Theoretical and practical advances in actuarial science : risk, ruin theory and life pensions

Doutoramento em Matemática Aplicada à Economia e GestãoEsta tese apresenta avanços teóricos e práticos na ciência atuarial. É composta por seis capítulos que consistem em seis artigos sobre temas da ciência atuarial, ou seja, tópicos em teoria do risco, teoria da ruína, fundos de pensão, dados de seguros e análise de risco. Começamos com text mining em big data em um estudo de caso sobre modelos de risco com dependência. Em seguida, apresentamos um programa de micro pensão pública no Brasil com duas bases de dados contendo 4.644.698 e 1.332.080 pessoas, respectivamente. Neste capítulo, apresentamos a heterogeneidade entre as Unidades Federativas e propomos um ajuste à idade de benefício. Em sequência, avançamos em um tópico teórico, estudamos medidas de ruína e dividendos no modelo de risco dual com renovação. Na primeira parte, calculamos a probabilidade de ruína, os dividendos esperados descontados, o valor do dividendo único, a probabilidade de dividendo, o número de ganhos para atingir uma determinada meta superior e o número de ganhos até a ruína. Finalizamos este capítulo trabalhando alguns exemplos numéricos ilustrativos. A seguir apresentamos uma função de penalização para o modelo de risco dual com renovação citado anteriormente, neste problema trabalhamos com um modelo em que os ganhos aleatórios ocorrem sob um processo de renovação. Apresentamos fórmulas, explicando como alcançá-las, resolvemos a equação integro-diferencial para casos positivos e negativos e, por fim, mostramos exemplos numéricos. Em seguida, trabalhamos em outro avanço prático, consideramos um modelo de risco com frequência e montante de sinistros dependentes para seguros de responsabilidade civil e habitacional, que são dois tipos de seguro, que ao contrário do seguro de automóvel, geralmente não são objeto de estudo. Para tal, trabalhamos com uma base de dados composta por 15.665 sinistros, sendo 966 de responsabilidade civil e 14.699 de seguros habitacionais. Neste capítulo, mostramos novos resultados na área e trabalhamos com modelos GAMLSS capazes de capturar a dependência entre a frequência e o montante dos sinistros. Apresentamos então o último artigo, uma modelagem de risco para commodities no Brasil, este é um trabalho aplicado para os preços à vista e futuros do Boi Gordo. Neste trabalho, usamos cinco métodos diferentes e comparamos 12 modelos distintos como forma de retratar e prever o comportamento da commodity BGI. Para finalizar, fazemos algumas observações finais na conclusão.This thesis presents theoretical and practical advances in actuarial science. It is composed of six chapters consisting of six papers on topics on actuarial science. That is, regarding risk theory, ruin theory, pension funds, insurance data, and risk analysis. It starts with text mining big data in the case study of research on risk models with dependence. Then we present a public micro pension programme in Brazil with two data sets containing 4,644,698 and 1,332,080 people, respectively. In this chapter, we present the heterogeneity among states and we set up a benefit age adjustment. In sequence, we advance in a theoretical topic, we study ruin and dividend measures in the renewal dual risk model. In the first part, we calculate ruin probability, expected discounted dividends, single dividend amount, dividend probability, number of gains to reach a given upper target, and number of gains down to ruin. We finalize this chapter working on some illustrative numerical examples. Thereafter we present a penalty function for the aforementioned renewal dual risk model, in this problem we work with a model that the random gains occur under a renewal process. We present the formula, explaining how to achieve it, we solve the integro-differential equation for both positive and negative cases and ultimately show numerical examples. Then we work on another practical advance, considering a risk model with dependent frequency and severity for liability and housing insurance that are two types of insurance, that, unlike automobile insurance, are not usually object of study. For this, we work on a database consisting of 15,665 claims, being 966 from liability and 14,699 from housing insurance. In this chapter, we show new results in the field and work on a GAMLSS model that can capture the dependence between claims frequency and severity. The last paper is then presented, a risk modelling for commodities in Brazil, it is an applied work for the Boi Gordo spot and future prices. In this work, we used five different methods and compared 12 different models as a way to portray and predict the BGI commodity’s behaviour. To finalize, we make some final remarks in the conclusion.info:eu-repo/semantics/publishedVersio

Applications of hidden Markov models in financial modelling

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.Various models driven by a hidden Markov chain in discrete or continuous time are developed to capture the stylised features of market variables whose levels or values constitute as the underliers of financial derivative contracts or investment portfolios. Since the parameters are switching regimes, the changes and developments in the economy as soon as they arise are readily reflected in these models. The change of probability measure technique and the EM algorithm are fundamental techniques utilised in the optimal parameter estimation. Recursive adaptive filters for the state of the Markov chain and other auxiliary processes related to the Markov chain are derived which in turn yield self-tuning dynamic financial models. A hidden Markov model (HMM)-based modelling set-up for commodity prices is developed and the predictability of the gold market under this setting is examined. An Ornstein-Uhlenbeck (OU) model with HMM parameters is proposed and under this set-up, we address two statistical inference issues: the sensitivity of the model to small changes in parameter estimates and the selection of the optimal number of states. The extended OU model is implemented on a data set of 30-day Canadian T-bill yields. An exponential of a Markov-switching OU process plus a compound Poisson process is put forward as a model for the evolution of electricity spot prices. Using a data set compiled by Nord Pool, we illustrate the vast improvements gained in incorporating regimes in the model. A multivariate HMM is employed as a framework in providing the solutions of two asset allocation problems; one involves the mean-variance utility function and the other entails the CVaR constraint. Finally, the valuation of credit default swaps highlights the important considerations necessitated by pricing in a regime-switching environment. Certain numerical schemes are applied to obtain approximations for the default probabilities and swap rates.Brunel Research Initiative and Enterprise Fund (BRIEF) and European Union (Marie Curie Fellowship

Stochastic and Numerical Analysis for Optimization Problems

There are three chapters of manuscripts in this dissertation and all of them are talking about a specific theme: stochastic control, but with completely different perspectives. In the first manuscript, we solve the optimal barrier strategy for dividend distribution in a complicated Lévy system. In this system, the capital of the company fluctuates with one-sided jumping Lévy noise and the macroeconomic indicator, interest rate, is al-lowed to go negative and move with possibly infinite jumping frictions in a short time. Both of them represent the current status of financial insurance markets. The main techniques used, Itô excursion theory and fluctuation identities of Lévy process, are purely probabilistic. In the second manuscript, we take a look at a classical Hamilton-Jacobian-Bellman equation related to dividend distribution. It needs some technical analysis to find the analytical solution but here we develop a novel numerical method: projected semismooth Newton with shooting-like method, which solves the approximate solution with desired superlinear convergence rate. It suggests a new way to solve numerically the constrained free-boundary variational inequality. In the last manuscript, we come up with a new economic model on efficiently utilizing the flexibility of renewable energy in its market. There are several pioneering considerations in this model. On the one hand, the electricity price and reservoir storage both fluctuate based on sophisticated stochastic models. On the other hand, we could sell the excessive hydropower to market while we could buy the electricity from market in case of low reservoir level. We propose a “trigger price” strategy to manage the flexibility of hydropower so as to optimize the net profit. In this manuscript, we find one explicit strategy in a classical setting and prove the existence and uniqueness of some intricate strategy in a complicated setting via viscosity solution technique

Applications of hidden Markov models in financial modelling

Various models driven by a hidden Markov chain in discrete or continuous time are developed to capture the stylised features of market variables whose levels or values constitute as the underliers of financial derivative contracts or investment portfolios. Since the parameters are switching regimes, the changes and developments in the economy as soon as they arise are readily reflected in these models. The change of probability measure technique and the EM algorithm are fundamental techniques utilised in the optimal parameter estimation. Recursive adaptive filters for the state of the Markov chain and other auxiliary processes related to the Markov chain are derived which in turn yield self-tuning dynamic financial models. A hidden Markov model (HMM)-based modelling set-up for commodity prices is developed and the predictability of the gold market under this setting is examined. An Ornstein-Uhlenbeck (OU) model with HMM parameters is proposed and under this set-up, we address two statistical inference issues: the sensitivity of the model to small changes in parameter estimates and the selection of the optimal number of states. The extended OU model is implemented on a data set of 30-day Canadian T-bill yields. An exponential of a Markov-switching OU process plus a compound Poisson process is put forward as a model for the evolution of electricity spot prices. Using a data set compiled by Nord Pool, we illustrate the vast improvements gained in incorporating regimes in the model. A multivariate HMM is employed as a framework in providing the solutions of two asset allocation problems; one involves the mean-variance utility function and the other entails the CVaR constraint. Finally, the valuation of credit default swaps highlights the important considerations necessitated by pricing in a regime-switching environment. Certain numerical schemes are applied to obtain approximations for the default probabilities and swap rates.EThOS - Electronic Theses Online ServiceBrunel Research Initiative and Enterprise Fund (BRIEF) : European Union (Marie Curie Fellowship)GBUnited Kingdo

Valuation, Empirical Analysis, and Optimal Exercise of Open-End Turbo Certificates

This dissertation analyzes Open-End Turbo Certificates (OETCs), a popular class of retail derivatives. OETCs can be exercised at any time at the investor’s discretion. In order to explain the existence of the certificates jump risk must be considered. We propose and implement an optimal stopping approach to price these securities, which further allows for determining optimal exercise thresholds. They result from the trade-off between benefits from downward jump protection and financing costs. We show that early exercise right has a significant impact on their values. In an empirical analysis pertaining to the years 2007 through 2009 it turns out that certificates which could be rationally held are very rare, although the degree by which the underlying exceeds the optimal exercise thresholds continually declines over the considered period. We suggest three lines of explanation: general market movement, jump risk perception by the market, and increased competition among issuers.Die vorliegende Dissertation behandelt Open-End Turbo Zertifikate (OETCs), eine populäre Klasse von Privatkundenderivaten, die jederzeit durch den Investor ausgeübt werden können. Um ihre Existenz rechtfertigen zu können, muss Sprungrisiko berücksichtigt werden. Zur Preisstellung des Produktes schlagen wir einen Optimal Stopping Ansatz vor und implementieren diesen. Dies erlaubt zudem die Berechnung optimaler Ausübungsschwellen, die aus dem Gegenspiel von Finanzierungskosten einerseits und Schutz gegen Abwärtssprünge andererseits entstehen. Wir zeigen, dass vorzeitige Ausübungsrechte in der Bewertung eine signifikante Rolle spielen. In einer empirischen Analyse für die Jahre 2007 bis 2009 zeigt sich schließlich, dass rationale Investoren nur sehr wenige OETCs halten sollten. Andererseits geht der Grad, um welchen das Underlying die optimale Ausübungsschwelle überschreitet, kontinuierlich zurück. Für diese Beobachtung lassen sich drei Begründungen anführen: allgemeine Marktbewegung, vom Markt wahrgenommenes Sprungrisiko und erhöhter Wettbewerb unter den Anbietern

Stochastics of Environmental and Financial Economics

Systems Theory, Contro