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\title{MA162: Finite mathematics}
\subtitle{Financial Mathematics}
\author{Paul Koester}
\institute{University of Kentucky}
\date{February 3, 2014}

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\begin{document}

\maketitle

\begin{frame}[t]{Loans}
\begin{itemize}
  \item An amount \$P is borrowed.  (P stands for principal, or present value)
  \item The loan is to be repaid by making \emph{regular} payments of size \$R and the end of each period for the next n periods.
  \item Interest rate is i per period.
  \item Then 
  \[ P = R \cdot \frac{1 - (1+i)^{-n}}{i} \]
  \item In Excel, P can be computed by \emph{=PV(i,n,R)}.
  \item In WeBWorK, P can be computed by \emph{R * PV(i,n)}.
\end{itemize}

\end{frame}

\begin{frame}[t]{Ex. 1: Car Loan}
\begin{itemize}
  \item Murray just purchased a car.  The price of the car was \(\$15,000.\)
  \item He makes a \(\$4000\) down payment takes out a car loan to cover the rest.
  \item He has to make payments at the end of each month for the next 4 years.  
  \item The interest on the loan is \(6\%\) APR compounded monthly.
  \item Determine the size of Murray's monthly payment.
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 1: Car Loan (Continued)}
\begin{itemize}
  \item What is the total amount of interest that Murray pays?
\vfill
  \item How much of Murray's first payment is due to interest?
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 1: Car Loan (Continued)}
\begin{itemize}
  \item It is now 2.5 years from the time Murray took out his car loan and Murray just made the \(30^{th}\) payment on his car.
  \item How much would he need to pay now in order to pay off the rest of his loan\footnote{assuming no ``early pay-off fees''}?  
\vfill
  \item What is the total amount of interest that Murray pays assuming he pays off the balance in full immediately after the \(30^{th}\) payment?  
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 2: Home-a-loan}
\begin{itemize}
  \item Norah has a 15 year home mortgage.
  \item She needs to pay \$2300 at the end of each month for the next 15 years.
  \item The interest on the loan is \(3.625\%\) APR compounded monthly.  
  \item She is having trouble affording the \$2300 per month.  
  \item To lower her monthly payment, she is going to refinance to a 30 year loan which has \(4.5\%\) APR compounded monthly.
  % P = 318,985
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 2: Home-a-loan}
\begin{itemize}
  \item Determine the size of her new monthly payment.
  %  1616.25
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 2: Home-a-loan}
\begin{itemize}
  \item Determine the total interest charges on the original loan.
  % 95,015
\vfill
  \item Determine the total interest charges on the new loan.
  % 262,865
\vfill
\end{itemize}
\end{frame}

%\begin{frame}[t]{Ex. 2: Home-a-loan}
%\begin{itemize}
%  \item Norah figures she can afford \$2000 each month.  
%  \item She decides to take the 30 year loan at \(4.5\%\) APR compounded monthly but she will pay \$2000 per month instead of the \$1616.25
%  per month.  In this way, she will pay off her home mortgage in fewer than 30 years.
%  \item How long will it take to pay off the loan if she makes monthly payments of \$2000?
%  % 243.53 monthly payments or 20 years and 3.5 months
%\vfill
%  \item Determine the total interest charges Norah pays in this situation.
%\vfill
%\end{itemize}
%\end{frame}
%
%\begin{frame}[t]{Ex. 2: Home-a-loan}
%\begin{itemize}
%  \item The previous calculation revealed that Norah needs to make 243.53 monthly payments.  But how can she make .53 of a payment?
%  \item In reality, she would most likely have to make a \alert{balloon payment}, in which case the \(243^{rd}\) payment would be a bit 
%  larger than the other payments.  
%  \item Assuming Norah makes 242 payments of \$2000 and pays off the remaining balance on the 243rd payment, 
%  determine the size of the 243rd payment.
%\vfill
%\end{itemize}
%\end{frame}

\begin{frame}[t]{Ex. 3: Chance to buy a ranch}
\begin{itemize}
  \item Blanch can't pass up the chance to buy a ranch.
  \item She will borrow \$400,000.
  \item She will pay back this loan by making quarterly payments at the end of each quarter for 30 years.
  \item Interest on the loan is 6.2\% APR compounded quarterly.  
  \vfill
  \item Determine the size of Blanch's quarterly payments.
  \vfill  % 7362.63
  \item Determine the interest charges on the loan.
  \vfill  % 483,515.65
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 3: Chance to buy a ranch}
\begin{itemize}
  \item Blanch suspects she can drastically cut her interest expenses if she is able to make quarterly payments that are larger than required.
  \item Supposing that Blanch pays twice her scheduled payment each month, determine how many payments Blanch needs to make before she pays 
    off the loan.
  % 35.5  -- Notice doubling payment seems to suggest pay it off twice as fast, so 60 instead of 120.  In fact, significantly better than 1/2!
  \vfill
  \item Determine Blanch's interest charges on the loan if she makes double payments.
\end{itemize}
\end{frame}


\begin{frame}[t]{Annuities}
\begin{itemize}
  \item A sequence of regular cash flows of \$R occurs at the end of each period for the next n periods.  (R stands for ``regular cash flow'')
  \item Interest rate is i per period.
  \item Then the present value, P, of this annuity is
  \[ P = R \cdot \frac{1 - (1+i)^{-n}}{i} \]
  \item In Excel, P can be computed by \emph{=PV(i,n,R)}.
  \item In WeBWorK, P can be computed by \emph{R * PV(i,n)}.
  \item P answers the question ``What is the value of this entire stream of cash flows evaluated at the beginning''

\end{itemize}
\end{frame}

\begin{frame}[t]{Annuities}
\begin{itemize}
  \item Then the accumulated value, or future value, F, of this annuity is
  \[ F = R \cdot \frac{(1 + i)^{n} - 1}{i} \]
  \item In Excel, P can be computed by \emph{=FV(i,n,R)}.
  \item In WeBWorK, P can be computed by \emph{R * FV(i,n)}.
  \item F answers the questions like ``If you save \$R at the end of each year for the next n years and interest is i per year, 
  then what is the value of your savings at the end?''
\end{itemize}
\end{frame}

\begin{frame}[t]{Annuities versus Loans}
\begin{itemize}
  \item Annuities and loans both involve level sized cash flows that are paid at regular time intervals
  \item Mathematically, they are treated the same
  \item Financially, the regular cash flows in a loan are being paid out, while the regular cash flows in an annuity are being received
\end{itemize}
\end{frame}

\begin{frame}{Ex 4: FV of Annuity}
\begin{itemize}
  \item Determine the accumulated value of a 8 year annuity with level cash flows of \$1200 at the end of each quarter,
  provided the cash flows earn \(6\%\) annual interest compounded quarterly.
  \vfill
  % Do simple upper estimate...
  \item Determine the present value of the above annuity.
\end{itemize}
\end{frame}


\end{document}


