Bear correlation is the correlation between two marginal asset distributions during the first asset negative returns.

Instead of computing a classical correlation between an asset and an index, it is better to compute the correlation during marginal asset distributions or conditional asset distributions. This will capture both assets non-normalities and co-dependence asymmetries. If the conditional correclation is high than the classical correlation, then the depndence between the asset the index is unstable and asset's extreme risks should be measured.

Correlation matrix between hedge funds and indices:

Bear correlation matrix: when the numbers below are higher than the number above, the assets are more correlated during equity market negative returns:

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