Revision as of 12:01, 18 November 2023 by Admin (Created page with "The following two annuities-immediate have the same present value at an annual effective interest rate of i, i > 0. #A ten-year annuity with annual payments of 475. #A perpetuity with annual payments of 400 in years 1-5, zero in years 6-10, and 400 in years 11 and beyond. Calculate i. <ul class="mw-excansopts"><li>10.65%</li><li>10.75%</li><li>10.85%</li><li>10.95%</li><li>11.05%</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 18'23
Exercise
The following two annuities-immediate have the same present value at an annual effective interest rate of i, i > 0.
- A ten-year annuity with annual payments of 475.
- A perpetuity with annual payments of 400 in years 1-5, zero in years 6-10, and 400 in years 11 and beyond.
Calculate i.
- 10.65%
- 10.75%
- 10.85%
- 10.95%
- 11.05%
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: B
[[math]]
\begin{aligned} & 475 a_{\overline{10} \mid i}=400\left(a_{5 \mid i}+v^{10} a_{\infty}\right) \\ & 475 \frac{1-v^{10}}{i}=400 \frac{1-v^5+v^{10}}{i} \\ & 475\left(1-v^{10}\right)=400\left(1-v^5+v^{10}\right) \\ & 875 v^{10}-400 v^5-75=0 \\ & v^5=\frac{400 \pm \sqrt{400^2+4(875)(75)}}{2(875)}=0.6 \\ & i=(1 / 0.6)^{1 / 5}-1=0.1076=10.76 \%\end{aligned}
[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.