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

S_t=S_t-1+W_t, 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 W_t is constant, and the distribution of W_t 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 W_t 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 S_t itself and the variable W_t, 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.

• Legal notice:
  

  Copyright (c) Pielaszek Research, all rights reserved.
  The above materials, including auxiliary resources, are subject to Publisher's copyright and the Author(s) intellectual rights. Without limiting Author(s) rights under respective Copyright Transfer Agreement, no part of the above documents may be reproduced without the express written permission of Pielaszek Research, the Publisher. Express permission from the Author(s) is required to use the above materials for academic purposes, such as lectures or scientific presentations.
  In every case, proper references including Author(s) name(s) and URL of this webpage: https://science24.com/paper/2470 must be provided.