Portfolio optimization and the random magnet problem

¹ Center for Polymer Studies and Department of Physics Boston University - Boston, MA 02215, USA
² Institut für Theoretische Physik, Universität zu Köln - D-50937 Köln, Germany
³ Department of Physics, Boston College - Chestnut Hill, MA 02167, USA

Received: 20 December 2001
Accepted: 17 May 2002

Abstract

Diversification of an investment into independently fluctuating assets reduces its risk. In reality, movements of assets are mutually correlated and therefore knowledge of cross-correlations among asset price movements are of great importance. Our results support the possibility that the problem of finding an investment in stocks which exposes invested funds to a minimum level of risk is analogous to the problem of finding the magnetization of a random magnet. The interactions for this “random magnet problem” are given by the cross-correlation matrix C of stock returns. We find that random matrix theory allows us to make an estimate for C which outperforms the standard estimate in terms of constructing an investment which carries a minimum level of risk.