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.

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.