FinancialRiskManagementHaibin Xie School of Banking and Finance, University of International Business and EconomicsOffice: Boxue708E-mail: hbxieamss.ac.cnTel: 010-64492533Extreme Value TheoryEVTandVaR1BaselRulesforBacktesting2ExtremeValueTheoryandVaRBasel Rules for BacktestingThe 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 slideThe Basel Penalty ZonesZoneNumber of ExceptionsPotential increase in KGreen0 to 40.00Yellow50.460.570.6580.7590.85Red101Appendix 1Why 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, 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 2VaR Parameters: To measure the VaR, we first need 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, however, at what horizon should one stop.VaR Parameters: Some rules for confidence level and horizon selectionThe 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 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 distribution.D.V.Gnedenko and EVT7 ; January 1, 1912 December 27, 1995Generalized Pareto DistributionThis has two parameters x (the shape parameter) and b (the scale parameter)By definition, we expect b to be positive.The cumulative distribution isGeneralized 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 Maximum Likelihood Estimator The observations, xi, are sorted in descending order. Suppose that there are nu observations greater than uWe choose x and b to maximizeMaximum Likelihood EstimatorConstraintsx 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 normalInappropriate value of u is chosenFrom parameters to tail of vBy definition:ThereforeAgain semi-parametricWhy power law?Extreme Value TheoryVaR Expected Short FallBlock Maxima ModelsDistribution of the largest variableAs n goes to infinity, and the support of r is -inf,infWe need to blow up the variable with a normalizationThe limiting distribution is Generalized Extreme Value DistributionBlock Maxima ModelsGeneralized Extreme Value DistributionVaR under GEV distributionAnything wrong?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 18Block Maxima ModelsEstimationBy definition of F*, we only have ONE observation to estimate three parametersWay-outApply GEV distribution to maximum returns within each blockMLESelection of nGEV is a limit property, n as large as possibleFor given T, g = T/n where g is the effective number of observations for parameter estimationBalance19Multiple period VaRUnder EVT the multiple period VaR is not just square root of time horizon.Why square root of time horizon?Under power lawFeller shows that tail risk is approximately additive, therefore:It is easy to see that 20Coherent Risk Measures1 Monotonicity: if X1X2,2 Translation invariance:3 Homogeneity:4 Subadditivity: ExerciseBased on a 90% confidence level, how many exceptions in backtesting a VaR would be expected over a 250-day trading year?a. 10b. 15c. 25d. 50A 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 exceptionsb. Over the past 250 days, the largest loss is $500 millionc. Over the past 250 days, the mean loss is $60 milliond. Over the past 250 days, there is no exceptionWhich of the following procedures is essential in validating the VaR estimates?a. stress-testingb. scenario analysisc. backtestingd. Once approved by regulators, no further validation is requiredThe Market Risk Amendment to the Basel Capital Accord defines the yellow zone as the following range of exceptions out of 250 observationsa. 3 to 7b. 5 to 9c. 6 to 9d. 6 to 10Extreme 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 estimationc. EVT estimates are subject to considerable model risk, and EVT results are ofen very sensitive to the precise assumptions maded. Because observed data in the tails of distribution is limited, EV estimates can be very sensitive to small sample effects and other biasesWhich 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 distributionb. 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 returnsc. EVT attempts to find the optimal point beyond which all values belong to the tail and then models the distribution of the tail separatelyd. 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 modelingReferences：Financial Risk Manager Handbook and Test Bank TestExtreme Value TheoryVaR