DUTANG Christophe

In this paper, we extend the non-cooperative one-period game of Dutang et al. (2013) to model a non-life insurance market over several periods by considering the repeated (one-period) game. Using Markov chain methodology, we derive general properties of insurer portfolio sizes given a price vector. In the case of a regulated market (identical premium), we are able to obtain convergence measures of long run market shares. We also investigate the consequences of the deviation of one player from this regulated market. Finally, we provide some insights of long-term patterns of the repeated game as well as numerical illustrations of leadership and ruin probabilities.

Generalized Linear Models with categorical explanatory variables are considered and parameters of the model are estimated with an original exact maximum likelihood method. The existence of a sequence of maximum likelihood estimators is discussed and considerations on possible link functions are proposed. A focus is then given on two particular positive distributions: the Pareto 1 distribution and the shifted log-normal distributions. Finally, the approach is illustrated on a actuarial dataset to model insurance losses.

The objective of this thesis is to analyze the interactions between economic agents operating in the retail insurance market. On the one hand, policyholders wishing to cover themselves against a risk of loss must explore the market in order to subscribe to a contract in line with their perception of the risk. On the other hand, insurers compete in a regulated market, imposing on them a certain level of capital in order to guarantee their solvency in a context of uncertainty about the risks underwritten. On the other hand, intermediaries offer their services in order to facilitate the interaction between consumers, who are averse to risk, and firms, which take risks. It is therefore in this context that we analyze the behavior of insurance actors from different perspectives. Chapters 1 and 2 of this thesis are the result of laboratory experiments, carried out using a web interface designed specifically for these studies. The results in Chapter 3 are based on a theoretical model and numerical simulations. Chapter 1 focuses on the relationship between honesty and honesty beliefs of economic agents. Using data collected in the laboratory, we show how uncertainty and the feeling of being in more or less advantageous conditions impact both the level of honesty and the belief in honesty towards others. In general, consumers overestimate the honesty of intermediaries. Thus, this result justifies their presence in the insurance market. On the other hand, we also show that the financial incentives offered to intermediaries distort honesty beliefs. The lower the incentive level, the more dishonest behavior is anticipated by consumers. In Chapter 2, we highlight the dilemma faced by the consumer in a market with multiple distribution channels. Should he explore by himself and choose among a large set of contracts or delegate part of his decision to an intermediary plus or minus search costs, we show that obfuscation related to a large amount of information and beliefs in the honesty of intermediaries are the main determinants of search and purchase decisions. We also show that obfuscation and intermediaries' attitudes are sources of inefficiency in decision making, in particular with respect to the characteristics of the insurance contracts purchased by consumers. In this sense, the identification of a focus effect supports the importance of the price level in decision making to the detriment of the risk environment and the level of coverage. The introduction of search costs in the exploration process, as well as the heterogeneity of honesty beliefs, justify the multi-channel distribution strategies adopted by insurers. An analysis of a repeated non-cooperative game is presented in Chapter 3 of this thesis where losses and consumer behavior are stochastic and insurers compete on price. In order to incorporate the regulators' constraints, we determine Nash equilibria under solvency constraints. We also analyze the sensitivity of equilibrium premiums to the parameters of the game, in particular when firms do not benefit from the same comparative advantages (i.e. reputation leading to different levels of customer retention, insurers' seniority leading to different capital stocks).

A parametric framework is proposed to model both attritional and atypical claims for insurance pricing. This model relies on a classical Generalized Linear Model for attritional claims and a non-standard Generalized Pareto distribution regression model for atypical claims. Maximum likelihood estimators (closed-form for the Generalized Linear Model part and computed with Iterated Weighted Least Square procedure for the Generalized Pareto distribution regression part) are proposed to calibrate the model. Two premium principles (expected value principle and standard deviation principle) are computed on a real data set of fire warranty of a corporate line-of-business. In our methodology, the tuning of the safety loading in the two premium principles is performed to meet a solvency constraint so that the premium caps a high-level quantile of the aggregate annual claim distribution over a reference portfolio.

eThis paper deals with the crucial problem of modeling policyholders' behaviours in life insurance. We focus here on the surrender behaviours and model the contract lifetime through the use of survival regression models. Standard models fail at giving acceptable forecasts for the timing of surrenders because of too much heterogeneity, whereas the competing risk framework provides interesting insights and more accurate predictions. Numerical results follow from using Fine & Gray model ([13]) on an insurance portfolio embedding Whole Life contracts. Through backtests, this framework reveals to be quite efficient and recovers the empirical lapse rate trajectory by aggregating individual predicted lifetimes. These results could be particularly useful to design future insurance product. Moreover, this setting allows to calibrate experimental lapse tables, simplifying the lapse risk management for operational teams.

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This short note focuses on the estimation procedure, an iterative weighted least square method, generally used for generalized linear models.

This video is the third example of the forthcoming paper 'A family of distributions tailored to skewed and fat tails' by Patrice Kiener. Four parameters K2, K3 and K4 distributions mentionned in the paper are used to analyze the distribution of the daily log-returns of eight financial assets (Gold bullion, Société Générale, Vivendi, EURUSD exchange rate, VIX, CAC40, DJIA, SP500) during the period January 2007 - December 2013. The data come from the extractData() dataset available in the R package FatTailsR. The video displays the plots of the empirical distributions and their estimates as well as the plot of the logit of the distribution function, which is used for the parameter estimation, over the whole period and for each year. The main results are: (1) The eight assets are perfectly described by K2, K3 and K4 distributions which exhibit high flexibility and excellent adjustment capabilities to the data. (2) The Value-at-Risk (VaR) and the Expected Shortfall (ES) are easily calculated as the formulas have closed forms. (3) Almost all distributions exhibit skewed and fat tails. (4) Over the whole period (seven years = 1818 points), the tail parameter k (k in latin, kappa in greek) varies in the range [2.8, 6]. The lowest value corresponds to DJIA, SP500 indices while the largest value corresponds to EURUSD exchange rate. (5) On a yearly period (about 260 points per year), the fluctuations of the k parameter are in the range [2.5, 10]. The asymmetry of the distribution, measured by parameters d (delta = distortion) or e (epsilon = eccentricity), is much more pronounced. For further information, please contact patrice.kiener@inmodelia.com or dutangc@gmail.com.

Insurance reserving is a key topic for both actuaries and academics. In the present paper, we present an efficient way to compute all the key indicators in a unified approach of the ruin theory and claim reserving methods. The proposed framework allows to derive closed-form formulas for both ruin theory and claim reserves indicators. A numerical illustration of these indicators is carried out on a real dataset from a private insurer.

Moments have been traditionally used to characterize a probability distribution. Recently, L-moments and trimmed L-moments are appealing alternatives to the conventional moments. This paper focuses on the computation of theoretical L-moments and TL-moments and emphasizes the use of combinatorial identities. We are able to derive new closed-form formulas of L-moments and TL-moments for continuous probability distributions. Finally, closed-form formulas for the L-moments for the exponential distribution and the uniform distribution are also obtained.

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In multivariate extreme value statistics, the estimation of probabilities of extreme failure sets is an important problem, with practical relevance for applications in several scientific disciplines. Some estimators have been introduced in the literature, though so far the typical bias issues that arise in application of extreme value methods and the non-robustness of such methods with respect to outliers were not addressed. We introduce a bias-corrected and robust estimator for small tail probabilities. The estimator is obtained from a second order model that is fitted to properly transformed bivariate observations by means of the minimum density power divergence technique. The asymptotic properties are derived under some mild regularity conditions and the finite sample performance is evaluated through an extensive simulation study. We illustrate the practical applicability of the method on a dataset from the actuarial context.

The goal of this paper is to give recent results in risk theory presented at the Conference "Journée MAS 2012" which took place in Clermont Ferrand. After a brief state of the art on ruin theory, we explore some particular aspects and recent results. One presents matrix exponential approximations of the ruin probability. Then we present asymptotics of the ruin probability based on mixing properties of the claims distribution. Finally, the multivariate case, motivated by reinsurance, is presented and some contemporary results (closed forms and asymptotics) are given.

We introduce a robust and asymptotically unbiased estimator for the coefficient of tail dependence in multivariate extreme value statistics. The estimator is obtained by fitting a second order model to the data by means of the minimum density power divergence criterion. The asymptotic properties of the estimator are investigated. The efficiency of our methodology is illustrated on a small simulation study and by a real dataset from the actuarial context.

In this paper, we formulate a noncooperative game to model a non-life insurance market. The aim is to analyze the e ects of competition between insurers through di erent indicators: the market premium, the solvency level, the market share and the underwriting results. Resulting premium Nash equilibria are discussed and numerically illustrated.

The purpose of this paper is to point out that an asymptotic rule "A+B/u" for the ultimate ruin probability applies to a wide class of dependent risk models, in discrete and continuous time. Dependence is incorporated through a mixing approach among claim amounts or claim inter-arrival times, leading to a systemic risk behavior. Ruin corresponds here either to classical ruin, or to stopping the activity after realizing that it is not pro table at all, when one has little possibility to increase premium income rate. Several special cases for which closed formulas are derived, are also investigated in some detail.

This paper deals with optimization methods solving the generalized Nash equilibrium problem (GNEP), which extends the standard Nash problem by allowing constraints. Two cases are considered: general GNEPs where constraint functions are individualized and jointly convex GNEPs where there is a common constraint function. Most recent methods are benchmarked against new methods. Numerical illustrations are proposed with the same software for a fair benchmark.

The generalized Nash equilibrium, where the feasible sets of the players depend on other players' action, becomes increasingly popular among academics and practitionners. In this paper, we provide a thorough study of theorems guaranteeing existence of generalized Nash equilibria and analyze the assumptions on practical parametric feasible sets.

We formulate a noncooperative game to model competition for policyholders among non-life insurance companies, taking into account market premium, solvency level, market share and underwriting results. We study Nash equilibria and Stackelberg equilibria for the premium levels, and give numerical illustrations.

Non-life actuarial science studies the various quantitative aspects of the insurance business. This thesis aims to explain from different perspectives the interactions between the different economic agents, the insured, the insurer and the market, in an insurance market. Chapter 1 highlights the importance of taking into account the market premium in the policyholder's decision to renew or not to renew his insurance contract with his current insurer. The need for a market model is established. Chapter 2 addresses this issue by using non-cooperative game theory to model competition. In the current literature, models of competition are always reduced to a simplistic optimization of premium volume based on a view of one insurer against the market. Starting from a one-period market model, a set of insurers is formulated, where the existence and uniqueness of the Nash equilibrium are verified. The properties of equilibrium premiums are studied to better understand the key factors of a dominant position of one insurer over the others. Then, the integration of the one-period game in a dynamic framework is done by repeating the game over several periods. A Monte-Carlo approach is used to evaluate the probability of an insurer being ruined, remaining leader, or disappearing from the game due to a lack of policyholders in its portfolio. This chapter aims at better understanding the presence of cycles in non-life insurance. Chapter 3 presents in depth the actual Nash equilibrium calculation for n players under constraints, called generalized Nash equilibrium. It provides an overview of optimization methods for solving the n optimization subproblems. This solution is done using a semi-smooth equation based on the Karush-Kuhn-Tucker reformulation of the generalized Nash equilibrium problem. These equations require the use of the generalized Jacobian for the locally Lipschitzian functions involved in the optimization problem. A convergence study and a comparison of the optimization methods are performed. Finally, chapter 4 deals with the calculation of the probability of ruin, another fundamental theme in non-life insurance. In this chapter, a risk model with dependence between the amounts or waiting times of claims is studied. New asymptotic formulas for the probability of ruin in infinite time are obtained in a broad framework of risk models with dependence between claims. In addition, explicit formulas for the probability of ruin in discrete time are obtained. In this discrete model, the dependence structure analysis allows to quantify the maximum deviation on the joint distribution functions of the amounts between the continuous and the discrete version.

Non-life actuarial science studies the various quantitative aspects of the insurance business. This thesis aims to explain from different perspectives the interactions between the different economic agents, the insured, the insurer and the market, in an insurance market. Chapter 1 highlights the importance of taking into account the market premium in the policyholder's decision to renew or not to renew his insurance contract with his current insurer. The need for a market model is established. Chapter 2 addresses this issue by using non-cooperative game theory to model competition. In the current literature, models of competition are always reduced to a simplistic optimization of premium volume based on a view of one insurer against the market. Starting from a one-period market model, a set of insurers is formulated, where the existence and uniqueness of the Nash equilibrium are verified. The properties of equilibrium premiums are studied to better understand the key factors of a dominant position of one insurer over the others. Then, the integration of the one-period game in a dynamic framework is done by repeating the game over several periods. A Monte-Carlo approach is used to evaluate the probability of an insurer being ruined, remaining leader, or disappearing from the game due to a lack of policyholders in its portfolio. This chapter aims at better understanding the presence of cycles in non-life insurance. Chapter 3 presents in depth the actual Nash equilibrium calculation for n players under constraints, called generalized Nash equilibrium. It provides an overview of optimization methods for solving the n optimization subproblems. This solution is done using a semi-smooth equation based on the Karush-Kuhn-Tucker reformulation of the generalized Nash equilibrium problem. These equations require the use of the generalized Jacobian for the locally Lipschitzian functions involved in the optimization problem. A convergence study and a comparison of the optimization methods are performed. Finally, chapter 4 deals with the calculation of the probability of ruin, another fundamental theme in non-life insurance. In this chapter, a risk model with dependence between the amounts or waiting times of claims is studied. New asymptotic formulas for the probability of ruin in infinite time are obtained in a broad framework of risk models with dependence between claims. In addition, explicit formulas for the probability of ruin in discrete time are obtained. In this discrete model, the dependence structure analysis allows to quantify the maximum deviation on the joint distribution functions of the amounts between the continuous and the discrete version.