extreme v alue theory as a risk managemen t to ol aul em ...janroman.dhis.org/finance/risk...

Extreme value theory as a risk management tool�

Paul Embrechts� Sidney I� Resnick and Gennady Samorodnitsky

Abstract

The �nancial industry� including banking and insurance� is undergoing ma�jor changes� The �re�insurance industry is increasingly exposed to catastroph�ic losses for which the requested cover is only just available� Due to an in�creasing complexity of �nancial instruments� sophisticated risk managementtools have to be put into place� The securitization of risk and alternative risktransfer highlight the convergence of �nance and insurance at the productlevel� Extreme value theory plays an important methodological role withinthe above�

� Introduction

Consider the time series in Table � of loss�ratios �yearly data� for earthquake insur�ance in California from �� till �� The data are taken from Jae and Russell��

��

Table �� California earthquake data

�A �rst version of this paper was presented by the �rst author as an invited paper at the

XXVIIIth International ASTIN Colloquium in Cairns and published in the conference proceedings

under the title �Extremes and Insurance�� Sidney Resnick and Gennady Samorodnitsky were

partially supported by NSF Grant DMS�� at Cornell University� Paul Embrechts gratefully

acknowledges the hospitality of the School of Operations Research and Industrial Engineering�

Cornell University during Fall ��

�

On the basis of these data� who would have guessed the �� value of � �� In�deed� on the ��th of January of that year the �� Richter scale Northridge earthquakehit California causing an insured damage of USD �� billion and a total damage ofUSD �� billion� making �� the year with the third highest loss burden �naturalcatastrophes and major losses� in the history of insurance The front�runners in thissad hit parade are �� the year of hurricane Andrew� and �� the year of thewinter storms Daria and Vivian� For details on these� see Sigma ��

The reinsurance industry experienced a rise in both intensity as well as magnitudeof losses due to natural and man�made catastrophes For the United States alone�Canter� Cole and Sandor �� estimate an approximate USD �� billion of capitalin the insurance and reinsurance industry to service a country that has USD �to �� trillion worth of property No surprise that the �nance industry has seizedupon this by oering �often in joint ventures with the �re�insurance world� properlysecuritized products in the realm of catastrophe insurance At an increasing pace�new products are being born Some of them only have a short life� and others arereborn under a dierent shape Some do not survive Examples include�

� CAT futures and PCS options �CBOT�� in these cases� securitization is achiev�ed through the construction of derivatives written on a newly constructedindustry wide loss�ratio index

� Convertible CAT bonds The Winterthur convertible hail�bond is an exampleThis European type convertible has an extra coupon payment contingent onthe occurrence of a well�de�ned cat�event� an excessive number of cars inWinterthur�s Swiss portfolio damaged within a hail storm over a speci�c timeperiod For details� see Schmock ��

Further interesting new products are the so�called multi�line� multi�year� high layer�infrequent event� products� credit lines� the catastrophe risk exchange �CATEX�For a brief review on some of these instruments� see Punter �� Excellentoverviews stressing more the �nancial engineering of such products are Doherty�� and Tilley �� This whole area of alternative risk transfer and securitiza�tion has become a major area of applied research in both the banking and insuranceindustry Actuaries are actively taking part in some of the new product develop�ment� and therefore have to consider the methodological issues underlying these andsimilar products

Also recently� similar methods are being introduced in the world of �nancethrough the estimation of Value�at�Risk �VaR� and the so�called shortfall� seeBassi� Embrechts and Kafetzaki �� and Embrechts� Samorodnitsky and Resnick�� Value At Risk for End�Users �� contains a recent summary of someof the more applied issues More generally� extremes matter eminently within theworld of �nance It is no coincidence that Alan Greenspan� Chairman of the Board ofGovernors of the FED� remarked at a research conference on risk measurement andsystemic risk �Washington� DC� �� November �� that �Work that characterizesthe statistical distribution of extreme events would be useful� as well�

For the general observer� extremes in the realm of �nance manifest themselvesmost clearly through stock market crashes or industry losses In Figure �� we have

�

plotted the events leading up to and including the �� crash for equity data �S�P�Extreme Value Theory �EVT� yields methods for quantifying such events and theirconsequences in a statistically optimal way See for instance McNeil �� foran interesting discussion on the �� crash example For a general equity bookfor instance� a risk manager will be interested in estimating the resulting down�siderisk which typically can be reformulated in terms of a quantile for a Pro�t�and�Loss�P � L� function

Figure �� Crash

Another area of risk�management research where EVT is increasingly playing animportant role is in credit risk management The interested reader may browseJP Morgan�s web site for information on Credit Metrics It is no coincidence thatbig investment banks are looking at actuarial methods for the sizing of reserves toguard against future credit losses Swiss Bank Corporation for instance introducedACRA �Actuarial Credit Risk Accounting� for credit risk management� see Figure In their risk measurement framework� they use the following de�nitions�

expected loss� the losses which must be assumed to arise on a continuingbasis as a consequence of undertaking particular business�

unexpected losses� the unusual� though predictable� losses which the Bankshould be able to absorb in the normal course of itsbusiness�

stress loss� the possible � although improbable � extreme scenarioswhich the Bank must be able to survive

EVT oers an important set of techniques for quantifying the boundaries betweenthese dierent loss�classes Moreover� EVT oers a scienti�c language for translatingmanagement guidelines on these boundaries into actual numbers A �nal examplewhere EVT oers help is in the area of modeling default probabilities and the es�timation of so�called diversi�cation factors in the management of bond portfoliosMany more examples could have been added

�

500 1000 1500 2000 2500 3000

Frequency of loan losses

Losses

Traditionalsystems

New Systems

Sta

tist

ical

mea

nExpected loss Unexpected loss Stress loss

Absorbed by net profit Absorbed by capital

Absorbed by ACRA reserveAbsorbed bynet profit

Absorbed bycapital

Figure � ACRA� Actuarial Credit Risk Accounting

It is our aim in this paper to review some of the basic tools from EVT relevantfor industry wide risk management Some examples towards the end of the papershould give the reader a better idea of the kind of answers EVT provides Most of thematerial covered in the paper �and indeed much more� is to be found in Embrechts�Kl�uppelberg and Mikosch �� The latter book also contains an extensive list offurther references For reference of speci�c results in this book we will occasionallyrefer to it as EKM �� where �� refers to a speci�c result

� The basic theory

As already �implicitly� stated in the introduction� the statistical analysis of extremesis key to many of the risk management problems related to insurance� reinsuranceand �nance In order to review some of the basic ideas underlying EVT� we discussthe most important results under the simplifying iid assumption� losses will beassumed to be independent and identically distributed Most of the results can beextended to much more general models In Example � a �rst indication of sucha generalization will be given

Throughout this paper� losses will always be denoted as positive� consequentlywe concentrate in our discussion below on one�sided distribution functions �dfs� forpositive random variables �rvs�

Given basic loss data

X�� X�� Xn iid with df F � ��

we are interested in the random variables

Xn�n � min �X�� Xn� � X��n � max �X�� Xn� � � �

�

Or indeed using the full set of so�called order statistics

Xn�n � Xn��n � � � � � X��n � ��

we may be interested in

kXr��

hr �Xr�n� ��

for certain functions hr� r � �� k and k � k�n� An important example corre�sponds to hr �

�k� r � �� k� ie we average the k largest losses X��n� � � � � Xk�n

Another important example would be to take k � n� hr�x� � �x� u�� where y� �max�� y�� for a given level u � � In this case we sum all excesses over u of losseslarger than u Typically we would normalize this sum by the number of such ex�ceedances yielding the so�called empirical mean excess function� see Example ��Most of the standard reinsurance treaties are of �or close to� the form �� The lastexample given corresponds to an excess�of�loss �XL�� treaty with priority u

In �classical� probability theory and statistics most of the results relevant forinsurance and �nance are based on sums

Sn �nX

r��

Xr �

Indeed the laws of large numbers� the central limit theorem �in its various degrees ofcomplexity�� re�nements like Berry�Ess�een� Edgeworth� saddle�point and normal�power approximations all start from Sn�theory Therefore� we �nd in our sum�toolkit such items like

� the normal distributions N ��

� the ��stable distributions� � � � � �

� Brownian motion�

� ��stable processes� � � � �

We are con�dent in our sum�toolkit when it comes to modeling�pricing�settingreserves of random phenomena based on averages Likewise we are con�dent instatistical techniques based on these tools when applied to estimating distributiontails �not too far� from the mean Consider however the following easy exercise

Exercise� It is stated that within a given portfolio� claims follow an exponential dfwith mean �� thousand dollars� say� We have now observed �� such claims withlargest loss �� Do we still believe in this model� What if the largest loss wouldhave been ��

�

Solution�

The basic assumption yields that

X�� X�� are iid with df P �X� � x� � �� e�x�� x � � �

Therefore� for Mn � max �X�� Xn��

P �M�� x� � �� P �X� � x��

� �� e�x��

��

From this� we immediately obtain

P �M��

P �M��

However� rather than doing the �easy� exact calculations above� consider the follow�ing asymptotic argument First� for all n � � and x � R�

P

�Mn

�� logn � x

�� P �Mn � ��x� logn��

�

��

e�x

n

�n

�

so that

limn��

P

�Mn

�� logn � x

�� e�e

�x

� �x� �

Therefore� use the approximation

P �Mn � x� � � x

�� logn

�to obtain

P �M��

P �M��

very much in agreement with the exact calculations above �

Suppose we were asked the same question� but now we would have much less speci�cinformation on F �x� � P �X� � x�� could we still proceed� This is exactly the pointwhere classical EVT enters In the above exercise� we have proved the following

Proposition � Suppose X�� Xn are iid with df F EXP�� then for x � R�

limn��

P ��Mn � logn � x� � �x� � �

�

Here are the key questions�

Q�� What is special about � Can we get other limits� possibly for otherdfs F �

Q � How do we �nd the norming constants � and logn in general� Thatis� �nd an and bn so that

limn��

P

�Mn � bn

an� x

�exists

Q�� Given a limit coming out of Q�� for which dfs F and norming constantsfrom Q � do we have convergence to that limit� Can one say somethingabout second order behaviour� ie speed of convergence�

The solution to Q� forms part of the famous Fisher�Tippett theorem

Theorem � �EKM�Theorem �� Suppose X�� Xn are iid with df F and�an�� bn� are constants so that for some non�degenerate limit distribution G�

limn��

P

�Mn � bn

an� x

�� G�x� � x � R �

Then G is of one of the following types�

� Type I �Fr�echet��

!��x� �

�� x � �

� � � �exp f�x��g � x � �

� Type II �Weibull��

"��x� �

��exp f��x��g � x � �

� � � �� x � �

� Type III �Gumbel��

�x� � exp�e�x

�� x � R � �

G is of the type H means that for some a � �� b � R� G�x� � H��x� b�a�� x � R�and the distributions of one of the above three types are called extreme value dis�tributions Alternatively� any extreme value distribution can be represented as

H��x� � exp

��

��

x� �

�

��

� x � R �

�

Figure �� Some examples of extreme value distributions H�� for � �� Fr�echet�� Gumbel� and � �� Weibull�

Here � R� � � R and � � � The case � � � � �� corresponds to the Fr�echet�Weibull��type df with � �� whereas by continuity � � corre�sponds to the Gumbel or double exponential�type df

In Figure �� some examples of the extreme value distributions are given Notethat the Fr�echet case �the Weibull case� corresponds to a model with �nite lower�upper� bound� the Gumbel is two�sided unbounded

Answering Q and Q� is much more complicated Below we formulate a completeanswer �due to Gnedenko� for the Fr�echet case This case is the most important forapplications to �re�insurance and �nance For a general df F � we de�ne the inverseof F as follows�

F��t� � inffx � R � F �x� � tg � � � t � � �

Using this notation� the p�quantile of F is de�ned as

xp � F��p� � � � p � � �

Theorem � �EKM �Theorem �� Suppose X�� Xn are iid with df F sat�isfying

limt��

�� F �tx�

�� F �x�� x�� x � � � � � � � ��

Then for x � ��

limn��

P

�Mn � bn

an� x

�� !��x�

where bn � � and an � F��

n

� The converse of this result also holds true �

�

A df F satisfying �� is called regularly varying with index �� denoted by F �� F � R�� An important consequence of the condition F � R�� is that fora rv X with df F �

EX�

�� for � � � �� for � � � �

��

In insurance applications� one often �nds ��values in the range �� whereas in�nance �equity daily log�returns say� an interval � � �� is common Theorem � isalso reformulated as� the maximal domain of attraction of !� is R�� ie

MDA�!�� R��

Dfs belonging to R�� are for obvious reasons also called of Pareto type Though wecan calculate the norming constants� the calculation of an depends on the tail of Fwhich in practice is unknown The construction of MDA �"�� is also fairly easy� themain dierence being that for F � MDA�"��

xF � supfx � R � F �x� � �g � �

The analysis of MDA� � is more involved It contains such diverse dfs as the ex�ponential� normal� lognormal� gamma For details� see Embrechts� Kl�uppelberg andMikosch �� Section ��

� Tail and quantile estimation

Theorem � is the basis of EVT In order to show how this theory can be put intopractice� consider for instance the pricing of an XL�treaty Typically� the priority�or attachment point� u is determined as a t�year event corresponding to a speci�cclaim event with claim size df F � say This means that

u � ut � F��

�

t

��

In our notation used before� ut � x��t Whenever t is large �typically the case inthe catastrophic �ie rare� event situation�� the following result due to Balkema�de Haan� Gnedenko and Pickands �see Embrechts� Kl�uppelberg and Mikosch ��Theorem ��b�� is very useful

Theorem Suppose X�� Xn are iid with df F Equivalent are�

i� F � MDA�H�� R�

ii� for some function � � R� � R��

limu�xF

sup��x�xF�u

��Fu�x��G��u��x��

��

where Fu�x� � P �X � u � x j X � u�� and the generalized Pareto df is givenby

G��x� � ��

��

x

�

��

� ��

for � � � �

It is exactly the so�called excess df Fu which both risk managers as well as reinsurersshould be interested in Theorem � states that for large u� Fu has a generalizedPareto df �� Now in order to estimate the tail F �u� x� for a �xed large value of uand all x � � consider the trivial identity

F �u� x� � F �u�F u�x� � u� x � � � ��

In order to estimate F �u� x�� one �rst estimates F �u� by the empirical estimator�F �u�

�b

�Nu

n

where Nu � # f� � i � n � Xi � ug In order to have a �good� estimator for F �u��we need u not too large� the level u has to be well within the data Given sucha u�value� we approximate F u�x� via �� by�

F u�x��b

� G��u��x�

for some estimators $ and $��u� depending on u For this to work well� we need u large�indeed in Theorem �ii u � xF � the latter being � in the Fr�echet case� A �good�estimator is obtained via a trade�o between these two con%icting requirementson u

The whole statistical theory developed in order to work out the above programruns under the name Peaks over Thresholds Method and is discussed in detail inEmbrechts� Kl�uppelberg and Mikosch �� Section �� McNeil and Saladin ��and references therein Software �S�plus� implementation can be found on

http��wwwmathethzch�mcneil�software

This maximum likelihood based approach also allows for modeling of the excessintensity Nu� as well as the modeling of time �or other co�variable� dependence inthe relevant model parameters As such� a highly versatile modeling methodology forextremal events is available Related approaches with application to insurance areto be found in Beirlant� Teugels and Vynckier �� Reiss and Thomas �� andthe references therein Interesting case studies using up�to�date EVT methodologyare McNeil �� Resnick �� and Rootz�en and Tajvidi �� The varioussteps needed to perform a quantile estimation within the above EVT context arenicely reviewed in McNeil and Saladin �� where also a simulation study is tobe found In the next section� we illustrate the methodology on real and simulateddata relevant for insurance and �nance

��

� Examples

�� Industrial �re insurance data

In order to highlight the methodology brie%y discussed in the previous sections� we�rst apply it to �� industrial �re insurance claims We show how a tail��t and theresulting quantile estimates can be obtained Clearly� a full analysis �as for instanceto be found in Rootz�en and Tajvidi �� for windstorm data� would require a lotmore work We have grouped the �gures towards the end of the example

Figure �� Log�histogram of the �re insurance data

Figure � contains the log�histogram of the data The right�skewness stresses thelong�tailed behaviour of the underlying data A useful plot in order to specify thelong�tailed nature of data is the so�called mean�excess plot as given in Figure � Init� the mean�excess function e�u� � E�X � u j X � u� is estimated by its empiricalcounterpart

en�u� ��

# f� � i � n � Xi � ug

nXi��

�Xi � u��

The Pareto df can be characterized by linearity �positive slope� of e�u� In general�long�tailed dfs exhibit an upwards sloping behaviour� exponential�type dfs haveroughly a constant mean excess plot� whereas short�tailed data yield a plot decreas�ing to � In our case� the upward trend clearly stresses the long�tailed behaviourThe increase in variability towards the upper end of the plot is characteristic of thetechnique� since towards the largest observation X��n� only few data points go intothe calculation of en�u� The main aim of our EVT�analysis is to �nd a �t of theunderlying df F �x� �or of its tail F �x�� by a generalized Pareto df� especially for thelarger values of x The empirical df F n is given in Figure � on a doubly logarith�mic scale This scale is used to highlight the tail region Here an exact Pareto dfcorresponds to a linear plot

�

��

��

��

��

��

��

�

�

�

�

Figure �� Mean�excess plot of the �re insurance data

� ��

�

�

Figure �� Empirical estimator of F on doubly logarithmic scale

Using the theory presented in Theorems and �� a maximum likelihood basedapproach yields estimates for the parameters of the extreme value df H�� and thegeneralized Pareto df G�� In order to start this procedure� a threshold value u hasto be chosen as estimates depend on the excesses over this threshold The estimatesof the key shape parameter as a function of u �alternatively� as a function of thenumber of order statistics used� is given in Figure � Approximate ��& con�denceintervals are given The picture shows a rather stable behaviour for values of ubelow �� say An estimate in the range �� results� which corresponds to an��value in the range �� It should be remarked that the �optimal� value ofthe threshold u to be used is di'cult �if not impossible� to obtain See Embrechts�Kl�uppelberg and Mikosch �� p �� and Beirlant� Teugels and Vynckier ��for some discussion We also would like to stress that in order to produce Figure ��a multitude of models �one for each u chosen� has to be estimated

For each given u� a tail��t for F u and F �as in �� can be obtained For theformer� in the case of u � �� an estimate $ � �� results A graphical represen�

��

Figure �� Maximum likelihood estimate of as a function of the threshold u �top��alternatively� as a function of the number of exceedances

tation of $F�� is given in Figure � Using the parameter estimates corresponding tou � �� in �� the tail��t of F on doubly logarithmic scale is given in Figure �

��

��

��

��

��

Figure �� Maximum likelihood �t of the mean excess tail df F u based on exceedancesabove u � ��

Though we have extended the generalized Pareto �t to the left of u � �� clearlyonly the range above this u�value is relevant The �tting method is only designedfor the tail Below u �where typically data are abundant� one could use a smoothversion of the empirical df From the latter plot� quantile estimates can be deduced

Figure �� contains as an example the estimate for the ��& quantile x� to�gether with the pro�le likelihood The latter can be used to �nd con�dence intervalsfor x� The ��& and ��& intervals are given

Figure �� contains the same picture but the �symmetric� con�dence intervals are

��

� ��

�

�

Figure �� Tail��t for F based on a threshold value of u � �� doubly logarithmicscale

� ��

�

�

� ��

�

�

Figure �� Tail��t with an estimate for x� and the corresponding pro�le likelihood

calculated using the Wald statistic Finally� the ��& quantile estimates acrossa whole range of models �depending on the threshold value� or number of exceedancesused� are given in Figure � Though the estimate of x� settles between �� and�� the ��& Wald intervals are rather wide� ranging from �� to about ��

The above analysis yields a summary about the high quantiles of the �re insur�ance data based on the information on extremes available in the data The analysiscan be used as a tool in the �nal pricing of risks corresponding to high layers �iecatastrophic� rare events� All the methods used are based on extremes and arefairly standard

��

� ��

�

�

Figure �� Estimate of x� with ��& Wald�statistic con�dence interval

Figure � � Estimates of the quantile x� as a function of the threshold u Thevertical line indicates the model corresponding to u � ��

�� An ARCH�example

To illustrate further some of the available techniques� we simulated an ARCH��time series of length �� The time series� called testarch� has the form

n � Xn

�� n��

�� n � � � ��

where fXng are iid N�� random variables In our simulation� we took

� � � � � � ��

From known results of Kesten �� see also Embrechts� Kl�uppelberg and Mikosch�� Theorem �� Goldie �� Vervaat ��

P �� x� cx�� x� � ��

��

and we get from Table � of de Haan et al �� that

� � ��

�see also Hooghiemstra and Meester ��

There are several reasons why we choose to simulate an ARCH process

� Despite the fact that the ARCH process is dependent� much of the classicalextreme value analysis applies with suitable modi�cations

� The ARCH process has heavy tails which matches what is observed in datasets emerging from �nance

� Although it is often plausible to model large insurance claims as iid� datafrom the �nance industry such as exchange rate data are demonstrably notiid Some of these examples have the property that the data look remarkablyuncorrelated but squares or absolute values of the data appear to have highcorrelations It is this property that the ARCH process and its cousins weredesigned to model See for instance Taylor �� for more details

To experiment with this ARCH data we took the �rst �� observations toform a data set shortarch which will be used for estimation Then based on theestimation� some model based predictions can be made and compared with actualdata in testarchnshortarch

Figure �� shows a time series plot of shortarch The plot exhibits the character�istic heavy tail appearance

Figure �� Time series plot of shortarch

The Hill estimator is a popular way of detecting heavy tails and estimatingthe Pareto index � for � � in �� See for instance Embrechts� Kl�uppelbergand Mikosch �� for an introduction Figure �� contains four views of the Hillestimator applied to shortarch

��

Figure �� Hill plots of shortarch

The Hill estimator is known to be a consistent estimator of � for the ARCHprocess �Resnick and St(aric(a �� In our case� the Hill estimator is trying toestimate � � �� A review of Figure �� yields an estimate of about � IfHk�n represents the Hill estimator when the sample is n and k upper order statisticsare used� ie

Hk�n �

��

k

kXj��

logXj�n � logXk�n

��

the usual methodology is to make a Hill plot�k�H��

k�n

�� k � n

� The upper

left graph is a Hill plot with some values for small and large k deleted to makethe picture scale attractively The upper right plot is the Hill plot in alt scale �see

Resnick and St(aric(a� �� where we plot

�� H��

)n�*�n

��

� The lower left

plot applies a smoother �Resnick and St(aric(a� �� and plots in alt scale

A supplementary tool for estimating the Pareto index is the QQ plot �see Em�brechts� Kl�uppelberg and Mikosch �� Section � �� and this has the addedadvantage that it allows simultaneous estimation of the constant c appearing in�� The method is sensitive to the choice of the number of upper order statisticsand some trial and error is usually necessary In Figure �� we give the QQ plotbased on the upper �� order statistics This gives estimates of � � �� andc � �� Applying this technique to the full testarch data produced estimatesof � � �� and c � �� when the number of upper order statistics was

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�

�

Figure �� QQ plot of shortarch

Based on these estimates� we experiment with some predictions and comparethese predictions with what is observed from the part of the data set testarch� calledplayarch obtained by removing the �� shortarch observations Thus the lengthof playarch is �� In the following table we give estimatedmarginal probabilities that the ARCH variable exceeds x for x � �� Notethat we are predicting values that are beyond the range of the data and have notbeen observed The second row gives the estimate �� based on the �tted valuesfor c and � In the third row we compute the empirical frequency that elements ofplayarch exceed x The last row gives the corresponding probabilities ��! �x� �� based on a normal distribution whose mean and variance are the sample meanand variance computed from shortarch One can see from the table the penaltypaid for ignoring extreme value analysis and relying on more conventional normaldistribution based analysis

x � �� P �X � x� �� bP �X � x� ��

�� ! �x� ��

The extreme value theory for the ARCH process is somewhat complicated bythe ARCH dependence structure not present for an iid sequence A quantity calledthe extremal index must be accounted for� see Embrechts� Kl�uppelberg and Mikosch�� Section �� From �� and de Haan et al �� Table � we have

P �max fi� � � � � ng � y� � exp

��

c �ny��

��

where the extremal index � � �� accounts for the eective reduction in samplesize due to dependence From this formula� estimates of upper quantiles can be

��

worked out The upper ��p& quantile xp would be

xp �

�� log�� p�

c �n

��

We give a few representative values�

y �� P �max fi� � � � � ng � y� � � � ��

p �� xp ��

�� Value�at�Risk� a word of warning

We already pointed out the similarity in estimating attachment points or retentionsin reinsurance and VaR calculations in �nance Both are statistically based methods�where the basic underlying risk measure corresponds to a quantile estimate bxp ofan unknown df Through the work of Artzner et al �� we know thata quantile based risk measure for general �non�normal� data fails to be coherent�ie such a measure is not subadditive creating inconsistencies in the construction ofrisk capital based upon them This situation typically occurs in portfolios containingnon�linear derivatives Further critical statements concerning VaR are to be foundin Danielsson� Hartmann and de Vries �� Garman �� Longin ��a�b�and C�ardenas et al ��

A much better� and indeed �almost� coherent risk measure is the conditionalVaR �or so�called mean excess�

E �X j X � bxp� � ��

For the precise formulation of the �almost� bove� see Artzner et al �� Weonly want to point out that the latter measure is well�known in insurance� butonly gradually is being recognized as fundamental to risk management It is oneof the �many� examples where an exchange of ideas between actuaries and �nanceexperts may lead to improved risk measurement Note that in the equation above�E �X j X � bxp� � e �bxp��bxp One could use the mean excess plot f�u� en�u�� u � �gin order to visualize the tail behavior of the underlying data and hence get someinsight on the coherent risk measure �� above

As a �nal example we have plotted in Figure �� the daily log�returns of BMWover the period January �� July � � �� together with the mean excess plotof the absolute values of the negative returns �hence corresponding to downside risk�In the latter plot� the heavy�tailed nature of the returns is very clear Also clearis the change in curvature of the plot around �� This phenomenon is regularlyobserved in all kinds of data One can look at it as a pictorial view of Theorem ��indeed� Smith �� indicates how to base on this observation a graphical toolto determine an initial threshold value for an extreme value analysis See also

�

��

��

��

��

��

�

��

��

��

��

�

�

Figure �� Time series and mean excess plots of BMW return data

Embrechts� Kl�uppelberg and Mikosch �� p �� for a discussion We would liketo warn the reader however that due to the intricate dependencies in �nance data�one should be careful in using these plots beyond the mere descriptive level

More work is needed to combine the ideas presented in this paper with detailedstatistical information on �nancial time series before risk measures such as condi�tional VaR �� can be formulated precisely and estimated reliably Once more�the interplay between statisticians� �nance experts and actuaries should prove to befruitful towards achieving this goal

References

Artzner� P� Delbaen� F� Eber� J�M and D Heath �� A characterization ofmeasures of risk Universit�e de Strasbourg preprint

Artzner� P� Delbaen� F� Eber� J�M and D Heath �� Thinking coherentlyRISK� ��

Bassi� F� Embrechts� P and M Kafetzaki �� Risk Management and quantileestimation In� Adler� R� Feldman� R and Taqqu� MS �Eds� A Practi�cal Guide to Heavy Tails� Statistical Techniques for Analysing Heavy�TailedDistributions Birkh�auser� Boston

Beirlant� J� Teugels� JLT and P Vynckier �� Practical analysis of extremevalues Leuven University Press� Leuven

Canter� MS� Cole� JB and RL Sandor �� Insurance derivatives� a new assetclass for the capital markets and a new hedging tool for the insurance industryJournal of Derivatives� ��

�

C�ardenas� J� Fruchard� E� Koehler� E� Michel� C and I Thomazeau �� VaR�One step beyond RISK ��

Danielsson� J� Hartmann� P and CG de Vries �� The cost of conservatismRISK ��

Doherty� NA �� Financial innovation in the management of catastrophe riskJoint Day Proceedings� XXVIIIth International ASTIN Colloquium and �thInternational AFIR Colloquium� Cairns �Australia��

Embrechts� P� Kl�uppelberg� C and T Mikosch �� Modelling extremal eventsfor insurance and nance Springer�Verlag� Berlin

Embrechts� P� Samorodnitsky� G and SI Resnick �� Living on the edgeRISK ��

Garman� M �� Taking VaR to pieces RISK �

Goldie� CM �� Implicit renewal theory and tails of solutions of random equa�tions Annals of Applied Probability ��

Haan� L de� Resnick� SI� Rootz�en� H and CG de Vries �� Extremal be�haviour of solutions to a stochastic dierence equation with applications toARCH processes Stoch Proc Appl ��

Hooghiemstra� G and L Meester �� Computing the extremal index of specialMarkov chains and queues Stochastic Process Appl ��

Jae� DM and T Russell �� Catastrophe insurance� capital markets anduninsurable risk Financial Institutions Center� The Wharton School� ��

Kesten� H �� Random dierence equations and renewal theory for products ofrandom matrices Acta Math ��

Longin� FM ��a� From value at risk to stress testing� the extreme value ap�proach ESSEC and CEPR Paris preprint

Longin� FM ��b� Beyond the VaR ESSEC and CEPR Paris preprint

McNeil� AJ �� Estimating the tails of loss severity distributions using extremevalue theory ASTIN Bulletin ��

McNeil� AJ �� On extremes and crashes RISK ��

McNeil� AJ and T Saladin �� The peaks over threshold method for estimatinghigh quantiles of loss distributions Proceedings of the XXVIIIth InternationalASTIN Colloquium� Cairns� ��

Punter� A �� A critical evaluation of progress to date In� Practical appli�cations of nancial market tools to corporate risk management InternationalRisk Management� p ��

Reiss� R�D and M Thomas �� Statistical Analysis of Extremal Values� Birk�h�auser� Basel

Resnick� SI �� Discussion of the Danish data on large �re insurance lossesASTIN Bulletin ��

Resnick� SI and C St(aric(a �� Tail index estimation for dependent data Avail�able as TR��psZ at http��wwworiecornelledu�trlist�trlisthtmlTo appear in Ann Applied Probability

Resnick� SI and C St(aric(a �� Smoothing the Hill estimator Adv AppliedProbability � � ��

Rootz�en� H and N Tajvidi �� Extreme value statistics and windstorm losses�a case study Scand Actuarial J ��

Schmock� U �� Estimating the value of the Wincat coupons of the WinterthurInsurance convertible bond Joint Day Proceedings� XXVIIIth InternationalASTIN Colloquium and �th International AFIR Colloquium� Cairns �Aus�tralia��

Sigma �� Natural catastrophes and major losses in �� Third highest lossburden in the history of insurance Swiss Reinsurance Company� ��

Sigma �� Natural catastrophes and major losses in �� high losses from man�made disasters� but not extremely costly losses from natural catastrophesSwiss Reinsurance Company� ��

Smith� RL �� Extreme value theory In� Ledermann� W �Chief Ed� Hand�book of Applicable Mathematics� Supplement� pp �� Wiley� Chichester

Taylor� SJ �� Modelling Financial Time Series� Wiley� Chichester

Tilley� JA �� The securitization of catastrophic property risks Joint Day Pro�ceedings� XXVIIIth International ASTIN Colloquium and �th InternationalAFIR Colloquium� Cairns �Australia��

Value at Risk for End�Users �� Risk Magazine� Supplement� March ��

Paul Embrechts Sidney I� Resnick and Gennady SamorodnitskyDepartment of Mathematics School of Operations ResearchETHZ and Industrial EngineeringCH � �� Zurich� Switzerland Cornell Universitye�mail� embrechts math�ethz�ch Rhodes Hall�ETC Building

Ithaca� New York �� USAe�mail� sid orie�cornell�edue�mail� gennady orie�cornell�edu

extreme v alue theory as a risk managemen t to ol aul em ...janroman.dhis.org/finance/risk...

Documents