FRM极值理论教学内容

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1、Financial Risk Management Haibin Xie School of Banking and Finance,University of International Business and Economics Office:Boxue708 E-mail: Tel:010-64492533Extreme Value TheoryEVT and VaR1Basel Rules for Backtesting2Extreme Value Theory and VaRBasel Rules for Backtesting The Basel Committee put in

2、 place a framework based on the daily backtesting of VaR.Having up to four exceptions is acceptable,which defines a green zone.If the number of exceptions is five or more,the bank falls into a yellow or red zone and incurs a progressive penalty,which is enforced with a higher capital charge.Roughly,

3、the capital charge is expressed as a multiplier of the 10-day VaR at the 99%level of confidence.The normal multiplier k is 3.After an incursion into the yellow zone,the multiplicative factor,k,is increased from 3 to 4,or plus factor described in the Table in the next slideThe Basel Penalty ZonesZone

4、Number of ExceptionsPotential increase in KGreen0 to 40.00Yellow50.460.570.6580.7590.85Red101Appendix 1 Why normal multiplier K=3By Chebyshev inequality:P(|x-|)1/2.Suppose symmetric distribution,we get P(x-)1/22,which determines the Max of VaR,VaRmx=.Let the confidence level be 0.99,we get 1/22=0.01

5、,from which,we get=7.071.Suppose the usual VaR is calculated under the assumption of normal distribution,we get VaRN=2.326.Thus,we need a multiplier if normal distribution is not satisfied.The multiplier,K=/2.36=3.03Appendix 2 VaR Parameters:To measure the VaR,we first need to define two quantitativ

6、e parameters:the confidence level and the horizonConfidence Level:The higher the confidence level,the greater the VaR measure!It is not clear,however,at what confidence level should one stopHorizon:The longer the horizon,the greater the VaR measure.It is not clear,however,at what horizon should one

7、stop.VaR Parameters:Some rules for confidence level and horizon selection The choice of the confidence level and horizon depend on the intended use for the risk measures.For backtesting purposes,a low confidence level and a short horizon is necessary;for capital adequacy purposes,a high confidence l

8、evel and a long horizon are required.In practice,these conflicting objectives can be accommodated by a complex rule,as is the case for the Basel market risk chargeExtreme Value Theory VaR is all about the tail behavior of loss distribution,A.K.A,we are only interested in some extreme value of a dist

9、ribution.D.V.Gnedenko and EVT7 ;January 1,1912 December 27,1995vv,v()()()()1()uF uyF uF yGyF u Generalized Pareto Distribution This has two parameters (the shape parameter)and (the scale parameter)By definition,we expect to be positive.The cumulative distribution is1/,()11 if 01 exp otherwiseGyyy Ge

10、neralized Pareto DistributionlWhen underling distribution of v is normal,we have .l increases as the tail of v gets heavierlFor most financial data,in 0.1,0.4lThe k-th moment of underling r.v.is finite if 01/kMaximum Likelihood Estimator uniiuv11/1)(11ln The observations,xi,are sorted in descending

11、order.Suppose that there are nu observations greater than u We choose and to maximizeMaximum Likelihood Estimator Constraints and are supposed to be positive,although not required to be positive by the definition of GPD.Negative indicates:Lighter tail of the underling distribution compared with norm

12、al Inappropriate value of u is chosenFrom parameters to tail of v By definition:Therefore Again semi-parametric,(|)1()P vuy vuGy ,()1()1()vP vxF uGxu 1/,()()1()1uunnxuP vxGxunn Why power law?widely so holdslaw power the why explains thereforetheory value Extreme where)Prob(law power the to scorrespo

13、nd this that see we Settingis than greater is variable the thaty probabilit cumulative the for estimator Our-11/1/1nnKKxxvuuxnnx vuuExtreme Value TheoryVaR 1)q1(nnuVaRisIt uVaR1nn1qsolvingby obtained is q level confidence when theVaR of estimate Theu/1uExpected Short Fall,()Expected shortfall is rel

14、ated to VaR by|It is easy to show that()()qqqqqqVaRVaRuESVaRE XVaRXVaRFyGy 1/1/,()()()(|)()1111()()1qqqqVaRqqqqVaRuqqP VaRvyVaRFyP vVaRy vVaRP vVaRyVaRuyGyVaRuVaRu Expected shortfall is related to VaR by()11qqqqVaRuVaRuESVaRBlock Maxima Models Distribution of the largest variable As n goes to infini

15、ty,and the support of r is-inf,inf We need to blow up the variable with a normalization The limiting distribution is Generalized Extreme Value Distribution,()()Pr()()nn nnFxrxF x,()0n nFx*()nnrrBlock Maxima Models Generalized Extreme Value Distribution VaR under GEV distribution Anything wrong?1/*()

16、exp1 0()expexp otherwiserF rr-ln(1-)1 0ln(-ln(1-)otherwiseqVaRqBlock Maxima Models is the distribution of the largest variable not the variable itself.The(1-q)th quantile of r is equivalent to(1-q)n th quantile of r(n)The correct VaR is 18*()F r,()()Pr()()nn nnFxrxF x-ln(1-)1 0ln(-ln(1-)otherwisenqV

17、aRnqBlock Maxima ModelsEstimation By definition of F*,we only have ONE observation to estimate three parameters Way-out Apply GEV distribution to maximum returns within each block MLESelection of n GEV is a limit property,n as large as possible For given T,g=T/n where g is the effective number of ob

18、servations for parameter estimation Balance19112(1)11,.,|,.,|,.,|,.,|,.,nnnggngngn mrrrrrrrrMultiple period VaR Under EVT the multiple period VaR is not just square root of time horizon.Why square root of time horizon?Under power law Feller shows that tail risk is approximately additive,therefore:It

19、 is easy to see that 201/Prob()vxKx1/12Prob(.)TvvvxTKx()qqVaR TT VaRCoherent Risk Measures 1 Monotonicity:if X1X2,2 Translation invariance:3 Homogeneity:4 Subadditivity:)()()(2121XXXX)()(21XXkXkX)()()()(XbbXExercise Based on a 90%confidence level,how many exceptions in backtesting a VaR would be exp

20、ected over a 250-day trading year?a.10 b.15 c.25 d.50 A large,international bank has a trading book whose size depends on the opportunities perceived by its traders.The market risk manager estimates the one-day VaR,at the 95%confidence level,to be$50 million.You are asked to be evaluate how good a j

21、ob the manager is doing in estimating the one-day VaR.Which of the following would be the most convincing evidence that the manager is doing a poor job,assuming that the losses are identical and independently distributed(i.i.d)?a.Over the past 250 days,there are eight exceptions b.Over the past 250

22、days,the largest loss is$500 million c.Over the past 250 days,the mean loss is$60 million d.Over the past 250 days,there is no exception Which of the following procedures is essential in validating the VaR estimates?a.stress-testing b.scenario analysis c.backtesting d.Once approved by regulators,no

23、further validation is required The Market Risk Amendment to the Basel Capital Accord defines the yellow zone as the following range of exceptions out of 250 observations a.3 to 7 b.5 to 9 c.6 to 9 d.6 to 10 Extreme value theory provides valuable insight about the tails of return distributions.Which

24、of the following statements about EVT and its applications is incorrect?a.The peaks over threshold,which then determines the number of observed exceedances;the threshold must be sufficiently high to apply the theory,but sufficiently low so that the number of observed exceedances is a reliable estima

25、te.b.EVT highlights that distributions justified by central limit theorem can be used for extreme value estimation c.EVT estimates are subject to considerable model risk,and EVT results are ofen very sensitive to the precise assumptions made d.Because observed data in the tails of distribution is li

26、mited,EV estimates can be very sensitive to small sample effects and other biases Which of the following statements regarding extreme value theory is incorrect?a.In contrast to conventional approaches for estimating VaR,EVT considers only the tail behavior of the distribution b.Conversational approa

27、ches for estimating VaR that assume that the distribution of returns follows a unique distribution for the entire range of values may fail to properly account for the fat tail of the distribution of returns c.EVT attempts to find the optimal point beyond which all values belong to the tail and then

28、models the distribution of the tail separately d.By smoothing the tail of the distribution,EVT effectively ignores extreme events and losses that can generally be labeled outliers.SummaryMain contents:Extreme Value TheoryKey notes：1.Extreme Value Theory 2.VaR and Parameters Selection 3.Basel Rules for Backtesting Homework:1.Read the related materials.2.Finish the Exercise!Arrangement for next time：Copulus and Multivariate modeling References：Financial Risk Manager Handbook and Test Bank TestExtreme Value TheoryVaR