An alternative Z-score measure for downside bank insolvency risk

ABSTRACT

We derive a Z-score measure reflecting downside bank insolvency risk, drawing on a Chebyshev inequality in terms of the lower semivariance. As then illustrated empirically for US banks, this may provide a useful alternative, or robustness check, to the more commonly used Z-score measure based on the standard Chebyshev inequality.

KEYWORDS:

• Bank
• insolvency risk
• z-score
• downside risk
• semivariance

JEL CLASSIFICATION:

• G21
• G28

Disclosure Statement

No potential conflict of interest was reported by the authors.

Notes

¹ For some recent papers using this methodology, see e.g. Pino and Sharma (20l9), Caiazza et al. (20l8), Berger et al. (20l6), Doumpos, Gaganis, and Pasiouras (20l5), Vazquez and Federico (20l5), Hakenes et al. (20l4), Berger, Goulding, and Rice (20l4), Delis, Hasan, and Tsionas (20l4), Fang, Hasan, and Marton (20l4), Fu, Lin, and Molyneux (20l4).

² The standard Chebyshev inequality states that for a random variable $X$ with finite mean $\mu$ and variance ${\sigma }^2$, it holds for any $k>0$ that $P\left\{\left|X-\mu \right|\ge k\right\}\le {\sigma }^2/k^2$ (see e.g. Ross l997, p. 396).

³ See also the Appendix for an analogous comparison of corresponding probability bounds when using an alternative interpretation of the traditional Z-score $Z$ that draws on the one-sided Chebyshev inequality instead (see Lepetit and Strobel 2015).

⁴ Stata code for the calculation of these measures is available from the authors on request.

⁵ For a discussion of different approaches to the construction of time-varying Z-score measures see Lepetit and Strobel (20l3).

⁶ The one-sided Chebyshev inequality states that for a random variable $X$ with finite mean $\mu$ and variance ${\sigma }^2$, it holds for any $a>0$ that $P\left\{X\le \mu -a\right\}\le \frac{{\sigma }^2}{{\sigma }^2+a^2}$ (see Ross l997, p. 414, or previously, Feller 1971, 152).