\chapter{Results}
\label{ch:results}

\section{Computational Results}
I specified a kernel function using the statistical program R 'epanechinikov' which are all scaled to have standard deviation equal to one, so the bandwidth parameter means roughly the same thing regardless of kernel function.


\subsection{Density Estimation}
\begin{figure}[htb]
\includegraphics[scale=0.4]{densities.png} 
\end{figure}

\newpage

\subsection{VaR Calculations}

The following is the list of percentiles for one-day $5\%$ VaR, the likelihood that a given portfolio's losses will exceed this certain amount.
\vspace{.5cm}
\\
\begin{tabular}[htb]{l|c c c c c c}
VaR &	S$\&$P	& NASDAQ	&	DOW	&	AT$\&$T	&	Microsoft	& PetroChina	\\
\hline
$5\%$ & 39.52153 & 81.48998 & 340.4551 & 1.064806 & 1.390841 & 8.99558
\end{tabular}
\vspace{.25cm}
\\

The above results can be interpreted as largest loss likely to be suffered on a portfolio solely this position over a holding period of one day.  For instance, an investment bank holding that position in the portfolio might report that its portfolio has a 1-day VaR of $\$39.52$ at the $95\%$ confidence level, if invested soley in Standard and Poors. 


\section{Conclusion}

The results presented in this paper are highly dependent on historical data.  The underlying method taken here was to simulate a density using kernel techniques and the lowest $5\%$ quantile of this distribution is used as an approximation to VaR. Current VaR calculations involve Monte Carlo simulations and Variance-Covariance matrices which account for more robust measures, as well as, more applicable to larger aggregate portfolios.  The methods taken here are soley for illustration of nonparametric techniques in a real world application.  


