SOCR EduMaterials Activities ApplicationsActivities Portfolio

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Revision as of 05:47, 3 August 2008

Portfolio Theory

An investor has a certain amount of dollars to invest into two stocks ( $I B M$ and $T E X A C O$ ). 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 $I B M$ and $R B$ is the monthly return of $T E X A C O$ . 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 $E (R A) = 0.010$ , $E (R B) = 0.013$ , $V a r (R A) = 0.0061$ , $V a r (R B) = 0.0046$ , and $C o v (R A, R B) = 0.00062$ . \\ We first want to minimize the variance of the portfolio. This will be: Failed to parse (syntax error): \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B) \\ \mbox{subject to} \ \ x_A+x_B=1

Or Failed to parse (syntax error): \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 $I B M$ and the remaining $58 \%$ into $T E X A C O$ the variance of the portfolio will be minimum and equal to: $V a r (0.42 R A + 0.58 R B) = 0.42 2 (0.0061) + 0.58 2 (0.0046) + 2(0.42)(0.58)(0.00062) = 0.002926.$ The corresponding expected return of this porfolio will be: $E (0.42 R A + 0.58 R B) = 0.42(0.010) + 0.58(0.013) = 0.01174.$ 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. \ $0.05 i n]$

@@ Line 1: / Line 1: @@
+== Portfolio Theory ==
 == Portfolio Theory ==
-An investor has a certain amount of dollars to invest into two stocks <math>IBM</math> and <math>TEXACO</math>.  A portion of the available funds will be invested into
+An investor has a certain amount of dollars to invest into two stocks
-IBM (denote this portion of the funds with <math>x_A</math> and the remaining funds
+(<math>IBM</math> and <math>TEXACO</math>).  A portion of the available funds will be invested into
+IBM (denote this portion of the funds with <math>x_A</math>) and the remaining funds
 into TEXACO (denote it with <math>x_B</math>) - so <math>x_A+x_B=1</math>.  The resulting portfolio
 will be <math>x_A R_A+x_B R_B</math>, where <math>R_A</math> is the monthly return of <math>IBM</math> and <math>R_B</math> is the
-monthly return of TEXACO.  The goal here is to
+monthly return of <math>TEXACO</math>.  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 <math>E(R_A)=0.010</math>, <math>E(R_B)=0.013</math>, <math>Var(R_A)=0.0061</math>, <math>Var(R_B)=0.0046</math>, and
-<math>Cov(R_A,R_B)=0.00062</math>.  We first want to minimize the variance of the portfolio.  This will be:
+<math>Cov(R_A,R_B)=0.00062</math>. \\
+We first want to minimize the variance of the portfolio.
+This will be:
 <math>
-\mbox{Minimize} \ \ mbox{Var}(x_A R_A+x_BR_B)
+\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B) \\
+\mbox{subject to} \ \ x_A+x_B=1
 </math>
+Or
 <math>
+\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
 </math>
+Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
+available funds that will be invested in each stock.  Substituting
+<math>x_B=1-x_A</math> into the equation of the variance we get
+<math>
+x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B)
+</math>
+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=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
+</math>
+and therefore
+<math>
+x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
+</math>
+The values of <math>x_a</math> and <math>x_B</math> are:
+<math>
+x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
+</math>
+and <math>x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58</math>.  Therefore if the investor invests
+<math>42 \%</math> of the available funds into <math>IBM</math> and the remaining <math>58 \%</math>
+into <math>TEXACO</math> the variance of the portfolio will be minimum and equal to:
+<math>
+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.
+</math>
+The corresponding expected return of this porfolio will be:
+<math>
+E(0.42R_A+0.58R_B)=0.42(0.010)+0.58(0.013)=0.01174.
+</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
+shown in the table and the graph below. \<math>0.05in]

SOCR EduMaterials Activities ApplicationsActivities Portfolio

From Socr

Revision as of 05:47, 3 August 2008

Portfolio Theory

Portfolio Theory

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