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 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 $86 million​ outstanding, which it also expects will be repaid today. It also has a 5% probability of not being repaid. Calculate the​ following:

a. The expected payoff of Bank A.

b. The expected payoff of Bank B.

c. The standard deviation of the overall payoff of Bank A.

d. The standard deviation of the overall payoff of Bank B.

Answer 1

Probability of Default = 5%

Probability of Repay = 95%

a. Expected Payoff of Bank A

Total Number of Loans = 86

Amount of Each Loan = $1 million

Expected Payoff = Total Number of Loans*Amount of Each Loan*(1-Probability of Default)

Expected Payoff = 86*1*0.95 = $81.7 million

b.

Expected Payoff of Bank B

Total Number of Loans = 1

Amount of each Loan = $86 million

Expected Payoff = Total Number of Loans*Amount of each Loan*(1-Probability of Default)

Expected Payoff = 1*86*0.95 = $81.7 million

c.

Probability	Payoff	Probability*Payoff
95%	$1million	$0.95 million
5%	$0	$0

Variance = (Payoff - Expected Payoff)² * Probability of Repay + (Payoff - Expected Payoff)² * Probability of Default

Variance = (1-0.95)² * 0.95 + (0-0.95)² * 0.05

Variance = 0.0475

Standard Deviation = (Variance)^(1/2) = (0.0475)^(1/2) = 0.2179

d.

Probability	Payoff	Probability*Payoff
95%	$86million	$81.7 million
5%	$0	$0

Variance = (Payoff - Expected Payoff)² * Probability of Repay + (Payoff - Expected Payoff)² * Probability of Default

Variance = (86-81.7)² * 0.95 + (0-81.7)² * 0.05

Variance = 351.31

Standard Deviation = (Variance)^(1/2) = (351.31)^(1/2) = 18.74