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.

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.