Bank insolvency risk, Z-score measures and unimodal returns: A refinement

We develop refined probability bounds for bank insolvency risk measures based on the Z-score, analogous to those given by Cantelli's inequality under the additional assumption of unimodality of returns, drawing on the one-sided Vysochanskii-Petunin inequality. Illustrating empirically for US banks, we argue that (i) unimodality of returns is not an overly restrictive assumption in this context, and (ii) the refined measures provide a less conservative alternative to insolvency probability bounds drawing on the (two-sided) Vysochanskii-Petunin inequality, particularly for banks with higher levels of insolvency risk.