A Quantitative Comparison of the Lee-Carter Model under Different Types of Non-Gaussian Innovations

Abstract

In the classical Lee-Carter model, the mortality indices that are assumed to be a random walk model with drift are normally distributed. However, for the long-term mortality data, the error terms of the Lee-Carter model and the mortality indices have tails thicker than those of a normal distribution and appear to be skewed. This study therefore adopts five nonGaussian distributions-Student's t-distribution and its skew extension (i.e., generalised hyperbolic skew Student's t-distribution), one finite-activity Le´vy model (jump diffusion distribution), and two infinite-activity or pure jump models (variance gamma and normal inverse Gaussian)-to model the error terms of the Lee-Carter model. With mortality data from six countries over the period 1900-2007, both in-sample model selection criteria (e.g., Bayesian information criterion, Kolmogorov-Smirnov test, Anderson-Darling test, Crame´r-von-Mises test) and out-of-sample projection errors indicate a preference for modelling the Lee-Carter model with non-Gaussian innovations.

Modelling longevity bonds: Analysing the Swiss Re Kortis bond

“…We find that using normally distributed innovations reduces the probability of the principal being reduced to 2.5%, whilst the probability of complete exhaustion of the principal was found to be 0.8%. This indicates that using non-normal distributions for time series innovations, as in Wang et al (2011), may be desirable, especially when trying to model extreme mortality scenarios. The Kortis bond is important and interesting because it is the prototype of a new breed of longevity bond indexed to the rates of improvement observed in mortality in different populations rather than in mortality rates directly (in the case of, say, the Vita bond) or in the survivorship of a cohort (in the case of the original survivor bond proposed by Blake and Burrows (2001) and marketed unsuccessfully by the European Investment Bank and BNP Paribas in 2004).…”

Mortality Modeling With Non‐Gaussian Innovations and Applications to the Valuation of Longevity Swaps

“…The NIG distribution is one of the most promising distributions for asset returns proposed in the prior literature, with several attractive theoretical properties and analytical tractability. It therefore has been used repeatedly for financial applications as the unconditional return distribution (Eberlein and Keller, ; Prause, ; Rydberg, ; Bølviken and Benth, ; Lillestøl, ) and for stochastic mortality modeling (Giacometti, Ortobelli, and Bertocchi, ; Wang, Huang, and Liu, ).…”

“…Giacometti, Ortobelli, and Bertocchi () employ the normal inverse Gaussian (NIG) distribution to model both the error distributions of the LC model, observing that the NIG distributional assumption for the residuals of the LC model is better than the Gaussian one for certain age groups. Wang, Huang, and Liu () fit the LC model with heavy‐tailed distributions to mortality rates from 1900 to 1999 and demonstrate that for applications of the LC model, the heavy‐tailed distributions appear to be the most appropriate choices for modeling long‐term mortality data. However, as proposed by Pitacco (), various disadvantages arise in connection with the LC model.…”

Two Factor Stochastic Mortality Modeling with Generalized Hyperbolic Distribution

“…Some authors try to add jump processes to model mortality, mostly incorporated in the Lee-Carter framework. See for example, Wang et al (2011), Giacometti et al (2009) and Hainaut and Devolder (2008). Over relatively short period of time, longevity has become a considerable risk in countries like Japan and Taiwan.…”