Revision as of 10:37, 18 November 2023 by Admin (Created page with "A perpetuity provides for continuous payments. The annual rate of payment at time t is <math display = "block"> \begin{cases} 1, \quad 0 \leq t < 10,\\ (1.03)^{t-10}, \quad t > 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. <ul class="mw-excansopts"><li>27.03</li><li>30.29</li><li>34.83</li><li>38.64</li><li>42.41</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 18'23
Exercise
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.
ABy Admin
Nov 18'23
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.