A bond of face value Rs. 1000 matures in 10 years and interest is paid annually. If the yield to maturity is 4% per annum and the present value of the bond is Rs. 838, find the coupon rate \((1.04)^{-10} \approx 0.676\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2.0% p.a.
Let the annual coupon (interest) be Rs. C. By the present value approach, the price of the bond is the present value of the 10 coupons plus the present value of the face value, both at the yield \(i = 0.04\):
\( \displaystyle V = C \times \frac{1 - (1+i)^{-n}}{i} + F(1+i)^{-n} \)
With \(F = 1000\), \(n = 10\) and \((1.04)^{-10} \approx 0.676\):
\( \displaystyle \frac{1 - 0.676}{0.04} = \frac{0.324}{0.04} = 8.1 \)
\(838 = 8.1C + 1000 \times 0.676\)
\(8.1C = 838 - 676 = 162\)
\(C = 20\)
\( \displaystyle \text{coupon rate} = \frac{20}{1000} \times 100\% = 2.0\% \text{ p.a.} \)
Note: It seems that there is a mistake in the original NTA question. It gave the interest (coupon) rate as 4% per annum and asked for the yield to maturity. A bond priced at Rs. 838, below its face value of Rs. 1000, must have a yield higher than its coupon rate of 4%, but every option is below 4% (the yield works out to about 6.2%), so the question cannot be answered as printed. For the benefit of students and to ensure that the question is correct and has a single valid answer, the question has been changed to give the yield to maturity as 4% per annum and to ask for the coupon rate.