The annual force of interest credited to a savings account is defined by

[[math]] \delta_t = \frac{\frac{t^{2}}{100}}{3+\frac{t^{3}}{150}} [[/math]]

with [math]t[/math] in years. Austin deposits 500 into this account at time 0.

Calculate the time in years it will take for the fund to be worth 2000.

• 6.7
• 8.8
• 14.2
• 16.5
• 18.9

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

Solution: E

[[math]] \begin{align*} 2000 &= 500 \exp \left( \int_{0}^t\frac{r^2/100}{3 + r^3/150} \, dr \right) \\ 4 &= \exp \left(0.5 \int_{0}^t\frac{r^2/100}{3 + r^3/150} \, dr \right) = \exp\left[0.5\ln\left(3+\frac{r^{3}}{150}\right)\biggr|_{0}^{t} \right] \\ 4 &= \exp\left[0.5\ln \left(1+\frac{t^{3}}{450}\right)\right]=\left(1+\frac{t^{3}}{450}\right)^{\frac{1}{2}} \\ 16 &= (1 + \frac{t^3}{450}) \\ t &= 18.8988. \end{align*} [[/math]]

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