Aggregation of risks

The problem of aggregation arises at three levels :

We will use the same methods for these three aggregation problems. Also, for the clarity of the exposition, we will only present the application of these methods to , which we will simply note below .

We have chosen to use the fast Fourier transformation method in case of independence and the Monte Carlo simulations with the use of copulas to characterize the dependencies.

C.2.1. Case of independence : The rapid Fourier transformation

This method makes it possible to estimate aggregate distributions of independent random variables. It is based on the characteristic functions of the various variables present.

Definition : Let Y be a random variable and its characteristic function. By definition,

It completely characterizes the distribution of Y.

The characteristic function of the sum of independent random variables is the product of their characteristic functions.

The risk aggregation by this method is done in three stages :

determination of the characteristic functions of the variables to be aggregated by application of the fast Fourier transformation

calculation of the characteristic function of the aggregated variable by the product of the characteristic functions of the variables concerned

inversion of this to obtain the aggregate distribution.

In the framework of the model, the distribution of will be determined as follows :

Characteristic functions of variables

We start by discretizing these variables ; the same discretization will be retained for all the variables and will have to cover the support of . Let L be the maximum value taken by , the discretization of the interval and h the step of the latter.

with and , with M being able to be written in the form .

We then calculate the vectors of the probabilities of the discretized variables that we always denote . For and ,

According to the right-hand equality, the distribution of is perfectly determined by that of and . This allows us to work directly on integer-valued variables.

Determination of Fourier transforms of variables

By definition, the fast Fourier transform of a random variable is an application which, to the probability vector defined by

with .

Conversely, the probability vector can be determined from the Fourier transform by the expression . (2)

The Fourier transform of the aggregated variable is then determined by product of the Fourier transforms of the variables .

(2) then allows us to invert it to finally obtain the distribution of and therefore that of .

C.2.2. Dependence between risks : The Monte Carlo random simulation - Copulas

This method makes it possible to generate uniform pseudo-random numbers on and to deduce therefrom, by various transformations or various algorithms, quantities according to a law defined in advance.

In general, there are two cases to be treated :

the pseudo-inverse function of the law to be generated is explicit. In this case, the inverse transformation method could be applied.

otherwise, mixed laws or pass / fail methods could be used.

There are many powerful and fast uniform random number generators, possessing a high period and good statistical properties. L'Ecuyer (2000) thus presents a combined generator with multiple recursion named MRG31k3p whose period is such (2 185 ) that it covers the needs of any extension of the model.

In the framework of our model, the distribution of will be determined in two steps :

we generate a q-sample of the vector taking into account the interdependence of the components, q being large enough to have a good estimate of the distribution of for any r retained

we form from this q-sample, another q-sample of and deduce its empirical distribution.

Vector generation

To simulate the vector , we will separate the joined distribution into two parts :

the first made up of marginal distributions

the second describing the structure of the dependence between the different components : their copula.

Definition : A copula is the distribution of a random vector whose marginal distributions are uniform over :

The idea is to transform the random vector into a uniform random vector having the same dependency structure as this one.

We retain here the Archimedean copulas, and more precisely, that of Gumbel, to characterize these dependencies. By definition, the copula Gumbel for a couple of random variables is given by : . The parameter may for example be secured from the rank coefficient of the couple (their Kendall) : .

An algorithm for generating uniform vectors from this type of copula is presented in Lindskog (1999).

SKLAR's theorem allows, by applying the pseudo-inverses of the marginal distribution functions, to generate vectors according to the law of .

C.3. Resolution of the optimization program

We seek here to maximize the RoRAC defined by equation (1). This maximization will be presented here only in the case of the use of Monte Carlo simulations.

The data of the problem :

we assume known the distribution of , a q-sample of this vector will have been previously generated

parameters and functions

A resolution algorithm

It consists in discretizing the set of admissible values of r, in calculating the value of RORAC by equation (1), in comparing the values obtained so as to define the set of values of r which maximize the RORAC. We will proceed as follows :

define the set of admissible values of r and discretize it

for each value of this discretization

- evaluate the functions , and

- then determine the empirical distribution of

- assess the capital requirement and therefore the RORAC .

finally, choose among all the values , those which maximize the RORAC.

Two questions arise at this level :

The function may not be regular, and the global optimum may have escaped segmentation. In this case, the latter may be considered to be too sensitive to a variation in parameters, and stability is a sought-after property.

We must try to minimize the number of RORAC evaluations. Indeed, the algorithm proposed above consisting of sweeping the set of admissible values of r should involve a fairly large calculation volume. The choice of algorithms making it possible to minimize the number of values studied depends on the shape of the curve of the RORAC with respect to r. When the latter is convex, we can for example use the Golden Section method to find the optimum.