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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.
| \(x_A\) | \(x_B\) | 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 |
