A family purchases a perpetuity-immediate that provides annual payments that decrease by 0.4% each year. The price of the perpetuity is 10,000 at an annual force of interest of 0.06.

Calculate the amount of the perpetuity’s first payment.

• 604
• 620
• 640
• 658
• 678

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

Solution: D

The equation of value is

[[math]] 10.000=X( v+ v^{2}0.996+ v^{3}0.996^{2}+\cdots)=X{\frac{e^{-0.06}}{1- v0.996}}=X{\frac{ v}{1-e^{-0.06}0.996}}=15.189X. [[/math]]

The solution is X = 10,000/15.189 = 658.37.

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