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\begin{center} Portfolio theory \end{center} \noindent An investor has a certain amount of dollars to invest into two stocks ($IBM$ and $TEXACO$). A portion of the available funds will be invested into IBM (denote this portion of the funds with $x_A$) and the remaining funds into TEXACO (denote it with $x_B$) - so $x_A+x_B=1$. The resulting portfolio will be $x_A R_A+x_B R_B$, where $R_A$ is the monthly return of $IBM$ and $R_B$ is the monthly return of $TEXACO$. The goal here is to find the most efficient portfolios given a certain amount of risk. Using market data from January 1980 until February 2001 we compute that $E(R_A)=0.010$, $E(R_B)=0.013$, $Var(R_A)=0.0061$, $Var(R_B)=0.0046$, and $Cov(R_A,R_B)=0.00062$. \\ We first want to minimize the variance of the portfolio. This will be: \begin{eqnarray*} \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B) \\ \mbox{subject to} \ \ x_A+x_B=1 \end{eqnarray*}