Question

Consider two local banks. Bank A has 100 loans​ outstanding, each for​ $1 million, that it expects will be repaid today. Each loan has a 5 % 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 $ 100 million outstanding that it also expects will be repaid today. It also has a 5 % probability of not being repaid. Calculate the following.

a. What is the expected payoff of each​ bank's loans?

b. How risky are each​ bank's loans? What is the standard deviation of the payoff of bank​ A's portfolio of​ loans? (Hint: the risk of default is independent across​ loans, so the variance of the payoff of a​ 100-loan portfolio is simply 100 times the variance of the payoff of a single​ loan.) What is the standard deviation of the payoff of bank​ B's loan?

Which bank faces less​ risk? Why?

Answer 1

a). Expected payoff for Bank A = number of loans*loan amount*probability of receiving the loan

= 100*1*(1-5%) = 95 million

Expected payoff for Bank B = loan amount*probability of receiving the loan = 100*(1-5%) = 95 million

b). Bank A variance: Variance for each loan = probability of receiving the loan*(1 - 0.95)^2 + probability of default*(0-0.95)^2

= 0.95*(1-0.95)^2 + 0.05*(0-0.95)^2 = 0.0475

Standard deviation = variance^0.5 = 0.0475^0.5 = 0.2179

The loans are independent of each other so standard deviation for the average loan = standard deviation for one loan/number of loans^0.5

= 0.2179/(100^0.5) = 0.02179

Standard deviation for Bank A portfolio = number of loans*standard deviation for one loan = 100*0.02179 = 2.179

Bank B variance = 0.95*(100-95)^2 + 0.05*(0-95)^2 = 475

Standard deviation for Bank B = 475^0.5 = 21.79

Bank A has lower risk compared to Bank B as its risk is diversified.