Tail Risk of Equity Market Indices: An Extreme Value Theory … · 2018. 7. 31. · Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 About the

Tail Risk of Equity Market Indices: An Extreme

Value Theory ApproachFebruary 2014

An EDHEC-Risk Institute Publication

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.

3An EDHEC-Risk Institute Publication

Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014

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

Table of Contents

4 An EDHEC-Risk Institute Publication


About the Authors

Lixia Loh is a senior research engineer at EDHEC-Risk Institute–Asia. Prior tojoining EDHEC Business School, she was a Research Fellow at the Centre forGlobal Finance at Bristol Business School (University of the West of England).Her research interests include empirical finance, financial markets risk, andmonetary economics. She has published in several academic journals, includingthe Asia-Pacific Development Journal and Macroeconomic Dynamics, and isthe author of a book, Sovereign Wealth Funds: States Buying the World (GlobalProfessional Publishing, 2010). She holds an M.Sc. in international economics,banking and finance from Cardiff University and a Ph.D. in finance from theUniversity 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.


Executive Summary



Value-at-risk (VaR) and conditional value-at-risk (CVaR) have become standard choices for risk measures in finance. Both VaR and CVaR are examples of measures of tail risk, or downside risk, because they are designed to exhibit a degree of sensitivity to large portfolio losses whose frequency of occurrence is described by what is known as the tail of the distribution: a part of the loss distribution away from the central region geometrically resembling a tail. In practice,VaR provides a loss threshold exceeded with some small predefined probability, usually 1% or 5%, while CVaR measures the average loss higher than VaR and is, therefore, more informative about extreme losses.

An interesting challenge is to compare tail risk across different markets. A stylised fact for asset returns is that they exhibit fat tails; that is, the frequency of observed extreme losses is higher than that predicted by the normal distribution. Usually, for practical purposes this frequency is calculated unconditionally while it is a well-known fact that in different market states the likelihood of getting an extreme loss varies, i.e. in more turbulent markets it is more likely to experience higher losses. As a result, tail risk would be affected by the temporal behaviour of volatility which is characterised by clustering: elevated levels of volatility are usually followed by similar volatility levels.

Apart from the dependence on the market state, a second more subtle challenge is that any downside risk measure (including VaR and CVaR) is sensitive to the tail of the portfolio loss distribution. CVaR, being the average of the extreme losses, is more sensitive to the way the relative frequency of extreme losses is reflected in the risk

model. Thus, a model such as the normal distribution underestimates this frequency and, therefore, underestimates tail risk aswell.

As a consequence, to compare tail risk across markets, we need to adopt a conditional measure which can take into account at least the clustering of volatility effect and also the tail behaviour of portfolio losses having explained away the dynamics of volatility. This decomposition into two components is important from a risk management perspective because the dynamics of volatility contribute to the unconditional tail thickness phenomenon and techniques do exist for volatility management. It is therefore important to understand how much residual tail thickness remains after explaining away the dynamics of volatility. The standard econometric framework taking into account the clustering of volatility effect is that of the Generalised Autoregressive Conditional Heteroskedastic (GARCH) model.

The academic literature on modelling VaR and CVaR indicates that a successful approach for modelling the high quantiles of the portfolio loss distribution is to combine a GARCH model with extreme value theory (EVT). The GARCH part is responsible for capturing the dynamics of volatility while EVT provides a model for the behaviour of the extreme tail of the distribution. The adopted EVT model is that of the Generalised Pareto Distribution (GPD). Not only does this approach allow reliable estimation of VaR and CVaR, but it also provides insight into the tail thickness through the fitted value of one of the GPD parameters known as the shape parameter. To measure tail risk, we choose VaR and

Executive Summary



CVaR at 1% tail probability which is a standard choice, but we also test other levels such as 2.5% and 5%.

Apart from tail risk, which is the focus of the research, we also test how the model performs at capturing the right tail of the return distribution which describes the upside potential. An appealing feature of EVT is that it can be applied independently for the left and the right tail. To measure the upside potential, we use quantities such as VaR and CVaR but translated for profits instead of losses: the intuition for that is that the tail risk of a short position is described by the upside potential of a corresponding long position.

Before running any comparisons, we first check if the GARCH-EVT model is statisticallyacceptable for application to the extreme quantiles of the returns data which would be in line with the academic literature. We run a VaR back-testing for 41 markets (22 developed and 19 emerging markets) covering periods of different length ranging from 13 to 62 years depending on data availability. The statistical tests indicate the VaR at 1% tail probability is reliably modelled through the GARCH-EVT model for all markets and both the left and the right tail.

In addition to the general GARCH-EVT model, we back-test several special cases. For all markets, in-sample analysis indicates that the important shape parameter of the GPD appears statistically insignificant almost at all times. The practical implication of this finding is that the residual tail is not too heavy.1 In the academic literature, empirical studies ignoring the clustering of volatility report a statistically significant

shape parameter which corresponds to aheavy tail with a power-type decay.2 The back-testing of the special cases of the base model with the shape parameter set to three distinct levels confirms out-of-sample that volatility clustering is the main factor for the thick tail of the unconditional return distribution.

As a consequence, any of the two tails of the return distribution can be described through only one parameter which is interpreted as the volatility of the extreme losses or profits, respectively. This parameter has a rather constant behaviour through time which indicates that the clustering of volatility is the most significant factor for the temporal variation of tail risk. Thus, techniques for dynamic hedging of volatility, such as those behind target volatility funds, indirectly control the dynamics of tail risk as well.

The developed and the emerging markets are compared cross-sectionally in terms of tail risk, upside potential, and forecasted volatility averaged in the period from January 2003 to June 2013, also in the bull market sub-period from January 2003 to June 2007, in the turbulent sub-period from July 2007 to June 2013 covering the financial crisis of 2008, as well as in the post-crisis period. The comparison reveals that over the entire period there appears to be no significant relationship between the average volatility and average residual tail risk suggesting that it may be possible 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

Executive Summary

1 - In a more technical language, the residual tail is exponential and has moments of any order.2 - Some higher-order moments are unbounded.



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 accumulatesundetected either unwillingly or through gaming of these tests.

Executive Summary


1. Introduction



Since its introduction in the 1990s, value-at-risk (VaR) has become a standard measure of risk in the practice of finance. It provides a threshold of the portfolio loss distribution such that losses higher than the threshold occur with a given probability, typical choices include 1% or5%. Another measure of risk, computing the average losses beyond VaR, which has gained popularity is conditional value-at-risk (CVaR). It is more informative than VaR and has better properties; see BIS (2011) by the Basel committee on banking supervision for an extended analysis of the application of VaR for risk measurement in the context of regulation.

An important component of a VaR- or a CVaR-based risk model is the probabilistic model underlying the portfolio P&L distribution. Both risk measures belong to the category of downside, or tail, risk measures indicating that the modelling of the tail has important consequences for the performance of the risk model. In fact, from a technical perspective, both risk measures need a reliable probabilistic model for the high quantiles of the loss distribution. One possible approach to such a model is the Extreme value theory (EVT); see Stoyanov et al. (2011) for an overview of other possible approaches. In a regulatory context, the Basel committee on banking supervision working paper, BIS (2011), has suggested employing fat tail distributions for the risk factor when stress testing for market risk.

Since the application of EVT in finance by Parkinson (1980) and Longin (1996), it hasplayed an increasing role in the estimation of the frequency of extreme events in finance. Studies on predictive

performance of various VaR methods have found the EVT-based method to be particularly accurate (Danielsson and de Vries, 1997; Pownall and Koedij, 1999; McNeil and Frey, 2000; Bekiros and Georgoutsos, 2005; Fernandez, 2005; Tolikas et al., 2007). EVT has also been used specifically to study the distribution of extreme stock returns (Jondeau and Rockinger, 2003; Gettinby et al., 2004; Longin, 2005; Tolikas and Gettinby, 2009).

There are two methods for defining extreme losses that arise from EVT with a corresponding limit model for their behaviour: the Block Maxima (BM) and the Peak-over-Threshold (POT) approaches. With the BM approach, extreme losses are obtained by taking the maxima of losses over certain blocks of observations. On the other hand, POT considers events as extreme when they exceed a chosen high threshold. While both approaches have advantages and disadvantages, the POT method seems to be preferred; see for example Embrechts et al. (1997) and also McNeil and Frey (2000) and Chavez-Demoulin et al. (2011) for a discussion on the threshold choice which turns out to be a critical parameter.

There are two ways in which EVT has been applied to VaR modelling: either directly on the return series or by first running a GARCH model to explain away the clustering of volatility effect. As far as theory is concerned, both approaches are valid. The direct method requires introducing a special parameter called the extremal index which is related to the temporal structure of the time series and describes the clustering of the extremes, see for example Longin (2000) for an

1. Introduction



empirical study. The conditional approach through GARCH relies on cleaning the clustering of the extremes first through a GARCH model and EVT is applied to the residuals time series, see McNeil and Frey (2000).

A big advantage of VaR over CVaR is the fact that VaR can be reliably back-tested. McNeil and Frey (2000) run a VaR back-testing comparison for five time series (3 stock indices, the USD/GBP exchange rate and gold) and conclude that a GARCH-EVT model yields better estimates of VaR and CVaR than unconditional EVT or the classical GARCH model with Student's t and normally distributed error terms. Fernandez (2005) runs a similar comparison for 13 stock indices and draws the same conclusion. Byström (2004) extend the methods by McNeil and Frey (2000) with both the BM and POT approaches to compare the performance of conditional EVT and find the two to perform similarly. Recent work of Furió and Climent (2013) adopted the McNeil and Frey (2000) approach and their results indicate that GARCH-EVT estimates are more accurate than the conventional GARCH models, assuming innovations have normal or Student's t distribution, for both in-sample and out-of-sample estimation. Further on, the superiority of GARCH-EVT is robust to changes in the GARCH model structure. The main goal of this paper is not to provide a comparison of relative performance of VaR models. Rather, we aim at drawing inference about the lower and the upper tail behaviour ofthe return distribution of different markets through the fitted parameters of a GARCH-EVT model.

Our paper differs from existing studies in a number of ways. First, we use a much larger global data set and examine left and the right tail of the market index returns in 22 developed and 19 emerging markets. Unlike previous studies, we compare the left and the right tails of different stock markets by carrying out out-of-sample analyses using both VaR- and CVaR-based tests over the full samples and in the period from Jan-2003 to Jun-2013 for which data is available for all markets. We consider three tail probability levels in the calculation of VaR and CVaR: 1%, 2.5%, and 5%.

Our conclusions can be classified in three groups. Firstly, the out-of-sample empirical results indicate that the GARCH-EVT model restricted with the shape parameter equal to zero is very successful in both the left and the right tail at 1% and 2.5% tail probability levels in the 2003-2013 period for almost all markets. This restricted model essentially uses an exponential tail and implies that the statistically significant power-tail behaviour reported by various authors using EVT without a GARCH-type structure is primarily caused by the clustering of volatility effect. Furthermore, because the estimated values of the only remaining parameter describing residual tail risk appear relatively constant through time, volatility turns out to be the most important factor driving the temporal variation of tail risk. As a consequence, techniques for dynamic hedging of volatility have an indirect control on the dynamics of tail risk.

Secondly, the developed and the emerging markets are compared cross-sectionally

1. Introduction



in terms of tail risk, upside potential, and forecasted volatility averaged in the period from January 2003 to June 2013 and also in the bull market sub-period from January 2003 to June 2007 and the turbulent sub-period from July 2007 to June 2013 covering the financial crisis of 2008 and the post-crisis period. The comparison reveals that over the entire period there appears to be no significant relationship between the average volatility and average residual tail risk suggesting that it may be possible 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.

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.

The paper is organised in the following way. Section 2 discusses EVT and the POT method. Sections 3 focuses on the GARCH-EVT model and Section 4 explains how VaR and CVaR can be calculated and forecasted through the model. Section 5 briefly describes the statistical tests and Section 6 discusses the data and the empirical results. Section 7 concludes.

1. Introduction


2. Extreme Value Theory



EVT finds application in problems related to rare events; it originated in areas other than finance. In finance, such problems can be the estimation of probabilities of extreme losses or estimation of a loss threshold such that losses beyond it occur with a predefined small probability, a quantity also known as a high quantile of the portfolio loss distribution. In fact, EVT provides a model for the extreme tail of the distribution which turns out to have arelatively simple structure described through the corresponding limit distributions.

Denote by X1, X2,…, Xn a sample of i.i.d. portfolio losses and the unknown cumulative distribution function (c.d.f.) of portfolio losses by F(x) = P(Xi ≤ x). We are interested in extreme losses which are described by the right tail of the loss distribution F. Denote by Mn = max(X1, X2,…, Xn) the maximal loss observed in a block of n observations. Since the portfolio losses are assumed to be i.i.d., the c.d.f. of the maximal loss can be expressed through F,

(2.1)

This formula provides a direct connection between the c.d.f. of the worst-case loss and the c.d.f. of portfolio losses but it hinges on knowing F explicitly.

An asymptotic approximation to the c.d.f. of the worst-case loss is provided by the Fisher-Tippett theorem, see for example (Embrechts et al., 1997, Chapter 3) which derives the Generalised Extreme Value (GEV) distribution.

Denote by (2.2)

the normalised maxima where bn > 0 and an are a sequence of normalising constants. The Fisher-Tippett theorem states that if Zn converges to some non-degenerate distribution as n increases indefinitely, P(Zn > x) —> Hξ(x), then this must be the GEV law defined by

(2.3)

where 1 + ξx > 0 and ξ is a shape parameter controlling the tail behaviour of Hξ(z). Depending on the sign of the shape parameter, the GEV is known under different names: the Frechet distribution (ξ > 0), the Gumbel distribution (ξ = 0), or the Weibull distribution (ξ < 0).

It is possible to completely characterise the set of portfolio loss distributions such that, at the limit, the worst-case losses behave according to a given Hξ in (2.3). The set of these distributions is called the maximum domain of attraction (MDA) of the given limit distribution. The accepted notation is X ∈ MDA(Hξ) which implies that the normalised maxima of X converge in distribution to Hξ.

There are three distinct MDAs corresponding to different values of the shape parameter ξ. Since EVT is concerned with rare events, the characterisations are in terms of the tail behaviour of the portfolio loss distribution; no other features of F are important.




The Fréchet MDA, ξ > 0 X ∈ MDA(Hξ) with ξ > 0 if and only if X has a tail decaydominated by a power function in the following sense,

The link between α and ξ is ξ = 1/α. It is possible to demonstrate that this MDA consists of fat-tailed distributions F that have unbounded moments of order higher than α, i.e. EXk < 1 if k < α. For applications in finance, it is safe to assume that volatility is finite which implies α > 2 and ξ < 1/2, respectively. For further detail, see (Embrechts et al., 1997, Section 3.3.1).

The Gumbel MDA, ξ = 0 The MDA of Hξ with ξ = 0 is much more diverse. A portfolio loss distribution belongs to the MDA of the Gumbel law if and only if

in which β(u) is a scaling function and can be chosen to be equal to the average excess loss provided that the loss exceeds the threshold x,

(2.4)

This choice of β(u) is also known as the mean excess function. This MDA is characterised in terms of excess losses that exhibit an asymptotic exponential decay and consists of distributions with a diverse tail behaviour: from moderately heavy-tailed such as the log-normal to light-tailed distributions such as the Gaussian or even distributions with bounded support having an exponential behaviour near the upper end of the support xF. For

further detail, see (Embrechts et al., 1997, Section 3.3.3).

The Weibull MDA, ξ < 0 The MDA of Hξ with ξ < 0 consists entirely of distributionswith bounded support and is, therefore, not interesting for modelling the behaviour of risk drivers. Distributions that belong to this MDA include for example the uniform and the beta distribution. For further detail, see (Embrechts et al., 1997, Section 3.3.2).

Finally, we should note that one distribution can be in only one MDA. There are examples of distributions that are not in any of the three MDAs but they are, however, rather artificial.

2.1. The Peak-over-Threshold MethodFor 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-samples (the blocks). However, in both the theoretical and the empirical literature, there is a preference for the peaks-over-threshold (POT) method which we describein this section, see for example (Embrechts et al., 1997, Section 6.5).

The POT method is related to another important limit result which leads to the Generalised Pareto Distribution (GPD). Suppose that we have selected a high loss threshold u and we are interested in the conditional probability distribution of the excess losses beyond u. We denote this distribution by Fu(x) which is expressed through the unconditional distribution in the following way,




(2.5)

where x > 0. Because we are interested in the extreme losses, we need to gain insight into the probability that the excesses beyond u, X — u, can exceed a certain loss level. Thus, (2.5) is re-stated in terms of the tail

(2.6)

The limit result states that as u increases towards the right endpoint of the support of the loss distribution denoted by xF, the conditional tail converges to the tail of the GPD which is defined by,

(2.7)

where 1 + ξx > 0 and β > 0 is a scale parameter. The limit results is (Embrechts et al., 1997, Chapter 3)

(2.8)

where β (u) is a scaling depending on the selected threshold u.

The limit result in (2.8) can be used to construct an approximation for the tail of the losses exceeding a high threshold u. If we denote by y = u + x and express x in terms of y in (2.6), we obtain (2.9)

after substituting the limit law ; for . For a fixed threshold u, note that F(u)

is a constant and the tail for y > u is determined entirely by the GPD tail .Finally, the MDAs of Hξ and Gξ,β are the same.

To apply (2.9) in practice, we need to choose a high threshold u and also to estimate the probability . In addition, we also need estimates of ξ and β(u). Regarding the choice of u, different strategies have been adopted in the academic literature. One general recommendation is to set it so that a given percentage of the sample are excesses. Chavez-Demoulin and Embrechts (2004) report that a 10% threshold provides a good trade-off between the bias and variability of the estimator of the important shape parameter 16 when the sample size is of about 1,000 observations. A similar guideline is provided by McNeil and Frey (2000).3 If the threshold is allowed to vary, then the probability can be estimated through the empirical c.d.f. as suggested for example in McNeil and Frey (2000). 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 are excesses, then the approximation in (2.9) becomes

(2.10)

where s = 1 — m/n and Xs,n is the s-th observation in the sample sorted in increasing order and and are estimates of ξ and β, respectively.

Regarding estimation, a variety of estimators can be employed to estimate ξ and β. We use the maximum likelihood


3 - An approach based on adaptive calibration of the threshold is adopted by some authors. Gonzalo andOlmo (2004) describe a method based on minimising the distance between the empirical and the tail of the GPD with parameters estimated through the maximum likelihood method. The suggested distance is theKolmogorov-Smirnov statistic.



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

(2.11)where

and (0, Σ) denotes a bivariate normal distribution. For additional details, see (Embrechts et al., 1997, Section 6.5). Since data do not exactly follow the GPD law but are in its MDA, we use the GPD log-likelihood and the result in (2.11) only as an approximation.4 In practice, the GPD is estimated from the sample Yi = Xs+i,n — Xs,n,where i = 1,…, n1 = n—s and s is defined as s = 1—m/n. Information about other estimators, such as the Hill and the Pickands estimator, and further detail on the relevance of the MLE are available in de Haan and Ferreira (2006).


4 - A technical comment is due regarding the approximate MLE method. The limit relationship in (2.8) involves the conditional probability P(X — u > xX > u).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 therefore the number kn, depends on the sample size n.The limit in (2.11) is with respect to the sample size n1 = n — kn. The approximate MLE makes sense only if kn —> 1 when n —> ∞ but so that kn/n —> 0. In fact, the growth rate of kn relative to n is very important. If F ∈ MDA(Hξ), then depending on the second-order terms of the tail expansion of F, it is possible to show formally that the approximate MLE leads to a result similar to (2.11) but with a possible asymptotic bias, see(de Haan and Ferreira, 2006, Theorem 3.4.2). Since kn/n —> 0, the choice of the 10% quantile as a high threshold can be regarded as a rule of thumb 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).





3. A Conditional EVT Model




To apply the BM or the POT method in practice, EVT requires the data to be i.i.d. which is a restrictive assumption. An appealing feature of EVT is that the i.i.d. hypothesis can be weakened without a change in the resulting limit theory. If the time series is assumed stationary and if it meets a couple of additional technical conditions, then the same distributions arise at the limit. The additional technical conditions include a specific form of asymptotic independence of the maxima over any two significantly big, non-overlapping time periods and also lack of asymptotic clustering of extremes, see (Embrechts et al., 1997, Section 4.4) for additional details. Processes that meet these conditions include, for example, the family of the ARMA processes with Gaussian noise.

A stylised fact about asset returns is that volatility tends to cluster: large returns in absolute value are followed by returns of similar magnitude. Although the excess losses of such time series exhibit clustering, the limit theory can still be extended to cover this case. For stationary processes of this type, assuming the same technical condition of asymptotic independence of maxima, the same limit distributions arise as possible models for the maxima, however, with an additional parameter called the extremal index. The extremal index is interpreted as the reciprocal of the average cluster size. This category include the ARCH and GARCH processes which are used as a model for volatility in financial econometrics.

Such extensions of EVT indicate that the method can be applied directly to more general stochastic processes allowing

for a time-dependent scale and perhaps shape parameter of the limit distribution. Applications in the context of the so-called self-exciting processes are suggested by McNeil et al. (2005).

Instead of applying the POT method to the time series directly, we prefer to build a model for the time-varying characteristics and apply EVT to the residuals of the model having explained away, at least partly, the temporal structure of the time series. For example, in line with McNeil and Frey (2000) we estimate a GARCH model to explain away the time structure of volatility. To make things simple, we fit a GARCH(1,1) model to the portfolio return time series as a general GARCH filter.

Denote the time series of portfolio losses by Xt. The GARCH(1,1) model is given by:

(3.1)

where ∈t = σtZt, the innovations Zt are i.i.d. random variables with zero mean, unit variance and marginal distribution function FZ(x) and K, a, and b are the positive parameters with a + b < 1. The model in (3.1) is fitted to the data and then the standardised residual is derived. If we assume that the data is generated by the model in (3.1), then the standardised residual is a sample from the distribution FZ. EVT is applied by fitting the GPD to the residual using approximate MLE.

Regarding the type of the GARCH model, Furió and Climent (2013) find no evidence of any difference in the conditional EVT estimated whether GARCH(1,1) or an asymmetric GARCH specification is




5 - The particular specifications are EGARCH(1,1) and TGARCH(1,1).6 - The pseudo-maximum likelihood method provides consistent and asymptotically normal estimator, see(Gourieroux, 1997, Chapter 4).7 - Jondeau and Rockinger (2003) consider samples of about 9,000 observations for the three markets while Furioand Climent (2013) use samples of about 6,000 observations.

applied for VaR estimation.5 Jalal and Rockinger (2008) perform a Monte Carlo study to analyse the impact of an incorrectly specified GARCH model for VaR modelling. The Monte Carlo study includes GARCH(1,1) with Gaussian and Student's t residuals, EGARCH(1,1) with Gaussian innovations, a switching regime volatility model, stochastic volatility with jumps, and a pure jump process for the data generating process. Jalal and Rockinger (2008) conclude that the two-step procedure of GARCH(1,1) and EVT is remarkably robust compared to a two-step parametric approach with the Gaussian or Student's t distribution.

Apart from the general robustness reported in empirical studies, we prefer the GARCHspecification rather than unconditional application for another reason. It is a well-known fact that a light-tailed distribution FZ, such as the Gaussian law, generates a fat-tailed Xt through the GARCH process. The ARCH(1) example in (Embrechts et al.,1997, Section 8.4) illustrates that a distribution of the error term which is in the MDA of the Gumbel distribution generates Xt in the MDA of the Fréchet distribution. The ARCH(1) process is a special case of the GARCH(1,1) process with β = 0 which can also be written as a GARCH(1,0) process. The example suggests that the clustering of volatility may be a significant contributor to the observed extreme events in the unconditional distribution of portfolio losses. As a result, the approach adopted here allows the time structure of volatility and the residual tail thickness due to factors other than volatility to be considered separately. In line with the empirical literature, for the estimation of the GARCH(1,1) model

in (3.1) we use the pseudo-maximum likelihood method assuming FZ follows the Gaussian distribution.6

Although very difficult to formally compare across empirical studies because different authors focus on different estimation methods, different market indices, and also different time periods, studies applying EVT unconditionally in the estimation of equity market tail risk generally report higher estimated ξ values than studies with a conditional EVT model. Jondeau andRockinger (2003) apply an unconditional EVT model through the block-of-maxima method and cover 20 equity market indices. For the S&P 500, FTSE 100, and Nikkei 225, for example, they report

= 0.282, 0.273, 0.265 and = 0.132, 0.33, 0.268 for the lower and the upper tails, respectively. They find that both the left and the right tails of the return distributions belong to the domain of attraction of the Fréchet law because

= 0 is rejected for all markets and for both tails. On the other hand, Furió and Climent (2013) use a conditional EVT model under three different GARCH specifications. For the same three markets, they report = 0.211, 0.083, 0.536 and = —0.1, 0.082, 0.006, respectively under the GARCH(1,1) and = 0.272, —0.037, 0.213 and

= —0.17, —0.047, 0.024 under the EGARCH(1,1) specifications. Based on the reported standard errors, which are higher than those in Jondeau and Rockinger (2003) because of shorter samples, it is not possible to reject = 0 for the upper tails of the three markets and also for the lower tail of Nikkei 225 index.7

Finally, we should note that reliable estimation of tail thickness is incredibly



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.



4. Risk Estimation with EVT



Apart from the probabilistic model, the other key component of a risk model is the measure of risk. We use two measures of risk: VaR and CVaR at tail probabilities of 1%, 2.5%, and 5%. In this section, we provide definitions and explicitly state the risk forecasts built through the probabilistic model.

The discussion below assumes that the random variable X describes portfolio losses and VaR and CVaR are defined for the right tail of the loss distribution which translates into the left tail of the portfolio return distribution. The same quantities for the right tail of the return distribution (left tail of the loss distribution) are obtained from the definitions below by considering — X instead of X; that is, the downside of a short position is the upside of the corresponding long position. The risk functionals are, however, multiplied by —1 to preserve the interpretation that negative risk means a potential for profit.

4.1. Value-at-RiskThe 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 horizon losses higher than it occur with a probability p. By construction, VaR is the negative of the the p-th quantile of the portfolio return distribution or the (1 — p)-th quantile of the portfolio loss distribution. In the industry, VaR is often defined in terms of a confidence level but we prefer to reserve the term confidence level for the context of statistical testing which we need in Section 6. Thus, to map the terms properly, in the industry we talk

about VaR at 95% and 99% confidence level which corresponds to VaR at 5% and 1% tail probability.

Formally, if we suppose that X describes portfolio losses, then VaR at tail probability p is defined as (4.1)

where F—1 denotes the inverse of the c.d.f. FX(x) = P(X ≤ x) which is also known as thequantile function of X.

As explained earlier, we employ EVT to estimate high quantiles of the loss distribution. To this end, we adopt the approximation of the tail in (2.9). Solving for the value of y yielding a tail probability of p, we get

(4.2)

The estimator is derived from (2.10) in the same way. Suppose that X1,n ≤ X1,n ≤… ≤ Xn,n denote the order statistics, then following (2.10) we get

(4.3)

where s = 1 — m/n and m denotes thenumber of observations that are considered excesses. The approximation in (4.3) is usually interpreted in the following way: the estimate of VaR equals the empirical quantile Xs,n, which is such that p < m/n, plus a correction term obtained through the GPD. In the implementation, we set m/n = 0.1 and, thus, in terms of quantiles the 99% quantile




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

As mentioned before, we assume that the portfolio loss distribution is dynamic and follows the GARCH(1,1) process. Under this assumption, the conditional VaR model is given by

(4.4)

where It denotes the information available at time t, is given in (4.3) and is calculated from the sample of the standardised residuals.

4.2. Conditional Value-at-RiskAn important criticism of VaR in the academic literature is that it is uninformative about the extreme losses beyond it. Indeed, the only information provided is the probability of losing more that VaR which is equal to the tail probability level p but should any such loss occur, there is no information about its possible magnitude. Conditional value-at-risk is constructed to overcome this deficiency: CVaR at tail probability p,CVaRp(X), equals the average loss providedthat the loss exceeds V aRp(X).

CVaR is formally defined as an average of VaRs,

(4.5)

and if we assume that the portfolio loss distribution has a continuous c.d.f. then CVaR can be expressed as a conditional expectation,

(4.6)

In the academic literature, CVaR is also known as average value-at-risk or expected shortfall.

Average value-at-risk corresponds directly to the quantity in (4.5) while expected shortfall is the quantity in (4.6). Although (4.5) is more general and average value-at-risk seems to be a better name for the quantity, we stick to the widely accepted CVaR; see for example Pflug and Römisch (2007) for further discussion.

Since CVaR integrates the entire tail, an asymptotic model for the tail in areas where no data points are available is even more important than for VaR. Assuming that ξ < 1, the expectation in (4.6) can be calculated explicitly through the GPD,

where

Plugging in from (4.3) and the corresponding estimates, we get

(4.7)

For derivations and further detail, see (McNeil et al., 2005, Section 7.2.3).


8 - The correction term is obtained from the GPD and could make sense for very small values of p as well; values that may extend beyond the available observations in the sample. For example, suppose that the sample contains 100 portfolio losses, n = 100, and set p = 0.001 which is the VaR corresponding to the 99.9% quantile. Then, X0.9 x n,n is the 90-th observation in the sorted sample and the empirical approximation to

would be the largest observation in the sample. As a consequence, the correction term in (4.3) allows us to go beyondthe available data points in the sample which emphasises a key advantage of EVT to the historical method.



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.



5. Back-testing and Statistical Tests



We estimate the risk model in a rolling time-window of 1,000 days and build forecasts for the VaR and CVaR at the three tail probabilities for both tails through the estimated model on a daily basis. Each day we verify if the realised return violates any of the forecasted quantile levels in the upper and the lower tail and count an exceedance if any such violation occurs. At the end of the back-testing, we have a sequence of indicators for each tail and each tail probability level. Using the sequence of indicators and the realised returns conditioned on a violation, we run three statistical tests for VaR and one for CVaR. We also calculate confidence bounds for the estimated ξ using (2.11).

5.1. VaR-based TestsThere are two standard tests that we run to check the relevance of the VaR risk model: Kupiec's test and Christoffersen's test. In addition, we describe another test based on an asymptotic result.

5.1.1. Kupiec's Test This test is directly related to the definition of VaR: at any given tail probability p and time window T, on average there would be p x T observations for which the realised return exceeds the forecasted VaR. Such observations are also known as VaR violations or VaR exceedances. Knowing the average number of violations is, however, insufficient. Kupiec's test provides a test statistic,

(5.1)

where N denotes the number of VaR violations in the sample. The asymptotic distribution is which can be used for a p-value or a confidence bound, see Kupiec (1995).

If the dynamics of the corresponding quantile of the return distribution is properly captured by the risk model, then the VaR exceedances should be independent events. In fact, the sequenceof the indicators marking the VaR exceedances should be indistinguishable from a sequence of tossing an unfair coin T times with a probability of success equal to p. If in practice the VaR violations are clustered, this would be an indication that there is a temporal structure of the empirical quantile which is not captured properly by the risk model. For example, the GARCH model is used to describe the temporal behaviour of volatility but we may be using an incorrect order, or it may be structurally incorrect, or there might be dynamics in the higher-order moments not reflected in the model.

5.1.2. Independence of ExceedancesChristoffersen's test concerns the independence of VaR violations. The test statistic is a likelihood ratio test similar to (5.1),

(5.2)

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




in one day while it was at i the previous day. The asymptotic distribution of the test statistic is the same, (1), see Christoffersen (1998).

It is possible to combine Kupiec's and Christoffersen's tests together into one 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.

5.1.3. A General Test for Lack of Memory If the risk model is realistic, the indicators marking VaR exceedances are i.i.d. following a Bernoulli distribution with a “successs" probability p. The probability of a run of k periods of no-exceedances before an exceedance occurs equals (1—p)kp and follows a geometric distribution. The geometric distribution is the only discrete distribution that has the lack-of-memory property

. 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 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.

5.2. A CVaR-Based TestA statistical test on CVaR can be based on the differences between the realised losses and the forecasted CVaR conditioned on the events of VaR exceedances, see (McNeil et al., 2005, Section 4.4.3). Consider the differences

(5.3)

in which 1{A} denotes the indicator of the event A. Because CVaR is the expected shortfall of the continuous loss distribution, EDt+1 = 0 and, therefore, Dt+1 forms a martingale difference series. Under the assumption of a GARCH model, the normalised differences Dt+1 = σt+1 should behave like a zero-mean i.i.d. sequence with a probability mass of (1 — p) at zero. We use the standard t-test to check if the conditional mean of the normalised differences,

is statistically different from zero.


9 - In fact, a more general limit result holds: the stochastic process counting the number of VaR exceedances in a given time period converges weakly a Poisson process for small tail probabilities; see, for example, Embrechts et al. (1997) for the exact limit law.





6. Data and Empirical Results



In this section, we describe the data and analyse the empirical results obtained using the methodologies and statistical tests outlined in the previous sections. We first proceed with a comparison of the GARCH-EVT VaR and GARCH-EVT CVaR across the markets with the full sample period. However, it turns out that the confidence bounds for are wide and = 0 cannot be rejected in-sample for all markets for almost all periods of estimation. To empirically check if the out-of-sample performance is affected by imposing the constraint ξ = 0 in the approximate MLE, we run a back-test for all markets with this constraint imposed at all times. A comparison to this constrained case would also reveal if the time variations in are significant or are an artefact of the rolling time-window estimation.

Apart from this constrained case, we also run two full back-tests for all markets imposing ξ = 0.1 and ξ = 0.2 to study the change in the out-of-sample performance of the risk models. This comparison makes sense because of the possible bias leading to underestimation of the true value of ξ due to the approximate MLE. This comparison is performed for the subsample period of 2003-2013 which covers the subprime crisis and the European debt crisis. Finally, we examine the differences in the estimated ξ and β in the context of developed and emerging markets.

6.1. DataThe daily stock price indices are obtained from Datastream. Due to availability of data, the starting sample period varies

for different countries. The sample period for all indices ends on 28 June 2013. New Zealand has the shortest sample size at 3,260 observations because the index was launched in March 2003 and data are available from January 2001. Table 1 presents the stock price indices used and the starting date for the sample for eachcountry. Data are organised into developed and emerging markets: there are 22 developed and 19 emerging markets. To carry out statistical estimation and the out-of-sample tests, we use log-returns.

6.2. Comparison of Tail Risk across Different MarketsWe use the approach outlined in the previous sections to compare the tail risk across different markets. Such a comparison is not simple for a number of reasons. First, tail risk depends on the particular risk measure; some risk measures are more sensitive to extreme losses than others, e.g. CVaR is more sensitive to the tail behaviour than VaR. Second, tail risk is dynamic. From (4.4) and (4.8) it becomes evident that the dynamics of volatility can be a big driver of the dynamics of tail risk. Third, although assumed constant, the tail behaviour of the error term in the GARCH process may also be time dependent. This effect, if present, would be partly captured through the estimated ξ.

We carry out the comparison in three steps. The objective of the first step is to assess whether the extreme losses in the unconditional distribution are only a result of the dynamics in volatility. We examine the quality of the VaR forecasts generated by the GARCH-EVT. The methodology takes into account the residual tail




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




thickness whether the latter assumes a light-tailed error distribution. The quality of the forecasts is assessed through the formal statistical tests outlined in Section 5 based on a long back-testing exercise.

If the statistical tests indicate that the risk models perform well at the 1% tail probability, then this would imply that the tail of the normalised residual is modelled properly at that tail probability level. As mentioned before, the GARCH setting allows for the dynamics of volatility to be explained, as well as for an estimation of the tail thickness due to factors other than volatility. In this setting, the behaviour of the fitted values of ξ would provide insight into the residual tail thickness. Conditional on this outcome, the objective of the second step is to compare the fitted values of the shape parameter ξ across markets and also the time series of forecasted values of VaR and CVaR. In addition, we compare the estimated β across the markets which have been neglected in most studies. Studies on tail risk focuses on the fitted ξ and not much attention has been paid to the estimated β.

Finally, a subtle warning is due regarding the use of the fitted values of ξ as an indicator of tail behaviour. EVT is an asymptotic theory and provides an approximation to the true tail of the return distribution. Therefore, the implications of the value of ξ for the tail thickness of the true distribution should be interpreted in the context of the the corresponding MDAs. For example, ξ > 0 implies indeed that the true tail exhibits a decay close to a power function and the value of ξ is

an indicator of the thickness of the tail of the portfolio loss distribution.

In contrast, the case ξ = 0 does not necessarily imply that the true tail is exponential. In fact, if the fitted ξ turns out to be exactly zero throughout the entire period of the back-testing, then this would be an indication that the error term might have been in the MDA of the Gumbel law in the entire period, which does not necessarily imply time invariant tail behaviour.

Although Jalal and Rockinger (2008) report a significant robustness of the GARCH-EVT model in cases of regime shifts in volatility and stochastic volatility with jumps, we should point out that in some periods the model may be misspecified. Any departures may get reflected in the residual and may eventually affect and . Thus, changes in the fitted values should be interpreted with care as they might be an indication of a misspecified volatility model or phenomena unaccounted for by the model.

6.2.1. GARCH-EVT with ξ Unconstrained and ξ = 0 This section analyses the performance of the risk model at 1% tail probability when the important ξ parameter is unconstrained in the approximate MLE and when it is constrained to zero using the full sample. In the following, the subscript L denotes left tail and R denotes right tail respectively. We split the discussion into two parts — the full sample periods and the 10-year period from Jan-2003 to Jun-2013.




Full sample period The left panel of Table 4 provides the results of the unconstrained case. In terms of the VaR-based tests, the performance of the risk model is remarkable for both the left and the right tail. The combined test for the left and the right tails combined fails onlyfor two countries (Argentina and the Philippines). The out-of-sample t-test for CVaR provided in the left panel of Table 5 rejects the model only for three countries for the left and the right tails combined. The average for all countries is available in the left panel of Table 5. The averages are close to zero; the three countries with the highest average L are Hungary (0.1269), Indonesia (0.1269), and Peru (0.1229) R all of them in the group of the emerging markets. The average R of any market is generally lower than the corresponding average L.

Time series plots of the estimated ξ of the left and the right tails and their 95% confidence bounds for 6 countries for the period from Jan-2003 to Jun-2013 are provided in Figure 1. These plots reflect the general properties of the time series of the estimated ξ for all countries: ξ = 0cannot be rejected for all countries almost at all times. This holds for both the bullish market before the financial crisis of 2008, the crisis itself, and the period that followed.10 Regarding differences between the left and the right tail behaviour, indeed L appears generally higher than R but because of the wide confidence bounds its is rarely possible to reject L = R.

Because there does not seem to be a significant variation across time for both

L and R, a reasonable goal is to study

the out-of-sample properties of the risk model with the restriction of L = R = 0. The VaR-based tests for the full samples are provided in the right panel of Table 4. The results imply no deterioration of the restricted risk model.

The right panel of Table 5 provides the CVaR-based statistics for the restricted model. Although the average forecasted CVaR (Avg CVaRf) and the average loss conditioned on the occurrence of VaR exceedances look similar, t-test for the left tail indicates a deterioration in out-of-sample performance for 8 markets (3 developed and 5 emerging). The same test for the right tail show results similar to the unrestricted risk model.

The period from Jan-2003 to Jun-2013 Since the performance of the restricted risk model does not deteriorate substantially over the full sample, we study the out-of-sample performance in the period from Jan-2003 to Jun-2013 which is the longest period for which data are available for all markets with the exception of New Zealand and Singapore. The VaR-based results are provided in Table 6 and the CVaR-based statistics are provided in Table 7. Both tables are split into three panels corresponding to different tail probabilities: left (1%), middle (2.5%), and right (5%).

First, we compare the differences in the performance of the risk model at 1% tail probability to the one for the full sample. In this 10-year period, we notice fewer rejections of the VaR-based tests. In fact, the combined test (KC-test) fails only for one country (the Philippines).


10 - Statistically significant negative values are difficult to interpret since the Weibull MDA contains only distributions with bounded support. The traditional assumption that the log-return distribution has unbounded support implies that statistically significant negative values are most likely an indication of a very slow rate of convergence to the Gumbel limit law combined with a negative bias caused by the fixed 10% threshold in the GPD estimation.



The t-test for the left-tail CVaR in Table 7 shows a rejection only for Egypt and the t-test for the right-tail CVaR rejects the model for four countries: Ireland, Japan, Singapore, and the US. These

results indicate that the rejection for the 10 countries in the full sample is due to unacceptable performance in time periods further back in time.


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.




Since EVT provides an asymptotic model for the tail, it is expected to work well for VaR and CVaR at low tail probabilities. Our results are consistent with other empirical papers indicating the GARCH-EVT model works well at the 1% tail probability level. It is, however, of practical importance to check how the performance of the model deteriorates for VaR and CVaR at higher tail probabilities. The middle and the right panels of Tables 6 and 7 provide results for 2.5% and 5%.

The combined KC-test shows rejections for 6 countries at the 2.5% level and 15 countries at the 5% level for the left and the right tails combined. Kupiec's test fails very rarely, most failures are caused by Christoffersen's test. A possible explanation is that at higher tail probabilities, dynamics in parameters other than volatility (e.g. higher-order moments) play a role. Since they are not captured by the model, they may cause exceedences to cluster. In contrast, the left- and the right-tail CVaRs get rejected for 6 countries at both the 2.5% and the 5% levels.

6.2.2. GARCH-EVT with ξ = 0.1 and ξ = 0.2 One possible explanation for the fact thatξL = ξR = 0 is statistically acceptable almost always on Figure 1 is that both L and R are underestimated because of the use of a fixed quantile as a threshold in the GPD estimation. Because the exact distribution of the data in the sample of the residual is unknown, it is difficult to provide an estimate of the bias but in view of the statistically significant positive estimates provided in the academic literature the bias is most likely negative for most time periods of estimation.

We repeat the back-testing of the restricted model with ξL = ξR = 0.1 and

L = R = 0.2 and check the performance of the risk model through the out-of-sample tests. Tables 8 and 9 contain the VaR- and CVaR-based tests, respectively, for the period from Jan-2003 to Jun-2013.

Tables 8 reveals that increasing the shape parameter to 0.1 leads to no rejections of the combined test for the left tail and only a couple of rejections of Kupiec's and Christoffersen's tests. As a consequence, no significant deterioration in the performance is registered compared to ξ = 0.

Regarding the right tail, Kupiec's test fails for 5 countries but the combined test fails in only two cases and the same conclusion follows. It is rather surprising that a similar conclusion can be drawn for the left tail of the restricted model with ξL = 0.2. It should be noted that an increase from 0 to 0.2 represents a substantial thickening of the left tail. Kupiec's test fails, however, across the board for the right tail implying that ξR = 0.2 is getting too high from an out-of-sample perspective.

The results in Table 9 reveal what may turn out to be a substantial flaw in the quantile-based tests for model adequacy. The case ξL = ξR = 0.1 already leads to rejections for 7 countries for the left tail and for 17 countries respectively for the right tail. Increasing the value of the shape parameter to 0.2 increases the number of rejections to 24 countries for the left tail and 30 countries for the right tail.

As consequence, although the case ξ = 0 is acceptable for almost all markets and both



tails, so would be other values in the interval [0, 0.1) which is not surprising because within GPD, a power-tail with a sufficiently small value for the shape parameter can approximate an exponential tail.

6.2.3. Tail Risk and Reward of Developed and Emerging Markets The analysis of the results for the period from 2003 to 2013 suggests that ξL = ξR = 0 is a statistically acceptable model 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.



of the residuals are thus determined by the fitted values L = R. Time series of estimated values and confidence bounds are provided in Figure 2 for the same 6 countries from Figure 1. The estimated values do not appear very noisy and do not seem to change behaviour from the pre-crisis to the post-crisis period. As a consequence, the average residual tail risk of the markets for the 10-year period can be captured by the average values of L

and R.

The practical implication of this is that the temporal variations in tail risk are almostcompletely captured by the dynamics of volatility. As a consequence, techniques for dynamic hedging of volatility are expected to be effective in managing the dynamics of tail risk.

The average tail risk of markets in the 10-year period is described by two quantities: the average volatility and the average L. Likewise, the average upside


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.



potential is described by the average volatility and average R. Table 2 provides the corresponding quantities for the individual markets averaged over the full period 2003-2013 and two half-periods of approximately equal size: the pre-crisis bull market period of Jan-2003 to Jun-2007 and the turbulent period of Jul-2007 to Jun-2013.

The confidence bounds in Figure 2 are too wide and cannot be used to draw any conclusions about any asymmetric tail behaviour. Nevertheless, the averaged L and R reported in the right panel of Table 7 suggest that the extremes in the left tail are more volatile than the extremes in the right tail. One approach to increase the statistical significance is to aggregate the results for the developed and emerging market groups.

Figure 3 provides scatter plots of the average volatilities and the values of L

and R of the 41 markets averaged across time. The developed markets are denoted by circles and the emerging ones by squares. The scatter plots illustrate that there is little to no correlation between the average volatility and the average L (

R). As a consequence, markets with high average volatility may have relatively low average L ( R) and vice versa. It is at this stage an open question to what degree the residual tail risk can be efficiently managed independently of volatility risk.

The developed and emerging markets seem to have different characteristics in terms of the aggregate volatility and aggregate residual risk. If we assume that both types of markets are characterised by a generic average volatility and average

βL and βR, we can test the hypothesis if the two types of markets have different parameter values.

Table 3 provides the average of the L,

R, and the corresponding volatilities across the markets belonging to each group. The aggregation is done for the full period from 2003 to 2013 and two half-periods. In all cases, the right tail has a lower scale parameter than the left tail indicating presence of tail asymmetry. The L and R of the developed markets do not change much while those of the emerging markets deteriorate in the post-crisis period, i.e. downside risk increases and upside potential decreases. On the other hand, the volatility of the generic emerging market stays relatively unchanged while that of the developed market increases dramatically.

In the pre-crisis period, the upside potential of the emerging market is much higher than that of the developed but this comes at the cost of higher volatility risk; the downside risk of both is statistically the same. In the post-crisis period, the volatilities of the two are statistically the same and the higher upside potential of the emerging market comes at the cost of higher downside risk.

Assuming currency risk has been completely hedged, the stylised description of the generic developed and emerging market suggests that volatility management is more important for the management of tail risk of a portfolio of developed equity markets, while the same problem for a portfolio of emerging equity markets appears more complex. From a portfolio construction perspective it is





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: 2003-2007

Turbulent Period: 2007-2013

Full Period: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



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.


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 Markets

EmergingMarkets

p-value

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


7. Conclusions



A stylised fact for asset returns is that they exhibit fat tails; that is, the frequency of observed extreme losses is higher than that predicted by the normal distribution. An interesting practical problem is to compare tail risk across different markets, which turns out to be challenging because of two reasons: (i) tail risk is dynamic and (ii) any downside risk measure requires a model for the tail behaviour. Tail risk dynamics are related at least to the dynamics of volatility,and possibly other factors. Furthermore, coming up with a model for the tail behaviour is complicated because the observations in the tail are rare events and the samples are short.

Our strategy of dealing with the two challenges is to adopt a GARCH model and an asymptotic description of the tail through the Generalised Pareto Distribution (GPD) arising from Extreme Value Theory (EVT). The GARCH model is supposed to explain away the clustering of volatility effect and the estimated shape parameter of the GPD provides insight into theresidual tail thickness.

We studied the out-of-sample behaviour of the GARCH-EVT model for 19 emerging and 22 developed equity markets over extended time periods. The VaR- and CVaR-based tests for the case of 1% tail probability indicated that, with a couple of exceptions, the model is statistically acceptable for all markets and both the left and the right tail, thus confirming other studies in the empirical literature.

A new finding is that the restricted model with the shape parameter set to zero is alsostatistically acceptable for the same tail probability level for most countries and

both tails with only a few rejections in the time period 2003-2013. Increasing the tail probability to 2.5% and 5% resulted in higher number of rejections mainly caused by failures of Christoffersen's test although the overall performance is quite acceptable at the 2.5% level across all markets. Even at the 5% tail probability, the average number of exceedances is within the confidence interval for both tails with only two exceptions.

There are two important conclusions to draw from these results. First, the reported strong significance of the power tail in unconditional EVT models can be attributed primarily to the clustering of volatility effect. Second, the increasing number of failures of Christoffersen's test when tail probability increases suggests that dynamics in characteristics other than volatility may start affecting those quantile levels. Overall, the restricted GARCH-EVT model has verygood out-of-sample performance at both 1% and 2.5% tail probabilities.

To check for possible consistent underestimation of the shape parameter, we report the out of-sample performance of two other versions of the restricted model with values for the shape parameter set to 0.1 and 0.2, respectively. It is rather surprising that the VaR-based tests do not strongly reject the case of ξL = 0.2 which represents a very substantial thickening of the tail from the base case of ξL = 0 which is otherwise strongly rejected by the CVaR-based t-test. The number of failures of the t-test increase even for ξL = 0.1 and even more so for the right tail indicating presence of tail asymmetry. Overall, the power of the VaR-based tests appears unsatisfactory which is a general concern

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 anydifference 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.

7. Conclusions



7. Conclusions


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.

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.

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.

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.

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.

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.

Appendices



Appendices


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About EDHEC-Risk Institute





The Choice of Asset Allocation and Risk ManagementEDHEC-Risk structures all of its research work around asset allocation and risk management. This strategic choice is applied to all of the Institute's research programmes, whether they involve proposing new methods of strategic allocation, which integrate the alternative class; taking extreme risks into account in portfolio construction; studying the usefulness of derivatives in implementing asset-liability management approaches; or orienting the concept of dynamic “core-satellite” investment management in the framework of absolute return or target-date funds.

Academic Excellence and Industry RelevanceIn an attempt to ensure that the research it carries out is truly applicable, EDHEC has implemented a dual validation system for the work of EDHEC-Risk. All research work must be part of a research programme, the relevance and goals of which have been validated from both an academic and a business viewpoint by the Institute's advisory board. This board is made up of internationally recognised researchers, the Institute's business partners, and representatives of major international institutional investors. Management of the research programmes respects a rigorous validation process, which guarantees the scientific quality and the operational usefulness of the programmes.

Six research programmes have been conducted by the centre to date:• Asset allocation and alternative diversification• Style and performance analysis• Indices and benchmarking• Operational risks and performance• Asset allocation and derivative instruments• ALM and asset management

These programmes receive the support of a large number of financial companies. The results of the research programmes are disseminated through the EDHEC-Risklocations in Singapore, which was established at the invitation of the Monetary Authority of Singapore (MAS); the City of London in the United Kingdom; Nice and Paris in France; and New York in the United States.

EDHEC-Risk has developed a close partnership with a small number of sponsors within the framework of research chairs or major research projects:• Core-Satellite and ETF Investment, in partnership with Amundi ETF• Regulation and Institutional Investment, in partnership with AXA Investment Managers• Asset-Liability Management and Institutional Investment Management, in partnership with BNP Paribas Investment Partners• Risk and Regulation in the European Fund Management Industry, in partnership with CACEIS• Exploring the Commodity Futures Risk Premium: Implications for Asset Allocation and Regulation, in partnership with CME Group

Founded in 1906, EDHEC is one of the foremost international

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of MBAs, EDHEC has for a number of years been pursuing

a strategy of international excellence that led it to set up EDHEC-Risk Institute in 2001.

This institute now boasts a team of 90 permanent professors,

engineers and support staff, as well as 48 research associates

from the financial industry and affiliate professors..




• Asset-Liability Management in Private Wealth Management, in partnership with Coutts & Co.• Asset-Liability Management Techniques for Sovereign Wealth Fund Management, in partnership with Deutsche Bank• The Benefits of Volatility Derivatives in Equity Portfolio Management, in partnership with Eurex• Structured Products and Derivative Instruments, sponsored by the French Banking Federation (FBF)• Optimising Bond Portfolios, in partnership with the French Central Bank (BDF Gestion)• Asset Allocation Solutions, in partnership with Lyxor Asset Management• Infrastructure Equity Investment Management and Benchmarking, in partnership with Meridiam and Campbell Lutyens• Investment and Governance Characteristics of Infrastructure Debt Investments, in partnership with Natixis• Advanced Modelling for Alternative Investments, in partnership with Newedge Prime Brokerage• Advanced Investment Solutions for Liability Hedging for Inflation Risk, in partnership with Ontario Teachers’ Pension Plan• The Case for Inflation-Linked Corporate Bonds: Issuers’ and Investors’ Perspectives, in partnership with Rothschild & Cie• Solvency II, in partnership with Russell Investments• Structured Equity Investment Strategies for Long-Term Asian Investors, in partnership with Société Générale Corporate & Investment Banking

The philosophy of the Institute is to validate its work by publication in international academic journals, as well as to make it available to the sector through its position papers, published studies, and conferences.

Each year, EDHEC-Risk organises three conferences for professionals in order to present the results of its research, one in London (EDHEC-Risk Days Europe), one in Singapore (EDHEC-Risk Days Asia), and one in New York (EDHEC-Risk Days North America) attracting more than 2,500 professional delegates.

EDHEC also provides professionals with access to its website, www.edhec-risk.com, which is entirely devoted to international asset management research. The website, which has more than 58,000 regular visitors, is aimed at professionals who wish to benefit from EDHEC’s analysis and expertise in the area of applied portfolio management research. Its monthly newsletter is distributed to more than 1.5 million readers.

EDHEC-Risk Institute:Key Figures, 2011-2012

Nbr of permanent staff 90

Nbr of research associates 20

Nbr of affiliate professors 28

Overall budget €13,000,000

External financing €5,250,000

Nbr of conference delegates 1,860

Nbr of participants at research seminars 640

Nbr of participants at EDHEC-Risk Institute Executive Education seminars 182




The EDHEC-Risk Institute PhD in FinanceThe EDHEC-Risk Institute PhD in Finance is designed for professionals who aspire to higher intellectual levels and aim to redefine the investment banking and asset management industries. It is offered in two tracks: a residential track for high-potential graduate students, who hold part-time positions at EDHEC, and an executive track for practitioners who keep their full-time jobs. Drawing its faculty from the world’s best universities, such as Princeton, Wharton, Oxford, Chicago and CalTech, and enjoying the support of the research centre with the greatest impact on the financial industry, the EDHEC-Risk Institute PhD in Finance creates an extraordinary platform for professional development and industry innovation.

Research for BusinessThe 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

School of Management to set up joint certified executive training courses in North America and Europe in the area of investment management.

As part of its policy of transferring know-how to the industry, EDHEC-Risk Institute has also set up ERI Scientific Beta. ERI Scientific Beta is an original initiative which aims to favour the adoption of the latest advances in smart beta design and implementation by the whole investment industry. Its academic origin provides the foundation for its strategy: offer, in the best economic conditions possible, the smart beta solutions that are most proven scientifically with full transparency in both the methods and the associated risks.


EDHEC-Risk Institute Publications and Position Papers

(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).

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).




• 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).




• 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).




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).

EDHEC-Risk Institute Position Papers (2011-2014)



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

EDHEC-Risk Institute393 promenade des AnglaisBP 3116 - 06202 Nice Cedex 3FranceTel: +33 (0)4 93 18 78 24

EDHEC Risk Institute—Europe 10 Fleet Place, LudgateLondon EC4M 7RBUnited KingdomTel: +44 207 871 6740

EDHEC Risk Institute—Asia1 George Street#07-02Singapore 049145Tel: +65 6438 0030

EDHEC Risk Institute—North AmericaOne Boston Place, 201 Washington StreetSuite 2608/2640, Boston, MA 02108United States of America Tel: +1 857 239 8891

EDHEC Risk Institute—France 16-18 rue du 4 septembre75002 Paris FranceTel: +33 (0)1 53 32 76 30

www.edhec-risk.com

Tail Risk of Equity Market Indices: An Extreme Value Theory … · 2018. 7. 31. · Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014 About the

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