A company owes 500 and 1000 to be paid at the end of year one and year four, respectively. The company will set up an investment program to match the duration and the present value of the above obligation using an annual effective interest rate of 10%. The investment program produces asset cash flows of X today and Y in three years.

Calculate X and determine whether the investment program satisfies the conditions for Redington immunization.

• X = 75 and the Redington immunization conditions are not satisfied.
• X = 75 and the Redington immunization conditions are satisfied.
• X = 1138 and the Redington immunization conditions are not satisfied.
• X = 1138 and the Redington immunization conditions are satisfied.
• X = 1414 and the Redington immunization conditions are satisfied.

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

Solution: A

The present value function and its derivatives are

[[math]] \begin{array}{l c r}{{P(i)=X+Y(1+i)^{-3}-500(1+i)^{-1}-1000(1+i)^{-4}}}\\ {{P^{\prime}(i)=-3Y(1+i)^{-4}+500(1+i)^{-5}-20,000(1+i)^{-5}}}\\ {{P^{\prime\prime}(i)=12Y(1+i)^{-5}-1000(1+i)^{-5}-20,000(1+i)^{-6}.}}\end{array} [[/math]]

The equations to solve for matching present values and duration (at i = 0.10) and their solution are

[[math]] \begin{array}{l c r}{{P(0.1)=X+0.7513Y-1137.56=0}}\\ {{P^{\prime}(0.1)=-2.0490Y+2896.91=0}}\\ {{Y=2896.91/2.0490=1413.82}}\\ {{X=1137.56-0.7513(1413.82)=75.36.}}\end{array} [[/math]]

The second derivative is

[[math]] P^{\prime\prime}(0.1)=12(1413.82)(1.1)^{-5}-1000(1.1)^{-5}-20,000(1.1)^{-6}=-1506.34. [[/math]]

Redington immunization requires a positive value for the second derivative, so the condition is not satisfied.

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