You are given the following information about a perpetuity with annual payments:

1. The first 15 payments are each 2500, with the first payment to be made three years from now.
2. Beginning with the 16th payment, each payment is k% larger than the previous payment.
3. Using an annual effective interest rate of 3.5%, the present value of the perpetuity is 115,000.

Calculate k

• 1.66
• 1.74
• 1.78
• 1.83
• 1.89

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

Solution: E

The value at time 17 of the payments beginning at time 18 is

[[math]] 2500\left(\frac{1+k}{1.035}+\frac{\left(1+k\right)^{2}}{0.035^{2}}+\cdots\right)=2500\frac{\frac{1+k}{1.035}}{1-\frac{1+k}{1.035}}=2500\frac{1+k}{0.035-k} [[/math]]

The total present value is

[[math]] \begin{array}{c}{{1\,15,000=2500(1.035^{-2})a_{\overline{15}|0.035}+2500v^{17}\,\frac{1+k}{0.035-k}}}\\ {{46(0.035-k)=10.751\,6(0.035-k)+0.55720(1+k)}} \\ k=\frac{1.61-0.3763\,1-0.55720}{46-10.751\,6+0.5720}=0.01889 \end{array} [[/math]]

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