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A tail-shape actuarial index based on equal level relationships between Value at Risk and Expected Shortfall

Georgios I. Papayiannis, Georgios Psarrakos

TL;DR

The paper tackles non-asymptotic tail-shape analysis by introducing the θ-index, defined through a probability-equal level VaR–FES relationship, linking VaR to a flexible mixture of ES and the mean via $FES_p(X; θ)$. It develops the theoretical properties of θ_p, including affine invariance, monotonicity under convex/concave transforms, a natural θ-order between risks, and a GP-tail characterization through the mean excess function. It then derives Euler risk allocations for θ through a level-dependent PELVaR representation that yields zero-sum component contributions and a practical route to VaR allocation using ES-based quantities. Finally, it validates the framework on standard loss models and the Danish fire dataset, demonstrating its diagnostic value for tail regimes and its potential to improve risk assessment and allocation in actuarial practice.

Abstract

We introduce a new actuarial tail-shape index, the $θ$-index, based on a probability equal level relationship between Value at Risk and Expected Shortfall. The index is defined at each tail probability level as the parameter value for which Value at Risk coincides with Flexible Expected Shortfall, that is a convex mixture of Expected Shortfall and the mean. This yields a level-dependent, scale-free measure of upper tail behaviour. We study basic theoretical properties of the $θ$-index and introduce a partial order for comparing loss distributions, characterized by the monotonicity of right-tail spread ratios. Additionally, the index leads to characterizations of the tail behaviour of a loss distribution as consistent to the generalized Pareto model, through a direct connection to the mean excess function. Moreover, we derive Euler risk contributions for the $θ$-index and use probability equal level relationships to compute Value at Risk allocations in a more stable way. Finally, the $θ$-index is examined as a diagnostic tool for distinguishing tail regimes and its capabilities are illustrated using the Danish fire insurance dataset.

A tail-shape actuarial index based on equal level relationships between Value at Risk and Expected Shortfall

TL;DR

The paper tackles non-asymptotic tail-shape analysis by introducing the θ-index, defined through a probability-equal level VaR–FES relationship, linking VaR to a flexible mixture of ES and the mean via . It develops the theoretical properties of θ_p, including affine invariance, monotonicity under convex/concave transforms, a natural θ-order between risks, and a GP-tail characterization through the mean excess function. It then derives Euler risk allocations for θ through a level-dependent PELVaR representation that yields zero-sum component contributions and a practical route to VaR allocation using ES-based quantities. Finally, it validates the framework on standard loss models and the Danish fire dataset, demonstrating its diagnostic value for tail regimes and its potential to improve risk assessment and allocation in actuarial practice.

Abstract

We introduce a new actuarial tail-shape index, the -index, based on a probability equal level relationship between Value at Risk and Expected Shortfall. The index is defined at each tail probability level as the parameter value for which Value at Risk coincides with Flexible Expected Shortfall, that is a convex mixture of Expected Shortfall and the mean. This yields a level-dependent, scale-free measure of upper tail behaviour. We study basic theoretical properties of the -index and introduce a partial order for comparing loss distributions, characterized by the monotonicity of right-tail spread ratios. Additionally, the index leads to characterizations of the tail behaviour of a loss distribution as consistent to the generalized Pareto model, through a direct connection to the mean excess function. Moreover, we derive Euler risk contributions for the -index and use probability equal level relationships to compute Value at Risk allocations in a more stable way. Finally, the -index is examined as a diagnostic tool for distinguishing tail regimes and its capabilities are illustrated using the Danish fire insurance dataset.

Paper Structure

This paper contains 8 sections, 13 theorems, 70 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main concepts and the $\theta$-index
  3. Theoretical properties of the $\theta$-index and characteri-zation results
  4. Marginal risk contributions with respect to VaR via $\theta$-index
  5. Tail behaviour analysis via $\theta$-index
  6. Illustration of $\theta$-index for standard loss models
  7. Empirical tail analysis of the Danish fire dataset
  8. Conclusions

Key Result

Proposition 1

For a random variable $X \in \mathcal{X}$ and any fixed $\theta \in(0,\infty)$, there exists a unique $p_{\theta} \in (0,1)$ such that where $p_{\theta} := \arg\max_{p \in(0,1)} \hbox{FES}_p(X)$.

Figures (7)

  • Figure 1: Illustration of $\theta_p(X)$ for loss distributions that maintain their shape pattern.
  • Figure 2: The $\theta$-index for some shape-varying loss distributions.
  • Figure 3: Illustration of the right spread ratio curve for couples of different risks
  • Figure 4: Histogram with smoothed density curve (left) and violin plot combined with boxplot (right) for the log-scaled Danish fire data.
  • Figure 5: Mean excess plot (left), $\theta$-index plot (middle) and Hill plot (right) for the Danish fire data.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Proposition 1
  • proof
  • Definition 1: $\theta$-index
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 1
  • ...and 33 more