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Benford's Law. Applications for Forensic Accounting, Auditing, and Fraud Detection. Wiley Corporate F&A

  • ID: 2213018
  • Book
  • 352 Pages
  • John Wiley and Sons Ltd
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Praise for Benford′s Law Applications for Forensic Accounting, Auditing, and Fraud Detection

"Mark Nigrini . . . took Benford′s original data and developed it into solid analytical methods to detect a variety of white–collar misdeeds: vendor, payroll, and sales tax frauds; fraudulent medical and insurance claims; check fraud and tax evasion among others. In other words, Dr. Nigrini has put Benford′s Law on the map. With the passage of time, he and others will find additional useful applications for this powerful tool."
From the Foreword by Dr. Joseph T. Wells, CFE, CPA

Apply Benford′s Law to detect fraud and to test the integrity of your data

Numbers play an important part in our lives from weather forecasts to the stock market, and from street addresses and phone numbers to baseball statistics and casino games. It would be impossible to take numbers out of our daily lives. The secret is out that there are patterns to the numbers that we come across every day, and more importantly, that deviations from these patterns are red flags for fraud, errors, and biases.

Author Mark Nigrini, a pioneer in forensic accounting, describes the mathematical foundations of Benford′s Law in a way that is easily understood by accounting and other business–related professionals. He then shows many examples of authentic and accurate data that conformed to Benford′s Law—and the fraudulent and invented numbers that did not.

Nigrini goes way beyond the first digits test and outlines a series of digit– and number–based tests called the Nigrini Cycle. With a companion website with data sets and Excel templates, these tests are based on the state–of–the–art tools, thinking, and research related to Benford′s Law.

The applications include:

  • The Madoff numbers

  • GM and Chrysler bankruptcy data

  • Numbers reported by Enron and AIG

  • Bill and Hillary Clinton′s tax returns

  • U.S. census population numbers

  • Corporate payments data

  • The Fibonacci numbers

  • Taxpayer interest and dividend income

Benford′s Law "gives Professor Nigrini a tool worthy of Sherlock Holmes," said Robert Burton, chief financial investigator for the district attorney′s office in Brooklyn, New York. In Benford′s Law, Nigrini shares his insights and passion for the topic in a way that is sure to cause fraudsters to have many sleepless nights in the years ahead.

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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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Mark J. Nigrini
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