Difference between revisions of "SOCR EduMaterials Activities ApplicationsActivities Portfolio"
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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 01:05, 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 \(x_A\)) 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.
| \(x_A\) | \(x_B\) | Risk (\(\sigma^2\)) | Return | Risk (\(\sigma\)) |
|---|---|---|---|---|
| 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 \(x_A+x_B=1\)). 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.
