An outstanding issue of Public Express Airlines debentures has a call provision

Question

An outstanding issue of Public Express Airlines debentures has a call provision attached. The total principal value of the bonds is $250 million, and the bonds have an annual coupon rate of 9 percent. The company is considering refunding the bond issue. Refunding means that the company would issue new bonds and use the proceeds from the new bond issuance to repurchase the outstanding bonds. The total cost of refunding would be 10 percent of the principal amount raised. The appropriate tax rate for the company is 35 percent

How low does the borrowing cost need to drop to justify refunding with a new bond issue?

I see the answer given by Sridevi Gandi, but I need help breaking down the last part of the solution.

How do I get from 16,250,000 = (22,500,000 – 250,000,000R)/R to thefinal answer?

Explanation / Answer

Answer: The company should refund when the NPV of refunding is greater than zero, so we need to find the interest rate that results in a zero NPV. The NPV of the refunding is the difference between the gain from refunding and the refunding costs.

The gain from refunding is the bond value times the difference in the interest rate, discounted to the present value. We must also consider that the interest payments are tax deductible, so the aftertax gain is:

NPV = PV(Gain) – PV(Cost)

The present value of the gain will be:

Gain = $250,000,000(.09 – R) / R

Since refunding would cost money today, we must determine the aftertax cost of refunding, which will be:

Aftertax cost = $250,000,000(.10)(1 – .35)

Aftertax cost = $16,250,000

So, setting the NPV of refunding equal to zero, we find:

0 = –$16,250,000 + $250,000,000(.09 – R) / R

0=-$16,250,000+($22,500,000-250,000,000R)/R

16250,000R=22500,000-250,000000R

266250000R=22500000

R=8.45%

Any interest rate below this will result in a positive NPV from refunding.

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