Question

A $1000 par value 4% bond with semiannual coupons matures at the end of 10 years. The bond is callable at $1050 at the ends of years 4 through 6, at $1025 at the ends of years 7 through 9, and at $1000 at the end of year 10. Find the maximum price that an investor can pay and still be certain of a yield rate of 5% convertible semiannually.

ANSWER: 922.05

Answer 1

In this question, the redemption values are not same at all possible redemption dates.

However, the modified coupon rates for each possible redemption period, i.e. year 4 - year 6, year 7-year 9, and year 10, which are 1.9%, 1.95% and 2% respectively, are smaller than = 5%/2 = 2.5%.

So the later redemption within each period is worse situation to the investor.

In other words, we need to compare the bond price assuming the bond is called at the end of the 6th, 9th or 10th year. If the bond is redeemed at the end of 6th year, then the price is 20a_{12 2.5}%+ 1050V_2.5% ¹² = 985.889.

And at the end of the 9th year, just replacing the term and the redemption amount on the above equation by 18 and 1025 yields the bond price 944.262.

At last, the bond price at maturity (at the end of the 10th year) is 922.054 (term is 20 and the redemption amount is 1000).

Thus, the smallest one among three prices obtained is the maximum price that an investor can pay to get i(2) = 5%. That is, $922.054.