Dynamical Optimization Theory of a Diversified Portfolio

TL;DR

This paper introduces a dynamical model for capital redistribution in diversified portfolios, showing it leads to power law distributions and improves growth rates over static strategies, with connections to directed polymers.

Contribution

It presents a novel dynamical redistribution model for portfolios, deriving power law distributions and demonstrating enhanced growth compared to static strategies.

Findings

Power law tails in capital distribution emerge naturally.

Dynamical strategy yields higher growth rates than buy-and-hold.

In large N limit, growth rate approaches expected stock return.

Abstract

We propose and study a simple model of dynamical redistribution of capital in a diversified portfolio. We consider a hypothetical situation of a portfolio composed of N uncorrelated stocks. Each stock price follows a multiplicative random walk with identical drift and dispersion. The rules of our model naturally give rise to power law tails in the distribution of capital fractions invested in different stocks. The exponent of this scale free distribution is calculated in both discrete and continuous time formalism. It is demonstrated that the dynamical redistribution strategy results in a larger typical growth rate of the capital than a static ``buy-and-hold'' strategy. In the large N limit the typical growth rate is shown to asymptotically approach that of the expectation value of the stock price. The finite dimensional variant of the model is shown to describe the partition function of directed polymers in random media.