FRM极值理论文件.ppt

The Basel Penalty Zones Zone Number of Exceptions Potential increase in K Green 0 to 4 0.00 Yellow 5 0.4 6 0.5 7 0.65 8 0.75 9 0.85 Red ≥10 1 .......... * Appendix 1 Why normal multiplier K=3 By Chebyshev inequality: P(|x-μ|λσ)≤1/λ2. Suppose symmetric distribution, we get P(x-μ-λσ)≤1/2λ2, which determines the Max of VaR, VaRmx=λσ. Let the confidence level be 0.99, we get 1/2λ2=0.01, from which, we get λ=7.071. Suppose the usual VaR is calculated under the assumption of normal distribution, we get VaRN=2.326σ. Thus, we need a multiplier if normal distribution is not satisfied. The multiplier, K=λσ/2.36σ=3.03 .......... * 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 .......... * 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 * Бори?с Влади?мирович Гнеде?нко; January 1, 1912 – December 27, 1995 .......... * 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 .......... * Generalized Pareto Distributio