What would you pay today for a twenty year, 8% semiannual coupon bond with a face amount of $10,000 if you desired a 10% annual return (coumpounded semiannually)?
To calculate the price of a bond, you can use the formula for the present value of a bond's cash flows. In this case, the bond has a face amount (also known as par value) of $10,000, a coupon rate of 8% semiannually, and a maturity of 20 years.
To determine the price of the bond, you need to discount each semiannual coupon payment and the face amount to the present value at a desired annual return rate.
Let's break down the steps to calculate the price of the bond:
Step 1: Determine the number of periods. Since the bond pays coupons semiannually for 20 years, there will be a total of 40 periods (20 years * 2 periods per year).
Step 2: Calculate the semiannual coupon payment. The coupon rate is given as 8% semiannually, so the coupon payment for each period can be calculated as: ($10,000 * 8%) / 2 = $400.
Step 3: Determine the discount rate per period. Since the desired annual return rate is 10% compounded semiannually, you need to calculate the semiannual discount rate. Divide the annual return rate by 2 to get the semiannual rate: 10% / 2 = 5%.
Step 4: Present value calculation. For each semiannual coupon payment, calculate the present value using the discount rate and add them all up. For the face amount, calculate the present value as the final payment. Add all the present values to find the total price of the bond.
Let's calculate the present value of each cash flow:
PV_coupon = $400 / ((1 + 0.05)^1) + $400 / ((1 + 0.05)^2) + ... + $400 / ((1 + 0.05)^40)
PV_face_amount = $10,000 / ((1 + 0.05)^40)
Price_of_bond = PV_coupon + PV_face_amount
Now you can use a financial calculator or spreadsheet software to calculate the present value of each coupon payment and the face amount. After determining these values, sum them up to get the price of the bond.
Note: The above calculation assumes that the bond pays coupons on time and the market interest rates are consistent with the yield required.