Estimation of foreseeable and unforeseeable risks in motor insurance

Estimation of foreseeable and unforeseeable risks in motor insurance

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This project works with the risk model introduced by  Li et al.(2015) and quests modelling, estimating and pricing insurance for risks brought in by innovative technologies, or some other emerging risks. The model considers together two different risks streams that arise together, however not clearly separated or observed. Specifically, we consider a risk surplus process where premiums are adjusted according to past claim frequencies, like in a Bonus-Malus system in motor insurance, where we consider a "classical" or "historical risk" stream and an "unforeseeable risk" one. Particularly, this latter stream represent unknown risks which can be of high uncertainty that, when pricing insurance (ratemaking and experience rating), suggest a sensitive premium adjustment strategy. When a claim arrives, it is not clear for the Actuary to observe which claim comes from one or the other stream, when modelling such risks it is of utmost importance to figure out the nature of both the probability of the occurrence of such claims and its amount. Subsequently, premium calculation must be estimated and fairily reflect these two risk streams. This is not an easy task. We propose here an estimation procedure for the distributions of the claim counts and the corresponding severities for both streams of risks, as well as the premium to be charged. We assume a Bayesian approach as used in credibility theory for premium calculation/estimation.

The model starts assuming a sum of two mixed counting processes, one representing the "historical risk" stream and the other the "unforeseeable risk" one. Concerning the first process, its randomised intensity parameter, has a classical behaviour, it is positive, whereas in the second process, we set a positive probability of the intensity parameter being zero.

Modelling two different streams of risks for the same portfolio can be done exclusively either on the claim count process like Dubey(1977) and Li et al.(2015) tried, or in the claim severity or on both. The introduction of the severity means that the stream difference may also affect this.

We deal the two components. We calculate Bayesian and credibility premia. Parameter estimation is not straightforward, we use the Expectation-Maximisation (EM) algorithm.

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