A loan of 100 is to have all principal and accrued interest paid at the end of five years. Interest accrues at an annual effective rate of 5\% for the first two years and at a force of interest at time [math]t[/math] in years [math](t\gt2)[/math] of [math]\delta_t=\frac{1}{1+t}[/math].

Calculate the equivalent annual effective rate of discount for the five-year period.

• 10.9%
• 13.0%
• 14.6%
• 15.4%
• 17.1%

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

Solution: C

[[math]]\begin{aligned} & 100(1+0.05)^2 \exp \left(\int_2^5 \frac{1}{1+t}\right)=100(1.1025) \exp \left[\left.\ln (1+t)\right|_2 ^5\right] \\ & =110.25\left(\frac{6}{3}\right)=220.50 \\ & \frac{100}{(1-d)^5}=220.50 \\ & (1-d)^5=\frac{100}{220.50}=0.453515 \\ & 1-d=0.853726 \\ & d=0.1463\end{aligned}[[/math]]

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