(4 marks, difficulty level: Easy) Your firm plans to buy a warehouse for $150,000. The bank offers you a 35-year loan with equal annual payments and an interest rate of 14% 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

BlackTom AI · Accepted Answer

We are given a loan amount (present value) of $90,000, an annual interest rate of 14%, and 35 years of equal annual payments. The goal is to find the annual loan payment.
First, recall the formula for the annual payment (annuity payment) on a loan: P = r * PV / (1 - (1 + r)^(-n)), where r is the annual interest rate (as a decimal), PV is the present value (loan amount), and n is the number of payments.
Here, r = 0.14, PV = 90,000, and n = 35. Compute the discount factor (1 + r)^(-n) = (1.14)^(-35).
Estimating (1.14)^35 via logarithms: ln(1.14) ≈ 0.1305, so 35 * 0.1305 ≈ 4.5675. Exponentiating, (1.14)^35 ≈ e^(4.5675) ≈ 96.0, hence (1.14)^(-35) ≈ 1/96.0 ≈ 0.0104.
Now plug into the formula: P ≈ (0.14 * 90,000) / (1 - 0.0104) = 12,600 / 0.9896 ≈ 12,734. Rounding to the nearest dollar gives about $12,730.
Next, evaluate each answer choice:
- $10,160: This is smaller than the computed payment. If you tried a rough check, it would imply a much higher PV or a much longer term than 35 years, neither of which aligns with the given data. The annuity factor at 14% for 35 years does not yield as low a payment as this.
- $9,915: This is even lower and would require an even larger present value, a higher interest rate, or more years of payments than stated; it is inconsistent with the provided loan terms.
- $12,730: This matches the calculation above. Using the exact annuity formula with PV = 90,000, r = 0.14, n = 35 yields a payment in the neighborhood of 12.73k, so this option is consistent with the math.
- $11,173: This is between the two lower options but still smaller than the computed 12,730. It would correspond to a smaller PV, a slightly lower rate, or a shorter term, none of which fit the given data.
In summary, the correct payment derived from the standard annuity formula for PV = 90,000 at 14% over 35 years is closest to $12,730, while the other numbers do not align with the calculated annuity factor or the stated loan terms.

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