Eric deposits 100 into a savings account at time 0, which pays interest at an annual nominal rate of [math]i[/math], compounded semiannually. Mike deposits 200 into a different savings account at time 0, which pays simple interest at an annual rate of [math]i[/math]. Eric and Mike earn the same amount of interest during the last 6 months of the 8th year.

Calculate [math]i[/math].

• 9.06%
• 9.26%
• 9.46%
• 9.66%
• 9.86%

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

Solution: C

Eric’s (compound) interest in the last 6 months of the 8th year is [math]100(1 + \frac{i}{2})^{15} \frac{i}{2}[/math].

Mike’s (simple) interest for the same period is [math]200 \frac{i}{2}[/math].

Thus

[[math]] \begin{align*} \left(1+{\frac{i}{2}}\right)^{\frac{5}{2}}{\frac{i}{2}} &= 200\frac{i}{2} \\ \left(1+{\frac{i}{2}}\right)^{\frac{15}{2}} &= 2 \\ 1+\frac{i}{2} &=1.04739 \\ i =0.09459 &=9.46\%. \end{align*} [[/math]]

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