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Partial comonotonicity and distortion riskmetrics

Muqiao Huang

Abstract

We establish a connection between dependence structures and subclasses of distortion riskmetrics under which the latter are additive. A new notion of positive dependence, called partial comonotonicity, is developed, which nests the existing concepts of comonotonicity and single-point concentration. For two random variables, being comonotonic with a third one does not imply that they are comonotonic; instead, this defines an instance of partial comonotonicity. Any specific instance of partial comonotonicity uniquely characterizes a class of distortion riskmetrics through additivity under this dependence structure. An implication of this result is the characterization of the Expected Shortfall using single-point concentration.

Partial comonotonicity and distortion riskmetrics

Abstract

We establish a connection between dependence structures and subclasses of distortion riskmetrics under which the latter are additive. A new notion of positive dependence, called partial comonotonicity, is developed, which nests the existing concepts of comonotonicity and single-point concentration. For two random variables, being comonotonic with a third one does not imply that they are comonotonic; instead, this defines an instance of partial comonotonicity. Any specific instance of partial comonotonicity uniquely characterizes a class of distortion riskmetrics through additivity under this dependence structure. An implication of this result is the characterization of the Expected Shortfall using single-point concentration.

Paper Structure

This paper contains 17 sections, 21 theorems, 60 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations, conventions, and some basic facts
  3. Risk functionals
  4. Tail events and concentration
  5. Set-indexed partial comonotonicity
  6. Defining $K$-concentration
  7. Partial comonotonic additivity
  8. Function-indexed partial comonotonicity
  9. Defining $g$-comonotonicity
  10. Relative strengths of function-indexed partial comonotonicity
  11. Equivalence to set-indexed partial comonotonicity
  12. On spectral risk measures
  13. Applications
  14. Additivity of well-known functionals
  15. Selection of spectral risk measures via $K$-additivity
  16. ...and 2 more sections

Key Result

Proposition 1

Let $K\in \mathcal{S}$ and suppose a random vector $\mathbf X$ is $K$-concentrated. Then there exists a collection of sets $\{A_p\}_{p\in K}$ such that

Figures (2)

  • Figure 1: An example where $Q_a^-$ and $\rho$ are not additive
  • Figure 2: $K$-concentration as a collection of Ordinal sums

Theorems & Definitions (58)

  • Definition 1
  • Example 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Proposition 3
  • ...and 48 more