A company has liabilities of 402.11 due at the end of each of the next three years. The company will match the duration of its liabilities by investing a total of 1000 in one-year and three-year zero-coupon bonds. The annual effective yield of both bonds is 10%.

Calculate the amount the company will invest in one-year bonds.

• 366
• 402
• 442
• 500
• 532

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

Solution: E

PV of liabilities is

[[math]]402.11(1/1.1 + 1/1.1^2 + 1/1.1^3)=1000.[[/math]]

Duration of liabilities is

[[math]]402.11(1/1.1 + 2/1.1^2 + 3/1.1^3)/1000 = 1.93653.[[/math]]

Let X be the investment in one-year bonds. To match duration, since zero-coupon bonds have duration = maturity, 1.93653 = [X + 3(1000 –X)]/1000. Then, 2X = 3000 – 1936.53 = 1063.47 and X = 532.

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