An insurance company has a liability of 750,000 due four years from now. To protect against interest rate risk, it decides to employ the technique of full immunization. Specifically, it decides to hold three assets, each providing a single cash flow as follows

1. Asset X provides a cash flow of A_X , exactly two years from now
2. Asset Y provides a cash flow of 250,000, exactly four years from now.
3. Asset Z provides a cash flow of A_Z , exactly five years from now.

The annual discount factor is v = 0.95

• 150,000
• 175,000
• 200,000
• 225,000
• 250,000

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

Solution: A

Since Asset Y provides a cash flow at the same time that the liability is due [math](t=4)[/math], we can apply its 250,000 value to reducing the liability amount from 750,000 to 500,000 . Then, we can establish the following two equations, both using [math]t=4[/math] as the reference point for all cash flows.

[[math]] 500,000=A_X v^{-2}+A_Z v=A_X(0.95)^{-2}+A_Z(0.95) [[/math]]

Second, taking the derivative (with respect to [math]v[/math] ) of both sides of the first equation, we have:

[[math]] \begin{aligned} & 0=-2 A_X v^{-3}+A_Z=-2 A_X(0.95)^{-3}+A_Z \\ & 0=-2 A_X(0.95)^{-2}+A_Z(0.95) \end{aligned} [[/math]]

Then, subtracting the second equation from the first equation yields:

[[math]] \begin{aligned} & 500,000=3 A_X(0.95)^{-2} \\ & A_X=150,416.67 \end{aligned} [[/math]]

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