Suppose you purchased your first home for $520,000. At the time of purchase, you could only afford to commit to a down-payment of $50,000, and you took a 30-year mortgage loan for the remaining amount, at a 5% interest rate.

In order to provide the loan, the lender required you to obtain private mortgage insurance (PMI) on their behalf. Assume that the PMI will cover losses up to 30% of whatever the loan principal was at origination.

It is now a year later, and although you have made the first 12 monthly payments as scheduled, owing to unfortunate circumstances you lose your job and default on the loan. If the lender were to foreclose on your property and sell it for $300,000, what would the lender’s loss of principal be, taking into consideration the protection of mortgage insurance?

Question

Suppose you purchased your first home for $520,000. At the time of purchase, you could only afford to commit to a down-payment of $50,000, and you took a 30-year mortgage loan for the remaining amount, at a 5% interest rate.

In order to provide the loan, the lender required you to obtain private mortgage insurance (PMI) on their behalf. Assume that the PMI will cover losses up to 30% of whatever the loan principal was at origination.

It is now a year later, and although you have made the first 12 monthly payments as scheduled, owing to unfortunate circumstances you lose your job and default on the loan. If the lender were to foreclose on your property and sell it for $300,000, what would the lender’s loss of principal be, taking into consideration the protection of mortgage insurance?

BlackTom AI · Accepted Answer

The lender’s loss equals the outstanding loan principal after 12 payments minus the foreclosure sale proceeds, reduced by the PMI payout (which caps at 30% of the original loan).

1) Relevant concepts
- Origination principal = purchase price − down payment.
- Monthly mortgage payment for fixed-rate loan: M = P * r / (1 − (1 + r)^−N).
- Outstanding balance after k payments: B_k = P*(1 + r)^k − M * (( (1 + r)^k − 1)/r).
- PMI covers losses up to 30% of the original loan principal.

2) Step-by-step reasoning and 3) intermediate calculations
- Origination loan principal P = $520,000 − $50,000 = $470,000.
- Monthly interest rate r = 5%/12 = 0.05/12 = 0.0041666667. Total months N = 360.
- Compute monthly payment M:
  (1 + r)^360 ≈ 4.46815, so 1 − (1 + r)^−360 = 1 − 1/4.46815 ≈ 0.77625.
  M = 470,000 * 0.0041666667 / 0.77625 ≈ $2,522.79 per month.
- Compute (1 + r)^12 ≈ 1.051162.
- Outstanding balance after 12 payments:
  B_12 = 470,000 * 1.051162 − 2,522.79 * ((1.051162 − 1)/0.0041666667)
       ≈ 494,044.14 − 2,522.79 * 12.27888
       ≈ 494,044.14 − 30,977.20
       ≈ $463,066.94.
- Shortfall before PMI = B_12 − sale proceeds = 463,066.94 − 300,000 = $163,066.94.
- PMI coverage = 30% of origination principal = 0.30 * 470,000 = $141,000.
- Net lender loss = shortfall − PMI = 163,066.94 − 141,000 = $22,066.94 ≈ $22,066.

4) Why this is correct
The calculation follows standard amortization to find the remaining principal after one year, subtracts the foreclosure sale proceeds, and then applies the PMI cap. PMI does not eliminate the entire shortfall—only up to 30% of the original principal—so the lender retains the residual loss.

5) Why the other choices are incorrect
- $0: Incorrect because PMI ($141,000) is less than the shortfall ($163,066.94), so there remains a loss.
- $163,066: This is the shortfall before PMI; it ignores the $141,000 PMI recovery.
- $7,066: No correct intermediate gives this; it would imply an incorrect PMI application or miscalculated outstanding balance.

6) Conclusion
After amortization, sale proceeds, and applying PMI coverage, the lender’s loss of principal is $22,066, matching the provided answer.

类似问题

Which of the following statements is CORRECT? Assume that the project being considered has normal cash flows, with one outflow followed by a series of inflows.

Suppose you are buying your first condo for $145,000, and you will make a $15,000 down payment. You have arranged to finance the remainder with a 30-year, monthly payment, amortized mortgage at a 6.5% nominal interest rate, with the first payment due in one month. What will your monthly payments be?

A $150,000 loan is to be amortized over 7 years, with annual end-of-year payments. Which of these statements is CORRECT?

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