An investor deposits 100 into a bank account at time 0. The bank credits interest at an annual nominal interest rate of i, compounded semi-annually. The total amount of interest credited in the twelfth year is twice the amount of interest credited in the fifth year.

Calculate i.

• 10.15%
• 10.24%
• 10.32%
• 10.41%
• 10.48%

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

Solution: A

[[math]] \begin{align*} 1{\mathrm{00}}{\bigg(}1+{\frac{i}{2}}{\bigg)}^{22}\left[\left(1+{\frac{i}{2}}\right)^{2}-1\right]=2{\bigg(}1{\mathrm00}{\bigg)}{\bigg(}1+{\frac{i}{2}}{\bigg)}^{8}\left[{\bigg(}1+{\frac{i}{2}}{\bigg)}^{2}-1\right] \\ 2=(1+i)^{14}\\ i = (2^{1/14}-1)2 = 0.1015 = 10.15\%. \end{align*} [[/math]]

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