Incorporating crossed classification credibility into the Lee–Carter model for multi-population mortality data

Recent developments in actuarial literature have shown that credibility theory can serve as an effective tool in mortality modelling, leading to accurate forecasts when applied to single or multi-population datasets. This paper presents a crossed classification credibility formulation of the Lee–Carter method particularly designed for multi-population mortality modelling. Differently from the standard Lee–Carter methodology, where the time index is assumed to follow an appropriate time series process, herein, future mortality dynamics are estimated under a crossed classification credibility framework, which models the interactions between various risk factors (e.g. genders, countries). The forecasting performances between the proposed model, the original Lee–Carter model and two multi-population Lee–Carter extensions are compared for both genders of multiple countries. Numerical results indicate that the proposed model produces more accurate forecasts than the Lee–Carter type models, as evaluated by the mean absolute percentage forecast error measure. Applications with life insurance and annuity products are also provided and a stochastic version of the proposed model is presented.     

The age pattern of transitory mortality jumps and its impact on the pricing of catastrophic mortality bonds

To value catastrophic mortality bonds, a number of stochastic mortality models with transitory jump effects have been proposed. Rather than modeling the age pattern of jump effects explicitly, most of the existing models assume that the distributions of jump effects and general mortality improvements across ages are identical. Nevertheless, this assumption does not seem to be in line with what we observe from historical data. In this paper, we address this problem by introducing a Lee–Carter variant that captures the age pattern of mortality jumps by a distinct collection of parameters. The model variant is then further generalized to permit the age pattern of jump effects to vary randomly. We illustrate the two proposed models with mortality data from the United States and English and Welsh populations, and use them to value hypothetical mortality bonds with similar specifications to the Atlas IX Capital Class B note that was launched in 2013. It is found that the features we consider have a significant impact on the estimated prices.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

Pitfalls and merits of cointegration-based mortality models

In recent years, joint modelling of the mortality of related populations has received a surge of attention. Several of these models employ cointegration techniques to link underlying factors with the aim of producing coherent projections, i.e. projections with non-diverging mortality rates. Often, however, the factors being analysed are not fully identifiable and arbitrary identification constraints are (inadvertently) allowed to influence the analysis thereby compromising its validity. Taking the widely used Lee–Carter model as an example, we point out the limitations and pitfalls of cointegration analysis when applied to semi-identifiable factors. On the other hand, when properly applied cointegration theory offers a rigorous framework for identifying and testing long-run relations between populations. Although widely used as a model building block, cointegration as an inferential tool is often overlooked in mortality analysis. Our aim with this paper is to raise awareness of the inferential strength of cointegration and to identify the time series models and hypotheses most suitable for mortality analysis. The concluding application to UK mortality shows by example the insights that can be obtained from a full cointegration analysis.     

Hedging mortality/longevity risks of insurance portfolios for life insurer/annuity provider and financial intermediary

In this paper, we propose two risk hedge schemes in which a life insurer (an annuity provider) can transfer mortality (longevity) risk of a portfolio of life (annuity) exposures to a financial intermediary by paying the hedging premium of a mortality-linked security. The optimal units of the mortality-linked security which maximize hedge effectiveness for a life insurer (an annuity provider) can be derived as closed-form formulas under the risk hedge schemes. Numerical illustrations show that the risk hedge schemes can significantly hedge the downside risk of loss due to mortality (longevity) risk for the life insurer (annuity provider) under some stochastic mortality models. Besides, finding an optimal weight of a portfolio of life and annuity business, the financial intermediary can reduce the sensitivity to mortality rates but the model risk; a security loading may be imposed on the hedge premium for a higher probability of gain to compensate the financial intermediary for the inevitable model risk.     

Estimation of time-varying mortality rates using continuous models for Daphnia magna

Structured population models that make the assumption of constant demographic rates do not accurately describe the complex life histories seen in many species. We investigated the accuracy of using constant versus time-varying mortality rates within discrete and continuously structured models for Daphnia magna. We tested the accuracy of the models we considered using density-independent survival data for 90 daphnids. We found that a continuous differential equation model with a time-varying mortality rate was the most accurate model for describing our experimental D. magna survival data. Our results suggest that differential equation models with variable parameters are an accurate tool for estimating mortality rates in biological scenarios in which mortality might vary significantly with age.