A perpetuity-immediate has monthly payments that increase by 5% every 12 payments. The initial 12 payments are 500 each. The present value of this perpetuity-immediate, using an annual effective interest rate of 8% is X.

Calculate X.

• 191,881
• 197,637
• 200,009
• 207,231
• 209,546

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

Solution: D

The present value of the first year's payments is:

[[math]] \begin{aligned} & 1.08^{\frac{1}{12}}-1=0.00643403 \\ & 500 a_{\overline{12} \mid 0.006434}=5756.43 \end{aligned} [[/math]]

This perpetuity can be thought of as a geometrically increasing perpetuity-due with first payment 5756.43. The present value is:

[[math]] \begin{aligned} & 5756.43\left[\frac{1}{0.08-0.05}\right] 1.08 \\ & =207,231.44 \end{aligned} [[/math]]

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