A company has liabilities of 402.11 due at the end of each of the next three years. The company will invest 1000 today to fund these payouts. The only investments available are one-year and three-year zero-coupon bonds, and the yield curve is flat at a 10% annual effective rate. The company wishes to match the duration of its assets to the duration of its liabilities.

Determine how much the company should invest in each bond.

• 366 in the one-year bond and 634 in the three-year bond.
• 484 in the one-year bond and 516 in the three-year bond.
• 500 in the one-year bond and 500 in the three-year bond.
• 532 in the one-year bond and 468 in the three-year bond.
• 634 in the one-year bond and 366 in the three-year bond.

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

Solution: D

The present value of the liabilities is 1000, so that requirement is met. The duration of the liabilities is

[[math]]402.1\,[1.1^{-1}+2(1.1)^{-2}+3(1.1)^{-3}]/1000=1.9365.[[/math]]

Let X be the investment in the one-year bond. The duration of a zero-coupon is its term. The duration of the two bonds is then

[[math]] [X + (1000-X)(3)]/1000 = 3-0.002X. [[/math]]

Setting this equal to 1.9365 and solving yields X = 531.75.

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