NEURO-FUZZY MODELING IN BANKRUPTCY PREDICTION * D. VLACHOS Y. A. TOLIAS

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1 Yugoslav Journal of Operatons Research 3 (23), Number 2, NEURO-FUZZY MODELING IN BANKRUPTCY PREDICTION * D. VLACHOS Department of Mechancal Engneerng Arstotle Unversty of Thessalonk, Thessalonk, Greece Y. A. TOLIAS Telecommuncatons Laboratory Department of Electrcal and Computer Engneerng Arstotle Unversty of Thessalonk, Thessalonk, Greece Communcated by Byron Papathanassou Abstract: For the past 3 years the problem of bankruptcy predcton had been thoroughly studed. From the paper of Altman n 968 to the recent papers n the '9s, the progress of predcton accuracy was not satsfactory. Ths paper nvestgates an alternatve modelng of the system (frm), combnng neural networks and fuzzy controllers,.e. usng neuro-fuzzy models. Classcal modelng s based on mathematcal models that descrbe the behavor of the frm under consderaton. The man dea of fuzzy control, on the other hand, s to buld a model of a human control expert who s capable of controllng the process wthout thnkng n a mathematcal model. Ths control expert specfes hs control acton n the form of lngustc rules. These control rules are translated nto the framework of fuzzy set theory provdng a calculus, whch can stmulate the behavor of the control expert and enhance ts performance. The accuracy of the model s studed usng datasets from prevous research papers. Keywords: Neuro-fuzzy, bankruptcy.. INTRODUCTION The ablty to predct frm bankruptces has been extensvely studed n the accountng lterature. Credtors, audtors, stockholders and senor managers all have a vested nterest n utlzng and developng a methodology or a model that wll allow them to montor the fnancal performance of a frm. There exst extensve studes n ths area usng statstcal approaches and Artfcal Intellgence, most of whch use * Presented at 6th Balkan Conference on Operatonal Research

2 66 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton fnancal ratos as nputs n a forecastng model. Dscrmnant Analyss, proposed by Altman [] has been used most frequently among statstcal approaches n bankruptcy predcton. Prevous emprcal results show that neural network models provde hgher predctve accuracy than statstcal methods. [2] [4] [8] [] Expert systems (ES) represent a feld of study wthn the Artfcal Intellgence, whch has earned the attenton and commtment from busness and nformaton systems managers, as well as knowledge engneers. They have become crtcal components n many products and servces, as well as n many decson-makng processes. There are dfferent types of ES dependng on the mathematcal tool used to nduce knowledge from the avalable data. One of the major categores of ES s rulebased ES where knowledge s expressed through "f... then..." rules. The frst attempt of usng expert systems for bankruptcy predcton was the one of Messer and Hansen [5]. The objectve of the proposed 'data-drven' method was to take frms of known classes (bankrupt/non-bankrupt) descrbed by a fxed set of attrbutes (fnancal ratos), and then to generate a producton system usng attrbutes whch correctly classfy all the frms of the sample. The rules at each stage (.e. the varable and the cut-off score) were defned by usng measures of entropy and selectng the mnmum entropy rule. A decson tree was derved from the producton system rules. Messer and Hansen's study was based on a sample of 23 frms (8 bankrupt and 5 non-bankrupt). The classfcaton accuracy of the producton system was encouragng n ths small case. Some other ES mplcatons are presented n the revew paper of Dmtras et al. [3]. The same paper summarzes most of the bankruptcy predcton methods. In ths paper, we present a rule-based expert system developed to predct the probablty of busness falure, where the rules are nduced through an approach, whch combnes neural networks and fuzzy logc, usually referred n lterature as neuro-fuzzy approach. Specfcally, we use the Adaptve Network-based Fuzzy Inference System (ANFIS), whch s a technque proposed by Roger Jang [7]. Secton 2 brefly revews a generalzed model for fuzzy nference systems. Secton 3 descrbes ANFIS, whle secton 4 presents the results and the performance evaluaton of ts mplementaton n bankruptcy predcton. Fnally, secton 5 advances some conclusons and recommendatons for ES developers and managers to proceed. 2. A GENERALIZED MODEL FOR FUZZY INFERENCE SYSTEMS Fuzzy f-then rules or fuzzy condtonal statements are expressons of the form IF A THEN B, where A and B are lngustc values of fuzzy lngustc varables. Each lngustc varable s characterzed by a unverse of dscourse and the membershp functons of ts values, defned on the same unverse of dscourse. The lngustc values are fuzzy sets. An example that clarfes the aforementoned defntons s the followng: IF HEIGHT IS TALL, THEN WEIGHT IS HEAVY.

3 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton 67 Here HEIGHT and WEIGHT are the lngustc varables whle TALL and HEAVY are the lngustc values of the lngustc varables, beng characterzed by approprately defned membershp functons. By usng fuzzy f-then rules we can capture and encode expert knowledge n the form of a fuzzy nference system. For more detals on the theory of fuzzy sets and the defnton and use of lngustc varables the nterested reader s referred to Zadeh's orgnal work [2], [3]. A fuzzy nference system (also known as fuzzy rule-based system) s composed of fve functonal blocks, a rule-base that contans a number of fuzzy f-then rules, a database that defnes the membershp functons of the fuzzy sets used by the fuzzy rules, a decson-makng subsystem that performs the nference operatons on the rules, a fuzzfcaton nterface that transforms crsp measurement to degrees of membershp to dfferent fuzzy sets and fnally, a defuzzfcaton nterface that transforms the fuzzy results nto a crsp output (e.g. a control sgnal, a predcted value, etc). The block dagram of a fuzzy nference system s shown n Fgure 2.. A fuzzy nference system performs the followng processng steps on the gven nputs:. Fuzzfcaton: Compare the nput varables wth the membershp functons that consttute the database on the premse part to obtan the membershp values of each lngustc label. 2. Determnaton of the frng strength of each rule by combnaton of the membershp values on the premse part. 3. Generaton of the consequent of each rule dependng of the frng strength. 4. Defuzzfcaton: Aggregaton of the consequents to produce a crsp output. Knowledge Base Database & Rule Base Input Fuzzfcaton Interface Defuzzfcaton Interface Output Decson Makng Unt Fgure 2.: The block dagram of a fuzzy nference system Dependng on the type of fuzzy reasonng and the fuzzy f-then rules that are used, there are three types of nference systems. The overall output n the frst type s the weghted average of each rule's crsp output nduced by the rules frng strength and output membershp functons. The output membershp functons must be nondecreasng functons n the unverse of dscourse. In the second type, the overall fuzzy output s derved by applyng the maxmum operator to the fuzzy outputs and the fnal crsp output s calculated usng an approprate defuzzfcaton method (area, bsector of

4 68 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton area, center of mass, and others). Fnally, the thrd type of fuzzy nference systems uses the Sugeno type approach [9] and the fuzzy output for each rule s a lnear combnaton of nput varables wth an addtonal constant term. The fnal output s the weghted average of each rule's output. 3. THE ADAPTIVE NETWORK-BASED FUZZY INFERENCE SYSTEM (ANFIS) The Adaptve Network-based Fuzzy Inference System (ANFIS) that was proposed by Roger Jang [7] s one of the most commonly used fuzzy nference systems. In ths secton we hghlght ANFIS's archtecture. For addtonal detals the reader s referred to [7]. ANFIS s a 5-layer feed-forward network n whch each node performs a partcular functon n ncomng sgnals as well as a set of parameters pertanng to the node. Let us suppose that the fuzzy nference system under consderaton has two nputs x and y and one output z. Suppose that the rule base contans the followng two Sugeno-type fuzzy f-then rules: R: IF x s A and R2: IF x s A 2 and s y s y 2 B, THEN f = p x+ q y+ r, B, THEN f = p x+ q y+ r These rules correspond to the thrd category of fuzzy nference systems mentoned n [7]. The archtecture of the equvalent ANFIS system s shown n Fgure 3.. x A A2 Π w Ν x y Σ f y B B2 Π w2 Ν x y Fgure 3.: The archtecture of ANFIS Every node that belongs to the frst layer O s characterzed by the membershp functon A ( x ), where A s the lngustc value of the nput varable x. µ Usually, the membershp functons are generalzed bell functons, of the form:

5 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton 69 O = µ A ( x; α, β, γ) = β 2 γ + x α The values { α, β, γ } consttute the parameter set that s adjusted usng the learnng algorthm. 2 The nodes O belongng to layer 2 calculate the frng strength of a rule by fndng the product of the membershp values of the nodes of layer that belong to a rule: 2 A B O = w = µ ( x) µ ( x), = 2,. (2) 2 Ths operaton s essentally the applcaton of the product T-norm on the premse membershps. 3 The nodes O belongng to layer 3 normalze the rules' frng strengths, accordng to the equaton: 3 w O = = w, = 2,. (3) w + w2 4 The nodes O n layer 4 multply the rules' frng strengths wth the consequent parameters { p, q, r }, 4 O = wf = w( px+ qy+ r). (4) Fnally, the sngle node that consttutes layer 5 calculates the overall output, a crsp value, whch s defned as follows: O 5 = wf w. (5) Learnng n ANFIS s mplemented by employng a hybrd learnng rule that combnes the gradent-descent method and the least squares estmate to dentfy the sets of parameters { α, β, γ } and { p, q, r }. The back propagaton learnng rule s appled to tune the parameters n the hdden layers and the parameters n the output layer are dentfed by the least squares method. Each epoch of ths hybrd procedure s composed of a forward and a backward pass. In the forward pass the premse parameters are kept fxed and the consequent parameters are estmated by the least squares method. In the backward pass, the consequent parameters are kept fxed and the premse parameters are calculated by gradent descent. Agan, for more detals the reader s referred to [9]. () 4. NUMERICAL INVESTIGATION The frst methodologcal step s the determnaton of nput varables (ratos) for the model. Snce the purpose of ths study was to examne the value of applyng

6 7 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton expert systems n bankruptcy predcton and not to evaluate the approprate nput varables, we used the smple fnancal ratos employed by Altman []. These ratos were: Workng Captal/Total Assets (WC/TA) Retaned Earnngs/Total Assets (RE/TA) Earnng before Interest and Taxes /Total Assets (EBIT/TA) Market Value of Equty/Total Debt (MVE/TD) Sales /Total Assets (S/TA) The sample under study conssts of frms that ether were n operaton or went bankrupt between 975 and 982. It was obtaned from Moody's Industral Manuals, and ncluded a total of 29 frms, out of whch 65 went bankrupt durng the perod and 64 non bankrupt frms matched on ndustry and year. Data used for the bankrupt frms were taken from the last fnancal statement ssued before the frms declared bankruptcy. The sample was chosen the same used by Wlson and Sharda [] and Rahman et al. [6] to make possble drect comparson. The numercal values of the entre data set are shown n Fgure 4.. The horzontal axs n the fgure represents the frm ndex. The frst 65 ndex numbers refer to bankrupt frms. WC / TA RE / TA EBIT / TA MVE / TD S/TA Fgure 4.: The numercal values of the entre dataset used n ths paper

7 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton 7 Ths method was mplemented n a Pentum II PC usng Matlab (The Mathworks, Natck, MA, USA). Frst, two data sets were generated; a tranng data set consstng of 74 vectors that were pcked n random and a testng data set consstng of the remanng 55 data vectors. For the tranng data set, the covarance matrces and the mean value vectors were calculated for the bankrupt and the non-bankrupt samples, Cb, Cnb, m b and, respectvely. m nb For every bankrupt/non bankrupt company vector, a covarance transform was appled n order to provde uncorrelated, zero mean tranng samples: yb/ nb= ( xb/ nb mb/ nb) C b/ nb (6) The dataset y that results from ths covarance transform s shown n Fgure Fgure 4.2: The dataset after the covarance transformaton

8 72 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton We used the transformed tranng data set to tran ANFIS and obtan a fuzzy nference system havng 5 nput varables (the transformed ratos). Its output shows the possblty of not gong bankrupt,.e. the Non Bankrupt Index (NBI). ANFIS requred tranng epochs to reach a Root-Mean-Squared-Error (RMSE) of.34. The plot of the tranng error per epoch s shown n Fgure 4.3. ANFIS produced 32 fuzzy rules of the form mentoned n Sec. 3. The system was evaluated usng the testng data set, by utlzng the followng approach: Snce we are not aware whether a testng vector belongs to bankrupt or nonbankrupt company, we need to generate two hypotheses for every testng vector. That s, H: The vector corresponds to a company that s not gong to get bankrupt, and H: The vector corresponds to a company that s gong to get bankrupt. RMSE Iteratons Fgure 4.3: The plot of the RMSE per tranng epoch For each hypothess, a covarance transformaton s appled usng the correspondng vectors and matrces: ytest / H = ( xtest mnb) Cnb (7) ytest / H = ( xtest m b) Cb (8) A dfferent value of the Non-Bankrupt Index s calculated by ANFIS for each vector n (7,8),.e., NBI test / H and NBI test / H. From these values we select the ones havng the lowest fuzzness, accordng to the rule: NBItest / H, f NBItest / H. 5 NBItest / H. 5 NBI = NBItest / H, f NBItest / H 5. > NBItest / H 5. (9)

9 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton 73 The results of the applcaton of the proposed scheme to the entre dataset are shown n Fgure Non Bankrupt Index Data Sample (company ndces) Fgure 4.4: The Non-Bankrupt Index for the entre data set From Fgure 4.4 we conclude that the entre dataset (tranng and test data) are correctly classfed by the Non-Bankrupt Index. So, ANFIS acheved % correct classfcatons, whle the results of prevous research were worse. Specfcally, dscrmnant analyss gves 85% correct classfcatons and the best neural network approach of Wlson and Sharda [] gves 93%. The paper of Rahman et al. [6] provdes analytc results of applyng dfferent methods n the same dataset. 5. CONCLUSIONS Ths paper has compared the predctve capablty of neuro-fuzzy technques wth that of prevous research paper wthn the context of forecastng frm bankruptces on the bass of a small number of fnancal ratos. In ths study the proposed system ANFIS outperformed both classcal methods and modern approaches. Although the method s tested wth a small sample of frms, the % correct classfcaton of bankrupt and non-bankrupt frms shows that ths s a robust and promsng approach n the predcton of frm stablty. Thus, the research on bankruptcy predcton wth neuro-fuzzy methods must contnue.

10 74 D. Vlachos, Y.A. Tolas / Neuro-Fuzzy Modelng n Bankruptcy Predcton REFERENCES [] Altman, E.I., "Fnancal ratos, dscrmnant analyss and the predcton of corporate bankruptcy", Journal of Fnance, (968) [2] Coats, P.K., and Fant, F., "Recognzng fnancal dstress patterns usng a neural network approach", Fnancal Management, 22(3) (992) 42 [3] Dmtras, A.I., Zanaks, S.H., and Zopounds, C., "A survey of busness falures wth an emphass on predcton methods and ndustral applcatons", European Journal of Operatonal Research, 9 (996) [4] Flecher, D., and Goss, E., "Forecastng wth neural networks-an applcaton usng bankruptcy data", Informaton and Management, 24 (993) [5] Messer, W., and Hansen, J., "Inducng rules for expert system development: an example usng default and bankruptcy data", Management Scence, 34 (988) 2. [6] Rahman, E., Sngh, S., Thammachote, T., and Vrman, R., "Bankruptcy predcton by neural network", n: R., Trpp, and E. Turban (eds.), Neural Networks n Fnance and Investng, Irwn, 996. [7] Roger Jang, J.S., ANFIS: "Adaptve network-based fuzzy nference system", IEEE Transactons on Systems, Man and Cybernetcs, 23 (3) (993) [8] Salchenberger, L., Mne Cnar, E., and Lash, N., "Neural networks: a new tool for predctng thrft falures", Decson Scences, 23 (992) [9] Takag T., and Sugeno, M., "Fuzzy dentfcaton of systems and ts applcaton to modelng and control", IEEE Transactons on Systems, Man and Cybernetcs, 5 (985) [] Wlson, R., and Sharda, R., "Bankruptcy predcton usng neural networks", Decson Support Systems, (994) [] Yoon, Y., Gumaraes, T., and Swales, G., "Integratng neural networks wth rule-based expert systems", Decson Support Systems, (994) [2] Zadeh, L.A., "Fuzzy sets", Informaton Control, 8 (965) [3] Zadeh, L.A., "Outlne of a new approach to the analyss of complex systems and decson processes", IEEE Transactons on Systems, Man and Cybernetcs, 3 () (973)