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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\OUTCOME SAVINGS}
\markboth{Outcome Savings}{Outcome Savings}



\author{Brian Winkel,
Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org}


\makeStitlePDFLaTex


\begin{abstract}
We ask students to build a model for a savings account and to determine the monthly deposit in a savings account in order meet a long term savings goal. 
\end{abstract}

\section*{SCENARIO DESCRIPTION}

You are going to build a mathematical model for savings. Be sure to state all your assumptions
\begin{enumerate}
  \item Build a mathematical model to determine the monthly deposit you will need to make in a fixed interest account in order to attain a set financial goal in 20 years.
  \item Apply this model to a situation in which you make an initial deposit of \$100 and you wish to attain a balance of \$100,000 in 20 years. Make a table of the monthly deposits for fixed interest rates going from 0.5\% to 20\% in increments of 0.5\%. 
  \item Modify some aspect of your model, e.g., change an assumption,  redo your analysis, and compare your results. Comment on the comparison.
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}

All of our analyses are in a Mathematica file, 1-40-T-Mma-OutcomeSavings-TeacherVersion.nb and an accompanying .pdf version for those who do not have access to Mathematica.

For Activities (1) and (2) one uses either the continuous model (\ref{eq:contconstant}), where $p(t)$ is the account balance at month $t$, $r$ is the annual interest rate, and $s$ is the monthly deposit,  

\begin{equation}\label{eq:contconstant}
p'(t) = \frac{r}{12} p(t) + s
\end{equation}

\noindent or the discrete model (\ref{eq:discconstant}),  where $R(n)$ is the account balance at month $n$, $r$ is the annual interest rate, and $s$ is the monthly deposit,  
 

\begin{equation}\label{eq:discconstant}
R(n+1)  = R(n) + \frac{r}{12} R(n) + s\, .
\end{equation}

For Activity (3) one natural thing is to increase the monthly amount of deposit, because as one grows older, presumably one's salary or earnings increases. So we presumed a 3\% annual increase in salary and hence a 3\% increase in the monthly deposit. This makes our models slightly different.

Here, one uses either the continuous model (\ref{eq:contincrease}), where $p(t)$ is the account balance at month $t$, $r$ is the annual interest rate,  $s$ is the monthly deposit, and we have effective 3\% increase in our monthly deposit each year,  

\begin{equation}\label{eq:contincrease}
p'(t) = \frac{r}{12} p(t) + s \left(1 + \frac{.03}{12} t\right)\, ,
\end{equation}

\noindent or the discrete model (\ref{eq:discincrease}),  where $R(n)$ is the account balance at month $n$, $r$ is the annual interest rate,   $s$ is the monthly deposit,  and we have effective 3\% increase in our monthly deposit each year,  

\begin{equation}\label{eq:discincrease}
R(n+1)  = R(n) + \frac{r}{12} R(n) + s\left(1+ \frac{.03}{12} n\right)\, .
\end{equation}


\vspace*{.3 true cm}

We offer a summary of our results based on our two approaches (discrete and continuous) modeling:

\vspace*{.2 true cm}
Continuous, fixed deposit

We need to deposit \$241.83 per month.

\vspace*{.2 true cm}
Discrete, fixed deposit

We need to deposit \$242.63 per month.

\vspace*{.2 true cm}
Continuous, deposit increases by 3\% per year

We need to deposit \$193.34 per month.

\vspace*{.2 true cm}
Discrete, deposit increases by 3\% per year

We need to deposit \$194.16 per month.

}

\end{document}
 
