Final answer:
The present value of a $3,000 bond with an 8% interest rate is $2,907.41 when the discount rate is 8% and $2,805.36 when the discount rate rises to 11%. These calculations are determined using the present value formula for each payment to be received from the bond, demonstrating the inverse relationship between bond prices and interest rates.
Step-by-step explanation:
Present Value Calculation of a Two-Year Bond
When calculating the present value of a bond, we need to consider both the interest payments and the principal that will be returned at maturity. For a $3,000 bond with an 8% interest rate, the bond will pay $240 in interest each year. To determine the present value of these payments when the discount rate is the same as the interest rate (8%), we can use the present value formula for each year:
- Year 1 Interest: Present Value = $240 / (1 + 0.08) = $222.22
- Year 2 Interest + Principal: Present Value = ($240 + $3,000) / (1 + 0.08)^2 = $2,685.19
Total present value when the discount rate is 8%: $222.22 + $2,685.19 = $2,907.41.
If the interest rates rise, causing the discount rate to change to 11%, the new calculations are as follows:
- Year 1 Interest: Present Value = $240 / (1 + 0.11) = $216.22
- Year 2 Interest + Principal: Present Value = ($240 + $3,000) / (1 + 0.11)^2 = $2,589.14
Total present value when the discount rate is 11%: $216.22 + $2,589.14 = $2,805.36.
Therefore, an increase in the discount rate from 8% to 11% decreases the present value of the bond. In financial terms, this is expressed as the inverse relationship between bond prices and interest rates.