Consider two local banks. Bank A has 86 loans outstanding, each for $1.0 million
ID: 3066096 • Letter: C
Question
Consider two local banks. Bank A has 86 loans outstanding, each for $1.0 million, that it expects will be repaid today. Each loan has a 4% probability of default, in which case the bank is not repaid anything. The chance of default is independent across all the loans. Bank B has only one loan of $ 86 million outstanding, which it also expects will be repaid today. It also has a 4% probability of not being repaid. Calculate the following:
a. The expected overall payoff of each bank.
b. The standard deviation of the overall payoff of each bank.( round to 2 decimal places)
Explanation / Answer
a. Expected pay-off for Bank A = 1 million x 0.96 x 86 = $ 82.56 million
Expected pay-off for Bank B = 86 million x 0.96 = $ 82.56 million
The expected pay-offs are the same for both banks.
b. For bank A, for each loan, the probability of default is 0.04 and no-default is 0.96
The receipt in case of default is 1 million and in case of default is 0
Mean receipt (or,pay-off) for each loan = 0.96 x 1 + 0.04 x 0 = 0.96 million
Variance of each loan for bank A = 0.96 x (1-0.96)2 + 0.04 x (0-0.96)2 = 0.0384
Standard deviation = sqrt (0.0384) = 0.196
Standard deviation of the average loan = 0.196 / sqrt(86) = 0.0211 million
For bank B, for the loan, the probability of default is 0.04 and no-default is 0.96
The receipt in case of default is 86 million and in case of default is 0
Mean receipt (or,pay-off) for bank B = 0.96 x 86 + 0.04 x 0 = 82.56 million
Variance of the loan for bank B = 0.96 x (86-82.56)2 + 0.04 x (0-82.56)2 = 284.01 million
Standard deviation = sqrt (284.01) = 16.85 million