Klobucnik, Jan / Miersch, David / Sievers, SoenkeJan Klobucnik[a], David Miersch[b], Soenke Sievers...

No. 13 / November 2015 Klobucnik, Jan / Miersch, David / Sievers, Soenke Predicting Early Warning Signals of Financial Distress: The Role of Accounting Volatility Measures

Predicting Early Warning Signals of Financial Distress: The Role of Accounting

Volatility Measures*

Jan Klobucnik[a], David Miersch[b], Soenke Sievers[c]

First version: March 1, 2013 This draft: November 18, 2015

Abstract This study proposes a simple accounting-based framework that allows to assess financial distress up to five years in advance for US listed firms over the period of 1980-2010. We jointly model financial distress by two of its key driving forces, (1) a declining profitability, modeled by independent stochastic processes, and (2) insufficient liquid assets, viewed as a minimum level of operating liquidity. The financial distress model incorporates important underlying (i) time-series (ii) industry-sector (iii) firm-level growth, and (iv) accounting volatility developments. Cross-sectional and longitudinal analyses show improvements in the discriminatory power and demonstrate incremental information content beyond state of the art accounting-based models. For example, we find a 11 percent (4 percent) higher ranking performance documented by the area under the receiver operating characteristic curve (AUROC) relative to the standard Altman’s z-score (Ohlson’s o-score) models. Consequently, this study might provide important ex ante warning signals for investors, regulators and practitioners. JEL classification: C63, C52, C53, G33, M41 Keywords: Financial distress prediction, probability of default, accounting information, stochastic processes, simulation Data Availability: Data used in this study are available from public sources identified in the paper. * Acknowledgments: The authors are grateful to Wayne Landsman, Gilad Livne, Mingyi Hung, Kirill Novoselov, Igor

Goncharov, Ken Peasnell, Steve Stubben, Florin Vasvari and Jon Tucker for their valuable comments. This paper has also benefited from the comments of seminar participants at the Hong Kong University of Science and Technology (2013), the Eighth Accounting Research Seminar in Basel (2013), the Frankfurt School of Finance & Management (2013), the Seventeenth Financial Reporting and Business Communication Conference in Bristol (2013) and the Accounting and Audit Convention in Cluj (2015). An earlier version of this research project circulated under the title “Bankruptcy Prediction Based on Stochastic Processes: A New Model Class to Predict Corporate Bankruptcies?” and is part of the first author's Ph.D. thesis (Klobucnik 2013). Any errors are our own.

[a] Cologne Graduate School in Management, Economics and Social Sciences (CGS), Albertus Magnus Platz, 50931 Cologne, Germany, e-mail: [email protected].

[b] (Corresponding Author) Cologne Graduate School in Management, Economics and Social Sciences (CGS), Albertus Magnus Platz, 50931 Cologne, Germany, e-mail: [email protected].

[c] Chair of International Accounting, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany, e-mail: [email protected].

mailto:[email protected]

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INTRODUCTION

This paper investigates a simple accounting-based framework that allows us to predict financial

distress up to five years in advance using a broad sample of US listed firms. Thus, it adds to the debate

of accounting-based vs. market-based bankruptcy prediction models, that centers on the question

whether market-based models outperform accounting-based models or vice (references include:

Agarwal and Taffler 2008; Das et al. 2009; Reisz and Perlich 2007; Xu and Zhang 2009).

Attributable to its academic and practical relevance, the two most widely applied classes to predict

financial distress1 are mainly driven by either employing accounting-information (references include:

Altman 1968; Beaver 1966, 1968b, 1968a; Ohlson 1980; Zmijewski 1984), or relying mostly on

market-information (references include: Bharath and Shumway 2008; Campbell et al. 2008; Hillegeist

et al. 2004; Vassalou and Xing 2004). Accounting-based prediction models made a major progress

since the 1960s given the multivariate approach by Altman (1968) and conditional probability models

as in Ohlson (1980), which are both best-known in research and practice today. 2 However,

accounting-based models are often criticized for their inherent limitations. As Vassalou and Xing

(2004) or Hillegeist et al. (2004) argue, accounting measures, according to their backward-looking

perspective, are less value-relevant to predict imminent financial distress. In addition, accounting

models are mainly static, single-period, and purely statistical models (Mensah 1984). They generally

exhibit limited theoretical grounding as they summarize the default conditions in their estimated

factor loadings (Agarwal and Taffler 2008). Consequently, the estimation results might be unstable

and not generalizable to other samples. In addition, several studies find evidence of a declining

predictive ability using accounting ratios (Beaver et al. 2005; Francis and Schipper 1999).

1 The literature has used a number of different metrics to proxy for financial distress (Balcaen and Ooghe

2006).Technically, financial distress is an early stage of business failure or bankruptcy, where the firm is unable to meet the terms of a loan obligation (Whitaker 1999). Separating different levels of financial strength is in line with Duffie et al. (2007), who merge Moody’s Default Risk Service database with CRSP/Compustat and other data sources and distinguish amongst others 1) bankruptcy, 2) default and 3) failure, where failure includes 1) bankruptcy, 2) default and adds any failures to meet exchange listing requirements. Jones and Hensher (2004) define their financial distress indicator in a similar three-stage classification model and matched financially distressed firms to a group that is unable to (i) pay listing fees, (ii) generate sufficient working capital, (iii) meet loan agreements (loan defaults) and (iv) to make loan repayments (with debt/equity restructurings). Given that our study uses CRSP and Compustat as data source (see section 4 for details) we measure financial distress broadly and our approach can be regarded as broader business failure model.

2 For exhaustive literature reviews we refer to Dimitras et al. (1996); Aziz and Dar (2006); Balcaen and Ooghe (2006); Ravi Kumar and Ravi (2007); Jackson and Wood (2013).

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Most importantly, however, the standard accounting models do not explicitly incorporate volatility

measures such as asset volatility, which is a major driving force for bankruptcy and thus prominently

highlighted in many studies investigating the performance of market- and accounting-based failure

models (see e.g., Hillegeist et al. 2004; Shumway 2001; Das et al. 2009). Naturally, more volatile

businesses correspond to higher default risk and neglecting this relation may cause lower model

performance. Consistent with these findings, Pástor and Veronesi (2003) provide evidence that (i) the

market volatility (measured by volatility of returns) increases with metrics of accounting volatility

(measured by volatility of profitability), (ii) the firm age is negatively related to uncertainty in

accounting measures, and (iii) more volatile firms are more likely to fail. Finally, accounting-based

models do not utilize the benefits of non-financial statement information contained in the market

prices of equity and debt (Campbell et al. 2008).

We address the previous shortcomings of accounting-based models by combining the research on

forecasting financial statement information using the equity valuation literature (refernces include:

Duffie and Lando 2001; Duffie et al. 2007; Favara et al. 2012; Garlappi et al. 2008; Garlappi and Yan

2011; Leland and Toft 1996; Pástor and Veronesi 2003, 2006; Schwartz and Moon 2000, 2001;

Anderson and Carverhill 2012) 3 with the well-established bankruptcy prediction literature. This

allows us to leave the static, backward-looking perspective of statistical models and instead offers a

multi-period, stochastically grounded, generalizable and forward looking approach.4

In contrast to the prior literature financial distress is triggered by the exogenous boundary condition

to meet short term obligations (i.e., viewed as down-and-out-barrier) and take into account the relative

information content of accounting volatility measures. Specifically, we estimate two key stochastic

processes (i) sales and (ii) costs. The stochastic processes are calibrated to track a firm’s underlying

3 Another early approach using a binomial process to model the adjusted cash position in order to predict failure is

found in Wilcox (1971, 1973). Selecting explanatory variables for a multivariate model motivated by firm valuation considerations is also found in Aziz et al. (1988). Finally, Emery and Cogger (1982) employ a structural stochastic prediction model linking the liquidity position of a firm to the net cash flow to measure a firm’s liquidity and solvency status. Contrary to our study the primary focus of Emery and Cogger (1982) is to provide measures on liquidity, not measuring the probability of financial distress.

4 Shumway (2001) emphasizes that static prediction models ignore the changing characteristics of firms through time. Although we use backward looking accounting data to initialize the stochastic processes to model future firm development for the large and anonymous dataset, one could also use other sources as expert knowledge to estimate forward looking parameters as the growth rate. One source could be analyst forecasts for the growth rate; however they are only available for around 26% of our sample.

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drift and uncertainty of future operating cash flow and liquidity position.5 Additionally, we include

industrial-, time- and firm-specific parameters to capture group and individual firm effects as

demonstrated to be important components (see e.g., Chava and Jarrow 2004; Opler and Titman

1994).6

We assess current state of the art accounting and market-based financial distress prediction models,

i.e., Altman (1968), Ohlson (1980), Bharath and Shumway (2008) and Campbell et al. (2008) by

comparing three different dimensions: (i) forecast accuracy, (ii) information content, and (iii)

additional statistical inference, where we use a large sample comprised of 41,664 observations

classified as financially distressed and 288,885 non-distressed firm quarter observations spanning

from 1980Q1-2010Q4.

The results confirm that a stochastically driven model improves the long-run discrimination ability,

measured in terms of the area under the receiver operating characteristic curve (AUROC), and carries

incremental information not covered by the original and re-estimated versions of the o-score and z-

score models. First, our stochastically approach is more accurate in distinguishing between non-

delisting and delisting firms than the prominent o-score or z-score models, even if re-estimated (e.g.,

AUROC=0.8122 compared to AUROC=0.7818 and AUROC=0.7327 for the o-score and z-score

models). Second, a combination of all three models has significantly higher explanatory power for

financial distress (e.g., the pseudo-R² increases about 47% by adding our stochastically driven

financial distress measure to the original o-score or z-score measures using a 1-year ahead prediction

horizon).

This paper contributes to the literature by separating financially distressed firms from their healthy

counterparts, which is of particular interest in a variety of contexts.7 Balcaen and Ooghe (2006) point

5 Referring to the financial distress drivers, Hilscher and Wilson (2013, p. 9) stated that „Firms are more likely to fail if

they are less profitable, have higher leverage, lower and more volatile past returns, and lower cash holdings.” 6 While this study picks up the idea that volatility in accounting numbers might be helpful for bankruptcy prediction, in

contrast to the above mentioned studies we do not simply add volatility as an additional variable, but consider the volatility measures in historical accounting numbers as components of the aforementioned stochastic processes that drives a firm’s operating performance and liquidity. While many financial statement variables have been shown to be useful in failure prediction a recurring and often confirmed result is that liquidity variables measured from accounting data have predictive power for bankruptcies up to five years into the future (Beaver 1966, 1968a; Gentry et al. 1985, 1987; Merwin 1942; Pompe and Bilderbeek 2005).

7 For example, specific fields of application and research that are worth mentioning include: First, the ongoing debates in the research literature about the relation between equity returns and a firm’s default risk, i.e. the “distress-puzzle” (see e.g., Campbell et al. 2008; Dichev 1998; Friewald et al. 2014; Kapadia 2011; Kim 2012; Vassalou and Xing 2004) or the explanation of credit risk premiums (see e.g., Correia et al. 2012; Das et al. 2009; Subrahmanyam et al. 2014).

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out that future research should put more weight on longer-term failure prediction. Most studies focus

on failure prediction over the next period, which can be, e.g. one month or one year (see e.g., Chava

and Jarrow 2004; Shumway 2001). However, it is important to predict potential failure earlier, since

the majority of loans agreements fail after their first year (Falkenstein et al. 2000). Campbell et al.

(2008, p. 2900) describe this point succinctly with the paraphrase that distress prediction “[…] may

not be very useful information if it is relevant only in the extremely short run, just as it would not be

useful to predict a heart attack by observing a person dropping to the floor clutching his chest”.

Information that a firm is approaching a difficult financial condition also allows for early preventive

and corrective actions of both management and stakeholders. While we acknowledge that some

studies investigate longer forecast horizons (Campbell et al. 2008; Duffie et al. 2007; Reisz and

Perlich 2007), this paper adds to this research line by also investigating longer forecast horizons of

up to five years.

The rest of this study is structured as follows. In the next section we give a brief overview of the

financial distress prediction literature and introduce the benchmarks. Section 3 introduces the model.

The subsequent section 4 presents the data and the model implementation while section 5 offers the

results from the empirical tests. Finally, section 6 summaries principal findings and concludes.

PREVIOUS RESEARCH

The role of accounting information in modern credit risk theory dates back to the 1930s and numerous

metrics and statistical tools have been proposed to predict the financial condition of

distressed/defaulted firms.8 Ravi Kumar and Ravi (2007) report in their literature review that the

Second, studies show that CEO compensations and executives turnovers are influenced by the presence of ex ante financial distress risk (see e.g., Chang et al. 2015; Gilson 1989). Third, financial distress also affects the dividend policy (DeAngelo and DeAngelo 1990) and corporate liquidity policy (Sufi 2009). Fourth, the use of financial distress models has been fully established in applied areas, e.g., bank regulation, rating and investing processes and to assess the creditworthiness of bond issuer and credit agreements. Fifth, financially distressed firms lose economic values, e.g., market shares, customers, suppliers and human resources, suffer in form of paid covenants penalties and/or higher cost of debt etc. (see Opler and Titman 1994; Purnanandam 2008).

8 Referring to the early studies, the pioneer work of Beaver (1966, 1968a, 1968b) using univariate analysis established a new strand on distress/default prediction literature. The univariate approaches lead to statistical-grounded techniques concentrated on the multivariate framework (references for the multivariate discriminant analysis are Altman (1968), Altman et al. (1977), Deakin (1972), critical commented by Eisenbeis (1977), Dambolena and Khoury (1980), Pompe and Bilderbeek (2005)) and conditional probability models (references include Martin (1977), Ohlson (1980) and the probit model by Zmijewski (1984)). Jackson and Wood (2013) identify a total of 25 alternative approaches throughout their literature review of 350 academic papers.

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majority of studies are based on accounting measures. To overcome many of the limitations of

accounting-based models the recent literature examines financial distress models including particular

information from the stock markets. Examples of prior research include the contingent claims

framework introduced by Black and Scholes (1973) and Merton (1974) and adapted, for example, by

Sobehart et al. (2000), Vassalou and Xing (2004), Hillegeist et al. (2004), Bharath and Shumway

(2008) or more recently by Charitou et al. (2013). As shown by Shumway (2001), Beaver et al. (2005),

Campbell et al. (2008), and Correia et al. (2012) combining market-related with accounting-related

variables can increase the predictive ability of the traditional ad-hoc accounting-based models.9 On

the other hand, the empirical findings of Agarwal and Taffler (2008), Reisz and Perlich (2007), Das

et al. (2009) and Xu and Zhang (2009) imply that the statistical performance between market-based

and accounting-based prediction models remains competing. While market-based variables reflect

additional, timelier information not captured by fundamental statements (Beaver et al. 2012), several

studies suggest that accounting measures are incrementally informative to predict bankruptcy (see

e.g., Hillegeist et al. 2004). Xu and Zhang (2009) summarize that “the option pricing theory-based

bankruptcy measure is more successful than the accounting variable-based measures alone, but it does

not subsume the accounting measures.” Moreover, Batta and Wan (2014) show that accounting-based

default models are less sensitive to stock market misvaluations than market-based default prediction

models.

The question then arises whether the benefits of market-based drivers could be incorporated for the

vast majority (i.e., more than 99%) of companies in the United States having no shares listed at the

NYSE/NASDAQ, i.e. private firms?10 Due to the predominance of private companies in the economy

9 However, in a related study Fu (2008) focuses on modeling future working capital and cash flow. Regarding the

working capital model she links the development of future working capital over time to estimates of expected net income and volatility of net income based on, e.g., the normal distribution. Similarly, regarding the cash flow model, she treats revenues as a random variable and estimates the distribution for revenues (expected revenue and volatility of revenue). However, this study does not combine various stochastic processes at the same time as in Schwartz and Moon (2001) or Pástor and Veronesi (2006) to forecast future company performance.

10 According to the latest (July 2015) statistics the World Federation of Exchanges reports 5,292 listed companies at the NASDAQ/NYSE (www.world-exchanges.org/statistics) which are related to 23,735,915 nonemployer establishments or 5,726,160 enterprises reported by the Statistics of U.S. Businesses 2012 (SUBS) (http://www.census.gov/econ/susb). For the definitions see http://www.census.gov/econ/susb/definitions.html. Contrary, the U.S. Internal Revenue Service (IRS) accounts 5.8 million active corporate tax returns (http://www.irs.gov/uac/Tax-Stats-2) and CRSP covers 26,500 active and inactive stocks listed at the NASDAQ/NYSE. The statistic provided by NASDAQ group reports 6,751 listed companies at the NASDAQ/NYSE/AMEX (see www.nasdaq.com/screening/company-list.aspx). Davis et al. (2007) reports that private firms employ over two-thirds of the US workforce. A further discussion of the relative importance of private vs. public firms is offered by e.g., Chen et al. (2011) and Hope et al. (2013). For example, Hope et al. (2013) compare the reporting quality between public and private firms using a large dataset of U.S. private companies.

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that are not listed on exchanges, market-based models cannot be employed for these firms as they

lack essential market data. At the same time, it is especially important to evaluate default risk for non-

listed firms, for example private firms and small borrowers, as they experience high risk of default.

Moreover, there is evidence that equity markets are not necessarily efficient. Prior studies such as

Sloan (1996) find that the market does not accurately reflect all information in the financial

statements. During times of volatile markets, as for example the dot-com bubble, relying on

accounting information rather than market data helps to assess the default probability (see e.g., Das

et al. 2009). Consequently and different from recent bankruptcy prediction models by Shumway

(2001), Chava and Jarrow (2004) and Campbell et al. (2008), we restrict our information set to

accounting data to potentially allow default prediction for non-listed firms. The aforementioned

studies present recent attempts to find the most accurate prediction model; however, they rely on

market data and therefore are not applicable for non-listed firms.

While Dimitras et al. (1996), Balcaen and Ooghe (2006), or Ravi Kumar and Ravi (2007) offer

extensive reviews of the last 70 years of bankruptcy prediction literature and related prediction

techniques, we focus on statistical accounting models, because they are still used in the accounting

practice. In particular, we compare our fundamental-based framework to Altman (1968) and Ohlson

(1980). While there are several studies, for example Begley et al. (1996) and Hillegeist et al. (2004),

which demonstrate that Ohlson’s model outperforms the Altman z-score, Agarwal and Taffler (2008)

find that the z-score model outperforms other statistical models. Moreover, Altman and Saunders

(1998) report that multivariate discriminant analysis models are by far the most widely used statistical

models. Hence, we use both z-score and o-score, which dominate the literature on statistical

bankruptcy prediction models, as there is no clear (ex ante) superiority of one over the other.11 Finally,

several studies (see e.g., Agarwal and Taffler 2008; Dichev 1998) confirm that the two statistical

models also perform well on more recent data.

11 Both models were estimated on data more than 30 years ago, hence they are completely out-of-sample for this purpose.

In addition, we employ the re-estimated coefficients for z- and o-score to address possible structural breaks over the last decades in the bankruptcy process. However, the empirical results for the updated coefficients are weaker. One reason may be that they are developed on a narrower definition of bankruptcy. This is in line with Begley et al. (1996), who do not find an improvement in predictive performance for the updated z- and o-score models and Hillegeist et al. (2004), who confirm this finding for the z-score model. Further, Beaver et al. (2012) and Beaver et al. (2005) find evidence for a significant decline in the predicative power of fundamental information. Balcaen and Ooghe (2006); Mensah (1984) also notes that non-stationary and data instability causes (1) lower predictive power, (2) fundamentally unstable factor loadings and (3) a sample-specific arbitrary character of the estimation results.

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2.1 Z-Score Model

The multivariate discriminant approach of Altman (1968) is a widely used accounting-benchmark

model scoring financial distress risk based on proxy variables capturing the financial liquidity, long-

term and short-term profitability of assets, the capital structure and the asset turnover of a firm. The

original z-score model based on a paired sample design of 66 firms from 1946-1965 and predicting

bankruptcy over the next year has the following form:

1.2 1.4 3.3 0.6 0.999WC RE EBIT MVE SALESz scoreTA TA TA BVL TA

− = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ (1)

where WC, RE, EBIT, MVE, SALES, TA and BVL correspond to working capital, retained earnings,

earnings before interest and taxes, market value of equity, sales, total assets and book value of total

liabilities, respectively. While Altman (1968) finds 95% accuracy for the small initial sample one

year before failure, subsequent tests until 2000 demonstrate the accuracy of 80-90% in distinguishing

solvent from insolvent firms (Altman 2000). Moreover, the model's predictive ability drops off

considerably for out-of-sample predictions (79%) and for longer horizons. Hence, to allow for sample

specific coefficients we re-estimate the z-score model using a standard logic regression design in a

growing window approach (henceforth z-scoreu).12

2.2 O-Score Model

Although still popular and widely employed, Altman’s z-score model is subject to several criticisms

because of the multiple discriminant analysis’ highly restrictive assumptions. These are, among

others, the multivariate normality of the independent variables and omitting of non-linear effects as

argued by Ohlson (1980). As response, Ohlson (1980) introduces an alternative model based on logit

analysis, which is commonly considered as less demanding (Balcaen and Ooghe 2006). He estimates

the conditional probability model on a dataset of 105 bankrupt and 1058 non-bankrupt firms for the

period 1970 to 1976 with the help of the non-linear maximum likelihood technique. As result, he

finds nine variables to be significant and defines the o-score model as:

12 The results and specification are given in the Appendix 1. We are aware of the revised z´-score for private firms (see

Altman 1983) that uses book value of equity and do not rely on market information. For the purposes of comparison and with reference to wide adaption in the prior bankruptcy literature, we include the original z-score model (1968) to avoid an unwanted look-ahead biases and generate the updated model as additional benchmark.

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[ ] [ ]1 2 0 & 0

1

1

1.320 0.407 log 6.030 1.430 0.076 price level

1.720 2.370 1 .830 0.285

0.521 | |

t tBVL TA NI NI

t t

t t

TA BVL WC CLo scoreGNP TA TA CA

NI FFOI ITA BVL

NI NINI NI

− −> < <

−

−

− = − − ⋅ + ⋅ − ⋅ + ⋅

− ⋅ + ⋅ − ⋅ + ⋅

−− ⋅ −

(2)

where TA, BVL, WC, CL, CA, NI and FFO correspond to total assets, book value of total liabilities,

working capital, current liabilities, current assets, net income and funds from operations, respectively,

and Ix represents the indicator function, which is equal to one if condition x is fulfilled and zero

otherwise. While a higher z-score corresponds to higher solvency, the relation is reversed for the o-

score. As with the z-score model we re-estimate the original o-score coefficients (henceforth o-scoreu)

on a quarterly basis, using a growing window and the logit regression approach, to capture economic

changes in the underlying informativeness of financial ratios.13

Next, in order of comparability we use a logistic transformation 11+𝑒𝑒𝑋𝑋

,𝑋𝑋 ∈ (𝑍𝑍,−𝑂𝑂) to transform the

original (re-estimated) z-score (z-scoreu) and o-score (o-scoreu) into financial distress probabilities z-

prob (z-probu) and o-prob (o-probu), which are used as benchmarks in the following.14

One of the major critiques of the class of statistical models is their limited theoretical underpinning.

Statistical models capture information from the default process in the estimated parameters. Hence,

the exact default mechanism remains hidden and the parameter estimates are confined to a specific

time and sample. The necessary re-estimation of the parameters consumes a lot of data. Chava and

Jarrow (2004), for example, consider the sample period 1962-1999, but have to use the period 1962

to 1990 for parameter estimation. Hence, they can only predict bankruptcies for the period 1991 to

1999. Moreover, Xu and Zhang (2009) point out that the predictive variables and the functional form

of the predictive relationship are not derived rigorously for statistical models. In this context, Platt

and Platt (1990) claim the discrepancy of within-sample and out-of-sample classification results.

Mensah (1984) notes that the structure of statistical models requires a stable economic environment

(e.g., such as the rate of inflation, interest rates) and shows that data instability reduces the

13 The results and specifications are given in the Appendix 1. 14 More exactly, the z-score would be transformed into the default probability z-prob with standard normal cumulative

distribution function Φ(-Z). However, Hillegeist et al. (2004) argue that this approximation is sensible. Moreover, it does not change the order and therefore has no impact on sorting based performance measures.

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discrimination power. In the next section we present a theoretical approach, which is based on the

current line of company valuation research to address these issues. The proposed framework explicitly

models the firm’s processes and the default mechanism using an intuitive and theoretically grounded

approach.

THE MODEL

In the following we provide a simple yet comprehensive non-linear model, where (1) the firm’s

operating performance and (2) the development of its operating liquidity are the major driving forces

for survival. The focus is on the firm’s ability to generate sufficient cash inflows from its operations,

which is critical in avoiding distress (Luoma and Laitinen 1991; Pompe and Bilderbeek 2005). It is

well established through strong evidence that cash flows and operating performance are important

variables for bankruptcy (references include: Altman 1968; Beaver 1966; Emery and Cogger 1982;

Jones and Hensher 2004; Shumway 2001). Moreover, Mossman et al. (1998) demonstrate that among

different models, the cash flow model discriminates most consistently two to three years before

bankruptcy. In order to model the operating cash flow, we build upon recent equity valuation models

(Duffie and Lando 2001; Garlappi et al. 2008; Garlappi and Yan 2011; Pástor and Veronesi 2003,

2006; Schwartz and Moon 2000, 2001; Anderson and Carverhill 2012). These studies model a firm’s

key profitability by stochastic processes to derive a value estimate for the firm similar to a discounted

cash flow setting. The most recent models used in Garlappi and Yan (2011) and Pástor and Veronesi

(2006) employ mean-reverting processes for the revenues and earnings, respectively. For example,

Garlappi and Yan (2011) build a simple equity valuation model to investigate the impact of

shareholder recovery and the risk structure of equity; however, they obtain their probability of default

estimates from Moody’s KMV. Pástor and Veronesi (2006) seek to explain the high valuations of

technology firms during the dot-com bubble rationally with respect for uncertainty of the average

profitability. However, their model neglects the probability of default and does not include costs

explicitly. Anderson and Carverhill (2012) simulate a well-designed structural equity valuation model

with mean-reverting revenues. Their study is based on a single benchmark firm and focuses on the

determinants of the optimal cash saving policy.15 While all of these models are related to financial

15 Contrary to the reported default rates from the latest Standard & Poor’s (2014) default study, their model calibrations

produce unreasonably high estimates for the default probability. For example, the single benchmark firm fail in 10.86

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distress, neither study benchmarks its implied distress probabilities to alternative bankruptcy

prediction models.

With reference to the prior literature, we apply mean-reverting (Ornstein-Uhlenbeck) processes for

the modeling of the cash-generating processes (Anderson and Carverhill 2012; Garlappi and Yan

2011; Klobucnik and Sievers 2013; Pástor and Veronesi 2006; Schwartz and Moon 2001). One major

advantage of the approach for this purpose is that it offers explicit modeling of volatility, both in the

sales and cost processes. It is well established in the literature as shown by Shumway (2001) or

Hillegeist et al. (2004) that volatility is a major driving force for bankruptcy. While structural models

(e.g., Black and Scholes 1973 and Merton 1974) are successful because they estimate asset volatility

from market data, which usually includes various firm relevant information, this study restricts to

accounting information. Consequently, the use and interaction of different stochastic processes

intends to mimic the market volatility by combining various sources of accounting volatility. In

particular, there are three independent sources of volatility: sales volatility, growth volatility, and cost

volatility. They correspond to variations in business and market characteristics, future prospects, and

production technology, respectively.16 The accounting volatility measures are discussed further in

section 3.2. Given the modeling of the operating performance and its volatility, the idea is to adopt

this framework to predict early warning signals through the firm’s cash flow from operations.

In order to determine a measure of financial distress, we follow the perspective of Beaver (1966)

viewing a firm’s liquid assets as a buffer against unexpected cash outflows. The buffer is

operationalized by the operating liquidity reserve (OLR), which is defined as the difference of current

assets (including cash and unused credit lines) and current liabilities and closely related to the net

working capital position.17 As stated above, the OLR level is increased (decreased) by the positive

(negative) stochastic cash flow from operations, i.e. sales minus costs. It has long been known

(Merwin 1942; Smith and Winakor 1935) and is well established (Altman 1968; Kahya 1997; Ohlson

1980) that the operating cash level is an important indicator for bankruptcy and is used, in terms of

percent with reference to a 5-year prediction horizon (Anderson and Carverhill 2012, Table 1). In comparison, Standard & Poor’s report a global five years cumulative default probability in the ‘BBB’ category of about 3.4 percent (1981-2014). The corresponding default rate using Fitch rating category is about 2.03 percent (1990-2014).

16 In a similar view, Pástor and Veronesi (2006) account for a firm-specific and an economic component (reflecting the economic cycle) as drivers of their stochastic earnings process.

17 In the literature the terms “net working capital” (see e.g., Bates et al. 2009) and “working capital” (see e.g. Altman 1968) are usually used as synonyms. However, in this study we speak of the “operating liquidity reserve” to stress the difference to the measurement of working capital.

11

the net working capital position, in both the z-score and o-score models. In his study Altman (1968)

finds that the working capital ratio is the only significant individual predictor for failure. Moreover,

there is evidence that the role of liquidity in the form of working capital has increased from the 1980s

on (Begley et al. 1996). In contrast Acharya et al. (2012) emphasis that the role of the cash savings

are associated with higher, not lower credit risk in the long-term. A positive OLR demonstrates the

firm’s ability to pay off its short-term liabilities. As a measure of available liquidity, this indicates

that suppliers and creditors could be satisfied from this amount. Anderson and Carverhill (2012) and

Sufi (2009) demonstrate that the corporate liquid assets are a function of the expected cash flow from

operation, whereby a higher profitability risk is accompanied with higher liquidity reserves. For the

model we assume that the liquid reserves evolves with the firm’s cash flow from operations (see e.g.,

Emery and Cogger 1982; Anderson and Carverhill 2012). In case the OLR has been exhausted and

drops below a certain level, the firm is technically in financial distress, analogous to structural models

in Black and Scholes (1973) and Merton (1974). But in contrast to the contingent claims framework

we trigger financial distress by the short term operating liquidity reserve.18 This allows firms to

continue operating with negative net worth (i.e., total assets minus total liabilities) as long as the

liquidity from the reserve remains, which is consistent with empirical findings (Reisz and Perlich

2007). The implicit assumption is that creditors are reluctant to provide additional funding when both

the firm’s current assets do not cover current liabilities and it loses money from its operations. Overall,

the larger the cash flows from operations and the operating liquidity reserve, the lower the probability

of financial distress predicted by the model, which is in line with the results in Kahya (1997) and

Anderson and Carverhill (2012).

In the following, we present a firm stochastic profitability process implemented in a simplified income

statement that is accumulated in the firm-specific operating liquidity reserve.19

3.1 Sales and Costs Processes

Banker and Chen (2006); Barth et al. (2001), among others, find improvements in the predictive

ability by disaggregating earnings into its major components. Accordingly, we model the operating

18 This is consistent with Wruck (1990, p. 421), who defines financial distress as “the situation were the cash flow is

insufficient to cover current obligations”. 19 In a related study Fu (2009) looks at future working capital and cash flow. However, the study links the development

of future working capital over time to estimates of expected net income and volatility of net income based on, e.g., the normal distribution. Hence, the analysis does not combine various stochastic processes at the same time to forecast future company performance and limit the scope of her study to a dataset of 183 small U.S manufacturing firms.

12

performance process by decomposed sales and cost processes. It is a management accounting

perspective where operating earnings are sales net of costs, which mimics the major cash flows of the

firm. The sales dynamics (S) are given by the stochastic differential equation:

( )( ) ( ) ( ) ( )S S

dS tg t dt t dz t

S tσ= + ⋅ (3)

where 𝑔𝑔(𝑡𝑡) is the sales growth rate, 𝜎𝜎𝑆𝑆(𝑡𝑡) the volatility of sales, and 𝑧𝑧𝑠𝑠 represents unanticipated

changes in sales, which follow a Wiener process. Hence, the development of the sales is driven by

the growth rate. This modeling approach is well established in the literature (Anderson and Carverhill

2012; Garlappi and Yan 2011; Pástor and Veronesi 2006; Schwartz and Moon 2001). For the sales

growth rate we assume a mean-reverting process:

( ) ( )( ) ( ) ( ) g gdg t g g t dt t dz tκ σ = − + ⋅ (4)

which pictures best the convergence to a long-run equilibrium growth rate �̅�𝑔 determined by market

specifications as competition. The volatility of sales growth 𝜎𝜎𝑔𝑔(𝑡𝑡) and the unanticipated changes in

growth rates 𝑧𝑧𝑔𝑔 describe the unpredictable components (i.e., systematic noise) of the growth rate. The

speed of convergence parameter 𝜅𝜅 captures the concept of adjusting to long-run equilibriums. With

the help of the term ln(2)/𝜅𝜅 the speed of convergence can be expressed as an intuitive half-life of the

process.20

Beside sales, the cost component has to be modeled. Following the approach in Klobucnik and Sievers

(2013) we abstract for simplicity from the separation of fixed and variable costs and model the selling,

general and administrative expenses and the cost of goods sold together in the cost ratio c with a

mean-reverting process:

( ) ( )( ) ( ) ( ) c cdc t c c t dt t dz tκ σ = − + ⋅ (5)

which also convergences to the long-run cost ratio 𝑐𝑐̅ with the speed parameter 𝜅𝜅 and takes the

volatility of the cost rate 𝜎𝜎𝑐𝑐(𝑡𝑡) and the unanticipated changes in costs 𝑧𝑧𝑐𝑐 into account.

20 Pástor and Veronesi (2003) interpret the speed of mean-reversion κ as component of learning mechanisms about the

long-term equilibrium.

13

3.2 The Accounting Volatility Measures

For the accounting volatility measures in the above processes we assume a deterministic behavior

consistent with the company valuation literature. Regarding the volatility of accounting information,

Dambolena and Khoury (1980) study the variability of financial ratios using a paired sample of 23

bankrupt and nonbankrupt firms in a multivariate discriminant approach. Their findings indicate that

bankrupt firms experience higher accounting volatilities even for longer forecast horizons of up to

five years. In contrast, Pompe and Bilderbeek (2005) show that the inclusion of accounting ratio

volatilities adds only limited additional power for bankruptcy prediction when simply included as a

factor in a linear statistical model using a sample including 1369 bankrupt firms. Analogous to

accounting volatility measures, Duffie and Lando (2001) show that accounting noise (i.e., uncertainty

of fundamental information) increase the default intensity in a structural valuation setting. Therefore,

besides the high importance of volatility measured by stock market data in contingent claims models,

there seems to be some indicative evidence that volatility inherent in accounting data might be

beneficial for bankruptcy prediction.21 It is often assumed that maturing firms’ businesses stabilize

over time (Nissim and Penman 2001; Pástor and Veronesi 2003). This means, that the abnormal part

for the volatility measures is faded away, thus the risk in a firm’s profitability becomes “normal”. We

will use the following three accounting volatilities to proxy the idiosyncratic profitability risk of (i)

sales volatility (ii) sales growth volatility and (iii) cost volatility, which converge deterministically to

a long-term volatility level 𝜎𝜎�𝑠𝑠, 𝜎𝜎�𝑔𝑔 and 𝜎𝜎�𝑐𝑐 with 𝜅𝜅 as the speed of convergence:

( ) ( )S S Sd t t dtσ κ σ σ = ⋅ − (6)

( ) ( ) ( ) g g g gd t t dt t dtσ κ σ σ κ σ = ⋅ − = − ⋅ (7)

( ) ( )c c cd t t dtσ κ σ σ = ⋅ − (8)

The long-run sales volatility and long-run cost volatility is assumed to decline to a normal noise level,

𝜎𝜎𝑆𝑆� and 𝜎𝜎�𝑐𝑐 , while the long-term growth rate volatility 𝜎𝜎�𝑔𝑔 converges to zero so that formula (7)

21 Additional evidence is provided by Altman et al. (1977) using “stability of earnings”, which is measured as the

standard error of the estimate around a 10-year trend in return on assets. In particular, this study notes that the stability of earnings is the second most important variable among a set of seven total explanatory variables considered. Earlier, Blum (1974) includes accounting variability measures, such as the standard deviation of net income, in a cash-flow framework to explain company failure. Donelson and Resutek (2015) study about earnings qualities show a strong relevance between earnings uncertainty and earnings expectations. Similar, Dichev and Tang (2009) emphasize substantial predictive improvements of long-term earnings implied by the volatility of earnings.

14

simplifies. By considering three sources of accounting volatility, the model attempts to capture the

different origins of volatility in a firm’s operations.

3.3 The Change in Operating Liquidity Reserve (OLR)

From the processes above, the firm’s earnings before interests, taxes, depreciation and amortization

EBITDA(t) as basis for the operating cash flow is derived as:

( ) ( )( ) ( )1EBITDA t c t S t= − ⋅ (9)

which is the gross profit margin (1 − 𝑐𝑐(𝑡𝑡)) multiplied by sales. From the earnings before interests,

taxes, depreciation and amortization the firm has to service its debt. Therefore, the interest positions

also play an important role for the firm’s liquidity. The net interest (interest income less interest

expense) is modeled as:

( ) ( )f LTDInterest t r cash t r LTD= ⋅ − ⋅ (10)

where cash is the interest earning cash position and LTD the interest-bearing long-term debt. Cash

earns interest at the risk-free rate 𝑟𝑟𝑓𝑓 while the firm has to pay 𝑟𝑟𝐿𝐿𝐿𝐿𝐿𝐿 for its long-term debt. We explicitly

take the financial activities into account because recent research shows that firms significantly

increased this position, e.g., to buffer the risk against future cash shortfalls (Bates et al. 2009; Lins et

al. 2010; Sufi 2009; Anderson and Carverhill 2012; Opler et al. 1999). Despite the importance of

excess cash holdings research argues that (un)used lines of credit are a second instrumental

component of corporates liquidity.22 Therefore, we explicitly model the amount of unused line of

credit available as additional liquidity buffer using the unique dataset of Sufi (2009). 23 By

22 Sufi (2009) notes that about 85% in his random sample of US firms have a line of credit between 1996 and 2003,

which on average representing 16% of total assets (where ca. 10% is the unused portion on line on credit). Lins et al. (2010) report an equivalent median credit line about 15% of total assets in his survey analysis of international companies. Yun (2009) stated that about 63% have lines of credit in his sample of 2,533 firms. Acharya et al. (2013) report a trade-off between credit lines and cash holdings for firms having a growing aggregate risk exposure.

23 Sufi (2009) provides an indicator variable that equals one if a firm has a line of credit and zero otherwise for the period 1996-2003. Additionally, he reports the handcollected data for the unused and used amount of credit lines disclosed in the 10-K filings. Specifically, we use this information to approximate a firms annually unused line of credits by the fitted values from linear (OLS) regression for each Fama-French 10-industry classification code:

[ ] [ ] [ ]ln (totalassets ) /0, 1, 2, 3,lineunused cash flow cash M B cashi i i iβ β β β= + + − + −

where cash flow is the operating cash flow of the firm (oancf), (total asset–cash) are the total assets of the firm adjusted for the cash position (at-che) and (M/B-cash) is the decile rank for the market-to-book value adjusted for the cash position. To limit the influence of outliers negative predicted values were set to zero and the results were truncated at the 95th

percentile. There are 2,534 / 46,653 firms (firm quarter) observation with a predicted unused line of credit.

15

incorporating unused credit lines we address firms operating with low or even negative working

capital (e.g., Dell Inc., Boeing Co.) but high operating performance. As result, the net cash flow from

operations CFFO that drives the changes in the operating liquidity reserve equals:

( ) ( ) ( )( )

( )( )

1 ( ) ( ) ( ) ( ) 0 |

( ) ( ) ( ) ( ) 0tax EBITDA t Interest t if EBITDA t Interest t

CFFO tEBITDA t Interest t if EBITDA t Interest t

− ⋅ + + > = + + ≤ (11)

where tax is the corporate tax rate. In order to arrive at the change in the OLR we assume no major

cash flows from investing or financing activities for simplicity. Consequently, there is no investing

in property, plant and equipment in the short-term and there is no change in the capital structure.24

The latter assumption could be extended in future research by explicitly looking at financing

constraints as for example in Anderson and Carverhill (2012). Finally, the change in OLR is driven

by the cash flow from operations:

( ) ( )OLR t CFFO t∆ = (12)

In order to implement the model, two adjustments are necessary. Compared with the daily frequency

of market data in market-based structural models, the highest available frequency of accounting data

is quarterly data. Moreover, the popular Black and Scholes (1973) and Merton (1974) model is based

on one stochastic process while the model proposed in this study employs different processes to

capture various dimensions. Hence, first, we discretize the stochastic processes as in Schwartz and

Moon (2001) to fit the quarterly frequency of accounting data. Second, due to the interaction of these

processes there is no closed form solution. Instead, we have to apply Monte-Carlo techniques to

simulate the cash-generating sample paths. Given this study aims to assess the predictive power for

bankruptcies for up to five years into the future, it is sufficient to simulate the OLR paths up to the

next twenty quarters. Similar to the structural models mentioned above, this study defines financial

distress as state-dependent criterion, where the firm’s current liabilities exceed its current assets plus

an optional liquidity reserve from unused credit lines for any quarter within the subsequent twenty

We thank Amir Sufi for making his dataset publicly available: http://faculty.chicagobooth.edu/amir.sufi/chronology.html.

24 However, by using selling, general and administrative expenses (item: xsgaq) and cost of goods sold (item: cogsq) from Compustat, we do consider research and development costs and amortization of tools and dies where the useful life is two years or less, which are part of investing activities. Moreover, we do take interest payments as part of the cash flow from operations into account, which can be regarded as financing activities in US GAAP. Including dividend payments as cash flow from financing activities and tax loss carry forward balances did not improve predictive power, therefore we do not consider these items in this model.

16

quarters, i.e. the OLR falls below a certain level b and the sample paths is classified as failed and no

longer simulated:

{ }( ) 1OLR k bI < = if ( ) | ( 1) bOLR k b OLR k< − ≥ , where ( )1, 20k t t∈ + + (13)

Consequently, the empirical probability of financial distress equals the inverse survival probability at

time T conditional on the simulated sample paths for the OLR:

{ }( )1 10,000

1( )N

OLR k bk N

P financial distress IN <

= =

= ∑ (14)

where N=10,000 is the number of Monte Carlo simulations. Regarding the financial distress boundary

b we need to answer when a firm enters the financial distress stage. For example, Reisz and Perlich

(2007) and Davydenko (2012) discuss findings for firm-specific boundaries. However, their

implication refers to asset-implied values and not to operating liquidity reserve specific boundaries.

While the OLR of solvent firms significantly differs by industry and period, the level for financially

distressed firms naturally converges to zero. In line with the findings of prior literature (see e.g.,

Brockman and Turtle 2003; Reisz and Perlich 2007), we trigger a financially distress event at an early

stage, i.e., before the current liabilities exceed the operating liquidity reserve. Therefore, we allow the

barrier b to be positive and discuss further evidence in the empirical analysis (section 4.2). The next

section describes the data and discusses the empirical estimation results.

17

DATA AND MODEL IMPLEMENTATION

4.1 The Data

Our sample consist of quarterly accounting data from Compustat from 1980Q1 to 2010Q4 for the US

market. 25 Thereby the results are comparable with prior research as the majority of studies are

conducted for the time period 1980-2003 as presented in Ravi Kumar and Ravi (2007). At the same

time, they are more up to date because the sample additionally covers the recent financial crisis. There

are three reasons for starting in 1980 in the literature. First, Dichev (1998) demonstrates that the

Compustat sample contains substantially more firms after 1980 and that business failures have

significantly increased after 1980. Second, the Bankruptcy Reform Act of 1978 changed the

institutional setting and economic risk factors of financially distressed firms (Hackbarth et al. 2015;

Hillegeist et al. 2004). Third, since both the z-score and the o-score models were developed before

1980, the out-of-sample perspective is guaranteed. While most studies use annual accounting data,

this study offers results on a more favorable quarterly frequency. Chava and Jarrow (2004) and

Campbell et al. (2008) demonstrate that the higher frequency of quarterly data, which contain more

timely information, can improve forecasting abilities of the models. We exclude (1) firms with SIC-

codes 6,000 to 6,999 (financial firms) and (2) firm quarter observations where essential accounting

information were unavailable. The equity price data and delisting codes are taken from CRSP daily

and monthly databases for NYSE/AMEX/NASDAQ/ARCA common stocks (share codes 10/11). To

include delisting firm quarter observations not in the date range of the CRSP/Compustat merged

databases (CCM) we follow the methodology of Beaver et al. (2007) for an extended merge

procedure.26 The accounting data is lagged by two months to ensure that the data are observable prior

to a delisting, i.e., it is assumed that financial statements are available by the end of the second month

after the firm’s fiscal quarter-end (Campbell et al. 2008; Correia et al. 2012).

This leaves a large and anonymous dataset of 330,549 firm quarter observations for 10,687 non-

financial firms from Compustat. While many bankruptcy studies work with small datasets, large

samples as in Chava and Jarrow (2004) offer more convincing results. Consistent with prior literature,

25 The risk-free rate and industry-classification portfolios were taken from Kenneth French’s library. 26 We would like to thank Richard Price for providing the associated SAS program on his website:

http://richardp.bus.usu.edu/research.

18

we winsorize the independent variables at the 1st and 99th percentile of their pooled distributions to

prevent extreme values from driving the results.27

There are numerous proxies to empirically define financial distress/failure in the literature, which

range from narrow definitions, e.g., actual filing for bankruptcy or credit spreads to a broader

definition relying on a subsequently deterioration of financial ratios such as EBITDA or cash flow

coverage.28 This study follows the recent literature and applies a broad empirical definition for three

main reasons. First, as defaults are rare events, using this broader (delisting) measure increases the

sample size and therefore the robustness of the findings. Second, certain difficult to predict cases such

as strategic defaults receive less weight. With this approach we concentrate on a severe financial

weakness and avoid firms that only declare bankruptcy for strategic reasons (relief of debt) or

unexpected events (natural disasters) from driving the results (Balcaen and Ooghe 2006). Finally, a

broader definition ensures to capture cases where firms have to counter serious financial problems,

but could avoid filing for bankruptcy (Hilscher and Wilson 2013) or willfully delay their filings. In

general, the broadly conceived financial distress definition may capture more firms than relying on

the legal definition of bankruptcy where the default occurs after a firm is already in a financial distress

situation.

Thus, this study adopts exchange-delisting codes from CRSP, which are common in the literature and

more reliable given the underlying delisting regulations and standards existing for US stock

exchanges. In addition, CRSP delisting information offers explicit dates and reasons for a delisting.

Regarding the economic impact after the delisting date, Macey et al. (2008) report that stocks delisted

from the NYSE and subsequently traded on the so-called “Pink-Sheets” nearly double their mean

volatility of closing prices, triple their average percentage spreads and simultaneously lose about 50

percent of their share price and daily trading volume. There are different sets of delisting codes related

to financial distress/business failure in the literature depending on the broadness of the definition.

Most studies use the definition of performance-related delistings (see for instance: Campbell et al.

27 Note that the values of sales and operating liquidity reserve do not need to be winsorized or trimmed given these

measures serve as initial parameters of the sample paths. However, the results are robust to alternative filtrations. 28 Examples for a broader definition include e.g., Andrade and Kaplan (1998) who define financial distress by (1) the

ability to meet contractual obligations and (2) debt restructuring. Wruck (1990) distinguishes between (1) stock-based definitions (i.e., poor stock price performance), (2) flow-based definitions (i.e., ability to meet current cash obligations) and the (3) legal definition (i.e., filing for bankruptcy protection). Platt and Platt (2002) search financially distressed companies according to (1) several years of negative net operating income (2) suspended dividends and (3) restructuring or layoffs. More recently Kapadia (2011) rely on an aggregated measure of business failure provided by Dun & Bradstreet.

19

2011; Caskey et al. 2012; Dichev 1998; Reisz and Perlich 2007; Shumway 2001). Performance

delistings are associated with negative changes for the firm and cover the CRSP delisting codes 400

and 550-585. Besides bankruptcy, performance delistings additionally include, among others,

insufficient capital, market capitalization or market-makers and non-payment of exchange fees

(Campbell et al. 2008; Dichev 1998). Similar to Xu and Zhang (2009) we consider all these cases as

financially distressed. By using this broad measure of distress we obtain 41,664 firm quarter

observations and 3,435 firms that were delisted for performance-related reasons. This yields a

cumulative distress rate of 32%, which is substantially higher than for example the 14% in Reisz and

Perlich (2007), who work with a smaller sample and a shorter sample period. Table 1 offers an

overview over the reasons and distributions of delistings and clearly show that performance delistings

can be associated with financial distress or bankruptcy.

We note that delistings as a proxy for bankruptcy have the drawbacks that they are only available for

exchange-listed firms and that they are sometimes not accurate with respect to the timing of

bankruptcy. However, Vassalou and Xing (2004), for example, argue that in their sample firms delist

two to three years prior to actual default. On the other hand, Dichev (1998) argues that some firms

that enter bankruptcy do not delist or delist significantly later. Franzen et al. (2007) find qualitatively

similar results when using performance delistings instead of legal definition using bankruptcy filings.

Nevertheless, this measure is adequate to make statements about cross-sectional variation in the

probability of distress and it is widely used (Campbell et al. 2011; Caskey et al. 2012; Dichev 1998;

Reisz and Perlich 2007; Shumway 2001).

------------------------- Please insert Table 1 about here -------------------------

4.2 Parameter Estimation

The stochastic prediction model has in total 17 parameters, which stem from the firms’ balance sheets

and income statements. However, the employed information set is the same as for the z-score and o-

score models. Remember that the two statistical models are based on EBIT and sales among others,

and in the same vein our model, for example, employs sales and costs to arrive at the EBITDA.

20

The estimation of the model initial input parameters is presented in Table 2. It employs intuitive and

straightforward estimations based on the firm- and industry-specific first two moments of the

preceding seven quarters to control for seasonal effects. Where there are less than seven preceding

quarters available, we restrict the estimation to the available information set (with a minimum of three

quarters) to keep as many observations as possible in the sample. This is another advantage of the

model compared to the statistical models, which demand more data to be initialized as argued above.

Hence, this model does not need first-stage regression to estimate coefficients (Hillegeist et al. 2004)

and no holdout sample approach to test its performance. Moreover, it is less sensitive to structural

breaks as it depends on (short-term) accounting data and estimates parameters firm-specifically.

The initial sales growth rate is determined as weighted average over the respective series of the

previous seven quarters. Similar to Campbell et al. (2008) we incorporate declining weights to

account for a decreasing information content of lagged accounting data.29 For the initial variable cost

as fraction of revenues we use the estimated coefficient from rolling regression of quarterly costs on

sales assuming no intercept. The initial volatility of sales for each firm is calculated as the standard

deviation of the previous seven quarters. For the initial volatility of sales growth and the initial

volatility of costs we use the standard deviation of residuals from regressions similar to the accrual

quality measure in Francis et al. (2005). For the long-term parameter of costs we estimate industry

medians over the Fama-French 48-Industry classification for two main reasons. First, prior studies

(e.g., Chava and Jarrow 2004; Hillegeist et al. 2004) clearly demonstrate that industry effects are

important for bankruptcy prediction (because of different levels of competition, etc.). Second,

industry-specific parameters provide more stable estimates and neutralize individual outliers in the

large dataset (Klobucnik and Sievers 2013). Both the long-term volatility of sales and the long-term

volatility of costs are set to 0.05, the degree of a long-run noise level.30

The estimation of the speed of convergence parameter deserves more explanation. In the economics

literature the standard approach to estimate convergences is to use the absolute convergence

29 In particular, the weights are reduced every quarter by about 1/3 and the estimated growth rate is reduced about 50

percent with respect to a declining weighted-average. Initializing the growth rate with an arithmetic mean calculation produces similar results but does not cover firms that rapidly lose sales growth (within 8 quarters) and become financially distressed.

30 Moreover, we confirm these values empirically by estimating growing window standard deviations over the whole sample period for maturing firms (age>10 years), which yields a median value of 0.14 (0.03) for the volatility of sales (long-term volatility of costs), respectively. However, the results are not sensitive to these parameters.

21

regression (Rogoff 1996). Similar to Altomonte and Pennings (2008) we use the following regression

to estimate the convergence: 31

,

1, ,1,

ln t ii i t i t i

t i

gg

gα β ε−

−

= + ⋅ +

(15)

where 𝑔𝑔𝑡𝑡,𝑖𝑖 is the corresponding sales growth rate for time t and firm i and 𝛽𝛽𝑖𝑖 yields the firm-specific

convergence. These firm-specific values are then pooled for each industry (Fama-French 48-Industry

classification) to define the median as industry-specific convergence parameters for reasons

mentioned above. The balance sheet positions are initialized as presented in table 2.

A final key parameter in our stochastic model is the boundary condition b, i.e., the OLR barrier to

classify a firm as financially distressed. As discussed in section 3.3, technically, a firm is in financial

distress if the current obligations exceeds the liquid funds (including the cash position and unused

credit lines). Empirically, the financially distressed firms in our sample enter a downward spiral

before the operating liquidity reserve is exhausted. Instead of setting an arbitrary level, we model a

down-and-out barrier b that is strictly positive. For each quarter, we estimate the initial distress barrier

b as the median OLR for financially distressed firms five years prior a corresponding delisting event

(estimated in a 5-year rolling window). If the initial OLR is below the prespecified barrier b we fund

the firm with additional interest-bearing debt by the required amount (this adjustment is necessary for

85,113 firm quarters). Prior research mainly focuses on the market-value of assets to specify a certain

default boundary condition (Davydenko 2012; Reisz and Perlich 2007). For example, Reisz and

Perlich (2007) estimate the mean (median) implied barrier level at 30.53% (27.58%) of the market-

value of assets (i.e., non-zero). Analogously, Davydenko (2012) empirically findings indicate a mean

(median) barrier level that equals 66.0% (61.6%) of the face value of debt. Given the boundary

condition is a critical parameter we conduct additional robustness checks in section 5.5. Figure 1 plots

31 Additionally, we use the approach of Klobucnik and Sievers (2013) to calculate the speed of convergence, i.e. the

kappas. Solving � 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−1𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−1

=𝑡𝑡−8

𝑖𝑖=𝑡𝑡−5�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−1

𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠𝑖𝑖−1

𝑡𝑡−4

𝑖𝑖=𝑡𝑡−1� ∙ 𝑒𝑒−4∙𝜅𝜅� for kappa yields an alternative estimator,

which is then pooled to medians for the same industry (three digit SIC codes). The estimated default probabilities and the other findings remain qualitatively the same. However the approach in this study is well established in the economics literature and therefore preferable.

22

the estimated barrier b over time. As shown, the barrier level reveals a positive relation to periods

with greater uncertainty.

The debt interest rate is defined as the interest expense divided by the book value of current and long-

term debt as in Francis et al. (2005). The corporate tax rate is set to 35% consistent with, among

others, Garlappi et al. (2008) and Pástor and Veronesi (2006). For the risk-free rate we choose the

short-term risk free rate represented by the 3-Month US-Treasury Bill rate.

------------------------- Please insert Figure 1 about here -------------------------


The summary statistics for the individual parameters are presented in Table 3 separating solvent and

distressed firms and the cross-sectional statistics. One can see that the sample firms are highly

divergent with respect to size (parameter 1: quarterly sales ranging from less than one million dollars

to more than four billion dollars), growth (parameter 2: quarterly sales growth ranging from less than

-13% to more than 14%) and profitability (parameter 7: sales margin ranging from 44% to more than

138%). Moreover, the substantially lower sales median of around 38 million dollars compared with a

mean of 280 million dollars demonstrates that there are few very large firms while the majority are

medium-sized firms. Finally, the initial OLR position (parameter 15) for solvent and financially

distressed firms differs significantly (the median for solvent (distressed) firms is 42 (12) million

dollars), which also illustrates the diversity of firms in the sample. Most importantly, we find

significant differences between solvent and distressed firms along the volatility of sales, sales growth

rate and variable costs (parameters 4, 5, and 9). Figure 2 (Panel A, B and C) highlights the

deterioration of the model parameters approaching the performance-related delisting.


------------------------ Please insert Figure 2 about here -------------------------

23

EMPIRICAL ANALYSIS

In this section we evaluate the performance of the distress prediction based on stochastic processes

approach (henceforth s-score) along several dimensions and benchmark it against the two prominent

statistical models, z-score and o-score, which rely on a similar set of accounting information. First,

we examine the forecast accuracy and construct validity among the main financial distress indicator,

the OLR. Second, we compare our model’s default rates with observed default rates with the help of

various summary statistics and investigate its early warning ability, i.e. the ability to indicate as early

as possible an increase in distress risk. Third, we evaluate the models’ ability to distinguish between

distressed and non-distressed firms by using the well-established concept of accuracy ratios. This

prediction-oriented test assesses the cross-sectional discriminatory power (Vassalou and Xing 2004).

Fourth, we employ information content tests and examine the ability to explain actual defaults with

logit and hazard models as in Hillegeist et al. (2004). Most standard approaches concentrate on short-

term default prediction for the next period, which is usually one year into the future. This study looks

at different forecasting horizons as Duffie et al. (2007) emphasize their importance. In particular, we

investigate longer forecast horizons of up to five years (=20 quarters) because, naturally, it is

favorable to predict potential bankruptcy as early as possible.

5.1 Operating Liquidity Reserve (OLR) Forecast Accuracy

One way to judge the construct validity of our model is to compare the predicted OLR with the

corresponding actual OLR boundary condition implied from the financial statements. Table 4 presents

the absolute and relative forecast accuracy for the s-score model. Panel A contains the accuracy of

the OLR predictions computed as the median of 10,000 sample paths for each of the forecast horizons

(i.e., 1-20 quarters ahead where 𝑂𝑂𝑂𝑂𝑂𝑂𝑘𝑘predict,𝑘𝑘 ∈ (𝑡𝑡 + 1, … 𝑡𝑡 + 20)). Similar to the methodology of

Weiss et al. (2008) we evaluate the model’s predictive performance by benchmarking the predictions

against the actual OLR (𝑂𝑂𝑂𝑂𝑂𝑂𝑠𝑠𝑐𝑐𝑡𝑡𝑎𝑎𝑠𝑠𝑠𝑠) and report the absolute and relative differences:

( ) predictab

actu lt ts

aOLR LRBias O−= (16)

predict actualt t

actualt

relOLR OLRBia

OLRs

=

−

(17)

24

Panel B provides the descriptive forecast accuracy for the distressed sample. The results show that

the median (mean) absolute one quarter ahead differences in the OLR are -0.16 (+1.91) million or

about -3% (+9%) for the distressed firm sample.32 As expected the bias increase with the forecast

horizon. First, the average estimated time to distress is about 4 quarters (i.e., the median time when

the sample paths violate the down-and-out-barrier). Accordingly, the model relies more on the early

prediction paths. Second, the OLR is clearly industry-specific. Figure 3 gives an unbiased view of the

model performance by plotting the historical industry-specific 1-, 2-, and 3-year ahead predictions for

randomly selected industries. The graphs emphasize the mimicking character of OLR predictions for

longer horizons.

Table 5 reports the expected deterioration of mean and median levels of the OLR for the quarter

preceding a delisting. As previously discussed, the OLR of a firm approaching a performance-related

delisting converges to zero preserving a positive liquidity reserve before delisting. Contrary, the mean

(median) OLR of solvent firms are about 12 (6) times greater.


------------------------ Please insert Figure 3 about here -------------------------


5.2 Summary Statistics and Correlations

In this section we present the summary statistics and correlations of the different measures. Table 6,

Panel A, presents the summary statistics for the three models and for Standard & Poor’s long-term

credit rating, which is used to provide an alternative distress risk measure.


32 Weiss et al. (2008) report for the accounting-based earnings prediction model following Abarbanell and Bushee (1997)

a pooled 1-year ahead median percentage forecast error of 0.329.

25

All three models show the ability to separate delisting firms from non-delisting firms as the average

estimated distress probabilities for delisting firms are significantly higher. Looking at the medians,

the s-prob model reveals an even clearer ability to distinguish non-delisting firms (0.03) from

delisting firms (0.70). Overall, the z-prob and s-prob measure yields the highest estimated

probabilities while the updated z-probu and o-probu presents the lowest. As noted in Hillegeist et al.

(2004) the results indicate that the re-estimated versions are not well calibrated and do not reflect

changes in the underlying accounting ratios. For example, Franzen et al. (2007) report a negative

trend in the relevant accounting ratios, which increase the probability of misclassification. Not

surprisingly, all measures experience misclassifications with solvent (delisting) firms having high

(low) probabilities in the 99% (1%) percentile. However, we acknowledge that the original o-prob

model performs better for the 99% percentile for solvent firms with a relatively low probability of

0.66 compared to 0.99 and 1.00 for the z-prob and the s-prob models. Table 6, Panel B, displays the

correlation measures for the different models. We focus on Spearman’s correlation measure (below

the diagonal) as the relationships are supposed to be monotonic but not necessarily linear, which is

what Pearson’s measure detects. While the original statistical models are strongly correlated (0.73),

which is consistent with the value of 0.65 reported in Hillegeist et al. (2004), they are only modestly

correlated with the s-prob model (0.48 and 0.53). This shows that the model proposed in this study

captures relevant aspects that are not included in the z-prob and o-prob models. The lower correlations

(0.37 and 0.25) with the updated statistical models confirm this finding. These promising descriptive

statistics results are to be confirmed in the following analyses.


Figure 4 offers the median estimated distress probabilities for the different models according to the

Standard & Poor’s rating class. The predicted probabilities of all models increase with a deteriorated

credit rating, as expected. Compared with the historic default rates from Standard & Poor’s, the s-

prob model pictures the growing inherent risk of credit ratings with a steep increase from rating class

“BB-” onwards to high probabilities levels in the substantial risk categories. However, while ratings

26

provide a reasonably good measure for rough classification of firms according to their risk of failure,

they are a poor measure of actual default probabilities as recently demonstrated by Hilscher and

Wilson (2013). Consequently, we do not consider ratings for the further analyses.


Figure 5 displays the development of the estimated default probabilities over the time period 1980Q1

to 2010Q4. In line with the samples in Chava and Jarrow (2004), Campbell et al. (2008) and Kapadia

(2011), the data displays a steep increase in the number of bankruptcies which occurred in the period

from 1984Q1-1992Q3. As demonstrated earlier, the s-prob model pictures the behavior of the actual

defaults with peaks in the mid-1980s and around 2000. The declining pattern in the median financial

distress probability after 1990 is also related to, among less subsequent delistings, higher cash

holdings and a downturn in the leverage ratio (Bates et al. 2009). The o-prob model also performs

well while the z-prob model does not display periods with low bankruptcy rates properly.

Interestingly, for the financial crisis 2007-2009 all models show a slight increase in the average

default probability. As seen in Table 6, the z-prob model yields the highest estimated probabilities

with on average more than 18% while the value is less than 8% for the o-prob model.33


Figure 6 demonstrates the evolution of the default probabilities in the quarters that precede the

delisting. One can clearly see the early warning ability of the s-prob model. Compared with the two

statistical models, the s-prob model’s estimated default probabilities start to increase substantially

already around five years before actual delisting compared with the three years for the z-prob and the

33 Carefully note that the z- and o-score models need a mapping function to transform the score into a distress probability

estimates (see section 4.2). Hence, the calibration of the models could be adjusted by including an empirical mapping function as in the ZETA score (Altman et al. 1977) to change the magnitude of default probabilities. However, the ZETA score is a proprietary product that is not publicly available. As the analyses in the following depend on the ranking by default probabilities, a monotone mapping function would not change the results as it does not influence the ranking.

27

two years for the o-prob. These early signals are economically highly relevant to avoid costly

financial distress (Altman 1984; Warner 1977). One reason for the better performance of the s-prob

model for longer horizons might be the incorporation of the accounting volatility measures, since,

e.g., Agarwal and Taffler (2008) also demonstrate that market-based models, which take volatility

measures into account, perform better for longer horizons than statistical models.

In sum, this section demonstrates how well the models are calibrated to describe observed default

rates. However, the absolute probability levels are less relevant for the following cross-sectional

discrimination tests, which therefore offer another dimension of model testing.

5.3 Accuracy

When evaluating credit risk models two types of errors influence a model’s quality. First, bankrupt

firms can be classified as non-bankrupt, i.e., a type I error (false negative). Second, non-bankrupt

firms might be measured as bankrupt, thereby making a type II error (false positive). Type I errors

are associated with higher costs, which is why they are commonly considered as less desirable

(Agarwal and Taffler 2008; Altman et al. 1977) while type II errors cause hypothetical loss of profits,

interests and fees.34 As the stochastic model yields expected default frequency estimates, there is no

cutoff value for classifying distressed and non-distressed firms. In contrast, the methodology for the

original z-score and o-score models focus on a specific cutoff by minimizing the type I and/or type II

error. Instead, all three models in this study are viewed as continuous measures (see Hillegeist et al.

2004, Reisz and Perlich 2007). This is desirable because corporate distress is not a well-defined

dichotomy in reality. Loan officers, for example, make continuous decisions at what rate to lend

(Hillegeist et al. 2004). Hence instead of directly counting the number of misclassified firms for a

specific cutoff point we evaluate the discriminatory capacity using a prevalent analytic accuracy

measure, the area under the receiver operating curve (AUROC) (see for references: Agarwal and

Taffler 2008; Caskey et al. 2012; Chava and Jarrow 2004; Reisz and Perlich 2007; Vassalou and Xing

2004). The ROC curve is constructed by varying the cutoff points and plotting the true positive rates

(correct classification of a financially distressed firm) versus the false positive rates (false

classification of a solvent firm). To compute the AUROC measure we follow the nonparametric

approach by Hanley and McNeil (1982) and DeLong et al. (1988) based on sorting the firms by their

34 For a discussion of the relative importance of a type I or type II error see for instance Altman et al. (1977), Altman

(1984), Andrade and Kaplan (1998) and Blöchlinger and Leippold (2006).

28

probability of default estimates (from high to low) and assessing the number of actual defaults in the

highest k-percentiles (k=1,…,100).35 The AUROC measure, which ranges from 0.5 (random model)

to a maximum of 1.0 (perfect discrimination model) answers the question of how accurate the model

is in predicting actual defaults and determining cross-sectional distress risk. The higher the curve rises

towards the top left corner point, the higher the area under the curve (AUROC) and the better the

discrimination power.

To obtain fundamental assessments of the model performances table 7 compares the AUROC results

inferred from predicting actual defaults over a 5-year horizon (i) by longitudinal analysis over years

(Panel A), (ii) by cross-sectional and bootstrapped analysis over the sample period 1980-2010 (Panel

B and C), and (iii) by the pairwise statistical comparison for the calculated AUROC across the

investigated models.


Overall, the s-prob model experiences the highest accuracy (AUROC=0.8122) followed by the o-

prob model (AUROC=0.7818) and the z-prob model (AUROC=0.7327). The most striking result is

the declining accuracy of the statistical models. While the o-prob model performed remarkably well

to the beginning of 1990’s, the accuracy of the z-prob, o-prob and updated z-probu and o-probu models

subsequently decline. One reason for this finding might be due to varying number of bankruptcies

and delistings over time, where a higher number corresponds to a lower accuracy ratio (see Table 1).

The findings also imply deterioration in the predictive power of accounting variables indicating

unstable coefficients and miscalibration, which is consistent with the findings of Beaver et al. (2005),

Hillegeist et al. (2004) and Begley et al. (1996) and explain the lower cross-sectional discrimination

power of the re-estimated models. However, the o-prob model clearly outperforms the z-prob model

in each dimension. Given these results, it is not surprising that the AUROC measures for the z-prob

model and the o-prob model are widespread and depend highly on the sample as well as the forecast

horizon in the literature. For example Agarwal and Taffler (2008, table 2) report an AUROC value of

35 For a detailed discussion of this measure please see Hanley and McNeil (1983) and Stein (2002). For an overview

of the estimation and implementation procedures using STATA see Pepe et al. (2009).

29

0.89 (1-year forecast horizon) compared to the AUROC of 0.7794 (1-year forecast horizon) and

0.6483 (5-year forecast horizon) reported by Reisz and Perlich (2007, Table 3). Chava and Jarrow

(2004, Table 2) estimate a bootstrapped median AUROC of 0.8662 (1-year forecast horizon) based

on yearly firm observations from 1991-1999 using a hazard model. Franzen et al. (2007, Table 7)

report a mean cumulative accuracy ratio of 0.465 (1-year forecast horizon) for the R&D-adjusted o-

prob model using the sample period 1980-2003, which yields an equivalent AUROC of 0.7325.36

Similar, Jackson and Wood (2013) find an area under the ROC curve of 0.7801 (1-year forecast

horizon) based on an annually sample for UK listed firms from 2000-2009. Vassalou and Xing (2004),

find a cumulative accuracy ratio of 0.59 (5-year forecast horizon) for the structural Merton model in

predicting actual defaults that equals an AUROC of 0.7950. However, their model requires market

data and therefore restricts the sample. For example, they exclude firms with negative book value of

equity and firms, which do not have stock market quotes available. Using quarterly data Das et al.

(2009) find accuracy ratios of around 0.60 for accounting and market-based models. Overall, the

reported accuracy ratios are comparable to the prior literature while this study uses a broader measure

of distress (more than 2,000 delisting firms over the whole sample period compared to around 300

for most studies as for example in Shumway 2001) and longer forecast horizons.

In Panel C the results are confirmed by employing 1,000 bootstrapped resamples and calculate the

descriptive statistics for the AUROC (Chava and Jarrow 2004; Reisz and Perlich 2007). In unreported

statistics we compare the AUROC from the s-prob model pairwise with the statistical models

following the U-statistic algorithm by DeLong et al. (1988). As expected by the bootstrapped results,

the performance of the s-prob model is significantly different better compared to the statistical models

(p-value < 0.0001). For the sake of completeness, we sort by (i) cash, (ii) quick ratio, and (iii) assets-

to-liability ratio, which are known to have predictive power for bankruptcy (Ravi Kumar and Ravi

2007), but find, as expected, lower accuracies ((i) AUROC=0.2665, (ii) AUROC=0.3683, (iii)

AUROC=0.2168).


36 Agarwal and Taffler (2008) note that the Cumulative Accuracy Profile (CAP) or Accuracy Ratio and the area under

the ROC Curve (AUROC) based on the same information set and are linear related by: 𝐶𝐶𝐶𝐶𝐶𝐶 = 2 ∙ (𝐶𝐶𝐴𝐴𝑂𝑂𝑂𝑂𝐶𝐶 − 0.5).

30

Figure 7 shows the AUROC of the models for different forecast horizons. It starts with six years

before delisting and shows the evolution of the estimated default probabilities until actual delisting at

time zero. While the s-prob and the o-prob model perform overall comparably, the z-prob and the re-

estimated o-probu and z-probu models are clearly inferior with regard to distinguishing delisting from

non-delisting firms. Looking at the figure in more detail, one can see that until up to the delisting

quarter, the s-prob model performs better than the o-prob model. This is consistent with Mossman et

al. (1998) and Reisz and Perlich (2007), who find that the accounting-based ratio models are the best

single model during the year immediately preceding bankruptcy. Generally, the accuracy naturally

increases the closer the firm is to actual performance-related delisting. As discussed above the values

of around 0.80 to 0.90 for one year before delisting are in line with prior studies.


Figure 8 shows the accuracy of the models over time. As before, the s-prob model and the o-prob

model outperform the z-prob model with regard to distinguishing delisting from non-delisting firms

after the initial sample years. In contrast, the z-prob model increase its discrimination power in

economic downturns (2000-2002, 2008-2009). The s-prob model and the statistical models show a

similar behavior over time that is also reflected in the correlation pattern. Overall, the accuracy of all

three models is decreasing until mid-1990s and increasing thereafter again. Interestingly, all models

perform remarkably well during the financial crisis. Finally, with reference to the model’s industry-

specific character and the findings of Chava and Jarrow (2004), table 8 provide the AUROC

performances across the Fama-French 10-Industry classifications. The results support the industry-

specific accuracy of the s-prob model. Being precise, the AUROC ranking performance shows that

10 out of 10 industry classifications are outperformed by the proposed stochastically framework.


31

In the next section we investigate the incremental information content of the individual and combined

distress probability estimates.

5.4 Test of Information Content

Following Campbell et al. (2008), Chava and Jarrow (2004) and Shumway (2001) we estimate a

multiperiod (or dynamic) logit model to evaluate the incremental explanatory power. We assume that

the marginal probability of bankruptcy or failure over the next period follows a logistic distribution

and is given by:

( ) ( ), 1, 1

111 exp i t

i t it PDP Y

α β −− − − ⋅

= =+

(18)

where Y is coded one if the firm delists in period t (and zero otherwise) and PD is the explanatory

variable, which is the default probability estimate (coded as logit score) for the different models. The

major criteria to compare the s-score model to the benchmark models is the pseudo-R² as in Shumway

(2001). Additionally we report the Vuong test statistic for strictly non-nested model selection and the

likelihood ratio statistic to compare the nested models.37 To examine the value of the models’ default

probabilities for longer forecast horizons we vary them up to five years prior to delisting as in

Campbell et al. (2008).

Table 9, Panel A, shows the results for the dynamic logit model estimation to determine the

information content of the models for a 1-year ahead prediction. Considered individually, all models

have substantial explanatory power for actual delistings as the coefficients are significantly different

from zero and strictly positive. Moreover, the pseudo-R²s range from 0.0450 to 0.0831 individually.

Overall, the magnitudes of the pseudo-R²s are consistent with the findings in Xu and Zhang (2009)

ranging from 0.07 to 0.16 or in Hillegeist et al. (2004) from 0.07 to 0.12 where both include market

information. The s-score and re-estimated z-scoreu models are more preferable in the 1-year forecast

horizon (pseudo-R2=0.0831 and pseudo-R2=0.0745) and perform comparably well while the o-score

and z-score are inferior in terms of the conveyed information (pseudo-R2=0.0689 and pseudo-

R2=0.0450). The findings are in line with the two mentioned studies. In contrast to Hillegeist et al.

37 The Vuong statistic is an established methodology to evaluate competing strictly nonnested models (see for references

Agarwal and Taffler 2008; Banker and Chen 2006; Dechow 1994; Dichev and Tang 2009; Hillegeist et al. 2004). For the implementation of a logit version for the Vuong test we follow Lechner (1991, table 3) and confirm the results using the SAS implementation of the Vuong test statistic.

32

(2004) the re-estimated z-scoreu performs significantly better than the original z-score. As Xu and

Zhang (2009) suggest, Table 9 also sets out the combination of the three measures (i.e., original and

re-estimated Model 6 and Model 7) to compare the incremental relevance carried by the distress

prediction models. For the combined models, the z-score explanatory parameter loses its significance

(the z-statistic declines from 1.79 to 1.34 referring to the results in model 8 vs. model 6). Looking at

the differences in pseudo-R2 of Model 6 and Model 7 versus the pseudo-R2 from the combined

versions (i.e., Model 8 and Model 9 including only the z-score and o-score explanatory variables),

the s-score seems to explain information not contained in the traditional measures.

Overall, the combined pseudo-R² of 0.1069 and 0.1164, which are significantly larger than the

pseudo-R² of the separated models reveal that all measures convey exclusive information.

Table 9, Panel B, Panel C, extent the forecast horizon of the dynamic logit model to three and five

years, respectively. As one expects, the pseudo-R²s decrease for longer forecast horizons. However,

the results demonstrate that the s-score model clearly outperforms the two statistical models for longer

horizons of three to five years. For a horizon of three years, for example, the s-score model displays

a pseudo-R² of 0.0489 compared to 0.0348 (0.0139) for the o-prob (z-prob) model.38 This is in line

with the findings above that the s-score model is superior in early prediction of defaults, which can

be due to the inclusion of volatility (Campbell et al. 2008). In all panels of Table 9 the combined

models (model 6 and model 7) offer the highest explanatory power. Surprisingly, the re-estimated

models and the z-score lose part of their explanatory power compared to the original o-score model

and become insignificant in the combined model versions referring to a 5-year forecast horizon. The

high explanatory power of the s-score for long-term financial distress predictions is confirmed by the

pairwise Vuong-statistic offered in Panel D. Overall, the results suggest that s-score model is

preferable compared to the original and re-estimated o-score and z-score prediction models in case

of long-term forecast horizons.


38 Recall from above that this literature is concerned with pseudo-R²s ranging from 0.07 to 0.12 for a single explanatory

variable for a one period ahead forecast (Hillegeist et al. 2004). Hence, a pseudo-R² value of about 0.05 for a three years ahead forecast is economic significant.

33

In order to test an alternative dimension of the predictive ability we assess the homogeneity of the

evaluated models in Table 10. Homogeneity measures are derived by ranking the related distress

probabilities into deciles (10=high probability of financial distress – 1=low probability of financial

distress) and compare the predictive ability. This assessment is closely related to the accuracy ratio

and offers additional insights in the models’ performances. Panel A1 reports the decile-based

correctly classified frequency of failed firms (separate and cumulative). The results suggest, that the

s-prob, o-prob and z-probu models identify about one-third of the financially distressed firm quarter

observations in the highest-probability decile. Panel A2 presents the ranking for the longer term 2-5

years ahead prediction horizon. Panel B addresses the question of equal classified financially

distressed firm quarter observations. For each probability decile we report the number and frequency

of correspondingly classified observations in relation to the s-prob results. As shown in Table 9 the

s-prob model uses information not enclosed in related models. The equally identified observations in

the highest decile range between 52.6%-66.4% stepwise increasing with each decile. In other words,

about one-third of the financially distressed observations in the highest s-prob probability decile are

individually not identified by related accounting models.


5.5 Robustness and Extensions

In this section, we report the robustness of the results laid out in the earlier sections. The question

arises as to whether the accuracy and information content hold when we use alternative (i) (market-

based) benchmark measures, and (ii) parametrizations for the s-score model.

(i) Market-based distress prediction models

First, we conduct all our ROC and information content analyses from section 5.2 to 5.4 using two

complementary market-based prediction models. In particular, we operationalize the research designs

for the contingent claim framework (EDF) following the sequential-iterations algorithm specified by

Crosbie and Bohn (2003) and Bharath and Shumway (2008) and the comprehensive information

34

approach (original c-prob and re-estimated c-probu) of Campbell et al. (2008) that incorporates both

market- and accounting covariates.39 As discussed above, we expect that market-information allows

for additional explanatory power regarding the financial distress prediction models. The summary

statistics, accuracy analysis and information content tests are tabulated in the corresponding tables

11, 12 and 13. The additional data requirements to calculate three alternative market-based distress

measures, (i) EDF, (ii) c-prob , and (iii) c-probu reduce the sample size relative to the accounting-

based sample from 330,549 (10,687) to 280,234 (10,143) firm quarter observations (firms) with

observable probability estimations and ensures comparability between the default probability

measures (Correia et al. 2012).40

However, the most surprising fact revealed in the cross-sectional AUROC analysis (see table 12,

Panel A) is the low accuracy of the contingent claims prediction model using the complete sample

spanning from 1980-2010 (AUROC=0.7629), which has multiple reasons. First, we recognize a low

prediction accuracy during the 1990’s increasing stepwise after the dot-com bubble 2000/2001. This

is in line with the findings of Beaver et al. (2005) indicating a decline in the value-relevance of

accounting ratios, while market information becomes more important in explaining default risk.

Second, the contingent-claim model does not compensate inefficient market-information (Bharath

and Shumway 2008; Das et al. 2009), particularly within a sample period enveloping market

downturns. An unreported longitudinal analysis shows a reduced explanation power of the EDF

model, especially during financial crisis 2008-2009. 41 Contrary, the c-prob and c-probu model

perform remarkably well over the complete sample period and clearly outperform the EDF, while the

s-prob model (AUROC=0.8121) predicts financially distressed events comparable to the original c-

prob model (AUROC=0.8109) 5-years prior to a delisting. While we acknowledge that the

performance of the c-probu is slightly better (AUROC=0.8398), recall that the s-prob model uses

accounting information only and could be employed in a private firm setting. Prior findings conducted

by Bauer and Agarwal (2014) imply a analogous ranking performance.42

39 The results and specification of the models are given in the Appendix 2 and 3. 40 For example we require at least 60 trading days of stock price returns prior to the month the implied asset volatility is

calculated, which is not available in 65,717 firm quarters with corresponding accounting distress measures. Note that the limiting market information are not been necessary in previous analyses.

41 In contrast, Correia et al. (2012) find that the “predictive power of the different models for future credit returns is greater for bust periods.”

42 Bauer and Agarwal (2014) analyze the receiver operating characteristic using the parametrization of Shumway (2001), Campbell et al. (2008) Model 1, the contingent claims framework following Bharath and Shumway (2008) and Taffler’s z-score model. The conducted AUROC comparison implies that Campbell et al. (2008) – parametrization is

35

Table 13 contains the corresponding information content analysis. As shown in Table 13, Panel A-C

(Model 6 vs. Model 8) the coefficient on the EDF model loses statistically significance (3-year and

5-year) when the s-score estimates are added to the a combined specification using the c-scoreu while

the s-score coefficient remains significantly for the 1-year and 3-year ahead prediction horizons. In

addition, the the Vuong statistic in Panel D demonstrates, that the EDF measure loses its relevance

as the forecast horizon is extended to 5-years ahead, while the c-scoreu model has the most

information content due to its skillful combination of accounting and market data.




(ii) Alternative Prediction Horizon and Specifications

Finally, with respect to the long-term prediction horizon of this study we extent the forecast horizon

of the re-estimation procedure (z-probu and o-probu) to predict financial distress 5-years prior to a

performance-related delisting.43 Balcaen and Ooghe (2006) point out that the horizon to re-estimation

or re-development on observations should meet the horizon of the predictive classification statements

(financially distressed/solvent) with reference to the parametric nature of the statistical framework.

For the sake of brevity, we have not presented these comprehensive results here (available upon

request). The corresponding results for the alternative z-probu and o-probu are (AUROC=0.7273 and

AUROC=0.7294). The findings indicate that an alternative estimation horizon to estimate the related

coefficients does not influence the accuracy of the statistical models. In addition, we test several

alternative specifications for the z- and o-score models reported in the previous literature (Begley et

superior to the contingent claims framework, even if their sample size covers the period 1985–2009 with 22,217 observations (Bauer and Agarwal 2014, table 5).

43 Traditionally, most studies define a short-term prediction horizon (i.e., less than 16 month in the future) to calibrate their distress/default prediction models. For example the original z-score model by Altman (1968) was developed defining bankruptcy firms having an average lead time of the financial statements of 7 ½ months. Ohlson (1980) report an average lead time between the date of the fiscal year of the last relevant report and bankruptcy of 13 months. According to Crosbie and Bohn (2003) the time horizon specification for the contingent-claims model is regularly set to T=1 year even if the outcomes are used in predict distress/default for different forecast horizon.

36

al. 1996; Hillegeist et al. 2004), but do not find significant improvements in the ability to correctly

classifying financially distressed firms compared to the updated model versions.

(iii) Alternative Distress Barrier

With reference to the asset/debt implied boundary condition we employ industry-specific and

leverage-specific boundary conditions and test Parisian option barriers (Reisz and Perlich 2007) to

indicate a firm as financially distressed, but do not find significantly improvements in the

discriminatory power. Finally, we conduct an practical asset-implied barrier that equals 30% of the

firm’s average OLR 2-years prior a performance-related delisting event and obtained slightly

improvements in the cross-sectional discrimination power for the s-prob model (AUROC=0.8130).

However, since this criterion is somewhat arbitrary, we do not enhance our model in this regard.

37

CONCLUSION

Projecting the financial health is a challenging and important task. Recent research by Shumway

(2001), Chava and Jarrow (2004) and Campbell et al. (2008) demonstrate the effort to develop more

refined models. However, these models mostly require market data, which are not available for the

majority of firms. In this study, we present a new approach to predict corporate bankruptcy based on

stochastic processes (s-score). The novel framework for bankruptcy prediction is based on recent

research in company valuation and has several advantages over standard statistical models. First, it is

theoretically well grounded and can address the problem of the backward-looking perspective of

accounting-based models. Second, it demands less data than the classical statistical models (such as

probit, logit and MDA approaches). Most importantly, however, it explicitly considers accounting

volatility, which is an important factor of early financial distress warning.

From a detailed examination we find the following results. First, the bankruptcy prediction based on

stochastic processes fits the distribution of historic default rates reasonably well and thus provides

early warning signals. Second, the bankruptcy prediction based on stochastic processes is more

accurate in distinguishing between non-delisting and delisting firms than the prominent z- or o-score

models. Third, this model significantly outperforms the two statistical models for longer time

horizons, which is probably due to the explicit incorporation of accounting volatility measures.

Fourth, while the prominent z- or o-score models perform comparably well, a combination of all three

models has a significantly greater explanatory power.

As a common critique to structural models, one possible drawback of our model is the strong

underlying assumptions (Xu and Zhang 2009). The assumptions about firms’ cash generating

operating processes might be too simplified. However, the model successfully addresses several

drawbacks of the statistical bankruptcy models and offers solid results. Obviously, the financial

distress prediction based on stochastic processes has many degrees of freedom. It can therefore be

regarded as a novel model class whose flexibility is a strength and an adequate response to the

complex process of financial distress. Future research on this topic could help to model more detailed

financing and investing policies of the firm and incorporate analysts’ forecasts to improve the

initializing parametrization.

38

Appendix

In this section we present the construction and estimation results of the re-estimated accounting-based

(Appendix 1) and market-based financial distress prediction models (Appendix 2: expected default

frequency (EDF) model, Appendix 3: c-score model of Campbell et al. (2008)).

Appendix 1 Summary statistics: Original and re-estimated Altman (1968) and Ohlson (1980)

Panel A: Updated Z-Score Model (z-probu) (firm quarters from 1980Q1 – 2010Q4) N Mean Median Std.dev. Constant 124 (3.24) (3.03) 0.58 X1=WC/TA 124 (1.87) (1.81) 0.36 X2=RE/TA 124 (0.15) (0.17) 0.14 X3=EBIT/TA 124 (10.60) (9.22) 3.82 X4=MVE/TL 124 (0.10) (0.02) 0.27 X5=SA/TA 124 0.08 0.03 0.48 Pseudo R2 124 0.17 0.17 0.01

Panel B: Updated O-Score Model (o-probu) (firm quarters from 1980Q1 – 2010Q4) N Mean Median Std.dev. Constant 124 (0.33) (0.27) 0.16 X1=SIZE 124 (0.45) (0.45) 0.09 X2=TLTA 124 1.85 1.80 0.35 X3=WCTA 124 (1.13) (1.26) 0.65 X4=CLCA 124 (0.04) (0.10) 0.21 X5=NITA 124 (0.70) (0.86) 0.59 X6=FULT 124 (0.59) (0.42) 0.70 X7=INTWO 124 1.30 1.14 0.45 X8=OENEG 124 0.31 0.41 1.41 X9=CHIN 124 (1.41) (1.41) 0.34 Pseudo R2 124 0.23 0.24 0.03 Panel A reports summary statistics for the updated z-score model (estimated by a growing window logit regression model, starting in firm quarter 1978Q1). For example, we estimate the updated coefficients to calculate the z-score in 1980Q1 using firm quarter observations (X1-X5) from 1978Q1-1979Q3 and delisting information (CRSP delisting codes 400, 500-585) from 1979Q4 (out-of-sample-approach). We list the mean, median and standard deviation of the estimated coefficients by summarizing the main sample period of 1980Q1-2010Q4 (=124 firm quarters). The dependent variable in the logit regression model equals one if a delisting occurs within the next 4 quarters. All estimated coefficients have the expected signs. We winsorize all measures at the 1st and 99th percentile to avoid outliers driving the results. The z-scores are transformed into probabilities using the standard logistic function (z-probu=1/(1+exp(-z-scoreu)). Panel B reports summary statistics for the updated o-score model (estimated by a growing logit regression model, starting in firm quarter 1978Q1). As in Panel A we ensure all measures are observable at that quarter over which the updated probabilities are calculated (out-of-sample-approach). The dependent variable in the logit regression model equals one if a delisting occurs within the next 4 quarters. We winsorize all measures at the 1st and 99th percentile to avoid outliers driving the results. Consistent with Begley et al. (1996), Grice and Dugan (2003) and Hillegeist et al. (2004) we find that the coefficients are not stable across different time periods. The o-scores are transformed into probabilities using the standard logistic function (o-probu=1/(1+exp(-o-scoreu)). WC/TA = working capital/total assets; RE/TA = retained earnings/total assets; EBIT/TA = earnings before interest and taxes/total assets; MVE/TL = market capitalization/total liabilities; SA/TA = sales/total assets; SIZE = ln(total assets/ lagged GNP level (with base year 1968)); TLTA= total liabilities/total assets; CLCA= current liabilities/current assets; NITA= net income/total assets; FULT=pretax income/total liabilities; INTWO = 1 if the net income in the last two quarter were negative and zero otherwise; OENEG= 1 if total liabilities > total assets; CHIN=(net incomet – net incomet-1)/(|net incomet|+|net incomet-1|); Pseudo-R2= defined by McFadden’s (1974)-Pseudo-R2: (1-L1/L0).

39

Appendix 2

Results of the re-estimated EDF Measure

The contingent claims-based framework is a widely used operationalization of a market-based

prediction model that requires (daily) market- and accounting observations of: (1) the total firm value

of assets (V=iteratively inferred), (2) the related volatility of the firm assets (𝜎𝜎𝑉𝑉=iteratively inferred),

(3) the face value of total debt maturing at time T (B=debt in current liabilities plus one-half of the

long-term interest bearing debt), (4) the value of equity (E=market value of equity from CRSP), (5)

the expected return on assets (μ=iteratively inferred), (6) the prediction horizon T, and (7) the risk-

free rate (r=1-year US-Treasury Bill Rate), as input parameters. The underlying assumptions of the

contingent claims framework are: (i) the total firm value of assets V follows a geometric Brownian

motion and (ii) the total debt, i.e., the total claims of a firm, maturing at time T.44 In order to receive

estimates for the firm value of assets V and volatility of assets 𝜎𝜎𝑉𝑉, we operationalize the sequential-

iterations algorithm as described by Crosbie and Bohn (2003), Vassalou and Xing (2004), Bharath

and Shumway (2008) and more recently in Jessen and Lando (2015) with a maximum of 15 iterations.

The value of equity, viewed as a European call option is obtained from:

1 2E( ; ) ( ) ( )rTV T VN d Be N d−= − (19)

2

1ln( / ) ( 0.5 )V

V

V B TdT

µ σσ+ +

= (20)

2

2 1ln( / ) ( 0.5 )V

vV

V B Td d TT

µ σσσ+ −

= − = (21)

where we require, similar to Bharath and Shumway (2008), a forecast horizon of T=1 year and at least

60 trading days over the previous 12 months. Finally, the probability of default EDF is given by:

2 2

2

P(V ) 1 ( ) ( d )

ln( / ) ( 0.5 )T

V

V

EDF B N d N

V B TNT

µ σσ

= < = − = −

+ −= −

(22)

44 To be precise, the theoretical underpinning of the Black-Scholes model requires additional assumptions, for example

lognormally distributed returns, constant volatilities and other related market frictions etc. (Agarwal and Taffler 2008).

40

where N(∙) is the cumulative standard normal distribution function. 45 We winsorize all model

parameters at the 1st and 99th percentile to avoid outliers driving the results.

45 Additionally, we compare our estimated EDF-measure with the aggregated default likelihood indicator presented in

Vassalou and Xing (2004). We thank the authors Maria Vassalou and Yuhang Xing to make their results available at link: http://maria-vassalou.com/research/data. Based on the merged 146,189 firm quarter observation with nonmissing information from 1980Q-1999Q4 we compare the AUROC of both measures. The findings indicate, as expected, that the performance of implied default probability following Bharath and Shumway (2008) and the default probability by Vassalou and Xing (2004) are not significantly different from each other (AUROCBharath/Shumway = 0.7650 vs. AUROCVassalou/Xing = 0.7651). The test is based on an indicator variable for financial distress that equals one if a delisting (indicated by CRSP delisting codes 400, 500-585) occurs within the next 20 quarters (i.e., a delisting firm quarter is determined if the firm experience a delisting within the next 20 quarters as defined in table 1).

41

Appendix 3

Results of the re-estimated C-Score Model No. 2 of Campbell et al. (2008)

Updated C-Score Coefficients (c-probu) (firm month from 1980|01 – 2010|12)

N Mean Median Std.dev. Constant 372 (14.86) (14.49) 2.77 X1=NIMTAAVG 372 (8.02) (6.51) 3.02 X2=TLMTA 372 1.49 1.44 0.75 X3=EXRETAVG 372 (0.58) (0.69) 1.56 X4=SIGMA 372 0.72 0.49 4.10 X5=RSIZE 372 (0.59) (0.59) 0.12 X6=CASHMTA 372 (1.58) (1.15) 0.93 X7=MB 372 0.07 0.07 0.03 X8=PRICE 372 (0.60) (0.60) 0.11 Pseudo R2 372 0.28 0.30 0.03 This table reports summary statistics for the re-estimated c-score model (estimated by a growing window logit regression model, starting in firm month 1976|01). For example, we estimate the updated coefficients to calculate the c-score in January of 1980 using firm quarter observations (X1-X8) as defined by the Appendix of Campbell et al. (2008) from January, 1976-November, 1979 and monthly delisting information (CRSP delisting codes 400, 500-585) from December, 1979 (out-of-sample-approach). We list the mean, median and standard deviation of the coefficients by summarizing the main sample period of 1980|01-2010|12 (=124 firm quarters). The dependent variable in the logit regression model equals one if a delisting occurs within the next month. Consistent with the alternative failure model (Model 1) of Campbell et al. (2008) and the re-estimated version of Bauer and Agarwal (2014) the coefficients have the expected signs. We winsorize all measures at the 1st and 99th percentile to avoid outliers driving the results. The c-scores are transformed into probabilities using the standard logistic function (c-probu=1/(1+exp(-c-scoreu)). Pseudo R2= defined by McFadden’s (1974)-Pseudo R2: (1-L1/L0).

42

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50

Table 1: Summary Delisting Statistics

Panel A: Performance-related Delisting Codes and Frequency (N= 330,549 firm quarters from 1980Q1 to 2010Q4)

CRSP delisting code Delisting reasons Unique

firms Firm

quarters %

400 Issue stopped trading as result of company liquidation. 2 20 0.05% 550 Delisted by current exchange - insufficient number of market makers. 156 1,466 3.52% 551 Delisted by current exchange - insufficient number of shareholders. 62 778 1.87% 552 Delisted by current exchange - price fell below acceptable level. 573 7,225 17.34% 560 Delisted by current exchange - insufficient capital, surplus, and/or equity 635 7,317 17.56%

561 Delisted by current exchange - insufficient (or non-compliance with rules of) float or assets. 346 4,003 9.61%

570 Delisted by current exchange - company request (no reason given). 186 2,441 5.86%

573 Delisted by current exchange - company request, deregistration (gone private). 44 665 1.60%

574 Delisted by current exchange - bankruptcy, declared insolvent. 369 4,938 11.85%

575 Delisted by current exchange - company request, offer rescinded, issue withdrawn by underwriter. 2 24 0.06%

580 Delisted by current exchange - delinquent in filing, non-payment of fees. 397 4,162 9.99%

581 Delisted by current exchange - failure to register under 12G of Securities Exchange Act. 37 431 1.03%

582 Delisted by current exchange - failure to meet exception or equity requirements. 94 1,107 2.66%

583 Delisted by current exchange - denied temporary exception requirement. 1 8 0.02%

584 Delisted by current exchange - does not meet exchange’s financial guidelines for continued listing. 456 6,179 14.83%

585 Delisted by current exchange - protection of investors and the public interest. 76 900 2.16%

400, 550-585 Delisting - all reasons 3,435 41,664 100.00%

This table lists the financial distress-related delisting reasons and their frequency and percentages of firm quarters in the sample. Generally, CRSP delisting codes 400-499 denote liquidations and 500-599 denote issues dropped from the stock exchange. For the purpose of this study performance-related delisting codes 400 and 550-585 are considered as described in section 4.1. In total there are 3,435 delisted firms (with 41,664 quarters) having distress-related information 20 quarters ahead of delisting. Note that after 1987 CRSP assigned the three digit delisting code “500” to the delisting category prior coded “5” (Shumway 1997).

(continued on next page)

51

(Table 1 continued)

Panel B: Distribution of delisting’s (N= 330,549 firm quarters from 1980Q1 to 2010Q4)

Year # Traded firm's

# Delisted firm's

(%) Delisting rate

# Firm quarters

# distressed firm

quarters

(%) Delisting firm

quarter rate 1980 1,652 3 0.2% 5,774 124 2.1% 1981 1,602 9 0.6% 5,519 141 2.6% 1982 2,238 6 0.3% 6,156 251 4.1% 1983 2,805 12 0.4% 8,908 752 8.4% 1984 2,954 55 1.9% 10,059 1,068 10.6% 1985 3,052 81 2.7% 10,303 1,172 11.4% 1986 3,080 125 4.1% 10,562 1,362 12.9% 1987 3,041 90 3.0% 10,410 1,544 14.8% 1988 3,148 89 2.8% 10,617 1,722 16.2% 1989 3,167 114 3.6% 11,083 1,823 16.4% 1990 3,122 141 4.5% 10,967 1,690 15.4% 1991 3,130 166 5.3% 11,035 1,573 14.3% 1992 3,101 197 6.4% 10,955 1,279 11.7% 1993 3,227 86 2.7% 11,363 1,362 12.0% 1994 3,415 105 3.1% 12,006 1,613 13.4% 1995 3,670 94 2.6% 12,739 1,802 14.1% 1996 3,805 105 2.8% 13,358 2,187 16.4% 1997 3,979 127 3.2% 13,771 2,542 18.5% 1998 4,023 208 5.2% 13,843 2,718 19.6% 1999 3,831 214 5.6% 13,306 2,459 18.5% 2000 3,606 179 5.0% 12,675 2,315 18.3% 2001 3,462 255 7.4% 11,905 1,769 14.9% 2002 3,327 204 6.1% 11,830 1,335 11.3% 2003 3,138 162 5.2% 11,355 974 8.6% 2004 3,000 55 1.8% 11,051 1,040 9.4% 2005 2,911 86 3.0% 10,541 964 9.1% 2006 2,802 37 1.3% 10,243 914 8.9% 2007 2,750 27 1.0% 9,882 1,030 10.4% 2008 2,640 93 3.5% 9,630 935 9.7% 2009 2,561 110 4.3% 9,476 680 7.2% 2010 2,487 63 2.5% 9,227 524 5.7%

This table provides the distribution and summary statistics for the sample period (1980-2010). Traded firms comprise all unique observable firms in the sample, the number of delisted firms is the total number of firms delisted in a specific year and the number of firm quarters equals the total number of observations (traded and financially distressed firms). A firm quarter is considered as delisting firm quarter if the company is delisted in the next 20 quarters ahead. Note that 137 firms are delisted after 2010 (e.g., 2011-2014). In total, the sample consists of 3,435 delisted firms with nonmissing data (i.e., delistings with CRSP delisting codes 400, 550-585) resulting in 41,664 financially distressed firm quarters.

52

Table 2: Estimation of Parameters

No. Label Description Measurement (abbreviations are Compustat mnemonics)

Sales dynamics 1 S = initial sales = quarterly firm sales

2 𝑔𝑔0 = initial sales growth rate = estimated as smoothed weighted average using the 7 lags of quarterly sales

growth rates: ( )71, where 0.7 ln

107

t saleqt itg and git it saleqittα α

=∑ ⋅ = =

−=

3 �̅�𝑔 = long-term sales growth rate = 0.0075 [per quarter]

4 𝜎𝜎𝑔𝑔,0 = initial volatility of the sales growth rate

= estimated by the root mean squared error (𝜎𝜎�𝜀𝜀) of a recursive least squares growing (starting in 1978Q1) firm-by-firm regression model (using a smoothed series of moving-averaged filtered quarterly sales ( )MA

saleqt

observations):

( )

;1

( ) ( )

ln1

and

15 1 2 3 4

MA MAg git itit

with firms i across firm quarters t

MAsaleqMA itwhere git MAsaleqit

MAsaleq saleq saleq saleq saleq saleqit it it it it it

α β ε= + +−

=

−

= + + + +− − − −

5 𝜎𝜎𝑆𝑆,0 = initial sales volatility = 1

7

27ln 010

saleqt gsaleqtt

∑ − − =

6 𝜎𝜎𝑆𝑆� = long-term volatility of sales = 0.05 [per quarter]

Cost dynamics

7 𝑐𝑐0 = initial variable cost rate = estimated by the slope coefficient (𝛽𝛽) of a recursive least squares rolling firm-by-firm regression model (with 7 lagged firm quarters and assuming a zero intercept): ; ( ) andc S wherec cogsq xsgaq S saleqit it it it it it itβ ε= + = + =

8 𝑐𝑐̅ = industry median long-term variable cost

= estimated as industry-specific median of costs (Fama-French (2015) 48-industry classification):

, 1980, , 2010481970

median for tffiT cogs xsgat t

salett= …

+∑=

9 𝜎𝜎𝑐𝑐,0 = initial volatility of variable costs

= estimated by the root mean squared error (𝜎𝜎�𝜀𝜀) of a recursive least squares rolling firm-by-firm regression model (with 7 lagged firm quarters):

;1cogsq xsgaqit itvk vk wherevkit it itit

saleqitα β ε

+= + + =−

10 𝜎𝜎𝑐𝑐� = long-term volatility of variable costs = 0.05 [per quarter]


53

(Table 2 continued)

Other parameters

11 κ = speed of adjustment

= estimated by the slope coefficient (𝛽𝛽) using an industry-specific (Fama-French (2015) 48-industry classification) least squares firm-by-industry regression model (re-estimated each year):

1

1 1 2ln ln1 2 2

2

1980, ..., 2010

sale saleit it

sale sale saleit it ititsale sale saleit it it

saleit

where t

α β ε

− −

−− − −= + ⋅ +−− − −

−

=

12 tax = tax rate = 0.35

13 𝑟𝑟𝑓𝑓 = risk-free rate = 3-Month US-Treasury Bill Rate

14 𝑟𝑟𝐿𝐿𝐿𝐿𝐿𝐿 = interest on debt = ( )xint

dlc dltt+

Balance sheet positions

15 OLR = operating liquidity reserve = actq+(unused credit line)a-lctq

16 cash = interest-bearing cash and cash equivalents = cheq

17 LTD = long-term interest-bearing liabilities = dlttq+(additional debt)

COMPUSTAT Quarterly data (q) Annual data (a) item number mnemonic description item

number mnemonic description

#1 xsgaq Selling, General, and Administrative Expenses

#12 sale Sales (Net)

#2 saleq Sales (Net) #15 xint Interest and Related Expense – Total

#30 cogsq Cost of Goods Sold #34 dlc Debt in Current Liabilities

#36 cheq Cash and Equivalents #41 cogs Cost of Goods Sold #37 rectq Receivables – Total #142 dltt Long-Term Debt – Total #38 invtq Inventories – Total #189 xsga Selling, General, and

Administrative Expenses

#39 acoq Current Assets – Other #40 actq Current Assets – Total (as sum of

cheq, acoq, invtq and rectq)

#45 dlcq Debt in Current Liabilities #46 apq Accounts Payable #49 lctq Current Liabilities – Total #51 dlttq Long-Term Debt – Total


54

(Table 2 continued) This table presents the estimators for the different model parameters on a quarterly (yearly) basis (using Compustat mnemonics for reference). aThe unused credit line is estimated by (Fama-French (2015) 10-industry classification using linear (OLS) regression:

[ ] [ ] [ ]ln (totalassets ) /0, 1, 2, 3,lineunused cash flow cash M B cashi i i iβ β β β= + + − + − as described in section 3.3.

55

Table 3: Initial Parameters

Univariate statistics (N=330,549 firm quarters from 1980Q1 to 2010Q4) No. Status N Mean Median Std.dev. 1% 99%

Sales dynamics 1 initial sales Solvent 288,885 311.85 46.89 1080.23 0.35 4738.00 Distressed 41,664 62.58 7.264 312.40 0.01 862.98 Full sample 330,549 280.43 38.04 1019.30 0.16 4379.00

2 initial sales growth rate Solvent 288,885 0.011 0.010 0.036 -0.104 0.126 Distressed 41,664 0.001 0.002 0.056 -0.154 0.148 Full sample 330,549 0.009 0.009 0.039 -0.126 0.138

3 long-term sales growth rate Solvent 288,885 0.008 0.008 0.000 0.008 0.008 Distressed 41,664 0.008 0.008 0.000 0.008 0.008 Full sample 330,549 0.008 0.008 0.000 0.008 0.008

4 initial sales volatility Solvent 288,885 0.199 0.144 0.187 0.031 1.054 Distressed 41,664 0.335 0.240 0.290 0.043 1.395 Full sample 330,549 0.216 0.153 0.208 0.031 1.207

5 initial volatility of the sales growth rate Solvent 288,885 0.041 0.028 0.045 0.005 0.263 Distressed 41,664 0.069 0.046 0.071 0.007 0.363 Full sample 330,549 0.045 0.030 0.050 0.005 0.318

6 long-term volatility of sales Solvent 288,885 0.050 0.050 0.000 0.050 0.050 Distressed 41,664 0.050 0.050 0.000 0.050 0.050 Full sample 330,549 0.050 0.050 0.000 0.050 0.050 Cost dynamics

7 initial variable cost rate Solvent 288,885 0.899 0.901 0.153 0.423 1.382 Distressed 41,664 1.020 0.978 0.186 0.545 1.382 Full sample 330,549 0.914 0.910 0.162 0.436 1.382

8 industry median long-term variable cost Solvent 288,885 0.903 0.904 0.034 0.791 0.956 Distressed 41,664 0.904 0.907 0.037 0.793 0.957 Full sample 330,549 0.903 0.904 0.035 0.791 0.956

9 initial volatility of variable costs Solvent 288,885 0.096 0.030 0.325 0.003 1.521 Distressed 41,664 0.318 0.078 0.714 0.005 3.775 Full sample 330,549 0.124 0.033 0.403 0.003 2.556

10 industry median long-term volatility of variable costs Solvent 288,885 0.050 0.050 0.000 0.050 0.050

Distressed 41,664 0.050 0.050 0.000 0.050 0.050 Full sample 330,549 0.050 0.050 0.000 0.050 0.050


56

(table 3 continued)

Other parameters 11 speed of adjustment Solvent 288,885 0.126 0.126 0.031 0.051 0.218 Distressed 41,664 0.124 0.124 0.030 0.054 0.205 Full sample 330,549 0.126 0.126 0.031 0.051 0.217

12 tax rate Solvent 288,885 0.350 0.350 0.000 0.350 0.350 Distressed 41,664 0.350 0.350 0.000 0.350 0.350 Full sample 330,549 0.350 0.350 0.000 0.350 0.350

13 risk-free rate Solvent 288,885 0.012 0.012 0.007 0.000 0.035 Distressed 41,664 0.012 0.012 0.006 0.000 0.025 Full sample 330,549 0.012 0.012 0.007 0.000 0.035

14 interest rate on debt Solvent 288,885 0.022 0.020 0.021 0.000 0.120 Distressed 41,664 0.028 0.023 0.024 0.000 0.120 Full sample 330,549 0.023 0.020 0.021 0.000 0.120 Balance sheet positions

15 operating liquidity reserve Solvent 288,885 190.123 41.885 620.028 1.943 2315.000 Distressed 41,664 39.148 12.068 173.580 1.681 498.370 Full sample 330,549 171.094 32.947 585.053 1.814 2153.375

16 interest-bearing cash and cash equivalents Solvent 288,885 91.070 11.039 288.082 0.005 1710.519 Distressed 41,664 17.509 1.325 103.798 0.000 285.000 Full sample 330,549 81.798 8.572 272.919 0.001 1562.591

17 interest-bearing liabilities Solvent 288,885 289.374 22.464 916.087 0.000 5469.689 Distressed 41,664 107.901 13.086 463.166 0.000 1850.109 Full sample 330,549 266.500 20.186 874.130 0.000 5098.000

18 unused credit line Solvent 288,885 17.500 0.000 69.398 0.000 360.704 Distressed 41,664 4.434 0.000 87.964 0.000 109.664 Full sample 330,549 15.853 0.000 72.132 0.000 338.400

This table provides the summary statistics for the main s-score model variables (see Table 2 for the calculation). All rates are quarterly growth rates and the balance sheet positions are in million dollars. There are 330,549 firm quarter observations over the period 1980Q1 to 2010Q4 (288,885 solvent firm quarters and 41,664 financially distressed firm quarters). Financially distressed firm quarters are indicated if the firm was delisted 20 quarter ahead. For the interest rate on debt 14,799 observations were set to zero as no debt expense was recorded. For the operating liquidity reserve 85,113 values below the barrier b were funded by additional external debt to allow the stochastic processes to be initialized as described in section 4.2. All differences are statistically significant (at the 1%-level) based on a t-test of the means between the solvent and financial distressed sample (we confirm the results by testing the differences of the medians with the Wilcoxon rank-sum test yielding the same results).

57

Figure 1: Distress Barrier Level Over Time

This figure plots the distress barrier b using the median operating liquidity reserve (OLR) in million USD five years prior to a corresponding delisting event, recursively estimated in a 5-year rolling window approach to avoid look-ahead bias.

58

Figure 2: Comparison of Initial Parameters

Panel A:

Panel B:


59

(figure 2 continued)

Panel C:

This figures shows the mean accounting volatility measures (see No. 9, No. 4, No. 5 in table 3), mean initial sales growth rate (No. 2), median income statement and balance sheet positions (No. 1, No. 15, No. 16, No. 17) and the other model parameters (No. 7, No. 11, No. 14) along with the trend line for the quarters before a delisting (N=41,664 financially distressed firm quarters). Note that the balance sheet positions of the OLR (OLR=actq+(unused credit line)-lctq) is neither winsorized nor adjusted for initialization of the s-score model to provide an unbiased view.

60

Table 4: S-prob model Forecast Accuracy Panel A: Operating liquidity reserve - forecast accuracy for complete sample (288,885 solvent and 41,664 financially distressed firm quarters from 1980Q1-2010Q4)

Absolute OLR forecast inaccuracy (in Millions of Dollars) (OLRpredict – OLRactual)

Relative OLR forecast inacurracy (OLRpredict - OLRactual)/ OLRactual

absolute relative Qtr ahead N Mean Median Std.dev. N Mean Median Std.dev.

1 310,934 17.24 0.64 138.31 310,934 0.58 0.03 7.84

2 295,991 34.45 1.81 213.97 295,991 1.19 0.08 14.41

3 284,136 50.71 3.04 280.85 284,136 1.74 0.12 20.62

4 276,218 66.65 4.34 339.59 276,218 2.27 0.17 26.83

5 263,455 81.18 5.58 395.65 263,455 2.79 0.21 33.40

6 254,291 95.67 6.97 445.01 254,291 3.28 0.25 39.39

7 245,568 110.30 8.36 496.55 245,568 3.74 0.29 44.31

8 239,556 126.37 10.00 550.37 239,556 4.21 0.34 49.74

9 229,670 140.45 11.29 602.29 229,670 4.65 0.38 55.39

10 222,391 154.74 12.73 651.64 222,391 5.11 0.41 62.35

11 215,291 168.93 14.18 704.12 215,291 5.58 0.45 69.21

12 210,421 184.98 15.82 767.51 210,421 6.10 0.48 76.43

13 202,166 198.30 16.99 845.43 202,166 6.54 0.51 82.32

14 195,972 213.38 18.50 952.35 195,972 7.12 0.55 109.62

15 189,943 228.41 19.91 1134.33 189,943 7.58 0.57 105.64

16 185,782 246.63 21.74 1447.15 185,782 8.22 0.61 130.33

17 178,734 260.94 23.08 1899.71 178,734 8.76 0.63 169.36

18 173,472 276.99 24.64 2289.29 173,472 9.33 0.66 185.39

19 168,305 293.50 26.19 2689.59 168,305 9.72 0.68 181.77

20 164,741 314.19 28.37 3137.10 164,741 10.28 0.72 203.26

(continued on the next page)

61

(table 4 continued)

Panel B: Operating liquidity reserve - forecast accuracy for distressed sample (41,664 financially distressed firm quarters from 1980Q1-2010Q4)

Absolute OLR forecast inaccuracy (in Millions of Dollars) (OLRpredict – OLRactual)

Relative OLR forecast inacurracy (OLRpredict - OLRactual)/ OLRactual

Qtr ahead N Mean Median Std.dev. N Mean Median Std.dev. 1 38,549 1.91 -0.16 59.79 38,549 0.09 -0.03 2.69

2 35,241 6.10 0.25 82.13 35,241 0.39 0.04 3.39

3 33,242 9.80 0.60 100.91 33,242 0.61 0.09 4.49

4 31,717 13.42 1.01 118.35 31,717 0.82 0.14 5.29

5 29,971 16.86 1.47 138.11 29,971 1.03 0.19 6.03

6 28,597 20.29 2.06 148.66 28,597 1.22 0.25 6.27

7 27,275 24.27 2.68 174.49 27,275 1.46 0.31 8.61

8 26,154 28.21 3.36 201.08 26,154 1.65 0.36 9.88

9 24,894 32.02 3.94 222.51 24,894 1.87 0.42 11.62

10 23,836 35.35 4.67 205.14 23,836 2.05 0.47 11.11

11 22,831 39.36 5.41 221.25 22,831 2.22 0.52 11.17

12 21,998 43.72 6.17 249.41 21,998 2.42 0.57 12.44

13 21,005 48.66 6.81 315.31 21,005 2.66 0.63 14.04

14 20,157 53.94 7.48 430.19 20,157 2.89 0.68 15.60

15 19,338 55.83 8.18 265.53 19,338 3.14 0.73 16.78

16 18,664 60.23 8.88 274.92 18,664 3.36 0.78 17.72

17 17,860 64.18 9.58 282.78 17,860 3.55 0.81 18.11

18 17,187 68.26 10.30 293.91 17,187 3.72 0.86 18.78

19 16,515 73.45 11.06 313.53 16,515 3.92 0.91 19.68

20 15,943 85.40 12.01 756.83 15,943 4.43 0.97 38.98

This table reports the forecasted model inaccuracy of the s-prob model. Panel A reports the forecast inaccuracy for the complete sample (i.e., solvent and financially distressed firm quarters). Panel B restrict the comparison to financially distressed firm quarters. For each of the twenty firm quarters predicted ahead we take the median over all 10,000 simulated (adjusted) OLR paths and compare the forecasted median OLR to the actual OLR (where OLR=actq+lineun-lctq). Next, we calculate the absolute and relative forecast inaccuracy. For example, we compare the actual OLR with the median predicted OLR simulated three quarters ago to yield the three quarters ahead absolute and relative forecast accuracy.

62

Figure 3: Selected Operating Liquidity Reserve-Predictions by Industry

This figure shows the median of OLR-prediction (1, 2, and 3 years ahead) compared to the actual OLR by four randomly selected industries (FFI48-No. 3, 17, 21, and 36) using the Fama-French (2015) 48-industry classification.

63

Table 5: Summary Statistics for OLR by Firm Quarter to Delist

Panel A: OLR by quarter to delisting (41,664 financially distressed firm quarters) OLR by quarter to delist

Qtr to delist Mean Median Std.dev. 1 -4.51 0.07 133.23 2 6.58 0.68 139.16 3 14.79 1.32 106.96 4 20.52 1.98 125.13 5 21.48 2.48 180.40 6 25.03 2.90 135.57 7 29.72 3.28 195.35 8 33.08 3.89 225.85 9 35.21 4.33 203.80

10 34.23 4.78 207.49 11 36.31 5.13 205.32 12 33.97 5.30 267.50 13 34.10 5.63 278.88 14 37.76 6.02 260.61 15 39.77 6.35 253.29 16 43.16 6.64 233.13 17 45.76 6.86 234.26 18 47.32 7.00 236.16 19 44.11 7.23 173.95 20 45.41 7.28 180.37

This table shows the distribution of the operating liquidity reserve (OLR=actq+lineun-lctq) for the twenty quarters ahead of firm’s delisting.

64

Table 6: Descriptive Statistics

Panel A: Univariate summary statistics (solvent vs. financially distressed firm quarters from 1980Q1-2010Q4)

Variable Status Mean Median Std.dev. 1% 99% N

Rating Solvent 10.34 11.00 3.63 2.00 17.00 56,991

Financially Distressed 14.37 14.00 2.29 8.00 21.00 3,827

Z-Prob Solvent 0.16 0.10 0.19 0.00 0.99 288,885


O-Prob Solvent 0.05 0.01 0.12 0.00 0.66 288,885


S-Prob Solvent 0.18 0.03 0.29 0.00 1.00 288,885


Z-Probu Solvent 0.03 0.02 0.05 0.00 0.18 288,885


O-Probu Solvent 0.02 0.00 0.05 0.00 0.23 288,885


Panel B: Correlations (Pearson above, Spearman rank below the diagonal)

Variable Financial Distressed Rating Z-Prob O-Prob S-Prob Z-Probu O-Probu

Financially Distressed 0.27 0.30 0.24 0.31 0.20 0.22 Rating 0.28 0.58 0.36 0.42 0.28 0.31 Z-Prob 0.26 0.60 0.61 0.58 0.57 0.47 O-Prob 0.26 0.57 0.73 0.53 0.46 0.66 S-Prob 0.24 0.45 0.48 0.53 0.44 0.38 Z-Probu 0.18 0.31 0.67 0.48 0.37 0.50

O-Probu 0.18 0.51 0.42 0.54 0.25 0.52

This table reports summary statistics for solvent vs. financially distressed observations by model. Panel A presents summary statistics of the evaluated model outcomes. RATING (=Standard & Poor's Credit Ratings, coded from 1 (“AAA”) to 21 (“D” or "SD") based on the Compustat item splticrm, Z-PROB (=original Altman Z-Score model form 1968). The z-score is transformed into probabilities by the logit function (1/(1+exp(z-score) ), O-PROB (=original Ohlson O-Score model 1 probabilities 1980), S-PROB are the probability outcomes from the financial distressed prediction model using stochastic processes, Z-PROBu (=updated z-score model by a logistic regression). The updated z-scores are transformed into probabilities by the standard logit transformation (1/(1+exp(-z-score) to allow the probabilities to increase with the absolute values of z-scores), O-PROBu (=updated o-score model probabilities by a logistic regression). The total number N of observations are 330,549 firm quarters / 10,687 firms over the sample period 1980Q1 to 2010Q4. FINANCIALLY DISTRESSED represents an indicator variable, which is one if delisting (indicated by CRSP delisting codes 400, 500-585) occurs within the next 20 quarters (i.e., a delisting firm quarter is determined if the firm experience a delisting within the next 20 quarters as defined in table 1). The differences in probabilities of the solvent and financial distressed samples are significant at the 1%-level using a t-test and the Wilcoxon rank sum test (two-tailed). Note that for the Rating variable the sample size is reduced to 56,991 firm quarters / 2,243 firms covered by Standard & Poor’s credit ratings and CRSP delistings. Panel B shows the Pearson correlation above and the Spearman rank correlation below the diagonal for the evaluated models.

65

Figure 4: Median Estimated Default Probability per S&P Rating Class

This figure shows the median estimated distress/default probability per Standard & Poor’s rating class. RATING CLASS (=Standard & Poor's Credit Ratings, based on the Compustat item splticrm). The non-investment grade starts with (“BB”). For this analysis the sample size is reduced to 60,562 firm quarter observations / 2,118 firms that are covered by the major rating agency Standard & Poor’s. The historic default rate as benchmark is from the 2013 Annual U.S. Corporate Default Study by Standard & Poor's Global Fixed Income Research.

66

Figure 5: Default Probabilities Over Time

This figure shows the median of the estimated distress probabilities for the different models through the period 1980Q1 to 2010Q4 together with the actual delisting rate (where DELISTING RATE = number of delisted firms (CRSP delisting codes 400, 500-585) / number of active firms per quarter). Note that the peak in the distress probability in the o-probu

model during 1982q1 – 1983q1 is attributable to a high proportion of young and small firms entering the sample in 1980-1982 (i.e., approximately an increase by around 700 new unique firms). This leads to a decrease (i) in the median sample firm size (i.e. total assets), and (ii) the firm’s average age. These two effects influence the size-coefficient of the o-probu model significantly.

67

Figure 6: Evolvement of Default Probabilities Before Delisting

This figure shows the evolvement of the median estimated default probabilities for the different models up to 40 quarters before delisting (from 1980Q1 to 2010Q4). The zero represents the quarterly end of an actual listing at one of the following stock exchanges (NYSE, AMEX, NASDAQ or NYSE ARCA) using CRSP delistings codes (dlstcd=400, 500-585).

68

Table 7: Comparative Receiver Operating Characteristic (ROC) and Area Under the Curve (AUROC)

Panel A: Receiver Operating Characteristic (ROC): Area under the Curve (AUROC) by years (from 1980-2010) # firm quarters S-Prob Z-Prob O-Prob Z-Probu O-Probu

Year solvent financially distressed AUROC SE AUROC SE AUROC SE AUROC SE AUROC SE

1980 5,650 124 0.8257 0.0212 0.8293 0.0185 0.8780 0.0158 0.7965 0.0218 0.8422 0.0174

1981 5,378 141 0.7574 0.0235 0.8374 0.0168 0.8720 0.0147 0.7600 0.0210 0.8009 0.0192

1982 5,905 251 0.8195 0.0140 0.7583 0.0188 0.8570 0.0132 0.7556 0.0153 0.8271 0.0141

1983 8,156 752 0.8363 0.0080 0.6742 0.0124 0.8170 0.0082 0.7414 0.0102 0.8131 0.0080

1984 8,991 1,068 0.8470 0.0065 0.7419 0.0094 0.8317 0.0065 0.7774 0.0082 0.8370 0.0064

1985 9,131 1,172 0.8279 0.0064 0.7640 0.0082 0.8266 0.0065 0.7614 0.0081 0.8079 0.0070

1986 9,200 1,362 0.8096 0.0062 0.7389 0.0079 0.8173 0.0060 0.7415 0.0075 0.7941 0.0065

1987 8,866 1,544 0.8037 0.0059 0.7294 0.0076 0.8037 0.0060 0.7365 0.0073 0.7945 0.0060

1988 8,895 1,722 0.8129 0.0054 0.7427 0.0072 0.7975 0.0060 0.7395 0.0070 0.7859 0.0061

1989 9,260 1,823 0.8268 0.0049 0.7562 0.0069 0.8024 0.0057 0.7524 0.0066 0.7963 0.0057

1990 9,277 1,690 0.8301 0.0051 0.7616 0.0070 0.8038 0.0058 0.7461 0.0071 0.8017 0.0055

1991 9,462 1,573 0.8210 0.0056 0.7415 0.0077 0.7925 0.0063 0.7379 0.0075 0.8080 0.0057

1992 9,676 1,279 0.8118 0.0063 0.7160 0.0087 0.7752 0.0072 0.7202 0.0085 0.7999 0.0065

1993 10,001 1,362 0.8100 0.0061 0.6955 0.0086 0.7627 0.0071 0.7302 0.0081 0.7924 0.0064

1994 10,393 1,613 0.8115 0.0055 0.7042 0.0077 0.7576 0.0065 0.7143 0.0074 0.7776 0.0062

1995 10,937 1,802 0.7852 0.0056 0.7018 0.0073 0.7460 0.0064 0.7144 0.0069 0.7563 0.0062

1996 11,171 2,187 0.7940 0.0051 0.6938 0.0067 0.7399 0.0060 0.7135 0.0064 0.7539 0.0057

1997 11,229 2,542 0.7943 0.0049 0.7426 0.0057 0.7589 0.0054 0.7305 0.0057 0.7628 0.0053

1998 11,125 2,718 0.8000 0.0046 0.7411 0.0055 0.7543 0.0052 0.7245 0.0055 0.7452 0.0053

1999 10,847 2,459 0.7980 0.0046 0.7157 0.0062 0.7574 0.0053 0.7272 0.0058 0.7522 0.0054

2000 10,360 2,315 0.8089 0.0048 0.7399 0.0061 0.7574 0.0055 0.7397 0.0060 0.7567 0.0055

2001 10,136 1,769 0.8130 0.0051 0.7794 0.0065 0.7867 0.0059 0.7590 0.0066 0.7664 0.0062

2002 10,495 1,335 0.8096 0.0058 0.7652 0.0074 0.7769 0.0069 0.7505 0.0077 0.7687 0.0070

2003 10,381 974 0.7776 0.0074 0.7308 0.0087 0.7480 0.0081 0.7196 0.0090 0.7524 0.0080

2004 10,011 1,040 0.8019 0.0065 0.7343 0.0086 0.7662 0.0075 0.7256 0.0088 0.7729 0.0076

2005 9,577 964 0.8231 0.0066 0.7502 0.0086 0.7711 0.0079 0.7414 0.0090 0.7764 0.0079

2006 9,329 914 0.8126 0.0072 0.7346 0.0095 0.7745 0.0083 0.7309 0.0096 0.7845 0.0080

2007 8,852 1,030 0.8308 0.0071 0.7542 0.0090 0.8016 0.0077 0.7745 0.0087 0.8092 0.0075

2008 8,695 935 0.8435 0.0072 0.7661 0.0088 0.7965 0.0079 0.7916 0.0085 0.8027 0.0077

2009 8,796 680 0.8283 0.0091 0.7550 0.0105 0.7666 0.0101 0.7656 0.0109 0.7895 0.0093

2010 8,703 524 0.8180 0.0106 0.7588 0.0116 0.7724 0.0109 0.7683 0.0123 0.7940 0.0106 (continued on the next page)

69

(Table 7 continued)

Panel B: Receiver Operating Characteristic (ROC): complete sample

Model: # solvent # financially distressed AUROC SE

S-Prob 288,885 41,664 0.8122 0.0011

Z-Prob 288,885 41,664 0.7327 0.0014 O-Prob 288,885 41,664 0.7818 0.0012 Z-Probu 288,885 41,664 0.7272 0.0014

O-Probu 288,885 41,664 0.7207 0.0013

Panel C: Receiver Operating Characteristic (ROC): bootstrapped samples

Model: MEAN MEDIAN MIN MAX SE S-Prob 0.8123 0.8123 0.8087 0.8160 0.0011 Z-Prob 0.7328 0.7328 0.7280 0.7366 0.0014 O-Prob 0.7819 0.7819 0.7771 0.7855 0.0012 Z-Probu 0.7272 0.7271 0.7225 0.7321 0.0014 O-Probu 0.7207 0.7207 0.7165 0.7253 0.0013

Panel D: Proportion of financially distressed firm quarters outperformed by best model (N=41,664 delisted firm quarters and N=330,549 total firm quarters)

Model: S-Prob Z-Prob O-Prob Z-ProbU O-ProbU

N 39,786 - 1,878 - -

% 95.49% 0.00% 4.51% 0.00% 0.00% This table shows the Receiver Operating Characteristic curve for the evaluated models. Panel A reports the area under Receiver Operating Characteristic curve (AUROC) by years. The area under the ROC curve, AUROC and SE is calculated following the nonparametric approach by DeLong et al. (1988), Hanley and McNeil (1982). By definition a firm quarter is considered as "financially distressed" if delisting occurs within the next 20 quarters (carefully note that CRSP delisting information are only available up to the end of year 2014). Panel B compares the AUROC for the whole sample (1980-2010). The equality of the model performances related to the AUROC is pairwise compared following the U-statistic algorithm by DeLong et al. (1988). The AUROC’s of the S-Prob model are significantly different from other models (pairwise comparison) at a p-value < 0.0001. In Panel C we confirm the statistical inference and sample independence of the ROC curve by employing 1,000 bootstrap replications of the original sample to obtain bootstrap standard errors (SE). Panel D compares the number of firm quarters outperformed by the best model per year.

70

Figure 7: ROC Analysis for Different Forecast Horizons

This figure shows the area under Receiver Operating Characteristic curve (AUROC) for different periods to delisting (in quarters) using the broad sample (1980Q1-2010Q4 with 330,549 firm quarter observations). Depending on the forecast horizon a binary variable is set to one if a delisting occurs within the next 1-24 quarter ahead and zero otherwise. The AUROC is calculated following the nonparametric approach by DeLong et al. (1988), Hanley and McNeil (1982).

71

Figure 8: ROC Analysis over Time

This figure shows the area under Receiver Operating Characteristic curve (AUROC) over time. The AUROC is calculated following the nonparametric approach by DeLong et al. (1988) and Hanley and McNeil (1982) for each single quarter (1980Q1-2010Q4) using a five-year forecast horizon.

.65

.7.7

5.8

.85

.9

1980

q119

81q1

1982

q119

83q1

1984

q119

85q1

1986

q119

87q1

1988

q119

89q1

1990

q119

91q1

1992

q119

93q1

1994

q119

95q1

1996

q119

97q1

1998

q119

99q1

2000

q120

01q1

2002

q120

03q1

2004

q120

05q1

2006

q120

07q1

2008

q120

09q1

2010

q1

subprime crisis S-ProbZ-Prob O-ProbZ-Prob_Upd O-Prob_Upd

72

Table 8: Receiver Operating Characteristic by Industry

Receiver Operating Characteristic (ROC): Area under the Curve (AUROC) by Fama and French 10-Industry classification

Model: S-Prob Z-Prob O-Prob Z-Probu O-Probu FFI10 Desc solvent distressed AUROC SE AUROC SE AUROC SE AUROC SE AUROC SE

1 NoDur 24,247 3,396 0.7878 0.0044 0.7321 0.0051 0.7547 0.0047 0.6959 0.0048 0.7010 0.0048 2 Durbl 10,547 1,733 0.8178 0.0055 0.7431 0.0071 0.7838 0.0060 0.7350 0.0068 0.7380 0.0064 3 Manuf 60,632 6,476 0.8376 0.0025 0.7619 0.0036 0.8164 0.0029 0.7541 0.0034 0.7420 0.0032 4 Enrgy 16,197 2,527 0.8353 0.0043 0.7162 0.0065 0.7771 0.0051 0.6759 0.0067 0.7798 0.0048 5 HiTec 67,159 9,581 0.8296 0.0021 0.7429 0.0030 0.8114 0.0023 0.7577 0.0028 0.7414 0.0028 6 Telcm 6,916 1,273 0.7721 0.0066 0.6728 0.0091 0.6746 0.0084 0.7189 0.0085 0.6841 0.0077 7 Shops 41,363 7,020 0.7924 0.0029 0.7515 0.0033 0.7623 0.0032 0.6943 0.0035 0.6787 0.0035 8 Hlth 26,646 2,910 0.8215 0.0041 0.7333 0.0055 0.7971 0.0042 0.7637 0.0048 0.7366 0.0049 9 Utils 822 62 0.9447 0.0118 0.8583 0.0365 0.8062 0.0377 0.8331 0.0326 0.8788 0.0193

10 Other 32,733 5,398 0.7724 0.0034 0.7050 0.0041 0.7445 0.0037 0.7014 0.0042 0.6865 0.0039 n/a n/a 1,623 1,288 0.7667 0.0089 0.6743 0.0104 0.6793 0.0099 0.7732 0.0086 0.6680 0.0099

This table reports the area under the ROC curve for Fama-French 10-industry classifications (2015). Note that if neither Compustat nor CRSP SIC codes were available we assign these firms to a mixed industry classification. There are 327,638 / 10,563 firm quarter observations / firms with a four digit SIC code classified in the Fama-French 10-industry definition (2015). Similar results are obtained, in untabulated analysis, by Fama-French 48-industry classifications (2015).

73

Table 9: Information content tests

Panel A: (1-year ahead prediction, firm quarters from 1980Q1-2010Q4, 327,451 solvent firm quarters vs. 3,098 financially distressed firm quarters) Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Model(9) Constant -4.616 -4.490 -3.839 -2.170 -2.615 -4.094 -2.542 -3.900 -1.730 (38.43)*** (37.98)*** (26.06)*** (7.85)*** (10.07)*** (25.15)*** (7.05)*** (25.31)*** (5.98)*** coeff S-Score 0.148 . . . . 0.112 0.093 . . (9.85)*** . . . . (6.11)*** (4.75)*** . . coeff Z-Score . 0.176 . . . 0.046 . 0.068 . . (7.27)*** . . . (1.34) . (1.79)** . coeff O-Score . . 0.302 . . 0.153 . 0.243 . . . (8.53)*** . . (3.02)*** . (4.96)*** . coeff Z-Scoreu . . . 0.752 . . 0.336 . 0.459 . . . (9.11)*** . . (3.10)*** . (4.49)*** coeff O-Scoreu . . . . 0.481 . 0.207 . 0.315 . . . . (7.89)*** . (2.64)*** . (4.19)*** Pseudo-R² 0.0831 0.0450 0.0689 0.0745 0.0685 0.1069 0.1164 0.0725 0.0939 LRa 2,132*** 1,155*** 1,768*** 1,913*** 1,758*** 2,743*** 2,989*** 1,861*** 2,411***


74

(Table 9 continued)

Panel B: (3-year ahead prediction, firm quarters from 1980Q1-2010Q4, 327,451 solvent firm quarters vs. 3,098 financially distressed firm quarters) Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Model(9) Constant -4.662 -4.654 -4.084 -2.836 -3.329 -4.227 -3.238 -4.086 -2.454 (32.65)*** (28.78)*** (20.99)*** (6.50)*** (8.84)*** (19.77)*** (6.19)*** (20.83)*** (5.57)*** coeff S-Score 0.126 . . . . 0.100 0.091 . . (6.13)*** . . . . (4.04)*** (3.54)*** . . coeff Z-Score . 0.115 . . . 0.004 . 0.004 . . (2.90)*** . . . (0.09) . (0.08) . coeff O-Score . . 0.242 . . 0.137 . 0.239 . . . (5.43)*** . . (2.01)** . (3.93)*** . coeff Z-Scoreu . . . 0.563 . . 0.234 . 0.365 . . . (4.70)*** . . (1.67)* . (2.74)*** coeff O-Scoreu . . . . 0.322 . 0.122 . 0.220 . . . . (4.20)*** . (1.33) . (2.50)** Pseudo-R² 0.0489 0.0139 0.0348 0.0318 0.0283 0.0577 0.0605 0.0348 0.0420 LRa 859*** 244*** 611*** 559*** 498*** 1,014*** 1,064*** 611*** 738***


75

(Table 9 continued)

Panel C: (5-year ahead prediction, firm quarters from 1980Q1-2010Q4, 327,451 solvent firm quarters vs. 3,098 financially distressed firm quarters)

Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Model(9) Constant -4.756 -4.854 -4.236 -3.317 -3.660 -4.337 -3.574 -4.227 -2.890 (27.77)*** (22.87)*** (17.37)*** (5.75)*** (7.77)*** (16.39)*** (5.42)*** (17.40)*** (5.03)*** coeff S-Score 0.111 . . . . 0.085 0.082 . . (4.28)*** . . . . (2.74)*** (2.58)*** . . coeff Z-Score . 0.071 . . . -0.023 . -0.028 . . (1.42) . . . (0.49) . (0.56) . coeff O-Score . . 0.223 . . 0.152 . 0.248 . . . (4.00)*** . . (1.81)* . (3.51)*** .

coeff Z-Scoreu . . . 0.450 . . 0.175 . 0.290

. . . (2.97)*** . . (1.08) . (1.85)*

coeff O-Scoreu . . . . 0.267 . 0.109 . 0.198 . . . . (2.94)*** . (1.03) . (1.97)** Pseudo-R² 0.0348 0.0052 0.0255 0.0192 0.0193 0.0417 0.0421 0.0262 0.0279 LRa 437*** 65*** 321*** 242*** 242*** 524*** 529*** 330*** 351***


76

(Table 9 continued)

Panel D: Vuong-test statistics

S-Score Z-Score O-Score Z-Scoreu O-Scoreu

Model(1) vs. Model(2) Model(3) Model(4) Model(5)

1-year ahead 12.97*** 4.87*** 3.17*** 5.00*** 3-year ahead 12.17*** 5.74*** 6.79*** 7.14*** 5-year ahead 10.60*** 3.82*** 6.51*** 5.82*** */**/*** asterisks refer to significance at a 10%/ 5%/ 1% level for a one sided t-test. Figures in brackets are the z-statistics. This table shows the coefficients, z-statistics (in parentheses, which accounts for firm dependence between firm quarter observations), McFadden’s (1974)-Pseudo-R2 and the likelihood ratio statistic (LR=2(L1-L0), where L1 is the maximized log likelihood for the model and L0 is the maximized log likelihood for the baseline model with a constant only using a dynamic logit models as in Chava and Jarrow 2004; Shumway 2001. Due to the panel structure of the data there are less independent observations than assumed by a standard logit regression model. The panels compare the contribution of the s-score estimation with the results from univariate and combined regressions results for 1, 3 and 5-year ahead predictions. The scores are calculated in the same way outlined in table 6 (i.e., original and updated z-score and o-score values (Altman 1968, Ohlson 1980)). The probabilities are converted into scores according to: 𝑠𝑠𝑐𝑐𝑠𝑠𝑟𝑟𝑒𝑒𝑖𝑖 = 𝑙𝑙𝑙𝑙[𝑝𝑝𝑟𝑟𝑠𝑠𝑟𝑟𝑖𝑖/(1 − 𝑝𝑝𝑟𝑟𝑠𝑠𝑟𝑟𝑖𝑖)]. We bound all scores between ±18.420680999. Panel D reports the Vuong LR test statistic results for strictly non-nested models (model(1) vs. model(2-5)). A positive LR test statistic indicates that the s-score model (i.e. model(1)) is preferable. If the LR test statistic is negative the compared model is favored. a in unreported likelihood ratio tests we show that the s-score model provides incremental information beyond (updated) combined z-scores/o-scores models. The difference between the models (comparing nested models Model(6) vs. Model(8) and Model(7) vs. Model(9)) is significant at the 1%- level for all forecast horizon (1, 3 and 5-year ahead predictions).

77

Table 10: Homogeneity

Panel A1: Number of delisting firm quarters by decile (20 qtr ahead) S-Prob Z-Prob O-Prob Z-Probu O-Probu

Decile Distressed firm quarters Distressed firm quarters Distressed firm quarters Distressed firm quarters Distressed firm quarters 10 14,874 14,874 13,647 13,647 14,510 14,510 14,386 14,386 11,835 11,835 9 9,618 24,492 7,840 21,487 8,323 22,833 5,990 20,376 7,149 18,984 8 5,828 30,320 4,848 26,335 5,417 28,250 4,067 24,443 5,387 24,371 7 3,671 33,991 3,329 29,664 3,709 31,959 3,530 27,973 4,017 28,388 6 2,482 36,473 2,504 32,168 2,736 34,695 3,230 31,203 3,239 31,627 5 1,850 38,323 2,186 34,354 2,206 36,901 2,835 34,038 2,814 34,441 4 1,361 39,684 1,838 36,192 1,658 38,559 2,579 36,617 2,590 37,031 3 840 40,524 1,645 37,837 1,305 39,864 2,247 38,864 2,094 39,125 2 673 41,197 1,637 39,474 1,065 40,929 1,863 40,727 1,678 40,803 1 467 41,664 2,190 41,664 735 41,664 937 41,664 861 41,664

Panel A2: Number of distressed firm quarters by decile (8-20 qtr ahead) of total 23,309 distressed firm quarters S-Prob Z-Prob O-Prob Z-Probu O-Probu

Decile Distressed firm quarters Distressed firm quarters Distressed firm quarters Distressed firm quarters Distressed firm quarters 10 7,397 7,397 5,894 5,894 6,701 6,701 6,379 6,379 5,353 5,353 9 4,933 12,330 4,025 9,919 4,580 11,281 2,879 9,258 3,651 9,004 8 3,501 15,831 2,803 12,722 3,102 14,383 2,377 11,635 2,971 11,975 7 2,271 18,102 2,085 14,807 2,329 16,712 2,231 13,866 2,348 14,323 6 1,641 19,743 1,775 16,582 1,804 18,516 2,077 15,943 2,010 16,333 5 1,278 21,021 1,517 18,099 1,448 19,964 1,944 17,887 1,840 18,173 4 923 21,944 1,281 19,380 1,148 21,112 1,781 19,668 1,772 19,945 3 576 22,520 1,176 20,556 918 22,030 1,579 21,247 1,472 21,417 2 463 22,983 1,171 21,727 746 22,776 1,363 22,610 1,230 22,647 1 326 23,309 1,582 23,309 533 23,309 699 23,309 662 23,309


78

(Table 10 continued)

Panel B: Homogeneity (distressed firm quarters identified by s-prob = distressed firm quarters identified by other models) S-Prob Z-Prob O-Prob Z-Probu O-Probu

Decile No. of

distressed firm quarters

Equal No. of distressed

firm quarters Percentage







10 14,874 9,068 61.0% 8,950 60.2% 9,876 66.4% 7,830 52.6% 9 24,492 16,806 68.6% 17,560 71.7% 16,913 69.1% 15,382 62.8% 8 30,320 21,908 72.3% 23,670 78.1% 21,449 70.7% 21,064 69.5% 7 33,991 25,784 75.9% 28,198 83.0% 25,211 74.2% 25,472 74.9% 6 36,473 29,091 79.8% 31,745 87.0% 28,799 79.0% 29,202 80.1% 5 38,323 32,129 83.8% 34,752 90.7% 32,127 83.8% 32,547 84.9% 4 39,684 34,787 87.7% 37,212 93.8% 35,254 88.8% 35,678 89.9% 3 40,524 36,996 91.3% 39,055 96.4% 37,974 93.7% 38,226 94.3% 2 41,197 39,085 94.9% 40,543 98.4% 40,309 97.8% 40,359 98.0% 1 41,664 41,664 100.0% 41,664 100.0% 41,664 100.0% 41,664 100.0%

This table reports the model’s power and their homogeneity compared to the s-prob model. Panel A1 reports the frequency of classified financially distressed firm quarters by estimated distress/default probability deciles (10=high probability of financial distress – 1=low probability of financial distress). In the highest decile the s-prob model identify 14,874 out of 41,664 financially distressed firm quarters. The corresponding frequencies are 13,647/14,510/14,386/11,835 for the z-prob/o-prob/z-probu/o-probu –models. A firm quarter is considered as distressed firm quarter if the company is delisted in the next 20 quarters ahead. For the long-term performance we split the financial distress sample into a second part (2-5 years before a financial distressed related delisting occurs with 23,309 financially distressed firm quarter observations). The results show that the s-prob model can identify 35.7% of the indicated financially distressed firm quarter in the highest probability decile. Panel B shows the absolute and relative proportion of equally identified financially distressed firm quarter observations for the corresponding probability deciles (10=high probability of financial distress – 1=low probability of financial distress).

79

Table 11: Robustness Check: Descriptive Statistics (Market Sample)

Panel A: Univariate summary statistics (Solvent vs. financially distressed firm quarters from 1980Q1-2010Q4)

Variable Status Mean Median Std.dev. 1% 99% N S-Prob Solvent 0.18 0.03 0.29 0.00 1.00 243,523 Financially

Distressed 0.58 0.71 0.37 0.00 1.00 36,711

EDF Solvent 0.05 0.00 0.16 0.00 0.84 243,523 Financially

Distressed 0.23 0.05 0.30 0.00 0.99 36,711

C-Prob Solvent 0.00 0.00 0.05 0.00 0.09 243,523 Financially

Distressed 0.05 0.00 0.16 0.00 0.93 36,711

C-Probu Solvent 0.00 0.00 0.01 0.00 0.02 243,523 Financially

Distressed 0.01 0.00 0.03 0.00 0.14 36,711

Panel B: Correlations (Pearson above, Spearman rank below the diagonal)

Variable Financially Distressed S-Prob EDF C-Prob C-Probu

Financially Distressed 0.41 0.30 0.20 0.28 S-Prob 0.36 0.39 0.24 0.35 EDF 0.31 0.52 0.38 0.42 C-Prob 0.36 0.60 0.71 0.76

C-Probu 0.40 0.65 0.67 0.72 This table reports summary statistics for solvent vs. financially distressed observations by model. Panel A presents summary statistics of the evaluated (market) model outcomes. EDF (Expected Default Frequency) is the default probability estimated monthly following the sequential-iterations algorithm specified by Crosbie and Bohn (2003) and Bharath and Shumway (2008). C-PROB is defined as the monthly-estimated default probability using the original coefficient from the model No. 2 by Campbell et al. (2008), where −𝑠𝑠𝑐𝑐𝑠𝑠𝑟𝑟𝑒𝑒 = −9.08 − 29.67 ∗ (𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝐶𝐶𝐶𝐶𝑁𝑁𝑁𝑁) + 3.36 ∗ (𝑁𝑁𝑂𝑂𝑁𝑁𝑁𝑁𝐶𝐶) − 7.35 ∗(𝐸𝐸𝑋𝑋𝑂𝑂𝐸𝐸𝑁𝑁𝐶𝐶𝑁𝑁𝑁𝑁) + 1.48 ∗ (𝑆𝑆𝑁𝑁𝑁𝑁𝑁𝑁𝐶𝐶) + 0.082 ∗ (𝑂𝑂𝑆𝑆𝑁𝑁𝑍𝑍𝐸𝐸) − 2.40 ∗ (𝐶𝐶𝐶𝐶𝑆𝑆𝐶𝐶𝑁𝑁𝑁𝑁𝐶𝐶) + 0.054 ∗ (𝑁𝑁𝐵𝐵) − 0.937 ∗(𝐶𝐶𝑂𝑂𝑁𝑁𝐶𝐶𝐸𝐸) ). C-PROBU

is the corresponding model No. 2 (re-estimated by a growing window logit regression model, starting with month 1976|01, see the Appendix 1 for a detailed description). We winsorize all model’s parameter at the 1st and 99th percentile. The c-scores are transformed into probabilities using the standard logistic function (c-prob=1/(1+exp(-c-score)). The total number N of observations are 280,234 firm quarters / 10,143 firms over the sample period 1980Q1 to 2010Q4 with nonmissing values for the (market) default probabilities. FINANCIALLY DISTRESSED represents an indicator variable, which is one if delisting (indicated by CRSP delisting codes 400, 500-585) occurs within the next 20 quarters (i.e., a delisting firm quarter is determined if the firm experience a delisting within the next 20 quarters as defined in table 1). The differences in probabilities of the solvent and financially distressed means are significant at the 1%-level using a t-test or the Wilcoxon rank sum test (two-tailed). Panel B shows the Pearson correlation above and the Spearman rank correlation below the diagonal for the evaluated models.

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Table 12: Robustness Check: Accounting vs. Market Models

Panel A: Receiver Operating Characteristic (ROC): complete sample

Model: # solvent # delisted AUROC SE S-Prob 243,523 36,711 0.8121 0.0012

EDF 243,523 36,711 0.7629 0.0013 C-Prob 243,523 36,711 0.8109 0.0012

C-Probu 243,523 36,711 0.8398 0.0010

Panel B: Receiver Operating Characteristic (ROC): bootstrapped samples

Model: MEAN MEDIAN MIN MAX

S-Prob 0.8121 0.8121 0.8084 0.8158

EDF 0.7630 0.7630 0.7590 0.7676

C-Prob 0.8109 0.8109 0.8072 0.8143

C-Probu 0.8398 0.8398 0.8365 0.8433

Panel A reports the area under Receiver Operating Characteristic curve (AUROC) for the s-prob model compared to (mixed)-market models. The EDF (Expected Default Frequency) equals the default probability estimated monthly following the sequential-iterations algorithm of Crosbie and Bohn (2003) and Bharath and Shumway (2008). We winsorize all model’s parameter at the 1st and 99th percentile. C-PROB is defined as the monthly-updated default probability using the original coefficient from the mixed-model No. 2 by Campbell et al. (2008). C-PROBu

is the corresponding model No. 2 (re-estimated by a growing window logit regression model, starting with month 1976|01). We winsorize all measures at the 1st and 99th percentile before computing the logits. The area under the ROC Curve (AUROC) and its standard deviation (SE) is calculated following the nonparametric approach by DeLong et al. (1988), Hanley and McNeil (1982). By definition a firm is considered as "delisted" if delisting occurs within the next 20 quarters ahead. In Panel B we confirm the statistical inference and sample independence of the ROC curve by employing 1,000 bootstrap replications of the original sample to obtain bootstrap standard errors (SE).

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Table 13: Robustness Check: Information content tests (Market Sample)

Panel A: (1-year ahead prediction, firm quarters from 1980Q1-2010Q4, 277,373 solvent firm quarters vs. 2,861 financially distressed firm quarters)

Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Constant -4.590 -3.576 -1.825 0.113 -2.453 -0.787 -2.032 -0.186 (37.55)*** (24.26)*** (7.68)*** (0.29) (8.35)*** (1.57) (7.88)*** (0.43) coeff S-Score 0.147 . . . 0.085 0.058 . . (9.60)*** . . . (4.27)*** (2.59)*** . . coeff EDF . 0.167 . . 0.069 0.044 0.079 0.042 . (7.53)*** . . (2.73)*** (1.77)* (3.12)*** (1.67)* coeff C-Score . . 0.359 . 0.212 . 0.265 . . . (11.35)*** . (4.56)*** . (6.09)*** . coeff C-Scoreu . . . 0.710 . 0.526 . 0.624 . . . (10.91)*** . (5.76)*** . (7.52)*** Pseudo R² 0.0814 0.0962 0.1229 0.1620 0.1542 0.1728 0.1358 0.1656 LRa 1,777*** 2,100*** 2,683*** 3,537*** 3,367*** 3,772*** 2,965*** 3,615***


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Panel B: (3-year ahead prediction, firm quarters from 1980Q1-2010Q4, 277,373 solvent firm quarters vs. 2,861 financially distressed firm quarters)

Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Constant -4.633 -3.903 -2.365 -0.875 -3.120 -1.541 -2.669 -1.056 (31.39)*** (18.87)*** (6.13)*** (1.67)* (6.38)*** (2.33)** (6.11)*** (1.81)* coeff S-Score 0.126 . . . 0.079 0.049 . . (5.88)*** . . . (2.91)*** (1.67)* . . coeff EDF . 0.112 . . 0.054 0.021 0.063 0.021 . (4.96)*** . . (1.92)* (0.76) (2.24)** (0.74) coeff C-Score . . 0.284 . 0.126 . 0.187 . . . (6.31)*** . (1.82)* . (2.90)*** . coeff C-Scoreu . . . 0.542 . 0.413 . 0.492 . . . (7.00)*** . (3.66)*** . (4.81)*** Pseudo R² 0.0478 0.0502 0.0543 0.0909 0.0770 0.0962 0.0638 0.0919 LRa 691*** 726*** 786*** 1,315*** 1,114*** 1,393*** 923*** 1,330***


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Panel C: (5-year ahead prediction, firm quarters from 1980Q1-2010Q4, 277,373 solvent firm quarters vs. 2,861 financially distressed firm quarters)

Variables: Model(1) Model(2) Model(3) Model(4) Model(5) Model(6) Model(7) Model(8) Constant -4.750 -4.204 -2.880 -1.453 -3.624 -1.843 -3.173 -1.483 (26.26)*** (15.73)*** (5.45)*** (2.22)** (5.44)*** (2.28)** (5.29)*** (2.07)** coeff S-Score 0.111 . . . 0.074 0.037 . . (3.99)*** . . . (2.12)** (0.97) . . coeff EDF . 0.086 . . 0.043 0.003 0.051 0.003 . (3.34)*** . . (1.30) (0.10) (1.53) (0.11) coeff C-Score . . 0.236 . 0.088 . 0.150 . . . (3.99)*** . (0.96) . (1.78)* . coeff C-Scoreu . . . 0.468 . 0.400 . 0.460 . . . (5.08)*** . (2.96)*** . (3.79)*** Pseudo R² 0.0333 0.0307 0.0319 0.0674 0.0486 0.0697 0.0381 0.0674 LRa 333*** 307*** 320*** 675*** 487*** 699*** 382*** 675***


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Panel D: Vuong-test statistics

S-Score EDF C-Score C-Scoreu Model(1) vs. Model(2) Model(3) Model(4)

1-year ahead -2.16** -9.62** -18.19** 3-year ahead 0.21 -1.17 -10.78*** 5-year ahead 1.81* 1.26 -8.60*** */**/*** asterisks refer to significance at a 10%/ 5%/ 1% level for a one sided t-test. This table shows the coefficients, z-statistics (in parentheses, which accounts for firm dependence between firm quarter observations), McFadden’s (1974)-Pseudo-R2 and the likelihood ratio statistic (LR=2(L1-L0), where L1 is the maximized log likelihood for the model and L0 is the maximized log likelihood for the baseline model with a constant only) using a dynamic logit models (Chava and Jarrow 2004; Shumway 2001). The sample is limited to nonmissing observations of the default probabilities using market models (in total there are 280,234 / 10,143 firm quarter / firm observations). Due to the panel structure of the data there are less independent observations than assumed by a standard logit regression model. The panels compare the contribution of the s-score estimation with the results from univariate and combined regressions results for 1, 3 and 5-year ahead predictions. The scores are calculated in the same way outlined in table 9. The probabilities are converted into scores according to: 𝑠𝑠𝑐𝑐𝑠𝑠𝑟𝑟𝑒𝑒𝑖𝑖 = 𝑙𝑙𝑙𝑙[𝑝𝑝𝑟𝑟𝑠𝑠𝑟𝑟𝑖𝑖/(1 − 𝑝𝑝𝑟𝑟𝑠𝑠𝑟𝑟𝑖𝑖)]. We bound all scores between ±18.420680999. Panel D reports the Vuong LR test statistic results for strictly non-nested models (model(1) vs. model(2-4)). A positive LR test statistic indicates that the s-score model (i.e. model(1)) is preferable. If the LR test statistic is negative the compared model is favored. a in unreported likelihood ratio tests we show that the s-score model provides incremental information beyond (updated) combined edf/c-scores models. The difference between the models (comparing nested models Model(5) vs. Model(7) and Model(6) vs. Model(8)) is significant at the 1%- level for all forecast horizon (1, 3 and 5-year ahead predictions).

Klobucnik, Jan / Miersch, David / Sievers, SoenkeJan Klobucnik[a], David Miersch[b], Soenke Sievers...

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