2. [Calibration of a mixed binomial model] Let Z ~ Beta(a, b). Consider a portfolio of m = 1000 similar obligors whose default indicator variables X; are conditionally independent given Z such that X; | Z = 2 ~ Bernoulli( z). Assume that for every obligor the exposure is EAD; = 1 and LGD; = 100%. (a) Show that the probability distribution of the portfolio loss Lm is 1000 B(k + a, 1000 - k + b) P(Lm = k) = k = 0, 1, 2, . . . , m. k B(a, b) (b) Suppose that the default probability for each obligor is p; = 1% and the default correlation paj = 0.005 for i * j. Find (calibrate) the parameters a and b that are consistent with those parameter values. (c) Show that the LPA distribution is 1 F(x) B(a, b) Jo 20-1(1 - 2) 6-1dz