Damped oscillations in the ratios of stock market indices

A stock market index is an average of a group of stock prices with weights. Different stock market indices derived from various combinations of stocks may share similar trends in certain periods, while it is not expected that there are fixed relations among them. Here we report our investigations on the daily index data of Dow Jones Industry Average (DJIA), NASDAQ, and S&P500 from 1971/02/05 to 2011/06/30. By analyzing the index ratios using the empirical mode decomposition, we find that the ratios NASDAQ/DJIA and S&500/DJIA, normalized to 1971/02/05, approached and then retained the values of 2 and 1, respectively. The temporal variations of the ratios consist of global trends and oscillatory components including a damped oscillation in 8-year cycle and damping factors of 7183 days (NASDAQ/DJIA) and 138471 days (S&P500/DJIA). Anomalies in the ratios, corresponding to significant increases and decreases of indices, only appear in the time scale less than an 8-year cycle. Detrended fluctuation analysis and multiscale entropy analysis of the components with cycles less than a half-year manifest a behavior of self-adjustment in the ratios, and the behavior in S&500/DJIA is more significant than in NASDAQ/DJIA. Copyright c © EPLA, 2012 Financial markets are a complex system, in which traders interact with one another and react to external information to determine the best prices for items. The emergence of econophysics as a new branch of statistical physics has advanced our understanding to financial activities in markets using the concepts and theories developed in physics [1–20]. Properties revealed from empirical data analysis of financial systems usually provide primary clues to understanding the underlying mechanisms and are essential for subsequent modelling [6–21]. These include financial stylized facts [2–6,22,23], such as fat tails in asset return distributions, absence of autocorrelations of asset returns, aggregational normality, asymmetry between rises and falls, volatility clustering [10], and phase clustering [18–20]. Successful empirical analysis and modelling of financial criticality have suggested possible physical pictures for financial crashes and stock market instabilities [11–17]. In this paper, we study the relations among the daily stock market indices of Dow Jones Industry Average (a)E-mail: [email protected] (DJIA), NASDAQ, and S&P500, from 1971/02/05 to 2011/06/30. The data were downloaded from Yahoo Finance (http://finance.yahoo.com/). The original lengths of the data of the three indices are different. In the preprocessing of the data, the three indices are aligned by removing three data points in DJIA and S&P500 (1973/9/26, 1974/10/7, and 1975/10/16) which do not exist in NASDAQ. Finally there are 10197 data points involved in the study. Figure 1(a) shows the daily index data of the three stock markets. Though the indices of DJIA, NASDAQ, and S&P500 are distinct, there is a remarkable feature that by keeping DJIA as a reference and multiplying the NASDAQ index by a factor of 5.2 and S&P500 by 8.5, the curves of the rescaled indices coincide very well in several epoches, except large deviations in NASDAQ for the periods 1999–2001 and 2009–2011, as shown in fig. 1(b). In the year 2011, DJIA is the average price of 30 companies (http://www.djaverages.com/), NASDAQ consists of 1197 companies (http://www.nasdaq.com/), and the S&P500 index is an average result of 500 companies (http://www.standardandpoors.com). Some companies,