Fraud Detection using Benford s Law

Transcription

1 Fraud Detection using Benford s Law The Hidden Secrets of Numbers James J.W. Lee MBA (Iowa,US), B.Acc (S pore), FCPA (S pore), FCPA (Aust.), CA (M sia), CFE, CIA, CISA, CISSP, CGEIT

2 Contents I. History II. III. IV. Introduction Real-world Application Case Studies V. Caveats VI. VII. Demo (using ACL) Demo (using Excel)

3 History Benford s Law

4 History In 1881, Professor Simon Newcomb observed that the first few pages of his Logarithm Book (digits 1, 2, 3) looked more worn and dirtier than the other pages (digits 7, 8, 9).

5 History However, Professor Simon Newcomb did not provide any statistical basis for his observation, and his discovery did not have any practical applications

6 Don t Believe?

7 Introduction Benford s Law

8 Introduction Who s Benford? In 1938, physicist Frank Benford, who was unaware of Newcomb s observation, also discovered same phenomena with his Logarithm Book used by scientists & engineers. Unlike Newcomb, Benford attempted to test his theory with empirical data.

9 Introduction Frank Benford analyzed 20,229 data sets, eg: baseball statistics, areas of rivers, molecular weights of atoms, electricity bills, stock market quotes, populations of towns, physical and mathematical constants. He discovered that appearance of each digits (1 9) is not equally distributed, instead some digits appear more frequently than others. Benford s Law formulated.

10 Introduction So, what s Benford s Law? Benford s law states the probability of obtaining digits 1 through 9 in each position of a number Eg: in number 3879 first position: 3 second position: 8 third position: 7 fourth position: 9 What s likelihood of getting 3 in First position?

11 Introduction What s Benford s Law? (cont d) Most people assume probability for each digit 1 9 in any position of a number is 1/9, i.e. each digit is equally likely to occur. This is not true. Benford s Law: in any given position of a number probability of 1 is higher than 2 probability of 2 is higher than 3 probability of 8 is higher than 9

12 Introduction Digit 1 st Position 2 nd Position 3 rd Position 4 th Position Digit 0 is excluded because it s not first digit. Source: Nigrini, 1996

13 Introduction How is Benford s Law possible? If a data entry begins with the digit 1, it has to double in size (100%) before it begins with the next digit digit 2. If a data entry begins with the digit 9, it only has to be increased by only 11% in order for the first digit to be digit 1 again. Hence, chances of digit 1 is more likely than digit 9. Can t Understand? See Demo.

14 Introduction Mathematical Formula for Benford s Law The probability of any number d from 1 through 9 being the first digit is. Log 10 (1 + 1/d)

15 OK, now you know Benford s Law So What?

16 Real-World Application

17 Real-World Application Both Benford and Newcomb did not illustrate any practical use for this observation. In 1992, Chartered Accountant Dr Mark Nigrini used Benford s Law for fraud detection and assessment of inefficiency.

18 Real-World Application Mark Nigrini s Analysis What Benford s Law can do? To detect 1. Fraud 2. Non-compliance 3. Inefficiency

19 Mark Nigrini s Analysis Why Benford s Law can detect fraud? Because culprits 1. Attempt to evenly distribute digits, subconsciously 2. Consistently select specific numeric sequences 3. Avoid repetitions of numbers

20 Mark Nigrini s Analysis How to apply Benford s Law? Identify data with frequency of digits deviated from expected pattern Investigate deviated data

21 Mark Nigrini s Analysis Where can Benford s Law be applied on? Purchases, Payroll and other payments Sales and other income Claims and Refunds Cash deposits / withdrawal Bank balances Account Payable and Receivable Etc

22 Mark Nigrini s Analysis Eg: Apply Benford s Law on Purchases data Investigate deviated data sets Suppliers Purchase value Frequency Objectives: To uncover 1. Fraud (duplicate payment / bogus supplier) 2. Non-compliance (avoid threshold) 3. Inefficiency (massive small payment)

23 Distortion Factor model: Data conformed with Benford s Law Benford Law is applicable Transaction-level data Large data sets Data with mean > median & positive skew Data resulted from mathematical combinatn of numbers Examples Sales, expenses Full year s transactions Most accounting numbers A/Rec ble (no. sold x price) A/Pay ble (no. bought x pr) Source: Durtschi, 2004, 24.

24 Distortion Factor model: Data not conformed with Benford s Law Benford Law not applicable Truly random data Data with limits (max/min) Artificially created / assigned data Examples Lottery no. (4D, TOTO) Max ATM withdrawal amount Telephone no., housing block no., cheque no., invoice no., postal codes

25 Case Studies

26 Case Study #1 State of Arizona v. Wayne James Nelson (CV ) Nelson, a manager in Office of Arizona State Treasurer, diverted nearly US$2m funds to a fictitious vendor. Amount in defaulted cheques appeared to be in random, no duplicated amount, no round numbers (i.e. each amount included cents).

27 Case Study #1 However, subconsciously, Nelson repeated some digits & digit combinations, and tendency toward the higher digits. Nelson was arrested, despite he argued that he only demonstrated absence of safeguards in new computer system.

28 Case Study #2 SME (Anna M. Rose and Jacob M. Rose, Aug 2003) SME Owner expanded his 1-store family-owned business into a 4-store chain He relinquished some control to store managers He concerned about bookkeeping errors or possibility of fraud He applied Benford s Law to analyze each store s expenditure data

29 Rate Rate Case Study #2 Result of his Benford analysis: First Digit Test Digits 5, 6, & 7 appear more, while digit 1 appears less, than expected Digit Benford Sample Second Digit Test Again, digits 6 & 7 appear more, while digit 0 didn t occur at all Digit Benford Sample

30 Rate Case Study #2 Result of his Benford analysis (cont d): First Two- Digits Test 56 & 67 appear more than expected Benford Sample Two Digit Pair

31 Case Study #2 Action taken after result of his Benford analysis: SME Owner vouched sample of expenditure starting with the 56 & 67 sequences He discovered payments to unfamiliar vendors Further investigation revealed vendors did not exist! Payments gone to personal account of culprit store manager!

32 Caveats Benford s Law

33 Caveats False positives Not necessarily fraud Certain types of fraud will not be detected using Benford s Law analysis My Law vs Benford s Law Substitute with your own expected distribution

34 Demo (ACL)

35 Demo (Excel)

36 Thank You