Z-score#
Introduction#
A measure of bank insolvency risk, defined as:
where \(\text{ROA}\) is the bank’s ROA, \(\text{CAR}\) is the bank’s capital ratio, and \(\sigma_{\text{ROA}}\) is the standard deviation of bank ROA.
The rationale behind Z-score is simple. A bank is insolvent when its loss \(-\pi\) exceeds equity \(E\), i.e., \(-\pi>E\). The probability of insolvency is \(P(-\pi>E)\).
If bank assets is \(A\), then \(P(-\pi>E)=P(-\frac{\pi}{A}>\frac{E}{A})=P(-ROA>CAR)\).
Assuming profits are normally distributed, then scaling \((\text{ROA}+\text{CAR})\) by \(\sigma_{\text{ROA}}\) yields an estimate of the distance to insolvency.
A higher Z-score implies that larger shocks to profitability are required to cause the losses to exceed bank equity.
References#
Laeven and Levine (2009), Bank governance, regulation and risk taking, Journal of Financial Economics, 93, 2, 259-275.
Houston, Lin, Lin and Ma (2010), Creditor rights, information sharing, and bank risk taking, Journal of Financial Economics, 96, 3, 485-512.
Beck, De Jonghe, and Schepens (2013), Bank competition and stability: cross-country heterogeneity, Journal of Financial Intermediation, 22, 2, 218-244.
Delis, Hasan, and Tsionas (2014), The risk of financial intermediaries, Journal of Banking & Finance, 44, 1-12.
Fang, Hasan, and Marton (2014), Institutional development and bank stability: Evidence from transition countries, Journal of Banking & Finance, 39, 160-176.
API#
- frds.measures.z_score(roa: float, capital_ratio: float, past_roas: ndarray) float[source]#
-
- Parameters:
roa (float) – the current bank ROA.
capital_ratio (float) – the current bank equity to asset ratio.
past_roas (np.ndarray) –
(n_periods,)array of past bank ROAs used to calculate the standard deviation.
- Returns:
The bank’s Z-score
- Return type:
float
Examples#
>>> from frds.measures import z_score
>>> import numpy as np
>>> roas = np.array([0.1,0.2,0.15,0.18,0.2])
>>> z_score.estimate(roa=0.2, capital_ratio=0.5, past_roas=roas)
18.549962900111296