Mean-Variance Portfolio Theory

The expected return on a portfolio of two assets is a simple-weighted average of the expected returns on the individual securities. The same is not necessarily true of the risk of the portfolio, as commonly measured by the standard deviation of the return. Markowitz [1959] showed the standard deviation of the return on a portfolio of two assets to be equal to

where σp is the standard deviation of the return on the portfolio;

σ2A is the variance of the return on security A;

σ2B is the variance of the return on security B; and

ρAB is the correlation coefficient between the returns on security A and security B.

The standard deviation of the portfolio is not a simple-weighted average of the standard deviation of each security: cross-product terms are involved and the weights do not usually add to one. In order to demonstrate this relationship, specific cases involving different degrees of co-movement between two securities have been considered.

The correlation coefficient is a standardised measure of two securities covariance, and has a maximum value of plus 1 and a minimum value of minus 1. A value of plus 1 means that two securities will always move in perfect unison, while a value of minus 1 means that their movements are exactly opposite to each other. These extreme cases, together with an intermediate value, are plotted in the following figures.

In the case of perfectly correlated assets, the risk and return on the portfolio of the two assets is a weighted average of the risk and return on the individual assets. This is illustrated in figure 3.1. Note that combinations of the two assets lie along the straight line connecting the two assets. Nothing has been gained by diversifying instead of purchasing the individual assets.

In the case where two securities are perfectly negatively correlated, it will always be possible to find some combination of these two securities that has zero risk. This is clearly shown by the lines in figure 3.2 below, and illustrates the power of diversification: the ability of combinations of securities to reduce risk. As shown, it is not uncommon for combinations of two securities to have less risk than either of the individual assets in the combination. The lower the correlation coefficient between assets, ceteris paribus, the greater the benefit of diversification.

Figure 3.3 on the following section plots both of these relationships on the same graph. These two curves, often called portfolio possibility curves, represent the limits within which all portfolios of these two securities must lie for intermediate values of the correlation coefficients.1 Such an intermediate correlation would produce a curve such as the solid black line in figure 3.3.


1This assumes short sales are not allowed.