An insurance company purchases a perpetuity-due providing a geometric series of quarterly payments for a price of 100,000 based on an annual effective interest rate of i. The first and second quarterly payments are 2000 and 2010, respectively.

Calculate i.

• 10.0%
• 10.2%
• 10.4%
• 10.6%
• 10.8%

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

Solution: D

The first payment is 2,000, and the second payment of 2,010 is 1.005 times the first payment. Since we are given that the series of quarterly payments is geometric, the payments multiply by 1.005 every quarter. Based on the quarterly interest rate, the equation of value is

[[math]] \begin{align*} 100,000=2,0000+2,000(1.005)v+2,000(1.005)^{2}v^{2}+2,000(1.005)^{3}v^{3}+\cdots={\frac{2,0000}{1-1.005v}} \\ 1-1.005v=2,000/100,000 \Rightarrow v=0.98/1.005. \end{align*} [[/math]]

The annual effective rate is [math]v^{-4}-1=\left(0.98\,/1.005\right)^{-4}-1=0.10601=10.69 \%. [/math]

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