기업회계 패턴분석에의 응용
Benford's Law, Scale Invarianceand Fraud Detection
최 원규
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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
TEAM SUPPLIER RECEIVE전산소모품 대원사무기기 11759가구집기 592전산소모품 상신전산 0청소용역 동양인력 695전산소모품 새론산업 3275전산소모품 엔피플 4179문구 29조명 한진산업사 397가구집기 대지가구㈜ 11838문구 ㈜주풍 7151문구 유삼홀로아트 1624문구 오피스넷 0청소용역 ㈜명신 51648청소용역 애드민 2191문구 ㈜프라택 907문구 서울문구판매 2318전산소모품 ㈜대화 1065조명 오스람코리아 6039전산소모품 대진실업 0조명 머큐리코리아 2335전산소모품 13484전산소모품 새론산업 3220전산소모품 대원사무기기 30011조명 머큐리코리아 25278문구 오피스넷 0청소용역 동양인력 822조명 오스람코리아 4710조명 한진산업사 2491문구 ㈜주풍 361전산소모품 ㈜대화 1735
대기산업 (주 )
진성 PLT
(주 )한성엠에스
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, 1881
.. 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 = ?
2,467
785
3.392169
2.89487
6.287039
1,000,0006
0.287039 1.93
1,930,000
Log Table
+ =
Log Table
II. Benford's Law – History (3). Frank Benford, 1920's
A GE Engineer, Observed
1 2 3 4 5 6 7 8 9River Area 31 16.4 10.7 11.3 7.2 8.6 5.5 4.2 5.1Population 33.9 20.4 14.2 8.1 7.2 6.2 4.1 3.7 2.2
Physical Constants 41.3 14.4 4.8 8.6 10.6 5.8 1 2.9 10.6#'s from News Articles 30 18 12 10 8 6 6 5 5
Specific heat 24 18.4 16.2 14.6 10.6 4.1 3.2 4.8 4.1Pressure 29.6 18.3 12.8 9.8 8.3 6.4 5.7 4.4 4.7H.P.Lost 30 18.4 11.9 10.8 6.7 5.1 4.1 2.8 3.21
Mol. Weight 26.7 25.2 15.4 10.8 6.7 5.1 4.1 2.8 3.2Drainage 27.1 23.9 13.8 12.6 8.2 5 5 2.5 1.9
Atomic Weight 47.2 18.7 5.5 4.4 6.6 4.4 3.3 4.4 5.5Design 26.8 14.8 14.3 7.5 8.3 8.4 7 7.3 5.6
Cost Data 32.4 18.8 10.1 10.1 9.8 5.5 4.7 5.5 3.1X-Ray Volts 27.9 17.5 14.4 9 8.1 7.4 5.1 5.8 4.8Am. League 32.7 17.6 12.6 9.8 7.4 6.4 4.9 5.6 3Addresses 28.9 19.2 12.6 8.8 8.5 6.4 5.6 5 5Death Rate 27 18.6 15.7 9.4 6.7 6.5 7.2 4.8 4.1
31.03 18.66 12.31 9.73 8.06 6.08 4.78 4.47 4.44Theory 30.1 17.61 12.49 9.69 7.92 6.69 5.8 5.12 4.58
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, MR2, MR3, ... MRN
First Digit D Determined by Fractional Part of Log(V)
Log(Vk )= Log(MRk) = Log(M) + k Log(R)
Investment : A Shift Process on Circle
Log(Vk+1
) = Log(Vk) + Log(R) mod 1
When Log(R) : Irrational (Mostly..), Log(Vk) Uniformly Fills the Unit
Circle. Thus, Scale Invariant.
Log1
Log(M)
Log(R)
Log(R)
P(D) = Log(D+1) – Log(D) = Log(1+1/D)
Log2
Log3Log4
Log5
Log10
IV. Analysis (2) - Generalization
From the Original Logarithm Book Problem,
Letlog(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
1Log4
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
1Log4
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 = (m+ 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 (2002/9 - 2003/3)
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 www.aitcorp.co.kr.SERI Forum, www.seri.org/forum/caats .
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