Linear Rational Insurance Model & Economic Scenario Generator

Linear Rational Insurance Model Abstract The aim of the first work is to provide a closed pricing formula for insurance contracts in a linear rational framework, which consists in assuming the existence of a linear drift diffusion process and a state price density which is a linear function of it. The main advantage of this process is that we can compute the conditional expectation of polyno- mials function of this diffusion: in fact a polynomial can be see as a linear combination of an enlarged set of variables with a linear drift, this can be proofed by means of the Itô’s formula. This result is very important because it allows us, under the hypothesis that the diffusion part is a martingale, to use all the results we have about linear drift diffusions for this new set of variables. As a consequence, we are able to give the price of three important life insurance contracts: the sur- vival and death benefit and the guaranteed annuity option (also called GAO). It is about the GAO that we can see the advantage of the framework we are using: actually the payoff of the GAO is not an affine or a polynomial function, so the only way to treat it is by performing a change of measure or a Monte Carlo simulation. We show that, under the assumption that the state space is compact, we are able to approximate the GAO payoff by a polynomial, which will allow us to find a closed formula for the price of this contract. The end of this work is dedicated to some numerical experiments which have the aim to point out the importance of the choice of the degree of the approximated polynomials in order to have reliable results. We show that a ten degree polynomial is able to estimate with a small error the Monte Carlo price of the GAO. This work extend the existing literature concerning polynomial models and their application in life insurance, proposing a pricing method also for liabilities which are not necessarily building blocks, but more complicated functions, like the guaranteed annuity option. Economic Scenario Generator The aim of this second work is to build an economic scenario generator with the intention of improving the portfolio allocation of Bpifrance. In order to do that, we have to pass through a different number of steps. The first thing is to study, by a principal component analysis, the present portfolio of Bpifrance, in order to find the variables which explains the most of its variability. A second step consists in selecting from the market the financial instruments that allows us to replicate the components we retained from the step before. This part is then completed by both an univariate and multivariate analysis of these assets, finding in this way the stylized facts that we need to take into account when choosing a model for the diffusion of the price of these financial factors. The third step, and last concerning our work, is to estimate the parameters of the models we retained and see if they are able to fit the empirical data and, as a consequence, if they could be used as a part of our future economic scenario generator. In order to achieve this point, we focus only on the diffusion of the equity indices, proposing also a model who takes into account the dependency on the inflation. We will see that on the basis of our data there is no evidence to link the return on equity indices on the realisations of this macroeconomic factor.