Calculate the effective annual interest rate corresponding to a nominal annual rate of 10% when interest is compounded continuously.

Introduction:
This question explores the concept of continuous compounding, which is a theoretical limit where interest is added at every instant. You are asked to find the effective annual rate corresponding to a nominal rate of 10% compounded continuously, a concept widely used in higher finance and mathematical modeling.

Given Data / Assumptions:

• Nominal annual interest rate r_nominal = 10% per annum
• Compounding type = continuously compounded interest
• We need the effective annual rate r_effective that gives the same growth if compounded once per year.

Concept / Approach:
For continuous compounding, the amount after time t years is given by:
A = P * e^(r_nominal * t)For one year (t = 1), this becomes:
A = P * e^(r_nominal)Thus, the effective annual rate is:
r_effective = e^(r_nominal) - 1where r_nominal is expressed as a decimal, and e is the base of natural logarithms, approximately 2.71828.

Step-by-Step Solution:
Step 1: Convert the nominal rate to decimal.r_nominal = 10% = 0.10Step 2: Compute the growth factor for one year.Growth factor = e^(0.10)This value is approximately 1.10517.
Step 3: Determine the effective rate.r_effective = 1.10517 - 1 = 0.10517r_effective ≈ 10.52%

Verification / Alternative check:
We know that for annual compounding at 10%, the effective rate is exactly 10%. For more frequent compounding, such as quarterly or monthly, the effective rate becomes slightly higher than 10%. Continuous compounding is the theoretical maximum frequency, so the effective rate must be somewhat above 10%, and a value near 10.5% fits this pattern well.

Why Other Options Are Wrong:

• 9.52%: This is less than the nominal rate and cannot result from compounding.
• 11.52% and 12.52%: These are too high for a 10% nominal rate, even with continuous compounding.
• 10.00%: This corresponds to annual compounding, not continuous compounding.

Common Pitfalls:
A key mistake is to use the standard discrete compounding formula (1 + r/m)^m for continuous compounding. Another issue is forgetting to convert the percentage to decimal before using the exponential function. Always use the formula A = P * e^(r * t) for continuous compounding, and then subtract 1 from the growth factor to obtain the effective rate.

Final Answer:
The effective annual interest rate corresponding to a 10% nominal rate with continuous compounding is approximately 10.52%.