A company is required to pay 50, 000(1.075)y in y years. The company invests 30,000 in a 28- year zero-coupon bond and 20,000 in a 35-year zero-coupon bond to Redington immunize its position against small changes in interest rates, based on an annual effective interest rate of 7.5%.

Calculate y.

• 30.80
• 31.32
• 31.50
• 31.68
• 32.20

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

Solution: A

For a [math]7.5 \%[/math] yield rate, the present value and Macaulay duration of the assets are, respectively, [math]30,000+20,000=50,000[/math] and [math]\frac{30,000(28)+20,000(35)}{30,000+20,000}=30.8[/math]

The present value and Macaulay duration, of the liabilities are, respectively, [math]\frac{50,000(1.075)^y}{(1.075)^y}=50,000[/math] and [math]y[/math].

Note that the present values of assets and liabilities already match. Since Macaulay durations must match, [math]y=30.8[/math].

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