A three-year bond has 8.0% coupon rate and face value of $1000. If the yield to maturity on the bond is 10%, calculate the price of the bond assuming that the bond makes semi-annual coupon interest payments.
To calculate the price of the bond, we need to find the present value of the future cash flows generated by the bond.
Step 1: Calculate the semi-annual coupon payment:
The coupon rate is given as 8.0%, so each year the bond will pay 8.0% of $1000, which is $80. Since the bond makes semi-annual coupon payments, the coupon payment for each period will be half of this, which is $40.
Step 2: Determine the number of periods:
Since the bond is a three-year bond, and the coupon payments are made semi-annually, there will be a total of 3 years * 2 semi-annual periods per year = 6 semi-annual periods.
Step 3: Calculate the present value of future cash flows:
Now, we need to calculate the present value of each semi-annual cash flow. The formula for present value is:
Present Value = Coupon Payment / (1 + Yield to Maturity / 2) ^ Periods
Using the values given:
Coupon Payment = $40
Yield to Maturity = 10% = 0.10
Periods = 6
Let's calculate the present value for each cash flow:
Period 1:
Present Value = $40 / (1 + 0.10 / 2)^1 = $40 / (1.05)^1 = $38.10
Period 2:
Present Value = $40 / (1 + 0.10 / 2)^2 = $40 / (1.05)^2 = $36.34
Period 3:
Present Value = $40 / (1 + 0.10 / 2)^3 = $40 / (1.05)^3 = $34.66
Period 4:
Present Value = $40 / (1 + 0.10 / 2)^4 = $40 / (1.05)^4 = $33.06
Period 5:
Present Value = $40 / (1 + 0.10 / 2)^5 = $40 / (1.05)^5 = $31.51
Period 6:
Present Value = $40 / (1 + 0.10 / 2)^6 = $40 / (1.05)^6 = $30.01
Step 4: Calculate the present value of the face value:
The face value of the bond is $1000, which will be received at the end of the last period. We need to calculate the present value of this cash flow as well:
Present Value = $1000 / (1 + 0.10 / 2)^6 = $1000 / (1.05)^6 = $747.26
Step 5: Calculate the bond price:
To get the bond price, you need to sum up the present values of all the cash flows:
Bond Price = Present Value of Cash Flows + Present Value of Face Value
Bond Price = $38.10 + $36.34 + $34.66 + $33.06 + $31.51 + $30.01 + $747.26
Bond Price ≈ $951.94
Therefore, the price of the bond is approximately $951.94.