The numbers below represent an APR and the corresponding effective annual rates when compounded semi-annually, monthly and daily. Identify the effective annual rate when compounded semi-annually. [Nothing is missing, you have enough information to figure it out.]

Question

BlackTom AI · Accepted Answer

The problem provides a single APR and the corresponding effective annual rates (EARs) for three compounding frequencies: semi-annually, monthly, and daily. Our task is to identify which of the given options is the EAR for semi-annual compounding.

Option A: 17.69% - This value is plausible as an EAR for a nominal APR, and it is higher than a lower-frequency EAR would typically be if the same APR were used. However, since semi-annual compounding yields fewer compounding events per year than monthly or daily, its EAR should be the smallest among the standard frequencies for a given APR. Therefore, 17.69% would correspond to a more frequent compounding (monthly or daily) rather than semi-annual, making this option unlikely for the semi-annual EAR.

Option B: 17.07% - This is a moderate EAR value. Because semi-annual compounding uses only two periods per year, its EAR tends to be lower than the EARs from monthly or daily compounding when the same nominal APR is applied. Among the provided options, 17.07% sits at a lower bound relative to the others (17.69% and 17.82%), making it consistent with the semi-annual case. It also remains clearly above 16% and well above any misinterpreted baseline, aligning with typical relationships between compounding frequencies and EARs.

Option C: 16.4% - This is the smallest value among the options. If we interpret it as an EAR, it would be unusually low for a scenario that includes semi-annual, monthly, and daily compounding with the same APR. In standard scenarios, all EARs for higher-frequency compounding (monthly, daily) exceed the semi-annual EAR, so 16.4% would likely correspond to a base APR that is not consistent with the other two higher EARs, making this option unlikely for the semi-annual EAR.

Option D: 17.82% - This is the largest EAR among the options. Such a high EAR would typically correspond to the daily compounding case (or the monthly case, depending on the APR), not the semi-annual case. Since semi-annual compounding yields the fewest periods per year, its EAR should not exceed the EARs of monthly or daily compounding with the same APR.

In summary, reasoning through the typical relationship between nominal APRs and their EARs under different compounding frequencies shows that the semi-annual EAR should be the relatively smaller value among the standard frequencies, while monthly and daily EARs would be larger. Among the given options, 17.07% best fits the expected position for the semi-annual EAR, with the other options either being too large for semi-annual or implausibly small for any of the listed frequencies.

The numbers below represent an APR and the corresponding effective annual rates when compounded semi-annually, monthly and daily. Identify the effective annual rate when compounded semi-annually. [Nothing is missing, you have enough information to figure it out.]

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