기업회계 패턴분석에의 응용

Benford's Law, Scale Invarianceand Fraud Detection

최 원규

Advanced Information Technologywgchoe@aitcorp.co.kr

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 .