$10,000,000 face value, zero coupon bonds due in 20 years, priced on the market to yield 8% compounded semiannually.
To calculate the price of the zero coupon bonds, we need to use the formula for present value of a bond. The formula is:
Price = Face Value / (1 + Yield/2)^(Number of Periods)
In this case, the face value is $10,000,000, the yield is 8% compounded semiannually, and the number of periods is 20 years (40 semiannual periods).
First, let's convert the yield to a decimal and divide it by 2 to reflect the semiannual compounding:
Yield = 8% / 100 = 0.08
Semiannual Yield = 0.08 / 2 = 0.04
Next, let's plug in the values into the formula:
Price = $10,000,000 / (1 + 0.04)^(40)
Using a calculator or spreadsheet software, let's calculate the value inside the parentheses:
Value inside parentheses = (1 + 0.04)^40 = 3.172152
Now, divide the face value by the value inside parentheses:
Price = $10,000,000 / 3.172152 = $3,150,037.53
Therefore, the price of the zero coupon bonds is approximately $3,150,037.53.