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ISSN Goda XV - br septembar 01. REVIzOR Časops za teorju prasu INSTITUT ZA EKONOMIKU I FINANSIJE Bul. M. Pupa 10 B/I, Beograd, e-mal: tel: ; fax: Izdavač -

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ISSN Goda XV - br septembar 01. REVIzOR Časops za teorju prasu INSTITUT ZA EKONOMIKU I FINANSIJE Bul. M. Pupa 10 B/I, Beograd, e-mal: tel: ; fax: Izdavač - Publsher INSTITUT ZA EKONOMIKU I FINANSIJE Bulevar Mhala Pupa 10 B/I Beograd tel. 011/ fax: Redacja Edtors Adrć, dr Mro Berdovć, mr Mlorad Cvetaovć, mr Mlvoje Dragojevć, Dragut Ivaš, dr Maro Ivaš, dr Željo Ivaševć, dr Mlorad Ljubsavljevć, dr Sežaa Ljutć, dr Brao Mlojevć, dr Mroslav Negovaovć, dr Mla Polć, dr Staslav Račevć, dr Boždar Slovć, dr Dragoslav Stašć, dr Mle Stašć, dr Mlova Staojevć, dr Ljubša Šarć-Jovaovć, dr Kata Izdavač savet Edtoral Board Bogetć, dr Pavle Flpovć, dr Mlova Ivaš, dr Željo Martć, dr Slavoljub Mlojevć, dr Mroslav Mumovc, dr Saša Pasze, dr Zbgew Petrovć, mr Mloš Popovć, dr Deja Rodć, dr Jova Stać, dr Slavoljub Staovć, mr Ljubomr Svčevć, Stamra Vasljevć, dr Mro Predsed saveta Edtoral Board Presdet dr Pavle M. Bogetć Glav odgovor ured Edtor--chef dr Mroslav M. Mlojevć Pomoć glavog ureda Edtor--chef Assstat Aa Krvčevć Letor Proofreader MSc Marja Rodć Tehč ured Techcal edtor Radomr Lazovć Časops zlaz tromesečo (mart, ju, septembar decembar) Godšja pretplata ,00 dara, PDV uračuat REVIZOR Isttute for ecoomcs ad face Audtor TABLE OF CONTENTS Audtor Joral for theory ad practce Edtorals 5 APPLICATION OF THE FIRST DIGIT LAW IN CREDIBILITY EVALUATION OF THE FINANCIAL-ACCOUNTING DATA BASED ON PARTICULAR CASES Tadeusz Grabńs, Marzea Farbaec, Bartłomej Zabłoc, Wacław Zając 7 The Role ad Resposblty of the Audt Commttee Jozefa Bee-Trvuac 5 The role of a Audt Commttee prevetg cybercrme Mloš Ilć 35 FEATURES CONTINUOUS INTERNAL AUDIT Mle Stašć 49 HYBRID SOURCES OF FINANCE OF A CORPORATION Maro Ivaš 67 SPECIFICS OF INCREASE AND DECREASE OF CAPITAL JOINT STOCK COMPANIES Slavša Vučurevć 79 corporate ratg research Tatjaa Mrvć 85 FINANCIAL REPORTING CRISES Dragut Dragojevć 95 audt frms serba 011. Mroslav M. Mlojevć 98 FINANCIAL APHORISMS Mloje A. Mleovć 109 GLOSSARY Maro Petrovć 111 INSTRUCTIONS FOR AUTHORS 115 4 Tadeusz Grabńs, Marzea Farbaec, Bartłomej Zabłoc, Wacław Zając Pregled člaa UDK: ; APPLICATION OF THE FIRST DIGIT LAW IN CREDIBILITY EVALUATION OF THE FINANCIAL-ACCOUNTING DATA BASED ON PARTICULAR CASES Smo Newcomb ( ), the most famous Amerca astroomer of the tmes otced that frst pages of lbrary logarthm tables are much more wor ad drty tha later pages. Hs cocluso was that umbers begg wth 1 were looed up more ofte tha those begg hgher dgts, to 9. I 1881 he publshed hs observatos ad coclusos a paper ettled Note o the Frequecy of Use of the Dfferet Dgts Natural Numbers The Amerca Joural of Mathematcs, but hs paper remaed uotced. Fra Beford ( ), physcst at Geeral Electrc, re-dscovered the law. Watchg log tables he came to the same cocluso as Newcomb ad 1938 publshed a artcle called The Law of Aomalous Numbers Proceedgs of the Amerca Phlosophcal Socety. From 199 tll 1934 Fra Beford has aalyzed more tha 0,000 samples from about 0 datasets cotag umbers observed evromet. Beford s Law vestgates the frequecy dstrbuto of dgts a gve data set. It states that f we radomly select a umber from a table of physcal costats or statstcal data, the probablty that the frst dgt wll be a 1 s about 0.301, rather tha 0.11 as we mght expect f all dgts were equally lely. The followg formula, whch was frst proposed by Newcomb, represets the probablty of the frst dgt beg a d : P 1 log 10 + d ( d ) 1, ( d 1,,,9). Table ad graph below you shows the frequecy of leadg dgt from 1 up to 9. As we ca see the frequecy that the frst dgt wll be 1 s 30.10%, whle for dgt 9 t s oly 4.58%. ABSTRACT Keywords: Beford s dstrbuto, Beford s law, frst dgt law, fraud The artcle presets the use of Beford s law to detect potetal fraud, errors ad other rregulartes the face data sets. Authors tae to accout twelve dfferet facal cases order to detfy the areas that requre detaled aalyss. Results have bee dscussed detals, cludg graphs ad avalable statstcs as well as created for the research purpose. The artcle cotas the essece of Beford s law, descrpto of the sets that follow metoed law, statstcal tests ad measures of complace two dstrbutos. I the last part of the paper authors focused o presetg areas where Beford s law ca be used as a dagostc tool detectg aomales a sequece of umbers. prof. dr hab. Tadeusz Grabńs sardc: mgr Marzea Farbaec; gr Bartłomej Zabłoc, mgr Wacław Zając, Eoomsa Aademja u Kraovu 7 Fgure 1 - Frequecy of leadg dgt. Table 1 - Frequecy of leadg dgt. Leadg dgt Frequecy % 17.61% % % 5 7.9% % % 8 5.1% % Total % Ths couter tutve result apples to a wde varety of fgures, from may real lfe sources of data, cludg electrcty blls, street addresses, stoc prces, populato umbers, death rates, legths of rvers, physcal ad mathematcal costats. Beford Law ca be aalyzed ot oly terms of the frst sgfcat dgt (F1 test), but also for the frst two sgfcat dgts (F test) ad the frst three sgfcat dgts (F3 test). The frst formula s used to calculate the sequece frequecy of dgts (-elemet combato of umbers) at the begg of the umber. Probablty P{D} that the umber starts wth the sequece {D}: 1 F1, F, F3 : P{ D} log D Probablty that the secod (D ) ad thrd (D 3 ) sgfcat dgt of a umber the decmal system s d: 8 D, D 3 : P d log , + d d ( 0,1,,,9). Probablty that the last dgt of a umber the decmal system s d: 1 L1 : Pd, d ( 0,1,, 9). 10 I ay case, a close coformty to Beford s Law ca geerally be expected, f certa codtos are fulflled (Ngr 000): 1. All data must be recorded the same physcal ut. Ths mples that all data must descrbe the same (factual) stuato. For example, both prces ad quattes should ot be captured oe data set.. The data set may ot cota ay heret mma or maxma. I partcular, ths meas that there may be o lmts that determe whether a data pot s or s ot cluded the vestgated data set. For example, a data set may ot cota oly accouts excess of 400, because smaller accouts are booed through aother accout. 3. The data set may ot clude ay assged umbers. Assged umbers serve the purpose of detfcato ad do ot therefore arse from ay atural process. Ba accouts, persoel ad telephoe umbers, thus most lely do ot coform to a Beford dstrbuto. 4. A data set should ted to have more small tha large umbers, whch also accords wth the atural developmet process. For example, t ca be expected geeral, that small voces are foud more ofte tha large oes. Noetheless, the data set does ot have to be dspersed wdely (Ram 1976), (Rodrguez 004). I case a Beford Set s costtuted, oe could observe three major characterstcs: 5. Scale varace: multplyg all values the Beford Set by ay costat yelds aother Beford Set, as demostrated by Pham (1961). Ths s partcularly relevat wth respect to currecy coversos. 6. Base varace: Hll (1995b) was able to provde evdece that Beford s Law apples ot oly to umercal systems wth a bass of 10, but all other umercal systems as well. 7. Ivarace mathematcal operatos: rasg to powers, multplcato ad dvso (Hammg 1970, p. 1616; Schatte 1988, p. 446; Boyle 1994, p. 883) as well as addto ad subtracto (Hammg 1970, p. 161; Schatte 1988, p. 448) of Beford Sets wth oe aother lead to a ew Beford Set. 9 BENFORD LAW USAGE Beford s s used the accoutg professo to detect fraud. Because data le tax returs ad chec regsters follow Beford s, audtors ca use t as a hgh-level chec of a data set. I case of ay aomales, t may be worth to pay more atteto to ths partcular area as potetal fraud. Credt card fraud Moey lauderg Computer truso Telecommucatos fraud Accoutg Fgure - Fraud Arzoa. Scetsts from Australa Natoal Uversty Caberra have dscovered atural pheomea: the depths of almost 50,000 earthquaes that occurred worldwde betwee 1989 ad 009, the brghtess of gamma rays that reach Earth as recorded by the Ferm space telescope follows Beford Law. Ths dcates that Frst Dgt Law usage does t have to be arrowed oly to facal feld. May sources dcate also followg areas: Electo fraud Verfcato of statstcal data Medcal ad scetfc fraud Computer data storage archtecture desg Computer graphcs modfcato detecto DETECTED FRAUD IN ARIZONA IN 1993 I Arzoa, 1993 Waye James Nelso was foud gulty of attempts to defraud the almost two mllo dollars from the state fud. Durg the process, Nelso tred to covce the court to hs ocece, clamg that the fuds were trasferred to a fae suppler. Ths would dcate the lac of the adequate securty the ew computer system. Because huma choces are ot radom, t s ulely that the magary value (veted by someoe) followed Beford s Law. Below are show some sgals that may dcate decepto ths case: He started wth smaller amouts ad started to crease t; Most of the amouts were just below $ ; The dgt patters were almost opposte to Beford s Law dstrbuto over 90% of the values had 9, 8, 7 as frst dgt; Noe of the chec amouts were duplcated; There were o roud umbers. 10 Nelso ucoscously repeated some of the umbers ad combatos of umbers. Amog the frst two dgts of the value of checs combatos: 87, 88, 93 ad 96th were used twce. I the last two dgts: 16, 67 ad 83 are reproduced. It has a tedecy to peel the hgher umbers, whch s cotrary to the Law of Beford. A total of 160 dgts were used 3 umbers. The umber of dvdual dgts ragg from 0 to 9 swtches as follows: 7, 19, 16, 14, 1, 5, 17,, ad 6. Facal Cotroller famlar wth Beford s Law, by examg these fgures at oce sees that they do ot overlap wth the model, ad therefore deserve deeper cosderato. Graph 1 - F1 test: Nelso vs. Beford. Graph - All dgts used by Nelso. 11 1 MEASURES OF FIT Table presets measures of compatblty betwee emprcal ad theoretcal dstrbuto, whch follows the Beford law. Tests that have bee tae to accout are summarzed below. Tabel - Measures of ft. Ow Aalyss. Test Equato Ch-square ( ) ( ) p p 1 1 χ Kolmogorov-Smrov 1 (KS1) ) 1,..., ( 1 f f x ma D D KS Kolmogorov-Smrov (KS) ) 1,..., ( f f x ma D D KS Kolmogorov-Smrov 3 (KS3) [ ] [ ] N f f D f f D D D V where N N V KS N N N N N N sup sup ) 1,..., ( ] 0,4 0,155 [ 3 1/ Z-test ) 1,..., ( ) (1 p p p p z M 1 c c c M M ( ) c c M 1 1 M 3 ( ) c c M 1 3 M 4 c c c M ) ( 100 M 5 M ANALYSIS We have aalyzed 1 dfferet facal data sets obtaed from varous ecoomc areas (trasport, cosmetc, photographc, pharmaceutcal etc.). Each of them was examed for complace wth frst dgt law. I order to aalyze data we have created Excel template cotag several macros usg VBA code. Thas to that tool we were able to compare results of each case usg the same measures. Complete aalyss result cotas most mportat dstrbutos: Ch-square test, Z-test, Kolmogorov-Smrov test. Beford Excel Aalyzer s dvded to two sheets: Bf_put Bf_output_h Frst sheet s used to operate o put data ad maage macro optos. Frst ad most mportat feature s cludg F ad F3 tests. Full aalyzg process taes a few mutes (depeds o amout of data, computer performace etc.). I order to perform quc aalyze we ca dsable those features. I some cases audtors have to focus o partcular type of data. Beford Excel Aalyzer allows to exclude some percetage of smallest or/ad largest values from populato. 13 Macro s desged by default to sp egatve values. But aother opto allows aalyzg absolute value from egatve umbers: It s also possble to set values for partcular o-complace parameters. Those wll be used bf_output_h sheet Z-Test table. Last opto allows maagg hstogram optos. There s possblty to chage umber of rejected m ad max values for each terval. The hstogram s prted after performg aalyze o bf_output_h sheet. I order to perform aalyze user has to push Aalyze! butto. Macro goes trough data ad cludes oly ettes whch meet the crtera set bf_put sheet. Frstly the detal tables for every cluded test are flled wth data. 14 Based o partal data detal tables for every test a summary table s composed. It cotas comprehesve formato, tag to accout the partculars. The summary table cludes the followg formato: umber of observatos for every test, M1-M5 meters, correlato coeffcet, Ch-square test, Z-test, Kolmogorov Smrov test ad statstcal parameters: mmal ad maxmum values, average value, stadard devato, urtoss ad sewess. Curret example was chose as the worst-fttg oe from the set of twelve data sets. It cotas teger values of sales voces of facal compay (compay ame s cofdetal). Followg fgure aother tool opto compares graphcally emprcal ad theoretcal data. As t s show some cases there are sgfcat devatos: Graph 3 - Beford test graphs. The table below cotas summarzed Z-test values whch measures complace of Beford tests wth emprcal values. Secod part of the table shows ra for partcular test result (the ras are customzable bf_put sheet). 15 16 Graph 4 - Z-test detals. Very useful mght become charts of the emprcal values of Z-test: Graph 5 - Z-test graphs. They ca be compared wth correspodg graphs of Beford dstrbuto: 17 Graph 6 - Comparso of Beford F3 test graph ad Z-test (F3) graph. Abormal values should tur o warg lght ad focus audtors specal atteto. Below aalyss cocers dataset cotag polygraph compay sales voces. Dataset s cosdered as best-fttg to Beford s Law (from 1 sets). As t s show for Ch-square test oly two from sx values cotas the crtcal set ( case of FV3 dataset all 6 values were statstcally sgfcat). Cosderg Z-test results t s cocluded that oly 10 cases values reach crtcal value (level of sgfcace 0,05) ad 3 cases o level of sgfcace 0,01. 18 The same summary prevous dataset showed respectvely 94 ad 76 ll-fttg values. Also graphs are t showg ay bgger dffereces betwee emprcal ad theoretcal values: Graph 7 - Beford test graphs. Table 3 cotas bary ra ch-square test where 1 states that statstc value s less tha crtcal oe. It s mpossble to declare whch sample s the best ad the worst ftted to Beford Law based oly o below table. I order to eable creatg such a cocluso, a ew statstc has bee bult. Table 3 - Evaluato of bary ra. Ow aalyss. Ch-square F1 F F3 D D3 Sum FV 1 0 FV FV 3 0 FV FV FV 6 0 FV FV 8 0 FV FV FV 11 0 FV 1 0 Sum Emprcal values of Ch-square, KS1, KS ad KS3 were dvded by crtcal values at 0,05 cofdece level ad averages of ew statstc were calculated. The data sets ad tests were arraged by decreasg values of the calculated averages. Wth ths orderg, cases that are the worst terms of ft to Beford s Law ca be foud at the top ad tests that are the most restrctve at the left of the Table 4. Table 4 - Quotet emprcal values to crtcal values. Ow aalyss. Ch-square F1 F F3 D D3 FV 1,3 3,05,84 3,37 6,96 FV 0,83 0,97 1,59 0,90 14,4 FV 3 1,6 46,04 14,57 34,69 13,59 FV 4,79,35,50 0,7 0,96 FV 5 0,93 0,97 1,0 0,65 0,4 FV 6,85 3,04 3,04 6,81 3,5 FV 7 1,06 0,93 1,08 0,55 0,43 FV 8 5,74 3,44 1,78 5,66 6,9 FV 9 1,84 1,04,44 0,61 33,36 FV 10 3,01,6 1,40,3 0,38 FV 11 8,57 1,3 1,68 11,18 68,18 FV 1 6,80 10,0 11,84 30,00 16,57 The results of the aalyss dcate that data sets FV5 ad FV7 are best suted to Beford Law whle the FV3, FV1 ad FV11 are worst ftted. Beford dstrbuto usually s dcated by a test D ad the by F1. Tests F ad F3 are more restrctve ad less lely to dcate the compatblty of the emprcal dstrbuto wth the Beford Law. These coclusos were draw based o all four tests. Table 5 - Rag of the cases. Ow aalyss. Ch-square KS1 KS KS3 FV 1 FV 3 FV 3 FV 3 FV 3 FV 1 FV 1 FV 1 FV 11 FV 4 FV 4 FV 11 FV 9 FV 11 FV 11 FV 4 FV 6 FV 10 FV 10 FV 10 FV 8 FV 6 FV 6 FV 6 FV FV 1 FV 1 FV 1 FV 1 FV 8 FV 8 FV 8 FV 4 FV 9 FV 9 FV 7 FV 10 FV 7 FV 7 FV 5 FV 7 FV 5 FV 5 FV 9 FV 5 FV FV FV 0 Table 6 - Rag of the tests. Ow aalyss. Ch-square D3 D F F1 F3 KS1 F3 D3 F F1 D KS F3 D3 F F1 D KS3 F3 D3 F F1 D Table 7 - Ch-square test. Ow aalyss. Ch-square test Average of emprcal value/ crtcal value Coeffcet of varato Kurtoss Sewess Number of observatos FV 1 44, ,4 616,7 43, ,0 FV 3 6,03 93,5 5,6 4,7 7418,0 FV 11, ,5 38,3 13, ,0 FV 9 7, ,6 1,6 3,1 575,0 FV 6 7, ,5 63,7 7,3 3119,0 FV 8 4,7067 7,7 5,5 5,6 107,0 FV 3, ,3 14,3 3,3 500,0 FV 1 3, ,9 373,1 61,1 3733,0 FV 4 1, ,5 4346,5 64, ,0 FV 10 1, ,8 393, 15,6 4884,0 FV 7 0, ,4 45, 4,7 7999,0 FV 5 0, ,9 51, 5, ,0 Table 7 combes the averages of emprcal values dvded by crtcal values, parameters of the dstrbuto: coeffcet of varato, urtoss, sewess ad sample sze. I order to verfy depedece betwee values, the correlato matrx was calculated ad preseted Table 8. Table 8 - Ch-square test - Correlato matrx. Ow aalyss. Correlato matrx Average emp./ crtc. Coeff. of varato Average emp./ crtc. Coeff. of varato Kurtoss Sewess No. of observatos 1,000-0,149 0,101 0,1 0,459-0,149 1,000 0,907 0,908 0,000 Kurtoss 0,101 0,907 1,000 0,993 0,59 Sewess 0,1 0,908 0,993 1,000 0,58 No. of observatos 0,459 0,000 0,59 0,58 1,000 1 As mght be expected there s a large correlato betwee coeffcet of varato, urtoss ad sewess of 0.9. What mght be terestg there s o correlato betwee dstrbuto parameters ad the degree of coformty wth Beford dstrbuto. There are traces of depedeces betwee sample sze ad the degree of match of 0.5, whch meas that larger sets are less ftted to Beford law. CONCLUSION As you already fd out, Beford`s Law ca be used as a tool for the aalyss of facal ad accoutg data. It has bee show that preseted macro gather ad cout the most mportat measures ad how they should be terpreted. But, durg the aalyss of data usg methodologes preseted the case of detecto, of devatos ad aomales s ot evdece crmal proceedgs. Ths s the premse that data should be subjected more careful wth detaled examato, because there s a hgh probablty that they were mapulated. However, Beford`s Law s ot oly used to chec the facal data. So far, t has bee used for: Detecto of false data or utetoal accoutg error; Detecto of tax fraud; Aalyss of stoc maret data (prces ad tradg of securtes); Aalyss of the prces of goods o Iteret auctos; Assessmet of the clcal effcacy of drugs; Desgg a storage archtecture for computers; Dstgushg betwee actual photographs from the graphcs geerated by computer programs (where the actual mages pxel value should be cosstet wth Beford dstrbuto); Aalyss of the accuracy of the estmates of clams surace compaes; Assess the relablty of fes ad facal pealtes whch are mposed court (correctly determed the sze of the fe should reflect the sze of the damage ad ot hover aroud arbtrarly set lmts the legslato), I 010, Polsh Mstry of Face aouced a teder for the applcato, whch betwee other wll use Beford s Law to aalyze the collected data. Ths s oe of the frst steps tae by the govermet to use ths Law practce. I the offer, the requremet s descrbed as: The use of Beford s Law (whch was based o a aalyss of the sequece umbers) to detect possble errors, potetal fraud or other rregulartes. Not so log ago t was sad that the emsso of toxc gases by Polad, exceedg the stadards accepted by the Europea Uo. Thas to Beford s law, we ca chec whether the data collected by the EU o the ssue of all EU coutres are le wth the orm ad do ot correspod to ay devatos. The same apples to agrcultural subsdes. It s ow that the sze of agrcultural subsdes grated accordace wth the amout held by the farmer s feld. So f the admstrato wll report more acres, farmer wll get more moey. Ths practce s ofte used by people who brbe the govermet worer, to fll the approprate box the adequate data. The farmer gets more ad whole socety losses. Usg Beford s Law, collected the collecto of formato, we are able to detect abormaltes related to the sze of grats awarded. I other words we are able to verfy the data. Aother example of the use of the Law may be the Cetral Statstcal Offce, whch collects most of the areas of lfe, ad some stes of publc ad prvate lfe. Tag to accout the fact that the use of Beford s Law s very broad, our research ad aalyss that curretly we serve to provde the tools for deep statstcal aalyss. These examples, methodology ad coclusos wll be gathered to oe coheret boo. There s, at ths pot, ot eve a sgle polsh publcato, whch would gather formato about the possbltes of usg Beford s Law. There are plety of artcles ad wors coected wth t, but f you would loo for a scetfc publcato - t does ot exst. The usefuless of Beford s Law fdg ufaress s justfed by the fact that geeral people ca ot fabrcate radom dat

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