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ABy Admin
Nov 17'23
Exercise
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.
ABy Admin
Nov 17'23
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.