DENUIT Michel

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A major challenge for modern societies, the loss of autonomy in the elderly, also known as dependence, is defined as a state of inability to perform all or part of the Acts of Daily Living (ADL) alone. It appears in the vast majority of cases under the effect of chronic pathologies related to aging. Faced with the significant costs associated with this condition, private insurers have developed a range of products to supplement public assistance. To quantify the risk, a multi-state model is used and the question arises of estimating the transition probabilities between the states (autonomy, death and one or more levels of dependence). Under the Markov hypothesis, these depend only on the current state, an assumption that is too restrictive to account for the complexity of the dependency process. In the more general semi-Markovian framework, these probabilities also depend on the time spent in the current state. In this thesis, we study the need for a semi-Markovian modeling of the process. We highlight the impact of the time spent in dependency on the probabilities of death. We also show that taking into account the diversity induced by the pathologies allows us to improve significantly the adequacy of the proposed model to the studied data. Moreover, we establish that the particular shape of the probability of death as a function of the time spent in dependency can be explained by the mixture of the groups of pathologies that constitute the population of dependent individuals.

In addition to risk aversion, decision-makers tend to be also downside risk averse. Besides the usual size for risk trade-off, this allows several other trade-offs to be considered. The decision to increase the level of self-protection generates five trade-offs each involving an unfavourable downside risk increase and an accompanying beneficial change. Five stochastic orders that correspond to these trade-offs are defined, characterised and used to prove comparative static theorems that provide information concerning the self-protection decision. The five stochastic orders are general in nature and can be applied in any decision model where downside risk aversion is assumed.

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In this paper, we propose new relational models linking some specific mortality experience to a reference life table. Compared to existing relational models which distort the forces of mortality, we work here on the age scale. Precisely, age is distorted making individuals younger or older before performing the computations with the reference life table. This is in line with standard actuarial practice, specifically with the so-called Rueff’s adjustments. It is shown that the statistical inference can be conducted with the help of a suitably modified version of the standard IRWLS algorithm in a Poisson GLM/GAM setting. A dynamic version of this model is proposed to produce mortality projections. Numerical illustrations are performed on Belgian mortality statistics.

The implementation of Solvency II leads actuaries to question the good adequacy between models and data. Therefore, this thesis aims to study several statistical approaches, often unknown to practitioners, allowing the use of multi-state methods to model and manage individual risks in insurance. Chapter 1 presents the general context of this thesis and positions its main contributions. We discuss the basic concepts related to the use of multi-state models in insurance and describe the classical inference techniques adapted to the data encountered, whether Markovian or non-Markovian. Finally, we present how these models can be used for credit risk management. Chapter 2 focuses on the use of non-parametric inference methods for the construction of incidence laws in LTC insurance. Since several input causes are likely to be involved and of interest to actuaries, we focus on a method used for estimating continuous-time Markovian multi-state models. We then compare these estimators to those classically used by survival analysis practitioners. This second approach can have significant biases because it does not allow to correctly understand the possible interaction between the causes. In particular, it includes an independence hypothesis that cannot be tested in the context of competing risks models. Our approach then consists in measuring the error made by practitioners when constructing incidence laws. A numerical application is then considered on the basis of data from a long term care insurer.

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The existing literature on savings, insurance, and portfolio choices under risk has revealed that quite often comparative statics results depend, among other things, upon the values of the coefficients of relative risk aversion and relative prudence. More specifically the benchmark values for these coefficients are, respectively, one and two. Recently, several papers investigated constraints on the higher degree extensions of the coefficients of relative risk aversion and of relative prudence. The present work provides a unified approach to this question based on the concept of elementary correlation increasing transformations, allowing for a better understanding of changes in risk in the multiplicative case.

This paper presents a general result on the random selection of an element from an ordered sequence of risks and uses this result to derive additive and cross risk apportionment. Preferences favoring an improvement of the sampling distribution in univariate or bivariate first-order stochastic dominance are those exhibiting additive or cross risk apportionment. The univariate additive and multiplicative risk apportionment concepts are then related to the notion of bivariate cross risk apportionment by viewing the single-attribute utility function of an aggregate position (sum or product of attributes) as a 2-attribute utility function. The results derived in the present paper allow one to further explore the connections between the different concepts of risk apportionment proposed so far in the literature.