Susan receives annual payments from a 20-year annuity-immediate. The payment in year 1 is 100 and in each succeeding year the payment is 90% of the prior year’s payment. Upon receipt of each payment, Susan invests the payment in a savings account earning interest at a 3% annual effective rate. Calculate the balance in the savings account immediately after Susan invests the last annuity payment.

• 696
• 717
• 739
• 1296
• 1335

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

Solution: D

[[math]] \begin{aligned} & F V=100\left[\frac{1-\left(\frac{0.9}{1.03}\right)^{20}}{0.03+0.10}\right](1.03)^{20} \\ & F V=1295.80 \end{aligned} [[/math]]

Alternatively,

[[math]] F V=100\left[(1.03)^{19}+0.9(1.03)^{18}+\cdots+(0.9)^{19}(1.03)^0\right]=100 \frac{(1.03)^{19}-(0.9)^{20}(1.03)^{-1}}{1-0.9(1.03)^{-1}}=1295.80 [[/math]]

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