An insurance company sells an annuity that provides 20 annual payments, with the first payment beginning one year from today and each subsequent payment 2% greater than the previous payment. Using an annual effective interest rate of 3%, the present value of the annuity is 200,000.

Calculate the amount of the final payment from this annuity.

• 11,282
• 16,436
• 16,765
• 19,784
• 24,162

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

Solution: B

The initial payment, X is

[[math]] \begin{aligned} 200,000 &=X\left({\frac{1}{1.03}}+{\frac{1.02}{1.03^{2}}}+\dots+{\frac{1.02^{19}}{1.03^{2}}}\right)=X{\frac{1/1.03-1.02^{20}\;/1.03^{21}}{1-1.02/1.03}}=17.7267 \\ X &= 11,282.42. \end{aligned} [[/math]]

The final payment is 11,282.42 (1.02)¹⁹ =16, 436.36.

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