In the presented thesis we deal with the generalized linear models framework in a claims reserving problem. Claims reserving in non-life insurance is firstly described and the considered class of models is introduced. Consequently, this branch of stochastic modelling is implemented in the reserving setup. For computation of the risk associated with claims reserving, we need a predictive distribution of future liabilities in order to evaluate risk measures such as Va- lue at Risk and Conditional Value at Risk. Since datasets in non-life insurance commonly consist of a small number of observations and estimation of predictive distributions can be complicated, we adopt a bootstrap method for this purpose. Model fitting, simulations and consequent measuring of the reserving risk are performed within the use of real-life data. Based on this, an analysis of fitted models and their comparison together with graphical outputs is included. 1

The determination of the sufficient amount of economic capital and its allocation to the business lines is the key issue for insurance companies. In this thesis we introduce two methods of aggregating economic capital. One is based on linear correlation and the second deals with copulas. A multitude of allocation principles have been proposed in the literature. We choose those which are the most used in practice and compare advantages and disadvantages of their application. The last chapter is devoted to the numerical examples of capital aggregation and allocation principles. 1

[pt] O objetivo desta dissertação é analisar a aplicação de metodologias de alocação de capacidade firme de usinas termelétricas através da teoria dos jogos cooperativos e suas conseqüências na cooperação entre os agentes. Mostra-se que não existe uma maneira ótima, única, de se fazer esta repartição, mas existem critérios para verificar se uma metodologia de repartição específica apresenta algum aspecto inadequado. Um desses critérios é a justiça. Mostra-se que este sentido de justiça equivale a pertencer ao chamado núcleo de um jogo cooperativo, onde não há subsídio de um subgrupo por outro. O cálculo da capacidade firme ou Capacidade de Suprimento de Carga será formulado como um problema de otimização linear e serão investigadas vantagens e desvantagens de distintos métodos de alocação (benefícios marginais, última adição, Nucleolus, Shapley). A aplicação desses métodos tem um crescimento exponencial de esforço computacional, o método de Aumann- Shapley abordado em seguida fornece para o problema de alocação de capacidade firme uma solução computacional mais eficiente, embora em sua descrição aparentemente o método aumente o esforço computacional. Em seguida foram realizados resultados numéricos com sistemas genéricos de pequeno porte. / [en] The objective of this work is to investigate the application of different methodologies of allocation of firm capacity rights among thermal plants using a game-theoretic framework and the consequences in the cooperation among the agents. It is shown that there is not an optimal and unique approach to make this allocation but there are criteria to verify if a given approach presents any inadequate aspect. One of these criteria is the justice, or fairness. It is shown that a one sense of justice is equivalent to the condition of the core of a cooperative game. The calculation of the firm capacity will be formulated as a linear program and advantages/disadvantages of different allocation methods (marginal allocation, incremental allocation, Nucleolus, Shapley) will be investigated. The complexities of these methods are exponential, so it will be shown that the Aumann-Shapley (AS) scheme to the problem of allocation of capacity rights will be more efficient. Numerical results about the difference allocations in these methods are presented in general smalls systems.

Mit der vorliegenden Dissertation wollen wir einen Beitrag zur Theorie der konvexen Risikomaße und ihrer Dynamik leisten. Im Kapitel 1 beschäftigen wir uns zunächst mit unbedingten konvexen Risikomaßen. Wir erläutern die Eigenschaften dieser Funktionale und geben einen Überblick über die Möglichkeiten ihrer Darstellung. Anschließend diskutieren wir das Problem ihrer Fortsetzbarkeit. Im Kapitel 2 erklären wir, wie sich die Darstellungssätze auf bedingte konvexe Risikomaße übertragen lassen, und untersuchen, unter welchen Voraussetzungen eine reguläre bedingte Darstellung existiert. Auf polnischen Räumen beweisen wir die Existenz auf den Klassen der halbstetigen Funktionen. Für das bedingte Average Value at Risk zeigen wir, dass durch eine zugehörige Familie von unbedingten AVaR-Risikomaßen eine reguläre bedingte Darstellung sogar auf der Klasse aller beschränkten Auszahlungsprofile gegeben ist. Im Kapitel 3 untersuchen wir die intertemporale Struktur von dynamischen konvexen Risikomaßen. Zunächst analysieren wir verschiedene Formen der Akzeptanz- und Ablehnungskonsistenz, welche einem zeitlich rückwärts gerichteten Ansatz der Risikobewertung entsprechen und in der Regel zur Konstruktion von dynamischen konvexen Risikomaßen zu einer vorgegebenen Filtration verwendet werden. Als Alternative formulieren wir einen vorwärts gerichteten Ansatz, bei dem jedes bedingte konvexe Risikomaß als eine Konsequenz aus der vorherigen Risikobewertung und der eingehenden Information konstruiert wird. Dann diskutieren wir Aktualisierungsvorschriften für konvexe Risikomaße. Wir überprüfen, inwieweit die vorgestellten Bedingungen der zeitlichen Konsistenz in ihrer starken und schwachen Form oder die Bedingung der Konsequenz als Aktualisierungskriterium geeignet sind. In diesem Zusammenhang diskutieren wir abschließend auch das Problem der Unsicherheitsreduzierung nach dem Erhalt von Zusatzinformation. / This thesis is a contribution to the theory convex risk measures and their dynamics. In chapter 1 we consider unconditional convex risk measures. At first, we explain the properties of these functionals and present different possibilities of their representation. Then we discuss the extension problem for convex risk measures. In chapter 2 we study conditional convex risk measures and their representations. We also analyze under which conditions these functionals admit a regular conditional representation. On polish spaces we prove existence on the classes of semicontinuous functions. For the conditional Average Value at Risk, we show that a regular conditional representation is given by a corresponding family of unconditional AVaR risk measures on the class of all bounded payoff functions. In Chapter 3 we investigate the intertemporal structure of dynamic convex risk measures. We begin by considering different conditions of acceptance and rejection consistency which correspond to a backward approach of dynamic risk evaluation and which are used for the construction of dynamic convex risk measures with respect to some given filtration. We also introduce an alternative forward approach where each conditional convex risk measure is constructed as a consequence of the initial risk evaluation and the incoming information. Then we discuss update rules for convex risk measures. We analyze whether the conditions of strong and weak consistency and the condition of consecutivity are appropriate update criteria. In this context, we finally discuss how uncertainty may be reduced after receiving some additional information.

Thema der Dissertation ist zum einen die Quantifizierung und zum anderen die endogene Modellierung von Finanzrisiken. Die mathematische Analyse führt unter anderem auf Zusammenhänge finanzmathematischer Probleme mit der Theorie großer Abweichungen, der Choquet-Theorie, der Theorie interagierender Teilchensysteme und der Theorie dynamischer Systeme. Die ersten zwei Kapitel der Arbeit beleuchten die Bemessung von Finanzrisiken aus zwei unterschiedlichen Perspektiven. In Kapitel 1 analysieren wir die Berechnung von Risikomaßen mittels Monte Carlo Methoden. In Kapitel 2 wird die Rolle von Information und Zeit bei der Bewertung von Finanzrisiken untersucht. Die Modellierung von Finanzrisiken auf Märkten interagierender Akteure wird in den beiden letzten Kapiteln der Arbeit in zwei Fallstudien betrachtet. In der ersten Fallstudie in Kapitel 3 befassen wir uns dabei mit dem Zusammenhang von Kreditrisiken und Ansteckungsprozessen von Firmen, die mit ihren Geschäftspartnern interagieren. In der zweiten Fallstudie in Kapitel 4 beleuchten wir die Marktinteraktion von eingeschränkt rationalen Investoren in einem evolutionären Marktselektionsmodell. / In this thesis, we study monetary measures and endogenous models of financial risk. The mathematical analysis identifies connections between problems in mathematical finance on the one hand and large deviations, Choquet-theory, interacting particle systems, and dynamical systems on the other hand. The first part of the thesis considers two aspects of the quantification of financial risk. In the first chapter, we focus on the calculation of risk measurements by Monte Carlo simulation. In the second chapter, we investigate a particular class of dynamic risk measures. In the second part we analyze two models of financial risk in economies with interacting agents. First, we focus in the third chapter on credit contagion of firms which interact with each other in a network of business partners. Second, we investigate in the fourth chapter the market interaction of investors with bounded rationality in an evolutionary selection market model.

Les précipitations extrêmes en France sont responsables de phénomènes d'inondations entraînant la perte de vies humaines et des millions d'euros en dégâts matériels. Mesurer le risque associé à ces événements météorologiques rares fait appel à la théorie statistique des valeurs extrêmes, qui propose plusieurs approches permettant d'évaluer des scénarios catastrophes. Cette thèse s'intéresse en particulier à trois mesures de risque faisant intervenir à la fois des lois de probabilité jointes et des méthodes de prédiction spatiale liées à la géostatistique.Dans un premier temps, plusieurs modèles spatiaux de valeurs extrêmes construits sur des données de maxima annuels sont évalués dans une étude comparative sous la forme d'un article. La comparaison des méthodes est menée en se servant de simulations construites à partir de données réelles de maxima annuels de précipitations en France et porte sur des critères liés aux deux mesures de risque que sont le niveau de retour centennal et le coefficient extrémal.Un modèle en particulier, le processus max-stable et hiérarchique de Reich et Shaby (2012) est étudié en détail et fait l'objet d'une implémentation sous la forme d'un package R dédié à la simulation et à l'estimation par cette méthode.Dans un second temps, les données journalières dépassant un seuil élevé sont modélisées dans un cadre spatial dans le but d'estimer une probabilité d'échec conditionnelle. Plusieurs estimateurs de cette mesure sont proposés en se concentrant d'une part sur des méthodes paramétriques liées aux processus Pareto et d'autre part sur deux approches non paramétriques. Les méthodes sont construites de sorte que la dépendance temporelle observable dans les valeurs journalière soit prise en compte lors de l'estimation.Tout au long de la thèse, les méthodes développées sont appliquées sur des données journalières de précipitations en France / Extreme precipitation in France are responsible for flooding events that cause people's deaths and billions of euros in material damage. Measuring the risk associated to these rare meteorological events is possible thanks to the extreme value theory which allows the estimation of such catastrophic scenarios. This thesis focus on three risk measures involving joint probabilities and spatial prediction methods related to geostatistics.In a first time, several spatial models for extreme values built on annual maxima are evaluated in a comparative study in the form of an article. This comparison is performed using simulated data from real annual maxima of precipitation in France. It is also based on two criteria linked to risk measures: the hundred years return level and the extremal coefficient. One particular model is presented in details: the one of Reich and Shaby (2012). This model is implemented under a R package entirely dedicated to its estimation and simulation procedures.In a second time, exceedances of spatial daily data are modelled in order to estimate a conditional failure probability. Several estimators of this measure are proposed, based on the one hand on parametric methods involving Pareto processes and on the other hand on non parametric approaches. The temporal dependence in extremes is also considered with care when estimating this probability.Along this thesis, the methods are applied on daily data of precipitation in France

Neste trabalho, aprofundamos o estudo sobre risco sistêmico via funções de agregação. Consideramos três carteiras diferentes como proxy para um sistema econômico, estas carteiras são consistidas por duas funções de agregação, baseadas em todos as ações do E.U.A, e um índice de mercado. As medidas de risco aplicadas são Value at Risk (VaR), Expected Shortfall (ES) and Expectile Value at Risk (EVaR), elas são previstas através do modelo GARCH clássico unido com nove funções de distribuição de probabilidade diferentes e mais por um método não paramétrico. As previsões são avaliadas por funções de perda e backtests de violação. Os resultados indicam que nossa abordagem pode gerar uma função de agregação adequada para processar o risco de um sistema previamente selecionado. / In this work, we deepen the study of systemic risk measurement via aggregation functions. We consider three different portfolios as a proxy for an economic system, these portfolios are consisted in two aggregation functions, based on all U.S. stocks and a market index. The risk measures applied are Value at Risk (VaR), Expected Shortfall (ES) and Expectile Value at Risk (EVaR), they are forecasted via the classical GARCH model along with nine distribution probability functions and also by a nonparametric approach. The forecasts are evaluated by loss functions and violation backtests. Results indicate that our approach can generate an adequate aggregation function to process the risk of a system previously selected.

Bakgrund Människor måste alltid fundera över risk och avkastning. Att omkring 80% av svenskarna äger någon form av fond skapar ett stort beroende av hur en extern aktör, portföljförvaltare, ser på begreppet och hur de hanterar portföljrisken mer precist. Det är därför intressant för alla investerare att förstå om och hur portföljrisk används och ses på utifrån förvaltarna som styr över vårt sparande. Är deras synsätt speglat i de befintliga teorierna och används den ofta kritiserade riskvariabeln beta i praktiken. Syfte: Syftet med magisteruppsatsen är att förklara och analysera hur institutionella investerare använder risk i portföljförvaltning, illustrera hur de i praktiken använder riskvariabler och om risk är nära relaterat till avkastning. Metod: Den här uppsatsen har sin utgångspunkt i den kvalitativa forskningsmetodiken för att kunna analysera hur portföljförvaltare ser på portföljrisk. Ett slumpmässigt urval av nio portföljförvaltare, oberoende av storlek och strategi, valde att ställa upp på intervjuer. De intervjuade fick fritt besvara frågorna för att skapa en så heltäckande bild som möjligt av de olika uppfattningarna inom portföljrisk. Slutsats: Analysen av det empiriska materialet visar att det är svårt att frambringa en enhetlig syn på portföljrisk och definition av densamma. De intervjuade skiljer sig åt i de flesta frågor förutom i kritiken mot betas värde som riskvariabel. Ingen använder beta som främsta riskmått, istället används riskvariabler som Value at Risk, tracking error och/eller variansen av avkastning. De statligt ägda fonderna använder sig av strategier där riskhantering kommer i främsta rummet och de ser även en stark koppling mellan risk och avkastning. Värdet av riskhantering skiljer sig åt bland de privata portföljförvaltarna eftersom några aktivt justerar och övervakar risknivån medan andra inte använder sig av risktänkande alls. Korrelationen mellan risk och avkastning är inte heller uppenbar då några anser att sambandet inte alltid är positivt eller linjärt. / Background: Humans have to constantly consider risk- and return tradeoffs. The fact that about 80% of the Swedish population owns some kind of mutual fund creates a great dependency on how an external part, a portfolio manager, views this tradeoff and especially how the concept of portfolio risk is looked upon. It becomes interesting for all investors to understand if and how portfolio risk is utilized and looked upon through the eyes of the mangers in charge over our savings. Do their view of risk and return translate to available theories and is the theoretically popular and much criticized beta measure used at all in practice. Purpose: The purpose of this master thesis is to describe and analyze how institutional investors apply the concepts of risk in portfolio management, to illustrate how they work with risk variables in practice and if risk is closely linked to return. Methodology: To be able to thoroughly analyze a few selected portfolio managers’ view on portfolio risk, this thesis has its foundation in the qualitative research approach. A random sample of nine mutual funds’ portfolio managers, independent of size and investment strategies, agreed to participate in face-to-face inter-views. The interviewees were allowed to answer freely in order to get the full picture of the different views of portfolio risk. Conclusion: The analysis of the empirical findings makes it clear that it is hard to find a unified view nor a unified definition of portfolio risk. The respondents differ a lot in their opinions in most issues except that they doubt beta being a good risk measure. No one is using beta as its main risk variable, instead risk variables such as Value at Risk, tracking error and variance of returns are used. The government operated funds have strategies putting risk management on the frontline and sees a strong connection between risk and return. The importance of risk management show a large divergence amongst the private portfolio managers since some respondents actively adjust and monitor the level of risk while other employ strategies that do not incorporate risk thinking at all. The correlation between risk and return is not apparent since some respondents do not believe the relation to be linear or positive at all times.

The thesis focuses on the analysis of various extensions of the classical multi-period single-item stochastic inventory problem. Specifically, we investigate two particular approaches of modeling risk in the context of inventory management: risk-averse models and robust formulations. We analyze the classical newsvendor problem utilizing a coherent risk measure as the objective function. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. We show that the structure of the optimal policy of the risk-averse model is similar to that of the classical expected value problem for both single and multi-period cases. The result carries over even when there is a fixed ordering cost. We expand our analysis to robust formulations of multi-period inventory problems. We consider both independent and dependent uncertainty sets and prove the optimality of base-stock policies for the general problem formulation. We focus on budget of uncertainty approach and develop a heuristic that can also be employed for a class of parametric dependency structures. We compare our proposed heuristic against alternative solution techniques.

In insurance risk theory, the surplus of an insurance company is modelled to monitor and quantify its risks. With the outgo of claims and inflow of premiums, the insurer needs to determine what financial portfolio ensures the soundness of the company’s future while satisfying the shareholders’ interests. It is usually assumed that the net profit condition (i.e. the expectation of the process is positive) is satisfied, which then implies that this process would drift towards infinity. To correct this unrealistic behaviour, the surplus process was modified to include the payout of dividends until the time of ruin. Under this more realistic surplus process, a topic of growing interest is determining which dividend strategy is optimal, where optimality is in the sense of maximizing the expected present value of dividend payments. This problem dates back to the work of Bruno De Finetti (1957) where it was shown that if the surplus process is modelled as a random walk with ± 1 step sizes, the optimal dividend payment strategy is a barrier strategy. Such a strategy pays as dividends any excess of the surplus above some threshold. Since then, other examples where a barrier strategy is optimal include the Brownian motion model (Gerber and Shiu (2004)) and the compound Poisson process model with exponential claims (Gerber and Shiu (2006)). In this thesis, we focus on the optimality of a barrier strategy in the more general Lévy risk models. The risk process will be formulated as a spectrally negative Lévy process, a continuous-time stochastic process with stationary increments which provides an extension of the classical Cramér-Lundberg model. This includes the Brownian and the compound Poisson risk processes as special cases. In this setting, results are expressed in terms of “scale functions”, a family of functions known only through their Laplace transform. In Loeffen (2008), we can find a sufficient condition on the jump distribution of the process for a barrier strategy to be optimal. This condition was then improved upon by Loeffen and Renaud (2010) while considering a more general control problem. The first chapter provides a brief review of theory of spectrally negative Lévy processes and scale functions. In chapter 2, we define the optimal dividends problem and provide existing results in the literature. When the surplus process is given by the Cramér-Lundberg process with a Brownian motion component, we provide a sufficient condition on the parameters of this process for the optimality of a dividend barrier strategy. Chapter 3 focuses on the case when the claims distribution is given by a univariate mixture of Erlang distributions with a common scale parameter. Analytical results for the Value-at-Risk and Tail-Value-at-Risk, and the Euler risk contribution to the Conditional Tail Expectation are provided. Additionally, we give some results for the scale function and the optimal dividends problem. In the final chapter, we propose an expectation maximization (EM) algorithm similar to that in Lee and Lin (2009) for fitting the univariate distribution to data. This algorithm is implemented and numerical results on the goodness of fit to sample data and on the optimal dividends problem are presented.