(4 marks, difficulty level: Easy) Your firm plans to buy a warehouse for $150,000. The bank offers you a 25-year loan with equal annual payments and an interest rate of 10% per year. The bank requires that your firm pay 40% of the purchase price as a down payment, so you can borrow only $90,000. What is the annual loan payment?

Question

(4 marks, difficulty level: Easy) Your firm plans to buy a warehouse for $150,000. The bank offers you a 25-year loan with equal annual payments and an interest rate of 10% per year. The bank requires that your firm pay 40% of the purchase price as a down payment, so you can borrow only $90,000. What is the annual loan payment?

BlackTom AI · Accepted Answer

We start by identifying the loan details: purchase price is $150,000, down payment is 40% of that, so down payment = 0.40 × 150,000 = 60,000. The amount to borrow is 150,000 − 60,000 = $90,000. The loan term is 25 years with equal annual payments, and the annual interest rate is 10%.

To find the annual loan payment (PMT) for a fixed-rate loan, we use the standard amortization formula: PMT = P × r / (1 − (1 + r)^(−n)), where P is the loan principal, r is the periodic interest rate, and n is the number of periods.

Plugging in the numbers: P = 90,000, r = 0.10, n = 25.
First compute (1 + r)^(−n) = 1.1^(-25). The value of 1.1^25 is about 10.828, so its reciprocal is roughly 0.0924. Therefore, 1 − (1.1)^(-25) ≈ 1 − 0.0924 = 0.9076.
Next, compute the numerator P × r = 90,000 × 0.10 = 9,000.
Finally, PMT ≈ 9,000 ÷ 0.9076 ≈ 9,923 (approximately). The more precise calculation with a calculator yields a payment near $9,915 to $9,920.

Now evaluate the answer choices:
- $12,730: This is significantly higher than the calculated payment; it would correspond to a much larger loan or a shorter term, so this option overestimates the necessary annual payment.
- $11,173: This is also higher than the computed figure; using the same formula, a payment around $11k would imply a higher loan amount, a higher interest rate, or a shorter term than specified, making this unrealistic for the given data.
- $9,915: This is the closest to the precise PMT derived from the formula. Given typical rounding in financial calculations, $9,915 matches the expected order of magnitude and aligns with a 25-year amortization at 10% on a $90,000 loan.
- $10,160: This value is somewhat higher than the precise calculation. It would occur if the rate or term were slightly different or if rounding pushed the PMT upward, but with 10% and 25 years it is a less accurate estimate.

In summary, the payment necessary to amortize a $90,000 loan over 25 years at 10% interest is about $9,915, with the other options deviating upward due to overestimate, misalignment with the formula, or inconsistent assumptions.

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