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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\[ \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B) \mbox{subject to} \ \ x_A+x_B=1 \] Or \( \mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B) \mbox{subject to} \ \ x_A+x_B=1 \) Therefore our goal is to find \(x_A\) and \(x_B\), the percentage of the available funds that will be invested in each stock. Substituting \(x_B=1-x_A\) into the equation of the variance we get \( x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B) \) To minimize the above exression we take the derivative with respect to \(x_A\), set it equal to zero and solve for \(x_A\). The result is\[ x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)} \] and therefore \( x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)} \) The values of \(x_a\) and \(x_B\) are\[ x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42. \] and \(x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58\). Therefore if the investor invests \(42 \%\) of the available funds into \(IBM\) and the remaining \(58 \%\) into \(TEXACO\) the variance of the portfolio will be minimum and equal to\[ Var(0.42R_A+0.58R_B)=0.42^2(0.0061)+0.58^2(0.0046)+2(0.42)(0.58)(0.00062) =0.002926. \] The corresponding expected return of this porfolio will be\[ E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174. \] We can try many other combinations of \(x_A\) and \(x_B\) (but always \(x_A+x_B=1\)) and compute the risk and return for each resulting portfolio. This is shown in the table and the graph below. \<math>0.05in]