A company has liabilities of 1000 and 300 due at the end of years two and four, respectively. The company develops an investment program that produces asset cash flows of X and Y at the end of years one and three, respectively. The investment portfolio is constructed to match the present value and duration of the company’s payment obligations based on an annual effective rate of interest of 5%.

Calculate X/Y.

• 2.14
• 2.75
• 3.42
• 4.05
• 4.91

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

Solution: B

[[math]] \begin{aligned} & P_L=1000(1+i)^{-2}+300(1+i)^{-4} \\ & P_L=1153.84 \\ & P_L^{\prime}=-2000(1+i)^{-3}-1200(1+i)^{-5} \\ & P_L^{\prime}=-2667.91 \\ & P_A=X(1+i)^{-1}+Y(1+i)^{-3} \\ & 1153.84=0.95238 X+0.86384 Y \\ & P_L^{\prime}=-X(1+i)^{-2}-3 Y(1+i)^{-4} \\ & -2667.91=-0.90703 X-2.46811 Y \end{aligned} [[/math]]

So, we have two equations and two unknowns. Solving simultaneously, we get:

[[math]] Y=953.57, \quad X=346.61, \quad \frac{Y}{X}=2.75 [[/math]]

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