A company has liabilities of 573 due at the end of year 2 and 701 due at the end of year 5. A portfolio comprises two zero-coupon bonds, Bond A and Bond B.

Determine which portfolio produces a Redington immunization of the liabilities using an annual effective interest rate of 7.0%.

• Bond A: 1-year, current price 500; Bond B: 6-years, current price 500
• Bond A: 1-year, current price 572; Bond B: 6-years, current price 428
• Bond A: 3-years, current price 182; Bond B: 4-years, current price 1092
• Bond A: 3-years, current price 637; Bond B: 4-years, current price 637
• Bond A: 3.5 years, current price 1000; Bond B: Not used

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

Solution: A

This solution uses Macaulay duration and convexity. The same conclusion would result had modified duration and convexity been used.

The liabilities have present value 573 /1.07² + 701/1.07⁵ = 1000. Only portfolios A, B, and E have a present value of 1000.

The duration of the liabilities is [2(573) /1.07² + 5(701)/1.07⁵]/1000 = 3.5.

The duration of a zero coupon bond is its term. The portfolio duration is the weighted average of the terms. For portfolio A the duration is [500(1) + 500(6)]/1000 = 3.5. For portfolio B it is [572(1) + 428(6)]/1000 = 3.14. For portfolio E it is 3.5. This eliminates portfolio B.

The convexity of the liabilities is [4(573)/1.07² + 25(701)/1.07⁵]/1000 =14.5. The convexity of a zero-coupon bond is the square of its term. For portfolio A the convexity is [500(1) + 500(36)]/1000 = 18.5 which is greater than the convexity of the liabilities. Hence portfolio A provides Redington immunization. As a check, the convexity of portfolio E is 12.25, which is less than the liability convexity.

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