An insurance company has a known liability of 1,000,000 that is due 8 years from now. The technique of full immunization is to be employed. Asset I will provide a cash flow of 300,000 exactly 6 years from now. Asset II will provide a cash flow of X, exactly y years from now, where y > 8. The annual effective interest rate is 4%.

Calculate X.

• 697,100
• 698,600
• 700,000
• 701,500
• 702,900

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

Solution: D

This solution uses time 8 as the valuation time. The two equations to solve are

[[math]] \begin{array}{l c r}{{P(i)=300,000(1+i)^{2}+X(1+i)^{8-y}-1,000,000=0}}\\ {{P^{\prime}(i)=600,0000(1+i)+(8-y)X(1+i)^{7-y}=0.}}\end{array} [[/math]]

Inserting the interest rate of 4% and solving:

[[math]] \begin{align*} 300,000(1.04)^{2}+X(1.04)^{8-y}-1,000,000=0 \\ 600000(1.04)+(8-y)X(1.04)^{7-y}=0 \\ X(1.04)^{-y}=[1.000,000-300,000(1.04)^{2}]/1.04^{8}=493,595.85 \\ 624,000+(8-y)(1.04)^{7}(493,595.85)=0 \\ y=8+624,000 /\left[493,595.85(1.04)^{7}\right]=8.9607 \\ X=493,595.85(1.04)^{\mathrm{8.9607}}=701,459. \end{align*} [[/math]]

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