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Portfolio theory

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\[ \mbox{Minimize} \ \ mbox{Var}(x_A R_A+x_BR_B) <br\> \mbox{subject to} \ \ x_A+x_B=1 \]