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A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization

NeurIPS 2024 · Learning theory and algorithms › Combinatorial and search optimization · approach: Constrained neural network optimization · 1 citations (OpenAlex)

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Reformulates m-sparse Sharpe ratio maximization as an m-sparse quadratic program and solves it with a proximal gradient algorithm whose KL-based analysis gives global optimality under conditions.

Problem

Maximizing the Sharpe ratio of a portfolio under an exact cardinality (m-sparse, l0) constraint, plus long-only and self-financing constraints, is nonconvex and fractional, and few existing methods handle it with guarantees.

Approach

Convert the m-sparse fractional SR problem into an equivalent m-sparse quadratic program (subtraction form) via a scaling argument, then apply a proximal gradient algorithm whose projection onto the m-sparse nonnegative set is closed-form; KL property of the semi-algebraic objective gives convergence and conditions for global optimality.

Key results

• On FF25EU mSSRM-PGA's Sharpe ratio is over 70% higher than 1/N for both T=60 and T=120 (e.g., 0.2712 vs 0.1574 at T=60).
• mSSRM-PGA achieves the best Sharpe ratio on all 6 datasets when T=60 and on 5 of 6 when T=120 among compared methods.
• Actual portfolio sparsity averages about 4.4-4.9 assets on FF25EU and 8-11 on FF100/FF100MEINV across m=10,15,20.

Measured effects reported in the paper

Surprising findings

• The 1/N strategy outperforms most competitors in most settings, despite its lack of optimization.
• The actual number of selected assets stays well below the sparsity level m and is stable across m values.

Limitations

• Cannot directly handle fractional models with nondifferentiable numerator and denominator.
• Global optimality guaranteed only under conditions (mv*<m or a gradient condition, plus EPER); otherwise only local optimality.
• Evaluated only on monthly Kenneth French portfolio datasets with no baseline runtime reported.

Experimental setup

Datasets: Kenneth R. French FF25 FF25EU FF32 FF49 FF100 FF100MEINV

Benchmarks: FF25 FF25EU FF32 FF49 FF100 FF100MEINV

Metrics: monthly Sharpe ratio cumulative wealth transaction cost sensitivity sparsity (support size)

Baselines: 1/N SPOLC SSPO S1 S2 S3 SSMP MAXER IPSRM-D PLCT

Builds on: Sharpe ratio SSPO Pang PLCT proximal gradient method

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