Practical Risk-Adjusted Performance Measurement For other titles in the Wiley Finance series please see www.wiley.com/finance Practical Risk-Adjusted Performance Measurement

Carl R. Bacon

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Library of Congress Cataloging-in-Publication Data

Bacon, Carl R. Practical risk-adjusted performance measurement / Carl R. Bacon. p. cm. Includes bibliographical references and index. ISBN 978-1-118-36974-6 (cloth) 1. Financial risk management. 2. Performance standards. 3. Risk management. I. Title. HD61.B33 2013 658.155Ðdc23 2012023317 A catalogue record for this book is available from the British Library.

ISBN 978-1-118-36974-6 (hardback) ISBN 978-1-118-39153-2 (ebk) ISBN 978-1-118-39152-5 (ebk) ISBN 978-1-118-39137-2 (ebk)

Set in 11/13pt Times by Aptara Inc., New Delhi, India Printed in Great Britain by TJ International Ltd, Padstow, Cornwall, UK To my parents

Contents

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 viii Contents

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 Contents ix

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

5Drawdown 97 Drawdown 97 x Contents

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 Contents xi

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 xii Contents

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 Contents xiii

Risk measures and the Global Investment Performance Standards (GIPSR ) 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

Preface

“Beauty is in the eye of the beholder.” Margaret Wolfe Hungerford (1855Ð1897), Molly Bawn 1878

The book I wanted to read on risk did not exist and this book attempts to fill that gap. There are many books and articles, perhaps hundreds, written on the subject of portfolio risk but for the most part they focus on ex-ante risk, tend to be highly academic with authors seemingly in a competition to present the material in as complex a language as possible and are typically devoid of worked examples. This book is written for risk and performance measurement practitioners from a buy side, asset management perspective, focusing on quantitative ex-post measures rather than the qualitative aspects of risk. Risk has an undeserved reputation within asset management firms for being an overly complex, mathematical subject. The purpose of this book is to simplify the subject and demonstrate with many practical examples that risk is perfectly straightforward and not as complicated as it might seem. In addition I wanted to document, with appropriate referencing, as many discrete ex-post risk measures as possible in a structured format, filling gaps, encouraging consistency, suggesting new measures and highlighting possible areas of confusion or misrepresentation. In truth many of these measures are rarely used in practice, often for good reason. This book will not recommend any particular risk measure, although it is difficult to disguise my preferences and prejudices. Risk like beauty is very much in the eye of the beholder and different risk measures will suit different investment strategies or investor concerns at different xvi Preface times. This book should provide enough information and insight for the reader to determine their own preferences. In terms of structure Chapter 1 is naturally an introduction to the subject of risk in the context of asset management firms. In Chapter 2 the foundations are laid introducing the descriptive statistics that will be used in later chapters. The following chapters are structured accord- ing to the type of risk measure being considered, simple measures in Chapter 3, regression measures in Chapter 4,drawdowninChapter 5, partial moments in Chapter 6, extreme risk in Chapter 7, risk measures for fixed income instruments in Chapter 8 and risk-adjusted returns in Chapter 9. In the penultimate Chapter 10 there is a discussion about which risk measures to use and finally in Chapter 11 their application in terms of risk control. The objective of this book is to provide a complete list of ex-post risk measures used by asset managers. Although some have little merit I’ve avoided censoring measures I dislike. If a risk measure is not included, perhaps it’s in continuous not discrete form, maybe I don’t fully understand it with enough confidence to write about it, or in a few rare cases I’ve determined that it literally has no merit. Acknowledgements

My thanks are owed to many that have contributed to this book, both directly and indirectly, working colleagues over many years, attendees at my various training courses and workshops which I hope will continue, attentive readers of my previous books that have spotted a number of errors and indeed made many good suggestions, numerous fellow GIPSR committee members that have been so insightful and of course the many authors that have laid the foundations of this subject. I’m particularly thankful to the diligent reviewers of this book: Kate Maryniak and Ralph Purtscher-Wydenbruck of StatPro, Jerry Pinto, CFA of CFA Institute, Philip Lawton, CFA, CIPM, PRM of Aite Group, Colin Morrison of Paradigm Investment Consulting Ltd and Dimitri Senik, CFA, FCCA. Of course all errors and omissions are my own

Carl R. Bacon CIPM Deeping St James April 2012 [email protected]

1 Introduction

“Money is like muck, not good except it be spread.” Francis Bacon (1561–1626)

DEFINITION OF RISK Risk means very different things to different audiences at different times; risk is truly in the eye of the beholder. In the context of portfolio man- agement the Oxford English Dictionary provides a surprisingly good definition of risk:

The potential impact of an event determined by combining the likelihood of the event occurring with the impact should it occur.

Risk is the combination of exposure and uncertainty. As Holton1 (2004) so eloquently points out it is not risky to jump out of an aircraft without a parachute, death is certain. Holton also points out that we can never operationally define risk; at best, we can operationally define our perception of risk. Another common and effective, but broader definition of risk is exposure to uncertainty.

Risk types Within asset management firms there are many types of risk that should concern portfolio managers and senior management. For convenience I’ve chosen to classify risk into five main categories:

Compliance Risk Operational Risk

1 Glyn A. Holton (2004) Defining Risk. Financial Analysts Journal 60(6). 2 Practical Risk-Adjusted Performance Measurement

Liquidity Risk Counterparty Risk Portfolio Risk These risks are ranked in my priority order of concern at the point in time I assumed the role of Director of Risk Control at an asset management firm in the late 1990s.2 What I didn’t appreciate fully then, but appreciated much later, is that priorities will vary through time; during the credit crisis I’m sure counterparty risk became the number one priority for many firms. Although a major concern of all asset managers, reputational risk does not warrant a separate category; a risk failure in any category can cause significant damage to a firm’s reputation. Compliance or regulatory risk is the risk of breaching a regulatory, client or internally imposed guideline, restriction or clear limit. I draw no distinction between internal or external limits; the breach of an internal limit indicates a control failure, which could just as easily have been a regulatory, or client mandated limit. Of course the financial impact of breaching limits can be significant; in August 1996 Peter Young of Morgan Grenfell Asset Management allegedly cost Deutsche Bank £300 to £400 million in compensation payments to investors in highly regulated authorised unit trusts. Peter Young used Luxembourg listed shell companies to circumvent limits on unlisted and risky holdings. Operational risk, often defined as a residual catch all category to include risks not defined elsewhere, actually includes the risk of human error, fraud, system failure, poor controls, management failure and failed trades. Risks of this type are more common but usually less severe. Nevertheless it is important to continuously monitor errors and near misses of all types, even those that do not result in financial loss. An increase in the frequency of errors regardless of size or sign may indicate a more serious problem that requires further investigation and corrective action. Although typically small in size, operational errors can lead to large losses. In December 2005 a trader at the Japanese brokerage firm Mizuho Securities made a typing error and tried to sell 610,000 shares at 1 yen apiece in recruiting company J-Com Co., which was debuting on the exchange, instead of an intended sale of one share at 610,000 yen, an example of fat-finger syndrome. Mizuho lost approximately 41 billion yen.

2 In truth I did not identify liquidity risk as a separate risk category at the time. Introduction 3

Liquidity risk is the risk that assets cannot be traded quickly enough in a market to change asset and risk allocations, realise profits or prevent losses. Perhaps liquidity risk has received less attention than it should in the past but it is capable of causing significant damage. Understanding liquidity risk in both normal and turbulent markets is a crucial element of effective risk control; the relatively recently identified phenomenon of crowded exits is a characteristic of those turbulent markets. Counterparty risk occurs when counterparties are unwilling or unable to fulfil their contractual obligations, at its most basic through corporate failure. Counterparty exposure could include profits on an OTC deriva- tives contract, unsettled transactions, cash management, administrators, custodians, prime brokers, and even with the comfort of appropriate collateral the failure to return stock that has been used for stock lending. Perhaps the most obvious example of counterparty risk is the failure of Lehman Brothers in September 2008. In the middle office of asset management firms we are most concerned with portfolio risk, which I define as the uncertainty of meeting client expectations. Is the portfolio managed in line with the client’s investment objectives? The consequences of not meeting client expectations can be quite severe. Early in 2001,3 the Unilever Superannuation Fund sued Merrill Lynch for damages of £130 million claiming negligence that Merrill Lynch had not sufficiently taken into account 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. I’m sure readers can quickly add to this brief list of risks and extend through various subdivisions, but I’m fairly certain any risk I’ve not mentioned so far can be allocated to one or more of the above categories. 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 portfolio managers in the pursuit of client objectives would constitute portfolio risks in this context.

3 A.F. Perold and R. Alloway (2003) The Unilever Superannuation Fund vs. Merrill Lynch. Harvard Business School Publishing. 4 Practical Risk-Adjusted Performance Measurement

Risk management v risk control It is useful to distinguish between the ways portfolio managers and risk professionals see risk. For this purpose, let us refer to portfolio managers as “risk managers” and to risk professionals as “risk controllers”. Then there is a clear distinction between risk management and risk control. As risk managers, portfolio managers are paid to take risk, and they need to take risk in order to achieve higher returns. For the risk manager “Risk is good”. Risk controllers on the other hand are paid to monitor risk; their role is to measure risk and make transparent to the entire firm how much risk is being taken by the portfolio manager (and often from their perspective to reduce risk). The risk controller’s objective is to reduce the probability or eliminate entirely a major loss event on their watch. For the risk controller “Risk is bad”. Risk managers’ and risk controllers’ objectives are in conflict leading to a natural tension between them. To resolve this conflict we need measures that assess the quality of return and answer the question, “Are we achieving sufficient return for the risk taken?”

Risk aversion It is helpful to assume that investors are risk averse, that is to say, that given portfolios with equal rates of return they will prefer the portfolio with the lowest risk. Investors will only accept additional risk if they are compensated by the prospect of higher returns.

Ex-post and ex-ante Risk is calculated in two fundamentally different ways, ex-post and ex- ante. Ex-post or historical risk is the analysis of risk after the event; it answers the question how risky has the portfolio been in the past. On the other hand ex-ante risk or prospective risk is forward looking, based on a snapshot of the current securities and instruments within the portfolio and their historical relationship with each other; it is an estimate or forecast of the future risk of the portfolio. Obviously the use of historical returns and correlations to forecast future risk is problematic, particularly for extreme, low probability events. Increasing the length of the historical track record or increasing the frequency of observations