Effectiveness of cardinality-return weighted maximum independent set approach for financial portfolio optimization
Keita Takahashi, Tetsuro Abe, Yasuhito Nakamura, Ryo Hidaka, Shuta Kikuchi, Shu Tanaka
TL;DR
This work addresses the limitations of the Markowitz mean-variance framework by proposing CR-WMIS, a graph-theoretic portfolio optimization that fuses the diversification power of MIS with the return-focused weighting of WMIS, formulated as a QUBO and solved by an Ising-machine. The market is represented as a graph with edges between highly correlated stocks, and the objective is encoded in $H_\mathrm{CR-WMIS} = A \sum_{i<j} f_{i,j} x_i x_j - B \sum_i x_i - \mu_R \sum_i r_i x_i$, with $A \gg B, \mu_R r_i$. A comprehensive five-year backtest on S&P 500 data (2019–2024) demonstrates that CR-WMIS achieves superior cumulative returns and robust risk metrics (MDD, VaR, CVaR, volatility) versus MIS, WMIS, and the index, particularly when expected returns are estimated via exponentially weighted averages (EWAvg) and when using equal-weight allocation. The results validate CR-WMIS as a practical, risk-aware alternative to mean-variance under non-normal returns and parameter uncertainty, while illustrating the viability of physics-inspired solvers for large-scale portfolio optimization.
Abstract
The portfolio optimization problem is a critical issue in asset management and has long been studied. Markowitz's mean-variance model has fundamental limitations, such as the assumption of a normal distribution for returns and sensitivity to estimation errors in input parameters. In this research, we propose a novel graph theory-based approach, the cardinality-return weighted maximum independent set (CR-WMIS) model, to overcome these limitations. The CR-WMIS model pursues the optimization of both return and risk characteristics. It integrates the risk diversification effect by selecting the largest number of weakly correlated stocks, a feature of the maximum independent set (MIS) model, with the weighting effect based on expected returns from the weighted maximum independent set (WMIS) model. We validated the effectiveness of the proposed method through a five-year backtesting simulation (April 2019 - March 2024) using real market data from the S&P 500. For this task, we employed a simulated-bifurcation-based solver for finding high-quality solutions to large-scale combinatorial optimization problems. In our evaluation, we conducted a comprehensive risk assessment, which has not been sufficiently explored in previous MIS and WMIS studies. The results demonstrate that the CR-WMIS model exhibits superiority in both return and risk characteristics compared to the conventional MIS and WMIS models, as well as the market index (S&P 500). This study provides a practical portfolio optimization method that overcomes the theoretical limitations of the mean-variance model, contributing to both the advancement of academic theory and the support of practical investment decision-making.