Question

what is the minimum-risk (standard deviation) portfolio of AT&T and Microsoft if the correlation between the two stocks is 0? 0.5? 1? -1? what do you notice about the change in the allocations between AT&T and Microsoft as the correlation coefficient moves from -1 to 0? to 0.5? to +1? why might this be? what is the standard deviation of each these minimum-risk portfolios?

Answer 1

So, since the portfolio standard deviation can be calculated as :

Standard deviation = root over of [(weight of AT&T)^2 * VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 VARIANCE OF MICROSOFT + 2* W1*W2* COVARIANCE OF AT&T AND MICROSOFT

When the correlation between the two stocks is -1,

standard deviation = root over [(weight of AT&T)^2 * VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 *VARIANCE OF MICROSOFT +  2*WEIGHT OF AT&T*WEIGHT MICROSOFT *AT&T*-1*STANDARD DEVIATION OF MICROSOFT AND STANDARD DEVIATION OF AT&T ]

Since, the correlation between the two stocks is 0,

The minimum risk :

= root over of [(WEIGHT OF  AT&T)^2 * VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 VARIANCE OF MICROSOFT + 2 * WEIGHT OF MICROSOFT * WEIGHT OF AT&T * 0* STANDARD DEVIATION OF AT&T* STANDARD DEVIATION OF MICROSOFT]

The standard deviation when the correlation is 0.5

= ROOT OVER OF[(WEIGHT OF AT&T)^2*VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 * VARIANCE OF MICROSOFT + 2 * 0.5*WEIGHT OF AT&T* WEIGHT OF MICROSOFT *STANDARD DEVIATION OF AT&T & STANDARD DEVIATION OF MICROSOFT]

When the correlation is + 1,

ROOT OVER OF[(WEIGHT OF AT&T)^2*VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 * VARIANCE OF MICROSOFT + 2 *WEIGHT OF AT&T * WEIGHT OF MICROSOFT * STANDARD DEVIATION OF AT&T & STANDARD DEVIATION OF MICROSOFT].

The lower the correlation between the two stocks in the portfolio, the higher the diversification benefits and the lower will be the standard deviation of the portfolio . So, the standard deviation of the portfolio will be the lowest, in case the correlation is -1, then the lowest will be when the correlation is 0, then will be 0.5 and the highest will be, when the correlation is +1.

The risk is minimized the most, when the correlation between the assets is -1.

standard deviation = root over [(weight of AT&T)^2 * VARIANCE OF AT&T + (WEIGHT OF MICROSOFT)^2 *VARIANCE OF MICROSOFT +  2*WEIGHT OF AT&T*WEIGHT MICROSOFT *AT&T*-1*STANDARD DEVIATION OF MICROSOFT AND STANDARD DEVIATION OF AT&T ]