# Bank Z-score¶

## API¶

function
z_score(roa, capital_ratio, past_roas)

Z-score

A measure of bank insolvency risk, defined as:

\text{Z-score} = \frac{\text{ROA}+\text{CAR}}{\sigma_{\text{ROA}}}

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.

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 (float)

The bank's Z-score

Examples
>>> from frds.measures.bank import z_score
>>> import numpy as np
>>> z_score(roa=0.2, capital_ratio=0.5, past_roas=np.array([0.1,0.2,0.15,0.18,0.2]))
18.549962900111296

References

Last update: August 5, 2021
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