coronado extreme value theory (evt) for risk managers 2001

8/3/2019 Coronado Extreme Value Theory (Evt) for Risk Managers 2001

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EXTREME VALUE THEORY (EVT) FOR RISK MANAGERS:PITFALLS AND OPPORTUNITIES IN THE USE OF EVT

IN MEASURING VaR

Prof. Mara Coronado, PhDDepartment of Finance

Facultad de Ciencias Econmicas y Empresariales.ICADE. Universidad P. Comillas de Madrid.Alberto Aguilera 23, 28015 Madrid. Spain.

Tel: +34 (9)1 542 28 00 (Ext 2254)Fax: +34 (9)1 541 77 96

E-mail: [email protected]

ABSTRACT

Extreme Value Theory, more widely used up until now in the actuarial field, is experiencing aboom in the financial field, especially with respect to risk management. In this way, its most recentapplication is to VaR, as a market risk measure.

Its appearance as a possible instrument for use with VaR can be explained as a consequence of twofactors:

On one hand, the assumption of normality of financial markets does not reflect the reality of thesituation and the VaR estimation methods on which they are based provide wrong estimates. Thesimulation methods (historical or Monte Carlo) arise as alternative methods; but given the difficultiesof these methods and their slowness, new solutions are sought: for instance, Extreme Value Theory(EVT) among others.

On the other hand, even though VaR is calculated with better methods (simulation), it still haslimitations, so this measure needs to be complemented with others.

In fact, there are two different applications, and therefore, we will analyse them separately. In thisway, there are authors that present it as a way of solving the problem of fat tails when calculating VaR;while other authors present it as a specific methodology for VaR estimation comparable with the classicmethodologies. In our opinion, the second proposition is a wrong one. Our conclusion is that the

application of EVT to VaR should come through the following two ways:

1. As a complementary (but never substitute) analysis for VaR, rather than as an approach forestimating VaR in a strict sense. By definition, VaR does not include all the aspects of market risk.With VaR, the extreme market movements, such as those that occurred in recent years, can not beestimated or predicted. It is not for this reason that VaR can be considered a bad measure, but it onlyneeds to be complemented with other measures. And it is here where EVT plays a fundamental role.Thus we show its applicability in the use of the complementary measures of VaR (stress testing,worst case scenario, CrashMetricsTM), being even an improvement of this complementary measures.

Financial support from Fundacin Caja de Madrid under research grant is acknowledged. I would also

like to thank participants at the Forecasting Financial Markets 2001 Conference (FFM2001),London,for their valuable comments and suggestions. All errrors are entirely my own.


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We think that the authors who consider that the methods based on this Theory can substitute VaR, orcombine VaR with stress testing, commit a basic error.

2. As a manner of improving the already existing methods of estimating VaR. Thus, for example: As an additional mechanism for solving the problem of fat tails when estimating VaR

(remaining the fact that it is not the only one).

The risk of modelling, in the variance-covariance matrix method and in the Monte Carlosimulation method could be quantified from the estimation process of the tail index and otherparameters of the distribution of the tails of the returns.

Indeed, the choice of processes used in the Monte Carlo simulation method could be based alsoon the results of the extreme values.

KEYWORDS: Value-at-Risk, Extreme Value Theory, Stress Testing, Historical Simulation, MonteCarlo Simulation, Tail Estimation, Risk Analysis.

1.- INTRODUCTIONExtreme Value Theory, more widely used up until now in the actuarial field, is

experiencing a boom in the financial field, especially with respect to risk management.In this way, its most recent application is to VaR, as a market risk measure.

Its appearance as a possible instrument for use with VaR can be explained as aconsequence of two factors:

On one hand, the assumption of normality of financial markets does not reflect thereality of the situation and the VaR estimation methods on which they are based

provide wrong estimates. The simulation methods (historical or Monte Carlo) ariseas alternative methods; but given the difficulties of these methods and theirslowness, new solutions are sought: for instance, Extreme Value Theory (EVT)among others.

On the other hand, even though VaR is calculated with better methods(simulation), it still has limitations, so this measure needs to be complemented withothers.

VaR is the maximum expected loss in the market value of an asset or portfolio,

during a given period of time and for a given confidence level, under normal marketconditions .

Conceptually and in agreement with the established definition, to calculate VaRsimply means to calculate a percentile and all the different methodologies of VaRestimation, in fact, are the classic statistical techniques of quartiles estimation,experiencing nowadays a renaissance in the context of VaR1.

1 Additional progress the field of the computer science or information technology is allowing for the

recover of such techniques for use in real time, as in the case of the Monte Carlo Simulation.


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However, although VaR is conceptually simple, its implementation in practice bythe banks is not so straightforward, since there are several alternative methodologies,each with their advantages and disadvantages, derived from the underlying assumptionsof such methodologies.

At present there are three methodologies for calculating VaR: the analytic methodof the variance-covariance matrix, the historical simulation and the Monte Carlosimulation, even though there are also many variants from each of them2. These threemethods are all based on a single approach: their starting point is the probabilitydistribution of the returns. Their only difference is the way the distribution ismodelled.

Very recently, some authors are discussing a fourth methodology to calculateVaR, based on Extreme Value Theory3. The approach is completely different. Thosemethods based on EVT start, not from the distribution of returns but from theprobability distribution of extreme returns. Thus, it allows us to estimate extreme

quartiles more precisely4.

We aim in this paper to analyse the real possibilities of applying Extreme ValueTheory (EVT) to estimate VaR as a market risk measure.

In fact there are two different proposed ways of applying this theory, and so wewill analyse them separately. In this manner, there are authors who present it as a wayof solving the problem of fat tails when estimating VaR (this will be analysed insections 3 and 4); while others present it as a methodology of VaR estimation in thestrict sense, comparing it with the classical methodologies (this will be analysed insection 5).

In our opinion, this second proposition is erroneous and the most directapplication of EVT to VaR (besides the one and others mentioned above that will alsobe presented) is as a complementary measure of VaR (section 6). In section 7 weconclude and propose further research.

2.- VALUE-AT-RISK (VAR): DEFINITION AND ESTIMATION METHODS

2 See Coronado (1999b), chapter 6

3 Even though, as we will conclude in this work, in our opinion such a Theory serves as a complementary

analysis to VaR, rather than as an approach to VaR estimation itself.

4 The result is that a specific bank with the same risk should have different capital depending only upon

the chosen method to calculate VaR. This presents an inconvenience not only for the bank (if the

requirements are higher than what is really needed) but also for a correct and trustworthy estimation ofbank solvency on the part of the supervisors. It can also produce the opposite effect of not assuring the

stability and solvency of the financial system, due to the fact that this situation would final lead to the

banks using the method giving a lower VaR and not the one that provides the best measurement of themarket risk for each case. So, it is important to establish the best methodologies to measure VaR, in a

way that they will be accepted by the supervisors not imposing their own extra requirements but at thesame time allowing a reasonable and responsible management of the banks risk and solvency.


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VaR is defined as:

The maximum expected loss (measured in monetary units) of an asset value(or a portfolio) over a given time period and at a given level of confidence (or

with a given level of probability), under normal market conditions.

VaR answers the following question: How much could I lose as a maximum, witha probability of x per cent, during a period of time y?

For instance, a portfolio manager could state that the monthly VaR is 20 millionpesetas with a probability of a 99 per cent; and that means that during the next monththere is only a one per cent probability of the loss being higher than 20 million pesetas.

A formal mathematical definition of VaR can be found in Coronado (1999b),

chapters 5 and 6.

We should emphasize the study of two key topics for estimating VaR in generaland the VaR of non-linear positions (e.g. options) in particular: the validity of thenormality assumption for the value or the return of an asset or portfolio (that includesthe study of both fat tails and volatility clustering) and the non-linear assumption ofthe assets for which we wish to estimate VaR.

And as a function of these two assumptions we can compare the different methodsof VaR estimation.

Even though we cannot present here a detailed description of each method 5, beforeanalysing the possible applications of EVT for estimating VaR (that is the aim of thepresent paper), we consider it convenient to compare briefly the advantages anddisadvantages of the three classical methodologies of VaR estimation 6.

The analytic method of the variance-covariance matrix7 is based on theassumption of the normal joint-distribution of the portfolio returns and on theassumption of a linear relation (ship a most a quadratic one) between the risk factors(or independent variables) of the market and the portfolio value.

There are two principal advantages of using these hypotheses for the VaR

estimation: simplicity in calculating VaR (what makes this method easilycomprehensible for all the people involved in risk management; as well as the simplicityof its practical implementation); and speed of these calculations, something veryimportant when working in real time.

Three are the main disadvantages of this method:

5 Go to Coronado (1999b), chapter 6.6 A more complete theoretical and practical comparison can be found on Coronado (1999a), pp. 44 and.7 It should be remembered that the famous RiskMetricsTM of J. P. Morgan is an analytic method of

variance-covariance matrix.


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1. The VaR estimation using the variance-covariance matrix givesoverestimated values of VaR for small confidence levels andunderestimated values of VaR for high confidence levels; and thus producessurplus or lacks in the required bank capital in order to face market risk (bysupervisory authorities), with the resulting repercussions for bank solvency.

This drawback derives from the assumption of normality of the portfolioreturns and thus the method being not able to allow for fat tails.2. The assumption of linearity means that this method is only applicable, in

theory, to linear portfolios; something which is not very useful given theextensive and growing use of non linear assets (mostly options) in the banksportfolios.

3. Indeed, extending the approximation of the portfolios value to a quadraticfunction (delta-gamma methods), with this method can not be reached anaccurate VaR estimation for non-linear portfolios. We should also take intoaccount that this extension implies reducing the simplicity of this method(and that was precisely one of its advantages) due o the required additional

assumptions (as a consequence of the loss of normality when applyingTaylors second-order series).

These last two disadvantages are especially true in the case of options near themoney and close to their date of maturity.

The Historical Simulation method is a non-parametric method, that doesnot depend on any assumption concerning the underlying probability distributionsand thus allows us to capture fat tails (and other non-normal characteristics) at thesame time as it eliminates the necessity of estimating and working with volatilitiesand correlations, and for the most part, avoids the risk of modelling. It is a globalvaluation method and so, eliminates the need of establishing approximations (as withthe calculations base on Taylors series) that introduces inaccuracies into thecalculations. It can be applied, then, to all types of linear and non-linear instruments.

All of these advantages give it a theoretical superiority compared to thevariance-covariance matrix method, especially in the case of calculating VaR of non-linear portfolios. This general theoretical superiority is also supported by abundantempirical evidence which we will comment upon in the next section were we will seehow this method is able to incorporate the fat tails; something which cannot be donewith the variance-covariance method.

However, the historical simulation method also has some disadvantages,fundamentally related to the characteristics of the historic database used and this impliesthat this method depends completely on a specific database used and ignores any otherevent that is not represented in the database. It is the inconvenience of assuming that thefuture will be similar to the past; or estimating VaR from a single path followed byprices (the one followed by the chosen specific historic prices). This disadvantage 8, as

8 Two other aspects not sufficiently clear in this method are: how the confidence of the VaRs estimation

decreases when the level of confidence used increases; and also the relation of that confidence with thelength of the data base used. The empirical evidence support is not conclusive in either of the cases.


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we will see, does not arise in the Monte Carlo simulation method, (but, of course,the Monte Carlo method presents other disadvantages).

The Monte Carlo Simulation method, parametric and non-parametric, is aglobal valuation method and thus eliminates the need to establish approximations

such as those based on Taylors series) that introduce some degree of inaccuracy intothe calculations. It can be applied then to all types of linear and non-linearinstruments.

Furthermore, in the case of the non-parametric Monte Carlo simulation method, asit does not depend on any assumption of the underlying probability distributions, for themost part it avoids the risk of modelling and allows to capture fat tails (and other non-normal characteristics) at the same time as eliminating the necessity of estimating andworking with volatilities and correlations. Simultaneously it avoids the disadvantages ofthe historical simulation method compared with the variance-covariance matrix one.

All of these advantages give the non-parametric Monte Carlo simulationmethod a theoretical superiority over the variance-covariance matrix method,especially in the case of the VaR estimation of non-linear portfolios.

Also, the parametric Monte Carlo simulation method, being applicable to non-linear positions and not requiring the normality assumption (although it requires thespecification of a specific stochastic process for the risk factors variables and therefore,presents a modelling risk), presents a theoretical superiority with respect to thevariance-covariance matrix method.

And not only compared to the variance-covariance matrix method, but also when

compared to the historical simulation method. Its advantage lies in the true randomfeature of the paths of future prices, while the prices generated by the historicalsimulation represent only one of the possible paths that could occur.

The Monte Carlo simulation offers a more realistic description of risk due to thefact that the distribution of the price variations presents a complete set of all the eventsand their probabilities.

Although this is the most complex method to understand, explain and implementof the three specified methods for VaR estimations, and also, although it is the slowestof the three, the Monte Carlo simulation method is the most powerful, flexible and

precise method of VaR estimation.

Its most important drawback, its slowness, is not so in reality, thanks todevelopments en computing, and it will be less and less so in the future thanks to theforthcoming new technologies that will help solve this problem.

This is the most appropriate method, in theory 9, for VaR estimation of non-linear portfolios.

9 There is also empirical evidence to support this affirmation. See (1999a), pp. 48; and Coronado (1999b),

section 6.6. and chapter 9.


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3.- THE PROBLEM OF FAT TAILS IN VaR ESTIMATION: A REVIEW OFTHE MOST RECENT CONTRIBUTIONS WITH SPECIAL MENTION TO THEEXTREME VALUE THEORY

Given the fact that VaR centres its attention on the behaviour of the tails of

financial returns, more specifically on the left tail, it is obvious that the issue of fat tailsis fundamental to our study. (It should also be taken into account the importance that fattails have in the choice of the confidence level in the VaR estimation).

So then, even though the topic of the validity of the normal assumption forfinancial returns (which include the study of fat tails as web as volatility clustering) hasbeen the focus of continuous research for the last 25 tears, it is now, since theintroduction of VaR as a measure of market risk, that research into this topic is stronglyemerging, with very important contributions at the theoretical and practical levels, andwith a broad field of study open for new lines of research.

For this reason, we have carried out an extensive coverage and analysis of themost recent contributions , starting from the classic empirical studies that havecorroborated what has already been demonstrated by the pioneer works ofMANDELBROT (1963) and FAMA (1965) regarding the characteristics shown byfinancial returns (fat tails and leptokurtosis, asymmetry, volatility clustering).

We will first establish a chronological list of the studies that, in our view, andafter presenting a commented synthesis, stand out for their importance. This list willinclude some traditional studies as well as other more recent ones.

Following this, we will identify the latest contributions, the most relevant ones

from the point of view of VaR, related to fat tails, with special mention to the EVTapplication.

Among the empirical works that have demonstrated that the returns present thepreviously mentioned characteristics rather than those of a normal distribution, it isnecessary to identify10:

PRAETZ (1972), BLATTBURG and GONEDES (1974), KON (1984), JORION(1988), JANSEN and de VRIES (1991), TUCKER (1992), GLOSTEN et al. (1993),KIM and KON (1994); LONGIN (1996), LONGIN (1997a, b), DANIELSSON andde VRIES (1997b), KLPELBERG et al. (1998), McNEIL (1998) y HUISMAN et

al. (1998) for the stock prices; ROGALSKI y VINSO (1978), KOEDIJDK et al. (1992),WILSON (1993), de VRIES (1994), JORION (1995c), ZANGARI (1996a),ZANGARI (1996d), HUISMAN et al. (1997), ALEXANDER and WILLIAMS(1997), VENKATARAMAN (1997), DANIELSSON and de VRIES (1997C),CORREDOR CASADO et al. (1998), LIM et al. (1998) y HULL and WHITE (1998)for the exchange rates; y BOLLERSLEV (1986), BOLLERSLEV (1987), GHOSE andKRONER (1993), RUIZ (1994), HARVEY et al. (1994), LONGERSTAEY et al.

10In black those studies deserving a special attention due to its current importance and interest and for

being directly applied to the VaR and the EVT. They are described in sections 3 and 4. Thebibliographic reference of the rest can be consulted in Coronado (1999b).


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(1996), HARVEY and SHEPARD (1996), DUAN (1997), CAMPBELL et al. (1997),KEARNS and PAGAN (1997) y LUCAS (1997) for all markets .

The fact that the distributions of financial returns present fat tails means that theextreme movements of prices occur more frequently than implied by a normal

distribution.And if, furthermore, leptokurtosis appears, it means that very small movements of

prices occur more frequently than predicted by a normal probability distribution.

So, as we move from a normal distribution to a leptokurtic one, a probabilitymass is added to the central part of the distribution and also to the tails; and at thesame time, a probability mass is taken from the intermediate areas of the distribution(between the middle and the tails). Therefore, the effect of leptokurtosis is an increaseof the probability of large movements and also the very small ones for the value of thevariable, and a decrease on the probability of moderate movements.

If in the theoretical world of normal distributions an equivalent movement to 10standard deviations is something that only occurs once every million, in real financialmarkets we know this is not so, which means we should investigate with whatfrequency these extreme movements could happen. As Robert Gumerlock, managingdirector the Swiss Bank Corporation says :

The troublesome thing about fat tail distributions is that they sever the link

between ordinary and extraordinary events. Under a purely normal distribution,

the extraordinary events are strictly governed by probabilities, policed by the

standard deviation. With fat tailed distributions, outliers can occur with

maddening frequency and no amount of analysis of the standard deviations canyield useful information about them.

In this way, the Swiss Bank Corporation has noticed that the yen-dollar / $ dailyreturns between 1988 and 1993 excels the three standard deviations in 7 times and the 5standard deviations in 2 times.

In the same way, Chew (1994)11 demonstrates such properties in 10-year German,Us and British Treasury bond markets, with respect to daily returns throughout the1988-1993 period as it is shown in Table 1, which illustrates that movements of threestandard deviations occurred with a frequency of 2.3% of the total in the case of UK

bonds (2.8% for US bonds and 1.5% for German bonds) when the normal distributionwould have estimated such events at only 0.3% of the total

The consequences, in terms of estimating VaR are very important: although,following the normal distribution we should concentrate on events provoked bymovements of 2 or 3 sigmas, however, basing our analysis on the empirical distributionwe should look with greater attention to movements of 4 or 6 sigmas. This couldunderestimate the VaR and, as a result, the capital requirements to cover suchlosses, putting at risk the solvency of the bank .

11.p. 64.


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s L $ DM Expected Probability

3 2,3% 2,8% 1,5% 0,300%

4 0,6% 0,4% 0,2% 0,006%

5 0,2% 0,4% 0,0%


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the collapse of the bonds market in 1994, and the Russian ruble crisis in August 1998,etc) have to be considered apart trough stress testing, 13 or EVT.

Among the five most recent contributions concerning the effect of fat tails onthe VaR estimation, the most interesting study is the EVT, which we mention in fifthplace and we develop in section 4. Due to limitations of space we cannot describe, noteven briefly, the other four contributions, in such a way that we only presentbibliographical references without having space to comment on the comparativeempirical studies of the VaR estimations 14. However, it is necessary to know that thereare other ways of solving the problem of fat tails in VaR estimations, without the needto use EVT.

3.1.- Students t:

WILSON (1993), pp. 37-43 and more recently, LUCAS (1997) propose tosubstitute the normal distribution for a Students t one.

On the other hand, Longerstaey et al. (1996)15, conscious of this fact and inresponse to criticism16 that the RiskMetricsTM model of J.P. Morgan for the VaRestimation, has received for having assumed the conditional normality hypothesis fordaily returns on financial assets, offer and describe two probability distributions whichallow for a more realistic modelling of the tails of the returns.

Thus, in addition to the standard RiskMetrics model (which assumesconditional normality) two other models are proposed: the mixture o combination ofnormal distributions and the generalized error distribution (GED)17.

3.2.- Mixture of normal distributions18

ZANGARI (1996d) has proposed this new approach. Another empirical test ofthis model for fifteen emerging economies appears in ZANGARI (1996a).

13 This topic will be analyzed in section 6.

14 A detailed description can be found in Coronado (1999b), part 5.3.1.

15

See Appendix B (pp. 235-242): Relaxing the assumption of conditional normality, from Riskmetrics Technical Document, Fourth Edition, (1996); although in this appendix they simplyreproduce previous works of Zangari, dealing with normal mixtures (Zangari (1996d)), as well as GED

(Zangari (1996a)) and an empirical comparison of Zangari (1996a).

16 The RiskMetrics model has been widely criticized not only due to its normality assumption, but thistopic goes beyond the principal aim of this article. See Coronado (1999b), section 6.1.3.

17 The use of this model applied to VaR was firstly described in Zangari (1996d); subsequently

Venkataraman (1997) improved it and its more recent use appears in Hull and White (1998).

18 The mixture of normal distributions was already proposed by KON, Stanley J. (1984): Models of

Stock Returns. A Comparison. Journal of Finance, Vol. 39, N1, March, pp. 147.165; who in additioncompared it with the Students t, but this author proposes a discrete distribution with a finite variance.


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Subsequently there appeared the innovation of VENKATARAMAN (1997). Thisauthor also applies the parametric methodology in order to estimate VaR starting from amixture of normal distributions, but he estimates the parameters using a quasi-Bayesianmaximum likelihood procedure19, that it is simpler for calculating than that proposed byZangari; although in his article he does not undertake a direct comparison between both

procedures (it is something that could be a topic for future research).

Finally, the most recent work on fat tails and VaR that also uses a mixture ofnormal distributions is that of HULL and WHITE (1998).

Their model is non-bayesian and conceptually is simpler than those of Zangariand Venkataraman. Also it adopts a slightly different approach to the treatment ofcorrelations.

That is to say, they use a mixture of normal distributions in this specific work but

they could use any other distribution (as they themselves admit) since they include thefat tails through a different procedure; and so, due to the fact that the authors do not giveit a specific name, we will call it the Hull and White transformation; and we willconsider it briefly after we study the other approach of J. P. Morgan announcedpreviously: the generalized error distribution.

3.3.- Generalized Error Distribution.

Another way of incorporating fat tails is trough the generalized error distribution(GED). This is a well-known distribution used by researchers in finance due to the largenumber of concrete forms that it can adopt (including the normal distribution), as afunction of the value taken by the parameter.

ZANGARI (1996a) proposes this distribution, combined with an EGARCH20

process of volatility estimation, incorporating this way, both fat tails and volatilityclustering.

In truth, NELSON (1991)21 was the first to propose the GED associated with aGARCH process but he did not apply it to VaR. This was first done by ZANGARI(1996a).

19 In truth, Hamilton, James (1991) was the first to suggest this procedure of estimation: A quasi-

Bayesian approach to estimating parameters for mixtures of normal distributions. Journal of Business &

Economic Statistics, Vol. 9, n 1, pp. 27-39.

20 Ruiz, Esther (1994): Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models.

Journal of Econometrics, Vol.63, pp. 289-306; she was the first to combine this probability

distribution with a stochastic volatility process in place of the GARCH process used by Zangari(1996a), for the modelling of financial returns; but she does not apply it directly to the VaR.

21 Nelson, Daniel B. (1991): Conditional Heteroskedasticity in Asset Returns. A New Approach.Econometrica, Vol. 59, N2, March, pp. 347-370.


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3. Weibulls distribution (type III):

F (Rx) =

0

0).1(exp 1

11

siRfor

siRforR

x

xx (7)

The parameter ( =1/k), is called the tail index, reflects the degree of thicknessof the tails of the distribution of returns. It measures the speed with which the tailapproaches to zero (that is, the velocity of decreasing the probability in the extreme of

the tail). The heavier the tail, the slower the velocity and smaller thetail index .A property of thetail index is that it is equivalent to the number of moments that

exist in a distribution (for example, if = 2, the first moment (mean) as much as thesecond (variance) exist, but the higher moments have an infinite value).

The tail index determines the type of distribution of the tails (of the extremereturns): If 0, it corresponds to a Weibull distribution (type III).

And if = 0, it corresponds to a Gumbel distribution (type I).The Gumbel distribution can be considered as a kind of transition of the Frchet

and Weibull distributions. For small values of, the distributions of type II and III arevery similar to the type I distribution.

GNEDENKO (1943) establishes the necessary and sufficient conditions so thateach concrete probability distribution belongs to one of the three types. For example,the normal and lognormal distributions lead to a Gumbel distribution for their extremesor tails; the Students t follows (its tails) a Frchet distribution where the parameter k isequivalent to the degrees of freedom; the tails of the Pareto stable distribution alsofollow a Frchet distribution where k is equivalent to its standard exponent; the uniformdistribution belongs to the domain of a Weibull distribution.


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These theoretical results show the level of generality reached by the ExtremeValue Theorem: all the mentioned probability distributions possess the same shapeof distribution for their tails (the generalized distribution of the extreme values)with the only difference being the value given to each tail index.

Logically, this Theorem (which marks the start of the Extreme Value Theory24 asone of the central themes in Probability Theory) and these distributions of the tails arenot recent, rather as we have seen, they are somewhat old. What is very recent is the factof the great recent advances achieved, related to the estimations of the parameter, thetail index, which as we have said, is the key to the topic of this paper: to estimate thedegree of thickness of the tails; and moreover, it is allowing us to apply EVT to VaRestimations (this being one of the most interesting contributions and, as we have said,one of the most recent).

We will now proceed to present these latest contributions based on EVT andaimed at solving the problem of fat tails in VaR estimation (section 4) and in some

cases, going a step further, in order to estimate VaR directly, as a further method ofestimation (section 5).

4.- A CRITICAL ANALYSIS OF THE STUDIES BASED ON EVT FORSOLVING THE PROBLEM OF FAT TAILS IN VaR ESTIMATION

We will follow a chronological order arriving at the most recent contribution25,that being HUISMAN et al. (1998)26.

4.1..- The work of LONGIN

LONGIN (1997b) was the first researcher to apply EVT to VaR estimation. Butno significant contribution has been found in estimating the degree of thickness of thetails, that is, the estimation of the tail index (arrived at through the maximum likelihoodmethod). He applies a method27 based on EVT for estimating the daily VaR of a longposition an another short position in the S&P 500 index with different confidence levelsand compares the results to those VaR estimations according to the normality of thereturns.

He obtains similar results to those obtained with the other distributions mentioned

in section 3 (students t distribution, for example): for conservative (small) confidence24 Those readers interested in a rigorous, extensive and complete study of the EVT can consult the

books published by Embrechts et al. (1997), Galambos et al. (1994), Beirlant et el. (1996) or Reiss et

al. (1997); but if one wishes to acquire a basic overview, then we recommend Bassi et al. (1997).

25 We refer here to the specific aspect of fat tails in VaR, not to the specific estimation of VaR. In this last

case the most recent study is by McNeil (1999).

26.Since Hill (1975) proposed his estimator for the tail index, there have been few advances with theresult that the disadvantages that characterise Hill estimator, have not been resolved by the middle of

the 1990s.

27. This specific meted is described in section 5.


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levels, no great difference exists between the VaRs; but for 95% levels of confidence,under the normality assumption he obtains a VaR of 2.93 dollars, much lower than thatobtained through the use of Frchet distribution: 5,72 dollars; which underlines the logicof the greater the confidence level (and thus, the lower the quantile chosen to estimateVaR) the greater the influence of the extreme values in the distribution of the assts

returns; and therefore, the assumption of normality is less appropriate .

LONGIN (1997a), in a later publication obtains the same conclusions, but for a99% level of confidence and comparing it with other models, not only the normal one.However, in this paper he does not even mention how is estimated.

4.2.- The work of DANELSSON et al.

DANELSSON and de VRIES (1997c) and DANELSSON et al. (1997b) domake contributions regarding the estimation of the tail index (they solve the problem bythe Hill estimator regarding the number of observations of the tails to be used for the

estimation) and also they apply them in DANIELSSON and de VRIES (1997b) forestimating VaR: using this estimator of the tail index they hardly obtain any errors inestimating the VaR for five hundred random portfolios, comprised of seven differenttypes or US shares (J. P. Morgan, 3M, McDonalds, Intel, IBM, Xerox y Exxon), fordifferent confidence levels (95%, 99,5%, 99,9% y 99,95%)28.

More recently, DANIELSSON and de VRIES (1997a) establish a new meted forestimating VaR based on EVT (which they call the Tail Kernel Method) which we willconsider in section five.

4.3.- The work of KLPERBERG et al.

KLPERBERG et al. (1998), although not contributing anything new concerningestimates of the degree of thickness of the tail (they use Hills classical estimator), theydo contribute with another method for estimating VaR (which will be described insection five) based in EVT and they compare the daily VaR for a 95% level ofconfidence for a position in the German DAX index, obtained according to their methodand according to the historical simulation method and the method based on thenormality assumption. Once more, the conclusions are an underestimation of VaR giventhe normality assumption.

4.4.- The work of McNEIL

McNEIL (1998), also uses a method of estimating VaR based on EVT and appliesit to BMW shares, but no comparison is made with other methods of VaR estimation,and so no conclusions can be obtained.

4.5.- The work of HUISMAN et al.The most recent contribution has been made by HUISMAN et al. (1998).

Previously, HUISMAN et al. (1997) has proposed and constructed a new estimator of

28. Upon presenting these levels of confidence, most of them higher than 99%, they concentrate onreally extreme or extraordinary events, which, in pure theory, is the principal use of EVT. We would

then like to emphasise (although we will develop it further in section 6) that rather than as a method ofestimating VaR, EVT serves as a complementary analysis, similar tostress testing.


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For their empirical study they use a sample of fortnightly returns of an 18-yearperiod (from January 1980 to July 1988) for two types of assets (the S&P 500 index andthe index for 10-year US Treasury bond).

The use of two types of assets with a different degree of thickness in the tails oftheir distribution of returns (as they themselves estimate, the left tail index for sharesis nearly half: 4,137 than for the bonds: 7,204; that is, the bonds have thinner tails)should have allowed them to obtain further conclusions about the danger of usingdistributions with fat tails when they are not present: something which no other authorcomments on (they all concentrate in contrast, on the danger of no including them whenthey really exist); and the authors, even when having results, they are not recognized(we will present this argument in the final part of this section).

v In the case of S&P 500 index, whose distribution of returns has thicker than normaltails and thicker tails than the bonds distribution, VaR+ (assuming Students t) is

calculated, and so is (assuming normality) and they are compared with the empiricalVaR (calculated according to historical simulation). They calculate 10 days VaRsand for confidence levels ranging from 95% to 99% increasing at 0,5%. Theconclusions reached are consistent with those we have described on a number ofoccasions:For all confidence levels, VaR+ (assuming Students t) practically coincides withthe empirical VaR; while VaR assuming normality, compared with the empiricalVaR, gives overestimated values for small levels of confidence (from 95% to96.5%) and underestimates for levels of confidence from 97% to 99%. Once again,we can deduce the necessity to avoid assuming normality in order to obtain moreaccurate VaR measures.

v In the case of the 10-year American Treasury bonds index, on the other hand, theresults are different and for all levels of confidence, the VaR+ (assuming Studentst) overestimates the empirical VaR; even though they do not recognize it (andindeed they arrive at quite distinct conclusions).It is important to remember that in this case, the tails were much thinner than in theS&P 500 index. This presents us with the danger of including fat tails when inreality they are not present; a theme, as we have said, ignored by those who onlyconcentrate on precisely the opposite: the danger of not including fat tails when inreality they are present.

What is surprising is that these authors, who were able to estimate the degree ofthickness of the tails, not only by estimating the kurtosis (they obtain values of 9,391 forthe S&P 500 and of 5,193 for the bonds, when in the case of a normal distribution, thevalue is 3), but also by estimating the tail index; they are not able to predict these resultsin the case of the bonds.

Although this could be a new line of research in the future, in our view, the errorcould be that they use an estimator for the tail index which, as they say themselves, isgood for small samples, while the sample that they use, although not having anexcessive frequency (fortnightly as opposed to daily) does have a long time period(eighteen years); for this reason, the tail index for bonds is 7,204 and not greater, (it is


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important to remember that for a normal distribution it would be 8) which would allowfor the prediction of these results.

We would like to remark that this work is one of our favourites, and reflectsperfectly what we consider to be the right application of EVT to VaR estimation.

5.- VaR ESTIMATION METHODS BASED ON EVT

The three classical methods : the variance-covariance methods, the historical andthe Monte Carlo simulation are all based on the same approach: they start from theprobability distribution of the returns. The only way they differ is the manner inwhich they model this distribution.

Another completely different approach is that of those methods which startfrom the probability distribution of extreme returns rather than from the distribution of

returns, that is to say, those methods based on EVT.We commented on such a Theory in the previous section and we discussed the

work based on it as a way of solving the fat tails problem in VaR estimations. But wealso mentioned that some of these authors go further and use such a Theory in order todevelop concrete methods of VaR estimation; such as31 LONGIN (1997b),DANELSSON and de VRIES (1997a) and KLPELBERG et al. (1998).

We are going to discuss such methods, but before, as we noted in section 2, wewant to comment that, while constituting the most recent topic of research, our opinionis that EVT serves as a complementary analysis to VaR (and in this way we will

analyse it in section six) more than an estimation approach to VaR in a strict sense.And, if VaR measures the maximum expected loss under normal market

conditions , this means that we concentrate (or should concentrate) on ordinary events(up to a 99% confidence level) and not on extraordinary or extreme events (stockmarket crash of 87, collapse of the 92 EMS; the 94 bond market crash, the December 94and January 95 Mexican peso crisis; the 97 south east Asia currency crisis; thedevaluation of the Russian rouble in august 98 and the Brazilian real In January 99, etc)which are those studied by EVT. This Theory is nor necessary for estimating ruttingquantities in risk management, such as the 10th, 5th and even 1st percentiles. EVT findsits place in the extreme quantities (for example the 0.1 percentile) and it has to remain

in its place.Therefor, in our view, the methods that we are going to describe here based

on this Theory, suffer from a basic weakness . For example DANELSSON and deVRIES (1997a) argue:

31 McNeil (1999), although using this Theory (specifically the POT (Peaks-Over-Threshold) method) for

estimating VaR, in fact what he estimates is the complementary measure to VaR (that is, in extremeconditions) and from there, he deduces VaR.


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VaR is designed to quantify the risk of a large loss, and hence extreme outcomes

are the main component of VaR estimates.

Or LONGIN (1997b), affirms:

Extreme movements are associated with both little tremors like marketadjustments or corrections during ordinary periods, and also earthquake-likestock market crashes, bond market collapses or foreign exchange crises observed

during extraordinary periods.

In conclusion, EVT serves a s a complementary analysis to VaR and not as asubstitute for VaR. This is, also, the spirit behind the phrase that the president of theFederal Reserve, Alan Greenspan, pronounced in the Joint Central Bank Conference"Risk Measurement and Systemic Risk", of the Board of Governors of the Federal

Reserve System , November 16, 1996 in Washington D.C.:

"Work that characterizes the statistical distribution of extreme events would beuseful, as well" .

And not only that, EVT can be very useful for improving the already existingestimation methods of VaR, rather than substituting them.

For example, the risk of modelling in the first (variance-covariance matrixmethod) and in the third (Monte Carlo simulation) methods we have exposed already,could be quantifies from the process of estimating the tail index and the otherparameters in the distribution of the tails of the returns.

Indeed, the choice of the processes used in the Monte Carlo simulation methodcould also be based on the results of the extreme values.

These can constitute two possible future research topics that can introduceimprovements or advances in the work carried out until now in this field.

Having made these qualifying points (that we will develop in section 6), we willnow explain the three methods for estimating VaR based on EVT.

It should be remarked that there are only 32 three empirical works that havebeen undertaken until now (we remain readers that this is also the most up-to-dateresearch topic), so their importance of being pioneers.

The general method based on EVT is comprised of eighth steps that we presentin Figure 1 and which we will comment on below. In fact, the three empirical workscited, follow these eight steps and only differ according to the data and the specifictechniques adopted at each step.

The choice of the length of time of the extreme returns (the third step) is veryimportant, given that the period chosen needs to be of sufficient length so that it

32

Without taking into account the work of McNeil (1999), which cannot be critiqued in the same way aswe have done to these three works.


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satisfies the condition of applicability of the Extreme Value Theorem (explained insection 3)

Given the fact that the theorem is asymptotic, the extreme returns need to beselected from sufficiently long periods, so that the exact distribution of the extreme

returns can be substituted for the asymptotic distribution without problems.As for the estimation of the distribution of extreme returns (the fifth step), it is

here where the three cited methods diverge: LONGIN (1997b) and KLPPELBERG etal. (1998) use the maximum likelihood method; DANELSSON and de VRIES (1997a)do not specify it. Moreover Longin obtains a Frchet distribution and Klppelbergobtains a generalized Pareto distribution. Danelsson and de Vries obtain a mixeddistribution between the empirical distribution (as in the historical simulation method)and a Kernel distribution; in fact, they are the only ones to give a specific name to theirown method, the tail Kernel method.

The only one to specify the test employed in evaluating the goodness of fit (sixthstep) is LONGIN, who uses the test developed by SHEPARD (1957)33.

In addition to the specific points that we made at the beginning of the section,having commented on the methods of VaR estimation based on EVT, we would like tohighlight two interesting features:

1. None of the authors compare this method to the Monte Carlo simulationmethod34: Longin does not compare it with any other method and Danelssonand de Vries and Klpperlberg et al. compare it to the parametric method(variance-covariance) and to the historical simulation method.

2. None of the authors apply it to non-linear portfolios or to options.

Both aspects are the two central and most debated issues in the topic of the VaRestimation35, and for that reason, such an analysis could constitute a future researchtopic that would broaden the results obtained in Coronado (1999a and b).

33 Shepard, L.K. (1957): Percentiles of the On Statistic. Annals of Mathematical Statistics, Vol. 28, pp.

259-268.

34 As we have said, this is the most powerful, flexible and accurate method in addition to being the most

appropriate method for estimating the VaR of non-linear portfolios.

35 See Coronado (1999a and b).


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Figure 1.- Method for estimating VaR based on EVTSource: Longin (1997b)

6.- LIMITATIONS OF VaR AND COMPLEMENTARY ANLISIS (STRESSTESTING, WORST CASE SCENARIO, CRASHMETRICS TM): THEAPPLICATION OF EVT.

The widespread acceptance of VaR at the international level as a measure ofmarket risk in financial institutions, by both managers and supervisors presents thedanger of enjoying an excessive popularity without the recognition of its limitations.

Besides the issue of whether or not the VaR is correctly calculated 36, it isnecessary to bear in mind the exact significance of VaR in order to avoid making wrongdecisions. It is necessary to make it clear what VaR measures and what it does not.

36

A problem, which we cannot treat here but we point to chapter 7 of Coronado (1999b), where wepresent the different systems, used for validating such calculations.

Choose the frequency of returns

Build the history of returns of the position

Choose the length of the selection period

Select extreme returns

Estimate the asymptotic distribution of extreme returns

Goodness-of-fit test of the hypothesis. Does the asymptotic extreme valuedistribution correctly describe the observed extreme returns?

The hypothesis is rejected The hypothesis is not rejected

Choose the extreme probability value

Compute the VaR of the position


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It is necessary to remember that VaR is calculated for a specific level ofconfidence and a given period of time, assuming normal market conditions . That isto say, by definition, VaR does not include all aspects of market risk. And we insistthat it is by definition. This is the result of the conceptual limitation of VaR, not acriticism that one should or can make (as some authors indicate). It is true that with VaR

one cannot estimate or predict losses due to extreme market movements, such as thosethat have occurred in recent years; however VaR was not designed or conceived for thispurpose37. For this reason in this section we analyse complementary, not alternative,measures to VaR38; and it is trough the application of such complementarymeasures to VaR that EVT plays a fundamental role, and where, in our opinion, ithas a place.

It is necessary to remember that the conceptual limitation of VaR is the precisereason why the Basle rules have imposed the controversial multiplier factor which makeit necessary to multiply the VaR in order to calculate the capital requirements of banks.The objective of this factor (fixed presently at three) is to take into account those factors

not contemplated by VaR, such as, for example, extreme situations. Moreover, thisfactor can be increased depending on the result of back testing. However, the choice ofthe multiplier factor has hardly been justified and some specialists, such asBOUDOUKH (1995) consider it arbitrary: "(...) The Bank of England wanted a scale

factor of 1 and the Bundesbank wanted a factor of 5 in such a way that they split".

In addition to this conceptual limitation of VaR, some authors criticize VaRfundamentally because of its lack of qualitative factors39.

The principal limitation of VaR is conceptual. VaR is not a model, it is aconcept; and it is this idea that managers and supervisors must have clear: VaR

measures the level of risk under certain assumptions.

If risk management should try to take into account all normal and extraordinary orextreme events, VaR, by definition, only concentrates on normal or ordinary events(that is, the maximum expected loss under normal market conditions).

This is the key idea when analysing the factors to take into account for the choiceof the confidence level for anyone wanting to estimate VaR. That is to say, it is not aquestion of choosing the maximum possible level of confidence but choosing a levelbetween 95 to 99% which is the interval that determines normal market conditions.Evidently, here the definition one uses for describing an extraordinary or extreme eventcomes into play. Aspects which are credible for one person, for another person are easy

37 Moreover, we should not forget that some portfolios are particularly sensitive to small price changes,

to which they are not hedged and so, can generate losses greater than those assumed for extrememarket movements for which have been covered. See Beder, T. S. (1996): Report Card on Value at

RisK: High Potential but Slow Starter.Bank Accounting & Finance, Vol. 10, N 1 (fall), pp. 14-25.

38 Artzner Et al. (1997 and 1999), present an alternative method to VaR, called ShortFall, which is basedon EVT.

39

Our opinion on such criticisms concerning the lack of qualitative factors has been presented in chapter8 at Coronado (1999b).


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to discard. Where should be the dividing line be drawn between what is typical andwhat is credible? Should one assume that it is possible than three earthquakes couldoccur in Tokyo in the next 10 days? Should one consider the possibility of the marketreaching a value of zero?

It is usual to consider movements which imply a VaR of up to 99 are not extremeones (in sense of crashes) and that from 99% they are extreme events (such as forexample the stock market crash of 87, collapse of the 92 EMS; the 94 bond marketcrash, the December 94 and January 95 Mexican peso crisis; the 97 south east Asiacurrency crisis; the devaluation of the Russian rouble in august 98 and the Brazilian realIn January 99, etc)

The fact that VaR does not take into account these events, does not mean that it isa bad measure but simply it is necessary to complement it with other measures that takethese events into account, and which we describe below; and it is here where the EVTplays a fundamental role.

As said the president of the Chase Manhattan Corporation, LABRECQUE40

(1998), p. 237: "In my view, value at risk is important but it cannot stand alone". Let uscontinue by studying these other complementary measures or analyses.

6.1.- COMPLEMENTARY ANALYSIS

VaR, which is calculated under normal market conditions, does not allow us toknow the consequences of extreme market movements. For this reason it is necessary tocalculate, in addition to the VaR, the possible losses that could result from the financial

institution being open to risk under extreme or less than frequent situations, howeverpossible. This analysis is called stress testing.

A special case half way between VaR and stress testing is known as the worst-case scenario. Such an analysis tries to answer the following question: What is theworst that can happen to the value of a portfolio during a specific period of time? Thatis, from the point of view of risk management, it is interesting to know not only thenumber of times in which maximum loss (provided by VaR) will be exceeded, but alsowhat will be the size of the loss. The analysis of the worst case scenario concentrates onthe analysis of the size of this loss, studying the distribution of the maximum loss.

The analysis of the worst case scenario is similar to stress testing but while thelatter is more associated with the analysis of improbable or extreme events, the analysisof the worst case scenario, concentrates on events that will occur with certainty.

In addition to these two complementary analyses, we are going to mention anotheranalysis: Crashmetrics TM, highly related to the previous two.

40 Labrecque, Thomas G. (1998). Risk Management: One Institutions Experience. Federal Reserve Bankof New York Economic Policy Review, Vol. 4, N 3, (October). Especial Issue, Proceedings of a

Conference: Financial Services at the Crossroads: Capital Regulation in the Twenty-First Century, pp.237-240.


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But, as MEZRICH (1998)43 has emphasized, these recommendations to choosescenarios with which to carry out stress testing are not always valid. He takes theexample of the two of the most recent shocks in currency markets: the devaluation of

the Mexican peso at the end of 94 and that of the Thai bath in July 97 and analyses whatwould have occurred with a short position of a one-year put on the peso on June 1st

1994 and another short position of a one-year put on the bath on the 14th February 1997.

He demonstrates how, in both cases, it would have been necessary to recommendor prove the consequences of a shock of more than 200% (compared to the 20%recommended by the Derivatives Policy Group), in order to have captured what reallyoccurred.

SCHACHTER (1998), has carried out, in our opinion, the best research on stresstesting in the management of market risk (not only with respect to the choice of crisesscenarios). On this specific theme, in addition to the two types of classic scenarios(historic and hypothetical), a third method of choosing scenarios is explained, which isan hybrid between the first two, based on the use of matrices; and it was used in thebank were the author worked (Chase Manhattan Corporation). Thus, it is argued that inthe Chase, the number and nature of hypothetical scenarios used in stress testing hadvaried in response to changes in macroeconomics and political predictions (the price ofcrude oil, the growth of GDP, trade deficit, etc). In this matrix approach, the designer,specifies the maximum and minimum movements of prices and rates based on thehistorical experience. Having determined this range, different steps are establishedwithin the range and the stress testing is carried out evaluating the portfolio for eachdifferent interval within this range.

In truth, each of these three possibilities of choosing crises scenarios (historical,hypothetical and the hybrid of matrices) has its advantages and disadvantages thatSchachter covers well. We recommend this study, although we consider it convenient topresent the advantages and disadvantages of the historical crises scenarios.

Compared to the advantage of being scenarios or crises which have reallyoccurred, 3 disadvantages of this method of choosing scenarios to make stress testingcan be identified:

1. The number of possible scenarios is limited because by definition it deals withinfrequent situations and moreover for some products it is difficult to obtainhistorical data: for example, few emerging markets had bond and stockmarkets in October 1987.

2. In addition, the extreme scenarios that have occurred might not be relevant tothe kind of portfolio belonging to a specific bank. Schachter, in the Chase,recognized that the amount of activity in shares was weak and the stockmarket crisis of 1987 as a crisis scenario is hardly valid.

43 Mezrich, Aliza E. (1998): Learning Curve. Stress Testing.Derivatives Week, (July 27), pp. 7-8.


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3. Having chosen the scenario or specific situation, it is necessary to identify thestart and finish dates of the scenario. This being evident in some of the cases(above all start dates) as in the example of the stock market crash of 17October 1987, but not so for other cases such as the start and finish dates ofthe bond market crisis of 1994.

In any case, in order that the analysis of extreme scenarios is acceptable, it shouldconsider at least these three types of crisis scenarios and in addition be dynamic andchanging.

In fact, as DEMBO (1997b) argues:

"The selection of adequate scenarios is the art of risk management. They are theessence of the risk measurement. Once the scenarios have been determined, one only

has to perform a series of mechanical calculations. Many quantitative techniques exist

which guide the selection of scenarios. These methods are good for generating the most

common events. The adequate choice of extreme scenarios is that which separates theartists from the technicians"

Among the authors that compare the different methods of calculating VaR underturbulent market conditions, we can highlight MORI et al. (1996a and b); SHIMIZUand YAMASHITA (1996); DIMSON and MARSH (1997) and KUPIEC (1998).

On the other hand, some authors do not propose it (stress testing) as acomplementary measure to VaR but rather argue that it should be included in the verycalculation of VaR. They are the ones who specifically defend the use of EVT as analternative to VaR more than as a complementary measure. These authors are discussed

in section 5.In our opinion, the authors that consider that the method based on this Theory can

substitute VaR or combine stress testing with VaR commit a basic weakness.

We repeat here the position of LONGIN (1997b):

"Extreme movements are associated with both little tremors market adjustmentsor corrections during ordinary periods, and also earthquake-like stock market crashes,

bond market collapses or foreign exchange crises observed during extraordinary

periods".

Clearly, according to this definition of extreme movement it would be anequivalent procedure if we apply EVT or the combination of VaR and stress testing;however, we do not agree with this definition.

WILMOTT himself, whose CrashMetricsTM methodology we analyse below, alsorejects the Longin definition and shares our position in his web page:

If value at risk is about normal market conditions, then CrashMetricsTM is the

opposite side of the coin, it is about fire sale conditions and the far-from-orderly

liquidation of assets in far-from-normal-conditions.


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What is obvious is that EVT is an adequate and appropriate tool for analysingextreme situations, complemeting, in this way, VaR.

LONGIN (1997a) define an alternative to stress testing in order to quantify theloss in extreme situations (which he calls Loss beyond the VaR).

To end this part, it is necessary to highlight the major disadvantage in theapplication of EVT, which consists of only having a limited number of extreme eventsto form the sample used for estimating the shape of the extreme returns distribution.With only one or two observations per year, the size of the sample necessary in order tooffer a statistical inference with a degree of confidence should be a number of years.

6.1.2. WORST CASE SCENARIO ANALYSIS

Worst case analysis is half way between VaR and stress testing.

In opposition to VaR, which identifies the average number of occasions thevariation in the value of the portfolio exceeds a specific loss, the worst case scenarioquantifies such losses.

In opposition to stress testing, more associated with the analysis of improbable orextreme events, the worst case analysis concentrates on events that will surely occur.

For a detailed study of this complementary measure see BOUDOUKH (1995) andLONGIN (1997a), who propose the study of the distribution of the maximum losses,making use precisely of EVT.

GONZLEZ MOSQUERA (1995) states that the worst case analysis should

allow for a better explanation of what scale factor should be applied to VaR in order tocalculate the capital requirements on the part of supervisors

6.1.3.- CRASHMETRICS TM

As its name indicates, the CrashMetricsTM methodology developed by WILMOTT(1998) is a dataset and methodology for estimating the exposure of a portfolio toextreme market movements or crashes.

It is based, as explained by the author in his web page(http://www.wilmott.com/crashm.htm) on the fact that the markets behave completely

differently when a crash occurs. As a result, an attempt to study the behaviour ofmarkets in extreme scenarios can only analyse the historical observations thatcorrespond to such events.

The only basic assumption is that the movements in markets have a limit and thenumber of crashes is limited too. There are 3 basic assumptions in this methodology:

Correlations and volatilities of assets are ignored The assumptions on the distributions of assets are minimal In order to estimate possible catastrophic losses as a consequence of extreme

events, it is not necessary to know what would occur on average or with acertain probability, but rather the worst scenario.


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7.- CONCLUSIONS AND FURTHER RESEARCH

The application of EVT to the area of VaR should proceed along 2 tracks:

1. As a complementary (but never substitute) analysis to VaR; more than as anapproach for estimating VaR itself. By definition VaR does not include all aspectsof market risk. With VaR the extreme market movements as those which hadoccurred in recent years can not be estimated or predicted VaR is no t a bad measurebut simply one that should be complemented by others. And it is here where EVTplays a fundamental role. Thus, we have shown its applicability in the use ofcomplementary measures to VaR (stress testing, worst case scenario,CrashMetricsTM), being even an improvement of these complementary measures.We think that the authors who consider that methods based on this EVT cansubstitute VaR or combine VaR with stress testing, commit a basic weakness. Inaddition, such authors do not compare this method with the Monte Carlo simulationmethod nor do they apply it to non-linear positions.

2. As a manner of improving already existing methods of estimating VaR. Thusfor example:

As an additional mechanism for solving the problem of fat tails in estimatingVaR (remember, not the only one). In this sense, the most interestingcontribution is that of Huisman et al. (1998) who propose a Students t, but inorder to obtain estimators of degrees of freedom of the Students t, they useEVT.

The risk of modelling in the variance-covariance method and in the Monte Carlosimulation method could be quantified from the estimation process of the tail

index and other parameters of the distribution of the tails of the returns. Indeed, the choice of processes used in the Monte Carlo simulation methods

could be based also on the results of the extreme values.

In any case, the major disadvantage in the application of EVT, consists of onlyhaving a limited number of extreme events to form the sample used for estimating theshape of the distribution of extreme returns. With only one or two observations per year,the size of the sample necessary in order to offer a statistical inference with a degree ofcertainty should be a number of years.

Finally, so much as VaR has represented an authentic revolution in the financialrisk management, the same cannot be said of EVT. This serves subareas of risk andgaps not covered by VaR. In these subareas, its importance is crucial. We cannot askfrom it more than it can offer, which, in fact, is quite a lot.

If the research done until now allow us to be optimistic in the case of linearpositions, it also makes it necessary to carry out research into the case of more complex,non-linear positions and to take into account the variable volatility.

It is essential that we continue to make improvements with this Theory in theMonte Carlo simulation as a method of estimating VaR.


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8.- REFERENCES

ARTZNER, Philippe et al. (1999): Coherent Measures of Risk. M athematical Finance, N 8(July), pp. 203-228.

ARTZNER, Philippe et al. (1997): Thinking Coherently. Risk, Vol. 10, N 11,

(November).BASLE COMMITTEE ON BANKING SUPERVISION (1996a): Amendment to theCapital Accord to Incorporate Market Risks. Basle Committee on Bank ing Supervision,Basle, (January).

BASLE COMMITTEE ON BANKING SUPERVISION (1996b): SupervisoryFramework for the Use of Backtesting in Conjunction with the Internal ModelsApproach to Market Risk Capital Requirements. Basle Committee on Bank ing Supervision,Basle, (January).

BASLE COMMITTEE ON BANKING SUPERVISION (1995): An Internal ModelBased Approach to Market Risk Capital Requirements. Basle Committee on Bank ingSupervision, Basle, (April).

BASSI, Franco et al. (1997): Survival Kit on Quantile Estimation. Working Paper,Department of Mathematics Swiss Federal Institute of Technology, Zrich.BOUDOUKH, Jacob et al. (1998): The Best of Both Worlds: A Hybrid Approach to

Calculating Value at Risk.Risk, Vol. 11, N 5 (May), pp.64-67.BOUDOUKH, Jacob (1995): Expect the Worst. Risk, Vol. 8, N 9, (September), pp. 100-

101.CORONADO, Mara (2000a): Comparing Different Methods for Estimating VaR for

Non-Linear Actual Portfolios: Empirical Evidence. In Proceedings of the 1st

International Research Conference in Financial Risk Management, organised bythe Global Association of Risk Professionals (GARP), London (June 2000); andIn Proceedings of the VII International Conference on Computational Finance

and Forecasting Financial Markets, organised by London Business School,London (May 2000).CORONADO, Mara (2000b): La tica de los mercados de derivados financieros:

Lecciones a aprender de las prdidas con derivados.Boletn de E studios E conmicos, Vol.LV, N 170, pp. 273-302, Agosto de 2000. Universidad Comercial de Deusto, Bilbao(Espaa).

CORONADO, Mara (1999a): Valor en Riesgo: Aplicacin a Carteras de Opciones enEntidades Financieras. A ctualidad Financiera, N 11, Noviembre, pp. 37-65.

CORONADO, Mara (1999b): Prediccin de Crisis Bancarias: el Value-at-Risk (VaR)como Medida del Riesgo de Mercado. (Bank Failures Prediction: VaR as a MarketRisk Measure, PhD Dissertation) Tesis Doctoral, Universidad P. Comillas de Madrid

(April 27th

).CHEW, Lillian (1994): Shock Treatment. R isk, Vol. 7, N 9, (September), pp. 63-70.CHRISTOFFERSEN (1998): Horizon Problems and Extreme Events in Financial Risk

Management. Federal Reserve Bank of N ew Y ork E conomic Policy, (October), pp. 109-118.DANELSSON, Jn et al. (1998a): The Cost of Conservatism: Extreme Returns, Value-

at-Risk and the Basle "Multiplication Factor".Risk. Vol. 11, N 1, (January), pp. 101-103.

DANELSSON, Jn et al. (1998b): The Value of Value at Risk: Statistical, Financial andRegulatory Considerations. Federal Reserve Bank of N ew Y ork E conomic Policy Review,Vol. 4, N 3, (October). Especial Issue, Proceedings of a Conference: FinancialServices at the Crossroads: Capital Regulation in the Twenty-First Century, pp. 107-

109.


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