  Measures of concordance have been widely used in insurance and risk
management to summarize non-linear dependence among risks modeled by random
variables, 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. We
characterize a whole class of such concordance measures arising from
correlations 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 such
measures are studied, that is, whether a given square matrix of such measures
of concordance can be realized for some random vector and how such a random
vector can be constructed. Compatibility and attainability of block matrices
and hierarchical matrices are also studied due to their practical importance in
insurance and risk management. In particular, a subclass of attainable block
Spearman's rho matrices is proposed to compensate for the drawback that
Spearman's rho matrices are in general not attainable for dimensions larger
than four. Another result concerns a novel analytical form of the Cholesky
factor of block matrices which allows one, for example, to construct random
vectors with given block matrices of van der Waerden's coefficient.
