\chapter{Details of Data}
\label{ch:datafiles}


\section{Financial Application}
\subsection{Setup of Problem}
In finance, there exists a need to analyze the distribution of returns of various indexes.  From these calculations, can obtain measurements of risk.  The value-at-risk (VaR) is a measurement which accounts for a confidence interval for the amount of loss one may expect.  There exists a percentage of certainty that the portfolio manager will not lose more than the value of the Var of the portfolio in the next $N$ days.

\subsection{Setup of Problem: VaR}
if a portfolio of stocks has a one-day $5\%$ VaR of $\$1$ million, there is a $5\%$ probability that the portfolio will decline in value by more than $\$1$ million over the next day, assuming markets are normal and there is no trading.
\vspace{.15cm}

The reason for assuming normal markets and no trading, and to restricting loss to things measured in daily accounts, is to make the loss observable. In some extreme financial events it can be impossible to determine losses, either because market prices are unavailable or because the loss-bearing institution breaks up.
\vspace{.25cm}
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.5]{VaR_graph.png} 
\caption{Illustration of the $10\%$ Value at Risk with normally distibuted portfolio value}
\end{center}
\end{figure}

\newpage

\section{Financial Definitions}
Collected historical daily last price for the following companies below, from 01/2005-11/2009.  The daily log return was calculated.
\begin{itemize}
\item \underline{Standard and Poor's 500 Index:} (S\&P) A capitalization-weighted index of 500 stocks.  The index is designed to measure performance of a broad domestic economy through changes in the aggregate market of 500 stocks representing all major industries.  
\item \underline{NASDAQ:} The composite index is a board-based capitalization-weighted index of stocks in all three tiers: Global Select, Global Market, and Capital Markets.
\item \underline{DOW Jones:} The industrial average is a price-weighted average of 30 blue chips stocks that are generally the leaders in their industry. 
\item \underline{AT\&T:}  Communications holding company, provides local and long-distance phone service, wireless and data communications, etc.  
\item \underline{Microsoft:} develops, manufactures, licenses, sells, and supports software products.  Offers operating system software, etc.  
\item \underline{PetroChina:}  Explores, develops, and produces crude oil and natural gas.  
\end{itemize} 


\newpage

\section{Goal}
Stochastic dynamics of stock prices is commonly described by a geometric (multiplicative) Brownian motion, which gives a log-normal density distributions for returns. However, numerous observations show that the tails of the PDF decays lower than the log–normal distribution predicts (the so-called ‘fat-tails’ effect).  Which does not provide a substantial estimate for for calculating VaR, basically the area under the lower left curve, when returns are high spreads.  

The following provides a histogram of the distribution of returns.  Each of the distributions used followed similar shape and symmetry by analyzing the following output.  The distributions of returns display a normal shape with outliers causing a relatively fat tails.  
  
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.5]{Allhistgraphs.png} 
\caption{Histogram for distribution of returns}
\end{center}
\end{figure}

