The Unlucky Investor's Guide to Options Trading. Julia Spina

      Covariance measures the direction of the linear relationship between two variables, but it does not give a clear notion of the strength of that relationship. Because the covariance between two variables is specific to the scale of those variables, the covariances between two sets of pairs are not comparable. Correlation, however, is a normalized covariance that indicates the direction and strength of the linear relationship, and it is also invariant to scale. For signals with standard deviations and covariance , the correlation coefficient ρ (rho) is given by the following:

      (1.20)

The correlation coefficient ranges from –1 to 1, with 1 corresponding to a perfect positive linear relationship, –1 corresponding to a perfect negative linear relationship, and 0 corresponding to no measured linear relationship. Revisiting the example pairs shown in Figure 1.9, the strength of the linear relationship in each case can now be evaluated and compared.

      For Figure 1.9(a), QQQ returns versus SPY returns, the correlation between these assets is 0.88, indicating a strong, positive linear relationship. For Figure 1.9(b), TLT returns versus SPY returns, the correlation between these assets is –0.43, indicating a moderate, negative linear relationship. And for Figure 1.9(c), GLD returns versus SPY returns, the correlation between these assets is 0.00, indicating no measurable linear relationship between these variables. According to the correlation values for the pairs shown, the strongest linear relationship is between SPY and QQQ because the magnitude of the correlation coefficient is largest.

      The correlation coefficient plays a huge role in portfolio construction, particularly from a risk management perspective. Correlation quantifies the relationship between the directional tendencies of two assets. If portfolio assets have highly correlated returns (either positively or negatively), the portfolio is highly exposed to directional risk. To understand how correlation impacts risk, consider the additive property of variance. For two random variables with individual variances and covariance , the combined variance is given by the following:

      (1.21)

      When combining two assets, the overall impact on the uncertainty of the portfolio depends on the uncertainties of the individual assets as well as the covariance between them. Therefore, for every new position that occupies additional portfolio capital, the covariance will increase portfolio uncertainty (high correlation), have little effect on portfolio uncertainty (correlation near zero), or reduce portfolio uncertainty (negative correlation).