A 1000-par value 30-year bond has an annual coupon rate of 7% paid semiannually. After an initial 10-year period of call protection, the bond is callable immediately following the payment of any of the 20th through the 59th coupons.

• If the bond is called before payment of the 40th coupon, the redemption value is 1250.
• If the bond is called immediately after the payment of any of the 40th through the 59th coupons, the redemption value is 1125.
• If the bond is not called, it will be redeemed at par.

To ensure that the bond will provide at least an annual nominal yield rate of 5% convertible semiannually, it must be assumed that the bond will be called or redeemed immediately after the payment of the nth coupon.

Calculate n.

• 20
• 39
• 40
• 59
• 60

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

During the first redemption period the modified coupon rate is 1000(0.035)/1250 = 2.80% which is larger than the desired yield rate. If redeemed during this period, bond sells at a premium and so the worst case for the buyer is the earliest redemption. The price if called at that time is

[[math]] 35a_{\overline{{{20}}}|0.025}+1250(1.025)^{-20}=35(15.5892)+762.84=1308.46\,. [[/math]]

During the second redemption period the modified coupon rate is 1000(0.035)/1125 = 3.11% which is also larger than the desired yield rate and the worst case for the buyer is again the earliest redemption. The price if called at that time is

[[math]] 35a_{\overline{{{40}}}|0.025}+1125(1.025)^{-40}=35(25.1028)+418.98=1297.58\,. [[/math]]

Finally, if the bond is not called, its value is

[[math]] 35a_{\overline{60}|0.025}+1000(1.025)^{-60}=35(30.9087)+227.28=1309.08 [[/math]]

The appropriate price is the lowest of these three, which relates to the bond being called after the 40th coupon is paid.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.