# 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

If you want diversification, then closer to zero is what you want.

Trivially, a positive correlation (+ x) with respect to some reference portfolio can be reversed (- x) by just shorting the reference portfolio.

• So to quantify that, the closer the absolute portfolio correlation to zero the better diversified the portfolio? – abu Jul 17 '18 at 15:49
• If you define diversified as uncorrelated, then you want the correlation to be zero. That tends to be what one looks for when trying to combine multiple trading strategies. However, if you want to minimize the variance (without taking anything else into consideration) then you need to have some additional constraint to avoid favoring a portfolio with a correlation of -1, as this is just telling you to remove all risk through a flat position. – delsim Jul 17 '18 at 16:16
• This is what I would like to achieve, with the assumption that diversified is uncorrelated. Thanks! – abu Jul 17 '18 at 21:19