Revision as of 11:44, 18 November 2023 by Admin (Created page with "A perpetuity-due with annual payments consists of ten level payments of X followed by a series of increasing payments. Beginning with the eleventh payment, each payment is 1.5% larger than the preceding payment. Using an annual effective interest rate of 5%, the present value of the perpetuity is 45,000. Calculate X. <ul class="mw-excansopts"><li>1,679</li><li>1,696</li><li>1,737</li><li>1,763</li><li>1,781</li></ul> {{soacopyright | 2023 }}")
ABy Admin
Nov 18'23
Exercise
A perpetuity-due with annual payments consists of ten level payments of X followed by a series of increasing payments. Beginning with the eleventh payment, each payment is 1.5% larger than the preceding payment.
Using an annual effective interest rate of 5%, the present value of the perpetuity is 45,000.
Calculate X.
- 1,679
- 1,696
- 1,737
- 1,763
- 1,781
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: A
The present value of the ten level payments is [math]X\ddot{a}_{\overline{\infty}|0.05} = 8.10782X[/math]. The present value of the remaining payments is
[[math]]
\begin{array}{l}{{X\biggl[1+1.05/1.1025+(1.05/1.1025)^{2}+(1.05/1.1025)^{3}+(1.05/1.1025)^{4}+(1.05/1.1025)^{5}\biggr]}}\\ {{=X[1+1.05^{-1}+1.05^{-2}+1.05^{-3}+1.05^{-5}]=X\frac{1-1.05^{-6}}{1-1.05^{-1}}=5.3295X.}}\end{array}
[[/math]]
Then, 45,000 = 8.10782X + 18.69366X = 26.80148X for X = 1679.
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.