Modeling and optimization
Modelization
Let us consider a non-life insurance company operating in m branches of activity, each branch itself being composed of homogeneous groups in the sense that the risks are identically distributed within each group.
We denote by n the total number of groups all branches combined and we will reason directly at the level of homogeneous risk groups.
For the sake of simplification, the model considered is for one period, and only incorporates the sole risk linked to the claim load of the period of occurrence.
We make the following assumptions as part of the model :
The security system (premium and capital) is held at the start of the period and must allow all payments to be made at the end of the period.
Suppose that the capital requirement at the end of the period is an amount sufficient to hold at the start of the period, being the return on the asset in which the capital is invested. For the sake of simplicity, we assume here that the capital is invested in the risk-free asset.
Premiums are determined so as to meet payments at the end of the period knowing that they are cashed at the start of the period and invested at the same rate .
Notations
Are :
, the average claim charge per group i contract over the period considered. The risk of group i is characterized by the distribution of ,
, the claims expense for group i over the period considered : where denotes the number of cases obtained by the insurer over this period.
, the company's overall claims charge over the same period
, the value at the end of the period of the total premium charged to group i
, the value at the end of the period of the total amount of premiums :
, the random variable representing the technical result of the company over the period considered
For any random variable , we denote
, its distribution function
, its survival function
The technical result
To write the technical result, we add to the previous assumptions the balance of the management costs and the management charge. In the absence of reinsurance, the company's result over the period considered is written :
That is , the premium requested on each contract of group i. We assume that this premium is the same for all contracts in this group. The total premium for group i is then written , and the result of the company over this period is :
B.1.1. Calculation of the technical premium
According to the choice justified in A. , the premium function applied to each contract of group i is given by :
where designates the parameter of loading applied to the group i. This parameter is set for each group from a certain value chosen at the company level. We will assume that the parameter applied to each group is a function of , where the function is set at the start so as to account in particular for the expertise of the company on the different groups. It is possible for example to use a function of the form where is a real constant known in advance and such that .
Under these assumptions, the parameter defines the loads applied to all groups and therefore the company's pricing policy.
For the following, we will denote the set of admissible values of .
We further assume that the number of contracts obtained by the insurer on group i depends on the level of the risk premium and therefore on the parameter . We thus note , where is a schematic of an insurance demand function within group i.
The premium being decreasing with respect to , the function must be increasing in .
For reasons of simplification, we will choose, for example, deterministic demand functions with constant elasticity with respect to the premium. This request function is of the form :
where denotes the elasticity of the number of contracts with respect to the level of the premium (except for the whole part) and corresponds to the situation prior to the period considered.
Taking into account these assumptions, the result of the company is a function of the parameter and is then written :
B.1.2. Calculation of the technical capital requirement (Risk Adjusted Capital RAC)
As indicated above, we choose as a measure of technical capital need, the principle of insufficiency.
By definition,
with
where is set in advance by company executives and reflects their risk aversion.
From where
denoting the pseudo-inverse function of .
By definition,
The capital requirement is then given by :
or
designates the indicator of the event .
It comes :
With the notations and assumptions made, the capital requirement will then be written :
with ,
,
B.2. Optimization program
We are looking here for all the pairs (safety load, capital) such that the return on equity (RoRAC) of the company over the period is maximum.
Definition
We define here the return on equity as the ratio between the expected result and the capital requirement. We will note it .
According to the model, it is given by :
therefore corresponds to the ratio between the mass of the safety load applied by the company and its capital requirement. It is defined by the choice of the parameter at the global level of the company.
The optimization program will therefore be as follows : which parameter should the company choose from the set of admissible values to maximize its RoRAC ?
The program : (1)
C. Aggregation of risks and resolution of the optimization program
C.1. Modeling of underwriting risks
For the purposes of the model, we analyze the risk group by group. We consider that each group is subject to three types of risks :
T1 : a load of small / medium claims that can be modeled by a bell law
T2 : major claims by risk
Q3 : major claims per event
We will note :
the average charge per contract for ordinary claims of group i
the average charge per contract for large claims affecting group i
the average charge per contract for claims arising from catastrophic events. By noting the random variable number of catastrophic events occurring during the period, we can write where is the average load per contract of claims resulting from event k. For the sake of simplicity, we assume that there is only one type of event likely to affect the company's portfolio.
The average claim charge per contract corresponds to the aggregation of these three risks
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