At an annual effective interest rate of i, i > 0%, the present value of a perpetuity paying 10 at the end of each 3-year period, with the first payment at the end of year 3, is 32.

At the same annual effective rate of i, the present value of a perpetuity paying 1 at the end of each 4-month period, with first payment at the end of 4 months, is X.

Calculate X.

• 31.6
• 32.6
• 33.6
• 34.6
• 35.6

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

Solution: B

For the first perpetuity,

[[math]] \begin{align*} 32 &=10\left( v^{3}+ v^{6}+\cdot\cdot\cdot\right)=10 v^{3}/\left(1- v^{3}\right)\\ 32-32 v^{3} &=10 v^{3} \\ v^{3} &=32/42. \end{align*} [[/math]]

For the second perpetuity

[[math]] X= v^{13}+ v^{23}+\cdots= v^{1/3}\,/\,(1- v^{1/3})=(32\,/\,42)^{1/9}\,/\,[1-(32\,/\,42)^{1/9}\,]=32.599. [[/math]]

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