You are given that the force of interest at time t (in years) is [math]\frac{1}{t+8}[/math]

Calculate the annual effective rate of interest in year 5.

• 6.2%
• 6.6%
• 7.4%
• 8.3%
• 9.5%

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

Solution: D

[[math]] \begin{aligned} & a(t)=\exp \left(\int_0^t \frac{1}{r+8} d r\right) \\ & =\exp \left[\left.\ln (t+8)\right|_0 ^t\right] \\ & =\exp [\ln (t+8)-\ln (8)] \\ & =\exp \left(\ln \frac{t+8}{8}\right)=\frac{t+8}{8} \\ & \frac{a(5)}{a(4)}=\frac{13 / 8}{12 / 8}=1.08333 \\ & i_5=0.08333\end{aligned} [[/math]]

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