On December 31, 2013, University Theatres issued $500,000 face value of bonds. The stated rate is 8%, and interest is paid semiannually on June 30 and December 31. The bonds mature in 15 years.
If required, round your answers to the nearest whole dollar. Follow the format shown in present value tables as you complete the requirements below.
Required:
a. Assuming the market rate of interest is 6%, calculate at what price the bonds are issued.
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To calculate the price at which the bonds are issued, we need to use the present value formula. The present value of a bond is the discounted value of the future cash flows, which in this case are the semiannual interest payments and the final face value payment at maturity.
The present value of a bond can be calculated using the present value tables or a financial calculator. Since you mentioned using the present value tables, let's use that method.
The present value formula for a bond is:
PV = (C/r) * [1 - (1/(1+r)^n)] + (F/(1+r)^n)
Where:
PV = Present value or price of the bond
C = Coupon payment (interest payment)
r = Market interest rate per period
n = Number of periods (in this case, number of semiannual periods)
F = Face value of the bond
Given data:
Face value (F) = $500,000
Stated rate = 8%
Market rate = 6%
Number of periods (n) = 15 years = 30 semiannual periods
Using the present value formula, let's calculate the price of the bonds:
Step 1: Calculate the semiannual coupon payment (C):
C = (Stated rate / 2) * Face value
C = (8% / 2) * $500,000
C = $20,000
Step 2: Calculate the present value (PV) using the present value tables:
PV = ($20,000 / 0.03) * [1 - (1/(1+0.03)^30)] + ($500,000 / (1 + 0.03)^30)
Using the present value table, find the factor for 30 periods and a market interest rate of 3%. According to the table, this factor is 0.37512.
PV = ($20,000 / 0.03) * [1 - (1/1.03^30)] + ($500,000/1.03^30)
PV = ($20,000 / 0.03) * [1 - (1/1.37512)] + ($500,000/1.03^30)
PV = ($20,000 / 0.03) * (1 - 0.72760) + ($500,000 / 1.03^30)
PV = ($666,666.67) * (0.27240) + ($500,000 / 1.03^30)
PV = $181,600 + $500,000 / 2.20809
PV = $181,600 + $226,624.20
PV = $408,224.20
Therefore, the bonds are issued at a price of approximately $408,224.