Question

For this question, use the following data table: AT&T Microsoft Expected Return 0.10 0.21 Standard Deviation 0.15 0.25 a. What is the minimum-risk (standard deviation) portfolio allocation of AT&T and Microsoft if the correlation between the two stocks is 0? 0.5? 1? -1? What is the standard deviation of each of these minimum-risk portfolios? b. What is the optimal combination of these two securities in a portfolio for each of the four given values of the correlation coefficient, assuming the existence of a money market fund that currently pays a risk-free 0.045?

Answer 1

Here we have the following information:

	Er	Standard deviation
AT&T	0.1	0.15
microsoft	0.21	0.25
Correlations	0	0.5	1	-1

a) The formula to find out the minimum variance weights for the stocks in Portfolio is under:

and

Here D and E are the Stock companies here AT&T = D and microsoft =E

and COV(rD,rE) =

and Formula for variance of portfolio is :

and Standard deviation = Square root of Variance = risk of the portfolio

So When Correlation = 0

Wmin. D = 0.25^2 - 0.15*0.25*0 / 0.25^2+0.15^2 - 0.15*0.25*0 = 0.0625/0.085 = 0.735 or 73.5%

Wmin E = 1-0.735= 0.265 or 26.5%

Standard Deviation =SQRT(0.735^2*0.15^2+0.265^2*0.25^2+2*0*0.25*0.15*0.735*0.265)= 0.129 or 12.9%

So When Correlation = 0.5

Wmin. D = 0.25^2 - 0.15*0.25*0.5/ 0.25^2+0.15^2 - 0.15*0.25*0.5 = 0.04375/0.06625 = 0.66 or 66%

Wmin E = 1-0.66 = 0.34 or 34%

Standard Deviation =SQRT(0.66^2*0.15^2+0.34^2*0.25^2+2*0.5*0.25*0.15*0.66*0.34)= 0.1595 or 16% approx.

So When Correlation = 1

Wmin. D = 0.25^2 - 0.15*0.25*1/ 0.25^2+0.15^2 - 0.15*0.25*1 = 0.025/0.0475 = 0.526 or 52.6%

Wmin E = 1-0.526 = 0.474 or 47.4%

Standard Deviation =SQRT(0.526^2*0.15^2+0.474^2*0.25^2+2*1*0.25*0.15*0.526*0.474) = 0.1974 or 19.74%

So When Correlation = -1

Wmin. D = 0.25^2 - 0.15*0.25*(-1)/ 0.25^2+0.15^2 - 0.15*0.25*(-1) = 0.1/0.1225 = 0.816 or 81.6%

Wmin E = 1-0.816 = 0.184or 18.4%%

Standard Deviation =SQRT(0.816^2*0.15^2+0.184^2*0.25^2+2*-1*0.25*0.15*0.816*0.184) = 0.0764 or 7.64%

B) risk free rate is 0.045

Formula for optimal weights:

where B is our AT&T and S is or MICROSOFT

When Correlation is 0

Wb =((0.1-0.045)*0.25^2-(0.21-0.045)*0.15*0.25*0)/((0.1-0.045)*0.25^2+(0.21-0.045)*0.15^2-(0.1-0.045+0.21-0.045)*0.15*0.25*0) = 0.48 or 48%

Ws = 1-0.48 = 0.52 or 52%

When Correlation is 0.5

Wb =((0.1-0.045)*0.25^2-(0.21-0.045)*0.15*0.25*0.5)/((0.1-0.045)*0.25^2+(0.21-0.045)*0.15^2-(0.1-0.045+0.21-0.045)*0.15*0.25*0.5)= 0.1136 or 11.36%

Ws = 1-0.1136 = 0.886 or 88.6%

When Correlation is 1

Wb =((0.1-0.045)*0.25^2-(0.21-0.045)*0.15*0.25*1)/((0.1-0.045)*0.25^2+(0.21-0.045)*0.15^2-(0.1-0.045+0.21-0.045)*0.15*0.25*1) = 2.5 or 250%

Ws = 1-2.5 = -1.5 or -150% ( you have to borrwow in s and invest in b)

When Correlation is -1

Wb =((0.1-0.045)*0.25^2-(0.21-0.045)*0.15*0.25*-1)/((0.1-0.045)*0.25^2+(0.21-0.045)*0.15^2-(0.1-0.045+0.21-0.045)*0.15*0.25*-1) = 0.625 or 62.5%

Ws = 1-0.625 = 0.375 or 37.5%

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