Predicting Bankruptcy via Cross-Sectional Earnings Forecasts

Transcription

1 Predicting Bankruptcy via Cross-Sectional Earnings Forecasts DIETER HESS MARTIN HUETTEMANN* March 2018 * For their insightful discussions and suggestions, we are grateful to Thomas Hartmann-Wendels, Ashok Kaul, Alexander Kempf, Roman Liesenfeld, Peter Limbach, William Liu, Tobias Lorsbach, Martin Meuter and Hartmut Jakob Stenz. The paper has also greatly benefitted from comments by seminar participants at the University of Cologne. We are grateful to Sudheer Chava for kindly providing us with his bankruptcy data. Contact: Dieter Hess, University of Cologne, Germany, and Centre for Financial Research (CFR), hess@wiso.uni-koeln.de. Martin Huettemann, Cologne Graduate School in Management, Economics and Social Sciences (CGS), Germany, huettemann@wiso.uni-koeln.de. 1

2 Predicting Bankruptcy via Cross-Sectional Earnings Forecasts Abstract We develop a model to predict bankruptcies, exploiting that a firm s overindebtedness (negative book equity) is a state of immediate financial distress. Accordingly, our key predictor of bankruptcy is the probability that future losses deplete the book equity. To calculate this probability, we use earnings forecasts and their standard deviations that we obtain from cross-sectional models. However, not all over-indebted firms finally turn bankrupt. Thus, in an expanded model, we add accounting variables that we find to discriminate between bankrupt and nonbankrupt firms. As these models solely require accounting data, we can provide bankruptcy predictions for a wide range of firms, including firms that have no access to capital markets. In strictly out-of-sample tests we show that our accounting model performs substantially better than alternatives of corporate failure risk that solely use accounting information. If we allow for stock market information, we significantly outperform all leading alternatives, including those that require market data. Keywords: bankruptcy prediction, probability of negative book equity, overindebtedness, mechanical earnings forecasts Data Availability: Data used in this study are available from public sources identified in the study. JEL Classification: G17, G33, M41, C25 2

3 1. Introduction General Motors, Lehman Brothers and WorldCom are only a few examples of bankruptcies with a huge impact on capital markets. Predicting corporate failures is critical for investors, managers, regulators and banks. For example, it enables investors to avoid specific securities, managers and regulators to take corrective actions or banks to decide whom to grant loans. The goal of this paper is to develop a model for bankruptcy prediction. Our overall assumption is that a firm s over-indebtedness is a good measure of bankruptcy. A firm is over-indebted if its book equity turns negative, i.e., if the value of its assets falls below the value of obligations it must service. In the U.S. a firm is not required to voluntarily file for bankruptcy when it becomes overindebted. However, over-indebtedness is a strong indication of financial distress, making it less likely for a firm to obtain further credit and ultimately to pay its debt when it comes due. We calculate next year s book equity as the sum of the current book equity and forecasted earnings (change in retained earnings). Thus, our key predictor is the probability that a firm s current book equity is not large enough to cover expected losses. To calculate this probability, we look at the distribution of a firm s future earnings. Our approach shares the use of the loss distribution with the value at risk concept. The value at risk is defined as that loss value which is not exceeded by a pre-specified probability. We, however, start with a specific value (book equity) and then measure the probability that a firm s loss exceeds this specific value. We use crosssectional linear regressions as proposed by Hou et al. (2012) and Li and Mohanram (2014) to predict the range of possible earnings (or losses). Thus, we use lagged accounting ratios of all firms to calculate earnings forecasts of an individual firm. These linear regressions provide us with earnings forecasts and the corresponding standard errors. By this, we can derive a closed formula for the probability that a firm s book equity becomes negative. To our best knowledge, this is the first study to incorporate cross-sectional earnings forecasts into bankruptcy prediction. A major advantage of our approach over distance-to-default models is that a firm s negative book equity is an economically intuitive predictor of bankruptcy. Thus, we can create models on the basis of an economic hypothesis relating to research on negative book equity. In addition, we can provide bankruptcy predictions for firms without access to the capital market as we use book equity instead of market equity. Furthermore, we do not require a closed theory such as option pricing theory that comes along with assumptions and restrictions. Instead, we say that a firm s 3

4 future book equity can be directly calculated as the current book equity plus the change in retained earnings. To forecast these earnings, we use cross-sectional models instead of time series. Thus, we exploit the history of all firms and a broader dataset. In contrast to time series models, crosssectional models can provide forecasts for firms with a short or even with no history. We present three bankruptcy prediction models: an over-indebtedness model, an accounting model and a market model. Our over-indebtedness model exploits the fact negative book equity is a good predictor of bankruptcy. Thus, it consists of a single variable, the probability that forecasted losses exceed the current book equity. Calculating this probability requires an earnings forecast and the volatility of that forecast, among other things. However, over-indebted firms do not inevitably turn bankrupt. Firms might intentionally operate on negative book equity, for example, due to tax avoidance. Thus, our accounting model adds further accounting-based independent variables to discriminate between negative book equity firms that keep operating and those that turn bankrupt. Haowen (2015) finds, for example, that non-bankrupt negative book equity firms tend to have a higher book leverage ratio, have more capital expenditures, pay less tax, have a lower profitability and a smaller size. In addition, our market model replaces book leverage ratio by market leverage ratio. It further adds two common market-based variables: the excess return and its standard deviation. For example, Shumway (2001) and Campbell, Hilscher and Szilagyi (2008) show that market variables raise performance. In contrast, Reisz and Perlich (2007) and Agarwal and Taffler (2008) demonstrate that accounting models show a similar performance as market models. We analyze the out-of-sample performance of our three models and compare them to leading alternatives: We estimate the models of Altman (1968), Ohlson (1980) and Shumway (2001) and the best model version of Merton s distance-to-default approach as outlined in Bharath and Shumway (2008). To eliminate the effects of different statistical methods we embed all these models into logistic regressions that exploit the firms entire histories. Our empirical results are manifold. First, we find justification for our overall approach to tie a firm s bankruptcy risk to over-indebtedness. In fact, we document that book equity and earnings diminish in the years before bankruptcy, with the most dramatic fall happening in the last year before bankruptcy. Second, we show that the probability of negative book equity is a good measure of bankruptcy by its own. Our one-variable over-indebtedness model already produces better results than the models of Altman (1968) and Ohlson (1980). We further provide evidence for the fact that the functional form that we use for the probability cannot be completely replaced 4

5 by a linear combination of the variables used to calculate the probability. The functional form remains significant even when we include all components as individual variables in the model. Third, we find strong differences in the means of certain variables for bankrupt versus nonbankrupt firms with negative book equity, respectively. While this validates, for example, Haowen s (2015) results for the market leverage ratio, the profitability and the size, it also suggests that we can improve our over-indebtedness model by including further accounting variables. In fact, we demonstrate that an augmented accounting model significantly outperforms all other existing models that solely rely on accounting information. Fourth, we show that performance can be further improved by adding stock market information: Our market model shows significantly better results than our accounting model. Importantly, it outperforms all leading alternatives of bankruptcy prediction, including those that use market information. By this, we support Shumway (2001), Beaver, McNichols and Rhie (2005) and Campbell, Hilscher and Szilagyi (2008) who demonstrate that market variables add explanatory power. However, it should be noted that the performance boost associated with models relying on market data comes at a high cost, as these models are limited to firms which have access to the capital market. Altman, Iwanicz-Drozdowska, Laitinen and Suvas (2017) point out that predicting bankruptcies of private firms is equally important. Managing large loan portfolios, for example, requires models that can assess small- and medium-sized firms as well. Therefore, it is all the more important to note that our accounting model substantially improves out-of-sample performance, coming close to alternative market models, but at the same time it does not restrict applicability to public firms alone. There are two other types of structural bankruptcy prediction models that are related to our approach as they use equity as an indication of bankruptcy. First, e.g., Vassalou and Xing (2004), Hillegeist, Keating, Cram and Lundstedt (2004), Bharat and Shumway (2008) and Charitou, Dionysiou, Lambertides and Trigeorgis (2013) use Merton s (1974) option pricing theory to compute default probabilities. They view the market value of equity as a call option on the market value of assets where the strike price is the book value of liabilities. Thus, if market assets fall below liabilities, market equity goes to zero and the firm goes bankrupt. The value of assets is assumed to follow a Geometric Brownian Motion. Option pricing models do not contain any earnings variable. Furthermore, they require restrictive assumptions, e.g., that a firm s assets follow a certain stochastic process or that it has just one single zero-coupon bond outstanding. Second, Feller (1968) develops the Gambler s Ruin Theory which says that a firm fails if its book equity turns negative. It assumes that a firm does not have access to the capital market and thus can meet 5

6 losses solely by selling assets. An extension to Feller s theory is Scott s (1976) perfect-access model which assumes that a firm goes bankrupt due to investors negative expectations. Accordingly, this approach uses market equity instead of book equity. Wilcox (1973, 1976), Santomero and Vinso (1977) and Vinso (1979), for example, apply these approaches by using equity information along with a mean and volatility estimate of earnings. However, earnings are modelled via time series that only exploit information of the firm itself. Moreover, future earnings are solely estimated via past earnings and not via further accounting variables. The paper proceeds as follows: The second section describes and motivates the variables that we use in our bankruptcy prediction models. In the third section we describe our sample selection, report descriptive statistics and explain our methodology. In the fourth section we present and discuss our results. The fifth section concludes. 2. Constructing measures of bankruptcy 2.1 Probability of negative book equity over-indebtedness model We exploit that a negative book equity is a strong indication of financial distress. Our most basic model just includes one variable, the probability that book equity turns negative. We call this model our over-indebtedness model. Let BkEq i,t denote the current book equity of a firm i. Book equity is equal to stockholder s equity (COMPUSTAT item SEQ). If SEQ is missing, we take the common equity (CEQ) plus the value of preferred stocks (PSTK). In case CEQ or PSTK are missing, book equity is evaluated as total assets (AT) minus total liabilities (LT) minus minority interest (MIB). Further let Earn i,t+12m denote the future earnings (change in retained earnings) for this firm for the subsequent twelve months. Earnings are equal to net income (NI) plus dividends payments (DVT). We calculate next year s book equity as the sum of the current book equity and the earnings. Then, over-indebtedness occurs if future losses (i.e., negative earnings) exceed the current book equity: Earn i,t+12m > BkEq i,t. (1) Thus, our key predictor is the probability that a firm s future earnings are smaller than the negative value of current book equity: 6

7 PNBE i,t = Prob(Earn i,t+12m < BkEq i,t ). (2) To calculate this probability, we use the mean of an individual firm s conditional earnings estimate, μ(earn i,t+12m ), which we derive from a rolling cross-sectional regression model in the spirit of Hou et al. (2012) and Li and Mohanram (2014). For a detailed description of this approach see Section 2.2. This regression model also provides us with a measure of uncertainty of earnings estimates, i.e., the standard deviation of a firm s conditional earnings estimate, σ(earn i,t+12m ). 1 Assuming normality of earnings we can use the means and the standard deviations of earnings estimates and directly calculate the probability that firms future earnings might fall under a given threshold, i.e., that (negative) earnings exceed current book values of equity: PNBE i,t = Prob(Earn i,t+12m < BkEq i,t ) (3) = Φ ( BkEq i,t + μ(earn i,t+12m ) ), σ(earn i,t+12m ) where Φ( ) is the cumulative distribution function of the standard normal distribution. 2 The probability of default depends on the sum of a firm s current book equity and its mean earnings estimate relative to the standard deviation of its earnings estimate. Option pricing models view market equity as a call option on the market value of firm s assets where the strike price is the market value of the firm s liabilities. Though this is a different setting than in our model, Vassalou and Xing (2004) and Hillegeist et al. (2004) get a formula with a similar structure. Their probability of default depends on the ratio of expected future market equity and asset volatility, where future market equity is calculated as current market assets minus current liabilities plus the expected asset changes according to a Geometric Brownian Motion. Our probability of a negative book equity and the default probability extracted from option models have 1 σ(earni,t+1 ) denotes the standard deviation of a predicted response of an individual firm for given data rather than the standard deviation of the estimated conditional mean. Thus, it yields the prediction interval rather than the confidence interval. In addition to the uncertainty in estimating the conditional mean, σ(earn i,t+1 ) also reflects the variability of an individual observation in this conditional distribution: σ(earn i,t+1 ) = σ (1 + x i (X X) 1 1 x i ), where σ = n i=1 (y i y ) i is the standard deviation of the residuals, x i is the explanatory vector of firm i, X is the data n 2 matrix, y i is the outcome of firm i and y i is the predicted outcome of firm i. 2 Linear regression estimates follow a t-distribution with n p degrees of freedom, with n denoting the number of observations and p the number of independent variables. Due to the large amount of observations n p is consistently far above 40. Then, the t-distribution is well approximated by a standard normal distribution. 7

8 in common that a lower volatility and a larger equity lead to the assessment that a default is less likely. 2.2 Earnings forecasts We use earnings forecasts for the subsequent twelve months to calculate PNBE. Following Hess, Meuter and Kaul (2017), who compare the performance of several cross-sectional models, we implement the RI model of Li and Mohanram (2014) on a per-share basis. 3 However, we differ from previous studies such as Hou et al. (2012) or Li and Mohanram (2014) who only make predictions at the end of next June, as we run the regressions every month. This ensures that the estimation is made promptly as soon as all information are at hand. Note that all predictor variables are lagged by three months. That is, we make predictions three months after the fiscal year-end. This approach avoids a look-ahead-bias by ensuring that information is not used before it is actually at hand. Following Hou et al., we employ a rolling regression technique based on windows including the most recent 10 years of accounting data. Every month we run the following crosssectional regression: E i,t+τ = α 0 + α 1 E i,t + α 2 NegE i,t + α 3 NegE i,t E i,t + α 4 BkEq i,t + α 5 AC i,t + ε i,t+τ, (4) where E i,t denotes the change in retained earnings per share of firm i at time t, NegE i,t is a dummy that takes the value of 1 if firm i shows negative earnings at time t and NegE i,t E i,t is an interaction term. BkEq i,t is the book value of equity per share, AC i,t are accruals per share and τ = 1,2. We calculate accruals following Hou et al. (2012): Up to 1988, accruals are the change in non-cash current assets (COMPUSTAT items ACT and CHE) minus the change in current liabilities (LCT) plus the change in short term debt (DLC) plus the change in taxes payable (TXP) excluding depreciation and amortization costs (DP). From 1988 onwards, we define accruals as income before extraordinary items (IB) minus cash flow from operations (OANCF). Missing values are set to zero. 3 We also tested the cross-sectional earnings forecast model of Hou et al. (2012) and the EP model of Li and Mohanram (2014) and all models with level earnings instead of per-share earnings. All results of this paper are robust regarding this method. 8

9 Using the coefficients from this regression we can easily calculate out-of-sample predictions for the two subsequent fiscal years. Weighting these predictions, we construct earnings forecasts (and corresponding standard deviations) with a horizon of 12-month-ahead Accounting model A central assumption in our basic one-variable model version is that an over-indebtedness directly leads to bankruptcy. However, firms with negative equity do not inevitably turn bankrupt. Instead they might intentionally operate on negative book equity, for example, to reduce taxes. To further discriminate between healthy and bankrupt firms, we introduce an extended model version, our accounting model, that adds further independent variables. In particular, we follow Haowen (2015) who finds that non-bankrupt negative book equity firms tend to have a lower leverage ratio, more capital expenditures, pay less tax, have a lower profitability and a smaller size than bankrupt negative book equity firms. Besides the probability that book equity turns negative, our accounting model therefore includes the following accounting-based measures: We use a dummy that takes the value 1 if the book equity is negative, and the value 0, otherwise (NegBkEq). Similarly, we add a dummy that is 1, if the earnings forecast is negative, and that is 0, otherwise (NegEarnFrc). Moreover, we use the book leverage ratio (BLR) calculated as the sum of long-term debt (COMPUSTAT item DLTT) and current debt (DLC) divided by total assets (AT). CAPXTA denotes capital expenditures (CAPX) divided by total assets (AT), TXT is paid taxes (TXT), EBITTA is calculated as earnings before interest and taxes (EBIT) divided by total assets and SIZE is measured by the logarithm of sales (SALE). Note that EBITTA is also used by Altman (1968) and SIZE is used by Ohlson (1980). 2.4 Market model There is an ongoing debate whether market variables consistently add explanatory power compared to models that solely exploit accounting variables. For example, Shumway (2001), Hillegeist et al. (2004), Beaver, McNichols and Rhie (2005) and Campbell et al. (2008) 4 As we make estimations three months after the fiscal year-end, the predictions for the next fiscal year-end have a weight of nine twelvth and the predicitions for the fiscal year-end after that have a weight of three twelvth. 9

10 demonstrate that market variables can improve accuracy. In contrast, for example Reisz and Perlich (2007) and Agarwal and Taffler (2008) show that accounting-based models have a similar performance. To test these hypotheses, we add two market variables that are taken from Shumway (2001) to our accounting model: the stock s past excess return (ER) which is the last year s stock s return minus last year s value-weighted index return and the standard deviation of the stock s return (STDER). Moreover, we replace the book leverage ratio by the market leverage ratio (MLR) calculated as the sum of long-term debt (COMPUSTAT item DLTT) and current debt (DLC) divided by the sum of long-term debt, current debt and market equity. Market equity is the fiscal year-end equity price (PRCC_F) multiplied by the number of common shares outstanding (CSHO). We call this augmented specification our market model. 3. Data and Method 3.1 Sample dataset We use bankruptcy information taken from Chava and Jarrow (2004) which is updated in Chava (2014) and Alanis, Chava and Kumar (2016). 5 This data includes bankruptcy events of firms traded on NYSE, AMEX or NASDAQ and spans from January 1964 to December Bankruptcy is defined as a petition for filing for Chapter 7 or Chapter 11. We make earnings forecasts three months after the fiscal year-end to ensure public availability of the information that we use. That is, we predict book equity 15 months after the fiscal year-end. Accordingly, we declare a firm to turn bankrupt during the subsequent year if the bankruptcy date lies between the last fiscal year-end plus three months (our estimation date) and the last fiscal year-end plus 15 months. Thus, the dependent variable equals one if the firm turns bankrupt during this period and zero otherwise. Note that our bankruptcy forecast horizon is 12 months for all firm-years. In contrast, many former studies use the fiscal year or calendar year as their horizon. As we have bankruptcies till the end of 2014, our sample only includes observations with a fiscal year-end before or equal to the end of September We are grateful to Sudheer Chava for kindly providing us with his bankruptcy data. 10

11 Table 1 summarizes information about our bankruptcy dummies. The first column shows the number of active firms in each year. The second column shows the number of firms that have a bankruptcy dummy equal to one and the third column the corresponding percentage of active firms that turn bankrupt. In our final sample there are 1490 bankruptcy events. The overall bankruptcy rate is 0.79% with a strong fluctuation over the years. Chava and Jarrow (2004) use a total of 464 for their sample period from 1963 till 1998 and Shumway (2001) uses 300 bankruptcies between 1962 and It is apparent that bankruptcies were rare until the late 1970s. The bankruptcy rate fluctuated largely with a high of 1.20% in 1985, a peak of 2.47% in [TABLE 1] Our initial sample includes all firms listed on NYSE, AMEX or NASDAQ in the intersection of the annual Compustat North America fundamentals files and the daily and monthly CRSP files between 1958 and We obtain earnings forecasts by utilizing a rolling regression technique. Requiring 10 years of data for these cross-sectional earnings regressions, in a first step, we obtain one-year ahead earnings forecasts for the years 1969 to The first earnings forecasts are made in 1968 for the year 1969 using accounting data from 1958 to 1967, and the last forecasts are made in 2013 for the year 2014 based on data from 2003 to Note that these forecasts are free of look-ahead bias as we only use information up to the point in time when these forecasts are made. We then use the resulting earnings forecasts to predict bankruptcies in a second step. To produce strictly out-of-sample forecasts, we estimate the parameters using only data from 1968 to 2002 and use the coefficients for predicting bankruptcies for the period from 2003 to Just like in the earnings regression, all our measures of bankruptcy are lagged to ensure that they are observable at the time we use them for estimation. We assume that the accounting and market information are available three months after the fiscal year-end. For bankruptcy predictions we use the earnings forecasts that are made three months after the fiscal year-end. Accordingly, we make our one-year bankruptcy predictions three months after the fiscal year-end. We delete observations with missing variables that are required in the earnings forecast model or in any of the bankruptcy prediction models. These include the variable sets of our overindebtedness model, our accounting model and our market model, as well as of the models used for benchmarking, i.e., of the Altman (1968), Ohlson (1980) and Shumway (2001) models and of 11

12 the distance-to-default (DD) model used by Bharath and Shumway (2008). 6 The appendix of this paper describes the variable construction for these bankruptcy prediction models. To reduce the effect of outliers, we winsorize all variables (except for the indicator variables and probabilities) annually at the 1st and 99th percentile. [TABLE 2] Table 2 provides summary statistics for all variables described above. Panel A shows the measures used to forecast bankruptcy and Panel B shows the measures used to forecast earnings. We report means, medians, standard deviations and certain percentiles of 189,251 firm-years with complete data availability for the period from 1968 to Most importantly, the overall firmyear average of the probability that losses exceed the current book equity (PNBE) is 10.7%. At the same time only 25% of all firm-years have a PNBE which is greater than 9.1%. For 1% of all firmyears PNBE is greater than 96.2%. Half of firms have a PNBE that is zero. By this, PNBE might be a good proxy for the probability of default, although the overall bankruptcy rate is only 0.79%. For 33.4% or all firm-years, the cross-sectional earnings models forecast negative earnings. The statistics for EBITTA, ER and STDER are consistent with former bankruptcy prediction studies, e.g., Shumway (2001). 3.2 Logistic regressions Following Shumway (2001), Chava and Jarrow (2004) and Campbell et al. (2008), we estimate the probability that a firm turns bankrupt by using a logistic regression. Thus, this probability follows a logistic distribution with parameters (α, β) and is equal to P t (y it = 1) = 1 1+exp( α βx i,t ), (5) where y it is the bankruptcy dummy that equals one if the firm fails in the following 12 months and zero, otherwise, and x i,t is the vector of explanatory variables that are known in t, i.e., three months after the end of the fiscal year. The higher α + βx i,t, the higher is the estimated probability of 6 DD models use Merton s (1974) option pricing theory. They have been shown to be a good predictor of bankruptcy by e.g., Hillegeist et al. (2004) and Vassalou and Xing (2004). We use the DD version model that Bharath and Shumway (2008) find to perform best. They call this best model Model 7. It comprises their naïve version of Merton s DD probability, the inputs of this probability as individual measures and the ratio of net income and total assets. 12

13 bankruptcy. To produce strictly out-of-sample forecasts, we estimate the parameters using only data from 1968 to 2002 and use the coefficients for predicting bankruptcies for the period from 2003 to Static models (for example Altman (1968), Ohlson (1980)) use one single observation per firm. Arbitrarily selecting only one firm-year might entail a sample selection bias. In contrast, we use the entire histories of the firms within the logistic regression. Hence, our estimation technique exploits more information and eliminates the sample selection bias. Note that applying such a technique to the static models of Altman (1968) and Ohlson (1980) already improves their performance as compared to using the estimation techniques suggested originally. 4. Results of Empirical Analysis 4.1 The evolution of book equity and earnings around bankruptcy We assume that over-indebtedness is a good indication of bankruptcy in the sense that the book equity of bankrupt firms diminishes in the years before bankruptcy and is finally depleted by losses in the year of bankruptcy. Table 3 reports means of variables related to book equity and earnings for bankrupt firms and for firms that never turned bankrupt, respectively. For bankrupt firms, we show means of these variables for each of the last five years before bankruptcy and of the year after bankruptcy. For example, year -1 denotes the year directly before bankruptcy and year 0 denotes the year directly after bankruptcy. To ensure that we investigate the same firms over time, we only include those 739 bankrupt firms with a history of at least five years before bankruptcy. 300 of these firms even submit a balance sheet in the year after bankruptcy. The nonbankrupt firms only include firms that have never filed for bankruptcy. We report statistics for the corresponding 168,297 firm-years. [TABLE 3] Only 2.7% of non-bankrupt firm-years have a negative book equity. In contrast, even five years before bankruptcy, already 3.9% of bankrupt firms have a negative book equity. This ratio increases monotonically to 29.4% in the year -1, i.e. directly before bankruptcy. Finally, 55.3% of bankrupt firms have a negative book equity in the year when they go bankrupt (year 0). 13

14 Accordingly, the average book equity five years before bankruptcy is and thus already much smaller than the average of firms that never turn bankrupt ( ). The mean book equity for bankrupt firms consistently declines from year -4 on. It experiences the most severe fall from year -2 with to year -1 with In the year after bankruptcy the average book equity is and thus negative. The same statements apply to the average book equity that is standardized by total assets. For the earnings there is a similar pattern. Only 32.8% of non-bankrupt firm-years report negative earnings. In contrast, the ratio of bankrupt firms with negative earnings rises from 44.8% in year -5 to 89.3% the year -1. The mean earnings five years before bankruptcy is and by this already much smaller than the mean of firms that never turn bankrupt (49.168). The average earnings for bankrupt firms have a downward trend from year -5 on and experience the most significant fall from in year -2 to in year -1. In the year after bankruptcy, the average earnings are and thus even more negative. Similar results apply to the mean of earnings that are standardized by total assets. Importantly, the average losses in the year before bankruptcy ( ) deplete the average book equity in the year of bankruptcy (82.754). On average, our probability of negative book equity (PNBE) is monotonically rising in the years before bankruptcy. In year -5, the mean of PNBE is 11.7% which is already higher the average PNBE of non-bankrupt firm-years (10.0%). In year -2 PNBE of bankrupt firms is 22.3% and in the year before bankruptcy it makes big jumps to 41.2% and finally to 60.6% in the year after bankruptcy. The results strongly support our overall assumption that book equity diminishes in the years before bankruptcy and finally turns bankrupt after bankruptcy. This depletion of book equity is explained by earnings that are negative even five years before bankruptcy and that further decrease in the years till bankruptcy. Especially in the year directly before bankruptcy book equity and earnings experience a dramatic fall. Losses exceed book equity in the year before bankruptcy. Accordingly, our variable PNBE which also incorporates the volatility of the earnings estimate consistently rises in the years before bankruptcy. Furthermore, we find differences of bankrupt firms and firms that never turn bankrupt already five years before the actual bankruptcy happens. That is, book equity and earnings are early warning signals for bankruptcy. 14

15 Amongst others, negative book equity is one important aspect for predicting bankruptcies. In Section 4.2 we motivate further indicators of bankruptcies that augment the explanatory power of negative book equity. 4.2 Profile analysis of bankrupt and non-bankrupt firms Table 4 provides a profile analysis for the variables of our models. We report the mean and the standard deviation of those measures for the group of non-bankrupt firm-years and the group of bankrupt firm-years, respectively. A bankrupt firm-year is an observation for which the fiscal year-end lies three to fifteen months before the bankruptcy, i.e. for which the bankruptcy dummy is equal to one. A non-bankrupt firm is an observation, for which the bankruptcy dummy is equal to zero. That is, in contrast to Section 4.1 a non-bankrupt firm-year might be an observation of a firm that turns bankrupt at a later point of time. The column labeled Diff shows the mean difference between healthy firm-years and bankrupt firm-years. We further report results of Welch s t-test on mean equality which is a two-sample test for the hypothesis that two populations have the same mean. Unlike the more common Student s t-test, Welch s t-test does not assume equal variances and equal sample sizes. [TABLE 4] This test is highly significant for all our bankruptcy model variables, that is the hypothesis that bankrupt and non-bankrupt firm-years have the same mean is rejected for all variables. Firms that are about to turn bankrupt differ from non-bankrupt firms in ways that we expect for most of our bankruptcy measures. Most importantly, bankrupt firms have an average probability of negative book equity (PNBE) of 41.3% which is significantly higher than 10.4% for the nonbankrupt group. Thus, PNBE has a huge power to discriminate between bankrupt and non-bankrupt firms. The untabulated median for the bankrupt group is 39.0% and thus close to its mean. Note, however, that the distribution of PNBE is extremely skewed for the non-bankrupt group. Its untabulated median is 0.0% and thus much smaller than its mean of 10.4%. Only 25% of nonbankrupt observations have a PNBE which is higher than 8.53% and only 10% have a PNBE which is higher than 45.34%. Furthermore, bankrupt firm-years have more often negative earnings forecasts, a higher leverage ratio, a lower amount of paid taxes, a lower profitability measured by 15

16 EBITTA, a smaller size, a lower excess return and a higher standard deviation of the return. Unexpectedly, bankrupt firms have higher capital expenditures divided by their total assets. These variables help to discriminate between bankrupt and non-bankrupt firm-years. By this, they can even increase the explanatory power of the probability of negative book equity (PNBE). In Section 4.3, we investigate if those variables are indeed significant predictors of bankruptcy. 4.3 Logistic regression results Table 5 reports the estimation results of several logistic regressions. Column 1 shows the parameter estimates for our over-indebtedness model, column 2 for our accounting model and column 3 for our market model. [TABLE 5] The coefficients of our over-indebtedness model confirm that PNBE is an extremely significant bankruptcy predictor by its own. All measures that we add in our accounting and our market model are statistically significant as well and, by this, help to increase the predictive power of PNBE. The fact that the market-based variables are significant supports the hypothesis that a combination of accounting- and market-based variables can improve the accuracy of bankruptcy prediction models. The signs of most coefficients are consistent with economic intuition: Firms with a higher PNBE are more likely to fail and firms with negative earnings forecasts (NegEarnFrc) are more likely to fail. The higher the book leverage ratio (BLR), the higher the market leverage ratio (MLR) and the higher the volatility of the return (STDER), the higher is the estimated probability of bankruptcy. The lower the tax (TXT), the lower the profitability (EBITTA) and the lower the excess return (ER), the higher is the estimated probability of bankruptcy. Unexpectedly, a higher capital expenditures (CAPXTA) and a larger size (SIZE) lead to the assessment that bankruptcy is more likely. 4.4 Out-of-sample results Table 6 assesses the out-of-sample predictive ability of different models. We compare our over-indebtedness, accounting model and market model to leading and well-known alternatives of 16

17 Altman (1968), Ohlson (1980) and of Shumway (2001). Furthermore, we estimate Merton s distance-to-default model in its best version as found by Bharath and Shumway (2008). Since our over-indebtedness model is univariate, there is no difference if one ranks the firms by the probability estimated by the logistic regression or directly by PNBE. Differences in the out-ofsample results compared to other studies like for example Shumway (2001) and Bharath and Shumway (2008) are because we use an augmented period of time. [TABLE 6] Panel A reports the goodness-of-fit deciles. Following Shumway (2001), we rank firms into deciles based on their fitted bankruptcy probability values for every year of our validation sample (2002 to 2013). That is, the firms that will most likely default in the subsequent year are sorted into the first decile and the firms with the lowest estimated default probabilities are assigned into the tenth decile. We report the percentages of bankrupt firms that fall into each of the ten probability deciles. A model is accurate if it yields high default probability estimates for bankrupt firm-years and thus assigns many bankrupt firms into first deciles. Our accounting model classifies 61.64% of all bankrupt firm-years into the highest default probability decile (decile one). That is, a bank can exclude 61.64% of all bankruptcies if it does not lend money to the 10% of firms with the highest expected default measures. By this, it significantly outperforms the models by Altman (51.03%), Ohlson (55.82%) that use accounting information as well. 7 Even our over-indebtedness model identifies 57.19% of bankruptcies correctly (in the first decile) and thus outperforms Altman and Ohlson. Given that it has only one variable, it performs surprisingly well. For the top two deciles (in aggregate) the correct predictions are 76.37% for our accounting model. That is, if a bank does not lend money to the 20% of firms with the highest default probabilities it excludes 76.37% of all bankruptcies. Again, our accounting model performs better than Altman (60.96%) and Ohlson (74.66%). Importantly, the accounting model achieves better out-of-sample performances than all other accounting-based models without the need for market variables. 7 The model version of Altman (1968) requires stock price information and is by this not purely accounting-based. We are aware of Altman s (1983) z -score for private firms that replaces market equity by book equity (applications e.g. in Altman (1993) and Altman (2017)). However, we use Altman (1968) model due to its wide adaption and acceptance. 17

18 If limit the scope of application of our models and, therefore, add market information, we can further significantly improve performance: Now, our market model classifies 75.34% of all bankrupt firm-years into the highest default probability decile (decile one). In contrast, Shumway (2001) only classifies 72.6% of bankrupt firms into the first decile and Bharath and Shumway (2008) 73.63%. For the top two deciles (in aggregate), the correct predictions made by our market models are 89.04% versus 84.93% for Shumway and 81.85% for Bharath and Shumway. Importantly, our market model significantly outperforms all leading alternatives including those that also exploit market information. Furthermore, the fact that our market model performs significantly better than our accounting model supports previous findings of Shumway (2001), Hillegeist et al. (2004) and Campbell et al. (2008) that market variables add explanatory power. An alternative evaluation measure of bankruptcy prediction models is the area under the ROC (receiver operating characteristic) curve, also referred to as AUC (see Sobehart and Keenan (2001)). The AUC is interpreted as the probability that a randomly chosen defaulting firm has a greater predicted probability of default than a randomly chosen surviving firm. A value of 0.5 indicates a random model with no predictive ability, and a value of 1.0 indicates perfect discrimination. To compute the AUC, we estimate the parameters for each model based on the training sample (1968 to 2002) and use these parameters to predict bankruptcies on our validation sample (2003 to 2013). The distribution of the AUC for the validation sample can be found in Panel B. The AUC results are consistent with the results reported via goodness-of-fit deciles: Our accounting model has an average AUC of and, thus, again outperforms Altman (1968) with an average AUC of and Ohlson (1980) with an average AUC of As our overindebtedness model has an average AUC of 0.773, it performs better than Altman. Those models that use a combination of accounting and market variables have a higher average AUC than accounting-based models. Again, our market model has the highest average AUC (0.907) followed by Shumway (2001) with an average AUC of and Bharath and Shumway (2008) with an average AUC of Note that Shumway has a better out-of-sample performance than Bharath and Shumway with respect to the AUC which stands in contrast to the results of Campbell, Hilscher and Szilagyi (2011). 18

19 These results support the conclusion that our market model performs best if one allows market information to be included in the model and that our accounting model performs best if one restricts the model to accounting information. 4.5 Functional Form of PNBE To calculate the probability of negative book equity we use a non-linear functional form with three inputs. To assess the importance of this rigid functional form we compare two models. Model 1 uses the same inputs as PNBE, i.e., the current book equity, the earnings forecast and the inverse of its standard deviation, but does not squeeze these variables into one variable. Model 2 comprises all covariates of model 1, but adds the non-linear combination of these variables, that is PNBE. Table 7 reports the results of these two models. Panel A contains the estimation results of the logistic regressions and Panel B reports the out-of-sample results. [TABLE 7] From the estimation results for model 2 we see that PNBE is a significant predictor of bankruptcy although all components that are required to construct PNBE are included as individual variables. This strongly suggests that the functional form that we use for constructing the probability provides value beyond the information that is contained in the individual variables used to calculate the it. Looking at the out-of-sample assessment, we see that model 2 classifies 56.85% of all bankruptcies into the highest probability decile and thus outperforms model 1 (48.97%). Accordingly, the average AUC of model 2 (0.8101) exceeds the mean AUC of model 1 (0.7225). This provides additional support for the notion that the functional form of PNBE is a valuable construct for predicting bankruptcy. 5. Conclusion We develop a new framework to predict bankruptcies, focusing on the economically intuitive idea that future losses might deplete the firm s book equity. Our new approach to calculate the probability that equity turns negative does not require stock market information, as for example Hillegeist et al. (2004) and Bharath and Shumway (2008). 19

20 So far, major improvements of bankruptcy prediction models have been achieved by the inclusion of market measures. We suggest an accounting model that achieves a performance improvement without excluding private firms. Among those models that require market information and thus can solely applied to public firms, we suggest a market model that outperforms leading alternatives of corporate risk failure. Thus, our accounting model performs best if one wants to predict a bankruptcy of a non-public firm, and our market model produces best results if one wants to predict a bankruptcy of a public firm. We make predictions for the subsequent twelve months. However, we demonstrate, that a firm s negative book equity can indicate financial distress earlier than twelve months before its bankruptcy. Thus, further research can create multi-period bankruptcy prediction models by using multi-period earnings forecast models which are described in Hou et al. (2012). Alternatively, one can use analysts' earnings forecasts instead of mechanical forecasts to model the changes of book equity. In addition, further research can aim at grasping the imperfect relation between firms with a negative book equity and bankrupt firms. On the one hand, one could add more variables that help to discriminate between bankrupt and non-bankrupt firms with negative book equity. On the other hand, one could relax the assumption that a negative book equity in its formal definition is the predictor of choice. For example, one can create a new indicator of bankruptcy by deleting those components that belong to the definition of book equity but do not have an influence on bankruptcy. Appendix: Construction of variables of earlier bankruptcy prediction models This appendix describes the construction of variables used by Altman (1968), Ohlson (1980) and Shumway (2001) and of Bharath and Shumway (2008). Altman (1968) suggests the Z-Score, a linear weighted sum of five ratios which best discriminates between failing and surviving firms: Z = β 0 + β 1 WCTA + β 2 RETA + β 3 EBITTA + β 4 METL +β 5 STA, where WCTA is working capital (COMPUSTAT item WCAP) divided by total assets (AT), RETA is retained earnings (RE) divided by total assets (AT), EBITTA is earnings before interest 20

21 and taxes (EBIT) divided by total assets (AT), METL is the market value of equity (PRCC_F multiplied with CSHO) divided by the book value of total debt (LT), STA is sales (SALE) divided by total assets (AT) and Z is the Z-Score. WCTA is a proxy for a firm s liquidity and RETA and EBITTA measure different aspects of profitability. METL is a widely used measure of leverage and STA describes the firm s efficiency to use assets in generating sales. The Z-score characterizes the financial strength of a firm by aggregating the above five accounting ratios into one figure via the estimated coefficients β 1,, β 5. Ohlson (1980) finds nine variables to be significant and defines his O-score as: O = β 0 + β 1 SIZE + β 2 TLTA + β 3 WCTA + β 4 CLCA + β 5 OENEG + β 6 NITA + β 7 FUTL + β 8 INTWO + β 9 CHIN, where SIZE is the logarithm of total assets (AT), TLTA is total liabilities (LT) over total assets (AT), WCTA is working capital (WCAP) over total assets (AT), CLCA is current liabilities (LCT) over current assets (ACT), OENEG is a dummy that is one if total liabilities (LT) exceed total assets (AT) and that is zero otherwise, NITA is net income (NI) over total assets (AT), FUTL is funds provided by operations 8 (PI plus DP) over total liabilities (LT), INTWO is a dummy that is one if the net income (NI) was negative for the last two years and that is zero otherwise, CHIN is the change in net income (NI) and O is the O-Score. WCTA and CLCA measure liquidity. NITA, FUTL, INTWO and CHIN capture different aspects of profitability and TLTA and OENEG describe the capital structure. In addition to selected financial ratios already used by Ohlson, Shumway (2001) adds two market variables which are the excess return and its standard deviation: S = β 0 + β 1 RSIZE + β 2 TLTA + β 3 NITA + β 4 ER + β 5 STDER, where RSIZE is the logarithm of market equity divided by value-weighted market equity of the index, ER is the excess return calculated as the difference of last year s return and last year s value-weighted index return, STDER is the standard deviation of the return and S is the S-Score. 8 Funds provided by operations are not reported anymore. We use an approximation by summing pretax income and depreciations and amortization. 21

22 ER is a measure of the profit of an investment, and STDER captures the variability of the excess return. RSIZE is an alternative measure of the firm s size. Bharath and Shumway (2008) expand on the distance-to-default models that e.g., Vassalou and Xing (2004) and Hillegeist et al. (2004) construct by applying Merton s (1974) option pricing theory. Merton s probability of bankruptcy is calculated as PD Merton = N ( ( ln(v/f) + (μ 0.5σ V 2 ) σ V )), where V is the market value of a firm s assets, σ V its standard deviation, μ is the expected return on assets, F is the market value of the firm s debt, and N( ) is the cumulative standard normal distribution function. Vassalou and Xing (2004) compute V and σ V numerically by applying an iterative procedure. However, Bharath and Shumway propose a naïve approach. They approximate the market value of debt by the book value of debt. Furthermore, the volatility of a firm s debt is approximated by σ F = σ E, where σ E is the volatility of the market return. An approximation of the volatility of the firm s assets is then given by σ V = E σ E+F E + F σ E+F F. The expected return on assets μ is approximated by last year s return on assets. And the market value of assets is approximated by the sum of the market value of equity and the book value of debt. The best model in Bharath and Shumway is constructed as: BS = β 0 + β 1 PD Merton + β 2 LNE + β 3 LNF + β 4 1/σ E + β 5 ER + β 6 NITA, where PD Merton is the probability constructed above, LNE is the logarithm of market equity E (PRCC_F multiplied with CSHO), LNF the logarithm of the book value of debt F calculated as current debt (DLC) plus one half of long-term debt (DLTT), 1/σ E is the inverse of the volatility of market equity, ER is the excess return calculated as the difference of last year s return and last year s value-weighted index return, and NITA as the ratio of net income (NI) and total assets (TA). 22