Diversifying Investments and Maximizing Sharpe Ratio: a novel QUBO formulation

TL;DR

This paper introduces a new QUBO formulation for portfolio optimization that balances maximizing the Sharpe Ratio with diversification, leveraging quantum annealing and hybrid approaches to handle complex, multi-objective financial problems.

Contribution

The paper presents a novel QUBO model for joint optimization of Sharpe Ratio and diversification, facilitating quantum and hybrid computing solutions for complex portfolio management.

Findings

QUBO formulation effectively models multi-objective portfolio optimization.

Hybrid approaches show promise for large-scale problem solving.

Trade-offs between Sharpe Ratio and diversification are demonstrated.

Abstract

The Portfolio Optimization task has long been studied in the Financial Services literature as a procedure to identify the basket of assets that satisfy desired conditions on the expected return and the associated risk. A well-known approach to tackle this task is the maximization of the Sharpe Ratio, achievable with a problem reformulation as Quadratic Programming. While the sole Sharpe Ratio could be efficiently optimized via classical solvers, in business scenarios it is common that multiple additional needs arise, which have to be integrated in the optimization model as either new constraints or objective function terms. Then, in general, the problem may become non-convex and hence could potentially be not efficiently solvable via classical techniques anymore. One example of such additional objective function term consists of maximizing a diversification measure penalizing portfolios holding significant portions of investments on assets belonging to the same sector, while favouring solutions that diversify over multiple sectors. The problem of optimizing both the Sharpe Ratio and a diversification term can be mapped to a QUBO and be solved via quantum annealing devices or Hybrid Computing approaches, which are expected to find high quality solutions. We propose a new QUBO formulation for the task described and provide the mathematical details and required assumptions, showing the ease of modeling the optimization as QUBO against the effort that would be required by classical strategies. We derive results via the available QUBO solvers, as well as discussing the behaviour of Hybrid approaches to tackle large scale problems in the near term. We finally elaborate on the results showing the trade-off between the observed values of the portfolio's Sharpe Ratio and diversification, as a natural consequence of solving a multi-objective optimization problem.