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Selection effect modification to the Lee-Carter model

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Abstract

Although other risk factors can be used, depending on feasibility, marketing, and data availability, age and gender are the two most common risk factors considered in life insurance products. Previous studies have shown that the newly insured, who passed certain health examinations, tend to have lower mortality rates than those already insured. Insurance companies often use select and ultimate tables to handle mortality discrepancies between the insured in different policy years (i.e., the selection effect). However, the selection effect is easily confused with mortality improvement, and its estimate is likely to be influenced by the annual reduction in mortality rates. In this study, we propose modifying the Lee-Carter model, including the selection effect and mortality improvement. We first use a simulation to evaluate the parameter estimation of the proposed approach and then apply it to experienced data from Taiwan’s largest insurance company, Cathay Life Insurance Company Ltd. The results of our simulation and empirical studies support the newly proposed approach, which provides stable and accurate estimates of the selection effect and mortality improvement. We also find that the size of the selection effect concerning policy year was larger than the difference in mortality rates between smokers and non-smokers; this is particularly noticeable for older age groups.

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Availability of data and material

The empirical data are the property of the Cathay Life Insurance Co. and are not open to the public.

Code availability

The code is open source and is written in R statistical software.

Notes

  1. In 2011, the European Court of Justice ruled that using gender to calculate premiums and benefits was inconsistent with the European Charter.

  2. Conversely, the selection effect would be overestimated if the insured were ex-smokers.

  3. Mortality improvement is an important factor for modeling mortality rates today; however, it was not considered until the late 1990s.

  4. IFRS 17 is an International Financial Reporting Standard, and is expected to be effective in 2023. In 2021, the Taiwan government decided to delay the effective date to 2026.

  5. The selection effect can also exist in health insurance products. For example, cancer insurance is a popular product in Taiwan, and insurance claims occur when the insured is diagnosed with cancer for the first time. Thus, we can plug the incidence rates into the population's proposed model to evaluate a selection effect. Note that the proportion of initial disease cases is defined as an incidence rate.

  6. We also tried singular value decomposition for the parameter estimation, and the results were similar.

  7. The estimates of parameters \({\alpha }_{x}\), \({\beta }_{x}\), \({\kappa }_{t}\), and \({C}_{xs}\) are derived from the “StMoMo” package. The estimates obtained from the “ilc” package are extremely similar to those from “StMoMo.” The \({\alpha }_{x}\) estimates from “ilc” are also negatively biased at the 1st iteration.

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Authors and Affiliations

  1. Department of Statistics, National Chengchi University, Taipei, 11605, Taiwan, ROC

    Jack C. Yue

  2. Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan, ROC

    Jack C. Yue

  3. Cathay Life Insurance Co., Ltd., Taipei, 10693, Taiwan, ROC

    Chao-Ting Lin, Yu-Lin Yang, Yi-Chun Chen & Wan-Chen Tsai

  4. Department of Risk Management and Insurance, Feng Chia University, Taichung, 40724, Taiwan, ROC

    Yin-Yee Leong

Authors
  1. Jack C. Yue
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Appendices

Appendix: Estimates and Monte Carlo Confidence Intervals of \({\alpha }_{x}\), \({\beta }_{x}\), and \({\kappa }_{t}\) (Simulation)

Bias of \(\alpha_{x}\) Estimates

See Fig. 7

Fig. 7
figure 7

Bias of Estimates for Parameter \(\alpha_{x}\)(1st and 15th iteration)

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Bias of βx Estimates

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Fig. 8
figure 8

Bias of Estimates for Parameter βx (1st and 15th iteration)

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8

Bias of \(\kappa_{t}\) Estimates

See Fig. 

Fig. 9
figure 9

Bias of Estimates for Parameter \(\kappa_{t}\)(1st and 15th iteration)

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9

Estimates of Selection Effect

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Fig. 10
figure 10

Estimates of Selection Effect (15th iteration)

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10

Parameters \(\alpha_{x}\),\(\beta_{x}\), and \(\kappa_{t}\) with the Monte Carlo 95% Confidence Intervals

See Tables

Table 8 Parameters \(\alpha_{x}\) and \(\beta_{x}\) with their Monte Carlo 95% Confidence Intervals
Full size table

8 and

Table 9 Parameters \(\kappa_{t}\) with their Monte Carlo 95% Confidence Intervals
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9

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Yue, J.C., Lin, CT., Yang, YL. et al. Selection effect modification to the Lee-Carter model. Eur. Actuar. J. 13, 213–234 (2023). https://doi.org/10.1007/s13385-022-00312-6

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  • DOI: https://doi.org/10.1007/s13385-022-00312-6

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