A company is required to pay 500,000 ten years from now and 500,000 fifteen years from now. The company needs to create an investment portfolio using 5-year and 20-year zero-coupon bonds, so that, using a 7% annual force of interest, the present value and Macaulay duration of its assets match those of its liabilities.

Calculate the amount invested today in each bond.

• 211,631 for the 5-year bond and 211,631 for the 20-year bond
• 217,699 for the 5-year bond and 217,699 for the 20-year bond
• 223,852 for the 5-year bond and 199,410 for the 20-year bond
• 229,857 for the 5-year bond and 205,540 for the 20-year bond
• 248,293 for the 5-year bond and 174,969 for the 20-year bond

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

Solution: C

Let [math]x[/math] and [math]y[/math] be the amount invested in the five and twenty year bonds respectively. To match the present values: [math]x+y=500,000 e^{-0.07(10)}+500,000 e^{-0.07(15)}=423,262[/math]. To match the durations, noting that the denominators of the durations for assets and liabilities are the same, [math]5 x+20 y=500,000(10) e^{-0.07(10)}+500,000(15) e^{-0.07(15)}=5,107,460[/math]. Subtracting five times the first equation from the second one gives [math]15 y=2,991,150[/math] for [math]y=199,410[/math] and [math]x=423,262-[/math] [math]199,410=223,852[/math].

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