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Combination of traditional and parametric insurance: calibration method based on the optimization of a criterion adapted to heavy tail losses

Olivier Lopez, Daniel Nkameni

TL;DR

The paper tackles protection against heavy-tailed losses by blending traditional indemnity with a parametric, covariate-driven tail component activated above a threshold $s$. It develops a tail-adapted optimization criterion and a two-step calibration procedure to cope with tail data scarcity, supported by theoretical convergence results. Empirical validation through simulated and real tornado data demonstrates that the proposed index-based hybrid contract can outperform a conventional capped indemnity while reducing claim-management costs. The approach offers a practical framework for insuring extreme risks, with potential extensions to high-dimensional covariates and alternative tail modeling to further improve robustness and applicability.

Abstract

In this paper, we address the problem of providing insurance protection against heavy-tailed losses, for which the expected loss may not even be finite. The product we study is based on a combination of traditional insurance up to a given limit and a parametric (or index-based) cover for larger losses. This second component of the coverage is computed from covariates available immediately after the loss occurs, allowing claim management costs to be reduced through rapid compensation. To optimize the design of this second component, we use a criterion adapted to extreme losses, that is, to loss distributions of Pareto type. We support the calibration procedure with theoretical results establishing its convergence rate, as well as empirical evidence from both a simulation study and a real-data analysis on tornado losses in the United States. We also propose a two-step optimization procedure as a potential solution to the issue of data scarcity in the tails of loss distributions. We conclude by empirically demonstrating that the proposed hybrid contract outperforms a traditional capped indemnity contract.

Combination of traditional and parametric insurance: calibration method based on the optimization of a criterion adapted to heavy tail losses

TL;DR

The paper tackles protection against heavy-tailed losses by blending traditional indemnity with a parametric, covariate-driven tail component activated above a threshold . It develops a tail-adapted optimization criterion and a two-step calibration procedure to cope with tail data scarcity, supported by theoretical convergence results. Empirical validation through simulated and real tornado data demonstrates that the proposed index-based hybrid contract can outperform a conventional capped indemnity while reducing claim-management costs. The approach offers a practical framework for insuring extreme risks, with potential extensions to high-dimensional covariates and alternative tail modeling to further improve robustness and applicability.

Abstract

In this paper, we address the problem of providing insurance protection against heavy-tailed losses, for which the expected loss may not even be finite. The product we study is based on a combination of traditional insurance up to a given limit and a parametric (or index-based) cover for larger losses. This second component of the coverage is computed from covariates available immediately after the loss occurs, allowing claim management costs to be reduced through rapid compensation. To optimize the design of this second component, we use a criterion adapted to extreme losses, that is, to loss distributions of Pareto type. We support the calibration procedure with theoretical results establishing its convergence rate, as well as empirical evidence from both a simulation study and a real-data analysis on tornado losses in the United States. We also propose a two-step optimization procedure as a potential solution to the issue of data scarcity in the tails of loss distributions. We conclude by empirically demonstrating that the proposed hybrid contract outperforms a traditional capped indemnity contract.

Paper Structure

This paper contains 24 sections, 5 theorems, 85 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Index based reinsurance
  3. A metric to optimize
  4. Price aversion
  5. Risk and index based compensation
  6. A first method of selection of an optimal payoff function $\phi$
  7. A new metric to optimize an index-based product
  8. Approximation of the criterion
  9. Alternative optimization for the parametric part
  10. Convergence rate for the statistical optimization
  11. Empirical analysis on simulated data
  12. Setting
  13. Results
  14. Empirical analysis on real data
  15. Data and setting
  16. ...and 9 more sections

Key Result

Theorem 3.1

Assume that $\sup_x |L'(x)|<\infty.$ Let where Under Assumption a-1 to assum_pdelta, with $\sup_{\theta}|\frak{R}_{\theta}(s)|\rightarrow 0$ when $s$ tends to infinity.

Figures (9)

  • Figure 1: Panel (a) shows the objective functions for the whole sample of size $m$ as a function of the parameter $\theta$ for both $\hat{\mathfrak{L}}$ and $\hat{\mathfrak{L}}^*$. Panel (b) shows the payout function $\phi_{\hat{\tilde{\theta}}}$ as a function of the real losses of policyholders. $\hat{\tilde{\theta}}$ is the optimal value of $\theta$ obtained from the maximization of $\hat{\mathfrak{L}}_\theta$.
  • Figure 2: Evolution of the estimates of the parameters $a$ (panel (a)) and $b$ (panel (b)) of equation \ref{['eq:simul_shape']} with sample size.
  • Figure 3: Panel (a) shows the errors incurred in the estimation of $\hat{\mathfrak{L}}$ by $\hat{\mathfrak{L}}_n$ (in one step) and $\hat{\mathfrak{L}}^*_n$ (in two steps) as a function of $n$ (see steps 7 and 8 of algorithm \ref{['alg:simul']}). Panel (b) shows the objective values obtained through $\hat{\mathfrak{L}} = \hat{\mathfrak{L}}_m$, $\hat{\mathfrak{L}}_n$, and $\hat{\mathfrak{L}}^*_n$. The evolution of the latter two is presented as a function of $n$.
  • Figure 4: Tornado losses per unit area at the starting locations of the tornadoes. A distinction is made between losses per unit area below and above the 85$^{\text{th}}$ percentile, which is the value chosen for $s$.
  • Figure 5: Objective function $\hat{\mathfrak{L}}^*$ with respect to $\theta_1$ and $\theta_2$ for $s=q_{0.85}(Y)$ (panel (a)) and for $s\in \{q_{0.84}(Y), q_{0.85}(Y), q_{0.86}(Y)\}$ (panel (b)).
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 8.1
  • Proposition 8.2
  • Lemma 8.3