Academic journal article Journal of Risk and Insurance

Mortality Modeling with Non-Gaussian Innovations and Applications to the Valuation of Longevity Swaps

Academic journal article Journal of Risk and Insurance

Mortality Modeling with Non-Gaussian Innovations and Applications to the Valuation of Longevity Swaps

Article excerpt

Abstract

This article provides an iterative fitting algorithm to generate maximum likelihood estimates under the Cox regression model and employs nonGaussian distributions-the jump diffusion (JD), variance gamma (VG), and normal inverse Gaussian (NIG) distributions-to model the error terms of the Renshaw and Haberman (2006) (RH) model. In terms of mean absolute percentage error, the RH model with non-Gaussian innovations provides better mortality projections, using 1900-2009 mortality data from England and Wales, France, and Italy. Finally, the lower hedge costs of longevity swaps according to the RH model with non-Gaussian innovations are not only based on the lower swap curves implied by the best prediction model, but also in terms of the fatter tails of the unexpected losses it generates.

Introduction

Longevity represents an increasingly important risk for defined benefit pension plans and annuity providers, because life expectancy is dramatically increasing in developed countries. In 2007, exposures to improved life expectancy amounted to $400 billion for pension funds and insurance companies in the United Kingdom and United States (see Loeys, Panigirtzoglou, and Ribeiro, 2007). Stochastic mortality models quantify mortality and longevity risks, which makes mortality risk management possible and provides the foundation for pricing and reserving. Among all stochastic mortality models, the Lee-Carter (LC) model, proposed in 1992, is one of the most popular choices because of its ease of implementation and acceptable prediction errors in empirical studies. Various modifications of the LC model have been extended by Brouhns, Denuit, and Vermunt (2002), Renshaw and Haberman (2003, 2006), Cairns, Blake, and Dowd (2006), Li and Chan (2007), Biffis, Denuit, and Devolder (2010), and Hainaut (2012) to attain a broader interpretation. Cairns, Blake, and Dowd (2006) propose a two-factor stochastic mortality model, the CBD model, in which a first factor affects mortality at all ages, whereas a second factor affects mortality at older ages much more than at younger ages. Modeling the number of deaths with the Poisson model, Cairns et al. (2009) classify and compare eight stochastic mortality models, including an extension of the CBD model, with mortality data from England and Wales and the United States. They find that an extension of the CBD model that incorporates the cohort effect fits the English and Welsh data best, whereas for the U.S. data, the Renshaw and Haberman (2006) (RH) model, which also allows for a cohort effect, provides the best fit (Cairns et al., 2009). In addition to the cohort effect, short-term catastrophic mortality events, such as the influenza pandemic in 1918 and the Tsunami in December 2004, may lead to much higher mortality rates. Using empirical data from 1900 to 1984, we find that the residuals in the RH model for England and Wales, France, and Italy exhibit leptokurticity. It is crucial to address such mortality jumps in age-period-cohort mortality models. The main goal of this study is to incorporate non-Gaussian innovations into the RH model.

To take heavy-tailed distributions into account in stochastic mortality models, Milidonis, Lin, and Cox (2011) use a Markov regime-switching model to analyze the 1901-2005 U.S. population mortality data and price mortality securities. In contrast, Biffis (2005) employs affine jump diffusions to model asset prices and mortality dynamics and thus addresses the risk analysis and market valuation of life insurance contracts. For Italian mortality data, Luciano and Vigna (2005) demonstrate that a diffusion process with a jump component (JD) provides a better fit than does a diffusion component in stochastic mortality processes. Cox, Lin, and Wang (2006) employ the JD process to model age-adjusted mortality rates for the United States and United Kingdom and to evaluate the first pure mortality security: the Swiss Re Vita bond. In addition, Lin and Cox (2008) combine a Brownian motion and a discrete Markov chain with a log-normal jump size distribution to price mortality-based securities in an incomplete market framework. …

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