Financial Risk Forecasting For other titles in the Wiley Finance Series please see www.wiley.com/finance The Theory and Practice of Forecasting Market Risk, with Implementation in R and Matlab Jo´ n Danı´elsson Financial Risk Forecasting A John Wiley and Sons, Ltd, Publication This edition first published 2011 Copyright 2011 Jo´ n Danı´elsson Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. 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Project management by OPS Ltd, Gt Yarmouth, Norfolk Typeset in 10/12pt Times Printed in Great Britain by CPI Antony Rowe, Chippenham, Wiltshire # Contents Preface xiii Acknowledgments xv Abbreviations xvii Notation xix 1 Financial markets, prices and risk 1 1.1 Prices, returns and stock indices 2 1.1.1 Stock indices 2 1.1.2 Prices and returns 2 1.2 S&P 500 returns 5 1.2.1 S&P 500 statistics 6 1.2.2 S&P 500 statistics in R and Matlab 7 1.3 The stylized facts of financial returns 9 1.4 Volatility 9 1.4.1 Volatility clusters 11 1.4.2 Volatility clusters and the ACF 12 1.5 Nonnormality and fat tails 14 1.6 Identification of fat tails 16 1.6.1 Statistical tests for fat tails 16 1.6.2 Graphical methods for fat tail analysis 17 1.6.3 Implications of fat tails in finance 20 1.7 Nonlinear dependence 21 1.7.1 Sample evidence of nonlinear dependence 22 1.7.2 Exceedance correlations 23 1.8 Copulas 25 1.8.1 The Gaussian copula 25 1.8.2 The theory of copulas 25 1.8.3 An application of copulas 27 1.8.4 Some challenges in using copulas 28 1.9 Summary 29 Contents 2 Univariate volatility modeling 31 2.1 Modeling volatility 31 2.2 Simple volatility models 32 2.2.1 Moving average models 32 2.2.2 EWMA model 33 2.3 GARCH and conditional volatility 35 2.3.1 ARCH 36 2.3.2 GARCH 38 2.3.3 The ‘‘memory’’ of a GARCH model 39 2.3.4 Normal GARCH 40 2.3.5 Student-t GARCH 40 2.3.6 (G)ARCH in mean 41 2.4 Maximum likelihood estimation of volatility models 41 2.4.1 The ARCH(1) likelihood function 42 2.4.2 The GARCH(1,1) likelihood function 42 2.4.3 On the importance of �1 43 2.4.4 Issues in estimation 43 2.5 Diagnosing volatility models 44 2.5.1 Likelihood ratio tests and parameter significance 44 2.5.2 Analysis of model residuals 45 2.5.3 Statistical goodness-of-fit measures 45 2.6 Application of ARCH and GARCH 46 2.6.1 Estimation results 46 2.6.2 Likelihood ratio tests 47 2.6.3 Residual analysis 47 2.6.4 Graphical analysis 48 2.6.5 Implementation 48 2.7 Other GARCH-type models 51 2.7.1 Leverage effects and asymmetry 51 2.7.2 Power models 52 2.7.3 APARCH 52 2.7.4 Application of APARCH models 52 2.7.5 Estimation of APARCH 53 2.8 Alternative volatility models 54 2.8.1 Implied volatility 54 2.8.2 Realized volatility 55 2.8.3 Stochastic volatility 55 2.9 Summary 56 3 Multivariate volatility models 57 3.1 Multivariate volatility forecasting 57 3.1.1 Application 58 3.2 EWMA 59 3.3 Orthogonal GARCH 62 3.3.1 Orthogonalizing covariance 62 3.3.2 Implementation 62 3.3.3 Large-scale implementations 63 vi Contents 3.4 CCC and DCC models 63 3.4.1 Constant conditional correlations (CCC) 64 3.4.2 Dynamic conditional correlations (DCC) 64 3.4.3 Implementation 65 3.5 Estimation comparison 65 3.6 Multivariate extensions of GARCH 67 3.6.1 Numerical problems 69 3.6.2 The BEKK model 69 3.7 Summary 70 4 Risk measures 73 4.1 Defining and measuring risk 73 4.2 Volatility 75 4.3 Value-at-risk 76 4.3.1 Is VaR a negative or positive number? 77 4.3.2 The three steps in VaR calculations 78 4.3.3 Interpreting and analyzing VaR 78 4.3.4 VaR and normality 79 4.3.5 Sign of VaR 79 4.4 Issues in applying VaR 80 4.4.1 VaR is only a quantile 80 4.4.2 Coherence 81 4.4.3 Does VaR really violate subadditivity? 83 4.4.4 Manipulating VaR 84 4.5 Expected shortfall 85 4.6 Holding periods, scaling and the square root of time 89 4.6.1 Length of holding periods 89 4.6.2 Square-root-of-time scaling 90 4.7 Summary 90 5 Implementing risk forecasts 93 5.1 Application 93 5.2 Historical simulation 95 5.2.1 Expected shortfall estimation 97 5.2.2 Importance of window size 97 5.3 Risk measures and parametric methods 98 5.3.1 Deriving VaR 99 5.3.2 VaR when returns are normally distributed 101 5.3.3 VaR under the Student-t distribution 102 5.3.4 Expected shortfall under normality 103 5.4 What about expected returns? 104 5.5 VaR with time-dependent volatility 106 5.5.1 Moving average 106 5.5.2 EWMA 107 5.5.3 GARCH normal 108 5.5.4 Other GARCH models 109 5.6 Summary 109 Contents vii 6 Analytical value-at-risk for options and bonds 111 6.1 Bonds 112 6.1.1 Duration-normal VaR 112 6.1.2 Accuracy of duration-normal VaR 114 6.1.3 Convexity and VaR 114 6.2 Options 115 6.2.1 Implementation 117 6.2.2 Delta-normal VaR 119 6.2.3 Delta and gamma 120 6.3 Summary 120 7 Simulation methods for VaR for options and bonds 121 7.1 Pseudo random number generators 122 7.1.1 Linear congruental generators 122 7.1.2 Nonuniform RNGs and transformation methods 123 7.2 Simulation pricing 124 7.2.1 Bonds 125 7.2.2 Options 129 7.3 Simulation of VaR for one asset 132 7.3.1 Monte Carlo VaR with one basic asset 133 7.3.2 VaR of an option on a basic asset 134 7.3.3 Options and a stock 136 7.4 Simulation of portfolio VaR 137 7.4.1 Simulation of portfolio VaR for basic assets 137 7.4.2 Portfolio VaR for options 139 7.4.3 Richer versions 139 7.5 Issues in simulation estimation 140 7.5.1 The quality of the RNG 140 7.5.2 Number of simulations 140 7.6 Summary 142 8 Backtesting and stress testing 143 8.1 Backtesting 143 8.1.1 Market risk regulations 146 8.1.2 Estimation window length 146 8.1.3 Testing window length 147 8.1.4 Violation ratios 147 8.2 Backtesting the S&P 500 147 8.2.1 Analysis 150 8.3 Significance of backtests 153 8.3.1 Bernoulli coverage test 154 8.3.2 Testing the independence of violations 155 8.3.3 Testing VaR for the S&P 500 157 8.3.4 Joint test 159 8.3.5 Loss-function-based backtests 159 8.4 Expected shortfall backtesting 160 8.5 Problems with backtesting 162 viii Contents 8.6 Stress testing 163 8.6.1 Scenario analysis 163 8.6.2 Issues in scenario analysis 165 8.6.3 Scenario analysis and risk models 165 8.7 Summary 166 9 Extreme value theory 167 9.1 Extreme value theory 168 9.1.1 Types of tails 168 9.1.2 Generalized extreme value distribution 169 9.2 Asset returns and fat tails 170 9.3 Applying EVT 172 9.3.1 Generalized Pareto distribution 172 9.3.2 Hill method 173 9.3.3 Finding the threshold 174 9.3.4 Application to the S&P 500 index 175 9.4 Aggregation and convolution 176 9.5 Time dependence 179 9.5.1 Extremal index 179 9.5.2 Dependence in ARCH 180 9.5.3 When does dependence matter? 180 9.6 Summary 181 10 Endogenous risk 183 10.1 The Millennium Bridge 184 10.2 Implications for financial risk management 184 10.2.1 The 2007–2010 crisis 185 10.3 Endogenous market prices 188 10.4 Dual role of prices 190 10.4.1 Dynamic trading strategies 191 10.4.2 Delta hedging 192 10.4.3 Simulation of feedback 194 10.4.4 Endogenous risk and the 1987 crash 195 10.5 Summary 195 APPENDICES A Financial time series 197 A.1 Random variables and probability density functions 197 A.1.1 Distributions and densities 197 A.1.2 Quantiles 198 A.1.3 The normal distribution 198 A.1.4 Joint distributions 200 A.1.5 Multivariate normal distribution 200 A.1.6 Conditional distribution 200 Contents ix A.1.7 Independence 201 A.2 Expectations and variance 201 A.2.1 Properties of expectation and variance 202 A.2.2 Covariance and independence 203 A.3 Higher order moments 203 A.3.1 Skewness and kurtosis 204 A.4 Examples of distributions 206 A.4.1 Chi-squared �2 � � 206 A.4.2 Student-t 206 A.4.3 Bernoulli and binomial distributions 208 A.5 Basic time series concepts 208 A.5.1 Autocovariances and autocorrelations 209 A.5.2 Stationarity 209 A.5.3 White noise 210 A.6 Simple time series models 210 A.6.1 The moving average model 210 A.6.2 The autoregressive model 211 A.6.3 ARMA model 212 A.6.4 Random walk 212 A.7 Statistical hypothesis testing 212 A.7.1 Central limit theorem 213 A.7.2 p-values 213 A.7.3 Type 1 and type 2 errors and the power of the test 214 A.7.4 Testing for normality 214 A.7.5 Graphical methods: QQ plots 215 A.7.6 Testing for autocorrelation 215 A.7.7 Engle LM test for volatility clusters 216 B An introduction to R 217 B.1 Inputting data 217 B.2 Simple operations 219 B.2.1 Matrix computation 220 B.3 Distributions 222 B.3.1 Normality tests 223 B.4 Time series 224 B.5 Writing functions in R 225 B.5.1 Loops and repeats 226 B.6 Maximum likelihood estimation 228 B.7 Graphics 229 C An introduction to Matlab 231 C.1 Inputting data 231 C.2 Simple operations 233 C.2.1 Matrix algebra 234 C.3 Distributions 235 C.3.1 Normality tests 237 C.4 Time series 237 x Contents C.5 Basic programming and M-files 238 C.5.1 Loops 239 C.6 Maximum likelihood 242 C.7 Graphics 243 D Maximum likelihood 245 D.1 Likelihood functions 245 D.1.1 Normal likelihood functions 246 D.2 Optimizers 247 D.3 Issues in ML estimation 248 D.4 Information matrix 249 D.5 Properties of maximum likelihood estimators 250 D.6 Optimal testing procedures 250 D.6.1 Likelihood ratio test 251 D.6.2 Lagrange multiplier test 252 D.6.3 Wald test 253 Bibliography 255 Index 259 Contents xi Preface The focus in this book is on the study of market risk from a quantitative point of view. The emphasis is on presenting commonly used state-of-the-art quantitative techniques used in finance for the management of market risk and demonstrate their use employing the principal two mathematical programming languages, R and Matlab. All the code in the book can be downloaded from the book’s website at www.financialrisk forecasting.com The book brings together three essential fields: finance, statistics and computer programming. It is assumed that the reader has a basic understanding of statistics and finance; however, no prior knowledge of computer programming is required. The book takes a hands-on approach to the issue of financial risk, with the reading material intermixed between finance, statistics and computer programs. I have used the material in this book for some years, both for a final year under- graduate course in quantitative methods and for master level courses in risk forecasting. In most cases, the students taking this course have no prior knowledge of computer programming, but emerge after the course with the ability to independently implement the models and code in this book. All of the material in the book can be covered in about 10 weeks, or 20 lecture hours. Most chapters demonstrate the way in which the various techniques discussed are implemented by both R and Matlab. We start by downloading a sample of stock prices, which are then used for model estimation and evaluation. The outline of the book is as follows. Chapter 1 begins with an introduction to financial markets and market prices. The chapter gives a foretaste of what is to come, discussing market indices and stock prices, the forecasting of risk and prices, and concludes with the main features of market prices from the point of view of risk. The main focus of the chapter is introduction of the three stylized facts regarding returns on financial assets: volatility clusters, fat tails and nonlinear dependence. Chapters 2 and 3 focus on volatility forecasting: the former on univariate volatility and the latter on multivariate volatility. The aim is to survey all the methods used for volatility forecasting, while discussing several models from the GARCH family in considerable detail. We discuss the models from a theoretical point of view and demonstrate their implementation and evaluation. This is followed by two chapters on risk models and risk forecasting: Chapter 4 addresses the theoretical aspects of risk forecasting—in particular, volatility, value- Preface