A bank is considering a 1-year loan of $1 million at a promised interest rate of 8%. The

probability of repayment is 97% and the probability of default is 3%. If default occurs, the

bank expects to recover 40% of the principal. What is the expected return on this loan?

Question

A bank is considering a 1-year loan of $1 million at a promised interest rate of 8%. The

probability of repayment is 97% and the probability of default is 3%. If default occurs, the

bank expects to recover 40% of the principal. What is the expected return on this loan?

BlackTom AI · Accepted Answer

First, restating the scenario and the choices helps frame what we’re evaluating.
Question: A bank considers a 1-year loan of $1,000,000 at a promised interest rate of 8%. Probability of repayment is 97% and probability of default is 3%. If default occurs, the bank recovers 40% of the principal. What is the expected return on this loan?
Answer options: 7.84%, 8.00%, 5.96%, 6.80%

Now, assess each option step by step.
Option 1: 7.84%.
- This would imply an expected profit of about $78,400 on a $1,000,000 loan. To test this, compute the expected cash flow: if repaid, you receive 1,080,000 (principal plus 8% interest) with probability 0.97; if defaulted, you recover 400,000 with probability 0.03. The total expected cash flow is 0.97*1,080,000 + 0.03*400,000 = 1,047,600 + 12,000 = 1,059,600. The implied return relative to the principal is (1,059,600 - 1,000,000) / 1,000,000 = 0.0596 = 5.96%, not 7.84%. So this option is incorrect.
Option 2: 8.00%.
- An 8.00% return would require an expected cash flow of 1,080,000 on a $1,000,000 investment. However, because there is a 3% chance of default with only 40% recovery of the principal, the expected cash flow is reduced: 0.97*1,080,000 + 0.03*400,000 = 1,059,600, which yields a 5.96% return, not 8%. This option miscalculates the risk-adjusted return and is incorrect.
Option 3: 5.96%.
- This matches the calculated expected return precisely. The 0.97 probability of full repayment yields 1,080,000 in that case, while the 0.03 probability of default yields a 400,000 recovery. Computing the expected cash flow: 0.97*1,080,000 + 0.03*400,000 = 1,059,600. The profit over the initial 1,000,000 is 59,600, giving a return of 59,600/1,000,000 = 5.96%. This aligns with the correct risk-adjusted expectation. This option is correct.
Option 4: 6.80%.
- A 6.80% return would correspond to an expected cash flow of 1,068,000. But as shown, the actual expected cash flow given the stated probabilities and recoveries is 1,059,600, which corresponds to 5.96%, not 6.80%. Therefore this choice is incorrect.

Summary of why the correct answer is correct and the others are not:
- The correct calculation accounts for both the high likelihood of repayment and the partial recovery in default, yielding an overall expected cash flow of 1,059,600 and an expected return of 5.96%.
- The other options fail because they overstate the return by ignoring the impact of the 3% probability of default and the 40% recovery, or they misapply the arithmetic of combining repayment and recovery probabilities.

BU.231.710.51.FA25 The Final Exam

A bank is considering a 1-year loan of $1 million at a promised interest rate of 8%. The probability of repayment is 97% and the probability of default is 3%. If default occurs, the bank expects to recover 40% of the principal. What is the expected return on this loan?

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