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.
It is commonly believed that there exists a strong negative association between corporate social performance (CSP) and firm risk.1 To investigate the structure of this relationship, we decompose the dynamics of large U.S. company stock returns into two components: Gaussian and non-Gaussian innovations. Our findings indicate that CSP affects firm risk mainly through the non-Gaussian risk channel. In particular, it significantly reduces the magnitude of extreme returns. We find no consistent nor robust effect of CSP on the frequency of price boom and crash probability as it varies widely across industries. Last, we find no statistically significant impact of CSP on standard Gaussian volatility risk.