Revision as of 08:51, 18 November 2023 by Admin (Created page with "A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and a charity receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and the charity’s share is K. Calculate K. <ul class="mw-excansopts"><li>24%</li><li>28%</li><li>32%</li><li>36%</li><li>40%</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 18'23
Exercise
A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and a charity receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and the charity’s share is K.
Calculate K.
- 24%
- 28%
- 32%
- 36%
- 40%
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: D
The present value of the perpetuity = X/i. Let B be the present value of Brian’s payments.
[[math]]
\begin{align*}
B=X a_{\overline{n}|i}=0.4{\frac{X}{i}} \\
K=0.36{\frac{X}{i}} \\
a_{\overline{n}|i}=\frac{0.4}{i}\Longrightarrow0.4=1-{\nu}^{n}\Longrightarrow{\nu}^{n}=0.6 \\
K=\nu^{2n}\,{\frac{X}{i}} \\
K = 0.36 \frac{X}{i}
\end{align*}
[[/math]]
Thus the charity’s share is 36% of the perpetuity’s present value.
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.