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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>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