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\begin{document}%
\normalsize%
\logo%
\rulename{Static Change Regression}%
\tblofcontents%
\ruledescription{Regresses the one-day price changes against the lagged level one-day change of the signal for the specified number of days, using coefficients estimated from the start of the data.}%
\howtotrade
{Given default parameter values, if the asset drift is 0.01 and the error is 0.05 (5\% daily volatility), this rule will take a $0.01 / (0.05)^2  = 4.0$ or 400\% position (leveraged).}
{
\begin{figure}[H]
\begin{multicols}{2}
  \centering
    \begin{subfigure}{\linewidth}
        \includegraphics[width=\linewidth]{\graphdir{market.png}}
        \caption{Market series data}
        \label{fig:01}
    \end{subfigure}
  \par
  \vspace{5mm}
  \begin{subfigure}{\linewidth}
    \includegraphics[width=\linewidth]{\graphdir{research.png}}
    \caption{Research series data}
    \label{fig:02}
  \end{subfigure}
  \par
  \begin{subfigure}{\linewidth}
    \includegraphics[width=\linewidth]{\graphdir{pa(StaticChangeRegression).png}}
    \caption{ Suggested volume to buy or sell}
    \label{fig:03}
  \end{subfigure}
  \par
  \vspace{5mm}
  \begin{subfigure}{\linewidth}
    \includegraphics[width=\linewidth]{\graphdir{pr(StaticChangeRegression).png}}
    \caption{Portfolio return}
    \label{fig:04}
  \end{subfigure}
  \end{multicols}
  \caption{Graphical depiction of the Static Change Regression algorithm. 20 Days of trading data is visualized in the graphs (\ref{fig:01}) A line chart showing changes in the market price for multiple trading days.(\ref{fig:02}) A chart displaying the research series data. (\ref{fig:03})Positive values indicate that buying the security by x\%. The negative values mean you are shorting the security by x\% (\ref{fig:04})Chart showing the portfolio return when using the Static Change Regression as the trading rule.}
  \label{fig:cps_graph}
\end{figure}
}

\ruleparameters{Kelly fraction}{1.0}{Amplitude weighting. 1.0 is maximum growth if regression is exact. <1.0 scales down positions taken.}{$\kellyfraction$}{Regression length}{50}{This is the number of days used to estimate the regression coefficients.}{$\lookbacklength$}%
\stoptable%

\section{Equation}
Below are the equations which govern how this trading rule determines a trading position.

\begin{equation}
\regressionprice_\currenttime = \amplitudecoefficient\left(\frac{\research_\currenttime}{\research_{\currenttime - 1}} - 1\right)+\constantc 
\label{eq1}
\end{equation}

The equation ($\ref{eq1}$) predicts the value of the price $\regressionprice_\currenttime$ at time $\currenttime$ using the ratio of the research value at time $\currenttime$ at $\research_\currenttime$ and the research value at time $\currenttime-1$. Since we are using a static approach the amplitude coefficient $\amplitudecoefficient$ remains constant. In order to calculate the resultant fractional portfolio allocation $\position_{\currenttime}$ we use the Kelly fraction to obtain the maximum results for the long run. 

\begin{equation}
\position_\currenttime = \kellyfraction \frac{\regressionprice_\currenttime}{ \rmserror_{\regressionprice}^{2}}  
\label{eq2}
\end{equation}

\\
where

$\research$: is the value of the research series.

$\regressionprice_\currenttime$: is the predicted price at time.$\currenttime$.

$\rmserror_\regressionprice$: is the standard error.

$\kellyfraction$: is the Kelly Fraction.

$\position$: is the resultant fractional portfolio investment.

The standard error $\rmserror_{\regressionprice}$ is calculated and included in equation (\ref{eq2}) to normalize the predicted price. 

\keyterms%
\furtherlinks%
\end{document}
