Essays on Capital Calculation in Insurance

In order to be able to bear the risk they are taking, insurance companies have to set aside a certain amount of cushion that can guarantee the payment of liabilities, up to a dened probability, and thus to remain solvent in case of bad events. This amount is named capital. The calculation of capital is a complex problem. To be sustainable, capital must consider all possible risk sources that may lead to losses among assets and liabilities of the insurance company, and it must account for the likelihood and the eect of these bad (and usually extreme) events that could occur to the risk sources. Insurance companies build models and tools in order to perform this capital calculation. For that, they have to collect data, build statistical evidence, build mathematical models and tools in order to eciently and accurately derive capital. The papers exposed in this thesis deal with three major diculties. First, the uncertainty behind the choice of a specic model and the quantication of this uncertainty in terms of additional capital. The use of external scenarios, i.e. opinions on the likelihood of some events happening, allows to build a coherent methodology that make the cushion more robust against wrong model specication. Second, the computational complexity in using these models in an industrialized environment, and numerical methods available for increasing their computational eciency. Most of these models cannot provide an analytical formula of capital. Consequently, one has to approximate it via simulation methods. Considering the high number of risk sources and the complexity of insurance contracts, these methods can be slow to run before providing a reasonable accuracy. This often makes these models unusable in practical cases. Enhancements of classical simulation methods are presented in the aim of making these approximations faster to run for the same level of accuracy. Third, the lack of reliable data and the high complexity of problems with long time horizons, and statistical methods for identifying and building reliable proxies in such cases. A typical example is life-insurance contracts that imply being exposed to multiple risks sources over a long horizon. Such contracts can in fact be approximated wisely by proxies that can capture the risk over time.

Semiparametric Bayesian Risk Estimation for Complex Extremes

Thomas Lugrin

Extreme events are responsible for huge material damage and are costly in terms of their human and economic impacts. They strike all facets of modern society, such as physical infrastructure and insurance companies through environmental hazards, banking and finance through stock market crises, and the internet and communication systems through network and server overloads. It is thus of increasing importance to accurately assess the risk of extreme events in order to mitigate them. Extreme value theory is a statistical approach to extrapolation of probabilities beyond the range of the data, which provides a robust framework to learn from an often small number of recorded extreme events. In this thesis, we consider a conditional approach to modelling extreme values that is more flexible than standard models for simultaneously extreme events. We explore the subasymptotic properties of this conditional approach and prove that in specific situations its finite-sample behaviour can differ significantly from its limit characterisation. For modelling extremes in time series with short-range dependence, the standard peaks-over-threshold method relies on a pre-processing step that retains only a subset of observations exceeding a high threshold and can result in badly-biased estimates. This method focuses on the marginal distribution of the extremes and does not estimate temporal extremal dependence. We propose a new methodology to model time series extremes using Bayesian semiparametrics and allowing estimation of functionals of clusters of extremes. We apply our methodology to model river flow data in England and improve flood risk assessment by explicitly describing extremal dependence in time, using information from all exceedances of a high threshold. We develop two new bivariate models which are based on the conditional tail approach, and use all observations having at least one extreme component in our inference procedure, thus extracting more information from the data than existing approaches. We compare the efficiency of these models in a simulation study and discuss generalisations to higher-dimensional setups. Existing models for extremes of Markov chains generally rely on a strong assumption of asymptotic dependence at all lags and separately consider marginal and joint features. We introduce a more flexible model and show how Bayesian semiparametrics can provide a suitable framework allowing simultaneous inference for the margins and the extremal dependence structure, yielding efficient risk estimates and a reliable assessment of uncertainty.

EPFL2018

Approche mathématique de l'impact des sites contaminés sur les eaux souterraines dans un contexte d'analyse de risque

Dominique Guyonnet

The primary objectives of this thesis are, firstly, to develop analytical solutions for estimating the impact of contaminated sites (in particular landfills, and to a lesser extent accidentally contaminated soils) on groundwater, and secondly to evaluate methods for taking into account the uncertainty relative to the different relevant parameters. These developments are performed with particular reference to a risk analysis framework. One of the distinctive features of such a framework is the incomplete and uncertain nature of the information that is generally available in practice. This has a direct influence on the choice of tools and approaches. It is proposed, in the introductory chapter, to define risk as the likelihood that a damage may occur, thus avoiding the more frequently used term probability. The latter refers to a strictly statistical description of the uncertainty, and assumes a degree of knowledge with respect to the various relevant parameters that is not necessarily available in practice. A presentation of several risk analysis models underlines their respective areas of application. Chapter 2 presents analytical solutions for estimating the impact of contaminated sites on groundwater, several of which are original. The mathematical development of the original solutions is presented in detail in the appendix. The succession of proposed solutions goes from the simpler to the more complex, starting with solutions for a one-layer domain and one-, two- and three-dimensional contaminant migration, and then passing to solutions for a multi-layer domain. These solutions are presented in terms of dimensionless variables that allow the equations to be written in a compact fashion, and independently of any system of units. The dimensionless variables serve to construct type curves that provide a visual analysis of solution behaviour with respect to variations of input parameter values. The proposed original solutions are of particular interest for estimating the potential impact of waste disposal sites, as they take into account the effect of the diffusive-dispersive flux on the impact on groundwater. This component of the mass flux is generally either omitted by existing risk analysis tools, or else it is not truly coupled with the mass balance in the groundwater (via the boundary conditions). The proposed solutions are applied to several practical examples. The example involving the original solution developed for estimating the impact of a stabilized waste-disposal site, illustrates the solution's usefulness as an aid for site conception and design. The example also shows how, by providing a picture of the site's global performance with respect to its potential impact, the model can bring the designer to question some of his choices. Another example helps to clarify the concept of "equivalence" between different barrier systems. An original solution is also proposed for the problem of early contaminant detection using inorganic trace elements. A comparison with another approach proposed in the literature illustrates the limits of the latter. Chapter 3 addresses the question of how to take imperfection of knowledge into account in the calculations. Two methods for accounting for uncertainty with respect to input parameter values are described and compared. The first method is the Monte Carlo method, which relates to a statistical and probabilistic framework. The second method is based on the theory of possibilities and on so-called fuzzy calculus. The representation of the main model parameters (hydraulic conductivity, porosity, etc.) using fuzzy numbers rather than probability distributions, is often more consistent with the basic nature of the information that is available in practice. The comparison between the two methods, underlines the conservative character of the fuzzy approach. This is related to the fact that the calculation based on fuzzy numbers considers all possible combinations of fuzzy input parameter values, but does not transmit (through multiplication) the degrees of unlikelihood of the different values. In the Monte Carlo calculation, on the other hand, a scenario that combines low-probability values of the input parameters, has all the less chance of being randomly selected. Nevertheless, when the mere possibility that an unfavourable scenario may occur becomes an element of decision, the fuzzy approach may seem preferable. This situation can appear in an environmental context, where human health is often at stake. The examples presented herein contribute to introducing an "impact framework" to the management of contaminated sites. Such a framework tends to define the acceptability of choices with respect to the management of these sites, on the basis of a comparison between emitted pollutant fluxes and fluxes considered acceptable for the local environment. They also underline the usefulness of having a certain degree of freedom with respect to the choice of the conceptual model underlying the mathematical model, in order to provide reasonable appropriateness between the calculation tool and the various situations that might appear in practice. Existing risk analysis models are often difficult to apply to specific sites, due to the rigidity of their conceptual models. In some cases, these models are better described as tools for ranking chemical substances as a function of their contamination potential. Some of the original analytical solutions proposed herein can be readily incorporated into the "vector" module of a risk analysis tool.

EPFL1998

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

This work deals with factorial models for multiple time series. Its core content puts it at the interface between statistics and finance. After a brief description of the historical link between the two sciences, it reviews the literature on factorial models that are close to the model introduced in this work called The dynamical factor analysis model for time series. This model makes the hypothesis that the observed time series are influenced by a common factor, difficult to define and impossible to measure. No a priori structure is put on the factor, at each point time the value of the factor is considered as a new parameter that has to be estimated. As a consequence of this fact, the number of parameters is large and it is not possible to provide the usual asimptotic properties of the estimations by letting the number of periods tend to infi- nity. Asymptotic results in our context have been obtained by increasing the number of time series. The model makes the hypothesis that there is a linear dependence between the time series and the factor with coefficients, that are not constant over time, but rather follow a smooth random walk. This mean that the linear structure of the model is evolving slowly from a period to another. All the information which is not contained in the factor and the coefficients is considered as white noise. Using the normal distribution makes the estimation more easy and opens a toolbox of statistical methods that have been developed for this kind of data. The model is a part of the family of state space models and the Kalman filter is an essential ingredient of our estimator. The effort was concentrated on the elaboration of the structure of the model. Its complexity was constrained by the difficulty of estimation. The final shape of the model does not allow an analytical solution of the optimization problem introduced by the maximum likelihood estimation of the parameters. Numerical solutions have been found and compared with the parameters for simulated data. Some others models have been developed as simpler versions of the dynamical factor model for time series. The case where the factor can be observed has been studied and a new method for the estimation has been provided and compared with the existing methods. A second study considers a latent factor model without noise. Two methods for the estimation of the factor have been provided. The last chapter contains a detailed description of the main statistical tools used during this work. The links with the previous chapters are highlighted followed by comments.

EPFL2004