An investor makes deposits into an account at the end of each year for ten years. The deposit in year one is 1, year two is 2 and so forth until the final deposit of 10 in year ten. The account pays interest at an annual effective rate of i. Immediately following the final deposit, the investor uses the entire account balance to purchase a perpetuity-immediate at an annual effective interest rate of i. The perpetuity makes annual payments of 10.

Calculate the purchase price of the perpetuity.

• 68.0
• 72.4
• 76.2
• 81.3
• 91.3

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

Solution: D

Equate the accumulated value of the deposits to the present value of the perpetuity:

[[math]] \begin{aligned} (Is)_{\overline{10}|} = \frac{\ddot{s}_{\overline{10}|}}{i} = \frac{10}{i} \\ \ddot{s}_{\overline{10}|}-10 = 10 \\ \ddot{s}_{\overline{10}|} = 20 (\textrm{using the BAII Plus}) \implies i = 12.3\% \end{aligned} [[/math]]

The PV of the perpetuity is 10/0.123 = 81.30.

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