Revision as of 10:35, 18 November 2023 by Admin (Created page with "Jack inherited a perpetuity-due, with annual payments of 15,000. He immediately exchanged the perpetuity for a 25-year annuity-due having the same present value. The annuity-due has annual payments of X. All the present values are based on an annual effective interest rate of 10% for the first 10 years and 8% thereafter. Calculate X. <ul class="mw-excansopts"><li>16,942</li><li>17,384</li><li>17,434</li><li>17,520</li><li>18,989</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 18'23
Exercise
Jack inherited a perpetuity-due, with annual payments of 15,000. He immediately exchanged the perpetuity for a 25-year annuity-due having the same present value. The annuity-due has annual payments of X. All the present values are based on an annual effective interest rate of 10% for the first 10 years and 8% thereafter.
Calculate X.
- 16,942
- 17,384
- 17,434
- 17,520
- 18,989
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: B
[[math]]
\begin{align*}
\mathrm{PV}_{\mathrm{perp.}}=\left[\frac{1}{0.1}+\frac{\frac{1}{0.08}-\frac{1}{0.1}}{1.1^{10}}\right](15,000)+15,000 \\
164, 457.87 + 15, 000 = 179, 457.87 \\
X\left(\ddot{a}_{\overline{{{10}}}|0.10}+\frac{\ddot{a}_{\overline{{{15}}}|0.08}}{1.10^{10}}\right)=179,458 \\
X\biggl(6.759+\frac{9.244}{1.10^{10}}\biggr)=179,458 \\
X = 17,384.
\end{align*}
[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.