Complements

Complements

 

 

 

 

Reinsurance

 

Given the wide variety of coverages and possible combinations, we have considered reinsurance as a given in the model. By varying it, we can compare the benefit-cost balance of different options.

 

Let us take the example of two classic forms of treaty: quota share and excess of claims. In both cases, we consider in our model the parameters of the treaties as given :

 

- transfer rate and commission rate for the quota share,        

- priority, ceiling and premium demanded by the market for excess claims.        

 

In order to find optimal coverage, we can study the impact of a variation in reinsurance programs on the evolution of the optimal security system (capital, security load) defined above.

 

From the point of view of calculation, the application of the retention rate for the quota does not pose any difficulty.

 

In the case of excess loss, the market premium can be considered as a given (known or estimated) and exogenous to the model. The impact of the reinsurance treaty on the claims burden must be the subject of a separate study , the results of which will be incorporated into the model via the distributions of the average claims charges per policy, as defined and used at paragraph C.1 . Thus, the following preliminary analysis is carried out : 

 

For T2 type claims, their expense is written as follows :

For a given portfolio volume, we model the burden of large claims of group i before reinsurance by a process composed of frequency / cost : where represents the charge of the jth large claim affecting group i, and the number of large claims.

The application of excess of loss reinsurance is made on a claim by claim basis. By noting the expense of the jth claim of group i after reinsurance, the overall claim expense of group i net reinsurance is written .

 

Given the level of reinsurance, we deduce the distribution, net of reinsurance, of the average claims charges per contract (ratio between and the volume of the portfolio).

 

For T3 type claims, we also model the overall claim load for all events combined by a process composed of frequency / cost : with , where denotes the impact on group i of the kth catastrophic event.

 

As reinsurance is applied by event, it is necessary to be able to distribute, for each event, the overall claim load before and after reinsurance and the reinsurance premium between the different groups.

 

As far as claims charges are concerned, this distribution can be made from a matrix giving, according to the level of the overall charge, its distribution between the different groups. Thus, denotes the claims expense net of reinsurance of group i relating to event k, and the catastrophic claims expense of group i, net of reinsurance, over the period considered is given by .

 

With regard to reinsurance premiums, the average contribution of each group to the expected claims charge on the reinsurance contract can be used as a distribution key.

 

Given the level of reinsurance, we deduce the net reinsurance distribution of the average claims charges per contract (ratio between and the volume of the portfolio).

 

 

D.2. Taking into account the uncertainty on the distribution of risk in the capital requirement

 

The simplified model presented in B.1. does not reflect the entire risk against which the capital must be defined.

 

Thus, a good formulation of the equation of the capital need would lead to consider a distribution charged with the result. The capital requirement would therefore be written as :

        with                   

or 

and   the random variable such as .

 

This formulation does not call into question the consistency, in the sense defined above, of the resulting security system. It can also help reduce the asymmetry between the risk measures used for premiums and for capital. On the other hand, it makes the calculations heavier and more complex.

 

 

D.3. Allocation of the security system by branch

 

If all the markets were perfect, and the apprehension of risk identical at all stages and by all agents, we would expect an identical RoRAC in each branch. This property does not come naturally from the assumptions made in our model, but it is important not to stray too far from it.

 

The actuarial literature (Schmock (1999) and Odjo (1999)) presents numerous capital allocation methods that take into account the risks specific to each branch and the contribution of each branch to the overall risk.

 

The definition of a fair and efficient method of allocating the security system should complement this model.

 

 

 

Conclusion

 

We have thus defined a coherent and optimal security system within the framework of a model integrating the risks of claims and their interdependencies, and in the presence of a reaction from policyholders to the pricing policy. In practice, the choice is often made of local or partial definitions of security systems when these can be biased or misleading. The pair of risk measures chosen in this presentation does not prevent another choice provided that the required properties are verified. For a good mastery of the tool, it would be useful to compare and analyze the results from different choices. We have justified the choice of coherent risk measures based on separate principles for capital and risk premium, but they should not be too divergent. The methods proposed for the resolution of the model compensate for the absence of analytical solution. In the case studied, these methods require a volume of computation that computers today allow to perform fairly quickly. However, the model considered is for one period and only incorporates the underwriting risk, an extension to several periods and integrating more risks (risks on provisions and investments in particular) would lead to a very rapid increase in complexity and calculation volume. Finally, it should be noted that the accuracy of the result depends on the constraints on the number of feasible scenarios, but also and above all on the quality of the inputs and assumptions used, which itself depends in particular on the quantity and quality of the data used for estimates.