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Portfolio Theory

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 xA) and the remaining funds into TEXACO (denote it with xB) - so xA + xB = 1. The resulting portfolio will be xARA + xBRB, where RA is the monthly return of IBM and RB 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(RA) = 0.010, E(RB) = 0.013, Var(RA) = 0.0061, Var(RB) = 0.0046, and Cov(RA,RB) = 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 xA and xB, the percentage of the available funds that will be invested in each stock. Substituting xB = 1 − xA 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 xA, set it equal to zero and solve for xA. 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 xa and xB 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(0.42RA + 0.58RB) = 0.422(0.0061) + 0.582(0.0046) + 2(0.42)(0.58)(0.00062) = 0.002926. The corresponding expected return of this porfolio will be: E(0.42RA + 0.58RB) = 0.42(0.010) + 0.58(0.013) = 0.01174. We can try many other combinations of xA and xB (but always xA + xB = 1) and compute the risk and return for each resulting portfolio. This is shown in the table and the graph below.

Failed to parse (unknown function\math): x_A<\math>  ! <math>x_B<\math> ! Risk (<math>\sigma^2<\math>)  ! Return ! Risk (<math>\sigma<\math>) |- | 1.00 | 0.00 | 0.006100 | 0.01000 | 0.078102 |- | 0.95 | 0.05 | 0.005576 | 0.01015 | 0.074670 |- | 0.90 | 0.10 | 0.005099 | 0.01030 | 0.071404 |- | 0.85 | 0.15 | 0.004669 | 0.01045 | 0.068329 |- | 0.80 | 0.20 | 0.004286 | 0.01060 | 0.065471 |- | 0.75 | 0.25 | 0.003951 | 0.01075 | 0.062859 |- | 0.70 | 0.30 | 0.003663 | 0.01090 | 0.060526 |- | 0.65 | 0.35 | 0.003423 | 0.01105 | 0.058505 |- | 0.60 | 0.40 | 0.003230 | 0.01120 | 0.056830 |- | 0.55 | 0.45 | 0.003084 | 0.01135 | 0.055531 |- | 0.50 | 0.50 | 0.002985 | 0.01150 | 0.054635 |- | 0.42 | 0.58 | 0.002926 | 0.01174 | 0.054088 |- | 0.40 | 0.60 | 0.002930 | 0.01180 | 0.054126 |- | 0.35 | 0.65 | 0.002973 | 0.01195 | 0.054524 |- | 0.30 | 0.70 | 0.003063 | 0.01210 | 0.055348 |- | 0.25 | 0.75 | 0.003201 | 0.01225 | 0.056580 |- | 0.20 | 0.80 | 0.003386 | 0.01240 | 0.058193 |- | 0.15 | 0.85 | 0.003619 | 0.01255 | 0.060157 |- | 0.10 | 0.90 | 0.003899 | 0.01270 | 0.062439 |- | 0.05 | 0.95 | 0.004226 | 0.01285 | 0.065005 |- | 0.00 | 1.00 | 0.004600 | 0.01300 | 0.067823 |}
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