In this article, I will describe the Altman Z-Score and its variations, a set of financial models developed by Professor Edward Altman to predict the likelihood of a company going bankrupt. The original Z-Score model, introduced in 1968, combines five key financial ratios to assess a company's financial health and provide investors with an indicator of potential bankruptcy risk. Subsequent adaptations, such as the Z'-Score for private companies (1998) and the Z"-Score for non-manufacturing and emerging firms (1995), have extended the model's applicability.

This article will also highlight two proprietary models, the ZETA Credit Risk Model (1977) and the Altman Z-Score Plus (2012), which incorporate additional variables and industry-specific factors to improve predictive accuracy. While these models are only available to subscribers and are not fully disclosed to the public, they represent important advancements in bankruptcy prediction.

This article will explain each model in detail, including the variables used, calculations, and interpretation of results. Real-world examples and a free Excel template will illustrate the practical application of the Z-Score models. The article will also discuss Altman's research supporting the models' accuracy and its limitations, as identified by studies that propose improvements, such as incorporating time-varying and industry-specific factors, expanding the sample, and carefully selecting prediction variables.

## Altman Z-Score Explained

The **Altman Z-Score** is a financial model created in 1968 by Edward I. Altman, a professor at New York University's (NYU) Stern School of Business. The model aims to evaluate a company's creditworthiness and predict the likelihood of bankruptcy within two years. By combining five key financial ratios, each with different weights, the Z-Score generates a single score that reflects a company's financial health, including its operational strength, liquidity, solvency, profit margins, and leverage.

It's important to note that the Altman Z-Score is not related to the statistical concept of the z-score, which measures the number of standard deviations an observation is from the mean of a distribution.

Since its introduction in 1968, the Altman Z-Score has undergone several revisions to accommodate different types of companies and/or improve its predictive power. While the original estimation was based on data from publicly held manufacturers, the Z-Score has since been modified after further research from Altman, including:

**Z-Score (1968)**: U.S. public manufacturing companies.**Z'-Score (1983)**: U.S. private manufacturing companies.**Z"-Score (1995)**: U.S. non-manufacturing and emerging firms (both public and private).**ZETA Credit Risk Model (1977) and Altman Z-Score Plus (2012)**: Proprietary risk models that incorporate additional variables and industry-specific factors to improve predictive accuracy. These models are only available to subscribers and not fully disclosed to the public.

These revisions have varied the weights assigned to each component and adjusted the cutoff ranges for interpreting the resulting score, as discussed in the following sections.

## Altman Z-Score for Public U.S. Manufacturing

In 1968, Edward I. Altman published a paper titled "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy," in which he developed a model to predict corporate bankruptcy using a combination of financial ratios and multiple discriminant analysis (MDA).

"**Discriminant analysis**" is a statistical method that classifies observations into predefined groups by considering multiple variables simultaneously. Altman found that this approach increases the statistical significance of findings compared to univariate analysis, which focuses on a single predictor.

The Altman Z-Score was derived from a study of 66 publicly traded manufacturing companies in the U.S., equally divided between those that had filed for bankruptcy under Chapter X of the National Bankruptcy Act between 1946-1965 and those that had not. To ensure comparability, Altman restricted the sample to manufacturing firms in the U.S. with total assets ranging from $1-$25 million.

### Altman Z-Score Formula

The original Altman Z-Score formula presented in the 1968 paper is as follows:

Z-Score [Original; Public U.S. Manufacturing] = (0.012 Ã— X_{1}) + (0.014 Ã— X_{2}) + (0.033 Ã— X_{3}) + (0.006 Ã— X_{4}) + (0.999 Ã— X_{5})

**where**:

- X
_{1}= Working Capital / Total Assets - X
_{2}= Retained Earnings / Total Assets - X
_{3}= Earnings Before Interest and Taxes (EBIT) / Total Assets - X
_{4}= Market Value of Equity / Total Liabilities - X
_{5}= Sales / Total Assets

Over time, many investors have modified the Altman Z-Score formula to make it more user-friendly:

Z-Score [Modified; Public U.S. Manufacturing] = (1.2 Ã— X_{1}) + (1.4 Ã— X_{2}) + (3.3 Ã— X_{3}) + (0.6 Ã— X_{4}) + (1.0 Ã— X_{5})

In this version, the first four variables (X_{1}-X_{4}) are entered as decimals (e.g., 0.25 for 25%), and the coefficient for X_{5} is rounded to 1.0 from 0.999. The X_{5} variable is still entered as the number of times (because it's a multiple), not as a percentage.

### Five X-Variables in the Altman Z-Score Explained

Below, we discuss the five X-variables in the Altman Z-Score, including their specific formulas, interpretations, and various nuances.

#### X_{1}: Working Capital / Total Assets (WC/TA)

The **Working Capital to Total Assets** (WC/TA) ratio provides insight into a company's short-term financial health and liquidity:

WC/TA Ratio = Working Capital / Total Assets

**where**:

- Working Capital = Current Assets - Current Liabilities

Here's how a higher/lower WC/TA ratio can be interpreted:

**Higher WC/TA Ratio**: Indicates better liquidity and a greater ability to meet short-term obligations. However, high positive working capital may suggest excessive inventory or underutilized cash, indicating that the company is not investing its excess cash efficiently.**Lower WC/TA Ratio**: May indicate that the firm is struggling to pay its bills, although negative working capital isn't always a bad sign, especially for companies with high inventory turnover. These companies can leverage their suppliers with favorable payment terms, allowing their current liabilities to outweigh their current assets.

Typically, a firm experiencing consistent operating losses will see a decrease in current assets relative to total assets. Altman found this ratio to be the most valuable among the three liquidity ratios he evaluated, with the current ratio and quick ratio being less useful and sometimes showing misleading trends for failing firms.

#### X_{2}: Retained Earnings / Total Assets (RE/TA)

The **Retained Earnings to Total Assets** (RE/TA) ratio measures the proportion of a company's assets financed through retained earnings rather than debt:

RE/TA Ratio = Retained Earnings / Total Assets

Here's how a higher/lower RE/TA ratio can be interpreted:

**Higher RE/TA Ratio**: Suggests a history of profitability, less reliance on borrowing, and the ability to fund operations and growth using accumulated profits. Firms with high RE, relative to TA, have financed their assets through retention of profits and have not used as much debt.**Lower RE/TA Ratio**: Implies that companies are more dependent on debt financing or dilution to fund their assets, as they have little to no retained earnings to continue operations.

Retained earnings represent the percentage of net earnings not paid out as dividends and can be used to operate the business, reinvest, or pay off debt. This ratio also indirectly considers the age of a firm, as younger firms tend to have lower RE/TA ratios due to having less time to accumulate profits. Consequently, younger firms may be more likely to be classified as bankrupt in this analysis, which aligns with Altman's observation that failure rates are higher for younger companies.

In fact, according to a Dun & Bradstreet (1994) study cited by Altman in his 2012 paper "Predicting Financial Distress of Companies - Revisiting the Z-Score and ZETA Models," approximately 50 percent of all firms that failed in 1993 did so within their first five years of existence.

It's important to note that Altman mentions the RE/TA ratio can be affected by corporate actions such as quasi-reorganizations (i.e., a method used by companies to eliminate accumulated deficits and restructure their balance sheets without having to file for bankruptcy) and stock dividend declarations. These actions can potentially bias the ratio, and appropriate adjustments should be made to the accounts when such events occur.

#### X_{3}: Earnings Before Interest and Taxes (EBIT) / Total Assets (EBIT/TA)

The **Earnings Before Interest and Taxes (EBIT) to Total Assets** (EBIT/TA) ratio assesses a company's ability to generate operating profits from its assets before considering interest and taxes:

EBIT/TA Ratio = EBIT / Total Assets

Here's how a higher/lower EBIT/TA ratio can be interpreted:

**Higher EBIT/TA Ratio**: Indicates greater profitability and more efficient asset utilization.**Lower EBIT/TA Ratio**: Suggests lower profitability and less efficient use of assets, which may signal potential financial distress.

The EBIT/TA ratio is similar to return on assets (ROA) but uses EBIT instead of net income in the numerator. The ratio measures the true productivity of a firm's assets, regardless of tax or leverage factors. In other words, the EBIT/TA ratio assesses the firm's capacity to generate profits exclusively from its operations and fund future growth, measuring how much profit the firm's assets are producing.

Altman found that the EBIT/TA ratio consistently outperforms other profitability measures, including cash flow, making it particularly relevant for studying corporate failure. A company's survival ultimately depends on the earning power of its assets, and insolvency occurs when total liabilities exceed the fair value of the firm's assets, which is determined by the assets' earning power.

#### X_{4}: Market Value of Equity / Total Liabilities (MVE/TL)

The **Market Value of Equity to Total Liabilities** (MVE/TL) ratio assesses how much a company's market value can decline before its liabilities exceed its assets, indicating potential insolvency:

MVE/TL Ratio = Market Value of Equity / Total Liabilities

**where**:

- Market Value of Equity = Stock Price Ã— Shares Outstanding

Here's how a higher/lower MVE/TL ratio can be interpreted:

**Higher MVE/TL Ratio**: Suggests stronger investor confidence in the company's financial health.**Lower MVE/TL Ratio**: Indicates weak investor confidence, as the market value of equity is low compared to liabilities.

The MVE/TL ratio incorporates a market value perspective into the model, reflecting the market's confidence in the company's financial position. Again, this ratio indicates how much the firm's assets (measured by market value of equity plus debt) can decrease in value before liabilities exceed assets, resulting in insolvency.

Here's an example Altman provided in his 1968 paper:

"For example, a company with a market value of its equity of $1,000 and debt of $500 could experience a two-thirds drop in asset value before insolvency. However, the same firm with $250 in equity will be insolvent if its drop is only one-third in value."

- Edward I. Altman in "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy," pg. 595

Altman found this ratio to be a more effective bankruptcy predictor than the more commonly used net worth to total debt (NW/TD) ratio (based on book values).

However, this ratio is somewhat flawed and should be considered with caution by investors, as it relies on market capitalization, which can be influenced by stock price fluctuations. In short, a high stock price can inflate this ratio, making it less reliable.

In later papers, Altman addresses this issue for privately held firms (Z'-Score) and non-manufactures (Z"-Score) by substituting the book value of equity (aka shareholders' equity) for the market value, as we'll discuss further below in this article.

#### X_{5}: Sales / Total Assets (S/TA)

The **Sales to Total Assets** (S/TA) ratio, also known as the asset turnover ratio, evaluates a company's efficiency in using its assets to generate sales:

S/TA Ratio = Sales / Total Assets

Here's how a higher/lower S/TA ratio can be interpreted:

**Higher S/TA Ratio**: Indicates better management of competition, more effective asset utilization, higher profitability, and a smaller reliance on investments to generate future sales.**Lower S/TA Ratio**: Suggests a failure to maintain or grow market share and a higher dependency on resources and investments for future growth.

The S/TA ratio demonstrates the sales generating ability of a firm's assets and provides insight into management's ability to handle competition. Altman discovered that despite being the least significant ratio individually and not appearing significant based on univariate statistical tests, it ranks second in its contribution to the model's overall discriminating ability due to its unique relationship with other variables.

However, it's important to note that asset turnover varies widely among industries, and Altman later introduced the Z"-Score model for non-manufacturers that no longer includes the S/TA ratio, as discussed later in this article.

### Altman Z-Score Findings

The results of Altman's original 1968 study were notable:

**One Year Before Bankruptcy**: The Altman Z-Score was found to be**95%**accurate in predicting bankruptcy one year before the event, with a Type II error (false negatives) of**3%**.**Two Years Before Bankruptcy**: When testing the model with data two years prior to bankruptcy, the accuracy dropped to**72%**, with a Type II error of**6%**.

Altman also provided a table in his 1968 paper to demonstrate how the model's accuracy becomes less reliable after the initial two-year horizon:

As Altman himself says:

"It is obvious that the accuracy of the model falls off consistently with the one exception of the fourth and fifth years, when the results are reversed from what would be expected. The most logical reason for this occurrence is that after the second year, the discriminant model becomes unreliable in its predictive ability, and, also, that the change from year to year has little or no meaning."

- Edward I. Altman in "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy," pg. 604

Thus, although the Altman Z-Score demonstrates high accuracy in predicting bankruptcy within a one to two-year timeframe, and may be useful for business credit evaluation, internal control procedures, and investment guidelines (as Altman suggests), investors should be cautious when using the model to assess a company's bankruptcy risk beyond two years, as its predictive power significantly diminishes after this point.

### How to Interpret the Altman Z-Score

In his 1968 paper, Altman established cut-off points for interpreting the Z-Score, which categorize companies into three risk zones:

**Safe Zone (Z-Score > 2.99)**: Companies with a Z-Score above 2.99 are considered to be in a safe financial position and have a low probability of bankruptcy.**Gray Zone (1.81 < Z-Score < 2.99)**: Companies with a Z-Score between 1.81 and 2.99 fall into a gray area, where there is some risk of bankruptcy, but the prediction is less certain. This zone is also known as the "zone of ignorance" because of the susceptibility to error classification.**Distress Zone (Z-Score < 1.81)**: Companies with a Z-Score below 1.81 are considered to be in financial distress and have a high probability of bankruptcy within the next two years.

The Altman Z-Score visual below can also be referenced:

Altman determined these cut-off points by analyzing the Z-Scores of the original sample of 66 companies. He observed that all companies with a Z-Score greater than 2.99 were clearly non-bankrupt, while those with a Z-Score below 1.81 were all bankrupt. The gray zone, between 1.81 and 2.99, contained a mix of both bankrupt and non-bankrupt firms.

More recently, in a 2019 lecture titled "50 Years of the Altman Score," Professor Altman noted that based on recent data, a Z-Score closer to 0, not 1.8, is the figure at which investors should worry about a company's financial strength. This suggests that the Altman Z-Score's distress zone has become more lenient over time, possibly due to changes in the business environment and financial reporting practices.

Lastly, it's important to note that these cut-off points are not absolute and may vary depending on the industry or economic conditions. However, they provide a useful starting point for assessing a company's financial health and bankruptcy risk using the Altman Z-Score model.

## Altman Z'-Score for Private U.S. Manufacturing

In 1983, Altman revised the original Z-Score model to create the **Altman Z'-Score**, which accommodates private U.S. manufacturing companies by revising the coefficients of the original formula and replacing the market value of equity with the book value of equity (aka shareholders' equity) in the X_{4} variable.

This revision was published in the first edition book of "Corporate Financial Distress: A Complete Guide to Predicting, Avoiding, and Dealing with Bankruptcy." The study examined a sample of 66 private U.S. manufacturing firms, with 33 firms in each of the bankrupt and non-bankrupt groups (similar to the original model).

### Altman Z'-Score Formula

The revised Altman Z'-Score formula for private U.S. manufacturing companies is shown below:

Z'-Score [Private U.S. Manufacturing] = (0.717 Ã— X_{1}) + (0.847 Ã— X_{2}) + (3.107 Ã— X_{3}) + (0.420 Ã— X_{4}) + (0.998 Ã— X_{5})

**where**:

- X
_{1}, X_{2}, X_{3}, and X_{5}remain the same as in the original model - X
_{4}= Book Value of Equity / Book Value of Total Liabilities

The change in X_{4} to book value of equity was necessary because private companies do not have a readily available market value for their equity. Additionally, the coefficients of the Z'-Score model have been adjusted to account for the change in the X_{4} variable and to optimize the model's performance for private U.S. manufacturing companies.

Notably, the new X_{4} variable, which uses book value instead of market value, has a lower coefficient (0.420) compared to the original model's X_{4} coefficient (0.600), indicating a slightly reduced impact on the Z'-Score.

### Altman Z'-Score Findings

In his July 2000 paper "Predicting Financial Distress of Companies: Revisiting the Z-Score and ZETAÂ® Models," Altman reassessed the Z-score based on 120 firms that defaulted on their publicly traded debt between 1997-1999. He found the Z-Score to still be impressive for public U.S. manufacturing firms, correctly identifying **94%** of the bankrupt companies (113 out of 120) when using a cutoff score of 2.67, and **84%** when using a more conservative cutoff of 1.81.

In comparison, the Z'-Score model, which was developed for private U.S. manufacturing companies, demonstrated a slightly lower accuracy rate of** 91%** in classifying bankrupt firms (30 out of 33). However, the Z'-Score model matched the original Z-Score model's Type II accuracy of **97%** in correctly classifying non-bankrupt companies.

Altman presents a table in his July 2000 paper summarizing these findings:

Thus, despite the Z'-Score model not being extensively tested on a secondary sample due to the lack of a comprehensive private firm database (according to Altman), the results suggest strong predictive capabilities for evaluating the financial well-being of private U.S. manufacturing companies.

### How to Interpret the Altman Z'-Score

Following adjustments to the formula and insights from Altman, the Altman Z'-Score's cut-off points or risk zones have been updated from those of the original Z-Score. Here's how they should be interpreted:

**Safe Zone (Z'-Score > 2.90)**: Companies with a Z'-Score above 2.90 are considered to be in a safe financial position and have a low probability of bankruptcy.**Gray Zone (1.23 < Z'-Score < 2.90)**: Companies with a Z'-Score between 1.23 and 2.90 fall into a gray area, where there's some risk of bankruptcy, but the prediction is less certain.**Distress Zone (Z'-Score < 1.23)**: Companies with a Z'-Score below 1.23 are considered to be in financial distress and have a high probability of bankruptcy within the next two years.

The Altman Z'-Score visual below can also be referenced:

Notably, because the revised model's gray area (aka "zone of ignorance") is wider than the original model (1.81 < Z-Score < 2.99), with a lower bound of 1.23 and an upper bound of 2.90, this suggests that the revised model may be somewhat less reliable than the original model, but only marginally.

## Altman Z"-Score for Non-Manufacturers (Public and Private)

In 1995, Altman introduced a revised version of the Z-Score model, known as the **Altman Z"-Score**, to accommodate non-manufacturing companies (both private and public). This model removes the X_{5} variable (Sales / Total Assets) to minimize the potential industry effect that might arise from the asset turnover ratio, making it more adaptable to a wider range of non-manufacturing industries.

### Altman Z"-Score Formula

The revised Altman Z"-Score formula for non-manufacturing companies (both private and public) is :

Z"-Score [Non-Manufacturing] = (6.56 Ã— X_{1}) + (3.26 Ã— X_{2}) + (6.72 Ã— X_{3}) + (1.05 Ã— X_{4})

**where**:

- X
_{1}, X_{2}, and X_{3}remain the same as in the original model - X
_{4}= Book Value of Equity / Book Value of Total Liabilities - The X
_{5}variable (Sales / Total Assets) is omitted

The coefficients for variables X_{1} to X_{4} in the Z"-Score model have changed compared to the Z and Z'-Score models, reflecting the unique financial characteristics of non-manufacturing companies. By removing the X_{5} variable, the Z"-Score model reduces the potential for industry-specific distortions in the bankruptcy prediction process. This makes it useful in industries where the type of asset financing varies greatly among firms and important adjustments, such as lease capitalization, are not made.

Although little is said about the effectiveness of the Z"-Score model, Altman elaborates on its effectiveness and how to interpret the Z"-Score when discussing a slightly modified version for emerging markets companies, as detailed below.

## Altman Z"-Score for Emerging Markets (Public and Private)

Altman further revised the Z"-Score model to accommodate companies in emerging markets (both public and private). The emerging market model, called the **Altman EMS** (Emerging Market Scoring) model, was thoroughly discussed in the 2005 paper "An Emerging Market Credit Scoring System for Corporate Bonds."

### Altman EMS Z"-Score Formula

The Altman EMS Z"-Score formula for emerging market companies (both public and private) is shown below:

Z"-Score [Emerging Markets] = (6.56 Ã— X_{1}) + (3.26 Ã— X_{2}) + (6.72 Ã— X_{3}) + (1.05 Ã— X_{4}) + 3.25

**where**:

- X
_{1}, X_{2}, X_{3}, and X_{4}remain the same as in the non-manufacturing model - The constant term of 3.25 is added to standardize the scores, so that a score of zero is equivalent to a bond rating of D (default). Thus, if a company's EMS is calculated to be 0 or less, its considered to be in default.

The EMS model combines financial statement analysis with a modified bond rating equivalent (BRE), which can then be compared to agency ratings (if any) and mapped to the U.S. bond rating scale. The model incorporates additional steps to adjust for the unique risks in emerging markets, such as:

- Foreign currency devaluation vulnerability.
- Industry characteristics.
- Competitive position.
- Special debt issue features.
- Sovereign spreads.

These adjustments are made to the initial BRE derived from the EMS score to arrive at a final modified rating.

### Altman EMS Z"-Score Findings

The EMS model has been successfully applied to assess the credit risk of companies in various emerging markets:

- In a test on Mexican companies during the post-peso crisis period (1994-1996), the model accurately predicted every defaulting firm's debt, while also indicating the successful restructuring of a few entities.
- When compared to agency ratings for 13 Mexican firms, the EMS model's modified BRE assigned higher ratings for seven firms, lower ratings for three firms, and the same rating for three firms.
- The model has been effectively used to assess companies in other emerging markets, such as Brazil, Argentina, and Southeast Asian countries, demonstrating its robustness and adaptability.

Therefore, the EMS model offers a valuable tool for assessing credit risk in emerging markets, where agency ratings may be limited or unavailable. By blending a quantitative credit scoring approach with qualitative adjustments tailored to emerging markets, the model provides a framework for analyzing the creditworthiness of companies operating in these regions.

### How to Interpret the Altman Z"-Scores

Like the Altman Z'-Score, the Altman Z"-Scores' cut-off points have been adjusted to reflect the updated formula and findings from Altman. Unlike previous models, these cut-off points and the interpretation of the Altman Z"-Score are the same for both the non-manufacturing and emerging market (both public and private) versions of the model, as described below:

**Safe Zone (Z"-Score > 2.60)**: Companies with a Z"-Score above 2.60 are considered to be in a safe financial position and have a low probability of bankruptcy.**Gray Zone (1.10 < Z"-Score < 2.60)**: Companies with a Z"-Score between 1.10 and 2.60 fall into a gray area, where there's some risk of bankruptcy, but the prediction is less certain.**Distress Zone (Z"-Score < 1.10)**: Companies with a Z"-Score below 1.10 are considered to be in financial distress and have a high probability of bankruptcy within the next two years.

The Altman Z"-Score visual below can also be referenced:

Compared to the original Altman Z and Z'-Score models, the Z"-Score's risk ranges are more conservative, with lower thresholds for the safe and distress zones, reflecting the unique financial characteristics of non-manufacturing and emerging market firms.

## Proprietary Altman Z-Score Models

In addition to his original Z-Score model, Altman developed two proprietary credit risk models: (1) the ZETA credit risk model in 1977 and (2) the Altman Z-Score Plus model in 2012. Both models build upon the original Z-Score approach with several enhancements and are still used by practitioners today to assess corporate credit risk.

### ZETA Credit Risk Model

In 1977, Altman and his colleagues developed a second-generation model called the **ZETA Credit Risk Model**. The ZETA model was introduced in the paper "ZETA Analysis: A New Model to Identify Bankruptcy Risk of Corporations" and later revisited in the 2012 paper "Predicting Financial Distress of Companies: Revisiting the Z-Score and ZETAÂ® Models."

The key improvements of the ZETA model include:

- Developed using a sample of 53 bankrupt and 58 non-bankrupt firms, with an average asset size of approximately $100 million. This makes it more applicable to the increasing size of bankrupt firms in the 1970s.
- Includes both manufacturing and retail firms in the sample. Appropriate adjustments were made to capitalize leases, a key factor for retailers.
- Data adjusted for recent changes in financial reporting standards as of the 1970s.
- Tested both linear and quadratic forms of discriminant analysis, while the original Z-Score model only used the linear form. The linear form was found to perform better in the ZETA model.

The final 7 variables/ratios selected for the ZETA model are as follows:

- X
_{1}= Return on Assets (EBIT / TA) - X
_{2}= Stability of Earnings (measured by a normalized measure of the standard error of estimate around a 5 to 10-year trend in X_{1}) - X
_{3}= Debt Service (EBIT / Total Interest Payments) - X
_{4}= Cumulative Profitability (Retained Earnings / TA) - X
_{5}= Liquidity (Current Assets / Current Liabilities) - X
_{6}= Capitalization (Market Value Equity / Total Capital) - X
_{7}= Size (Total assets)

The model takes the form: Z = V_{1}X_{1} + V_{2}X_{2} + â€¦ + V_{7}X_{7}, where V_{1}-V_{7} are the discriminant coefficients. The exact coefficients are proprietary.

The ZETA model is still used today by practitioners to assess bankruptcy risk. However, it's only available to subscribers of "ZETA Services, Inc," according to Altman. The full specifications, coefficients, and ongoing research are provided to subscribers.

In tests, the ZETA model was over **90%** accurate in classifying bankrupt firms 1 year prior to bankruptcy and **70%** accurate up to 5 years prior. This is significantly more accurate than the original Z-Score model over long timeframes. Furthermore, the inclusion of retail firms in the sample did not negatively affect the model's performance, which suggests the model is applicable to both manufacturers and retailers.

The ZETA model also uses information about the likelihood of a company belonging to a bankrupt or non-bankrupt group (prior probabilities) and the costs of misclassifying a company to determine optimal cutoff scores for classifying companies. This allows the model to be customized for different decision-makers who may have different tolerances for misclassification errors.

In summary, the ZETA credit risk model is a proprietary model that enhances the original Z-Score approach. Despite its greater predictive accuracy, the full model specification is not publicly available, limiting its application for the average retail investor.

### Altman Z-Score Plus

The **Altman Z-Score Plus**, available at Altman Z-Score Plus, is a comprehensive credit risk assessment model (in the form of a web application and mobile app) that builds upon Altman's original Z-Score model from the 1960s. Developed in collaboration with Business Compass LLC in 2012, the Z-Score Plus expands the original model's capabilities to cater to the needs of modern, global businesses.

Key features of the Z-Score Plus include its applicability to a wider range of companies, such as non-U.S. firms (including those in emerging markets like China), non-manufacturing companies (both public and private), and privately-held industrial manufacturing firms. The model calculates the appropriate Z-Score (Z, Z', or Z") based on the company's characteristics and financial data.

In addition to the Z-Score, the Altman Z-Score Plus provides a more in-depth analysis of a company's credit risk by offering several supplementary metrics. These include:

- A probability of default estimate for the next 1 to 10 years.
- A percentile ranking comparing the company's bankruptcy likelihood to others in the same industry.
- A bond-rating equivalent (BRE) that benchmarks the company's Z-Score against average scores for different bond rating classes (AAA to D).

While the Altman Z-Score Plus seems to offer an enhanced model for evaluating credit risk across a broad spectrum of companies worldwide, it's important to reiterate that the model's full details are proprietary and not publicly available. Access to the model may also be costly for some investors.

## Altman Z-Score Model (Z, Z', and Z") Calculation Examples

To demonstrate the calculation of the Altman Z-Score, we'll use Virgin Galactic (SPCE) as our example, an aerospace and spaceflight company focused on developing and operating a commercial spaceline for space tourism and research missions.

Virgin Galactic's business model has faced significant challenges, as the market for space tourism has not developed as quickly as anticipated due to the high costs and technical difficulties involved. The company's share price has plummeted (at the time of writing this article), and it's now trading as a penny stock. The limited customer base and the need to repeatedly launch humans into space have put pressure on Virgin Galactic's financial stability, making it an interesting case for applying the Altman Z-Score to assess its bankruptcy risk.

We'll use Virgin Galactic's fiscal year (FY) 2023 financial data to calculate the various Altman Z-Scores (Z, Z', and Z") to assess the company's financial distress and bankruptcy risk. Technically, only the Altman Z"-Score, for public U.S. non-manufacturing companies would be applicable (because Virgin Galactic is more of an aerospace and defense company), but for demonstration purposes we'll show the calculation of all of the non-proprietary Altman Z-Score models as well.

Now, to apply the Altman Z-Score(s), investors can follow these steps:

- Gather the necessary financial data for the company, specifically the current year's income statement and balance sheet.
- Calculate the relevant ratios.
- Calculate the Altman Z-Score formula(s) and interpret the results.

Investors can use the spreadsheet linked below to calculate the Altman Z-Score:

### Step #1: Gather Necessary Financial Data

To calculate the Altman Z-Score for any company, you'll need to gather relevant financial data from the income statement and balance sheet. The cash flow statement is not required for the Altman Z-Score calculation(s). Financial statements can be sourced from the company's 10-K annual report, which can be found via the SEC or the company's investor relations pages.

###### Income Statement

The income statement is a financial statement that summarizes a company's revenues, expenses, and profits (or losses) over a specific period.

From the income statement, you'll need to locate these two financials:

**Revenues**: To calculate the asset turnover ratio (S/TA).**Earnings Before Interest and Taxes (EBIT)**: To calculate the EBIT/TA ratio.

Here's Virgin Galactic's income statement with these two financials outlined:

Thus, in FY 2023, Virgin Galactic reported revenues of $6,800T and EBIT of -$531,509T.

###### Balance Sheet

The balance sheet is a financial statement that provides a snapshot of a company's assets, liabilities, and shareholders' equity at a specific point in time.

From the balance sheet, you'll need to locate these six financials:

**Current Assets**: To calculate the WC/TA ratio.**Total Assets**: For the WC/TA, RE/TA, EBIT/TA, and S/TA ratios.**Current Liabilities**: Also to calculate the WC/TA ratio.**Total Liabilities**: For the MVE/TL and BVE/TL ratios.**Retained Earnings**: To calculate the RE/TA ratio.**Total Shareholders' Equity**: To calculate the BVE/TL ratio.

Here's Virgin Galactic's balance sheet with these six financials outlined:

Thus, in FY 2024, Virgin Galactic reported current assets of $950,829T, total assets of $1,179,517T, current liabilities of $185,660T, total liabilities of $674,041T, retained earnings of -$2,126,132T, and total shareholders' equity of $505,476T.

Notably, Virgin Galactic shows its retained earnings as an "accumulated deficit" on its balance sheet. This term refers to retained earnings that have become negative after cumulative losses surpassed cumulative profits. Essentially, it signifies that the company has experienced more losses than profits since its inception, suggesting a heavy reliance on external financing.

### Step #2: Calculate and Interpret the Altman Z-Score Ratios

Now that we've gathered the relevant financials for the Altman Z-Scores, the next step is to calculate the ratios necessary for completing the Altman Z-Score calculations (across all of the non-proprietary models, for demonstration purposes).

###### X_{1}: Working Capital / Total Assets (WC/TA)

WC/TA compares a company's working capital (current assets - current liabilities) to its total assets. Here's Virgin Galactic's WC/TA calculation for FY 2023:

WC/TA [SPCE; FY 2023] = ($950,289T - $185,660T) / ($1,179,517T) --> **0.65 or 65%**

A WC/TA of **65%** indicates that Virgin Galactic has a relatively high proportion of working capital to total assets, suggesting a strong liquidity position and ability to meet short-term obligations.

###### X_{2}: Retained Earnings / Total Assets (RE/TA)

RE/TA compares a company's cumulative profits reinvested in the business (retained earnings) to its total assets. Here's Virgin Galactic's RE/TA calculation for FY 2023:

RE/TA [SPCE; FY 2023] = (-$2,126,132T) / ($1,179,517T) --> **-1.80 or -180%**

An RE/TA of **-180%** suggests that Virgin Galactic has accumulated significant losses over its lifetime, which may indicate a history of unprofitability and potential financial distress.

###### X_{3}: Earnings Before Interest and Taxes (EBIT) / Total Assets (EBIT/TA)

EBIT/TA compares a company's earnings before interest and taxes (EBIT) to its total assets. Here's Virgin Galactic's EBIT/TA calculation for FY 2023:

EBIT/TA [SPCE; FY 2023] = (-531,509T) / ($1,179,517T) --> **-0.45 or -45%**

An EBIT/TA of **-45%** indicates that Virgin Galactic's operating efficiency is poor, as it generates negative EBIT relative to its total asset base, which may signal financial distress.

###### X_{4} (Version #1): Market Value of Equity / Total Liabilities (MVE/TL)

MVE/TL compares a company's market value of equity (stock price Ã— shares outstanding) to its total liabilities. Note that this ratio is only necessary for the original Z-Score model. Here's Virgin Galactic's MVE/TL calculation for FY 2023:

MVE/TL [SPCE; FY 2023] = ($2.45/share Ã— 337,262T shares outstanding) / ($674,041T) --> **1.23 or 123%**

A MVE/TL of **123%** suggests that Virgin Galactic's market value of equity exceeds its total liabilities, indicating that the market perceives the company as having sufficient value to cover its obligations, despite its current financial challenges.

###### X_{4} (Version #2): Book Value of Equity / Total Liabilities (BVE/TL)

BVE/TL compares a company's book value of equity (aka shareholder's equity) to its total liabilities, and is used for the Z' and Z" model calculations. Here's Virgin Galactic's BVE/TL calculation for FY 2023:

BVE/TL [SPCE; FY 2023] = ($505,476T) / ($674,041T) --> **0.75 or 75%**

A BVE/TL of **75%** indicates that Virgin Galactic's book value of equity is lower than its total liabilities, suggesting that the company's assets may not be sufficient to cover its obligations if they were to be liquidated at book value.

###### X_{5}: Sales / Total Assets (S/TA)

S/TA compares a company's net sales revenue to its total assets. Note that this ratio is omitted in the Z"-Score model. Here's Virgin Galactic's S/TA calculation for FY 2023:

S/TA [SPCE; FY 2023] = ($6,800T) / ($1,179,517T) --> **0.01 or 0.01x**

An S/TA of **0.01x** suggests that Virgin Galactic is generating very low sales relative to its total asset base, indicating poor asset turnover and potential inefficiencies in utilizing its resources to generate revenue.

### Step #3: Calculate and Interpret the Altman Z-Score Models

Now that we've gathered the relevant financials and calculated the ratios/variables necessary for the Altman Z-Score models, the final step is to calculate and interpret all of the non-proprietary models using the provided formulas.

We'll begin with the Altman Z''-Score model, which is technically the only appropriate model for Virgin Galactic, a U.S. publicly traded non-manufacturing company. Then, we'll demonstrate how to calculate the other Altman Z-Scores (for demonstration purposes only).

###### Altman Z"-Score Calculation

First, here's the completed calculation for the Altman Z"-Score for Virgin Galactic in FY 2023:

Altman Z"-Score [SPCE; FY 2023] = (6.56 Ã— 0.65) + (3.26 Ã— -1.80) + (6.72 Ã— -0.45) + (1.05 Ã— 0.75) --> **-3.86**

A Z"-Score of **-3.86** indicates that Virgin Galactic is in the "Distress" zone (Z" < 1.10), suggesting a high likelihood of financial distress or bankruptcy risk. This is consistent with our observations of their income statement and balance sheet.

For illustration purposes, the visual below displays Virgin Galactic's Z"-Scores from FY 2021 to FY 2023:

As you can see, Virgin Galactic's Altman Z"-Score has been showing a declining trend from FY 2021 to 2023.

###### Altman EMS Z"-Score Calculation

Here's the completed calculation for the Altman EMS Z"-Score for Virgin Galactic in FY 2023, if we are to assume it's a non-manufacturing emerging markets company:

Altman EMS Z"-Score [SPCE; FY 2023] = (6.56 Ã— 0.65) + (3.26 Ã— -1.80) + (6.72 Ã— -0.45) + (1.05 Ã— 0.75) + 3.25 --> **-0.61**

An EMS Z"-Score of **-0.61** suggests that Virgin Galactic, if considered an emerging markets company, would be in the "Distress" zone (EMS Z"-Score < 1.10), again indicating a high probability of bankruptcy within the next two years.

###### Altman Z'-Score Calculation

Here's the completed calculation for the Altman Z'-Score for Virgin Galactic in FY 2023, if we are to assume it's a U.S. private manufacturing company:

Altman Z'-Score [SPCE; FY 2023] = (0.717 Ã— 0.65) + (0.847 Ã— -1.8) + (3.107 Ã— -0.45) + (0.42 Ã— 0.75) + (0.998 Ã— 0.01) --> **-2.14**

A Z'-Score of **-2.14** indicates that Virgin Galactic, if considered a private manufacturing company, would be in the "Distress" zone (Z' < 1.23), once more suggesting a high probability of bankruptcy within the next 1-2 years.

###### Altman Z-Score Calculation

Here's the completed calculation for the Altman Z-Score for Virgin Galactic in FY 2023, if we are to assume it's a U.S. publicly traded manufacturing company:

Altman Z-Score [SPCE; FY 2023] = (1.2 Ã— 0.65) + (1.4 Ã— -1.80) + (3.3 Ã— -0.45) + (0.6 Ã— 1.23) + (1.0 Ã— 0.01) --> **-2.49**

A Z-Score of **-2.49** suggests that Virgin Galactic, if considered a public manufacturing company, would also be in the "Distress" zone (Z < 1.81), yet again indicating a high likelihood of bankruptcy within the next 1-2 years.

## Limitations of the Altman Z-Score

While the Altman Z-Score has been widely used as a tool to assess a company's financial health and predict the likelihood of bankruptcy, it's essential to recognize its limitations. Despite its usefulness, the model has several drawbacks that investors should consider when applying the Z-Score to their decision-making process, as described in the list below:

**Limited Applicability Across Industries**: The Altman Z-Score was originally developed based on a sample of manufacturing companies in the U.S., which means its effectiveness may vary when applied to other sectors or industries, although to its credit the Z"-Score addresses this issue to some extent by removing the X_{5}asset turnover variable. Regardless, retail and restaurant businesses, for instance, often operate with negative working capital as part of their normal business model. In these cases, a negative working capital ratio could signify strong cash flow management rather than potential insolvency.**Unsuitability for Financial Services Companies**: The Altman Z-Score and other balance sheet-based models are not recommended for use with financial companies due to the opacity of their balance sheets and frequent use of off-balance sheet items. The model's assumptions and weightings may not accurately reflect the financial realities of these companies.**Inadequacy for Early-Stage and High-Growth Companies**: The Z-Score model may not be suitable for evaluating early-stage companies that are experiencing rapid growth but are currently unprofitable. These companies often receive low scores despite their financial health, as the model does not account for their growth potential and future prospects.**Dependence on Historical Financial Data**: The Z-Score model relies heavily on historical financial information, which may not always be a reliable indicator of a company's future performance. Recent changes in a company's operations, management, or industry trends that could significantly impact its financial health are not captured by the model, potentially limiting its predictive power.**Absence of Qualitative Factors**: The Altman Z-Score is entirely based on quantitative financial ratios and does not consider qualitative factors that can impact a company's performance and risk of bankruptcy. Important aspects such as management quality, competitive landscape, technological disruptions, and regulatory changes are not accounted for by the model.**Vulnerability to One-Time Events**: The Altman Z-Score can be significantly influenced by one-time events or non-recurring items, such as write-offs or restructuring charges. These events can lead to substantial fluctuations in the model's input variables, resulting in a distorted Z-Score that may not accurately represent the company's true financial condition.**Potential for Financial Manipulation**: The accuracy of the Z-Score heavily depends on the reliability of the financial data used as inputs. Companies faced with financial distress may be tempted to misrepresent their financial statements, leading to misleading Z-Scores. The model's effectiveness is only as good as the quality and integrity of the underlying financial information.**Overemphasis on Accounting Ratios**: Modern academic default and bankruptcy prediction models place greater emphasis on market-based data (e.g., KMV Model, Ohlson O-Score, etc.) rather than the accounting ratios predominant in the Altman Z-Score. This shift suggests that the Z-Score may not be as effective as more contemporary models that incorporate market information.**Lack of Direct Cash Flow Consideration**: The Altman Z-Score does not directly consider the cash flow statement, which is important for understanding a company's ability to meet its financial obligations. The model only hints at cash flow through the use of the working capital to total assets (WC/TA) ratio, which generally does not provide a comprehensive picture of a company's cash flow health.

In conclusion, while the Altman Z-Score offers a quick and straightforward assessment of a company's financial health and may have high predictive power in specific contexts, it should not be the sole criterion for assessing financial distress or bankruptcy risk. Investors should use the Z-Score alongside thorough financial statement analysis, other financial ratios, and qualitative assessments to assess the company's future direction.

### Independent Studies on the Altman Z-Score

While the Altman Z-Score model has been widely used for decades to predict corporate bankruptcies, several academic studies have identified limitations and suggested enhancements to improve its accuracy and applicability. These papers argue that the original Z-Score's static, single-period approach can be significantly improved by incorporating time-varying and industry-specific factors, expanding the estimation sample, and carefully selecting prediction variables.

Key insights from two notable studies that evaluate the Z-Score's statistical properties and propose alternative models are summarized below.

###### Forecasting Bankruptcy More Accuracy: A Simple Hazard Model (2001)

In the 2001 paper "Forecasting Bankruptcy More Accurately: A Simple Hazard Model," Shumway argues that the Altman Z-Score model, along with other static models like Zmijewski's 1984 paper "Methodological Issues Related to the Estimation of Financial Distress Prediction Models," are inappropriately estimated using static single-period bankruptcy data.

Key findings from this paper include:

- Static models fail to control for each firm's period at risk, ignoring the fact that some firms file for bankruptcy after many years of being at risk while others fail in their first year.
- Static models cannot incorporate time-varying covariates or macroeconomic variables that change over time for each firm.
- Static models introduce unnecessary selection bias by choosing when to observe each firm's data, usually using data from only one year prior to bankruptcy and ignoring data on healthy firms that eventually go bankrupt.

Shumway proposes a "**hazard model**," which is a type of model that estimates the probability of an event (like bankruptcy) occurring over time, that can more accurately account for time, incorporate time-varying covariates, and utilize all available data on each firm to produce more efficient out-of-sample forecasts.

Empirically, Shumway's model correctly classifies **75%** of bankruptcies in the top decile (the 10% of firms with the highest predicted bankruptcy risk) compared to only **63%** for Altman's model and **43%** for Zmijewski's model. In the top two deciles (the 20% riskiest firms), Shumway's model identifies **86%** of bankruptcies compared to **78%** for Altman and **58%** for Zmijewski.

In conclusion, Shumway's paper highlights significant limitations in the Altman Z-Score and other static bankruptcy prediction models, and demonstrates that a hazard model approach can substantially improve accuracy by incorporating time-varying information and using all available data.

###### Bankruptcy Prediction with Industry Effects (2004)

In the 2004 paper "Bankruptcy Prediction with Industry Effects," Chava and Jarrow use an expanded bankruptcy database to investigate potential improvements to the Altman Z-Score and other bankruptcy prediction models.

Key findings from this paper include:

- Adding industry effects significantly improves explanatory power and accuracy. Estimating separate coefficients for each industry group improves in-sample fit and out-of-sample accuracy, especially for private firm models. The area under the "
**ROC curve**" (a measure of model accuracy where 1 indicates perfect prediction and 0.5 indicates no predictive power) increases from**0.7513 to 0.7646**for private firms and from**0.9022 to 0.9101**for public firms when industry effects are included. - The study verifies Shumway's (2001) finding that hazard models are superior to static models like Altman's Z-Score. Shumway's model has an area under the ROC curve of
**0.9113**compared to**0.8662**for Altman's model and**0.7392**for Zmijewski's model. - Expanding the model to include financial firms leads to reductions in accuracy compared to models estimated on non-financial firms only. The area under the ROC curve drops from
**0.9101**to**0.8724**when financial firms are included, suggesting bankruptcy prediction is more challenging for these firms. - Shifting to monthly observation intervals dramatically improves forecasting accuracy compared to the annual observation intervals typically used in the literature. A public firm hazard model achieves an area under the ROC curve of
**0.9449**with monthly data versus**0.8675**with yearly data.

In summary, Chava and Jarrow's paper confirms the superiority of hazard models over static models like the Altman Z-Score, and identifies industry effects and higher-frequency data as important enhancements that can further improve bankruptcy prediction accuracy.

## The Bottom Line

The Altman Z-Score is a widely-used model developed by Edward I. Altman in 1968 to predict the likelihood of corporate bankruptcy within two years. It combines five key financial ratios, each weighted differently, to generate a single score that reflects a company's financial health.

The original Z-Score model was designed for publicly traded U.S. manufacturing companies, with subsequent adaptations (Z'-Score and Z"-Score) to accommodate private companies, non-manufacturers, and emerging markets companies. Proprietary versions, such as the ZETA Credit Risk Model and Altman Z-Score Plus, incorporate additional variables and industry-specific factors to improve predictive accuracy.

Altman's extensive research and testing found the Z-Score models to be highly accurate in predicting bankruptcy within one to two years, with the original model correctly identifying 95% of bankrupt U.S. manufacturing companies one year before the event. However, the models' accuracy diminishes beyond the two-year horizon.

Interpretation of the Z-Scores involves comparing the calculated score against established cut-off points, which categorize companies into "Safe," "Gray," or "Distress" zones based on their bankruptcy risk. Generally, a lower Z-Score indicates a higher risk of bankruptcy, while a higher Z-Score suggests a lower risk. The specific range for each zone varies across the different Z-Score models, but typically falls between 0 and 3, with a score above 3 considered safe, a score between 1.8 and 3 in the gray area, and a score below 1.8 indicating distress or high bankruptcy risk.

While widely adopted, the Altman Z-Score has limitations, including limited applicability across industries, unsuitability for financial services companies, inadequacy for early-stage and high-growth firms, dependence on historical data, absence of qualitative factors, vulnerability to one-time events and financial manipulation, overemphasis on accounting ratios, and lack of direct cash flow consideration.

Independent studies have proposed enhancements, such as incorporating time-varying and industry-specific factors, expanding the estimation sample, carefully selecting prediction variables, and using hazard models instead of static models like the Z-Score.

Overall, the Altman Z-Score provides a relatively useful tool for assessing a company's financial health and bankruptcy risk, but should be used alongside other financial analysis techniques and not as the sole criterion for decision-making.