\chapter{Introduction}
\label{ch:intro}


To obtain an estimate for the distribution of the returns for a given asset or a portfolio.  Once obtained, to use the distribution to calculate risk measurement for a given time horizon.  
\vspace{.25cm}

Probability density estimation goes hand in hand with nonparametric estimation of the cumulative distribution.  The density function provides a better visual summary of how the random variable is distributed across its support.  Skewness, kurtosis, disperseness are just a few properties can ascertained from the density plot.
\vspace{.25cm}

\begin{figure}[htb]
\includegraphics[scale=0.25]{SPXlastgraph.png} 
\includegraphics[scale=0.25]{SPXgraph2.png}  
\caption{Standard and Poor Daily Returns}
\end{figure}
\vspace{.25cm}

Unlike using density function, which places probability mass $\dfrac{1}{n}$ on each observation, the paper focuses on the kernel density estimator that more fairly spreads out the probability mass of each observation, not arbitrarily in a fixed interval, but smoothly around the observation, typically in a symmetric way.
\vspace{.25cm}

The data for this project will illustrate the usage of basic kernels, to estimate the distribution of returns on a certain asset, as well as a returns for a portfolio.  Using this curve estimation, a certain key risk measure, Value at Risk, which will ultimately provide a single quantitative number summarizing the total risk in  portfolio of financial assets, or on a single asset.  This measurement is widely used for managers when determining the possible losses for the market risks during certain scenarios.



