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Measures of concordance have been widely used in insurance and riskmanagement to summarize non-linear dependence among risks modeled by randomvariables, which Pearson's correlation coefficient cannot capture. However,popular measures of concordance, such as Spearman's rho and Blomqvist's beta,appear as classical correlations of transformed random variables. Wecharacterize a whole class of such concordance measures arising fromcorrelations of transformed random variables, which includes Spearman's rho,Blomqvist's beta and van der Waerden's coefficient as special cases.Compatibility and attainability of square matrices with entries given by suchmeasures are studied, that is, whether a given square matrix of such measuresof concordance can be realized for some random vector and how such a randomvector can be constructed. Compatibility and attainability of block matricesand hierarchical matrices are also studied due to their practical importance ininsurance and risk management. In particular, a subclass of attainable blockSpearman's rho matrices is proposed to compensate for the drawback thatSpearman's rho matrices are in general not attainable for dimensions largerthan four. Another result concerns a novel analytical form of the Choleskyfactor of block matrices which allows one, for example, to construct randomvectors with given block matrices of van der Waerden's coefficient.

Compatibility and attainability of matrices of correlation-based measures of concordance