You have the option of purchasing one of the following two annuities-immediate:

1. The first annuity makes annual payments of 1000 for 20 years.
2. The second annuity is a perpetuity that also has annual payments. The payment in each of the first 10 years is 600. Beginning in year 11, the payments increase to 1200, and remain at 1200 forever.

At an annual effective interest rate of i > 0 , both annuities have a present value of X.

• 8700
• 8750
• 8800
• 8850
• 8900

Calculate X.

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

Solution: B

[[math]] \begin{aligned} & 1000 \cdot a_{20 \mid i}=\frac{600}{i}+\frac{600 v^{10}}{i} \\ & 5\left(\frac{1-v^{20}}{i}\right)=\frac{3}{i}\left(1+v^{10}\right) \\ & 5-5 v^{20}=3+3 v^{10} \\ & 0=5 v^{20}+3 v^{10}-2 \\ & \text { Let } x=v^{10}=\frac{-3 \pm \sqrt{9+4(5)(2)}}{2(5)}=\frac{-3 \pm 7}{10}=0.4 \Rightarrow i=9.59582 \% \\ & X=\frac{600}{0.0959582}(1+0.4)=8753.8\end{aligned} [[/math]]

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