Robust logistic regression for insurance risk classification

Risk classification is an important part of the actuarial process in Insurance companies. It allows for the underwriting of the best risks, through an appropriate choice of classification variables, and helps set fair premiums in rate-making. Logistic regression is one of the sophisticated statistical methods used by the banking industry to select credit rating variables. Extending the method to insurance risk classification seems natural. But Insurance risks are usually classified in a larger number of classes than good and bad, as is usually the case in credit rating. Here we consider a model generalization to extend the use of logistic regression to insurance risk classification. Since insurance data presents catastrophic losses and heavy tail claim distributions, robust estimation will be important. A new robust regression estimator for the logistic model, both in the binary and multinomial response cases, is proposed. Its asymptotic properties are also studied

On credibility and robustness with the Kalman filter

Bühlmann (1967) gave a formal Bayesian derivation of the credibility ratio estimators that actuaries had been using for many years. Since then various generalizations of Bühlmann's model have appeared in the literature, each relaxing the i.i.d. assumptions in its own way. The introduction of weights is due to Bülhmann & Straub (1970) and that the regressors to Hachemeister (1975), but the first comprehensive actuarial application of the Kalman filter is due to de Jong & Zehnwirth (1983). More recent efforts have concentrated on the robustification of these estimators, as they provedı to be extremely sensitive to large claims. Kremer (1991) studies a robust regression credibility model and Künsch (1992) tackles the weighted case. Following Kremer (1994) we propose here a robust Kalman filter credibility model

On double periodic non-homogeneous poisson processes

Non-homogenous Poisson processes with periodic claim intensity rate are proposed as the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long term trends, illustrated by a double-beta intensity function. Here periodicity does not repeat the exact same short term pattern every year, but lets its peak intensity vary over a longer period. This model reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model are discussed in detail

On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model

We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company's survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.Comment: 23 pages, 3 figure

ON DOUBLE PERIODIC NON-HOMOGENEOUS POISSON PROCESSES

Non-homogenous Poisson processes with periodic claim intensity rate are proposed as the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long term trends, illustrated by a double-beta intensity function. Here periodicity does not repeat the exact same short term pattern every year, but lets its peak intensity vary over a longer period. This model reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model are discussed in detail.

Robust Logistic Regression for Insurance Risk Classification

Risk classification is an important part of the actuarial process in Insurance companies. It allows for the underwriting of the best risks, through an appropriate choice of classification variables, and helps set fair premiums in rate-making. Logistic regression is one of the sophisticated statistical methods used by the banking industry to select credit rating variables. Extending the method to insurance risk classification seems natural. But Insurance risks are usually classified in a larger number of classes than good and bad, as is usually the case in credit rating. Here we consider a model generalization to extend the use of logistic regression to insurance risk classification. Since insurance data presents catastrophic losses and heavy tail claim distributions, robust estimation will be important. A new robust regression estimator for the logistic model, both in the binary and multinomial response cases, is proposed. Its asymptotic properties are also studied.

ON THE TIME VALUE OF RUIN IN THE DISCRETE TIME RISK MODEL

Using an approach similar to that of Gerber and Shiu (1998), a recursive formula is given for the expected discounted penalty due at ruin, in the discrete time risk model. With it the joint distribution of three random variables is obtained; time to ruin, the surplus just before ruin and the deficit at ruin. The time to ruin is analyzed through its probability generating function (p.g.f.). The joint distribution for the compound binomial model is derived in Cheng et al. (2000) using martingale techniques and a duality argument. Here we find a recursive formula for the p.g.f. of ruin time T; the discounted moments of the deficit at ruin and the surplus just before ruin. A detailed discussion is given in the case u = 0 and when the claim size in a unit time is geometrically distributed.

Bayesian Credibility for GLMs

We revisit the classical credibility results of Jewell and B\"uhlmann to obtain credibility premiums for a GLM using a modern Bayesian approach. Here the prior distributions can be chosen without restrictions to be conjugate to the response distribution. It can even come from out-of-sample information if the actuary prefers. Then we use the relative entropy between the "true" and the estimated models as a loss function, without restricting credibility premiums to be linear. A numerical illustration on real data shows the feasibility of the approach, now that computing power is cheap, and simulations software readily available

Modeling and analysis of random and stochastic input flows in the chemostat model

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

The effect of social programs and exposure to professionals on the educational aspirations of the poor

Investment in human capital is an important tool for reducing poverty. However, the poor may lack the capacity to aspire, which often results in underinvestment in their children’s education. This paper studies the effect of a social program on the educational aspirations of the poor, and explores the role of exposure to educated professionals as a possible channel for increasing aspirations. First, using differences-in-differences, we show that beneficiary parents of the Mexican antipoverty program PROGRESA have higher educational aspirations for their children of a third of a school year than do non-beneficiary parents. This effect corresponds to a 15% increase in the proportion of parents who aspire for their children to finish college. Then, we exploit the design of the program whose requirements cause its target population to have different levels of mandated exposure to doctors and nurses. Our triple difference estimate shows that, educational aspirations for children from high-exposure households (relative to low- exposure households) in treatment villages (relative to control villages) were a third of a school year higher six months after the start of the program (relative to before its introduction). These results suggest that the change in aspirations is driven by exposure to highly educated professionals.social programs, educational aspirations, poverty, educational aspirations