For a given interest rate i > 0, the present value of a 20-year continuous annuity of one dollar per year is equal to 1.5 times the present value of a 10-year continuous annuity of one dollar per year.

Calculate the accumulated value of a 7-year continuous annuity of one dollar per year.

• 5.36
• 5.55
• 8.70
• 9.01
• 9.33

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

Solution: D

[[math]] \overline{{{a}}}_{\overline{{{20}}|}}=1.5\overline{{{a}}}_{\overline{{{10}}|}},\;\;\;\frac{1-e^{-20\delta}}{\delta}=1.5\frac{1-e^{-10\delta}}{\delta},\;\;\;e^{-20\delta}-1.5e^{-10\delta}+0.5=0.[[/math]]

Let [math]X = e^{-10\delta}[/math]. We then have the quadratic equation

[[math]] X^2 -1.5X + 0.5 = 0 [[/math]]

with solution [math]X = 0.5[/math] for [math]\delta = \ln0.5\,/\,(-10)=0.069315. [/math]

Then, the accumulated value of a 7-year continuous annuity of 1 is

[[math]] \overline{{{s}}}_{\overline{{{7}}}|}=\frac{e^{7(0.069315)}-1}{0.069315}=9.01\,. [[/math]]

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