Final answer:
To determine the price paid for a $1,000 face value bond, the present value of the bond's future cash flows is calculated based on the market interest rate of 4%. Adding up the present value of each payment results in a total present value of $1,044.53, which rounds to $1,045.
Step-by-step explanation:
To determine how much you would pay for a $1,000 face value bond that has a 5% annual yield and compounds annually with a market interest rate of 4%, we need to calculate the present value of the bond. The bond will pay $50 (5% of $1,000) annually and will mature in five years. The present value (PV) of the bond is the sum of the present values of all future cash flows, including the interest payments and the face value repaid at maturity.
The formula for the present value of each cash flow (CF) is F / (1 + r)^n, where r is the market interest rate (4% in this case), and n is the period number. So we will calculate the present value of each of the five $50 payments, plus the present value of the $1,000 payment at the end of five years.
PV = $50 / (1 + 0.04) + $50 / (1 + 0.04)^2 + $50 / (1 + 0.04)^3 + $50 / (1 + 0.04)^4 + $50 / (1 + 0.04)^5 + $1,000 / (1 + 0.04)^5
Calculating these gives us:
- $50 / 1.04 = $48.08
- $50 / 1.0816 = $46.23
- $50 / 1.1249 = $44.45
- $50 / 1.1699 = $42.74
- $50 / 1.2167 = $41.10
- $1,000 / 1.2167 = $821.93
Adding these amounts together gives us a total present value of:
$48.08 + $46.23 + $44.45 + $42.74 + $41.10 + $821.93 = $1,044.53
Therefore, you would pay $1,045 to purchase this bond (rounded to the nearest dollar), which means the correct answer is (b) $1,045.