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\title{MA162: Finite mathematics}
\subtitle{Financial Mathematics}
\institute{University of Kentucky}
\date{January 29, 2014}

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\begin{document}

\maketitle

\begin{frame}{Last Time}
\begin{itemize}
  \item \textbf{Compound Interest, Compounded m times per year}\\
  If \$P is invested for t years at annual interest rate r compounded m times per year, then A (the accumulated value)
  is \[ A = P \left(1 + \frac{r}{m}\right)^{mt} \]
  \item Equivalently, letting \(i = r/m\) (interest per period) and \(n = mt\) (total number of periods) then 
  \[ A = P\left(1 + i \right)^n\]
  \item \textbf{Continuously Compounded interest}\\
  If \$P is invested for t years at continuously compounded rate of \(\delta\) per year, then 
  \[ A = P \, e^{\delta t}\]
\end{itemize}
\end{frame}

\begin{frame}{Last Time}
\begin{itemize}
  \item Consider annual compound interest r compounded m times per year.  The \emph{annual effective rate of interest}, \(r_eff\) 
  is obtained by solving this equation for \(r_{eff}:\) 
  \[  \left(1 + \frac{r}{m}\right)^{m} = (1 + r_{eff})\]
  \item Consider continuously compounded interest rate of \(\delta\). The \emph{annual effective rate of interest}, \(r_eff\) 
  is obtained by solving this equation for \(r_{eff}:\) 
  \[  e^{\delta} = (1 + r_{eff})\]
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 1: Continuous Compound Interest}
Find the present value of \$2700 invested for 12 years at continuously compounded rate of 2\%.

\end{frame}

\begin{frame}[t]{Ex. 2: Effective Rate}
Determine the continuously compounded rate of interest if the effective rate of interest is 6\%

\end{frame}

\begin{frame}[t]{Ex. 3: Dealing with Varying Rate}
\begin{itemize}
  \item John borrowed \(\$3500\) on January 29, 2010.  He is to repay the loan on January 29, 2014. 
  \item His loan is an adjustable rate loan, meaning that the interest rate \emph{can} vary at certain specific times. 
  \item The original interest rate was \(3\%\) APR compounded quarterly.
  \item On July 29, 2013, the interest rate was changed to \(5\%\).
  \item Assuming no more changes in the interest rate, how much must he pay back?
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 4: Dealing with multiple cash flows}
\begin{itemize}
  \item Kendall opened a savings account 6 years ago, by invested \(\$5000\) in an account which 
  earns \(4\%\) APR compounded semi-annually.
  \item 5 years ago, she deposited an extra \$2000.
  \item 2 years ago, she withdrew some amount, call it X.
  \item Her current balance is \(\$4361.\)
  \item Determine the size of X.
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 5: Dealing with lots of cash flows}
\begin{itemize}
  \item Loraine won the Ohio Lottery.  She can choose between accepting \$9,000,000 today or she can accept \$600,000
  at the end of each year for the next 30 years.
  \item Assume discount rate of 3\% per year, compounded annually.  
  \item Which option is ``better''?

\end{itemize}
\end{frame}

\begin{frame}[t]{Geometric Sum}
Given a real number \(a \neq 1\) and a positive integer \(n\),
\[ a + a^2 + a^3 + \ldots + a^{n} = \frac{a - a^{n+1}}{1 - a} \]
\vfill
Consequently,
\[ \frac{1}{1+i} 
+ \left(\frac{1}{1+i}\right)^2 
+ \left(\frac{1}{1+i}\right)^3 
+ \ldots 
+ \left(\frac{1}{1+i}\right)^n 
= \frac{ \frac{1}{1+i} - \left(\frac{1}{1+i}\right)^{n+1}}{1 - \frac{1}{1+i}}
\]
and with a bit of algebra, this simplifies down to 
\[ \frac{1 - (1+i)^{-n}}{i} \]
\vfill

\end{frame}

\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. 6: 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 \(6\%\) compounded monthly.
  \item Determine the size of Murray's monthly payment.
\end{itemize}
\end{frame}

\begin{frame}[t]{Ex. 6: Car Loan}
\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}


\end{document}


