A perpetuity pays 1 at the beginning of each three-month period. Another perpetuity pays X at the beginning of each four-year period. Using an annual effective interest rate of i, each perpetuity has a present value of 300.

Calculate X.

• 15.41
• 15.61
• 15.91
• 16.21
• 16.41

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

Solution: B

Let j equal the quarterly rate and let k equal the four-year rate:

[[math]] \begin{aligned} & 300=\frac{1}{j}(1+j) \\ & j=\frac{1}{299} \\ & \left(1+\frac{1}{299}\right)^4=(1+k)^{\frac{1}{4}} \\ & k=0.054875 \\ & 300=\frac{X(1+k)}{k} \\ & X=15.61\end{aligned} [[/math]]

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