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Tail Risk of Equity Market Indices: an Extreme Value Theory Approach An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach February 2014 Institute 2 Printed in France, February 2014. Copyright EDHEC 2014. The opinions expressed in this survey are those of the authors and do not necessarily reflect those of EDHEC Business School. Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 Table of Contents Executive Summary .................................................................................................5 1. Introduction ............................................................................................................9 2. Extreme Value Theory .......................................................................................13 3. A Conditional EVT Model .................................................................................19 4. Risk Estimation with EVT .................................................................................23 5. Back-testing and Statistical Tests .................................................................27 6. Data and Empirical Results .............................................................................31 7. Conclusions .........................................................................................................43 Appendices ..............................................................................................................47 References ...............................................................................................................55 About EDHEC-Risk Institute ................................................................................59 EDHEC-Risk Institute Publications and Position Papers (2011-2014) ........63 An EDHEC-Risk Institute Publication 3 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 About the Authors Lixia Loh is a senior research engineer at EDHEC-Risk Institute–Asia. Prior to joining EDHEC Business School, she was a Research Fellow at the Centre for Global Finance at Bristol Business School (University of the West of England). Her research interests include empirical finance, financial markets risk, and monetary economics. She has published in several academic journals, including the Asia-Pacific Development Journal and Macroeconomic Dynamics, and is the author of a book, Sovereign Wealth Funds: States Buying the World (Global Professional Publishing, 2010). She holds an M.Sc. in international economics, banking and finance from Cardiff University and a Ph.D. in finance from the University of Nottingham. Stoyan Stoyanov is professor of finance at EDHEC Business School and head of research at EDHEC Risk Institute–Asia. He has ten years of experience in the field of risk and investment management. Prior to joining EDHEC Business School, he worked for over six years as head of quantitative research for FinAnalytica. He has designed and implemented investment and risk management models for financial institutions, co-developed a patented system for portfolio optimisation in the presence of non-normality, and led a team of engineers designing and planning the implementation of advanced models for major financial institutions. His research focuses on probability theory, extreme risk modelling, and optimal portfolio theory. He has published over thirty articles in leading academic and practitioner-oriented scientific journals such as Annals of Operations Research, Journal of Banking and Finance, and the Journal of Portfolio Management, contributed to many professional handbooks and co-authored three books on probability and stochastics, financial risk assessment and portfolio optimisation. He holds a master in science in applied probability and statistics from Sofia University and a PhD in finance from the University of Karlsruhe. 4 An EDHEC-Risk Institute Publication Executive Summary An EDHEC-Risk Institute Publication 5 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 Executive Summary Value-at-risk (VaR) and conditional value- model. Thus, a model such as the normal at-risk (CVaR) have become standard distribution underestimates this frequency choices for risk measures in finance. Both and, therefore, underestimates tail risk as VaR and CVaR are examples of measures of well. tail risk, or downside risk, because they are designed to exhibit a degree of sensitivity As a consequence, to compare tail risk across to large portfolio losses whose frequency of markets, we need to adopt a conditional occurrence is described by what is known as measure which can take into account at least the tail of the distribution: a part of the loss the clustering of volatility effect and also distribution away from the central region the tail behaviour of portfolio losses having geometrically resembling a tail. In practice, explained away the dynamics of volatility. VaR provides a loss threshold exceeded with This decomposition into two components some small predefined probability, usually is important from a risk management 1% or 5%, while CVaR measures the average perspective because the dynamics of loss higher than VaR and is, therefore, more volatility contribute to the unconditional informative about extreme losses. tail thickness phenomenon and techniques do exist for volatility management. It is An interesting challenge is to compare tail therefore important to understand how risk across different markets. A stylised fact much residual tail thickness remains after for asset returns is that they exhibit fat tails; explaining away the dynamics of volatility. that is, the frequency of observed extreme The standard econometric framework losses is higher than that predicted by the taking into account the clustering of normal distribution. Usually, for practical volatility effect is that of the Generalised purposes this frequency is calculated Autoregressive Conditional Heteroskedastic unconditionally while it is a well-known (GARCH) model. fact that in different market states the likelihood of getting an extreme loss The academic literature on modelling varies, i.e. in more turbulent markets it VaR and CVaR indicates that a successful is more likely to experience higher losses. approach for modelling the high quantiles As a result, tail risk would be affected by of the portfolio loss distribution is to the temporal behaviour of volatility which is combine a GARCH model with extreme characterised by clustering: elevated levels value theory (EVT). The GARCH part is of volatility are usually followed by similar responsible for capturing the dynamics volatility levels. of volatility while EVT provides a model for the behaviour of the extreme tail of Apart from the dependence on the market the distribution. The adopted EVT model is state, a second more subtle challenge is that of the Generalised Pareto Distribution that any downside risk measure (including (GPD). Not only does this approach allow VaR and CVaR) is sensitive to the tail of reliable estimation of VaR and CVaR, but it the portfolio loss distribution. CVaR, being also provides insight into the tail thickness the average of the extreme losses, is more through the fitted value of one of the GPD sensitive to the way the relative frequency parameters known as the shape parameter. of extreme losses is reflected in the risk To measure tail risk, we choose VaR and 6 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 Executive Summary CVaR at 1% tail probability which is a shape parameter which corresponds to a standard choice, but we also test other heavy tail with a power-type decay.2 The levels such as 2.5% and 5%. back-testing of the special cases of the base model with the shape parameter set to three Apart from tail risk, which is the focus of distinct levels confirms out-of-sample that the research, we also test how the model volatility clustering is the main factor for performs at capturing the right tail of the the thick tail of the unconditional return return distribution which describes the distribution. upside potential. An appealing feature of EVT is that it can be applied independently As a consequence, any of the two tails of the for the left and the right tail. To measure return distribution can be described through the upside potential, we use quantities only one parameter which is interpreted as such as VaR and CVaR but translated for the volatility of the extreme losses or profits, profits instead of losses: the intuition for respectively. This parameter has a rather that is that the tail risk of a short position constant behaviour through time which is described by the upside potential of a indicates that the clustering of volatility is corresponding long position. the most significant factor for the temporal 1 - In a more technical variation of tail risk. Thus, techniques for language, the residual tail is exponential and has moments Before running any comparisons, we first dynamic hedging of volatility, such as those of any order. 2 - Some higher-order check if the GARCH-EVT model is statistically behind target volatility funds, indirectly moments are unbounded. acceptable for application to the extreme control the dynamics of tail risk as well. quantiles of the returns data which would be in line with the academic literature. The developed and the emerging markets We run a VaR back-testing for 41 markets are compared cross-sectionally in terms of (22 developed and 19 emerging markets)
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