A Two-Factor Model for Stochastic Mortality With Parameter Uncertainty: Theory and Calibration

By Andrew J. G. Cairns, David P. Blake, Kevin Dowd. 2006

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In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly. A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.

Subfields: Demography, General Health Professions, Health

View Paper: https://www.semanticscholar.org/paper/99b05fbc83adb7bc8488c2451e3114a73ac3cfb8

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Citations in the corpus

Mortality surface by means of continuous time cohort models (2013, methodology): “Continuous-time stochastic mortality models for single generation were considered by a number of researchers, including Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Cairns et al. (2006a), Schrager (2006) and Luciano and Vigna (2008).”
Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives (2011, methodology): “The logit transform was chosen based on Cairns et al (2006) [14] and observation of the data.”
On the pricing of longevity-linked securities (2010, methodology): “…questionable.4 Moreover, it is not clear whether annuity prices offer an adequate starting point when pricing longevity derivatives, and it is unclear “how different transforms for different cohorts and terms to maturity relate to one another and form a coherent whole” (cf. Cairns et al. (2006a)).”
Multi-population mortality models: A factor copula approach (2015, neutral): “Hua Chen, Richard MacMinn and Tao Sun Pricing and Hedging Pricing methodologies Arbitrage free pricing (Cairns et al. 2006b, Bauer et al. 2010).”
A comparative study of parametric mortality projection models (2011, methodology): “The 4 structures M5–M8 are due to Cairns et al. (2008a,b,c; 2009). We note that a re-parameterised version of M5 features prominently in Cairns et al. (2006). The 4 extensions, M5*–M8* are motivated by Plat (2009) who specifically introduces M6*. We note that the Cairns et al. (2009) structures, den”
Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities (2006, background): “For further discussion of this risk premium, the reader is referred to Cairns, Blake, Dawson & Dowd (2005) and Cairns, Blake & Dowd (2005a) .”
Demographic Issues in Longevity Risk Analysis (2006, neutral): “Actuarial (Cox, Lin, and Wang, 2006; Cairns, Blake, and Dowd, 2006 ), economic (Turner, 2006; Brown and Orszag, 2006), and financial issues (Blake et al., 2006; Milevsky, Promislow, and Young, 2006) are addressed elsewhere in this issue of the Journal of Risk and Insurance.”
Understanding, modelling and managing longevity risk: key issues and main challenges (2012, methodology): “In recent contemporary studies, many authors have introduced stochastic models to capture the cohort effect (see e.g. Cairns et al. (2006, 2009)); this subsection, briefly presents some of them.”
Stochastic mortality under measure changes (2010, positive): “The calibration approach is adopted, for example, by Cairns et al. (2006a), who restrict their attention to the case f 0.”
Measuring Basis Risk in Longevity Hedges (2011, methodology): “12 Calibration of q-forward hedges is described in Cairns et al. (2009) and Coughlan (2009). In particular, Cairns et al.”
Pricing Mortality Securities with Correlated Mortality Indexes (2011, positive): “As highlighted in Cairns, Blake, and Dowd (2006), a model that takes into account parameter uncertainty is able to generate more reliable forecasts.”
The New Life Market (2013, neutral): “(2011b) and Cairns et al. (2011b).56 Suppose the aim is to forecast the mortality of the members of a small pension plan, but both the quantity and quality of the mortality data are poor.”
A step-by-step guide to building two-population stochastic mortality models (2015, methodology): “It is therefore warranted in future research to include parameter uncertainty in the model-building procedure, possibly by using Bayesian approaches (see Cairns et al., 2006, 2011) or bootstrapping methods (see Li, 2010; Renshaw and Haberman, 2008).”
Modelling and management of longevity risk: Approximations to survivor functions and dynamic hedging (2011, methodology): “…and cohort effects, and models that work with the log of the death rate, log m(t, x), (e.g. Lee and Carter, 1992) rather than logit q(t, x) as in Cairns et al. (2006a).5 Additionally, most of the commonly used stochastic mortality models share the characteristic with the CBD two-factor model…”