Functional portfolio optimization in stochastic portfolio theory

TL;DR

This paper introduces a practical convex optimization method for designing rank-based portfolios in stochastic portfolio theory, leveraging exponential concavity to improve stability and computational feasibility, with empirical validation on US stock data.

Contribution

It develops a fully implementable convex optimization framework for functionally generated portfolios using exponential concavity, including stability analysis and empirical testing.

Findings

The optimization problem admits a unique solution.

The method demonstrates stable portfolio performance in empirical tests.

Discretization achieves accurate approximations with manageable errors.

Abstract

In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets, and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We prove an existence and uniqueness result for our optimization problem and provide a stability estimate in terms of a Wasserstein metric of the input measure. Then, we formulate a discretization which can be implemented numerically using available software packages and analyze its approximation error. Finally, we present empirical examples using CRSP data from the US stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs.