What tax dodgers should know
Tuesday, January 22If you write number «9» too often in your tax return you will arouse suspicion.
Common sense is often a difficult thing to deal with – especially in math. We should think that stock prices should all start with a 1 as with 2, or with any other number. At a probability of 11.1%, each of the nine numbers should be the first digit of a stock price. In addition, we would assume that no single number is preferred in population figures, and in physical or mathematical constants. As a matter of fact, in many numbers that have social or physical meaning, however, numbers are not equally distributed.
The first person who made this observation was Simon Newcomb, a Canadian physicist who found the first numbers in log tables to be worn more than those in the back. He concluded that his colleagues dealt more often with numbers that start with 1, or 2 than those beginning with 8, or 9. The seemingly fanciful assumption sank into oblivion, and was only rediscovered in 1938 by a physicist named Fran Benford.
Benford, however, went one step further. He tested his assumption with data from different subject areas – results of the American baseball league, atom weights of chemical elements, experimental data in scientific journals, and numbers in articles of «Reader’s Digest» magazine. Permanently he got the same result: While more than 30% of numbers start with 1, merely 5% start with 9.
You will come to the same conclusion when analyzing closing prices of stocks at the New York Stock Exchange. 33.3% of stocks have a value that starts with the number 1. Only 16.4% of securities start with a 2, and only 6.1% with 9.Incidentally, this phenomenon does not depend on currencies. You would find the same distribution of numbers when calculating with Euros, Yen, or Swiss Francs.
Benford’s distribution of numbers is everywhere. After a census of population in 1990, it turned out that the population figures of 3000 counties obeyed it, as well as the length of rivers, and numerical data in scientific papers.
The distribution remained curiosity until conclusive evidence had been found. It was only in 1995 that the mystery was solved by a Theodore Hill, a professor of mathematics at the Georgia Institute of Technology. His complete demonstration would be a bit of a stretch, but the phenomenon may be illustrated as follows: Suppose a stock is valued at USD 100.00 and appreciates at a 10% annual rate. For 88 months – until its value reaches USD 200 – the stock price begins with 1.The stock only has a leading 2 for 52 months until the stock climbs to USD 300. The stock price will start with 9 only for 12 months because this is how long it takes it to rise from 900 to 1000. Again it will take 88 months for the stock to reach the 2000 mark, and during this period of time the stock price will begin with 1.This roughly complies to the distribution Benford observed. In mathematic terms, it is a logarithmic distribution. The frequency f of a digit d (1,2, … 9) is calculated by the formula f = log (1 + 1/d).For d = 1 we get f = 30.1%, for d = 9 we get f = 4.6%.
After this phenomenon was taken from a mere curiosity to the level of a mathematical law, professionals started to look for practical applications. A professor of accounting subjected 170 000 tax returns to a deeper analysis. He examined whether the numbers in the tax forms obeyed the Bredford distribution. Not only did they obey it, and this is how he was able to detect erroneous tax returns. Since that time, American tax authorities have used this approach in order to identify tax fraud. And computer scientists want to avail the fact that small numbers occur more often as the first digit by adapting a computer’s architecture.
