A company faces the following liabilities at the end of the corresponding years:

Year	1	2	3	4
Liability	5,766	X	15,421	7,811

The company can invest in the following three annual coupon bonds redeemable at par:

	Term (in years)	Annual coupon rate
Bond A	1	1%
Bond B	3	5%
Bond C	4	7%

The company invests in each bond so that the asset and liability cash flows are exactly matched.

Calculate X.

• 1221
• 1245
• 1290
• 1318
• 1375

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

Solution: A

Let [math]a, b[/math], and [math]c[/math] represent the face values of the three bonds. One, two, three, and four years from now, respectively: the 1-year bond provides payments of [math]1.01 a, 0,0,0[/math]; the 3-year bond provides payments of [math]0.05 \mathrm{~b}, 0.05 \mathrm{~b}, 1.05 \mathrm{~b}, 0[/math]; and the 4-year bond provides payments of [math]0.07 c, 0.07 c, 0.07 c, 1.07 c[/math]. The total payments one, two, three, and four years from now must match the liabilities. Therefore, we have

[[math]] \begin{aligned} & 1.01 a+0.05 b+0.07 c=5766 \\ & 0.05 b+0.07 c=X \\ & 1.05 b+0.07 c=15421 \\ & 1.07 c=7811 \end{aligned} [[/math]]

Note that to find [math]X[/math], we do not need the first equation. Solving the fourth equation for [math]c[/math] yields [math]c=\frac{7811}{1.07}=7300[/math]. Substituting this value of [math]c[/math] into the third equation and solving for [math]b[/math] yields

[[math]] b=\frac{15421-0.07(7300)}{1.05}=14200 \text {. } [[/math]]

Finally, substituting these values of [math]b[/math] and [math]c[/math] into the second equation yields [math]X=0.05(14200)+0.07(7300)=1221[/math]

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