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Background risk model in presence of heavy tails under dependence

Dimitrios G. Konstantinides, Charalampos D. Passalidis

TL;DR

The paper investigates dependence in heavy-tailed risk settings and introduces Generalized Tail Asymptotic Independence (GTAI) to capture second-order joint extremal behavior. It proves a bi-variate max-sum equivalence for randomly weighted sums when primary variables lie in $\mathcal{D}\cap \mathcal{L}$ and weights are bounded, leading to asymptotic ruin results in a bi-dimensional renewal framework. It also analyzes tail distortion risk measures under a Background Risk Model with multivariate regular variation, deriving asymptotic expressions for $\rho_g[\mathbf{\Theta X(w)}|\mathbf{\Theta X(w)}>VaR_p]$ and connecting them to marginal VaRs via Breiman-type results. Together, these results extend multivariate heavy-tailed analysis to dependent components and provide practical tools for joint tail risk assessment and distortion-based risk quantification in risk management.

Abstract

In this paper, we examine two problems on applied probability, which are directly connected with the dependence in presence of heavy tails. The first problem, is related to max-sum equivalence of the randomly weighted sums in bi-variate set up. Introducing a new dependence, called Generalized Tail Asymptotic Independence, we establish the bi-variate max-sum equivalence, under a rather general dependence structure, when the primary random variables follow distributions from the intersection of the dominatedly varying and the long tailed distributions. On base of this max-sum equivalence, we provide a result about the asymptotic behavior of two kinds of ruin probabilities, over a finite time horizon, in a bi-variate renewal risk model, with constant interest rate. The second problem, is related to the asymptotic behavior of the Tail Distortion Risk Measure, in a static portfolio, called Background Risk Model. In opposite to other approaches on this topic, we use a general enough assumption, that is based on multivariate regular variation.

Background risk model in presence of heavy tails under dependence

TL;DR

The paper investigates dependence in heavy-tailed risk settings and introduces Generalized Tail Asymptotic Independence (GTAI) to capture second-order joint extremal behavior. It proves a bi-variate max-sum equivalence for randomly weighted sums when primary variables lie in and weights are bounded, leading to asymptotic ruin results in a bi-dimensional renewal framework. It also analyzes tail distortion risk measures under a Background Risk Model with multivariate regular variation, deriving asymptotic expressions for and connecting them to marginal VaRs via Breiman-type results. Together, these results extend multivariate heavy-tailed analysis to dependent components and provide practical tools for joint tail risk assessment and distortion-based risk quantification in risk management.

Abstract

In this paper, we examine two problems on applied probability, which are directly connected with the dependence in presence of heavy tails. The first problem, is related to max-sum equivalence of the randomly weighted sums in bi-variate set up. Introducing a new dependence, called Generalized Tail Asymptotic Independence, we establish the bi-variate max-sum equivalence, under a rather general dependence structure, when the primary random variables follow distributions from the intersection of the dominatedly varying and the long tailed distributions. On base of this max-sum equivalence, we provide a result about the asymptotic behavior of two kinds of ruin probabilities, over a finite time horizon, in a bi-variate renewal risk model, with constant interest rate. The second problem, is related to the asymptotic behavior of the Tail Distortion Risk Measure, in a static portfolio, called Background Risk Model. In opposite to other approaches on this topic, we use a general enough assumption, that is based on multivariate regular variation.
Paper Structure (7 sections, 9 theorems, 103 equations)

This paper contains 7 sections, 9 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Concepts and motivation.
  3. Notations.
  4. Generalized tail asymptotic independence
  5. Application on bi-dimensional renewal risk model
  6. Tail distortion risk measures
  7. Conclusion

Key Result

Lemma 2.1

Under the Assumption ass.KP.2, we obtain that all the products $\Theta_{1}X_{1},\,\ldots,\,\Theta_{n}X_{n}$, $\Delta_{1}Y_{1},\,\ldots,\,\Delta_{m}Y_{m}$, are $GTAI$.

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.1
  • Example 2.1
  • Example 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • ...and 15 more