Let S be the accumulated value of 1000 invested for two years at a nominal annual rate of discount d convertible semiannually, which is equivalent to an annual effective interest rate of i.

Let T be the accumulated value of 1000 invested for one year at a nominal annual rate of discount d convertible quarterly.

[math]S/T = (39/38)^4.[/math]

Calculate [math]i[/math]

• 10.0%
• 10.3%
• 10.8%
• 10.9%
• 11.1%

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

Solution: C

[[math]] \begin{align*} \ {\frac{\left(1-d/2\right)^{-4}}{\left(1-d/4\right)^{-4}}}=\left({\frac{39}{38}}\right)^{4}\Rightarrow{\frac{1-d/2}{1-d/4}}={\frac{38}{39}}\Rightarrow39-39(d/2)=38-38(d/4) \\ d(39/2 -38/4) = 39-38 \\ d = 1/(19.5-9.5)=0.1 \\ 1+i=\left(1-d/\,2\right)^{-2}=95^{-2}=1.108\Longrightarrow i=10.8 \% \end{align*} [[/math]]

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