on bankruptcy information systems

European Journal of Operational Research 56 (1992) 67-79 67 North-Holland

Theory and Methodology

On bankruptcy information systems *

D a n i e l E . O ' L e a r y

Graduate School of Business, Uni~'ersity of Southern California, Los Angeles, California 90089-1421, USA

Received December 1989; revised March 1990

Abstract: Bankruptcy prediction information systems provide a noisy signal. Thus, these information systems are not perfectly reliable. This paper investigates a means for including the reliability of the signal into a basic bankruptcy prediction model. Then the implications of integrating reliability into the model are explored. The results indicate that the model is highly sensitive to reliability. As a result, the bankruptcy decision models should include information system reliability or else decision outcomes can be suboptimal (for example, loans made when they should not be or not made when they should be). The results also indicate that some heuristics that may be used by decision makers to account for reliability are likely to lead to quite misleading results.

Keywords: Bankrupty prediction, Bayesian models

1. Introduction

The bank loan process is characterized by determining if a loan should be given, based in part, on an evaluation of whether the firm will become bankrupt or not. Evaluations of bankruptcy generally are done using financial statement analysis (Beaver [1966]), the output of statistical models (Ohlson [1980]), expert opinion (Elmer et al. [1988]) or some combination. Unfortunately, each of these types of 'bankruptcy information systems' has limited reliability.

While interpreting financial information and model output financial analysts must cope with their own imperfect recognition and detection capabilities. Information on liquidity and information on debt may be interpreted to yield con- flicting conclusions as to the likelihood of

* The author would like to acknowledge the comments of the anonymous referees on earlier versions of this paper.

bankruptcy. To the extent that the signals are divergent, there is likely to be less reliability in the report.

A number of statistical models of the bankruptcy process have been developed (e.g., Zavgren [1983]). By the very nature of statistical models there is no such thing as the 'perfect ' prediction model. Thus, there is a lack of perfect reliability in any statistical model that may be chosen.

In addition, neither the financial analyst or the statistical model directly observe the develop- ment of the financial information, but instead depend on furnished financial reports, from par- ties that may try to bias the loan decision. Since accounting systems are not without error the data that is used to analyze the bankruptcy possibili- ties are not perfectly reliable.

The extent of a lack of confidence in the reliability of the report of the likelihood of bankruptcy can impact the behavior of bank management. Unfortunately, although bankruptcy in-

0377-2217/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved

68 D.E. O'Leary / On bankruptcy information systems

formation systems are not perfectly reliable, decision models and studies of bankruptcy do not take into account the inferential impact of the lack of reliability. Thus, bank management that recognizes the lack of reliability is likely to use some heuristic adjustment of bankruptcy information. However, such heuristic approaches are found in this paper to yield substantially non-optimal solutions.

Accordingly, the purpose of this paper is, first, to introduce the reliability of the bankruptcy information system into bankruptcy analysis, second, to determine the impact of the reliability of the system on the optimal strategies for loan evaluation and, third, to determine the ability of certain heuristic strategies to incorporate reliability.

This paper

This paper proceeds as follows. Section 2 dis- cusses some general issues of model ing bankruptcy, including the growing importance of bankruptcy decision making and the impact of reliability on those decisions.

Section 3 summarizes the basic decision theory model on which this paper is based. Section 4 investigates the Type I and Type II error struc- ture of the model, defining the nature of a Type I and Type II error bound; a characteristic used later in the results on reliability.

Section 5 introduces reliability into the model of Section 3. Section 6 investigates when that resulting model is independent of reliability. Sec- tions 7 and 8 study the impact of the introduction of reliability on loan granting behavior. Section 9 investigates the impact of using heuristics rather than the analytic appoach.

The results in Sections 5 through 9 are summarized in Section 10, where the relationship to the decision theory model, the reliability decision theory approach and some implementation issues are discussed. Finally, Section 11 presents a brief summary of the paper and some extensions.

2. Modeling bankruptcy decisions

Modeling bankruptcy decisions has generated many different operations research, statistical and artificial intelligence models. Financial planning

using mathematical programming has been discussed by McBride et al. [1990]. A wide range of statistical approaches have been developed by a number of authors, including Altman [1968], Alt- man et al. [1977] and Frydman et al. [1985]. While more recently, artificial intelligence/computer science approaches have been generated by Shaw and Gentry [1988], Duchessi et al. [1988] and Elmer [1988].

Importance of bankruptcy decision making

Bankruptcy decision making has long been of interest to virtually anyone granting credit to any other firm or individual. Substantial literatures have been developed addressing this set of issues (e.g., Zavgren [1983], Frydman et al. [1985] and Duchessi et al. [1988]).

However, as there becomes more of a 'world' economy there also is increasing international concern with such decisions. For example, the 'European Community 1992' (EC-1992) issues are designed to eliminate many of the protectionist restrictions so that goods can flow from country to country in Europe (e.g., Magee [1989] and Friberg [1989]). Although there are many barriers to be encountered before its full implementation, EC-1992 is still expected to have a substantial impact on how business is done in and with Europe. One of the anticipated changes is that there will be structural changes in some busi- nesses, because of greater competition. As a result, some companies will be bankrupt or discon- tinue operations. Because of fewer restrictions, it also may mean a more open flow of capital between countries. With this flow of capital may come more competition for the same investments and alternative approaches to decision making for those alternative investments. Thus, there likely will be additional concern over loans, in terms of bankruptcy decisions, and different approaches to determining if a firm is expected to become bankrupt (i.e., bankruptcy decision making). As a result, there is now international concern over the processes of bankruptcy decision making.

Reliability

Most bankruptcy models use data generated by or for the particular firm, such as accounting

D.E. O'Leary / On bankruptcy inJormation systems 69

data. Typically, the models use this data in the form of ratios such as the current ratio (current asse t s /cur ren t liabilities), in order to character- ize financial aspects such as profitability, liquidity, leverage and growth. Ratio accounting data may be useful in bankruptcy prediction because ratios are typically used in debt covenants in order to restrict management ' s efforts. By not maintaining appropriate accounting ratios management can put the firm in default. Simply vio- lating the ratios may not automatically make the firm bankrupt, instead, such ratios can be indica- tive of performance that will lead to bankruptcy. At any rate, the heavy use of accounting data in such models brings out the importance of reliability of the information system, since the models are dependent on the quality of the data.

Because of these concerns there are some po- tential problems with the reliability of the accounting numbers. Accounting systems are an area where there is substantial opportunity for errors to cascade. An error made in recording a transaction may lead to errors in the particular account and in errors for the particular day. Those errors can combine with other errors.

In addition, statistical models are never pre- fectly reliable. The quality of the models is measured by such devices as 'correlation' or 't-statis- tics.' In a similar manner, certain artificial intelli- g e n c e / c o m p u t e r science approaches, such as 'inductive learning' (e.g., Shaw and Gentry [1988]) are measurably unreliable.

3. Bankruptcy information systems and a basic model of bankruptcy investigation

This paper assumes that before a loan is made, the decision maker predicts whether or not the firm will go bankrupt. If the loan is not issued because the model predicts bankruptcy and the firm does not go bankrupt then there can be a Type I error. However, if the bank makes a loan because it predicted nonbankruptcy and the firm goes bankrupt, there can be a Type II error. Thus, bank management must find some 'middle ground' between the risks and the corresponding costs associated with two types of errors.

This paper couches its analysis in a classic model of bankruptcy investigation that is based on an economic trade-off of Type I and II errors. The model assumes that a firm that receives the

Table 1

Costs in the bankruptcy problem

B = Bankrupt NB = Not Bankrupt

Predict B 0 C

Predict NB L - !

loan either goes bankrupt (B) or does not go bankrupt (NB). The model assumes that either it is predicted that the firm will go bankrupt (PB) or that the firm will not go bankrupt (PNB).

If the bank makes a loan to a firm and the firm does not go bankrupt then it incurs a benefit of receiving interest ( I ) on the loan. If the bank makes the loan and the firm goes bankrupt (B) then the bank incurs an expected loss of the amount of the loan or less (L). However, if the bank does not make the loan and the firm does not go bankrupt then the bank incurs a opportunity cost (C). Without loss of generality, it is assumed that I >_ C. These costs are summarized in Table 1.

Let P be the probability of bankruptcy and 1 - P the probability of not being bankrupt. The bank is interested in making loans as long as the benefits of a making a loan are greater than or equal to the costs. Thus, a critical point occurs when ( l - P ) * I > _ ( 1 - P ) * C + P * L , so the critical probability at which the bank will make the loan is Pcr <- ( I - C ) / ( L + I - C).

Let Pr(a) be the probability of a and let the Pr(a and b) be represented as Pr(a, b). Let P r ( a l b ) be the probability of a given b. Let z refer to the prediction made using a bankruptcy information system that is perfectly reliable.

Assuming that banks predict whether or not the firm requesting the loan will go bankrupt, the real interest is not in the prior probability, P = Pr(B). Instead, the concern is in the quality of the recommendations made by the information system and the resulting posterior probabilities, Pr(B I z = PB) and Pr(BI z = PNB). If the posterior probability is above the critical point then the manager would not issue the loan.

While commenting on Ohlson [1980], Watts and Zimmerman [1986] use a version of Bayes' Theorem (1) for z = PB or PNB to develop (2)

Pr(BI z)

= [Pr( z IB)Pr(B)]

/ [ P r ( z IB )P r (B) + Pr( z I N B ) P r ( N B ) ] .

(1)


Thus,

P r ( B I z ) = P / [ P + ( 1 - P ) * L ( z ) ] , (2)

where L(z) = Pr(z [NB) /Pr (z ]B). L(z) is called a likelihood ratio. Further,

L(z = PB) = Pr(PB [ N B ) / P r ( P B ]B)

= Pr(PB I N B ) / [ 1 - Pr(PNB IB)]

and thus, L(z = PB) is a function of the Type I and II error distributions.

reliability of the information that relates to the probability of those errors. Accordingly, this is a limited model. Yet it allows us to focus on the impact of reliability. Whereas most approaches to reliability of the information are purely qualita- tive, this model allows us to quantitatively study the characteristics of reliability. As we will see later in the paper, reliability has a major impact on cost benefi t decisions associated with bankruptcy analysis.

Example

In a particular application the situation specific parameters would be used. However, as an example, as noted in Watts and Zimmerman [1986], the data suggests that it is generally 35 times more costly to make a loan that subse- quently defaults than it is to reject a loan that will not default. Assume that I = 2C. Thus, the critical point is 1 /36 or 0.0277. For the economy in general, Pr(B) is approximately 0.02 and Pr(NB) is approximately 0.98 (Watts and Zimmerman [1986]). Thus, using (2), L(PB) must be less than 0.714 in order for Pr(B JPB) to be greater than the critical value.

Information used in bankruptcy decision process

A wide range of additional information may be used in this bankruptcy information system. For example, Shaw and Gentry [1988] delineate economic characteristics (size, market, and diversifi- cation), competitive position in industry, financial characteristics, management (quality, experience and depth), availability of funds, ability to repay loan, value of collaterial and experience at a previous bank. The relative importance of these sources of information may differ from model to model and expert to expert (Shaw and Gentry [1988]). This paper does not assume any particular importance or weight on individual sources of information. However, it is assumed that ulti- mately, this information can be distilled into the cost information in order to establish a critical point and a set of probabilities to establish L(PB).

Limitations of the model

The model developed in this paper is limited to the study of Type I and II errors and the

4. Bankruptcy information systems

In this paper it is assumed that a bankruptcy information system can be characterized by cu- mulative probability distributions for Type I and II errors (for example, Ohlson [1980]). Discrete 'cut-off points', i, define points on the distribu- tion where different probabilities Pr(zi = PB INB) and P r ( z i = P N B I B ) are enumerated, for i between 0 and 1. For notational simplicity, through- out most of the paper the subscript will be dropped, but, reference to a given z implies a prediction that occurs at a cut-off point i. An example of such a system is given in Table 2.

There are at least two desirable characteristics of these systems. First, it is desirable to have a 'Type I error bound' of Pr(z = PB ]B) > Pr(z = PBINB). This bound derives its importance from the desire to have a system that has a higher probability associated with predicting bankruptcy when the firms are bankrupt than with predicting bankruptcy when the firms are not bankrupt. Second, it is desirable to have 'Type II error

Table 2 Example of a bankruptcy information system "

Cut-off Pr(PB t NB), Pr(PNB [B), Pr(Type I error) Pr(Type II error)

0.0095 0.470 0.000 0.02 0.287 0.076 0.04 0.167 0.143 0.06 0.118 0.200 0.08 0.093 0.257 0.10 0.072 0.267 0.20 0.033 0.448 0.30 0.0175 0.486 0.40 0.0107 0.571 0.50 0.0063 0.676

a Source: Ohlson [1980].

D.E. O'Leao' / On bankruptcy information systems 7 l

bound' of Pr(z = PNB I NB) >_ Pr(z = PNB I B). This bound derives its importance from the desire to have a system that has a higher probability associated with predicting not bankrupt when the firms are not bankrupt then with predicting not bankrupt when the firms are bankrupt.

The following theorem indicates that the existence of a Type I error bound implies a Type II error bound, and conversely.

Theorem 1. A bankruptcy information system has a Type I error bound if and only if it has a Type H error bound.

Remark on proofs. All proofs are included in the Appendix.

In terms of (2), Type I and Type II error bound properties derive their importance from the likelihood ratio, L(z) . In the case of a Type I error bound, L(PB) < 1, while, in the case of a Type II error bound L(PNB)>_ 1. The behavior of the likelihood ratio is important in ascertaining the effect of the introduction of reliability into (2). These implications are discussed later in the paper.

5. Reliability of reports of bankruptcy prediction

Unfortunately, the approach developed in (1) and (2) does not take into consideration the reliability of the bankruptcy information system. Mathematically, the distinction between the output of the model used by the bankruptcy information system (the ' report ' ) and the output of a model with perfect fit can be introduced into Pr(BI z) by introducing it into the only compo- nent in (2) that includes the variable z. This is done in Lemma 1. Let z # represent the recom- mendation from the bankruptcy information system that is not perfectly reliability. Let z ' be 'not Z ~ '

T a b l e 3

S y m m e t r i c r e l i ab i l i t y a

z = P B z = P N B

z # = PB r 1 - r

z # - P N B 1 - r r

a z a s s u m e s p e r f e c t re l i ab i l i ty , z # a s s u m e s pos s ib ly less t h a n

p e r f e c t re l i ab i l i ty .

The factors in L(z #) that relate to Pr(z# [ - ) reflect what Schum and DuCharme [3] refer to as the reliability of the reported evidence. If Pr(z # ]NB and z) = 1 and Pr(z # INB and z ' ) = 0 and if P r ( z# lB and z ) = l and P r ( z # l B and z ' ) = 0, then L(z ) = L(z #) and the report would be 100% reliable. The model in Lemma 1 would reduce to (1). However, if Pr(z # tNB and z ' ) and Pr(z # IB and z ' ) are not zero then that indicates that there is nonzero probability that the reported value is dependent on either the state of the process or the actual value of the process or both.

This paper will investigate two special cases of Pr(z#] • ). First, it is assumed that the reported version of z, z #, is not dependent on whether the firm is NB or B and that Pr(z# ] z) is symmetric, i.e., P r ( z # l z ) = P r ( z # ' l z ') = r. Here the system's inaccuracies are independent of the state of the firm. The example above of cascading errors in an accounting system can occur independent of state of the firm. Further, statistical errors also generally will be independent of the firm state. In classic gaming theory reliability is assumed symmetric. Second, a more general case is where we assume that the reported version of z, z #, is not dependent on whether the firm is NB or B and that P r ( z # f z ) is asymmetric, i.e., Pr(z # I z) = r 1 does not equal Pr(z #' ] z ' ) = (1 - r2). These two distributions are summarized in Tables 3 and 4. These two cases are summarized in the following two theorems;

Lemma 1. (Based on Schum and DuCharme [1971])

L ( z #) = [Pr(z # INB and z ) P r ( z INB)

+ P r ( z # INB and z ' ) P r ( z ' INB)]

/ [ P r ( z # IB and z ) P r ( z IB)

+ P r ( z # IB and z ' ) P r ( z ' l B ) . ]

T a b l e 4

A s y m m e t r i c r e l i ab i l i t y a

z = P B z = P N B

Z # r 1 r 2

z # = P N B 1 - r 1 1 - r 2

a z a s s u m e s p e r f e c t r e l i ab i l i ty , Z # a s s u m e s pos s ib ly less t h a n

p e r f e c t r e l i ab i l i ty .


Table 5 Impact of reliability a

r L(PB) Pr(BI z # = PB)

1.0 0.1986 0.0932 0.9 0.2987 0.0639 0.8 0.4195 0.0464 0.7 0.5683 0.0347 0.6 0.7559 0.0263 0.5 1.0000 0.0200 0.4 1.3305 0.0151 0.3 1.8032 0.0112 0.2 2.5349 0.0079 0.1 3.8193 0.0053 0.0 6.6612 0.0035

a This table assumes symmetric reliability. Furthermore, Pr(B) = 0.02, Pr(PB [NB) = 0.174, Pr(PNB [B) = 0.876, from Ohlson [1980].

Theorem 2. I f refiabifity is symmetric, then

L ( z #) = L ( r , z )

= [r * P r (z INB) + ( 1 - r ) * P r ( z ' I N B ) ]

/ [ r * P r ( z l B ) + (1 - r ) * P r ( z ' l B ) ] .

Lemma 2. l f L ( z ~) = 1, then (BI z #) : P(B).

This can occur if the Type I error bound is such that Pr(PB I B) = Pr(PB I NB). Thus, if a cut- off point is chosen such that all firms are predicted bankrupt, then L ( z #) = 1, independent of the level of reliability.

However, the reliability level also may cause L ( z ~) to equal one. Thus, the question then becomes "At what reliability levels does L ( z ~) = 1?" If reliability is symmetric, then L ( z #) = 1 at r = 0.5. For symmetric reliability, r = 0.5 is the level at which there is maximum uncertainty about the reliability of the reported bankruptcy prediction; it is completely unreliable. Since the system is completely unreliable there is no reason to revise the prior probabilities based on the report from the system. For example, in the case of a classic coin flip with a 'fair' coin, there is maximum uncertainty (reliability) as to the outcome of a flip of the coin. This discussion leads to the following theorem.

Theorem 4. Assume that reliability is symmetric. I f r = 0.5, then Pr(BI z #) = P(B).

Theorem 3. I f reliabifity & asymmetric, then

L ( z #) = L ( r l , r 2, z )

= [rl * P r ( z l g n ) +r2 * P r ( z ' l N B ) ]

/ [ r 1 * Pr(z [B) + r 2 * P r ( z ' IB)].

As seen in the example in Table 5, L ( z ~) is highly sensitive to reliability. Small changes in reliability can have a major impact on L ( z ~) and, thus, Pr(BI z#).

6. The level of reliability for which Pr(B[z #) is independent of z #

If reliability is asymmetric, then L ( z #) = 1 for the case of r t = r 2 = c, 1 > c > 0. In that case, there is no difference between the reliability associated with Pr(z I NB) and Pr(z I B) or Pr (z ' I NB) and Pr(z ' I B). Accordingly, this also is a case of maximum unreliability. Thus, there is no reason to revise the prior probabilities. As a result, under conditions of maximum uncertainty of reliability, the reported level from the process, z #, does not impact Pr(BI z#), as summarized in the following theorem.

Theorem 5. Assume that reliability is asymmetric. I f r 1 = r 2 = c for 1 > c > O, then Pr(BI z #) = P(B).

One special case of the value of L ( z #) is of particular interest, because for that value, Pr(B[ z #) is independent of z ~. This means that the decision is independent of the recommenda- tion of the model. In particular, if L takes on a value of 1, as noted in the following lemma the posterior probability is equal to the prior probability that the firm is bankrupt.

7. Impact of reliability; Symmetric case

Assuming a symmetric reliability model, L ( z #) and Pr(B[ z #) are a function of the reliability and the relationship between the likelihoods Pr(z ] - ). In some cases, P r (BIz #) increases as reliability decreases and in other cases Pr(B] z #) decreases as reliability decreases, depending on the whether

D.E. O'Leary / On bankruptcy information systems 73

there is a Type I error bound. The results assuming a Type I error bound are summarized in Lemma 3 and Theorem 6.

Lemma 3 (Type I error bound). I f

Pr (z I NB) < Pr (z I B) and r" > r ' ,

then L(r" , z )<_L( r ' , z).

Theorem 6 (Type I error bound). I f

P r ( z I N B ) _ < P r ( z I B ) and r" > r ' ,

then Pr(Bl(r" , z)) > P r (Bl ( r ' , z)).

The existence of a Type I error bound makes a major assumption about the information system. It assumes that the system has higher probabilities associated with predicting firms are bankrupt when they are bankrupt than other types of systems. Thus, in those systems, if they are more reliable, the system will lead to higher probability about whether a firm is bankrupt than if the same system was less reliable. In the case of a system with a Type I error bound there is less uncertainty in the bankrupty prediction with a system that has a higher reliability. In general, this might be referred to as making an accurate system more reliable.

As a result, assuming a Type I error bound, indicates that by not including reliability in the model, i.e., assuming perfect reliability, Pr(BI z) overstates the value Pr(B] z #) that would be computed when the proper reliability is used. Thus, depending on the critical value, Pcv, a loan may not be made even if the value of Pr(BI z #) that incorporates the reliability is less than the critical value. Thus, loans may not be made when they should be made.

Similar results can be developed for the case where there is no Type 1 error bound (and more generally, when, P r (z INB) >_ Pr(z lB)) and are summarized in Lemma 4 and Theorem 7. In contrast to Theorem 6, the following result indicates that in the situation where there is no Type 1 error, when reliability is not accounted for too many loans may be made.

Lemma 4 (No Type I error bound). I f

P r ( z I N B ) > _ P r ( z l B ) and r " > r ' ,

then L(r" , z)>_ L( r ' , z).

Theorem 7 (No Type I error bound). / f

Pr( z I NB) >_ Pr( z I B) and r" > r ' ,

then e r ( B l ( r ' , z ) ) _< e r ( B ) I ( r ' , z ) ) .

The lack of existence of a Type 1 error bound assumes that the system has higher probabilities associated with predicting firms are not bankrupt when they are bankrupt than Type I systems. Thus, in those systems, if they are more reliable, the system will lead to a lower probability about whether a firm is bankrupt than if the same system was less reliable. Thus, in the case of a system with no Type I error bound there is more uncertainty in the bankruptcy prediction with a system that has a higher reliability. In general, such a situation might be referred to as making an inaccurate system more reliable, so that the system's predictions are even more inaccurate.

These results, assuming no Type 1 error bound, indicate that by not including reliability in the model, i.e., assuming perfect reliability, Pr(B[ z) understates Pr(B] z #) the value that would be computed by including the proper reliability. Thus, depending on the critical value, f~,, a loan may be made even if the value of Pr(B I(z #, r)) that incorporates the reliability is more than the critical value. Thus, loans may be made when they should not be made.

8. Impact of reliability; Asymmetric case

Assuming an asymmetric reliability model, L ( z #) and Pr(B] z #) are a function of the reliability values and the likelihoods Pr(z l- ). In the case of perfect reliability r 1 = 1 and r 2 = 0 and, thus, the ratio r J r 2 approaches infinity. With the introduction of less than perfect reliability that ratio would decrease. As a result, determining what occurs when we move from a model without reliability to one with reliability can be assessed in the asymmetric model by examining behavior of the model as a function of the ratio r l / r 2. These results are summarized in Lemma 5 and Theorem 7.

Lemma 5 (Type I error bound). I f

Pr (z I NB) _< Pr (z I B) and r~'/r~' > r ; / r2 ,


then vv t t v L ( r l , r2, z) <L(r~, r2, z ) .

Theorem 8 (Type I error bound). I f

Pr(z I NB) < Pr(z I B) and r~'/r~' > r~/r~,

then

Pr(B [(r~', r2', z))_> Pr(B I(r~, r~, z)) .

These results, assuming a Type I error bound, indicate that by not including reliability in the model, i.e., assuming perfect reliability, Pr(B]z) overstates the value Pr(B[z#)) that would be computed using proper reliability measures. Thus, depending on the critical value, P~v, a loan may not be made even if the value of Pr(BI z #) that incorporates the reliability is less than the critical value. Thus, loans may not be made when they should be made.

The opposite situation occurs with Pr(zlNB) > Pr(z I B). This results are summarized in Lemma 6 and Theorem 8.

Lemma 6 (No Type I error bound). I f

Pr(z INB) _> Pr(z [B) and r~'/r~' < r~/r~,

then

L(r~', " z) > L( ' ' z) ?'2, r I , r 2 , •

Theorem 9 (no Type I error bound). I f

Pr(z [NB) >_ Pr(z [B) and r~'/r½' < r~/r~,

then

Pr(B [(r~', r~', z))_< Pr(Bl(r ; , r~, z)) .

These results, assuming no Type I error bound, indicate that by not including reliability in the model, i.e., by assuming perfect reliability, Pr(Bfz) overstates the value Pr(Btz~)), that would be computed assuming proper reliability measures. Thus, depending on the critical value, P~v, a loan may be made even if the value of Pr(B] z #) that incorporates the reliability is more than the critical value. Thus, loans may be made when they should not be made.

9. Heuristic approaches

As Schum and DuCharme [1971, p. 111] note that " . . . a reduction in reliability of evidence from a given source acts to decrease the inferential impact or discriminative power of evidence from that source". As a result, bank management may try to account for the clear lack of reliability in the bankruptcy information system by using any of a number of different heuristics. These heuristics can include the following:

Table 6 Example : C o m p u t a t i o n of Pr(B [PB); wi th and wi thou t re l iabi l i ty a

Cut-off d Pr(PB [NB) d Pr(PB IB) ~ Perfect reliab., Imper fec t reliab., Pr(BI z = PB) b Pr(B] z = PB) c

r = 1 r ~ 0.84

0.0095 0.470 1.000 0.042 0.034

0.02 0.287 0.924 0.062 0.043

0.04 0.167 0.857 0.095 0.052 0.06 0.118 0.800 0.121 0.056

0.08 0.093 0.743 0.140 0.057 0.10 0.072 0.733 0.172 0.060 0.20 0.033 0.552 0.254 0.057 0.30 0.0175 0.514 0.375 0.057 0.40 0.0107 0.429 0.443 0.052

0.50 0.0063 0.324 0.512 0.045

a This table assumes symmetr ic rel iabil i ty. The resul ts also assume tha t P r ( B ) = 0.02 and r = 0.84. Note tha t this bankrup tcy

in format ion system has a Type I e r ror bound. h Pr(B [ P B ) w i t h o u t the use of rel iabil i ty, i.e., (2). c Pr(B IPB) wi th the use of rel iabil i ty, i.e., T h e o r e m 1. J Source for first th ree columns: Ohlson [1980].

D.E. O'Leat3' / On bankruptcy information systems

Table 7 Example : Compar i son of analyt ic and heur is t ic approach a to adjust ing for re l iabi l i ty

75

Cut-off Pr(BI z = PB) b Pr(BI z = PB) * r Pr(B[ z # = PB) c E r r o r d

Perfect r = 1 Heur i s t i c r = 0.84

0.0095 0.042 0.035 0.034 2.9 0.02 0.062 0.052 0.043 20.9

0.04 0.095 0.080 0.052 53.9

0.06 0.121 0.102 (I.056 82.1 0.08 (1.140 0.1 l 8 0.057 107.0

0.10 (1.172 0.132 0.060 120.0

0.2(I (I.254 0.213 0.057 273.7

I).30 0.375 0.315 (I.(157 452.6

0.40 0.443 (I.372 0.052 615.4

0.50 0.512 0.43(I 0.045 855.5

" A s s u m i n g Pr(B) = 0.02 and r = 0.84.

b Pr(B I P B ) w i t h o u t the use of rel iabil i ty, i.e., (2).

Pr(B I P B ) w i t h the use of rel iabil i ty, i.e., T h e o r e m I. ,1 The di f ference be tween Pr (BIPB) * r, i.e., the model from (2) t imes the rel iabi l i ty level, and P r (BIPB) c o m p u t e d from T h e o r e m

1, d ivided by the la t ter quanti ty.

• do nothing, i.e., use the existing model (2) or • increase the result of model (2), e.g., 10%

for a 90% reliability. Such a heuristic is a monotonic heuristic since it indicates always adding or subtracting from some base estimate.

Since the appropriateness of these heuristics is difficult to judge, a priori, there is a need to compare the results between the model that includes the heuristic and a model that uses the optimal approach, in order to compare the effect.

The results generated by Ohlson [1980] will be used to illustrate the impact of such heuristics on bankruptcy decision making. For purposes of the example, symmetric reliability is assumed and reliability is measured using goodness of fit of the statistical model. Based on Table 6, it is easy to see that the first heuristic, of doing nothing, is an inappropriate manner to include reliability. The results are summarized in Table 6.

In addition, the results in Table 7 indicate that the monotonic heuristic of decreasing P(BIPB) as a linear function of reliability also is inappropriate. Using that heuristic the example has error rates of from 2.9% to over 855%. In addition, each heuristic substantially over estimates the actual value of Pr(Bt z # = PB).

10. Bankruptcy investigation with reliability

This paper has developed an approach to bankruptcy investigation that is different, from

that of other approaches: it includes the reliability of the system on which bankruptcy judgments are made. Once that reliability is accounted for there is substantial impact with even small move- ments away from perfect reliability.

Relationship to dec&ion theory approach

The approach discussed in this paper derives from the basic decision theory model summarized in Section 3. That model is a critical value model, where the critical value is determined as a function of the costs and benefits of making a loan. The model indicates that a loan should be granted if the value of (2) is less than or equal to that critical value and not granted if it is greater than that critical value.

Table 2 provides one information system for the generation of probabilities for that decision theory model. However, the example system summarized in Table 2 does not account for reliability. Reliability has a substantial impact on that information system, as noted in noted in Tables 5 and 6. Further, that impact is not predictable using heuristics instead of the appropriate analyt- ics, as summarized in Table 7.

A substantial difference occurs with the introduction of reliability into the decision theory model. The results in Sections 7 and 8 indicate that accounting for reliability has a substantial impact on the critical value decision process. By building the notion of the reliability of a ' repor t '


of evidence into (2), we impact the value of (2). Since the process is a critical value process this means that we also impact whether or not we grant the loan.

The reliability-based decision theory approach

The first step to the approach discussed in this paper is to gather information to establish the costs necessary to establish the critical value. The critical value furnishes the reference point for (2).

Second, in order to use the results developed in this paper, the bankruptcy information system must be evaluated to determine if it has a Type I or Type II error bound as discussed in Section 3. The existence of a Type I or II error bound impacts the value of L(PB). This is important since it is through L(PB) that reliability is integrated into the model. For the remainder of this section it is assumed the system has a Type I error bound. The example summarized in Table 2 provides such a system.

Third, the approach to representation of reliability must be chosen. Although this paper has developed specific results for the case of symmetric and asymmetric reliabilities, Lemma 1 allows for more general versions. For the remainder of this discussion, however, it is assumed that symmetric reliability is appropriate. As seen in Table 5, reliability has a tremendous impact on the values of L(PB) and thus (2): a change in reliability from 1.0 to 0.9 yields about a 50% change in the value of L(PB).

Fourth, the value for (2) can then be computed and compared to the critical value. In addition, under the assumptions made so far in this section, Lemma 3 and Theorem 6 can be invoked. Those results provide an important implications. If perfect reliability (i.e., r = 1) and a Type I information system are assumed, then the cut-off point Pcv will always be higher than an approach that incorporates less than perfect reliability. This indicates that by not accounting for reliability, a cut-off approach may lead to not making loans that should be made.

Some implementation concerns

To prove useful in solving actual bankruptcy problems, the parameters in the model (probabilities, costs and benefits) need to be measured.

Although implementation problems may be ex- tensive they are not impossible to overcome.

There has been research on assessing probabilities associated with financial processes. Felix [1976] explored a number of ways to establish prior probabilities. Further, Wright [1987] has initiated research to analyze approaches to reduce the risk of misspecified prior probability distributions. In addition, estimates of reliability parameters can be developed from previous experience.

Use can be made of symmetric and asymmetric reliability assumptions because these models have been investigated (Sections 7 and 8), appear eas- ier to understand, and can provide estimates of the behavior of more general models.

Even if reliability is not integrated in the actual models, the results of Section 7 and 8 indicate the expected behavior that we need to account for in models of bankruptcy analysis. Fur- ther, the results of Section 9 indicate some 'inappropriate' heuristics. If the system is a Type I error system then we know that (2) will be too high. Without actual reliability estimates, the only issue is how high is too high. In any case, we can do a sensitivity analysis around the cut-off value to assess the sensitivity of the value to (2).

11. Summary and extensions

This paper integrated reliability into the bankruptcy prediction problem. It specified desirable characteristics of bankruptcy information system in terms of Type I and Type II errors. Then the paper analyzed the behavior of systems that meet and those that do not meet the require- ments of Type I and Type II systems. The paper found that the introduction of reliability into the bankruptcy prediction model has a substantial impact on the bankruptcy prediction. In addition, it was found that using some heuristics, instead of the analytic approach, are likely to be quite misleading.

The results of this paper can be extended in at least three directions. First, this problem formu- lation can be extended to a fuzzy set model, since firms may face varying degrees of financial diffi- culty. Second, this problem can be extended to a multiple period model. If the firm is not predicted bankrupt this period there is still a nonzero


probability that the firm will be predicted to be bankrupt next period, based in part on the performance of the firm this period. Third, the problem can be extended to specific areas of firm failure, e.g., commercial loan noncompliance (Chesser [1974]).

Further, this approach could form the analytic

basis for behavioral research on the use of expertise in bankruptcy problems, e.g., in the area of expert systems. Reliability could be used as a measure of the degree of expertise. Alternatively, behavioral research on the actual response to reliability issues of loan officers could be exam- ined.

Appendix: Proofs

Proof of Theorem 1. Assume there is Type I error bound Pr(PB I B) > Pr(PB I NB) then that indicates that 1 - Pr(PNB I B) >_ 1 - Pr(PNB I NB), thus Pr(PNB I B) < Pr(PNB I NB). A similar approach is used when Type II error bounds are assumed.

Proof of Lemma 1. Based on Schum and Du Charme [1971].)

L ( z # ) = e r ( z # I N B ) / P r ( z # IB)

= [Pr(z # and N B ) / P r ( N B ) ] / [ e r ( z # and B ) / P r ( B ) ]

= [{Pr(z # and z and NB) + Pr(z # and z' and NB)} /P r (NB) ]

/ [ { P r ( z " and z and B) + Pr(z * and z' and B)} /P r (B) ]

= [{Pr(z # INB and z ) P r ( z INB)Pr(NB)

+ Pr( z # I NB and z ') Pr( z ' [NB) Pr(NB) } /Pr( NB)]

/ [ { e r ( z ~ IB and z ) P r ( z IB)Pr(B) + Pr(z # IB and z ' ) e r ( z ' [B)Pr (B)} /Pr (B) ]

= [Pr(z # ]NB and z ) P r ( z INB) + Pr (z # INB and z ' ) P r ( z ' l N B ) ]

/ [ P r ( z # IB and z ) P r ( z IB) + Pr (z # IB and z ' ) P r ( z ' l B ) ] .

Proof of Theorem 2. By Lemma 1.

L ( z ~) = [Pr(z # [NB and z ) P r ( z INB) + Pr (z # [NB and z ' ) P r ( z ' l N B ) ]

/ [ P r ( z # IB and z ) P r ( z [B) + P r ( z # IB and z ' ) P r ( z ' l B ) ] .

If we assume that the reported version of z, z#, is not dependent on whether the system is NB or B, then

L ( z ~) = [ e r ( z # I z ) e r ( z I NB) + e r ( z ~ I z ' ) e r ( z ' l N B ) ]

/ [ e r ( z # 1 z ) e r ( z I B ) + e r ( z ~ l z ' ) e r ( z ' l B ) ] .

Since reliability is symmetric r = Pr(z ~ I z) and 1 - r = Pr(z ~ I z ').


Proof of Theorem 3. Similar to T h e o r e m 2, except that asymmetric reliability is assumed, so r i = Pr(z # ] z) and r 2 = Pr (z ~ ]z ' ) .

P roof of Lemma 2. P r ( B l z ~ ) = P ( B ) / [ P ( B ) + ( 1 - P ( B ) ) , L(z~)] . If L (z ~ ) = 1 , then P(r, z ) = P ( B) / [P ( B) + (1 - P(B))] = P(B).

Proof of Theorem 4. If r = 0.5, then

L ( r , z ) = [0.5 * P r ( z INB) + 0.5 * P r ( z ' I N B ) ] / [ 0 . 5 * P r ( z IB) + 0.5 * P r ( z ' IB)] = 1.

By Lemma 1, P r (B] ( r , z ) ) = P ( B ) .

Proof of Theorem 5. If r 1 = r 2 = c, then

L ( r l, r 2, z ) = [k * P r ( z I N B ) + k * P r ( z ' l N B ) ] / [ k * P r ( z I B ) + k * P r ( z ' [ B ) ] = 1.

By Lemma l, Pr(B ] z ~) = P ( B ](r 1, r2, z ) ) = P(B).

Proof of Lemma 3. Proof by contradict ion. Assume that r ' > r ' and L(r", z ) > L(r ' , z). Let Pl = Pr(z ]NB) and P2 = Pr(z ]B).

[ r ' * Pl + (1 - r " ) * (1 - p l ) ] / [ r " * p2+ (1 -r ') • (1 -P2)] > [ r ' * Pl + ( 1 - r ' ) * ( 1 - - p l ) ] / [ r ' * p2-t-(1-r') • ( 1 - P 2 ) ] ;

[ r " * p , + (a - r " ) * (1 - p , ) ] * [ r ' * P2 + (1 - r ' ) * (1 - P 2 ) ]

> [ r ' * Pl + (1 -- r ' ) * (1 - -P l ) ] * [r" * P2 + (1 - r") • (1 - p 2 ) ] ;

r ' * p2 * (1 - p , ) * (1 - r " ) + r" * p l * (1 -P2) * (1 - r ' )

> r ' * Pl * (1 - P 2 ) * (1 - r " ) + r" * P2 * (1 - P l ) * (1 - r ' ) ;

r' * P2 + r" * Pl > r' * Pl + r" * P2;

r ' * ( P z - P l ) > r " * ( P z - P , ) .

But r" > r', so there is a contradict ion and L(r", z) < L(r ' , z).

Proof of Theorem 6. If Pr (z INB) < Pr(z ]B) and r" > r', then L(r", z) <_ L(r ' , z). Thus, Pr(B ](r", z)) > Pr(B I(r ' , z)).

P roof of Lem ma 4. Similar to Lemma 3.

P roof of Theorem. Similar to T h e o r e m 6.

P roof of Lemma 5. P roof by contradict ion. Assume that r " > r ' and L(r~', r½', z)>_ L(r(, r~, z). Let Pl = Pr (z INB) and P2= Pr(z IB).

[r[' * Pl

> [r;

* p,

t !

> [ r I * P l + r 2

r; * P2 * (1 - P l )

+ r~' * (1 - p l ) ] / [ r i ' * P2 + r~'(1 - P 2 ) ]

• Pl + r~ * (1 - p l ) ] / [ r ; * P2-4-r~ * ( 1 - P 2 ) ] ;

+ r;' * (1 - p i ) ] * [r; * P2 + r; * (1 - P 2 ) ]

• ( l - P l ) ] * [r~' * P2+r~' * (1 - P 2 ) ] ;

. . . . ( 1 - P 2 ) * r2 • r2 +El * Pl *


> r~' * P2 * (1 -Pt) * r~ + r ; * p , * (1 - P 2 ) * r~'; vt v ! vv It v I v ,

rl * Pl * r2 + rl * P2 * r2 > r~ * P2 * r2 + r( * p~ * r 2 , t vv ! tv t¢ t v! v ,

rl * P2 * r2 - - r l * Pl * r2 > rl * P2 * r 2 - rl * Pl * r2,

r~ * r~' * ( P z - P , ) >r~' * r~ * ( P z - P l )

r ; / r ~ * ( P z - - P l ) > r ~ ' / r j * ( P z - P , ) .

It It ? But r~'/r~' > r~/r~, so t h e r e is a c o n t r a d i c t i o n and L ( r I , r2, 7.) < L(r~, r e, z) .

Proof of Theorem 8. I f P r ( z INB) < P r ( z ]B) and r~'/r~ > r ; / r 2, t h e n L(r~', r½', z ) < L ( r ; , r ' , z) . Thus , tv tv ! !

Pr (B ] ( r I , r2, Z ) ) > P r (B I ( q , r2, Z)).

Proof of Lemma 6. S imi l a r to L e m m a 5.

Proof of Theorem 9. S imi l a r to T h e o r e m 8.

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