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Estimating corporate bankruptcy forecasting models by maximizing discriminatory power

January 2022
Review of Quantitative Finance and Accounting 58(8)

January 2022
58(8)

DOI:10.1007/s11156-021-00995-0

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CC BY 4.0

Authors:

Zenon Taoushianis

University of Southampton

In this paper, we estimate coefficients of bankruptcy forecasting models, such as logistic and neural network models, by maximizing their discriminatory power as measured by the Area Under Receiver Operating Characteristics (AUROC) curve. A method is introduced and compared with traditional logistic and neural network models, using out-of-sample analysis, in terms of discriminatory power, information content and economic impact while we forecast bankruptcy one year ahead, two years ahead but also financial distress, which is a situation that precedes firm bankruptcy. Using US public firms over the period 1990-2015, in all, we find that training models to maximize AUROC, provides more accurate out-of-sample forecasts relative to training them with traditional methods, such as maximizing the log-likelihood function, highlighting the benefits arising by using models with maximized AUROC. Among all models, however, a neural network trained with our method is the best performing one, even when we compare it with other methods proposed in the literature to maximize AUROC. Finally, our results are more pronounced when we increase the forecasting difficulty, such as forecasting financial distress. The implementation of our method to train bankruptcy models is robust in various settings and therefore well-justified.

This figure shows the yearly distribution of bankruptcies over the sample period 1990-2015

…

This figure presents a sample of dij's of three models. The top plot presents the dij's generated by a logistic model trained to maximize the log-likelihood function, given by Eq. (6). The middle plot presents the same dij's generated by a logistic model but maximizng AUROC using our proposed ε-smoothed function given by Eq. (5) and setting the parameter γ = 0. The bottom plot presents the same dij's generated from a logistic model but trained to maximize AUROC using our proposed ε-smoothed function given by Eq. (5) and setting the parameter γ = 0.3

…

This figure shows the yearly distribution of bankruptcies over the sample period 1990–2015

…

The function max(0,z) is a surrogate function aiming to maximize the AUROC. However, this function is non-differentiable when z = 0. Thus, we replace it with the differentiable ε-smoothed function, hεz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h}_{\varepsilon }\left(z\right)$$\end{document}

…

This figure summarizes the work in our study and specifically how we train the models. The input vector, x, which can be financial and market data, enter the bankruptcy model (logistic or neural network model). The output of the model is the probability of bankruptcy, p(β), which depends on the coefficients imposed by the model and enters the merit function along with the target, t. The merit function can be the log-likelihood function (LL) or the AUROC, which is the one we propose in this study to use as the optimization criterion to obtain model coefficients. At each iteration, the optimization algorithm updates the coefficients until the merit function is optimized. For training we use data from the period 1990–2006

…

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https://doi.org/10.1007/s11156-021-00995-0

1 3

ORIGINAL RESEARCH

Estimating corporate bankruptcy forecasting models

bymaximizing discriminatory power

ChrisCharalambous1· SpirosH.Martzoukos2· ZenonTaoushianis3

Accepted: 2 June 2021 / Published online: 19 June 2021

Abstract

In this paper, we estimate coeﬃcients of bankruptcy forecasting models, such as logistic

and neural network models, by maximizing their discriminatory power as measured by the

Area Under Receiver Operating Characteristics (AUROC) curve. A method is introduced

and compared with traditional logistic and neural network models, using out-of-sample

analysis, in terms of discriminatory power, information content and economic impact

while we forecast bankruptcy one year ahead, two years ahead but also ﬁnancial distress,

which is a situation that precedes ﬁrm bankruptcy. Using US public ﬁrms over the period

1990–2015, in all, we ﬁnd that training models to maximize AUROC, provides more accu-

rate out-of-sample forecasts relative to training them with traditional methods, such as

maximizing the log-likelihood function, highlighting the beneﬁts arising by using models

with maximized AUROC. Among all models, however, a neural network trained with our

method is the best performing one, even when we compare it with other methods proposed

in the literature to maximize AUROC. Finally, our results are more pronounced when we

increase the forecasting diﬃculty, such as forecasting ﬁnancial distress. The implementa-

tion of our method to train bankruptcy models is robust in various settings and therefore

well-justiﬁed.

Keywords Bankruptcy Forecasting· Discriminatory Power· AUROC· Optimization·

Economic Beneﬁts

JEL classiﬁcation C18· C61· C45· C53· G33

* Zenon Taoushianis

z.taoushianis@soton.ac.uk

Chris Charalambous

bachris@ucy.ac.cy

Spiros H. Martzoukos

baspiros@ucy.ac.cy

1 Department ofBusiness andPublic Administration, School ofEconomics andManagement,

University ofCyprus, P.O. Box20537, 1678Nicosia, CY, Cyprus

2 Department ofAccounting andFinance, School ofEconomics andManagement, University

ofCyprus, P.O. Box20537, 1678Nicosia, CY, Cyprus

3 Department ofBanking andFinance, Southampton Business School, University ofSouthampton,

SouthamptonSO171BJ, UK

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C.Charalambous et al.

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1 Introduction

1.1 Background andmotivation

Increased attention has been paid in recent years for the development of powerful bank-

ruptcy forecasting models, mainly for two reasons. First, the recent global ﬁnancial crisis

in 2007–2009 has left banks to experience huge losses from their credit portfolios and con-

sequently their lending policies and decision-making processes have been seriously criti-

cized from regulators, investors and other stakeholders. Second, since the reform of Basel

Accord in 2006, banks can develop their own internal models to assess credit risks and pro-

tect themselves through the capital reserves that should withhold to face potential losses.

Thus, for a matter of bank viability,1 ﬁnancial stability and investor protection, it would be

of great interest to develop powerful bankruptcy forecasting models, which is the aim of

this paper.

One of the most signiﬁcant measures to evaluate the performance of bankruptcy fore-

casting models is their ability to discriminate bankrupt from healthy ﬁrms. It has been

shown that models with higher discriminatory power are associated with higher economic

beneﬁts for a bank (Bloechlinger etal. 2006; Agarwal et al. 2008). Furthermore, Bauer

etal. (2014) show that even small diﬀerences in the discriminatory power among bank-

ruptcy forecasting models yield superior bank economic performance. In addition, com-

mercial vendors and industry experts, such as Moody’s KMV, extensively use discrimina-

tory power as an integral part of their validation processes, especially when comparing

their newly developed models with existing ones (see for instance the RiskCalc 3.1 model

in Dwyer et.al. 2004). As it is stated in their paper:

“The greatest contribution to proﬁtability, eﬃciency and reduced losses comes from

the models’ powerful ability to rank-order ﬁrms by riskiness so that the bank can

eliminate high risk prospects.”

Beyond that, Moody’s KMV provides ample explanatory documentation on how to use

various discriminatory power measures in practice (see for instance Keenan etal. 1999 and

Sobehart etal. 2000) and it is also extensively used in academic research to compare vari-

ous bankruptcy forecasting models. This extensive use from practitioners and academics

alike, in fact, highlights the importance of using discriminatory power as a leading meas-

ure to evaluate the performance of bankruptcy forecasting models.

Despite the empirical evidence on the economic beneﬁts arising by using models with

higher discriminatory power, it is somewhat surprising that a common practice in bank-

ruptcy forecasting is to use discriminatory power only ex-post as an indication of model

performance, rather than obtaining model coeﬃcients directly by maximizing discrimina-

tory power. Exceptions include Miura etal. (2010) and Kraus etal. (2014) in the related

area of credit scoring which we discuss and compare with our method. We contribute to

this limited literature by introducing a method that we use to train bankruptcy forecasting

models, such as logistic and neural network models and comparing these models with tra-

ditional logistic and neural network models, such as models which maximize the log-like-

lihood function. Ultimately, our goal in this study is to highlight the importance of using

models which are trained to maximize discriminatory power.

1 See for instance Papakyriakou etal. (2019) for the consequences of bank failures around the globe.

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Estimating corporate bankruptcy forecasting models by…

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To measure discriminatory power, we use the Area Under Receiver Operating Charac-

teristics curve (AUROC or AUC). This is a widely used statistic that has been employed by

many studies recently to compare discriminatory power of various bankruptcy forecasting

models (including Chava etal. 2004; Campbell etal. 2008; Tinoco etal. 2013; Filipe etal.

2016 and many others). Furthermore, it has been used in related areas, such as mortgage

default prediction (Fitzpatrick etal. 2016) and generally when assessing the performance

of credit scoring models (see for instance Lessmann etal. 2015). Moreover, the AUROC is

an appealing measure because it is easy to interpret and compute empirically and impor-

tantly, it does not depend on cut-oﬀ values, such as those needed when constructing the

standard confusion matrices. Instead, the AUROC simply summarizes discriminatory

power in a single number, thus it is easy to compare across various models, without using

such cut-oﬀ values, which is the main reason it is has received considerable attention in

bankruptcy studies. Due to these reasons, we select AUROC as the optimization criterion

and we develop a method which seeks to maximize AUROC.

For our main analysis we collect annual ﬁnancial data and daily equity prices for a large

sample of U.S. public bankrupt and healthy ﬁrms and construct variables to make one-year

forecasts, two-year forecasts and ﬁnally, we forecast ﬁnancial distress which is a situation

prior to the formal bankruptcy ﬁling, over the period 1990–2015. We keep approximately

70% of the whole sample as a training set and evaluate the performance of the models in

the testing set using three distinct type of tests, following Bauer etal. (2014); 1) AUROC

analysis 2) Information content tests 3) Economic performance, when banks use various

bankruptcy forecasting models in a competitive loan market.

1.2 Main ﬁndings

First, we employ standard statistical analysis to select few predictive variables, from a pool

of variables, which individually exhibit high discriminatory power, have low correlation

from each other and are statistically signiﬁcant. This is to eliminate insigniﬁcant variables

that may add noise and helps us constructing parsimonious models. When we consider

only ﬁnancial variables, we ﬁnd that several ﬁnancial variables related to ﬁrm leverage,

proﬁtability, liquidity and coverage, are signiﬁcant predictors of bankruptcy. When we also

consider market-based variables in the analysis, however, the model with both ﬁnancial and

market variables outperforms the model with only ﬁnancial variables, consistent with prior

research (Shumway 2001; Chava etal. 2004; Campbell et al. 2008; Wu et al. 2010 and

Tinoco etal. 2013). These two selected sets of variables (ﬁnancial variables and ﬁnancial

with market variables) are the inputs to all models (logistic and neural networks trained to

maximize AUROC and the log-likelihood function).

We begin our analysis by evaluating and comparing the out-of-sample performance of

logistic and neural network models trained with our method, to those trained to maximize the

log-likelihood function, one year ahead, two years ahead and ﬁnally, when we forecast ﬁnan-

cial distress. Overall, we ﬁnd that our proposed method yields logistic and neural network

models which outperform, out-of-sample, traditional logistic and neural network models.

The results with respect to the three testing approaches suggest that models with maximized

AUROC 1) Signiﬁcantly outperform the traditional models in terms of their ability to dis-

criminate bankrupt from healthy ﬁrms, 2) They provide signiﬁcantly more information about

future bankruptcy-ﬁnancial distress relative to traditional methods 3) Banks using models with

maximized AUROC earn superior returns on a risk-adjusted basis relative to banks that use

traditional models to forecast bankruptcy-ﬁnancial distress. From all models, however, our

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C.Charalambous et al.

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neural network is the best performing one. In addition, the results are more pronounced in the

case of ﬁnancial distress.

Next, we compare our method with other methods proposed in the literature to maximize

AUROC. Using our proposed neural network model as a representative since it is the best

performing model in all tests, we ﬁnd that it outperforms the alternative AUROC maximiza-

tion methods proposed by Miura etal. (2010) and Kraus etal. (2014). This result is more pro-

nounced when forecasts are performed two years ahead and to the case of ﬁnancial distress.

Finally, we compare, out-of-sample, the discriminating ability of logistic and neural

networks trained by maximizing the AUROC, to the models trained with the traditional

approach but this time, the input variables are constructed using quarterly data. In this way,

we update the models as new information becomes available with higher frequency. In all,

ﬁndings advocate the implementation of our estimation method since it provides better pre-

diction performance, in terms of discriminatory power, relative to traditional estimation

methods.

Our paper has implications in the way bankruptcy analysis is conducted and aims

towards better decision-making through more accurate bankruptcy forecasts. First, our

study can be viewed as a way towards improving the general practice of bankruptcy fore-

casting by providing an alternative estimation technique to obtain model coeﬃcients rela-

tive to obtaining them with traditional methods. To this end, our proposed estimation

method signiﬁcantly improves performance, out-of-sample, especially when we increase

the forecasting diﬃculty, such as forecasting bankruptcy two years in the future but also

forecasting ﬁnancial distress, which is a situation before the formal ﬁrm bankruptcy. In

addition, our paper provides an extended methodological framework to commonly used

traditional bankruptcy models such as Altman (1968); Ohlson (1980) but also more recent

ones, such as Campbell etal. (2008) and many other similar models, by introducing a new

optimization method to obtain their coeﬃcients and increase forecasting accuracy. Finally,

the advantage of our method is that it works well using any modelling approach where

the output is a probability, thus it retains the same interpretability with the outputs of tra-

ditional estimation methods. This is also in contrast with the methodologies proposed by

Miura etal. (2010) and Kraus etal. (2014) as these methods cannot be used by logistic and

neural network models which are two of the most popular bankruptcy forecasting models

(se for instance Kumar etal. 2007 and references therein). To the best of our knowledge,

this is the ﬁrst time such extensive work is performed to compare maximizing AUROC

with the traditional maximizing the log-likelihood function for bankruptcy forecasting

models and highlighting the beneﬁts arising by AUROC-maximized models.

The remainder of the paper proceeds as follows: In Sect.2 we discuss data collection,

in Sect.3 we present the methodology to maximize AUROC as well as the three distinct

type of tests we use to evaluate performance, in Sect.4 we discuss the results and Sect.5

concludes.

2 Data

2.1 Sample

Our sample consists of 11,096 non-ﬁnancial U.S. ﬁrms from which 422 ﬁled for bank-

ruptcy under Chapter 7 or Chapter 11 between 1990 and 2015. We have a total of

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Estimating corporate bankruptcy forecasting models by…

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97,1332 ﬁrm-year observations with non-missing data to forecast bankruptcies using the

corresponding data which are lagged by one or two years for our one or two year ahead

forecasts respectively but also, to forecast ﬁnancial distress.3 Bankrupt ﬁrms and the

date of their bankruptcy ﬁling were identiﬁed from BankruptcyData, which is a compre-

hensive database containing corporate bankruptcy and distressed information for ﬁrms

in the US. Table1 reports the frequency of bankrupt and healthy ﬁrms (i.e. non-bank-

rupt ﬁrms) collected each year over the sample period spanning the years 1990–2015.

Since our main “distress” event is bankruptcy, we treat exits unrelated to bankruptcy as

non-bankrupt observations4 (i.e. healthy ﬁrms) and we report them in Table1. In particu-

lar, Table1 breaks down the healthy ﬁrms, each year, into three categories; 1) active ﬁrms

are ﬁrms that survived during the year, 2) ﬁrms that stopped ﬁling information due being

merged or acquired (M&As) and 3) ﬁrms that stopped ﬁling information for other reasons

including conversion to private company, engaged in a levarage buyout etc. The delisting

reasons was found in COMPUSTAT using the DLRSN variable which provides codes for

each delisting reason.

Next, Fig.1 presents graphically the yearly number of bankruptcies, to visualize the

variation of the number of bankruptcies over our sample period. Figure1 shows that bank-

ruptcies peak in three major time-periods: 1) during the 1990–1991 US crisis, 2) during

the dot-com bubble occurred around 2000 and 3) during the ﬁnancial crisis period when

bankruptcies peaked in 2009. Overall, the plot shows that the sample period we use cap-

tures the prevailing market conditions with higher (lower) number of bankruptcies during

crisis (normal) periods.

2.2 Variables construction

We collect annual ﬁnancial data and market (equity) data from Compustat and CRSP

respectively and we construct several variables based on related studies in the literature.

For example, in our analysis we consider variables used in traditional corporate bankruptcy

studies, such as Altman (1968), Ohlson (1980), Zmijewski (1984) but also in more recent

studies, such as Shumway (2001), Chava etal. (2004), Campbell etal. (2008) etc.

First, we construct ﬁnancial ratios capturing aspects of a ﬁrm’s ﬁnancial performance,

such as leverage, proﬁtability, liquidity, coverage, activity, cash ﬂows, as presented in panel

A of Table2. A limitation of ﬁnancial variables is that by their nature look backwards

and the quality of information they carry depends on accounting practices (Hillegeist etal.

2004; Agarwal etal. 2008). Market variables, instead, constructed from equity prices, are

forward-looking since they carry market perceptions about the prospects of the ﬁrm. For

publicly traded ﬁrms it would be more appropriate to incorporate market variables in the

2 Possibly, there are two main reasons that make our sample to diﬀer from other studies, such as Campbell

etal. (2008). First, we exclude ﬁnancial ﬁrms with SIC codes in the range of 6000 which is something com-

monly done in the literature (corresponds to more than 8,000 unique ﬁrms over our sample period). Second,

we delete observations with missing data whereas Campbell etal. (2008) replace missing data with sample

averages.

3 We present more details about the ﬁnancial distress case in Sect.4.5.

4 In other words, unhealthy ﬁrms that did not ﬁle for bankruptcy but eventually exited the sample, are con-

sidered as healthy observation. Failure to account for this, it will overestimate the predictive performance.

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C.Charalambous et al.

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models. To this end, we collect daily equity prices from CRSP for the entire ﬁscal year and

several market-based variables are constructed, as reported in panel B of Table2. Annu-

alized volatility of daily equity returns (VOLE) refers to the ﬂuctuations of ﬁrm’s equity

value returns, expecting to be higher for bankrupt ﬁrms. Next, excess return (EXRET)

refers to the diﬀerence between ﬁrm’s annualized equity return and the annualized value-

weighted return of a portfolio with NYSE, AMEX, NASDAQ stocks, expecting to be lower

Table 1 Yearly distribution of

bankrupt and non-bankrupt (i.e.

healthy) ﬁrms

This table shows the yearly distribution of bankrupt and healthy ﬁrms

(i.e. non-bankrupt ﬁrms) over the sample period 1990–2015. Bankrupt

ﬁrms were identiﬁed in the BankruptcyData database. The remaining

ﬁrms which did not ﬁle for bankruptcy and have the relevant data in

COMPUSTAT and CRSP databases, are considered as healthy ﬁrms.

Healthy ﬁrms are those who survived each year (active ﬁrms) or

exited for reasons other than bankruptcy such as, mergers and acquisi-

tions (M&A’s) or Other reasons (Others). Reasons for company dele-

tion are obtained from COMPUSTAT using the DLRSN variable (01 is

the code for M&A’s and Other reasons include, for instance, leverage

buyouts, conversion to private company with codes 06 and 09 respec-

tively, etc.)

Year Bankrupt

Firms

Non-Bankrupt Firms Total

Active Firms M&A’s Others

1990 22 3215 85 12 3334

1991 25 3208 57 6 3296

1992 17 3239 46 6 3308

1993 20 3295 62 2 3379

1994 10 3487 112 4 3613

1995 14 3873 144 4 4035

1996 14 4186 167 5 4372

1997 13 4520 233 4 4770

1998 19 4675 286 2 4982

1999 27 4554 351 2 4934

2000 20 4336 318 6 4680

2001 21 4317 236 28 4602

2002 14 4115 118 30 4277

2003 15 3790 135 38 3978

2004 13 3500 115 12 3640

2005 15 3433 148 18 3614

2006 10 3432 163 4 3609

2007 14 3298 215 6 3533

2008 20 3288 136 5 3449

2009 31 3196 94 6 3327

2010 6 3046 139 2 3193

2011 9 2985 128 8 3130

2012 12 2920 118 6 3056

2013 12 2873 106 7 2998

2014 12 2872 85 14 2983

2015 17 2900 114 10 3041

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Estimating corporate bankruptcy forecasting models by…

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for bankrupt ﬁrms. Further, we consider the relative size of the ﬁrm (RSIZE), the logarithm

of stock price (LOGPRICE) and the Market-to-Book ratio (MB), expecting a negative asso-

ciation with bankruptcy risk. Finally, we include three ﬁnancial variables scaled by ﬁrm’s

market value. More precisely, Campbell etal. (2008) show that scaling ﬁnancial variables

with a market-based measure of ﬁrm’s value i.e. market equity + liabilities (MTA), com-

pared to total assets as reported in the balance sheet, increases the predictive accuracy

of bankruptcy forecasting models. These variables are cash over MTA (CASHMTA), net

income over MTA (NIMTA), expecting a negative association with bankruptcy risk and

lastly, total liabilities over MTA (TLMTA). Following common practice, we winsorize the

variables between 1st and 99th percentile to avoid problems induced by outliers.

2.3 Variables selection

Table 2 presents an extensive list of variables that previous studies found to be signiﬁ-

cant predictors of bankruptcy risk. Out of these variables, a smaller set should be selected

in order to construct parsimonious models with few variables but with high forecasting

power. We establish a three-step approach to select the most powerful variables (see for

instance Altman etal. 2007 and Filipe etal. 2016) and summarized in the following three

steps:

Step 1: Removing variables with low discriminating ability (as a cut-oﬀ, we use

AUROC equal to 0.60). The idea of this step is to qualify the variables that individually

exhibit a satisfactory ability to discriminate bankrupt from healthy ﬁrms.

Step 2: Removing highly correlated variables using the Variance Inﬂation Factor (VIF)

criterion. The idea of this step is to remove the variables that are highly correlated with

others, since multicollinearity may yield misleading results regarding the signiﬁcance

of the variables in the ﬁnal model. Beyond that, we end up with variables that provide

Fig. 1 This ﬁgure shows the yearly distribution of bankruptcies over the sample period 1990–2015

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C.Charalambous et al.

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Table 2 List of ﬁnancial and market variables

This table shows all ﬁnancial ratios and market variables that we consider to construct the bankruptcy prediction models. From these, only a set of variables are selected

according to a three-step procedure described in the text

Panel A: Financial Ratios (Compustat)

Variable Detailed Description Compustat Item

NITA Net Income/Total Assets NI/AT

EBITTA Earnings Before Interests and Taxes/Total Assets EBIT/AT

RETA Retained Earnings/Total Assets RE/AT

CASHTA Cash and Short-Term Investments/Total Assets CHE/AT

WCTA Working Capital/Total Assets WCAP/TA

STD TA Debt in Current Liabilities/Total Assets DLC/AT

TLTA Total Liabilities/Total Assets LT/AT

CLCA Current Liabilities/Current Assets LCT/ACT

EBITCL Earnings Before Interests and Taxes/Current Liabilities EBIT/LCT

NICL Net Income/Current Liabilities NI/LCT

CFOTA Operating Cash Flows/Total Assets OANCF/AT

CFOTL Operating Cash Flows/Total Liabilities OANCF/LT

SLTA Sales/Total Assets SALE/AT

LOGASSETS Natural logarithm of Total Assets LOG(AT)

Panel B: Market Variables (CRSP)

VOLE Annualized volatility of daily equity returns

EXRET Annualized equity return minus the value-weighted return of NYSE, AMEX, NASDAQ stocks

LOGPRICE Natural logarithm of the stock price, at the ﬁscal-year end

RSIZE Natural logarithm of ﬁrm’s market capitalization over the total market capitalization of NYSE, AMEX, NASDAQ stocks

MB Firm’s market capitalization over book value of equity (Market-to-Book ratio)

TLMTA Total Liabilities/ (Market Capitalization + Total Liabilities)

NIMTA Net Income/ (Market Capitalization + Total Liabilities)

CASHMTA Cash and Short-Term Investments/ (Market Capitalization + Total Liabilities)

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Estimating corporate bankruptcy forecasting models by…

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diﬀerent information and explain bankruptcy uniquely. We use 5 as cut-oﬀ (variables with

VIF ≥ 5 are removed).

Step 3: Performing a stepwise multivariate logistic regression to the remaining vari-

ables in order to obtain the most signiﬁcant variables from a statistical point of view (we

use a signiﬁcance level of α = 5%). The logistic regression program estimates coeﬃcients

assuming independent observations, which is an invalid assumption, since the data con-

tains information for ﬁrms over multiple periods. In such case, an appropriate correction

measure which we adopt in our study, is to use clustered robust standard errors (also used

by Filipe etal. 2016).

Using the three-step approach, we develop two types of models. The ﬁrst one is a “pri-

vate ﬁrm” type of model, including only ﬁnancial variables. We further develop a “pub-

lic ﬁrm” type of model, including both ﬁnancial and market variables. For example, the

private ﬁrm model includes ﬁve ﬁnancial variables (TLTA, STDTA, NITA, CASHTA,

EBITCL), while the public ﬁrm model includes six variables (TLTA, STDTA, LOGPRICE,

CASHMTA, NIMTA, EXRET). Notice that two ﬁnancial-based variables (CASHTA and

NITA) are replaced with CASHMTA and NIMTA. Generally, the majority of variables that

are found to be signiﬁcant for the public ﬁrm model are market variables, which is consist-

ent with the perception that market-based variables are better bankruptcy risk measures,

due to their forward-looking nature. These two sets of variables are the inputs to all models

(i.e. used in the models which we train to maximize AUROC and in the models trained to

maximize the log-likelihood function). For simplicity, these two sets of variables are also

used in the models when forecasting bankruptcy two years ahead and when forecasting

ﬁnancial distress.

2.4 Descriptive statistics

Table3 reports descriptive statistics for the accounting and market variables that we ﬁnd

to be signiﬁcant predictors of bankruptcy. As expected, bankrupt ﬁrms are more levered on

average relative to healthy ﬁrms (TLTA and STDTA for bankrupt ﬁrms are higher), they are

also less proﬁtable (NITA and NIMTA are lower for bankrupt ﬁrms). Furthermore, bankrupt

Table 3 Descriptive statistics for the selected variables

This table reports descriptive statistics for the ﬁnancial and market variables which enter in the ﬁnal mod-

els, one year prior to bankruptcy, for both bankrupt and healthy ﬁrm observations. The deﬁnition of the

variables can be found in Table2

TLTA STDTA NITA CASHTA EBITCL LOG-

PRICE

EXRET CASH-

MTA

NIMTA

Bankrupt

Firms

Mean 0.854 0.181 −0.382 0.107 −0.578 0.528 −0.224 0.0684 −0.250

Median 0.825 0.099 −0.211 0.045 −0.153 0.560 −0.349 0.032 −0.172

St.Dev 0.327 0.183 0.450 0.157 1.289 1.146 0.876 0.100 0.257

Healthy

Firms

Mean 0.486 0.048 −0.042 0.190 0.07 2.293 0.205 0.121 −0.022

Median 0.478 0.014 0.031 0.101 0.153 2.474 0.106 0.063 0.022

St.Dev 0.253 0.083 0.254 0.217 1.322 1.266 0.684 0.162 0.149

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C.Charalambous et al.

1 3

ﬁrms are more constrained in terms of cash available (CASHTA and CASHMTA are lower)

as opposed to healthy ﬁrms. Going to the market variables, it is evident that the stock price

of bankrupt ﬁrms (LOGPRICE) on average is lower than healthy ﬁrms, possibly due to

their deteriorating ﬁnancial position that is priced by investors, leading to a depreciation

of their stock prices at the year prior to bankruptcy. Finally, bankrupt ﬁrms exhibit lower

and negative market performance relative to the market (EXRET is lower one year prior to

bankruptcy), as opposed to healthy ﬁrms.

3 Methodology

3.1 Measuring discriminatory power

Discriminatory power refers to the ability of a model to discriminate bankrupt from healthy

ﬁrms. According to a cut-oﬀ score, ﬁrms whose bankruptcy score exceeds the cut-oﬀ are

classiﬁed as bankrupt and healthy otherwise. Therefore, a way to measure the discriminat-

ing ability of a model is, for a given cut-oﬀ score, to count the true forecasts (percentage

of bankrupt ﬁrms correctly classiﬁed as bankrupt) and the false forecasts (percentage of

healthy ﬁrms incorrectly classiﬁed as bankrupt). Doing this process with multiple cut-oﬀs,

we get a set of true and false forecasts. A graph made from this set is the ROC curve with

false forecasts on the x-axis and true forecasts on the y-axis. A perfect model would always

(never) make true (false) forecasts and thus its ROC curve would pass through the point

(0,1). Generally, the closer the ROC curve to the top-left corner, the better the discrimina-

tory power of the model.

The ROC curve provides a graphical way to visualize discriminatory power. A quan-

titative assessment of the discriminatory power is given by the Area under ROC curve

(AUROC) which is calculated as follows5:

where

I(x)

is an indicator function, deﬁned to be 1 if x is true and 0 otherwise,

and

denote the bankruptcy scores of a model for the i-th bankrupt ﬁrm and for the j-th healthy

ﬁrm observation respectively. Finally, n is the number of bankrupt ﬁrms and m is the num-

ber of healthy ﬁrm observations. Note that Eq.(1) is discontinuous and non-diﬀerentiable.

3.2 Maximizing discriminatory power

In this section, we present a methodology to maximize the discriminatory power (AUROC)

when the bankruptcy score, s, is a probability, meaning that the model has a probabilistic

(1)

�

AUROC =

∑

i=1

∑

j=1

(

−sj

H>0

)

5 For further explanation, refer to Hanley etal. (1982) and Sobehart etal. (2001).

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Estimating corporate bankruptcy forecasting models by…

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response function6 which is the case of popular bankruptcy forecasting models such as

logistic and neural network models.

Ideally, we should have used Eq.(1) directly as the objective function in the optimiza-

tion. However, traditional gradient-based optimization methods cannot be used to maxi-

mize Eq.(1) directly because it is discontinuous and non-diﬀerentiable. For this reason, we

introduce a surrogate function that seeks to maximize the discriminatory power. We deﬁne:

as the diﬀerence between the probability of bankruptcy for the i-th bankrupt ﬁrm,

𝛽)

and the probability of bankruptcy for the j-th healthy ﬁrm observation, p

,𝛽

)

conditional on the predictor variables in X which could be a set of ﬁnancial and market

variables. From Eq. (1), to obtain the coeﬃcients, β, that maximize the discriminatory

power of a model we would like as many as possible

di,j’s

to be positive because AUROC

increases in this way. A way to achieve this is through the minimization of the following

surrogate merit function:

where 0 ≤ γ ≤ 1. The above merit function ignores the terms where

di,j(𝛽)

𝛾

(meaning that

the diﬀerence in bankruptcy probabilities between the i-th bankrupt ﬁrm and j-th healthy

ﬁrm observation is relatively high, as speciﬁed by the parameter γ) and penalizes the terms

where

di,j(𝛽)

≤

𝛾

. In other words, the parameter

𝛾

can be considered as a parameter which

controls the magnitude of the

di,j’s

that are to be penalized. For instance, if γ = 0, we penal-

ize only the negative

di,j’s

(i.e. only the cases where the model assigned a higher prob-

ability of bankruptcy for a healthy ﬁrm than a bankrupt ﬁrm) while if γ = 1, we penalize all

di,j’s

Based on the optimality conditions of minimizing F(β), at the optimal solution, a num-

ber of

di,j’s

must satisfy the condition

di,j

= γ.7 Hence, by selecting γ (close) to zero, we

force a number of

di,j’s

to be close to zero in absolute terms. In that case, a small change of

the input data can easily induce

di,j’s

to change signs which in turn will cause a change in

the AUROC. This may be particularly evident in the case of out-of-sample data. That is, by

training a model to produce

di,j’s

close to zero, may yield a model with poor generalization

ability and consequently the out-of-sample AUROC will be very sensitive. On the other

hand, selecting γ (close) to one, coeﬃcient estimates can blow up and provide unreason-

able results. Thus, theoretically, the parameter value should be in between 0 and 1 (we

explain later in this section how we compute the parameter empirically).

However, the surrogate function in Eq.(3) is non-diﬀerentiable when

z=𝛾−di,j(𝛽)=0

To overcome this problem and thus being able to use traditional gradient-based optimiza-

tion algorithms, we replace the term

max (0, z)

with a diﬀerentiable function. Note that, we

can minimize F(β) given by Eq.(3) using linear programming provided that the response

function is linear with respect to the coeﬃcients, β. Here, the probability is a non-linear

function and as such we should use non-linear optimization algorithms to obtain the coef-

ﬁcients. We replace the term

max (0, z)

by the following ε-smoothed diﬀerentiable approxi-

mation,

h𝜀(z)

(2)

di,j

(𝛽)=si

,𝛽

)

−s

,𝛽

)

=pi

,𝛽

)

−p

,𝛽

)

(3)

(𝛽)=

nm ∑

i=1

∑

j=1max

(

0, 𝛾−di,j(𝛽)

)

6 A good choice for the probabilistic response function that is usually used in bankruptcy-related studies

and also adopted in our study, is the logistic function; p = 1/(1 + exp(β*Χ)).

7 This draws on results from Charalambous (1979).

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308

C.Charalambous et al.

1 3

where ε is a small positive number close to zero. Here we set ε = 0.001. The ε-smoothed

function

h𝜀(z)

, which we graphically present in Fig.2, is a shifted version of the smoothed

function used previously by Charalambous etal. (2007) to value call options. As can be

(4)

𝜀(z)=

⎧

⎪

⎨

⎪

⎩

0, z≤−𝜀∕2

2𝜀(z+𝜀∕2)2,−𝜀∕2<z≤𝜀∕

z,z>𝜀

∕2

Fig. 2 The function max(0,z) is a surrogate function aiming to maximize the AUROC. However, this func-

tion is non-diﬀerentiable when z = 0. Thus, we replace it with the diﬀerentiable ε-smoothed function,

h𝜀(z)

Bankruptcy ModelMerit Function

Logistic Model

NN Model

Maximizing LL

Maximizing AUROC

Input

xp(β)

Output

Target

Fig. 3 This ﬁgure summarizes the work in our study and speciﬁcally how we train the models. The input

vector, x, which can be ﬁnancial and market data, enter the bankruptcy model (logistic or neural network

model). The output of the model is the probability of bankruptcy, p(β), which depends on the coeﬃcients

imposed by the model and enters the merit function along with the target, t. The merit function can be the

log-likelihood function (LL) or the AUROC, which is the one we propose in this study to use as the optimi-

zation criterion to obtain model coeﬃcients. At each iteration, the optimization algorithm updates the coef-

ﬁcients until the merit function is optimized. For training we use data from the period 1990–2006

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Estimating corporate bankruptcy forecasting models by…

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seen from the graph, the

h𝜀(z)

function has similar properties with the

max (0, z)

except that

h𝜀(z)

is diﬀerentiable when z = 0.

Hence, the merit function to be minimized is replaced by:

The next step is to estimate the coeﬃcients, β, by training the model to minimize F(β)

given by Eq.(5). Figure3 summarizes the work in our study.

Consider that we have N training input samples (i.e. observations). Each input sample,

[

,…,x

kn]

, is associated with a known target,

, where n = 1,2,…, N and k is the

number of variables. In the context of bankruptcy forecasting, the input sample

can be

information characterizing the n-th ﬁrm, such as ﬁnancial and market information, whereas

is an indicator variable which equals 1 if the corresponding ﬁrm-observation goes bank-

rupt and 0 otherwise. The inputs enter the bankruptcy model (logistic or neural network

model) to produce the probability of bankruptcy which is a function of the coeﬃcients

imposed by the model. The output of the bankruptcy model, p(β), with the associated tar-

get, t, are used in the merit function. Traditionally, the log-likelihood function is used to

obtain the coeﬃcients. In this study, we propose another way to obtain coeﬃcients and spe-

ciﬁcally we use the merit function given by Eq.(5) which is optimized in order to obtain

the coeﬃcients of the bankruptcy models and consequently the probability of bankruptcy.

Note that the target, t, is indirectly used in the merit function given by Eq.(5) in order to

identify the bankrupt and healthy ﬁrms and to estimate their probability of bankruptcy.

In this paper, the training sample spans the period 1990–2006. To solve the problem, we

formulate a nonlinear unconstrained optimization process using MATLAB. Speciﬁcally,

we use the fminunc command and the trust-region optimization algorithm to obtain the

coeﬃcients of the logistic and neural network models. At each iteration, the optimization

algorithm updates the coeﬃcients and the probability of bankruptcy (as shown in Fig.3)

until the merit function we propose is optimized.

As far as the parameter γ is concerned, we compute it empirically based on validation-a

straightforward and easy to implement approach, which makes use only the training data

to determine the parameter γ while the testing data remain intact. Also, validation is a fre-

quently used method implemented by many studies to determine parameters underlying the

models. We further divide our training sample into training (70%) and validation (30%)

sets.8 We train the models by choosing from the set of parameter values γ = {0, 0.1, 0.2, …,

1} and keep the value that gives the highest AUROC on the validation set. For instance,

using our private and public ﬁrm models we ﬁnd that γ equals 0.3 and 0.1 respectively,

consistent with our conjecture that the γ parameter should be between 0 and 1. Then we

merge the training and validation sets, to train the models as explained before and test their

performance on the testing set 2007–2015.

We further illustrate the role of γ by providing an example using our data to provide

an idea of how our method works and why it increases AUROC. First, we estimate the

coeﬃcients of a logistic model by maximizing the log-likelihood function and we cal-

culate the

di,j’s

. Second, we estimate the coeﬃcients of a logistic model by minimizing

(5)

(𝛽)=

nm ∑n

i=1

∑m

j=1h𝜀

(

𝛾−di,j(𝛽)

)

8 We also use these two sets of data to compute the optimal number of neurons for the neural networks. We

consider one, two, three and four neurons, starting also from various initial coeﬃcient values and we select

the number of neurons that performs the best (in terms of AUROC) in the validation set. We ﬁnd that the

optimal number of neurons is two. We also use a logistic transfer function in the hidden and output layer.

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310

C.Charalambous et al.

1 3

F(β) given by Eq.(5) and we calculate the

di,j’s

. Figure4 shows a sample of those

di,j’s

produced by logistic regression i.e. by the model trained to maximize the log-likelihood

function (top plot) and by maximizing AUROC with the ε-smoothed function, setting

γ = 0 (middle plot) and γ = 0.3 (bottom plot). Recall that we would like as many as pos-

sible of

di,j’s

to be greater than zero. Hence, they should lie above the solid straight line.

For the logistic regression, some lie above and some below. Using the ε-smoothed func-

tion, we want to make as many as possible negative

di,j’s

to move above the straight line.

Setting γ = 0, we observe that all

di,j’s

are close to zero. Some cases, 21 in particular,

that were negative according to the logistic regression became positive (denoted with

green crosses) and one case that was positive became negative (denoted with a red star),

highlighting the limitation of producing

di,j’s

that are close to zero. Setting γ = 0.3, not

only more

di,j’s

that were negative according to logistic regression became positive (59

in particular), but now the majority lie well above the solid straight line, several also

passing the γ parameter which are the points that lie above the dashed line. Notice now

that none of the

di,j’s

that were positive according to logistic regression became nega-

tive because the higher value of γ, causes

di,j’s

to be well above zero and as a conse-

quence, AUROC will not be sensitive.

Fig. 4 This ﬁgure presents a sample of dij’s of three models. The top plot presents the dij’s generated by a

logistic model trained to maximize the log-likelihood function, given by Eq.(6). The middle plot presents

the same dij’s generated by a logistic model but maximizng AUROC using our proposed ε-smoothed func-

tion given by Eq.(5) and setting the parameter γ = 0. The bottom plot presents the same dij’s generated from

a logistic model but trained to maximize AUROC using our proposed ε-smoothed function given by Eq.(5)

and setting the parameter γ = 0.3

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Finally, as a benchmark, we obtain the coeﬃcients of the logistic and neural networks

models by maximizing the log-likelihood function, LL. Assuming that we have N training

samples, LL is deﬁned as follows:

where p

𝛽)

is the bankruptcy probability of the n-th observation, given the input vec-

tor of variables,

and the coeﬃcients,

𝛽

3.3 Information content tests

We further consider information content tests, also employed by related studies (see for

instance Hillegeist etal. 2004; Agarwal etal. 2008; Charitou etal. 2013; Bauer etal. 2014).

In such tests the out-of-sample bankruptcy probabilities produced by various models, such

as by models with maximized AUROC, enter as inputs to logistic regression models and

we are interested to assess their explanatory power. In particular, we estimate the following

panel logit speciﬁcation:

where

pi,t

is the probability of bankruptcy at time t, that the i-th ﬁrm will go bankrupt the

next year and Yi, t+1 is the status of the i-th ﬁrm the next year (1 if it goes bankrupt and 0

if it is solvent). The variable of interest is

probi,t

, which is the out-of-sample bankruptcy

probability of the i-th ﬁrm at time t, produced by a model, for instance with maximized

AUROC. Finally,

𝛽

is the coeﬃcient estimate and

is the baseline hazard rate that is only

time-dependent and it is common to all ﬁrms at time t. Similar with prior studies, we proxy

the baseline hazard rate with the actual bankruptcy rate at time t.

The speciﬁcation in Eq.(7) is equivalent with the hazard model speciﬁcations used in

related bankruptcy studies, such as Hillegeist etal. (2004); Agarwal et al. (2008); Bauer

etal. (2014) etc. Speciﬁcally, Shumway (2001) argues that a panel logit model, like the

one in Eq.(7), is equivalent with a hazard rate model and therefore standard log-likelihood

maximization procedures can be used to estimate the logit model in Eq.(7), with a minor

adjustment that we explain below.

The model in Eq.(7) represents a multi-period logit model as it includes observations

for each ﬁrm across time. However, the inclusion of multiple ﬁrm-year observations per

ﬁrm yields understated standard errors because the log-likelihood objective function,

which is maximized to estimate the multi-period logit model, assumes that each observa-

tion is independent from each other. This is a wrong assumption since ﬁrm observations

at time t + 1 cannot be independent from ﬁrm observation at time t. Failing to address this

econometric issue, could lead to wrong inference regarding the signiﬁcance of the individ-

ual coeﬃcients. Similar with Filipe etal. (2016), we use clustered-robust standard errors

to adjust for the number of ﬁrms in the sample but also for heteroskedasticity (Huber 1967

and White 1980).

(6)

(𝛽)=

∑N

n=1

tnln

(

xn,𝛽

))

(

1−tn

)

(

1−pn

(

xn,𝛽

))

(7)

(

Yi,t+1=1

probi,t

)

=pi,t=e

+𝛽∗prob

i,t

1+e

at+𝛽∗probi,t

a∗Rate

+𝛽∗prob

i,t

1+e

a∗Ratet+𝛽∗probi,t

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C.Charalambous et al.

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3.4 Economic analysis ofbankruptcy models

The analysis so far addressed the forecasting accuracy of the bankruptcy models. But how

accuracy is economically beneﬁcial for banks? In particular, Bauer et al. (2014) show

that even small diﬀerences in the AUROCs between the models aﬀect the proﬁtability of

a bank. Similar ﬁndings are found in Charalambous etal. (2020). Therefore, it would be

interesting to investigate the eﬀect of using models with maximized AUROC, on bank eco-

nomic performance. Here, we follow the approach of Agarwal etal. (2008) and Bauer etal.

(2014) to examine it by assuming a loan market worth $100 billion and banks compete to

grant loans to individual ﬁrms. Each bank uses a bankruptcy model to evaluate the credit

worthiness of their customers.

3.4.1 Calculating credit spreads

We estimate the models using data spanning the years 1990–2006 (70% of the sample).

We sort ﬁrm-customers from this sample in 10 groups of equal size and a credit spread is

calculated according to the following rule; Firms in the ﬁrst group, which are ﬁrms with

the lowest bankruptcy risk, are given a credit spread, k and ﬁrms in the remaining groups

are given a credit spread, CSi, obtained from Blochlinger etal. (2006) and it is deﬁned as

follows:

where p(Y = 1|S = i) and p(Y = 0|S = i) is the average probability of bankruptcy and non-

bankruptcy respectively, for the i-th group, with i = 2, 3, …,10 and LGD is the loan loss

upon default. Following Agarwal et al. (2008), the average probability of bankruptcy for

the i-th group is the actual bankruptcy rate for that group, deﬁned as the number of ﬁrms

that went bankrupt the following year divided by the number of ﬁrms in the group. Fur-

thermore, k = 0.3% and LGD = 45%.

3.4.2 Granting loans andmeasuring economic performance

To evaluate economic performance, we assume that banks compete to grant loans to pro-

spective ﬁrm-customers between the period 2007–2015. Each bank uses a bankruptcy

model that has been estimated in the period 1990–2006. The bank sorts those customers

according to their riskiness and rejects the bottom 5% with highest risk. The remaining

ﬁrms are classiﬁed in 10 groups of equal size and ﬁrms from each group are charged a

credit spread that has been obtained from the period 1990–2006. Finally, the bank that

charges the lowest credit spread for the customer (i.e. for the ﬁrm-year observation) is

granting the loan. Two measures of proﬁtability are used. The ﬁrst one, Return on Assets

(ROA) is deﬁned as Proﬁts/Assets lent and the second one, Return on Risk-Weighted

Assets (RORWA) takes into consideration the riskiness of the assets, deﬁned as Proﬁts/

Risk-Weighted Assets. Risk-Weighted Assets are obtained from formulas provided by the

Basel Committee on Banking Supervision (2006).

(8)

p(Y=1|S=i)

p(Y=0|S=i)

LGD +

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Estimating corporate bankruptcy forecasting models by…

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4 Results

In this section, we present the out-of-sample comparisons between models with maximized

AUROC and models with maximized log-likelihood (traditional models). We use the bank-

ruptcy years 1990–2006 for training and keep the bankruptcy years 2007–2015 as the test-

ing set. We start our analysis by comparing their performance in terms of discriminatory

power, information content and economic beneﬁts, when forecasting bankruptcy one year

ahead.9 Next, using the same tests, we compare their performance by forecasting bank-

ruptcy two years ahead and then when our sample consists of ﬁnancially distressed ﬁrms.

An additional analysis is performed in this section where we compare our methodology

with other methods proposed in the literature to maximize AUROC using the same analysis

as before. Finally, we provide out-of-sample discriminatory power comparisons, when we

use quarterly data.

Table 4 AUROC results-Out of sample (2007–2015)

This table reports AUROC results for logistic and neural network models in two cases; First, when their

coeﬃcients are estimated by maximizing the AUROC and second, when their coeﬃcients are estimated by

maximizing the log-likelihood function (denoted as LL). The models are trained in the period 1990–2006

and the table reports results in the out-of-sample period 2007–2015. In the models, we use either ﬁnancial

variables (Private Firms Models) or ﬁnancial and market data (Public Firms Models)

Panel A: AUROC estimation

Models Private ﬁrms model Public ﬁrms models

Models with maximized AUROC

Neural Network 0.9332 0.9508

Logistic 0.9221 0.9470

Models with maximized log-likelihood

Neural Network 0.9138 0.9440

Logistic 0.8991 0.9425

Panel B: DeLong (1988) test statistic for diﬀerences in AUROCs

Private Firms Model Public Firms Model

Neural Network with max. AUROC vs

Neural Network with max. LL

2.04 1.72

Logistic model with max. AUROC vs

Logistic model with max. LL

2.43 1.68

9 For instance, we train the models in the bankruptcy years 1990–2006 using the corresponding variables

lagged by one year (1989–2005). To forecast bankruptcy in the bankruptcy years 2007–2015, we apply the

models using as inputs, the variables lagged by one year (2006–2014). For example, if a ﬁrm goes bankrupt

within 2010, we use its latest ﬁnancial information in 2009 as inputs to the models so that by the beginning

of 2010, investors have the information available to predict bankruptcy within 2010. For the two year ahead

forecasts, we use its ﬁnancial information in 2008 (but the ﬁrm is reported as bankrupt in 2010).

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C.Charalambous et al.

1 3

4.1 AUROC results

Table4 shows the out-of-sample performance (2007–2015) of the models with maximized

AUROC versus the models with maximized log-likelihood, in terms of discriminatory

power.

Overall, models that are trained to maximize AUROC perform better out-of-sample

compared to models trained to maximize the log-likelihood function, indicating that the

function we introduced performs well out-of-sample in discriminating ﬁrms that will go

bankrupt the next year. The eﬀect by maximizing the AUROC, as expected, is more pro-

nounced in the case of “private ﬁrms model” where only limited information is available

(i.e. ﬁnancial information), hence there is more space to improve the performance. For

instance, the AUROC of the logistic and neural network model trained to maximize the

log-likelihood function (LL) are 0.8991 and 0.9138 respectively where those trained to

maximize AUROC are 0.9221 and 0.9332 respectively. DeLong tests indicate that AUROC

diﬀerences are statistically signiﬁcant at the 5% level (2.43 and 2.04 respectively). In con-

trast, the eﬀect is less pronounced in the case of “public ﬁrms model”, since the inclusion

of market data in addition to ﬁnancial data, further increases the forecasting power of the

models. In fact, the AUROCs of models trained to maximize the log-likelihood function

are quite high and speciﬁcally, 0.9425 for the logistic10 model and 0.9440 for the neural

network. This improves to 0.9470 and 0.9508 respectively when maximizing AUROC. Dif-

ferences are statistically signiﬁcant at the 10% level.

From the results in this section, we suggest using the neural network model trained

to maximize AUROC since it is the best-performing model which is consistent with the

notion that neural networks outperform simpler modeling approaches (Zhang etal. 1999;

Kumar etal. 2007; Lessmann et al. 2015). For the user interested in simpler models, we

suggest the implementation of the logistic model but trained to maximize AUROC.

4.2 Information content results

In this section we report the results from information contest tests. We compare the infor-

mation contained in out-of-sample bankruptcy probabilities produced by models where

the AUROC is maximized versus where the log-likelihood is maximized. Models 1 and

2 include the out-of-sample (2007–2015) bankruptcy probabilities produced by a neural

network (Prob1) and a logistic model (Prob2) respectively, obtained by maximizing the

AUROC. Models 3 and 4 include the bankruptcy probabilities produced by a neural net-

work (Prob3) and a logistic model (Prob4) respectively, obtained by maximizing the log-

likelihood function.

Table 5 reports the results of logit models that include the out-of-sample bankruptcy

probabilities as explanatory variables but also the annual bankruptcy rate (Rate) as the

baseline hazard rate.

10 We acknowledge that such AUROC values are relatively high but can happen, even rarely, as they have

been observed in prior research. Chava and Jarrow use Shumway’s (2001) model with ﬁve accounting and

market variables and report that AUROC can reach up to 0.9421. More recently, Charalambous etal. (2020)

show that incorporating market-based information in accounting models, yields an AUROC that can reach

up to 0.9449. Moreover, Aﬁk etal. (2016) use the simple Black–Scholes-Merton default prediction model

and show that its AUROC can reach up to 0.9636.

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Estimating corporate bankruptcy forecasting models by…

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Panel A reports results from four logit regression models. Models 1–4 in the ﬁrst four

columns refer to the models where the corresponding bankruptcy probability which is

included as predictor (Prob1-Prob4) is generated with ﬁnancial data only. We re-estimate

the four logit regressions, of which their results are presented in the next four columns of

Table4 but this time, the corresponding bankruptcy probability is generated with ﬁnancial

and market data.

According to the results, the bankruptcy probabilities in all cases are highly statisti-

cally signiﬁcant, indicating that they carry signiﬁcant information in predicting bank-

ruptcy one year ahead, (coeﬃcient estimates are signiﬁcant at the 1% signiﬁcance

Table 5 Information content test results-Out of sample (2007–2015)

This table reports results from information content tests. Panel A shows estimation of four logit models.

Models 1 and 2 include out-of-sample (2007–2015) bankruptcy probabilities produced by a neural network

and a logistic model respectively, whose coeﬃcients are obtained by maximizing AUROC, with ﬁnancial

data as inputs. Models 3 and 4 include bankruptcy probabilities produced by a neural network and a logistic

model respectively, whose coeﬃcients are obtained by maximizing the log-likelihood function, with ﬁnan-

cial data as inputs (private ﬁrm models). We re-estimate the four logit models 1–4 as explained before and

report results in the next four columns but this time, the out-of-sample bankruptcy probabilities are gener-

ated with ﬁnancial and market data as inputs to the neural network and logistic models (public ﬁrm models)

All logit regression models include the Rate as proxy for the baseline hazard rate, which is the prior year

bankruptcy rate in our sample. The last two rows of the panel reports log-likelihood and pseudo-R2 for each

model. Panel B reports Vuong test statistics for diﬀerences in the log-likelihoods between the models

Private ﬁrms model Public ﬁrms model

Variable Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4

Panel A: Logit models estimation

Prob1 0.069

(0.004)

0.195

(0.012)

Prob2 0.066

(0.004)

0.182

(0.008)

Prob3 0.695

(0.028)

0.511

(0.055)

Prob4 0.296

(0.041)

0.271

(0.026)

Rate -1.120

(0.387)

-1.061

(0.387)

-0.879

(0.387)

-0.218

(0.371)

-1.070

(0.470)

-1.431

(0.454)

-0.789

(0.469)

-0.370

(0.402)

Constant -8.018

(0.306)

-8.553

(0.341)

-5.716

(0.183)

-5.521

(0.181)

-20.324

(1.085)

-17.456

(0.688)

-5.673

(0.220)

-5.540

(0.192)

LL -601.35 -624.16 -687.86 -774.96 -554.91 -561.34 -680.25 -728.26

Pseudo-R229.05% 26.36% 18.84% 8.56% 34.53% 33.77% 19.74% 14.07%

Panel B: Vuong test statistics for diﬀerences in log-likelihoods

Vuong test stat

Private Firms Model

Model 1 vs Model 3 5.75

Model 2 vs Model 4 7.37

Public Firms Model

Model 1 vs Model 3 4.37

Model 2 vs Model 4 8.07

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C.Charalambous et al.

1 3

level). More importantly, bankruptcy probabilities produced by models with maximized

AUROC (Prob1 and Prob2) contain signiﬁcantly more information than bankruptcy

probabilities produced by models with the log-likelihood being maximized (Prob3

and Prob4). This is especially evident by the substantially higher pseudo-R2 of models

1 and 2 (29.05% and 26.36% respectively) compared to models 3 and 4 (18.84% and

8.56% respectively) for the private ﬁrms case. Similarly, pseudo-R2 of models 1 and 2

(34.53% and 33.77% respectively) is substantially higher than models 3 and 4 (19.74%

and 14.07% respectively) in the case of public ﬁrms models.

In panel B, we use the Vuong (1989) test-statistic to test for diﬀerences in model

log-likelihood values between various (non-nested) models. Results show that, in the

case of private ﬁrms, the log-likelihoods of models 1 and 2 are signiﬁcantly diﬀerent

than models 3 and 4 (test-statistics are 5.75 and 7.37 respectively) whereas Vuong test-

statistics are 4.37 and 8.07 respectively for public ﬁrm models.

Overall, our results suggest that models with maximized AUROC provide probability

estimates that contain signiﬁcantly more information about bankruptcies over the next year

compared to models which are trained to maximize traditional functions, such as the log-

likelihood function, even when the increase in AUROC is relatively small (as in the case

Table 6 Economic performance results

This table reports economic results for four banks in a competitive loan market worth $100 billion. Bank 1

uses a neural network model with maximized AUROC for estimating the bankruptcy score of its customers.

Bank 2 uses a logistic model with maximized AUROC. Banks 3 and 4 are competitors to banks 1 and 2

respectively, using a neural network and a logistic model respectively, but by maximizing the log-likelihood

function. The table reports results in two cases; When the inputs to the models include ﬁnancial data (pri-

vate ﬁrm models) and when the inputs include ﬁnancial and market data (public ﬁrm models)

The banks sort prospective customers (2007–2015) and reject the 5% of ﬁrms with the highest risk. The

remaining ﬁrms are classiﬁed in 10 groups of equal size and for each group, a credit spread is calcu-

lated, as described in the main text. The bank that classiﬁes the ﬁrm to the group with the lowest spread

is ﬁnally granting the loan. Market share is the number of loans given divided by the number of ﬁrm-

years, Revenues = (market size)*(market share)*(average spread), Loss = (market size)*(prior probability

of bankruptcy)*(share of bankruptcies)*(loss given default). Proﬁt = Revenues-Loss. Return on Assets is

proﬁts divided by market size*market share and Return on Risk-Weighted-Assets is proﬁts divided by Risk-

Weighted Assets, obtained from formulas provided by the Basel Accord (2006). The prior probability of

bankruptcy is the bankruptcy rate for ﬁrms between 1990–2006 and equals 0.42%. Loss given default is

45%

Private ﬁrms model Public ﬁrms model

Bank1 Bank2 Bank3 Bank4 Bank1 Bank2 Bank3 Bank4

Credits 12,689 3,723 4,095 7,459 12,136 6,049 4,111 5,311

Market Share (%) 44.20 12.97 14.26 25.98 42.27 21.07 14.32 18.50

Bankruptcies 6 14 7 42 9 4 7 28

Bankruptcies/Credits (%) 0.047 0.38 0.17 0.56 0.074 0.066 0.17 0.53

Average Spread (%) 0.34 0.46 0.36 0.54 0.35 0.42 0.38 0.81

Revenues ($M) 151.30 59.77 51.06 139.28 148.64 88.01 54.52 150.70

Loss($M) 8.57 20.01 10.00 60.02 12.86 5.72 10.00 40.01

Proﬁt($M) 142.73 39.76 41.06 79.26 135.78 82.29 44.52 110.69

Return on Assets (%) 0.32 0.31 0.29 0.31 0.32 0.39 0.31 0.60

Return on RWA (%) 2.24 1.27 1.52 0.91 1.98 1.53 1.68 1.20

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Estimating corporate bankruptcy forecasting models by…

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of our public ﬁrm models as shown in Table4). From all models, however, our proposed

neural network model is the best performing one.

4.3 Economic performance results

So far, we have considered discriminatory power and information contest tests to assess

model performance. However, a bank is generally interested in the economic beneﬁts aris-

ing by using bankruptcy forecasting models in the decision-making process of granting

loans to individual ﬁrms. Following Agarwal etal. (2008) and Bauer et al. (2014), we

consider a loan market worth $100 billion and four banks are competing to grant loans

to prospective ﬁrm customers. We hypothesize that banks 1 and 2 are more sophisticated

banks, training models by maximizing AUROC (a neural network and a logistic model

respectively) whereas banks 3 and 4 are more “naïve”, training models by maximizing the

log-likelihood function (a neural network and a logistic model respectively). In Table6 we

report the results, for both private and public ﬁrm models.

Clearly, banks 1 and 2 which use models with maximized AUROC, respectively, man-

age loan portfolios with higher quality relative to banks 3 and 4. This is evident by the

lower concentration of bankruptcies they attract. In particular, the bankruptcy rate of

bank’s 1 portfolio, which uses a neural network trained to maximize AUROC is 0.047%

and 0.074% when using the private and public ﬁrms model respectively. In contrast, the

bankruptcy rate of bank’s 3 portfolio, which uses a neural network trained to maximize the

log-likelihood is 0.17% when using the private and public ﬁrms model respectively. Simi-

larly, bank 2 which uses a logistic model trained to maximize AUROC manages a credit

portfolio with bankruptcy rate equal to 0.38% and 0.066% when using the private and pub-

lic ﬁrms model whereas for the bank using models to maximize the log-likelihood function

(bank 4) the rates are 0.56% and 0.53% respectively.

Consequently, banks 1 and 2 achieve superior economic performance11 compared to

banks 3 and 4 respectively, on a risk-adjusted basis. For example, considering the private

ﬁrms model, bank 1 which uses a neural network model with maximized AUROC, earns

2.24% relative to bank 3 which uses a traditional neural network model (1.52%). Also,

bank 2 which uses a logistic model trained to maximize AUROC, earns 1.27% relative to

bank 4 which uses a traditional logistic model (0.91%). In the case of public ﬁrms model,

similar insights are obtained; bank 1 earns higher risk-adjusted returns relative to bank 3

(1.98% and 1.68% respectively). This is also the case for banks 2 and 4 (1.53% and 1.20%

respectively). Again, the neural network trained to maximize AUROC, provides the higher

economic beneﬁts for the bank using it (bank 1).

4.4 Forecasting bankruptcy two years ahead

In this section, we evaluate and compare the performance of the models by increasing the

forecasting horizon to two years. This is a more challenging problem because the charac-

teristics of bankrupt ﬁrms are less pronounced relative to one year prior to bankruptcy and

therefore is more diﬃcult to forecast bankruptcy. Also, identifying the signs of the crisis

11 Results are robust with respect to diﬀerent speciﬁcations for LGD (0.4–0.7) and k (0.002–0.004), sug-

gesting that models with maximized AUROC, outperform the traditional approaches.

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C.Charalambous et al.

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Table 7 Forecasting bankruptcy two years ahead; Maximizing AUROC vs maximizing LL

This table reports out-of-sample performance in terms of discriminatory power, information content and economic impact, of models trained to maximize AUROC versus

models trained to maximize the log-likelihood function, when forecasting bankruptcy two years ahead. Panel A reports the performance when ﬁnancial variables are used in

the models (private ﬁrms models) and when ﬁnancial and market variables are used in the models (public ﬁrms models). Panel B reports DeLong (1988) test statistics for dif-

ferences in AUROCs between the models of interest whereas panel C reports Vuong (1989) test statistics for diﬀerences information content between the models of interest

Panel A: Performance, out-of-sample, 2007–2015

Private Firms Model Public Firms Model

Model AUROC Info. Cont Econ. Ben AUROC Info. Cont Econ. Ben

Models with maximized AUROC

Neural Network 0.8678 16.20% 1.36% 0.8864 18.27% 1.57%

Logistic 0.8503 13.83% 0.82% 0.8664 15.24% 0.56%

Models with maximized log-likelihood

Neural Network 0.8441 13.66% 0.81% 0.8571 9.53% 0.90%

Logistic 0.8113 2.10% 0.50% 0.8558 4.33% 0.97%

Panel B: DeLong (1988) test statistic for diﬀerences in AUROC

Private Firms Model Public Firms Model

Neural Network with max.

AUROC vs Neural Net-

work with max. LL

2.18 3.17

Logistic model with max.

AUROC vs Logistic model

with max. LL

7.15 1.28

Panel C: Vuong (1989) test statistic for diﬀerences in information content

Private Firms Model Public Firms Model

Neural Network with max.

AUROC vs Neural Net-

work with max. LL

2.34 6.13

Logistic model with max.

AUROC vs Logistic model

with max. LL

7.15 7.33

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Estimating corporate bankruptcy forecasting models by…

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earlier, although diﬃcult, can help the management of the ﬁrm to take remedial actions to

correct the adverse situation and avoid bankruptcy. We perform the same analysis as before

and we provide the results in Table7.

As expected, performance has dropped relative to when forecasting bankruptcy one year

ahead due to the increased diﬃculty of the problem. However, models which are trained

to maximize AUROC outperform those trained to maximize the log-likelihood function.

Starting from the AUROC, which is the focus of this paper to improve, we document out-

of-sample that it is signiﬁcantly higher, especially for the neural networks case. Two years

prior to bankruptcy, the neural network with maximized AUROC achieves an AUROC

equal to 0.8678 and 0.8864 for the private and public ﬁrms models respectively whereas

for the neural network trained to maximize the log-likelihood function, these are 0.8441

and 0.8571 respectively. DeLong tests indicate that the diﬀerences are signiﬁcant at the 5%

and 1% level respectively (test statistics are 2.18 and 3.17 respectively). A logistic model

with accounting data as predictors (private ﬁrms model) trained to maximize AUROC, has

signiﬁcantly higher discriminatory power than a competing logistic model trained to maxi-

mize the log-likelihood function (0.8503 vs 0.8113 respectively). Diﬀerences in AUROCs

are statistically signiﬁcant at the 1% level according to the DeLong test (test statistic is

3.54). For the public ﬁrms case, an enhancement is achieved (0.8664 vs 0.8558) but the

diﬀerence is not statistically signiﬁcant (DeLong test statistic is 1.28).

The summary of the remaining tests is that the neural network with maximized AUROC

performs signiﬁcantly better in terms of information content and provides higher economic

beneﬁts relative to a competing neural network which is trained by maximizing the log-

likelihood function. The former is also the best performing model in all tests. Finally, a

logistic model with maximized AUROC provides signiﬁcantly more information than a

logistic with maximized log-likelihood function, albeit no economic beneﬁts are achieved

in this case.

4.5 Forecasting ﬁnancial distress

In this section, we change the event and instead of forecasting bankruptcy, we forecast

ﬁnancial distress. There are several reasons as to why ﬁrm stakeholders should be inter-

ested in models forecasting ﬁnancial distress more accurately. First, ﬁnancial distress is a

state prior to bankruptcy ﬁling and therefore forecasting the early signs of the crisis may

help, for instance the management of the ﬁrm, to take corrective measures in order to avoid

further deterioration that may ultimately lead to bankruptcy in which case the ﬁrm loses

most of its value (see for instance Asquith etal. 1994; Glover 2016). Second, forecast-

ing the early signs of the crisis is more challenging to accomplish not only because it is

the starting point of the crisis but also because ﬁnancial distress is not a formal event like

bankruptcy, thus we need to construct a ﬁnancial distress indicator. In this study, we follow

Keasey etal. (2015) and Gupta etal. (2018) and we consider a ﬁrm as ﬁnancially distressed

if the following conditions are satisﬁed; 1) Earnings Before Interest, Tax and Depreciation

and Amortization (EBITDA) is less than ﬁnancial expenses (i.e. interest payments) for two

consecutive years 2) Total Debt is higher than the Net Worth of the ﬁrm for two consecu-

tive years and 3) The ﬁrm experiences negative Net Worth growth between two consecu-

tive years. The ﬁrm is classiﬁed as ﬁnancially distressed in the year immediately following

these three events. For forecasting purposes, we use the data two years before ﬁnancial

distress. For example, when the conditions are satisﬁed for the years t and t-1, then the ﬁrm

is considered as ﬁnancially distressed in the year t and we construct the variables at t-2 to

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C.Charalambous et al.

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Table 8 Forecasting ﬁnancial distress; Maximizing AUROC vs maximizing LL

This table reports out-of-sample performance in terms of discriminatory power, information content and economic impact, of models trained to maximize AUROC versus

models trained to maximize the log-likelihood function, when forecasting ﬁnancial distress. Panel A reports the performance when ﬁnancial variables are used in the models

(private ﬁrms models) and when ﬁnancial and market variables are used in the models (public ﬁrms models). Panel B reports DeLong (1988) test statistics for diﬀerences in

AUROCs between the models of interest whereas panel C reports Vuong (1989) test statistics for diﬀerences information content between the models of interest

Panel A: Performance, out-of-sample, 2007–2015

Private ﬁrms model Public ﬁrms model

Model AUROC Info. Cont Econ. Ben AUROC Info. Cont Econ. Ben

Models with maximized AUROC

Neural Network 0.9175 32.82% 1.14% 0.9000 28.21% 1.05%

Logistic 0.8982 26.84% 0.36% 0.8824 24.28% 0.56%

Models with maximized log-likelihood

Neural Network 0.8956 25.89% 0.44% 0.8870 22.23% 0.97%

Logistic 0.8897 12.70% -0.34% 0.8753 12.60% 0.28%

Panel B: DeLong (1988) test statistic for diﬀerences in AUROC

Private Firms Model Public Firms Model

Neural Network with max.

AUROC vs Neural Net-

work with max. LL

5.42 5.99

Logistic model with max.

AUROC vs Logistic model

with max. LL

3.23 2.24

Panel C: Vuong (1989) test statistic for diﬀerences in information content

Private Firms Model Public Firms Model

Neural Network with max.

AUROC vs Neural Net-

work with max. LL

7.62 7.32

Logistic model with max.

AUROC vs Logistic model

with max. LL

10.27 9.31

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Estimating corporate bankruptcy forecasting models by…

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predict ﬁnancial distress. Following these conditions, we generate an extensive database

with 1,929 ﬁnancially distressed ﬁrms. In Table8 we report the out-of-sample results from

this exercise.

Starting from the private ﬁrms model, logistic and neural network models trained to

maximize AUROC, signiﬁcantly outperform models trained to maximize the log-like-

lihood function (0.9175 vs 0.8956 for neural networks and 0.8982 vs 0.8897 for logistic

models). Diﬀerences in AUROCs are statistically signiﬁcant at the 1% level (DeLong test

statistics are 5.42 and 3.29 for neural networks and logistic models respectively). Similar

results are found with respect to the public ﬁrms model (0.9000 vs 0.8870 for neural net-

works and 0.8824 vs 0.8753 for logistic models). Diﬀerences in AUROCs are statistically

signiﬁcant at the 1% and 5% level respectively (DeLong test statistics are 5.99 and 2.24 for

neural network and logistic models respectively).

Regarding the remaining tests, we ﬁnd that the both models which are trained to maxi-

mize AUROC provide signiﬁcantly more information and there is more gain by banks

using them relative to using models trained to maximize the log-likelihood function. Over-

all, from this test we conclude that our methodology can help the interested parties to

improve bankruptcy forecasts, considering the harder nature of the problem, either using

a neural network or a logistic model. Once again, the neural network constructed to maxi-

mize AUROC is the best performing one.

4.6 Comparing our methodology withother methodologies

In this section we use the same analysis as before to compare our proposed methodology

with other methods of AUROC maximization proposed in the bankruptcy literature and the

advantages (shortcomings) of our method (other methods) are discussed.

We consider two other approaches proposed by Miura et al. (2010) and Kraus et al.

(2014), to maximize AUROC of credit scoring models. Miura etal. (2010) suggest a sig-

moid function as an approximation of Eq.(1). Speciﬁcally, they maximize the following

objective function12:

where

i,j(𝛽)=𝛽T

(

B−Xj

)

. However, unlike the function that we introduced previ-

ously, it treats all

di,j’s

in the same way, whereas our function, give more emphasis on the

“bad” cases, for example, when a healthy ﬁrm has higher bankruptcy score than a bankrupt

ﬁrm. Further, the authors consider only a linear response function (the output is a linear

score) and unlike our method, it cannot be used by models which employ probabilistic

response functions such as logistic models and highly non-linear models such as neural

networks. Instead, our methodology works well with logistic and neural networks, which

are the among the most popular bankruptcy models, because they allow for a probabilistic

response function. Finally, Kraus etal. (2014) suggest using directly Eq.(1) as the objec-

tive function and implementing derivative-free methods (such as Nelder et al. 1965) to

optimize the coeﬃcients. The optimization algorithm that is used, however, assumes that

(9)

(𝛽)=

∑

i=1

∑

j=1

1+exp

[

−d

i,j

(𝛽)∕𝜎

]

12 The authors, in the original speciﬁcation, set the tuning parameter σ = 0.01 or 0.1. Here, we use σ = 1

because the original speciﬁcations performed poorly. Further, they constrain the norm of coeﬃcients to be

1. Again, we ﬁnd that this speciﬁcation performs poorly.

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Table 9 Forecasting bankruptcy one year ahead; Our methodology vs alternative methodologies

This table reports out-of-sample performance in terms of discriminatory power, information content and economic impact, using our methodology to maximize AUROC on

a neural network versus alternative methods to maximize AUROC as proposed by Miura etal. (2010) and Kraus etal. (2014), denoted as KK (2014), when forecasting bank-

ruptcy one year ahead. Panel A reports the performance when ﬁnancial variables are used in the models (private ﬁrms models) and when ﬁnancial and market variables are

used in the models (public ﬁrms models). Panel B reports DeLong (1988) test statistics for diﬀerences in AUROCs between the models of interest whereas panel C reports

Vuong (1989) test statistics for diﬀerences information content between the models of interest

Panel A: Performance, out-of-sample, 2007–2015

Private ﬁrms model Public ﬁrms model

Model AUROC Info. Cont Econ. Ben AUROC Info. Cont Econ. Ben

Our methodology

Neural Network 0.9332 29.05% 2.21% 0.9508 34.53% 1.92%

Alternative methodologies

Miura etal. (2010) 0.9188 17.59% 1.55% 0.9471 26.76% 1.80%

KK (2014) 0.9136 8.57% 1.23% 0.9473 25.44% 1.64%

Panel B: DeLong (1988) test statistic for diﬀerences in AUROC

Private Firms Model Public Firms Model

Our methodology vs Miura

etal. (2010)

1.55 1.15

Our methodology vs KK

(2014)

1.99 1.18

Panel C: Vuong (1989) test statistic for diﬀerences in information content

Private Firms Model Public Firms Model

Our methodology vs Miura

etal. (2010)

5.72 5.24

Our methodology vs KK

(2014)

8.32 5.50

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Estimating corporate bankruptcy forecasting models by…

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Table 10 Forecasting bankruptcy two years ahead; Our methodology vs alternative methodologies

This table reports out-of-sample performance in terms of discriminatory power, information content and economic impact, using our method to maximize AUROC on a neu-

ral network versus alternative methods to maximize AUROC as proposed by Miura etal. (2010) and Kraus and Kuchenhoﬀ (2014), denoted as KK (2014), when forecasting

bankruptcy two years ahead. Panel A reports the performance when ﬁnancial variables are used in the models (private ﬁrms models) and when ﬁnancial and market variables

are used in the models (public ﬁrms models). Panel B reports DeLong (1988) test statistics for diﬀerences in AUROCs between the models of interest whereas panel C reports

Vuong (1989) test statistics for diﬀerences information content between the models of interest

Panel A: Performance, out-of-sample, 2007–2015

Private Firms Model Public Firms Model

Model AUROC Info. Cont Econ. Ben AUROC Info. Cont Econ. Ben

Our methodology

Neural Network 0.8678 16.76% 1.42% 0.8864 18.92% 1.47%

Alternative methodologies

Miura etal. (2010) 0.8479 10.97% 1.39% 0.8605 12.85% 0.97%

KK (2014) 0.8488 5.08% 0.87% 0.8595 7.47% 0.84%

Panel B: DeLong (1988) test statistic for diﬀerences in AUROC

Private Firms Model Public Firms Model

Our methodology vs Miura

etal. (2010)

1.83 2.90

Our methodology vs KK

(2014)

1.75 3.01

Panel C: Vuong (1989) test statistic for diﬀerences in information content

Private Firms Model Public Firms Model

Our methodology vs Miura

etal. (2010)

3.63 5.02

Our methodology vs KK

(2014)

6.69 7.27

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C.Charalambous et al.

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Table 11 Forecasting ﬁnancial distress; Our methodology vs alternative methodologies

This table reports out-of-sample performance in terms of discriminatory power, information content and economic impact, using our method to maximize AUROC on a neu-

ral network versus alternative methods to maximize AUROC as proposed by Miura etal. (2010) and Kraus and Kuchenhoﬀ (2014), denoted as KK (2014), when forecasting

ﬁnancial distress. Panel A reports the performance when ﬁnancial variables are used in the models (private ﬁrms models) and when ﬁnancial and market variables are used

in the models (public ﬁrms models). Panel B reports DeLong (1988) test statistics for diﬀerences in AUROCs between the models of interest whereas panel C reports Vuong

(1989) test statistics for diﬀerences information content between the models of interest

Panel A: Performance, out-of-sample, 2007–2015

Private Firms Model Public Firms Model

Model AUROC Info. Cont Econ. Ben AUROC Info. Cont Econ. Ben

Our methodology

Neural Network 0.9175 32.82% 1.12% 0.9000 28.21% 1.07%

Alternative methodologies

Miura etal. (2010) 0.8982 3.97% 0.70% 0.8825 10.53% 0.71%

KK (2014) 0.8964 7.75% 0.47% 0.8827 9.89% 0.71%

Panel B: DeLong (1988)

test statistic for diﬀer-

ences in AUROC

Private Firms Model Public Firms Model

Our methodology vs

Miura etal. (2010)

6.14 6.33

Our methodology vs

KK (2014)

7.69 6.19

Panel C: Vuong (1989)

test statistic for diﬀer-

ences in information

content

Private Firms Model Public Firms Model

Our methodology vs

Miura etal. (2010)

15.55 13.93

Our methodology vs

KK (2014)

14.39 12.65

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325

Estimating corporate bankruptcy forecasting models by…

1 3

the objective function is continuous, which is not the case for Eq.(1). Also, this approach

while is easy to implement, ignores information provided by the gradient which could

increase the accuracy of the coeﬃcients after the optimization process and thus we believe

that using speciﬁcations with diﬀerentiable functions is a better choice.13 Table9 presents

the out-of-sample results by forecasting bankruptcy one year ahead (we use our neural net-

work model which consistently outperformed the competing models).

Overall, we ﬁnd that the neural network trained with our method has higher discrimina-

tory power than the other models but the diﬀerences is AUROCs are not statistically sig-

niﬁcant. Despite that, the neural network provides signiﬁcantly more information relative

to Miura etal. (2010) and KK (2014) which is also economically beneﬁcial.

When we increase the complexity of the problem, however, by forecasting bankruptcy

two years ahead and forecasting ﬁnancial distress, our neural network model consistently

outperforms the competing methods and diﬀerences in performance are statistically signiﬁ-

cant. Results regarding forecasting bankruptcy two years ahead, reported in Table10, show

that the neural network model more accurately discriminates bankrupt from healthy ﬁrms

Table 12 Forecasting bankruptcy and ﬁnancial distress using quarterly data

This table reports AUROC results for logistic and neural network models, when their coeﬃcients are esti-

mated by maximizing the AUROC (reported in the second and third columns respectively) and when their

coeﬃcients are estimated by maximizing the log-likelihood function (denoted as LL and reported in the

fourth and ﬁfth columns respectively)

The models are trained in the period 1990–2006 and the table reports the prediction performance, in terms

of AUROC, in the out-of-sample period when predicting bankruptcy one quarter ahead (panel A), four

quarters ahead (panel B), eight quarters ahead (panel C) but also ﬁnancial distress (panel D). In the models,

we incorporate either ﬁnancial variables (Private Firms Model) or ﬁnancial and market data (Public Firms

Model) constructed using quarterly data

Maximizing AUROC Maximizing LL

Logistic Neural Net Logistic Neural Net

Panel A: 1 quarter ahead

Private Firm Model 0.9255 0.9201 0.9150 0.9111

Public Firm Model 0.9589 0.9611 0.9513 0.9472

Panel B: 4 quarters ahead

Private Firm Model 0.9040 0.9186 0.8927 0.9050

Public Firm Model 0.9444 0.9456 0.9431 0.9300

Panel C: 8 quarters ahead

Private Firm Model 0.8128 0.8280 0.7825 0.7918

Public Firm Model 0.8485 0.8523 0.8398 0.8258

Panel D: Financial Distress Case

Private Firm Model 0.8648 0.8871 0.8558 0.8797

Public Firm Model 0.8566 0.8733 0.8544 0.8568

13 We use the optimization toolbox in MATLAB. For Kraus etal. (2014) we use the fminsearch command

while for Miura etal. (2010) we use the fminunc command with the trust-region algorithm.

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326

C.Charalambous et al.

1 3

two years prior to bankruptcy according to AUROCs (0.8678 vs 0.8479 for the Miura etal.

2010 and 0.8488 for KK 2014 in the private ﬁrm model case and 0.8864 vs 0.8605 and

0.8595 in the public ﬁrm model case). Diﬀerences are statistically signiﬁcant at the 10%

level using the private ﬁrms model and at the 1% level using the public ﬁrms model.

In the remaining tests, we document that the neural network provides signiﬁcantly more

information about future bankruptcies than the Miura etal. (2010) and KK (2014) methods

and the better performance is associated with higher economic beneﬁts for the bank which

uses our proposed method.

Finally, in Table11 we report our results when forecasting ﬁnancial distress. Consistent

with previous results, our method achieves signiﬁcantly higher discriminatory power than

the other methods (0.9175 vs 0.8982 for the Miura etal. 2010 and 0.8964 for KK 2014

in the private ﬁrm model case and 0.9000 vs 0.8825 and 0.8827 in the public ﬁrm model

case). Diﬀerences are statistically signiﬁcant at the 1% level.

In the remaining tests we conﬁrm previous ﬁndings; our method provides signiﬁ-

cantly more information and the better performance overall is economically beneﬁcial

for the bank using our method as opposed to the competition.

4.7 Forecasting using quarterly data

Public ﬁrms issue ﬁnancial information each quarter, thus investors can update their risk

assessments more frequently as new information becomes available. In this section, we

perform the same analysis as before but this time, the input variables to the models are

constructed using quarterly data and we make predictions one, four and eight quarters

ahead. Overall, the AUROC results reported in Table12, are qualitatively similar with

the results reported in the case where yearly data are used. More speciﬁcally, training

the bankruptcy models with our proposed method, in all cases, improves the out-of-

sample performance in terms of discriminatory power as opposed to training the models

to maximize the traditional log-likelihood function.

5 Conclusions

The goal of this paper is to propose an alternative method to estimate the coeﬃcients

of bankruptcy forecasting models and speciﬁcally logistic and neural network models

which are the most popular bankruptcy models used in prior research. In particular, we

suggest those interested in forecasting bankruptcy, to obtain the coeﬃcients by maxi-

mizing the discriminatory power as measured by the Area Under ROC curve (AUROC).

In this study, a method is introduced and we highlight the beneﬁts arising, out-of-sam-

ple, by using models which are trained to maximize AUROC over models trained with

traditional methods, such as optimizing the log-likelihood function. Overall, we ﬁnd that

models trained to maximize AUROC outperform traditional methods, out-of-sample, in

terms of discriminatory power, information content and economic impact. Our results

hold when we test the method in diﬀerent settings, such as forecasting bankruptcy one

year ahead which is the most common horizon, forecasting bankruptcy two years ahead

and forecasting ﬁnancial distress which make forecasts more diﬃcult (using yearly and

quarterly data). Thus, forecasting bankruptcy accurately well in advance, which would

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327

Estimating corporate bankruptcy forecasting models by…

1 3

be beneﬁcial for the ﬁrm to take corrective measures, requires a more sophisticated esti-

mation method, such as maximizing the AUROC function by using our method. From

all models, the neural network trained with our method is the best performing one.

Next, we compare our method with alternative methods proposed in the literature

and we provide both theoretical as well as empirical justiﬁcations as to why our method

should be preferred. As expected, the results are more pronounced when we increase the

forecasting diﬃculty, such as forecasting ﬁnancial distress.

Our results have implications to the way bankruptcy forecasting is performed. Our

proposed estimation approach provides, to those interested to forecast bankruptcy, a sig-

niﬁcant advancement over traditional methods that can be used by logistic and neural

networks for better bankruptcy analysis and possibly can be extended to areas such as in

credit risk analysis.

Declarations

Conﬂict of interest he authors declare no conﬂicts of interest.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,

which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long

as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-

mons licence, and indicate if changes were made. The images or other third party material in this article

are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the

material. If material is not included in the article’s Creative Commons licence and your intended use is not

permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly

from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Applicability of bankruptcy prediction models in the western Hungarian food retail sectorCsődelőrejelző modellek alkalmazhatósága a nyugatmagyarországi élelmiszer-kiskereskedelmi szektorbanЗастосовність моделей прогнозування банкрутства в секторі роздрібної продовольчої продукції західної Угорщини

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