What a Bank Z-Score Can and Cannot Tell Us: A Distribution-Free Failure Bound, the Small-Sample Bias of Estimated Z, and Evidence From 162 Banking Systems and India, 2000–2021

The bank Z-score, (ROA + equity/assets)/sd(ROA), is the most widely used summary of banking-system insolvency risk. It counts how many standard deviations of return on assets a bank's buffer can absorb. I prove two results that govern its interpretation. First, without any distributional assumption, the probability that losses exhaust the buffer is at most 1/(1+Z²) (the one-sided Chebyshev or Cantelli bound), and this bound is attained. At Z = 5 the bound is 3.8%, while a normal model gives 2.9 in ten million; a Student-t(4) return distribution gives 0.1%. Second, when the standard deviation is estimated from T years of normal returns, the estimated Z overstates the true Z on average by the exact factor c(T) = √((T−1)/2)·Γ((T−2)/2)/Γ((T−1)/2): 25% at T = 5 and 9% at T = 10. Using the World Bank Global Financial Development Database (last updated September 2022), the median Z across 162 banking systems in their latest year (2017–2021) is 15.34; India's is 19.44, rank 109 of 162. India's Z rose from 12.89 in 2000 to 19.00 in 2020 mainly because capital buffers grew, even while ROA turned negative in 2017–2018. In a panel of 129 economies, a 10% lower Z goes with a 0.22-point larger rise in the NPL ratio over the next three years, with country and year fixed effects.