Fund X receives a deposit of 1000 at time 0. Fund X accumulates at a nominal rate of interest k, compounded semiannually.

Fund Y receives a deposit of 921.90 at time 0. Fund Y accumulates at a nominal rate of discount, also equal to k, compounded semiannually. At the end of 5 years, the accumulated amount in Fund X and the accumulated amount in Fund Y are both equal to P.

Calculate P.

• 1820
• 1970
• 2100
• 2240
• 2370

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

Solution: E

The accumulated values for Funds [math]X[/math] and [math]Y[/math] are [math]1000\left(1+\frac{k}{2}\right)^{10}[/math] and [math]921.90\left(1-\frac{k}{2}\right)^{-10}[/math] respectively. Equating them and solving for [math]k[/math] :

[[math]] \begin{aligned} & 1000\left(1+\frac{k}{2}\right)^{10}=921.90\left(1-\frac{k}{2}\right)^{-10} \\ & 0.9219=\left[\left(1+\frac{k}{2}\right)\left(1-\frac{k}{2}\right)\right]^{10}=\left(1-\frac{k^2}{4}\right)^{10} \\ & 1-\frac{k^2}{4}=0.9919 \\ & k^2=0.0324 \\ & k=0.18 \end{aligned} [[/math]]

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