Mike is looking for a loan. He is willing to pay no more than an effective rate of 8.000% annually. Which, if any, of the following loans meet Mike’s criteria? Loan X: 7.815% nominal rate, compounded semiannually Loan Y: 7.724% nominal rate, compounded monthly Loan Z: 7.698% nominal rate, compounded weekly

Convert each nominal rate to an effective annual rate using (1 + nominal rate ÷ compounding periods) raised to the number of periods, then subtract 1. Loan X gives (1 + 0.07815 ÷ 2)² − 1 ≈ 7.968%, and Loan Z gives (1 + 0.07698 ÷ 52)⁵² − 1 ≈ 7.996%; both are at or below 8.000%. Loan Y’s effective annual rate is about 8.003%, so it exceeds Mike’s limit. The best answer is “X and Z.”

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