Amy invested $2500 in a bank account at an interest rate of 2.5% per annum, compounded monthly. The effective rate of interest for her investment is closest to

Question

BlackTom AI · Accepted Answer

Effective annual rate (EAR) is given by (1 + r/m)^m − 1 where r is the nominal annual rate and m is the number of compounding periods per year.

1) Relevant knowledge points and concepts  
- Nominal rate r = 2.5% = 0.025 (per year).  
- Compounding monthly ⇒ m = 12.  
- EAR accounts for intra-year compounding and will be slightly larger than the nominal rate when m > 1.

2) Complete reasoning process step by step  
Apply the EAR formula: EAR = (1 + r/m)^m − 1. Substitute r = 0.025 and m = 12:  
EAR = (1 + 0.025/12)^12 − 1.

3) Calculations with intermediate steps  
Compute the monthly rate: 0.025/12 = 0.0020833333333333333.  
So 1 + monthly rate = 1.0020833333333333.  
Raise to the 12th power. Using series expansion or direct calculation:  
(1.0020833333)^12 ≈ 1 + 12(0.0020833333) + 66(0.0020833333)^2 + 220(0.0020833333)^3 + ...  
= 1 + 0.025 + 0.000286278 + 0.000001989 + ... ≈ 1.0252883.  
Therefore EAR ≈ 1.0252883 − 1 = 0.0252883 = 2.52883%. Rounded to two decimal places gives 2.53%.

4) Why the correct answer is correct  
Monthly compounding increases the effective yield above the nominal 2.50% by the accumulation of interest on interest each month; the precise formula yields about 2.5288%, which matches option B when rounded.

5) Why other options are incorrect  
A (20.4%) and C (12.44%) are far too large given a 2.5% nominal rate; they arise from miscalculation or misinterpretation. D (2.50%) is the nominal rate, not the effective rate with monthly compounding—EAR must be slightly higher.

6) Conclusion confirming the final answer  
Thus the effective annual rate is approximately 2.53%, confirming answer B.

Amy invested $2500 in a bank account at an interest rate of 2.5% per annum, compounded monthly. The effective rate of interest for her investment is closest to单项选择题

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