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An EDHEC-Risk Institute Publication
Tail Risk of Equity Market
Indices: An Extreme
Value Theory Approach
February 2014
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Table of Contents
Executive Summary.................................................................................................. 5
1. Introduction.............................................................................................................9
2. Extreme Value Theory........................................................................................13
3. A Conditional EVT Model..................................................................................19
4. Risk Estimation with EVT..................................................................................23
5. Back-testing and Statistical Tests..................................................................27
6. Data and Empirical Results..............................................................................31
7. Conclusions..........................................................................................................43
Appendices...............................................................................................................47
References................................................................................................................55
About EDHEC-Risk Institute.................................................................................59
EDHEC-Risk Institute Publications and Position Papers (2011-2014).........63
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
About the Authors
Lixia Loh is a senior research engineer at EDHEC-Risk Institute–Asia. Prior to
joining EDHEC Business School, she was a Research Fellow at the Centre for
Global Finance at Bristol Business School (University of the West of England).
Her research interests include empirical finance, financial markets risk, and
monetary economics. She has published in several academic journals, including
the Asia-Pacific Development Journal and Macroeconomic Dynamics, and is
the author of a book, Sovereign Wealth Funds: States Buying the World (Global
Professional Publishing, 2010). She holds an M.Sc. in international economics,
banking and finance from Cardiff University and a Ph.D. in finance from the
University of Nottingham.
Stoyan Stoyanov is professor of finance at EDHEC Business School and head of
research at EDHEC Risk Institute–Asia. He has ten years of experience in the field
of risk and investment management. Prior to joining EDHEC Business School,
he worked for over six years as head of quantitative research for FinAnalytica.
He has designed and implemented investment and risk management models
for financial institutions, co-developed a patented system for portfolio
optimisation in the presence of non-normality, and led a team of engineers
designing and planning the implementation of advanced models for major
financial institutions. His research focuses on probability theory, extreme
risk modelling, and optimal portfolio theory. He has published over thirty
articles in leading academic and practitioner-oriented scientific journals such
as Annals of Operations Research, Journal of Banking and Finance, and the
Journal of Portfolio Management, contributed to many professional handbooks
and co-authored three books on probability and stochastics, financial risk
assessment and portfolio optimisation. He holds a master in science in applied
probability and statistics from Sofia University and a PhD in finance from the
University of Karlsruhe.
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Executive Summary
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Executive Summary
Value-at-risk (VaR) and conditional valueat-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
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An EDHEC-Risk Institute Publication
model. Thus, a model such as the normal
distribution underestimates this frequency
and, therefore, underestimates tail risk as
well.
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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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.
1 - In a more technical
language, the residual tail is
exponential and has moments
of any order.
2 - Some higher-order
moments are unbounded.
Before running any comparisons, we first
check if the GARCH-EVT model is statistically
acceptable 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 a
heavy 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
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Executive Summary
dispersed than the upside. In the pre-crisis
period, residual downside risk of both
types of markets was similar, the emerging
markets, however, enjoyed a better upside
potential but also much higher volatility.
In the crisis and post-crisis period, both
types of markets had similar average
volatility, the emerging markets, however,
had a better upside potential which came
at the cost of higher residual tail risk.
Finally, as a by-product of the back-testing
of the three special cases of the base model,
our results illustrate a remarkable weakness
of the standard VaR-violation tests for
model adequacy in that they are unable
to detect a significant thickening of the tail
for the residual which is otherwise detected
by the CVaR-based test. The standard tests
have become a common tool for model
validation and the lack of power could
pose systemic risks if tail risk accumulates
undetected either unwillingly or through
gaming of these tests.
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1. Introduction
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
1. Introduction
Since its introduction in the 1990s, valueat-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% or
5%. Another measure of risk, computing
the average losses beyond VaR, which has
gained popularity is conditional value-atrisk (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 has
played an increasing role in the
estimation of the frequency of extreme
events in finance. Studies on predictive
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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 ChavezDemoulin 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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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 backtesting 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 of
the 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 CVaRbased 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
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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 subperiod 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 postcrisis 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
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An EDHEC-Risk Institute Publication
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.
2. Extreme Value Theory
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
2. Extreme Value Theory
Denote by
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 a
relatively simple structure described
through
the
corresponding
limit
distributions.
(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).
(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.
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An EDHEC-Risk Institute Publication
(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 nondegenerate distribution as n increases
indefinitely, P(Zn > x) —> Hξ(x), then this
must be the GEV law defined by
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,
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.
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
2. Extreme Value Theory
The Fréchet MDA, ξ > 0 X ∈ MDA(Hξ)
with ξ > 0 if and only if X has a tail decay
dominated 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. EXk < 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 heavytailed such as the log-normal to lighttailed 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 distributions
with 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
Method
For the purposes of statistical work, the
approach behind GEV gives rise to the
block maxima (BM) method. To fit the GEV
distribution, we need observations on the
maximal losses Mn calculated from subsamples (the blocks). However, in both the
theoretical and the empirical literature,
there is a preference for the peaks-overthreshold (POT) method which we describe
in 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,
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
2. Extreme Value Theory
(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
3 - An approach based on
adaptive calibration of the
threshold is adopted by some
authors. Gonzalo and
Olmo (2004) describe a
method based on minimising
the distance between the
and the tail
empirical
of the GPD with parameters
estimated through the
maximum likelihood method.
The suggested distance is the
Kolmogorov-Smirnov statistic.
(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.
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An EDHEC-Risk Institute Publication
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 tradeoff 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)
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
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.
(2.9)
Regarding estimation, a variety of
estimators can be employed to estimate
ξ and β. We use the maximum likelihood
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
2. Extreme Value Theory
estimator (MLE) which is rationalised by
the uniform convergence in (2.8). Under
the assumption that data are distributed
exactly according to the GPD, then given a
sample of i.i.d. observations Y = (Y1,…,Yn1)
the log-likelihood function equals
and can be maximised numerically. The
ML estimator
,
where D = (—1/2, ∞) x (0, ∞), satisfies the
following asymptotic property
4 - A technical comment is due
regarding the approximate MLE
method. The limit relationship
in (2.8) involves the conditional
probability P(X — u > xX > 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 secondorder 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).
where
(2.11)
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 loglikelihood 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).
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2. Extreme Value Theory
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3. A Conditional EVT Model
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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, nonoverlapping 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
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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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
3. A Conditional EVT Model
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 twostep 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.
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 Furio
and Climent (2013) use samples
of about 6,000 observations.
Apart from the general robustness reported
in empirical studies, we prefer the GARCH
specification rather than unconditional
application for another reason. It is a wellknown 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 and
Rockinger (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
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
3. A Conditional EVT Model
difficult. Heyde and Kou (2004) consider
6 different methods and demonstrate
that a sample of 5,000 observations may
be insufficient to discriminate between
exponential and power-type tails. It
is nevertheless important to find out
through an out-of-sample risk backtesting 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.
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4. Risk Estimation with EVT
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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-Risk
The VaR of a random variable X describing
portfolio losses at a tail probability p,
VaRp(X), is implicitly defined as a loss
threshold such that over a given time
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
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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 the
quantile 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 the
number 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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
4. Risk Estimation with EVT
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.
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 beyond
the available data points in
the sample which emphasises
a key advantage of EVT to the
historical method.
4.2. Conditional Value-at-Risk
An 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 provided
that 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 valueat-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).
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4. Risk Estimation with EVT
Under the assumption of a GARCH(1,1)
process for the portfolio loss distribution,
the counterpart of (4.4) for CVaR equals
(4.8)
is given in (4.7) and
where
is estimated from the sample of the
standardised residuals.
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5. Back-testing and
Statistical Tests
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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 backtesting, 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 Tests
There 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,
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(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 sequence
of 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 Exceedances
Christoffersen'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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
5. Back-testing and Statistical Tests
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.
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.
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 (interexceedance 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 Test
A 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.
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5. Back-testing and Statistical Tests
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6. Data and Empirical Results
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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 backtest 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. Data
The daily stock price indices are obtained
from Datastream. Due to availability of
data, the starting sample period varies
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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 each
country. 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 Markets
We 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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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
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
Developed Markets
Austria
Germany
Hong Kong
Ireland
DAX 30
Jan-65
12,651
Hang Seng Index
Jan-70
11,347
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
IBEX 35
Jan-87
6,910
OMX Stockholm Index
Jan-87
6,910
SMI
Jul-88
6,521
Spain
Sweden
Switzerland
US
S&P 500 Index
Jan-64
12,913
UK
FTSE 100
Jan-84
7,695
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
Prague SE Index
Apr-94
5,016
Emerging Markets
Argentina
Czech Republic
Egypt
Egypt Hermes Financial
Jan-95
4,825
Budapest SE Index
Jan-91
5,868
CNX 500 Index
Jan-91
5,868
Indonesia
IDX Composite Index
Apr-83
7,890
Malaysia
FTSE Bursa Malaysia KLCI
Jan-80
8,738
Hungary
India
Mexico
Peru
Philippines
IPC Index
Jan-88
6,650
Lima SE General Index
Jan-91
5,868
Philippines SE Index
Jan-86
7,172
Poland
Warsaw General Index
Jan-93
5,346
Russia
MICEX Index
Oct-97
4,108
KOSPI
Jan-75
10,043
South Korea
Taiwan
Taiwan SE Weighted Index
Jan-71
11,084
Thailand
SET
May-75
9,957
BIST National 100
Jan-98
6,650
Turkey
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6. Data and Empirical Results
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
34
An EDHEC-Risk Institute Publication
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.
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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.
Full sample period The left panel
of Table 4 provides the results of the
unconstrained case. In terms of the VaRbased 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 only
for 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 = 0
cannot 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 outof-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 outof-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).
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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.
36
An EDHEC-Risk Institute Publication
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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 that
ξL = 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-ofsample 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
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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.
38
An EDHEC-Risk Institute Publication
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
Figure 3: The average estimated L, R, and volatility of the developed and emerging markets for the period 2003-2013. The
assumed model is the restricted GARCH-EVT with ξ = 0.
of the residuals are thus determined by
the 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 almost
completely 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
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
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 precrisis 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
40
An EDHEC-Risk Institute Publication
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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
Table 2: The characteristics of the developed and the emerging markets obtained by averaging over the corresponding periods. New
Zealand and Singapore are excluded because of insufficient observations.
Bull Market:
2003-2007
Avg
Avg
Turbulent Period:
2007-2013
Avg
Avg
Avg
Full Period:
2003-2013
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
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
Emerging
Chile
0.0881
0.5892
0.4755
0.1361
0.6343
0.4844
0.1155
0.6150
0.4806
Czech Republic
0.1691
0.6405
0.5358
0.2430
0.6686
0.5111
0.2114
0.6566
0.5217
Egypt
0.2519
0.6398
0.6529
0.2645
0.7654
0.4954
0.2591
0.7116
0.5628
Hungary
0.2061
0.5444
0.5894
0.2724
0.5439
0.5268
0.2440
0.5441
0.5536
Indonesia
0.2023
0.7039
0.5696
0.2279
0.7665
0.5026
0.2169
0.7397
0.5313
India
0.2116
0.6726
0.4240
0.2372
0.7116
0.4696
0.2263
0.6949
0.4501
Mexico
0.1729
0.6520
0.5573
0.2030
0.6713
0.5263
0.1901
0.6631
0.5396
Malaysia
0.1200
0.6271
0.6757
0.1235
0.7409
0.5872
0.1220
0.6921
0.6251
Philippines
0.1964
0.6487
0.6631
0.2118
0.6995
0.5350
0.2052
0.6778
0.5899
Poland
0.1768
0.5552
0.6009
0.2123
0.6367
0.4901
0.1971
0.6018
0.5376
Peru
0.1583
0.6068
0.6192
0.2567
0.6286
0.5807
0.2146
0.6193
0.5972
Russia
0.3006
0.6873
0.5347
0.3231
0.7283
0.5095
0.3135
0.7108
0.5203
South Korea
0.2269
0.6513
0.5012
0.2232
0.6891
0.4518
0.2248
0.6729
0.4730
Taiwan
0.1903
0.6317
0.5379
0.2135
0.7014
0.4615
0.2035
0.6715
0.4942
Thailand
0.1975
0.6040
0.5476
0.2078
0.7119
0.5256
0.2034
0.6657
0.5350
Turkey
0.3338
0.6130
0.6095
0.2785
0.5891
0.5820
0.3022
0.5994
0.5938
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
6. Data and Empirical Results
Table 3: The stylised characteristics of the developed and the emerging markets obtained by averaging of the stand-alone market
characteristics over the corresponding periods and the p-value of the t-test that the averages are equal.
Developed
Markets
Emerging
Markets
p-value
Full Period:
L
0.6125
0.6588
0
2003-2013
R
0.4794
0.5433
0
0.1973
0.2242
0.0444
0.6016
0.6298
0.1153
0.4848
0.5746
0
0.1493
0.2092
0
Bull Market:
2003-2007
L
R
Turbulent Period:
2007-2013
L
L
0.6207
0.6805
0
0.4754
0.5198
0.0011
0.2333
0.2354
0.8715
clear that the degree to which portfolio
volatility can be managed depends
critically on the correlation matrix and,
likewise, the management of the volatility
of the extremes would depend on the
way they are jointly dependent. These are
topics in a multivariate context and go
beyond the scope of this paper.
42
An EDHEC-Risk Institute Publication
7. Conclusions
An EDHEC-Risk Institute Publication
43
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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 the
residual 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 also
statistically acceptable for the same tail
probability level for most countries and
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An EDHEC-Risk Institute Publication
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 very
good 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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
7. Conclusions
bearing in mind the wide application of
these tests in model validation.
Finally, we used the values L and R of
the restricted model to check if there is any
difference in the downside and the upside
of the developed and the emerging markets
and if there is any relationship between
this parameter and the volatility parameter.
Over the entire period, there appears to be
no significant relationship between the
average volatility and the average residual
tail risk. Overall, developed markets have
statistically significant lower tail risk and
volatility than the emerging markets but
also lower upside potential. Both types of
generic markets exhibit tail asymmetry in
the dispersion of the extremes.
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
7. Conclusions
46
An EDHEC-Risk Institute Publication
Appendices
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Appendices
Table 4: P-values of VaR based statistics of the out-of-sample performance of the GARCH-EVT model covering the full samples of
all markets. Exc denotes number of exceedances, Ku-test, Ch-test, and KC-test denote Kupiec's, Christoffersen's and the combined
tests and Geo-test denotes the geometric distribution test. Rejection at 5% level is marked in bold.
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Appendices
Table 5: CVaR based statistics of the out-of-sample performance of the GARCH-EVT model covering the full samples of all
markets. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of
VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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. Kutest, Ch-test, and KC-test denote Kupiec's, Christoffersen's and the combined tests and Geo-test denotes the geometric distribution
test. Rejection at 5% level is marked in bold.
50
An EDHEC-Risk Institute Publication
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Appendices
Table 7: CVaR based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013 and p-values
of the t-test. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of
VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
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.
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Appendices
Table 9: CVaR-based statistics of the out-of-sample performance of the GARCH-EVT model in the period 2003-2013 and p-values
of the t-test. Avg denotes average, CVaRf denotes forecasted CVaR, and C-Loss denotes observed loss conditioned on the events of
VaR exceedances. The averaging is performed over the entire sample of estimated values. Rejection at 5% level is marked in bold.
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Appendices
54
An EDHEC-Risk Institute Publication
References
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
References
• Bekiros, S. D. and D. A. Georgoutsos. 2005. Estimation of value-at-risk by extreme value
and conventional methods: A comparative evaluation of their predictive performance.
International Financial Markets, Institutions and Money 15: 209-228.
• BIS. 2011. Messages from the academic literature on risk management for the trading
book. Working Paper 19, Basel Committee on Banking Supervision, Bank for International
Settlements.
• Byström, H. N. 2004. Managing extreme risks in tranquil and volatile markets using
conditional extreme value theory. International Review of Financial Analysis 13(2): 133- 152.
• Chavez-Demoulin, V. and P. Embrechts. 2004. Smooth extremal models in finance.
Journal of Risk and Insurance 71(2): 183-199.
• Chavez-Demoulin, V., P. Embrechts, and S. Sardy. 2011. Extreme-quantile tracking for
financial time series. Swiss Finance Institute. Research Paper Series 11-27.
• Christoffersen, P. 1998. Evaluating interval forecasts. International Economic Review
39: 841-862.
• Danielsson, J. and C. G. de Vries. 1997. Beyond the sample: Extreme quantile and
probability estimation. Working Paper, London School of Economics.
• de Haan, L. and A. Ferreira. 2006. Extreme Value Theory: An Introduction. Springer-Verlag.
• Embrechts, P., C. Klüppelberg and C. Mikosch. 1997. Modelling extremal events for
insurance and finance. Springer-Verlag.
• Fernandez, V. 2005. Risk management under extreme events. International Review of
Financial Analysis 14: 113-148.
• Furió, D. and F. J. Climent. 2013. Extreme value theory versus traditional GARCH approaches
applied to financial data: A comparative evaluation. Quantitative Finance 13(1): 45-63.
• Gettinby, G., C. Sinclair, D. Power, and R. Brown. 2004. An analysis of the distribution
of extreme share returns in the UK from 1975 to 2000. Journal of Business Finance and
Accounting 31(5&6): 607-645.
• Gonzalo, J. and J. Olmo. 2004. Which extreme values are really extreme? Journal of
Financial Econometrics 2(3): 349-369.
• Gouriéroux, C. 1997. ARCH-models and Financial Applications. Springer, New York.
• Heyde, C. C. and S. G. Kou. 2004. On the controversy over tailweight of distributions.
Operations Research Letters 32: 399-408.
• Jalal, A. and M. Rockinger. 2008. Predicting tail-related risk measures: The consequences
of using GARCH filters for non-GARCH data. Journal of Empirical Finance 15: 868-877.
• Jondeau, E. and M. Rockinger. 2003. Testing for differences in the tails of stock-market
returns. Journal of Empirical Finance 10: 559-581.
• Kupiec, P. 1995. Techniques for verifying the accuracy of risk management models.
Journal of Derivatives 2: 73-84.
56
An EDHEC-Risk Institute Publication
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
References
• Longin, F. 1996. The asymptotic distribution of extreme stock market returns. Journal
of Business 69(3): 383-408.
• Longin, F. 2000. From value at risk to stress testing: The extreme value approach. Journal
of Banking and Finance 24: 1097-1130.
• Longin, F. 2005. The choice of the distribution of asset returns: How extreme value
theory can help? Journal of Banking and Finance 29: 1017-1035.
• McNeil, A. and R. Frey. 2000. `Estimation of tail-related risk measures for heteroscedastic
financial time series: An extreme value approach. Journal of Empirical Finance 7(3-4):
271-300.
• McNeil, A., R. Frey, and P. Embrechts. 2005. Quantitative Risk Management. Princeton.
• Parkinson, M. 1980. The extreme value method for estimating the variance rate of
returns. Journal of Business 53(1): 61-65.
• Pflug, G. and W. Römisch. 2007. Modeling, measuring and managing risk. World Scientific.
• Pownall, R. A. and K. G. Koedij. 1999. Capturing downside risk in financial markets: The
case of the Asian crisis. Journal of International Money and Finance 18: 853-870.
• Stoyanov, S. V., S. T. Rachev, B. Racheva-Iotova and F. J. Fabozzi. 2011. Fat-tailed models
for risk estimation. Journal of Portfolio Management 37(2): 107-117.
• Tolikas, K. and G. D. Gettinby. 2009. Modelling the distribution of the extreme share
returns in Singapore. Journal of Empirical Finance 16: 254-263.
• Tolikas, K., A. Koulakiotis and R. A. Brown. 2007. Extreme risk and value-at-risk in the
German stock market. European Journal of Finance 13(4): 373-395.
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
References
58
An EDHEC-Risk Institute Publication
About EDHEC-Risk Institute
An EDHEC-Risk Institute Publication
59
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
About EDHEC-Risk Institute
Founded in 1906, EDHEC is one
of the foremost international
business schools. Accredited by
the three main international
academic organisations,
EQUIS, AACSB, and Association
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..
The Choice of Asset Allocation
and Risk Management
EDHEC-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 Relevance
In 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.
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An EDHEC-Risk Institute Publication
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-Risk
locations 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
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
About EDHEC-Risk Institute
• 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
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
About EDHEC-Risk Institute
The EDHEC-Risk Institute PhD in
Finance
The 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 Business
The Institute’s activities have also given
rise to executive education and research
service offshoots. EDHEC-Risk's executive
education programmes help investment
professionals to upgrade their skills with
advanced risk and asset management
training across traditional and alternative
classes. In partnership with CFA Institute,
it has developed advanced seminars based
on its research which are available to CFA
charterholders and have been taking
place since 2008 in New York, Singapore
and London.
In 2012, EDHEC-Risk Institute signed two
strategic partnership agreements with
the Operations Research and Financial
Engineering department of Princeton
University to set up a joint research
programme in the area of risk and
investment management, and with Yale
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An EDHEC-Risk Institute Publication
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 knowhow 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)
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
EDHEC-Risk Institute Publications
(2011-2014)
2014
• Badaoui, S., Deguest, R., L. Martellini and V. Milhau. Dynamic Liability-Driven Investing
Strategies: The Emergence of a New Investment Paradigm for Pension Funds? (February).
• Deguest, R., and L. Martellini. Improved Risk Reporting with Factor-Based Diversification
Measures (February).
2013
• Loh, L., and S. Stoyanov. Tail Risk of Asian Markets: An Extreme Value Theory Approach
(August).
• Goltz, F., L. Martellini, and S. Stoyanov. Analysing statistical robustness of crosssectional volatility. (August).
• Loh, L., L. Martellini, and S. Stoyanov. The local volatility factor for asian stock markets.
(August).
• Martellini, L., and V. Milhau. Analysing and decomposing the sources of added-value
of corporate bonds within institutional investors’ portfolios (August).
• Deguest, R., L. Martellini, and A. Meucci. Risk parity and beyond - From asset allocation
to risk allocation decisions (June).
• Blanc-Brude, F., Cocquemas, F., Georgieva, A. Investment Solutions for East Asia's
Pension Savings - Financing lifecycle deficits today and tomorrow (May)
• Blanc-Brude, F. and O.R.H. Ismail. Who is afraid of construction risk? (March)
• Loh, L., L. Martellini, and S. Stoyanov. The relevance of country- and sector-specific
model-free volatility indicators (March).
• Calamia, A., L. Deville, and F. Riva. Liquidity in european equity ETFs: What really
matters? (March).
• Deguest, R., L. Martellini, and V. Milhau. The benefits of sovereign, municipal and
corporate inflation-linked bonds in long-term investment decisions (February).
• Deguest, R., L. Martellini, and V. Milhau. Hedging versus insurance: Long-horizon
investing with short-term constraints (February).
• Amenc, N., F. Goltz, N. Gonzalez, N. Shah, E. Shirbini and N. Tessaromatis. The EDHEC
european ETF survey 2012 (February).
• Padmanaban, N., M. Mukai, L . Tang, and V. Le Sourd. Assessing the quality of asian
stock market indices (February).
• Goltz, F., V. Le Sourd, M. Mukai, and F. Rachidy. Reactions to “A review of corporate
bond indices: Construction principles, return heterogeneity, and fluctuations in risk
exposures” (January).
• Joenväärä, J., and R. Kosowski. An analysis of the convergence between mainstream
and alternative asset management (January).
• Cocquemas, F. Towar¬ds better consideration of pension liabilities in european union
countries (January).
64
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
EDHEC-Risk Institute Publications
(2011-2014)
• Blanc-Brude, F. Towards efficient benchmarks for infrastructure equity investments
(January).
2012
• Arias, L., P. Foulquier and A. Le Maistre. Les impacts de Solvabilité II sur la gestion
obligataire (December).
• Arias, L., P. Foulquier and A. Le Maistre. The Impact of Solvency II on Bond Management
(December).
• Amenc, N., and F. Ducoulombier. Proposals for better management of non-financial
risks within the european fund management industry (December).
• Cocquemas, F. Improving Risk Management in DC and Hybrid Pension Plans (November).
• Amenc, N., F. Cocquemas, L. Martellini, and S. Sender. Response to the european
commission white paper "An agenda for adequate, safe and sustainable pensions"
(October).
• La gestion indicielle dans l'immobilier et l'indice EDHEC IEIF Immobilier d'Entreprise
France (September).
• Real estate indexing and the EDHEC IEIF commercial property (France) index (September).
• Goltz, F., S. Stoyanov. The risks of volatility ETNs: A recent incident and underlying
issues (September).
• Almeida, C., and R. Garcia. Robust assessment of hedge fund performance through
nonparametric discounting (June).
• Amenc, N., F. Goltz, V. Milhau, and M. Mukai. Reactions to the EDHEC study “Optimal
design of corporate market debt programmes in the presence of interest-rate and
inflation risks” (May).
• Goltz, F., L. Martellini, and S. Stoyanov. EDHEC-Risk equity volatility index: Methodology
(May).
• Amenc, N., F. Goltz, M. Masayoshi, P. Narasimhan and L. Tang. EDHEC-Risk Asian index
survey 2011 (May).
• Guobuzaite, R., and L. Martellini. The benefits of volatility derivatives in equity portfolio
management (April).
• Amenc, N., F. Goltz, L. Tang, and V. Vaidyanathan. EDHEC-Risk North American index
survey 2011 (March).
• Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, L. Martellini, and S. Sender. Introducing
the EDHEC-Risk Solvency II Benchmarks – maximising the benefits of equity investments
for insurance companies facing Solvency II constraints - Summary - (March).
• Schoeffler, P. Optimal market estimates of French office property performance (March).
• Le Sourd, V. Performance of socially responsible investment funds against an efficient
SRI Index: The impact of benchmark choice when evaluating active managers – an update
(March).
An EDHEC-Risk Institute Publication
65
Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
EDHEC-Risk Institute Publications
(2011-2014)
• Martellini, L., V. Milhau, and A.Tarelli. Dynamic investment strategies for corporate
pension funds in the presence of sponsor risk (March).
• Goltz, F., and L. Tang. The EDHEC European ETF survey 2011 (March).
• Sender, S. Shifting towards hybrid pension systems: A European perspective (March).
• Blanc-Brude, F. Pension fund investment in social infrastructure (February).
• Ducoulombier, F., Loh, L., and S. Stoyanov. What asset-liability management strategy
for sovereign wealth funds? (February).
• Amenc, N., Cocquemas, F., and S. Sender. Shedding light on non-financial risks – a
European survey (January).
• Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Ground
Rules for the EDHEC-Risk Solvency II Benchmarks. (January).
• Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing
the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments
for Insurance Companies facing Solvency II Constraints - Synthesis -. (January).
• Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing
the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments
for Insurance Companies facing Solvency II Constraints (January).
• Schoeffler.P. Les estimateurs de marché optimaux de la performance de l’immobilier
de bureaux en France (January).
2011
• Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. A long horizon perspective on the
cross-sectional risk-return relationship in equity markets (December 2011).
• Amenc, N., F. Goltz, and L. Tang. EDHEC-Risk European index survey 2011 (October).
• Deguest,R., Martellini, L., and V. Milhau. Life-cycle investing in private wealth
management (October).
• Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of indexweighting schemes (September).
• Le Sourd, V. Performance of socially responsible investment funds against an
Efficient SRI Index: The Impact of Benchmark Choice when Evaluating Active Managers
(September).
• Charbit, E., Giraud J. R., F. Goltz, and L. Tang Capturing the market, value, or momentum
premium with downside Risk Control: Dynamic Allocation strategies with exchange-traded
funds (July).
• Scherer, B. An integrated approach to sovereign wealth risk management (June).
• Campani, C. H., and F. Goltz. A review of corporate bond indices: Construction principles,
return heterogeneity, and fluctuations in risk exposures (June).
• Martellini, L., and V. Milhau. Capital structure choices, pension fund allocation decisions,
and the rational pricing of liability streams (June).
66
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
EDHEC-Risk Institute Publications
(2011-2014)
• Amenc, N., F. Goltz, and S. Stoyanov. A post-crisis perspective on diversification for risk
management (May).
• Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of indexweighting schemes (April).
• Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. Is there a risk/return tradeoff across
stocks? An answer from a long-horizon perspective (April).
• Sender, S. The elephant in the room: Accounting and sponsor risks in corporate pension
plans (March).
• Martellini, L., and V. Milhau. Optimal design of corporate market debt programmes in
the presence of interest-rate and inflation risks (February).
An EDHEC-Risk Institute Publication
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
EDHEC-Risk Institute Position Papers
(2011-2014)
2012
• Till, H. Who sank the boat? (June).
• Uppal, R. Financial Regulation (April).
• Amenc, N., F. Ducoulombier, F. Goltz, and L. Tang. What are the risks of European ETFs?
(January).
2011
• Amenc, N., and S. Sender. Response to ESMA consultation paper to implementing
measures for the AIFMD (September).
• Uppal, R. A Short note on the Tobin Tax: The costs and benefits of a tax on financial
transactions (July).
• Till, H. A review of the G20 meeting on agriculture: Addressing price volatility in the
food markets (July).
68
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Tail Risk of Equity Market Indices: An Extreme Value Theory Approach — February 2014
Notes
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An EDHEC-Risk Institute Publication
69
For more information, please contact:
Carolyn Essid on +33 493 187 824
or by e-mail to: [email protected]
EDHEC-Risk Institute
393 promenade des Anglais
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France
Tel: +33 (0)4 93 18 78 24
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London EC4M 7RB
United Kingdom
Tel: +44 207 871 6740
EDHEC Risk Institute—Asia
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#07-02
Singapore 049145
Tel: +65 6438 0030
EDHEC Risk Institute—North America
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Suite 2608/2640, Boston, MA 02108
United States of America
Tel: +1 857 239 8891
EDHEC Risk Institute—France
16-18 rue du 4 septembre
75002 Paris
France
Tel: +33 (0)1 53 32 76 30
www.edhec-risk.com