# Z-score¶

## Introduction¶

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.

## 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)