Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

Abstract

This paper assesses the hedge effectiveness of an index-based longevity swap and a longevity cap. Although swaps are a natural instrument for hedging longevity risk, derivatives with non-linear pay-offs, such as longevity caps, also provide downside protection. A tractable stochastic mortality model with age dependent drift and volatility is developed and analytical formulae for prices of these longevity derivatives are derived. Hedge effectiveness is considered for a hypothetical life annuity portfolio. The hedging of the life annuity portfolio is comprehensively assessed for a range of assumptions for the market price of longevity risk, the term to maturity of the hedging instruments, as well as the size of the underlying annuity portfolio. The model is calibrated using Australian mortality data. The results provide a comprehensive analysis of longevity hedging, highlighting the risk management benefits and costs of linear and nonlinear payoff structures.

Pricing participating longevity-linked life annuities: a Bayesian Model Ensemble approach

“…For pricing purposes, we need to consider the market price of longevity risk. Since the underlying longevity index is not an existing tradable asset in a liquid market, we use a distortion operator to create an equivalent risk-adjusted probability distribution for or to compute the fair value of the derivative security, an approach recommended when pricing long-term contracts [ 3 , 36 ]. 14 To be more specific, we use the flexible risk-neutral simulation approach proposed by Boyer and Stentoft [ 6 ] using the classical Wang transform as a risk measure [ 70 – 72 ].…”

Pricing longevity derivatives via Fourier transforms

“…The valuation of longevity-linked derivatives with non-linear payoff structures (e.g., longevity options) has received little attention in the literature compared to that of longevity derivatives with a linear payoff but is attracting increasing interest. Some exceptions are Lin and Cox (2007), who study the pricing of a longevity call option linked to a population longevity index for older ages, Cui (2008), who discusses the valuation of longevity options (floors and caps) using the Equivalent Utility Pricing Principle, Dawson et al (2010), who derive closed-form Black-Scholes-Merton-type prices for European swaptions, Boyer and Stentoft (2013), who price European and American type survivor options using a risk neutral simulation approach, Wang and Yang (2013), who price survivor floors using an extension of the Lee-Carter model, Yueh et al (2016), who develop valuation models for mortality calls and puts-employing the jump-diffusion model developed by Cox et al (2006)-, Bravo andde Freitas (2018) and Bravo (2019Bravo ( , 2020, who discuss the valuation of longevity options embedded in longevity-linked life annuities, Fung et al (2019), who derive closed-form solutions for the price of longevity caps under a two-factor Gaussian mortality model resembling the Black-Scholes formula for option pricing when the underlying stock price follows a geometric Brownian motion, and Li et al (2019) as well as Cairns and Boukfaoui (2019), who discuss the valuation of K-options and call-spreads, respectively.…”

“…In the literature, several continuous-time stochastic mortality models have been proposed for modelling the dynamics of mortality rates-see, for example, Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Cairns et al (2006aCairns et al ( ,b, 2008, Biffis and Millossovich (2006), Schräger (2006), Miltersen and Persson (2006), Ballotta and Haberman (2006), Luciano and Vigna (2008), Bravo (2007Bravo ( , 2011, Zhu and Bauer (2011), Luciano et al (2012), Fung et al (2019) and references therein. 3 The task of modelling (and managing) longevity risk is challenging since the nature of the risk addressed is multivariate, encompassing mortality trend uncertainty, longevity diffusion risk, mortality jump risk, as well as model and parameter risk.…”

Pricing Survivor Bonds with Affine-Jump Diffusion Stochastic Mortality Models

“…In this paper, we use a risk-neutral valuation approach to incorporate the market price of longevity risk. 1 In the actuarial, financial, and demographic literature, several single and multiple-population continuous-time stochastic mortality models have been proposed for modelling the dynamics of mortality rates (see, e.g., [7,[13][14][15][16][17] and references therein), along with several individual discrete-time extrapolative models (see, e.g., [18][19][20][21] and references therein) and, more recently, model combinations [22-24, 36-39, 44]. In this paper, we follow [7] and use a non-mean reverting square-root jump-diffusion Feller process combined with a Poisson process with double asymmetric exponentially distributed jumps [27] to account for both negative (e.g., medical breakthroughs) and positive jumps (e.g., pandemics) of different sizes.…”