기업회계 패턴분석에의 응용 Benford's Law, Scale Invariance and Fraud Detection 최 원규 Advanced Information Technology

Contents I. Audit Risk and Risk-Based Sampling II. Benford's Law – History III. Examples IV. Analysis V. Application in Audit Analysis VI. Summary

I. Audit Risk and Risk-Based Sampling 감사업무 - 회계장부의 적합성, 수치의 정확성, 재산의 보전 확인 - 고의적인 누락, 부정 또는 실수, 프로세스의 문제점 적발 Sampling 필요성 - 시간, 비용의 제약으로 전체 데이터를 검증할 수 없음 - Sampling : 일부 데이터를 선정하여 검증 - Sampling 을 잘못할 경우 부정을 적발하지 못함 (Audit Risk) Sampling 방법 - Risk Ranking : 사전 정의된 Risk 수준에 의해 - Risk Indicator : 데이터의 부정여부 를 판별하도록 해 주는 장치 Risk Indicator 의 개발이 중요

II. Benford's Law – History (1). Motivation Benford's Law - “ 숫자 " 에 숨어있는 법칙 숫자들의 첫번째 숫자의 분포는 ? D = 1,5,6,3,4,2,3,1,7,1,5,..... First Guess – Uniform ??? Second Guess – No Regular Pattern??

II. Benford's Law – History (2). Simon Newcomb, Newcomb observed that the first few pages of his logarithm book were more worn and dirtier than the other pages... Newcomb theorized that scientists spent more time dealing with numbers begin with 1,2 and 3, than others. Logarithm book for fast multiplication Example. 2,467 * 785 = ? 2, ,000, ,930,000 Log Table + = Log Table

II. Benford's Law – History (3). Frank Benford, 1920's A GE Engineer, Observed Discovered : P(D) = log( 1 + 1/D )

III. An Experiment – Investment Problem (1) Invest $1,000, with interest rate 5.40%

III. An Experiment – Investment Problem (2), Scale Invariance Invest $2,000, with interest rate 5.40% Benford's Law is Scale Invariant !!

IV. Analysis (1) – Investment Problem Value of Investment V, Face Value M, rate R, over N periods, V = M, MR, MR 2, MR 3,... MR N First Digit D Determined by Fractional Part of Log(V) Log(V k )= Log(MR k ) = Log(M) + k Log(R) Investment : A Shift Process on Circle Log(V k+1 ) = Log(V k ) + Log(R) mod 1 When Log(R) : Irrational (Mostly..), Log(V k ) Uniformly Fills the Unit Circle. Thus, Scale Invariant. Log1 Log(M) Log(R) P(D) = Log(D+1) – Log(D) = Log(1+1/D) Log2 Log3 Log4 Log5 Log10

IV. Analysis (2) - Generalization From the Original Logarithm Book Problem, Let log(x) = n1 + a1, log(y) = n2+a2, and z = x*y Then, log(z) = (n1+n2) + a1 + a2, First Digit of z is determined by fractional part of (a1+a2) a1 a2

IV. Analysis (3) a1 a2 첫번째 글자가 3 보다 같거나 작은 경우 (1) 0 < a1+a2 < log(4) (2) 1 < a1+a2 < 1 + log(4) log(4) 1+log(4) 1 Area = Log(4) 첫글자가 D 와 같거나 작을 확률 = Log(D+1) 첫글자가 정확히 D 일 확률 = Log(D+1)-Log(D) = Log(1+1/D)

IV. Analysis (4) - Extensions 일반적인 Commutative, Power-Law Process 에도 성립 : Z = (X*Y) n a1 a2 log(4)/2 Area = Log(4) n=2 Benford's Law : Invariant Measure for (Generalized) Multiplication

IV. Analysis (5) - Discussion Simple “Proof” of Benford's Law in the case of Multiplication R = P * Q (or R = A/B ) And either P or Q covers the unit circle (spans one decade) Multiplication Process - Process Measured by “Rates” - Stock Price : Geometric Brownian, dS = (  + dB) S - Purchase Order : Purchase Amount = unit price * quantity - Expenses : Prop. To Total Expenses (Total Assets) - Revenue, Consumer Demand : Proportional to GDP Growth What is the true “Physical” or “Sociological” reason? Benford's Law : True “Law” of the Nature, or an Artifact ?

IV. Analysis (6) – Application in Fraud Detection Fraud Loss Amounts 6% of Revenue in US Deviation from Benford's Law may indicate Fraud - Expense Limit, Fraudulent Purchase Orders, Hidden Controls,.. Fraud Detection is, An Information Battle - Availability – Separation of Duties, Hidden Rules - Processing Power, Analysis Techniques Fraud Detection Techniques - Atomic / Local - Relational - Global : Benford's Law, Scaling, Statistics, Local volatility,... Human activities are not random – ex. Key-tab password generation Difficult to Beat Global tests – Requires lots of fraudulent records Proactive Measures : Discourage Fraudsters !

V. Application in Audit Analysis (1) 강남구청 지출 데이터, 22,400 건 (2002/ /3)

V. Application in Audit Analysis(2) – Benford's Law First Digit

V. Application in Audit Analysis (3) – Benford's Law, Test for Scale Invariance Amount * 2

V. Application in Audit Analysis (4) – First Two Digits

V. Application in Audit Analysis (5) – Data Beginning with “49” Fraud or Inefficiency?

V. Application in Audit Analysis (6) Anomalies may be a sign of Fraud... - Numbers created/entered by a human operator is not scale-free - Near Upper Limit (say, 100,000) records - Fake records, copy of previous ones - Amount, Sum, Deviation,... and Scale Invariance Discourage Fraudsters - Hidden Scales - Many Artificially Created Records (ex. ERP Error correction) Anomalies may be a sign of unnatural, sub-optimized business behavior

V. Application in Audit Analysis (7) – Another Scale-Invariance Indicator

VI. Summary ● Benford's Law Explains Frequency of the First Digit of Numbers ● Applicable to Wide Variety of Natural/Social Processes ● Mathematical modeling through Dynamical Systems Theory ● Successfully applied to Audit Analysis ● CAATs(Computer Assisted Audit Techniques) ● Scale-Free Property : Feature or Not-a-Feature? ● Scale Invariance is Everywhere For more information, please visit AIT Homepage SERI Forum,