ESTIMATION AND UNIQUENESS OF THE GOMPERTZ FORCE OF MORTALITY IN TERMS OF THE MODAL AGE AT DEATH

The initial level of mortality and the rate at which mortality rises with age are generally expressed in terms of the Gompertz force of mortality (hazard function). In their paper, James W. Vaupel and others define the Gompertz force of mortality as the rate at which mortality rises with age and the modal age at death. In this paper we estimate the Gompertz force of mortality and prove uniqueness theorem.

The Negative Impact of COVID-19 on Life Insurers

Understanding COVID-19 induced mortality risk is significant for life insurers to better analyze their financial sustainability after the outbreak of COVID-19. To capture the mortality effect caused by COVID-19 among all ages, this study proposes a temporary adverse mortality jump model to describe the dynamics of mortality in a post-COVID-19 pandemic world based on the weekly death numbers from 2015 to 2021 in the United States. As a comparative study, the Lee-Carter model is used as the base case to represent the dynamics of mortality without COVID-19. Then we compare the force of mortality, the survival probability and the liability of a life insurer by considering COVID-19 and those without COVID-19. We show that a life insurer's financial sustainability will deteriorate because of the higher mortality rates than expected in the wake of COVID-19. Our results remain unchanged when we also consider the effect of interest rate risk by adopting the Vasicek and CIR models.

Modelling Frontier Mortality Using Bayesian Generalised Additive Models

Mortality rates differ across countries and years, and the country with the lowest observed mortality has changed over time. However, the classic Science paper by Oeppen and Vaupel (2002) identified a persistent linear trend over time in maximum national life expectancy. In this article, we look to exploit similar regularities in age-specific mortality by considering for any given year a hypothetical mortality ‘frontier’, which we define as the lower limit of the force of mortality at each age across all countries. Change in this frontier reflects incremental advances across the wide range of social, institutional and scientific dimensions that influence mortality. We jointly estimate frontier mortality as well as mortality rates for individual countries. Generalised additive models are used to estimate a smooth set of baseline frontier mortality rates and mortality improvements, and country-level mortality is modelled as a set of smooth, positive deviations from this, forcing the mortality estimates for individual countries to lie above the frontier. This model is fitted to data for a selection of countries from the Human Mortality Database (2019). The efficacy of the model in forecasting over a ten-year horizon is compared to a similar model fitted to each country separately.

Human mortality at extreme age

We use a combination of extreme value statistics, survival analysis and computer-intensive methods to analyse the mortality of Italian and French semi-supercentenarians. After accounting for the effects of the sampling frame, extreme-value modelling leads to the conclusion that constant force of mortality beyond 108 years describes the data well and there is no evidence of differences between countries and cohorts. These findings are consistent with use of a Gompertz model and with previous analysis of the International Database on Longevity and suggest that any physical upper bound for the human lifespan is so large that it is unlikely to be approached. Power calculations make it implausible that there is an upper bound below 130 years. There is no evidence of differences in survival between women and men after age 108 in the Italian data and the International Database on Longevity, but survival is lower for men in the French data.

Does the Risk of Death Continue to Rise Among Supercentenarians?

A study published in the Society of Actuaries’ 2017 Living to 100 Monograph suggests, in contrast to previous research, that the risk of death after age 110 years increases with age. By fitting a Gompertz model to estimated central death rates for the oldest old, the authors challenge existing theory and empirical research indicating a deceleration of mortality at older ages and the emergence of a plateau. We argue that their results are inconclusive for three reasons: (1) the data selection was arbitrary; (2) the statistical analysis was inappropriate; and (3) the presentation of the results is misleading and inadequate. We therefore claim that the hypothesis that the human force of mortality increases after age 110 has not been proved.

The Annual Premium of Life Insurance on The Joint-Life Status based on The 2011 Indonesian Mortality Table

Life insurance is insurance that protects against risks to someone's life. Joint Life Insurance is insurance where the life and death rules are a combination of two or more factors, such as husband-wife or parent-child, and if the first death occurs, then the premium payment process is stopped. The annual premium is the premium paid annually. In this study, the annual premium is calculated continuously with the equivalence principle based on the 2011 Indonesian Mortality Table. The calculation shows that the amount of annual premiums for 2 (two) and 3 (three) people is not much different. The factors that influence the annual premium amount are the duration insurance period, age at signing the policy, interest rates, life chances, force of mortality, and the number of benefits.

Analytical model construction of optimal mortality intensities using polynomial estimation

The aim of this paper is to describe a non-parametric technique as a means of estimating the instantaneous force of mortality which serves as the underlying concept in modeling the future lifetime. It relies heavily on the analytic properties of life table survival functions 𝒍𝒙+𝒕. The specific objective of the study is to estimate the force of mortality using the Taylor series expansion to a desired degree of accuracy. The estimation of the continuous death probabilities has aroused keen research interest in mortality literature on life assurance practice. However, the estimation of 𝝁𝒙 involves a model dependent on deep knowledge of differencing and differential equation of first order. The suggested method of approximation with limiting optimal properties is the Newton’s forward difference model. Initiating Newton’s process is an important level in terms of theoretical work which produces parallel results of great impact in the study of mortality functions. The paper starts from an assumption that 𝒍𝒙 function follows a polynomial of least degree and hence gives an answer to a simple model which overcomes points of singularity. Keywords: polynomials, contingency, analyticity, basis, differential, mortality, modeling

Stochastic Mortality Modelling for Dependent Coupled Lives

Broken-heart syndrome is the most common form of short-term dependence, inducing a temporary increase in an individual’s force of mortality upon the occurrence of extreme events, such as the loss of a spouse. Socioeconomic influences on bereavement processes allow for suggestion of variability in the significance of short-term dependence between couples in countries of differing levels of economic development. Motivated by analysis of a Ghanaian data set, we propose a stochastic mortality model of the joint mortality of paired lives and the causal relation between their death times, in a less economically developed country than those considered in existing studies. The paired mortality intensities are assumed to be non-mean-reverting Cox–Ingersoll–Ross processes, reflecting the reduced concentration of the initial loss impact apparent in the data set. The effect of the death on the mortality intensity of the surviving spouse is given by a mean-reverting Ornstein–Uhlenbeck process which captures the subsiding nature of the mortality increase characteristic of broken-heart syndrome. Inclusion of a population wide volatility parameter in the Ornstein–Uhlenbeck bereavement process gives rise to a significant non-diversifiable risk, heightening the importance of the dependence assumption in this case. Applying the model proposed to an insurance pricing problem, we obtain the appropriate premium under consideration of dependence between coupled lives through application of the indifference pricing principle.