Zmijewski Score: A Three-Ratio Probit Check on Financial Distress

The Zmijewski score is the shortest of the classic bankruptcy models. It asks three questions about a company: is it making money on its assets, how much of those assets are funded by liabilities, and can its current assets cover its current dues? Mark Zmijewski published it in 1984 in the Journal of Accounting Research, in a paper titled “Methodological Issues Related to the Estimation of Financial Distress Prediction Models”. If you have a balance sheet and a profit and loss statement open, you can work it out in about two minutes.

That speed is the appeal. The catch, as we’ll see, is that a quick number is not the same as a reliable one, especially when the model is applied to Indian companies decades after it was built.

Where the Zmijewski score comes from

Zmijewski’s paper was as much about method as about prediction. As a University of Twente master’s thesis by Bouwmeester summarises it, he looked at two biases that creep into bankruptcy studies. The first is choice-based sample bias: bankruptcies are rare, so researchers tend to stuff their samples with far more failed firms than exist in real life. The second is sample selection bias: failing companies often have missing data, and if you only keep firms with complete numbers, the firms you drop are not random.

To build the model he used firms listed on the American and New York stock exchanges between 1972 and 1978, leaving out financial, service and public administration businesses. A firm counted as bankrupt if it filed a bankruptcy petition in that period. The estimation sample held 40 bankrupt and 800 non-bankrupt firms, with a separate prediction sample of 41 bankrupt and 800 non-bankrupt firms. That 40-to-800 mix is far closer to real life than the 50-50 samples many earlier studies used.

He fitted the model with probit regression. If that sounds technical, the idea is simple. Instead of producing a score you compare with fixed zones, like the Altman Z-score does, a probit model produces a number that converts into a probability between 0 and 1. The Ohlson O-score does something similar with a logistic function; probit uses the cumulative normal distribution instead.

The Zmijewski score formula

The version reproduced in the Bouwmeester thesis is:

X = −4.336 − 4.513 × (Net income / Total assets) + 5.679 × (Total liabilities / Total assets) − 0.004 × (Current assets / Current liabilities)

Some papers round these to −4.3, −4.5, 5.7 and 0.004, which is how the Indian re-estimation study in Financial Innovation prints them. The thesis labels the second ratio total debt to total assets, while the Indian paper uses total liabilities to total assets. For a quick screen, use total liabilities so you catch everything the company owes.

Once you have X, you convert it to a probability using the standard normal distribution. In Excel or Google Sheets that’s =NORM.S.DIST(X, TRUE). The cut-off is 0.5: a probability of 0.5 or more puts the firm in the bankrupt group, below 0.5 in the healthy group. Because the normal distribution is centred on zero, a probability of 0.5 is the same as X = 0. So in practice you can read the raw score this way: negative means the model leans safe, positive means it leans distressed.

Debt does almost all the work in the Zmijewski score. The liquidity term carries a coefficient of just 0.004, so even a big change in the current ratio barely moves the answer.

What each ratio is really telling you

Return on assets (net income / total assets)

This is the profitability check. The coefficient is negative, so higher profits push the score down, towards safety. A loss pushes it up. A company earning 6% on its assets knocks about 0.27 off its score; one losing 5% adds about 0.23.

Leverage (total liabilities / total assets)

This is the heavyweight. With a coefficient of 5.679, every 10 percentage points of extra leverage adds about 0.57 to X. Work out the break-even and it’s striking: for a company with zero profit, the score crosses zero when liabilities reach roughly 76% of assets (4.336 divided by 5.679). That single ratio decides most outcomes.

Liquidity (current assets / current liabilities)

This is the current ratio, and its weight is tiny. Doubling a current ratio from 1.6 to 3.2 changes X by less than 0.01. We wouldn’t ignore liquidity when studying a company, but you should know this model mostly does.

Worked example 1: a steady manufacturer

Take a hypothetical mid-sized Indian manufacturer. All figures are made up for illustration.

• Total assets: ₹1,000 crore
• Net income: ₹60 crore
• Total liabilities: ₹450 crore
• Current assets: ₹400 crore; current liabilities: ₹250 crore

The ratios are 0.06, 0.45 and 1.6. Plug them in:

X = −4.336 − (4.513 × 0.06) + (5.679 × 0.45) − (0.004 × 1.6) = −4.336 − 0.271 + 2.556 − 0.006 = −2.058

NORM.S.DIST(−2.058, TRUE) gives about 0.020, or roughly a 2% modelled probability. Comfortably on the safe side.

Worked example 2: a stretched balance sheet

Now a hypothetical company in trouble:

• Total assets: ₹800 crore
• Net loss: ₹40 crore
• Total liabilities: ₹680 crore
• Current assets: ₹200 crore; current liabilities: ₹260 crore

Ratios: −0.05, 0.85 and about 0.77.

X = −4.336 + (4.513 × 0.05) + (5.679 × 0.85) − (0.004 × 0.77) = −4.336 + 0.226 + 4.827 − 0.003 = 0.714

That converts to a probability of about 0.76, well above the 0.5 cut-off. Now try a thought experiment. Suppose the same company somehow doubled its current assets to ₹400 crore, lifting its current ratio to about 1.54. The score only falls to roughly 0.711 and the probability stays at about 0.76. That’s the liquidity coefficient at work, or rather not at work.

Worked example 3: finding the edge

Take a break-even company with zero profit and a current ratio of 1. At 70% leverage, X is about −0.365, a probability of roughly 0.36. Push leverage to 80% and X becomes about 0.203, a probability of roughly 0.58. The firm has crossed the line without its profits or liquidity changing at all. If you track a company over several quarters, watching which side of that 76% leverage mark it sits on will tell you most of what this model will say.

How well does it predict?

According to the Bouwmeester thesis, Zmijewski’s model classified 99% of the estimation sample correctly, and the paper didn’t report a hold-out accuracy. That headline needs care. With 800 healthy firms and only 40 failures in the sample, a model can score very high overall just by calling most companies healthy.

That problem shows up when the model leaves home. When the thesis tested it on Dutch and Belgian public and large private firms, the Zmijewski model correctly identified 38.03% of the bankrupt firms and 100% of the non-bankrupt ones. In other words, it rarely raised a false alarm but missed most of the failures. The thesis also found no statistically significant difference between the Zmijewski, Altman and Ohlson models in that sample, and concluded that one fiscal year before the event was the best horizon for prediction.

The Zmijewski score on Indian companies

The study most relevant to us is by Bhanu Pratap Singh and Alok Kumar Mishra, published in Financial Innovation in 2016. They took 208 Indian manufacturing companies, half of them defaulted and half not, with 130 used to estimate the models and 78 held out to test them. Distressed firms were identified from references to the Board for Industrial and Financial Reconstruction (BIFR) among companies registered as sick between 2006 and 2014.

With the original 1984 coefficients, Zmijewski’s model got 71.538% of the estimation sample right. It caught 98.462% of the distressed firms but correctly identified only 44.615% of the healthy ones. On the hold-out sample, overall accuracy was 79.487%, with 97.436% of distressed firms and 61.538% of healthy firms correctly classified. So on Indian data the model behaved the opposite way to the Dutch test: it caught almost every failure but flagged a lot of healthy firms too.

When the authors re-estimated the coefficients on Indian data, the numbers moved a lot. The leverage coefficient fell from 5.7 to 0.586, the profitability coefficient went from −4.5 to −13.797, and the constant changed from −4.3 to −1.222. The current ratio was no longer statistically significant. Accuracy on the estimation sample rose to 89.231%, though hold-out accuracy stayed at 79.487%. The authors’ conclusion was blunt: the coefficients are sensitive to time period and financial conditions, and researchers should recalculate models on recent data before relying on them.

On Indian manufacturing data, the original model caught nearly every company that went sick, but it also flagged more than half of the healthy ones in the estimation sample.

Using the Zmijewski score in practice

Here’s how we’d actually use it when reading results.

• Treat it as a leverage alarm. Because the liabilities ratio dominates, a positive score mostly tells you the balance sheet is heavily funded by others. That’s worth knowing, but it’s not a verdict.
• Use annual, audited numbers. The original work and the later tests used yearly accounts. Quarterly profit is noisy, and a single bad quarter can swing return on assets.
• Skip banks, NBFCs and insurers. Zmijewski excluded financial firms, and their balance sheets are built on borrowed money by design. A lender will almost always look “distressed” on this model.
• Cross-check with other models. Run the Altman Z-score and Ohlson O-score alongside it. If all three point the same way, take it seriously. If only Zmijewski is worried, look at what’s driving the leverage.
• Watch the trend. One reading tells you little. A score climbing towards zero over three years tells you much more.

For the broader process of reading a company’s numbers before you get to models like this, our guide on how to analyse quarterly results of a company walks through the basics.

Limits you should keep in mind

The model was built on US industrial companies from the 1970s. Accounting rules, interest rates, and the way Indian companies fund themselves are all different now. Its liquidity term is so small that a company sliding into a cash crunch can look no worse on the score. And it says nothing about promoter pledges, related-party dealings or contingent liabilities, which are often where trouble in Indian companies first shows up.

There’s also the definition problem. Zmijewski defined distress as filing for bankruptcy, so the model predicts a legal event, not general weakness. A company can be in bad shape for years without filing.

The bottom line

The Zmijewski score is a fast, three-ratio test that mostly measures how much of a company’s assets are funded by liabilities, adjusted a little for profitability. It’s useful as a first filter and as one input alongside other distress models. Use it to decide which balance sheets need a closer look, not to decide which stocks to sell. The Indian evidence suggests the original coefficients flag too many healthy firms here, so a positive score is a reason to dig deeper, not proof of anything.

For background on the wider field, the Wikipedia overview of bankruptcy prediction lists the main models and the original Zmijewski citation.

Sources: Zmijewski (1984), Journal of Accounting Research; Bouwmeester, University of Twente master’s thesis; Singh and Mishra (2016), Financial Innovation. All company figures in the worked examples are hypothetical.

Disclaimer: This article is for information and education only and is not investment advice. Please consult a SEBI-registered adviser before making any investment decision.