A STUDY OF BENFORD S LAW, WITH APPLICATIONS TO THE ANALYSIS OF CORPORATE FINANCIAL STATEMENTS

Transcription

1 The Pennsylvania State University The Graduate School Eberly College of Science A STUDY OF BENFORD S LAW, WITH APPLICATIONS TO THE ANALYSIS OF CORPORATE FINANCIAL STATEMENTS A Thesis in Statistics by Juan C. Chang c 2017 Juan C. Chang Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2017

2 ii The thesis of Juan C. Chang was reviewed and approved by the following: Donald St. P. Richards Professor of Statistics Thesis Adviser James L. Rosenberger Professor of Statistics Aleksandra B. Slavković Professor of Statistics Associate Head for Graduate Studies, Department of Statistics Signatures on file in the Graduate School.

3 iii Abstract We consider in this thesis the numerical phenomenon, known as Benford s Law, which asserts numerical values for the empirical probabilities of first digits appearing in many lists of numbers. We introduce Benford s Law through motivating explanations and examples, and we explain why this numerical phenomenon can be applied to many different data sets. We also apply Benford s Law to the financial statements of three companies to test whether data derived from those statements follow Benford s Law. In the early part of the thesis, we introduce extensions of Benford s Law to calculating the empirical frequencies of specific digits and each sequence of digits. We motivate Benford s Law by compound growth processes, the scale-invariance of many randomly occurring processes, and the Central Limit Theorem. We also introduce extensions of Benford s Law beyond the first digit phenomenon to calculating the empirical frequencies of the second, third, and any given digit or sequence of digits. We apply Benford s Law to data drawn from the financial statements of three corporations: Bernard L. Madoff Investment Securities LLC, Toshiba Corporation, and Valeant Pharmaceuticals International, Inc., each of which has received widespread scrutiny in recent years. We apply Pearson s chi-square and the discrete Kolmogorov- Smirnov goodness-of-fit statistics to test the hypotheses that the data obtained from the statements of each of these three corporations follows Benford s Law. Finally, we provide in an appendix the software code, from the statistical package R, which was used to carry out the analyses of the data drawn from the corporate financial statements.

4 iv Table of Contents List of Tables v List of Figures vi Acknowledgments vii Chapter 1. Introduction Chapter 2. Motivating Explanations & Examples Introduction An Explanation of Compound Growth Processes The Scale-Invariance Property The Central Limit Theorem Benford s Law for a Collection of Digits Chapter 3. Application of Benford s Law to Madoff s Data Introduction Data Applications of Benford s Law to Madoff s Data Chapter 4. Analysis of Corporate Financial Statements Introduction Toshiba Corporation Data Applications of Benford s Law to Toshiba s Data Valeant Pharmaceuticals International, Inc Data Applications of Benford s Law to Valeant s Data Chapter 5. Conclusions Appendix A. Results from Additional Hypothesis Tests A.1 Country Population Analysis A.2 Manipulated Data Analysis Appendix B. R Code for Hypothesis Tests B.1 Country Population and Manipulated Data Analysis B.2 Bernard Madoff Analysis B.3 Toshiba Corporation Analysis B.4 Valeant Pharmaceuticals International, Inc. Analysis Bibliography

5 v List of Tables 2.1 The empirical probabilities of occurrences of the first, second, third, and fourth digit, according to Benford s Law Monthly investment gains and losses from Madoff s reports Percentages of the first digits of Madoff s data versus the expected Benford probabilities The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Madoff s data The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Madoff s data between December, 1990 and December, The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Toshiba Corporation from The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Toshiba Corporation from The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Valeant Pharmaceuticals International, Inc. from The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for Valeant Pharmaceuticals International, Inc. from 2014 and A.1 The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for the observed data presented in Chapter A.2 The results of Pearson s chi-square and the Kolmogorov-Smirnov goodnessof-fit tests for the manipulated data presented in Chapter

6 vi List of Figures 2.1 Frequencies of the first digit for: (a) Benford s Law (Predicted); (b) Country populations, including regions, economic groups, and total world population for 2015 (Observed) [28]; (c) Country populations multiplied by π (Manipulated) The Comparative Consolidated Balance Sheets for Toshiba Corporation from the Fiscal Year 2015 [26]. All reported figures are in Japanese Yen The Comparative Consolidated Balance Sheets for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16] The Consolidated Statements of (Loss) Income for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16] The Consolidated Statements of Comprehensive (Loss) Income for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16] The Consolidated Statements of Cash Flows for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16]

7 vii Acknowledgments I would like to thank my family and friends for their endless love and support. You have each played an essential role in shaping me into the person that I am today. I owe all of my accomplishments to you. I would like to express my deepest gratitude to my friends, colleagues, and professors in the Department of Statistics at The Pennsylvania State University for their constant support and encouragement throughout the past three years. In particular, I would like to thank my thesis adviser, Dr. Donald Richards. Thank you for dedicating your time and effort, and instilling your confidence in me. Your continuous guidance has allowed me to achieve the goals that I had set forth for myself. For this, I am truly grateful.

8 1 Chapter 1 Introduction The history of Benford s Law begins over 50 years prior to its discovery by Frank Benford. The astronomer Simon Newcomb was the first to observe the numerical phenomenon of the leading digit. A graduate of Harvard University, Newcomb was widely recognized for his work on planetary theories and variations in the values of astronomical constants. In 1881, Newcomb noticed that the first pages of his logarithm tables were more worn out than the later pages. For example, pages that began with the number 1 or 2 were dirtier than those that began with an 8 or 9. This curious phenomenon led Newcomb to write a paper, Note on the Frequency of Use of the Different Digits in Natural Numbers. In his paper, Newcomb writes, That the ten digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener 1 than any other digits, and the frequency diminished up to 9 (Newcomb [18]). Newcomb s paper introduced the idea that digits are not equally likely to occur, and he extended this idea to calculating the probabilities of occurrence for the first and second digits, individually. In addition, Newcomb noted that the numerical value of a specific quantity is dependent upon the scale in which it is used, and he suggested that the proper scale in which to study the occurrence of digits is the ratio of measurements.

9 2 Simon Newcomb s discoveries of leading digit phenomena were the first observations of what would later be called Benford s Law. In 1938, the physicist Frank Benford wrote a paper entitled, The Law of Anomalous Numbers, in which, similar to Newcomb, he observed the worn out pages of logarithm tables. Unaware of Newcomb s paper, Benford extended his research by gathering over 20,000 observations, from a variety of sources, to compute the frequency of occurrence for each leading digit (Benford [4]). Benford s data set included values such as the surface areas of rivers, populations of U.S. cities, and entries from a mathematical handbook. The results of this experiment would eventually be known as Benford s Law. Benford demonstrated that the first digit law emerges for virtually any group of numbers in a data set. Exceptions include data sets that are arbitrary and contain restrictions, such as lottery numbers, telephone numbers, or gas prices. Unlike Newcomb s paper, which received little attention after its publication, Benford s paper was recognized almost immediately. The results discovered by Benford were published in a physics journal and preceding an important article by Bethe, Rose and Smith (Miller [17, p. 7]). The proximity of Benford s article to Bethe s caused the former to catch the attention of several phyicists. Shortly thereafter, numerous academics began to discuss the results of Benford s Law and so its popularity grew. Benford s Law has extended beyond a mathematical result, to a variety of diverse disciplines such as accounting, computer science, economics, engineering, financial auditing, and statistics. Benford s Law initially had little or no impact on real-world

10 3 problems. But, with the emergence of computer technology and the capability of analyzing large data sets, interest in Benford s Law has grown across a wide range of applications including fraud detection, computer design, and data mining. In this thesis, we investigate some aspects of the theory and applications of Benford s Law. Motivating explanations are introduced in order to explain why diverse data sets satisfy the first digit law. An explanation of the role of the Central Limit Theorem in comprehending the pervasiveness of Benford s Law is also discussed. Next, we apply Benford s Law to some corporate financial statements. In recent years, there have been several industrial and financial corporations that have fallen under the scrutiny of the U.S. Securities and Exchange Commission (SEC) due to irregular accounting practices. We study three such companies, analyzing whether data drawn from their financial statements follow Benford s Law. Two goodness-of-fit tests, Pearson s chi-square test and the discrete Kolmogorov- Smirnov test, are implemented in our analysis to test for Benford s Law. We are interested in testing a variety of digits in this portion of the study. These digits include the first digit, the first two digits, the second digit, and the first three digits of the designated data sets. This introduction concludes with a summary of the remaining chapters of this thesis. Chapter 2 introduces several motivating explanations for Benford s Law. We discuss compound growth processes and an example pertaining to interest rates, and we explain the scale-invariance property and its relationship to differing units of measurement. In addition, we explain the relationship between the Central Limit Theorem and

11 4 Benford s Law, and we introduce Benford s Law as it pertains to a collection of digits extending beyond the first. In Chapter 3, we discuss the infamous financier Bernard Madoff; his financial firm, Madoff Investment Securities LLC; and the Ponzi scheme which he operated until We apply Pearson s chi-square and the discrete Kolmogorov-Smirnov tests to analyze the first digit, the first two digits, and the second digit of Madoff s monthly financial reports during the period This analysis tests whether or not there is statistically significant evidence that Madoff s data follow Benford s distribution. This application, as well as the applications in Chapter 4, illustrate how Benford s Law applies to the analysis of accounting data, a subject which is covered in detail by Cleary and Thibodeau [8, pp ] and Nigrini [19, pp. 1-23]. In Chapter 4, we extend the analysis from Chapter 3 to study corporate financial statements. We introduce two companies associated recently with accounting irregularities: Toshiba Corporation and Valeant Pharmaceuticals International, Inc. We apply the goodness-of-fit tests introduced in Chapter 3 to analyze consolidated financial statements of both companies, seeking to determine whether or not their data follow Benford s Law. In Chapter 5, we present the conclusions of our analyses. We conclude that Madoff s and Toshiba s data contain statistically significant departures from Benford s Law. On the other hand, our analysis of the data on Valeant Pharmaceuticals International, Inc. does not find statistically significant evidence of departures from Benford s distribution.

12 5 Chapter 2 Motivating Explanations & Examples 2.1 Introduction Frank Benford s research led him to discover that the first digit law pertained to more than logarithmic tables, solely. By extracting numbers from a variety of sources, Benford demonstrated that the frequency of occurrence for the leading digit of these values were consistent amongst one another. Benford observed that the empirical proportion for the appearance of the number 1 as the first digit was approximately equal to log 10 (2/1). Similarly, the empirical proportion for the appearance of the number 2 as the first digit was approximately equal to log 10 (3/2). This pattern continued for all other integers and led to the following definition: Definition In order for a data set to satisfy Benford s Law, the probability of observing the integer d as the leading digit is approximately, P (D 1 = d) = log 10 ((d + 1)/d), (2.1) where 1 d 9. It is important to note that the integer 0 is inadmissible as a leading digit. However, 0 becomes admissible when testing for digits beyond the first. In addition, Benford s

13 6 Law pertains not only to the first digit, probabilities of occurrence exist for each particular digit and each sequence of digits, and these results will be explained in detail in Section 2.5. The remainder of this chapter discusses several motivating explanations, all drawn from Miller [17, pp. 3-16], which describe why diverse data sets satisfy Benford s Law. 2.2 An Explanation of Compound Growth Processes Benford s paper introduces the geometric foundation of the first digit law. This explanation proposes that a process which grows at a constant growth rate tends to stay longer at lower digits than at higher digits. Stock prices, for example, demonstrate this behavior. Consider the price of a stock which is increasing at a constant compound rate of r%. By the well-known formula for compound interest, the number of years, n d, that the price of the stock takes to increase from d dollars to d + 1 dollars equals n d = log 10((d + 1)/d). (2.2) log 10 (1 + r) Hence, the amount of time the stock price will take to increase from $1 to $2 is longer than the time it will take to increase from $9 to $10. Indeed, substituting d = 1 in (2.2), the number of years the stock price will take to increase from $1 to $2, at an annual compound rate of r = 5%, is years. By contrast, substituting d = 9 in (2.2), the number of years the stock price will take to increase from $9 to $10, at an annual compound rate of r = 5%, is 2.16 years.

14 7 Let n be the amount of time necessary for $1 to increase to $10, with growth occurring at the constant compound rate of r%. Again by the well-known formula for compound interest rates, we have n = (log 10 (10)/(log 10 (1+r)). Then by Equation (2.2), the proportion of time that the stock price has leading digit d is equal to / log 10 ((d + 1)/d) log10 (10) log 10 (1 + r) log 10 (1 + r) = log ( ) 10((d + 1)/d) d + 1 = log log 10 (10) 10, (2.3) d which is precisely Benford s Law. Consequently, the formula in Equation (2.2) equals the probability of observing d as the first digit, as mentioned at the beginning of this chapter. This result remains the same for any value of r because (2.3) does not depend on r. This example demonstrates that Benford behavior arises from compound growth processes. Other examples that exhibit similar compound growth behavior include nuclear chain reactions and population growth. 2.3 The Scale-Invariance Property Another motivating explanation amplifying the ubiquity of Benford s Law is the scale-invariance property. Scale-invariance refers to the consistency of a mathematical approach despite changes in scales of size, currency, or other measures or variables. In 1961, the mathematician, Roger Pinkham, became the first academic to study this relationship (Nigrini [19, p. 31]). Pinkham proposed that a law which measured the proportion of frequencies in digits should be universal. If the values for areas measured

15 8 Figure 2.1: Frequencies of the first digit for: (a) Benford s Law (Predicted); (b) Country populations, including regions, economic groups, and total world population for 2015 (Observed) [28]; (c) Country populations multiplied by π (Manipulated). in square meters follow Benford s Law, for example, then we can expect that converting the values to square feet will result in measurements which also follow Benford s Law. The following example studies the population of all countries and economic groups, as well as the world population, for the year The data were obtained from the World Bank [28] and analyzed in R, the statistical software system [22]. All of the leading digits were extracted from the data and their frequencies were calculated. In addition, each of the values were multiplied by π = in order to test the scale-invariance property. The leading digits of the manipulated data were extracted and their frequency of occurrences were computed. The frequency of each observed digit was then plotted

16 9 side-by-side together with the predicted frequencies arising from Benford s Law. The resulting histograms for this example are provided in Figure 2.1. As is evident from Figure 2.1, the proportions at which each digit occurs, for the observed and the manipulated data, follow Benford s Law closely. For the observed data, the number 1 appears as the leading digit approximately 29% of the time, for the manipulated data the corresponding number is approximately 34%, and for Benford s Law, the corresponding number is approximately 30%. As d increases, the difference in densities decreases. These observations, although not as rigorous as statistical hypothesis testing, are convincing, particularly so when similar patterns are obtained from other scalings of the data. In fact, when we apply Pearson s chi-square goodness-of-fit test and the discrete Kolmogorov-Smirnov test, we fail to reject the null hypothesis that the observed and the manipulated data follow Benford s Law. The results of these analyses are provided in Appendix A, where the smallest p-value for each of these tests is calculated to be This example further underscores the ubiquity of Benford s Law. 2.4 The Central Limit Theorem This section relates the Central Limit Theorem to Benford s Law. The Central Limit Theorem (Hogg and Tanis [14, p. 256]) states:

17 Theorem Let W 1,..., W n be a random sample from a population with finite mean and variance, and let W = W W n. Then, as n, 10 W E(W ) Var(W ) converges to N(0, 1), the standard normal distribution. We now follow Miller [17, pp ] in providing an explanation of the relationship between the Central Limit Theorem and Benford s Law. Definition For t 0, we define S(t) to be the unique number such that t = 10 k S(t), where 1 S(t) < 10 and k Z. Also, we define S(0) = 0. The function S(t) is called the significand of t. Definition If x is a nonnegative real number then x (mod 1) is defined to be the fractional part of x. If x < 0 then x (mod 1) is defined to be 1 ( x) (mod 1). In studying the leading digit of a random variable X, we need only to study the significand S(X). The values of X which have leading digit d are {x : d S(x) < d + 1 {x : log 10 (d) log 10 (S(x)) < log 10 (d + 1). Definition A random variable X is Benford if P (S(X) t) = log 10 t for all 1 t < 10.

18 Written in terms of its significant digits, the random variable X is Benford if and only if its significant digits D 1,..., D n satisfy P (D 1 = d 1,..., D n = d n ) = log 10 (1 + ) 1 n. j=1 10n j d j 11 for all n N, all d 1 {1,..., 9, and all d j {0,..., 9 for j 2. Suppose that the random variable X follows Benford s Law. Then the probability that X has leading digit d is log 10 ((d + 1)/d) log 10 (d + 1) log 10 (d), which is exactly the length of the interval [log 10 (d), log 10 (d + 1)). Indeed, the following result from Berger and Hill [5, p. 44] shows that if X follows Benford s Law then X can be transformed using the log 10 function into a random variable which is uniformly distributed on [0, 1). Theorem If the random variable X is distributed according to Benford s Law then (log 10 X ) (mod 1) is uniformly distributed on [0, 1).

19 Proof. For t [0, 1), the inequality (log 10 X ) (mod 1) t means that either X = 0 or there exists k Z such that k log 10 X < k + t. Therefore, 12 P ((log 10 X ) (mod 1) t) = P (X = 0) + P ((log 10 X ) [k, k + t)) k= = P (X = 0) + P (k log 10 X < k + t) k= = P (S(X) 10 t ). Since X is Benford then P (S(X) 10 t ) = log 10 (10 t ) = t. Therefore, for all t [0, 1), P ((log 10 X ) (mod 1) t) = t. Therefore (log 10 X ) (mod 1) is uniformly distributed on [0, 1). Theorem provides a connection between Benford s Law and the uniform distribution. In order to apply the Central Limit Theorem to explain the ubiquity of Benford s Law, we need to connect the uniform distribution with the normal distribution, and this is provided in the next result. Let Z be a normally distributed random variable with mean µ and variance σ 2. The following result (see Miller [17, p. 16]) demonstrates that, as σ 2, Z (mod 1) converges in distribution to a random variable which is uniformly distributed on [0, 1). Theorem Let Z N(µ, σ 2 ) and let Y = Z (mod 1). Then Y U[0, 1) as σ 2.

20 Proof. For t [0, 1), the inequality Z (mod 1) t means that there exists some k Z such that k Z < k + t. Therefore, 13 {Y t = {k Z < k + t. k= Hence, P (Y t) = P (k Z < k + t) = k= k= [P (Z < k + t) P (Z < k)]. Therefore, F Y (t) = [P (Z < k + t) P (Z < k)]. k= To differentiate F Y (t) with respect to t, we apply the criterion for interchanging integral and derivative provided by Burkill and Burkill [6, p. 290]) to obtain the probability density function, f Y (t) d dt F Y (t) = k= d [P (Z < k + t) P (Z < k)], (2.4) dt

21 14 for all t [0, 1). However, k= d [P (Z < k + t) P (Z < k)] = dt = k= k= = 1 σ 2π d P (Z < k + t) dt f Z (k + t) k= [ exp 1 2 ( k + t µ σ ) 2 ]. (2.5) We now split the latter summation into the following: 1 σ 2π ( 1 k= [ exp 1 2 ( k + t µ σ ) 2 ] + k=0 [ exp 1 2 ( ) k + t µ 2 ] ). (2.6) σ Next, observe that 1 k= [ exp 1 ] 2σ2 (k + t µ)2 = = [ exp 1 ] 2σ2 ( k + t µ)2, [ exp 1 ] 2σ2 (k t + µ)2. (2.7) k=1 k=1 Define [ a k = exp 1 2σ2 (k t + µ)2 ], for k 1. Then we have a k+1 a k = exp [ 1 (k + 1 t + µ) 2] 2σ 2 exp [ 1 (k t + µ) 2] 2σ 2 = exp[ 1 2σ2 (2k 2t + 2µ + 1)],

22 15 which converges to zero as k. Therefore, by the Ratio Test, the series (2.7) converges. It is easy to see that the same approach applies to establish convergence of the second series in (2.6). Therefore, the series (2.5) converges absolutely. Therefore, by (2.4), f Y (t) exists for all t [0, 1) and f Y (t) = k= [ 1 σ 2π exp 1 2 ( k + t µ σ ) 2 ], 0 t < 1. Further, it is clear that f Y (t) is continuous on [0,1). Next, the characteristic function of Y is ( ) E e 2πinY = = e 2πint f Y (t)dt [ ] e 2πint f Z (k + t) dt. k= Before we proceed we must justify the interchange of integral and summation, i.e., 1 0 e 2πint [ ] f Z (k + t) dy = k= k= 1 0 e 2πint f Z (k + t)dt. (2.8) Since 1 k= 0 e 2πint f Z (k + t) dt = = = 1 k= 0 k+1 k= k f Z (k + t)dt f Z (t)dt f Z (t)dt = 1,

23 16 then by Fubini s Theorem (Burkill and Burkill [6, p. 290]), (2.8) is valid. Therefore, ( ) E e 2πinY = = = 1 k= 0 k+1 k= k= k e 2πint f Z (k + t)dt e 2πin(z k) f Z (z)dz k+1 e 2πink e 2πinz f Z (z)dz. k, let z = k + t Since e 2πink = 1 for all n, k then ( ) E e 2πinY = = k= k+1 k e 2πinz f Z (z)dz e 2πinz f Z (z)dx = exp (2πinµ 12 ) σ2 (2πn) 2 = f Y (n). Therefore, by the Fourier Inversion Theorem (Folland [11, p. 244]), f Y (t) = = = k= k= k= f Y (k)e 2πikt exp (2πikµ 12 ) σ2 (2πk) 2 e 2πikt exp (2πik(t + µ) 12 ) (2πσ)2 k 2. By Jacobi s identity (Whittaker and Watson [27, p. 463]), for τ, z C with Im(τ) > 0, k= ( ) exp πik 2 τ + 2πikz = k=1 ( exp πiτk 2) cos(2πkz).

24 17 Setting τ = 2iπσ 2 and z = t + µ in Jacobi s identity, we obtain f Y (t) = k=1 ( exp 2π 2 σ 2 k 2) cos (2πk(t + µ)), (2.9) for t [0, 1). Since ( exp 2π 2 σ 2 k 2) cos(2πk(t + µ)) exp ( 2π 2 σ 2 k 2), then lim σ k=1 exp ( 2π 2 σ 2 k 2) cos(2πk(t + µ)) = ( lim exp 2π 2 σ 2 k 2) σ k=1 0 = 0. k=1 Hence, in (2.8), we can interchange limit and summation. Therefore, for all t [0, 1), f Y (t) 1 as σ 2. Our proof is now complete. Now, let X 1,..., X n be positive, independent and identically distributed random variables. Consider the product, X = X 1 X n. Then, W = log 10 (X) = n log 10 (X i ). i=1

25 Assuming that log 10 (X) has finite mean and variance then, by the Central Limit Theorem 18 as stated earlier, W E(W ) Var(W ) N(0, 1), as n. Since Var(W ) = nvar(log 10 (X 1 )) as n, then by Theorem 2.4.6, Y = W (mod 1) converges in distribution to U[0, 1) as n. This explains why products of large numbers or randomly chosen numbers, after the transformation x (log 10 x) (mod 1) is imposed, conforms to Benford s Law. 2.5 Benford s Law for a Collection of Digits As mentioned in the introduction to this chapter, Benford s Law extends beyond the first digit phenomenon. Newcomb and Benford each noticed that numerical phenomena existed for any given digit, and they derived the empirical probabilities of occurrence for the first and second digits. Benford went even further by providing empirical probabilities of occurrence for any digit. Recall, from definition 2.1.1, that the empirical probability that the first digit is d 1 is P (D 1 = d 1 ) = log 10 ((d 1 + 1)/(d 1 ),

26 19 where 1 d 1 9. The empirical probability that the second digit is d 2 is P (D 2 = d 2 ) = 9 ) 1 log 10 (1 +, (2.10) 10d 1 + d 2 d 1 =1 where 0 d 2 9 The empirical probability of observing the integer 0 as the second digit, for example, can be computed using (2.10). The calculation for this example is given below: P (D 2 = 0) = 9 ) 1 log 10 (1 + 10d 1 + d 2 d 1 =1 = log 10 ( ( + log log 10 ( = ) ( + log ) ( + log ) ( + log ) ( + log ) ( + log ) ( + log ) ) ) Similar to the formulas (2.1) and (2.10), the empirical probabilities of occurrences for the third and fourth digits are, respectively, P (D 3 = d 3 ) = 9 d 1 =1 d 2 =0 9 1 log 10 ( d d 2 + d 3 ), (2.11) where 0 d 3 9, and P (D 4 = d 4 ) = 9 9 d 1 =1 d 2 =0 d 3 =0 9 1 log 10 ( d d d 3 + d 4 ), (2.12)

27 20 Table 2.1: The empirical probabilities of occurrences of the first, second, third, and fourth digit, according to Benford s Law. The Frequency of Occurrence for the First, Second, Third, and Fourth Digit Integer d 1 d 2 d 3 d where 0 d 4 9. Further, this pattern continues for all remaining digits. For later digits, the empirical distribution of occurrence becomes nearly uniform. For example, the difference P (D 1 = 1) P (D 2 = 9) is much greater than the difference P (D 4 = 1) P (D 4 = 9), and moreover the latter difference is close to zero. These empirical probabilities are provided in Table 2.1, for the first, second, third, and fourth digits. Table 2.1 displays the change in frequencies of each integer as a particular digit increases. It is evident that as d i increases, the empirical probabilities converge to

28 This convergence in uniformity is the reason why the analysis of the first digit phenomenon is more prevalent in the literature. In addition to analyzing specific digits, it is also possible to derive the empirical frequencies of occurrences for sequential digits. From the work of Nigrini [19, pp. 5-6], the empirical probabilities for the first two digits and the first three digits are, respectively, ) 1 P (D 1 = d 1, D 2 = d 2 ) = log 10 (1 +, (2.13) 10d 1 + d 2 and ) 1 P (D 1 = d 1, D 2 = d 2, D 3 = d 3 ) = log 10 ( , (2.14) d d 2 + d 3 where 1 d 1 9, 0 d 2 9, 0 d 3 9. As is evident from equations (2.13) and (2.14), the set of possible values from the collection of digits increases sharply. Therefore, in order to perform statistical testing to analyze whether a given distribution follows Benford s Law, we will need commensurately larger sample sizes. In conclusion, we see that generalizations of Benford s Law applies to digits beyond the first. Moreover, as the ith digit increases, the empirical distributions of the ith digit converges to uniformity. Last, we shall use the empirical frequencies given above to analyze the occurrence of sequence of digits in some financial data sets.

29 22 Chapter 3 Application of Benford s Law to Madoff s Data 3.1 Introduction Prior to discussing the details of Benford s Law as it applies to Madoff s data, it is essential to provide background on the formerly well-respected financier. Bernard Madoff was a stockbroker and investment advisor who founded in 1960 the Wall Street firm, Bernard L. Madoff Investment Securities LLC [25]. The firm executed transactions over-the-counter from retail brokers. Over-the-counter (OTC) trading is a financing technique which facilitates liquidity, or the ability to quickly convert a portfolio to cash with little or no loss in value, and where transaction prices are not necessarily published for the public. In OTC trading, financial securities are traded through a dealer network, as opposed to trading on a centralized exchange, such as the New York Stock Exchange (NYSE) or the National Association of Securities Dealers Automated Quotations (NASDAQ). Thus, OTC trading differs from the purchasing of stocks in companies on the NYSE or the NASDAQ because of the absence of publiclypublished transaction prices. On the contrary, OTC securities transactions are made through market-makers who carry an inventory of securities in order to facilitate trading in a timely manner (Dodd [10]). On December 11, 2008 Bernard Madoff was arrested for operating a Ponzi scheme that turned out to be one of the most extensive cases of fraud in United States history.

30 23 Ponzi schemes are run by a central operator, who use money from new investors to pay promised returns to earlier investors [21]. For many years, Madoff promised uniform and consistently positive investment returns to his clients. It was relatively easy for Madoff s malfeasance to go unnoticed as he was, for over 50 years, an active and prominent member of the financial industry. Madoff claimed that his Ponzi scheme began as early as 1990; however, federal investigators suspected that the fraud may have begun as early as the mid-1980s (Collins [9, p. 443]). It was also discovered that the majority of the victims of Madoff s scheme were members of his own synagogue. When a fraudster preys upon members of an identifiable group, such as a religious or ethnic community, it is known as an affinity scheme [1]. Madoff was charged with eleven counts of fraud, money laundering, perjury, and theft. In total, he cheated his clients out of approximately $20 billion and was sentenced subsequently to 150 years in prison (Picard [20]). 3.2 Data The data used for this study consists of the monthly investment gains and losses as reported by Bernard L. Madoff Investment Securities LLC to clients from December, 1990 to December, The data comes from information provided by the Fairfield Greenwich Group [13, p. 8]. This data set is provided in Table 3.1, and it contains a total of 217 data points.

31 24 Table 3.1: Monthly investment gains and losses from Madoff s reports We will apply Benford s Law to Madoff s data, and we will also apply statistical tests to infer whether Madoff s reported investment returns deviate from Benford s Law. 3.3 Applications of Benford s Law to Madoff s Data This section begins with the overriding question: Did Madoff fabricate the monthly investment returns reported to his clients between December, 1990 to December, 2008? In order to address this question, Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit statistics are applied to test whether the first digit, the first two digits, and the second digit of the reported returns conform to Benford s Law. The results of each test are then compared with each other. In analyzing Madoff s data, it is assumed (under the null hypothesis) that his financial statements are free of material misstatement. The conclusion that the data do not follow Benford s Law can be drawn if there is statistically significant evidence to

32 25 Table 3.2: Percentages of the first digits of Madoff s data versus the expected Benford probabilities First Digit Percentage of Observed Values Benford Probabilities reject the claim. For Pearson s chi-square and the Kolmogorov-Smirnov tests, the following hypotheses were proposed: H 0 : The data in Table 3.1 follow Benford s Law H A : The data in Table 3.1 do not follow Benford s Law The goal of this study is to assess the strength of the evidence against the null hypothesis, H 0. Therefore, if a test rejects H 0, then there is statistically significant evidence that the returns in Table 3.1 do not follow Benford s Law, and then we infer that Madoff fabricated the returns reported to his clients. The hypothesis test begins by extracting all of the first digits from the data set. Recall that when testing Benford s Law for the first digit, only the observed values 1 through 9 are considered. It must be noted that in this part of the study, only 216 of the 217 observations were used because it was reported in December, 2002 that Madoff had no gains or losses, the return for that month being 0.00%. Once the first digits were extracted, the number of times each digit appeared in the data were tallied.

33 26 Table 3.3: The results of Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit tests for Madoff s data Digits p-values for Goodness-of-Fit Tests for Benford s Law Observed values of Observed values of p-values of the the chi-square test the discrete K-S test chi-square test statistic statistic p-values of the discrete K-S test D (D 1, D 2 ) D < 0.05 Table 3.2 compares the percentages of the observed values with the hypothesized Benford probabilities. The comparisons of the observed and the expected percentages provide a preliminary indication of the results to be expected. However, this visual information is insufficient for reaching statistically significant conclusions. Consequently, we applied Pearson s chi-square and the Kolmogorov-Smirnov test statistics using the package BenfordTests on R to test whether the values for the various observed digits follow Benford s distribution [15]. The result of each test is provided in Table 3.3. When testing with a level of significance α = 0.05, Pearson s chi-square test fails to reject the null hypothesis for the first digit with a p-value of Therefore, the chi-square test provides only weak evidence against the null hypothesis. Although the p-value of the chi-square test for the first digit is fairly close to the level of significance α = 0.05, the Kolmogorov-Smirnov test, which is known to be a more powerful test, reaches the opposing conclusion. Indeed, the Kolmogorov-Smirnov test, with a p-value of , rejects the null hypothesis. This shows that we have

34 27 statistically significant evidence against the claim that the first digit of Madoff s data follows Benford s distribution. We apply a similar analysis to the first two digits and to the second digit of Madoff s data. As shown in Table 3.3, the chi-square test again fails to reject H 0. However, on applying the Kolmogorov-Smirnov tests to (D 1, D 2 ) and D 2, the null hypothesis is rejected. It must be noted that the results for the first two digits must be taken with some caution. The sample size of 217 may not be sufficiently large for the accurate calculation of p-values. As presented in Chapter 2.5, when performing statistical inference on the first two digits of Benford s Law, the integers form the set of possible values; therefore, a sample size greater than 217 is needed in order to draw accurate conclusions. Despite this drawback, the results for (D 1, D 2 ) remain consistent with the remainder of the analysis. When we apply the chi-square goodness-of-fit test to the first digit, the first two digits, and the second digit of Madoff s data, we obtain p-values which provide weak evidence against the claim that the data follow Benford s Law. However, when we apply the more powerful Kolmogorov-Smirnov test, we find that there is statistically significant evidence against H 0. Therefore, we have found strong evidence to support the claim that Madoff s data do not follow Benford s Law. Further, an analysis on the first nine years of the data was performed. In the preceding analysis, we discovered that Madoff s data did not follow Benford s Law. These results are consistent with the reports found on Bernie Madoff. Thus, we can claim that

35 28 Table 3.4: The results of Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit tests for Madoff s data between December, 1990 and December, Digits p-values for Goodness-of-Fit Tests for Benford s Law Observed values of Observed values of p-values of the the chi-square test the discrete K-S test chi-square test statistic statistic p-values of the discrete K-S test D the investment returns for the years were, indeed, fraudulent. The purpose of the following test is to detect if fraudulent activity could have been sought out sooner. Therefore, we analyze the years The results of this analysis are demonstrated in Table 3.4. It appears that the first nine years of Madoff s data demonstrate deviation from Benford s Law. The p-value of the chi-square test is and the p-value of the Kolmogorov-Smirnov test is These results provide statistically sufficient evidence that the data between December, 1990 and December, 1999 do not follow Benford s Law. This demonstrates that if these statistical tests had been performed on Madoff s data in early 2000, the deviation from Benford s Law could have been detected nearly nine years prior to its discovery.

36 29 Chapter 4 Analysis of Corporate Financial Statements 4.1 Introduction In the previous chapter, we analyzed the returns reported by Madoff Investment Securities and found statistically significant evidence that those returns were fabricated. In this chapter, we will apply analytical methods similar to those introduced in Chapter 3. Here, we will evaluate the financial statements of Toshiba Corporation and Valeant Pharmaceuticals International, Inc. by investigating the annual reports for the years and , respectively. The data in this section were gathered through the use of PDF-scraping, a method designed to reliably extract data from PDFs into CSV and Excel files. Tabula, an openly available tool for PDF-scraping, was used for this section of our study. 4.2 Toshiba Corporation Toshiba Corporation is a Japanese conglomerate founded in 1875 that invests in the sectors of electronics and energy. Global sales of Toshiba s products, which include information technology and communications equipment and systems, electronic components and materials, power systems, and other appliances related to energy, infrastructure, and storage, have grown immensely throughout the past several decades. In recent years, Toshiba has had the reputation of being one of Japan s highest quality

37 30 and premier companies. At the end of the fiscal year 2015, the company reported net worldwide sales of over $63 billion, while employing more than 200,000 people (Carpenter [7]). In 2015, Toshiba was involved in an accounting scandal that found their profits to have been inflated by approximately $1.2 billion over the previous seven years (Carpenter [7]). Inappropriate accounting practices and overstated profits were discovered by investigators in several Toshiba business units. The global financial crisis of 2008 sparked enormous losses in Toshiba s operations, and this was followed by accounting misconduct. It has been reported that the chief executive officer at that time forced divisions that could not meet profit targets to achieve specific goals as a challenge. Achievement of these goals became realistically unattainable, which led to the use of irregular accounting practices. Some of the inappropriate accounting techniques used throughout those years included booking future profits early, pushing back losses, and pushing back charges (Fukase [12]). A back charge is a billing made to collect an expense acquired in a previous billing statement [3]. It is believed that Toshiba s interest in overstating profits was to maintain their current shareholder base and to ensure that their stock price remained high. Toshiba was eventually fined 7.37 billion yen ($60 million) for its fraudulent financial reports, the largest fine for a Japanese company due to accounting-related violations (Fukase [12]). In addition, individual investors, foreign and domestic, who suffered major losses have sued Toshiba, demanding compensation for losses provoked by the decline of Toshiba shares.

38 Data Bearing in mind that it is now known that Toshiba s annual reports for contained accounting irregularities, we wish to determine whether such irregularities could have been detected earlier by applications of Benford s Law. Therefore, the data used in this section are drawn from the complete annual reports for Toshiba Corporation for the fiscal years The software package, Tabula was used for the purpose of PDF-scraping to collect the data. For Toshiba, the fiscal year generally begins on April 1 and ends on March 31 of the following calendar year. The annual reports were obtained from Toshiba s company website, under their investor relations section [26]. The comparative consolidated balance sheets, which includes the assets, liabilities, and equity are of primary interest to us. Amiram, et al. [2, p. 32], performed a similar study in which they applied Benford s Law to public companies. In their paper, the balance sheets of each company were used for the purpose of obtaining their data. Therefore, we have chosen to analyze Toshiba s balance sheet for this study. An example of the consolidated balance sheet is provided in Figure 4.1. A total of 396 data points were obtained from the sixteen annual reports covered by this study. We note that the units for each of the values in Toshiba s annual reports are in Japanese Yen. As mentioned in Chapter 2 during our explanation of the scaleinvariance property, the adherence to Benford s Law is not dependent on the units at which the data are measured. Therefore, we can expect that our analysis will result in the same conclusions, irrespective of whether the data are expressed in yen or dollars.

39 Figure 4.1: The Comparative Consolidated Balance Sheets for Toshiba Corporation from the Fiscal Year 2015 [26]. All reported figures are in Japanese Yen. 32

40 Statistical methods similar to those depicted in Chapter 3 will be applied in this section in order to test whether Toshiba s annual reports remain consistent with Benford s Law Applications of Benford s Law to Toshiba s Data We begin this study by proposing the following hypotheses: H 0 : The data for Toshiba Corporation follow Benford s Law H A : The data for Toshiba Corporation do not follow Benford s Law Recall, that the goal of this analysis is to assess the strength of the evidence against the null hypothesis, H 0. If the test rejects H 0, then we have found statistically significant evidence that the Toshiba data do not follow Benford s Law. The chi-square and the Kolmogorov-Smirnov tests were again applied to test the hypotheses above. However, unlike in Chapter 3 where the first digit, the first two digits, and the second digit were tested, this chapter analyzes the first digit, the first two digits, and the first three digits to see whether they follow Benford s distribution. The purpose of this change was to calculate an actual p-value using the discrete Kolmogorov-Smirnov test, as opposed to relying on an estimate. The result of each test is shown in Table 4.1. Noting that the chi-square goodness-of-fit test for the first digit has a p-value of , we fail to reject H 0. Therefore, the chi-square test statistic provides only weak evidence against the claim that the Toshiba data follow Benford s distribution. On the contrary, the Kolmogorov-Smirnov test has p-values substantially smaller than 0.05 for tests on the first digit, the first two digits, and the first three digits. Therefore, the Kolmogorov-Smirnov test provides strong evidence to support the claim that Toshiba s data do not follow Benford s Law.

41 34 Table 4.1: The results of Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit tests for Toshiba Corporation from Digits p-values for Goodness-of-Fit Tests for Benford s Law Observed values of the chi-square test statistic p-values of the chi-square test Observed values of the discrete K-S test statistic p-values of the discrete K-S test D (D 1, D 2 ) (D 1, D 2, D 3 ) It is important to note that the chi-square test was not implemented when analyzing the first two digits and the first three digits due to an inability in meeting the conditions needed to adequately approximate Benford s distribution. Generally, the sample size, n, must be large enough so that np i 5, i = 1, 2,..., k, where p i is the probability that the event i will occur and k corresponds to all possible values of (D 1, D 2 ) and (D 1, D 2, D 3 ) (Hogg and Tanis [14, p. 409]). In each case there were several instances in which this condition was violated, causing the observed values of the chi-square test statistic to be inflated. As a result, this caused inaccurate calculation of the p-value, often leading to the rejection of H 0. Our analysis of the data for finds weak evidence against the claim that Toshiba s data follow Benford s Law. However, we are also interested in knowing whether the same results hold for the period since it was in 2008 when Toshiba was discovered to have overstated its profits. The results of the analysis for the period are provided in Table 4.2.

42 35 Table 4.2: The results of Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit tests for Toshiba Corporation from Digits p-values for Goodness-of-Fit Tests for Benford s Law Observed values of the chi-square test statistic p-values of the chi-square test Observed values of the discrete K-S test statistic p-values of the discrete K-S test D (D 1, D 2 ) (D 1, D 2, D 3 ) Similar to the previous analysis, when testing for D 1 using the data for , the chi-square test fails to reject H 0. On the other hand, for D 1, (D 1, D 2 ), and (D 1, D 2, D 3 ), the Kolmogorov-Smirnov test rejects H 0 at the significance level of α = These results remain consistent with the reports of the company at the time of the scandal. Thus, we have statistically significant evidence against the claim that Toshiba s data for follow Benford s Law, and we therefore may infer that Toshiba s consolidated financial reports for that period were fabricated. 4.3 Valeant Pharmaceuticals International, Inc. Valeant Pharmaceuticals International, Inc. is a Canadian multinational pharmaceutical company, which develops generic pharmaceuticals, over-the-counter products, and medical devices. The growth of the company over the past decade was due, largely, to a series of mergers and acquisitions. During that period, the growth strategy of Valeant turned away from investments in research and development and toward aggressive marketing and rapid price increases on current products. An analysis from 2015 revealed that for drugs whose prices had

43 36 risen between 300% and 1,200% in the previous two years, about nine of the top nineteen drugs belonged to Valeant (Surowiecki [24]). In addition, Valeant was accused of using eerie accounting practices to obscure the performance of their newly acquired companies. In 2015, a group of Congressional officials criticized Valeant over its overpricing of drugs, and since then there has been an ongoing criminal investigation. Since the start of the investigation, Valeant s market value has decreased by nearly 90% and, according to its 2015 annual report, the company now is in debt by approximately $30 billion (Roumeliotis [23]). Our goal in this study is to test whether or not data drawn from Valeant s annual reports conform to Benford s Law Data For this study we will analyze the annual reports of Valeant for the fiscal years The data were compiled from archive filings provided by Morningstar, Inc., an investment research firm that provides reports on thousands of corporate companies [16]. Similar to the Toshiba analysis, data were drawn from the comparative consolidated balance sheets of Valeant. The PDF-scraping software tool, Tabula, was used to extract data from the annual reports. It is important to note that the fiscal year for Valeant begins on January 1 and ends on December 31 of the same calendar year, and the units of each value are in U.S. dollars. A sample comparative consolidated balance sheet is provided in Figure 4.2. A total of 272 data points were compiled throughout the eight year period.

44 37 Figure 4.2: The Comparative Consolidated Balance Sheets for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16] Applications of Benford s Law to Valeant s Data As mentioned above, the goal of this study is to test whether Valeant s data conforms to Benford s distribution. We will again apply the chi-square test and Kolmogorov- Smirnov statistics to test our hypotheses: H 0 : The data for Valeant follow Benford s Law H A : The data for Valeant do not follow Benford s Law

45 38 Table 4.3: The results of Pearson s chi-square and the Kolmogorov-Smirnov goodness-of-fit tests for Valeant Pharmaceuticals International, Inc. from Digits p-values for Goodness-of-Fit Tests for Benford s Law Observed values of the chi-square test statistic p-values of the chi-square test Observed values of the discrete K-S test statistic p-values of the discrete K-S test D (D 1, D 2 ) (D 1, D 2, D 3 ) Similar to the Toshiba analysis, we will apply our proposed goodness-of-fit tests to the first digit, the first two digits, and the first three digits. Due to the violation of the conditions necessary for the chi-square test, the chi-square statistic will not be applied for (D 1, D 2 ) and (D 1, D 2, D 3 ) because the sample size is not large enough to make accurate calculations of the p-value. Instead, we will continue to rely on the Kolmogorov-Smirnov statistic. The results of each test are shown in Table 4.3. As is evident from Table 4.3, the chi-square test and the Kolmogorov-Smirnov test each fail to reject H 0 for all digits analyzed. Each p-value for these tests fall above the significance level of α = 0.05, so we fail to reject the null hypothesis that the Valeant data follow Benford s Law. Although our analysis finds no statistically significant evidence that Valeant used fraudulent accounting techniques during , media reports state that prosecutors and the SEC were not suspicious of Valeant until Therefore, we believe it is also beneficial to apply our tests to the data for those fiscal years.

46 39 Figure 4.3: The Consolidated Statements of (Loss) Income for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16]. In this portion of the study, more information was needed to run the analysis for the period. Therefore, further PDF-scraping of Valeant s annual reports was required for the purpose of attaining an adequate sample size. In addition to the comparative consolidated balance sheets, the consolidated statements of (loss) income, the consolidated statements of comprehensive (loss) income, and the consolidated statements of cash flows were analyzed. Examples of the new consolidated statements are

47 Figure 4.4: The Consolidated Statements of Comprehensive (Loss) Income for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16]. 40

48 Figure 4.5: The Consolidated Statements of Cash Flows for Valeant Pharmaceuticals International, Inc. from the Fiscal Year 2015 [16]. 41