Trevor has assets at time 2 of A and at time 9 of B. He has a liability of 95,000 at time 5. Trevor has achieved Redington immunization in his portfolio using an annual effective interest rate of 4%.

• 0.7307
• 0.9670
• 1.0000
• 1.0132
• 1.3686

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

Solution: D

[[math]] \begin{array}{l l}{{P_{A}=A(1+i)^{-2}+B(1+i)^{-9}}}\\ {{P_{L}=95,000(1+i)^{-5}}}\\ {{P_{L}^{\prime}=-2A(1+i)^{-9}-9B(1+i)^{-10}}}\end{array} [[/math]]

Set the present values and derivatives equal and solve simultaneously.

[[math]] \begin{align*} 0.92456A + 0.70259B &= 78, 083 \\ -1.7780.21-6.0801B &=-375,400 \\ B &=\frac{78,083(1.7780/0.92456)-375,400}{0.70259(1.7780/0.92456)-6.0801}=47,630 \\ A &=[78,083-0.70259(47,630)]/0.92456=48,259 \\ \frac{A}{B} &= 1.0132 \end{align*} [[/math]]

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