The effective anuual rate of interest corresponding to a nominal rate of `8%` per annum payable half-yearly is :

Options

A. `8%`
B. `8.64%`
C. `8.16%`
D. `6.09%`

Answer

To find the effective annual rate of interest corresponding to a nominal rate of 8% per annum payable half-yearly, we can follow these steps:

### Step 1: Identify the nominal interest rate and compounding frequency
The nominal interest rate (i) is given as 8% per annum. Since it is compounded half-yearly, we divide this rate by 2 to find the interest rate for each half-year.

\[ \text{Half-yearly interest rate} = \frac{8\%}{2} = 4\% \]

### Step 2: Calculate the effective annual rate using the formula
The effective annual rate (EAR) can be calculated using the formula for compound interest:

\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1 \]

where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year,
- \( t \) is the number of years.

In this case:
- \( r = 0.08 \) (8% as a decimal),
- \( n = 2 \) (since it is compounded half-yearly),
- \( t = 1 \) (for one year).

Substituting these values into the formula:

\[ \text{EAR} = \left(1 + \frac{0.08}{2}\right)^{2 \times 1} - 1 \]

### Step 3: Simplify the expression
Calculating the expression inside the parentheses:

\[ \text{EAR} = \left(1 + 0.04\right)^{2} - 1 \]

\[ \text{EAR} = (1.04)^{2} - 1 \]

### Step 4: Calculate \( (1.04)^{2} \)
Now, we calculate \( (1.04)^{2} \):

\[ (1.04)^{2} = 1.0816 \]

### Step 5: Subtract 1 to find the effective rate
Now, we subtract 1 from the result:

\[ \text{EAR} = 1.0816 - 1 = 0.0816 \]

### Step 6: Convert to percentage
Finally, convert the decimal back to a percentage:

\[ \text{EAR} = 0.0816 \times 100 = 8.16\% \]

Thus, the effective annual rate of interest corresponding to a nominal rate of 8% per annum payable half-yearly is **8.16%**.

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