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\documentclass{simiode}

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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
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\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
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% the document. See this file for examples. This has been done for you in the document below.   
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% This option was created  by Eric Sullivan, Carroll College, Helena MT.
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\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}

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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\FINANCE - SAVINGS AND LOANS}




\markboth{Finance - Savings and Loans}{Finance - Savings and Loans}






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

\makeStitlePDFLaTex



\begin{abstract}
 We describe two situations,  one in which we are saving for a purpose and  one in which we are borrowing for a purpose.  In the first case we ask for discrete and continuous model of the situation and in the second case we ask that the results of the model be used to examine some ``what if'' situations.  
\end{abstract}





\section*{SCENARIO DESCRIPTION}


We offer two situations. The first needs a savings model and the second needs a loan model.

\subsection*{Savings}


Build two mathematical models to answer the given question for the following bank investment problem in two different ways (a) discrete difference equation and (b) continuous differential equation.

Invest \$10,000 initially and add \$2,400 each year in an interest bearing account that pays annual interest according to the following schemes: (a) [for the discrete difference equation approach] compounded annually at 3.2\% or (b) [for the continuous differential equation approach] compounded continuously at 3.15\% interest. State all your assumptions used in the model building process.  Build your model very carefully and be sure to define your variables and units.

Compare your results over a reasonable period of time and critique the faithfulness to reality of each of your approaches.

\subsection*{Loan}

Use a continuous differential equation approach to determine how long it will take to pay off a business debt of \$150,000 which was given at an annual interest rate of 6.7\% if we pay \$1,900 per month. State all your assumptions used in the model building process.  Build your model very carefully and be sure to define your variables and units. Show your analysis in detail. 

In a reverse scenario, suppose you wanted to pay off the loan in  15 years. How much would your monthly payment be to do this?

Finally, do some sensitivity analysis in which you consider different interest rates (lower and higher) as well as different loan periods, i.e. time until the loan is paid off.


\teacher{
\section*{NOTES FOR TEACHERS}

We offer up a complete analysis in a Mathematica notebook with full details - in both notebook format (1-006-T-Mma-FinancingAnalysis-Teacherversion.nb) and pdf format (1-006-T-Mma-FinancingAnalysis-Teacherversion.pdf) However, we highlight some issues here.

This is a good in-class activity followed by a homework write-up. It is always best to permit students to attempt to build the model rather than just ``tell" them the difference or differential equation which would do the job. To this end there can be some difficulty in getting all students to start doing something and to come to a consensus on final models.  There will be the usual ``goof'' of misrepresenting interest rate as a decimal and the attendant outrageous sums involved when someone writes 3.2\% as 3.2 and not .032. However, this is part of the feedback necessary to improve a model and some mention should be given to this cycling back to improve, even at this early, formative stage.

There is the issue of unreal with regard to continuously depositing and this needs some discussion for there are ways to model discrete ``dump''ing of money, e.g., the Delta Dirac or Impulse function, but this may not be the time to go there and the difficulty in modeling continuous application or deposit of money should be regarded as a weakness of the continuous model.  

Often, students want to push further and examine, say, college loan situations or home financing -- the former because they have such loans and the latter because they hear their parents talk about such things or they are actually thinking of buying a condominium or small home rather than rent for this next period in their lives. Another area of interest is credit card interest and paying off a credit card. This usually opens up a whole world of discussion and exchange as many students carry such credit card balances and sometimes students can get carried away with ``preaching'' restraint to each other.

The teacher might need to let this happen. However, there will need to be time devoted to complications and more realistic issues, e.g., in a mortgage model add the income deduction for mortgage interest payments in order to help students figure out the real cost of the loan and this may include income bracket issues as well. Other comparisons can enter the discussion. For example, is it better to rent a cheap ``student'' place now and save for home purchase; what are the realistic numbers in the region they need to know about, e.g., San Francisco CA or Oswego NY?

Students are always fascinated by money and thus such scenarios are an easy way to draw them into modeling wherein they can answer questions that are important to them. We highly recommend such investigations.
}

\end{document}
