A portfolio consists of two bonds. Bond A is a three-year 1000 face amount bond with an annual coupon rate of 6% paid annually. Bond B is a one-year zero-coupon bond. Both bonds yield an annual effective rate of 4%.

Calculate the percentage of the portfolio to invest in Bond A to obtain a Macaulay duration of two years.

• 44.5%
• 45.6%
• 50.0%
• 54.4%
• 55.5%

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

Solution: D

The price of Bond [math]\mathrm{A}[/math] is [math]60\left(1.04^{-1}+1.04^{-2}+1.04^{-3}\right)+1000\left(1.04^{-3}\right)=1055.50[/math], while the Macaulay duration of Bond A is

[[math]]\frac{60\left[1.04^{-1}+2\left(1.04^{-2}\right)+3\left(1.04^{-3}\right)\right]+3(1000)\left(1.04^{-3}\right)}{1055.50}=2.838[[/math]]

. Note that the one-year zero-coupon bond has duration 1.

Let [math]w[/math] denote the proportion of wealth to invest in Bond A; then, [math]1-w[/math] is the proportion of wealth invested in Bond B. Then [math]2=2.838 w+1(1-w)[/math], or [math]w=0.5440[/math].

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