SOCR EduMaterials Activities ApplicationsActivities Portfolio

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(→Portfolio Theory)
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<math>x_B=1-x_A</math> into the equation of the variance we get  
<math>x_B=1-x_A</math> into the equation of the variance we get  
<math>
<math>
-
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B)
+
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
</math>
</math>
 +
<br>
To minimize the above exression we take the derivative with respect to  
To minimize the above exression we take the derivative with respect to  
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>.  The result is:
<math>x_A</math>, set it equal to zero and solve for <math>x_A</math>.  The result is:
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| 0.067823
| 0.067823
|}
|}
 +
 +
For the above calculations short selling was not allowed (<math>0 \le x_A \le 1</math> and
 +
<math>0 \le x_B \le 1</math>, in addition to <math>x_A+x_B=1</math>).  We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph).  Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return.

Revision as of 05:59, 3 August 2008

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.

xA xB Risk (σ2) Return Risk (σ)
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

For the above calculations short selling was not allowed (0 \le x_A \le 1 and 0 \le x_B \le 1, in addition to xA + xB = 1). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return.

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