Portfolio optimization and the random magnet problem

– 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. Challenging optimization problems are encountered in many branches of science. Typical examples include the traveling salesman problem [1–3] and the traveling tourist problem [4]. Another type of optimization problem occurs when system parameters are not accurately known and only estimates are available, such as in the problem of finding the least risky investment in the stock market which earns a given return. Such an investment is called an optimal portfolio and was introduced by Markowitz [5]. It has been suggested [6] that the calculation of an optimal portfolio has an analogy in pure physics: finding the ground state of a random magnet. However, the portfolio optimization problem is more intricate due to the fact that many “system” parameters such as correlations are not known with any degree of accuracy, but can only be estimated from empirical data. Two relevant pieces of information are necessary for an investor to judge the quality of an investment: the investor must know (i) the expected relative change in price (“return”), and (ii) the uncertainty of the return (“risk”), usually measured by the standard deviation of the returns over some preselected time intervals. Given two investments with the same return, the investment with smaller risk is preferred. One way to reduce risk is to diversify the investment, i.e., to buy stocks of not one, but of N different companies [5,7]. Diversifying the investment would work best if the fluctuations of stock prices were completely uncorrelated; the risk would then decrease with N as 1/ √ N . In reality, the price fluctuations of different stocks are correlated. The challenging optimization problem is to choose the fraction of money to be invested into each stock mi where i runs over all N stocks, in such a way as to minimize the effect of correlations on risk of the N -stock portfolio. We define the return Gi as the relative