Revision as of 23:09, 17 November 2023 by Admin (Created page with "Fund <math>\mathrm{X}</math> accumulates at a force of interest of <math>\delta_t=\frac{2}{1+2 t}</math>, where <math>t</math> is measured in years, <math>0 \leq t \leq 20</math>. Fund <math>\mathrm{Y}</math> accumulates at an annual effective interest rate of <math>i</math>. An amount of 1 is invested in each fund at time <math>t=0</math>. After 20 years, Fund X has the same value as Fund <math>\mathrm{Y}</math>. Calculate the value of Fund Y after five years. <ul cla...")
ABy Admin
Nov 17'23
Exercise
Fund [math]\mathrm{X}[/math] accumulates at a force of interest of [math]\delta_t=\frac{2}{1+2 t}[/math], where [math]t[/math] is measured in years, [math]0 \leq t \leq 20[/math]. Fund [math]\mathrm{Y}[/math] accumulates at an annual effective interest rate of [math]i[/math]. An amount of 1 is invested in each fund at time [math]t=0[/math]. After 20 years, Fund X has the same value as Fund [math]\mathrm{Y}[/math].
Calculate the value of Fund Y after five years.
- 2.46
- 2.53
- 2.60
- 2.67
- 2.74
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 17'23
Solution: B
[[math]]
\begin{aligned} & e^{\int_0^{20} \frac{2}{1+2 t} d t}=e^{\left.\ln (1+2 t)\right|_0 ^{20}}=41 \\ & 41=(1+i)^{20} \\ & i=0.204035 \\ & (1+0.204035)^5=2.53\end{aligned}
[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.