Calculate the effective annual rate (EAR) corresponding to the annual percentage rate (APR) of 8.14% when it is compounded 10 times a year.

Question

BlackTom AI · Accepted Answer

We start by restating the problem clearly: calculate the effective annual rate (EAR) given APR = 8.14% compounded 10 times per year.

Option-by-option analysis:
Option A: 8.26% - This value is close to the nominal APR but does not account for compounding. EAR must be higher than the APR when compounding happens within the year, so 8.26% cannot be correct for an APR of 8.14% with 10 compounding periods per year.

Option B: 8.12% - This is actually lower than the stated APR, which would be inconsistent with any positive compounding within the year. Since compounding increases the effective yield, EAR cannot be less than APR in this scenario, so this option is incorrect.

Option C: 8.44% - This option reflects the effect of intra-year compounding. To verify, compute EAR = (1 + APR/n)^{n} − 1 with APR as a decimal and n = 10. Here APR/n = 0.0814/10 = 0.00814, so (1.00814)^{10} − 1. The natural approach is to approximate using a logarithm or a calculator: ln(1.00814) ≈ 0.0081069, multiply by 10 gives ≈ 0.081069, and e^{0.081069} ≈ 1.0844, subtract 1 gives ≈ 0.0844, i.e., 8.44%. This matches the third option exactly when expressed as a percentage.

Option D: 8.48% - This is slightly higher than the value derived from the exact formula above. While the difference is small, precise compounding at 10 periods per year with APR 8.14% does not reach 8.48%; the calculation shows it sits closer to 8.44%, making this option inaccurate.

Summary reasoning: The correct approach is to apply the EAR formula for discrete compounding, converting APR to a decimal, dividing by the number of periods, raising to the power of the number of periods, and subtracting one. Doing so yields approximately 8.44%, which aligns with option C. The other options either neglect the compounding effect (8.26%), fall below the APR due to a misinterpretation, or overshoot the result (8.48%), revealing their inaccuracies.

Calculate the effective annual rate (EAR) corresponding to the annual percentage rate (APR) of 8.14% when it is compounded 10 times a year.

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