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The aim of this paper is to construct a dynamic programming algorithm for pricing variable annuities with GLWB under a stochastic mortality framework. Although our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes, in the numerical implementation of the algorithm we model the former as a non mean reverting square root process, and the latter as an exponential Lévy process. In this way we get a tractable and flexible stochastic model for efficient pricing and risk management of the GLWB. Another contribution of our paper is the verification, through backward induction, of the bang-bang condition for the set of discrete withdrawal strategies of the model. This result is particularly remarkable as in the insurance literature either the existence of optimal bang-bang controls is assumed or it requires suitable conditions. We present extensive numerical examples and compare the results obtained for different market parameters and policyholder behaviours.

In this paper we focus on the initiation option featured in many Guaranteed Lifelong Withdrawal Benefit variable annuity contracts, granting their owner the right to decide the age at which lifetime withdrawals should begin. Such contracts have been successfully analysed using a PDE approach. The latter method is elegant, but becomes less viable when the valuation model is more involved and other guarantees are considered. As an alternative, we exploit the Least Squares Monte Carlo approach and model the interaction of the initiation option with lapses and other riders.

In this paper we propose a methodology for valuing future annuity contracts based on the Least-Squares Monte Carlo approach. We adopt, as first step, a simplified computational framework where just one risk factor is taken into account, and then we extend it introducing other sources of risk. We give a brief description of the valuation procedure and provide some numerical illustrations. Furthermore, to test the efficiency of the proposed methodology, we compare our results with those obtained by applying a straightforward and time-consuming approach based on nested simulations. Finally, we present some possible applications in the context of de-risking strategies for pension plans and in the valuation of guaranteed annuity options.

In this paper we study how policyholders and equityholders contribute to the formation of a life insurance company issuing participating contracts. The structure of these contracts is stylized and features a guaranteed rate of return and a terminal bonus, as in the pioneering model by Briys and de Varenne (1994, 1997). Policyholders aim at maximizing their preferences by choosing the leverage ratio and the guaranteed level, while being subject to regulatory constraints of fair valuation and solvency. We provide conditions under which non trivial contracts exist and analyze their properties.

In this paper we assess the joint impact of biometric and financial risk on the market valuation of life insurance liabilities. We consider a stylized, contingent claim based model of a life insurance company issuing participating contracts and subject to default risk, as pioneered by Briys and de Varenne (1994, 1997), and build on their model by explicitly introducing biometric risk and its components, namely diversifiable and systematic risk. The contracts considered include pure endowments, deferred whole life annuities and guaranteed annuity options. Our results stress the predominance of systematic over diversifiable risk in determining fair participation rates. We investigate the interaction of contract design, market regimes and mortality scenarios, and show that, particularly for lifelong benefits, usually offered participation rates may not be sustainable even under moderate longevity improvements.

In this paper we consider a variable annuity which provides guarantees at death and maturity financed through the application of a state-dependent fee structure, as defined first in Bae and Ko (2013) and extensively analysed in Bernard et al. (2014) and MacKay et al. (2017). We propose a quite general valuation model for such guarantees, along the lines of Bacinello et al. (2011). We then analyse numerically the interaction between fee rates, death/maturity guarantees, fee thresholds and surrender penalties under alternative model assumptions and policyholder behaviours. This allows us to get also some interesting insights into the model risk. Since the assumptions adopted in the numerical analysis are not at all trivial, we resort to Monte Carlo and Least Squares Monte Carlo methods (LSMC) for the numerical implementation of the valuation model. In particular, special care is needed in the application of LSMC, due to the shape of the surrender region. We are able to stem the numerical errors arising in the regression step by using suitable arrangements of the LSMC valuation algorithm.

In this paper we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different levels of rationality in the policyholder’s withdrawal behaviour can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours, and different types of Lévy processes.

Variable Annuities (VA) package living and death benefits, eventually aiming at constructing a post-retirement income, while offering a number of possible guarantees in respect of financial and mortality/longevity risks. In many circumstances, the policyholder keeps access to her fund, and can in particular cash money beyond the benefits already set. Apart from financial and mortality/longevity risks, the insurer is therefore exposed also to the risk of unexpected decisions of the policyholder. We suggest to adopt a comprehensive approach to these different risk sources, and we describe a unifying valuation framework of the guarantees under quite general assumptions.

This article gradually highlights the main problems that an insurance company has to face when issuing complex life insurance contracts with embedded options, and presents the basic principles that can be applied in order to price them. The article is based on some conferences that the author gave as invited speaker at the Austrian Workshop on Asset Liability Management in Insurance (Wien, 2004), the Workshop on Life Insurance Fair Valuation (Lyon, 2005), the Annual Meeting of the Swiss Association of Actuaries (Lausanne, 2006), the 9th Spanish-Italian Congress of Financial and Actuarial Mathematics (Alcalá de Henares, 2006), and at the University of Ljubljana (2007).