A Study of Statistical Methods for Modelling Longevity and Climate Risks

Abstract

In recent years, two pivotal risks have emerged and taken a significant position in modern actuarial science: longevity risk and climate risk. Longevity risk, or the risk of individuals living longer than expected, poses a severe challenge to both private insurance companies and public pension systems, potentially destabilizing financial structures built on assumptions of life expectancy. On the other hand, climate risk, associated with fluctuations and extreme conditions in weather, has substantial implications for various sectors such as agriculture, energy, and insurance, particularly in the era of increasing climate change impacts. The Society of Actuaries (SOA) has recognized the growing importance of these risks, advocating for innovative research and solutions to manage them effectively. Furthermore, statistical modelling plays an indispensable role in understanding, quantifying, and managing these risks. The development of sophisticated and robust statistical methods enables practitioners and researchers to capture complex risk patterns and make reliable predictions, thereby informing risk management strategies. This thesis, composed of four distinct projects, explores statistical methods for modelling longevity and weather risk, contributing valuable insights to these fields.

The first part in this thesis studies the statistical methods for modelling longevity risk, and in particular, modelling mortality rates. In the first chapter, we study parameter estimation of the Lee-Carter model and its multi-population extensions. Although the impact of outliers on stochastic mortality modelling has been examined, previous studies on this topic focus on how outliers in the estimated time-varying indexes may be detected and/or modelled, with little attention being paid to the adverse effects of outliers on estimation robustness, particularly that pertaining to age-specific parameters. In this chapter, we propose a robust estimation method for the Lee-Carter model, through a reformulation of the model into a probabilistic principal component analysis with multivariate t-distributions and an efficient expectation-maximization algorithm for implementation. The proposed method yields significantly more robust parameter estimates, while preserving the fundamental interpretation for the bilinear term in the model as the first principal component and the flexibility of pairing the estimated time-varying parameters with any appropriate time-series process. We also extend the proposed method for use with multi-population generalizations of the Lee-Carter model, allowing for a wider range of applications such as quantification of population basis risk in index-based longevity hedges. Using a combination of real and pseudo datasets, we demonstrate that the superiority of the proposed method relative to conventional estimation approaches such as singular value decomposition and maximum likelihood.

Next, we move onto parameter estimation of the Renshaw-Haberman model, a cohort-based extension to the Lee-Carter model. In mortality modelling, cohort effects are often taken into consideration as they add insights about variations in mortality across different generations. Statistically speaking, models such as the Renshaw-Haberman model may provide a better fit to historical data compared to their counterparts that incorporate no cohort effects. However, when such models are estimated using an iterative maximum likelihood method in which parameters are updated one at a time, convergence is typically slow and may not even be reached within a reasonably established maximum number of iterations. Among others, the slow convergence problem hinders the study of parameter uncertainty through bootstrapping methods. In this chapter, we propose an intuitive estimation method that minimizes the sum of squared errors between actual and fitted log central death rates. The complications arising from the incorporation of cohort effects are overcome by formulating part of the optimization as a principal component analysis with missing values. Using mortality data from various populations, we demonstrate that our proposed method produces satisfactory estimation results and is significantly more efficient compared to the traditional likelihood-based approach.

The third part of this thesis continues our exploration of the efficient computational algorithm of the Renshaw-Haberman model. Existing software packages and estimation algorithms often rely on maximum likelihood estimation with iterative Newton-Raphson methods, which can be computationally intensive and prone to convergence issues. In this chapter, we present the R package RHals, offering an efficient alternative with an alternating least squares method for fitting a generalized class of Renshaw-Haberman models, including configurations with multiple age-period terms. We extend this method to multi-population settings, allowing for shared or population-specific age effects under various configurations. The full modelling workflow and functionalities of RHals are demonstrated using mortality data from England and Wales.

Lastly, we turn to modelling climate risk in the final chapter of the thesis. The use of weather index insurances is subject to spatial basis risk, which arises from the fact that the location of the user's risk exposure is not the same as the location of any of the weather stations where an index can be measured. To gauge the effectiveness of weather index insurances, spatial interpolation techniques such as kriging can be adopted to estimate the relevant weather index from observations taken at nearby locations. In this chapter, we study the performance of various statistical methods, ranging from simple nearest neighbor to more advanced trans-Gaussian kriging, in spatial interpolations of daily precipitations with data obtained from the US National Oceanic and Atmospheric Administration. We also investigate how spatial interpolations should be implemented in practice when the insurance is linked to popular weather indexes including annual consecutive dry days (CDD) and maximum five-day precipitation in one month (MFP). It is found that although spatially interpolating the raw weather variables on a daily basis is more sophisticated and computationally demanding, it does not necessarily yield superior results compared to direct interpolations of CDD/MFP on a yearly/monthly basis. This intriguing outcome can be explained by the statistical properties of the weather indexes and the underlying weather variables.