
Portfolio Risk and ReturnMost investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away. This principle is presented on the Diversification page. First, the computation of the expected return, variance, and standard deviation of a portfolio must be illustrated. Once again, we will be using the probability distribution for the returns on stocks A and B.
From the Expected Return and Measures of Risk pages we know that the expected return on Stock A is 12.5%, the expected return on Stock B is 20%, the variance on Stock A is .00263, the variance on Stock B is .04200, the standard deviation on Stock S is 5.12%, and the standard deviation on Stock B is 20.49%. Portfolio Expected ReturnThe Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The weights reflect the proportion of the portfolio invested in the stocks. This can be expressed as follows:
where
For a portfolio consisting of two assets, the above equation can be expressed as
Portfolio Variance and Standard DeviationThe variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient. The Covariance between the returns on two stocks can be calculated using the following equation:
where
The Correlation Coefficient between the returns on two stocks can be calculated using the following equation:
where
Using either the correlation coefficient or the covariance, the Variance on a TwoAsset Portfolio can be calculated as follows:
The standard deviation on the porfolio equals the positive square root of the the variance.
Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. This is the essence of Diversification, by forming portfolios some of the risk inherent in the individual stocks can be eliminated.
© 2002  2010 by Mark A. Lane, Ph.D.
