Analysis of Mortality Models with Covariates Missing at Random

Demographic indicators such as mortality rates play a very important role in health, financial and pension policies. Therefore, the accuracy of mathematical models in estimating mortality rates is an important challenge. One of the tasks of actuaries is to construct a suitable mortality model for the available data so that these mortality models can calculate mortality for different ages and longevity, as well as the different information available to individuals on retirement plans. Missing data is a problem that may be faced by actuaries when they are analyzing the real data. Missing data can occur for a variety of reasons, such as unanswered or censored. The presence of missing data can pose a threat to the accuracy of the data analysis results. The purpose of this study is to model the mortality in a retirement plan. In this regard, it is assumed that data are available at the individual level, including date of birth, date of joining the retirement plan, date of completion of the observation, and reason for discontinuation (usually death or right censoring). Information on covariate variables such as gender, benefits or size of pension, demographic geography or health status will also be available. More precisely, this study aims to model the mortality in a retirement plan based on missing data and access to information from various covariate variables, to carefully analyze the structure of different models, to estimate and finally to investigate the financial implications for different mortality experiences containing missing data.

In this article, we deal with a pension plan in which each member's future life expectancy is modeled using parametric survival models incorporating covariates which may be missing for some individuals. Likelihood-based techniques estimate parameters, and in this regard, an algorithm is proposed that can perform the estimation task in the best possible way. One of the necessary features to check the adequacy of the statistical model, especially when the data contains missing values, is identifiable. If not identifiable, it can be claimed that the statistical model is not a full rank and is not a suitable model for the data. It is worth noting that the Jacobin matrix needs to be calculated to verify identifiability. As mentioned, in the analysis of mortality models with the presence of missing values, the maximum likelihood method can be used. In such cases, an estimation error may often occur when fitting the model, which can be reduced by modeling from a larger population. For this reason, hybrid retirement plans that remain homogeneous are often used. This proposed method can also be useful for calculating financial quantities based on pension factors. In fact, in this proposed method, different data sets with equal or similar death experiences are combined, sample size increases and risk of parameter decreases, which also leads to a reduction in capital requirement. Socio-economic variables such as the level of benefits and geographical characteristics of the population are also considered more if interest rates are low. 

First, complete data are analyzed and modeled for observations of members of a retirement plan, which includes survival time and ancillary variables for each individual. Estimation of parameters is obtained using the maximum likelihood method. however, when the data is missing, it is not easy to estimate the parameters with the maximum likelihood method. In this case, the model parameters are estimated by the maximum likelihood method which are calculated using the proposed algorithm; then, statistical indicators such as identifiability of parameters are calculated to evaluate the performance of the proposed structure and algorithm. Furthermore, the financial effects, in particular the annuity factors, and the mis-estimation risk capital requirements for the mortality experience which includes the maximum covariates variables are calculated and compared with the individual segments when the data are missing. In addition, it can be seen that when the two statistical variables are not observed together, the model is not identifiable according to the data. 

It was found that if the data are missing, the statistical model is not always identifiable using the maximum likelihood, and data combination from two or more experiments can avoid identifiable barriers. The methods proposed in this paper can be useful for actuaries when calculating financial committees based on annuity factors. These methods may combine different datasets with equal or similar mortality experiences, increase sample size, and reduce parameter risk, thus, reducing capital requirements. Socio-economic variables such as the level of benefits and geographical characteristics of the population are given more attention if the interest rate is low.JEL-Classification: C13, C24, C51