FRM极值理论实用教案

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1、Extreme Value TheoryEVT and VaR1Basel Rules for Backtesting2Extreme Value Theory and VaR第1页/共28页第一页，共29页。Basel Rules for Backtesting The Basel Committee put in place a framework based on the daily backtesting of VaR. Having up to four exceptions is acceptable, which defines a green zone. If the numb

2、er 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, 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

3、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 slide第2页/共28页第二页，共29页。The Basel Penalty Zones第3页/共28页第三页，共29页。Appendix 1 Why normal multiplier K=3 By Chebyshev inequality: P(|x-|)1/2. Suppose symmetric distr

4、ibution, 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, from which, we get =7.071. Suppose the usual VaR is calculated under the assumption of normal distribution, we get VaRN. Thus, we need a multiplier if normal distribution is not sa

5、tisfied. The multiplier, K=第4页/共28页第四页，共29页。Appendix 2 VaR Parameters: To measure the VaR, we first need to define two quantitative parameters: the confidence level and the horizon Confidence Level :The higher the confidence level, the greater the VaR measure! It is not clear, however, at what confi

6、dence level should one stop Horizon:The longer the horizon, the greater the VaR measure. It is not clear, however, at what horizon should one 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

7、risk measures. For backtesting purposes, a low confidence level and a short horizon is necessary; for capital adequacy purposes, a high confidence level 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 mark

8、et risk charge第5页/共28页第五页，共29页。Extreme 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 distribution.D.V.Gnedenko and EVT6 ; January 1, 1912 December 27, 1995vv,v()( )( )( )1( )uF uyF uF yGyF u 第6页/共28页第六页，共29页。Generalized

9、 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 第7页/共28页第七页，共29页。Generalized Pareto DistributionlWhen underling distribution of v is normal, we have .l

10、 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/k第8页/共28页第八页，共29页。Maximum Likelihood Estimator uniiuv11/1)(11ln The observations, xi, are sorted in descending order. Suppose that there are nu observations greater than u

11、We choose and to maximize第9页/共28页第九页，共29页。Maximum 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 normal Inappropriate value of u is chosen第10

12、页/共28页第十页，共29页。From parameters to tail of v By definition: Therefore Again semi-parametric,(|)1( )P vuy vuGy ,()1( )1()vP vxF uGxu 1/,()()1()1uunnxuP vxGxunn 第11页/共28页第十一页，共29页。Why power law? widely so holdslaw power the why explains thereforetheory value Extreme where)Prob(law power the to scorresp

13、ond this that see we Settingis than greater is variable the thaty probabilit cumulative the for estimator Our-11/1/1nnKKxxvuuxnnx vuu第12页/共28页第十二页，共29页。Extreme Value TheoryVaR 1)q1 (nnuVaRisIt uVaR1nn1qsolvingby obtained is q level confidence when theVaR of estimate Theu/1u第13页/共28页第十三页，共29页。Expecte

14、d Short Fall,()Expected shortfall is related 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()11qqqqVaRuVaRuESVaR第14页/共28页第十四页，共29页。Block Maxima

15、 ModelsDistribution of the largest variable As n goes to infinity, and the support of r is -inf,infWe need to blow up the variable with a normalization The limiting distribution is Generalized Extreme Value Distribution,( )( )Pr()( ( )nn nnFxrxF x,( )0n nFx *( )nnrr第15页/共28页第十五页，共29页。Block Maxima Mo

16、delsGeneralized Extreme Value DistributionVaR under GEV distribution Anything wrong?1/*()exp1 0( )expexp otherwiserF rr-ln(1- )1 0ln(-ln(1- ) otherwiseqVaRq第16页/共28页第十六页，共29页。Block Maxima Models is the distribution of the largest variable not the variable itself.The (1-q)th quantile of r is equivale

17、nt to (1-q)n th quantile of r(n)The correct VaR is 17*( )F r,( )( )Pr()( ( )nn nnFxrxF x- ln(1- )1 0ln(- ln(1- ) otherwisenqVaRnq第17页/共28页第十七页，共29页。Block Maxima Models Estimation By definition of F*, we only have ONE observation to estimate three parameters Way-out Apply GEV distribution to maximum

18、returns within each block MLE Selection of nGEV is a limit property, n as large as possible For given T, g = T/n where g is the effective number of observations for parameter estimationBalance18112(1) 11 ,.,|,.,|,.,|,.,|,.,nnnggngngn mrrrrrrrr第18页/共28页第十八页，共29页。Multiple period VaR Under EVT the mult

19、iple 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 is easy to see that 191/Prob()vxKx1/12Prob(.)TvvvxTKx( )qqVaR TT VaR第19页/共28页第十九页，共29页。Coherent Risk Measures 1 Monotonicity:

20、 if X1X2, 2 Translation invariance: 3 Homogeneity: 4 Subadditivity: )()()(2121XXXX)()(21XXkXkX)()()()(XbbX第20页/共28页第二十页，共29页。Exercise Based on a 90% confidence level, how many exceptions in backtesting a VaR would be expected over a 250-day trading year? a. 10 b. 15 c. 25 d. 50第21页/共28页第二十一页，共29页。 A

21、 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 job the manager is doing in estimating the one-da

22、y 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 days, the largest loss is $500 million c.

23、 Over the past 250 days, the mean loss is $60 million d. Over the past 250 days, there is no exception第22页/共28页第二十二页，共29页。 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 further

24、validation is required第23页/共28页第二十三页，共29页。 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第24页/共28页第二十四页，共29页。 Extreme value theory provides valuable insight about the

25、tails of return distributions. Which 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

26、 observed exceedances is a reliable estimate. 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 o

27、bserved data in the tails of distribution is limited, EV estimates can be very sensitive to small sample effects and other biases第25页/共28页第二十五页，共29页。 Which of the following statements regarding extreme value theory is incorrect? a. In contrast to conventional approaches for estimating VaR, EVT consi

28、ders only the tail behavior of the distribution b. Conversational approaches 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 fi

29、nd the optimal point beyond which all values belong to the tail and then 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.第26页/共28页第二十六页，共29页。SummaryMain contents: Ext

30、reme 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第27页/共28页第二十七页，共29页。感谢您的观看(gunkn)！第28页/共28页第二十八页，共29页。NoImage内容(nirng)总结Extreme Value Theory。D.V.Gnedenko and EVT。 。Extreme Value TheoryVaR。感谢您的观看(gunkn)第二十九页，共29页。