Iterative Adjustment of Survival Functions by Composed Probability Distortions

The Geneva Risk and Insurance Review, 2012, 37, (156–179) r 2012 The International Association for the Study of Insurance Economics 1554-964X/12 www.palgrave-journals.com/grir/ Iterative Adjustment of Survival Functions by Composed Probability Distortions Alexis Bienvenüe and Didier Rullière Laboratoire SAF, Institut de Science Financière et d’Assurances, Université de Lyon, Université Lyon 1, EA 2429, 50 Avenue Tony Garnier, Lyon F-69007, France. E-mails: ; We introduce a parametric class of composite probability distortions that can be combined to converge to a target survival function. These distortions respect analytic invertibility and stability, which are shown to be relevant in many actuarial fields. We study the asymptotic impact of such distortions on hazard rates. The paper provides an estimation methodology, including hints for initialisation. Some applications to survival data bring results for catastrophic event impact modelling. We also obtain accurate parametric representations of the mortality trend over years. Finally, we suggest a prospective mortality simulation model that comes naturally from the above analysis. The Geneva Risk and Insurance Review (2012) 37, 156 – 179. doi:10.1057/grir.2011.7; published online 20 December 2011 Keywords: probability distortions; mortality; iterated compositions; hyperbolic transform; risk measure; survival function transformation Introduction In an insurance company, many problems may occur when analysing data mortality. First, it may be necessary to use a reference mortality table, especially when there is a lack of data at some ages, or when the construction of a whole mortality table is excluded. In this case, the reference mortality table lies on a population with a specific risk, distinct from that of the insurance company. These differences of risk-exposed population require an adaptation of one table given the other, which can be expressed as a parametric deformation. Second, a precise representation of mortality over ages shows some local phenomena, leading to a non-monotone hazard rate, which may require a relatively complex parametric shape. Third, the analysis of the evolution of mortality rates over time requires a model that can stay reliable after years. A large amount of literature deals with these problems. To adapt a mortality insurance table given a reference one, one may use Proportional Hazard Alexis Bienvenüe and Didier Rullière Iterative Adjustment of Survival Functions by Composed Probability Distortions 157 transform or Wang transform.1 Heligman and Pollard2 studied the precise structure of mortality as a function of age. Lee and Carter3 described the evolution of mortality over time, and many other authors suggest different parametric representations of mortality and its evolution.4 Nevertheless, these classical parametric solutions have several drawbacks:  These solutions do not improve data adequation, and adding parameters is relatively tricky. This way, considering Wang transforms,1 the use of several successive transforms does not extend the class of transformed survival functions; the adaptation of one table given another with a single parameter may remain insufficiently accurate, and parameters adjunction could denature such a transform. Among other models, such as those of Heligman and Pollard2 or Lee and Carter,3 potential extensions may lead to very different expressions depending on the number of parameters that we wish to add, and the convergence properties of such transformation when increasing the number of parameters are unknown.  The use of several parameters in order to fit data may cause important estimation problems, this estimation being numerically feasible only in the presence of initial values sufficiently close to the solution. Adding parameters or introducing a prospective framework requires the knowledge of initial values that may be hard to obtain.  Practical simulations of random death dates are sometimes generated from easily invertible survival functions in order to speed up simulations. This choice leads away from previously presented classical models to favour simple, easily invertible laws. The good representation of mortality tables is then reduced with the use of laws having few parameters, such as that of Gompertz. Thus, parametric inverse distribution functions are sometimes used to obtain stochastic simulations, but the adequacy of a set of mortality tables will not be able to exceed a given precision. Many parametric expressions have been suggested to deal with each of those problems, but they assume different forms, and it is interesting to look for a common parametric form, which may be used for probability distortions, for static and prospective mortality tables, and for inverse distribution function intended for stochastic simulations. Moreover, depending on the desired accuracy, the choice of the number of parameters, without modifying the nature of the adjustment, is a question of great importance that is difficult to solve with classical tools. 1 2 3 4 See Wang (1996). Heligman and Pollard (1980). Lee and Carter (1992). See Pitacco (2004). The Geneva Risk and Insurance Review 158 Trying to give a helpful tool for all the issues we have introduced, it is natural to suggest the use of probability distortions, and to consider the composition of these distortions. Composed distortions allow us to get accurate and easily invertible adjustments of survival functions, with the possibility of increasing the number of parameters in order to converge to a target law. This choice can be useful to many issues, such as pricing or risk measuring. In this paper, we show how our distortions modify random variables (Proposition 1, linked with Accelerated Failure Time models), hazard rates (Proposition 2) and stop-loss premiums in the regular variation case (Proposition 3). The main finding of this paper is to establish that some particular distortions reduce the number of parameters (Theorem 1), that these distortions allow an initial survival function to converge to any target survival function (Theorem 2) and that accurate initialisation values can be given for parameter estimation (Proposition 4). The paper is structured as follows. In the section “Probability distortions and constraints”, we introduce some general uses of probability distortions in the actuarial field, and the more specific constraints that we have chosen for our distortions. In the section “Transformations”, we deal with the general form of these distortions. Some initial results on distorted random variables are given here. In particular, the sub-section ‘Conversion functions’ gives specific examples of distortions, mainly smoothed and composed versions of a basic class of angle functions. The estimation problem and the convergence demonstration of chosen distortions to any survival function target is explained in the section “Estimation and convergence of iterative (...truncated)