model insolventa de tradus

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IMA Journal of Management Mathematics(2007)18, 269295

doi:10.1093/imaman/dpl017

Advance Access publication on December 27, 2006

Early discovery of individual firm insolvency

AMY ( WENXUAN) DIN G

Department of Information and Decision Sciences (M/C 294), University of Illinois, 601

South Morgan Street, Chicago, IL 60607, USA

[Received on 11 January 2006; accepted on 21 November 2006]

This paper proposes a new methodology for the early discovery of individual firm insolvency without

employing any other firms data. The proposed individual-level model can be applied to different firms,

regardless of industry type or asset size, and thereby overcomes the sample selection problem commonly

found in aggregate-level prediction models. Unlike many previous studies, which assume that the dis-

tributions of variables involved do not change over time and that the variables follow a single known

distribution, the proposed model can capture each individual firms potential multiple data-generatingprocesses and determine the actual distributions exhibited in its own data. Thus, it captures each individ-

ual firms intrinsic heterogeneity. An empirical study illustrates the greater predictive power of this model

compared with the current conventional methods. Specifically, the predictive accuracy of the proposed

model is 92.65% and 77.45% for 2 and 5 years prior to actual bankruptcy, respectively. Moreover, the

proposed model is adaptive and simple to implement.

Keywords: risk analysis; financial modelling; adaptive learning; business bankruptcy.

1. Introduction

More and more companies have filed for bankruptcy recently (Redmond, 2005), making predicting a

firms poor performance and providing early warning of the potential risk for bankruptcy increasingly

important to lenders, investors and managers. Since conventional bankruptcy prediction models have

failed in many bankruptcy cases (Altman, 2003; Balcaen & Ooghe, 2006), a new methodology is pro-

posed. A review of research on predicting business failure in the past 35 years suggests that most existing

models are based on cross-sectional statistical methods and suffer from two major flaws: potential sam-

ple selection bias and problems with data aggregation (Balcaen & Ooghe, 2006). Because prior models

are aggregate-level models, they require a large cross-sectional sample of firms to provide sufficient data

for model estimation (also called model training in data mining and machine learning) and generally

assume that the variables of interest follow a known, single distribution that does not change over time.

According to this paradigm, the obtained parameters, threshold and/or resulting estimated model are

applied to holdout samples to predict a firms bankruptcy risk.

However, these models fail to take into account that even if two firms belong to the same industry

type and have similar asset sizes, each has its own characteristics and the selected variables of interest

pertaining to each individual firm may follow different probability distributions. Moreover, a firms

risk of insolvency changes over time due to changes in the competitive nature of the market, corporate

strategy and/or technology adoptions. Therefore, the values of the parameters estimated from a group

of sample firms may not account for each firms intrinsic heterogeneity accurately, even if the firms

observed characteristics, such as financial ratios, are incorporated. In addition, almost all aggregate-level

Email: [email protected]

c The authors 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.


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models use a single observation (e.g. one annual account) for each firm in the estimation samples and

ignore past information about each firms performance. This situation indicates that a firms time-series

performance behaviour is also ignored (Dirickx & van Landeghem, 1994; Kahya & Theodossiou, 1999;

Theodossiou, 1993). The classification results of aggregate-level models therefore largely depend on

sample selection, as well as on the choice of the various assumed probability distributions used in anestimation (e.g. Balcaen & Ooghe, 2006; Benos & Papanastasopoulos, 2005; Cookeet al., 1988; Simon,

1997; Zmijewski, 1984).

To overcome the sample selection problem and capture each individual firms intrinsic heterogeneity

and time-series behaviour, I develop a new model-building methodology that provides a more accurate

early prediction of a firms insolvency. The proposed method uses only a single firms time-series data

to compute the likelihood of its insolvency, without considering any other firms information, which

avoids the sample selection problem completely. In addition, instead of assuming known probability

distributions of the variables of interest (BarNiv & McDonald, 1992; Beneish, 1997; Campbell, 1996;

Merton, 1974; Shumway, 2001; Zmijewski, 1984), the proposed model estimates them on the basis

of each individual firms own data. That is, the model identifies an unknown number of probability

distributions of the individual firms profitability ratio through adaptive learning. Because each firms

earning trend and profitability may change over time due to constantly changing markets, different firms

may have different distributions of profitability, and the underlying distribution exhibited in a firms

own data may not follow a single distribution but rather exhibit multiple modes. Thus, the proposed

model implements a real-time estimation of each individual firms possible multiple data-generating

processes to determine the actual distributions exhibited in its data. In the meantime, the model updates

the underlying estimated distribution with the firms new data in each time period and thus captures

the firms past and current behaviour. Finally, unlike most current models that use extensive sets of

financial ratios as part of their predictor variables, the proposed model only uses two pieces of critical

information from the individual firm as model variablesthe firms operating earnings and interest

payments for debtto make a prediction.

Managerially, the proposed model is simple to implement and requires only a single firms informa-

tion, which is easily available, unlike Merton models, e.g. that suffer implementation difficulty due to

the invisibility of the firms value process and computational complexity (Das & Sundaram, 2000). Eco-

nomically, the cost of collecting samples for estimation of aggregate-level models also can be avoided

with the proposed model.

To validate my model, I conduct an empirical study in which I apply the model to a real-world

data set and compare it with three conventional aggregate-level models. The empirical results show that

the proposed model predicts insolvency 25 years prior to actual bankruptcy with high accuracy (i.e.

92.65% predictive accuracy for 2 years ahead and 77.45% accuracy for 5 years in advance), with a prob-

ability cut-off point of 0.5. For the comparisons with conventional aggregate-level models, I select three

benchmark models: the Z-score model with a multivariate discriminant analysis (Altman, 1968, 2000;

Altman et al., 1977), the well-known machine learning decision tree algorithm C4.5 (Quinlan, 1993)

and the conditional probability models such as the probit model (BarNiv & McDonald, 1992; Beneish,1997; Campbell, 1996; Zmijewski, 1984). The Z-score model is widely used by both academics and

practitioners, and many other studies have treated it as a benchmark for comparison with a new and

improved model (Altman & Narayanan, 1997; Balcaen & Ooghe, 2006; Holmen, 1988). I choose C4.5

because it is popularly used in machine learning and data mining literature for classification tasks (Quin-

lan, 1993). Recently, academic studies also have applied conditional probability models (e.g. probit/logit

analyses) and other statistical models, such as hazard models, to forecast bankruptcy (Dimitras et al.,

1996; Jones, 1987; Shumway, 2001; Zavgren, 1983; Zmijewski, 1984). However, Shumway (2001)


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EARLY DISCOVERY OF INDIVIDUAL FIRM INSOLVENCY 271

demonstrates that the hazard model performs better than the Z-score model but does not outperform a

conditional probability model (probit/logit model). Therefore, I use the probit model as another bench-

mark for comparison in this study.

The comparison results show that the overall predictive accuracy 1 year prior to actual bankruptcy

for the proposed model is 95.10%, which is much better than each of the three benchmark approaches(Z-score: 70.59%, C4.5: 78.85%, probit: 81.73%) with a probability cut-off point of 0.5 for both the

proposed model and the probit model. The proposed model signals a firms vulnerability to insolvency

as early as 5 years prior to actual bankruptcy with 77.45% accuracy (Z-score: 59.80%, C4.5: 66.35%,

probit: 67.31%). The prediction results generated by the proposed model remain similar if different

classification cut off points are used. Therefore, the proposed method, which uses only an individ-

ual firms data, can provide a more accurate signal of the firms risk of potential bankruptcy far in

advance.

The remainder of this paper is organized as follows: Section 2 provides a brief review of the literature

on bankruptcy prediction models. Section 3 describes the model formulation and learning procedure.

I describe the empirical performance of the proposed method in Section 4 and offer some discussion of

the managerial implications in Section 5.

2. Literature review

The management and finance literature include many models for predicting a firms insolvency, which

can be grouped into two broad categories according to the analysis they adopt (Benos &

Papanastasopoulos, 2005). The first category adopts a fundamental analysis with econometric methods

of model estimation. Examples in this category include Z-score models that use multivariate discrimi-

nant analysis, conditional probability models such as logit/probit, classification models using decision

trees, models that employ Bayesian reasoning and hazard models using survival functions and linear

regression (e.g. Altman, 1983, 2000; BarNiv & McDonald, 1992; Breiman et al., 1984; Diamond,

1976; Eidleman, 1995; Jarrow & Turnbull 1995; Sarkar & Sriram, 2001; Shumway, 2001). The ob-

ject of these models is to find and estimate selected financial ratios/variables that are important for

assessing or forecasting a firms potential risk of insolvency from the cross-sectional data (i.e. from

a group of similar firms). The working procedure of these models can be summarized as a three-step

framework: Step 1 collects two sets of sample firms, of which set one is a group of financially healthy

firms and set two is a group of failed firms matched to set one along dimensions such as asset size,

growth and industry type. Step 2 involves model estimation through statistical regression or very so-

phisticated econometric techniques to obtain coefficients or parameters (i.e. weights attached to the

selected independent variables). Before the estimation, important factors that may affect the firms

performance, such as the firms financial ratios, are identified to indicate significant differences be-

tween the two sets of sample firms. A model incorporating these factors as independent variables is

then estimated on the basis of the sample data of the two groups, and the parameters of the model

are obtained after the estimation. Step 3 uses the estimated model and obtained parameters to clas-

sify the firms in the test sample as bankrupt or not. The features of this category are that the pa-

rameters of interest are estimated and determined by aggregate cross-sectional data, and that during

model estimation in Step 2, the variables of interest are assumed to follow a single known distribu-

tion. For example, in the popular Z-score model, the variables in every group are assumed to follow a

multivariate normal distribution, and the covariance matrices are assumed to be equal for every prior

defined group. Even though empirical studies show that defaulted firms in particular violate the nor-

mality assumption (Benos & Papanastasopoulos, 2005), in the past 30 years, Z-score and conditional


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272 WX. DING

probability models, such as logit/probit, have dominated the literature on business bankruptcy prediction

(Balcaen & Ooghe, 2006). Recently, Shumway (2001) proposed a hazard model to capture a firms

past performance using its time-series data. His research indicates that the hazard model performs better

than the Z-score model but does not outperform the logit model in terms of insolvency prediction.

The second category of models is referred to as Merton, or structural, models, which adopt con-tingency claim analyses. The object of Merton models is to view corporate liabilities as contingent

claims on the assets of the firms (Black & Cox, 1976; Black & Scholes, 1973; Collin-Dufresne &

Goldstein, 2001; Geske, 1977; Longstaff & Schwartz, 1995; Merton, 1974). In a typical Merton model,

the determination of whether a firm will default depends on the values of two variables: the firms

forward asset, an option value of the assets of the firm at time T, and the firms outstanding debt at

time T, which is an estimated face value of a single debt payment at time T. The firm defaults if

the value of the firms forward assets is less than the promised debt repayment at time T. To use the

Merton models, researchers must determine the current market value of the firms assets, the volatil-

ity of the assets, the firms forward assets, its outstanding debt and debt maturity. Due to the volatility

of options, the market value and volatility of the firm are estimated from the stocks market value,

volatility and the book value of liabilities. Obtaining these values requires statistical estimation of a

group of sample firms, called reference entities (Black & Cox, 1976; Collin-Dufresne & Goldstein,

2001; Geske, 1977; Hullet al., 2004; Longstaff & Schwartz, 1995). From this point of view, the Merton

models are also aggregate-level models. Similarly, they assume that the variable of interest (i.e. the value

of the firm, projected to a given future date T) follows a single known distribution of a lognormal

diffusion process with constant volatility during the estimation (Merton, 1974). According to the lit-

erature, Merton models suffer important practical weaknesses (Das & Sundaram, 2000); specifically,

they are difficult to implement because the firms value process is unobservable and computationally

complex.

Methodologically, models in these two categories can all be considered aggregate-level models and

share two common potential drawbacks, though they adopt different analyses and select different vari-

ables for model building. First, the parameters of interest in the models are statistically estimated by

cross-sectional data and are not specific to an individual firm. Therefore, the prediction decision for

a firm is dominated exogenously by parameters obtained from a group of firms (e.g. Grice & In-

gram, 2001; Platt & Platt, 1991; Sarkar & Sriram, 2001). Simon (1997) points out that such data

are too aggregated to reveal much about each individual firms own characteristics. In addition, the

estimation samples consist of a group of firms in which each firm is represented by a single observa-

tion, which alone is unable to account for the firms time-series performance behaviour because insol-

vency takes time and represents a process (Balcaen & Ooghe, 2006). Heckman (1979) and Zmijewski

(1984) both examine the effects of sample selection on model estimation and show that aggregate-

level models are very sensitive to sample selection. For example, if firms in the estimation sample set

do not represent the same type of industries as those in the test sample set, aggregate-level models

are inappropriate, because the prediction results may not be accurate. Second, these types of mod-

els subjectively assume that the distributions of the variables involved in the model do not changeover time but remain the same for future samples. Thus, the known probability distributions of the

model variables get applied during model estimation, which may be too strong and incorrect, as I show

subsequently.

This article presents a new methodology that can resolve these two issues. I show that this method

provides a more accurate early prediction of insolvency than the current aggregate-level models, and,

even more important, it can adapt to each individual firm by incorporating the individual firms time-

series performance to capture its own characteristics.


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3. Model formulation

To construct the model, I follow the definition of financial distress offered by Wruck (1990) and that

provided by corporate finance theory (Ross et al., 2004), according to which financial distress1 is a

situation in which (1) a firms operating cash flow is insufficient to meet its current obligations (e.g.

trade credits, interest expenses) or can meet them only with difficulty or (2) a firm has a negative networth (i.e. the value of assets is less than the values of debts). A recent report by Clarke & Dean (2001)

and Clarke et al. (2003) on corporate collapse analysis concludes that the ultimate cause of a firms

failure seems to be a lack of cash to pay debts on time, which seems like a good starting place for senior

managers who need to assess the health of their own firms. Lawson (2002) summarizes the suggestions

of various insolvency experts and senior managers, and notes out that cash flow is of critical importance

and that a firms on-time payment ability is key to its ability to remain solvent. If a firms periodical (e.g.

monthly, quarterly, yearly) earnings before taxes are insufficient to pay basic debts in a timely manner,

the firm is in trouble and its managers may need to start thinking of rescue plans. Glen (2004) also

reviews empirical evidence regarding a sample of more than 6000 real sector firms in 41 countries and

their ability to service debt for the period 19942001. His results show that cash flow must be available

to make interest payments; otherwise, debt pushes the firm from vulnerability to insolvency. Previousresearch also indicates that the only statistically significant variables in predicting insolvency are those

that measure the firms profitability and leverage (Shumway, 2001). Therefore, in the proposed model,

instead of assuming the availability of complete information about the individual firms operation, I use

only two pieces of information as model variables: operating earnings and interest payments for debt.

Instead of assuming known probability distributions of the firms earning trends or other factors, the

model estimates them. Instead of considering all financial ratios pertaining to the firms performance,

such as working capital, total assets, retained earnings, market value equity, book value, sale level and

so forth, I am only interested in whether the firm can make its interest payments on time.

Let a firms total assets be TAj and its operating earnings after depreciation but before interest

expenses and income taxes be OEj at time j (where j = 1, 2, . . . ,J. The unit j can be a month,quarter or year). The firms profitability rate j is defined as j= OEj /TAj , which measures the trueproductivity of the firms assets, independent of any tax and interest expenses. Suppose each time the

firm has to pay income taxes and interest payments for securing all its short- and long-term debts or

loans (if the firm has debts), the earnings, Rj , after these payments in each time period will be

Rj= OEj Taxj IntExpj , (1)

which can be rewritten as Rj+ Taxj= OEj IntExpj , where Taxj denotes the income taxes the firmneeds to pay at time j and IntExpj represents the timely required expenses that the firm must pay to

secure all its different short- and long-term debts. If the firm is a financial services institution, IntExp jcould be the periodic (e.g. monthly) interest expenses that the firm must pay on customers deposits,

its own short- and long-term debts and any other borrowings. Obviously, the income taxes and timely

required expenses are nonnegative, such that Taxj 0 and IntExpj 0. To meet timely payments, the

firm must have enough operating earnings to pay them, i.e. OEj IntExpj 0.When OEj IntExpj > 0, the firm has the capacity to meet its timely required payments. If OEj

IntExpj= 0, the firms earnings at time j are just enough to cover the timely required payments for

1According to bankruptcy literature (Zmijewski, 1984), financial distress also may be defined as the act of filing a petition for

bankruptcy such that a firm is identified as bankrupt if it has filed a bankruptcy petition and nonbankrupt if it has not. Therefore,

I use the terms financially distress, insolvency and bankruptcy interchangeably.


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274 WX. DING

securing the debts, and Rj+ Taxj= 0. Because the income taxes are nonnegative (i.e. Tax j 0), thisscenario implies nonpositive earnings after all payments (i.e. Rj 0); so the firms earnings surplus at

time j equals 0.

However, when OEj IntExpj < 0, then Rj+ Taxj < 0, which implies that Rj < 0 (becauseTaxj 0). In other words, the firm loses money in its business. When a firm experiences consistentoperating losses, i.e. OEj IntExpj < 0 occurs continually (e.g. several months or quarters), the firmis unlikely to have the capacity to meet timely interest payments, not including the payment of income

taxes plus the repayment of principals. Therefore, OE j IntExpj < 0 may signal that a firm is likelyto be financially distressed, and the firms operating income is insufficient to satisfy current obligations

such as interest expenses.

Because a firms ultimate existence is based on the earning power of its assets, the firms profitability

rate, j (= OEj /TAj , for j= 1, 2, . . . ,J), appears instead of OEj in the model to capture the trueproductivity of the firms assets, independent of any taxes and interest expenses. Previous studies have

shown that profitability rate, j , can be very helpful in assessing the firms performance (Altman, 2000).

Hence, I rewrite the signalling indicator OEj IntExpj 0 asj IntExpj

TAj 0(for j= 1, 2, . . . ,J).

Because the firms profitability ratej changes over time, a firm may face a risk that the value ofj

will fall below the value of IntExpj

TAj, suggesting that the firm could be financially distressed. Therefore,

one must calculate the probability such that the value ofj is less than the value of IntExpj

TAjat time

j+ 1, given the firms previous data records about j and IntExpjTAj for j= 1, 2, . . . ,J. Ifj follows adistribution f(), the risk of financial distress a firm may face at time j + 1 can be predicted using (2):

P

< j 0.0045 class 0 [92.2%]


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Default class: 1 (bankrupt)

Evaluation on training data (100 items):

Rule Size Error Used Wrong Advantage- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1 1 3.2% 43 0 (0.0%) 0 (0|0) 12 1 3.6% 4 0 (0.0%) 0 (0|0) 15 2 5.4% 48 1 (2.1%) 17 (17|0) 04 3 7.8% 2 0 (0.0%) 2 (2|0) 0

Tested training data (100), errors 2 (2.0%)


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B.2 Calibration performance for the probit model

The size of the calibration sample is 100.

Prediction Frequency

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Correct 96Actually Bankrupt

but predicted nonbankrupt 3

Actually Healthy

but predicted bankrupt 1

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


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