Diversification and Stochastic Dominance: When All Eggs Are Better Put in One Basket

Paper: arXiv 2507.16265

Authors: Léonard Vincent

Abstract

Diversification is usually viewed as a reliable way to reduce risk, yet it can dramatically fail for heavy-tailed losses with infinite mean: pooling independent losses of this type may increase tail risk at every threshold. We study this reversal by comparing a diversified portfolio (a weighted average) of risks to a “one-basket” benchmark that concentrates the full exposure on a single component chosen at random according to the same weights. In the iid case, the benchmark reduces to a single risk, recovering the classical comparison between a single risk and a diversified portfolio. Our main result – the one-basket theorem – provides new sufficient conditions under which the diversified portfolio has larger tail probabilities for all thresholds (first-order stochastic dominance) than this benchmark. The theorem enables weight-specific verification of the stochastic dominance relation and yields new applications, notably to averages of infinite-mean discrete Pareto risks. We further show that these failures of diversification are boundary cases of a general phenomenon: diversification always increases the likelihood of exceeding thresholds near zero, and under specific conditions this local effect extends to all thresholds, yielding first-order stochastic dominance.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework and new theoretical conditions (the one-basket theorem) under which portfolio diversification worsens risk via first-order stochastic dominance for infinite-mean losses. While the theoretical derivations and literature connections are exceptionally complete and well-articulated, the paper is purely mathematical with no empirical backtesting or real-world dataset validations.

Research Flowchart

  flowchart TD
    A[Research Goal: Evaluate portfolio diversification vs. one-basket concentration for heavy-tailed, infinite-mean losses] --> B[Inputs: Independent risks X_i & portfolio weights w_i]
    B --> C[Methodology: Compare diversified portfolio ∑w_i X_i to One-Basket Benchmark X_I via First-Order Stochastic Dominance]
    C --> D[Analytical Derivation: Evaluate tail probabilities P & threshold behavior near zero]
    D --> E[Outcome 1: One-Basket Theorem - Identifies conditions where concentration dominates diversification across ALL thresholds]
    D --> F[Outcome 2: Local Reversal - Diversification always increases risk near zero and extends globally under heavy tails]

Cited References (35)

1. Portfolio Selection (1952)
2. Universal portfolios (1991)
3. A theoretical investigation of randomized asset allocation strategies (1998)
4. Comparison Methods for Stochastic Models and Risks (2002)
5. Correlation and Dependence in Risk Management: Properties and Pitfalls (2002)
6. The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee (2004)
7. New majorization theory in economics and martingale convergence results in econometrics (2005)
8. Actuarial theory for dependent risks: measures, orders and models (2006)

6 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.