FRM极值理论ppt课件

1、1Financial Risk ManagementHaibin Xie School of Banking and Finance, University of International Business and EconomicsOffice: Boxue708E-mail: Tel: 2Extreme Value TheoryEVT and VaR1Basel Rules for Backtesting2Extreme Value Theory and VaR3Basel Rules for Backtesting The Basel Committee put in place a

2、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, th

3、e 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 slide4The Basel Penalty Zon

4、esZoneNumber of ExceptionsPotential increase in KGreen0 to 40.00Yellow50.460.570.6580.7590.85Red1015Appendix 1 Why normal multiplier K=3 By 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, w

5、e 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=2.326. Thus, we need a multiplier if normal distribution is not satisfied. The multiplier, K=/2.36=3.036Appendix 2 VaR Parameters: To measure the VaR, we first nee

6、d to define two quantitative 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,

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

8、ital 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 market risk charge7Extreme Value Theory VaR is all about the tail behavior of loss distribution, A.K.A, we ar

9、e only interested in some extreme value of a distribution. D.V.Gnedenko and EVT7 ; January 1, 1912 December 27, 1995vv,v()( )( )( )1( )uF uyF uF yGyF u 8Generalized Pareto Distribution This has two parameters (the shape parameter) and (the scale parameter) By definition, we expect to be positive. Th

10、e cumulative distribution is1/,( )11 if 01 exp otherwiseGyyy 9Generalized 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/k10Maximum Likelihood Es

11、timator uniiuv11/1)(11ln The observations, xi, are sorted in descending order. Suppose that there are nu observations greater than u We choose and to maximize11Maximum Likelihood Estimator Constraints and are supposed to be positive, although not required to be positive by the definition of GPD. Neg

12、ative indicates: Lighter tail of the underling distribution compared with normal Inappropriate value of u is chosen12From parameters to tail of v By definition: Therefore Again semi-parametric,(|)1( )P vuy vuGy ,()1( )1()vP vxF uGxu 1/,()()1()1uunnxuP vxGxunn 13Why power law? widely so holdslaw powe

13、r the why explains thereforetheory value Extreme where)Prob(law power the to scorrespond this that see we Settingis than greater is variable the thaty probabilit cumulative the for estimator Our-11/1/1nnKKxxvuuxnnx vuu14Extreme Value TheoryVaR 1)q1 (nnuVaRisIt uVaR1nn1qsolvingby obtained is q level

14、confidence when theVaR of estimate Theu/1u15Expected 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(

15、)11qqqqVaRuVaRuESVaR16Block Maxima Models Distribution of the largest variable As n goes to infinity, 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 *( )nn

16、rr17Block Maxima Models Generalized Extreme Value Distribution VaR under GEV distribution Anything wrong?1/*()exp1 0( )expexp otherwiserF rr-ln(1- )1 0ln(-ln(1- ) otherwiseqVaRq18Block Maxima Models is the distribution of the largest variable not the variable itself. The (1-q)th quantile of r is equ

17、ivalent 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- ) otherwisenqVaRnq19Block Maxima ModelsEstimation By definition of F*, we only have ONE observation to estimate three parameters Way-out Apply GEV distribution to maximum returns with

18、in 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 observations for parameter estimation Balance19112(1) 11 ,.,|,.,|,.,|,.,|,.,nnnggngngn mrrrrrrrr20Multiple period VaR Under EVT the multiple period VaR is not just

19、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 201/Prob()vxKx1/12Prob(.)TvvvxTKx( )qqVaR TT VaR21Coherent Risk Measures 1 Monotonicity: if X1X2, 2 Translation invariance: 3 Homogen

20、eity: 4 Subadditivity: )()()(2121XXXX)()(21XXkXkX)()()()(XbbX22Exercise 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. 5023 A large, international bank has a trading book whose size depends on the opportun

21、ities 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-day VaR. Which of the following would be the most convincing evidence that the man

22、ager 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. Over the past 250 days, the mean loss is $60 million d. Over the past 250 days,

23、 there is no exception24 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 validation is required25 The Market Risk Amendment to the Basel Capital Accord defines the yellow

24、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 1026 Extreme value theory provides valuable insight about the tails of return distributions. Which of the following statements about EVT and its applications is incorrect? a. The peaks over thresh

25、old, 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 estimate. b. EVT highlights that distributions justified by central limit theorem can be used for

26、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 limited, EV estimates can be very sensitive to small sample effects and other biases27 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 b