Up- and Down-Correlations in Normal Variance Mixture Models

We study conditional correlations between pairs of risks in normal variance mixture models, which are widely used in risk management and finance. In particular, we examine up- and down-correlations defined as the conditional correlation between the sum of risks and an individual component, conditionally on the sum being large or small, respectively. We provide a characterization of conditional covariances and use this to derive explicit formulas for up- and down-correlations in important cases of the normal variance mixture model such as the normal, the Student-t, the logistic, and the Laplace model. Based on invariance properties of betas, we also provide a graphical test for assessing the validity of a normal variance mixture model