Converting Kendall’s Tau Before Building a Covariance Matrix

Summary

The document asks whether Kendall rank correlation can be used directly in place of Pearson correlation when constructing a covariance matrix by scaling correlations with return standard deviations. The response offers a conversion from Kendall’s tau to Pearson correlation under a bivariate normal copula assumption: apply the sine transformation involving half of pi times tau, then use the resulting correlation with the standard deviations.

The key caveat is that this relationship depends on the assumed copula. The answer does not establish that the conversion applies to ordinary correlation matrices in general, nor does it provide a procedure for ensuring a converted matrix is valid for portfolio or risk calculations. Thus, Kendall’s tau should not simply be treated as interchangeable with Pearson correlation when forming covariances; the distributional assumptions and matrix-level implications need consideration. The exchange is a concise pointer to the relevant mathematical relationship, not a full treatment of robust covariance estimation or empirical validation.

Key ideas

• Kendall’s tau is not automatically interchangeable with Pearson correlation in a covariance matrix.
• Under a bivariate normal copula assumption, a sine transformation relates tau to Pearson correlation.
• The conversion depends on the copula assumption and may not apply to general correlation matrices.
• Covariance construction also requires considering whether the resulting matrix is appropriate for the intended analysis.

Tags

# Using Kendall rank correlation to construct a covariance matrix?


# Using Kendall rank correlation to construct a covariance matrix?












I am wondering if it's mathematically 'correct' to employ a correlation matrix based on Kendall-correlation when constructing a covariance matrix?

I.e., is it wrong to multiply standard deviations of e.g. returns with the Kendall-correlation-matrix to form a covariance matrix?

That is, I am only changing the correlation estimates to be based on Kendall's tau instead of the standard Pearson linear correlation coefficient when 'constructing' my covariance matrix.

## Answer by simzoor (score 3)

https://quant.stackexchange.com/a/47140

A first hint:

To convert Kendall's $\tau$ to the Pearson correlation coefficient $\rho$, one could use the relationship: $$\rho = \sin\Bigl(\frac{\pi}{2}\tau\Bigr)$$

But keep in mind that this only holds for the bivariate normal copula assumption, I don't know if this also holds to convert "plain vanilla" correlation matrices, see McNeil et al. (2005: Proposition 5.29).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.