Statistical Support of Forensic Auditing ALAN D. OLINSKY Oepartmeut of Mathematics Bryant CoUege Smithfield, Rhode Island 02917 PAUL M. MANGIAMELI Department of Management Science and information Systems University of Rhode Island Kingston, Rhode Island 02881 SHAW K. CHEN Department of Management Science and Information Systems University of Rhode Island Kingston, Rhode Island 02881 During a three year period, $4.38 million of gold was stolen from the vaults of Sammartino's House of Diamonds. A civil suit be- tween the gold's owner and its insurance company centered on when the gold was stolen. The time of the loss was critical to the insurer's contention that it was not liable because the gold was missing prior to the inception date of the policy. The insurer's at- torneys retained an accounting firm to conduct a forensic audit (a financial audit to investigate fraud). The key to this investiga- tion was reconstructing a daily gold inventory balance by sam- pling sales invoices. Since the audit's sampling methodology was open to question, we were retained to use various statistical tech- niques to verify the process. To test the results of the forensic au- dit, we constructed two formal hypotheses: The first to test for nonrandomness in the selected sample observations; the second to test if the amount of sales dollars selected in the sample was proportional to the amount of sales dollars in the monthly popu- lation. To accomplish this, we utilized a one-sample runs test, time-series charts, regression analysis, and interval estimations. We found that the sampled invoice transactions exhibited no evi- dence of nonrandomness or bias and appeared to be representa- tive of the population. Hence, these results supported the inven- tory reconstruction obtained by the accounting firm. Ciipyrishi «:> !'>%, Inslitule lor tlperations Research STATISTICS—SAMPLING and Ihe Management Scit-ncm jUDFCIAL/LEGAL OO9Z-3l02/%/2bO6/O(»5S01.25 This paper was reftTit'd. INTERFACES 26: 6 November-December 1996 (pp. 95-104)
Transcript
Statistical Support of Forensic Auditing
A L A N D . OLINSKY Oepartmeut of MathematicsBryant CoUegeSmithfield, Rhode Island 02917
PAUL M . MANGIAMELI Department of Management Science and informationSystems
University of Rhode IslandKingston, Rhode Island 02881
SHAW K. CHEN Department of Management Science and InformationSystems
University of Rhode IslandKingston, Rhode Island 02881
During a three year period, $4.38 million of gold was stolen fromthe vaults of Sammartino's House of Diamonds. A civil suit be-tween the gold's owner and its insurance company centered onwhen the gold was stolen. The time of the loss was critical to theinsurer's contention that it was not liable because the gold wasmissing prior to the inception date of the policy. The insurer's at-torneys retained an accounting firm to conduct a forensic audit(a financial audit to investigate fraud). The key to this investiga-tion was reconstructing a daily gold inventory balance by sam-pling sales invoices. Since the audit's sampling methodology wasopen to question, we were retained to use various statistical tech-niques to verify the process. To test the results of the forensic au-dit, we constructed two formal hypotheses: The first to test fornonrandomness in the selected sample observations; the secondto test if the amount of sales dollars selected in the sample wasproportional to the amount of sales dollars in the monthly popu-lation. To accomplish this, we utilized a one-sample runs test,time-series charts, regression analysis, and interval estimations.We found that the sampled invoice transactions exhibited no evi-dence of nonrandomness or bias and appeared to be representa-tive of the population. Hence, these results supported the inven-tory reconstruction obtained by the accounting firm.
Ciipyrishi «:> !'>%, Inslitule lor tlperations Research STATISTICS—SAMPLINGand Ihe Management Scit-ncm jUDFCIAL/LEGALOO9Z-3l02/%/2bO6/O(»5S01.25This paper was reftTit'd.
After the largest gold theft in the his-tory of the State of Rhode Island, the
gold's owner, Putnam Resources, tried torecover its losses from its insurance com-pany in a civil trial. Amid rumors of theftand fraud, Sammartino's House of Dia-monds, Inc., a manufacturer and retailer offine jewelry located in Cranston, Rhode Is-land, had its business seized by ShawmutBank of Boston on August 3, 1987 whenthe bank, a major creditor, found thatSammartino's did not have the funds tocover its debts. One year later, a federalgrand jury indicted Walter Sammartino,the owner, on 16 counts of defraudingShawmut Bank of $8 million. In addition,an estimated $4.38 million dollars in goldconsigned to the company was missingfrom the House of Diamonds' vaultIRakowsky and Howard 1988]. As the trialbegan, a plea bargain was struck with Mr.Sammartino pleading guilty to four of the16 counts of bank fraud. As part of thedeal, the remaining 12 counts weredropped, and Mr. Sammartino was not in-dicted on any counts stemming from thegold theft [Boston Globe 19891. On May 18,1989, Mr. Sammartino was fined $50,000and sentenced to serve seven and one halfyears; five of those years were suspendedand two and one half years were to beserved in prison. Although federal authori-ties have identified several people involvedin the gold theft, including members of thePatriarca crime family, no one has been ac-cused of the crime [UPI 19891.All That Glitters Is Not Gold
Although the criminal justice portion ofthis episode ended in mid-1989, a civil suitfiled on August 4, 1987 by Putnam Re-sources of Greenwich, Connecticut was still
active. Putnam owned the gold that wasmissing from the Sammartino vaults. Leas-ing gold is a common practice in the RhodeIsland jewelry industry: The leasing com-pany sends gold to a jewelry manufacturerwhile still retaining ownership. As the rawgold is turned into jewelry, the manufac-turer becomes the owner of the metal usedand pays the leasing company for it. Peri-odically Putnam sent its auditors, Emst andWhinney, to check their accounts againstthe amount of gold in the vaults. Unfortu-nately, the auditors only counted andweighed the gold bars; they did not assaythe gold. It was subsequently discoveredduring examination at the civil trial that the"gold" bars were actually base metal barselectro-plated with gold by Sammartino'sproduction supervisor.
The court awarded judgments againstWalter Sammartino and Sammartino'sHouse of Diamonds to Shawmut Bank andPutnam Resources. As neither Mr.Sammartino nor the House of Diamondshad assets to cover these judgments,Shawmut sued its insurer and collected itsloss. Putnam also sued its insurer, Lloyd'sof London. Lloyd's originally paid $2 mil-lion, but then, in mid-February 1990, re-fused in federal court to pay the outstand-ing balance and demanded its original $2million back IHarrop 1990a].The Heart of the Case
Lloyd's of London's case hinged uponwhether the gold had been stolen before orafter the insurance pohcy went into effecton September 11, 1986. From May 14, 1984to September 10, 1986, Sammartino's re-ceived 40,398 fine troy ounces (FTO) ofgold of which accounting records of goldbeing processed could account for only
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21,300 FTO (Figure 1). Supposedly, the re-maining 19,098 FTO should have been ininventory in Sammartino's vaults. If, infact, 19,098 FTO was in the vaults throughSeptember 10, 1986, when the policy wentinto effect, then Putnam's claim againstLloyd's would be substantiated. If most ofthe gold was stolen before that time, thenLloyd's counterclaim would be correct.
It should be noted that no one physicallytook most of the missing gold. The employ-ees of Sammartino's House of Diamondshad made jewelry out of the gold and soldit to customers. The business kept meticu-lous records of these sales transactions. Thetheft occurred when Mr. Sammartino, in-stead of paying Putnam, kept the money to
Beginning inventory 5/14/84Gold received
Total gold available
Gold out for processing
Gold unaccounted for
pay off debts to underworld figures.Five witnesses who would have known
the dates of the robbery invoked the FifthAmendment against self-incrimination. Asixth witness, Stephen Saccoccia, disap-peared before the courts could serve a sub-poena. (After the trial, he was found andextradited from Switzerland. He was laterconvicted and sentenced to 50 years forlaundering over $180 million for thePatriarca crime family.) A seventh witness,Charles Harrison, a Sammartino ware-houseman who electroplated the fake goldbars, although fearing for the safety of hisfamily, did testify but gave conflictingdates as to when the gold was stolen.Sometimes he said it occurred in July 1986
KnownKnown
Known
Known
Retail Sales
Ending Inventory 9/10/86
Unknown
Unknown
To/a/of both is 19,098
Putnam claims that Sammartino'sending inventory should he J 7.000
Figure 1: Gold flows in fine troy ounces al Sammartino's House of Diamonds for the period 5/14/84 to 9/10/86 show the lotal gold available to be 40,398 ounces. The forensic audit had to deter-mine the value of retail sales and of the ending inventory on September 10,1986, the date theinsurance policy took effect. The retail sales represent how much gold was stolen before the pol-icy went into effect while the ending inventory on that date was the gold that Putnam claimedunder the policy.
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and other times he testified to dates of Oc-tober or November 1986 [Harrop 1990bl.Given that Lloyd's issued the insurance onSeptember 11, 1986, its lawyers had to findother means to determine the date of thetheft.The Forensic Audit
Lloyd's retained the accounting firm ofSansiveri, Ryan, Sullivan, and Company todetermine when the gold was stolen. Itconducted a forensic audit to recreate theinventory record. To do this, it took a san:\-ple of Sammartino's House of Diamonds'sales receipts and randomly chose one ofthe jewelry items listed on each receipt. Re-ferring to this item's manufacturing specifi-cation, it determined the amount of goldneeded to make the item. If the sample re-flected overall retail sales, it could thus es-tablish the average sales dollars needed forone fine troy ounce (FTO) of gold. Giventhat the daily sales for Sammartino's wereknown, it could use this average to deter-mine how much gold had been used inmaking jewelry (and thus how much hadbeen stolen) and how much was actually inthe vaults and owned by Putnam.
To get a sample of the invoice records,the accounfing firm sent several auditors toShawmut Bank in Boston which held thesales invoices. These files of invoices werestored chronologically in marked boxes.The auditors would pick a box in a haphaz-ard fashion and then reach into the box andpull out a manila folder that contained allthe sales receipts from a given day. Fromthis folder, they would select an individualsales receipt in an arbitrary fashion. If thisreceipt had more than one item on it, theywould randomly select one of these itemsand find the corresponding manufacturing
specificafions. The auditors confinued inthis fashion until they had reviewed 7,750invoices totaling approximately $2 million(23.34 percent of population dollar salesover the period May 14,1984 to June 25,1987).
They thus reconstructed the inventory(Figure 2). The accounting firm's forensicaudit showed a large discrepancy betweenwhat it found should have been left andthe gold that Putnam had shipped andthought was in Sammartino's vault. The fo-rensic audit showed that over 8 thousandFTO of the total of 11.2 thousand FTO ofgold had been stolen before the policy tookeffect on September 11, 1986 (Figure 3).
The auditors presented the results toJames Campise, the attorney representingLloyd's in early 1990. Although he had noobjection to the method used to measurethe amount of the gold theft, he questionedthe sampling methodology. He thought thesampling procedure was not truly randombut biased in some way. If it were, the av-erage sales dollars to FTO of gold might bewrong, thus leaving the forensic audit's in-ventory reconstruction open to question. Inessence, Mr. Campise felt that the forensicaudit contained a sampling error. Whatwas needed was independent statisticalverification that the results of the forensicaudit were in fact based on random sam-pling and representafive of total sales. Theaccounting firm asked us to do this work.Because he might have to present the statis-Hcal support to the jury, Mr. Campiseasked us to make our report visually com-pelling by using charts and graphs that thejury members could easily understand.With this request in mind, we set about de-veloping our statistical analysis.
Figure 2: This graph compares, at key dates, the amount of gold that should have been on handaccording to the leasing company's records with the amount of gold on hand as evidenced by theforensic audit's inventory reconstruction. This chart was presented as evidence during the trial.
Statistical SupportWe had about one month before the trial
started. If the sampling method were bi-ased, Lloyd's case would be badly dam-aged. We constructed two formal hypothe-ses to test the results of the forensic audit.The first hypothesis was designed to testfor nonrandomness in the selected sampleobservations.Hypothesis 1: There is no pattern of bias inthe sample taken by the forensic auditors.Specifically, the sequence of ratios of sales
dollars to FTO of gold for each transactionin the forensic audit's sample was random.
Since we would ultimately make infer-ences in the form of confidence-interval es-timates, we had to first test the sample forevidence of nonrandomness or bias. Wesubjected the sequence of ratios of salesdollars to FTO of gold to a Wald-Wolfowitzone-sample runs test. This is a nonparamet-ric test for runs above and below the me-dian of a sequence of values (in this case,our ratios) that are arranged in temporal
Figure 3; Based upon Figure 2, this chart shows the cumulative loss of gold when comparing theamount of gold on hand according lo the leasing company's records with the amount of gold onhand according to the forensic audit. If this forensic audit is correct, most of the missing gold wasstolen before the policy was issued.
order. Too many runs or too few runswould indicate nonrandomness in the se-lection of the sample. Eor example, if theauditors had selected invoices with largedollar amounts early in the selection pro-cess (a distinct possibility if they wanted tospeed up the sampling process), this wouldhave caused a pattern of ratios that theruns test would detect.
Eor this test, we needed the total numberof runs, the number of individual values
below the median, and the number of indi-vidual values above the median. As long asthe number of values below the medianand the number above both equal or ex-ceed 10, it is appropriate to use a z scorefor testing fWatson et al. 1994]. The one-sample runs test was not significant(2 = -0.5687 and p = 0.5696), and there-fore, our first major hypothesis was not re-jected. There was no evidence of a patternof nonrandomness.
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The second hypothesis we introducedwas as follows:Hypothesis 2: For all the months involvedin the sampling process, the amounts ofsales selected in the sample were propor*tional to the amounts of sales in the popu-lafion on a monthly basis.
If we were to develop inferences basedupon the ratio of sales dollars to FTO ofgold, it was critical that the sampled salesbe truly representative of the populationsales. We used two different statisticalmethods to test this hypothesis. Since wehad available the actual population dollarsales figures for each day of the period, wefirst developed a time-series chart to com-pare the total sales population with thesampled sales for each month of the twoyears under consideration.
We carefully examined the time-serieschart (Figure 4) and found that the amountof sales in the sample for the variousmonths increased and decreased along withthe amount of total sales. We also dividedeach month's sales by the total sales for thetwo-year time period to find the percent ofsales attributable to each month (Figure 5)and found that the sample mimics the pop-ulation very well. These results supportedour second major hypothesis.
In addition to testing these two hypothe-ses, we also tested the accounfing firm's es-timate of the ratio of sales dollars to FTO.We first created a scatterplot of sales dol-lars to FTO of gold which showed a linearrelationship between the two variables. Wetherefore deemed it appropriate to run ahnear regression with sales dollars as thedependent variable and FTO of gold as theindependent variable. We forced the result-ing regression line through the origin so
that its slope would be a ratio estimate. Asthis procedure is independent of themethod used in the forensic audit, our ratiowould not necessarily be the same as deter-mined in the forensic audit.
The regression line, forced through theorigin, fit the data quite well: R", that is thenumber of FTO of gold (independent vari-able), explaining 80.55 percent of the vari-ability in sales dollars (dependent variable).In other words, only t9.45 percent of thevariability in sales dollars was due to otherfactors. The slope of this line was found tobe $844.30, which is our independent esti-mate of the rafio of sales dollars to FTO ofgold. In other words, $844.30 received fromcustomers was due to the sale of one finetroy ounce of gold. Since we used a differ-ent stafistical technique, we did not expectthis estimate to be identical to the point es-timate of $882.80 determined by the foren-sic audit. The auditors calculated their ratioby dividing the total sales dollars in thesample by the total number of FTO of gold.Using a regression approach, we found theratio by determining the slope of the linebased on all 7,750 transactions in the sam-ple. Because we used all the values, ratherthan just two numbers, we probably ob-tained a more accurate ratio than the foren-sic auditors did.
Next, we used the estimated slope of theregression model to construct a 95 percentconfidence interval for our ratio. We com-pared this confidence interval with the con-fidence interval based upon the approachthe auditors used. If the two confidence in-tervals failed to overlap, we would behevethe sample was biased.
Using our regression estimate of $844.30,we constructed a 95 percent confidence in-
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OLINSKY, MANGIAMELI, CHEN
Sales ($ Thousands)
700
600
500
I 1 1 1 r
6 7 8 9 10 1 1 1 2 1
84
Month
^ Population + Sample
Figure 4: In Ihis comparison of the recorded total sales ($ thousands) with the sampled sales ($thousands) over the period from June 1984 to June 1986, the two lines seem to parallel each other,indicating that the sample represents the total sales accurately.
terval for the ratio, yielding a range from$802.12 to $886.48. Our range includes theforensic audit's ratio estimate of $882.80. Inaddition, a 95 percent confidence interval,based on the methods of Scheaffer,Mendenhall, and Ott 11979] for the forensicaudit's ratio ranged from $812.27 to$953.33. This confidence interval has amuch wider range than the confidence in-terval from the fitted regression model.However, the point estimate of the regres-sion slope, $844.30, is within the confidenceinterval of the forensic audit. Given that the
forensic auditor's ratio fell within the re-gression model's confidence interval andthat the regression model's ratio fell withinthe auditor's confidence interval, we con-cluded that the forensic audit estimate of$882.80 represents the population ratio.
The results of the time-series analysisand the overlapping confidence intervalssupported our second hypothesis. The sam-ple does indeed represent the entire popu-lation.
In that they supported both hypotheses,our findings supported the forensic audi-
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0.12
0.1
6 7 8 9 1 0 1 1 1 2 184 I
Month
"^Population + Sample
Figure 5: This comparison of the percent of sales in the population with the percent of sales inthe sample from June 1984 to June 1986 shows that they Irack together very well, again indicatingthat the sample accurately represents the population.
tor's inventory reconstruction. This meantthat over 70 percent of the gold had beenstolen before the policy took effect on Sep-tember 11, 1986.Conclusions
Armed with the forensic auditor's resultsand our statistical support, Mr. Campisewent to court and successfully defendedLloyd's of London. On March 2,1990, thejury ruled that Lloyd's was not liable asmost of the gold had been stolen before thepolicy date. Putnam was ordered to repaythe $2 million advance plus interest to
Lloyd's [UPI 1990].The accounting firm of Sansiveri, Ryan,
Sullivan, and Company, which conductedthe forensic audit, had been far too haphaz-ard in sampling methods. Although no onecan ever establish whether the case wouldhave been determined in Lloyd's favorwithout our statistical support, Mr.Campise clearly had far firmer groundupon which to build his case because ofour work. Even though we were brought inafter Sansiveri, Ryan, Sullivan, and Com-pany conducted the forensic audit, we used
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statistical techniques to verify that its re-sults were, in fact, random and representa-tive of total sales.
In retrospect, we should have conductedadditional statistical tests, such as a Chi-square test. Two factors stopped us fromperforming more formal tests. Because Mr.Campise wanted visually compelling sup-port, we used approaches that providedcharts and graphs that the jury could easilyunderstand and appreciate (a Chi-squaretest would not have done this). Also, wehad very little time in which to conduct ourstudies. With this rush we honestly did notthink of using any statistical tests that werenot visual in nature, even though they aremore powerful. Clearly, had the accountingfirm employed trained statisticians fromthe beginning instead of bringing them inat the last minute to "bail them out," theywould have used correct sampling meth-ods. This would have saved them time,money, and anxiety. In such critical areasof litigation as tax fraud and forensic audit-ing, proper statistical methodology plays acrucial role.
Harrop, Froma 1990a, "Lloyd's reneged in goldloss, court told," The Providence Journal-Bulle-tin, Eebruary 7, Vol. 18, No. 33, Sec. B, pp. 1.
Harrop, Froma 1990b, "Witness testifies hefaked gold," The Providence journal-Bulletin.February 16, Vol. 18, No. 41, Sec. B, pp. ].
Rakowsky, Judy and Howard, Peter E. 1988,"Jewelry dealer Sammartino is indicted byUS," The Providence journal-Bulletin. September23, Vol. 159, No. 186, Sec. A, pp. I.
Scheaffer, R. L.; Mendenhall, W.; and Ott, L.1979, Elementary Survey Sampling, second edi-tion, Duxbury Press, North Scituate,Massachusetts.
United Press International 1989, "Judge sentencesex-jeweler for bilking bank," May 18, RegionalNews.
United Press International 1990, "Lloyd's earns $2million judgment," March 4, Financial News.
Watson, C; Billingsley, P.; Croft, D.; andHuntsberger, D. 1994, Statistics for Managementand Economics, fifth edition, AUyn and Bacon,Boston, Massachusetts.
Daniel J. Ryan, CPA, Partner, Sansiveri,Ryan, Sullivan, and Company, CertifiedPublic Accountants, 55 Dorrance Street,Providence, Rhode Island 02903, writes,"Our court testimony, as to when the goldwas missing, depended heavily on our in-ventory reconstruction which in turn de-pended to a great degree on our sample ofinvoice transactions. Although we tookgreat care in obtaining a very large sampleof invoice transactions in a random fashion,our attorneys were concerned about thisrandomness issue being challenged in court.
"Therefore, it was important for us tohave supporting documentation that oursample of invoice transactions showed noevidence of nonrandomness or bias. Weturned to Dr. Alan Olinsky, professor ofmathematics at Bryant College, for his pro-fessional expertise.
"The efforts documented in Dr. Olinsky'spaper submitted to your journal resulted inthe required supporting documentation.We felt confident of the legal standing ofour inventory reconstruction by virtue ofhis work as summarized in his paper.Lloyd's of London was also most pleasedwith the resulting court decision in theirfavor."
