A perpetuity-immediate pays 20 for 10 years, decreases by 1 per year for 19 years, and then pays 1 per year thereafter. At an annual effective interest rate of 6%, the present value is equal to X.

Calculate X.

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

Solution: C

[[math]] \begin{aligned} 2000 & =X(1.08)^{14}+X(1.03)(1.08)^{13}+X(1.03)^2(1.08)^{12}+\cdots+X(1.03)^{14} \\ & =X \frac{1.08^{14}-1.03^{15} / 1.08}{1-1.03 / 1.08} \\ & =32.284 X \\ X & =61.95\end{aligned} [[/math]]

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