A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and a charity receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and the charity’s share is K.

Calculate K.

• 24%
• 28%
• 32%
• 36%
• 40%

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

Solution: D

The present value of the perpetuity = X/i. Let B be the present value of Brian’s payments.

[[math]] \begin{align*} B=X a_{\overline{n}|i}=0.4{\frac{X}{i}} \\ K=0.36{\frac{X}{i}} \\ a_{\overline{n}|i}=\frac{0.4}{i}\Longrightarrow0.4=1-{\nu}^{n}\Longrightarrow{\nu}^{n}=0.6 \\ K=\nu^{2n}\,{\frac{X}{i}} \\ K = 0.36 \frac{X}{i} \end{align*} [[/math]]

Thus the charity’s share is 36% of the perpetuity’s present value.

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