Je remets ici un lien vers un vieux papier puisque plusieurs personnes m’en ont récemment demandé une version électronique. La version officielle du papier est en ligne sur le site du Bulletin Français d’Actuariat (http://www.institutdesactuaires.com/bfa/, ou ici pour le papier tel qu’il a été publié). Pour ceux qui souhaitent aller plus loin, je mets à leur disposition la base (brute, xls) qui a servie à l’étude. Il s’agit de la version expurgée des noms des assurés, ainsi que de (certaines) informations sur les contrats.
Insurability is a major issue for risk managers in the insurance industry. Zajdenweber (1996) mentioned that business interruption is hardly insurable, using extreme value results: the right tail of the distribution should be modeled using some Pareto distribution with parameter 1, which has none finite moment. Since the expected value in tails is infinite, on a theoretical point of view, it becomes impossible to assess the price of that risk, and to hedge it using standard insurance covers. As we shall see, the use of more advanced results in extreme value theory (a wide survey will be proposed) may let us think that the assumption of very fat tails may be not relevant. For instance, we will propose a test to see if a distribution has a finite mean. We shall also discuss at the end the use of the pure premium as a criteria to assess whether a risk is or not insurable.
Parution d’un article sur l’histoire des mesures de risques, dans la revue Risques, en partant des débats suite à la vaccination, jusqu’à l’adoption de la VaR (Value-at-Risk, i.e un quantile statistique) sous la pression de JP Morgan. En particulier, je voulais rappeler dans ce débat le lien avec la “probabilité de ruine“, étudiée en actuariat depuis plus d’un siècle correspondant à la principale mesure de risque car explicitement liée à la solvabilité d’une compagnie d’assurance.
Publication de Tails of Archimedean Copulas, coécrit avec Johan Segers, dans le Journal of Multivariate Analysis, enfin… en ligne sur http://sciencedirect.com/science…
A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.
Le papier sur l’estimation de quantile par noyau beta, coécrit avec Abder Oulidi, est accepté pour publication dans Statistics and Computing, http://link.springer.com/…
In this paper we propose several nonparametric estimators of quantiles based on Beta kernel and applied to transformed data by the generalized Champernowne distribution initially fitted to the data. A Monte-Carlo based study will show that those estimators improve the efficiency of a traditional ones, not only for light tailed distributions, but also heavy tails, when the probability level is close to 1. We also compare these estimators with the Extreme Value Theory Quantile applying to Danish data on large fire insurance losses.
The paper on Dynamic dependence ordering for Archimedean copulas and distorted copulas should appear soon in Kybernetika.
This paper proposes a general framework to compare the strength of the dependence in survival models, as time changes, i.e. given remaining lifetimes X, to compare the dependence of X given X>t, and X given X>s, where s>t. More precisely, analytical results will be obtained in the case the survival copula of X is either Archimedean or a distorted copula. The case of a frailty based model will also be discussed in details.
Parution d’un article sur l’histoire des tables de mortalité. Le but était d’expliquer la problématique de la construction des nouvelles tables, i.e. des tables prospectives, faites en prenant en compte la dynamique des taux de mortalité, avec les améliorations observées depuis une cinquantaine d’années (si ce n’est plus).
Les tables de mortalité sont un élément essentiel en actuariat puisque légalement, le seul rôle reconnu aux actuaires est précisément dans la certification des tables. Mais de plus, historiquement, les tables de mortalité sont intéressantes puisqu’elle ont permis aux actuaires de participer au développement de la théorie des probabilités. Sur un exemple simple, on peut ainsi voir une élégante application des calculs de lois conditionnelles.
Parution d’un court article sur la Théorie de la Crédibilité dans la revue Risques.
La théorie de la crédibilité est une jolie application de l’approche bayésienne de la statistique, puisque l’on cherche à déterminer la probabilité d’appartenir à une classe de risque conditionnellement à une expérience de sinistralité observée. La théorie développée par Hans Bühlmann propose d’ailleurs de jolies applications, en particulier sur les jeux de fléchettes, où l’on compare la moyenne sur un sous-groupe homogène (ou supposé l’être) avec la moyenne de la population totale. En faisant intervenir des variance intra et des variances inter, on arrive à un résultat relativement simple et élégant.
The paper on the pricing of catastrophe options just appeared in the Proceedings of the Actuarial and Financial Mathematics Conference.
In complete markets, pricing financial products is easy (at least from a theoretical point of view). In incomplete markets (e.g. when the underlying process has jumps with random size, such as an insurance loss process), the price is no longer unique. So on the one hand, it becomes difficult to provide a tractable price of insurance-linked derivatives. On the other hand, when facing catastrophic losses, using the pure premium as a price might not be relevant (e.g. for solvency issues). Both financial market and (re)insurance industry have proposed techniques to price identical hedging products that can be related (e.g. Esscher transform and more generally distorted risk measures in insurance, Gerber-Shiu transform in finance). In this paper, we focus on indifference utility techniques, assuming that stock prices have jumps,related to major catastrophic losses, and thus, partial hedging should then be possible.
La conférence cette année se tiendra les 5 et 6 février (site) a Bruxelles.
Value-at-Risk, despite being adopted as the standard risk measure in finance, suffers severe objections from a practical point of view, due to a lack of convexity, and since it does not reward diversification (which is an essential feature in portfolio optimization). Furthermore, it is also known as having poor behavior in risk estimation (which has been justified to impose the use of parametric models, but which induces then model errors). The aim of this paper is to chose in favor or against the use of VaR but to add some more information to this discussion, especially from the estimation point of view. Here we propose a simple method not only to estimate the optimal allocation based on a Value-at-Risk minimization constraint, but also to derive— empirical—confidence intervals based on the fact that the underlying distribution is unknown, and can be estimated based on past observations.
Je met en ligne un (court) papier sur l’augmentation du nombre de naissance les jours de pleine lune, Lunar cycles and birth rates: from a full Moon to a first quarter Moon effect
Objective The purpose of this study was to examine the influence of the lunar cycle on the frequency of deliveries.
Method This was a time series of 29,385,552 births across 470 lunar cycles that were identified from the French National Institute of Statistics from 1968 to 2005.
Results Using nonparametric smoothing techniques to remove patterns, t-tests validated a significant difference in the number of births on a short window around the full Moon, stress by a significant drop after the first quarter Moon.
Conclusion An analysis of 37 years of data demonstrated a small but statistically significant influence of the lunar cycle on deliveries.
The paper on Convergence of Archimedean Copulas, with Johan Segers, just appeared, in Statistics and Probability Letters.
Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions. No extra differentiability conditions on the generators are needed.
Exposé sur Pricing catastrophe options in incomplete markets, à la conférence Actuarial and Financial Mathematics Conference (interplay between Finance and Insurance), à Bruxelles.
Cet exposé présentait la problématique de la valorisation d’options sur indices catastrophes (en marché incomplets). Une version détaillée apparaîtra dans les Proceedings.
In February 2005, opening the conference on Climate change: a global, national
and regional challenge, chairman Dennis Tirpak pointed out that there is no longer any doubt that the Earth’s climate is changing […] globally, nine of the past 10 years have been the warmest since records began in 1861“. He
singled out the heatwave that gripped western Europe in 2003 as an example:
Europe’s worst natural disaster in 50 years killed as many as 40; 000 people
and inflicted an estimated 30 billion dollars in damage. Hence, the summer of
2003 will be remembered for the extreme heat, and the approximately 40,000
heat-related deaths over western Europe (from 30,000 up to more than 52,000
depending on the source). More specifically, the period 1-15 August 2003 was
the most intense heat of the summer. The report of Pirard et al. (2005) states
that Europe experienced an unprecedented heat wave in the Summer 2003. In France, it was the warmest summer recorded for 53 years in terms of minimal, maximal and average temperature and in terms of duration” […]
Conférence Insurance and Adaptation to Climate Change, Paris, Mars 2007. The paper appeared in the Geneva Papers.
The IPCC 2007 report noted that both the frequency and strength of hurricanes, floods and droughts have increased during the past few years. Thus, climate risk, and more specifically natural catastrophes, are now hardly insurable: losses can be huge (and the actuarial pure premium might even be infinite), diversification through the central limit theorem is not possible because of geographical correlation (a lot of additional capital is required), there might exist no insurance market since the price asked by insurance companies can be much higher than the price householders are willing to pay (short-term horizon of policyholders), and, due to climate change, there is more uncertainty (and thus additional risk). The first idea we will discuss in this paper, about insurance markets and climate risks, is that insurance exists only if risk can be transferred, not only to reinsurance companies but also to capital markets (through securitization or catastrophes options). The second one is that climate is changing, and therefore, not only prices and capital required should be important, but also uncertainty can be very large. It is extremely difficult to insure in a changing environment.
The paper was presented in a conference, in Paris, in 2007
When working on river floods—annual river levels maxima—, two approaches are usually considered: one inspired from Emil Gumbel where annual maxima are supposed to be i.i.d. and distributed according to Gumbel’s distribution, and one inspired from Edwin Hurst where annual maxima are strongly dependent, and exhibit long range memory. This paper tries to solve this apparent paradox by deriving a dynamic model inspired from financial models, which does not take into account annual maxima only but also threshold exceedances. It studies the implications of such a paradox in terms of return period—a notion valid as long as the data are i.i.d—and of extremal events.
"sendo l'intento mio scrivere cosa utile a chi la intende…"
Je remets ici un lien vers un vieux papier puisque plusieurs personnes m’en ont récemment demandé une version électronique. La version officielle du papier est en ligne sur le site du Bulletin Français d’Actuariat (http://www.institutdesactuaires.com/bfa/, ou ici pour le papier tel qu’il a été publié). Pour ceux qui souhaitent aller plus loin, je mets à leur disposition la base (brute, xls) qui a servie à l’étude. Il s’agit de la version expurgée des noms des assurés, ainsi que de (certaines) informations sur les contrats.
Parution d’un 
The 
s la revue Risques.
Modeling the 2003 heatwave in Paris (
Conférence Insurance and Adaptation to Climate Change, Paris, Mars 2007. The paper appeared in the 