A bank offers a nominal interest rate of 10% per annum compounded continuously. What is the corresponding effective annual interest rate, correct to two decimal places?

Introduction:
This question involves continuous compounding, a concept used in higher level finance and mathematics. Instead of compounding monthly or quarterly, continuous compounding assumes interest is added at every instant. The goal is to convert a nominal rate of 10% compounded continuously into an equivalent effective annual rate that represents the actual percentage increase over one full year.

Given Data / Assumptions:

• Nominal annual rate with continuous compounding, r_nominal = 10% per year
• Interest is compounded continuously, not at discrete intervals
• We want the effective annual interest rate, r_effective
• No withdrawals or additional deposits occur during the year

Concept / Approach:
For continuous compounding, the amount A after time t (in years) with principal P and nominal rate r is given by: A = P * e^(r * t) For one year, t = 1, so: A = P * e^r The effective annual rate r_effective is the overall percentage increase in one year: r_effective = (A - P) / P = e^r - 1 Here e is the mathematical constant approximately equal to 2.71828.

Step-by-Step Solution:
Step 1: Convert the nominal rate from percent to decimal. r = 10% = 0.10 Step 2: Use the continuous compounding formula for one year. A = P * e^(0.10) Step 3: Express the effective rate as: r_effective = e^0.10 - 1 Step 4: Approximate e^0.10. e^0.10 ≈ 1.1052 Step 5: Subtract 1 to get the decimal effective rate. r_effective ≈ 1.1052 - 1 = 0.1052 Step 6: Convert to a percentage. r_effective ≈ 0.1052 * 100 = 10.52%

Verification / Alternative check:
As a sense check, we know that with annual compounding at 10%, the effective rate is 10%. With continuous compounding at the same nominal 10%, the effective rate must be slightly higher than 10% but not dramatically larger. The value 10.52% lies just above 10%, which is reasonable and consistent with expectations for continuous compounding at this rate.

Why Other Options Are Wrong:
9.52%: This is below the nominal rate and therefore cannot be the effective rate when compounding continuously at 10%. 10.52%: This matches the correct computation using e^0.10 - 1. 11.52%: This is too high for a nominal rate of 10% and would require a larger r value. 12.52%: This is even further from the correct effective rate and does not match the continuous compounding formula. 10.25%: This might come from a rough or incorrect approximation but does not match the precise calculation.

Common Pitfalls:
Learners sometimes confuse continuous compounding with very frequent discrete compounding and try to use (1 + r / m)^m directly with a large m instead of using e^r. Another frequent error is to substitute r in percent form (10) rather than decimal form (0.10) into the exponent. It is also easy to forget to subtract 1 when converting from the growth factor e^r to the effective interest rate, which would incorrectly suggest the growth factor itself is the rate.

Final Answer:
The effective annual interest rate corresponding to a nominal 10% per year compounded continuously is approximately 10.52%.