Ž.Journal of Empirical Finance 72000271–300
r locate r econba
Estimation of tail-related risk measures for
heteroscedastic financial time ries:an extreme
value approach
Alexander J.McNeil a,),Rudiger Frey b,1¨
a
Department of Mathematics,Swiss Federal Institute of Technology,ETH Zentrum,
CH-8092Zurich,Switzerland b Swiss Banking Institute,Uni Õersity of Zurich,Plattenstras 14,CH-8032Zurich,Switzerland Abstract
Ž.We propo a method for estimating Value at Risk VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return ries.Our approach combines
pudo-maximum-likelihood fitting of GARCH models to estimate Ž.the current volatility and extreme value theory EVT for estimating the tail of the innovation distribution of the GARCH model.We u our method to estimate conditional Ž.Žquantiles VaR and conditional expected shortfalls the expected size of a return exceeding .VaR ,this being an alternative measure of tail risk with better theoretical properties than the quantile.Using backtesting of historical daily return ries we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility.With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.q 2000Elvier Science B.V.All rights rerved.
波若JEL classification:C.22;G.10;G.21
Keywords:Risk measures;Value at risk;Financial time ries;GARCH models;Extreme value theory;Backtesting
Corresponding author.Tel.:q 41-1-632-61-62;fax:q 41-1-632-15-23.
Ž.Ž.E-mail address:hz.ch A.J.McNeil ,freyr@isb.unizh.ch R.Frey .1Tel.:q 41-1-6
34-29-57;fax:q 41-1-634-49-03.
0927-5398r 00r $-e front matter q 2000Elvier Science B.V.All rights rerved.
Ž.PII:S 0927-53980000012-8
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A.J.McNeil,R.Frey r Journal of Empirical Finance72000271–300
1.Introduction
The large increa in the number of traded asts in the portfolio of most
Ž
financial institutions has made the measurement of market risk the risk that a financial institution incurs loss on its trading book due to adver market .
病毒性感冒和细菌性感冒区别movements a primary concern for regulators and for internal risk control.In particular,banks are now required to hold a certain amount of capital as a cushion against adver market movements.According to the Capital Adequacy Directive
Ž.Ž.
by the Bank of International Settlement BIS in Basle,Basle Comittee,1996the risk capital of a bank must be sufficient to cover loss on the bank’s trading portfolio over a10-day holding period in99%of occasions.This value is usually
Ž.
referred to as Value at Risk VaR.Of cour,holding period and confidence level may vary according to application;for purpos of internal risk control most financial firms also u a holding period of one day and a confidence level of 95%.From a mathematical viewpoint VaR is simply a quantile of the Profit-and-Ž.
Loss P&L distribution of a given portfolio over a prescribed holding period.
Alternative measures of market risk have been propod in the literature.In two
Ž.
洗洁剂recent papers,Artzner et al.1997,1999show that VaR has various theoretical deficiencies as a measure of market risk;they propo the u of the so-called expected shortfall or tail conditional expectation instead.The expected shortfall measures the expected loss given that the loss L exceeds VaR;in mathematical
w<x
terms it is given by E L L)VaR.From a statistical viewpoint the main challenge in implementing one of the risk-measures is to come up with a good estimate for the tails of the underlying P&L distribution;given such an estimate both VaR and expected shortfall are fairly easy to compute.
In this paper we are concerned with tail estimation for financial return ries. Our basic idealisation is that returns follow a stationary time ries model with stochastic volatility structure.There is strong empirical support for stochastic
Ž.
我的牙齿volatility in financial time ries;e for instance Pagan1996.The prence of stochastic volatility imp
lies that returns are not necessarily independent over time. Hence,with such models there are two types of return distribution to be consid-ered—the conditional return distribution where the conditioning is on the current volatility and the marginal or stationary distribution of the process.
Both distributions are of relevance to risk managers.A risk-manager who wants to measure the market risk of a given portfolio is mainly concerned with the possible extent of a loss caud by an adver market movement over the next day Ž.
or next few days given the current volatility background.His main interest is in the tails of the conditional return distribution,which are also the focus of the prent paper.The estimation of unconditional tails provides different,but comple-mentary information about risk.Here we take the long-term view and attempt to
Žassign a magnitude to a specified rare adver event,such as a5-year loss the size
.
of a daily loss which occurs on average once every5years.This kind of
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A.J.McNeil,R.Frey r Journal of Empirical Finance72000271–300273 information may be of interest to the risk manager who wishes to perform a scenario analysis and get a feeling for the scale of worst ca or stress loss.
In a referee’s report the concern was raid that the u of conditional return distributions for market risk measurement might lead to capital requirements that fluctuate wildly over time and are therefore difficult to implement.Our answer to this important point is threefold.First,while it is admittedly impossible for a financial institution to rapidly adjust its capital ba to changing market condi-tions,the firm might very well be able to adjust the size of its exposure instead. Moreover,besides providing a basis for the determination of risk capital,measures of market risk are also employed to give the management of a financial firm a better understanding of the riskiness of its portfolio,or parts thereof.We are convinced that the riskiness of a portfolio does indeed vary with the general level of market volatility,so that the current volatility background should be reflected in the risk-numbers reported to management.Finally,we think that the economic problem of defining an appropriate risk-measure for tting capital-adequacy standards should be parated from the statistical problem of estimating a given measure of market risk,which is the focus of the prent paper.
Schematically the existing approaches for estimating the P&L distribution of a portfolio of curities can be divided into three groups:the non-parametric
Ž.
historical simulation HS method;fully parametric methods bad on an econo-metric model for volatility dynamics and the assumption of conditional normality Že.g.J.P.Morgan’s Riskmetrics and most models from the ARCH r GARCH .Ž.
大蒜种植
family;and finally methods bad on extreme value theory EVT.
In the HS-approach the estimated P&L distribution of a portfolio is simply given by the empirical distribution of past gains and loss on this portfolio.The method is therefore easy to implement and avoids A ad-hoc-assumptions B on the form of the P&L distribution.However,the method suffers from some rious drawbacks.Extreme quantiles are notoriously difficult to estimate,as extrapolation beyond past obrvations is impossible and extreme quantile estimates within sample tend to be very inefficient—the estimator is subject to a high variance. Furthermore,if we ek to mitigate the problems by considering long samples the method is unable to distinguish between periods of high and low volatility.
Econometric models of volatility dynamics that assume conditional normality, such as GARCH-models,do yield VaR estimates which reflect the current volatility background.The main weakness of this approach is that the assumption of conditional normality does not em to hold for real data.As shown,for
Ž.
instance,in Danielsson and de Vries1997b,models bad on conditional normality are therefore not well suited to estimating large quantiles of the P&L-distribution.2
2Note that the marginal distribution of a GARCH-model with normally distributed errors is usually fat-tailed as it is a mixture of normal distributions.However,this matters only for quantile estimation
Ž.
over longer time-horizons;Duffie and Pan1997.
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A.J.McNeil,R.Frey r Journal of Empirical Finance72000271–300
The estimation of return distributions of financial time ries via EVT is a
Ž
topical issue which has given ri to some recent work Embrechts et al.,1998, 1999;Longin,1997a,b;McNeil,1997,1998;Danielsson and de Vries,1997b,
.
1997c;Danielsson et al.,1998.In all the papers the focus is on estimating the Ž.Ž. unconditional stationary distribution of ast returns.Longin1997b and McNeil Ž.
1998u estimation techniques bad on limit theorems for block maxima. Longin ignores the stochastic volatility exhibited by most financial return ries and simply applies estimators for the iid-ca.McNeil us a similar approach but shows how to correct for the clustering of extremal events caud by stochastic
Ž.
volatility.Danielsson and de Vries1997a,b u a miparametric approach bad
Ž.
on the Hill-estimator of the tail index.Embrechts et al.1999advocate the u of a parametric estimation technique which is bad on a limit result for the excess-distribution over high thresholds.This approach will be adopted in this paper and explained in detail in Section2.2.
EVT-bad methods have two features which make them attractive for tail estimation:they are bad on a sound statistical theory;they offer a parametric form for the tail of a distribution.Hence,the methods allow for some extrapola-tion beyond the range of the data,even if care is required at this point.However, none of the previous EVT-bad methods for quantile estimation yields VaR-estimates which reflect the current volatility background.Given the conditional heteroscedasticity of most financial data,which is well documented by the considerable success of the models from the ARCH r GARCH family,we believe this to be a major drawback of any kind of VaR-estimator.
In order to overcome the drawbacks of each of the above methods we combine ideas from all three approaches.We u GARCH-modelling and pudo-maxi-mum-likelihood estimation to obtain estimates of the conditional volatility.Statis-tical tests and exploratory data analysis confirm that the e
rror terms or residuals do form,at least approximately,iid ries that exhibit heavy tails.We u historical Ž.
simulation for the central part of the distribution and threshold methods from Ž.
EVT for the tails to estimate the distribution of the residuals.The application of
Ž.
the methods is facilitated by the approximate independence over time of the residuals.An estimate of the conditional return distribution is now easily con-structed from the estimated distribution of the residuals and estimates of the conditional mean and volatility.This approach reflects two stylized facts exhibited by most financial return ries,namely stochastic volatility and the fat-tailedness of conditional return distributions over short time horizons.
Ž.
In a very recent paper Barone-Adesi et al.1998have independently propod an approach with some similarities to our own.They fit a GARCH-model to a financial return ries and u historical simulation to infer the distribution of the residuals.They do not u EVT-bad methods to estimate t
he tails of the distribution of the residuals.Their approach may work well in large data ts—they u13years of daily data—where the empirical quantile provides a reasonable quantile estimator in the tails.With smaller data ts threshold methods
五行缺水火()A.J.McNeil,R.Frey r Journal of Empirical Finance 72000271–300275
from EVT will give better estimates of the tails of the residuals.During the revision of this paper we also learned that the central idea of our approach —the application of EVT to model residuals —has been independently propod by Ž.Diebold et al.1999.
We test our approach on various return ries.Backtesting shows that it yields better estimates of VaR and expected shortfall than unconditional EVT or GARCH-modelling with normally distributed error terms.In particular,our analy-Ž.sis contradicts Danielsson and de Vries 1997c ,who state that A an unconditional approach is better suited for VaR estimation than conditional volatility forecasts B Ž.page 3of their paper .On the other hand,we e that models with a normally distributed conditional return distribution yield very bad estimates of the expected shortfall,so that there is a real need for working with leptokurtic error distribu-tions.We also study quantile estimation over longer time-horizons using simula-Žtion.This is of interest if we want to obtain an estimate of the 10-day VaR as .required by the BIS-rule from a model fitted to daily data.
2.Methods
Ž.Let X ,t g Z be a strictly stationary time ries reprenting daily obrva-t tions of the negative log return on a financial ast price.3We assume that the dynamics of X are given by
X s m q s Z ,1Ž.t t t t Žwhere the innovations Z are a strict white noi independent,t .identically distributed with zero mean,unit variance and marginal distribution Ž.function F z .We assume that m and s are measurable with respect to G ,Z t t t -1the information about the return process available up to time t y 1.
Ž.Ž.Let F x denote the marginal distribution of X and,for a horizon h g N ,X t Ž.let F x denote the predictive distribution of the return over the X q ...q X <G t q 1t q h t next h days,given knowledge of returns up to and including day t .We are interested in estimating quantiles in the tails of the distributions.For 0-q -1,an unconditional quantile is a quantile of the marginal distribution denoted by
x s inf x g R :F x G q ,
Ä4Ž.q X 3
In the prent paper we test our approach on return ries generated by single asts only.However,
the method obviously also applies to the time ries of profits and loss generated by portfolios of financial instruments and can therefore by ud for the estimation of market risk measures in a portfolio context.
()A.J.McNeil,R.Frey r Journal of Empirical Finance 72000271–300
276and a conditional quantile is a quantile of the predictive distribution for the return over the next h days denoted by
x t h s inf x g R :F x G q .
Ž.Ž.Ä4q X q ...q X <G t q 1t q h t We also consider an alternative measure of risk for the tail of a distribution known as the expected shortfall.The unconditional expected shortfall is defined to be <S s E X X )x ,
q q and the conditional expected shortfall to be h h t t <S h s E
X X )x h ,G .Ž.Ž.ÝÝq t q j t q j q t j s 1j s 1
We are principally interested in quantiles and expected shortfalls for the 1-step
predictive distribution,which we denote respectively by x t and S t .Since
q q <F x s P s Z q m F x G Ä4
Ž.X <G t q 1t q 1t q 1t t q 1t s F x y m r s ,
戏曲背景
Ž.Ž.Z t q 1t q 1the measures simplify to
x t s m q s z ,2Ž.q t q 1t q 1q t <S s m q s E Z Z )z ,3Ž.q t q 1t q 1q where z is the upper q th quantile of the marginal distribution of Z which by q t assumption does not depend on t .
To implement an estimation procedure for the measures we must choo a Ž.specific process in the class a particular model for the dynamics of the conditional mean and volatility.Many different models for volatility dynamics have been propod in the econometric literature including models from the Ž.ŽARCH r GARCH family Bollerslev et al.,1992,HARCH process Muller et ¨
.Ž.al.,1997and stochastic volatility models Shephard,1996.In this paper,we u Ž.the parsimonious but effective GARCH 1,1process for the volatility and an Ž.AR 1model for the dynamics of the conditional mean;the approach we propo extends easily to more complex models.
In estimating x t with GARCH-type models it is commonly assumed that the q innovation distribution is standard normal so that a quantile of the innovation
y 1Ž.Ž.distribution is simply z s F q ,where F z is the standard normal d.f.A q GARCH-type model with normal innovations can be fitted by maximum likeli-Ž.hood ML and m and s can be estimated using standard 1-step forecasts,
春旦t q 1t q 1t Ž.so that an estimate of x is easily constructed using 3.This is clo in spirit to
q Ž.the approach advocated in RiskMetrics RiskMetrics,1995,but our empirical finding,which we will later show,is that this approach often underestimates the
