A payment of 7 is deposited into an account at the end of each year for 45 years. The account earns an annual effective interest rate of i. Immediately after the 45th deposit of 7, the account has accumulated to X, which is used to purchase a perpetuity-immediate. At an annual effective interest rate of 1.2i, the perpetuity will make annual payments of 75.60.

Calculate X.

• 1050
• 1120
• 1200
• 1260
• 1340

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

Solution: C

[[math]]\begin{aligned} & 7 s_{\overline{45}|i}=\frac{75.6}{1.2 i} \\ & \frac{7\left[(1+i)^{45}-1\right]}{i}=\frac{75.6}{1.2 i} \\ & 7(1+i)^{45}-7=63 \\ & (1+i)^{45}=10 \\ & i=0.0525 \\ & X=7 s_{\overline{45} \mid 0.0525}=1200\end{aligned}[[/math]]

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