Your bank issues you a loan at 9% APR interest, compounded quarterly.

What is the effective interest rate for the entire loan if you pay it back after two months?

Round your answer to a percentage with two decimal places and do not use the “%” sign, i.e. if the answer is 1.23% then write 1.23

Question

Your bank issues you a loan at 9% APR interest, compounded quarterly.

What is the effective interest rate for the entire loan if you pay it back after two months?

Round your answer to a percentage with two decimal places and do not use the “%” sign, i.e. if the answer is 1.23% then write 1.23

BlackTom AI · Accepted Answer

We start by identifying the given: a loan with 9% APR, compounded quarterly, and the repayment occurs after two months. To find the effective interest rate over a partial period, we use the formula for effective rate over t months when compounding m times per year: (1 + r/m)^{m*(t/12)} − 1, where r is the annual interest rate.

Here, r = 0.09, m = 4 (quarterly), and t = 2 months. Compute the quarterly rate: r/m = 0.09/4 = 0.0225. The exponent becomes m*(t/12) = 4*(2/12) = 8/12 = 2/3.

So the effective rate over two months is (1 + 0.0225)^{2/3} − 1. Evaluate step by step: 1 + 0.0225 = 1.0225. Raise to the power 2/3 (approximately 0.6667). Using natural logs for a quick approximation, ln(1.0225) ≈ 0.022242, multiply by 2/3 to get ≈ 0.014828. Exponentiate: e^{0.014828} ≈ 1.01493. Subtract 1 to get ≈ 0.01493, which is about 1.493%.

Rounding to two decimal places as requested yields 1.49. In decimal terms, the two-month effective rate is approximately 0.0149, corresponding to 1.49%. If one were to think in terms of simple interest for two months at 9% annual, the result would be 9%*(2/12) = 1.5%, but because compounding occurs quarterly, the exact adjusted rate is slightly lower, hence 1.49% rather than 1.50%.

Therefore, the correct rounding result for the two-month effective rate is 1.49.

COMM_V 298 101 102 103 2025W1 COMM 298 2025 Winter Term 1: Midterm Extra Practice

Your bank issues you a loan at 9% APR interest, compounded quarterly. What is the effective interest rate for the entire loan if you pay it back after two months? Round your answer to a percentage with two decimal places and do not use the “%” sign, i.e. if the answer is 1.23% then write 1.23

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