A perpetuity provides for continuous payments. The annual rate of payment at time t is

[[math]] \begin{cases} 1, \quad 0 \leq t \lt 10,\\ (1.03)^{t-10}, \quad t \gt 10 \end{cases} [[/math]]

Using an annual effective interest rate of 6%, the present value at time t = 0 of this perpetuity is x.

Calculate x.

• 27.03
• 30.29
• 34.83
• 38.64
• 42.41

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

Solution: A

Present value for the first 10 years is

[[math]]{\frac{1-\left(1.06\right)^{-10}}{\ln\left(1.06\right)}}=7.58 [[/math]]

Present value of the payments after 10 years is

[[math]] \left(1.06\right)^{-10}\int_{0}^{\infty}\left(1.03\right)^{s}\left(1.06\right)^{-s}d s={\frac{0.5584}{\ln\left(1.06\right)-\ln\left(1.03\right)}}=19.45 [[/math]]

Total present value = 27.03

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