Quantitative relations between risk, return and firm size

B. Podobnik; D. Horvatic; A. M. Petersen; H. E. Stanley

doi:doi:10.1209/0295-5075/85/50003

EPL, 85 (2009) 50003
DOI: 10.1209/0295-5075/85/50003

Quantitative relations between risk, return and firm size

B. Podobnik^{1, 2, 3}, D. Horvatic⁴, A. M. Petersen¹ and H. E. Stanley¹

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

bp@phy.hr

received 28 October 2008; accepted in final form 9 February 2009; published March 2009
published online 18 March 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 $\sigma (R)$ , as a measure of risk, decreases with the MC as a power law $\sigma (R)\sim (MC)^{- \beta}$ . For both the Nasdaq Composite and the S& P 500, we find that the average growth rate $\langle R\rangle$ decreases faster than $\sigma (R)$ with MC, implying that the return-to-risk ratio $\langle R\rangle /\sigma (R)$ also decreases with MC. For the S& P 500, $\langle R\rangle$ and $\langle R\rangle /\sigma (R)$ also follow power laws. For a 20-year time horizon, for the Nasdaq Composite we find that $\sigma (R)$ vs. MC exhibits a functional form called a volatility smile, while for the NYSE Composite, we find power law stability between $\sigma (r)$ and MC.

PACS
02.50.Ey - Stochastic processes.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
89.90.+n - Other topics in areas of applied and interdisciplinary physics.

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