Two 15-year par value bonds, X and Y, each pay an annual coupon of 200 at the end of the year. The face amount of Bond X is one-half the face amount of Bond Y. At an annual effective yield of i, the price of Bond X is 2695.39 and the price of Bond Y is 3490.78.

Calculate the coupon rate for Bond X.

• 6.3%
• 7.4%
• 8.8%
• 10.0%
• 11.4%

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

Solution: D

Let F be the face amount of Bond X. Then,

[[math]] 2695.39=200a_{\overline{15}|}+F_{V}^{15}\mathrm{~and~3490.78}=200a_{\overline{15}|}+F_{V}^{15}. [[/math]]

Subtract the first equation from the second to obtain [math]795.39 = Fv^{15}.[/math] Then for bond X,

[[math]] 2695.39=200a_{\overline{15}|}+795.39\Rightarrow a_{\overline{15}|}=(2695.39-795.39)/200=9.5. [[/math]]

This implies [math]i =0.0634[/math]. Then

[[math]] 9.5=(1- v^{15})/0.0634\Longrightarrow v^{15}=1-0.0634(9.5)=0.3977 [[/math]]

and [math]F = 795.39 / 0.3977 = 2000[/math]. The coupon rate is 200/2000 = 10.0%.

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