Benford's law, as accounting professionals use today to detect frauds and explained in the book by Mark J. Nigrini was given by Simon Newcomb in the year 1881. Later, in the mid - twentieth century, the theory saw a variety of applications across disciplines especially when Franc Benford popularised it in his paper titled The Law of Anomalous Numbers in 1938. Soon in the 1970s, the analogy saw its applications in macroeconomics as pointed out by Hal R. Varian.
The mathematical law simply states that if one tries to count the frequencies of numerals from 1 to 9 as first digits in a series of numbers, the number 1 would approximately occur 30.1% of the times given the total frequency and possibility of occurrence declines as the number increases. To share precisely, the number 9 would occur less than 5% of the total frequency. The reason for such occurrences of possibilities is that the frequencies of the numbers when counted as appearing as first digits, tend to follow a logarithmic scale with base 10.
In this blog, the simple experiment on the monthly average index prices for BSE BANKEX between 2003 and 2018 is carried out based on Benford's law to determine the anomalies, if any. Such experiments have been done widely in academia to study the unusual trends in asset prices of certain companies which were either unlisted due to malpractices or have been charged against manipulating the books of accounts besides explicitly questioning the general character of equity markets (which is that markets are a leading indicator of the economy).
Case of BSE BANKEX
The BSE BANKEX, is among the component indices used to compute the final free float market capitalization for BSE’s mainboard index - BSE SENSEX. As a component index, it consists of 10 banking companies taken from both the public and private sector. For the purpose of experiment, the index prices were sampled as on December 15, 2023. The visualisation for illustrating the behaviour of index price is given below wherein,
Figure 1: Benford’s test for average monthly prices of BSE BANKEX
Source: Author’s own calculations Courtesy: Bombay Stock Exchange (BSE)
The X - axis represents the first digits from 1 to 9 and the Y - axis represents the extent to which the digit has occurred in the series (a percentage based on total frequency of digits). Furthermore, the coloured bars represent the observed occurrences of digits and the curve represents the expected frequencies based on Benford's law. Comparing the observed and expected occurrences, it is understood that the frequencies do not conform with the law. This was concluded based on the test results based on Kolmogorov - Smirnov stat for more than 400 samples (as total frequency in this case is larger than 400). So, what does this anomaly tell us about the behaviour of the index all this time since its launch in 2003? The occurrence of number 1 can either be due to index value touching new highs or new lows over the period. Both the cases, however, require further investigation focusing on the number of banks expanding (in case its booming economy) or contracting (in case of recessionary economy) alongside other non - financial sectors such as construction, tourism and hospitality, medicine and healthcare, etc.
As a standalone case, the law can be applied to develop a cobweb model to study the cycle of individual asset prices within the index considering the technical factor that the components within index get rebalanced periodically. In order to give an idea, the exercise can be done to derive the market equilibrium using the buy and sell transactions of individual scrips constituting the index (accounting for periodical rebalancing that takes place based on the stock performance) and later applying the Benford’s law to test for anomalies, if any in the model. This shall also give an idea about anomalies in market transactions.
To expand the application to national accounts, the law can even be experimented to choose a base year. The base year for national accounts and other economic indicators undergo revision to reflect on the present conditions. In order to consider any given year as a base year, it is important that it is a normal year. To provide an idea about the application of law to the choice of base year, one can easily identify the years of recession and boom based on the behaviour of volumes and count the frequencies for which the volumes have been range bound to weigh the higher evidence. Accordingly, based on the planned development cycle in the economy, the base year can be revised.
Karan Agarwal
