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.