You are given the following information about a company's liabilities:

• Present value: 9697
• Macaulay duration: 15.24
• Macaulay convexity: 242.47

The company decides to create an investment portfolio by making investments into two of the following three zero-coupon bonds: 5-year, 15-year, and 20-year. The company would like its position to be Redington immunized against small changes in yield rate.

The annual effective yield rate for each of the bonds is 7.5%.

Determine which of the following portfolios the company should create.

• Invest 3077 for the 5-year bond and 6620 for the 20-year bond.
• Invest 6620 for the 5-year bond and 3077 for the 20-year bond.
• Invest 465 for the 15-year bond and 9232 for the 20-year bond.
• Invest 4156 for the 15-year bond and 5541 for the 20-year bond.
• Invest 9232 for the 15-year bond and 465 for the 20-year bond.

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

Solution: A

Let x, y, and z represent the amounts invested in the 5-year, 15-year, and 20-year zero-coupon bonds, respectively. Note that in this problem, one of these three variables is 0. The present value, Macaulay duration, and Macaulay convexity of the assets are, respectively,

[[math]] x+y+z=9697,\quad{\frac{5x+15y+20z}{x+y+z}}=15.24 [[/math]]

We are given that the present value, Macaulay duration, and Macaulay convexity of the liabilities are, respectively, 9697, 15.24, and 242.47. Since present values and Macaulay durations need to match for the assets and liabilities, we have the two equations

[[math]] x+y+z,{\frac{5x+15y+20z}{x+y+z}},{\frac{5^{2}x+15^{2}y+20^{2}z}{x+y+z}} [[/math]]

Note that 5 and 15 are both less than the desired Macaulay duration 15.24, so z cannot be zero. So try either the 5-year and 20-year bonds (i.e. y = 0), or the 15-year and 20-year bonds (i.e. x = 0). In the former case, substituting y = 0 and solving for x and z yields

[[math]] x={\frac{(20-15.24)969^{7}}{20-5}}=3077.18, \quad z=\frac{(15.24-5)9697}{20-5}=6619.82\,. [[/math]]

We need to check if the Macaulay convexity of the assets exceeds that of the liabilities. The Macaulay convexity of the assets is

[[math]] \frac{5^{2}(3077.18)+20^{2}(6619.82)}{9697}=281.00 [[/math]]

which exceeds the Macaulay convexity of the liabilities, 242.47. The company should invest 3077 for the 5-year bond and 6620 for the 20-year bond. Note that setting x = 0 produces y = 9231.54 and z = 465.46 and the convexity is 233.40, which is less than that of the liabilities.

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