Measuring Tail Dependence in Non-Normal Hedge Fund Returns
Summary
The document considers how to measure relationships between hedge fund returns when distributions are skewed and heavy-tailed, with particular interest in how funds behave together during market declines. It explains why Pearson correlation may be insufficient: it summarizes linear co-movement around average returns and can miss dependence concentrated in extreme outcomes.
One suggested approach is to use the chi and chi-bar measures from extreme value analysis to characterize dependence in upper or lower tails. Another is rank correlation, which does not require a linear relationship, while a t-copula can be used to examine tail dependence through its degrees of freedom. These are methodological suggestions rather than an empirical comparison: the document reports no fund data or results. The appropriate measure depends on whether the analysis targets general monotonic association or joint extremes, and the excerpt does not provide implementation details or discuss estimation uncertainty.
Key ideas
- Pearson correlation can understate dependence in joint extreme returns.
- Chi and chi-bar measures can describe dependence in distribution tails.
- Rank correlation offers a measure of association that does not rely on linearity.
- A t-copula provides one framework for examining tail dependence.
- The document gives candidate methods but no empirical results or implementation guidance.
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Full text
# Measuring correlation between random variables when they are not normally distributed?
# 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?
## Answer by phdstudent (score 2, accepted)
https://quant.stackexchange.com/a/41650
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"
## Answer by xiaomy (score 2)
https://quant.stackexchange.com/a/41668
You can use rank correlation in lieu of Pearson correlation to remove that linearity basis. And if tail dependence is of particular interest, one way to look at it is using a t-copula and check the degrees of freedom.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.