BENSUSAN Harry

In this paper, we introduce a new structured financial product: the so-called Life Nominal Chooser Swaption (LNCS). Thanks to such a contract, insurers could keep pure longevity risk and transfer a great part of interest rate risk underlying annuity portfolios to financial markets. Before the issuance of the contract, the insurer determines a confidence band of survival curves for her portfolio. An interest rate hedge is set up, based on swaption mechanisms. The bank uses this band as well as an interest rate model to price the product. At the end of the first period (e.g. 8 to 10 years), the insurer has the right to enter into an interest rate swap with the bank, where the nominal is adjusted to her (re-forecasted) needs. She chooses (inside the band) the survival curve that better fits her anticipation of future mortality of her portfolio (during 15 to 20 more years, say) given the information available at that time. We use a population dynamics longevity model and a classical two-factor interest rate model %two-factor Heath-Jarrow-Morton (HJM) model for interest rates to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology. This structure enables insurers and financial institutions to remain in their initial field of expertise.

In this paper, we introduce a new structured financial product: the so-called Life Nominal Chooser Swaption (LNCS). Thanks to such a contract, insurers could keep pure longevity risk and transfer a great part of interest rate risk underlying annuity portfolios to financial markets. Before the issuance of the contract, the insurer determines a confidence band of survival curves for her portfolio. An interest rate hedge is set up, based on swaption mechanisms. The bank uses this band as well as an interest rate model to price the product. At the end of the first period (e.g. 8 to 10 years), the insurer has the right to enter into an interest rate swap with the bank, where the nominal is adjusted to her (re-forecasted) needs. She chooses (inside the band) the survival curve that better fits her anticipation of future mortality of her portfolio (during 15 to 20 more years, say) given the information available at that time. We use a population dynamics longevity model and a classical two-factor interest rate model %two-factor Heath-Jarrow-Morton (HJM) model for interest rates to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology. This structure enables insurers and financial institutions to remain in their initial field of expertise.

This thesis is divided into three parts. The first part consists of chapters 2 and 3 in which we consider models that describe the evolution of an underlying asset in the world of stocks as well as the evolution of interest rates. These models, which use Wishart processes, belong to the affine class and generalize the multi-dimensional Heston models. We study the intrinsic properties of these models and focus on the valuation of vanilla options. After recalling some valuation methods, we introduce approximation methods providing closed formulas of the asymptotic smile. These methods facilitate the calibration procedure and allow an interesting analysis of the parameters. The second part, from chapter 4 to chapter 6, studies mortality and longevity risks. We first recall the general concepts of longevity risk and a set of issues underlying this risk. We then present a model of individual mortality that takes into account age and other characteristics of the individual that are explanatory of mortality. We calibrate the mortality model and analyze the influence of certain individual characteristics. Finally, we introduce a microscopic population dynamics model that allows us to model the evolution over time of a population structured by age and traits. Each individual evolves over time and is likely to give birth to a child, change characteristics and die. This model accounts for the possibly stochastic evolution of individual demographic rates over time. We also describe a micro/macro link that provides this microscopic model with good macroscopic properties. The third part, concerning chapters 7 and 8, focuses on applications of the previous modeling. The first application is a demographic one since the microscopic population dynamics model allows us to make demographic projections of the French population. We also set up a demographic study of the pension problem by analyzing the solutions of an immigration policy and a reform of the retirement age. The second application concerns the study of life insurance products combining longevity and interest rate risks that have been studied in detail in the first two parts of the thesis. First, we study the basic risk generated by the heterogeneity of annuity portfolios. In addition, we introduce the Life Nominal Chooser Swaption (LNCS) which is a risk transfer product for life insurance products: this product has a very interesting structure and allows an insurance company holding an annuity portfolio to transfer its entire interest rate risk to a bank.

In French: This thesis is divided into three parts. The first part consists of chapters 2 and 3 in which we consider models that describe the evolution of an underlying asset in the world of stocks as well as the evolution of interest rates. These models, which use Wishart processes, belong to the affine class and generalize the multi-dimensional Heston models. We study the intrinsic properties of these models and focus on the valuation of vanilla options. After recalling some valuation methods, we introduce approximation methods providing closed formulas of the asymptotic smile. These methods facilitate the calibration procedure and allow an interesting analysis of the parameters. The second part, from chapter 4 to chapter 6, studies mortality and longevity risks. We first recall the general concepts of longevity risk and a set of issues underlying this risk. We then present a model of individual mortality that takes into account age and other characteristics of the individual that are explanatory of mortality. We calibrate the mortality model and analyze the influence of certain individual characteristics. Finally, we introduce a microscopic population dynamics model that allows us to model the evolution over time of a population structured by age and traits. Each individual evolves over time and is likely to give birth to a child, change characteristics and die. This model accounts for the possibly stochastic evolution of individual demographic rates over time. We also describe a micro/macro link that provides this microscopic model with good macroscopic properties. The third part, concerning chapters 7 and 8, focuses on applications of the previous modeling. The first application is a demographic one since the microscopic population dynamics model allows us to make demographic projections of the French population. We also set up a demographic study of the pension problem by analyzing the solutions of an immigration policy and a reform of the retirement age. The second application concerns the study of life insurance products combining longevity and interest rate risks that have been studied in detail in the first two parts of the thesis. First, we study the basic risk generated by the heterogeneity of annuity portfolios. In addition, we introduce the Life Nominal Chooser Swaption (LNCS) which is a risk transfer product for life insurance products: this product has a very interesting structure and allows an insurance company holding an annuity portfolio to transfer its entire interest rate risk to a bank.