A perpetuity costs 77.1 and makes end-of-year payments. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, ...., n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%.

Calculate n.

• 17
• 18
• 19
• 20
• 21

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

Solution: C

[[math]] \begin{align*} 77.1 &=\nu\Big(I a\Big)_{{\overline{n}|i}}^{}+\frac{n\nu^{n+1}}{i} \\ &=\nu\left[\frac{\ddot{a}_{\overline{{{n}}}}-n\nu^{n}}{i}\right]+\frac{n\nu^{n+1}}{i} \\ &=\frac{a_{\overline{n}|i}}{i}-\frac{n\nu^{n+1}}{i}+\frac{n\nu^{n+1}}{i} \\ &=\frac{a_{\overline{n}|i}}{i}=\frac{1-\nu^{n}}{i^{2}}=\frac{1-\nu^{n}}{0.01\ 1\vert025} \\ 0.85003 &=1-\nu^{n} \\ 1.105^{-n} &=0.14997 \\ n &= -\frac{\ln(0.14997)}{\ln(1.105)} = 19 \end{align*} [[/math]]

To obtain the present value without remembering the formula for an increasing annuity, consider the payments as a perpetuity of 1 starting at time 2, a perpetuity of 1 starting at time 3, up to a perpetuity of 1 starting at time n + 1. The present value one period before the start of each perpetuity is 1/i. The total present value is

[[math]](1/i)(v+v+v^2+\cdots + v^n) = (1/i)a_{\overline{n}|i}.[[/math]]

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