The development of corporate default prediction models began with univariate ratio analysis pioneered by [4], who demonstrated that financial ratios exhibited significant discriminant power between failing and non-failing firms. [1] advanced this field by applying Multiple Discriminant Analysis (MDA) to a sample of manufacturing firms, combining liquidity, cumulative profitability, operating efficiency, solvency, and asset turnover into a single composite metric known as the original Z-Score.

Because the original 1968 Z-Score required market capitalization to calculate the equity-to-debt ratio (X4), its application was limited to publicly traded companies. Therefore, [2] re-estimated the discriminant function for private manufacturing companies, establishing the Z’-Score model:

where X1 is Working Capital/Total Assets, X2 is Retained Earnings/Total Assets, X3 is EBIT/Total Assets, X4 is Book Value of Equity/Total Liabilities, and X5 is Sales/Total Assets.

The interpretation of the model is based on the linear relationship between the five financial ratios and the resulting weighted Z’-Score. The weights of the ratios differ from each other, beginning from 0.420 (X4) to 3.107 (X3). The linear model is simple, transparent and intuitively easy to interpret, calculate, and use. The interpretation of the Z’-Score is based on the threshold values for the score. If Z’ > 2.9, the target firm is expected to have good financial stability, so that there is only a minimal chance that it will fail (“Safe Zone”). If Z’ is between 1.23 and 2.9, it is interpreted that the risk of facing financial difficulties soon is low (“Gray Area”). Finally, if Z’ is less than 1.23, the firm is likely to face financial challenges in the near future (“Distress Zone”).

Because of the lack of a private company database, Altman did not test the performance of the Z’-Score model on a secondary sample. However, he analyzed the accuracy of a four-variable Z’-Score model that excluded the Sales/Total Assets ratio, X5, from the revised model. Altman excluded X5 because of a potential industry effect that makes the score report financial difficulties more likely to take place in low asset-turnover industries, when this kind of industry-sensitive variable (Asset Turnover) is included in the model. Thus, Altman then estimated the four-variable Z’-Score model without Asset Turnover ratio ([2]). The Z’-Score is designed especially for non-manufacturing firms and emerging markets.

Based on a literature survey, [3] showed that the Z-Score model (for publicly traded firms), the Z’-Score model (for private manufacturing firms), and the Z’-Score model (for private and publicly traded manufacturing and non-manufacturing firms) have been widely adapted for different purposes. Typically, Z-Score was either used as a failure prediction proxy or assessed in terms of predictive ability, serving as the measure of distress or of financial strength. In several studies, Altman’s original model was modified or verified, including re-estimation, or it was used for the robustness check. [15] noted that Z-Score is also used for the evaluation of the costs and benefits of covenants in bonds, the choice of debt type (bank versus non-bank, private or public), and the relationship between investment and internal funds.

[3] concluded that the broad use of the Z-Score model for measuring financial distress and performing robustness checks indicates its acceptability as a reasonable, simple, and consistent measure of financial distress. For private firms, Altman’s Z’-Score model is one of the most influential bankruptcy prediction models in financial analysis. In spite of the emergence of such advanced methods as machine learning, survival analysis, and advanced credit-risk models, the Z’-Score remains widely used in both academic research and practice. Recent literature demonstrates that researchers continue to employ the model extensively as a benchmark or explanatory variable. Studies on unlisted SMEs also re-affirm that linear discriminant models like Z’-Score may perform competitively when calibrated properly, mainly referring to re-estimation of the coefficients. This re-estimation however does not always make the revised model differ significantly from the original Z’-Score model ([8]).

In conclusion, the Z’-Score remains one of the most widely used bankruptcy prediction models globally. Its popularity is sustained by transparency, interpretability, and low implementation costs. Most financial distress studies use the model as an accepted indicator of financial distress rather than evaluating its validity. The literature is therefore heavily weighted toward application rather than critical assessment. In this study, the approach is very different from prior empirical failure studies on Z’-Score. While the predictive accuracy, classification thresholds, and statistical properties of distress models have been thoroughly investigated, a major gap persists in the literature: existing studies treat ratio components as exogenous statistical inputs rather than endogenous outcomes of fundamental firm operations. In this study, fundamental parameters such as internal rate of return, corporate growth rate, revenue lag, the fraction of debt in external capital raised, and the ratios of current assets and current liabilities are used to assess Z’-Score and its five financial ratios.

This study does not follow the mainstream of the empirical research on Z’-Score model, which is concentrated on the predictive accuracy, classification thresholds, and statistical properties of the model. The purpose of the study is to analyze the Z’-Score and its five financial ratios mathematically, utilizing an approach where they are in a steady framework described by fundamental economic variables. This kind of approach makes it possible to assess the sensitivity of the financial ratios and Z’-Score to these economic variables. The approach belongs to the family of steady-state models, which focus on the financial ratios as generated by the internal rate of return (true profitability), a growth rate, and a lag structure that distributes the revenue contributions of expenditures over future periods ([10]; [6]; [5]).

In this research line, this approach is related to the studies that examine the potential bias in the reported return on investment ratio (ROI) as an approximation of IRR ([7]; [16]; [10]; [14]; [13]; [6]; [5]). [16] concluded that ROI is not an accurate measure of IRR, and that the error is neither constant nor systematic. ROI equals IRR only if the growth rate of the firm matches its internal rate of return. [6] conclude the analytical findings of this research and characterize them as generally pessimistic. Studies analyze the behavior of firms investing usually in consecutive projects under alternative assumptions regarding depreciation policy and growth. The findings indicate that, unless very restrictive assumptions are set, ROI does not accurately measure IRR, the errors do not follow systematic patterns, and the resulting biases can be significant.

This study however does not concentrate on the relationship between ROI and IRR. It has a wider scope, following the approach by [10] to analyze a wider set of financial ratios. Following [11], this approach adopts a model structure similar to that of the research line to derive long-term (steady) financial ratios. In this study, the five financial ratios standing in Z’-Score model are derived. Accordingly, the results describe the steady-state values that financial ratios are assumed to converge over time.

The framework investigates revenue flows at the firm level by applying an infinite geometric lag structure between expenditures and the resulting revenue stream. In a steady-state setting, all flows grow at the same constant rate, and the ratios of the flows stay constant. In addition to the IRR and the growth rate, the model has the fraction of debt in external capital raised, and the ratios of current assets and liabilities as fundamental parameters. These parameters make it possible to derive approximations for the five financial ratios in the Z’-Score model. For the sake of simplicity, the framework excludes taxes, interest expenses, and dividends.

The main part of the basic concepts of the framework is derived following the presentation in [11]. This steady-state framework assumes that each periodic expenditure will generate an infinite flow of revenues in a constant relation to the expenditure. The revenue contributions generated by expenditure follow an infinite geometric distribution with lag parameter q. Thus, based on the infinite geometric distribution, the average lag between expenditure and generated revenue flow is q/(1 − q).

The steady-state assumption makes it possible to derive the revenue of the firm in period t when applying the infinite geometric lag function in the following way:

In Equation (2), R_t is revenue, M_t is expenditure, Kis the level of the revenue flow in a relation to expenditure, g is the steady growth rate of expenditure, and q is the lag parameter. The sum of lagged revenue contributions in (2) converges for infinite t if q < 1 + g. In practice, the growth rate is typically positive but can have a large variety depending on the growth strategy of the firms.

Using Equation (2), the internal rate of return (IRR) can be included into the model to solve the level K as follows:

In Equation (3), r is the constant IRR. The sum of the lagged contributions in (3) converges if q < 1+r.

Substitution of K in (3) into (2) gives the steady revenue-expenditure ratio such as:

Equation (4) shows that the revenue of the firm in period t exactly covers the periodic expenditure if r = g. It also shows that the firm can finance its expenditure by revenue and does not need external capital. However, if g > r, it follows that R/M < 1, and the firm needs external capital M-R to cover total expenditure. If g < r, it leads to that R/M >1.

Equations (2) - (4) together show that the change in the external capital can be presented as follows:

Equation (5) shows that ∆C_t = 0 if r = g, > 0 if r < g, and < 0 if r > g following R/M in Equation (4).

The steady-state model makes it possible to transform flow concepts to stock concepts and vice versa in the following way:

Thus, Equation (6) shows that the external capital of the firm (stock) can be derived from the change (flow) as:

Equation (7) shows that the stock of external capital is a function of parameters g, q, and r. In this framework, it is assumed that the external capital may consist of debt and equity. Let us denote the share of debt in the increase of external capital by B. If g > r, D_t = B ∙ C_t refers to the debt capital in the balance sheet while S_t = (1 − B) ∙ C_t refers to the (external) equity invested.

The asset side of the balance sheet can be derived assuming an expensing method for total expenditure. The most neutral assumption is that expenditures expire by the same rate as revenues are accumulated over periods ([11]). In that case, the lag distribution of expenses also follows the lag parameter q. Therefore, the periodic expense is calculated as follows:

The sum in Equation (8) converges if q < 1 + g.

Expenses E_t can be used to calculate assets according to the accounting identity using the following procedure ([11]):

Substituting E_t in (8) into (9) leads to the following result for the assets at the end of the period t:

The geometric sum in Equation (10) converges if q < 1 + g.

The periodic profit of the firm can be derived from subtracting expenses from the revenue, such as:

In (11), ∆P_t refers to the periodic profit that is a function of r, g, and q.

The periodic profit in (11) can through Equation (6) be transformed to retained earnings, leading to:

Equation (12) indicates that retained earnings P_t are closely related to the growth rate g but also related to r and q. If r < 0, P_t is negative.

Finally, current assets and current liabilities are estimated by assuming that they maintain constant proportions of total revenue and total expenditure, respectively. Specifically, current assets (e.g., cash and cash equivalents, accounts receivable, and inventories) are assumed to be proportional to revenue, with a constant coefficient (b1). Accordingly, current assets can be estimated based on revenue using Equation (6), as follows:

In Equation (13), CA denotes current assets, which are assumed to vary proportionally with revenue.

Similarly, current liabilities (e.g., short-term debt and accounts payable) are assumed to be proportional to total expenditure, with a constant coefficient (b2). Accordingly, current liabilities can be estimated based on expenditure as follows:

In Equation (14), CL denotes current liabilities, which are assumed to vary proportionally with expenditure.

The five financial ratios in Z’-Score model can be derived from the concepts in Equations (2) - (14). First, let us derive Working Capital Ratio (X1) that reflects the liquidity of the firms. In this framework, this financial ratio is approximated by the ratio of the difference between current assets and current liabilities divided by total assets. The use of Equations (13), (14), and (10) leads to the following result for the ratio X1:

Equation (15) shows that current assets (CA) depend on profitability r whereas current liabilities (CL) are independent of r but closely associated with growth g.

If b1 = b2 = b, then X1 is simplified to the following ratio:

In this simplified form, the sign of X1 depends on the difference between r and g. If g > r, X1 is negative, referring to that current liabilities exceed current assets. If g = r, X1 is equal to zero.

Second, let us derive Retained Earnings to Total Assets Ratio (X2) that measures cumulative profitability. This ratio can be derived by dividing retained earnings in Equation (12) by total assets in (10), which leads to:

This ratio (17) is independent of the lag parameter q but closely related to ratio between internal rate of return r and growth rate g.

The third ratio X3 measures periodic profitability and is defined as Earnings Before Interest and Taxes (EBIT) divided by total assets. This ratio can be calculated by dividing EBIT (profit) (11) by total assets (10) on the beginning of the year basis as follows:

Equation (18) clearly shows that X3 gives an approximation of IRR that is biased due to the effect of growth rate g. However, if r and g are close to each other, this bias is negligible.

The fourth ratio X4 refers to the financial structure of the firm. This ratio is defined as the ratio of Book Value of Equity to Total Liabilities. In this framework, total liabilities are derived by multiplying external capital in (7) by the fraction of debt B. Then, book value of equity is derived by subtracting total liabilities from total assets in (10). The financial ratio X4 is finally obtained in the following way:

In Equation (19), D_t refers to total debt (liabilities). This ratio X4 is sensitive to the difference between g and r.

The fifth ratio X5 refers to asset turnover being dependent on industry or technology. This ratio can be derived simply by solving revenue (sales) from Equations (2) and (3) and dividing it by total assets in (10), which leads to the following result:

Equation (20) shows that X5 is independent of g but is strongly related to the revenue lag q, reflecting industry effects.

The primary objective of the numerical part is to demonstrate empirically the theoretical results on the effects of model parameters on the five financial ratios X1 - X5 and Z’-Score. The main attention is concentrated on the sensitivity of the ratios and the score, to these parameters. This sensitivity is assessed in two different contexts with respect to performance: average active firm and average bankruptcy (distressed) firm. The numerical cases presented in the study are profiles assembled from separate parameter medians. Thus, the calculated financial ratios and Z’-Scores considered here do not need to be equal to the medians of observed ratios and scores reported by the firms.

The empirical data used to approximate median values of parameters are extracted from the set of Finnish limited companies in ORBIS database from a period before the COVID pandemic. The sampling was not random, since all companies in the database fulfilling the limitations set by the study were included in the sample. The latest financial statements in the data are from years 2015 and 2016. In Orbis database, firms are classified into six classes: active, active (insolvency proceedings), bankruptcy, dissolved, dissolved (merger or take-over), and liquidation. Bankruptcy code generally refers to a code or status indicating a company’s bankruptcy-related situation. However, active code means that the company is classified as active and has not been assigned an inactive status, such as bankruptcy, liquidation, or dissolution. In this study, the sample consists of only active (non-failed) and bankruptcy (failed) firms.

The sample firms are Finnish industrial limited companies which in the last reporting year report the total of the balance sheet more than EUR 1 million. In addition, it was required that the firm had in the database financial statements at least from ten last years. The initial sample included in total 12,969 active industrial firms. The number of firms, however, decreased finally to 4099 firms because of the technical limitation that the growth rate g must in the long run exceed the profitability rate r. This limitation is used to ensure that the empirical results are consistent with the theoretical results. In the same way, the sample of bankruptcy firms included originally 552 firms but the number of firms fulfilling the technical limitation g > r, was only 278. Thus, the percent of firms fulfilling the limitation was higher for bankruptcy than for active firms. The data also included missing values for a number of companies, leading to the final estimation sample for active firms of 3232 firms and for bankruptcy of 254 firms. The size of the final sample is not decisive here because only the medians of the parameters are used in the numerical experiments.

The sample of non-failed firms is more positively skewed than the sample of failed firms. The median number of employees in the non-failed group is only 16, whereas the average number is as high as 111.58. Failed (bankruptcy) firms are generally very small, since the median number of employees is only 12 and the average number is 14.12. Failed firms are largely concentrated on industries such as wholesale & retail trade, manufacturing, and construction, whereas non-failed firms are more evenly distributed across different industries. In all, 74.1% of the failed firms belong to one of these three industries, while the percentage is only 48.3% for non-failed firms.

The model parameters r, g, and q are approximated using ten-year time series following the methods used by [11] and [12]. First, the growth estimate g is calculated by dividing the sum of net sales (revenues) from the last five years by the sum of revenues from the first five years, taking the fifth root of the result, and subtracting 1 (unity). Second, the approximation of the lag parameter q is more complicated. Thus, only a simple estimation method is applied. The first step was to approximate the average lag between expenditures and revenues by dividing the sum of balance sheet totals over a ten-year period by the sum of revenues for the same period. The result of this approximation indicates how long revenues on average remain tied up in the balance sheet, reflecting revenue lag. The approximated value of parameter q is then calculated by dividing this average lag by lag plus unity according to the average of distributed lag function q/(1 − q). This parameter is closely associated with the concept of capital intensity that measures the amount of capital employed relative to revenue.

Third, a rough estimate of profitability r (IRR) is calculated simply as a weighted average by dividing the sum of the profits of the accounting period over the last ten years by the sum of balance sheet totals for the same period. It is clear that accounting policy in terms of depreciation method affects the reported accounting rate of return (ARR), making it deviate from the economic rate of return (IRR). However, in the long run, the deviations at least partly cancel each other so that the weighted average of ARRs can be used to overcome accounting policies that may distort reported profits ([9]; [14]). Fourth, parameter b1 is calculated by dividing the sum of current assets over the ten-year period by the corresponding sum of total revenue. Fifth, the parameter b2 is approximated by dividing the ten-year sum of current liabilities by the sum of total expenditure. Finally, the fraction B of debt in external financing raised is approximated by dividing the ten-year change of liabilities by the change of the sum of liabilities and equity capital.

Table 1 presents statistics for the approximate values of parameters in non-failed (active) and failed (bankruptcy) firms. In the experiments, only the median values of the parameters are used. The median liquidity parameters b1 and b2 are relatively equal in both groups of firms. The median fraction of debt B in external capital raised is very close to unity for both non-failed (0.9847) and failed (0.9885) firms, referring to the minor role of equity finance in the sample. The median IRR or r is positive but quite low for non-failed firms (0.0241), and negative for failed firms (−0.0457). The median growth rate g for non-failed firms is relatively close to ten percent (0.0944), but significantly lower for failed firms (0.0632). The median revenue lag parameter q is higher in non-failed firms (0.4053) than in failed firms (0.3409). These values refer to average revenue lags of 0.6815 and 0.5172 years, respectively.

Table 1. Statistics of the parameter values in non-failed (n = 3232) and failed (254) Finnish firms.

Table 2 presents the values of financial ratios X1 - X5, and Z’-Score calculated by the model using the median parameter values in Table 1. The weighted financial ratios show that Sales to total assets ratio X5 plays an important role in Z’-Score. However, in spite of the high weight (3.107), the positive effect of EBIT to total assets X3 on Z’-Score is low in non-failed firms whereas in failed firms, the effect is negative and strong. Retained earnings to total assets ratio X2 and Book value of equity to total liabilities X4 show a relatively significant effect on Z’-Score, positively in non-failed firms and negatively in failed firms. The effect of Working capital to total assets ratio X1 is weak, particularly in failed firms where the ratio is close to zero. The Z’-Scores calculated using the values of the financial ratios X1 - X5 refer in non-failed firms to Gray zone (2.0315 < 2.90) and in failed firms to Distress zone (0.8551 < 1.23).

Table 2. Calculated financial ratios X1 to X5 and Z’-Score for non-failed and failed firms.

Table 3 presents the partial derivatives and elasticities of Working capital to total assets ratio X1 with respect to the five model parameters. These results indicate that X1 is strongly sensitive, especially to the liquidity parameters b1 and b2. The elasticities of X1 for the sample of failed firms are high because the value of X1 is in this sample close to zero. Table 4 shows the results for Retained earnings to assets ratio X2. This financial ratio is very sensitive to internal rate of return rbut also to growth rate g. The partial derivative of the ratio with respect to g is however negative for non-failed firms, but positive for failed firms, indicating different roles of the parameter in these samples.

Table 5 presents the results for Earnings before interest and tax to total assets ratio X3. This important profitability ratio is very sensitive to internal rate of return r. It is also slightly affected by growth g. The signs of the partial derivative with respect to g in non-failed and failed firms are again different due to different signs of IRR. Table 6 shows the findings for Book value of equity to total liabilities ratio X4. This ratio is most sensitive to r and g followed by B. For this ratio also, the derivatives show a different sign in non-failed and failed firms. For non-failed firms,

Table 3. Partial derivatives and elasticities of working capital to total assets (X1).

Table 4. Partial derivatives and elasticities of retained earnings to total assets (X2).

Table 5. Partial derivatives and elasticities of earnings before interest and tax to total assets (X3).

Table 6. Partial derivatives and elasticities of book value of equity to total liabilities (X4).

the derivative is negative, whereas for failed firms it is positive. Table 7 presents the results for Sales to total assets ratio X5. This ratio is negatively very sensitive to revenue lag q, and slightly to internal rate of return r.

Table 7. Partial derivatives and elasticities of sales to total assets (X5).

The mathematical formulas for partial derivatives of Z’-Score are presented in Table A1. Since Z’-Score is a linear weighted function, also partial derivatives are weighted sums of the derivatives of the financial ratios X1 - X5. Table 8 presents the partial derivatives and elasticities of Z’-Score with respect to the parameters of the model. The results in Table 8 indicate for both groups of firms that profitability r clearly plays the central role in the sensitivity of the score. In addition to profitability, also revenue lag q and growth rate g have a significant impact on Z’-Score. The partial derivative of q is negative in non-failed firms but positive in

Table 8. Partial derivatives and elasticities of Altman’s Z’-Score.

failed firms.

The sensitivity of Z’-Score is related to the difference between g and r: the narrower is the difference, the higher sensitivity. Table 9 presents the estimated partial derivatives and elasticities when it is assumed that g = 0.04 in non-failed firms and g = 0.01 in failed firms. In these exemplary cases, the difference between g and r is narrower, when compared with the baseline case. Because of this narrower difference, the partial derivatives and elasticities of r and g are in absolute terms significantly higher. The decrease in the growth rate g also leads to that in non-failed firms Z’-Score rises significantly closer to Safe zone (still being in Gray zone), whereas in failed firms the score falls very low in Distress zone.

Table 9. Effect of lower growth rate on partial derivatives and elasticities of Z’-Score.

Table A2 presents examples of numerical ceteris paribus effects of liquidity (current assets) parameter b1 on Z’-Score. These examples show that the effect of b1 is too weak in non-failed firms to rise alone the score from Gray zone to Safe zone, even if the level of current assets increases very significantly. Similarly, even if b1 has fallen very notably, the score stays on Gray zone. In failed firms, the effect of b1 on Z’-Score is also weak. Thus, b1 should rise to a very high level to move the score of failed firms from Distress zone to Gray zone.

Table A3 presents similar numerical examples for parameter b2 that refers to the level of current liabilities. The effect of b2 is weak similarly to b1, and even a significant decrease in b2 down to zero cannot increase the score of the non-failed firms from Gray zone to Safe zone. In the same way, even a significant increase in b2 does not make the score move to Distress zone. For the distressed firms, b2 should fall to a level close to zero to increase the Z’-Score from Distress zone to Gray zone.

Table A4 presents examples of the score effects of parameter B, which refers to the fraction of debt in external capital risen. The baseline value of B is in both groups of firms very high, so that the limits to increase it are narrow and do not change the zone for Z’-Score. However, when B falls very low in the sample of failed firms, it makes the score to rise from the Distress zone to Gray zone. However, generally the effect of B on Z’-Score is weak.

Examples of numerical effects of profitability r on Z’-Score are presented in Table A5. Z’-Score is very sensitive to this parameter. If in non-failed firms, profitability falls from the baseline level to a relatively low, negative level, the score falls from Gray zone to Distress zone. However, a small increase in r from the baseline profitability makes the score rise from Gray zone to Safe zone. Similarly, a small increase in r that can still be negative makes failed firms to move from Distress zone to Gray zone. If r is increased to a relatively low positive level, Z’-Score is rising to Safe zone. It can make this movement to Safe zone for a lower profitability rate than non-failed firms can. Thus, generally, even a small change of rcan make significant changes in Z’-Score.

Table A6 presents the results for the score effects of growth rate g. For the non-failed firms, the effect on Z’-Score is more significant, the closer growth rate g is profitability r. When g is close to r, the firm moves in Gray zone towards Safe zone. For high values of g, the effect on score is weak and even a very high growth rate g does not lead the score to fall to Distress zone. For failed firms, growth g has an opposite effect on the score. Thus, the score is higher, the higher is g. When g is relatively high, Z’-Score moves from Distress zone to Gray zone.

Finally, the numerical effects of revenue lag parameter q on Z’-Score are presented in Table A7. For both groups of firms, Z’-Score is higher, the lower is revenue lag q. If q in non-failed firms rises significantly from baseline to a higher level, the score moves from Gray zone to Distress zone. If q is significantly lower than baseline, Z’-Score moves to Safe zone. In failed firms, the score moves from Distress zone to Gray zone when g falls to a relatively low level. However, if the firm wants to move further to Safe zone, it should considerably shorten the revenue lag q.