Julien Trufin

On a risk measure inspired from the ruin probability and the expected deficit at ruin

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities.

Sarmanov Family of Bivariate Distributions for Multivariate Loss Reserving Analysis

The correlation among multiple lines of business plays a critical role in aggregating claims and thus determining loss reserves for an insurance portfolio. We show that the Sarmanov family of bivariate distributions is a convenient choice to capture the dependencies introduced by various sources, including the common calendar year, accident year, and development period effects. The density of the bivariate Sarmanov distributions with different marginals can be expressed as a linear combination of products of independent marginal densities. This pseudo-conjugate property greatly reduces the complexity of posterior computations. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer.

Impact of underwriting cycles on the solvency of an insurance company

Abstract This paper studies the solvency of an insurance firm in the presence of underwriting cycles. A small or medium-size insurance company with a price-taker position in the market is considered. Its premium income is assumed to obey an autoregressive process with cycles. Specifically, the premium income for a specific calendar year is influenced by the market experience for the last couple years. Under this classical AR(2) dynamics governing the premium income, an explicit expression for the ultimate ruin probability is derived, using a martingale approach, in the lighttailed claims case. Furthermore, the logarithmic asymptotic behavior of the ultimate ruin probability as well as the typical path to ruin are investigated. Then a comparison is made with the classical case where the same company operates on a market without such cycles. Asymptotically, the presence of market cycles is shown to increase the risk for the company. Numerical illustrations are performed on Canadian motor insurance market data and support the theoretical analysis.

Beyond the Tweedie Reserving Model: The Collective Approach to Loss Development

This article proposes a new loss reserving approach, inspired from the collective model of risk theory. According to the collective paradigm, we do not relate payments to specific claims or policies, but we work within a frequency-severity setting, with a number of payments in every cell of the run-off triangle, together with the corresponding paid amounts. Compared to the Tweedie reserving model, which can be seen as a compound sum with Poisson-distributed number of terms and Gamma-distributed summands, we allow here for more general severity distributions, typically mixture models combining a light-tailed component with a heavier-tailed one, including inflation effects. The severity model is fitted to individual observations and not to aggregated data displayed in run-off triangles with a single value in every cell. In that respect, the modeling approach appears to be a powerful alternative to both the crude traditional aggregated approach based on triangles and the extremely detailed individual reserving approach developing each and every claim separately. A case study based on a motor third-party liability insurance portfolio observed over 2004–2014 is used to illustrate the relevance of the proposed approach.

Collective Loss Reserving with Two Types of Claims in Motor Third Party Liability Insurance

Abstract This paper adopts the new loss reserving approach proposed by Denuit and Trufin (2016), inspired from the collective model of risk theory. But instead of considering the whole set of claims as a collective, two types of claims are distinguished, those claims with relatively short development patterns and claims requiring longer developments. In each case, the total payment per cell is modelled by means of a Compound Poisson distribution with appropriate assumptions about the severities. A case study based on a motor third party liability insurance portfolio observed over 2004–2014 is used to illustrate the approach proposed in this paper. Comparisons with Chain-Ladder are performed and reveal significant differences in best estimates as well as in Value-at-Risk at high probability levels.

Updating Mechanism for Lifelong Insurance Contracts Subject to Medical Inflation

This paper proposes a practical way for ex-post indexing of level premiums in lifelong medical insurance contracts, in order to take into account observed medical inflation. We show that ex-post indexing can be achieved by considering only premiums, without explicit reference to reserves. This appears to be relevant in practice as reserving mechanisms may not be transparent to policyholders and as some insurers do not compute contract-specific reserves, managing the whole portfolio in a collective way. The present study originates from a proposal for indexing lifelong medical insurance level premiums in Belgium. As an application, we study the impact of various indexing mechanisms on a typical medical insurance portfolio on the Belgian market.

A note on compound renewal risk models with dependence

Over the last decade, there have been a significant amount of research works on compound renewal risk models with dependence. These risk models assume a dependence relation between interclaim times and claim amounts. In this paper, we pursue their investigation. We apply change of measure techniques within the compound renewal risk models with dependence to obtain exact expressions for the Gerber-Shiu discounted penalty function. We propose a more general approach than the usual one based on the random walk associated to the risk process as it is presented in the literature. More refined, our method keeps the embedded information in the sequence of claim amounts and interclaim times and enables us to derive an exact expression for the Gerber-Shiu discounted penalty function. Simulation is one of the advantages of change of measure techniques since we can find a new probability measure under which ruin occurs almost surely. In this paper, we investigate the importance sampling method based on change of measure techniques to compute several ruin measures. Numerical illustrations are carried out for specific bivariate distributions of the interclaim time and the claim amount to approximate interesting ruin measures.