Although the portfolio expected return is a straightforward, weighted average of returns on the individual securities, the portfolio standard deviation is not the simple, weighted average of individual security standard deviations. To take a weighted average of individual security standard deviations would be to ignore the relationship, or covariance, between the returns on securities. This covariance, however, does not affect the portfolio's expected return.
Covariance is a statistical measure of the degree to which two variables (e.g., securities returns) move together. Positive covariance shows that, on average, the two variables move together. Negative covariance suggests that, on average, the two variables move in opposite directions. Zero covariance means that the two variables show no tendency to vary together in either a positive or negative linear fashion. Covariance between security returns complicates our calculation of portfolio standard deviation. Still, this dark cloud of mathematical complexity contains a silver lining - covariance between securities provides for the possibility of eliminating some risk without reducing potential return.
The calculation of a portfolio's standard deviation, op, is complicated and requires illustration. We therefore address it in detail in Appendix A at the end of this chapter. As explained in Appendix A, for a large portfolio the standard deviation depends primarily on the "weighted" «»variances among securities. The "weights" refer to the proportion of funds invested in each security, and the covariances are those determined between security returns for all pairwise combinations of securities.
An understanding of what goes into the determination of a portfolio's standard deviation leads to a startling conclusion. The riskiness of a portfolio depends much more on the paired security covariances than on the riskiness (standard deviations) of the separate security holdings. This means that a combination of individually risky securities could still constitute a moderate- to low-risk portfolio as long as securities do not move in lockstep with each other. In short, low covariances lead to low portfolio risk.