Three asset-liability cash flows are given in the following table where a positive amount is an asset cash flow and a negative amount is a liability due at the corresponding time

t (in years)	0	1	2	3
X	102,400	−192,000	0	100,000
Y	158,400	−342,000	100,000	100,000
Z	−89,600	288,000	100,000	-300,000

Determine which set of cash flows is Redington immunized for an annual effective interest rate of i = 25%.

• X only
• Y only
• Z only
• X, Y, and Z
• The correct answer is not given by (A), (B), (C) or (D)

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

Solution: A

Let [math]h(i)[/math] be the present value of the cash flows. For Redington immunization, the value of the function and its first derivative at [math]25 \%[/math] must be zero and the second derivative must be positive. [math]\mathrm{X}[/math] is immunized because:

[[math]] \begin{aligned} & h(0.25)=102,400-192,000 / 1.25+100,000 / 1.25^3=0 \\ & h^{\prime}(0.25)=192,000 / 1.25^2-100,000(3) / 1.25^4=0 \\ & h^{\prime \prime}(0.25)=-192,000(2) / 1.25^3+100,000(3)(4) / 1.25^5=196,608\gt0 \end{aligned} [[/math]]

[math]\mathrm{Y}[/math] is not immunized because:

[[math]] \begin{aligned} & h(0.25)=158,400-342,000 / 1.25+100,000 / 1.25^2+100,000 / 1.25^3=0 \\ & h^{\prime}(0.25)=342,000 / 1.25^2-100,000(2) / 1.25^3-100,000(3) / 1.25^4=-6,400 \neq 0 \end{aligned} [[/math]]

[math]\mathrm{Z}[/math] is not immunized because

[[math]] h(0.25)=-89,600+288,000 / 1.25+100,000 / 1.25^2-300,000 / 1.25^3=51,200 \neq 0 [[/math]]

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