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# optimum Portfolio and Capital market Line

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Please disregard the short sale of stock. Instead, consider that you have purchased 100 shares of stock A.

Assume the following variance-covariance matrix:

Stock A B E(r)
A 0.09 12%
B 0.20 0.25 18%

There is unlimited borrowing and lending at the risk free rate of 8%.

a. What is the optimal weight of the two assets?
b. Write the equation of the capital market line.
c. Create a portfolio with 20% return. Describe that portfolio in terms of the holdings of stock A, B and the risk free asset.

#### Solution Preview

See the attached file for complete solution. The text here may not be copied exactly as some of the symbols / tables may not print. Thanks
I have also enclosed the Excel sheet with different weight combinations to show that the highest slop from risk free rate to the minimum variance portfolio will be for stock B.

Assume the following variance-covariance matrix:

Stock A B E(r)
A 0.09 12%
B 0.20 0.25 18%

There is unlimited borrowing and lending at the risk free rate of 8%.

a. What is the optimal weight of the two assets?

The standard deviation of portfolio

Where x1 and x2 are proportions of two investments
sp2 = x12*0.09 + x22 0.25+2*x1*x2*0.20------------1
Since, x1+x2=1, we have x2=1-x1 put this value in 1 to get
sp2 = x12*0.09 + (1-x1)2 0.25+2*x1*(1-x1)*0.20
sp2 = -x12*0.06 + 0.25-x1*0.1
sp ...

#### Solution Summary

This posting illustrates how to build the optimum portfolio if we are given the variance-covariance matrix for the stocks (two in this case). The solution is arrived mathematically and to test whether we have reached the correct solution, various portfolios with different weights are built in Excel file and the optimum portfolio is shown on the graph.

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