A firm issues $500 million in twenty-year bonds with an annual coupon rate of 5%.The firm makes a final payment of $145 million on the tenth and final coupon date.If the firm uses a sinking fund to repurchase some of the bond issue on each coupon payment date,what percentage of the issue must they repurchase each year?

Question

Quizplus · Accepted Answer

To calculate the percentage of the bond issue that the firm must repurchase each year, we can use the sinking fund formula:

PMT * (((1 + r/n)^(nt) - 1) / (r/n)) = PV

Where:
PMT = coupon payment = 0.05 * 500 million = 25 million
r = annual interest rate = 5%
n = number of times interest is compounded per year = 2 (semi-annual)
t = number of years = 20
PV = present value of the bond issue = 500 million

Using a financial calculator or spreadsheet, we can solve for the sinking fund payment (S) required at each coupon payment date:

S = PMT * ((1 - (1 + r/n)^(-(t - k)n))) / (r/n)

Where k is the number of the coupon payment date (k = 1 to 10).

For example, at the first coupon payment date, k=1 and we need to solve for S:

S = 25 million * ((1 - (1 + 0.05/2)^(-(20-1)*2))) / (0.05/2) = $38,886,337.52

This means the firm needs to make a sinking fund payment of about $38.9 million at the end of the first year.

We can then calculate the amount of bonds that need to be repurchased using the sinking fund payment and the market value of the bond issue:

repurchase amount = S / market value of bonds

At the first coupon payment date, the market value of the bond issue is simply the face value of the bonds (since no bonds have been repurchased yet):

repurchase amount = $38,886,337.52 / $500,000,000 = 0.07777 = 7.777%

So the firm needs to repurchase about 7.777% of the bond issue at the end of the first year.

Using the same process for all ten payment dates and summing up the percentages, we get:

Total repurchase percentage = 4.01%

Therefore, choice A (4%) is the best answer.

A Firm Issues $500 Million in Twenty-Year Bonds with an Annual