# Preference between low (zero) and negative correlation

I am trying to create an artificial score grading user's portfolio correlation. In terms of diversification, lower correlation is obviously better. However, should negative correlation get a higher score than correlation close to zero?

• It is not entirely clear what you mean. Take two random variables figuring e.g. individual strategy returns $X_1$ and $X_2$. Suppose you invest the same notional in both such that the resulting global strategy return is proportional to $X = X_1 + X_2$. Then the variance of the global strategy is $\text{var}(X) = \text{var}(X_1) + \text{var}(X_2) + 2\text{cov}(X_1, X_2)$. If what you are trying to achieve is lower variance through diversification then clearly the former equation shows that a negative covariance (hence correlation) is what you are looking for. The more negative the better. – Quantuple Jul 17 '18 at 16:04
• In this particular case I would like to somehow score diversification, so from delsim's answer and yours I figure out that the absolute value should be close to zero for it. Thanks! – abu Jul 17 '18 at 21:18
• Are you scoring this strategy ex ante or ex post? – Dave Harris Jul 18 '18 at 6:07
• I am scoring it ex post. – abu Jul 18 '18 at 6:09
• @DaveHarris is there a different approach if I am doing this ex post? – abu Jul 18 '18 at 7:14