The following deposits are made into a fund at the beginning of each year:

Year	Deposit
1	100
2	100
3	150
4	150
5	150

The fund earns an annual effective interest rate of 13%. At the end of the tenth year, the fund is used to purchase a perpetuity-immediate with semiannual payments of X. The perpetuity earns an annual effective interest rate of 12%.

Calculate X.

• 99
• 102
• 105
• 180
• 204

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

Solution: A

[[math]] \begin{aligned} & {\left[100 \ddot{s}_{\overline{5} \mid 0.13}+50 \ddot{s}_{\overline{3} | 0.13}\right](1.13)^5=1703.81} \\ & \frac{x}{1.12^{1 / 2}-1}=1703.81 \\ & X=99.33 \end{aligned} [[/math]]

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