Question: 2. We define the loss variable L by L = EAD x SEV X L, where L = 1p with E[1p] = DP (default probability),

2. We define the loss variable L by L = EAD x SEV X L, where L = 1p with E[1p] = DP (default probability), EAD is the exposure at default and SEV is the random severity with E[SEV] = LGD (loss fraction given default). (a) Assuming SEV and 1p are independent, show that (i) var(1p) = DP(1 - DP); (ii) var(SEV1p) = var(SEV) x DP + LGD' x DP(1 - DP). Hint var(X) = E[X2] - E[X]2. (b) Suppose 1p, and 1p2 are correlated, show that P[1D2 = 1/1p, = 1] = p2 + cov(1D, , 1D2) where pi = P[1D, = 1], i = 1, 2. [3]

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