A perpetuity-immediate with annual payments is priced at X based on an annual effective interest rate of 10%. The amount of the first payment is 14,000. Each payment, from the second through the twentieth, is 4% larger than the previous payment. The 21st payment and each subsequent payment will be 1% larger than the previous payment.

Calculate X.

• 185,542
• 191,834
• 206,540
• 208,508
• 212,823

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

The PV of the first twenty payments is:

[[math]] 14,000\left(\frac{1-\left(\frac{1.04}{1.10}\right)^{20}}{0.10-0.04}\right)=157,337.48 [[/math]]

The PV of the remaining payments starting at time 21 is:

[[math]] 14,000(1.04)^{19}(1.01)\biggl(\frac{1}{0.10-0.01}\biggr)(1.10)^{-20}=49.202.44 [[/math]]

Total equals 206,539.92.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.