Analysis of a coherent and optimal security system for a P&C insurance company
This paper defines a coherent security system for a P&C insurance company, comprising premium and capital loading. It then analyzes the optimal conditions of this system taking into account the market reaction to the company's pricing policy. Starting from a modeling over a period of underwriting risks, their aggregation and then the resolution of the optimization program use numerical and simulation techniques. The approaches used are based on rapid Fourier transformations in the case of risk independence, and on Monte Carlo simulations using the copula method in the case of interdependence.
Keywords : aggregation, capital, safety loading, copulas, Fourier, risk measure, Monte Carlo, RoRAC
A non-life insurance company is subject to the risk of exceptional fluctuations in its results, and of default with regard to the commitments it has entered into. To face this risk, it has, in addition to regulatory constraints, a security system comprising mainly, but not exclusively, three quantifiable instruments: capital, reinsurance, premium loading. The company must set the parameters of its security system at the start of the period; its operation will then depend on the realization of the risks. Two properties of the security system must be sought : consistency and optimality.
We will seek here to define a coherent and optimal pair (capital, safety loading) ; the reinsurance conditions will be considered for simplicity as given. In the first part, the definition of the rules of consistency will be part of an axiomatic process. In the second part, optimality will be sought through the objective of maximizing the return on capital, taking into account a reaction of policyholders in relation to the tariff level applied : the level of premiums conditions the volume of the portfolio according to a demand function. .
The company is faced with a large number of risks (underwriting, provisions, investment, credit risks, etc.) and over a long period. We are only interested here in the only underwriting risk that we will develop from a simplified modeling over a period. The aggregation of these risks uses numerical and simulation techniques which will be exposed in the third part.
We will discuss in the fourth part some additions that can be added to the basic model.
A / Risk measures
We call risk, a function X which associates reality with a state of nature . We can thus formalize the cumulative claim cost covered by an insurance guarantee, or the result of a company over a given period. By convention, the loss load will be noted below as being a positive quantity, and the loss result as a negative quantity.
We call a risk measure, a function associating with a risk X, a positive reality . We can use such functions to determine, using the examples mentioned in the previous paragraph, the risk premium of a contract, or the capital requirement of a company.
A.1. Required properties of the security system
We assume here that the consistency of the security system is defined by :
- the intrinsic consistency of the risk measures used to calculate the risk premiums and the capital requirement,
- respect for the role assigned to each instrument,
With regard to the role attributed to instruments, the premium charge must ensure the remuneration of the shareholder and limit the risks linked to fluctuations in claims and measurement uncertainty, the capital must in the second place prevent ruin or reduce its amplitude. We will see that this conception leads us to use risk measures based on distinct principles for these two instruments.
Note (respectively ) the risk measure used to determine the capital requirement (respectively the risk premium of a contract).
Artzner et al. (1998) define the consistency property as the combination of the following four properties :
For any two risks X and Y
P1 : Invariance by translation
, for any constant c.
If we add (resp. Subtract a certain amount c in the profit center accounts, the capital requirement decreases (resp. Increases) by the same amount.
Note that in the case of the measure of the premium, the relation would become , where c would then be interpreted as a certain charge.
P2 : Sub-additivity
The merger of two profit centers does not create any additional risk. On the contrary, diversification tends to reduce overall risk. Furthermore, this property enables decentralized management of the capital requirement in the various profit centers without running the risk of an overall requirement greater than the sum of the individual needs of the entities. Above all, if this property was not respected, an agent, by artificially creating two companies, could find himself with a reduced capital requirement.
P3 : Positive homogeneity
Just as a merger does not create additional risk , so a merger without diversification does not reduce the overall need.
Note that within the framework of the premium, this property induces an invariance by change of monetary unit (Partrat (2000)).
P4 : Monotony
If the losses incurred with risk X are always greater than those obtained with Y, the capital requirement for X must be greater than that for Y.
Note that in the context of the premium, we write :
According to Artzner et al. (1998), a risk measure verifying the properties P1, P2, P3, P4 is consistent.
Some authors have criticized this body of properties. With regard to positive homogeneity in particular, it can be argued that a variation in the risk scale can lead to a more than proportional effect on the need for capital or on the premium. This may arise in particular from market constraints, such as the difficulty of reinsuring.
Regarding monotony, some authors (Wang (1996)) studying prime principles put forward the following comparison property :
P4 bis : Stochastic dominance of order 2
The measure of risk X is less than that of Y if :
such as
This property makes it possible to take into account a risk hierarchy which escapes the property of monotony. However, it is of less interest when looking only at the tail of the distribution.
Other desirable properties have been defined in the actuarial literature (in particular by certain authors already cited) :
P5 : Lower bound property
This property is required in the absence of cross-subsidy between branches and cycle effects.
P6 : Upper bound property
, or in the case of the premium :
P7 : The measurement is a function of the uncertainty on the parameters
By writing that the distribution of X depends on a random parameter , this property is written :
P8 : Conservatism
where denotes the negative part of , .
It is defined by Artzner in particular for determining the capital requirement, and results in only negative values being taken into account.
P9 : The capital requirement must take into account the possible magnitudes of the ruin
A.2. Properties of the main risk measures
The following classic formulations are encountered in practice :
(*)
(**)
(***)
where denotes the standard deviation of X and its variance.
However, we show that :
(*) does not respect in particular P4, P4 bis,
(**) does not respect in particular P2, P3, P4, P4 bis,
(***) does not respect in particular P1 and P4.
The principle of the probability of ruin is defined by . In the case of continuous variables, we have .
This principle does not respect P2 and P9. Thus, non-compliance with P9 has led some authors to define other measures for regulatory needs and to use in particular the Unexpected Policyholder Deficit (Butsic (1994)). Artzner et al. (1997) demonstrated that P2 was not respected in the general case. However, Embrechts et al. (1999) showed that on the class of elliptical distributions, the Value at Risk respected the coherence properties P1, P2, P3 and P4.
Artzner et al. (1998) studied the properties of the expected shortfall principle defined as follows : with .
The demonstration of the coherence properties presented in Artzner et al. (1997) is based on a method of generalized scenarios under the assumption of a finite set of states of nature. We also show that in the case of continuous variables of finite expectation, this risk measure respects the consistency properties (Odjo (1999)). We note that it also respects the properties P5, P6, P8, P9.
Wang (1996) studied a class of risk measures based on distortion operators .
Definition :
We call a distortion operator, any map , increasing and concave such that and .
The measure of risk X based on the distortion operator is written where denotes the survival function of X.
As recalled in Wirsch et al. (1999), this risk measure respects the properties P1, P2, P3. The P4 property is immediate. is therefore a consistent risk measure. It also checks the properties P4 bis, P5, P6, P7.
For , we find the prime principle defined by the Ph-transform (Wang (1997)).
Note that Wirsch and Hardi (1999) wrote the Unexpected Shortfall as a risk measure based on distortion operators in some cases.
A.3. Choice of risk measures for the security system
Risk measures based on distortion operators therefore have good properties. Among these measures, our choice must respect the following criteria :
Given the role assigned to each instrument, it seems natural to use a risk measure attached to the entire risk distribution in the case of the premium calculation, and a measure focused on the tail distribution in the case of measuring the capital requirement. Note in this regard that the Unexpected Shortfall is only interested in the tail of the distribution and that there are a large number of distortion functions affecting the entire distribution (beta operators, the principle of absolute deviation de Dennenberg, etc.). However, if the choice of risk measures based on different principles can be justified, it is desirable that they diverge little ; now this property depends on the form of the distributions to which these two measures are applied.
The simplicity of calculation and interpretation are among the qualities sought.
Finally, the nature of the portfolio (direct insurance / reinsurance, personal risks / industrial risks, etc.) and the form of risk aversion may lead to favoring a particular distortion operator. It is thus noted that the Ph-transform leads to overloading the high slices and that it can be particularly useful for the great risks.
We retain under these conditions for our application the principle of insufficiency (Expected Shortfall) for the measurement of the capital requirement and the principle of the Ph-Transform for the risk premium.
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