Consider two local banks. Bank A has 

110110 

loans​ outstanding, each for 

​$1.31.3 

​million, that it expects will be repaid today. Each loan has a 

6 %6% 

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 

$ 143$143 

million​ outstanding, which it also expects will be repaid today. It also has a 

6 %6% 

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.

Question content area bottom

Part 1

**a.** The expected overall payoff of each bank.

The expected overall payoff of Bank A is 

​$enter your response here 

million. ** ​(Round to two decimal​ places.)

Let's break down the problem step by step to calculate the expected overall payoff for each bank.

### For Bank A:

- **Number of loans:** 110 loans
- **Loan amount per loan:** $1.3 million
- **Probability of default (not repaid):** 6% or 0.06
- **Probability of repayment (successful):** 94% or 0.94

The expected payoff for each loan can be calculated as follows:
\[
\text{Expected payoff per loan} = (\text{Loan amount}) \times (\text{Probability of repayment})
\]
\[
\text{Expected payoff per loan} = 1.3 \times 0.94 = 1.222 \text{ million dollars}
\]

Since there are 110 loans, the expected overall payoff for Bank A is:
\[
\text{Expected payoff for Bank A} = 1.222 \times 110 = 134.42 \text{ million dollars}
\]

### For Bank B:

- **Number of loans:** 1 loan
- **Loan amount:** $143 million
- **Probability of default (not repaid):** 6% or 0.06
- **Probability of repayment (successful):** 94% or 0.94

The expected payoff for Bank B is:
\[
\text{Expected payoff for Bank B} = 143 \times 0.94 = 134.42 \text{ million dollars}
\]

Thus, the expected overall payoff for both Bank A and Bank B is **$134.42 million**.

Would you like me to proceed with the standard deviation calculations for part (b) or have any questions about this part? 

Here are 5 follow-up questions related to this:

1. What is the formula for calculating the variance for multiple loans?
2. How does diversification of loans affect risk in this scenario?
3. How would the expected payoff change if the probability of default increased?
4. What impact does loan amount have on the expected payoff and risk?
5. Can we model this scenario using the binomial distribution for Bank A's loans?

**Tip:** In such problems, diversification (spreading loans) typically reduces the overall risk, as the outcome is averaged over a large number of independent loans.

Consider two local banks. Bank A has 110 loans outstanding, each for $1.3 million, that it expects will be repaid today. Each loan has a 6% 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 $143 million outstanding, which it also expects will be repaid today. It also has a 6% 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.

This problem explores expected payoff and risk (standard deviation) for two local banks with different loan structures. Bank A has 110 small loans, each with a 6% default probability, while Bank B has a single large loan. Learn how to calculate expected values and standard deviations using probability concepts. Suitable for Grades 11-12 or college-level students.

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