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.
API¶
estimate(roa, capital_ratio, past_roas)
¶
Z-score
Parameters:
Name | Type | Description | Default |
---|---|---|---|
roa | float | the current bank ROA. | required |
capital_ratio | float | the current bank equity to asset ratio. | required |
past_roas | np.ndarray | (n_periods,) array of past bank ROAs used to calculate the standard deviation. | required |
Returns:
Name | Type | Description |
---|---|---|
float | float | The bank's Z-score |
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
Source code in src/frds/measures/z_score.py
def estimate(roa: float, capital_ratio: float, past_roas: np.ndarray) -> float:
r"""Z-score
Args:
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:
float: The bank's Z-score
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
"""
return (roa + capital_ratio) / np.std(past_roas)
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.