# The Silicon Valley Bank Failure: Application of Benford’s Law to Spot Abnormalities and Risks

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## Abstract

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## 1. Introduction

## 2. Remembrance—A Dive into the Past

## 3. Benford’s Law

**Figure 1.**Graphic depicting Benford’s law, with the initial values, δ, on the x-axis and their related chance on the y-axis. In the picture above, P(δ) is constrained on the y-axes, while δ is denied on the x-axes. The following ogives were created using the points shown in Table 1.

- It detects fraud by examining discrepancies in the election (Moore and Colley 2022) dataset.
- It is employed to look up pricing digits.
- It is used to authenticate genetic sequences (Waples 2023).
- It is employed to inspect for mistakes in scientific publications (Salemi et al. 2002).

## 4. Methodology

- Data acquisition from real world situations$$\mathcal{D}\left(n\right)\leftarrow {\xi}_{real-world}\forall n\in {\mathbb{Z}}^{+}$$
- If the data are a time series, the first and foremost thing would be to calculate the Lyapunov Exponent. The rapidity at which two extremely close pathways separate is represented mathematically by the Lyapunov exponent. Works by Tolle and Budzien (Tolle et al. 2000) have shown that dynamical systems follow Benford’s law. Diep and Desgranges (Diep and Desgranges 2021) have developed the proposition that Stock Prices seem to follow a statistical physics dynamical approach as a part of their system. The basic criterion for a system to be dynamic is to have chaoticity in its domains. A system can be proved chaotic if the Largest Lyapunov Exponent is greater than zero (${\lambda}_{max}>0$). Rosenstein (Rosenstein et al. 1993) along with his co-authors give an efficient algorithm to calculate the Largest Lyapunov Exponent. If, ${\lambda}_{max}>0$ then the time series of the data subjected towards validation is free from morphological fallacies that can be detected by Benford’s law. In such a case, we should retract the Benford’s Law application. If not, then the further steps (3–7) must be considered. In general, the morphing can affect the trend of any series (time-series to be specific), but it is not exact terms. If the Lyapunov Exponent is found as being greater than 0, it is a clear implication of chaos and proves the dynamic nature present in the time series. Such a series can be, in no way, affected by the data morphing, as it is governed by a set of statistics, set by the responsible. But, if the converse is observed (${\lambda}_{max}<0$), it means the series is neither dynamic nor chaotic, and might be infringed by data morphing.
- Subjecting the real-world data (Dutta et al. 2023f; Dutta and Kumar 2022) to Chi Squared Test, Mantissa Arc Test, and the Mean Absolute Deviation Test, to test its applicability on Benford’s law and similar dynamics.$$\mathcal{D}\left(n\right)=\left\{{\mathbb{x}}_{1},{\mathbb{x}}_{2},{\mathbb{x}}_{3},{\mathbb{x}}_{4},\dots ,{\mathbb{x}}_{n}\right\}$$$${\mathsf{\Delta}}_{MAD}=\frac{1}{n}\sum _{i=1}^{n}\left|{\mathbb{x}}_{i}-\frac{1}{n}\left(\sum _{i=1}^{n}{\mathbb{x}}_{i}\right)\right|$$If$$\underset{\mathbb{R}}{\mathrm{min}}\left(Conformal\text{}Range\right)\le {\mathsf{\Delta}}_{MAD}\le \underset{\mathbb{R}}{\mathrm{max}}\left(Conformal\text{}Range\right)$$
- Render the first digits of each data points from the dataset and compute their probability of occurrence in the main dataset.$${\mathcal{D}}_{f}\left(n\right)=\left\{{\mathbb{x}}_{f1},{\mathbb{x}}_{f2},{\mathbb{x}}_{f3},{\mathbb{x}}_{f4},\dots ,{\mathbb{x}}_{fn}\right\}$$$${\mathbb{x}}_{fi}=\left(\frac{\u2308{\mathbb{x}}_{i}\u2309\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n}-\u2308{\mathbb{x}}_{i}\u2309\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n-1}}{{10}^{n-1}}\right)=\left(\frac{\u230a{\mathbb{x}}_{i}\u230b\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n}-\u230a{\mathbb{x}}_{i}\u230b\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n-1}}{{10}^{n-1}}\right)=\left(\frac{\left[{\mathbb{x}}_{i}\right]\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n}-\left[{\mathbb{x}}_{i}\right]\mathrm{m}\mathrm{o}\mathrm{d}{10}^{n-1}}{{10}^{n-1}}\right)$$
- If the deviation of these probabilities from that of the Benford’s Law is within acceptable ranges (see works by Mark J Nigrini (Nigrini 2012)), the data from the real world is devoid of data morphing.$${Result}_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d}}=\left\{\begin{array}{c}P\left(\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}-\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\right)\le 0.5,\underset{\mathbb{R}}{\mathrm{min}}\left(Conformal\text{}Range\right)\le {\mathsf{\Delta}}_{MSE}\le \underset{\mathbb{R}}{\mathrm{max}}\left(Conformal\text{}Range\right)\\ P\left(\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}-\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\right)0.5,\underset{\mathbb{R}}{\mathrm{min}}\left(Conformal\text{}Range\right){\mathsf{\Delta}}_{MSE}\underset{\mathbb{R}}{\mathrm{max}}\left(Conformal\text{}Range\right)\end{array}\right.$$
- Otherwise, there may be chances of data morphing.
- The last step would be to cross validate the results from the Benford’s Law using that of the Zipf’s Law; if the mean squared error turns out to be less than 0.5, it signifies that the data is free of data morphing, as per the facts and statistics.$${Result}_{\mathrm{z}\mathrm{i}\mathrm{p}\mathrm{f}}=\left\{\begin{array}{c}P\left(\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}-\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\right)\le 0.5,{\mathsf{\Delta}}_{MSE}<0.5\\ P\left(\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}-\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\right)>0.5,{\mathsf{\Delta}}_{MSE}\ge 0.5\end{array}\right.$$

## 5. Results and Discussions

**−0.251**. Thus, there might be chances of morphological fallacies. Thus, the validatory steps of the framework are brought into action. To apply Benford’s Law in any dataset, we must perform some tests to confirm whether the collection (Kanti Kumar et al. 2023) is fit. We have conducted three tests on the display, namely.

- Mean Absolute Deviation Test: The direct deviation from the origin. Suppose we have a dataset, $\left\{{\mathbb{x}}_{1},{\mathbb{x}}_{2},{\mathbb{x}}_{3},{\mathbb{x}}_{4},\dots ,{\mathbb{x}}_{n}\right\}$. It is mean absolute deviation (Yitzhaki and Lambert 2011) is$$\frac{1}{n}\sum _{i=1}^{n}\left|{\mathbb{x}}_{i}-\frac{1}{n}\left(\sum _{i=1}^{n}{\mathbb{x}}_{i}\right)\right|$$To adopt Benford’s law appropriately, a collection must meet the mean absolute deviation requirement within a particular range. Table 2 shows the conformity range (Campbell-Meiklejohn et al. 2012) from a paper by Mark. J. Nigrini.
- Mantissa Arc Test: We may use this investigation to find the centroid of a set of mantissas that have wholly or partially distributed together around a unit circle. The pivot point, or the average vector, is the vector that results if the mantissa of integers is spread evenly over the unit circle with central coordinates (0, 0).
- ${\chi}^{2}$Test: If there is a discrepancy between conceptual stats and the actual data, it can be determined using Pearson’s chi-squared test (Cash 1979), a fitting experiment. Assuming that the null hypothesis is true as n approaches infinity, the ${\chi}^{2}$ dispersion is as follows:$${Z}^{2}=\sum _{i=1}^{k}\frac{{\left({z}_{i}-{m}_{i}\right)}^{2}}{{m}_{i}}=\sum _{i=1}^{k}\left(\frac{{{z}_{i}}^{2}}{{m}_{i}}-n\right)$$

#### 5.1. The SVB Stock Price Opening Value

**0.67412459 (>0.5)**, which is not in the acceptable range.

#### 5.2. The SVB Stock Price Closing Value

**0.672226388 (>0.5)**, which is not in the acceptable range.

#### 5.3. The SVB Stock Price Highest Value

**0.665344714 (>0.5)**, which is not in the acceptable range.

## 6. Conclusions

#### 6.1. Policy Recommendation

#### 6.2. Limitations and Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The contrast between actual and theoretical probabilities, taking into consideration opening price of the Silicon Valley Bank stocks.

**Figure 4.**The contrast between actual and theoretical probabilities taking into consideration the daily opening price of the Silicon Valley Bank stocks as per the Zipfian Distribution.

**Figure 5.**The contrast between actual and theoretical probabilities, taking into consideration closing price of the Silicon Valley Bank stocks.

**Figure 6.**The contrast between actual and theoretical probabilities taking into consideration the daily closing price of the Silicon Valley Bank stocks as per the Zipfian Distribution.

**Figure 7.**The contrast between actual and theoretical probabilities, taking into consideration highest price of the Silicon Valley Bank stocks.

**Figure 8.**The contrast between actual and theoretical probabilities taking into consideration the daily highest price of the Silicon Valley Bank stocks as per the Zipfian Distribution.

**Table 1.**Numbers’ likelihood of occurring according to Benford’s law. The general distribution of numbers is supposed to follow this likelihood table if not tampered with in most cases.

Digit | Probability |
---|---|

1 | 0.301029 |

2 | 0.176091 |

3 | 0.124938 |

4 | 0.096910 |

5 | 0.079181 |

6 | 0.066946 |

7 | 0.057991 |

8 | 0.051152 |

9 | 0.045757 |

Conformity Range | First Digits |
---|---|

Close Conformity | 0.000–0.006 |

Acceptable Conformity | 0.006–0.012 |

Marginal Conformity | 0.012–0.015 |

Non-conformity | Above 0.015 |

**Table 3.**Stock opening price for Silicon Valley Bank. The following segment has been scrapped from the whole dataset to make demonstrating it easier.

Date | Stock Opening Price (USD) |
---|---|

26 October 1987 | 0.751429 |

27 October 1987 | 0.751429 |

28 October 1987 | 0 |

29 October 1987 | 0.751429 |

30 October 1987 | 0.751429 |

2 November 1987 | 0.722997 |

3 November 1987 | 0.722997 |

4 November 1987 | 0.701772 |

5 November 1987 | 0.701772 |

6 November 1987 | 0 |

… | … |

3 March 2023 | 280.339996 |

6 March 2023 | 284.829987 |

7 March 2023 | 280.390015 |

8 March 2023 | 266.859985 |

9 March 2023 | 176.550003 |

10 March 2023 | 106.040001 |

13 March 2023 | 106.040001 |

14 March 2023 | 106.040001 |

15 March 2023 | 106.040001 |

16 March 2023 | 106.040001 |

Digit | Frequency |
---|---|

1 | 2031 |

2 | 2231 |

3 | 1152 |

4 | 1318 |

5 | 1032 |

6 | 375 |

7 | 183 |

8 | 196 |

9 | 268 |

Date | Stock Closing Price (USD) |
---|---|

26 October 1987 | 0.751429 |

27 October 1987 | 0.751429 |

28 October 1987 | 0 |

29 October 1987 | 0.751429 |

30 October 1987 | 0.751429 |

2 November 1987 | 0.722997 |

3 November 1987 | 0.722997 |

4 November 1987 | 0.701772 |

5 November 1987 | 0.701772 |

6 November 1987 | 0 |

… | … |

3 March 2023 | 284.410004 |

6 March 2023 | 283.040009 |

7 March 2023 | 267.390015 |

8 March 2023 | 267.829987 |

9 March 2023 | 106.040001 |

10 March 2023 | 106.040001 |

13 March 2023 | 106.040001 |

14 March 2023 | 106.040001 |

15 March 2023 | 106.040001 |

16 March 2023 | 106.040001 |

Digit | Frequency |
---|---|

1 | 2022 |

2 | 2226 |

3 | 1157 |

4 | 1337 |

5 | 1010 |

6 | 392 |

7 | 179 |

8 | 194 |

9 | 269 |

Date | Stock Highest Price (USD) |
---|---|

26 October 1987 | 0.84353 |

27 October 1987 | 0.84353 |

28 October 1987 | 0 |

29 October 1987 | 0.84353 |

30 October 1987 | 0.84353 |

2 November 1987 | 0.772647 |

3 November 1987 | 0.772647 |

4 November 1987 | 0.751429 |

5 November 1987 | 0.751429 |

6 November 1987 | 0 |

… | … |

3 March 2023 | 285.5 |

6 March 2023 | 286.519989 |

7 March 2023 | 283.079987 |

8 March 2023 | 271.01001 |

9 March 2023 | 177.749893 |

10 March 2023 | 106.040001 |

13 March 2023 | 106.040001 |

14 March 2023 | 106.040001 |

15 March 2023 | 106.040001 |

16 March 2023 | 106.040001 |

Digit | Frequency |
---|---|

1 | 2013 |

2 | 2202 |

3 | 1188 |

4 | 1296 |

5 | 1052 |

6 | 415 |

7 | 191 |

8 | 182 |

9 | 247 |

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## Share and Cite

**MDPI and ACS Style**

Dutta, A.; Voumik, L.C.; Kumarasankaralingam, L.; Rahaman, A.; Zimon, G.
The Silicon Valley Bank Failure: Application of Benford’s Law to Spot Abnormalities and Risks. *Risks* **2023**, *11*, 120.
https://doi.org/10.3390/risks11070120

**AMA Style**

Dutta A, Voumik LC, Kumarasankaralingam L, Rahaman A, Zimon G.
The Silicon Valley Bank Failure: Application of Benford’s Law to Spot Abnormalities and Risks. *Risks*. 2023; 11(7):120.
https://doi.org/10.3390/risks11070120

**Chicago/Turabian Style**

Dutta, Anurag, Liton Chandra Voumik, Lakshmanan Kumarasankaralingam, Abidur Rahaman, and Grzegorz Zimon.
2023. "The Silicon Valley Bank Failure: Application of Benford’s Law to Spot Abnormalities and Risks" *Risks* 11, no. 7: 120.
https://doi.org/10.3390/risks11070120