excans:790af9503f

Exercise

Nov 20'23

Answer

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.