Abstract
Stochastic mortality models have been developed for a range of applications from demographic projections to financial management. Financial risk based models built on methods used for interest rates and apply these to mortality rates. They have the advantage of being applied to financial pricing and the management of longevity risk. Olivier and Jeffery (2004) and Smith (2005) proposed a model based on a forward-rate mortality framework with stochastic factors driven by univariate gamma random variables irrespective of age or duration. We assess and further develop this model. We generalize random shocks from a univariate gamma to a univariate Tweedie distribution and allow for the distributions to vary by age. Furthermore, since dependence between ages is an observed characteristic of mortality rate improvements, we formulate a multivariate framework using copulas. We find that dependence increases with age and introduce a suitable covariance structure, one that is related to the notion of ax minimum. The resulting model provides a more realistic basis for capturing the risk of mortality improvements and serves to enhance longevity risk management for pension and insurance funds.
Highlights
A variety of empirical studies across many developed nations show that mortality trends have been improving stochastically; see, e.g., CMI (2005), Luciano and Vigna (2005), Blake et al (2006), Liu (2008)and Blackburn and Sherris (2013)
Numerous stochastic mortality models proposed in the literature apply extensions of interest rate term structure modelling, known as short-rate models
Bauer and Ruß (2006) demonstrate that, if mortality risk can be traded through securities such as longevity bonds and swaps, the techniques developed in financial markets for pricing bonds and swaps can be adapted for mortality risk
Summary
A variety of empirical studies across many developed nations show that mortality trends have been improving stochastically; see, e.g., CMI (2005), Luciano and Vigna (2005), Blake et al (2006), Liu (2008). Biffis (2005), Russo et al (2010) and Blackburn and Sherris (2013) develop a variety of affine frameworks extended from interest rate term structure modelling. Risks 2019, 7, 61 apply a forward-rate mortality framework with stochastic factors driven by univariate gamma random variables irrespective of age or duration. In addition to restricting the stochastic factors to identical gamma distributions, another critical assumption is that of independence amongst these factors across age. We examine these assumptions using England and Wales female mortality data for 1960 to 2009; a dataset used in Cairns (2007). Since dependence between ages is an observed characteristic of mortality rate improvements, we further generalize the model by formulating a multivariate distribution that incorporates age dependence using copulas.
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