Common scaling behavior in finance and macroeconomics

In order to test whether scaling exists in finance at the world level, we test whether the average growth rates and volatility of market capitalization (MC) depend on the level of MC. We analyze the MC for 54 worldwide stock indices and 48 worldwide bond indices. We find that (i) the average growth rate 〈r〉 of the MC and (ii) the standard deviation σ(r) of growth rates r decrease both with MC as power laws, with exponents αw = 0.28 ± 0.09 and βw = 0.12 ± 0.04. We define a stochastic process in order to model the scaling results we find for worldwide stock and bond indices. We establish a power-law relationship between the MC of a country’s financial market and the gross domestic product (GDP) of the same country. PACS. 89.65.Gh Economics; econophysics, financial markets, business and management – 89.65.-s Social and economic systems – 89.75.Da Systems obeying scaling laws – 02.50.Ey Stochastic processes Many real-world systems exhibit self-similarity, where fluctuations exhibit similar statistical properties at all scales [1–3]. As a result, these systems can be classified by the scale-free functions that characterize the underlying dynamics. In recent decades, methods from statistical physics have been extended into finance and macroeconomics, where for example, references [4,5] analyzed the fluctuations in the gross domestic product (GDP) of 152 countries and found that (i) the broad central region of the distribution of annual growth rates decays according to a Laplace distribution; and (ii) the standard deviation scales as a power law of GDP with a scaling exponent β ≈ 0.15. These scaling results were also obtained for several other macroeconomic variables [5], as well as for microeconomic variables, such as the number of employees and the sales of companies [6]. The same scaling properties (i) and (ii) have found by analyzing growth dynamics of university research [7], and recently by analyzing annual growth rates for many companies comprising different financial indices [8]. In finance, the probability density function (pdf) of logarithmic returns of stock price exhibits a power-law decay in the far tails described by a inverse quartic law [9,10]. Recent contributions to the field india e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] e e-mail: [email protected] cate that the far tails of the pdf of growth rates, of both microeconomic and macroeconomic variables, exhibit approximately inverse cubic power laws [5,11]. In order to further investigate the scaling properties in finance and macroeconomics, first we analyze Bloomberg data on the stocks comprising a common stock index, the Nasdaq Composite Index [12]. For each stock we know the stock price and market capitalization (MC) at the end of each year, whereMC is calculated as the number of stocks outstanding multiplied by the price of the stock. We define an annual logarithmic growth rate