Which of the following is an expression for the present value of a perpetuity with annual payments of 1, 2, 3, ..., where the first payment will be made at the end of n years, using an annual effective interest rate of i?

• [[math]]\frac{\ddot{a}_{\overline{n}|}-nv^n}{i}[[/math]]
• [[math]]\frac{n-a_{\overline{n}|}}{i}[[/math]]
• [[math]]\frac{v^n}{d}[[/math]]
• [[math]]\frac{v^n}{d^2}[[/math]]
• [[math]]\frac{v^n}{di}[[/math]]

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

Solution: D

One way to view these payments is as a sequence of level immediate perpetuities of 1 that are deferred n-1, n, n+1,... years. The present value is then

[[math]] v^{n-1}/\,i+ v^{n}\ /\,i+ v^{n+1}/i+\cdots=\left( v^{n-2}/i\,\right)\left( v+ v^{2}+ v^{3}+\cdots\right)= v^{n-2}/i^2. [[/math]]

Noting that only answers C, D, and E have this form and all have the same numerator,

[[math]] v^{n-2}/i^2\ = \, v^{n}/(vi)^{2}\,=\, v^{n}\,/\,d^2. [[/math]]

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