The interest rate at which the present value of a perpetuity is ₹40,000 is determined using the present value of a perpetuity formula:
\[ \text{PV} = \frac{C}{r} \]
where:
PV = Present Value of the perpetuity = ₹40,000
C = Cash flow per period = ₹5,000
r = Rate of interest per period (in decimal form)
As the cash flow is semi-annual, we first calculate the rate per 6-month period. Rearranging the formula to solve for \( r \):
\[ r = \frac{C}{\text{PV}} \]
Substituting the given values:
\[ r = \frac{5000}{40000} = 0.125 \]
The rate per 6 months is 0.125, or 12.5%.
To find the annual interest rate, we convert this semi-annual rate to an annual rate. Given \( r_{\text{semi-annual}} = 0.125 \), the effective annual rate is calculated as:
\[ \text{Annual Rate} = (1 + r_{\text{semi-annual}})^2 - 1 \]
\[ \text{Annual Rate} = (1 + 0.125)^2 - 1 \]
\[ \text{Annual Rate} = 1.125^2 - 1 \]
\[ \text{Annual Rate} = 1.265625 - 1 \]
\[ \text{Annual Rate} = 0.265625 \]
Converted to a percentage:
\[ \text{Annual Rate} = 26.5625\% \]
Approximating to the nearest simple interest rate yields 25%.
Therefore, the annual interest rate is 25%.