Question

The rate of interest (per annum), at which the present value of a perpetuity of ₹5,000 payable at the end of every 6 months will be ₹40,000 is:

• A. 20%
• B. 25%
• C. 15%
• D. 30%

Answer 1

The Correct Option is B

To find the rate of interest at which the present value of a perpetuity is ₹40,000, we use the formula for the present value of a perpetuity:

\[ \text{PV} = \frac{C}{r} \]

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)

The cash flow here is payable every 6 months, so we first define the rate per 6-month period. Rearranging the formula to solve for \( r \):

\[ r = \frac{C}{\text{PV}} \]

Substitute the known 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 need to convert this semi-annual rate to an annual rate. Given \( r_{\text{semi-annual}} = 0.125 \), the effective annual rate is calculated as:

\[ \text{Annual Rate} = (1 + r_{\text{semi-annual}})^2 - 1 \]

\[ \text{Annual Rate} = (1 + 0.125)^2 - 1 \]

\[ \text{Annual Rate} = 1.125^2 - 1 \]

\[ \text{Annual Rate} = 1.265625 - 1 \]

\[ \text{Annual Rate} = 0.265625 \]

Convert to percentage:

\[ \text{Annual Rate} = 26.5625\% \]

Rounding to the nearest simple interest rate, we approximate to 25%.

Thus, the rate of interest per annum is 25%.

Answer 2

The formula for the present value (PV) of a perpetuity is given by:

PV = \(\frac{C}{r}\),

where C is the periodic payment and r is the interest rate per period. Here, PV = 40,000 and C = 5,000.

Substituting these values:

40,000 = \(\frac{5,000}{r}\).

Simplify to find r:

r = \(\frac{5,000}{40,000}\) = 0.125 (per 6 months).

Since r = 0.125 is the rate for 6 months, the annual rate is:

r_annual = 0.125 × 2 = 0.25 = 25%.

Thus, the correct answer is 25%.