From around the nineteenth until the beginning of the twenty-first century, mortality rates show a declining trend. However, recent data on the United States population shows that the rate of decline started to slow down in the 2010s. Insurance companies need to be prepared in both ways: either mortality rates continue to decline, or there will be a turning point, and mortality rates start to increase. In this paper, we aim to get the whole picture of the mortality trend of Indonesian males, detect the possibility of a turning point in the mortality rates, and forecast mortality rates in the future. To reach this aim, we propose adjustments to the Makeham mortality model by including period and cohort information of the population via quadratic function. We also propose using the Bayesian method to estimate the parameters for the Indonesian old-aged males' population, where some adjustments were made in determining the priors, and the estimates were sampled from the posterior distribution using the Gibbs sampling algorithm. We found that our forecasting accuracy is satisfactory by considering the mean absolute percentage error values and coefficient of determination (R 2 ). We found that mortality rates are declining in the long term, but the probability of a turning point in the future is statistically significant. We identified two risks, longevity risk because of more centenarians in the future and mortality risk before their children complete compulsory education.
Mortality rates are important in conducting the pricing and valuation of life insurance policies. Raw values are usually wiggly to plot, and practitioners often graduate them to obtain smoothness. Current mortality models have problems related to the goodness of fit, interpretability, and usability without implementing other actuarial assumptions for fractional ages. This study proposes a mixture of Pareto, log-logistic, and two Weibull distributions with eleven parameters to graduate mortality rates. Lifespan covered are whole life, including childhood, adolescence, senescence, and the late elderly's phase. We adjusted the parameterization to improve the ease of model's interpretability right after obtaining the value of estimates. Prior distributions of the parameters and sampling model form for the data are also proposed to estimate the parameters' value using the Bayesian method with Gibbs sampling. High values of coefficient of determination produced by model fit into several data support the graphical evidence to show the model's goodness of fit and best fit occurs for the life table of Israeli males in 1987. Gelman-Rubin statistic is also very close to one and shows fast convergence in estimating the parameters. Based on the results, obtaining the best and worst estimates of newborn survival probabilities is possible. We also showed that this model could be implemented on annual and abridged mortality rates.
Several Indonesian life insurance companies recently faced financial problems due to inadequate pricing and idealistic investment expectation. Growing market and insurtech implementation might lead to worse conditions in the future. The current mortality table and investment return assumption are too ideal, so more conservative assumptions are required to get a more reasonable annual pure premium range. This research estimated complete life tables from abridged life tables by truncated Heligman-Pollard and Makeham model, when a lognormal stochastic process estimated annual investment return. Parameters for mortality models and return distribution are estimated using Bayesian method with Metropolis-Hasting's algorithm. Data from the abridged life table was bootstrapped due to insufficient number for statistical parametric modeling. Good accuracy for estimated abridged mortality rates was reached by referring to the Mean Absolute Percentage Error (MAPE) metric for both males and females, also for the young ages group (new-born to twenty-nine years old) and old ages group (thirty to eighty-four years old). The parameters were satisfactory to estimate the complete life table and extrapolate annual mortality rates calculation until age ninety-nine. A log-normal distribution was found to fit the monthly inflation rates satisfactorily. Assuming that investment return is close to the inflation rates, the annual investment return is anticipated for both profitable and losing situations. Therefore, insurance companies can win the customers' decisions without compromising their financial stability.
Since the first case in November 2019, COVID-19 has spread fast, infected many people, and taken many lives. Once COVID-19 cases are confirmed, and hospital treatments are required, they will need high costs yet still be at risk of death. Therefore, insurance companies and social service providers need a reliable model to estimate the expenses and ensure their financial health. This study discusses the modeling of time-until-release and time-until-death since cases are confirmed, using the South Korean data. Doubledecrement model is assumed to follow Weibull and log-logistic hazard for time-until-release and time-until-death, respectively. Considered risk factors are sex, symptoms, age group, and the month when the case is confirmed. The nonlinear survival regression model is proposed, with the Solver function in Microsoft Excel for parameter estimation. The results suggest that virus lifespan and mortality risk get lower over time. Time-until-death is also lower, implying that we have less time to save lives from mortality risk.More frequent testing with faster results and not waiting for the symptoms to occur is needed for people under thirty years old due to shorter time-until-death and those at sixty and above due to higher case fatality rates. Full dose vaccination must be prioritized for ages sixty and above to save as many lives as possible.
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