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An unexpected stochastic dominance: Pareto distributions, dependence, and diversification

Yuyu Chen, Paul Embrechts, Ruodu Wang

Abstract

We find the perhaps surprising inequality that the weighted average of independent and identically distributed Pareto random variables with infinite mean is larger than one such random variable in the sense of first-order stochastic dominance. This result holds for more general models including super-Pareto distributions, negative dependence, and triggering events, and yields superadditivity of the risk measure Value-at-Risk for these models.

An unexpected stochastic dominance: Pareto distributions, dependence, and diversification

Abstract

We find the perhaps surprising inequality that the weighted average of independent and identically distributed Pareto random variables with infinite mean is larger than one such random variable in the sense of first-order stochastic dominance. This result holds for more general models including super-Pareto distributions, negative dependence, and triggering events, and yields superadditivity of the risk measure Value-at-Risk for these models.
Paper Structure (8 sections, 5 theorems, 27 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stochastic dominance for super-Pareto risks
  3. Super-Pareto distributions and weak negative association
  4. The main result
  5. Proof of Theorem \ref{['thm:1']}
  6. A few remarks
  7. Conclusion
  8. Proofs of all other results

Key Result

Proposition 1

A random variable $X$ with essential infimum $z_X\in \mathbb{R}$ is super-Pareto if and only if the function $g: x\mapsto 1/\mathbb{P}(X>x)$ is strictly increasing and concave on $[z_X,\infty)$. If further $X$ is regular, then $z_X> 0$ and $g(x) \le x /z_X$ for $x \ge z_X$.

Figures (1)

  • Figure 1: An illustration of $\{\theta_1Y_1+\theta_2Y_2\ge x\} = \{ Y_2\ge \delta\}\cup \{\theta_1Y_1+\theta_2Y_2\ge x;Y_2< \delta\}$, where $x=5$, $\theta_1 = 2/5$, $\theta_2 =3/5$, and $\delta=23/3$

Theorems & Definitions (15)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Remark 1: EVT
  • Remark 2: Stable distributions
  • Remark 3: Notions of negative dependence
  • ...and 5 more