A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. The annual effective rate of interest is 8%.

Calculate X.

• 54
• 64
• 74
• 84
• 94

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

Solution: A

The value of the perpetuity after the fifth payment is 100/0.08 = 1250. The equation to solve is

[[math]] \begin{array}{c}{{1250=X( v+1.08 v^{2}+\cdots+1.08^{24} v^{25})}}\\ {{=X( v+ v+\cdots+ v)=X(25)/1.08}}\\ {{X=50(1.08)=54.}}\end{array} [[/math]]

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