17 publications classées par:
type de publication
: Revue avec comité de lecture
Articles Dufresne F. & Niederhauser E. (1997). Some analytical approximations of stop-loss premiums. Bulletin de l'Association Suisse des Actuaires, 25-47. [abstract]
This paper presents and compares five analytical formulas for the approximation of stop-loss premiums. Two of them, based on the inverse Gaussian distribution, are not widely known. The authors also suggest a technique which improves the precision of these approximations for portfolios.
Dufresne F. (1996). An Extension of Kornya's Method with Application to Pension Funds. Bulletin de l'Association Suisse des Actuaires, 171-181. [abstract]
A simple extension of the method of Kornya is derived. The extended method applies to the convolution of triatomic distributions with nonnegative support while the original method is restricted to diatomic distributions. This way, the algorithm can be applied in the calculation of the distribution of the total claims of a pension fund where only death and disability of active members are considered.
Dufresne F. (1995). The Efficiency of the Swiss Bonus-malus System. Bulletin de l'Association Suisse des Actuaires, 29-42. [abstract]
The efficiency of the (new) Swiss bonus-malus system is analyzed according to three efficiency measures. In addition, the analytical form of the stationary distribution of the system, which is involved in two of the efficiency measures, is obtained as a byproduct of its recursive calculation scheme.
Dufresne F. & Gerber H.U. (1993). The Probability of Ruin for the Inverse Gaussian and Related Processes. Insurance: Mathematics and Economics, 12(1), 9-22. [url] [abstract]
We consider a family of aggregate claims processes that contains the gamma process, the Inverse Gaussian process, and the compound Poissonprocess with gamma or degenerate claim amount distribution as special cases. This is a one-parameter family of stochastic processes. It is shown how the probability of ruin can be calculated for this family. Extensive numerical results are given and the role of the parameter is discussed.
Dufresne F. & Gerber H.U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance: Mathematics and Economics, 10(1). [url] [abstract]
The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distribution of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.
Dufresne F. & Gerber H.U. (1991). Rational ruin problems? A note for the teacher. Insurance: Mathematics and Economics, 10(1), 21-29. [url] [abstract]
It is shown how user friendly examples of ruin theory problems can be constructed, i.e., examples where all the parameters are integers or rational numbers. Tables containing more than 200 such examples are provided.
Dufresne F., Gerber H.U. & Shiu E.S.W. (1991). Risk theory with the gamma process. ASTIN Bulletin, 21(2), 177-192. [pdf] [abstract]
The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.
Dufresne F. & Gerber H.U. (1989). Three methods to calculate the probability of ruin. ASTIN Bulletin, 19(1), 71-90. [pdf] [abstract]
The first method, essentmlly due to GOOVAERTS and DE VYLDER, uses the connection between the probabiliy of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.
Dufresne F. & Gerber H.U. (1988). The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 7(3), 193-199. [url] [abstract]
In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).
Dufresne F. & Gerber H.U. (1988). The probability and severity of ruin for combinations of exponential claim amount distribution and their translations. Insurance: Mathematics and Economics, 7(2). [url] [abstract]
In the classical compound Poisson model of the collective theory of risk let ?(u, y) denote the probability that ruin occurs and that the negative surplus at the time of ruin is less than ? y. It is shown how this function, which also measures the severity of ruin, can be calculated if the claim amount distribution is a translation of a combination of exponential distributions. Furthermore, these results can be applied to a certain discrete time model.
Actes de conférence (partie) Viquerat S. & Dufresne F. (2008). How to get rid of round-off errors in recursive formulas. Insurance: Mathematics and Economics.
Stoica D. & Dufresne F. (2004). Evaluating the distribution of the discounted value of cash flows. Insurance: Mathematics and Economics.
Stoica D. & Dufresne F. (2003). Recursive calculation of moments and spproximation of the accumulated value of cash flows. Insurance: Mathematics and Economics.
Thèses TOUKOUROU Y., Dufresne F. (Dir.) (2016). ASSET LIABILITY MANAGEMENT AND JOINT MORTALITY MODELLING IN OLD-AGE INSURANCE. Université de Lausanne, Faculté des hautes études commerciales. [pdf] [abstract]
Studying old-age population is an active field of research in actuarial science. Espe- cially in the current context of aging population in OECD countries, the actuary has a leading role in managing the related risks. In this thesis, the old-age challenges are adressed considering the pension fund asset liability management (ALM) study as well as the modelling of joint mortality. The thesis contains three chapters.¦Chapter 1: Asset Liability Management for Pension Funds: A Survey¦Before examining a specific model of ALM for pension funds, it is of interest to review the different methodologies discussed in the literature. In this chapter, the analysis is conducted in two steps. At first, the reader is introduced to the different features of a pension fund. The Swiss system is discussed as an example. In partic- ular, we describe the Swiss three pillars system, emphasize the future reforms and discuss its implications. A brief comparison with some selected OECD countries shows that the Swiss pension funds perform quite well. Secondly, we identify two types of risks in a pension fund: the financial risks and the demographic risks. The ALM framework provides the theoretical background for managing these risks. Considering some key ALM methods, the chapter analyses the advantages and disadvantages of the models.¦Chapter 2: On Integrated Chance Constraints in ALM for Pension Funds¦The goal of this chapter is to discuss a concrete ALM model using the stochastic programming framework. In this respect, we discuss the role of integrated chance constraints (ICC) as quantitative risk constraints in ALM for pension funds. We de- fine two types of ICC: the one period integrated chance constraint (OICC) and the multiperiod integrated chance constraint (MICC). As their names suggest, the OICC covers only one period whereas several periods are taken into account with the MICC. A multistage stochastic linear programming model is therefore developed for this purpose and a special mention is paid to the modeling of the MICC.¦Based on a numerical example, we firstly analyse the effects of the OICC and the MICC on the optimal decisions (asset allocation and contribution rate) of a pension fund. By definition, the MICC is more restrictive and safer compared to the OICC. Secondly, we quantify this MICC safety increase. The results show that although the optimal decisions from the OICC and the MICC differ, the total costs are very close, showing that the MICC might represent a good alternative.¦Chapter 3: On Bivariate Lifetime Modeling in Life Insurance Applications¦Mortality has an important impact on the pension fund population. Chapter 3 pro- poses a model that describes the lifetimes within a married couple. Insurance and annuity products covering several lives require the modelling of the joint distribu- tion of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are indepen- dent. However, extensive research over the past decades suggests otherwise. In this chapter, a copula approach is used to model the dependence between lifetimes within a married couple using data from a large Canadian insurance company. As a novelty, the age difference and the gender of the elder partner are introduced as an argument of the dependence parameter. Maximum likelihood techniques are thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several products available on the life insurance market, the paper concludes with practical illustrations.
Gaille S., Dufresne F. (Dir.) (2010). Improving longevity and mortality risk models. Université de Lausanne, Faculté des hautes études commerciales.
Viquerat S., Dufresne F. (Dir.) (2010). On the efficiency of recursive evaluations with applications to risk theory. Université de Lausanne, Faculté des hautes études commerciales. [pdf] [abstract]
On the efficiency of recursive evaluations with applications to risk theoryCette thèse est composée de trois essais qui portent sur l'efficacité des évaluations récursives de la distribution du montant total des sinistres d'un portefeuille de polices d'assurance au cours d'un période donnée. Le calcul de sa fonction de probabilité ou de quantités liées à cette distribution apparaît fréquemment dans la plupart des domaines de la pratique actuarielle.C'est le cas notamment pour le calcul du capital de solvabilité en Suisse ou pour modéliser la perte d'une assurance vie au cours d'une année. Le principal problème des évaluations récursives est que la propagation des erreurs provenant de la représentation des nombres réels par l'ordinateur peut être désastreuse. Mais, le gain de temps qu'elles procurent en réduisant le nombre d'opérations arithmétiques est substantiel par rapport à d'autres méthodes.Dans le premier essai, nous utilisons certaines propriétés d'un outil informatique performant afin d'optimiser le temps de calcul tout en garantissant une certaine qualité dans les résultats par rapport à la propagation de ces erreurs au cours de l'évaluation.Dans le second essai, nous dérivons des expressions exactes et des bornes pour les erreurs qui se produisent dans les fonctions de distribution cumulatives d'un ordre donné lorsque celles-ci sont évaluées récursivement à partir d'une approximation de la transformée de De Pril associée. Ces fonctions cumulatives permettent de calculer directement certaines quantités essentielles comme les primes stop-loss.Finalement, dans le troisième essai, nous étudions la stabilité des évaluations récursives de ces fonctions cumulatives par rapport à la propagation des erreurs citées ci-dessus et déterminons la précision nécessaire dans la représentation des nombres réels afin de garantir des résultats satisfaisants. Cette précision dépend en grande partie de la transformée de De Pril associée.
Stoica D., Dufresne F. (Dir.) (2007). Essays on the treatment of cash flows under stochastic interest rates. Université de Lausanne, Faculté des hautes études commerciales. [abstract]
Abstract¦This paper shows how to calculate recursively the moments of the accumulated and discounted value of cash flows when the instantaneous rates of return follow a conditional ARMA process with normally distributed innovations. We investigate various moment based approaches to approximate the distribution of the accumulated value of cash flows and we assess their performance through stochastic Monte-Carlo simulations. We discuss the potential use in insurance and especially in the context of Asset-Liability Management of pension funds.