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

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.

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

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.

Paper Structure

This paper contains 28 sections, 24 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Diversification as a source of risk
  3. On the mixture model
  4. The content of this paper
  5. Preliminaries
  6. The classical view on diversification
  7. A reminder on convex order
  8. Convex order in our setting
  9. Main results
  10. The case of a single risk
  11. The general case
  12. Discussion
  13. Illustrations and connections to prior work
  14. Diversification and dependence
  15. Randomized reinsurance
  16. ...and 13 more sections

Key Result

Lemma 2.1

(Closure under increasing transformations) If $X \leq_{\text{st}} Y$ and $f$ is any increasing function, then $f(X) \leq_{\text{st}} f(Y)$. In particular, scaling by a positive constant preserves $\leq_{\text{st}}$. (Closure under convolution) Let $X_1, \dots, X_m$ and $Y_1, \dots, Y_m$ be two sets

Theorems & Definitions (46)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2: One-basket theorem
  • Corollary 4.1
  • Example 5.1: Non-identically distributed Pareto
  • Proposition 5.1: Discrete Pareto
  • Remark 5.1
  • Definition 6.1
  • ...and 36 more