Your company has just taken out a 1-year installment loan for $72,500 at a nominal rate of 14.0% but with equal end-of-month payments. What percentage of the 2nd monthly payment will go toward the repayment of principal?

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BlackTom AI · Accepted Answer

Question restatement: A company borrowed 72,500 with a 1-year installment loan, nominal annual rate 14.0%, with equal end-of-month payments. What percentage of the 2nd monthly payment goes toward principal?

First, determine the monthly interest rate i. With a nominal annual rate of 14% compounded monthly, i = 0.14 / 12 = 0.0116666667 (about 1.1667% per month).

Next, compute the monthly payment P for 12 months on a loan of 72,500 at rate i. The standard formula is P = L * i / (1 - (1 + i)^(-n)), where L = 72,500, n = 12.

Estimate (1 + i)^12 ≈ exp(12 * ln(1 + i)). Since i ≈ 0.0116667, ln(1 + i) ≈ 0.01159, so 12 * ln(1 + i) ≈ 0.1391, giving (1 + i)^12 ≈ 1.149. Therefore (1 + i)^(-12) ≈ 1 / 1.149 ≈ 0.869.

Denominator becomes 1 - (1 + i)^(-12) ≈ 1 - 0.869 = 0.131. Numerator L * i ≈ 72,500 * 0.0116667 ≈ 846.0. Thus P ≈ 846.0 / 0.131 ≈ 6,466 per month.

Now, determine the balance after the first payment to find interest portion of the second payment. Beginning balance after payment 1, B1 = L*(1 + i) - P. Compute L*(1 + i) ≈ 72,500 * 1.0116667 ≈ 73,422.1. Subtract P ≈ 6,466 → B1 ≈ 66,956.1.

Interest in month 2 equals B1 * i ≈ 66,956.1 * 0.0116667 ≈ 781.0.

Principal portion of the second payment = P - interest ≈ 6,466 - 781 ≈ 5,685.

Finally, the percentage of the second payment that goes toward principal is (principal portion) / (total payment) ≈ 5,685 / 6,466 ≈ 0.879, i.e., about 87.9%. Rounding aligns with the provided option around 88%.

Notes on the missing answer options: The prompt shows an exact answer guess of 88.02% but does not supply alternative choices to compare. The calculation above supports a principal portion around 88% of the second payment. If there were other options, they would need to be evaluated similarly by computing the second-month interest and principal portions from the amortization schedule.

fin_221_120251_242735 Submit: Test 1 (Chapters 1 & 5)

Your company has just taken out a 1-year installment loan for $72,500 at a nominal rate of 14.0% but with equal end-of-month payments. What percentage of the 2nd monthly payment will go toward the repayment of principal?

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