On December 31, 2008, University Theatres issued $500,000 face value of bonds. The stated rate is 6 percent, and interest is paid semiannually on June 30 and December 31. The bonds mature in 10 years.
Required:
If required, round your answers to the nearest whole dollar. Follow the format shown in present value tables as you complete the requirements below.
a. Assuming the market rate of interest is 4 percent, calculate at what price the bonds are issued.
b. Assuming the market rate of interest is 8 percent, calculate at what price the bonds are issued.
To calculate the price at which the bonds are issued, we need to find the present value of the bond's future cash flows. The present value is calculated by discounting the future cash flows using the market rate of interest.
a. Assuming the market rate of interest is 4 percent:
The bond pays semiannual interest at a rate of 6 percent. Since it has a 10-year maturity, there will be 20 semiannual interest payments. The face value of the bond is $500,000.
To find the present value of the bond, we need to calculate the present value of each semiannual interest payment and the present value of the face value payment at maturity, and then sum them up.
Step 1: Calculate the present value of the semiannual interest payments.
Each semiannual interest payment is calculated using the formula: Interest Payment = Face Value * Stated Rate / 2
In this case, the interest payment is: $500,000 * 6% / 2 = $15,000
The present value of each semiannual interest payment can be calculated using the present value of an ordinary annuity table. We look for the factor closest to 20 periods (the number of semiannual periods) and the interest rate of 2% (half the market rate of interest).
From the table, we find that the factor is approximately 17.193
Present value of each interest payment = Interest Payment * Factor = $15,000 * 17.193 = $257,895
Step 2: Calculate the present value of the face value payment at maturity.
The face value payment is $500,000, which will be received at the end of 10 years. To find the present value of this amount, we use the present value of a single sum table. We look for the factor closest to 20 periods and the interest rate of 2%.
From the table, we find that the factor is approximately 0.672
Present value of face value payment = Face Value * Factor = $500,000 * 0.672 = $336,000
Step 3: Calculate the total present value.
Total present value = Present value of interest payments + Present value of face value payment
Total present value = $257,895 + $336,000 = $593,895
So, the bonds are issued at approximately $593,895.
b. Assuming the market rate of interest is 8 percent:
We follow the same steps as above, but substitute the interest rate of 8% instead of 4%.
Step 1: Calculate the present value of the semiannual interest payments.
Interest Payment = $15,000
Using the present value of an ordinary annuity table for 20 periods at 4%, the factor is approximately 14.877
Present value of each interest payment = $15,000 * 14.877 = $223,155
Step 2: Calculate the present value of the face value payment at maturity.
Using the present value of a single sum table for 20 periods at 4%, the factor is approximately 0.456
Present value of face value payment = $500,000 * 0.456 = $228,000
Step 3: Calculate the total present value.
Total present value = $223,155 + $228,000 = $451,155
So, the bonds are issued at approximately $451,155.