Évaluation et allocation du risque dans le cadre de modèles avancés en actuariat

Dans cette thèse, on s’intéresse à l’évaluation et l’allocation du risque dans le cadre de modèles avancés en actuariat. Dans le premier chapitre, on présente le contexte général de la thèse et on introduit les différents outils et modèles utilisés dans les autres chapitres. Dans le deuxième chapitre, on s’intéresse à un portefeuille d’assurance dont les composantes sont dépendantes. Ces composantes sont distribuées selon une loi mélange d’Erlang multivariée définie à l’aide de la copule Farlie-Gumbel-Morgenstern (FGM). On évalue le risque global de ce portefeuille ainsi que l’allocation du capital. En utilisant certaines propriétés de la copule FGM et la famille de distributions mélange d’Erlang, on obtient des formules explicites de la covariance entre les risques et de la Tail-Value-at-Risk du risque global. On détermine aussi la contribution de chacun des risques au risque global à l’aide de la régle d’allocation de capital basée sur la Tail-Value-at-Risk et celle basée sur la covariance. Dans le troisième chapitre, on évalue le risque pour un portefeuille sur plusieurs périodes en utilisant le modèle de Sparre Andersen. Pour cette fin, on étudie la distribution de la somme escomptée des ladder heights sur un horizon de temps fini ou infini. En particulier, on trouve une expression ferme des moments de cette distribution dans le cas du modèle classique Poisson-composé et le modèle de Sparre Andersen avec des montants de sinistres distribués selon une loi exponentielle. L’élaboration d’une expression exacte de ces moments nous permet d’approximer la distribution de la somme escomptée des ladder heights par une distribution mélange d’Erlang. Pour établir cette approximation, nous utilisons une méthode basée sur les moments. À l’aide de cette approximation, on calcule les mesures de risque VaR et TVaR associées à la somme escomptée des ladder heights. Dans le quatrième chapitre de cette thèse, on étudie la quantification des risques liés aux investissements. On élabore un modèle d’investissement qui est constitué de quatre modules dans le cas de deux économies : l’économie canadienne et l’économie américaine. On applique ce modèle dans le cadre de la quantification et l’allocation des risques. Pour cette fin, on génère des scénarios en utilisant notre modèle d’investissement puis on détermine une allocation du risque à l’aide de la règle d’allocation TVaR. Cette technique est très flexible ce qui nous permet de donner une quantification à la fois du risque d’investissement, risque d’inflation et le risque du taux de change.In this thesis, we are interested in risk evaluation and risk allocation blems using advanced actuarial models. First, we investigate risk aggregation and capital allocation problems for a portfolio of possibly dependent risks whose multivariate distribution is defined with the Farlie-Gumbel-Morgenstern copula and with mixed Erlang distributions for the marginals. In such a context, we first show that the aggregate claim amount has a mixed Erlang distribution. Based on a top-down approach, closed-form expressions for the contribution of each risk are derived using the TVaR and covariance rules. These findings are illustrated with numerical examples. Then, we propose to investigate the distribution of the discounted sum of ascending ladder heights over finite- or infinite-time intervals within the Sparre Andersen risk model. In particular, the moments of the discounted sum of ascending ladder heights over a finite- and an infinite-time intervals are derived in both the classical compound Poisson risk model and the Sparre Andersen risk model with exponential claims. The application of a particular Gerber-Shiu functional is central to the derivation of these results, as is the mixed Erlang distributional assumption. Finally, we define VaR and TVaR risk measures in terms of the discounted sum of ascending ladder heights. We use a moment-matching method to approximate the distribution of the discounted sum of ascending ladder heights allowing the computation of the VaR and TVaR risk measures. In the last chapter, we present a stochastic investment model (SIM) for international investors. We assume that investors are allowed to hold assets in two different economies. This SIM includes four components: interest rates, stocks, inflation and exchange rate models. First, we give a full description of the model and we detail the parameter estimation. The model is estimated using a state-space formulation and an extended Kalman filter. Based on scenarios generated from this SIM, we study the risk allocation to different background risks: asset, inflation and exchange rate risks. The risk allocation is based on the TVaR-based rule

Mathematical Perspectives on Insurance for Low-Income Populations

In this thesis, insurance solutions for low-income populations and their capacity for poverty reduction are considered. Classical risk theory techniques are adopted to the study of the trapping probability, where "trapping" refers to the event at which an economic entity falls below the poverty line and into an area of poverty, from which it is difficult to escape without external help. In the poverty setting, the trapping probability mimics an insurer's probability of ruin. Studying two household-level capital processes that align with risk processes with deterministic investment and (i) random-valued and (ii) multiplicative claims, explicit trapping probabilities are derived. The ability of low-income insurance strategies to reduce trapping probabilities is assessed, with a particular focus on government subsidy schemes. For those closest to the poverty line, insurance without subsidies increases their probability of trapping in both the random-valued and multiplicative cases, in line with the existing literature. The governmental cost of social protection is reduced under subsidisation schemes, with a premium payment barrier strategy additionally ensuring the increased risk associated with insurance purchase is mitigated. Purchase of insurance for multiplicative losses is found to be more affordable than for random-valued losses. A stochastic dissemination model is proposed for the extension of the problem to the group setting, in line with the prevalence of risk sharing and group-based insurance schemes across low-income communities. Consideration of risk sharing suggests that the impact of loss and premium payment is shared throughout a homogeneous group, mitigating the severity of negative wealth transaction events. Subsidisation is also found to support both the insured and the uninsured, further highlighting the benefit of governmentally supported schemes. In the second part of the thesis, the existence of lifetime dependence and the influence of socioeconomic features on its structure are considered. Analysis is undertaken on data sets from Ghana and Egypt, with dependence induced through joint stochastic mortality and copula models, respectively. The impact of dependence on the pricing of a reversionary annuity is derived through implementation of the indifference pricing principle. In general, pricing under the dependence assumption decreases the indifference price of the annuity. In visualising both data sets, dependence is observed to be lower in this alternative socioeconomic environment than previously observed in the existing empirical literature, supporting suggestion of socioeconomic influences on bereavement processes. Studying the existence of pairwise dependence within relationships beyond the classical husband-wife case, dependence within child-parent relationships is also found to be significant. Accounting for this existence, even where reduced, is critical to improving the accuracy of insurance product pricing and to mitigate the mortality risks faced by insurers, particularly given the uncertain nature of the low-income financial environment

Quantitative Methods for Economics and Finance

This book is a collection of papers for the Special Issue “Quantitative Methods for Economics and Finance” of the journal Mathematics. This Special Issue reflects on the latest developments in different fields of economics and finance where mathematics plays a significant role. The book gathers 19 papers on topics such as volatility clusters and volatility dynamic, forecasting, stocks, indexes, cryptocurrencies and commodities, trade agreements, the relationship between volume and price, trading strategies, efficiency, regression, utility models, fraud prediction, or intertemporal choice

Pairwise versus mutual independence: visualisation, actuarial applications and central limit theorems

Accurately capturing the dependence between risks, if it exists, is an increasingly relevant topic of actuarial research. In recent years, several authors have started to relax the traditional 'independence assumption', in a variety of actuarial settings. While it is known that 'mutual independence' between random variables is not equivalent to their 'pairwise independence', this thesis aims to provide a better understanding of the materiality of this difference. The distinction between mutual and pairwise independence matters because, in practice, dependence is often assessed via pairs only, e.g., through correlation matrices, rank-based measures of association, scatterplot matrices, heat-maps, etc. Using such pairwise methods, it is possible to miss some forms of dependence. In this thesis, we explore how material the difference between pairwise and mutual independence is, and from several angles. We provide relevant background and motivation for this thesis in Chapter 1, then conduct a literature review in Chapter 2. In Chapter 3, we focus on visualising the difference between pairwise and mutual independence. To do so, we propose a series of theoretical examples (some of them new) where random variables are pairwise independent but (mutually) dependent, in short, PIBD. We then develop new visualisation tools and use them to illustrate what PIBD variables can look like. We showcase that the dependence involved is possibly very strong. We also use our visualisation tools to identify subtle forms of dependence, which would otherwise be hard to detect. In Chapter 4, we review common dependence models (such has elliptical distributions and Archimedean copulas) used in actuarial science and show that they do not allow for the possibility of PIBD data. We also investigate concrete consequences of the 'nonequivalence' between pairwise and mutual independence. We establish that many results which hold for mutually independent variables do not hold under sole pairwise independent. Those include results about finite sums of random variables, extreme value theory and bootstrap methods. This part thus illustrates what can potentially 'go wrong' if one assumes mutual independence where only pairwise independence holds. Lastly, in Chapters 5 and 6, we investigate the question of what happens for PIBD variables 'in the limit', i.e., when the sample size goes to infi nity. We want to see if the 'problems' caused by dependence vanish for sufficiently large samples. This is a broad question, and we concentrate on the important classical Central Limit Theorem (CLT), for which we fi nd that the answer is largely negative. In particular, we construct new sequences of PIBD variables (with arbitrary margins) for which a CLT does not hold. We derive explicitly the asymptotic distribution of the standardised mean of our sequences, which allows us to illustrate the extent of the 'failure' of a CLT for PIBD variables. We also propose a general methodology to construct dependent K-tuplewise independent (K an arbitrary integer) sequences of random variables with arbitrary margins. In the case K = 3, we use this methodology to derive explicit examples of triplewise independent sequences for which no CLT hold. Those results illustrate that mutual independence is a crucial assumption within CLTs, and that having larger samples is not always a viable solution to the problem of non-independent data

Risks

This book is a collection of feature articles published in Risks in 2020. They were all written by experts in their respective fields. In these articles, they all develop and present new aspects and insights that can help us to understand and cope with the different and ever-changing aspects of risks. In some of the feature articles the probabilistic risk modeling is the central focus, whereas impact and innovation, in the context of financial economics and actuarial science, is somewhat retained and left for future research. In other articles it is the other way around. Ideas and perceptions in financial markets are the driving force of the research but they do not necessarily rely on innovation in the underlying risk models. Together, they are state-of-the-art, expert-led, up-to-date contributions, demonstrating what Risks is and what Risks has to offer: articles that focus on the central aspects of insurance and financial risk management, that detail progress and paths of further development in understanding and dealing with...risks. Asking the same type of questions (which risk allocation and mitigation should be provided, and why?) creates value from three different perspectives: the normative perspective of market regulator; the existential perspective of the financial institution; the phenomenological perspective of the individual consumer or policy holder

Copulae and correlation products

This thesis studies copula applications to correlation products. There are six self-contained but related projects in this research, with the following objectives: 1) to review reduced-form approaches to model the default process of a single-name obligor and their extension to model the joint distribution of defaults in a portfolio of obligors; 2) to set up the CDO market; 3) to introduce copulae and to provide a justification of copulae as modelling tools; 4) to provide with our view regarding the most suitable copula when modelling complex correlation products; 5) to prepare a time-inhomogeneous intensity model for valuing cash-flow CDOs, which explicitly incorporates the credit rating of the firms in the collateral portfolio as the indicator of the likelihood of default; 6) to prepare a pricing model for CDOs of EDSs. We found strong evidence that the Clayton copula is a suitable tool when modelling correlation products with Li’s Survival model. The Clayton copula had some important consequences: we noticed a redistribution of losses from the Junior note to the Mezzanine and Senior notes; in addition, it picked up some extra risk in the Senior note, and finally, when compared with the Normal copula, it overestimated the fair compensation of the Senior and the Mezzanine notes and underestimated the fair compensation of the Junior note. We also found the Clayton copula was very adaptable into the dynamic copula framework of Schonbucher and Schubert. Modelling the notes of cash-flow CDOs with copulae and time-inhomogeneous transition matrices has not been an easy task. This is because the computation of the transition matrices for arbitrary periods of time was based on an annual transition matrix. In addition, this matrix, as most of the empirical annual transition matrices, was not compatible with a continuous Markov process since it did not admit a valid generator. Therefore, we computed a modified version of a true generator. Following this, we successfully applied one method, originally advanced by JLT (1997), to calibrate the adjusted matrix to the S&P’s probabilities of default. Finally, we described how to simulate the credit rating migration of one single credit, and how to join n -credit rating migrations via the Normal copula. Modelling the collateral credit risk in this way is very powerful, since it allowed us to take into account quality trigger linked to the rating-performance of the collateral and to keep the model of the joint credit rating migrations, totally separate with copulae. For example, when there are performance triggers linked to the collateral average rating, our Rating Transition Copula model perfectly captures the diversion of cash from the interest waterfall to the principal waterfall for the benefit of the Senior and Mezzanine notes. To price single-name Equity Default Swaps and CDOs of Equity Default Swaps, we extended the GARCH option framework of Duan (1995). Volatility of the underlying equity price is the critical factor affecting option prices, and in our EDS model, the variance of the equity return followed a nonlinear GARCH in mean. When pricing single-name EDS, we proposed two nonlinear GARCH in mean (NGARCH-M): normal and /-Student NGARCH-M model. As a benchmark, we assumed that the equity returns moved accordingly to the standard homoskedastic lognormal process of Black & Scholes and priced the single-name EDS with the Rubinstein and Reiner model for binary barrier options. The problem we found with this approach was that the implied volatilities for very deep out-of-the-money put options were not available. When the volatility was modelled as a GARCH process, it was not possible to derive the future distribution of the underlying equity. Therefore, our model relied on Monte Carlo simulations. To ensure that the simulated option price did not violate rational option pricing bounds, we used the empirical martingale simulation originally advanced by Duan and Simonato, coupled with the standard variance reduction technique. To address the issue of how to price a basket of EDSs, we resorted to the concept of copula. With copulae, we were able to decouple the pricing problem: keeping the aspect of modelling the marginal distribution of the equity returns via NGARCH-M, totally separate from addressing the dependence problem

Risk Management for the Future

A large part of academic literature, business literature as well as practices in real life are resting on the assumption that uncertainty and risk does not exist. We all know that this is not true, yet, a whole variety of methods, tools and practices are not attuned to the fact that the future is uncertain and that risks are all around us. However, despite risk management entering the agenda some decades ago, it has introduced risks on its own as illustrated by the financial crisis. Here is a book that goes beyond risk management as it is today and tries to discuss what needs to be improved further. The book also offers some cases

Complexity in Economic and Social Systems

There is no term that better describes the essential features of human society than complexity. On various levels, from the decision-making processes of individuals, through to the interactions between individuals leading to the spontaneous formation of groups and social hierarchies, up to the collective, herding processes that reshape whole societies, all these features share the property of irreducibility, i.e., they require a holistic, multi-level approach formed by researchers from different disciplines. This Special Issue aims to collect research studies that, by exploiting the latest advances in physics, economics, complex networks, and data science, make a step towards understanding these economic and social systems. The majority of submissions are devoted to financial market analysis and modeling, including the stock and cryptocurrency markets in the COVID-19 pandemic, systemic risk quantification and control, wealth condensation, the innovation-related performance of companies, and more. Looking more at societies, there are papers that deal with regional development, land speculation, and the-fake news-fighting strategies, the issues which are of central interest in contemporary society. On top of this, one of the contributions proposes a new, improved complexity measure