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Modeling lower-truncated and right-censored insurance claims with an extension of the MBBEFD class

Selim Gatti, Mario V. Wüthrich

TL;DR

The paper tackles the challenge of modeling lower-truncated and right-censored insurance claims by extending Bernegger's MBBEFD exposure-curve framework to a richer Bernegger class that accommodates unimodal and skewed densities. It develops two main exposure-family directions—logarithmic and exponentially linked—that yield closed-form densities, means, and right-censoring masses, enabling tractable maximum likelihood estimation on observed $(d,M)$-bounded claims. Through a series of explicit models (MBBEFD, power/sine/quadratic variants) and a real property-claims dataset, the authors demonstrate that the Bernegger class can closely approximate observed densities and can outperform classical gamma or log-normal fits in some cases, though in this dataset the log-normal model often provides the strongest simple benchmark. They also show the Bernegger class is closed under common transformations arising from deductible or maximal-cover adjustments, and discuss extrapolation challenges when the observed interval is not nested in the new range, highlighting practical implications for actuarial forecasting and pricing.

Abstract

In general insurance, claims are often lower-truncated and right-censored because insurance contracts may involve deductibles and maximal covers. Most classical statistical models are not (directly) suited to model lower-truncated and right-censored claims. A surprisingly flexible family of distributions that can cope with lower-truncated and right-censored claims is the class of MBBEFD distributions that originally has been introduced by Bernegger (1997) for reinsurance pricing, but which has not gained much attention outside the reinsurance literature. Interestingly, in general insurance, we mainly rely on unimodal skewed densities, whereas the reinsurance literature typically proposes monotonically decreasing densities within the MBBEFD class. We show that this class contains both types of densities, and we extend it to a bigger family of distribution functions suitable for modeling lower-truncated and right-censored claims. In addition, we discuss how changes in the deductible or the maximal cover affect the chosen distributions.

Modeling lower-truncated and right-censored insurance claims with an extension of the MBBEFD class

TL;DR

The paper tackles the challenge of modeling lower-truncated and right-censored insurance claims by extending Bernegger's MBBEFD exposure-curve framework to a richer Bernegger class that accommodates unimodal and skewed densities. It develops two main exposure-family directions—logarithmic and exponentially linked—that yield closed-form densities, means, and right-censoring masses, enabling tractable maximum likelihood estimation on observed -bounded claims. Through a series of explicit models (MBBEFD, power/sine/quadratic variants) and a real property-claims dataset, the authors demonstrate that the Bernegger class can closely approximate observed densities and can outperform classical gamma or log-normal fits in some cases, though in this dataset the log-normal model often provides the strongest simple benchmark. They also show the Bernegger class is closed under common transformations arising from deductible or maximal-cover adjustments, and discuss extrapolation challenges when the observed interval is not nested in the new range, highlighting practical implications for actuarial forecasting and pricing.

Abstract

In general insurance, claims are often lower-truncated and right-censored because insurance contracts may involve deductibles and maximal covers. Most classical statistical models are not (directly) suited to model lower-truncated and right-censored claims. A surprisingly flexible family of distributions that can cope with lower-truncated and right-censored claims is the class of MBBEFD distributions that originally has been introduced by Bernegger (1997) for reinsurance pricing, but which has not gained much attention outside the reinsurance literature. Interestingly, in general insurance, we mainly rely on unimodal skewed densities, whereas the reinsurance literature typically proposes monotonically decreasing densities within the MBBEFD class. We show that this class contains both types of densities, and we extend it to a bigger family of distribution functions suitable for modeling lower-truncated and right-censored claims. In addition, we discuss how changes in the deductible or the maximal cover affect the chosen distributions.
Paper Structure (23 sections, 16 theorems, 146 equations, 10 figures, 14 tables)

This paper contains 23 sections, 16 theorems, 146 equations, 10 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Exposure curves and their resulting densities
  3. From Exposure curves to distributions
  4. Flexibility of the point mass in the right-censoring point
  5. The Bernegger class of distributions
  6. The class of MBBEFD exposure curves and densities
  7. Properties of the MBBEFD class
  8. Extension of Bernegger's idea using non-bell shaped densities
  9. Logarithmic linked exposure family
  10. Exponentially linked exposure family
  11. Real dataset example
  12. The MBBEFD example
  13. Power logarithmic linked exposure example
  14. Sine logarithmic linked exposure example
  15. Quadratic exponentially linked exposure example
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $G:[0,1] \rightarrow \mathbb{R}$ be a non-decreasing, concave, and twice continuously differentiable function with $G(0) = 0, \, G(1) = 1, \, G'(0) > 0$. The function $F_Z: [0,1] \rightarrow \mathbb{R}$ defined by is a distribution function on $[0,1]$. Furthermore, this distribution has as density for $z \in [0,1)$, and a point mass in 1 given by Finally, the mean of $Z \sim F_Z$ is equal t

Figures (10)

  • Figure 1: Lower-truncated and right-censored claim with $d=2000$ and $M=5000$.
  • Figure 2: Swiss Re and Lloyd's exposure curves (lhs) and the resulting densities (rhs).
  • Figure 3: Histogram (lhs) and empirical density (rhs) of the claims $Z_i$, only showing the claims strictly smaller than $1$, and the point mass in $1$ is $3.4 \%$.
  • Figure 4: The MBBEFD example: densities of the random variable $Z|_{\{Z<1\}}$.
  • Figure 5: Power logarithmic linked exposure example: densities of the random variable $Z|_{\{Z<1\}}$.
  • ...and 5 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Remark 2.6
  • Example 3.1: Swiss Re and Lloyd's exposure curves
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • ...and 12 more