Minimum Variance Portfolio Weight



Minimum Variance Portfolio Weight

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Portfolio Variance with Weights and Standard Deviations

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Optimal Portfolio (Weight in Bonds)

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Covariance and Correlation Coefficient

Covariance (rS , rB ) = σS σB ρS,B

Or

ρS,B = Covariance (rS , rB ) / (σS σB )

Correlation Coefficient = ρS,B

Portfolio Problem

Use the following two portfolios to answer parts one to four.

|Portfolio |Expected Return |Standard Deviation |

|Bond Portfolio |6% |10% |

|Stock |13% |30% |

If the correlation coefficient (() is -0.40 for these two risky assets, what is the minimum variance portfolio you can construct? (Hint: What percent of your wealth is invested in each portfolio?)

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Weight in the stock is 1 minus weight in the bond or 1 – 0.8226 = 0.1774

82.26% in Bond, and 17.74% in Stock

What is the expected return on the MVP (minimum variance portfolio) in part one using your allocation of wealth to bonds and stocks?

E(rMVP) = (0.8226) x (0.06) + (0.1774) x (0.13) = 0.0724 or 7.24%

What is the standard deviation of the minimum variance portfolio?

σ2 = (.1774)2 (0.30)2 + (0.8226)2 (0.10)2 + 2 (.8226) (.1774) (0.30) (0.10) (-0.4)

σ2 = 0.002833 + 0.006766 - 0.003593 = 0.006097

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Hint, must be less than 10% the standard deviation of the bond portfolio.

What is the optimal portfolio of these two assets if the risk-free rate is 3% (i.e., how do you allocate of your wealth in stocks and bonds)?

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And the weight in the stocks is 1 – 0.7414 or 0.2586 or 25.86%

The expected return on the optimal portfolio is:

E(rOPT) = (0.7414) x (0.06) + (0.2586) x (0.13) = 0.0781 or 7.81%

And the standard deviation of the optimal portfolio is:

σ2 = (.2586)2 (0.30)2 + (0.7414)2 (0.10)2 + 2 (.2586) (.7414) (0.30) (0.10) (-0.4)

σ2 = 0.006017 + 0.005497 - 0.004601 = 0.006913

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