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(→Portfolio Theory)
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We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)  
We can try many other combinations of <math>x_A</math> and <math>x_B</math> (but always <math>x_A+x_B=1</math>)  
and compute the risk and return for each resulting portfolio.  This is  
and compute the risk and return for each resulting portfolio.  This is  
-
shown in the table and the graph below. \<math>0.05in]
+
shown in the table and the graph below.  
 +
 
 +
{| class="wikitable"
 +
|-
 +
! <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
 +
|}

Revision as of 05:49, 3 August 2008

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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