Quantitative relations between risk, return and firm size

We analyze —for a large set of stocks comprising four financial indices— the annual logarithmic growth rate R and the firm size, quantified by the market capitalization MC. For the Nasdaq Composite and the New York Stock Exchange Composite we find that the probability density functions of growth rates are Laplace ones in the broad central region, where the standard deviation σ(R), as a measure of risk, decreases with the MC as a power law σ(R)∼ (MC)−β . For both the Nasdaq Composite and the S&P500, we find that the average growth rate 〈R〉 decreases faster than σ(R) with MC, implying that the return-to-risk ratio 〈R〉/σ(R) also decreases with MC. For the S&P500, 〈R〉 and 〈R〉/σ(R) also follow power laws. For a 20-year time horizon, for the Nasdaq Composite we find that σ(R) vs. MC exhibits a functional form called a volatility smile, while for the NYSE Composite, we find power law stability between σ(r) and MC. Copyright c © EPLA, 2009 Estimations of risk and return are of interest in both finance and physics [1–12]. Both risk and return (growth rate) generally decrease with increasing firm size [4,10,12], but what are the functional dependences? Does risk or return decay faster? The distribution of returns is nonGaussian, but what is the functional dependence? To address these questions, we quantify the relationship between the average growth rate of stock price and firm size, between risk and firm size, and between return-to-risk ratio and firm size. We analyze Bloomberg data on the stocks comprising four common stock indices, the New York Stock Exchange (NYSE) Composite Index, the S&P 500, the Nasdaq Composite Index, and the FTSE All-Share Index [13]. For each stock we know the stock price and market capitalization (MC) for 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 Rt ≡ ln (St/St−1), where St and St−1 are the stock prices in two consecutive years. We do not mix data from different market indices, in contrast to some other investigations [4], which enables us to investigate possible differences among these markets. (a)E-mail: [email protected] Previous studies found that the broad central region of the probability density function (pdf) of returns exhibits slow convergence over a time scale ∆t from a truncated stable Levy distribution (∆t < 30 minutes) to a Gaussian distribution, in agreement with the central-limit theorem [1,14,15]. Here we test this convergence, finding that on the 1-year time scale, the pdf of returns R assumes a form that is between the stable Levy and the Gaussian distributions. The form of the pdf for annual returns R is important, since in practice, financial contracts are commonly based on times scales that are greater than 3 months into the future. We first analyze 6679 stocks comprising the Nasdaq Composite Index covering the 6-year time horizon, January 1, 2002 to January 1, 2008 (for which the index content is available). In order to assess the common properties of the market, we aggregate all 11836 pairs (Rt,MCt) into one common data set. In fig. 1(a) we show P (R) for two different ranges of MC (MC<7.8×107 andMC>8.4×109). Each P (R) displays a) an approximately double-exponential (Laplace) form, P (R) = 1 √ 2 σ(R) e−( √ 2|R−〈R〉|/σ(R)), (1) b) the standard deviation σ(R) depends on the MC, where smaller σ(R) corresponds to larger MC values.