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¶

>>> 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

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.

Bug report | Sponsor me