minimum variance portfolio example

8/2/2019 Minimum Variance Portfolio Example

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Determining the Minimum Variance Portfolio

Consider the two risky securities listed below.

ExpectedReturn StandardDeviation

Stocks (S) 15.0% 21.0%

Bonds (B) 6.0% 11.0%

Stock/Bond Portfolios

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

18.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%

Standard Deviation

Expec

ted

Return

1) If the correlation between these two securities is 0.15, what are the portfolio weightsfor the minimum variance portfolio created by combining these two securities?

The general formula for the portfolio weight that gives the minimum variance

portfolio is given by (see attached appendix for the derivation of this formula):

=+

=

BSBS

BSBMVP

S

Cov

Covw

,

22

,

2

2

Here the covariance equals SB = 0.15(0.21)(0.11) = 0.003465, so the formula

gives:

%47.828247.01753.011

%53.171753.0)003465.0(2)11.0()21.0(

003465.0)11.0(

22

2

====

==+

=

MVP

S

MVP

B

MVP

S

ww

w


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0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

2) If the correlation between these two securities is 0.0, what are the portfolio weightsfor the minimum variance portfolio created by combining these two securities?

With a correlation of zero, the formula for the portfolio weights in the minimum

variance portfolio simplifies to:

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

18.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%

Standard Deviation

Expected

Return

%47.787847.02153.011

%53.210121.00441.0

0121.0

)11.0()21.0(

)11.0(22

2

22

2

====

=+

=+

=+

=

MVP

S

MVP

B

BS

BMVP

S

ww

w

3) If the correlation between these two securities is -1.0, what are the portfolio weightsfor the minimum variance portfolio created by combining these two securities?

With a correlation of -1.0, the formula for the portfolio weights in the minimumvariance portfolio simplifies to:

%62.656562.03438.011

%38.343438.011.021.0

11.0

====

==

+

=

+

=

MVP

S

MVP

B

BS

BMVP

S

ww

w


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0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

18.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%

Standard Deviation

Expected

Return

Note that this represents a perfect hedge portfolio. If you plug these weights into

Rule 2*, you will find that the resulting standard deviation is zero.


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Appendix: Minimizing the Variance Function

According to Rule 2*, the variance of a portfolio of two risky securities is given by:

12

2

1121

2

2

2

1

2

21

2

2

2

1

2

1

2

1211

2

2

2

1

2

1

2

1

2

222

)1(2)1(

COVwCOVwwww

or

COVwwww

+++=

++=

To minimize this function, take the first derivative with respect to w1 and set equal to

zero, or:

12

2

2

2

1

1222

1

12

2

212

2

2

2

11

12112

2

21

2

2

2

11

1

2

2

22]422[0

:givesgrearrangin

422220

COV

COVw

or

COVCOVw

COVwCOVwww

+

=

++=

++==

minimum variance portfolio example

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