Final answer:
The duration of the bond can be calculated using the present value of each cash flow and their weights.
Step-by-step explanation:
The duration of a bond measures its sensitivity to changes in interest rates. It shows how much the bond's price will change in response to a 1% change in interest rates. To calculate the duration, we need to find the present value of each cash flow and multiply it by the time it will be received. In this case, the bond has a $1,000 face value and a semiannual coupon of 7%.
The semiannual coupon payment is $1,000 * 7% / 2 = $35. We can calculate the present value of each cash flow by discounting it at the yield to maturity, which is 10% / 2 = 5% for each semiannual period.
The present value of the face value at the end of five years is $1,000 / (1 + 5% )^10 = $1,000 / (1.05) ^10 = $613.91. The present value of each coupon payment is $35 / (1.05)^1 + $35 / (1.05)^2 + $35 / (1.05)^3 + $35 / (1.05)^4 + $35 / (1.05)^5 = $32.96 + $31.42 + $29.93 + $28.45 + $26.92 = $149.68
The weight of each cash flow is the present value of the cash flow divided by the sum of the present values of all cash flows. The weight of the face value is $613.91 / ($613.91 + $149.68) = 0.8045 and the weight of each coupon payment is $149.68 / ($613.91 + $149.68) = 0.1955. Duration can be calculated by multiplying the weight of each cash flow by the time it will be received, summing the results and dividing by the bond price. In this case, the duration is (0.8045 * 10) + (0.1955 * (0.5 + 1 + 1.5 + 2 + 2.5)) / $964 = 7.92 years.