Measuring correlation between random variables when they are not normally distributed?

I want to perform some analysis on portfolio that consists of hedge funds (thus fund of hedge funds) In particular, I want to know the relationship between the funds during the downmarket.

The problem complicating this analysis is that hedge funds are not normally distributed. In fact, they are normally highly skewed and have fat tails. If they are normally distributed, then I could just use their pearson correlation coefficient. Since they are not, I think I have to use some sort of skewness, kurtosis, etc. measures.

How would I be performing the analysis if the underlying funds are not normally distributed?

• Pearson correlation is defined for arbitrary continuous bivariate random variables. Normality is not required to compute or use Pearson. (Standard errors based on Fisher's 'r-z' transform do lean heavily on normality assumptions, but that is perhaps only tangential here.) – steveo'america Sep 10 '18 at 23:04
• You may be interested in Atillio Meucci's "Fully flexible probabilities" where you can do things like correlations weighting observations based on other conditioning information. A weighting scheme that may be interesting here is one based on the VIX. – user25064 Sep 11 '18 at 13:34

You should probably look into Poon, Rockinger and Tawn (2003). In particular check how they build the $\chi$ and $\bar{\chi}$ measures of correlation which account for extreme events in up or down markets.
From their paper: "The conventional dependence measure, the Pearson correlation $\rho$, is constructed as an average of deviations from the mean. It makes no distinction between large and small realizations, and it does not distinguish between positive and negative returns. It assumes a linear relationship and a multivariate Gaussian distribution, which might lead to a significant underestimation of the risk from joint extreme events. Here we illustrate how two distribution-free dependence measures, $\chi$ and $\bar{\chi}$ , may be used to identify the type of extremal dependence structure"