A financial manager is comparing two investment options. Option 1 offers an 8% annual interest rate compounded quarterly. Option 2 offers a 7.8% annual interest rate compounded monthly. Which option offers a higher effective annual interest rate (EAR)?
- AOption 2, with an EAR of approximately 8.12%
- BOption 1, with an EAR of approximately 8.24%
- COption 2, with an EAR of approximately 8.08%
- DOption 1, with an EAR of approximately 8.16%
Show answer & explanationAnswer & explanation
Correct answer: B. Option 1, with an EAR of approximately 8.24%
To compare, we must calculate the Effective Annual Interest Rate (EAR) for both options using the formula EAR = (1 + r/n)^n - 1, where r is the nominal annual rate and n is the number of compounding periods per year. For Option 1: r = 0.08, n = 4 (quarterly). EAR = (1 + 0.08/4)^4 - 1 = (1 + 0.02)^4 - 1 = (1.02)^4 - 1 ≈ 1.082432 - 1 ≈ 0.082432 or 8.24%. For Option 2: r = 0.078, n = 12 (monthly). EAR = (1 + 0.078/12)^12 - 1 = (1 + 0.0065)^12 - 1 ≈ 1.080849 - 1 ≈ 0.080849 or 8.08%. Comparing the two, Option 1 (8.24%) offers a higher EAR than Option 2 (8.08%).
Why the other options are wrong
- A. This is a plausible distractor, perhaps from a calculation error for Option 2 or incorrect comparison.
- C. This correctly calculates EAR for Option 2 but incorrectly states it's higher.
- D. This is a plausible distractor, perhaps from a calculation error for Option 1.
Effective Annual Interest Rate (EAR)
The Effective Annual Interest Rate (EAR) is the actual annual rate of return earned on an investment or paid on a loan, taking into account the effect of compounding over a year. It allows for a standardized comparison of different interest rates with varying compounding periods.
- Formula: EAR = (1 + r/n)^n - 1, where r is the nominal rate and n is compounding periods per year.
- Higher compounding frequency (larger 'n') generally leads to a higher EAR for a given nominal rate 'r'.
- EAR is always greater than or equal to the nominal rate for positive interest rates.
Memory trick: EAR: Rate 'r' divided by 'n', raised to 'n', then minus one for the true gain.