Paper: arXiv 2610.32959

Authors: An Chen, Christian Laudagé, Shihao Zhu

Abstract

We study optimal collective investment subject to a Value-at-Risk (VaR)-type constraint that accounts for uncertainty about future regulatory requirements. The constraint incorporates several possible wealth thresholds and imposes a minimum weighted average probability of meeting them. We characterize the optimal terminal wealth, which combines regions of constant wealth at regulatory thresholds with Merton-type investment regions. Unlike a classical VaR constraint with only one threshold, our formulation generates protection against several thresholds. For two possible thresholds, we identify when both are protected and when protection concentrates on the larger one. Numerical results show that the regulatory constraint can substantially reshape the optimal payoff and generate large welfare losses relative to unconstrained investment when the required compliance probability is high, even the smaller wealth threshold is high, or initial capital is limited. In heterogeneous investor pools, welfare losses depend strongly on the composition of the risk aversions and the Pareto weights governing risk sharing. Losses are particularly pronounced when highly risk-averse investors dominate the pool. Moreover, investors with smaller capital contributions can bear disproportionately large shares of the total welfare loss, while larger contributors may bear less than their capital-proportional share. Thus, regulatory uncertainty not only reduces aggregate welfare but can also create substantial disparities in how its costs are distributed among investors.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly mathematical framework using martingale methods for optimal portfolio choice under a novel VaR-type constraint. While the empirical section is numerical rather than data-driven, it is well-executed and provides strong insights. The core contribution lies in extending VaR constraints to multiple regulatory thresholds and analyzing welfare distribution in heterogeneous investor pools.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Collective Investment under Regulatory Uncertainty] --> B{Methodology: Stochastic Control & VaR-type Constraint};
    B --> C[Inputs: Regulatory Uncertainty (Multiple Wealth Thresholds), VaR Compliance Probability, Investor Risk Aversions, Initial Capital];
    C --> D{Computational Process: Characterize Optimal Terminal Wealth & Analyze Welfare Impacts};
    D --> E[Key Finding 1: Optimal Terminal Wealth = Constant Wealth at Regulatory Thresholds + Merton-type Regions];
    E --> F[Key Finding 2: Regulatory Constraint Reshapes Payoff, Causes Substantial Welfare Losses (High Compliance Prob, High Thresholds, Limited Capital)];
    F --> G[Key Finding 3: Welfare Losses & Distribution Depend on Investor Heterogeneity (Risk Aversion, Pareto Weights, Capital Contribution)];