Question

1) Consider two local banks. Bank A has 90 loans? outstanding, each for? $1.0 million, that it expects will be repaid today. Each loan has a 7% 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 $90 million? outstanding, which it also expects will be repaid today. It also has a 7% 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.

*

2) Consider two local banks. Bank A has 100 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 $100 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 overall payoff of each
bank.

**b.** The standard deviation of the overall payoff
of each bank.

Answer #1

The expected payoffs are the same, but bank A is less risky.

Expected payoff is the same for both banks

Bank B= ($100 million* 0.95)= $95 million

Bank A= ($1 million* 0.95)* 100= $95 million

Bank B

Variance=(100- 95)^2 * 0.95(0- 95)^2 * 0.05 475 =475

Standard Deviation= sqrt 475= 21.79

Bank A

Variance of each loan= (1- 0.95)^2 * 0.95(0- 95)^2 * 0.05 475 =0.0475

Standard Deviation of each loan= route 0.0475= 0.2179

Now the bank has 100 loans that are all independent of each other so the standard deviation of the average loan is

0.2179/sqrt 100= 0.02179.

But the bank has 100 such loans so the standard deviation of the portfolio is

100* 0.02179= 2.179

This is lower than Bank B.

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