Answer: To find the market price of the bond, we can use the present value formula for a bond. The present value of the bond is the sum of the present values of all the future cash flows (coupon payments and the maturity value) discounted at the yield rate.
Given:
Maturity value (F) = $1000,
Coupon rate = 9% (expressed as a decimal, 0.09),
Yield rate (r) = 8% (expressed as a decimal, 0.08),
Number of periods (n) = 20 years (but since coupons are paid semiannually, there will be 40 coupon payments).
Now, let's calculate the present value of each cash flow:
Present value of each coupon payment:
Since coupons are paid semiannually, the coupon payment will be half of the annual coupon rate and discounted for each period:
Coupon payment = (Coupon rate / 2) * Maturity value = (0.09 / 2) * $1000 = $45.
Now, we need to find the present value of each semiannual coupon payment, discounted at the yield rate (compounded semiannually):
PV of each coupon payment = Coupon payment / (1 + (r/2))^t,
where t is the number of periods remaining until the payment (1 to 40).
Present value of the maturity value (final payment):
The maturity value will be received after 40 semiannual periods. We calculate its present value as follows:
PV of maturity value = Maturity value / (1 + (r/2))^n,
where n is the number of periods remaining until maturity (40).
Calculate the total present value by summing the present values of all cash flows:
Total present value = Sum of PV of coupon payments + PV of maturity value.
Now, let's calculate the total present value using the given values:
PV of each coupon payment = $45 / (1 + (0.08/2))^t.
PV of maturity value = $1000 / (1 + (0.08/2))^40.
Total present value = Sum of PV of coupon payments + PV of maturity value.
Total present value = ($45 / (1 + 0.04))^1 + ($45 / (1 + 0.04))^2 + ... + ($45 / (1 + 0.04))^40 + ($1000 / (1 + 0.04))^40.
Now, calculate the total present value using the formula for the sum of a geometric series or use a financial calculator or spreadsheet software. After evaluating the sum, the market price of the bond should be rounded to the nearest cent.