Today we are looking at a more complex financial analysis model – the Altman Z-Score Analysis. But I promise, there’s a reward at the end of the article.

The Z-score formula for predicting bankruptcy was published in 1968 by Edward Altman, Assistant Professor in the field of Finance in the New York University.

In the 1960s, Edward Altman noted that the academic community is moving towards the elimination of ratio analysis as an analytical technique in the assessment of business enterprises. He did the primary research for his work in 1967, and several later studies commented on the Z-Score model and its effectiveness, including a 1995 adaptation of the model for companies operating in an emerging market. The author developed a second-generation model (ZETA), which he defined in 1976. We will look at this model in a separate article.

Today’s article is organized as follows:

- Development of the Z-Score model
- GoPro Case Study

## Development of the Z-Score model

### Traditional Ratio Analysis

William Beaver wrote some of the most famous works in the field of ratio analysis and bankruptcy classification in the 1960s. His study of one variable, describing a set of various bankruptcy predictors, set the basis for attempts at multi-variable analysis. Beaver found that several indicators can distinguish between bankrupt and non-bankrupt companies up to 5 years before the bankruptcy. He asked the question about using a multi-variable analysis. The Altman Z-Score model did precisely that.

Although these works lead to the increased emphasis on company performance and trends, the adaptation of results to potential bankruptcy predictions is both theoretically and practically controversial. In almost every case, the methodology is one-sided and emphasizes on single precursors of future problems. The analysis of the ratios presented in this way is susceptible to misinterpretation and is potentially misleading. For example, we can predict a company with poor returns and solvency for potential bankruptcy, yet because of its high liquidity, the situation may not be as critical. Also, business performance is strictly individual. The reason for these shortcomings of models is the usage of single-variable analysis. The appropriate development of the studies mentioned above was the combination of several measurements in a single, predictive model.

### Discriminant Analysis

After considering the nature of the problem and the purpose of the analysis, Multiple Discriminant Analysis (MDA) was chosen as a suitable statistical method, even though it is not as popular as regression analysis.

MDA, in its purest form, attempts to derive a linear combination of characteristics that best distinguish previously specified groups. If a particular entity, such as a corporation has features (financial ratios) that may be inherent in all groups in the analysis, the MDA defines a set of discriminant coefficients. When we apply these coefficients to the actual ratios, there is already a basis for classifying the corporation into one of the mutually exclusive groups. The MDA technique has the advantage of taking into account the complete profile of company characteristics as well as the interrelation of these properties. A single-variable study, on the other hand, can only report the features of the object sequentially. Another plus of MDA is the reduction of the analytical space, i.e., of the number of independent variables to G-1 dimensions, where G equals the number of initial groups. This analysis deals with two groups consisting of bankrupt and non-bankrupt firms. Therefore the study is transformed in its purest form to one dimension.

The discriminant function is of the type:

**Z****=V**_{1}X_{1}** + V**_{2}X_{2 }** + ……………………….. + V _{n}X_{n}**

This function transforms the values of the individual variables into a single discriminant result or **Z ****score** used to classify an object, where V_{1} to V_{n} are discriminant coefficients, and X_{1} to X_{n} are independent variables. MDA calculates the discriminant factor using a detailed list of financial ratios in the potential bankruptcy assessment, where some of the measurements will have a high degree of connectivity with each other. Since the discriminant analysis does not include this, it is usually performed by carefully selecting the predictive variables (ratios). MDA has the advantage that it can provide extensive information with a relatively small number of measurements. This information can easily show differences between groups, but whether these differences are essential or not is the more critical aspect of the analysis.

Perhaps the most crucial advantage of MDA in dealing with classification problems is the potential to analyze the entire set of variables at the same time rather than sequentially. More importantly, a combination of ratios can be examined together to avoid possible ambiguities and erroneous classifications.

The Z-Score model is a linear analysis, with weights attached to the five characteristics so that a cumulative result can be reached to form the basis for classifying the business in one of the two groups – bankruptcy and non-bankruptcy.

### Original Z-Score function from 1968

The initial selection is made up of 66 corporations – 33 companies in each group. The bankruptcy group (Group 1) consists of companies that filed for bankruptcy between 1946 and 1965. A 20-year period was not the best choice as the average ratios change over time. Realizing that this group is not entirely homogeneous (due to differences in production and size), Altman tried to make a more precise selection of non-bankruptcy firms. Group 2 consists of 33 companies selected based on production asset size. Also, Group 2 companies had to still exist at the time of the analysis, while the information gathered for these companies was for the same years as that collected for the bankrupt companies. For the initial test, Altman collected the data from financial statements dating back to one year before bankruptcy. The limited diversity in production asset sizes in Group 1 led to the decision to eliminate from the initial selection small companies with assets under USD 1 million, and big companies with assets exceeding USD 25 million. Moreover, the bankruptcy of large asset companies was quite rare until 1966.

There is a discussion about whether the Z-Score model is sufficiently robust to predict the bankruptcy of businesses whose assets are significantly above USD 25 million. The ZETA model, however, was used to analyze companies with substantial assets and is relevant for both small and large businesses. We will look further into the ZETA model in a future article.

#### Components used in the model

Once the groups were defined, and the companies selected, the Balance Sheets and the Income Statements were prepared. Due to a large number of variables considered to be significant indicators of corporate problems, they chose 22 ratios for evaluation. Altman classified the variables into five standard categories: liquidity, profitability, leverage, solvency, and activity. We have to notice that due to the limited information available Altman did not consider including in his analysis ratios based on cash flow values.

To reach a final set of variables, Altman used the following procedures:

- Observing the statistical significance of various alternative functions, determining the relative contribution of each independent variable;
- Evaluation of the interrelationships between the associated variables;
- Monitoring the accuracy of predictions of different profiles;
- Assessment of the analyzer.

From the initial list of 22 variables, five made the selection because they were doing the best in predicting corporate bankruptcy.

The final discriminant function is as follows:

**Z**** = 0.012 * X**_{1}** + 0.014 * X**_{2}** + 0.033 * X**_{3}** + 0.006 * X**_{4}** + 0.999 * X**_{5}

Let’s take a better look at the variables X_{1} to X_{5}.

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

The Working Capital to Total Assets (WC/TA) ratio measures the company’s liquid assets compared to its size. We define Working Capital as the difference between current assets and current liabilities. Typically, a company experiencing frequent operating losses will see a contraction of current assets compared to total assets. Of the three ratios for liquidity, this has proven to be the most prized. The other two tested ratios for liquidity were the current ratio and the quick ratio. However, these turned out to be less useful in Altman’s model.

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

Retained Earnings is a measure that represents a company’s total reinvested earnings for its entire lifecycle. We should note that Retained Earnings as a measure is susceptible to manipulation through corporate reorganizations and dividends. The company’s age is implicitly a part of this ratio. For example, a relatively new company would have a low RE / TA ratio because it did not have time to accumulate profits. We can, therefore, argue that this analysis discriminates somewhat against a new firm, and its chances of being classified in the bankrupt group are relatively more substantial than those of another older company. However, the risk of bankruptcy is much higher in the early years of the company. Also, the ratio measures the leverage effect. Companies with high Retained Earnings to Total Assets ratio have funded their assets by retaining profits and therefore, not taken as many debts.

##### X_{3} – Earnings Before Interest and Tax over Total Assets (EBIT/TA)

This ratio is a measure of the productivity of the company’s assets, eliminating factors such as taxes and leverage. Because the firm bases its whole existence on the return on its assets, this ratio is particularly suited to analyze corporate bankruptcy.

##### X_{4} – Market Capitalization over Total Liabilities (MVC/TL)

This ratio indicates how much company assets may be impaired if the value of the debt exceeds the value of the assets. For listed companies, we use the market value of ordinary and preferred shares or market capitalization. For private enterprises, we use the accounting value of the share capital. This ratio also considers the market price, which is neglected by most bankruptcy research models.

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

The Sales over Total Assets ratio is a standard financial ratio illustrating the ability of company assets to generate sales. This is a measure of the management’s ability to cope with competitive conditions.

#### Model Clarification

You may have noticed that in the original formula the distribution of the weights doesn’t look quite right. Due to the specificity of the original computer format, we must calculate the variables X_{1} to X_{4} as an absolute value. For example, we include an X_{1} at 10% as 10 instead of 0.10. We only express variable X_{5} differently – ratio at 70% is presented as 0.70. We can see the extremely high discriminant factor of X_{5}. This apparent discrepancy is due to the shape of the different variables. Over the years, many researchers have found that a more convenient model representation is:

**Z ****= 1.2 * X**_{1}** + 1.4 * X**_{2}** + 3.3 * X**_{3}** + 0.6 * X**_{4}** + 1.0 * X**_{5}

Using this formula, we can input the more common format for expressing percentages (0.10 for 10%) for the first four variables. The results for the respective companies and their classification in groups remain unchanged. But we have to note that the industry commonly accepts this format.

#### Revaluation of the model

The initial selection of 33 companies in each of the two groups is examined using information gathered from a set of financial statements before the bankruptcy. Because we derive the discriminant coefficients and distribution in the groups from this selection, we expect a high percentage of successful classification. This is because companies are classified using a discriminant function, based on individual measurements for the firms. In the table below, you can observe the classification matrix for the initial selection.

The model is extremely accurate, classifying 95% of the selection correctly. Type 1 error (false positives – a firm is classified as non-bankrupt when it goes into bankruptcy) occurs only in 6%, while Type 2 error (false negatives – a firm is ranked as bankrupt when it doesn’t go into bankruptcy), even less – 3%. The results are, therefore, encouraging, but we should keep in mind that the model tends to increase the results of the bankruptcy firms, i.e., more companies are expected to go bankrupt.

### Adapting the model for private companies

Perhaps the most common question regarding the application of the model is what needs to be done to implement it for private companies. Analysts, auditors, and the firms themselves are concerned that the original model applies only to public enterprises. To be exact, the Z-Score is a model for public companies, and any modifications would not have a scientific value. For example, the most apparent change would be to replace the market value of the shares by the book value of the shares and then to recalculate V_{4}X_{4}. Back then, analysts had limited choices and followed this procedure due to the lack of alternatives.

#### Revaluation of the model

Instead of modifying one of the model variables to compute the Z-Score, it is preferable to completely revise the model by replacing the market values of the shares with their book values for X_{4}. It is easy to assume that all coefficients will change, not only the parameter of the new variable and that the classification criterion and values will also change.

The result of the revalued Z-Score model with the new X_{4} variable is:

**Z****’ = 0.717 * X****1 + 0.847 * X****2 + 3.107 * X****3 + 0.420 * X****4 + 0.998 * X****5**

The formula now looks different from the original model. Note, for example, that the coefficient for X_{1} has decreased from 1.2 to 0.7. The new X_{4} variable has also changed its ratio from 0.6 to 0.42, i.e., they have less impact on the Z-score result.

### Adapting the model for service organizations

The next modification of the Z-Score model analyzes the characteristics and accuracy of a model without the variable X_{5} – Sales to Total Assets. We do this to minimize the potential effect that would occur when such a variable depends on the type of production, and we include the asset turnover. Altman used this model to assess the financial health of non-US corporations. In particular, Altman, John Hatzell, and Matthew Peck applied this expanded Z’’-Score model to Mexican companies holding euro bonds denominated in US dollars, in 1995. In such cases, we use the book value of the shares for X_{4}. The classification results are identical to those of the 5-variable Z’-Score model, the new Z’’-Score model is:

**Z****’’ = 6.56 * X****1 + 3.26 * X****2 + 6.72 * X****3 + 1.0****5 * ****X****4**

This model is particularly useful for businesses where the type of asset financing differs greatly across firms within the analysis.

### Adapting the model for companies in Emerging Markets

We can initially analyze Emerging Market companies in a way similar to the traditional analysis of US corporations. It is challenging to build a model for an emerging market based on a selection of companies in that country because of the lack of experience there. To deal with this problem, in 1995, Altman, Hartzell, and Peck changed the original Z-Score model of Altman to create an Emerging Market Model (EMM).

For the relative value analysis, they added the corresponding US corporate credit spreads to the sovereign bond spreads. Back then, rating agencies ranked very few of the Mexican companies. Risk assessments such as the one provided by the EMM were often the only reliable indicator of credit risk used by Mexican investors. Altman, Hartzell, and Peck report that modified ratings prove to be accurate in predicting both bankruptcies and improvements.

The modified Z’’-Score bankruptcy model for emerging markets is:

**Z****’’ = ****3.25 + ****6.56 * X****1 + 3.26 * X****2 + 6.72 * X****3 + 1.0****5 * ****X****4**

## GoPro Case Study

To get a better practical view of the Altman Z-Score model, we will take a look at my favorite action camera manufacturer, GoPro (NASDAQ: GPRO).

In the early 2000s, GoPro practically created the action camera market and defined it as we know it today. And if we look at their Z-Score historically, it was doing great for a new company up until February 2015 when the market capitalization of the company started to drop. GoPro had some financial struggles during the whole year 2015, which was not helped by the release of the Karma drone in 2016.

Since then, the Z-Score model has pointed out that GoPro is in the risk of going bankrupt.

Below is the calculation of the Z-Score model for GoPro, as of 31 December 2018.

Since GoPro is a listed company, we use its market capitalization as of 31 December 2018 (0.60 bln US dollars), instead of the book value of its share capital (740 thousand US dollars). We also use the Z-Score brackets and formula for public companies, as outlined above.

As you can see, the Z-Score model classifies GoPro as a **bankrupt** company.

However, if we look at current stats, the Z-Score of GoPro is around 1.86 as of today (source: gurufocus). Such value puts the company barely in the Grey Zone. Based on the Z-Score model for the past few years, the company should be approaching bankruptcy right about now, but it’s not. At least, not yet.

I selected GoPro for our case study because the company is a great example. Bankruptcy can still follow. However, looking at GoPro illustrates how important it is to never trust one analysis model 100%, without corroborating it with additional information. If we look at the media, we see that since GoPro released their Hero 7 Black in the autumn of 2018, they have seen increases in sales and even recorded profit in Q3 of 2018, which the company last showed in 2017. Whether this means they will beat the odds and become a Z-Score false positive, remains to be seen.

## Conclusion

The Altman Z-Score model has been the subject of numerous research papers, and mostly they have concluded that it is an accurate measure for the susceptibility of businesses to going bankrupt. But, as our look at GoPro showed, it is not always enough to conclude.

OK, this was a long article. I hope it was understandable and beneficial. Thank you for reading and keep exploring the vast world of financial analysis!

This was Altman’s Z-Score analysis.

Almost forgot, here you can download the Excel spreadsheet that we used to calculate the Z-score for GoPro here:

## Dobromir Dikov

FCCA, FMVA, Founder of Magnimetrics

Hi! I am a finance professional with 10+ years of experience in audit, controlling, reporting, financial analysis and modeling. I am excited to delve deep into specifics of various industries, where I can identify the best solutions for clients I work with.

In my spare time, I am into skiing, hiking and running. I am also active on Instagram and YouTube, where I try different ways to express my creative side.

The information and views set out in this publication are those of the author(s) and do not necessarily reflect the official opinion of Magnimetrics. Neither Magnimetrics nor any person acting on their behalf may be held responsible for the use which may be made of the information contained herein. The information in this article is for educational purposes only and should not be treated as professional advice. Magnimetrics and the author of this publication accept no responsibility for any damages or losses sustained in the result of using the information presented in the publication.

## 2 Comments

## Nick · 07/07/2021 at 13:54

Hi, the 3rd formula is related to Private Service Companies. Altman relates the 3rd formula to Non – Manufacturer companies. So they are both Public and Private Non Manufacturer companies. Other observations – Intangible assets have been omitted from the model – ideally Total Tangible Assets should be used instead of Total Assets. Book Value of Equity should be reduced by Intangible Assets. See Altman Z score plus website for confirmation

## Dobromir Dikov · 07/07/2021 at 16:22

Hi Nick, thank you for the comments. You are correct about removing the Intangibles assets, I guess I missed that while trying to keep the example simple.