You are given the following information regarding two annuities with annual payments:

1. Annuity X is a 20-payment annuity-immediate which provides an initial payment of 2500 and each subsequent payment is 5% larger than the preceding payment.
2. Annuity Y is a 30-payment annuity-due which provides an initial payment of k and each subsequent payment is 4% larger than the preceding payment.

Using an annual effective interest rate of 4%, Annuity X and Annuity Y have the same present value.

Calculate k.

• 1,758
• 1,828
• 1,901
• 2,078
• 2,262

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

Solution: A

[[math]] P V_X=2500\left[\frac{1-\left(\frac{1.05}{1.04}\right)^{20}}{0.04-0.05}\right]=52,732.61 [[/math]]

Annuity [math]\mathrm{Y}[/math] has the same increasing percentage as interest rate, so:

[[math]] \begin{aligned} & P V_Y=30 k \\ & P V_X=P V_Y \implies k=1757.75 \end{aligned} [[/math]]

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