An insurance company purchases a perpetuity-due at an annual effective yield rate of 12.5% for 9450. The perpetuity provides annual payments according to the repeating three-year pattern 100, X, 100, 100, X, 100, 100, X, 100, ... .

Calculate X.

• 2950
• 2963
• 3321
• 3344
• 3359

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

Solution: B

Split this into three perpetuities with payments 3 years apart. Find the three-year interest rate: [math]1.125=(1+j)^{\frac{1}{3}}, j=0.423828[/math].

The present value of the three perpetuities, starting at times 0,1 , and 2 is:

[[math]] \begin{aligned} & \frac{100(1.423828)}{0.423828}+\frac{X(1.423828)}{0.423828(1.125)}+\frac{100(1.423828)}{0.423828(1.125)^2}=9450 \\ & X=2963.19 \end{aligned} [[/math]]

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