Abstract
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&P 500, 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&P 500, ⟨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.