Radmila SOUSEDÍKOVÁ 1, Jaroslav DVOŘÁČEK 2, Igor SAVIČ 3

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1 APPLICATION OF DISCRIMINATE ANALYSIS TO PREDICTION OF COMPANY FUTURE ECONOMIC DEVELOPMENT VYUŽITÍ DISKRIMINAČNÍ ANALÝZY PRO PREDIKCI BUDOUCÍHO VÝVOJE FIRMY Radmla SOUSEDÍKOVÁ, Jaroslav DVOŘÁČEK, Igor SAVIČ 3 RNDr. Ph.D., Isttute of Ecoomcs ad Cotrol Systems, Faculty of Mg ad Geology, VŠB Techcal Uversty of Ostrava 7. lstopadu 5, Ostrava-Poruba, tel. (+40) e-mal: radmla.sousedkova@vsb.cz prof. Ig. CSc., Isttute of Ecoomcs ad Cotrol Systems, Faculty of Mg ad Geology, VŠB Techcal Uversty of Ostrava 7. lstopadu 5, Ostrava-Poruba, tel. (+40) e-mal: jaroslav.dvoracek@vsb.cz Abstract 3 Mgr., Cybe Idustral LTD, C/O Columbus Tradg-Parters GMBH Redger str. 8, Bayreuth, Germa, tel. (+49) e-mal: gor.savc@cybe-ole.com] The paper takes to accout applcatos of dscrmate aalyss as regards predcto of future ecoomc developmet of compaes. A assumpto of multvarate ormalty of dscrmators has bee tested ad outlers detfed. A outler reducto of orgal data fles brgs data dstrbuto closer to multvarate ormalty, ad substatally mproves dscrmate fucto classfcato abltes. Abstrakt Čláek se zabývá využtím dskrmačí aalýzy pro predkc budoucího vývoje frmy. Je testová předpoklad vícerozměré ormalty dskrmátorů a jsou detfkováy vybočující hodoty. Redukce původích datových souborů o vybočující data přspívá k přblížeí rozložeí dat vícerozměré ormaltě a vede k podstatému zlepšeí klasfkačí schopost dskrmačí fukce. Key words: Default predcto, dscrmate aalyss, multvarate ormalty, outlers INTRODUCTION The tal comprehesve studes of modellg future developmet of compaes date back to the thrtes of the tweteth cetury. These works take to accout aalyses of facal ratos as regards successful ad faled compaes. Ramser ad Foster (93) compared ratos for successful ad faled busesses ad could prove that corporate face of successful frms demostrate better rato fgures tha t would be the case wth those dstress or bakrupt. What more, Ftzpatrck (93) tred to detfy those corporate face ratos that mght serve the purpose of predctg future ecoomc developmets. Smth ad Wakor (935) aalyzed book keepg records of bakrupt compaes, detfyg dces of mmet default. A study of cardal mportace s that of Beaver (966) who aalyzed the utlsato of facal ratos for predctg future ecoomc developmet of compaes. Beaver desged hs study to be a bechmark for future vestgatos to alteratve predctors of falure. He succeeded provdg a emprcal verfcato of the predctve ablty of facal statemet accoutg data, amely the facal ratos, for the predcto of corporate busess falure. All the studes that have bee metoed above demostrate the ablty of facal ratos to predct bakruptcy, ad they try to detfy a sgle rato or a group of such ratos whose values would dcate mmece of default. Nevertheless, coclusos of these studes are ot uambguous, because each study eemplfes dfferet facal ratos as dcators of default mmece. The perod from the thrtes to the ed GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

2 of the stes of the 0th cetury ca be qualfed as the perod of oe-dmesoal aalyses of facal ratos, whe the dvdual ratos are aalyzed ther solato ad ther teracto s dsregarded. Altma (968) bult o the estg results ad could overcome lmtatos of oe-dmesoal approaches by combg several facal ratos for hs developmets of predcto models. He solved problems of the capablty of specfc ratos to predct default. He could provde for objectve quatfcato of each specfc rato by defg weghts that should be attached to these selected ratos ad how these weghts should be objectvely establshed. Altma s ovatve hs utlzato of the multple dscrmate aalyses by developg a mult-dmesoal model of a corporate busess default. I hs later studes, Altma adapted hs model several tmes ad alog wth Halderma ad Naryaa (977) developed a secod geerato of predcto models for corporate busess falure. Altma s aware of defceces of models that utlze lear dscrmate aalyss (Altma et. al., 98), especally as regards o-coformty to the assumpto multvarate ormal dstrbuto of dscrmators each class, ad equalty of class covarace matrces. The mpossble fulflmet of multdmesoal ormalty resulted the utlzato of other dstrbuto fuctos that would descrbe the put data dstrbuto. Ohlso (980) was the frst oe who used a logstc regresso aalyss for creatg predcto models of corporate busess defaults. Sce the begg of the etes, studes have bee performed that utlze euro etworks for modellg of corporate falures (Odom ad Sharda, 990; Tam, 99). DISCRIMINATE ANALYSIS The dscrmate aalyss eables the evaluato of dffereces betwee two or more subject groups that are charactersed by a certa umber of features. Such evaluato provdes a base of classfcato that bulds o t. If the dscrmate aalyss s appled for predctg future ecoomc footg of compaes, the subjects are partcular frms that are structured to two groups of prosperous frms ad those jeopardy of default. Each frm s charactersed by a defte umber of quattatve varables called dscrmators. The choce of dscrmators plays a major role as regards predcto abltes of fal modellg. The dscrmator calculato data ca be draw from commo accoutg documetato Balace Sheet, Proft ad Loss Statemet ad t provdes for calculatos of ratos or dees. There are two ways for choosg the dscrmators. The frst way takes advatage of a mathematcal approach to the problem, ad takes a lot of varous ratos ad dees to accout, ad so we are ot sure whch of them are effectve for classfyg a partcular frm as prosperous or threateed by default. We ca dstgush partcular dscrmator effceces by assessg the chagg values of the Mahalaobs dstace, D M, betwee mea values of both classes, that have bee effected by addg or removg a defte dscrmator. The dvdual steps of dscrmator addto or removal ca be drected by varous decso-makg crtera (for eample the Wlks Crtero, λ). The secod way has bee based o eperece, kowledge, ad tuto of researchers, whe the choce of dscrmators s supported by a theoretcal model for solvg the gve task. Ths paper s vestgato has opted for ths secod way ad the dscrmator selecto has bee based o the assumpto that bakruptcy s caused by dsrupted crculato of captal. That s why, 8 facal ratos were chose, amely,. Quck labltes/total assets. Curret assets de 3. Curret lqud assets/curret assets 4. Sales de 5. Total assets de 6. Recevables/Curret assets 7. Ide, Loa captal/total assets 8. Equty captal/total assets.. Model The dscrmate aalyss model cossts a lear combato of varables, so called dscrmators that best dstgush betwee prosperous ad default compaes. Ths s the lear dscrmate fucto formula: D d X d X d X, () m m GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

3 where Number of frms the class, m Number of dscrmators, D Dscrmate score for frm,, X Dscrmator value for frm,, ( j,, m ), j j d Lear dscrmate coeffcet for dscrmator, j, (for j,, m ). 3 Ths formula combes several frm s characterstcs (dscrmators) to a sgle multvarate score, D, whose value s betwee ad, ad dcates facal health of the frm. The dscrmate score low value, D, marks bad facal health of the frm. A correct applcato of the lear dscrmate aalyss asks for observato of the followg requremets: - Dscrmators evdece multvarate ormalty of dstrbuto, - Classes of prosperous frms ad those jeopardy of default have the same covarace matrces... Testg ormalty of dscrmators I the followg, we cocetrate o multvarate ormalty of dscrmators. The multvarate ormalty of depedet varables (dscrmators) should be verfed by a approprate statstcal test. The test of multvarate ormalty s dffcult. Although the radom vector partcular costtuets of all dscrmators evce oe-dmesoal ormalty, the assocated desty of probablty does ot ecessarly have multvarate ormal dstrbuto. As such, the oe-dmesoal ormalty of partcular dscrmators s a ecessary but suffcet prerequste of vector multvarate ormalty of all dscrmators. Ths fact ca serve as a tool of practcal verfcato of multvarate ormalty of dscrmators. The oe-dmesoal ormalty of partcular dscrmators s verfed frst. If, at least, oe dscrmator does ot evdece oe-dmesoal ormalty, t s obvous that o multvarate ormalty ests, ad the testg s termated. If all features evdece ormalty, t s ecessary to cotue testg of multvarate ormalty. For eample, the ormalty of partcular dscrmators ca be verfed by tests of skewess ad kurtoss. The testg crtero s defed by the followg formula [6]: C Eĝ ĝ ĝ, () Dĝ D ĝ where - Sample sze, - Value of the dscrmators tested for frm,, - Mea value of the dscrmator tested, ĝ Sample skewess, (3) Dĝ - Varace of sample skewess, (4) ĝ 4 - Sample kurtoss (5) 6 Eĝ 3 - Epected value of sample kurtoss, (6) D ĝ - Varace of sample kurtoss. (7) If ormalty s evdeced, the value, C, has a asymptotc dstrbuto,, wth degrees of freedom. If t s verfed that C, t s ecessary to reject hypotheses that the sample s ormally dstrbuted [6]. GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

4 The ormalty of partcular dscrmators ca be also assessed graphcally by meas of rakt plots, whch eable comparso of each dscrmator dstrbuto wth ormal dstrbuto. The rakt plots are eecuted by plottg the quatles of the stadard ormal dstrbuto, u, o the plot s horzotal as, ad the order statstcs, (dscrmator values structured herarchcally ascedg order), o ts vertcal oe. If a ( ) dscrmator s dstrbuto equals ormal dstrbuto, we ca observe a lear depedece,, o u ( ) p. For the class of prosperous frms, as well as for those jeopardy of default, the tests of oe-dmesoal ormalty for all eght dscrmators were performed. The value of the testg crtero, C, for partcular dscrmators (see Tab. ), ad ther comparsos wth the quatle, 0,95( ) 5, 99, make t obvous that oe of the dscrmator dstrbutos s ormal. Dscrmator Tab. Test crtero values for the ormalty verfcato of partcular dscrmators Prosperous frms p Test crtero value, C Frms jeopardy of default Quck labltes/total assets Curret assets de Curret lqud assets/curret assets Sales de Total assets de Recevables/Curret assets Ide, Loa captal/total assets Equty captal/total assets Also rakt plots of all dscrmators for both classes of frms evdece the fact that o dscrmator dstrbuto s ormal. The followg Fgs.,, provde for a comparso of rakt plots of sample dscrmators for the class of proftable busesses. The dstrbuto of the dscrmator, Equty captal/total assets (see Fg. ) appromates the ormal dstrbuto most closely. I cotrast to ths, the dstrbuto of the dscrmator, Sales de (see Fg. ) evdeces the worst result as regards the dstrbuto ormalty. 4 Fg. Rakt plot for the Equty captal/total assets (prosperous frms). GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

5 5 Fg. Rakt plot for the Sales de (prosperous frms). Aalogcally, for the class of frms jeopardy of default, Fg. 3 provdes for the rakt plot of the dscrmators, Total assets de, whch shows almost a lear depedece close to ormalty. I cotrast to ths, Fg. 4 testfes that the rakt plot for the dscrmator, Ide, Loa captal/total assets, evdeces a proouced devato from the ormal dstrbuto Fg. 3 Rakt plot for the Total assets de (default frms). Fg. 4 Rakt plot for the Ide, Loa captal/total assets (default frms). The above gve data are ot surprsg. For the ecoomc data, a ormalty devato s rather the rule tha the ecepto. The majorty of research works (clusve Altma) do ot test multvarate ormalty ad assume that models are suffcetly robust for provdg ratoal appromatos eve wthout meetg the prerequste of put data ormal dstrbuto. GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

6 Havg ths md, let us try to mprove the data dstrbuto by ecluso of outlers, ad let us compare the results of the dscrmate aalyss before ad after ths modfcato of the data..3. Idetfcato of outlers Two approaches wll be take for the outler detfcato:. Oe-dmesoal approach: Partcular dscrmator outlers are detfed by so called er feces.. Multvarate approach: Outlers are detfed by ther Mahalaobs dstace from the mea value data. Outler detfcato by er feces: The outlers are all dscrmator values that le outsde the terval, * D * H B ; B, (8) B B * ~ D 5 K ~ ~ 75 5, (9) * ~ D 75 K ~ ~ 75 5, (0) 3,6 K,5, () where - Sample sze, ~ 5 - Lower quartle, ~ 75 - Upper quartle [6]. The followg Tab. gves umbers of dscrmator outlers for both classes of frms. 6 Dscrmator Tab. Number of dscrmator outlers Prosperous frms Number of outlers Default frms Quck labltes/total assets 5 Curret assets de 8 Curret lqud assets/curret assets 0 9 Sales de 7 3 Total assets de Recevables/Curret assets 0 Ide, Loa captal/total assets 4 6 Equty captal/total assets 0 7 Frms that have oe or more dscrmator outlers were ecluded from the vestgato. The orgal samples of 93 prosperous ad 93 default compaes were reduced to 76 prosperous ad 74 default compaes. The outler detfcato by the Mahalaobs dstace: The outlers are all multvarate data of the formula, d T S m j, () - Dscrmator vector of partcular frms j,, j S, - Sample mea vector of partcular dscrmators, - Iverse co-varace matr, GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

7 7 m - Quatle of dstrbuto,, wth degrees of freedom, m where m - Number of dscrmators, α - Sgfcace level, - Number of frms of partcular class [6]. The Mahalaobs dstace detfed fve prosperous ad eght default frms as havg outlyg dscrmator values. As such, the orgal samples were reduced to 88 prosperous ad 85 default compaes. The data dstrbuto mprovemet ca be demostrated by a Q-Q plot. The plot llustrates depedece of * order statstcs, C ( C values structured herarchcally ascedg order) o values, C, where T S j C, (3) * a C, (4) a b where 0,5m a, m 0,5 ) m b. m I case of multvarate ormalty, the depedece of C o The Fg. 5 provdes for the Q-Q plot of 93 prosperous frms * C should be a lear oe. Fg. 5 Q-Q Plot of prosperous frms Orgal sample of 93 frms The followg Fg. 6 gves the Q-Q Plot for the reduced sample of 76. The reducto was made by removg frms that had some dscrmator values detfed by er feces as outlers. GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

8 Fg. 6 Q-Q Plot of prosperous frms Reduced sample of 76 frms The comparso of these two plots makes t obvous that removal of outlers mproved cosderably the data dstrbuto. The almost lear depedece of the plot Fg. 6 provdes for the possble cocluso that the reduced sample of prosperous frms closely appromates ormal dstrbuto. 3 DISCRIMINATE ANALYSIS OUTCOME COMPARISON For the samples reduced by outler elmato, the lear dscrmate fuctos were calculated. These provded for the classfcato of frms as prosperous or jeopardy of default. The results of ths classfcato were compared wth those, whch had bee the outcome of workg wth orgal o-reduced samples. The comparso of results gves Tab. 3. Tab. 3 Outcome comparso of workg wth orgal ad reduced samples Classfcato success [%] Prosperous frms Bakrupt frms Frms total 8 Orgal sample (93 prosperous frms + 93 jeopardy of falure) Sample reduced by er feces (76 prosperous frms + 74 jeopardy of falure) Sample reduced by Mahalaobs dstace (88 prosperous frms + 75 jeopardy of falure) It s obvous from the aforemetoed results that appromato to multvarate ormalty mproves the dscrmate fucto classfcato abltes cosderably. Cocerg the appromato method specfcty,.e. the dscrmate outler elmato, the reducto by er feces led to better results. Regardg the multvarate ormalty, the samples of whch the outlers were elmated by er feces demostrated better results whch were corroborated by the Q-Q Plots. 4 CONCLUSION The lear dscrmate aalyss represets a method of predctg future ecoomc developmet of frms that s ofte appled. The choce of dscrmators has a cosderable mpact o the predcto abltes of the fal model. The selecto of partcular dscrmators was based o the dea that bakruptcy s caused by dsrupted crculato of captal. The correct applcato of dscrmate aalyss asks, apart from other requremets, for fulflmet of the prerequste of the dscrmator ormal dstrbuto. The majorty of research that has bee coducted the feld does ot test the multvarate ormalty ad assumes that models are suffcetly robust, provdg for realstc results eve wthout meetg the dstrbuto ormalty requremets. The testg of the multvarate ormalty of dscrmators was performed the way, whch frst tested oedmesoal ormalty of dscrmators by tests of skewess ad kurtoss. The oe-dmesoal ormalty was also assessed by rakt plots. The testg of the aforemetoed dscrmators led to the cocluso that oe of the dscrmators tested had ormal dstrbuto, whch was the reaso why the requremet of multvarate ormal dstrbuto of all dscrmators could ot be met. Such results are ot surprsg because ecoomc data ofte devate from ormal dstrbuto. That was the reaso why the data dstrbuto was modfed. The samples of prosperous ad default frms were adapted two ways. The oe-dmesoal way cossted aalysg partcular dscrmators ad ecludg the outlyg dscrmator values by er feces. The multvarate way started wth the elmato of outlers by the Mahalaobs dstace. The mprovemet of the data dstrbuto s testfed by Q-Q Plots. The reduced samples were subject to a dscrmate aalyss whch provded for the classfcato of frms as successful or faled. It s obvous from the results obtaed that the appromato to multvarate ormalty mproves cosderably the classfcato abltes of the dscrmate fucto. REFERENCES [] ALTMAN, E. I. Facal ratos, dscrmate aalyss ad predcto of corporate bakruptcy. The Joural of Face. 968, XXIII. Nr. 4, pp GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN

9 [] ALTMAN, E., HALDERMAN, R. & NARAYANAN, P. ZETA Aalyss: A New Model to Idetfy Bakruptcy Rsk of Corporatos. Joural of Bakg ad Face. 977, I. Nr., pp [3] ALTMAN, E. I. et al. Applcato of Classfcato Techques Busess, Bakg ad Face. (Cotemporary Studes Ecoomc ad Facal Aalyss). Greewch, Co. : Ja Press, pp. [4] BEAVER, W. Facal Ratos as Predctors of Falure. Joural of Accoutg Research. 966, IV. Emprcal Research Accoutg: Selected Studes 966, pp. 7-. [5] FITZPATRICK, P. J. A Comparso of ratos of successful dustral eterprses wth those of faled frms. Certfed Publc Accoutat, 93. October, November, December, pp , pp , pp [6] Melou, M. & Mltký, J. Statstcká aalýza epermetálích dat (Statstcal Aalyss of Epermetal Data). Praha : Academa, 004. ISBN [7] ODOM, M. D. a R. SHARDA. A Neural etwork model for bakruptcy predcto. IJCNN Iteratoal Jot Coferece o Neural Networks. 990, I-III. pp. B63-B68. [8] OHLSON, J. Facal ratos ad the probablstc predcto of bakruptcy. Joural of Accoutg Research. 980, XVIII. Nr., pp [9] RAMSER, J. R. & FOSTER, L. O. A Demostrato of Rato Aalyss. Bullet No , Bureau of Busess Research, Uversty of Illos, Urbaa. [0] SMITH, R. F. ad A. H. WINAKOR. Chages the facal structure of usuccessful dustral corporatos. Urbaa: Uversty of Illos, 935. [] TAM, K. Y. Neural etwork models ad predcto of bak bakruptcy. Omega-Iteratoal Joural of Maagemet Scece. 99, IXX. Nr. 5, pp RESUMÉ Čláek se zabývá využtím dskrmačí aalýzy pro predkc budoucího vývoje frmy. Predkčí schopost výsledého modelu je výzamě ovlvěa volbou vstupích proměých dskrmátorů. Výběr použtých dskrmátorů byl založe a ázoru, že příčou bakrotu je porušeí koloběhu kaptálu. Korektí aplkace dskrmačí aalýzy vyžaduje mmo jé splěí předpokladu vícerozměré ormalty rozděleí dskrmátorů. Větša výzkumých prací však vícerozměrou ormaltu etestuje a předpokládá, že modely jsou dostatečě robustí a dávají rozumé apromace bez splěí tohoto předpokladu. Negatví výsledky testováí jedorozměré ormalty jedotlvých dskrmátorů pomocí testu kombace škmost a špčatost vedly k závěru, že emůže být splěa a vícerozměrá ormalta rozložeí všech dskrmátorů. Proto bylo rozložeí dat upraveo pomocí vyloučeí vybočujících hodot. Datové soubory zdravých frem frem ohrožeých bakrotem byly redukováy jedak pomocí vtřích hradeb, jedak pomocí Mahalaobsovy vzdáleost od středích hodot. Na redukovaé soubory pak byla aplkováa dskrmačí aalýza a provedea klasfkace frem a úspěšé a eúspěšé. Z výsledků klasfkace je zřejmé, že saha o přblížeí rozložeí dat vícerozměré ormaltě vede k podstatému zlepšeí klasfkačí schopost dskrmačí fukce. GeoScece Egeerg Volume LVIII (0), No.4 p. -9, ISSN