FRM极值理论.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,FRM极值理论,Extreme Value Theory,EVT and VaR,1,Basel Rules for Backtesting,2,Extreme Value Theory and VaR,2,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 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,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 slide,3,The Basel Penalty Zones,4,Appendix 1,Why normal multiplier K=3,By Chebyshev inequality:P(|x-,|,),1/,2,.Suppose symmetric distribution,we get,P(x-,-,),1/2,2,which determines the Max of VaR,VaR,mx,=,.Let the confidence level be 0.99,we get,1/2,2,=0.01,from which,we get=7.071.Suppose the usual VaR is calculated under the assumption of normal distribution,we get VaR,N,=2.326,.Thus,we need a multiplier if normal distribution is not satisfied.The multiplier,K=/2.36=3.03,5,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 confidence 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 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 market risk charge,6,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 EVT,7,;,January 1,1912 December 27,1995,7,大家有疑问的，可以询问和交流,可以互相讨论下，但要小声点,8,Generalized Pareto Distribution,This has two parameters,x(,the shape parameter)and,b(,the scale parameter),By definition,we expect,b,to be positive.,The cumulative distribution is,9,Generalized Pareto Distribution,When underling distribution of v is normal,we have .,increases as the tail of v gets heavier,For most financial data,in 0.1,0.4,The k-th moment of underling r.v.is finite if,10,Maximum Likelihood Estimator,The observations,x,i,are sorted in descending order.Suppose that there are,n,u,observations greater than,u,We choose,x,and,b,to maximize,11,Maximum Likelihood Estimator,Constraints,x,and,b,are supposed to be positive,although,x,not required to be positive by the definition of GPD.,Negative,x,indicates:,Lighter tail of the underling distribution compared with normal,Inappropriate value of u is chosen,12,From parameters to tail of v,By definition:,Therefore,Again semi-parametric,13,Why power law?,14,Extreme Value Theory,VaR,15,Expected Short Fall,16,Block 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,17,Block Maxima Models,Generalized Extreme Value Distribution,VaR under GEV distribution,Anything wrong?,18,Block 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,19,19,Block Maxima Models,Estimation,By definition of F*,we only have ONE observation to estimate three parameters,Way-out,Apply GEV distribution to maximum returns within each block,MLE,Selection 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,Balance,20,20,Multiple 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 is easy to see that,21,21,Coherent Risk Measures,1 Monotonicity:if X1X2,2 Translation invariance:,3 Homogeneity:,4 Subadditivity:,22,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.50,23,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 job 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 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,24,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 required,25,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 10,26,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 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 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 observed data in the tails of distribution is limited,EV estimates can be very sensitive to small sample effects and other biases,27,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 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 find 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.,28,