Find the effective annual rate of 5 percent per annum compound interest paid half yearly.

Options

A. `1.025%`
B. `6.0625%`
C. `5.062%`
D. None of these

Answer

To find the effective annual rate of a 5 percent per annum compound interest paid half yearly, we can follow these steps:

### Step 1: Determine the nominal interest rate and the compounding frequency.
The nominal interest rate is given as 5% per annum. Since the interest is compounded half-yearly, we need to divide this rate by 2.

**Calculation:**
\[ \text{Half-yearly interest rate} = \frac{5\%}{2} = 2.5\% \]

### Step 2: Use the formula for effective annual rate (EAR).
The formula to calculate the effective annual rate when compounding occurs multiple times a year is:
\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1 \]
Where:
- \( r \) = nominal annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years

In our case:
- \( r = 0.05 \) (5% as a decimal)
- \( n = 2 \) (since it's compounded half-yearly)
- \( t = 1 \) (for one year)

### Step 3: Substitute the values into the formula.
Substituting the values into the formula gives us:
\[ \text{EAR} = \left(1 + \frac{0.05}{2}\right)^{2 \times 1} - 1 \]

### Step 4: Simplify the expression.
Calculating the inside of the parentheses:
\[ \text{EAR} = \left(1 + 0.025\right)^{2} - 1 = (1.025)^{2} - 1 \]

### Step 5: Calculate \( (1.025)^{2} \).
Now, we calculate \( (1.025)^{2} \):
\[ (1.025)^{2} = 1.050625 \]

### Step 6: Subtract 1 to find the effective annual rate.
Now, we subtract 1 from the result:
\[ \text{EAR} = 1.050625 - 1 = 0.050625 \]

### Step 7: Convert to percentage.
To express this as a percentage, we multiply by 100:
\[ \text{EAR} = 0.050625 \times 100 = 5.0625\% \]

### Final Answer:
The effective annual rate of 5 percent per annum compound interest paid half yearly is **5.0625%**.

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