Scienze economiche e statistiche

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.