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Insurer's surplus model with varying risk parameter and delayed reporting

Wojciech Otto 

WydziaƂ Nauk Ekonomicznych Uniwersytetu Warszawskiego (WNE UW), Dluga 44/50, Warszawa 00-241, Poland

Abstract

The discrete-time model of the insurer's surplus process St typically assumes:

St=St-1+Wt, t=1,2,...,

where W1,W2,..., are i.i.d. random variables, representing yearly premium less yearly aggregate claims, and the initial surplus S0 is fixed. The model is intended to produce answers concerning the event of ruin (probability of ruin, time of ruin, deficit when ruin occurs etc.). Typically it is assumed that the premium component of Wt is constant, and the distribution of Wt is known.

In real life however, premium is written in advance to cover claims over the coming exposure period that are often reported and paid a number of periods later. The inadequacy is even more obvious in the case of the continuous-time model, where the time elapsed between receiving premium and paying (eventually) compensations is totally neglected. In order to restore correspondence of the model to real life processes, we could change the interpretation of variables involved. Premium inflow less claims outlays appearing in the model as Wt could be interpreted as corresponding to accounting concepts of premium earned (premium written less increment of the premium reserve) less claims occurred (claims paid plus increment of the outstanding claims reserve). This leads to interpreting the surplus as the amount of free assets, and consequently the ruin as insolvency. Under this interpretation the surplus model is meaningful for practice, as in fact it focuses on phenomena of crucial importance for all involved parties: shareholders, tax authority, policyholders, and insurance supervision. However, the problem arises when we take into account that:
I. outstanding claims amount is a random variable, and the corresponding reserve is in fact its point predictor, based on information available at the accounting date.
Additional problem that makes predictions complex is that:
II. in real life the risk parameter (characterising the claims process) is not fixed, so its predictions are needed as well for premium setting as for reserving purposes.

The paper concerns on incorporating the two above mentioned complications into the model of the insurers surplus. It is shown that (at the cost of certain simplifying assumptions) the incorporation could be presented as such reinterpretation of the surplus St itself and the variable Wt, that leaves classical relationships between these variables unaffected. So, in a way the paper is focused on "calibrating" of the above elements of the model to the empirical evidence.

Techniques used in the paper resembles in general those used by Scheike (1992), who introduced the notion of fair premium for the claim process with dependent increments, applying to this purpose the Doob-Meyer decomposition for sub-martingales. Assumptions have been chosen so as to enable casting the model into the state-space form, which allows for explicit expressions of premium and reserves as based on predictors of respective claim payments. Although using Kalman filtering techniques for reserving purposes is not a new idea, their application for restoring the correspondence of the simple surplus model to real life processes is, to my best knowledge, new.

 

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Presentation: oral at Symposium on Econo- and Sociophysics 2004, by Wojciech Otto
See On-line Journal of Symposium on Econo- and Sociophysics 2004

Submitted: 2004-10-20 20:39
Revised:   2009-06-08 12:55