SEARCH

SEARCH BY CITATION

Keywords:

  • Extreme value theory;
  • Risk management;
  • Fat-tailed distributions;
  • Value-at-risk;
  • Systemic risk;
  • Asset allocation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

Extreme value theory is concerned with the study of the asymptotic distribution of extreme events, that is to say events which are rare in frequency and huge in magnitude with respect to the majority of observations. Statistical methods derived from it have been employed increasingly in finance, especially for risk measurement. This paper surveys some of those main applications, namely for testing different distributional assumptions for the data, for Value-at-Risk and Expected Shortfall calculations, for asset allocation under safety-first type constraints, and for the study of contagion and dependence across markets under conditions of stress.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

Uncertainty and risk1 are present in the economy and the global financial system in at least two dimensions, namely temporal (with time accounting for any change to the present state of nature) and cross-sectional (relating mainly to imperfect knowledge of the states of nature). This is not just a matter of theoretical importance, as events have repeatedly shown. Several crises have overtaken the financial markets in the past 20 years: the stock market crash of October 1987 (when the S&P 500 index fell by more than 20% on just 1 day, 19th October, known as Black Monday), the Asian financial crisis of 1997–1998, the hedge fund crisis of 1998 and the financial crisis that erupted in 2007. Forecasting risk in order to prevent negative events impacting on personal investments or, more broadly, the problem of risk management, has thus become a major concern for both financial institutions and market regulators and due to developments in the theoretical framework and remarkable improvements in computational tools, the field is now gaining ground as an independent scientific discipline.

The primary task of risk management is to quantify risk. Ideally, the best and most informative risk measure for the returns on a financial asset is given by the whole tail of the distribution of those returns. In general, this distribution is

  1. unknown (we just know the time series of the returns and can only guess at the exact distribution);
  2. difficult to deal with (it contains, in a certain sense, too much information, while a tractable risk measure should synthesise the wealth of data that characterizes the entire probability distribution).

This second point has led to the adoption of several methods of risk measurement in the last 60 years, the main ones being i) variance, in the framework of portfolio selection as described by Markowitz (1952); ii) Value-at-Risk (VaR for short), first used after the 1987 crash, then developed and publicized by J. P. Morgan in 1994 and afterwards forming the core of international regulation under the first pillar of the Basel II Accord and iii) Expected Shortfall (ES)2, probably a better measure of risk than VaR as it takes into account the whole of the tail of the distribution and also possesses the properties required for a coherent risk measure as defined by Artzner et al. (1999).

Although the effectiveness (or even the appropriateness) of these three quantities (especially the first two) as risk measures has been widely questioned, they still represent the standard against which other measures are compared. They summarize the information contained in the tail of the distribution in a single number and so are much easier to deal with, providing a suitable tool for decision making.

On the other hand, the first problem, namely the lack of knowledge concerning the distribution of returns, leaves unsolved the question of exactly how risk measures should be computed.

Since VaR and ES only deal with extreme quantiles, disregarding the centre of the distribution, extreme value theory (EVT) may prove to be an effective tool for obtaining reliable estimates. In the field of probability, it is widely used to study the distribution of extreme realizations of a given distribution function, or stochastic processes that satisfy suitable assumptions. The foundations of the theory were laid by Fisher and Tippett (1928) and Gnedenko (1943), who demonstrated that the distributions of the extreme values of an independent and identically distributed sample from a cumulative distribution function F, when adequately rescaled, can converge (and indeed do converge for the majority of known cumulative distribution functions F) towards one out of only three possible distributions.

The crucial element of this finding is that the type of asymptotic distribution of extreme values does not depend on the exact cumulative distribution function F of returns. The precise form of F can thus be ignored and a non-parametric or a semi-parametric method can be used to estimate VaR. This is important, given that the whole tail of the distribution of returns is unknown and that, although financial time series usually exhibit skewed and fat-tailed distributions, there is no complete agreement on what distribution would fit them best.

In principle, EVT-based estimates of VaR should be more accurate and reliable than the usual ones because EVT concentrates directly on the tails of the distribution. This avoids a major flaw of parametric approaches, i.e. that their estimates are somehow biased by the credit given to the central part of the distribution, thus underestimating extremes and outliers, which are precisely what is of interest when calculating VaR.

The third and final reason why EVT is especially promising in risk measurement is that it allows each of the two tails of the distribution to be tackled independently, in a flexible approach that takes the skewness (typical feature of financial time series) of the underlying distribution into account.

These three main advantages of an EVT approach to risk management are summarized by DuMouchel (1983) as: ‘Letting the tails speak for themselves’. This is a very fitting description, as risk management focuses primarily on avoiding large unexpected losses and sudden crashes rather than on long sequences of medium-sized losses (mainly as a consequence of recent market crises and the empirical observation that a few extreme movements in the market have a greater effect on the final position of a portfolio than the sum of many small movements).3

Nowadays, the most popular application of EVT to finance is for the estimation of VaR and ES, but it is not the only possible one, and this paper presents a critical survey of all the main financial applications.4 The paper is organized as follows. Sections 'A Primer of EVT' and 'Multivariate EVT' give a concise presentation of univariate and multivariate EVT, highlighting some of the methodological issues of relevance to empirical studies. The other sections examine four major applications of EVT to finance: to discriminate between different distributional assumptions for the data (Section 'Testing Distributional Assumptions'), to estimate risk measures such as VaR and ES (Section 'Market Risk: VaR and ES Estimation'), to allocate assets optimally under safety-first constraints (Section 'Asset Allocation') and to detect dependence between financial markets (Section 'Dependence Across Markets: Correlation and Contagion'). For completeness, Section 'Further Applications' lists other possible applications of EVT to finance. Section 'Conclusions' concludes with an evaluation of the pros and cons of EVT.

2. A Primer of EVT

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

For readers unfamiliar with EVT, this section and the next give a brief presentation of the main theoretical underpinnings of EVT, starting with a review of the univariate version, then moving on to the multivariate.

Technical details can be found in the extended version of this paper,5 while for a thorough presentation of the theory, we refer the reader to the specialist literature, such as Beirlant et al. (2004), Coles (2001), de Haan and Ferreira (2006), Embrechts et al. (1997), Falk et al. (2004), Finkenstädt and Rootzén (2004), Galambos (1987), Kotz and Nadarajah (2000), Leadbetter et al. (1983), Reiss and Thomas (1997) and Resnick (1987, 2007).

Given an unknown distribution F (e.g. the returns on some type of financial asset), EVT only models the tails of F, without making any specific assumption concerning the centre of the distribution. There are three different approaches to EVT: two parametric and the third non-parametric. The two parametric approaches differ as to the meaning (albeit a complementary one) that they assign to the notion of ‘extreme value’.

Consider N independent and identically distributed random variables Xi, i = 1, …, N, representing positive losses (it being customary in the EVT literature to focus on the upper tail of the distribution, rather than the lower tail) and denote by F their distribution.

  1. The first parametric approach, the block maxima method, divides a given sample of N observations into m subsamples of n observations each (n-blocks) and picks the maximum Mk (k = 1, …, m) of each subsample, a so-called block maximum. The set of extreme values of F is then identified with the sequence inline image of block maxima and the distribution of this sequence is studied. The main result of EVT is that, as m and n grow sufficiently large (ideally, as inline image under some additional assumption), the limit distribution of (adequately rescaled) block maxima belongs to one of three different families (in technical jargon, F is in the maximum domain of attraction of one of these families). Which one it belongs to depends on the behaviour of the upper tail of F, whether it is power-law decaying (the fat-tailed case, the most important for financial applications), exponentially decaying (e.g. if F is the normal distribution), or with upper bounded support (the least relevant case for finance). The three asymptotic distributions of block maxima can be written in a unified manner by means of the generalized extreme value (GEV) distribution,6 a parametric expression depending on a real parameter, known as the shape parameter, that we denote by ξ. The three cases just mentioned correspond, respectively, to ξ > 0 (called the Fréchet case), ξ = 0 (Gumbel) and ξ < 0 (Weibull).
  2. The second parametric approach, the threshold exceedances method, defines extreme values as those observations that exceed some fixed high threshold u. This method models the distribution of the exceedances over u, that is to say, the random variables Yj = Xju, calculated for those observations Xj (returns, in our case) that exceed u, i.e. such that Xj > u. The main result of EVT following this approach is that as the threshold u tends to infinity (or to the right end point of the support of F, if that point is finite), the distribution of the positive sequence inline image, appropriately scaled, belongs to a parametric family, the generalized Pareto distribution (GPD),7 whose main parameter is the same shape parameter ξ as the corresponding GEV distribution. In other words, financial returns whose block maxima follow a GEV distribution with a certain value ξ0 of the parameter ξ are such that, for a sufficiently high threshold u, the exceedances over u follow a GPD with ξ = ξ0.
  3. Both the approaches above are parametric (or semi-parametric), as they fit a parametric model, usually via maximum likelihood estimation, to the upper tail of the distribution, although neglecting what happens at the centre of the distribution. If one does not want to fit a model to the tail either, the shape parameter ξ can be directly estimated using a non-parametric approach. Several estimators can accomplish this task, but the most frequently used by far is the Hill estimator (Hill, 1975), which only works in the Fréchet case (ξ > 0). Note that when considering fat-tailed distributions, the inverse of ξ, namely α = 1/ξ, is often the quantity of interest for estimation. This quantity is known as the tail index of the distribution and is the exponent of the power law that governs the decaying of the tail.

Any of these three approaches to EVT entails choosing an adequate cut-off between the central part of the distribution and the upper tail, i.e. a point separating ordinary realizations from extreme realizations of the random variable. When working with threshold exceedances, the cut-off is induced by the threshold u, while in the block maxima method, it is implied by the number m of blocks (or the amplitude n of each block). Even when using the non-parametric approach, a cut-off must be fixed in order to compute the Hill estimator.

This is a very problematic aspect of the statistical methods of EVT, as the estimated value of the shape parameter can vary considerably depending on the chosen cut-off. Indeed, there is a trade-off between bias and variance of the estimates of the shape parameter ξ. For instance, with threshold exceedances, if u is set too low, many ordinary data are taken as extreme, yielding biased estimates. By contrast, an excessively high threshold gives scant extreme observations, too few to obtain efficient estimates. In both cases, the resulting estimates are flawed and may lead to erroneous conclusions when assessing risk.

An optimal cut-off cannot be selected once and for all as it depends on the time series at hand. The literature suggests three main ways to cope with this issue:

  1. employing graphical methods (known as Hill plots when the Hill estimator is used) that display the estimated values of ξ as a function of the cut-off in order to find some interval of candidate cut-off points that yields stable estimates of ξ (corresponding to an approximately horizontal line in the Hill plot);
  2. making Monte Carlo simulations and then choosing the cut-off that minimizes a statistical quantity (the mean squared error), yielding a trade-off between bias and variance of the estimates;
  3. implementing algorithms, based for instance on the bootstrap method, that endogenously pick out the cut-off best suited to the data at hand.

Another important issue raised by the practical implementation of EVT is that for the theory to work, the data must be independent and identically distributed, whereas most financial time series do not satisfy this requirement. Therefore, using EVT without properly considering the dependence structure of the data yields incorrect estimates, possibly resulting in unexpected losses or in excessively conservative positions (both to be avoided for risk management purposes).

Two main approaches are usually employed to take data dependence into consideration.

  1. If the time series is strictly stationary (as is sometimes assumed for financial time series), then an additional parameter can be estimated, the extremal index, which accounts for the clustering of extremal values due to dependence.
  2. Alternatively, the dependence structure can be explicitly modelled, fitting some GARCH-type model to the data. If the standardized residuals exhibit a roughly independent and identically distributed structure, EVT can then be applied to them rather than directly to the data. This is the same as implementing a two-step procedure that filters the data with econometric tools and is suited to deal with conditional heteroskedasticity before applying EVT methods.

The latter approach works well when using EVT for estimating quantile-based measures of risk, such as VaR or ES, and it seems to be sufficiently robust to yield good estimates even when the GARCH model is mis-specified to some extent.

Finally, when EVT is applied to some data, the very choice of the dataset may be an issue owing to the dichotomy inscribed in the theory: on the one hand, EVT requires a lot of data as its results are asymptotic, but, on the other hand, it necessarily encounters a scarcity of data because it concentrates on the tails of the distribution and extreme events are, by definition, rare. Several practical remedies to this antinomy are found in empirical studies, such as using high frequency data (Hauksson et al., 2001; Lux, 2001; Werner and Upper, 2004), expanding the time window as much as possible (for instance Bali, 2007, considers data spanning one century, although for forecasting purposes, this approach is not advisable as the inclusion of very old data in the sample could damage the estimates), jointly modelling extreme values from both the upper and the lower tail, or pooling different data series in a single one (Byström, 2007).

3. Multivariate EVT

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

An important issue relating to EVT is the extension of its main results to a multivariate setting. Indeed, the parametric approaches to EVT described in Section 'A Primer of EVT' can be generalized to a multivariate framework, but these extensions are computationally burdensome and, in practice, the distributions considered are mainly bivariate.

From a theoretical point of view, when studying extremes of multivariate time series, the dependence between the extreme values of the different components plays a crucial role. The common notion of correlation, which is useful for the normal distribution, is often inadequate to explain the dependence between extremes of multivariate time series so that the coefficient of upper tail dependence is usually preferred. In the case of a bivariate series, this coefficient measures the conditional probability of one component of the series exceeding a given quantile, provided that the other component exceeds the same quantile, as this quantile tends to one.8 If the coefficient of upper tail dependence equals zero, the two components of the bivariate time series are asymptotically independent; otherwise, they are asymptotically dependent. The extremal dependence structure is typically different from the dependence we find at the centre of the distribution, since asymptotical independence can occur even if the components of the distribution are not linearly independent. For example, a bivariate normal distribution with correlation ρ ≠ 1 is asymptotically independent, although it is not linearly independent unless ρ = 0.

Multivariate time series can be modelled explicitly via multivariate EVT (MEVT) using the block maxima method or the threshold exceedances approach. In both cases, copula functions play a fundamental role. Recall that a k-dimensional copula is a joint distribution function defined on [0, 1]k with standard uniform marginal distributions.9 According to Sklar's well-known theorem, for every joint distribution function F with margins F1, …, Fk, there exists a copula C such that F(x1, …, xk) = C(F1(x1), …, Fk(xk)), for all inline image. Moreover, if the margins are continuous, C is unique. Thus, copulas allow the dependence structure of a joint distribution to be disentangled from its marginal behaviour.10

As in the univariate setting, the multivariate block maxima method is based on the partition of a given sample of independent and identically distributed observations, with joint cumulative distribution F, into subsamples (or blocks), and it models the asymptotic distribution of the (appropriately rescaled) maxima of these blocks. In the multivariate case, however, the observations are vectors instead of scalars and the distribution of (rescaled) component-wise taken maxima is modelled with EVT. If this distribution converges to a non-degenerate joint cumulative distribution H as the amplitude of the sample grows to infinity, then we say that F is in the maximum domain of attraction of the multivariate extreme value distribution H.

In fact, a deeper connection can be established between univariate and multivariate block maxima methods. A fundamental result of multivariate EVT indicates that the asymptotic distribution of multivariate extremes is a combination of the asymptotic distributions of extreme values of the margins (as described in the univariate framework) and the asymptotic behaviour of the copula. More precisely, a joint cumulative distribution function F, with copula C and continuous margins Fj, j = 1, …, k, is in the maximum domain of attraction of H, with copula C0 and margins Hj that follow GEV distributions, if and only if:

  1. Fj is in the maximum domain of attraction of Hj, for all j = 1, …, k;
  2. C is in the copula domain of attraction of C0, meaning that, for all (u1, …, uk) ∈ [0, 1]k, inline image.11

The copula C0 is called an extreme value copula and can be proven to satisfy the equality inline image for all t > 0 and u ∈ [0, 1]k.12

The threshold exceedances method can similarly be extended to the multivariate framework by decoupling the margins from the dependence structure embedded in the copula function. The aim is to model the upper tail of the joint distribution F, computed at points exceeding some vector of high thresholds u = (u1, …, uk). Under the assumption that F is in the maximum domain of attraction of a multivariate extreme value distribution, for all xjuj (j = 1, …, k) F(x) = C(F1(x1), …, Fk(xk)) can be approximated by inline image, where C0 is the limiting extreme value copula of C and the margins inline image, j = 1, …, k, have functional forms derived from GPDs.13

For both the multivariate block maxima method and the multivariate threshold exceedances method, model parameters can be estimated via maximum likelihood. In the threshold exceedances approach, the likelihood function is usually computed with censored data, meaning that for each observation x, the only relevant information carried by the components xj that are below the corresponding threshold uj is exactly that xj < uj, irrespective of their actual values.14

The multivariate methods outlined in this section, while inheriting the issues discussed in Section 'A Primer of EVT' concerning their univariate analogs, additionally suffer from a major problem relating to extreme value copulas. Indeed, in general, we do not know the copula C of the distribution F at hand, nor its limiting copula C0. Therefore, in practice, C0 is simply approximated by means of some parametric copula function that is manageable enough for implementation, like the logistic copula model,15 and consistent with the main features of F (e.g. a copula function allowing for asymptotic dependence when the data are asymptotically dependent).

4. Testing Distributional Assumptions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

Pioneering studies by Mandelbrot (1963, 1967) and Fama (1963, 1965) questioned the adequateness of the assumption of normality for the distribution of financial time series and were thus either ignored or regarded as heterodox (the normality assumption being successfully employed in the theories of Markowitz and Sharpe and, later, in the option pricing model of Black and Scholes). There is now considerable evidence that the normal distribution is too thin-tailed to fit financial data from many different markets adequately. Extreme observations are much more likely to occur than under the normality assumption. This suggests that distributions with power law tails are better suited to modelling real data than the Gaussian exponentially decaying tail.

A close study has been made of fat-tailed distributions in order to find appropriate ones to fit financial data.16 The main candidates are the Student t distribution and stable laws, and a lively debate on the advantages of one or the other has ensued:

  • The Student t distribution combines fat tails (observed in practice) with the existence of variance (assumed in many economic and financial models);17 skewed versions of the t distribution have also been proposed to take into account the empirical evidence of skewness in several financial time series.
  • Stable laws, first suggested by Mandelbrot as an appropriate modelling tool, allow for both heavy tails and skewness; furthermore, for any fixed value of the characteristic index (one of the parameters of the characteristic function of stable laws), they form a closed family with respect to addition; on the other hand, non-normal stable laws have infinite variance.

The issue is not merely theoretical as different distributional assumptions can lead to dramatically different estimates of expected risk.18 However, it is difficult to make a comparative study of the two distributional assumptions: stable and t distributions are not nested and it is difficult to construct a direct test that rejects one in favour of the other.

Since these different distributional assumptions are strictly related to the amount of heaviness in the tails, EVT looks very promising. Indeed, it offers a remarkably elegant and effective solution to the problem of directly comparing the two families: it nests both models, since stable laws and the Student t are both in the maximum domain of attraction of a Fréchet-type extreme value distribution, with the tail index α equal to the characteristic index of the stable law and to the degrees of freedom of the t distribution, respectively.

In practice, therefore, one can proceed as follows:

  1. assuming that the distribution of the data belongs to the domain of attraction of a Fréchet-type extreme value distribution, and estimates the tail index α of the distribution (or estimates the shape parameter ξ, which immediately yields the tail index as its inverse);
  2. if α is significantly lower than two, then conclude in favour of the stable law assumption; on the contrary, if α is greater than two, the Student t assumption is more appropriate.

This EVT approach was first employed by Koedijk et al. (1990) to evaluate how heavy-tailed were the bilateral foreign exchange rates of the European Monetary System. Using weekly data on the spot exchange rates of eight currencies quoted against the US dollar, roughly during the period 1971–1987, the authors find point estimates for α that are below two for most exchange rates and at 5% significance the hypothesis α < 2 is never rejected, while the hypothesis α > 2 is rejected on a few occasions. On the other hand, distributional assumptions of normal and of a mixture of normal distributions are definitely rejected. The authors also study the stability of α with respect to the creation of the European Monetary System and the effects of aggregation over time.

Lux (2000) uses the approach described above, based on the estimation of the tail index, to obtain a definite conclusion about the finiteness of the second moment of the distribution of German stock returns as previous papers record diverging results. He concludes that when using algorithms for endogenous selection of the optimal cut-off, there is evidence for heavy tails with finite variance.

Vilasuso and Katz (2000) use daily aggregate stock market index prices for several countries from 1980 to 1997 in order to test the hypothesis that the returns follow a stable distribution. They conclude that there is scant support for this assumption, while Student t and ARCH processes seem to be more suitable models.

Longin (2005), applying the method to logarithmic daily percentage returns on the S&P 500 index for the period 1954–2003, draws a similar conclusion, rejecting both the normal and the stable law hypotheses but not the Student t distribution and ARCH processes and thus providing reasonable tools for unconditional and conditional modelling of the US stock market.19

All in all, the main contribution of EVT to this stream of research is the possibility of performing a direct comparison between the Student t distribution and stable laws. In this respect, the evidence of the empirical studies mentioned seems to point in favour of the Student t as far as stock markets are concerned.

5. Market Risk: VaR and ES Estimation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

While in the first decade after EVT was recognized as a valuable tool of financial studies (roughly, 1990s) its main financial application was in testing different distributional assumptions, with the new millennium VaR calculations have become pre-eminent. There are two main reasons for this: on the one hand, the growing interest in risk management and the emergence of VaR as the standard measure of risk confirmed and reinforced by the resolutions of the Basel Committee; and on the other hand, the fact that both VaR and EVT are concerned with the tails of the distribution, disregarding its central part, so that EVT seems a very natural approach to VaR estimation.

This section first reviews the definitions and main properties of VaR and ES, the main alternative to VaR (Section 'VaR and ES: Definition and Main Properties'), recalling how both measures can be estimated via EVT (Section 'An EVT Approach to VaR and ES Calculation'). There follows a comparison of EVT with standard approaches to VaR calculation (Section 'Comparative Studies') and the presentation of some useful methods of back-testing (Section 'Back-testing').

5.1 VaR and ES: Definition and Main Properties

The standard references on VaR are the reviews by Duffie and Pan (1997) and Pearson and Smithson (2002) and the books by Dowd (1998, 2002) and Jorion (1997). Major criticisms of VaR as an effective risk measure have been advanced by Daníelsson (2002), Artzner et al. (1999) and Acerbi (2004), who suggest more suitable measures (ES and coherent risk measures in general).

After the 1987 crash, several companies introduced risk measures similar to VaR, until J. P. Morgan brought VaR to the attention of a wider audience in the mid-1990s and the Basel II Accord established it as the basis for market risk measurement in financial institutions.

VaR is defined as (the opposite of) the minimum loss that can occur at a given (high) confidence level, for a pre-defined time horizon. Regulatory norms for market risk set the time horizon (the period in which a bank is supposed to be able to liquidate its position) at 10 days and the confidence level at 99%.

In symbols, the VaR at the α confidence level, where inline image, for a time horizon of 1 time unit (the same unit that determines the frequency of the data employed, e.g. one day) is given by

  • display math

where X is a random variable standing for some random return and inline image is the generalized inverse of the cumulative distribution function of X.20 Note that owing to the minus sign in its definition, VaR represents losses as positive amounts. When directly considering a time series Y of positive losses, the previous definition is replaced by

  • display math

The main advantages of VaR are:

  • It is easy to understand and interpret from a financial viewpoint, thus providing an effective tool for risk management purposes.
  • It focuses on the tails (actually, on one tail) of the distribution only, thus capturing the occurrence of huge rare losses.

However, VaR has at least two major drawbacks:

  • It deals with the cut-off between the centre and the tail of the distribution, rather than with the tail itself, since it provides information about the minimum loss that will occur with a certain low frequency 1 − α, completely disregarding what happens farther in the tail.
  • It does not seem to behave entirely as a sensible risk measure, as examples can be provided in which the VaR of a portfolio of investments is greater than the sum of the VaRs of the single investments, contradicting the acknowledged role of diversification in lowering the level of risk21 (mathematically, this means that VaR is not a convex function, i.e. it can happen that inline image for some portfolio weights w1, w2 ∈   ]0, 1[).

A great variety of approaches to VaR computation have been proposed in the literature. The main alternatives can be divided roughly into three families:22

  • non-parametric methods, the most common being historical simulation (HS);
  • semi-parametric methods, such as EVT and CAViaR23;
  • parametric methods, such as the RiskMetrics approach and GARCH models.

HS is based on the empirical distribution obtained from the data: the α confidence level VaR is identified with (the opposite of) the 1 − α empirical quantile. While HS avoids the problem of choosing a distribution for the time series considered, it does have two important shortcomings:24 it cannot predict the occurrence of future extreme events, unless they have some ancestor in the time series, and it cannot provide reliable estimates of high confidence level VaR, which would require too many data.

By contrast, fully parametric methods use an explicit distributional assumption to model the time series. The model is fitted to the data and, by means of the estimated parameters, allows the VaR to be calculated at any specified confidence level. The main problem with this approach is that the actual distribution of the data at hand is unknown (often a major problem that needs to be tackled). Parametric methods can therefore lead to inaccurate estimates of VaR, potentially yielding either severe losses (if the actual risk level is underestimated) or conservative positions (in the case of overestimation).

Semi-parametric methods try to address this issue by combining the advantages of non-parametric and parametric methods. In this sense, EVT can be regarded as a semi-parametric approach to VaR as it avoids imposing a given distribution on the whole set of data (thus reducing model risk) but focuses on one tail, trying to model its asymptotic behaviour (and therefore not running into the main difficulties of wholly non-parametric methods either).

Irrespective of the approach chosen to calculate VaR, it is generally impossible to avoid the two flaws mentioned above, i.e. the inability to control what happens far away in the tail and the lack of convexity. Further risk measures have been proposed as a solution, the best-known being ES.

ES at the confidence level α is defined as the average of the VaRs which are greater than inline image,25 i.e.

  • display math

Thus, by definition, ES at the confidence level α takes account of what happens in the tail, beyond the corresponding VaRα. Moreover, it can be shown that ES is convex,26 making it a valuable candidate for a sensible risk measure.27

5.2 An EVT Approach to VaR and ES Calculation

The definitions of VaR and ES are essentially concerned with extreme quantiles of the distribution and EVT provides a natural solution to the problem of their estimation.

A simple EVT estimate of VaR can be obtained by applying the following algorithm:28

  1. Assuming that threshold exceedances of the data are asymptotically distributed according to a GPD, fix a high threshold u and fit the GPD to the exceedances over u, obtaining estimates inline image and inline image of the shape parameter ξ and the scale parameter β, respectively, that enter the expression of the GPD.
  2. Estimate the tail probability by means of inline image, for xu, where inline image, while inline image is the ratio between the number Nu of exceedances observed in the given sample and the size N of the sample itself and stands for an estimator of inline image.
  3. Invert the previous formula to obtain an estimate of the α quantile, i.e.
  • math image

From the given estimate of inline image, one can obtain an estimate of inline image as well, namely

  • math image

The previous arguments rely on the application of a GPD-based approach to model threshold exceedances. If, in turn, the other main parametric method is employed, i.e. the block maxima method, inline image, can be estimated by means of the αn confidence level VaR for the block maxima Mn, taking into account the asymptotic equality inline image.29 Otherwise, in a non-parametric approach, using the Hill estimator VaR can be estimated by means of30

  • math image

where Xk denotes the kth-order statistic (the same one chosen as a cut-off when estimating the shape parameter ξ with the Hill estimator) and α indicates the confidence level at which VaR is calculated.

The previous EVT approaches to VaR and ES estimation are unconditional, as they do not take into account the dependence structure of the time series under consideration. The use of unconditional measures is in order for low frequency data, where the effect of stochastic volatility is less perceivable than with daily (or intra-day) data. Moreover, it is practically unavoidable when dealing with multivariate time series with a large number of components. In this case, the modelling of volatility may be computationally expensive.

In numerous cases, however, conditional estimates of risk measures are of interest, as they explicitly model the stochastic nature of conditional volatility, yielding a more accurate assessment of risk. Indeed, they are generally much more precise than their unconditional counterparts, although they also show an undesirable variability over time that prevents their use as a benchmark for regulatory purposes.

When conditional VaR has to be estimated, i.e. VaR conditional on past information, considering the possible heteroskedasticity of the data, appropriate techniques have to be integrated in the previous scheme. For instance, a common and effective approach is based on the two-step procedure advocated by McNeil and Frey (2000), who first fit a GARCH-type model to the data and then apply EVT to the standardized residuals.31 VaR is then computed for the standardized residuals according to the unconditional scheme presented above, and substituting it for the unconditional quantile Zt in the equation Xt = μt + σt Zt governing the dynamics of returns a conditional estimate of VaR is obtained.

5.3 Comparative Studies

The first influential studies applying EVT to VaR computation are those of Pownall and Koedijk (1999), Daníelsson and de Vries (1997b, 2000), Longin (2000), McNeil and Frey (2000) and Neftci (2000).

Pownall and Koedijk (1999), studying the crisis of Asian markets, provide a conditional approach to VaR calculation based on EVT and find that it yields an improvement in VaR estimation compared with the technique employed by RiskMetrics.

Longin (2000), working with data from the S&P 500 index, draws a comparison between EVT and four standard methods of calculating VaR, namely HS, modelling with normal distribution, modelling with GARCH processes and the exponentially weighted moving average (EWMA) process for the variance employed by RiskMetrics. In the computation of VaR in an EVT setting, the paper first considers long and short positions, for which the lower and the upper tail of the distribution, respectively, are of interest.

McNeil and Frey (2000) apply their two-step procedure to obtain conditional VaR and ES estimates for S&P 500 and DAX indices, BMW share price, the US dollar/British pound exchange rate and the price of gold.32 Comparing the estimates provided by the two-step method with those obtained from unconditional EVT, GARCH modelling with conditional normal innovations and GARCH modelling with conditional t innovations, they conclude that, on the whole, the conditional approach to VaR provided by EVT outperforms the others.33

Neftci (2000) compares the EVT approach to VaR calculation with the standard one based on the normal distribution, dealing with several interest rates and exchange rates. He concludes that ‘the results, applied to eight major risk factors, show that the statistical theory of extremes and the implied tail estimation are indeed useful for VaR calculations. The implied VaR would be 20% to 30% greater if one used the extreme tails rather than following the standard approach.’34

Much more research comparing EVT-driven estimates of VaR to other approaches (both standard and new ones) has been done since these seminal papers appeared. The reader is referred, for instance, to Bao et al. (2006), Bali (2007), Bekiros and Georgoutsos (2005), Brooks et al. (2005), Gençay and Selçuk (2004), Ho et al. (2000), Kuester et al. (2006), Lee and Saltoğlu (2002), Manganelli and Engle (2004) and Tolikas et al. (2007).

Overall, an analysis of the literature shows that consensus has been reached on the following main conclusions:

  • EVT-based estimates of VaR outperform estimates obtained with other methodologies for very high quantiles, namely for α ≥ 0.99;
  • the farther one moves into the tails, i.e. the greater α, the better EVT estimates are;
  • when α < 0.99 (α = 0.95, typically), evidence is mixed and EVT can perform worse than other techniques.

Thus, we are prompted to question the absolute reliance that some preliminary studies placed on EVT methods and recognize both their advantages and their limitations.

Daníelsson and de Vries (2000) obtain an important result in this respect by comparing two different EVT estimators with HS and RiskMetrics on data from US stocks. They conclude that, at a 95% confidence level, RiskMetrics performs best because, when estimating VaR0.95, it does not go far enough in the tails for EVT to work efficiently. On the other hand, from the 99% confidence level upwards, EVT is more appropriate as extreme events are effectively involved.

The results of Gençay and Selçuk (2004) point to a similar conclusion. They compare EVT with HS and normal and Student t distribution modelling using daily stock market data from several emerging economies. Tolikas et al. (2007), studying the DAX index, reach the same conclusion, although they find that when a sufficiently large amount of data are available, HS can yield comparable results (which is somewhat at variance with current opinion on HS).

Bekiros and Georgoutsos (2008a) add an important qualification to the previous conclusions. Their findings from a study of market data summarized by the Cyprus Stock Exchange general index agree with the ones just described. However, when they turn to daily returns on the US dollar/Cyprus pound exchange, the performance of EVT methods is much worse than in other studies, firmly ruling out EVT as a viable candidate for estimating ‘low’ confidence level (α < 0.98) VaR. The authors put this down to the relatively scant tail fatness of the time series of exchange returns at hand compared with stock market data.

This is interesting because it highlights a feature that should be underscored in any comparative study, namely the (possible) dependence of the outcome of a comparison on the kind of data considered. The vast majority of papers applying EVT to finance employ time series from the stock market. The second most frequent source of data is probably exchange rates, but there are articles dealing with almost any kind of data, from equity returns and interest rates35 to energy and commodity market data36 and up to credit derivatives data.37 Most of these papers converge towards the conclusions described.

Note, furthermore, that the previous conclusions roughly hold for both conditional and unconditional VaR estimates. The extensive analysis of Kuester et al. (2006) shows that unconditional models for VaR can actually perform quite well if α = 0.99, while yielding unacceptable estimates for lower values of α (0.975 and 0.95). This is the case not only for EVT, but also for HS and a skewed t-based parametric approach, while normal and (symmetric) t distributions perform even worse. These findings point to the need for a conditional approach to VaR estimation. Comparing 13 conditional models on data from the NASDAQ Composite Index, the authors conclude that the two-step EVT procedure and filtered HS (FHS),38 both with normal (N-EVT, for short) and skewed t (ST-EVT) innovations for the GARCH filter, ‘are always among the best performers among all conditional models and for all values of 0 < λ < 0.1 [0.9 < α < 1, in our notation]. Moreover, the ST-EVT delivers virtually exact results for all 0 < λ < 0.1 [0.9 < α < 1], while N-EVT is competitive for 0 < λ < 0.025 [0.975 < α < 1] and then weakens somewhat as λ increases towards 0.1 [α decreases toward 0.9].’39

5.4 Back-testing

In order to keep things as straightforward as possible and to highlight common paths and conclusions, in the previous section, we omit to address an important issue which is now developed.40 Briefly, studies comparing different techniques for VaR estimation have to choose some performance criterion on which to rank them.

The most common criterion employed with methods for unconditional VaR estimation is the violation ratio.41 This method is described in detail by Kupiec (1995), Christoffersen (1998), McNeil and Frey (2000) and Bali (2007). A violation is said to occur at time t + 1 if the observed realization xt+1 of the random variable X is greater than the estimated Value-at-Risk inline image.42 Tests relating to the violation ratio can be carried out for both in-sample and out-of-sample performance of VaR estimates. In practice, in order to test, for instance, the out-of-sample performance

  1. consider a rolling window of n observations {xtn+ j, …, xt+ j−1}, for j = 1, …, p, and calculate inline image on the basis of the data in the corresponding time window;
  2. consider then the sequence of p observations {xt+1, …, xt+ p} and define a counter of overall violations occurring in this sample by inline image, where IA denotes the indicator function of a set A; the violation ratio is then defined as κ = V/p (number of violations occurring in a sample of p observations);
  3. test the null hypothesis that κ = 1 − α, exploiting the fact that under this hypothesis, the probability of observing V violations in a sample of p realizations can be computed by means of a binomial distribution with parameters p and 1 − α.

A suitable test, proposed by Kupiec (1995), to check whether the equality κ = 1 − α is statistically tenable is the likelihood ratio statistic

  • display math

which is known as the test for unconditional coverage and is asymptotically χ2 (1) distributed.43

Because the unconditional coverage test does not take into account the volatility clustering characterizing financial time series, it can be a misleading criterion for the accuracy of VaR estimates if considered on its own.

Another test, the test for conditional coverage developed by Christoffersen (1998), considers both unconditional accuracy and independence of violations (since unconditional VaR estimates tend to exhibit a violation pattern which is flawed by clustering effects) and it reads

  • display math

where LRuc denotes Kupiec's statistic and LRind stands for another likelihood ratio statistic, devoted to testing the null hypothesis of serial independence against the alternative hypothesis of Markov dependence of the first order. Since LRind converges in distribution to a inline image, LRcc converges to a inline image distribution.

Unconditional and conditional coverage tests are the back-testing methods most frequently used in the reviewed papers. The major problems with these tests are their limited power (i.e. their poor ability to minimize type II errors)44 and the fact that they consider only the frequency of tail losses, disregarding their magnitude. During the 1990s, other useful back-testing procedures were introduced, which can be subsumed in roughly two main classes (Dowd, 2002): tests based on the size of losses and procedures based on forecast evaluation.

The former class includes the back-tests proposed by Crnkovic and Drachman (1995, 1996), which test the difference between the profit and loss distribution predicted by the risk model and the empirical profit and loss distribution.45 By suitably mapping the empirical profit and loss distribution to the quantiles of the predicted distribution, two hypotheses are tested, namely that the reclassified observations are uniformly distributed and that they are independent of each other. Berkowitz (2001) further transforms the reclassified observations to obtain a normal distribution under the null hypothesis. Thus, both the distributional and the independence assumptions can be tested at the same time by means of a single, powerful likelihood ratio test.

Forecast evaluation approaches, introduced by Lopez (1998, 1999), do not provide econometric tests, but allow a ranking of different risk models according to performance as measured by some loss function. They can therefore be used even if scant data are available, unlike the previous approaches to back-testing. Moreover, different specifications of the loss function can be chosen, reflecting different concerns with the risk models at hand. For instance, Lopez suggests a constant loss function that takes into account only the frequency of tail losses, or a quadratic specification that also weighs the magnitude of tail losses.46 Blanco and Ihle (1998) propose another useful specification to match the interest in the size of losses, one which uses a loss function that lends itself more naturally to an economic interpretation.47

6. Asset Allocation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

A portfolio selection problem based on the maximization of returns subject to risk constraints can be formally written as

  • display math(1)

where wi are the unknown weights of the portfolio to be determined, μ w is the expected value of the portfolio, ρj are risk measures and Rj are upper bounds on risk. When j = 1 and inline image, we recover the classical mean variance framework of Markowitz (1952).

As Bensalah (2002, p. 5) points out ‘there is no final answer or optimal measure of risk’, the optimality of a given allocation being conditional on the actual correspondence of its underlying assumptions to the investor's risk preference. A plausible profile of risk preference, alternative to the one based on variance, is the safety-first criterion, a concept introduced by Roy (1952) and developed by Arzac and Bawa (1977), which is based on a constraint limiting downside risk. As Jansen et al. (2000) argue, a similar profile is both relevant in practice, given that circumstances can impose an asymmetric treatment of upside and downside risk, and psychologically sensible, as a lot of experimental evidence for loss aversion is available.

The safety-first approach to portfolio selection calls for accurate estimates of the probability of failure. Insofar as failure can be considered a rare event, EVT may provide suitable tools for accurate calculations. Indeed, it has been used in the context of portfolio selection with limited downside risk by Jansen et al. (2000). Faced with the problem of choosing between investing in a mutual bond fund or a mutual stock fund, the authors find that an assumption of tail fatness is plausible and employ EVT (precisely, the Hill estimator) to calculate the risk associated with each portfolio. On the same lines, Susmel (2001) uses EVT to model the tails of Latin American emerging markets and studies the effects of diversification for a US investor including those markets in his portfolio, based on a safety-first criterion.

As remarked by Hyung and de Vries (2007), downside risk criteria for portfolio selection, like the safety-first approach, often yield corner solutions, i.e. portfolio combinations in which only the least risky assets are chosen (e.g. bonds versus stocks). The authors argue that this is due to a focus on a first-order asymptotic expansion of the tail distribution of stock returns, while a second-order expansion provides more balanced portfolio combinations, in which some proportion of risky assets appears in the optimal mix. Hyung and de Vries (2007) present empirical evidence of this theoretical insight by computing optimal portfolios for the problems considered using the new approach on the data of Jansen et al. (2000) and Susmel (2001) and show that in this way, corner solutions appear less frequently.

The framework considered in this section extends the use of EVT for VaR calculation to the problem of portfolio selection, i.e. problem (1) with risk measure ρ given by VaR.48 In this mean-VaR setting, Consigli (2002), for instance, solves the asset allocation problem with data comprising the Argentinean crisis of July 2001, evaluating downside risk by means of both EVT and a jump-diffusion model. Both methods yield accurate estimates of the tail risk in the cases analysed; the latter seems to be more accurate, while the former provides more stable estimates.

A detailed analysis of the mean-VaR portfolio selection problem is given also by Bensalah (2002), who considers both the direct problem of maximizing returns, subject to a constraint on the VaR, that is problem (1) with j = 1 and inline image and its dual, i.e. the problem of minimizing VaR subject to a constraint which imposes a lower bound to the expected return. Moreover, Bensalah makes a comparison between different ways of calculating VaRα, namely HS, normal VaR and EVT. Considering a portfolio of two fixed-income securities (a 1-year treasury bill and a 5-year zero coupon bond) and several highly conservative values for the confidence level α (0.99, 0.999, and 0.9999), he concludes that HS and normal VaR yield the same allocation irrespective of the given α, investing the whole capital in the short-term security, both in the maximum return and in the minimum risk problem. This is no longer true with EVT-based calculations of VaR, which have two important features:

  • portfolio composition changes as the confidence level α grows;
  • the riskier asset (in terms of duration) is given non-zero weight.

More specifically, the weight given to the riskier asset is an increasing function of α; it is also greater in the minimum risk problem than in the maximum return one.

Bensalah concludes therefore that an EVT approach, being tailored on extreme events and responding differently to different confidence levels, is better suited to regulatory purposes and to coping with periods of market stress. In the latter circumstances, EVT could offer an alternative to the ‘flight to quality’, thus reducing behaviour risk (i.e. the risk of a financial crisis being exacerbated if everybody reacts the same way).

Finally, to make effective computation possible when dealing with multiple assets, Bensalah (2002) proposes an algorithm. The aim is to solve the problem of dimensionality in portfolio selection since, as the number of assets grows, a brute-force approach to the calculation of the optimal portfolio soon becomes intractable, even numerically. The proposed algorithm allows the number m of portfolios among which to search to be fixed in advance and it is based on m iterations of the following step: randomly generate the weight w1 of the first asset from a uniform distribution with support on the interval [0, 1] and set w2 = 1 − w1;49 then estimate the extreme value distribution of the resulting portfolio and the corresponding risk and return. After m iterations, m portfolios are obtained and the optimal one in the sample can be selected.50

Bradley and Taqqu (2004a) improve on this algorithm by proposing a two-step procedure, which uses, in the first step, Bensalah's sampling algorithm to generate a sensible starting point for the application of an incremental trade algorithm at the second step. Along the same lines as Bensalah (2002), Bradley and Taqqu (2004a) study the risk-minimization problem comparing normal and extreme value distributions in the computation of VaR and ES as risk measures. Making both simulation and empirical analyses, they find that:

  1. the optimal allocation is quantile-based, i.e. depends on α, confirming the findings of Bensalah (2002);
  2. at standard confidence levels (α = 0.975, 0.99), the optimal allocation can differ from the one obtained under the hypothesis of normal distribution, but the extra risk taken under the normality assumption is not particularly large, making it a viable alternative to EVT, especially given its ease of implementation; however, when moving to extreme quantiles (α = 0.999, 0.9999), the difference between the two approaches can no longer be ignored;
  3. in the simple case of two assets, as α changes, the weight given to each asset follows strictly the ratio of marginal risks between the two corresponding markets; specifically, under asymptotic independence, the optimal portfolio diversifies more into the riskier asset for very high confidence levels α.

The last point leads to the question of diversification and the issue of asymptotic dependence, as the driving factor is the ratio of marginal risks rather than linear dependence.51 Diversification is widely acknowledged as a leading rule of portfolio selection (several hedge funds successfully employ it), but the benefit of diversification can be seriously hampered when the markets used to diversify risk away exhibit some form of dependence. In agreement with Poon et al. (2004), Bradley and Taqqu (2004a) find for 12 international equity markets that most of them are asymptotically independent and the few cases of asymptotic dependence are related by geographical proximity.

This explains why a portfolio selection problem, which in principle is a multivariate problem, can reasonably be studied by means of univariate techniques (applied to some function of the allocation vector, typically the portfolio return).

Bradley and Taqqu (2004b) make a direct comparison of the performance of univariate and multivariate techniques for an asset allocation problem, concluding that multivariate EVT yields accurate results if the copula employed allows for asymptotic independence, reflecting a chief characteristic of the data at hand. Multivariate EVT models with a copula that prescribes asymptotic dependence instead overstate the risk of a properly diversified portfolio.52 The use of a univariate approach, which is easier to implement, therefore seems advisable particularly when extreme quantiles are not of interest.

7. Dependence Across Markets: Correlation and Contagion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

The issue of dependence among financial time series discussed in connection with the problem of portfolio selection can be addressed specifically when studying correlation and contagion among markets. Two approaches have been taken, one to study the dependence between different countries for a given financial sector, the other to look also at the dependence among different financial markets of the same country.

An excellent example of the former approach is provided by Longin and Solnik (2001), who employ multivariate EVT to test the validity of the common belief that correlation between stock markets increases during volatile periods. They use multivariate threshold exceedances, modelling dependence by means of the effective, though parsimonious, logistic model, in which a single parameter accounts for dependence and this parameter is related to correlation by a simple formula. Fitting a bivariate model to pairs of monthly equity index returns for the US and each of the other four G5 countries (United Kingdom, France, Germany and Japan), the authors find that extreme correlation is statistically different from the correlation estimated by means of a multivariate normal model. Moreover, the correlation pattern of threshold exceedances is asymmetric, as it depends on the chosen threshold both in size and in sign. Indeed, correlation of exceedances is increasing in the absolute value of the threshold, if the threshold is negative; otherwise, it is decreasing. Longin and Solnik therefore conclude that ‘[ … ] the probability of having large losses simultaneously on two markets is much larger than would be suggested under the assumption of multivariate normality. It appears that it is a bear market, rather than volatility per se, that is the driving force in increasing international correlation’.53

Bekiros and Georgoutsos (2008b) make a complementary study of the correlation of extreme returns between seven Asia-Pacific stock markets and the US. They use a bivariate logistic copula to model threshold exceedances, like Longin and Solnik (2001), in order to produce a ranking of Asia-Pacific countries in three broad categories of risk and check whether these countries form a distinct block with respect to Europe and the US. They find that this is not so and that US investors can benefit from diversifying their portfolios with assets from Asian countries. This remains true even during crisis periods, as a sensitivity analysis conducted by estimating the model with two separate sets of data, one including the 1987 crash and the other subsequent to it, yields similar results. In particular, the fact that these results are close to the correlation estimated via standard unconditional and conditional (GARCH model) methods provides evidence against contagion during the crises of the 1980s and 1990s.

The decision of these authors to measure extreme dependence by means of the correlation coefficient is anyway questionable. Poon et al. (2004) instead use the coefficient of upper tail dependence and a complementary measure developed by Ledford and Tawn (1996) to assess dependence across markets.54 More precisely, they analyse daily returns on the stock indices of the G5 countries by means of bivariate EVT. As non-zero estimates of the coefficient are obtained only for 13 out of 84 possible pairs of countries, the assumption of asymptotic dependence is inappropriate in the vast majority of cases, resulting in an overestimation of systemic risk in international markets.

For policy purposes, particular interest attaches to studies of currency crisis contagion and cross-country dependence in the banking sector.

The first issue is considered for instance by Haile and Pozo (2008), who use EVT to set appropriate thresholds to discriminate between crisis and normal periods and who conclude that currency crises, as experienced in the 1990s, are indeed contagious and that the main route of contagion is via trade. Moreover, regional proximity also plays a significant role (neighbourhood effects channel). Similar results on the contagious nature of currency crises are obtained by Garita and Zhou (2009), who indicate EVT as a suitable tool to detect contagion. Using an exchange market pressure (EMP) index to measure currency crises, as is standard in the literature, they choose as a dependence measure a quantity introduced by Huang (1992) and first applied by Hartmann et al. (2004), namely the expected value of the number k of extreme events (the EMP index of k countries exceeding a high VaR threshold, in the case of Garita and Zhou), conditional on the fact that at least one such event has occurred (k ≥ 1). Their findings rule out currency crisis contagion as a global phenomenon, confirming it instead to be a regional one. The authors also conclude that financial openness and a monetary policy aimed at price stability can reduce the probability of currency crises.

As far as the issue of dependence in the banking sector is concerned, an important reference is Hartmann et al. (2005), who study banking sector stability, considering separately the daily returns on stock prices (during the period 1992–2004) of two groups of 25 major banks in the US and the Eurozone. The main tool of analysis is multivariate EVT (reduced, with appropriate manipulation, to a univariate setting), and dependence is studied by means of two different measures: one that accounts for bank contagion risk, considering multivariate extreme spillovers; the other measuring aggregate banking systemic risk with reference to a benchmark (e.g. stock market indices). The first indicator provides evidence of bank spillover risk being lower in the Eurozone than in the US, probably owing to weak cross-boarder linkages of European countries. The second measure, on the other hand, yields similar effects of macro shocks on both European and American banking sector stability. Moreover, structural stability tests detect an increase (albeit very gradual) in systemic risk, both for Europe and the US, in the second half of the 1990s.

Increasing cross-country interconnections during the first years of the new millennium have been detected by Chan-Lau et al. (2007) in their study on contagion risk related to the role of London as a hub of the world banking sector, though ‘home bias’ is still crucial, i.e. the risk of contagion among local banks is high compared with cross-border risk. Based on a variant of multivariate EVT, the authors provide a detailed account of contagion risk among the major UK banks and with respect to foreign banks (e.g. Barclays is the most prone to contagion from foreign banks, while HSBC provides the highest contagion risk towards them).

The work of Pais and Stork (2009) also deals with contagion in the banking sector. Following the recent financial crisis, the authors inquire into the contagion risk between Australian and New Zealand banking and real estate sectors. Using EVT, they find that, as a consequence of the financial crisis, the probability of extreme negative returns has increased in both sectors, as well as the probability of inter-sector contagion.

This takes us to the second aspect of dependence, namely dependence across different kinds of financial activities (cross-asset or inter-sector dependence). An interesting insight in this field is provided by Hartmann et al. (2004), who study contagion risk between stock markets and government bond markets of the G5 countries. Instead of estimating full-sample correlation, which is biased towards normality and is not the actual quantity of interest, they focus directly on the expected number of crashes in a given market (stocks or bonds), conditional on the event that one crash has occurred. Defining crash levels on an historical basis (20% and 8% losses for stocks and bonds, respectively, based on weekly data over the period 1987–1999), the authors use a non-parametric approach to multivariate EVT to estimate the conditional expected value. Extreme cross-border linkages within the same asset class turn out to be stronger for stock markets than for bond markets, while bivariate stock/bond co-crashes, though not displaying particularly high probabilities per se, are considerable when compared with the unconditional univariate probabilities of crashes in a single market and, in any case, they are higher than the probabilities of extreme co-movements estimated with a multivariate normal distribution.

Finally, Bekiros and Georgoutsos (2008a) study the dependence between stock and foreign exchange markets in the case of Cyprus. In order to dissociate the correlation structure from the marginal distributions and model the asymptotic dependence, they use a tail dependence measure for the return exceedances based on the logistic copula. Using multivariate EVT to estimate the dependence of extreme returns, they conclude that low correlation can be found between stock market daily returns and exchange rates with US dollar, even during crisis periods. The bear-market conditions characterizing the period covered by the data used for back-testing do not make any difference to the estimated extreme correlation, suggesting that the results of Longin and Solnik (2001) do not apply in this context.

8. Further Applications

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

Four main applications of EVT to finance are discussed in the foregoing pages. Without taking the investigation further on this occasion, it is nonetheless interesting to consider briefly some other possible applications of the theory.

Section 'Dependence Across Markets: Correlation and Contagion' examines the question of contagion in relation to currency crises, although detecting and dating currency crises is itself a major issue. EVT has been employed for this purpose by, for instance Pozo and Amuedo-Dorantes (2003), Lestano and Jacobs (2007) and Ho (2008). Pontines and Siregar (2007, 2008) provide some important critical observations and Haile and Pozo (2006) study the connection between the exchange rate regime and the probability of currency crises occurring, concluding that the announced regime has an impact, while the observed regime does not.

Another important field of application on which attention is converging is the measurement of operational risk, mainly because it has been explicitly taken into consideration since the Basel II Accord. A major advantage of EVT in this field is its ability to model extreme events, such as large unexpected losses caused by human error. On the other hand, a serious limitation is the scarcity of data, which hinders asymptotic theories like EVT. Work in this area has been done by Moscadelli (2004), Dutta and Perry (2006), Allen and Bali (2007), Jobst (2007), Abbate et al. (2008), Chapelle et al. (2008) and Serguieva et al. (2009). A critique of the use of EVT in quantifying operational risk is provided by Sundmacher and Ford (2007).

Finally, there are a few other analyses, which cannot be subsumed in the previous taxonomy. For instance, several articles are devoted to setting optimal margin levels for futures contracts with different underlyings, for example Longin (1999) (silver futures contracts traded on COMEX), Dewachter and Gielens (1999) (NYSE composite futures), Cotter (2001) (stock index futures selected from European exchanges) and Byström (2007) (CDS index futures). Byström (2006) uses EVT to calculate the likelihood of failure in the banking sector, comparing his outcomes with Moody's and Fitch ratings. Markose and Alentorn (2005) use the GEV distribution to model risk neutral probability and obtain a closed-form solution for the price of a European option.

9. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

The present paper surveys the main applications of EVT to finance. From a theoretical viewpoint, EVT shows some considerable advantages, which make it particularly suitable:

  1. It offers tools, with strong theoretical underpinnings, to model extreme events, which are of paramount interest in finance, especially in the context of risk measurement, given the importance of extreme events to the overall profitability of a portfolio.
  2. it provides a variety of such tools, ranging from non-parametric methods to point processes, thus guaranteeing a flexible approach to the modelling of extreme events, which can be adjusted to the particular features of the problem at hand.
  3. The fact that the vast majority of standard distributions, even though displaying very different tail behaviour, can equally be modelled by EVT increases flexibility.
  4. Furthermore, the flexibility and accuracy of modelling are enhanced by the fundamental characteristic of EVT, namely its exclusive consideration of the tail of the distribution of the data, disregarding ordinary observations (the centre of the distribution).
  5. They are also enhanced by the capability of EVT to model each tail of the distribution independently.
  6. Finally, the availability of parametric approaches allows for projections and forecasting of extreme events.

Some drawbacks, though, have to be pointed out as well:

  1. The most problematic one is probably the dependence of the parameters on the choice of the cut-off (i.e. the delimitation of the subsample employed to estimate the extreme quantiles), given that there is not yet complete agreement on how such a choice should be made.
  2. Moreover, the basic theory of extreme values assumes that the data are not serially correlated; when this assumption is violated, some alternative approaches are available, but there is no agreement on which is the most suitable.
  3. Multivariate EVT is admittedly not as straightforward as its univariate counterpart and can still encounter severe computational limitations.
  4. EVT involves an unavoidable trade-off between its asymptotic nature (always requiring large amounts of data) and its interest in extreme events (which, by definition, are rare); therefore, the choice and preparation of the dataset can be crucial when applying EVT.

Despite these drawbacks, EVT has aroused considerable interest in the literature on finance. Its most important application in this field, both for its role in financial regulation and for the number of contributions to the research, is in the estimation of quantile-based risk measures, such as VaR and ES. Many papers deliver comparative analyses of the accuracy of different methods for VaR calculation and they agree that EVT is a valuable candidate when calculating VaR at high confidence levels (namely equal to or greater than 99%). The great degree of accuracy displayed by EVT-based estimates of VaR for several different markets probably makes risk measurement one of its most important and well-acknowledged contributions in the field of finance.

One role of EVT that is related to its use for risk management is in asset allocation, associated with the concept of the ‘safety-first investor’. The importance of taking the investor's risk profile into account is permeating financial practice and, for investors who are particularly interested in avoiding extreme shocks, i.e. huge and rare losses, EVT offers a suitable tool to model them accurately.

Finally, portfolio selection naturally entails the consideration of a multivariate setting. In this setting, the problem of systemic risk and the issue of contagion across markets in extreme events are especially relevant. These topics have been highlighted by the recent crisis and deserve particular attention, as the dependence pattern in a multivariate time series can be different in normal times and under stress conditions, i.e. extremal dependence can differ from ordinary correlation. This has an impact on diversification effects and has to be explicitly modelled and taken into account. Multivariate EVT offers suitable statistical tools to do this.

The list of possible applications of EVT to finance is longer, of course, but these examples suffice to show the contributions that the theory can bring to the literature and practice of finance. Those contributions are based on the very definition of EVT, namely its ability to model accurately the distribution of extreme events, which are the main concern of modern risk management. This takes us back to the well-known maxim of DuMouchel quoted at the beginning of this paper and which is key to EVT: ‘Let the tails speak for themselves’.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References

I am grateful to Giuseppe Grande and Antonio Di Cesare for helpful comments and to Christine Stone for substantial help in the revision of a previous version. I would also like to thank two anonymous referees for suggestions that improved and enriched the paper. The usual disclaimer applies. The views expressed are those of the author and do not involve any responsibility of the Bank of Italy.

Notes
  1. 1

    It is well known that the two concepts do not necessarily coincide. Anyway, we will not discuss here the variety of definitions and interpretations, the notion of risk has undergone in the financial literature.

  2. 2

    See Section 'Market Risk: VaR and ES Estimation' for the definition of VaR and ES.

  3. 3

    ‘Whilst it is hard to determine the holding period of an average investor, we know that portfolio performance, and therefore the investor's outcomes, are dependent on a few days of trading. Average daily returns are near zero but we know on any one day that extreme and large scale returns can occur.’ (Cotter, 2007, p. 1).

  4. 4

    Given our focus on financial applications, we will not survey applications of EVT to other fields relevant to Economics, like auction theory.

  5. 5
  6. 6

    Introduced by Jenkinson (1955), the GEV distribution (or Jenkinson-Von Mises representation of extreme value distributions) reads Hξ,μ,σ(x) = exp {− [1 + ξ (x − μ)/σ ]−1/ξ}, for ξ ≠ 0 and 1 + ξ (x − μ)/σ > 0, while, for ξ = 0, it is Hξ,μ,σ(x) = exp {− exp (−(x − μ)/σ)}.

  7. 7

    The GPD (see Balkema and de Haan, 1974, and Pickands, 1975) reads Gξ,β(x) = 1 − (1 + ξ x/β)−1/ξ, if ξ ≠ 0, inline image, otherwise, where x ≥ 0 if ξ ≥ 0 and 0 ≤ x ≤ − β/ξ if ξ < 0.

  8. 8

    Formally, given two random variables X1 and X2 with cumulative distribution functions F1 and F2, respectively, the coefficient of upper tail dependence of X1 and X2 is inline image, provided that this limit exists and is finite, where Farrow denotes the generalized inverse of F, i.e. the function defined as inline image for all q ∈ [0, 1].

  9. 9

    See, for instance McNeil et al. (2005) for more details on copula functions.

  10. 10

    A similar way to separate the dependence structure from the margins is via dependence functions (see e.g. Reiss and Thomas, 1997). On the relation between copulas and dependence functions, see McNeil et al. (2005, Theorem 7.45).

  11. 11

    See Galambos (1987) and McNeil et al. (2005).

  12. 12

    See McNeil et al. (2005, Theorem 7.44).

  13. 13

    More precisely, inline image, with xjuj and ζj = 1 − Fj(uj).

  14. 14

    The contribution of observation x to the likelihood function is given by the partial derivative of inline image with respect to the components xj of x such that xjuj, evaluated at the point y = (y1, …, yk) defined as yi = xi if xiui, yi = ui otherwise. See McNeil et al. (2005), Longin and Solnik (2001, Appendix 'A Primer of EVT'), Bradley and Taqqu (2004b).

  15. 15

    The logistic (or Gumbel) copula model is inline image, with 0 ≤ α, δ ≤ 1 and γ ≥ 1.

  16. 16

    A reference book on this topic is Rachev (ed.) (2003).

  17. 17

    In this section, we always assume that the parameter ν (degrees of freedom) of the Student t distribution is greater than two, i.e. that this distribution has finite variance.

  18. 18

    Lux (1996) compares the expected frequency of daily returns exceeding 20% (in absolute value) for different estimates of the tail index α. For α = 1.7 (corresponding to a stable law), the expected frequency is once in 7 years, while α = 3.0 (Student t) yields once in 75 years.

  19. 19

    These conclusions corroborate previous findings from Jansen and de Vries (1991) and Loretan and Phillips (1994), who, analysing US stock market returns, concluded for the existence of finite second moments (possibly finite third and fourth moments as well, but not higher then the fourth one, in any case).

  20. 20

    Recall that the generalized inverse of F is defined as inline image .

  21. 21

    It has been shown that this pitfall can be avoided in some cases, e.g. when dealing with portfolios that are linear combinations of a fixed set of underlying elliptically distributed risk factors (McNeil et al., 2005, Theorem 6.8).

  22. 22

    This classification, up to minor taxonomic details, is quite standard. See for instance Manganelli and Engle (2004) and Gençay and Selçuk (2004).

  23. 23

    The conditional autoregressive VaR (CAViaR) model was introduced by Engle and Manganelli (1999). It is an econometric approach to the calculation of VaR that models directly the distribution of the quantile over time, rather than the whole distribution of the data.

  24. 24

    Rachev et al. (2008, p. 188).

  25. 25

    ES is well-defined for random variables X with finite mean inline image (a weaker condition may be required; see Acerbi and Tasche, 2002).

  26. 26

    See Acerbi et al. (2001) and Acerbi and Tasche (2002).

  27. 27

    ES is a coherent risk measure in the meaning of Artzner et al. (1999), i.e. it satisfies the following four axioms: monotonicity, positive homogeneity, subadditivity and translation invariance. In any case, it is interesting to note that when heavy-tailed distributions are considered, asymptotic approximations based on the regular variation property yield approximate values for inline image and inline image that are proportional to one another, with a factor of proportionality that only depends on α (Daníelsson et al., 2011).

  28. 28

    See for instance McNeil et al. (2005, p. 283), or Gençay and Selçuk (2004, p. 298).

  29. 29

    See Longin (2000), where the explicit derivation of a GEV-based estimator of VaR is provided using this equality.

  30. 30

    See Daníelsson and de Vries (1997a).

  31. 31

    See also Kuester et al. (2006).

  32. 32

    See Section 'An EVT Approach to VaR and ES Calculation' for a description of the two-step procedure.

  33. 33

    ‘In 11 out of 15 cases, our approach is closest to the mark. On two occasions, GARCH with conditional t innovations is best and on one occasion, GARCH with conditional normal innovations is best. In one further case, our approach and the conditional t approach are joint best. On no occasion does our approach fail (lead to rejection of the null hypothesis), whereas the conditional normal approach fails 11 times; unconditional EVT fails three times.’ (McNeil and Frey, 2000, p. 290).

  34. 34

    Neftci (2000, p. 14).

  35. 35

    Bali (2003) analyses daily observations for the annualized yields on 3-month, 6-month, 1-year and 10-year US Treasury securities; Neftci (2000) includes in his dataset 3-month, 2-year, 5-year and 7-year interest rates from the US; Werner and Upper (2004) study Bund futures returns.

  36. 36

    See, e.g. Byström (2005) and Chan and Gray (2006) for electricity markets, Krehbiel and Adkins (2005) for an application to the NYMEX energy complex, and Marimoutou et al. (2009) for an application to the oil market.

  37. 37

    Byström (2007) uses data from the iTraxx Europe CDS index market to infer margin calculations for a hypothetical exchange-traded CDS index futures market.

  38. 38

    FHS, which is proposed by Barone-Adesi et al. (1998), shares the first step, i.e. the GARCH filtering of the data, with the two-step procedure of McNeil and Frey (2000), but at the second step employs HS rather than EVT.

  39. 39

    Kuester et al. (2006, p. 79). The authors perform their analysis for different time windows (1000, 500, 250), obtaining evidence in favour of skewed-t EVT and FHS approaches outperforming the other ones (in particular for a time window of 250 observations) or at least being comparable with them.

  40. 40

    Another interesting path that can be followed in the literature on EVT-based estimation of VaR is the problem of time aggregation and the square root scaling rule, suggested by the Bank of International Settlements but criticized by some academics. On this topic see, for instance, McNeil and Frey (2000), who propose a Monte Carlo simulation approach, and Jansen et al. (2000), who discuss the use of a ‘tail index-root’ rule, rescaling through a root of the time horizon as in the square root rule, but with the index of the root given by the tail index of the distribution. See also Diebold et al. (1998) and Daníelsson and de Vries (1997b).

  41. 41

    Interesting remarks on the main issues of back-testing VaR estimates can be found in Pearson and Smithson (2002).

  42. 42

    For simplicity, we assume a sample of positive losses.

  43. 43

    Gençay and Selçuk (2004) argue that a violation ratio κ less than the expected value 1 − α is not necessarily better than a violation ratio exceeding that value, if we are not committed to policy purposes. For risk managers, the closeness to the value 1 − α should be the most important because, while κ > 1 − α is not a good case (given that it indicates an underestimation of the actual riskiness of a position), κ < 1 − α is not a good scenario either, as it hints at an excessively conservative position.

  44. 44

    Type II errors occur when the null hypothesis of a statistical test is accepted while the alternative is true.

  45. 45

    To this end, the authors advocate using Kupier's statistic instead of the Kolmogorov–Smirnov statistic. The former is better at taking account of extreme values of the distribution, while the latter attaches more weight to values of the distribution that are around the median.

  46. 46

    Denoting by xt and inline image the (positive) loss observed at time t and the forecast VaR for time t, respectively, the loss function reads inline image if inline image, Lt = 0 otherwise. The former specification suggested by Lopez is obtained by picking inline image, while the latter is given by inline image.

  47. 47

    inline image.

  48. 48

    The importance of VaR for asset allocation is demonstrated by Bali (2007), who derives a modified version of the CAPM, relating the downside risk of a portfolio, as measured by VaR, to the expected return of the portfolio itself.

  49. 49

    When considering n different assets, w2 is randomly generated from a uniform distribution with support [0, 1 − w1 ], w3 from a uniform distribution with support [0, 1 − w1w2 ] , and so on. Finally, one sets wn = 1 − w1w2 − ⋯ − wn−1.

  50. 50

    It is easy to programme the algorithm so that different portfolios are generated.

  51. 51

    Another valuable approach to the problem of portfolio diversification is presented in Hyung and de Vries (2005). They study how the benefit of diversification influences downside risk, as measured by VaR, finding, both theoretically and empirically, that the downside risk of heavy-tailed distributions declines more rapidly than that of normal distributions as portfolio diversification increases.

  52. 52

    On the other hand, Poon et al. (2004, p. 601) conclude that ‘if asset returns are asymptotic-dependent, the risk estimation is not sensitive to the wrong assumption for the tail dependence and the methods used to estimate large event risk’. This is a further justification for employing simple but effective models such as the logistic one when asymptotic dependence has been ascertained.

  53. 53

    Longin and Solnik (2001, p. 670).

  54. 54

    See Section 'Multivariate EVT' for the definition of the coefficient of upper tail dependence.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. A Primer of EVT
  5. 3. Multivariate EVT
  6. 4. Testing Distributional Assumptions
  7. 5. Market Risk: VaR and ES Estimation
  8. 6. Asset Allocation
  9. 7. Dependence Across Markets: Correlation and Contagion
  10. 8. Further Applications
  11. 9. Conclusions
  12. Acknowledgements
  13. References
  • Abbate, D., Farkas, W. and Gourier, E. (2008) Operational risk quantification using extreme value theory and copulas: from theory to practice. Working Paper.
  • Acerbi, C. (2004) Coherent representations of subjective risk-aversion. In G. Szegö (ed.), Risk Measures for the 21st Century (pp. 147207). New York: Wiley.
  • Acerbi, C., Nordio, C. and Sirtori, C. (2001) Expected shortfall as a tool for financial risk management. Working Paper.
  • Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance 26: 14871503.
  • Allen, L. and Bali, T.G. (2007) Cyclicality in catastrophic and operational risk measurements. Journal of Banking and Finance 31: 11911235.
  • Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9: 203228.
  • Arzac, E.R. and Bawa, V.S. (1977) Portfolio choice and equilibrium in capital markets with safety-first investors. Journal of Financial Economics 4: 277288.
  • Bali, T.G. (2003) An extreme value approach to estimating volatility and value-at-risk. Journal of Business 76(1): 83108.
  • Bali, T.G. (2007) A generalized extreme value approach to financial risk measurement. Journal of Money, Credit and Banking 39(7): 1613–1649.
  • Balkema, A.A. and de Haan, L. (1974) Residual life at great age. Annals of Probability 2: 792804.
  • Bao, Y., Lee, T.-H. and Saltoğlu, B. (2006) Evaluating predictive performance of value-at-risk models in emerging markets: a reality check. Journal of Forecasting 25(2): 101128.
  • Barone-Adesi, G., Bourgoin, F. and Giannopoulos, K. (1998) Don't look back. Risk 11: 100103.
  • Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes: Theory and Applications. Chichester (England): Wiley.
  • Bekiros, S.D. and Georgoutsos, D.A. (2005) Estimation of value-at-risk by extreme value and conventional methods: a comparative evaluation of their predictive performance. Journal of International Financial Markets, Institutions and Money 15: 209228.
  • Bekiros, S.D. and Georgoutsos, D.A. (2008a) Extreme returns and the contagion effect between the foreign exchange and the stock market: evidence from Cyprus. Applied Financial Economics 18: 239254.
  • Bekiros, S.D. and Georgoutsos, D.A. (2008b) The extreme-value dependence of Asia-Pacific equity markets. Journal of Multinational Financial Management 18: 197208.
  • Bensalah, Y. (2002) Asset allocation using extreme value theory. Bank of Canada Working Paper 2002–2.
  • Berkowitz, J. (2001) Testing density forecasts, with applications to risk management. Journal of Business and Economic Statistics 19: 465474.
  • Blanco, C. and Ihle, G. (1998) How good is your VaR? Using backtesting to assess system performance. Financial Engineering News 11: 12.
  • Bradley, B.O. and Taqqu, M.S. (2004a) An extreme value theory approach to the allocation of multiple assets. International Journal of Theoretical and Applied Finance 7(8): 10311068.
  • Bradley, B.O. and Taqqu, M.S. (2004b) Asset allocation when guarding against catastrophic losses: a comparison between the structure variable and joint probability methods. Quantitative Finance 4: 619636.
  • Brooks, C., Clare, A.D., Dalle Molle, J.W. and Persand, G. (2005) A comparison of extreme value theory approaches for determining value-at-risk. Journal of Empirical Finance 12: 339352.
  • Byström, H.N.E. (2005) Extreme value theory and extremely large electricity price changes. International Review of Economics and Finance 14: 4155.
  • Byström, H.N.E. (2006) Using extreme value theory to estimate the likelihood of banking sector failure. The European Journal of Finance 12(4): 303312.
  • Byström, H.N.E. (2007) Back to the future: futures margins in a future credit default swap index futures market. The Journal of Futures Markets 27(1): 85104.
  • Chan, K.F. and Gray, P. (2006) Using extreme value theory to measure value-at-risk for daily electricity spot prices. International Journal of Forecasting 22: 283300.
  • Chan-Lau, J.A., Mitra, S. and Ong, L.L. (2007) Contagion risk in the international banking system and implications for London as a global financial center. IMF Working Paper WP/07/74.
  • Chapelle, A., Crama, Y., Hübner, G. and Peters, J.-P. (2008) Practical methods for measuring and managing operational risk in the financial sector: a clinical study. Journal of Banking and Finance 32: 10491061.
  • Christoffersen, P.F. (1998) Evaluating interval forecasts. International Economic Review 39: 841862.
  • Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer.
  • Consigli, G. (2002) Tail estimation and mean–VaR portfolio selection in markets subject to financial instability. Journal of Banking and Finance 26: 13551382.
  • Cotter, J. (2001) Margin exceedances for European stock index futures using extreme value theory. Journal of Banking and Finance 25: 14751502.
  • Cotter, J. (2007) Extreme risk in Asian equity markets. MPRA Paper 3536, University Library of Munich, Germany.
  • Crnkovic, C. and Drachman, J. (1995) A universal tool to discriminate among risk measurement techniques. Mimeo, Corporate Risk Management Group, JP Morgan.
  • Crnkovic, C. and Drachman, J. (1996) Quality control. Risk 9: 139143.
  • Daníelsson, J. (2002) The emperor has no clothes: limits to risk modelling. Journal of Banking and Finance 26: 12731296.
  • Daníelsson, J. and de Vries, C.G. (1997a) Tail index and quantile estimation with very high frequency data. Journal of Empirical Finance 4: 241257.
  • Daníelsson, J. and de Vries, C.G. (1997b) Value-at-Risk and extreme returns. In P. Embrechts (ed.), Extremes and Integrated Risk Management (pp. 85106). London: Risk Waters Group.
  • Daníelsson, J. and de Vries, C.G. (2000) Value-at-Risk and extreme returns. Annales d'Économie et de Statistique 60: 239270.
  • Daníelsson, J., Jorgensen, B.N., Samorodnitsky, G., Sarma, M. and de Vries, C.G. (2011) Fat tails, VaR and subadditivity. Working Paper.
  • Dewachter, H. and Gielens, G. (1999) Setting futures margins: the extremes approach. Applied Financial Economics 9: 173181.
  • de Haan, L and Ferreira, A. (2006) Extreme Value Theory. An Introduction. New York: Springer.
  • Diebold, F.X., Schuermann, T. and Stroughair, J.D. (1998) Pitfalls and opportunities in the use of extreme value theory in risk management. In A.-P.N. Refenes, J.D. Moody and A.N. Burgess (eds.), Advances in Computational Finance (pp. 312). Amsterdam: Kluwer. Reprinted in Journal of Risk Finance 1: 30–36.
  • Dowd, K. (1998) Beyond Value-at-Risk: The New Science of Risk Management. Chichester: Wiley.
  • Dowd, K. (2002) Measuring Market Risk. New York: Wiley.
  • Duffie, D. and Pan, J. (1997) An overview of value-at-risk. Journal of Derivatives 7: 749.
  • DuMouchel, W.H. (1983) Estimating the stable index α in order to measure tail thickness: a critique. The Annals of Statistics 11(4): 10191031.
  • Dutta, K. and Perry, J. (2006) A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital. Working Papers 06-13, Federal Reserve Bank of Boston.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer.
  • Engle, R.F. and Manganelli, S. (1999) CAViaR: conditional value-at-risk by quantile regression. NBER, Working Paper 7341.
  • Falk, M., Hüsler, J. and Reiss, R.-D. (2004) Laws of Small Numbers: Extremes and Rare Events. Basel: Birkhäuser.
  • Fama, E.F. (1963) Mandelbrot and the stable Paretian hypothesis. Journal of Business 35: 42029.
  • Fama, E.F. (1965) The behavior of stock market prices. The Journal of Business 38: 34105.
  • Finkenstädt, B. and Rootzén, H. (2004) Extreme Values in Finance, Telecommunications and the Environment. Boca Raton, Florida: Chapman & Hall/CRC.
  • Fisher, R.A. and Tippett, L.H.C. (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proceedings of the Cambridge Philosophical Society 24: 180190.
  • Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics. Melbourne: Krieger.
  • Garita, G. and Zhou, C. (2009) Can open capital markets help avoid currency crises?. DNB Working Papers 205, Netherlands Central Bank, Research Department.
  • Gençay, R. and Selçuk, F. (2004) Extreme value theory and value-at-risk: relative performance in emerging markets. International Journal of Forecasting 20: 287303.
  • Gnedenko, B.V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Annals of Mathematics 44: 42353.
  • Haile, F.D. and Pozo, S. (2006) Exchange rate regimes and currency crises: an evaluation using extreme value theory. Review of International Economics 14(4): 554570.
  • Haile, F.D. and Pozo, S. (2008) Currency crisis contagion and the identification of transmission channels. International Review of Economics and Finance 17: 572588.
  • Hartmann, P., Straetmans, S. and de Vries, C. (2004) Asset market linkages in crisis periods. The Review of Economics and Statistics 86(1): 313326.
  • Hartmann, P., Straetmans, S. and de Vries, C. (2005) Banking system stability: a cross-Atlantic perspective. Working Paper Series 527, European Central Bank.
  • Hauksson, H.A., Dacorogna, M., Domenig, T., Müller, U. and Samorodnitsky, G. (2001) Multivariate extremes, aggregation and risk estimation. Quantitative Finance 1: 7995.
  • Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3: 1163l174.
  • Ho, L.-C., Burridge, P., Cadle, J. and Theobald, M. (2000) Value-at-risk: applying the extreme value approach to Asian markets in the recent financial turmoil. Pacific-Basin Finance Journal 8: 249275.
  • Ho, T.-K. (2008) Extremal analysis of currency crises in Taiwan. Applied Economics 40: 11751186.
  • Huang, X. (1992) Statistics of bivariate extreme values. Ph.D. thesis, Tinbergen Institute Research Series, Erasmus University Rotterdam.
  • Hyung, N. and de Vries, C.G. (2005) Portfolio diversification effects of downside risk. Journal of Financial Econometrics 3: 107125.
  • Hyung, N. and de Vries, C.G. (2007) Portfolio selection with heavy tails. Journal of Empirical Finance 14: 383400.
  • Jansen, D.W., Koedijk, K.G. and de Vries, C.G. (2000) Portfolio selection with limited downside risk. Journal of Empirical Finance 7: 247269.
  • Jansen, D.W. and de Vries, C.G. (1991) On the frequency of large stock returns: putting booms and busts into perspective. The Review of Economics and Statistics 73(1): 1824.
  • Jenkinson, A.F. (1955) The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal Meteorology Society 87: 145158.
  • Jobst, A. (2007) Operational risk – the sting is still in the tail but the poison depends on the dose. IMF Working Papers 07/239, International Monetary Fund.
  • Jorion, P. (1997) Value-at-Risk: The New Benchmark for Controlling Market Risk. Chicago: McGraw-Hill.
  • Koedijk, K.G., Schafgans, M.M.A. and de Vries, C.G. (1990) The tail index of exchange rate returns. Journal of International Economics 29: 93108.
  • Kotz, S. and Nadarajah, S. (2000) Extreme Value Distributions: Theory and Applications. London: Imperial College Press.
  • Krehbiel, T. and Adkins, L.C. (2005) Price risk in the NYMEX energy complex: an extreme value approach. The Journal of Futures Markets 25(4): 309337.
  • Kuester, K., Mittnik, S. and Paolella, M.S. (2006) Value-at-risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics 4(1): 5389.
  • Kupiec, P.H. (1995) Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives 3: 7384.
  • Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983) Extremes and Relates Properties of Random Sequences and Processes. Berlin: Springer.
  • Ledford, A.W. and Tawn, J.A. (1996) Statistics for near independence in multivariate extreme values. Biometrika 83: 169187.
  • Lee, T.-H. and Saltoğlu, B. (2002) Assessing the risk forecasts for Japanese stock market. Japan and the World Economy 14: 6385.
  • Lestano, J.P. and Jacobs, A.M. (2007) Dating currency crises with ad hoc and extreme value-based thresholds: East Asia 1970–2002. International Journal of Finance and Economics 12: 371388.
  • Longin, F.M. (1999) Optimal margin level in futures markets: extreme price movements. The Journal of Futures Markets 19(2): 127152.
  • Longin, F.M. (2000) From value-at-risk to stress testing: the extreme value approach. Journal of Banking and Finance 24: 10971130.
  • Longin, F.M. (2005) The choice of the distribution of asset returns: how extreme value theory can help?. Journal of Banking and Finance 29: 10171035.
  • Longin, F.M. and Solnik, B. (2001) Extreme correlation of international equity markets. The Journal of Finance 56(2): 649676.
  • Lopez, J.A. (1998) Methods for evaluating value-at-risk estimates. Federal Reserve Bank of New York. Economic Policy Review 4(3): 119124.
  • Lopez, J.A. (1999) Regulatory evaluation of value-at-risk models. Journal of Risk 1: 3764.
  • Loretan, M. and Phillips, P. (1994) Testing the covariance structure of heavy-tailed time series. Journal of Empirical Finance 1: 211248.
  • Lux, T. (1996) The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks. Applied Financial Economics 6: 463475.
  • Lux, T. (2000) On moment condition failure in German stock returns: an application of recent advances in extreme value statistics. Empirical Economics 25: 641652.
  • Lux, T. (2001) The limiting extremal behaviour of speculative returns: an analysis of intra-daily data from the Frankfurt Stock Exchange. Applied Financial Economics 11: 299315.
  • Mandelbrot, B.B. (1963) The variation of certain speculative prices. Journal of Business 36: 394419.
  • Mandelbrot, B.B. (1967) The variation of some other speculative prices. Journal of Business 40: 393413.
  • Manganelli, S. and Engle, R. (2004) A comparison of value-at-risk models in finance. In G. Szegö (ed.), Risk Measures for the 21st Century. New York: Wiley.
  • Marimoutou, V., Raggad, B. and Trabelsi, A. (2009) Extreme value theory and value-at-risk: application to oil market. Energy Economics 31: 519530.
  • Markose, S.M. and Alentorn, A. (2005) The generalized extreme value (GEV) distribution, implied tail index and option pricing. Discussion Papers, University of Essex, Department of Economics.
  • Markowitz, H. (1952) Portfolio selection. Journal of Finance 7: 7791.
  • McNeil, A.J. and Frey, R. (2000) Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance 7: 271300.
  • McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management. Princeton, New Jersey: Princeton University Press.
  • Moscadelli, M. (2004) The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee. Temi di discussione del Servizio Studi n. 517, Banca d'Italia.
  • Neftci, S.N. (2000) Value-at-risk calculations, extreme events, and tail estimation. Journal of Derivatives 7(3): 2338.
  • Pais, A. and Stork, P.A. (2009) Contagion risk across real estate firms and banks: an Australian and New Zealand analysis. Working Paper.
  • Panaretos, J. and Tsourti, Z. (2003) Extreme value index estimators and smoothing alternatives: a critical review. MPRA Paper 6390, University Library of Munich, Germany.
  • Pearson, N.D. and Smithson, C. (2002) VaR: the state of play. Review of Financial Economics 11: 175189.
  • Pickands, J. III (1975) Statistical inference using extreme order statistics. Annals of Statistics 3: 119131.
  • Pontines, V. and Siregar, R. (2007) The Yen, the US dollar, and the trade weighted basket of currencies: does the choice of anchor currencies matter in identifying incidences of speculative attacks?. Japan and the World Economy 19: 214235.
  • Pontines, V. and Siregar, R. (2008) Fundamental pitfalls of exchange market pressure-based approaches to identification of currency crises. International Review of Economics and Finance 17: 345365.
  • Poon, S.-H., Rockinger, M. and Tawn, J. (2004) Extreme value dependence in financial markets: diagnostics, models, and financial implications. The Review of Financial Studies 17(2): 581610.
  • Pownall, R.A.J. and Koedijk, K.G. (1999) Capturing downside risk in financial markets: the case of the Asian Crisis. Journal of International Money and Finance 18: 853870.
  • Pozo, S. and Amuedo Dorantes, C. (2003) Statistical distributions and the identification of currency crises. Journal of International Money and Finance 22: 591609.
  • Rachev, S.T. (ed.) (2003) Handbook of Heavy Tailed Distributions in Finance. Amsterdam: Elsevier.
  • Rachev, S.T., Stoyanov, S.V. and Fabozzi, F.J. (2008) Advanced Stochastic Models, Risk Assessment and Portfolio Optimization: The Ideal Risk, Uncertainty and Performance Measures. Hoboken, New Jersey: Wiley.
  • Reiss, R.-D. and Thomas, M. (1997) Statistical Analysis of Extreme Values with Applications in Finance, Hydrology and Other Fields. Basel: Birkhäuser.
  • Resnick, S.I. (1987) Extreme Values, Regular Variation and Point Process. New York: Springer.
  • Resnick, S.I. (2007) Heavy-Tail Phenomena. New York: Springer.
  • Roy, A.D. (1952) Safety first and the holding of assets. Econometrica 20: 431449.
  • Serguieva, A., Bozhkov, S. and Yu, K. (2009) Catastrophic and operational risk measurement for financial institutions. Working Paper.
  • Sundmacher, M. and Ford, G. (2007) When statistics fail: extreme events in financial institutions. Working Paper.
  • Susmel, R. (2001) Extreme observations and diversification in Latin American emerging equity markets. Journal of International Money and Finance 20: 971986.
  • Tolikas, K., Koulakiotis, A. and Brown, R.A. (2007) Extreme risk and value-at-risk in the German stock market. The European Journal of Finance 13(4): 373395.
  • Vilasuso, J. and Katz, D. (2000) Estimates of the likelihood of extreme returns in international stock markets. Journal of Applied Statistics 27(1): 119130.
  • Werner, T. and Upper, C. (2004) Time variation in the tail behavior of bund future returns. The Journal of Futures Markets 24(4): 38733898.
  • White, H. (2000) A reality check for data snooping. Econometrica 68(5): 10971126.