Quantitative relations between risk, return and firm size

¹  Center for Polymer Studies and Department of Physics, Boston University - Boston, MA 02215, USA
²  Department of Physics, Faculty of Civil Engineering, University of Rijeka - 51000 Rijeka, Croatia
³  Zagreb School of Economics and Management - 10000 Zagreb, Croatia
⁴  Department of Physics, University of Zagreb - 10000 Zagreb, Croatia

Corresponding author: bp@phy.hr

Received: 28 October 2008
Accepted: 9 February 2009

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 , as a measure of risk, decreases with the MC as a power law . For both the Nasdaq Composite and the S& P 500, we find that the average growth rate decreases faster than with MC, implying that the return-to-risk ratio also decreases with MC. For the S& P 500, and also follow power laws. For a 20-year time horizon, for the Nasdaq Composite we find that  vs. MC exhibits a functional form called a volatility smile, while for the NYSE Composite, we find power law stability between and MC.