Mathematical and Statistical Methods for Actuarial Sciences and Finance

Checking financial markets via Benford’s law: the

S&P 500 case

Marco Corazza, Andrea Ellero and Alberto Zorzi

Abstract.In general, in a given financial market, the probability distribution of the first signif-

icant digit of the prices/returns of the assets listed therein follows Benford’s law, but does not

necessarily follow this distribution in case of anomalous events. In this paper we investigate the

empirical probability distribution of the first significant digit of S&P500’s stock quotations.

The analysis proceeds along three steps. First, we consider the overall probability distribution

during the investigation period, obtaining as result that it essentially follows Benford’s law,

i.e., that the market has ordinarily worked. Second, we study the day-by-day probability distri-

butions. We observe that the majority of such distributions follow Benford’s law and that the

non-Benford days are generally associated to events such as the Wall Street crash on February

27, 2007. Finally, we take into account the sequences of consecutive non-Benford days, and

find that, generally, they are rather short.

Key words:Benford’s law, S&P 500 stock market, overall analysis, day-by-day analysis,

consecutive rejection days analysis

1 Introduction

It is an established fact that some events, not necessarily of an economic nature,

have a strong influence on the financial markets in the sense that such events can

induce anomalous behaviours in the quotations of the majority of the listed assets.

For instance, this is the case of the Twin Towers attack on September 11, 2001.

Of course, not all such events are so (tragically) evident. In fact, several times the

financial markets have been passed through by a mass of anomalous movements which

are individually not perceptible and whose causes are generally unobservable.

In this paper we investigate this phenomenon of “anomalous movements in fi-

nancial markets” in a real stock market, namely the S&P 500, by using the so-called

Benford’s law. In short (see the next section for more details), Benford’s law is the

probability distribution associated with the first significant digit^1 of numbers belong-

ing to a certain typology of sets. As will be made clear in section 2, it is reasonable to

(^1) Here significant digit is meant as not the digit zero.
M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010