Sparse spanning portfolios and under-diversification with second-order stochastic dominance

ArXiv ID: 2402.01951 “View on arXiv”

Authors: Unknown

Abstract

We develop and implement methods for determining whether relaxing sparsity constraints on portfolios improves the investment opportunity set for risk-averse investors. We formulate a new estimation procedure for sparse second-order stochastic spanning based on a greedy algorithm and Linear Programming. We show the optimal recovery of the sparse solution asymptotically whether spanning holds or not. From large equity datasets, we estimate the expected utility loss due to possible under-diversification, and find that there is no benefit from expanding a sparse opportunity set beyond 45 assets. The optimal sparse portfolio invests in 10 industry sectors and cuts tail risk when compared to a sparse mean-variance portfolio. On a rolling-window basis, the number of assets shrinks to 25 assets in crisis periods, while standard factor models cannot explain the performance of the sparse portfolios.

Keywords: Sparse Portfolios, Stochastic Spanning, Tail Risk, Asset Allocation, Greedy Algorithm

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical constructs including second-order stochastic dominance, submodular optimization, and asymptotic recovery proofs (high math complexity), while also presenting large-scale empirical backtests on equity datasets with rolling-window analysis and tail risk metrics (high empirical rigor).

  flowchart TD
    A["Research Goal: Determine if relaxing<br>sparsity constraints improves<br>investment opportunity sets"] --> B["Methodology: Greedy Algorithm &<br>Linear Programming for Sparse<br>Second-Order Stochastic Spanning"]
    B --> C["Large Equity Datasets<br>(Rolling Windows)"]
    C --> D["Computational Process:<br>Estimate Optimal Sparse Portfolios<br>from 1 to 100+ Assets"]
    D --> E["Key Findings 1:<br>45 assets optimal; 10 sectors<br>captured; no benefit beyond 45"]
    D --> F["Key Findings 2:<br>25 assets in crises; cuts tail risk<br>vs. Mean-Variance; unexplained<br>by standard factor models"]