Estimate 2-year probit bankruptcy probability, leverage pressure, and insolvency default thresholds across 3 core accounting metrics.
| Factor | Metric Description | Raw Ratio | Coefficient | Term Contribution |
|---|
Model Characteristics: Zmijewski's model is prized in quantitative hedge funds and banking risk departments for its elegance: with just three financial ratios, it achieves predictive accuracy comparable to models with twice as many inputs while avoiding collinearity and overfitting.
Simulated 2-year probit default probability (%) across varied liabilities and bottom-line earnings.
| Liabilities \ Net Income | -$400M NI | -$200M NI | Base NI | +$200M NI | +$400M NI |
|---|
Zmijewski used probit analysis, which assumes an unobservable latent variable \(X^*\) driven by underlying financial indicators. The observable bankruptcy event occurs when \(X^* > 0\):
Where \(\Phi(X)\) represents the standard normal cumulative distribution function.
Calibrated on a matched sample of 40 bankrupt and 800 non-bankrupt industrial manufacturing firms:
Where \(\text{ROA} = \text{Net Income} / \text{Total Assets}\), \(\text{FINL} = \text{Total Liabilities} / \text{Total Assets}\), and \(\text{LIQ} = \text{Current Assets} / \text{Current Liabilities}\).
Prior studies (Altman 1968, Deakin 1972) paired 1 bankrupt firm with 1 non-bankrupt peer (50/50 sample composition), introducing severe econometric estimation bias because real-world corporate bankruptcies represent less than 1% to 2% of the population. Mark Zmijewski introduced weighted exogenous sampling maximum likelihood (WESML) estimators to produce unbiased parameters.
Test your mastery of Zmijewski probit modeling, default probabilities, and credit distress screening.
1. What statistical probability function does the Zmijewski model utilize to calculate default probability?
2. What critical X-Score value corresponds to an exact 50% probability of bankruptcy (Φ(0))?
3. Which ratio in the Zmijewski model has the highest positive coefficient (+5.679), representing the strongest driver of bankruptcy?
4. What major econometric contribution did Mark Zmijewski's 1984 research introduce to bankruptcy literature?
Developed by Mark E. Zmijewski in 1984, the Zmijewski X-Score is a parsimonious probit regression model that estimates the probability of bankruptcy using three core financial ratios: Return on Assets (ROA), Financial Leverage (Total Liabilities / Total Assets), and Liquidity (Current Assets / Current Liabilities).
Unlike the Altman Z-Score's linear index or the Ohlson O-Score's logit function, Zmijewski uses a probit function. The calculated X-Score is passed into the cumulative standard normal distribution function: P(Default) = Phi(X). A score of X = 0 corresponds to exactly a 50% probability of bankruptcy.
Zmijewski published his model in the Journal of Accounting Research to correct choice-based sampling bias and sample selection bias that plagued prior distress studies (such as Altman 1968), where bankrupt companies were artificially oversampled relative to their actual frequency in the broader economy.
A company with an X-Score greater than 0 has a default probability exceeding 50% and is considered in the High Distress Zone. An X-Score between -0.842 and 0 corresponds to default probabilities between 20% and 50% (Watchlist / Grey Zone), while scores below -0.842 indicate safe and solvent operations.
Yes. You can export complete balance sheet inputs, 3-factor term decompositions, probit default probabilities, and the 5x5 leverage vs. profitability sensitivity table as a formula-protected CSV spreadsheet.