Prediction of outstanding liabilities in non-life insurance.

Abstract

A fully time-continuous approach is taken to the problem of predicting the total liability of a non-life insurance company. Claims are assumed to be generated by a non-homogeneous marked Poisson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Poisson processes. By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner. Special attention is given to the case where the claim intensity per risk unit is a stationary stochastic process. A theory of continuous linear prediction is instrumental.

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

“…Works in this direction are due to Waters and Papatriandafylou (1985), Arjas (1989), Neuhaus (1992), and Norberg (1993). More recently, Mikosch (1995a) and(1995b) proposed to describe delayed claims in terms of Poisson shot noise processes.…”

Asymptotic results for perturbed risk processes with delayed claims

“…These liabilities refer to incidents which have already occurred and will become the subject of claims in the future, but have not yet been reported to the insurer. Stochastic models for this kind of claims, which require payments over several years until they are finally settled, have been proposed by Waters and Papatriandafylou (1985), Arjas (1989), Neuhaus (1992), Norberg (1993), Klüppelberg and Mikosch (1995) and Brémaud (2000). In particular, Klüppelberg and Mikosch (1995) propose the shot-noise process as a natural model for delay in claim settlement.…”

Semiparametric model for prediction of individual claim loss reserving

“…Recently, the individual claim loss model has attracted a great deal of interest in the literature, which was proposed firstly by Norberg (1986Norberg ( , 1993 and Jewell (1989Jewell ( , 1990) (see also Hachemeister (1980) and Haastrup and Arjas (1996)) in an attempt to lay down a comprehensive architecture of claims, and has been extensively investigated by Taylor et al (2006) and Larsen (2007). This individual claim loss model is fundamentally distinct from the conventional models based on the grouped or aggregated claims, referred to as the aggregate claim model by Taylor et al (2006).…”