Minimum Variance Portfolio Weight
Weight BOND 


2
S
2
s
  s B  S , B

  B2  2 s B  S , B

Portfolio Variance with Weights and Standard Deviations
 P2  wS2 S2  wB2 B2  2wS wB S B S , B
Optimal Portfolio (Weight in Bonds)
Er   r    Er   r    
Er   r    Er   r    Er   r  Er   r   
B
B
f
2
S
2
S
f
S
f
S
2
B
f
B
S
B
f
S, B
S
Covariance and Correlation Coefficient
Covariance (rS , rB ) = σS σB ρS,B
Or
ρS,B = Covariance (rS , rB ) / (σS σB )
Correlation Coefficient = ρS,B
f
S
B
S,B
Portfolio Problem
Use the following two portfolios to answer parts one to four.
Portfolio
Bond Portfolio
Stock
Expected Return
6%
13%
Standard Deviation
10%
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?)
0.30
Weight BOND 

 .30.10 0.4
0.102

 0.8226
.302  0.102  2  .300.10 0.4 0.124
2


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
  0.006097  0.078082  7.8082%
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)?
Wb 
0.06 - 0.030.302  0.13  0.030.300.10 0.4
0.06 - 0.030.302  0.13  0.030.102  0.06  0.03  0.13  0.030.300.10 0.4
Wb 
0.0027  0.0012
0.0039

 0.7414  74.14%
0.0027  0.001  0.00156 0.00526
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
  0.006913  0.083145  8.3145%