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Bankruptcy prediction for US listed companies between 2010-2016
Student: Bart Blok
Student number: 10419381
University of Amsterdam, Department of Business and Economics
Specialization: Economics and Finance
Date: 2 February 2016
Statement of Originality
This document is written by student Bart Blok (student number: 10419381) who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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1. Introduction
Bankruptcy prediction models are meaningful tools to investors, creditors, asset managers, rating agencies and even for companies themselves. Since 1968, a lot of models have been developed to assist shareholders by evaluating bankruptcy risk. Since the financial crisis of 2008, financial assessment and bankruptcy prediction proved to be even more relevant than before.
Accounting-based models are models that describe the relationship between accounting ratios that best differentiate between matched samples of bankrupt and non-bankrupt companies. In the existing literature, these models (Altman, 1968., Ohlson, 1980., Zmijewsky, 1984) are often criticized for their theoretical shortcomings. First, according to Agarwal and Taffler (2008, p. 1542) accounting data measure past performance. Therefore they may not be very informative about the future status of the company. Secondly, Hillegeist et al. (2004, p. 6) acknowledge that accounting data are subject to conservatism. Conservatism might hinder a true representation of asset values, since they are often understated relative to their market values. This causes leverage measures to be overstated, which in turn creates an upward bias for any accounting-based probability measure. Third, financial statements are prepared using the going-concern principle, which assumes that companies will not go bankrupt. Therefore they are, by design, limited to predict a future that may violate this assumption. Furthermore, Hillegeist et al. (2004, p.6) claim that the absence of a measure of asset volatility in accounting-based models. is a very important deficiency. Volatility is a crucial variable in bankruptcy prediction because it captures the likelihood that the value of a company’s assets will decline to such an extent that it won’t be able to repay its debt (Hillegeist et a.l, 2004, p.6). Last, Agarwal and Taffler (2008, p.1542) claim that accounting numbers are subject to manipulation by management.
Compared to accounting-based models, market-based models are theoretically more attractive for several reasons. First, in these models a volatility measure is included. Secondly, according to Agarwal and Taffler (2008, p.1542) market variables are unlikely to be influenced by a company’s accounting policy. Third but more importantly, market-based models draw on the option-pricing framework of Black and Scholes (1973) and Merton (1974). From this perspective, equity is viewed as a call option on the company’s assets with a strike price equal to the face value of a company’s debt. The probability of going bankrupt is the probability that the call option is worthless at maturity. The option-pricing framework assumes “efficient” markets, in which stock prices “fully reflect” all publicly available information (Fama,
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1970, p.383). In other words, stock prices reflect all information in the financial statements plus information outside the financial statements, including predictions about a company’s future earnings. Therefore, market-based models seem to be more appropriate for prediction purposes.
However, Bharath and Shumway (2008, p.1340) claim that the performance of a market-based model depends on how realistic its assumptions are. Sloan (1997, p.289) finds that stock prices do not fully reflect information about future earnings. Furthermore, market-based models assume the normality of stock returns and assume that a company has only one zero-coupon bond. More importantly, Bauer and Agarwal (2014, p.433) state that some key variables are unobservable and need to be approximated introducing potentially large errors. In the end, whether an accounting-based model or a market-based model performs better is an empirical question.
In a study on publicly listed industrial companies in the UK, Agarwal and Taffler (2008, p.1542) find that the UK z-score model from Taffler (1983) was marginally more accurate than a carefully developed market-based model. However, the difference was not significant. In sharp contrast with this finding were the results of a study from Bauer and Agarwal (2014, p.438). Due to a larger sample period, they found that the UK z-score model from Taffler (1983) produced significantly lower accuracy rates than the market-based model from Bharath and Shumway (2008).
The benefit of a market-based model is that it can be applied to any publicly traded company, because its probability measure is computed independently with use of the theoretical BSM option-pricing framework. This is not the case for an accounting-based model. Such a model is estimated in a first-stage regression and is as a result, specific to the characteristics of the sample used in the regression. Characteristics include for example the time-horizon of the estimates, the type of industry, accounting rules and economic conditions. Consequently, an accounting-based model can only appropriately be applied to the population from which it was developed. For this reason, it would be incorrect to simply assume that the results of Bauer and Agarwal’s study also hold for companies in the United States. A comparison between the z-score model of Altman and a market-based model in the US could potentially yield different results. For this reason, I decided to compare the accounting-based z-score model of Altman (1968) to the market-based model of Hillegeist et al. (2004). My research question is:
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Which model is the most accurate predictor for industrial business failure in the US? There are three ways to assess the discriminatory power of bankruptcy prediction models. Bauer and Agarwal (2014) state that the evaluation procedure depends upon the context. In this thesis, the primary interest is to identify the most accurate model. Therefore, I will try to identify the model with the highest accuracy ratio with use of the ROC method. Additionally, I will assess whether different models carry information incremental to each other through use of information content tests.
The paper proceeds as follows. In the next section, the Altman z-score model is discussed. The third section talks about market-based models. The fourth section describes the hypotheses. The fifth section describes the data and method used. In the sixth section the results are presented and the seventh section concludes.
2. Atlman z-score model
Financial ratios for liquidity, profitability, leverage, solvency and activity ratios are practical tools to assess the financial stability of companies. Until 1967, the dominant methodology was univariate analysis and practitioners focused on individual signs of impending difficulties (Altman et al., 2014, p.3). Therefore, ratio analyses could be confusing and was easily interpreted in a wrong way. Academics seemed to be moving forward to the elimination of the ratio-analysis, but Altman bridged the gap between the ratio analysis and the more rigorous statistical techniques (Altman, 1968, p. 589). Altman’s z-score model was the first accounting-based model and ever since a lot of other accounting-based models were developed. These models describe the relationship between ratios that best differentiate between matched samples of bankrupt and non-bankrupt companies.
Before explaining the model, it should be mentioned that Altman’s z-score model is not solely an accounting-based approach, because the model also utilizes the market value of equity. The z-score model of Altman was estimated using a matched sample of sixty-six manufacturing companies in the period 1945-1964. He used multiple discriminant analysis (MDA), which is a statistical technique that uses the characteristics of an observation to classify it into one of several a priori groupings (Altman, 1968, pp.591-592). When applied for bankruptcy prediction, the characteristics are several financial ratios and the two groupings are a bankrupt and a non-bankrupt group.
The classification is based on the z-score of a company. If the z-score is below a predetermined cut-of it is classified as bankrupt. A company with a z-score above the cut-of is classified as non-bankrupt. Thus, the lower the z-score, the higher the chance of being classified as bankrupt. To find the relationship that best
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discriminates, Altman considered twenty-two ratios in different combinations. It should be mentioned that he did not only looked at the statistical significance of each independent variable, but considered also the relative importance of the variables to the contribution of the model as whole. The univariate insignificance of the sales/total assets ratio is therefore justified, because the importance of MDA is its ability to separate groups using multivariate measures (Altman, 1968, p.597). He found that the combination of five ratios described below performed best:
𝑧 = 1.2
!"!"+ 1.4
!"!"+ 3.3
!"#$!"+ 0.6
!!!"
+ 1.0
!"#$!!"
,
where WC is working capital, TA is total assets, RE is retained earnings, EBIT is earnings before interest and taxes, 𝑉! is market value of equity and TL is total liabilities.
2.1 The ratio’s
The working capital/total assets ratio is a measure of the net liquidity of a company relative to its total assets. A company that experiences operating losses for a longer period, will see this ratio decrease and becomes less liquid. The positive coefficient illustrates that a company with a low ratio will have a lower z-score and has a higher probability of being classified as bankrupt.
The retained earnings/total assets ratio is a measure of a company’s cumulative profitability over time relative to its total assets. A younger company typically has a lower value for this ratio since it has had less time to built up cumulative profit. Therefore the ratio is very realistic as the incidence of failure is much higher in a company’s earlier years (Altman, 1968, p.595). The positive coefficient shows that, for this ratio too, the lower the ratio the higher the chance of being classified as bankrupt.
The earnings before interest and taxes/total assets ratio is a measure of the true productivity of the company’s assets without considering any tax or leverage factors. Since a company’s ultimate existence is based on the earning power of its assets, this ratio appears to be particularly appropriate for studies dealing with corporate failure (Altman, 1968, p.595). Again, the positive sign of the coefficient shows that the lower the ratio the higher the chance of being classified as bankrupt.
The market value of equity/book value of total debt ratio is a measure of a company’s solvency. It shows how much the company’s assets can decline in value before the liabilities exceed the assets and the company becomes insolvent. Assets are measured by market value of equity plus debt. For example, a company with a market value of equity of $1000 and with $500 debt can bear a drop of two-thirds in
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assets before it becomes insolvent. In contrast, a company with a market value of equity of $250 and debt of $500 can only bear a drop of one-third. And again, the positive coefficient for this ratio is what was expected.
The last ratio is the sales/total assets ratio. It is a measure of the ability of a company’s assets to generate sales. In another way, it could be seen as a measure of the management’s capability in dealing with competitive conditions. And also for this last ratio the coefficient was positive, illustrating that a company with a lower sales/total assets ratio has a higher chance of being classified as bankrupt.
2.2 Validation
After the establishment of the model, Altman applied it to the same matched-sample of sixty-six companies (1968, p.599). This procedure is called an in-sample “prediction”. All companies with a z-score greater than 2.99 were clearly classified as non-bankrupt, while all those with a z-score less than 1.81 went bankrupt within the following year. Altman defined the area between these two points as the “grey zone” because in this zone misclassifications could be observed (1968, p.602-603). For out-of-sample predictions it would be necessary to determine a stricter classification rule, or a so-called optimal cut-off point. Naturally, the point at which the least companies were misclassified should lie between these two values and appeared to be 2.675. With use of his optimal cut-off, Altman reported a not 95% accuracy rate on his first in-sample test (1968, p.599).
To validate his model, Altman performed two additional out-of-sample tests. The first test was on a sample of twenty-five bankrupt companies whose asset-size range was the same as that of the initial bankrupt group. Remarkably, this test produced an even higher (96%) accuracy rate, for which two possible reasons were provided. First, the model is something less than optimal. Second, no upward search bias is present in Altman’s validation. This bias is normally present in any empirical study and inherent in the process of intensive searching for the best combination of explanatory variables (Altman, 1968, pp.600-601). The second test was, in particular, more rigorous since it was performed on a group of sixty-six companies that experienced temporarily earning problems, but actually did not go bankrupt. Altman reported a very impressive 79% accuracy rate on this test (1968, pp.602). These results seemed to confirm the reliability of Altman’s model. However, subsequent tests for prediction up to five years before failure were, as one would expect, less impressive. The accuracy of the model decreased from 95% in year one, to 72% in year two, to 48% in year three and further (1968, p.604). From Altman’s study it can
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be concluded that his model was a reliable predictor for bankruptcy up to two years before bankruptcy.
2.3 Application to other populations.
As stated before, accounting-based models are specific to the characteristics of the sample used in the first-stage regression. In other words, when applied to a population with different characteristics the results are expected to be biased. A population with different characteristics can be a population from a different time with different economic conditions, accounting rules, industries or asset sizes.
Macroeconomic conditions like inflation, interest rates, credit availability and the state of the economy are constantly changing and affect the financial condition of companies. Therefore, the coefficients of the ratios and ranking in their relative contribution to the model are expected to change over time. For example, in times with an increasing acceptance of relatively high corporate debt levels (this was the case in the 1980’s), a given level of debt may not be associated with the same likelihood of bankruptcy as before (Begley et al. 1996, pp.267-268). This hypothesis was confirmed by Mensah. He found that the accuracy and structure of predictive models differ across different economic environments (Mensah, 1980, p.393).
Additionally, the Bankruptcy Reform Act of 1978 allowed for greater strategic use. Delaney (1992, pp.9-11) states that bankruptcy was not anymore only about bad management or poor financial health. Creative lawyers came up with new ways to stretch the bankruptcy code to handle specific problems. Among others, reasons for filing were avoiding cleaning up toxic waste dumps, eliminating unions, or reducing exposure in asbestos-related legislation (Delaney, 1992, p.9-11). Begley et al. (1996, p.276) claim that if such reasons are uncorrelated with the financial variables in accounting-based models, these models are expected to yield additional misclassifications.
Furthermore, Mensah states that different prediction models seem appropriate for companies in different industrial sectors even for the same economic environment (1980, p.393). In particular, the Altman z-score model is expected to lack from a industry-effect due to its inclusion of the sales/total assets ratio. This ratio varies wide among industries and may therefore have an effect on the boundary between bankrupt and non-bankrupt companies (Altman, 1983, p.108). Altman recognized this problem and estimated a model without a sales/total assets ratio for non-manufacturing companies (1983, p.124). Unfortunately, this model was estimated only for private companies.
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Last, when applying the original Altman model to a group with different asset-sizes an extra bias is expected to be present because the boundary between bankrupt and non-bankrupt companies is different for small and large companies (Altman et al. 2014, p.8). The original Altman z-score model was estimated on a sample of companies with assets ranging from 1 - 25 million US dollars. At the time, there was not enough data on small companies and for large companies bankruptcy was quite rare (Altman, 1968, p.593).
For all above reasons it seems wise to re-estimate the model to improve the predictive accuracy. However, both Hillegeist et al. (2004, pp.22)) and Begley et al. (1996, pp.276-278) found that a re-estimation of the Altman model did not improve the predictive accuracy of the model. Therefore, the original model seemed still quite robust.
3. Market-based models
Market-based models draw on the option-pricing framework of Black and Scholes (1973) and Merton (1974). In these models, equity is viewed as a call option on the company’s assets with a strike price equal to the face value of a company’s debt. When the value of the assets is below the face value of liabilities, the call option is left unexercised and the bankrupt firm is turned over to its debtholders (Hillegeist et al., 2004, p.6). In other words, the probability of going bankrupt is the probability that the call option is worthless at maturity.
Before explaining how this probability can be derived, it should be mentioned that the performance of such a probability is limited by the simplifying assumptions on which its underlying model is built. The BSM-model assumes “efficient” markets and normally distributed returns on the underlying assets (Black and Scholes, 1974, p.640). Additionally, the risk-free rate and volatility are known and constant. In the BSM-model the value of an option can be calculated with use of four variables (underlying asset price, strike price, risk-free rate and time-to-maturity) that are easily observed, and one variable (volatility) that can be estimated. However, in the case of bankruptcy prediction the implementation is a bit problematic. The market value of assets (𝐴), the volatility of the assets (𝜎!) and the expected return on the assets (𝜇!)
are not easily observable and must be approximated, introducing potentially large errors (Bauer and Agarwal, 2014, p.433). Furthermore, a market-based probability measure assumes that the face value of debt is equal to a zero-coupon bond, while in practice most companies have different types of debt with different maturities.
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3.1 The model
The first market-based probability measure was called the Merton Distance to Default model (Bharath and Shumway, 2008, p.1344). The formula for the distance to default is stated below.
𝐷𝐷 =
𝑙𝑛 𝐴 𝐷 + 𝜇
!− 0.5 ∗ 𝜎
!!𝑇
𝜎
!∗ √𝑇
With A = the market value of assets, 𝜎! = the volatility of A, D = the face value of debt and 𝜇! = the
expected return on assets.
In essence, the model subtracts the face value of debt from the market value of assets and divides the difference by the volatility of the assets, scaled to reflect the horizon of the forecast. The resulting z-score or DD is the number of standard deviations that the company is away from default. Thus the smaller the standard deviation, the larger the probability of default. The corresponding implied probability of default can be calculated by use of the DD in the cumulative normal density function N(.).
𝑃 = 𝑁(−𝐷𝐷)
The DD measure can be calculated by simultaneously solving two non-linear functions. However, Bharath and Shumway (2008, p.1356) prove that the true value of the DD model lies in its functional form rather than in the simultaneous solvation of the model. Therefore, they suggest a more simple “naïve” version that preserves the functional form, but avoids any solving or estimation procedures in its calculation (2008, p.1347). They approximate the volatility of the market value of debt in the following way:
𝜎
!= 0.05 + 0.25 ∙ 𝜎
!,
with 𝜎! = volatility of the market value of equity.
The rational for this approximation is that companies that are close to bankruptcy have very risky debt, and the risk of their debt is correlated with their equity risk. Bharath and Shumway (2008, p.1347) include the five percentage points to represent term structure volatility, and the 0.25 factor to allow for volatility associated with default risk. Accordingly, the total volatility of a company can be approximated by the
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weighted average of the volatility of debt and the volatility of equity in the following way:
𝜎
!=
!!!!𝜎
!+
!!!!𝜎
!.
Note that this formula implicitly implies that 𝐸 + 𝐷 = 𝐴. Additionally, it should be mentioned that Bharath and Shumway’s (2008, p.1348) modeling is not particularly well motivated but their objective was to construct a very easy to calculate predictor that might have significant predictive power. Remarkably, their predictor seemed to be very reliable. In subsequent test with hazard-models and out-of-sample forecasts it performed even better than the original DD model (Bharath and Shumway, 2008, pp.1354-1358).
The third and last unobserved key variable that needs to be estimated is the expected return on the assets (𝜇!). There are a couple of ways to do this and all methods seem to have their shortcomings. However, Agarwall and Taffler (2008, p.1546) show that the exact specification do not have an material impact on the forecasting ability of the model.
4. Hypotheses
The Altman z-score model and a market-based model will be compared using data of forty-eight US industrial and service companies between 2010 and the time of writing. Initially, I wanted to study bankruptcy for US industrial companies for the two most recent years 2014 and 2015. However, bankruptcy is a rare event and therefore I had to increase the timeframe and I also decided to include fourteen service companies.
I expect the z-score to be biased for three reasons. First, because of the inclusion of fourteen services companies, I expect that the industry effect will be present. Second, as discussed in section 2.3, since the development of the z-score model accounting rules as well as macro-economic conditions have changed. Third, the asset range of companies in my sample is very large. It includes companies with assets of 3000 USD but also companies with assets of 2800.000 USD. The boundary between bankrupt and non-bankrupt companies is different for small and large companies, and therefore, the model is expected to be biased.
For the market-based model, these kind of biases are not expected since it is not a descriptive model but draws on a fundamental theory. Although, some key variables need to be estimated, I think that the use of market information and especially the inclusion of a volatility measure will lead the market-based model to
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have more predictive ability compared to the Altman z-score model. Therefore, my null and alternative hypotheses are:
H0: The market-based model and the Altman z-score model have equal predictive accuracy.
H1: The market-based model has more predictive accuracy than the Altman z-score model.
5. Data and Method
5.1 Sample selection
All financial data was retrieved from COMPUSTAT North America. The COMPUSTAT database contains data that goes back to 1992 for companies that are or were listed on the S&P1500. To find companies that went bankrupt, I searched for companies that were deleted from COMPUSTAT between 2010 and the time of writing. This resulted in a list of 35 companies. Somehow, four companies that should have been deleted in the years 1992-1995 were included in the list. These were deleted. To be included in the final sample, the company and the data had to meet the following requirements. First, all data required for the models should be available. Second, the bankruptcy filing date should be prior to the data date. This is to ensure that financial statement data treated as available prior to bankruptcy was not released following the bankruptcy filing. Third, the data had to be older than six months prior to filing. I decided to set this requirement because just after the disclosure of the financial statements most filing for bankruptcies do not come as a sudden surprise. Therefore it would be more interesting to use data older than six months prior to filing. Fourth, the data had to be no older than eighteen months prior to filing. This criterion was also applied by Begley et al. (1996, p.270) and is to ensure that the data is still relatively current. The number of companies that met these four requirements was twenty-five.
5.2 Matching
To include non-bankrupt companies in my sample, I searched for matches on fiscal year, SIC code and on asset size, exactly in that order. In the case of Trailer Bridge (SIC: 4213), a very close match on asset size seemed impossible. Therefore, I decided to relax the SIC code by one digit. This yielded a match with Universal Truckload Services (SIC: 4210). The relaxation implies that the business is somewhat less comparable, but it seemed more reasonable than a match with an incomparable asset size. For Corinthian Colleges, I wasn’t able to find a suitable
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match. Relaxing SIC would yield a closest match with a company that had assets six times smaller or a company with assets eight times larger. Therefore, I decided to delete Corrinthian Colleges from the sample. This means that my final sample consisted of fourty-eight companies. The table with company names, their assets and the difference in assets with their matches is presented in the table in the appendix. The largest difference in asset size was 82% and the standard deviation of the difference is 0.24.
5.3 Estimation of variables
The unobserved variables for the market-based models (the market value of assets, the volatility of assets and the expected return on assets) are estimated following the methods of Bharath and Shumway and Hillegeist et al. (section 3.1). That is, the market value of assets is approximated by the market value of equity plus total liabilities.
𝐴 = 𝐸 + 𝐷
The volatility of assets is approximated by a weighted average of the volatility of equity and the volatility of debt.
𝜎
!=
𝐸
𝐸 + 𝐷
𝜎
!+
𝐷
𝐸 + 𝐷
𝜎
!The volatility of equity (𝜎!) is estimated using the historical stock prices. This
estimate will be more accurate with more observations. Therefore I gathered daily stock prices for one year prior to the disclosure of a company’s financial statements. With the prices I calculated the daily returns. To find the annualized volatility I multiplied the stocks volatility with the square root of the number of observations. For most companies, the number of observations was 250 as this is typically the number of trading days per year. The expected return of assets is calculated by:
𝜇
!=
𝐴
!+ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠 − 𝐴
!!!𝐴
!!!and bounded between the risk-free rate and 100%. For the risk-free rate I used the rate on 20-year treasury bills. For the face value of debt I used total liabilities.
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5.4 Receiver Operations Characteristics
To compare the results of the models, the z-scores are transformed into probabilities using a logistic transformation:
𝑃 =
1
1 + 𝑒
!!!"#$%Since the z-score model is estimated using an MDA technique this is not necessarily correct, but McFadden (1976, p.519-520) shows that the MDA and logit approaches are closely related under normality assumptions.
The first method to evaluate the performance of the models is called Receiver Operations Characteristics (ROC). This method is typically used in the field of medicine but Bauer and Agarwall (2014, p.436) state that ROC is also very suitable to assess bankruptcy prediction models. The ROC-curve is presented in a graph with on the x-axis the percentage of companies with highest default risk (x) and on the y-axis the actual percentage of failed companies (y). The curve was constructed in the following way. First, for each fiscal year the probabilities were sorted from high to low. Second, for each percentage of companies with highest default risk, I calculated the percentage of actually failed companies (actually failed companies divided by total companies in the sample). In the end, all years were cumulated together. The area under the curve is an indication of the accuracy of the model. The higher the area the more accurate the model. The area under the curve is approximated in the following way:
𝐴𝑈𝐶 = 𝑆𝑢𝑚 𝑜𝑓 0.5 𝑥
!− 𝑥
!∗ 𝑦
!− 𝑦
!+ ⋯ + 0.5 𝑥
!!− 𝑥
!""∗ 𝑦
!""− 𝑦
!! To judge whether the difference between the area under the curve’s is significant, Hanley and McNeil (1983, p.840) suggest the following normally distributed test statistic:𝑧 =
!"#!!!"#!!" !"#! !! !" !"#! !!!!"# !"#! !" !"#!
Where se(AUC) are the standard errors of the area’s and r is the correlation between the two area’s.
The r can be obtained from a table provided by Hanley and McNeil (1983, p.841). To obtain the r from the table, the average AUC and the average correlation of r1 and r2 are needed. The r1 is the correlation between the probabilities of the two models for
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failure group. Furthermore, Hanley McNeil (1982, p.31) suggest the following calculation for the standard deviation of the area under the curve:
𝑠𝑒(𝐴𝑈𝐶) =
𝐴𝑈𝐶 1 − 𝐴𝑈𝐶 + 𝑛
!− 1 𝑄
!− 𝐴𝑈𝐶
!+ (𝑛
!"− 1)(𝑄
!− 𝐴𝑈𝐶
!)
𝑛
!𝑛
!"Where 𝑛!= the number of failures, 𝑛!"= the number of non-failures, 𝑄!=!!!"#!"# 𝑎𝑛𝑑 𝑄!=!!"#
!
!!!"# According to Engelmann et al. (2003, p.2) the accuracy ratio is just a linear transformation of the area under the ROC-curve. Therefore, the accuracy ratio can be calculated in the following way.
𝐴𝑅 = 2 ∗ (𝐴𝑈𝐶 − 0.5)
5.5 Information Content Tests
In addition to the ROC method, I will use information content tests to evaluate the performance of the models. Hillegeist et al. (2004, p.19) state that prediction oriented tests are less valid tests for two reasons. First, a prediction-oriented test typically involves a predetermined cut-off value that is used to classify it into the bankrupt or non-bankrupt group (e.g the validation technique of Altman). However, a decision-maker is typically not faced with a dichotomous decision but with a continuous decision. In other words, rather than providing a loan or no loan, a creditor will typically charge a higher interest for a higher probability of bankruptcy. Second, in prediction oriented tests the total error rate is typically used to assess the accuracy of a model. However, when comparing models by total error rates one assumes that the costs of the misclassification of a type I and a type II error are equal. This assumption is not very realistic. According to Hillegeist et al. (2004, p.19) this implies that, for example, a regulator will decide whether or not to take over a company without taking into consideration that the cost of allowing a failing company to continue (type I error) may be many times greater than the cost of shutting down a solvent company (type II error). Agarwal and Taffler agree with this view and acknowledge that error costs are context specific (2008, p.1545). Therefore, I will assess the models not only with prediction oriented tests but also with information content tests.
An information content test is a test to check if an independent variable (or set of variables) contains information to explain the variation in a dependent variable. The dependent variable is in the case of bankruptcy a dummy variable that equals 1 if the company went bankrupt and equals 0 otherwise. The independent variable is
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the calculated probability by either the z-score model or the market-based model. Hence, to compare the two models I will conduct two information content tests. Following the existent literature I will use a logit model.
6. Results
6.1 Summary statistics
Graph 1 shows the lead time of the twenty-four bankrupt companies. The average lead time was 12.25 months which means that on average companies went bankrupt after 12.25 months.
Table 1 reports descriptive statistics for the accounting variables, the ratio’s used in the z-score model and the probability of the models. It also reports t-tests of the null hypothesis that the sample means of the bankrupt companies are equal to the sample means of the non-bankrupt companies.
Table 1. Bankrupt Companies Non-Bankrupt Companies
Variable mean Std. dev min max mean Std. dev min max t-test Working Capital 11 140 -300 497 94 197 -31 786 -1.69* Retained Earnings -303 333 1083 - 105 38 592 1616 2175 -2.46** -EBIT -32 86 -316 90 36 59 -26 218 - 3.20*** Total Assets 510 700 3 2827 490 654 3 2822 0.10 Market Equity 97 111 0 406 485 637 3 2775 -2.94*** Debt 461 723 4 3224 191 257 4 994 1.72* Sales 726 1837 2 8814 488 748 0 2929 0.59 0 1 2 3 4 5 6 7 8 6 7 8 9 10 11 12 13 14 15 16 17 18 Fre que n cy Months before 1iling Graph 1. Lead time bankrupt companies
16 WC/TA -0.30 0.99 -4.40 0.50 0.12 0.30 -0.91 0.53 -1.97* Ret.Earnings/TA -2.89 7.89 39.20 0.20 -1.26 5.42 - 26.31 - 0.87 -0.83 EBIT/TA -0.22 0.66 -2.90 0.30 0.02 0.32 -1.28 0.78 -1.59 Equity/Debt 0.59 1.00 0.00 4.00 5.05 7.22 0.17 34.92 -3.00*** Sales/TA 0.92 0.73 0.00 3.10 0.98 0.89 0.00 3.30 -0.26 P(z) 0.58 0.31 0.07 1.00 0.18 0.29 0.00 1.00 4.61*** P(m) 0.34 0.19 0.01 0.68 0.05 0.11 0.00 0.43 6.28***
All accounting variables are in thousands US dollars. * is significant at the 10% level (two-tailed), ** is significant at the 5% level (two-tailed), ***is significant at the 1% level (two-tailed), WC=working capital, TA=total assets, EBIT=earnings before interest and taxes, P(z)=probability of the z-score model, P(m)= probability of the market-based model
From table 1 it becomes clear that non-bankrupt companies have higher average working capital, EBIT, market equity and lower average liabilities as would be expected. However, surprisingly non-bankrupt companies seem to have lower average retained earnings and sales. Furthermore, it can be concluded that the variables EBIT and market equity seem to contain the most significant information to differentiate bankrupt and non-bankrupt companies. Both their t-tests report 1% significance levels. The t-test for retained earnings reports a 5% significance level, the t-tests for working capital and debt report significances at the 10% level. The sales and total assets are not significant at all. For total assets this is no surprise since the matching was based on assets. The very high correlation (table 2) between the assets and sales (0.83) might explain the insignificance of the sales variable. More importantly, from the five ratio’s used in the Altman z-score model, only the market equity/debt ratio is able to discriminate significantly (1% level) between bankrupt and non-bankruptcy. In contrast, Altman reported 1% significant levels for the first four ratios. This change in univariate significance implies that the explanatory power of these ratio’s has been decreased and therefore the z-score model is expected to produce lower accuracy rates than originally reported.
Furthermore, both models seem to contain discriminatory power. The average probabilities for the bankrupt group are higher than the average probabilities of the non-bankrupt group (both extremely significant). The average probability of the z-score model is higher than the average probability of the market-based model for both bankrupt and non-bankrupt companies.
17 Table 2. Correlations
WC RE EBIT TA Mark.
Equity
Debt Sales P (z) P (m) Failure
WC 1.00 RE 0.31 1.00 EBIT 0.40 0.49 1.00 TA 0.71 0.19 0.25 1.00 Mark. Equity 0.77 0.57 0.49 0.62 1.00 Debt 0.44 -0.14 0.01 0.86 0.23 1.00 Sales 0.49 -0.05 0.06 0.83 0.35 0.93 1.00 P(z) -0.30 -0.41 -0.55 -0.34 -0.42 -0.18 -0.30 1.00 P(m) -0.21 -0.29 -0.37 -0.06 -0.41 0.17 0.07 0.53 1.00 Failure -0.24 -0.34 -0.43 -0.01 -0.40 0.25 0.09 0.56 0.68 1.00
WC=working capital, RE=retained earnings, EBIT=earnings before interest and taxes, TA=total assets, P(z)=probability of the z-score model, P(m)= probability of the market-based model
Table 2 shows a correlation of 0.53 between the two models indicating that the two models are carrying information incremental to each other. Furthermore, it can be concluded that both the models have a relatively high correlation with the actual outcome of failure indicating that both models have explanatory power. However, the market model has a higher correlation compared to the z-score model.
6.2 Tests of predictive ability
In graph 2 both the ROC curves for the different models are presented. Additionally, I presented the ROC curve for a random model, a forty-five degrees line. The area under the curve (AUC) for the market based model is clearly greater than that of the z-score model.
18 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,00 11,00 21,00 31,00 41,00 51,00 61,00 71,00 81,00 91,00 101,00 % o f f ai le d c om p an ie s % of companies Graph 2. ROC-curve
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Table 3 shows the AUC, the standard error of the AUC, the accuracy ratio (AR), the correlation between the area’s (r), and the corresponding test statistic for the difference between the AUC’s.
Table 3. z-score model market model
AUC 0.635 0.660 se(AUC) 0.00335 0.00329 AR 0.2700 0.3192 Average(AUC) 0.647 r1 0.071 r2 0.058 Average(r) 0.065 r 0.050 test statistic -5.371
AR is the accuracy ratio. The 𝑟! is the correlation between the probabilities of the two models for the
failure group and the 𝑟! is the correlation between the probabilities for the no-failure group. The r is
obtained from the table provided by Hanley and McNeil (1983). To obtain the r, the average AUC and the average correlation of 𝑟! and 𝑟! are needed.
It should be mentioned that Hanley and McNeil’s (1983, p.841) table only reports for average AUC values of 0.70 and higher. The actual average AUC is 0.647 but I decided to treat it as 0.70, so I was still able to calculate a test-statistic. The test statistic looks extremely significant and that is mainly due to the very low standard errors of the AUC. The AUC of the market model is higher than the AUC of the z-score model. Consistently, the accuracy ratio of the market model is higher than that of the z-score model, though both are not very high (0.2700 and 0.3192). From the ROC curves and the test-statistic it can be concluded that the market-based model has a higher predictive accuracy than the score model. This implies that the z-score model is overestimating the probability of default since the z-z-score model produces higher average probabilities for both bankrupt and non-bankrupt companies (table 2).
6.3 Tests of information content
To test for information content, two logistic regressions were executed. In table 4, the results are presented.
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Pmarkt is the probability of the market-based model and paccount is the probability of the z-score model.
From table 4 it becomes clear that both models carry information about the probability of bankruptcy. However, the market model has a higher pseudo-𝑅!
(0.4175) compared to the accounting model (0.2577) indicating that it carries more information. Additionally, the higher coefficient of the market model implies that the market model is more likely to predict actual bankruptcy than the z-score model.
7. Conclusion
To test the alternative hypothesis that the market-based model has more predictive accuracy than the Altman z-score model two kind of tests were performed. Both the results from the prediction oriented and information content tests have shown that the market-based model has more predictive accuracy. Therefore, the null hypotheses is rejected. In addition, the combination of t-tests for differences in sample means and the ROC method has proved that the Altman z-score model is over-estimating the probability of bankruptcy for bankrupt as well as non-bankrupt companies. Three reasons could be causing this bias. The inclusion of service companies in the sample, the large asset range of the companies in the sample and the change of accounting rules and macro-economic conditions since the development of the model. Thereby, already two limitations of this research are discussed. Future research could prove the validity of the results of this study, through relying on a greater and more homogeneity sample.
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