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) tail risk, upside potential, and forecasted covering periods of different length ranging volatility averaged in the period from from 13 to 62 years depending on data January 2003 to June 2013, also in the availability. The statistical tests indicate bull market sub-period from January 2003 the VaR at 1% tail probability is reliably to June 2007, in the turbulent sub-period modelled through the GARCH-EVT model from July 2007 to June 2013 covering for all markets and both the left and the the financial crisis of 2008, as well as in right tail. the post-crisis period. The comparison reveals that over the entire period there In addition to the general GARCH-EVT appears to be no significant relationship model, we back-test several special cases. between the average volatility and average For all markets, in-sample analysis indicates residual tail risk suggesting that it may that the important shape parameter of be possible for the two quantities to be the GPD appears statistically insignificant managed separately. Overall, developed almost at all times. The practical implication markets have lower tail risk and volatility of this finding is that the residual tail is than the emerging markets, but also lower not too heavy.1 In the academic literature, upside potential. Both kinds of markets empirical studies ignoring the clustering of exhibit tail asymmetry in the dispersion of volatility report a statistically significant the extremes; the downside being more

An EDHEC-Risk Institute Publication 7 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Executive Summary

dispersed than the upside. In the pre-crisis period, residual downside risk of both types of markets was similar, the emerging markets, however, enjoyed a better upside potential but also much higher volatility. In the crisis and post-crisis period, both types of markets had similar average volatility, the emerging markets, however, had a better upside potential which came at the cost of higher residual tail risk.

Finally, as a by-product of the back-testing of the three special cases of the base model, our results illustrate a remarkable weakness of the standard VaR-violation tests for model adequacy in that they are unable to detect a significant thickening of the tail for the residual which is otherwise detected by the CVaR-based test. The standard tests have become a common tool for model validation and the lack of power could pose systemic risks if tail risk accumulates undetected either unwillingly or through gaming of these tests.

8 An EDHEC-Risk Institute Publication 1. Introduction

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1. Introduction

Since its introduction in the 1990s, value- performance of various VaR methods at-risk (VaR) has become a standard have found the EVT-based method to be measure of risk in the practice of finance. particularly accurate (Danielsson and de It provides a threshold of the portfolio Vries, 1997; Pownall and Koedij, 1999; loss distribution such that losses higher McNeil and Frey, 2000; Bekiros and than the threshold occur with a given Georgoutsos, 2005; Fernandez, 2005; probability, typical choices include 1% or Tolikas et al., 2007). EVT has also been 5%. Another measure of risk, computing used specifically to study the distribution the average losses beyond VaR, which has of extreme stock returns (Jondeau and gained popularity is conditional value-at- Rockinger, 2003; Gettinby et al., 2004; risk (CVaR). It is more informative than Longin, 2005; Tolikas and Gettinby, 2009). VaR and has better properties; see BIS (2011) by the Basel committee on banking There are two methods for defining supervision for an extended analysis of the extreme losses that arise from EVT with application of VaR for risk measurement a corresponding limit model for their in the context of regulation. behaviour: the Block Maxima (BM) and the Peak-over-Threshold (POT) approaches. An important component of a VaR- or a With the BM approach, extreme losses CVaR-based risk model is the probabilistic are obtained by taking the maxima of model underlying the portfolio P&L losses over certain blocks of observations. distribution. Both risk measures belong On the other hand, POT considers events to the category of downside, or tail, risk as extreme when they exceed a chosen measures indicating that the modelling high threshold. While both approaches of the tail has important consequences have advantages and disadvantages, the for the performance of the risk model. In POT method seems to be preferred; see fact, from a technical perspective, both for example Embrechts et al. (1997) and risk measures need a reliable probabilistic also McNeil and Frey (2000) and Chavez- model for the high quantiles of the loss Demoulin et al. (2011) for a discussion on distribution. One possible approach to the threshold choice which turns out to such a model is the Extreme value theory be a critical parameter. (EVT); see Stoyanov et al. (2011) for an overview of other possible approaches. In There are two ways in which EVT has a regulatory context, the Basel committee been applied to VaR modelling: either on banking supervision working paper, directly on the return series or by first BIS (2011), has suggested employing fat running a GARCH model to explain away tail distributions for the risk factor when the clustering of volatility effect. As far stress testing for market risk. as theory is concerned, both approaches are valid. The direct method requires Since the application of EVT in finance by introducing a special parameter called Parkinson (1980) and Longin (1996), it has the extremal index which is related to the played an increasing role in the temporal structure of the time series and estimation of the frequency of extreme describes the clustering of the extremes, events in finance. Studies on predictive see for example Longin (2000) for an

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1. Introduction

empirical study. The conditional approach Our paper differs from existing studies in through GARCH relies on cleaning the a number of ways. First, we use a much clustering of the extremes first through a larger global data set and examine left and GARCH model and EVT is applied to the the right tail of the market index returns residuals time series, see McNeil and Frey in 22 developed and 19 emerging markets. (2000). Unlike previous studies, we compare the left and the right tails of different stock A big advantage of VaR over CVaR is the markets by carrying out out-of-sample fact that VaR can be reliably back-tested. analyses using both VaR- and CVaR- McNeil and Frey (2000) run a VaR back- based tests over the full samples and in testing comparison for five time series the period from Jan-2003 to Jun-2013 for (3 stock indices, the USD/GBP exchange which data is available for all markets. We rate and gold) and conclude that a consider three tail probability levels in the GARCH-EVT model yields better estimates calculation of VaR and CVaR: 1%, 2.5%, of VaR and CVaR than unconditional and 5%. EVT or the classical GARCH model with Student's t and normally distributed Our conclusions can be classified in error terms. Fernandez (2005) runs a three groups. Firstly, the out-of-sample similar comparison for 13 stock indices empirical results indicate that the and draws the same conclusion. Byström GARCH-EVT model restricted with the (2004) extend the methods by McNeil and shape parameter equal to zero is very Frey (2000) with both the BM and POT successful in both the left and the right approaches to compare the performance tail at 1% and 2.5% tail probability levels of conditional EVT and find the two to in the 2003-2013 period for almost all perform similarly. Recent work of Furió markets. This restricted model essentially and Climent (2013) adopted the McNeil uses an exponential tail and implies that and Frey (2000) approach and their results the statistically significant power-tail indicate that GARCH-EVT estimates are behaviour reported by various authors more accurate than the conventional using EVT without a GARCH-type structure GARCH models, assuming innovations is primarily caused by the clustering of have normal or Student's t distribution, volatility effect. Furthermore, because the for both in-sample and out-of-sample estimated values of the only remaining estimation. Further on, the superiority of parameter describing residual tail risk GARCH-EVT is robust to changes in the appear relatively constant through GARCH model structure. The main goal of time, volatility turns out to be the most this paper is not to provide a comparison important factor driving the temporal of relative performance of VaR models. variation of tail risk. As a consequence, Rather, we aim at drawing inference about techniques for dynamic hedging of the lower and the upper tail behaviour of volatility have an indirect control on the the return distribution of different dynamics of tail risk. markets through the fitted parameters of a GARCH-EVT model. Secondly, the developed and the emerging markets are compared cross-sectionally

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1. Introduction

in terms of tail risk, upside potential, risks if tail risk accumulates undetected and forecasted volatility averaged in either unwillingly or through gaming of the period from January 2003 to June these tests. 2013 and also in the bull market sub- period from January 2003 to June 2007 The paper is organised in the following and the turbulent sub-period from July way. Section 2 discusses EVT and the 2007 to June 2013 covering the financial POT method. Sections 3 focuses on the crisis of 2008 and the post-crisis period. GARCH-EVT model and Section 4 explains The comparison reveals that over the how VaR and CVaR can be calculated and entire period there appears to be no forecasted through the model. Section 5 significant relationship between the briefly describes the statistical tests and average volatility and average residual Section 6 discusses the data and the tail risk suggesting that it may be possible empirical results. Section 7 concludes. for the two quantities to be managed separately. Overall, developed markets have lower tail risk and volatility than the emerging markets, but also lower upside potential. Both kinds of markets exhibit tail asymmetry in the dispersion of the extremes; the downside being more dispersed than the upside. In the pre-crisis period, residual downside risk of both types of markets was similar, the emerging markets, however, enjoyed a better upside potential but also much higher volatility. In the crisis and post- crisis period, both types of markets had similar average volatility, however, the emerging markets had a better upside upside potential which came at the cost of higher residual tail risk.

12 An EDHEC-Risk Institute Publication 2. Extreme Value Theory

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2. Extreme Value Theory

EVT finds application in problems related Denote by to rare events; it originated in areas other (2.2) than finance. In finance, such problems

can be the estimation of probabilities the normalised maxima where bn > 0 of extreme losses or estimation of a loss and an are a sequence of normalising threshold such that losses beyond it occur constants. The Fisher-Tippett theorem

with a predefined small probability, a states that if Zn converges to some non- quantity also known as a high quantile of degenerate distribution as n increases

the portfolio loss distribution. In fact, EVT indefinitely, P(Zn > x) —> Hξ(x), then this provides a model for the extreme tail of must be the GEV law defined by the distribution which turns out to have a relatively simple structure described through the corresponding limit distributions. (2.3)

Denote by X1, X2,…, Xn a sample of where 1 + ξx > 0 and ξ is a shape i.i.d. portfolio losses and the unknown parameter controlling the tail behaviour

cumulative distribution function (c.d.f.) of Hξ(z). Depending on the sign of the of portfolio losses by F(x) = P(Xi ≤ x). We shape parameter, the GEV is known under are interested in extreme losses which different names: the Frechet distribution are described by the right tail of the loss (ξ > 0), the Gumbel distribution (ξ = 0), or

distribution F. Denote by Mn = max(X1, the Weibull distribution (ξ < 0). X2,…, Xn) the maximal loss observed in a block of n observations. Since the portfolio It is possible to completely characterise losses are assumed to be i.i.d., the c.d.f. the set of portfolio loss distributions such of the maximal loss can be expressed that, at the limit, the worst-case losses

through F, behave according to a given Hξ in (2.3). The set of these distributions is called the maximum domain of attraction (MDA) of (2.1) the given limit distribution. The accepted

notation is X ∈ MDA(Hξ) which implies This formula provides a direct connection that the normalised maxima of X converge

between the c.d.f. of the worst-case loss in distribution to Hξ. and the c.d.f. of portfolio losses but it hinges on knowing F explicitly. There are three distinct MDAs corresponding to different values of the An asymptotic approximation to the c.d.f. shape parameter ξ. Since EVT is concerned of the worst-case loss is provided by the with rare events, the characterisations Fisher-Tippett theorem, see for example are in terms of the tail behaviour of (Embrechts et al., 1997, Chapter 3) which the portfolio loss distribution; no other derives the Generalised Extreme Value features of F are important. (GEV) distribution.

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2. Extreme Value Theory

The Fréchet MDA, ξ > 0 X ∈ MDA(Hξ) further detail, see (Embrechts et al., 1997, with ξ > 0 if and only if X has a tail decay Section 3.3.3). dominated by a power function in the following sense, The Weibull MDA, ξ < 0 The MDA of Hξ with ξ < 0 consists entirely of distributions with bounded support and is, therefore, not interesting for modelling the The link between α and ξ is ξ = 1/α. It behaviour of risk drivers. Distributions that is possible to demonstrate that this MDA belong to this MDA include for example consists of fat-tailed distributions F the uniform and the beta distribution. For that have unbounded moments of order further detail, see (Embrechts et al., 1997, higher than α, i.e. EXk < 1 if k < α. For Section 3.3.2). applications in finance, it is safe to assume that volatility is finite which implies Finally, we should note that one α > 2 and ξ < 1/2, respectively. For further distribution can be in only one MDA. detail, see (Embrechts et al., 1997, Section There are examples of distributions that 3.3.1). are not in any of the three MDAs but they are, however, rather artificial. The Gumbel MDA, ξ = 0 The MDA of

Hξ with ξ = 0 is much more diverse. A 2.1. The Peak-over-Threshold portfolio loss distribution belongs to the Method MDA of the Gumbel law if and only if For the purposes of statistical work, the approach behind GEV gives rise to the block maxima (BM) method. To fit the GEV distribution, we need observations on the

maximal losses Mn calculated from sub- in which β(u) is a scaling function and samples (the blocks). However, in both the can be chosen to be equal to the average theoretical and the empirical literature, excess loss provided that the loss exceeds there is a preference for the peaks-over- the threshold x, threshold (POT) method which we describe (2.4) in this section, see for example (Embrechts et al., 1997, Section 6.5). This choice of β(u) is also known as the mean excess function. This MDA is The POT method is related to another characterised in terms of excess losses that important limit result which leads to the exhibit an asymptotic exponential decay Generalised Pareto Distribution (GPD). and consists of distributions with a diverse Suppose that we have selected a high loss tail behaviour: from moderately heavy- threshold u and we are interested in the tailed such as the log-normal to light- conditional probability distribution of the tailed distributions such as the Gaussian excess losses beyond u. We denote this or even distributions with bounded distribution by Fu(x) which is expressed support having an exponential behaviour through the unconditional distribution in near the upper end of the support xF. For the following way,

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2. Extreme Value Theory

after substituting the limit law ; for . For a fixed threshold u, note that F(u) (2.5) is a constant and the tail for y > u is determined entirely by the GPD tail . where x > 0. Because we are interested Finally, the MDAs of Hξ and Gξ,β are the in the extreme losses, we need to same. gain insight into the probability that the excesses beyond u, X — u, To apply (2.9) in practice, we need to choose can exceed a certain loss level. Thus, a high threshold u and also to estimate (2.5) is re-stated in terms of the tail the probability . In addition, we also need estimates of ξ and β(u). Regarding the choice of u, different strategies have (2.6) been adopted in the academic literature. One general recommendation is to set it so that a given percentage of the The limit result states that as u increases sample are excesses. Chavez-Demoulin towards the right endpoint of the support and Embrechts (2004) report that a 3 - An approach based on adaptive calibration of the of the loss distribution denoted by xF, the 10% threshold provides a good trade- threshold is adopted by some conditional tail converges to the off between the bias and variability of authors. Gonzalo and Olmo (2004) describe a tail of the GPD which is defined by, the estimator of the important shape method based on minimising the distance between the parameter 16 when the sample size is empirical and the tail of about 1,000 observations. A similar of the GPD with parameters estimated through the guideline is provided by McNeil and Frey maximum likelihood method. 3 The suggested distance is the (2.7) (2000). If the threshold is allowed to vary, Kolmogorov-Smirnov statistic. then the probability can be estimated where 1 + ξx > 0 and β > 0 is a scale through the empirical c.d.f. as suggested parameter. The limit results is (Embrechts for example in McNeil and Frey (2000).

et al., 1997, Chapter 3) For instance, suppose that X1, X1, …, Xn is a sample of i.i.d. portfolio losses. If u is chosen such that exactly m observations (2.8) are excesses, then the approximation in (2.9) becomes where β (u) is a scaling depending on the selected threshold u. (2.10)

The limit result in (2.8) can be used to where s = 1 — m/n and Xs,n is the s-th construct an approximation for the tail of observation in the sample sorted in the losses exceeding a high threshold u. If increasing order and and are we denote by y = u + x and express x in estimates of ξ and β, respectively. terms of y in (2.6), we obtain (2.9) Regarding estimation, a variety of estimators can be employed to estimate ξ and β. We use the maximum likelihood

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estimator (MLE) which is rationalised by the uniform convergence in (2.8). Under the assumption that data are distributed exactly according to the GPD, then given a

sample of i.i.d. observations Y = (Y1,…,Yn1) the log-likelihood function equals

and can be maximised numerically. The ML estimator , where D = (—1/2, ∞) x (0, ∞), satisfies the following asymptotic property

4 - A technical comment is due regarding the approximate MLE method. The limit relationship (2.11) in (2.8) involves the conditional probability P(X — u > xX > u). where In a finite sample, the event in the condition X > u implies working with a certain number of higher order statistics. Denote by the order statistics that satisfy the condition. The threshold, and and (0, Σ) denotes a bivariate normal therefore the number kn, depends distribution. For additional details, see on the sample size n. The limit in (2.11) is with respect (Embrechts et al., 1997, Section 6.5). Since to the sample size n1 = n — kn. The approximate MLE makes sense data do not exactly follow the GPD law only if kn —> 1 when n —> ∞ but but are in its MDA, we use the GPD log- so that kn/n —> 0. In fact, the growth rate of kn relative to n is likelihood and the result in (2.11) only as very important. If F ∈ MDA(Hξ), an approximation.4 then depending on the second- In practice, the GPD is order terms of the tail expansion estimated from the sample Yi = Xs+i,n — Xs,n, of F, it is possible to show formally that the approximate where i = 1,…, n1 = n—s and s is defined MLE leads to a result similar to (2.11) but with a possible as s = 1—m/n. Information about other asymptotic bias, see estimators, such as the Hill and the (de Haan and Ferreira, 2006,

Theorem 3.4.2). Since kn/n —> 0, Pickands estimator, and further detail on the choice of the 10% quantile the relevance of the MLE are available in as a high threshold can be regarded as a rule of thumb de Haan and Ferreira (2006). only for samples of size close to 1,000 observations and a higher quantile should be used for larger samples. See also the comments in McNeil and Frey (2000) for a motivation of the 10% quantile and the discussion in (Embrechts et al., 1997, pp 341).

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18 An EDHEC-Risk Institute Publication 3. A Conditional EVT Model

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3. A Conditional EVT Model

To apply the BM or the POT method in for a time-dependent scale and perhaps practice, EVT requires the data to be i.i.d. shape parameter of the limit distribution. which is a restrictive assumption. An Applications in the context of the so-called appealing feature of EVT is that the i.i.d. self-exciting processes are suggested by hypothesis can be weakened without a McNeil et al. (2005). change in the resulting limit theory. If the time series is assumed stationary and if Instead of applying the POT method to the it meets a couple of additional technical time series directly, we prefer to build a conditions, then the same distributions model for the time-varying characteristics arise at the limit. The additional technical and apply EVT to the residuals of the conditions include a specific form of model having explained away, at least asymptotic independence of the maxima partly, the temporal structure of the time over any two significantly big, non- series. For example, in line with McNeil and overlapping time periods and also lack Frey (2000) we estimate a GARCH model of asymptotic clustering of extremes, see to explain away the time structure of (Embrechts et al., 1997, Section 4.4) for volatility. To make things simple, we fit a additional details. Processes that meet GARCH(1,1) model to the portfolio return these conditions include, for example, time series as a general GARCH filter. the family of the ARMA processes with Gaussian noise. Denote the time series of portfolio losses

by Xt. The GARCH(1,1) model is given by: A stylised fact about asset returns is that volatility tends to cluster: large returns in absolute value are followed by returns of (3.1) similar magnitude. Although the excess

losses of such time series exhibit clustering, where ∈t = σtZt, the innovations Zt are the limit theory can still be extended to i.i.d. random variables with zero mean, cover this case. For stationary processes unit variance and marginal distribution

of this type, assuming the same technical function FZ(x) and K, a, and b are the condition of asymptotic independence positive parameters with a + b < 1. The of maxima, the same limit distributions model in (3.1) is fitted to the data and arise as possible models for the maxima, then the standardised residual is derived. however, with an additional parameter If we assume that the data is generated by called the extremal index. The extremal the model in (3.1), then the standardised index is interpreted as the reciprocal of the residual is a sample from the distribution

average cluster size. This category include FZ. EVT is applied by fitting the GPD to the the ARCH and GARCH processes which are residual using approximate MLE. used as a model for volatility in financial econometrics. Regarding the type of the GARCH model, Furió and Climent (2013) find no evidence Such extensions of EVT indicate that the of any difference in the conditional method can be applied directly to more EVT estimated whether GARCH(1,1) or general stochastic processes allowing an asymmetric GARCH specification is

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3. A Conditional EVT Model

applied for VaR estimation.5 Jalal and in (3.1) we use the pseudo-maximum

Rockinger (2008) perform a Monte likelihood method assuming FZ follows the Carlo study to analyse the impact of an Gaussian distribution.6 incorrectly specified GARCH model for VaR modelling. The Monte Carlo study Although very difficult to formally compare includes GARCH(1,1) with Gaussian and across empirical studies because different Student's t residuals, EGARCH(1,1) with authors focus on different estimation Gaussian innovations, a switching regime methods, different market indices, and also volatility model, stochastic volatility different time periods, studies applying with jumps, and a pure jump process for EVT unconditionally in the estimation of the data generating process. Jalal and equity market tail risk generally report Rockinger (2008) conclude that the two- higher estimated ξ values than studies step procedure of GARCH(1,1) and EVT is with a conditional EVT model. Jondeau and remarkably robust compared to a two-step Rockinger (2003) apply an unconditional parametric approach with the Gaussian or EVT model through the block-of-maxima Student's t distribution. method and cover 20 equity market indices. For the S&P 500, FTSE 100, and 5 - The particular specifications Apart from the general robustness reported Nikkei 225, for example, they report are EGARCH(1,1) and TGARCH(1,1). in empirical studies, we prefer the GARCH = 0.282, 0.273, 0.265 and = 0.132, 6 - The pseudo-maximum likelihood method provides specification rather than unconditional 0.33, 0.268 for the lower and the upper consistent and asymptotically application for another reason. It is a well- tails, respectively. They find that both normal estimator, see (Gourieroux, 1997, Chapter 4). known fact that a light-tailed distribution the left and the right tails of the return 7 - Jondeau and Rockinger (2003) consider samples of FZ, such as the Gaussian law, generates a distributions belong to the domain of about 9,000 observations for fat-tailed X through the GARCH process. attraction of the Fréchet law because the three markets while Furio t and Climent (2013) use samples The ARCH(1) example in (Embrechts et al., = 0 is rejected for all markets and for both of about 6,000 observations. 1997, Section 8.4) illustrates that a tails. On the other hand, Furió and Climent distribution of the error term which is (2013) use a conditional EVT model under in the MDA of the Gumbel distribution three different GARCH specifications. For

generates Xt in the MDA of the Fréchet the same three markets, they report = distribution. The ARCH(1) process is a 0.211, 0.083, 0.536 and = —0.1, 0.082, special case of the GARCH(1,1) process 0.006, respectively under the GARCH(1,1) with β = 0 which can also be written as a and = 0.272, —0.037, 0.213 and GARCH(1,0) process. The example suggests = —0.17, —0.047, 0.024 under the that the clustering of volatility may be a EGARCH(1,1) specifications. Based on the significant contributor to the observed reported standard errors, which are higher extreme events in the unconditional than those in Jondeau and Rockinger distribution of portfolio losses. As a (2003) because of shorter samples, it is not result, the approach adopted here allows possible to reject = 0 for the upper tails the time structure of volatility and the of the three markets and also for the lower residual tail thickness due to factors other tail of Nikkei 225 index.7 than volatility to be considered separately. In line with the empirical literature, for Finally, we should note that reliable the estimation of the GARCH(1,1) model estimation of tail thickness is incredibly

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3. A Conditional EVT Model

difficult. Heyde and Kou (2004) consider 6 different methods and demonstrate that a sample of 5,000 observations may be insufficient to discriminate between exponential and power-type tails. It is nevertheless important to find out through an out-of-sample risk back- testing if we can reject the exponential tail in a conditional GARCH-based model. If the Gumbel MDA turns out to be statistically acceptable, then the statistical significance of the power tails in the unconditional models can be attributed primarily to the clustering of volatility effect.

22 An EDHEC-Risk Institute Publication 4. Risk Estimation with EVT

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4. Risk Estimation with EVT

Apart from the probabilistic model, the about VaR at 95% and 99% confidence other key component of a risk model is level which corresponds to VaR at 5% and the measure of risk. We use two measures 1% tail probability. of risk: VaR and CVaR at tail probabilities of 1%, 2.5%, and 5%. In this section, Formally, if we suppose that X describes we provide definitions and explicitly portfolio losses, then VaR at tail probability state the risk forecasts built through the p is defined as probabilistic model. (4.1)

The discussion below assumes that the where F—1 denotes the inverse of the c.d.f.

random variable X describes portfolio FX(x) = P(X ≤ x) which is also known as the losses and VaR and CVaR are defined for quantile function of X. the right tail of the loss distribution which translates into the left tail of the portfolio As explained earlier, we employ EVT return distribution. The same quantities to estimate high quantiles of the loss for the right tail of the return distribution distribution. To this end, we adopt the (left tail of the loss distribution) are approximation of the tail in (2.9). Solving obtained from the definitions below by for the value of y yielding a tail probability considering — X instead of X; that is, the of p, we get downside of a short position is the upside of the corresponding long position. The risk functionals are, however, multiplied by —1 to preserve the (4.2) interpretation that negative risk means a potential for profit. The estimator is derived from (2.10) in the

same way. Suppose that X1,n ≤ X1,n ≤… ≤ Xn,n denote the order statistics, then 4.1. Value-at-Risk following (2.10) we get The VaR of a random variable X describing portfolio losses at a tail probability p, VaRp(X), is implicitly defined as a loss threshold such that over a given time (4.3) horizon losses higher than it occur with a probability p. By construction, VaR is where s = 1 — m/n and m denotes the the negative of the the p-th quantile of number of observations that are the portfolio return distribution or the considered excesses. The approximation in (1 — p)-th quantile of the portfolio loss (4.3) is usually interpreted in the following distribution. In the industry, VaR is often way: the estimate of VaR equals the

defined in terms of a confidence level but empirical quantile Xs,n, which is such that we prefer to reserve the term confidence p < m/n, plus a correction term obtained level for the context of statistical testing through the GPD. In the implementation, which we need in Section 6. Thus, to map we set m/n = 0.1 and, thus, in terms of the terms properly, in the industry we talk quantiles the 99% quantile

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4. Risk Estimation with EVT

equals the 90% quantile (X(0.9xn)) plus the 8 corresponding correction term. (4.6)

As mentioned before, we assume that the In the academic literature, CVaR is portfolio loss distribution is dynamic and also known as average value-at-risk or follows the GARCH(1,1) process. Under expected shortfall. this assumption, the conditional VaR model is given by Average value-at-risk corresponds directly to the quantity in (4.5) while expected shortfall is the quantity in (4.6). Although (4.4) (4.5) is more general and average value- at-risk seems to be a better name for the where It denotes the information available quantity, we stick to the widely accepted at time t, is given in (4.3) and CVaR; see for example Pflug and Römisch is calculated from the sample of the (2007) for further discussion. standardised residuals. Since CVaR integrates the entire tail, an 8 - The correction term is obtained from the GPD asymptotic model for the tail in areas and could make sense 4.2. Conditional Value-at-Risk for very small values of p where no data points are available is even as well; values that may An important criticism of VaR in more important than for VaR. Assuming extend beyond the available the academic literature is that it is observations in the sample. that ξ < 1, the expectation in (4.6) can be For example, suppose that uninformative about the extreme losses the sample contains 100 calculated explicitly through the GPD, portfolio losses, n = 100, beyond it. Indeed, the only information and set p = 0.001 which is provided is the probability of losing the VaR corresponding to the 99.9% quantile. Then, more that VaR which is equal to the X0.9 x n,n is the 90-th observation in the sorted tail probability level p but should any sample and the empirical such loss occur, there is no information approximation to where would be the about its possible magnitude. Conditional largest observation in the sample. As a consequence, value-at-risk is constructed to overcome the correction term in (4.3) this deficiency: CVaR at tail probability allows us to go beyond the available data points in p,CVaRp(X), equals the average loss provided the sample which emphasises a key advantage of EVT to the that the loss exceeds V aRp(X). Plugging in from (4.3) and the historical method. corresponding estimates, we get CVaR is formally defined as an average of VaRs,

(4.7) (4.5) For derivations and further detail, see and if we assume that the portfolio loss (McNeil et al., 2005, Section 7.2.3). distribution has a continuous c.d.f. then CVaR can be expressed as a conditional expectation,

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Under the assumption of a GARCH(1,1) process for the portfolio loss distribution, the counterpart of (4.4) for CVaR equals

(4.8)

where is given in (4.7) and is estimated from the sample of the standardised residuals.

26 An EDHEC-Risk Institute Publication 5. Back-testing and Statistical Tests

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5. Back-testing and Statistical Tests

We estimate the risk model in a rolling where N denotes the number of VaR time-window of 1,000 days and build violations in the sample. The asymptotic forecasts for the VaR and CVaR at the distribution is which can be three tail probabilities for both tails used for a p-value or a confidence bound, through the estimated model on a daily see Kupiec (1995). basis. Each day we verify if the realised return violates any of the forecasted If the dynamics of the corresponding quantile levels in the upper and the lower quantile of the return distribution is tail and count an exceedance if any such properly captured by the risk model, violation occurs. At the end of the back- then the VaR exceedances should be testing, we have a sequence of indicators independent events. In fact, the sequence for each tail and each tail probability of the indicators marking the VaR level. Using the sequence of indicators exceedances should be indistinguishable and the realised returns conditioned on a from a sequence of tossing an unfair coin violation, we run three statistical tests for T times with a probability of success equal VaR and one for CVaR. We also calculate to p. If in practice the VaR violations are confidence bounds for the estimated ξ clustered, this would be an indication using (2.11). that there is a temporal structure of the empirical quantile which is not captured properly by the risk model. For example, 5.1. VaR-based Tests the GARCH model is used to describe the There are two standard tests that we run temporal behaviour of volatility but we to check the relevance of the VaR risk may be using an incorrect order, or it may model: Kupiec's test and Christoffersen's be structurally incorrect, or there might test. In addition, we describe another test be dynamics in the higher-order moments based on an asymptotic result. not reflected in the model.

5.1.1. Kupiec's Test 5.1.2. Independence of Exceedances This test is directly related to the definition Christoffersen's test concerns the of VaR: at any given tail probability p and independence of VaR violations. The test time window T, on average there would be statistic is a likelihood ratio test similar to p x T observations for which the realised (5.1), return exceeds the forecasted VaR. Such observations are also known as VaR violations or VaR exceedances. Knowing the average number of violations is, (5.2) however, insufficient. Kupiec's test provides a test statistic, where the indices 0,1 denote exceedance

and no exceedance respectively, pi denotes the probability of observing an exceedance conditional on state i in the

(5.1) previous time period, and Tij denotes the number of days in which state j occurred

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5. Back-testing and Statistical Tests

in one day while it was at i the previous 5.2. A CVaR-Based Test day. The asymptotic distribution of the A statistical test on CVaR can be based on test statistic is the same, (1), the differences between the realised losses see Christoffersen (1998). and the forecasted CVaR conditioned on the events of VaR exceedances, see It is possible to combine Kupiec's and (McNeil et al., 2005, Section 4.4.3). Christoffersen's tests together into one Consider the differences

test. If we denote LR = LRK + LRC, then (2). We calculate the p-values of the two tests and also that of the combined one. (5.3)

5.1.3. A General Test for Lack of in which 1{A} denotes the indicator Memory of the event A. Because CVaR is the If the risk model is realistic, the indicators expected shortfall of the continuous loss

marking VaR exceedances are i.i.d. distribution, EDt+1 = 0 and, therefore, Dt+1 following a Bernoulli distribution with a forms a martingale difference series. Under 9 - In fact, a more general limit result holds: the “successs" probability p. The probability the assumption of a GARCH model, the stochastic process counting the number of VaR of a run of k periods of no-exceedances normalised differences Dt+1 = σt+1 should exceedances in a given time before an exceedance occurs equals behave like a zero-mean i.i.d. sequence period converges weakly a Poisson process for small (1—p)kp and follows a geometric with a probability mass of (1 — p) at zero. tail probabilities; see, for example, Embrechts et al. distribution. The geometric distribution We use the standard t-test to check if the (1997) for the exact limit law. is the only discrete distribution that conditional mean of has the lack-of-memory property the normalised differences, . If τ measures the time until a VaR-exceedance occurs, then this property implies that the probability that this time exceeds k + n periods provided that n periods have already elapsed does not depend on the is statistically different from zero. time elapsed. For small tail probabilities, the distribution of the time intervals between consecutive exceedances (inter- exceedance times) asymptotically converge to an exponential distribution.9 We apply the chi-square goodness-of-t test to test if the inter-exceedance times follow a geometric distribution for the three tail probabilities. This test can be regarded as a general test for model adequacy similar to the combined test above because we use the theoretically correct probability level without estimating it.

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30 An EDHEC-Risk Institute Publication 6. Data and Empirical Results

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6. Data and Empirical Results

In this section, we describe the data and for different countries. The sample period analyse the empirical results obtained for all indices ends on 28 June 2013. New using the methodologies and statistical Zealand has the shortest sample size at tests outlined in the previous sections. 3,260 observations because the index We first proceed with a comparison of was launched in March 2003 and data the GARCH-EVT VaR and GARCH-EVT are available from January 2001. Table 1 CVaR across the markets with the full presents the stock price indices used and sample period. However, it turns out that the starting date for the sample for each the confidence bounds for are wide country. Data are organised into developed and = 0 cannot be rejected in-sample and emerging markets: there are 22 for all markets for almost all periods of developed and 19 emerging markets. To estimation. To empirically check if the carry out statistical estimation and the out-of-sample performance is affected out-of-sample tests, we use log-returns. by imposing the constraint ξ = 0 in the approximate MLE, we run a back- test for all markets with this constraint 6.2. Comparison of Tail Risk across imposed at all times. A comparison to this Different Markets constrained case would also reveal if the We use the approach outlined in the time variations in are significant or are previous sections to compare the tail an artefact of the rolling time-window risk across different markets. Such a estimation. comparison is not simple for a number of reasons. First, tail risk depends on the Apart from this constrained case, we also particular risk measure; some risk measures run two full back-tests for all markets are more sensitive to extreme losses than imposing ξ = 0.1 and ξ = 0.2 to study the others, e.g. CVaR is more sensitive to the change in the out-of-sample performance tail behaviour than VaR. Second, tail risk is of the risk models. This comparison dynamic. From (4.4) and (4.8) it becomes makes sense because of the possible evident that the dynamics of volatility can bias leading to underestimation of the be a big driver of the dynamics of tail risk. true value of ξ due to the approximate Third, although assumed constant, the tail MLE. This comparison is performed for behaviour of the error term in the GARCH the subsample period of 2003-2013 process may also be time dependent. This which covers the subprime crisis and the effect, if present, would be partly captured European debt crisis. Finally, we examine through the estimated ξ. the differences in the estimated ξ and β in the context of developed and emerging We carry out the comparison in three markets. steps. The objective of the first step is to assess whether the extreme losses in the unconditional distribution are only a result 6.1. Data of the dynamics in volatility. We examine The daily stock price indices are obtained the quality of the VaR forecasts generated from Datastream. Due to availability of by the GARCH-EVT. The methodology data, the starting sample period varies takes into account the residual tail

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Table 1: The equity market indices used in the study together with starting date and number of daily observations. Stock Market Index Starting date No. of obs Developed Markets Austria ATX Jan-86 7,169 Australia S&P/ASX 200 Index Jun-92 5,500 Belgium BEL 20 Jan-90 6,129 Canada S&P/TSX Composite Index Jan-69 11,608 Denmark OMX Copenhagen Index Jan-96 4,565 Finland OMX Helsinki Index Jan-87 6,911 France CAC 40 Jul-87 6,777 Germany DAX 30 Jan-65 12,651 Hong Kong Hang Seng Index Jan-70 11,347 Ireland Irish SE Overall Index Jan-87 6,911 Italy FTSE MIB Index Jan-98 4,042 Japan NIKKEI 225 Index Jan-50 16,500 New Zealand NZX 50 Index Jan-01 3,260 Netherlands AEX Index Jan-83 7,955 Norway OSLO Exchange All Share Index Jan-83 7,955 Portugal PSI-20 Jan-93 5,346 Singapore Strait Times Index Sep-99 3,608 Spain IBEX 35 Jan-87 6,910 Sweden OMX Stockholm Index Jan-87 6,910 Switzerland SMI Jul-88 6,521 US S&P 500 Index Jan-64 12,913 UK FTSE 100 Jan-84 7,695 Emerging Markets Argentina Argentina Merval Index Nov-89 6,173 Brazil Brazil Bovespa Index Jan-93 5,346 China Shanghai SE Composite Index Jan-91 5,868 Chile Santiago SE General Index Jan-87 6,911 Czech Republic Prague SE Index Apr-94 5,016 Egypt Egypt Hermes Financial Jan-95 4,825 Hungary Budapest SE Index Jan-91 5,868 India CNX 500 Index Jan-91 5,868 Indonesia IDX Composite Index Apr-83 7,890 Malaysia FTSE Bursa Malaysia KLCI Jan-80 8,738 Mexico IPC Index Jan-88 6,650 Peru Lima SE General Index Jan-91 5,868 Philippines Philippines SE Index Jan-86 7,172 Poland Warsaw General Index Jan-93 5,346 Russia MICEX Index Oct-97 4,108 South Korea KOSPI Jan-75 10,043 Taiwan Taiwan SE Weighted Index Jan-71 11,084 Thailand SET May-75 9,957 Turkey BIST National 100 Jan-98 6,650

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6. Data and Empirical Results

thickness whether the latter assumes a an indicator of the thickness of the tail of light-tailed error distribution. The quality the portfolio loss distribution. of the forecasts is assessed through the formal statistical tests outlined in In contrast, the case ξ = 0 does not Section 5 based on a long back-testing necessarily imply that the true tail is exercise. exponential. In fact, if the fitted ξ turns out to be exactly zero throughout the If the statistical tests indicate that the entire period of the back-testing, then risk models perform well at the 1% tail this would be an indication that the error probability, then this would imply that term might have been in the MDA of the the tail of the normalised residual is Gumbel law in the entire period, which modelled properly at that tail probability does not necessarily imply time invariant level. As mentioned before, the GARCH tail behaviour. setting allows for the dynamics of volatility to be explained, as well as Although Jalal and Rockinger (2008) for an estimation of the tail thickness report a significant robustness of the due to factors other than volatility. In GARCH-EVT model in cases of regime shifts this setting, the behaviour of the fitted in volatility and stochastic volatility with values of ξ would provide insight into jumps, we should point out that in some the residual tail thickness. Conditional on periods the model may be misspecified. this outcome, the objective of the second Any departures may get reflected in the step is to compare the fitted values of the residual and may eventually affect shape parameter ξ across markets and and . Thus, changes in the fitted values also the time series of forecasted values should be interpreted with care as they of VaR and CVaR. In addition, we compare might be an indication of a misspecified the estimated β across the markets which volatility model or phenomena have been neglected in most studies. unaccounted for by the model. Studies on tail risk focuses on the fitted ξ and not much attention has been paid to 6.2.1. GARCH-EVT with ξ the estimated β. Unconstrained and ξ = 0 This section analyses the performance Finally, a subtle warning is due regarding of the risk model at 1% tail probability the use of the fitted values of ξ as an when the important ξ parameter is indicator of tail behaviour. EVT is an unconstrained in the approximate MLE and asymptotic theory and provides an when it is constrained to zero using the approximation to the true tail of the return full sample. In the following, the subscript distribution. Therefore, the implications of L denotes left tail and R denotes right the value of ξ for the tail thickness of the tail respectively. We split the discussion true distribution should be interpreted into two parts — the full sample periods in the context of the the corresponding and the 10-year period from Jan-2003 MDAs. For example, ξ > 0 implies indeed to Jun-2013. that the true tail exhibits a decay close to a power function and the value of ξ is

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Full sample period The left panel the out-of-sample properties of the risk

of Table 4 provides the results of the model with the restriction of L = R = 0. unconstrained case. In terms of the VaR- The VaR-based tests for the full samples based tests, the performance of the risk are provided in the right panel of Table 4. model is remarkable for both the left and The results imply no deterioration of the the right tail. The combined test for the restricted risk model. left and the right tails combined fails only for two countries (Argentina and the The right panel of Table 5 provides the Philippines). The out-of-sample t-test CVaR-based statistics for the restricted for CVaR provided in the left panel of model. Although the average forecasted Table 5 rejects the model only for three CVaR (Avg CVaRf) and the average loss countries for the left and the right tails conditioned on the occurrence of VaR combined. The average for all countries exceedances look similar, t-test for the is available in the left panel of Table 5. left tail indicates a deterioration in out- The averages are close to zero; the three of-sample performance for 8 markets (3

countries with the highest average L are developed and 5 emerging). The same test Hungary (0.1269), Indonesia (0.1269), and for the right tail show results similar to 10 - Statistically significant Peru (0.1229) all of them in the group the unrestricted risk model. negative values are difficult R to interpret since the of the emerging markets. The average R Weibull MDA contains only distributions with bounded of any market is generally lower than the The period from Jan-2003 to support. The traditional corresponding average . Jun-2013 assumption that the L log-return distribution has Since the performance of the restricted risk unbounded support implies that statistically significant Time series plots of the estimated ξ of model does not deteriorate substantially negative values are most the left and the right tails and their 95% over the full sample, we study the out- likely an indication of a very slow rate of convergence confidence bounds for 6 countries for the of-sample performance in the period to the Gumbel limit law combined with a negative period from Jan-2003 to Jun-2013 are from Jan-2003 to Jun-2013 which is the bias caused by the fixed provided in Figure 1. These plots reflect longest period for which data are available 10% threshold in the GPD estimation. the general properties of the time series for all markets with the exception of New of the estimated ξ for all countries: ξ = 0 Zealand and Singapore. The VaR-based cannot be rejected for all countries results are provided in Table 6 and the almost at all times. This holds for both the CVaR-based statistics are provided in bullish market before the financial crisis Table 7. Both tables are split into three of 2008, the crisis itself, and the period panels corresponding to different tail that followed.10 Regarding differences probabilities: left (1%), middle (2.5%), and between the left and the right tail right (5%).

behaviour, indeed L appears generally higher than R but because of the wide First, we compare the differences in the confidence bounds its is rarely possible to performance of the risk model at 1% tail

reject L = R. probability to the one for the full sample. In this 10-year period, we notice fewer Because there does not seem to be a rejections of the VaR-based tests. In fact, significant variation across time for both the combined test (KC-test) fails only for

L and R, a reasonable goal is to study one country (the Philippines).

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The t-test for the left-tail CVaR in Table results indicate that the rejection for the 7 shows a rejection only for Egypt and 10 countries in the full sample is due to the t-test for the right-tail CVaR rejects unacceptable performance in time periods the model for four countries: Ireland, further back in time. Japan, Singapore, and the US. These

Figure 1: The fitted shape parameter ξ of the Generalised Parreto Distribution for selected market indices. The countries are: Hungary, Ireland, Japan, Singapore, the UK, and the US.

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Since EVT provides an asymptotic model We repeat the back-testing of the for the tail, it is expected to work well for restricted model with ξL = ξR = 0.1 and VaR and CVaR at low tail probabilities. Our L = R = 0.2 and check the performance results are consistent with other empirical of the risk model through the out-of- papers indicating the GARCH-EVT model sample tests. Tables 8 and 9 contain the works well at the 1% tail probability level. VaR- and CVaR-based tests, respectively, It is, however, of practical importance to for the period from Jan-2003 to Jun-2013. check how the performance of the model deteriorates for VaR and CVaR at higher Tables 8 reveals that increasing the shape tail probabilities. The middle and the right parameter to 0.1 leads to no rejections panels of Tables 6 and 7 provide results for of the combined test for the left tail and 2.5% and 5%. only a couple of rejections of Kupiec's and Christoffersen's tests. As a consequence, The combined KC-test shows rejections no significant deterioration in the for 6 countries at the 2.5% level and 15 performance is registered compared to countries at the 5% level for the left and ξ = 0. the right tails combined. Kupiec's test fails very rarely, most failures are caused by Regarding the right tail, Kupiec's test fails Christoffersen's test. A possible explanation for 5 countries but the combined test fails is that at higher tail probabilities, dynamics in only two cases and the same conclusion in parameters other than volatility (e.g. follows. It is rather surprising that a similar higher-order moments) play a role. Since conclusion can be drawn for the left tail they are not captured by the model, they of the restricted model with ξL = 0.2. It may cause exceedences to cluster. In should be noted that an increase from 0 contrast, the left- and the right-tail CVaRs to 0.2 represents a substantial thickening get rejected for 6 countries at both the of the left tail. Kupiec's test fails, however, 2.5% and the 5% levels. across the board for the right tail implying

that ξR = 0.2 is getting too high from an 6.2.2. GARCH-EVT with ξ = 0.1 out-of-sample perspective. and ξ = 0.2 One possible explanation for the fact that The results in Table 9 reveal what may

ξL = ξR = 0 is statistically acceptable almost turn out to be a substantial flaw in the always on Figure 1 is that both L and R quantile-based tests for model adequacy. are underestimated because of the use of The case ξL = ξR = 0.1 already leads to a fixed quantile as a threshold in the GPD rejections for 7 countries for the left tail estimation. Because the exact distribution and for 17 countries respectively for the of the data in the sample of the residual right tail. Increasing the value of the shape is unknown, it is difficult to provide an parameter to 0.2 increases the number of estimate of the bias but in view of the rejections to 24 countries for the left tail statistically significant positive estimates and 30 countries for the right tail. provided in the academic literature the bias is most likely negative for most time As consequence, although the case ξ = 0 is periods of estimation. acceptable for almost all markets and both

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tails, so would be other values in the 6.2.3. Tail Risk and Reward of interval [0, 0.1) which is not surprising Developed and Emerging Markets because within GPD, a power-tail The analysis of the results for the period

with a sufficiently small value for the from 2003 to 2013 suggests that ξL = ξR shape parameter can approximate an = 0 is a statistically acceptable model exponential tail. for almost all countries. The two tails

Figure 2: The fitted scale parameter β of the GPD for selected market indices with ξ = 0. The countries are: Hungary, Ireland, Japan, Singapore, the UK, and the US.

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Figure 3: The average estimated L, R, and volatility of the developed and emerging markets for the period 2003-2013. The assumed model is the restricted GARCH-EVT with ξ = 0.

of the residuals are thus determined by The practical implication of this is that the

the fitted values L = R. Time series of temporal variations in tail risk are almost estimated values and confidence bounds completely captured by the dynamics of are provided in Figure 2 for the same 6 volatility. As a consequence, techniques countries from Figure 1. The estimated for dynamic hedging of volatility are values do not appear very noisy and do expected to be effective in managing the not seem to change behaviour from the dynamics of tail risk. pre-crisis to the post-crisis period. As a consequence, the average residual tail risk The average tail risk of markets in the of the markets for the 10-year period can 10-year period is described by two

be captured by the average values of L quantities: the average volatility and the and R. average L. Likewise, the average upside

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potential is described by the average βL and βR, we can test the hypothesis if volatility and average R. Table 2 provides the two types of markets have different the corresponding quantities for the parameter values. individual markets averaged over the full

period 2003-2013 and two half-periods Table 3 provides the average of the L, of approximately equal size: the pre- R, and the corresponding volatilities crisis bull market period of Jan-2003 to across the markets belonging to each Jun-2007 and the turbulent period of group. The aggregation is done for the Jul-2007 to Jun-2013. full period from 2003 to 2013 and two half-periods. In all cases, the right tail The confidence bounds in Figure 2 are has a lower scale parameter than the left too wide and cannot be used to draw any tail indicating presence of tail asymmetry.

conclusions about any asymmetric tail The L and R of the developed markets behaviour. Nevertheless, the averaged L do not change much while those of the and R reported in the right panel of Table emerging markets deteriorate in the 7 suggest that the extremes in the left tail post-crisis period, i.e. downside risk are more volatile than the extremes in the increases and upside potential decreases. right tail. One approach to increase the On the other hand, the volatility of the statistical significance is to aggregate the generic emerging market stays relatively results for the developed and emerging unchanged while that of the developed market groups. market increases dramatically.

Figure 3 provides scatter plots of the In the pre-crisis period, the upside

average volatilities and the values of L potential of the emerging market is much and R of the 41 markets averaged across higher than that of the developed but this time. The developed markets are denoted comes at the cost of higher volatility risk; by circles and the emerging ones by the downside risk of both is statistically squares. The scatter plots illustrate that the same. In the post-crisis period, the there is little to no correlation between volatilities of the two are statistically the

the average volatility and the average L ( same and the higher upside potential of R). As a consequence, markets with high the emerging market comes at the cost of average volatility may have relatively low higher downside risk.

average L ( R) and vice versa. It is at this stage an open question to what degree Assuming currency risk has been the residual tail risk can be efficiently completely hedged, the stylised managed independently of volatility risk. description of the generic developed and emerging market suggests that volatility The developed and emerging markets management is more important for the seem to have different characteristics management of tail risk of a portfolio of in terms of the aggregate volatility and developed equity markets, while the same aggregate residual risk. If we assume that problem for a portfolio of emerging equity both types of markets are characterised by markets appears more complex. From a a generic average volatility and average portfolio construction perspective it is

40 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

6. Data and Empirical Results

Table 2: The characteristics of the developed and the emerging markets obtained by averaging over the corresponding periods. New Zealand and Singapore are excluded because of insufficient observations.

Bull Market: Turbulent Period: Full Period: 2003-2007 2007-2013 2003-2013

Avg Avg Avg Avg Avg Avg Avg Avg Avg Developed Austria 0.1455 0.6709 0.5271 0.2832 0.6242 0.4538 0.2242 0.6442 0.4852 Australia 0.1051 0.6300 0.4716 0.1950 0.6219 0.4400 0.1565 0.6254 0.4535 Belgium 0.1379 0.5782 0.4634 0.2215 0.6142 0.4794 0.1857 0.5988 0.4726 Canada 0.1151 0.5944 0.4440 0.1874 0.6082 0.4194 0.1564 0.6023 0.4299 Denmark 0.1328 0.6561 0.4927 0.1955 0.6014 0.4731 0.1686 0.6248 0.4815 Finland 0.2015 0.6583 0.5688 0.2486 0.6038 0.5195 0.2284 0.6271 0.5407 France 0.1640 0.5232 0.4694 0.2499 0.6016 0.4865 0.2131 0.5680 0.4792 Germany 0.1862 0.5288 0.4493 0.2323 0.6209 0.4704 0.2126 0.5815 0.4614 Hong Kong 0.1583 0.5973 0.5683 0.2676 0.5751 0.5009 0.2208 0.5846 0.5298 Ireland 0.1424 0.7368 0.4927 0.2712 0.6552 0.4601 0.2160 0.6902 0.4741 Italy 0.1391 0.5843 0.4725 0.2729 0.6229 0.4393 0.2156 0.6064 0.4535 Japan 0.1902 0.5646 0.4688 0.2506 0.5980 0.4221 0.2248 0.5837 0.4421 Netherlands 0.1713 0.5381 0.4592 0.2273 0.6099 0.4920 0.2033 0.5791 0.4780 Norway 0.1733 0.6905 0.4453 0.2480 0.5987 0.4501 0.2160 0.6381 0.4480 Portugal 0.1121 0.5910 0.5493 0.2123 0.6340 0.5575 0.1694 0.6156 0.5540 Spain 0.1473 0.5332 0.4685 0.2655 0.6512 0.4774 0.2149 0.6006 0.4736 Sweden 0.1629 0.6415 0.5137 0.2321 0.6205 0.4898 0.2024 0.6295 0.5000 Switzerland 0.1440 0.6505 0.4321 0.1862 0.6341 0.5128 0.1681 0.6411 0.4783 UK 0.1286 0.5883 0.4084 0.2078 0.6250 0.4724 0.1739 0.6093 0.4450 US 0.1278 0.4767 0.5306 0.2102 0.6929 0.4917 0.1749 0.6003 0.5084 Emerging Argentina 0.2877 0.6613 0.6450 0.2882 0.6978 0.6127 0.2880 0.6822 0.6266 Brazil 0.2666 0.6190 0.4499 0.2737 0.6199 0.5035 0.2707 0.6195 0.4806 China 0.2171 0.6178 0.7284 0.2760 0.7252 0.5195 0.2508 0.6792 0.6089 Chile 0.0881 0.5892 0.4755 0.1361 0.6343 0.4844 0.1155 0.6150 0.4806 Czech Republic 0.1691 0.6405 0.5358 0.2430 0.6686 0.5111 0.2114 0.6566 0.5217 Egypt 0.2519 0.6398 0.6529 0.2645 0.7654 0.4954 0.2591 0.7116 0.5628 Hungary 0.2061 0.5444 0.5894 0.2724 0.5439 0.5268 0.2440 0.5441 0.5536 Indonesia 0.2023 0.7039 0.5696 0.2279 0.7665 0.5026 0.2169 0.7397 0.5313 India 0.2116 0.6726 0.4240 0.2372 0.7116 0.4696 0.2263 0.6949 0.4501 Mexico 0.1729 0.6520 0.5573 0.2030 0.6713 0.5263 0.1901 0.6631 0.5396 Malaysia 0.1200 0.6271 0.6757 0.1235 0.7409 0.5872 0.1220 0.6921 0.6251 Philippines 0.1964 0.6487 0.6631 0.2118 0.6995 0.5350 0.2052 0.6778 0.5899 Poland 0.1768 0.5552 0.6009 0.2123 0.6367 0.4901 0.1971 0.6018 0.5376 Peru 0.1583 0.6068 0.6192 0.2567 0.6286 0.5807 0.2146 0.6193 0.5972 Russia 0.3006 0.6873 0.5347 0.3231 0.7283 0.5095 0.3135 0.7108 0.5203 South Korea 0.2269 0.6513 0.5012 0.2232 0.6891 0.4518 0.2248 0.6729 0.4730 Taiwan 0.1903 0.6317 0.5379 0.2135 0.7014 0.4615 0.2035 0.6715 0.4942 Thailand 0.1975 0.6040 0.5476 0.2078 0.7119 0.5256 0.2034 0.6657 0.5350 Turkey 0.3338 0.6130 0.6095 0.2785 0.5891 0.5820 0.3022 0.5994 0.5938

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6. Data and Empirical Results

Table 3: The stylised characteristics of the developed and the emerging markets obtained by averaging of the stand-alone market characteristics over the corresponding periods and the p-value of the t-test that the averages are equal. Developed Emerging p-value Markets Markets

Full Period: L 0.6125 0.6588 0

2003-2013 R 0.4794 0.5433 0 0.1973 0.2242 0.0444

Bull Market: L 0.6016 0.6298 0.1153

2003-2007 R 0.4848 0.5746 0 0.1493 0.2092 0

Turbulent Period: L 0.6207 0.6805 0

2007-2013 L 0.4754 0.5198 0.0011 0.2333 0.2354 0.8715

clear that the degree to which portfolio volatility can be managed depends critically on the correlation matrix and, likewise, the management of the volatility of the extremes would depend on the way they are jointly dependent. These are topics in a multivariate context and go beyond the scope of this paper.

42 An EDHEC-Risk Institute Publication 7. Conclusions

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7. Conclusions

A stylised fact for asset returns is that they both tails with only a few rejections in exhibit fat tails; that is, the frequency of the time period 2003-2013. Increasing the observed extreme losses is higher than that tail probability to 2.5% and 5% resulted predicted by the normal distribution. An in higher number of rejections mainly interesting practical problem is to compare caused by failures of Christoffersen's test tail risk across different markets, which although the overall performance is quite turns out to be challenging because of two acceptable at the 2.5% level across all reasons: (i) tail risk is dynamic and (ii) any markets. Even at the 5% tail probability, the downside risk measure requires a model for average number of exceedances is within the tail behaviour. Tail risk dynamics are the confidence interval for both tails with related at least to the dynamics of volatility, only two exceptions. and possibly other factors. Furthermore, coming up with a model for the tail There are two important conclusions to behaviour is complicated because the draw from these results. First, the reported observations in the tail are rare events and strong significance of the power tail in the samples are short. unconditional EVT models can be attributed primarily to the clustering of volatility Our strategy of dealing with the two effect. Second, the increasing number of challenges is to adopt a GARCH model failures of Christoffersen's test when tail and an asymptotic description of the tail probability increases suggests that dynamics through the Generalised Pareto Distribution in characteristics other than volatility may (GPD) arising from Extreme Value Theory start affecting those quantile levels. Overall, (EVT). The GARCH model is supposed to the restricted GARCH-EVT model has very explain away the clustering of volatility good out-of-sample performance at both effect and the estimated shape parameter 1% and 2.5% tail probabilities. of the GPD provides insight into the residual tail thickness. To check for possible consistent underestimation of the shape parameter, We studied the out-of-sample behaviour of we report the out of-sample performance of the GARCH-EVT model for 19 emerging and two other versions of the restricted model 22 developed equity markets over extended with values for the shape parameter set time periods. The VaR- and CVaR-based to 0.1 and 0.2, respectively. It is rather tests for the case of 1% tail probability surprising that the VaR-based tests do not

indicated that, with a couple of exceptions, strongly reject the case of ξL = 0.2 which the model is statistically acceptable for all represents a very substantial thickening

markets and both the left and the right of the tail from the base case of ξL = 0 tail, thus confirming other studies in the which is otherwise strongly rejected by the empirical literature. CVaR-based t-test. The number of failures

of the t-test increase even for ξL = 0.1 and A new finding is that the restricted model even more so for the right tail indicating with the shape parameter set to zero is also presence of tail asymmetry. Overall, the statistically acceptable for the same tail power of the VaR-based tests appears probability level for most countries and unsatisfactory which is a general concern

44 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

7. Conclusions

bearing in mind the wide application of these tests in model validation.

Finally, we used the values L and R of the restricted model to check if there is any difference in the downside and the upside of the developed and the emerging markets and if there is any relationship between this parameter and the volatility parameter. Over the entire period, there appears to be no significant relationship between the average volatility and the average residual tail risk. Overall, developed markets have statistically significant lower tail risk and volatility than the emerging markets but also lower upside potential. Both types of generic markets exhibit tail asymmetry in the dispersion of the extremes.

An EDHEC-Risk Institute Publication 45 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

7. Conclusions

46 An EDHEC-Risk Institute Publication Appendices

An EDHEC-Risk Institute Publication 47 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Appendices

Table 4: P-values of VaR based statistics of the out-of-sample performance of the GARCH-EVT model covering the full samples of all markets. Exc denotes number of exceedances, Ku-test, Ch-test, and KC-test denote Kupiec's, Christoffersen's and the combined tests and Geo-test denotes the geometric distribution test. Rejection at 5% level is marked in bold.

48 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Appendices

Table 5: CVaR based statistics of the out-of-sample performance of the GARCH-EVT model covering the full samples of all markets. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.

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Appendices

Table 6: P-values of VaR-based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013. Ku- test, Ch-test, and KC-test denote Kupiec's, Christoffersen's and the combined tests and Geo-test denotes the geometric distribution test. Rejection at 5% level is marked in bold.

50 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Appendices

Table 7: CVaR based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013 and p-values of the t-test. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.

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Appendices

Table 8: P-values of VaR-based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013. Exc denotes number of exceedances, Ku-test, Ch-test, and KC-test denote Kupiec's, Christoffersen's and the combined tests and Geo-test denotes the geometric distribution test. Rejection at 5% level is marked in bold.

52 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Appendices

Table 9: CVaR-based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013 and p-values of the t-test. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.

An EDHEC-Risk Institute Publication 53 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

Appendices

54 An EDHEC-Risk Institute Publication References

An EDHEC-Risk Institute Publication 55 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

References

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References

58 An EDHEC-Risk Institute Publication About EDHEC-Risk Institute

An EDHEC-Risk Institute Publication 59 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

About EDHEC-Risk Institute

Founded in 1906, EDHEC is one The Choice of Asset Allocation Six research programmes have been of the foremost international and Risk Management conducted by the centre to date: business schools. Accredited by EDHEC-Risk structures all of its research • Asset allocation and alternative the three main international academic organisations, work around asset allocation and risk diversification EQUIS, AACSB, and Association management. This strategic choice is • Style and performance analysis of MBAs, EDHEC has for a applied to all of the Institute's research • Indices and benchmarking number of years been pursuing programmes, whether they involve • Operational risks and performance a strategy of international excellence that led it to set up proposing new methods of strategic • Asset allocation and derivative EDHEC-Risk Institute in 2001. allocation, which integrate the alternative instruments This institute now boasts a team class; taking extreme risks into account • ALM and asset management of 90 permanent professors, in portfolio construction; studying the engineers and support staff, as well as 48 research associates usefulness of derivatives in implementing These programmes receive the support of from the financial industry and asset-liability management approaches; a large number of financial companies. affiliate professors.. or orienting the concept of dynamic The results of the research programmes “core-satellite” investment management are disseminated through the EDHEC-Risk in the framework of absolute return or locations in Singapore, which was target-date funds. established at the invitation of the Monetary Authority of Singapore (MAS); the City of London in the United Kingdom; Academic Excellence Nice and Paris in France; and New York in and Industry Relevance the United States. In an attempt to ensure that the research it carries out is truly applicable, EDHEC has EDHEC-Risk has developed a close implemented a dual validation system for partnership with a small number of the work of EDHEC-Risk. All research work sponsors within the framework of must be part of a research programme, research chairs or major research projects: the relevance and goals of which have • Core-Satellite and ETF Investment, in been validated from both an academic partnership with Amundi ETF and a business viewpoint by the Institute's • Regulation and Institutional advisory board. This board is made up of Investment, in partnership with AXA internationally recognised researchers, Investment Managers the Institute's business partners, and • Asset-Liability Management and representatives of major international Institutional Investment Management, institutional investors. Management of the in partnership with BNP Paribas research programmes respects a rigorous Investment Partners validation process, which guarantees the • Risk and Regulation in the European scientific quality and the operational Fund Management Industry, in usefulness of the programmes. partnership with CACEIS • Exploring the Commodity Futures Risk Premium: Implications for Asset Allocation and Regulation, in partnership with CME Group

60 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

About EDHEC-Risk Institute

• Asset-Liability Management in Private The philosophy of the Institute is to Wealth Management, in partnership validate its work by publication in with Coutts & Co. international academic journals, as well as • Asset-Liability Management to make it available to the sector through Techniques for Sovereign Wealth Fund its position papers, published studies, and Management, in partnership with conferences. Deutsche Bank • The Benefits of Volatility Derivatives Each year, EDHEC-Risk organises three in Equity Portfolio Management, in conferences for professionals in order to partnership with Eurex present the results of its research, one in • Structured Products and Derivative London (EDHEC-Risk Days Europe), one Instruments, sponsored by the French in Singapore (EDHEC-Risk Days Asia), and Banking Federation (FBF) one in New York (EDHEC-Risk Days North • Optimising Bond Portfolios, in America) attracting more than 2,500 partnership with the French Central professional delegates. Bank (BDF Gestion) • Asset Allocation Solutions, in EDHEC also provides professionals with partnership with Lyxor Asset access to its website, www.edhec-risk.com, Management which is entirely devoted to international • Infrastructure Equity Investment asset management research. The website, Management and Benchmarking, which has more than 58,000 regular in partnership with Meridiam and visitors, is aimed at professionals who Campbell Lutyens wish to benefit from EDHEC’s analysis and • Investment and Governance expertise in the area of applied portfolio Characteristics of Infrastructure Debt management research. Its monthly Investments, in partnership with Natixis newsletter is distributed to more than 1.5 • Advanced Modelling for Alternative million readers. Investments, in partnership with Newedge Prime Brokerage EDHEC-Risk Institute: • Advanced Investment Solutions for Key Figures, 2011-2012 Liability Hedging for Inflation Risk, Nbr of permanent staff 90 in partnership with Ontario Teachers’ Nbr of research associates 20 Pension Plan Nbr of affiliate professors 28 • The Case for Inflation-Linked Overall budget €13,000,000 Corporate Bonds: Issuers’ and Investors’ External financing €5,250,000 Perspectives, in partnership with Nbr of conference delegates 1,860 Rothschild & Cie Nbr of participants 640 • Solvency II, in partnership with Russell at research seminars Nbr of participants at EDHEC-Risk Investments 182 Institute Executive Education seminars • Structured Equity Investment Strategies for Long-Term Asian Investors, in partnership with Société Générale Corporate & Investment Banking

An EDHEC-Risk Institute Publication 61 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

About EDHEC-Risk Institute

The EDHEC-Risk Institute PhD in School of Management to set up joint Finance certified executive training courses in The EDHEC-Risk Institute PhD in Finance North America and Europe in the area of is designed for professionals who aspire investment management. to higher intellectual levels and aim to redefine the investment banking and asset As part of its policy of transferring know- management industries. It is offered in two how to the industry, EDHEC-Risk Institute tracks: a residential track for high-potential has also set up ERI Scientific Beta. ERI graduate students, who hold part-time Scientific Beta is an original initiative positions at EDHEC, and an executive track which aims to favour the adoption of the for practitioners who keep their full-time latest advances in smart beta design and jobs. Drawing its faculty from the world’s implementation by the whole investment best universities, such as Princeton, industry. Its academic origin provides the Wharton, Oxford, Chicago and CalTech, foundation for its strategy: offer, in the and enjoying the support of the research best economic conditions possible, the centre with the greatest impact on the smart beta solutions that are most proven financial industry, the EDHEC-Risk Institute scientifically with full transparency in PhD in Finance creates an extraordinary both the methods and the associated platform for professional development and risks. industry innovation.

Research for Business The Institute’s activities have also given rise to executive education and research service offshoots. EDHEC-Risk's executive education programmes help investment professionals to upgrade their skills with advanced risk and asset management training across traditional and alternative classes. In partnership with CFA Institute, it has developed advanced seminars based on its research which are available to CFA charterholders and have been taking place since 2008 in New York, Singapore and London.

In 2012, EDHEC-Risk Institute signed two strategic partnership agreements with the Operations Research and Financial Engineering department of Princeton University to set up a joint research programme in the area of risk and investment management, and with Yale

62 An EDHEC-Risk Institute Publication EDHEC-Risk Institute Publications and Position Papers (2011-2014)

An EDHEC-Risk Institute Publication 63 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

EDHEC-Risk Institute Publications (2011-2014)

2014 • Badaoui, S., Deguest, R., L. Martellini and V. Milhau. Dynamic Liability-Driven Investing Strategies: The Emergence of a New Investment Paradigm for Pension Funds? (February). • Deguest, R., and L. Martellini. Improved Risk Reporting with Factor-Based Diversification Measures (February).

2013 • Loh, L., and S. Stoyanov. Tail Risk of Asian Markets: An Extreme Value Theory Approach (August). • Goltz, F., L. Martellini, and S. Stoyanov. Analysing statistical robustness of cross- sectional volatility. (August). • Loh, L., L. Martellini, and S. Stoyanov. The local volatility factor for asian stock markets. (August). • Martellini, L., and V. Milhau. Analysing and decomposing the sources of added-value of corporate bonds within institutional investors’ portfolios (August). • Deguest, R., L. Martellini, and A. Meucci. Risk parity and beyond - From asset allocation to risk allocation decisions (June). • Blanc-Brude, F., Cocquemas, F., Georgieva, A. Investment Solutions for East Asia's Pension Savings - Financing lifecycle deficits today and tomorrow (May) • Blanc-Brude, F. and O.R.H. Ismail. Who is afraid of construction risk? (March) • Loh, L., L. Martellini, and S. Stoyanov. The relevance of country- and sector-specific model-free volatility indicators (March). • Calamia, A., L. Deville, and F. Riva. Liquidity in european equity ETFs: What really matters? (March). • Deguest, R., L. Martellini, and V. Milhau. The benefits of sovereign, municipal and corporate inflation-linked bonds in long-term investment decisions (February). • Deguest, R., L. Martellini, and V. Milhau. Hedging versus insurance: Long-horizon investing with short-term constraints (February). • Amenc, N., F. Goltz, N. Gonzalez, N. Shah, E. Shirbini and N. Tessaromatis. The EDHEC european ETF survey 2012 (February). • Padmanaban, N., M. Mukai, L . Tang, and V. Le Sourd. Assessing the quality of asian stock market indices (February). • Goltz, F., V. Le Sourd, M. Mukai, and F. Rachidy. Reactions to “A review of corporate bond indices: Construction principles, return heterogeneity, and fluctuations in risk exposures” (January). • Joenväärä, J., and R. Kosowski. An analysis of the convergence between mainstream and alternative asset management (January). • Cocquemas, F. Towar¬ds better consideration of pension liabilities in european union countries (January).

64 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

EDHEC-Risk Institute Publications (2011-2014)

• Blanc-Brude, F. Towards efficient benchmarks for infrastructure equity investments (January).

2012 • Arias, L., P. Foulquier and A. Le Maistre. Les impacts de Solvabilité II sur la gestion obligataire (December). • Arias, L., P. Foulquier and A. Le Maistre. The Impact of Solvency II on Bond Management (December). • Amenc, N., and F. Ducoulombier. Proposals for better management of non-financial risks within the european fund management industry (December). • Cocquemas, F. Improving Risk Management in DC and Hybrid Pension Plans (November). • Amenc, N., F. Cocquemas, L. Martellini, and S. Sender. Response to the european commission white paper "An agenda for adequate, safe and sustainable pensions" (October). • La gestion indicielle dans l'immobilier et l'indice EDHEC IEIF Immobilier d'Entreprise France (September). • Real estate indexing and the EDHEC IEIF commercial property (France) index (September). • Goltz, F., S. Stoyanov. The risks of volatility ETNs: A recent incident and underlying issues (September). • Almeida, C., and R. Garcia. Robust assessment of hedge fund performance through nonparametric discounting (June). • Amenc, N., F. Goltz, V. Milhau, and M. Mukai. Reactions to the EDHEC study “Optimal design of corporate market debt programmes in the presence of interest-rate and inflation risks” (May). • Goltz, F., L. Martellini, and S. Stoyanov. EDHEC-Risk equity volatility index: Methodology (May). • Amenc, N., F. Goltz, M. Masayoshi, P. Narasimhan and L. Tang. EDHEC-Risk Asian index survey 2011 (May). • Guobuzaite, R., and L. Martellini. The benefits of volatility derivatives in equity portfolio management (April). • Amenc, N., F. Goltz, L. Tang, and V. Vaidyanathan. EDHEC-Risk North American index survey 2011 (March). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, L. Martellini, and S. Sender. Introducing the EDHEC-Risk Solvency II Benchmarks – maximising the benefits of equity investments for insurance companies facing Solvency II constraints - Summary - (March). • Schoeffler, P. Optimal market estimates of French office property performance (March). • Le Sourd, V. Performance of socially responsible investment funds against an efficient SRI Index: The impact of benchmark choice when evaluating active managers – an update (March).

An EDHEC-Risk Institute Publication 65 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

EDHEC-Risk Institute Publications (2011-2014)

• Martellini, L., V. Milhau, and A.Tarelli. Dynamic investment strategies for corporate pension funds in the presence of sponsor risk (March). • Goltz, F., and L. Tang. The EDHEC European ETF survey 2011 (March). • Sender, S. Shifting towards hybrid pension systems: A European perspective (March). • Blanc-Brude, F. Pension fund investment in social infrastructure (February). • Ducoulombier, F., Loh, L., and S. Stoyanov. What asset-liability management strategy for sovereign wealth funds? (February). • Amenc, N., Cocquemas, F., and S. Sender. Shedding light on non-financial risks – a European survey (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Ground Rules for the EDHEC-Risk Solvency II Benchmarks. (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints - Synthesis -. (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints (January). • Schoeffler.P. Les estimateurs de marché optimaux de la performance de l’immobilier de bureaux en France (January).

2011 • Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. A long horizon perspective on the cross-sectional risk-return relationship in equity markets (December 2011). • Amenc, N., F. Goltz, and L. Tang. EDHEC-Risk European index survey 2011 (October). • Deguest,R., Martellini, L., and V. Milhau. Life-cycle investing in private wealth management (October). • Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of index- weighting schemes (September). • Le Sourd, V. Performance of socially responsible investment funds against an Efficient SRI Index: The Impact of Benchmark Choice when Evaluating Active Managers (September). • Charbit, E., Giraud J. R., F. Goltz, and L. Tang Capturing the market, value, or momentum premium with downside Risk Control: Dynamic Allocation strategies with exchange-traded funds (July). • Scherer, B. An integrated approach to sovereign wealth risk management (June). • Campani, C. H., and F. Goltz. A review of corporate bond indices: Construction principles, return heterogeneity, and fluctuations in risk exposures (June). • Martellini, L., and V. Milhau. Capital structure choices, pension fund allocation decisions, and the rational pricing of liability streams (June). 66 An EDHEC-Risk Institute Publication Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

EDHEC-Risk Institute Publications (2011-2014)

• Amenc, N., F. Goltz, and S. Stoyanov. A post-crisis perspective on diversification for risk management (May). • Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of index- weighting schemes (April). • Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. Is there a risk/return tradeoff across stocks? An answer from a long-horizon perspective (April). • Sender, S. The elephant in the room: Accounting and sponsor risks in corporate pension plans (March). • Martellini, L., and V. Milhau. Optimal design of corporate market debt programmes in the presence of interest-rate and inflation risks (February).

An EDHEC-Risk Institute Publication 67 Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

EDHEC-Risk Institute Position Papers (2011-2014)

2012 • Till, H. Who sank the boat? (June). • Uppal, R. Financial Regulation (April). • Amenc, N., F. Ducoulombier, F. Goltz, and L. Tang. What are the risks of European ETFs? (January).

2011 • Amenc, N., and S. Sender. Response to ESMA consultation paper to implementing measures for the AIFMD (September). • Uppal, R. A Short note on the Tobin Tax: The costs and benefits of a tax on financial transactions (July). • Till, H. A review of the G20 meeting on agriculture: Addressing price volatility in the food markets (July).

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Notes

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An EDHEC-Risk Institute Publication 69

For more information, please contact: Carolyn Essid on +33 493 187 824 or by e-mail to: [email protected]

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