Consider a loan with a quoted annual interest rate and monthly compounding. Which statement about the effective annual rate (EAR) is correct?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

EAR = (1 + r_nom/m)^m − 1, so the effective annual rate is the annualized rate that yields the same accumulated amount over one year as the quoted (nominal) annual rate with m intra-year compounding.

1) Relevant concepts:
- Quoted (nominal) annual rate r_nom (also called APR in some contexts) is reported per year but divided across m compounding periods.
- Periodic rate = r_nom / m.
- Effective annual rate (EAR) = total percentage growth over one year after intra-year compounding: (1 + periodic rate)^m − 1.

2) Step-by-step reasoning:
- If you deposit $1, the balance after one period (length = 1/m year) is 1 + r_nom/m.
- After m periods (one year) the accumulated balance is (1 + r_nom/m)^m.
- The total interest earned in one year = accumulated balance − 1 = (1 + r_nom/m)^m − 1.
- By definition, EAR is that total interest expressed as an annual rate: EAR = (1 + r_nom/m)^m − 1.

3) Calculation example (intermediate steps):
- Let r_nom = 12% = 0.12, m = 12 (monthly). Periodic rate = 0.12/12 = 0.01.
- Accumulated factor = (1 + 0.01)^12 = 1.01^12 ≈ 1.126825.
- EAR = 1.126825 − 1 = 0.126825 ≈ 12.6825%.

4) Why option (a) is correct:
- It precisely states the meaning of EAR: the annualized rate that produces the same interest over one year as the stated nominal rate with intra-year compounding. This matches the formula and reasoning above.

5) Why the other choices are incorrect:
- (b) Confuses EAR with the periodic rate: r_nom/m is the periodic rate, not the EAR.
- (c) False: EAR can be computed for any compounding frequency m (including monthly); it is not limited to annual compounding.
- (d) False: With intra-year compounding EAR is generally higher than the quoted nominal APR (compounding increases the effective rate); it is not always lower.

6) Conclusion:
The correct choice is (a). This matches the provided answer.

Consider a loan with a quoted annual interest rate and monthly compounding. Which statement about the effective annual rate (EAR) is correct?[Fill in the blank]单项选择题

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