We define the loss variable L by L = EAD SEV L, where L...

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We define the loss variable L by L = EAD × SEV × L, where L = 1p with E[15] = 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) × DP + LGD² × DP(1 - DP). Hint var (X) = E[X²] - E[X]². (b) Suppose 1p₁ and 1p₂ are correlated, show that where pi = P[1D₂ = 1|1D₁ = 1] = P₂ + P[1D, 1], i = 1,2. = cov(1D₁, 1D₂) P1 We define the loss variable L by L = EAD × SEV × L, where L = 1p with E[15] = 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) × DP + LGD² × DP(1 - DP). Hint var (X) = E[X²] - E[X]². (b) Suppose 1p₁ and 1p₂ are correlated, show that where pi = P[1D₂ = 1|1D₁ = 1] = P₂ + P[1D, 1], i = 1,2. = cov(1D₁, 1D₂) P1

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