SOCR EduMaterials Activities ApplicationsActivities Portfolio - Revision history

IvoDinov at 23:01, 8 July 2009

2009-07-08T23:01:33Z

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IvoDinov at 05:06, 8 July 2009

2009-07-08T05:06:02Z

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2009-07-07T16:09:33Z

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IvoDinov at 18:29, 3 August 2008

2008-08-03T18:29:59Z

IvoDinov at 18:29, 3 August 2008

2008-08-03T18:29:28Z

IvoDinov at 18:27, 3 August 2008

2008-08-03T18:27:25Z

@@ Line 277: / Line 277: @@
 <center>[[Image: Christou_tangent_point_5_stocks.jpg|600px]]</center>
 <br>
 ===References===

@@ Line 7: / Line 7: @@
 : 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>
+: <math>\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)</math>
 \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
-</math>
+: <math>\mbox{subject to} \ \ x_A+x_B=1</math>
 : Or
-: <math>
+: <math>\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B) </math>
-\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B)
+: <math>\mbox{subject to} \ \ x_A+x_B=1</math>
 <br>
-* Therefore our goal is to find <math>x_A</math> and <math>x_B</math>, the percentage of the
+* 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>

@@ Line 194: / Line 194: @@
 * 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.  <br>
-* If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities.  The investor can have very large expected return but this will be associated with very large risk.  The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative.  The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks.  The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed.  <br>
+* If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities.  The investor can have very large expected return but this will be associated with very large risk.  The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A</math> or <math>x_B</math> can be negative.  The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks.  The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed.  <br>
 <center>[[Image: Christou_ibm_texaco_short_sales.jpg|600px]]</center>

@@ Line 6: / Line 6: @@
 * 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 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 <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
+: 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>
 \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)
@@ Line 265: / Line 264: @@
 : <math> \bar R_k - R_f = z_k \sigma_k^2 + \sum_{j=1, j \ne k}^n z_j \sigma_{kj}
 </math>
-* We have one equation like (2) for each <math>i=1, \cdots, n</math>.  Here they are:
+* We have one such equation for each <math>k=1, \cdots, n</math>.  Here they are:
 : <math>
 \bar R_1 - R_f = z_1 \sigma_1^2 + z_2 \sigma_{12} + z_3 \sigma_{13} + \cdots + z_n \sigma_{1n}

@@ Line 5: / Line 5: @@
 * 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 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 <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
+: Using market data from January 1980 until February 2001 we compute
-<math>Cov(R_A,R_B)=0.00062</math>.  We first want to minimize the variance of the portfolio. This will be:
+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>
 \mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B)

@@ Line 43: / Line 43: @@
 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.
+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>
@@ Line 193: / Line 193: @@
 <br>
-* 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.  <br>
+* 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.  <br>
 * If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities.  The investor can have very large expected return but this will be associated with very large risk.  The constraint here is only <math>x_A+x_B=1</math>, since either <math>x_A<math> or <math>x_B</math> can be negative.  The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks.  The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed.  <br>

← Older revision		Revision as of 18:29, 3 August 2008
Line 1:		Line 1:
−	== [[SOCR_EduMaterials_ApplicationsActivities \| SOCR Applications Activities]] -	+	== [[SOCR_EduMaterials_ApplicationsActivities \| SOCR Applications Activities]] - Portfolio Theory ==
−	Portfolio Theory ==

	===Description===		===Description===

@@ Line 351: / Line 351: @@
 ===References===
 * The materials above was partially taken from ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.
-* [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site]
+* [http://www.socr.ucla.edu/htmls/app/ SOCR Applications Site]
 {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_Portfolio}}