The present value of a perpetuity paying 1 every two years with first payment due immediately is 7.21 at an annual effective rate of i. Another perpetuity paying R every three years with the first payment due at the beginning of year two has the same present value at an annual effective rate of i + 0.01.

Calculate R.

• 1.23
• 1.56
• 1.60
• 1.74
• 1.94

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

Solution: D

For the first perpetuity,

[[math]] \begin{align*} \frac{1}{\left(1+\dot{l}\right)^{2}-1}+1 &= 7.21 \\ \frac{1}{6.21} &= \left(1+i\right)^{2}-1 \\ i &= 0.0775 \end{align*} [[/math]]

For the second perpetuity,

[[math]] \begin{align*} R\left[\frac{1}{\left(1.0775+0.01\right)^{3}-1}+1\right]\left(1.0875\right)^{-1}=7.21 \\ 1.286139 R = 7.21(1.0875) 0.286139\\ R=1.74 \end{align*} [[/math]]

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