Portfolio Risk and Return

Most 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 Return

The 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

• E[R_p] = the expected return on the portfolio,
• N = the number of stocks in the portfolio,
• w_i = the proportion of the portfolio invested in stock i, and
• E[R_i] = the expected return on stock i.

For a portfolio consisting of two assets, the above equation can be expressed as

Portfolio Variance and Standard Deviation

The 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

• s₁₂ = the covariance between the returns on stocks 1 and 2,
• N = the number of states,
• p_i = the probability of state i,
• R_1i = the return on stock 1 in state i,
• E[R₁] = the expected return on stock 1,
• R_2i = the return on stock 2 in state i, and
• E[R₂] = the expected return on stock 2.

The Correlation Coefficient between the returns on two stocks can be calculated using the following equation:

where

• r₁₂ = the correlation coefficient between the returns on stocks 1 and 2,
• s₁₂ = the covariance between the returns on stocks 1 and 2,
• s₁ = the standard deviation on stock 1, and
• s₂ = the standard deviation on stock 2.

Using either the correlation coefficient or the covariance, the Variance on a Two-Asset 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.