Question

The effective rate equivalent to a nominal rate of 8% per annum compounded semi-annually is:

• A. 8.20%
• B. 8.24%
• C. 8%
• D. 8.16%

Answer 1

The Correct Option is D

The problem requires calculating the effective annual rate (EAR) given a nominal rate of 8% per annum compounded semi-annually. Here's how to solve it:

1. Identify the nominal annual rate (APR):APR = 8%
2. Determine the number of compounding periods per year:n = 2 (since it's semi-annually)
3. Convert the nominal rate to a decimal for calculations:r = \(\frac{8}{100} = 0.08\)
4. Use the formula for effective annual rate: \[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \]
5. Substitute the known values into the formula: \[ \text{EAR} = \left(1 + \frac{0.08}{2}\right)^2 - 1 \]
6. Simplify the expression: \[ \text{EAR} = \left(1 + 0.04\right)^2 - 1 = (1.04)^2 - 1 \]
7. Calculate the exponent: \[ (1.04)^2 = 1.0816 \]
8. Subtract 1: \[ 1.0816 - 1 = 0.0816 \]
9. Convert the effective rate back to percentage: \(\text{EAR} = 0.0816 \times 100 = 8.16\%\)

Thus, the effective annual rate equivalent to a nominal rate of 8% compounded semi-annually is 8.16%.