Scienze economiche e statistiche

It is well known that optimal risk sharing is an argument that deserves both theoretical and practical interest. It originally appears in the context of reinsurance problems, but now is widely used in a variety of financial and economical applications. The problem concerning the existence of individually rational Pareto optimal allocations, namely optimal solutions, is generally treated in the literature by considering the usual requirement of completeness over decision makers’ preferences. In this thesis we present several conditions for the existence of optimal solutions in a modern preference-based approach provided that agents’ preferences are expressed by not necessarily total preorders and by considering a topological context. We prove the equivalence between optimality and maximality with respect to a coalition preorder traducing the problem of finding optimal solutions to that of guaranteeing the existence of maximal elements for a not necessarily total preorder. In this framework a "folk theorem" is of help since it guarantees the existence of a maximal element for an upper semicontinuous preorder on a compact topological space. We study the functional approaches representing optimal risk sharing identified with the so called multi-objective maximization problem and the supconvolution problem, with the aim of incorporating functional representations of not necessarily total preorders, essentially expressed by order preserving functions and multi-utility representations. We use these two notions in order to guarantee the existence of optimal solutions, and to this aim we appropriately refer to well known results in mathematical utility theory (for example, Rader’s theorem). The case of individual preferences expressed by translation invariant total preorders is also considered, completing fundamental results from the literature also extended to the case of comonotone super-additive and positively homogeneous utility functions. When comonotone allocations are considered, we limit the research of maximal elements with respect to the coalition preorder to the set of comonotone allocations, provided that monotonicity conditions with respect to second order stochastic dominance are imposed to the individual preorders. In all our framework, we deal with risks belonging to some space of nonnegative random variables on a common probability space and, as a natural application of all our considerations, we consider the Choquet Integral when the topology L∞ is considered. Come noto, il problema di risk sharing è un argomento che interessa sia aspetti teorici che applicativi. Originariamente introdotto in contesti di riassicurazione, attualmente è ampiamente utilizzato in una varietà di applicazioni finanziarie ed economiche. Il problema legato all’esistenza di allocazioni Pareto ottimali ed individualmente razionali, definite soluzioni ottime, è generalmente trattato in letteratura considerando l’usuale assioma di completezza sulle preferenze degli agenti. In questa tesi presentiamo diverse condizioni per l'esistenza di soluzioni ottime in un moderno approccio di preferenza caratterizzato dall'espressione delle preferenze individuali per mezzo di preordini non necessariamente totali e considerando un contesto topologico. Viene dimostrata l’equivalenza tra ottimalità e massimalità rispetto ad un preordine di coalizione, traducendo così il problema di trovare soluzioni ottime nel garantire l’esistenza di elementi massimali per un preordine non necessariamente totale. In questo quadro di riferimento, un "folk theorem" è di aiuto in quanto garantisce l’esistenza di un elemento massimale per un preordine superiormente semicontinuo definito su uno spazio topologico compatto. Vengono studiati approcci funzionali legati al problema di risk sharing, identificati con il problema di massimizzazione multi-obiettivo ed il problema di sup-convoluzione, con l’obiettivo di incorporare rappresentazioni funzionali di preordini non necessariamente totali, essenzialmente definite da funzioni order preserving e rappresentazioni di multi-utilità. Queste due notazioni vengono utilizzate in modo da garantire l’esistenza di soluzioni ottime, e a questo scopo ci riferiamo in modo appropriato a ben noti risultati in teoria dell’utilità (ad esempio, il teorema di Rader). Il caso di preferenze individuali espresse da preordini totali invarianti per traslazioni è anche considerato, a completamento di fondamentali risultati presenti in letteratura ed estesi anche al caso di funzioni di utilità che soddisfino alle proprietà di comonotona super-additività e positiva omogeneità. Quando si considerano allocazioni comonotone, ci limitiamo alla ricerca di elementi massimali rispetto al preordine di coalizione nell’insieme delle allocazioni comonotone, purchè vengano imposte condizioni di monotonia sui preordini individuali rispetto alla dominanza stocastica di secondo ordine. In tutto il nostro contesto di riferimento affrontiamo il caso di rischi appartenenti a spazi di variabili aleatorie non-negative definite su un comune spazio di probabilità e come naturale applicazione consideriamo l’integrale di Choquet nel caso venga considerata la topologia L∞.

The past twenty years have seen a massive proliferation in insurance-linked derivative products. The public, indeed, has become more aware of investment opportunities outside the insurance sector and is increasingly trying to seize all the benefits of equity investment in conjunction with mortality protection. The competition with alternative investment vehicles offered by the financial industry has generated substantial innovation in the design of life products and in the range of benefits provided. In particular, equity-linked policies have become ever more popular, exposing policyholders to financial markets and providing them with different ways to consolidate investment performance over time as well as protection against mortality-related risks. Interesting examples of such contracts are variable annuities (VAs). This kind of policies, first introduced in 1952 in the United States, experienced remarkable growth in Europe, especially during the last decade, characterized by “bearish” financial markets and relatively low interest rates. The success of these contracts is due to the presence of tax incentives, but mainly to the possibility of underwriting several rider benefits that provide protection of the policyholder’s savings for the period before and after retirement. In this thesis, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. This option meets medium to long-term investment needs, while providing adequate hedging against market volatility and longevity-related risks. Indeed, based on an initial capital investment, it guarantees the policyholder a stream of future payments, regardless of the performance of the underlying policy, for his/her whole life. In this work, we propose a valuation model for the policy using tractable financial and stochastic mortality processes in a continuous time framework. We have analyzed the policy considering two points of view, the policyholder’s and the insurer’s, and assuming a static approach, in which policyholders withdraw each year just the guaranteed amount. In particular, we have based ourselves on the model proposed in the paper “Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities” by M. C. Fung, K. Ignatieva and M. Sherris (2014), with the aim of generalizing it later on. The valuation, indeed, has been performed in a Black and Scholes economy: the sub-account value has been assumed to follow a geometric Brownian motion, thus with a constant volatility, and the term structure of interest rates has been assumed to be constant. These hypotheses, however, do not reflect the situation of financial markets. In order to consider a more realistic model, we have sought to weaken these misconceptions. Specifically we have taken into account a CIR stochastic process for the term structure of interest rates and a Heston model for the volatility of the underlying account, analyzing their effect on the fair price of the contract. We have addressed these two hypotheses separately at first, and jointly afterwards. As part of our analysis, we have implemented the theoretical model using a Monte Carlo approach. To this end, we have created ad hoc codes based on the programming language MATLAB, exploiting its fast matrix-computation facilities. Sensitivity analyses have been conducted in order to investigate the relation between the fair price of the contract and important financial and demographic factors. Numerical results in the stochastic approach display greater fair fee rates compared to those obtained in the deterministic one. Therefore, a stochastic framework is necessary in order to avoid an underestimation of the policy. The work is organized as follows. Chapter 1. This chapter has an introductory purpose and aims at presenting the basic structures of annuities in general and of variable annuities in particular. We offer an historical review of the development of the VA contracts and describe the embedded guarantees. We examine the main life insurance markets in order to highlight the international developments of VAs and their growth potential. In the last part we retrace the main academic contributions on the topic. Chapter 2. Among the embedded guarantees, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. We analyze a valuation model for the policy basing ourselves on the one proposed by M. Sherris (2014). We introduce the two components of the model: the financial market, on the one hand, and the mortality intensity on the other. We first describe them separately, and subsequently we combine them into the insurance market model. In the second part of the chapter we describe the valuation formula considering the GLWB from two perspectives, the policyholder’s and the insurer’s. Chapter 3. Here we implement the theoretical model creating ad hoc codes with the programming language MATLAB. Our numerical experiments use a Monte Carlo approach: random variables have been simulated by MATLAB high level random number generator, whereas concerning the approximation of expected values, scenario- based averages have been evaluated by exploiting MATLAB fast matrix-computation facilities. Sensitivity analyses are conducted in order to investigate the relation between the fair fee rate and important financial and demographic factors. Chapter 4. The assumption of deterministic interest rates, which can be acceptable for short-term options, is not realistic for medium or long-term contracts such as life insurance products. GLWB contracts are investment vehicles with a long-term horizon and, as such, they are very sensitive to interest rate movements, which are uncertain by nature. A stochastic modeling of the term structure is thus appropriate. In this chapter, therefore, we propose a generalization of the deterministic model allowing interest rates to vary randomly. A Cox-Ingersoll-Ross model is introduced. Sensitivity analyses have been conducted. Chapter 5. Empirical studies of stock price returns show that volatility exhibits “random” characteristics. Consequently, the hypothesis of a constant volatility is rather “counterfactual”. In order to consider a more realistic model, we introduce the stochastic Heston process for the volatility. Sensitivity analyses have been con- ducted. Chapter 6. In this chapter we price the GLWB option considering a stochastic process for both the interest rate and the volatility. We present a numerical comparison with the deterministic model. Chapter 7. Conclusions are drawn. Appendix. This section presents a quick survey of the most fundamental concepts from stochastic calculus that are needed to proceed with the description of the GLWB’s valuation model.

The change in mortality trends experienced over the last decades leads to the use of projected mortality tables in order to avoid underestimation of the future liabilities and costs in long term insurance products such as life annuities and pension funds. Although the projected mortality tables aim to capture the dynamic structure of mortality in the future, the future mortality trend itself is random and systematic deviations from the projected mortality might take place. Being a non-pooling risk, the impact of this ``uncertainty risk'' on the insurance portfolios can be dramatic due to the fact that the severity resulting from it increases as the size of the portfolio. For this reason, a proper modelling of uncertainty risk in mortality trends is required. In this work the uncertainty risk modelling in mortality trends has been studied. In this aspect, the two stochastic models in the literature, scenario based and dynamic models have been adopted and assessed their level of capturing the uncertainty in mortality trends. One of the models, the static model, has been extended to the continuous case with the allowance of the multiple cohorts in the portfolio. As defining the model, two approximation methods has been adopted to define the distribution of total number of deaths in the portfolio. Bayesian inferential procedure has been used in updating the random variables representing the uncertainty risk to the experience in the portfolio.