6. Let V be the random loss on a portfolio over a certain time horizon. We...

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6. Let V be the random loss on a portfolio over a certain time horizon. We use the Pareto distribution. 1- (1 + ) -1/4 G,B(y) = 1- to model the random loss beyond u, where the probability that V > u+y conditional on V > u is 1G,s(y). Let F(v) be the cumulative distribution function for the random loss variable V. (a) Explain why the unconditional probability that V > xs given by - P[V>x] = [1 F(u)][1 - G,p(x - u)]. [2] (b) Suppose 1 - F(u) is estimated by empirical data to be nu/n so that P[V> x] = = nu n 1+ B - -1/ where n is the total number of scenarios and n is the number of scenarios that the loss exceeds u. Note that x = u+y. Recall that VaR with a confidence level of q is given by F(VaR) = 9, where F(u) = 1 - P[V > u]. Show the detailed derivation of finding VaR in terms and other parameters of the Pareto distribution. of q, u, Nu 6. Let V be the random loss on a portfolio over a certain time horizon. We use the Pareto distribution. 1- (1 + ) -1/4 G,B(y) = 1- to model the random loss beyond u, where the probability that V > u+y conditional on V > u is 1G,s(y). Let F(v) be the cumulative distribution function for the random loss variable V. (a) Explain why the unconditional probability that V > xs given by - P[V>x] = [1 F(u)][1 - G,p(x - u)]. [2] (b) Suppose 1 - F(u) is estimated by empirical data to be nu/n so that P[V> x] = = nu n 1+ B - -1/ where n is the total number of scenarios and n is the number of scenarios that the loss exceeds u. Note that x = u+y. Recall that VaR with a confidence level of q is given by F(VaR) = 9, where F(u) = 1 - P[V > u]. Show the detailed derivation of finding VaR in terms and other parameters of the Pareto distribution. of q, u, Nu