Chapter 7 Lecture Notes - 8-07

Chapter 7 Lecture Notes - 8-07 - Optimal Risky Portfolios...

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Optimal Risky Portfolios BKM Chapter 7 Mean standard deviation diagrams Mean variance analysis: 2 risky assets Mean variance analysis: 3 or more risky assets Many risky assets and the risk-free asset
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Two Risky Assets; No risk-free asset (This material in pages 7-11 goes with the “Basic Mean/Variance Two Asset Model” Spreadsheet) Let’s kick off our analysis with portfolios of two risky assets, without any risk- free assets. Consider assets X and Y. X has an E(R) of 10% and a variance of 0.0049, which implies a standard deviation of 0.07 or 7%. Asset Y has an E(R) of 20% and a variance of 0.0100, which implies a standard deviation of 0.10 or 10%. Let’s vary the weights (proportions of X and Y) of this portfolio, and observe what happens to the E(R p ) and σ (R p ) of the portfolio Case 1 : Correlation ( ρ X,Y ) = 1 E(R p ) = w X (0.10)+w Y (0.20), and σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (1)(0.07)(0.10) Each value of w X (and hence w Y ), gives us one point in the mean-standard deviation space . Mean-Standard Deviation Diagram Correlation =1 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Standard Deviation Mean Case 2 : Correlation ( ρ X,Y ) = 0.5 E(R p ) = w X (0.10)+w Y (0.20), and 2
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σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (0.5)(0.07)(0.10) Once again, each value of w X (and hence w Y ), gives us one point in the mean- standard deviation space . Mean-Standard Deviation Diagram Correlation =0.50 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Standard Deviation Mean Case 3 : Correlation ( ρ X,Y ) = 0.0 E(R p ) = w X (0.10)+w Y (0.20), and σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (0.0)(0.07)(0.10) Mean-Standard Deviation Diagram Correlation =0.0 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Standard Deviation Mean Case 4 : Correlation ( ρ X,Y ) = -0.5 3
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E(R p ) = w X (0.10)+w Y (0.20), and σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (-0.5)(0.07)(0.10) Mean-Standard Deviation Diagram Correlation =-0.5 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Standard Deviation Mean Case 5 : Correlation ( ρ X,Y ) = -1.0 E(R p ) = w X (0.10)+w Y (0.20), and σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (-1.0)(0.07)(0.10) Mean-Standard Deviation Diagram Correlation =-1.0 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% Standard Deviation Mean Points to be noted from this exercise : 1. The end-points of the diagram are the two assets X and Y themselves, which is simply saying that a portfolio of 100% X and 0% Y is just the 4
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asset X. Conversely, a portfolio of 0% X and 100% Y is just the asset Y. So, whatever the correlation, the end-points remain rooted to their spot. This means that we are not allowing for any shorting of either asset. 2. When the correlation is +1.0, the mean-standard deviation diagram is simply a straight line. For those mathematically oriented, when the correlation is 1.0, we have: σ 2 (R p ) = w X 2 (0.0049)+w Y 2 (0.0100)+2w X w Y (1)(0.07)(0.10) = (0.07w X ) 2 + (0.1w Y ) 2 +2(0.07w X )(0.10w Y ) = (0.07w X +0.1w Y ) 2 , by the (a+b) 2 =a 2 +b 2 +2ab formula σ (R p ) = (0.07w X +0.1w Y ), which is a linear combination of the standard deviations of the two assets. If we have a correlation of –1.0, a similar logic results in σ (R p ) = (0.07w X - 0.1w Y ) (try this!), which is also linear. Hence we have two straight lines meeting on the Y-axis (return axis), which we see in Case 5.
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