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Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection

ISBN: 978-1-118-15285-0
330 pages
April 2012


A powerful new tool for all forensic accountants, or anyone who analyzes data that may have been altered

Benford's Law gives the expected patterns of the digits in the numbers in tabulated data such as town and city populations or Madoff's fictitious portfolio returns. Those digits, in unaltered data, will not occur in equal proportions; there is a large bias towards the lower digits, so much so that nearly one-half of all numbers are expected to start with the digits 1 or 2. These patterns were originally discovered by physicist Frank Benford in the early 1930s, and have since been found to apply to all tabulated data. Mark J. Nigrini has been a pioneer in applying Benford's Law to auditing and forensic accounting, even before his groundbreaking 1999 Journal of Accountancy article introducing this useful tool to the accounting world. In Benford's Law, Nigrini shows the widespread applicability of Benford's Law and its practical uses to detect fraud, errors, and other anomalies.

  • Explores primary, associated, and advanced tests, all described with data sets that include corporate payments data and election data
  • Includes ten fraud detection studies, including vendor fraud, payroll fraud, due diligence when purchasing a business, and tax evasion
  • Covers financial statement fraud, with data from Enron, AIG, and companies that were the target of hedge fund short sales
  • Looks at how to detect Ponzi schemes, including data on Madoff, Waxenberg, and more
  • Examines many other applications, from the Clinton tax returns and the charitable gifts of Lehman Brothers to tax evasion and number invention

Benford's Law has 250 figures and uses 50 interesting authentic and fraudulent real-world data sets to explain both theory and practice, and concludes with an agenda and directions for future research. The companion website adds additional information and resources.

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Table of Contents

Foreword xi

Preface xiii

About the Author xix

Chapter 1: Introduction and Mathematical Foundations 1

Benford's Expected Digit Frequencies 5

Defining the First and First-Two Digits 6

Digit Patterns of U.S. Census Data 8

Logging on to Benford's Law 10

General Significant Digit Law 13

Log and Behold, the Census Data 13

Love at First Sight 15

Mantissa Test and Census Data 19

Number of Records and Benford's Law Tests 20

When Should Data Conform to Benford's Law? 21

Conclusions 23

Chapter 2: Theorems, Truisms, and a Little Trivia 25

Digits of Corporate Payments Data 26

Digits of Lake Data 28

Scale Invariance Theorem 31

Mean Absolute Deviation 34

Scale Invariance and Census Data 34

Scale Invariance and Corporate Payments 35

Scale Invariance and Lake Data 36

A Level Playing Field Becomes Benford 38

Multiplication by 1/X 42

All Distributions Lead to Benford 43

Getting a Gripf on Benford and Zipf 46

Conclusions 50

Chapter 3: More Formulas and Facts, and a Little Fibonacci 51

Fibonacci Numbers 51

Lucas Numbers 53

Back to Square One 55

3nþ1 Problem 58

Ultimate Uniform Distribution 60

Benford Embraces Other Bases 62

Nigrini's Second Last Theorem 65

Conclusions 69

Chapter 4: Primary Benford’s Law Tests 71

Corporate Payments Data 72

Data Profile 72

First Come, First Served 74

Playing Second Fiddle 75

First-Two Digits Test 78

Running the Digit Tests in Excel 80

Running the Digit Tests in Access 83

Conclusions 87

Chapter 5: Advanced Benford’s Law Tests 89

Summation Test 90

Running the Summation Test in Excel 94

Running the Summation Test in Access 95

Second-Order Test 97

An Analysis of Payments Data 102

An Analysis of Journal Entry Data 104

An Analysis of Census Data 107

Running the Second-Order Test in Excel 108

Excel, Thanks a Million(s) 110

Scale Invariance and the Second-Order Test 113

Conclusions 114

Chapter 6: Associated Benford’s Law Tests 117

Number Duplication Test 117

An Analysis of Payments Data 119

An Analysis of Census Data 121

Running the Number Duplication Test in Excel 122

Running the Number Duplication Test in Access 126

Last-Two Digits Test 129

An Analysis of Payments Data 130

An Analysis of Census Data 131

An Analysis of Election Results 132

Running the Last-Two Digits Test in Excel 135

Running the Last-Two Digits Test in Access 136

Distortion Factor Model 138

Distortion and the Census Data 145

Conclusions 146

Chapter 7: Assessing Conformity to Benford's Law 149

Z-Statistic 150

Chi-Square Test 153

Kolmogorov-Smirnoff Test 157

Mean Absolute Deviation Test 158

The Logarithmic Basis of Benford's Law 160

Creating a Perfect Synthetic Benford Set 163

Mantissa Arc Test 165

Conclusions 169

Chapter 8: Examples of Fraudulent Data 171

The Inside Story 172

The Vendor Who Never Was 174

Not Paying Attention 175

Funny Money 177

The Heart of the Matter 181

Going the Extra Mile 182

Laugh All the Way to the Bank 184

Culture Shock 187

Having a Bad Hair Day 189

An Unclean Bill of Health 191

Turning the Table on Tax Evasion 193

Conclusions 196

Chapter 9: Fraudulent Financial Statements, Part I 199

Number Crunching 200

Wrong Numbers 205

A Look at Enron's and AIG’s Numbers 207

Figuring Out the Controllers 210

Conclusions 213

Chapter 10: Fraudulent Financial Statements, Part II 215

Digital Yoga by Absaroka 216

Can’t See the Forest for the Trees 218

Digit a Little Deeper into Papua New Guinea 221

The Good, the Bad, and the Ugly 227

Dig a Little Deeper 233

There Are More Questions than Answers 237

Conclusions 244

Chapter 11: Madoff and Other Ponzi Schemes 247

The Madoff Claims 248

Don't Bank on Kaupthing 249

The Whole Ball of Waxenberg 252

General Motors Demoted to Private 253

Chrysler Unable to Dodge Bankruptcy 255

Discussion of the Claims Results 258

A Review of the Madoff Returns 259

Apple, Dell, Berkshire, and Benford 262

Discussion of the Returns Results 265

Chapter 12: Earth Science and Income Tax Applications 267

Still Waters Run Deep 268

The Lay of the Lake 274

For a Few Dollars Less 281

A Clean Bill of Clinton 286

Conclusions 290

Chapter 13: Future Directions and Conclusions 293

My Law 295

Insights into Number Invention 300

Lehman's Charitable Gifts 306

The Bottom Line 310

Glossary of Selected Terms 315

References 323

Index 327

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Author Information

MARK J. NIGRINI, PhD, is a professor at The College of New Jersey where he teaches forensic accounting courses. His research involves advanced theoretical work on Benford's Law and the legal process surrounding fraud convictions. Nigrini is also the author of Forensic Analytics (Wiley), which describes tests to detect fraud, errors, estimates, and biases in financial data. He has been published in national media including the Wall Street Journal and has published papers on Benford's Law and accounting in academic and professional journals. Nigrini regularly presents professional seminars for accountants and auditors in North America, Europe, and Asia with recent events in Singapore, Switzerland, and New Zealand.
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