# Reliability of Financial Information from the Perspective of Benford’s Law

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Research Methodology

#### 3.1. Research Model

#### 3.2. Research Hypotheses

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

**Hypothesis**

**5.**

#### 3.3. Data Collection

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BL | Benford’s Law |

BSE | Bucharest Stock Exchange |

IFRS | International Financial Reporting Standards |

## References

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**Figure 1.**Accounts receivable. The figure shows the conformity to Benford’s Law of Accounts Receivable in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 12.9442 (p-value 0.11) before IFRS implementation and 10.1538 (p-value 0.25) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0415 based on Kolmogorov-Smirnov is lower than the test value of 0.0532 at 5% level for 654 observations; after IFRS implementation, the AMS of 0.0206, based on Kolmogorov-Smirnov, is lower than the test value of 0.0611 at 5% level for 495 observations; this shows the conformity to Benford’s Law of Accounts Receivable in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0143 before IFRS implementation and 0.0135 after IFRS implementation is lower than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 1.39% before IFRS implementation and 0.92% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 2.**Accounts payable. The figure shows the conformity to Benford’s Law of Accounts Payable in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 7.1375 (p-value 0.52) before IFRS implementation and 6.6219 (p-value 0.58) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0491 based on Kolmogorov-Smirnov is lower than the test value of 0.0554 at 5% level for 602 observations; after IFRS implementation, the AMS of 0.0228, based on Kolmogorov-Smirnov, is lower than the test value of 0.0642 at 5% level for 449 observations this shows the conformity to Benford’s Law of Accounts Payable in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0109 before IFRS implementation and 0.0095 after IFRS implementation is lower than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 1.58% before IFRS implementation and 0.87% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 3.**Positive amounts of Net income (Profit). This figure shows the conformity to Benford’s Law of positive amounts of Net Income in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 8.7040 (p-value 0.37) before IFRS implementation and 7.3784 (p-value 0.50) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0328 based on Kolmogorov-Smirnov is lower than the test value of 0.0560 at 5% level for 589 observations; after IFRS implementation, the AMS of 0.0348, based on Kolmogorov-Smirnov, is lower than the test value of 0.0698 at 5% level for 380 observations; this shows the conformity to Benford’s Law of positive amounts of Net Income in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the Mean Absolute Deviation (MAD) of 0.0113 before IFRS implementation and 0.0111 after IFRS implementation is lower than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 0.49% before IFRS implementation and 0.81% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 4.**Negative amounts of Net income (Loss). The figure shows the conformity to Benford’s Law of negative amounts of Net Income in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 7.9736 (p-value 0.44) before IFRS implementation and 9.6632 (p-value 0.29) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0860 based on Kolmogorov-Smirnov is lower than the test value of 0.1410 at 5% level for 93 observations; after IFRS implementation, the AMS of 0.0685, based on Kolmogorov-Smirnov, is lower than the test value of 0.1158 at 5% level for 138 observations; this shows the conformity to Benford’s Law of negative amounts of Net Income in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the Mean Absolute Deviation (MAD) of 0.0277 before IFRS implementation and 0.0266 after IFRS implementation is bigger than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 2.41% before IFRS implementation and 1.89% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 5.**Positive cash-flows from operating activities. The figure shows the conformity to Benford’s Law of positive Cash-flows from operating activities in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 6.9925 (p-value 0.54) before IFRS implementation and 6.4150 (p-value 0.60) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0330 based on Kolmogorov-Smirnov is lower than the test value of 0.0717 at 5% level for 360 observations; after IFRS implementation, the AMS of 0.0435, based on Kolmogorov-Smirnov, is lower than the test value of 0.0701 at 5% level for 376 observations; this shows the conformity to Benford’s Law of positive Cash-flows from operating activities in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0137 before IFRS implementation and 0.0110 after IFRS implementation is lower than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 1.12% before IFRS implementation and 0.83% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 6.**Negative cash-flows from operating activities. The figure shows the conformity to Benford’s Law of negative Cash-flows from operating activities in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 12.461 (p-value 0.13) before IFRS implementation and 10.3184 (p-value 0.24) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.1095 based on Kolmogorov-Smirnov is bigger than the test value of 0.1089 at 5% level for 156 observations; after IFRS implementation, the AMS of 0.0949, based on Kolmogorov-Smirnov, is lower than the test value of 0.1184 at 5% level for 132 observations; this shows the conformity to Benford’s Law of negative Cash-flows from operating activities in accordance with Kolmogorov-Smirnov test only after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0244 before IFRS implementation and 0.0214 after IFRS implementation is bigger than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 2.44% before IFRS implementation and 2.29% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 7.**Depreciation expenses. The figure shows the nonconformity to Benford’s Law of Depreciation Expenses in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 30.9911 (p-value 0.00) before IFRS implementation and 27.6322 (p-value 0.00) after IFRS implementation is bigger than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0708 based on Kolmogorov-Smirnov is bigger than the test value of 0.0542 at 5% level for 630 observations; after IFRS implementation, the AMS of 0.0669, based on Kolmogorov-Smirnov, is bigger than the test value of 0.0646 at 5% level for 443 observations; this shows the nonconformity to Benford’s Law of Depreciation Expenses in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0201 before IFRS implementation and 0.0241 after IFRS implementation is bigger than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 1.47% before IFRS implementation and 2.59% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 8.**Revenues. The figure shows, in accordance with Chi-square test, the conformity to Benford’s Law of Revenues before IFRS implementation and the nonconformity to Benford’s Law of Revenues after IFRS implementation. Score based on Chi-Square of 5.9008 (p-value 0.66) before IFRS implementation is lower than the critical value of 15.5073 at 5% level and score based on Chi-Square of 20.4081 (p-value 0.01) after IFRS implementation is bigger than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0226 based on Kolmogorov-Smirnov is lower than the test value of 0.0519 at 5% level for 686 observations; after IFRS implementation, the AMS of 0.0588, based on Kolmogorov-Smirnov, is lower than the test value of 0.0599 at 5% level for 516 observations; this shows the conformity to Benford’s Law of Revenues in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0079 before IFRS implementation is lower than 0.015 and score based on the MAD of 0.0181 after IFRS implementation is bigger than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 0.74% before IFRS implementation and 1.37% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

**Figure 9.**Income tax expenses. The figure shows the conformity to Benford’s Law of Income Tax Expenses in accordance with Chi-square test, both before and after IFRS implementation. Score based on Chi-Square of 10.0816 (p-value 0.26) before IFRS implementation and 10.7521 (p-value 0.22) after IFRS implementation is lower than the critical value of 15.5073 at 5% level. Data set columns represents the observed percentages of the first digits and BL columns represents Benford’s Law expected percentages of the first digits. Before IFRS implementation, the absolute maximum score (AMS) of 0.0291 based on Kolmogorov-Smirnov is lower than the test value of 0.0559 at 5% level for 591 observations; after IFRS implementation, the AMS of 0.0404, based on Kolmogorov-Smirnov, is lower than the test value of 0.0681 at 5% level for 399 observations; this shows the conformity to Benford’s Law of Income Tax Expenses in accordance with Kolmogorov-Smirnov test, both before and after IFRS implementation. Score based on the mean absolute deviation (MAD) of 0.0113 before IFRS implementation and 0.0163 after IFRS implementation is lower than 0.0150 (MAD bigger than 0.0150 shows the nonconformity to Benford’s Law in accordance with Nigrini [64]). Standard deviation of differences between data set and Benford’s Law distributions is 0.86% before IFRS implementation and 1.61% after IFRS implementation. Z-Score is calculated for each digit, similarly with the studies of Lacina et al. [28], Carslaw [40], Costa et al. [62] and Santos et al. [63].

Digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Percentage | 0.30103 | 0.17609 | 0.12494 | 0.09691 | 0.07918 | 0.06695 | 0.05799 | 0.05115 | 0.04576 |

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**MDPI and ACS Style**

Jianu, I.; Jianu, I.
Reliability of Financial Information from the Perspective of Benford’s Law. *Entropy* **2021**, *23*, 557.
https://doi.org/10.3390/e23050557

**AMA Style**

Jianu I, Jianu I.
Reliability of Financial Information from the Perspective of Benford’s Law. *Entropy*. 2021; 23(5):557.
https://doi.org/10.3390/e23050557

**Chicago/Turabian Style**

Jianu, Ionel, and Iulia Jianu.
2021. "Reliability of Financial Information from the Perspective of Benford’s Law" *Entropy* 23, no. 5: 557.
https://doi.org/10.3390/e23050557