

EXHIBIT 3.1 Sharpe, alternative, skew-adjusted and adjusted Sharpe ratios

EXHIBIT 3.2 Revised Sharpe ratio

Interpretation

SKEWNESS–KURTOSIS RATIO

Note

ALTERNATIVE ADJUSTED SHARPE RATIOS

Note

SMOOTHING-ADJUSTED SHARPE RATIO

EXHIBIT 3.3 Smoothing-adjusted Sharpe ratio

MAD RATIO

GINI RATIO

RELATIVE RISK

Interpretation

Caution

TRACKING ERROR (OR TRACKING RISK, RELATIVE RISK, ACTIVE RISK)

RELATIVE SKEWNESS

RELATIVE KURTOSIS

INFORMATION RATIO

GEOMETRIC INFORMATION RATIO

Caution

Note

EXHIBIT 3.4 Arithmetic and geometric information ratios

MODIFIED INFORMATION RATIO

Caution

ADJUSTED INFORMATION RATIO

SKEW-ADJUSTED INFORMATION RATIO

EXHIBIT 3.5 Relative skewness, kurtosis and adjusted information ratio

NOTES

CHAPTER 4 Regression Analysis

REGRESSION EQUATION

REGRESSION ALPHA

REGRESSION BETA

REGRESSION EPSILON

EXHIBIT 4.1 Regression alpha and beta

CAPITAL ASSET PRICING MODEL (CAPM)

Caution

BETA (β) (SYSTEMATIC RISK OR VOLATILITY)

JENSEN'S ALPHA (JENSEN'S MEASURE OR JENSEN'S DIFFERENTIAL RETURN OR EX-POST ALPHA)

ANNUALISED ALPHA

EXHIBIT 4.2 CAPM beta and Jensen's alpha

BULL BETA (β+)

BEAR BETA (β−)

Note

BETA TIMING RATIO

EXHIBIT 4.3Beta timing ratio

Note

MARKET TIMING

SYSTEMATIC RISK

CORRELATION

R2 (OR COEFFICIENT OF DETERMINATION)

Caution

EXHIBIT 4.4 Correlation and R2

SPECIFIC (OR RESIDUAL) RISK

EXHIBIT 4.5 Specific risk, systematic risk and total risk

THE GEOMETRY OF RISK

EXHIBIT 4.6 Relative risk and correlation

TREYNOR RATIO (REWARD TO VOLATILITY)

MODIFIED TREYNOR RATIO

APPRAISAL RATIO (OR TREYNOR–BLACK RATIO)

Caution

MODIFIED JENSEN

FAMA DECOMPOSITION

SELECTIVITY

DIVERSIFICATION

NET SELECTIVITY

FAMA–FRENCH THREE-FACTOR MODEL

Note

THREE-FACTOR ALPHA (OR FAMA–FRENCH ALPHA)

CARHART FOUR-FACTOR MODEL

FOUR-FACTOR ALPHA (OR CARHART'S ALPHA)

TYPES OF ALPHA

MULTI-FACTOR MODELS

Caution

NOTES

CHAPTER 5. Drawdown

AVERAGE DRAWDOWN

MAXIMUM DRAWDOWN

LARGEST INDIVIDUAL DRAWDOWN

RECOVERY TIME (OR DRAWDOWN DURATION)

Caution

DRAWDOWN DEVIATION

ULCER INDEX

PAIN INDEX

CALMAR RATIO (OR DRAWDOWN RATIO)

Note

MAR RATIO

STERLING RATIO

STERLING−CALMAR RATIO

Caution

BURKE RATIO

MODIFIED BURKE RATIO

MARTIN RATIO (OR ULCER PERFORMANCE INDEX)

PAIN RATIO

EXHIBIT 5.1 Drawdown (peak to trough definition)

EXHIBIT 5.2 Drawdown (continuous uninterrupted definition)

ACTIVE (OR RELATIVE) DRAWDOWN

EXHIBIT 5.3 Active Pain and active Ulcer

NOTES

CHAPTER 6. Partial Moments

DOWNSIDE RISK (OR SEMI-STANDARD DEVIATION)

Note

DOWNSIDE POTENTIAL

PURE DOWNSIDE RISK

HALF VARIANCE (OR SEMI-VARIANCE)

UPSIDE RISK (OR UPSIDE UNCERTAINTY)

MEAN ABSOLUTE MOMENT

OMEGA RATIO (Ω)

BERNARDO AND LEDOIT (OR GAIN–LOSS) RATIO

d RATIO

OMEGA–SHARPE RATIO

Interpretation

SORTINO RATIO

REWARD TO HALF-VARIANCE

DOWNSIDE RISK SHARPE RATIO

DOWNSIDE INFORMATION RATIO

SORTINO–SATCHELL RATIO

Interpretation

KAPPA RATIO

UPSIDE POTENTIAL RATIO

Interpretation

Note

VOLATILITY SKEWNESS

Interpretation

VARIABILITY SKEWNESS

FARINELLI–TIBILETTI RATIO

Interpretation

EXHIBIT 6.1 Downside and upside partial moments

Caution

GAIN–LOSS SKEWNESS

DOWNSIDE SKEWNESS AND KURTOSIS

SORTINO RATIO WITH HIGHER ORDER MOMENTS

EXHIBIT 6.2 Downside skew and kurtosis

NOTES

CHAPTER 7. Prospect Theory

PROSPECT RATIO

Note

NEW PROSPECT RATIO

OMEGA–PROSPECT RATIO

Note

EXHIBIT 7.1 Prospect ratios

NOTES

CHAPTER 8. Extreme Risk

EXTREME EVENTS

EXTREME VALUE THEORY

VALUE AT RISK (VaR)

Note

Caution

RELATIVE VaR

EX-POST VaR

POTENTIAL UPSIDE (GAIN AT RISK)

PERCENTILE RANK

VaR CALCULATION METHODOLOGY

PARAMETRIC VaR

EXHIBIT 8.1 Parametric VaR

MODIFIED VaR

EXHIBIT 8.2 Modified VaR

HISTORICAL SIMULATION (OR NON-PARAMETRIC)

MONTE CARLO SIMULATION

Note

WHICH METHODOLOGY FOR CALCULATING VaR SHOULD BE USED?

VaR INTERPRETATION

FREQUENCY AND TIME AGGREGATION

TIME HORIZON

WINDOW LENGTH

REWARD TO VaR

REWARD TO RELATIVE VaR

DOUBLE VaR RATIO

CONDITIONAL VaR (EXPECTED SHORTFALL, TAIL LOSS, TAIL VaR OR AVERAGE VaR)

Caution

UPPER CVaR OR CVaR+

LOWER CVaR OR CVaR−

TAIL GAIN (EXPECTED GAIN OR EXPECTED UPSIDE)

CONDITIONAL SHARPE RATIO (STARR RATIO OR REWARD TO CONDITIONAL VaR)

EXHIBIT 8.3 Ex-post VaR and conditional VaR

MODIFIED SHARPE RATIO (REWARD TO MODIFIED VaR)

EXHIBIT 8.4 Parametric and modified VaR

TAIL RISK

Note

TAIL RATIO

RACHEV RATIO (OR R RATIO)

Interpretation

GENERALISED RACHEV RATIO

EXHIBIT 8.5 Tail risk and Rachev ratio

DRAWDOWN AT RISK

Caution

CONDITIONAL DRAWDOWN AT RISK

REWARD TO CONDITIONAL DRAWDOWN

Note

GENERALISED Z RATIO

NOTES

CHAPTER 9. Fixed Income Risk

PRICING FIXED INCOME INSTRUMENTS

REDEMPTION YIELD (YIELD TO MATURITY)

WEIGHTED AVERAGE CASH FLOW

DURATION (EFFECTIVE MEAN TERM, DISCOUNTED MEAN TERM OR VOLATILITY)

MACAULAY DURATION

MACAULAY–WEIL DURATION

MODIFIED DURATION

Note

EXHIBIT 9.1 Duration

PORTFOLIO DURATION

EFFECTIVE DURATION (OR OPTION-ADJUSTED DURATION)

EXHIBIT 9.2 Effective duration

DURATION TO WORST

CONVEXITY

MODIFIED CONVEXITY

EFFECTIVE CONVEXITY

EXHIBIT 9.3 Convexity

PORTFOLIO CONVEXITY

BOND RETURNS

EXHIBIT 9.4 Estimated bond returns. ACTUAL PRICE CHANGE

USING MODIFIED DURATION AND MODIFIED CONVEXITY

USING EFFECTIVE DURATION AND EFFECTIVE CONVEXITY

DURATION BETA

REWARD TO DURATION

NOTES

CHAPTER 10 Miscellaneous Risk Measures

UPSIDE CAPTURE RATIO (OR UP CAPTURE INDICATOR)

Interpretation

DOWNSIDE CAPTURE RATIO (OR DOWN CAPTURE INDICATOR)

Note

UP/DOWN CAPTURE (OR CAPTURE RATIO)

EXHIBIT 10.1 Capture ratios

UP NUMBER RATIO

DOWN NUMBER RATIO

UP PERCENTAGE RATIO

DOWN PERCENTAGE RATIO

PERCENTAGE GAIN RATIO

BATTING AVERAGE (OR RELATIVE BATTING AVERAGE)

HURST INDEX (OR HURST EXPONENT)

Interpretation

EXHIBIT 10.2 Hurst index. PORTFOLIO

BENCHMARK

RELATIVE HURST INDEX (OR ACTIVE HURST)

EXHIBIT 10.3 Relative Hurst index. ARITHMETIC EXCESS RETURNS

GEOMETRIC EXCESS RETURNS

BIAS RATIO

Note

Caution

EXHIBIT 10.4 Bias ratio. PORTFOLIO

BENCHMARK

ACTIVE SHARE

Interpretation

K RATIO

Note

EXHIBIT 10.5 Slope of cumulative return

EXHIBIT 10.6 K ratio

NOTES

CHAPTER 11 Risk-Adjusted Return

M2

M2 EXCESS RETURN

DIFFERENTIAL RETURN

GH1 (GRAHAM AND HARVEY 1)

GH2 (GRAHAM AND HARVEY 2)

CORRELATION AND RISK-ADJUSTED RETURN M3

RETURN ADJUSTED FOR DOWNSIDE RISK

EXHIBIT 11.1

Note

ADJUSTED M2

Interpretation

SKEW-ADJUSTED M2

OMEGA EXCESS RETURN

EXHIBIT 11.2M2, skew-adjusted M2 and adjusted M2

NOTES

CHAPTER 12. A Periodic Table of Risk Measures

PERIODIC TABLE DESIGN

FILLING THE PERIODIC TABLE

NOTATION

NOTES

CHAPTER 13 Risk-Adjusted Performance Fees

PERFORMANCE FEES

ASYMMETRIC OR SYMMETRIC

PERFORMANCE FEES IN PRACTICE

Note

Caution

NOTES

CHAPTER 14 Performance Dashboards

EFFECTIVE DASHBOARDS

DATA VISUALISATION TOOLS

NOTES

CHAPTER 15 Manager Selection

ASSET MANAGER SELECTION

MANAGER EVALUATION

PORTFOLIO EVALUATION

MONITORING AND CONTROL

Caution

NOTES

CHAPTER 16 The Four Dimensions of Performance

EX-POST RETURN (THE TRADITIONAL DIMENSION)

EX-POST RISK (THE NEGLECTED DIMENSION)

EX-ANTE RETURN (THE UNKNOWN DIMENSION)

EX-ANTE RISK (THE “SEXY” DIMENSION)

RISK EFFICIENCY RATIO

PERFORMANCE EFFICIENCY

EX-ANTE RISK STANDARDS

CONSISTENCY IN CALCULATIONS AND COMPARISON

DISCLOSURE

RECOGNITION OF ADHERENCE TO BEST PRACTICE

MORE ROBUST INTERNAL PROCESS AND CONTROL

NOTES

CHAPTER 17. Which Risk Measure to Use?

WHY MEASURE EX-POST RISK?

WHICH RISK MEASURES TO USE?

HEDGE FUNDS

SMOOTHING

Caution

OUTLIERS

Caution

DATA MINING

Caution

RISK MEASURES AND THE GLOBAL INVESTMENT PERFORMANCE STANDARDS (GIPS®)

FUND RATING SYSTEMS

WHICH MEASURES ARE ACTUALLY USED?

WHICH RISK MEASURES SHOULD REALLY BE USED?

COMMON ERRORS TO AVOID

NOTES

CHAPTER 18. Risk Control

REGULATIONS IN THE INVESTMENT RISK AREA

Caution

RISK CONTROL STRUCTURE

RISK MANAGEMENT

Glossary of Key Terms

APPENDIX A Composite Internal Risk Measures

Bibliography

Index. A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

R

S

T

U

V

W

Y

Z

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In the middle office of asset management firms, we are most concerned with portfolio risk, which I define as the uncertainty of meeting asset owner10 expectations. Is the portfolio of assets managed in line with the asset owner's investment objectives? The consequences of not meeting asset owner expectations can be quite severe. Early in 2001,11 the Unilever Superannuation Fund sued Merrill Lynch for damages of £130 million claiming negligence in that Merrill Lynch had not sufficiently considered the risk of underperformance. Ultimately the case was settled out of court for an undisclosed sum, believed to be £70 million, the perception to many being that Unilever won.

Credit risk (or issuer risk) as opposed to counterparty risk is a type of portfolio risk. Credit risk or default risk is the investor's risk of a borrower failing to meet their financial commitments in full. The higher the risk of default the higher the rate of interest investors will demand to lend their capital. Therefore, the reward or returns in terms of higher yields must offset the increased risk of default. Similarly, market, currency and interest rate risks taken by asset managers in the pursuit of asset owner objectives would constitute portfolio risks in this context.

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