Your investment account generates a return of 13.10% (APR). Interest is compounded quarterly (once every 3 months). What is the effective annual rate (EAR) for this account?

Enter your answer in units of percent (not decimals), round your answer to two decimal places and omit percent signs (i.e., enter 20.214% as 20.21).

Question

Your investment account generates a return of 13.10% (APR).  Interest is compounded quarterly (once every 3 months).  What is the effective annual rate (EAR) for this account?

Enter your answer in units of percent (not decimals), round your answer to two decimal places and omit percent signs (i.e., enter 20.214% as 20.21).

BlackTom AI · Accepted Answer

Use the formula EAR = (1 + APR/m)^m − 1 with m = 4 (quarterly compounding).

1) Knowledge points:
- APR is the nominal annual rate; EAR accounts for intra-year compounding.
- For m compounding periods per year, the periodic rate = APR/m and EAR = (1 + APR/m)^m − 1.

2) Step-by-step calculation:
- Convert APR to decimal: APR = 13.10% = 0.1310.
- Periodic (quarterly) rate = APR / m = 0.1310 / 4 = 0.03275.
- Add 1: 1 + periodic rate = 1.03275.
- Raise to the 4th power: (1.03275)^4.
  - First square: (1.03275)^2 = 1.03275 × 1.03275 = 1.0665725625.
  - Square that result to get the fourth power: (1.0665725625)^2 = 1.1375770308850391 (shown with intermediate multiplication).
- Subtract 1 to get EAR (decimal): EAR = 1.1375770308850391 − 1 = 0.1375770308850391.

3) Convert to percent and round:
- EAR = 0.137577... = 13.7577030885%.
- Rounded to two decimal places: 13.76.

4) Why this is correct:
- Quarterly compounding causes interest to be earned on previously credited interest within the year, so the effective yearly return is higher than the nominal APR. The formula above exactly captures that effect.

5) Why common alternatives are incorrect:
- Using APR directly (13.10) ignores intra-year compounding.
- Using continuous compounding (e.g., e^{APR}−1) would give a different (typically slightly higher) EAR and is incorrect unless the problem specifies continuous compounding.
- Using the wrong m (e.g., m = 12 or m = 2) yields incorrect periodic rates and a wrong EAR.

Conclusion: The effective annual rate is 13.76, which matches the provided correct answer.

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