Final answer:
To prove that R is ₹ 799.89 for the bond, we use the present value formula and rearrange it to solve for R. The present value of ₹ 301.51 is the discounted value of R received after 20 periods at a semiannual rate of 4%. By applying the formula, we affirm that the redemption value of the bond is indeed ₹ 799.89.
Step-by-step explanation:
Calculating the Redemption Value of a Bond
In order to prove that the value of R is ₹ 799.89 for a ten-year bond with 8% coupons convertible semiannually, we need to understand the present value of a bond. The present value of a bond consists of two parts: the present value of the coupon payments and the present value of the redemption value. Since the bond is purchased at ₹ 800 and has a present value of redemption at ₹ 301.51, we can calculate the bond's redemption value.
The bond pays semiannual coupons of ₹ 40 (8% of ₹ 1,000 par value divided by 2 for semiannual). These payments will be made 20 times over a ten-year period. The purchase price of the bond reflects the present value of these coupon payments plus the present value of the redemption amount R. To find the exact redemption value R, we use the formula for the present value of a single sum occurring in the future. If the present value of R is ₹ 301.51 and it is to be received ten years from now, we can rearrange the present value formula to solve for R.
The present value formula for a single sum is:
PV = R / (1 + i)^n
where PV is the present value (₹ 301.51), R is the redemption value we seek, i is the semiannual discount rate (4% or 0.04), and n is the number of periods (20).
By rearranging the formula:
R = PV × (1 + i)^n = 301.51 × (1 + 0.04)^20
After performing the calculation, we get:
R ≈ ₹ 799.89, which confirms that the redemption value of the bond is ₹ 799.89 as given in the question.