Piscopo, Gabriella (2009) Variable Annuities and Embedded Options: models and tools for fair valuation and solvency appraisal. [Tesi di dottorato] (Unpublished)

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The objective of this thesis is to provide an analysis of the Variable Annuity (VA) products. In particular, we focus on the actuarial and financial valuation of some embedded guarantees in VAs, derive No-arbitrage pricing models and study the impact of mortality risk. We have decided to deal with this products in the light of significant international success obtained by VAs and we believe that perspectives of their development in Italy market and throughout Europe and Asia are favourable. One of the reasons of this success is the presence of guarantees which offer partial protection against the downside movements of the interest rates or the equity market, an attractive feature for the individual retirement security. The shift from defined benefit to self-directed defined contribution plans and the reform of the Social Retirement System in many countries, so that it includes personal accounts, have encouraged the proliferation of new kind of products. Owing to the long term horizon of their commitments, pension funds are exposed to important financial risk due to the volatility of interest rates and equity markets. At this regard, VAs were first introduced by insurance companies in the 1970s in the United States to compete with mutual funds. Over the years, many practical and academic contributions have been offered for describing the VAs and the guarantees embedded. Most of the earlier literature (e.g., Rentz Jr. (1972) and Green (1973)) is constituted by empirical works dealing with product comparisons rather than pricing and hedging issues. It was not until recently that some guarantees were discussed by practitioners ( e.g., J.P.Morgan (2004), Lehman Brothers (2005), Milliman (2007)); they highlight the growing opportunities to introduce VAs in new markets. Recently, the academic literature has shown a fervent interest to the topic too (cfr. Bauer et al. (2006), Chen et al. (2008), Coleman et al. (2006), Dai (2008), Holz (2006), Milevsky and Panyagometh (2001), Milevsky and Posner (2001), Milevsky M.A and Promislow S.D (2001), Milevsky and Salisbury(2002)., Milevsky and Salisbury (2006), Nielsen and Sandmann (2003)). Bauer et al. offer the first universal general framework in which any design of options and guarantees currently offered within Variable Annuities can be modeled. Besides the valuation of a contract assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, they are able to price contracts with different embedded options. Milevsky und Posner (2001) price various types of guaranteed minimum death benefits. They present closed form solutions for this option in case of an exponential mortality law and numerical results for the more realistic Gompertz-Makeham law. They find that in general these guarantees are overpriced in the market. In Milevsky und Salisbury (2006), the authors price GMWB options. Besides a static approach, where deterministic withdrawal strategies are assumed, they calculate the value of the option in a dynamic approach. Here, the option is valuated under optimal policyholder behavior. They show that under realistic parameter assumptions optimally at least the annually guaranteed withdrawal amount should be withdrawn. Furthermore, they find that such options are usually underpriced in the market. This result is in contrast with the common belief that the guarantees embedded in variable annuity policies are overpriced (see Clements (2004)). This thesis aims at following this literature by proposing some theoretical and practical innovative works. Our original contributions lie in: 1) describe how the value of Guaranteed Minimum Death Benefit (GMDB) options evolves over time and in the presence of mortality changes and produce an application to Italian data; 2) study the insurance surplus over time for a portfolio of Variable Annuities with GMDB Options and offer a model that can be used for an evaluation of the adequacy of solvency; 3) develop a sensitivity analysis for the value of Guaranteed Lifelong Withdrawal Benefit (GLWB) options under the hypothesis of a static withdrawal strategy; 4) decompose a VA with a GLWB option into a life annuity plus a portfolio of Quanto Asian Put Options, with decreasing strikes and increasing expiration dates, and verify that this product is underpriced on US market.

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