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\title{Report I - SF2980 Risk Management}
\author{Henrik Hult 751023-nnnn \and Bj\"orn L\"ofdahl 990101-nnnn \and No One Else 990101-nnnn\and No One Else 990101-nnnn}
\date{25 November, 2013}                                           % Activate to display a given date or no date

\begin{document}
\maketitle
\section*{Project  -- Total Returns}
\subsection*{Objectives}
({\it Instructions: give the project formulation and highlight the objectives})
\\
In this project we consider a 5-year investment in a portfolio of dividend-paying stocks. The yearly portfolio returns given by $S_{t+1}/S_t$ and the dividends $D_{t+1}$ paid at time $t+1$ are modeled as
\begin{align*}
	\frac{S_{t+1}}{S_t} = e^{\mu + 0.2 X_{t+1}}, \qquad \frac{S_{t+1}}{S_t} = 0.05 e^{-0.05^2/2 +0.05 Y_{t+1}},
\end{align*}
where $X_1, \dots, X_5, Y_1, \dots, Y_5$ are independent standard normally distributed random variables. 

The investment strategy is to start at time $0$ with an initial capital of $1$ million dollars and to reinvest all dividend payments in the stock portfolio. The objectives are to
\begin{itemize}
\item[(a)] determine the function $f$ such that the value $V_5$ after five years can be written as
\begin{align*}
	V_5 = f(\mu, X_1, \dots, X_5, Y_1, \dots, Y_5), 
\end{align*}
\item[(b)] simulate independent outcomes of the random vector $(X_1, \dots, X_5, Y_1, \dots, Y_5)$ and determine the empirical distribution of $V_5$. We will investigate how the distribution varies with $\mu$ and determine the smallest value of $\mu$ for which the probability that $V_5$ exceeds the value of a risk free investment of $1$ million dollars in a 5-year zero-coupon bond with zero-rate $5\%$ per year, is $0.75$. 
\end{itemize}

\subsection*{Mathematical Background}
({\it Instructions: provide a mathematical background by explaining the concept of the empirical distribution and explain how to decide the sample size})

\subsubsection*{The empirical distribution}
The empirical distribution of a sample $Z_1, \dots, Z_n$ of independent and identically distributed random variables or vectors is$\ldots$

\subsubsection*{Selecting the sample size}
({\it Explain how to select the sample size used in this project.})

\subsubsection*{...other relevant topics...}

\subsection*{Results}

({\it Explain your derivation of the objective (a) and display and comment on your simulation results in (b). Include tables and figures as needed and refer to them as follows. In Table \ref{tab:sampletab} we see an example of a sample table. In Figure \ref{fig:samplefig1} we see an example of sample graphics with a single plot, whereas in Figure \ref{fig:samplefig2} we see an example with two plots.}) 

\begin{table}[!ht]
\center
\begin{tabular}{rrr|r}
Name & Street & Number & Phone\\ 
\hline
Henrik & Lindstedtsv. & 25 & 790 6911\\
Bj\"orn & Lindstedtsv. & 25 & 790 7316\\
\hline
\end{tabular}
\caption{Sample table. Use informative captions.} \label{tab:sampletab}
\end{table}

 \begin{figure}[!ht]
 \center
  \includegraphics[scale=.5]{normdenseffport.eps}
  \caption{ Sample graphics. Use informative captions}
  \label{fig:samplefig1}
\end{figure}


 \begin{figure}[!ht]
 \center
  \includegraphics[scale=.35]{normdenseffport.eps}  \includegraphics[scale=.35]{normdenseffport.eps}
  \caption{\emph{Left:} Sample graphics 1. \emph{Right:} Sample graphics 2.}
  \label{fig:samplefig2}
\end{figure}


\subsection*{Summary}
({\it Summarize your results and state your conclusions. })

\pagebreak

\section*{Project -- Pension Savings}

\subsection*{Objectives}
({\it Instructions: give the project formulation and highlight the objectives})
\\

\subsection*{Mathematical Background}
({\it Instructions: provide a mathematical background by explaining the concept of the empirical distribution, explain how to decide the sample size. You may cut and paste from Project 7})



\subsection*{Results}

({\it Explain your derivation of the objective (a) and display and comment on your simulation results in (b)}) 


\subsection*{Summary}
({\it Summarize your results and conclusions. })




\end{document}  