Final answer:
The present value of a bond is calculated by discounting future cash flows back to their present value using the present value formula. Adjusting the discount rate to match market rates affects the present value, as higher rates decrease it. A bond with a lower interest rate than the market rate would be priced based on what an equivalent investment at the market rate would yield in a year.
Step-by-step explanation:
To calculate the present value of a two-year bond with a face value of $3,000 and an interest rate of 8%, we must discount the future cash flows both from the annual interest payment and the principal repayment. The bond will pay $240 in interest each year, and then return the face value of $3,000 at the end of the second year. When using the same interest rate as the discount rate (8%), the present value of each payment can be calculated using the present value formula: PV = FV / (1 + r)^n, where PV is present value, FV is future value, r is the discount rate and n is the number of periods.
If the market interest rate rises to 11%, the discount rate used in the present value calculation must be increased, which will decrease the present value of the bond's future cash flows. This reflects a higher opportunity cost of capital, meaning each future payment is worth less in today's dollars given that the alternative investments now yield a higher return.
Let's illustrate with calculations:
- When the discount rate is 8%, the present value of the first interest payment ($240) would be $240 / (1 + 0.08)^1, and the second payment (interest + principal = $3,240) would be $3,240 / (1 + 0.08)^2.
- When the discount rate is 11%, the present value of the first interest payment would be $240 / (1 + 0.11)^1, and the second payment would be $3,240 / (1 + 0.11)^2.
Similarly, for a bond with a lower interest rate than market rates, such as one paying $1,080 in its final year with market rates at 12%, the bond's price would be set at no more than the amount that, if invested at the current market rate of 12%, would grow to $1,080 in one year. That amount is calculated to be $964 since $964(1 + 0.12) = $1080.