Question

A principal amount is charged a nominal annual interest rate of 10%. If interest is compounded continuously, what happens to the final amount after one year?

• A. higher than the amount obtained when interest is compounded monthly
• B. lower than the amount obtained when interest is compounded annually
• C. equal to 1.365 times the principal amount
• D. equal to the amount obtained using an effective interest rate of 27.18%

Hint:

Continuous compounding always yields the maximum amount for the same nominal interest rate.

Answer 1

The Correct Option is A

Step 1: Continuous compounding formula.
\[ A = Pe^{rt} \] With \(r = 0.10\), \(t = 1\): \[ A_{\text{continuous}} = Pe^{0.1} \approx P(1.10517) \]

Step 2: Monthly compounding.
\[ A_{\text{monthly}} = P\left(1 + \frac{0.10}{12}\right)^{12} \] \[ = P(1.10471) \]

Step 3: Compare amounts.
\[ 1.10517 > 1.10471 \] Thus, continuously compounded amount is slightly higher than monthly compounding.

Step 4: Eliminate wrong options.
- (B) False — continuous compounding is highest.
- (C) False — 1.365 corresponds to 36.5% interest, unrelated to 10%.
- (D) False — 27.18% is unrelated and much larger.
Conclusion: Continuous compounding gives more than monthly compounding.