CHAPTER 6 • BONDS
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i) If it is called after 15 years, P = 1000 + (30 - 35)/35olo:(B5 - $ 1 0 0 0 " $ 91 - 9 6 = $908.04 ii) If it matures after 20 years, P = 1000+ ( 3 0 - 3 5)*4oi0 035 =$1000-$106.78 = $893.22 The purchase price to guarantee a return of j2 = 7% is the lower of these two answers, or $893.22. If the investor pays $893.22 and the bond runs the full 20 years to maturity, the investor's yield will be exactly j2 = 7%. If the investor pays $893.22 and the bond is called after 15 years, the investor's return will exceed j2 = 7%. If the investor pays $908.04 for the bond, however, it will yield j2 = 7% only if the bond is called after 15 years. If the bond runs to its full maturity, the rate of return will be less than j 2 = 7%. Solution b
Again we calculate the price of the bonds at the two different dates using a yield of j2 = 5% and formula (16). i) If it is called after 15 years, P = 1000 + (30 -25)^ol0.o2s = $1000 -$104.65 = $1104.65. ii) If it matures after 20 years, P = 1000 + (30-25>4oi0.025 - $1000 + $125.51 = $1125.51. The purchase price to guarantee a return of j 2 = 5% is $1104.65. If the bond is called after 15 years, the investor's yield will be exactly _/2 = 5%. If the bond is called any time after 15 years, or if the bond is held to maturity, the investor's yield will exceed j2 = 5%.
In the above example, we have shown, in effect, that the investor must assume that the issuer of the bond will exercise his call option to the disadvantage of the investor and must calculate the price accordingly. The example above also illustrates a useful principle for bonds that are callable at par. That is,
If the yield rate is less than the coupon rate (if the bond sells at a premium), then you use the earliest possible call date in your calculation. If the yield rate is greater than the coupon rate (if the bond sells at a discount), then you use the latest possible redemption date in your calculation. These examples can be explained logically. For a bond purchased at a premium, the earliest call date is the worst for the investor since she gets only $1000 for\her bond whenever it is called. On the other hand, for a bond purchased at a discount, the earliest call date is die best for the investor, since at that early call date she will get a full $1000 for a bond that she values at something less than $1000. Thus, the most unfavourable situation is the latest possible redemption. Unfortunately, these guidelines cannot be applied often since, when a bond is called early, it is usually done so at a premium. In that case, we are forced to calculate all possible purchase prices and pay the lowest price calculated.