# Multiple Population Mortality Jointly Forecasting in China Using PCF Model

Authors: Ming Zhao; Wei Wu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Based on the data of population mortality in China since 1994, this paper studies the changing trend of sex differences of mortality by proposing and implementing some novel approaches. First of all, this paper proposes a new method using the Poisson common factor (PCF) model to forecast the mortality jointly by both sexes. The study finds that the PCF model effectively captures the common trend of mortality between two sexes and uses additional factors to reflect differences of between two sexes, which can reduce the errors caused by low quality or large fluctuation of the mortality in China. Meanwhile, the forecasting values of mortality based on the PCF model can abide by the human biological law well, and the sex ratio of mortality converges to a fixed constant in the long run without increasing too much statistical error. Second, this paper improves the parameter estimation method of PCF model, and the innovative two-step method is used to estimate the model, which can make the maximum likelihood estimator converge more easily. Finally, as an application of the novel methods which are proposing in this study, we measure the longevity risk of pension by using the PCF model and find that the PCF model can make up the underestimation of longevity risk from traditional models and provide more scientific information to the sponsor of pension plan.


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## Body

## 1. Introduction

With the extension of life expectancy and the continuous increase of ageing population, governments around the world have begun to pay attention to the impact of longevity risk on the financial sustainability of pension plan. In response to the negative influence of longevity risk and ageing population, most OECD countries have raised the official state pension age in different extent depending on the difference in life expectancy between sexes [1, 2]. Since the founding of New China, the life expectancy has increased significantly. Moreover, life expectancy of the female has increased higher than that of male, but the retirement age of female is lower than that of male. That means coordinating the adjustment scheme of the retirement age of two sexes well will help to the implementation of delayed retirement policy and the reduction of society resistance [3, 4]. For China, the situation is that the effective accumulation of population mortality data is insufficient, and its quality is poor. What is worse, there are serious omissions in the number of neonatal and elderly deaths in census years. So the establishment of the longevity risk model in line with the various characteristics of population life span is an important guarantee for related subject to manage longevity risk. Therefore, it is of great theoretical and realistic significance to explore the sex differences in China’s population mortality rate, take the common trend between sexes into account in the stochastic mortality model, reasonably forecast the mortality of male and female, and then provide the scientific basis for pension reform.There are many studies on stochastic mortality models at home and abroad, among which the extrapolation methods proposed by Lee and Carter [5] and their extensions have become the dominant [5]. However, these methods are all based on the single population group [6]. Since stochastic mortality modelling is systematic work, the isolated consideration of the single population (such as a single country or a single sex) will result in unreasonable crossover or deviation of mortality among different populations over time. By jointly modelling two or more population groups and considering the common trend of mortality changes between different groups, the rationality of long-term forecast for mortality can be effectively improved. Moreover, the forecast for gender-specific mortality is supposed to show common biological characteristics. There are widespread sex differences in population mortality worldwide. Generally, the average life expectancy of the female is higher than that of male, and the gap is supposed to remain unchanged for a certain period, which could be explained by human biological characteristics [7]. However, if the results obtained from the mortality model built on single sex population are that the mortality of male decreases rapidly and is lower than that of female at a certain extent, these are contrary to the biological characteristics. Similarly, in the long term, it is also unreasonable that the forecast results obtained are that the sex ratio of male and female population mortality varies greatly over time [8]. Therefore, establishing a gender-based stochastic mortality model is an important means to ensure that the population mortality forecasting is long-term reasonable and in line with the biological characteristics. Li and Lee [6] extend the single-population Lee-Carter model method and construct a multipopulation stochastic mortality model for the first time, namely the common factor model (CFM). The method combines the common trend factor with the additional factor to describe the common trend and difference characteristics of mortality changes in related population groups. In this method, the common factor describes the main long-term mortality changes in all population groups. In contrast, the additional factor describes the short-term deviation between each population group and the main long-term trend. As long as the cycling effect of the additional factor tends to a constant over time, the mortality ratios of different populations will converge to a constant, and certain undesirable characteristic such as mortality crossover can also be avoided. Based on the framework of CFM method, some scholars establish a common cycle effect model to describe the common trends of different population mortality in the cycle [9]. Li [10] establishes the Poisson common factor model (PCFM), relaxes the homoscedasticity assumption of logarithmic mortality in the CFM method, and directly models death toll as a Poisson process, which provides a strict statistical framework for data analysis, parameter estimation and model selection [10]. Yang et al. [11] add the cohort effect to the PCFM model under the assumption that the death toll obeys the Poisson distribution. According to their study, the direct parameterisation of the cohort effect can improve the goodness-of-fit of model, reduce the additional demand for cycle effect parameters, and provide consistent mortality forecast for multiple populations [11]. Enchev et al. [12] pointed that the PCFM contains both common factors and additional factors, when its parameters are estimated by using the maximum likelihood method, will meet the problem of unable convergence or the convergence result being locally optimal due to a large number of parameters [12]. Especially, for countries with poor mortality data quality, the problem is more remarkable. In view of the large fluctuation of population mortality data in China sampling survey years and the lack of mortality data in some years, Siu et al. [13] use the Gaussian condition space model to fill and correct the data under the Bayesian framework. However, this method is only for a single population and does not consider the relationship between populations [13].Through the analysis of current research, it is found that (1) there are more studies on the sex joint model of population mortality in developed countries, while there are fewer studies on countries with less effective accumulation and poor quality of mortality data; (2) there are few studies on the consistent change trend of sex differences in population mortality in China, which cannot support the establishment of a sex joint model for population mortality; (3) there are many types of research on the application of single population stochastic mortality model in China, while there are few studies on the joint modelling of stochastic mortality from the perspective of multipopulation. Based on the analysis above, this paper focuses on the following three aspects: (1) studying the changing trend of sex differences in population mortality since the founding of New China and providing a basis for establishing a multi-population stochastic mortality model; (2) forecasting China’s population mortality from the perspective of sex union, exploring the applicability of the Poisson common factor model in China’s population mortality data, and comparing it with the single-population mortality model; (3) in order to overcome the convergence problem of model parameter estimation under the maximum likelihood method, this paper presents a two-stage parameter estimation method: the first stage uses the maximum likelihood method and the second stage uses the ordinary least square method.


## 2. Stochastic Mortality Model

### 2.1. Lee-Carter Model

The classic stochastic mortality method used to forecast the mortality of a single population is the Lee-Carter model as follows:(1)lnmxt=ax+bxkt+εxt,in which mxt is the mortality of population aged x in the year t, ax indicates a certain population part, in which the mortality of different ages and age groups are unrelated to the year t, βx represents the interaction effect between mortality and the year t in different age groups, kt measures the changing trend of mortality in the year t, and εxt is the error term. To ensure the uniqueness of the results, additional restraint conditions are attached as follows:(2)∑tkt=0,∑xβx=1.


### 2.2. Poisson Common Factor (PCF) Model

Suppose the rear population of the mortality rate obeys the Poisson distribution. Li [10] proposes the Poisson common factor model (PCFM) as follows:(3)dx,t,i∼Poissonex,t,imx,t,i,(4)lnmx,t,i=αx,i+BXKt+∑j=1nbx,i,jkt,i,j+εx,t,j.In equation (3), dx,t,i, ex,t,i, and mx,t,i are the number of deaths, the number of population exposure, and the mortality rate at age x in the calendar year t of sex i. One may set i = 1 for female and i = 2 for male. In equation (4), the parameter αx,i represents the factor for the mortality rate of the population at different ages that are unrelated to the year t, BxKt is the common factor for the changing trend of population mortality of both sexes. The common factor indicates the main long-term trend in mortality change for both sexes. To be specific, Bx is the common factor of age effect, Kt is the common factor of period effect; ∑j=1nbx,i,jkt,i,j are j additional factors indicating the short-term variations in mortality change of single sex population. Specifically, bx,i,j is jth additional factor of age effect, and kt,i,j is jth additional factor of period effect.To ensure the uniqueness of the results, the PCF model must meet the following constraints:(5)∑xBx=1,∑tKt=0,∑xbx,i,j=1,∑tkt,i,j=0,Kt, the common factor of period effect, is usually modelled by the Random Walk Model (RW Model) with drift term as follows:(6)Kt=μ+Kt−1+εt,where μ is the drift term, a fixed constant and εt is the error term, obeying the following distribution as εt∼N0,σ2. Drift term is usually negative, indicating an overall mortality improvement over time.Moreover, in order to ensure the consistency of the sex differences in mortality of different populations, the mean-reversion model should be established for the additional factorkt,i,j, commonly by an AR(p) model as follows:(7)kt,i,j=ao,i,jkt−1,i,j+a2,i,jkt−2,i,j+...+ap,i,jkt−p,i,j+wt,i,j,where a0,i,j,a1,i,j,...,ap,i,j, and ωt,i,j are the parameters of the model; ωt,i,j is the error term, obeying the following distribution as ωt,i,j∼N0,vi,j2.


### 2.3. Method of Parameter Estimation

There are three main parameter estimation methods for stochastic mortality models: singular value decomposition (SVD), ordinary least square (OLS), and maximum likelihood estimation (MLE). The analysis of the single population mortality in China shows that OLS and MLE methods have high goodness-of-fit. In order to maximize the capture of the main trend factors in the population mortality changes, relax the homoscedasticity assumption in the traditional stochastic mortality model and avoid the difficulty of convergence caused by the excessive parameterisation of the model, this paper uses a two-stage method to estimate the parameters of the PCF model.In the first stage, MLE is used for parameter estimation of common factors. According to the hypothesis that the death toll obeys Poisson distribution, the likelihood function is constructed as follows:(8)lα,B,K=∑x,t,idx,t,iax,i+BxKt−ex,t,ieax,i+BxKt+constant.Starting from the following initial values of parameters,a^x,i0=0, B^x0=1, and K^t0=0, the iterative parameters are updated according to the following steps:(9)α^xv+1=α^xv+∑tdx,t−d^x,tv∑td^x,tv,B^xv+1=B^xv,K^tv+1=K^tv,(10)K^xv+2=K^xv+1+∑tdx,t−d^x,tv+1β^xv+1∑td^x,tvβ^xv+12,α^xv+2=α^xv+1,B^xv+2=B^xv+1,(11)B^xv+3=B^xv+2+∑tdx,t−d^x,tv+2K^xv+2∑td^x,tv+2K^xv+22,α^xv+3=α^xv+2,K^xv+3=K^xv+2.In the second stage, OLS is used for parameter estimation of the additional factors. According to the parameter estimation results obtained in the first stage, the residual sum of squares can be further obtained, as shown in(12)∑x,tεx,t,i2=∑x,tlnmx,t,i−α^x,i−B^xK^t−∑j=1nbx,i,jkt,i,j2,where α^x,i, B^x, and K^t are constants, which are the parameter estimates obtained in the first stage. Through the constraint conditions of OLS method and PCF model, the parameter estimation results of the first additional factor can be obtained as follows:(13)k^t,i,1=∑xlnmx,t,i−α^x,i−B^xK^t,b^x,i,1=∑tk^t,i,1lnmx,t,i−α^x,i−B^xK^t∑tk^t,i,12.Similarly, the parameter estimation results of any additional factor can be obtained. The most common defect of PCF, overparameterisation, can be avoided by adding birth year effect factors or reducing the number of additional factors as much as possible [14].


### 2.4. Method of Mortality Forecasting

Using the recently observed data as a starting point will help avoid significant deviations in the early forecasting period [15]. When t>s, the forecasting of future mortality is expressed as(14)m^x,t,i=m˙x,s,iexpB^xK^t−Ks+∑j=1nb^x,i,jk^t,i,j−ks,i,j,where K^t and k^t,i,j are the mean estimation results, m˙x,s,i is the forecast base year, and m^x,t,i is the forecast mortality rate.The forecast value of the sex ratio of mortality (female mortality divided by male mortality) can be further obtained:(15)m^x,t,2m^x,t,1=m˙x,s,2m˙x,s,1exp∑j=1nbx,2,jk^t,2,j−ks,2,j−bx,1,jk^t,1,j−ks,1,j.If the forecast additional period effect factorsk^t,1,j and k^t,2,j have the mean recovery property, the sex ratio of mortality will converge to a constant.


## 2.1. Lee-Carter Model

The classic stochastic mortality method used to forecast the mortality of a single population is the Lee-Carter model as follows:(1)lnmxt=ax+bxkt+εxt,in which mxt is the mortality of population aged x in the year t, ax indicates a certain population part, in which the mortality of different ages and age groups are unrelated to the year t, βx represents the interaction effect between mortality and the year t in different age groups, kt measures the changing trend of mortality in the year t, and εxt is the error term. To ensure the uniqueness of the results, additional restraint conditions are attached as follows:(2)∑tkt=0,∑xβx=1.


## 2.2. Poisson Common Factor (PCF) Model

Suppose the rear population of the mortality rate obeys the Poisson distribution. Li [10] proposes the Poisson common factor model (PCFM) as follows:(3)dx,t,i∼Poissonex,t,imx,t,i,(4)lnmx,t,i=αx,i+BXKt+∑j=1nbx,i,jkt,i,j+εx,t,j.In equation (3), dx,t,i, ex,t,i, and mx,t,i are the number of deaths, the number of population exposure, and the mortality rate at age x in the calendar year t of sex i. One may set i = 1 for female and i = 2 for male. In equation (4), the parameter αx,i represents the factor for the mortality rate of the population at different ages that are unrelated to the year t, BxKt is the common factor for the changing trend of population mortality of both sexes. The common factor indicates the main long-term trend in mortality change for both sexes. To be specific, Bx is the common factor of age effect, Kt is the common factor of period effect; ∑j=1nbx,i,jkt,i,j are j additional factors indicating the short-term variations in mortality change of single sex population. Specifically, bx,i,j is jth additional factor of age effect, and kt,i,j is jth additional factor of period effect.To ensure the uniqueness of the results, the PCF model must meet the following constraints:(5)∑xBx=1,∑tKt=0,∑xbx,i,j=1,∑tkt,i,j=0,Kt, the common factor of period effect, is usually modelled by the Random Walk Model (RW Model) with drift term as follows:(6)Kt=μ+Kt−1+εt,where μ is the drift term, a fixed constant and εt is the error term, obeying the following distribution as εt∼N0,σ2. Drift term is usually negative, indicating an overall mortality improvement over time.Moreover, in order to ensure the consistency of the sex differences in mortality of different populations, the mean-reversion model should be established for the additional factorkt,i,j, commonly by an AR(p) model as follows:(7)kt,i,j=ao,i,jkt−1,i,j+a2,i,jkt−2,i,j+...+ap,i,jkt−p,i,j+wt,i,j,where a0,i,j,a1,i,j,...,ap,i,j, and ωt,i,j are the parameters of the model; ωt,i,j is the error term, obeying the following distribution as ωt,i,j∼N0,vi,j2.


## 2.3. Method of Parameter Estimation

There are three main parameter estimation methods for stochastic mortality models: singular value decomposition (SVD), ordinary least square (OLS), and maximum likelihood estimation (MLE). The analysis of the single population mortality in China shows that OLS and MLE methods have high goodness-of-fit. In order to maximize the capture of the main trend factors in the population mortality changes, relax the homoscedasticity assumption in the traditional stochastic mortality model and avoid the difficulty of convergence caused by the excessive parameterisation of the model, this paper uses a two-stage method to estimate the parameters of the PCF model.In the first stage, MLE is used for parameter estimation of common factors. According to the hypothesis that the death toll obeys Poisson distribution, the likelihood function is constructed as follows:(8)lα,B,K=∑x,t,idx,t,iax,i+BxKt−ex,t,ieax,i+BxKt+constant.Starting from the following initial values of parameters,a^x,i0=0, B^x0=1, and K^t0=0, the iterative parameters are updated according to the following steps:(9)α^xv+1=α^xv+∑tdx,t−d^x,tv∑td^x,tv,B^xv+1=B^xv,K^tv+1=K^tv,(10)K^xv+2=K^xv+1+∑tdx,t−d^x,tv+1β^xv+1∑td^x,tvβ^xv+12,α^xv+2=α^xv+1,B^xv+2=B^xv+1,(11)B^xv+3=B^xv+2+∑tdx,t−d^x,tv+2K^xv+2∑td^x,tv+2K^xv+22,α^xv+3=α^xv+2,K^xv+3=K^xv+2.In the second stage, OLS is used for parameter estimation of the additional factors. According to the parameter estimation results obtained in the first stage, the residual sum of squares can be further obtained, as shown in(12)∑x,tεx,t,i2=∑x,tlnmx,t,i−α^x,i−B^xK^t−∑j=1nbx,i,jkt,i,j2,where α^x,i, B^x, and K^t are constants, which are the parameter estimates obtained in the first stage. Through the constraint conditions of OLS method and PCF model, the parameter estimation results of the first additional factor can be obtained as follows:(13)k^t,i,1=∑xlnmx,t,i−α^x,i−B^xK^t,b^x,i,1=∑tk^t,i,1lnmx,t,i−α^x,i−B^xK^t∑tk^t,i,12.Similarly, the parameter estimation results of any additional factor can be obtained. The most common defect of PCF, overparameterisation, can be avoided by adding birth year effect factors or reducing the number of additional factors as much as possible [14].


## 2.4. Method of Mortality Forecasting

Using the recently observed data as a starting point will help avoid significant deviations in the early forecasting period [15]. When t>s, the forecasting of future mortality is expressed as(14)m^x,t,i=m˙x,s,iexpB^xK^t−Ks+∑j=1nb^x,i,jk^t,i,j−ks,i,j,where K^t and k^t,i,j are the mean estimation results, m˙x,s,i is the forecast base year, and m^x,t,i is the forecast mortality rate.The forecast value of the sex ratio of mortality (female mortality divided by male mortality) can be further obtained:(15)m^x,t,2m^x,t,1=m˙x,s,2m˙x,s,1exp∑j=1nbx,2,jk^t,2,j−ks,2,j−bx,1,jk^t,1,j−ks,1,j.If the forecast additional period effect factorsk^t,1,j and k^t,2,j have the mean recovery property, the sex ratio of mortality will converge to a constant.


## 3. Mortality Forecasting by Both Sexes

The age-specific and gender-specific mortality data published by the National Bureau of Statistics of China from 1994 to 2016 and PCF model and Lee-Carter model are selected to forecast the factors respectively.


### 3.1. Parameter Estimation Results of Common Factors

According to equations (9) and (10), we can calculate the parameter estimates of age effect common factor and period effect common factor, the results are shown in Figures 1 and 2.Figure 1
Estimated common factors of the age effect.Figure 2
Estimated common factors of period effect.Figure1 shows the trend of age effect factors, indicating the age distribution of mortality changes estimated by PCF model and the Lee-Carter model. In this Figure, the common factors obtained by PCF model parameter estimation show a trend of decreasing slightly at first, then increasing, and then decreasing with age, which means that the population mortality rate decreases slowly in the two age groups under 15 years old and over 60 years old, while the decline rate is fast in the middle age group. The age effect factors of the male and female populations obtained by the Lee-Carter model have the same trend as the common factor as a whole, and the effect factor of the female population is closer to the common factor, which is mainly due to the low mortality rate and strong stability of the female population. Therefore, the improvement of the common factor method on the forecasting of the mortality of male is bigger than that of female, and especially, it can effectively improve the increasing trend of the male population effect factor of the age group over 80 years old. Moreover, the male and female factors obtained by the Lee-Carter model have high volatility. In contrast, the common factor is more stable, and the capture of the trend of mortality reduction is more robust.Figure2 is the trend chart of the period effect factor, which shows the distribution of mortality changes estimated by PCF model and Lee-Carter model in the calendar year. In this Figure, the slope of the common factor decline obtained by the parameter estimation of the PCF model is the largest, and the slope remains unchanged throughout the period, which means that the population mortality rate shows a significant continuous downward trend on the time axis. The slope of period effect factor of female population obtained by Lee-Carter model is less than common factor, but the changing trend is close; The slope of period effect factor of the male population is the smallest, and the slope decreases further after 2005, and the decline rate of mortality slows down. Therefore, the common period effect factor can effectively improve the slowdown trend of mortality of male after 2005 and solve the problem of large fluctuation of female population period effect factor.


### 3.2. Parameter Estimation Results of Additional Factors

On the selection of additional factors, in order to prevent overparameterisation of the model, and combined with the analysis of the fitting residual, we setj = 1, which contains a sex additional factor. According to equation (13), the additional age effect factor and additional period effect factor of the male and female population can be obtained, respectively. The results are shown in Figures 3 and 4.Figure 3
Estimated additional factors of the age effect.Figure 4
Estimated additional factors of period effect.Figure3 is a trend chart of additional age effect factor, showing the sex differences in population mortality trends in age distribution. Similar to the trend of common factors, the additional factor also shows a trend of increasing first and then decreasing, but there are certain differences in the trend of common factors with different ages. The additional factors of the population over 80 years old show a significant downward trend, while the common factors show a slow downward trend. Moreover, the additional age effect factor of the female population aged 5–30 fluctuates greatly because the mortality rate of the female population in this age group is low. In the 1% sample survey year, the number of deaths at some ages is 0, resulting in a serious distortion of the population mortality data. However, with the increase of age, the increase of mortality rate, and the decrease of the number of deaths at 0, the trend of additional age effect factor of women is gradually approaching that of men.Figure4 is a trend chart of period effect factors, which shows the sex differences in the temporal distribution of population mortality trends. Contrary to the changing trend of common factors, the additional factors generally show a trend of increasing over time. The upward slope of the male population is greater than that of the female population. If dividing the time into two stages: 1994–2005 and 2006–2016, therefore, sizes of training and testing data sets are 2400 and 2200, respectively.From 1994 to 2005, the period effect factor value of male population fluctuates stably in the negative 15 and that of female population fluctuates stably in the negative 10. From 2006 to 2016, the period effect factors of male and female population show strong consistency (the period effect factor of male population is slightly higher than that of female population), and the values fluctuate steadily around 10. Thus, both men and women, period effect factors have the characteristics of mean reversion, but in different periods show different mean levels. In summary, AR (p) model can be established to forecast the changing trend of period effect additional factors in the future.


### 3.3. Goodness of Fitting

According to equations (15) and (16), the logarithmic mortality data within the sample can be fitted, then we compare the data 2000 and 2010 census years, as shown in Figures 5 and 6. Overall, the fitting values of logarithmic mortality rates obtained by both the PCF model and the Lee-Carter model show a decreasing trend at first then increasing, which is consistent with the trend of real values and fluctuates slightly around the real values. Among them, in the age group with low population mortality (10–25 years old), the deviation between the fitting value and the real value is large because the fitting results based on the linear model does not have good smoothness at the inflexion point, then resulting in large fluctuations. Since this paper only discusses the goodness-of-fit of mortality, it is no longer necessary to smooth the mortality. If there is a need to make a morality table, the data can be further trimmed. By comparing the fitting values of the two models, it can be seen that the fitting values of logarithmic mortality of males in 2000 obtained by PCF model are slightly larger than those obtained by Lee-Carter model, while the fitting values of males in 2010 are slightly smaller than those obtained by Lee-Carter model.Figure 5
Comparison of logarithmic mortality of male in 2000.Figure 6
Comparison of logarithmic mortality of male in 2010.In order to test the fitting effect of the PCF model and the Lee-Carter model, the mean absolute error (MAE) and the mean absolute percentage error (MAPE) are selected for calculation, and the results are shown in the Table. MAE describes the absolute difference between actual and predicted values, while MAPE describes the absolute difference between all actual and predicted values as a percentage of all actual values.The reasons for using these two criteria to test the goodness-of-fit are as follows: first of all, in a large number of existing literature, these two indicators have been used to evaluate the goodness-of-fit of the random mortality model. For example, Chen and Millossovich [16] used MAPE to test the goodness-of-fit of the mortality model [10]. Second, in order to avoid overparameterisation problem of the PCF model, a two-step parameter estimation method using the maximum square estimation method for additional factors is proposed, which leads to the problem that the whole likelihood function and the maximum likelihood values of the parameters of the model cannot be obtained and therefore cannot meet the definition requirements of BIC (Bayesian information criterion) and AIC (Akaike information criterion), so AIC and BIC cannot be used. Finally, various methods can be used to estimate the parameters of the random mortality model, but MAE and MAPE do not have excessive restrictions to estimate the parameters in the early stage, the detected values can also be compared with the detected values of other same criteria in the future.Since the PCF model takes the biological characteristics of human beings into account, there are many parameters, and the two-stage parameter estimation method can capture the most effective common factor at the cost of certain goodness-of-fit. As can be seen from Table1, the PCF model has no such obvious sacrifice to goodness-of-fit, and its result is close to the Lee-Carter models in comparison of MAE and MAPE. The MAE and MAPE values calculated by PCFM model are only slightly larger than those calculated by Lee-Carter model, where the MAE gap and the MAPE gap are 0.0003 and 0.0097 in the male population, 0.0003 and 0.0171 in the female population, 0.0009 and 0.0107 in sex ratio.Table 1
Test of fitting results based on the PCF model and Lee-Carter model.
MAEMAPEPCF-M0.01080.3471LC-M0.01050.3374PCF-F0.00970.4085LC-F0.00940.3914PCF-sex ratio0.32010.5464LC-sex ratio0.31920.5357Note: PCF-M and PCF-F represent the fitting results of mortality rate of male and female population based on PCF model; LC-M and LC-F represent the fitting results of mortality rate of male and female population based on Lee-Carter model; PCF-Sex Ratio and LC-Sex Ratio represent the population sex ratio (female/male) based on PCF model and Lee-Carter model.The pursuit of absolute fitting for a single parameter often leads to the distortion of the whole model. The PCF model can improve the overfitting of Lee-Carter model effectively, which has a positive effect on mortality prediction and satisfies the sex difference caused by biological factors, it makes the model more reasonable in the long-term forecasting and provides a scientific basis for the pension system with long-term evaluation needs.


### 3.4. Forecasting

Table2 shows the male mortality, the female mortality, and the sex ratio of mortality forecast by the PCF model and Lee-Carter model from 2020 to 2050. For newborn infants, the mortality forecast by PCF model is higher than that forecast by Lee-Carter model, which effectively improves the overestimation of mortality caused by underreporting of neonatal death in China. The mortality of male infants forecast by PCF model is higher than that of female infants, while the Lee-Carter model result is just the opposite. Compared with developed countries with low mortality, the forecasting of PCF model is more reasonable. For the young and middle-aged population, the mortality forecast by the PCF model is higher than that of the Lee-Carter model in the early stage (2020) but lower than that of the Lee-Carter model in the later stage (2030–2050). For example, the mortality forecast by the PCF model and the Lee-Carter model is 0.65% and 0.56% in 2020, 0.32%, and 0.37% in 2030, 0.16% and 0.25% in 2040, 0.08% and 0.17% in 2050, respectively. Over time, the mortality rate of the young and middle-aged population forecast by PCF model decreases rapidly, which is also consistent with the trend of population mortality in developed countries in the early stage. At the advanced age, especially near the terminal age, the decline rate of mortality forecast by the PCF model begins to slow down, and the mortality rate is lower than that forecast by the Lee-Carter model. At this stage, because the population age is close to the biological limit, without major scientific breakthroughs, it is difficult for the lifespan of the human being to decline as rapidly as young and middle-aged. Moreover, the population sex ratio obtained by the PCF model can converge to a constant, while that obtained by Lee-Carter model does not have this feature. The population sex ratio obtained by the PCF model converges to 1.19 at the age of 0, 2.27 at the age of 10, 1.96 at the age of 60 and 1.11 at the age of 100. The sex ratio obtained by the PCF model is consistent with the changing trend of historical mortality data in China. Not only does the sex difference meet the consistency requirements but also the sex gap increases first and then decreases with the increase of age.Table 2
Forecast value based on the PCF model and Lee-Carter model (unit: %).
2020203020402050PCFLCPCFLCPCFLCPCFLCMale05.124.222.461.621.180.620.570.24100.320.270.190.190.110.130.070.09300.650.560.320.370.160.250.080.17608.878.154.936.292.774.861.563.75100353.80289.57296.51203.56247.46143.11206.52100.60Female04.304.472.061.530.990.520.480.18100.140.140.080.100.050.060.030.04300.260.260.130.140.060.070.030.04604.444.392.492.991.402.040.791.39100321.24315.17268.23259.67223.86213.94186.83176.27Sex ratio01.190.941.191.061.191.191.191.33102.271.892.272.002.272.132.272.22302.502.132.442.702.443.452.444.35602.001.851.962.081.962.381.962.701001.100.921.110.781.110.671.110.57For the reason that in some available papers, the authors use the word consistent to describe the high degree of similarity between predicted outcomes and empirical facts. Li [10], for example [10], points out in his paper that the increase in life expectancy of 65-year-old women in 2050 seems to account for a large part of the improvement in life expectancy at birth, this is consistent with the increasing importance of improving mortality in old age. Therefore, this paper also uses the word consistent to describe the high degree of similarity between predicted results and empirical facts.In summary, PCF model can well reflect the biological characteristics of human beings in forecasting population mortality, is consistent with the historical law of sex differences in population mortality in China, and is also consistent with the current empirical facts in developed countries at the same time.


## 3.1. Parameter Estimation Results of Common Factors

According to equations (9) and (10), we can calculate the parameter estimates of age effect common factor and period effect common factor, the results are shown in Figures 1 and 2.Figure 1
Estimated common factors of the age effect.Figure 2
Estimated common factors of period effect.Figure1 shows the trend of age effect factors, indicating the age distribution of mortality changes estimated by PCF model and the Lee-Carter model. In this Figure, the common factors obtained by PCF model parameter estimation show a trend of decreasing slightly at first, then increasing, and then decreasing with age, which means that the population mortality rate decreases slowly in the two age groups under 15 years old and over 60 years old, while the decline rate is fast in the middle age group. The age effect factors of the male and female populations obtained by the Lee-Carter model have the same trend as the common factor as a whole, and the effect factor of the female population is closer to the common factor, which is mainly due to the low mortality rate and strong stability of the female population. Therefore, the improvement of the common factor method on the forecasting of the mortality of male is bigger than that of female, and especially, it can effectively improve the increasing trend of the male population effect factor of the age group over 80 years old. Moreover, the male and female factors obtained by the Lee-Carter model have high volatility. In contrast, the common factor is more stable, and the capture of the trend of mortality reduction is more robust.Figure2 is the trend chart of the period effect factor, which shows the distribution of mortality changes estimated by PCF model and Lee-Carter model in the calendar year. In this Figure, the slope of the common factor decline obtained by the parameter estimation of the PCF model is the largest, and the slope remains unchanged throughout the period, which means that the population mortality rate shows a significant continuous downward trend on the time axis. The slope of period effect factor of female population obtained by Lee-Carter model is less than common factor, but the changing trend is close; The slope of period effect factor of the male population is the smallest, and the slope decreases further after 2005, and the decline rate of mortality slows down. Therefore, the common period effect factor can effectively improve the slowdown trend of mortality of male after 2005 and solve the problem of large fluctuation of female population period effect factor.


## 3.2. Parameter Estimation Results of Additional Factors

On the selection of additional factors, in order to prevent overparameterisation of the model, and combined with the analysis of the fitting residual, we setj = 1, which contains a sex additional factor. According to equation (13), the additional age effect factor and additional period effect factor of the male and female population can be obtained, respectively. The results are shown in Figures 3 and 4.Figure 3
Estimated additional factors of the age effect.Figure 4
Estimated additional factors of period effect.Figure3 is a trend chart of additional age effect factor, showing the sex differences in population mortality trends in age distribution. Similar to the trend of common factors, the additional factor also shows a trend of increasing first and then decreasing, but there are certain differences in the trend of common factors with different ages. The additional factors of the population over 80 years old show a significant downward trend, while the common factors show a slow downward trend. Moreover, the additional age effect factor of the female population aged 5–30 fluctuates greatly because the mortality rate of the female population in this age group is low. In the 1% sample survey year, the number of deaths at some ages is 0, resulting in a serious distortion of the population mortality data. However, with the increase of age, the increase of mortality rate, and the decrease of the number of deaths at 0, the trend of additional age effect factor of women is gradually approaching that of men.Figure4 is a trend chart of period effect factors, which shows the sex differences in the temporal distribution of population mortality trends. Contrary to the changing trend of common factors, the additional factors generally show a trend of increasing over time. The upward slope of the male population is greater than that of the female population. If dividing the time into two stages: 1994–2005 and 2006–2016, therefore, sizes of training and testing data sets are 2400 and 2200, respectively.From 1994 to 2005, the period effect factor value of male population fluctuates stably in the negative 15 and that of female population fluctuates stably in the negative 10. From 2006 to 2016, the period effect factors of male and female population show strong consistency (the period effect factor of male population is slightly higher than that of female population), and the values fluctuate steadily around 10. Thus, both men and women, period effect factors have the characteristics of mean reversion, but in different periods show different mean levels. In summary, AR (p) model can be established to forecast the changing trend of period effect additional factors in the future.


## 3.3. Goodness of Fitting

According to equations (15) and (16), the logarithmic mortality data within the sample can be fitted, then we compare the data 2000 and 2010 census years, as shown in Figures 5 and 6. Overall, the fitting values of logarithmic mortality rates obtained by both the PCF model and the Lee-Carter model show a decreasing trend at first then increasing, which is consistent with the trend of real values and fluctuates slightly around the real values. Among them, in the age group with low population mortality (10–25 years old), the deviation between the fitting value and the real value is large because the fitting results based on the linear model does not have good smoothness at the inflexion point, then resulting in large fluctuations. Since this paper only discusses the goodness-of-fit of mortality, it is no longer necessary to smooth the mortality. If there is a need to make a morality table, the data can be further trimmed. By comparing the fitting values of the two models, it can be seen that the fitting values of logarithmic mortality of males in 2000 obtained by PCF model are slightly larger than those obtained by Lee-Carter model, while the fitting values of males in 2010 are slightly smaller than those obtained by Lee-Carter model.Figure 5
Comparison of logarithmic mortality of male in 2000.Figure 6
Comparison of logarithmic mortality of male in 2010.In order to test the fitting effect of the PCF model and the Lee-Carter model, the mean absolute error (MAE) and the mean absolute percentage error (MAPE) are selected for calculation, and the results are shown in the Table. MAE describes the absolute difference between actual and predicted values, while MAPE describes the absolute difference between all actual and predicted values as a percentage of all actual values.The reasons for using these two criteria to test the goodness-of-fit are as follows: first of all, in a large number of existing literature, these two indicators have been used to evaluate the goodness-of-fit of the random mortality model. For example, Chen and Millossovich [16] used MAPE to test the goodness-of-fit of the mortality model [10]. Second, in order to avoid overparameterisation problem of the PCF model, a two-step parameter estimation method using the maximum square estimation method for additional factors is proposed, which leads to the problem that the whole likelihood function and the maximum likelihood values of the parameters of the model cannot be obtained and therefore cannot meet the definition requirements of BIC (Bayesian information criterion) and AIC (Akaike information criterion), so AIC and BIC cannot be used. Finally, various methods can be used to estimate the parameters of the random mortality model, but MAE and MAPE do not have excessive restrictions to estimate the parameters in the early stage, the detected values can also be compared with the detected values of other same criteria in the future.Since the PCF model takes the biological characteristics of human beings into account, there are many parameters, and the two-stage parameter estimation method can capture the most effective common factor at the cost of certain goodness-of-fit. As can be seen from Table1, the PCF model has no such obvious sacrifice to goodness-of-fit, and its result is close to the Lee-Carter models in comparison of MAE and MAPE. The MAE and MAPE values calculated by PCFM model are only slightly larger than those calculated by Lee-Carter model, where the MAE gap and the MAPE gap are 0.0003 and 0.0097 in the male population, 0.0003 and 0.0171 in the female population, 0.0009 and 0.0107 in sex ratio.Table 1
Test of fitting results based on the PCF model and Lee-Carter model.
MAEMAPEPCF-M0.01080.3471LC-M0.01050.3374PCF-F0.00970.4085LC-F0.00940.3914PCF-sex ratio0.32010.5464LC-sex ratio0.31920.5357Note: PCF-M and PCF-F represent the fitting results of mortality rate of male and female population based on PCF model; LC-M and LC-F represent the fitting results of mortality rate of male and female population based on Lee-Carter model; PCF-Sex Ratio and LC-Sex Ratio represent the population sex ratio (female/male) based on PCF model and Lee-Carter model.The pursuit of absolute fitting for a single parameter often leads to the distortion of the whole model. The PCF model can improve the overfitting of Lee-Carter model effectively, which has a positive effect on mortality prediction and satisfies the sex difference caused by biological factors, it makes the model more reasonable in the long-term forecasting and provides a scientific basis for the pension system with long-term evaluation needs.


## 3.4. Forecasting

Table2 shows the male mortality, the female mortality, and the sex ratio of mortality forecast by the PCF model and Lee-Carter model from 2020 to 2050. For newborn infants, the mortality forecast by PCF model is higher than that forecast by Lee-Carter model, which effectively improves the overestimation of mortality caused by underreporting of neonatal death in China. The mortality of male infants forecast by PCF model is higher than that of female infants, while the Lee-Carter model result is just the opposite. Compared with developed countries with low mortality, the forecasting of PCF model is more reasonable. For the young and middle-aged population, the mortality forecast by the PCF model is higher than that of the Lee-Carter model in the early stage (2020) but lower than that of the Lee-Carter model in the later stage (2030–2050). For example, the mortality forecast by the PCF model and the Lee-Carter model is 0.65% and 0.56% in 2020, 0.32%, and 0.37% in 2030, 0.16% and 0.25% in 2040, 0.08% and 0.17% in 2050, respectively. Over time, the mortality rate of the young and middle-aged population forecast by PCF model decreases rapidly, which is also consistent with the trend of population mortality in developed countries in the early stage. At the advanced age, especially near the terminal age, the decline rate of mortality forecast by the PCF model begins to slow down, and the mortality rate is lower than that forecast by the Lee-Carter model. At this stage, because the population age is close to the biological limit, without major scientific breakthroughs, it is difficult for the lifespan of the human being to decline as rapidly as young and middle-aged. Moreover, the population sex ratio obtained by the PCF model can converge to a constant, while that obtained by Lee-Carter model does not have this feature. The population sex ratio obtained by the PCF model converges to 1.19 at the age of 0, 2.27 at the age of 10, 1.96 at the age of 60 and 1.11 at the age of 100. The sex ratio obtained by the PCF model is consistent with the changing trend of historical mortality data in China. Not only does the sex difference meet the consistency requirements but also the sex gap increases first and then decreases with the increase of age.Table 2
Forecast value based on the PCF model and Lee-Carter model (unit: %).
2020203020402050PCFLCPCFLCPCFLCPCFLCMale05.124.222.461.621.180.620.570.24100.320.270.190.190.110.130.070.09300.650.560.320.370.160.250.080.17608.878.154.936.292.774.861.563.75100353.80289.57296.51203.56247.46143.11206.52100.60Female04.304.472.061.530.990.520.480.18100.140.140.080.100.050.060.030.04300.260.260.130.140.060.070.030.04604.444.392.492.991.402.040.791.39100321.24315.17268.23259.67223.86213.94186.83176.27Sex ratio01.190.941.191.061.191.191.191.33102.271.892.272.002.272.132.272.22302.502.132.442.702.443.452.444.35602.001.851.962.081.962.381.962.701001.100.921.110.781.110.671.110.57For the reason that in some available papers, the authors use the word consistent to describe the high degree of similarity between predicted outcomes and empirical facts. Li [10], for example [10], points out in his paper that the increase in life expectancy of 65-year-old women in 2050 seems to account for a large part of the improvement in life expectancy at birth, this is consistent with the increasing importance of improving mortality in old age. Therefore, this paper also uses the word consistent to describe the high degree of similarity between predicted results and empirical facts.In summary, PCF model can well reflect the biological characteristics of human beings in forecasting population mortality, is consistent with the historical law of sex differences in population mortality in China, and is also consistent with the current empirical facts in developed countries at the same time.


## 4. Measurement of Longevity Risk of Pension

The actuarial present value of continuous life annuity can be used to measure the longevity risk of pension as follows:(16)APV=a¯x=a¯x:ω−x≈12+∑t=1ω−x−1tpxvt+12ω−xpxvω−x,where ω is the terminal age, tpx is the probability of people aged x living through t years, v is the discount factor with the expression as v=1+i−1, and i is the discount rate.Three age scenarios of retirement pensions are set: Scenario 1 is 60 years old for male and 55 years old for female; Scenario 2 is 60 years old for male and 60 years old for female; Scenario 3 is 65 years old for male and 60 years old for female. The reasons for choosing these three scenarios are as follows: the retirement age for both men and women under Scenario 1 maintain the current retirement age policy unchanged, which is 60 for men and 55 for women. In the case of Scenario 2, the retirement age is raised only for women, since the average life expectancy of women is higher than that of men, so that the retirement age for women may be appropriately raised if only considered at an academic level. In this scenario, it is postponed to the same retirement age as men. In the Scenario 3, the retirement age for both men and women has been raised, taking into account issues of social equity and preferential treatment of women’s benefits, in conjunction with the academic and sociological aspects, in the context of a sustained increase in life expectancy for both sexes. The number of years of postponement maintained the five years of postponement for women in Scenario 2. Assuming that the discount rate is 2.5%, and the terminal age is 100 years old, the actuarial present value of pension annuities under the three Scenarios can be calculated according to equation (16). The results are shown in Tables 3 and 4. Tables 3 and 4 are the actuarial present value of pension annuity based on PCF model and Lee-Carter model, respectively.Table 3
The actuarial present value of pension annuity based on the PCF model.
Year2020Year2030Year2040Year2050Scenario 1Male 603.694.344.875.28Female 555.496.036.446.75Total9.1810.3711.3112.03Scenario 2Male 603.694.344.875.28Female 604.354.865.265.57Total8.049.2110.1310.85Scenario 3Male 652.773.363.864.25Female 604.354.865.265.57Total7.128.239.129.82Table 4
The actuarial present value of pension annuity based on the Lee-Carter model.
Year2020Year2030Year2040Year2050Scenario 1Male 603.834.204.564.88Female 555.505.866.166.41Total9.3310.0610.7111.29Scenario 2Male 603.834.204.564.88Female 604.354.704.995.23Total8.188.909.5410.11Scenario 3Male 652.903.253.583.89Female 603.774.354.865.26Total6.677.608.449.15A comparison of Tables3 and 4 shows that, first, under the assumptions of Scenario 1, the total actuarial present value of life annuities derived from the PCF model increases from 9.18 in 2020 to 12.03 in 2050; However, the Lee-Carter model increases from 9.33 to 11.29, which is less than the former. Second, if the retirement age is gradually raised to 60 for women alone, the total actuarial present value of the annual life annuity in Scenario 2 will decrease significantly, the PCF model shows that the decreases in 2020, 2030, 2040 and 2050 are 1.14, 1.16, 1.18 and 1.18, respectively, compared with the Lee-Carter model, which shows the decreases in 2020, 2030, 2040, and 2050 are 1.15, 1.16, 1.17, and 1.18, respectively, there is no obvious difference. Over time, the impact of postponing retirement on the relief of pension liabilities is increasing. Third, if the retirement age are gradually pushed back to 65 for men and 60 for women, the actuarial present value of the annual living annuity in Scenario 3 will continue to decline. Compared with Scenario 1, the effect of policy regulation decreases year by year, which are 2.06, 2.14, 2.19, and 2.21 for PCF model, 2.66, 2.46, 2.27, and 2.14 for Lee-Carter model, respectively, this shows that the effect of policy regulation is further remarkable. Finally, by looking at the total actuarial present value of pensions on the diagonal of the table, the sum under the PCF model is 9.18 in 2020 in Scenario 1 and 9.21 in 2030 in Scenario 2, in Scenario 3, 9.12 in 2040. The total values of Lee-Carter model are 9.33, 8.90, and 8.44, and the decrease range is significantly larger than that of PCF model. The values of the three scenarios in three different years are close (the numerical relationship on other diagonals also conforms to the above the law), indicating that Scenario 2 can effectively delay the pension solvency pressure in Scenario 1 by ten years and Scenario 3 by 20 years based on Scenario 1.Above all, the actuarial present value of pension annuity calculated by PCF model has the following advantages: (1) the actuarial present value of pension annuity calculated by PCF model is high, which can effectively deal with the common defects of high mortality rate and underestimation of longevity risk in current stochastic mortality model. (2) PCF model effectively considers the consistency of mortality changes between sexes, and the calculated actuarial present value of pension annuity shows a stable sex difference, while the sex difference in Lee-Carter model increases gradually. (3) With the passage of time, the rate of population mortality decreases to a certain extent, and the speed begins to slow down. The expansion trend of the actuarial present value of the survival annuity calculated by the PCF model is decreasing, which has the same trend as the change of mortality rate.


## 5. Conclusion and Future Work

This paper explores the changing trend of sex difference in population mortality since the founding of New China, uses the Poisson common factor model (PCF model) to forecast the mortality rate of male and female, then compares the results with those of single population model, and discusses the scientific nature of the method used to forecast the population mortality. The conclusions are as follows.The sex combined stochastic mortality model can simultaneously reflect the common trend of mortality changes and the individual differences of mortality changes. By taking the consistency of sex differences in population mortality into account, the sex combined stochastic mortality model established can effectively capture the common trend of population mortality changes of different sexes and reduce the bias caused by poor quality or large fluctuations of gender-specific population mortality data. At the same time, the use of additional factors can reflect the characteristics of sex differences.The PCF model better reflects the biological characteristics of people in the forecasting of the population mortality in China, and there is no excessive sacrifice to the goodness-of-fit in this model. The sex ratio of population mortality forecast by the PCF model converges to a fixed value in the long term. The convergence result is in line with the biological characteristics of human beings. Moreover, the error value of the forecasting results does not increase significantly compared with that of the Lee-Carter model.The method of exchanging less sacrifice goodness-of-fit to more consistency between the population mortality data obtained and the biological laws makes the model more reasonable in long-term forecasting. It provides a scientific basis for the pension system, which needs long-term evaluation. Also, as for the parameter estimation method, the two-stage parameter estimation effectively overcomes the over-parameterisation problem of PCF model and can better make the calculation process converge, so that the realisation of model parameter estimation becomes more concise and efficient.The actuarial present value results of pension annuity based on the mortality data of the PCF model make up for the defects of the underestimation of longevity risks and have stronger stability, which is beneficial to provide the scientific basis for pension decision-making. The PCF model effectively considers the consistency of mortality changes between sexes, and the actuarial present value of pension annuity calculated by the model shows a stable sex difference In contrast the sex difference obtained from the Lee-Carter model increases gradually. Also, with time, the population mortality rate decreases to a certain extent and then the speed begins to slow down. The expansion trend of the actuarial present value of the life annuity calculated by the PCF model is also decreasing, which has the same trend with the change of mortality.This paper assumes that the number of deaths follows a Poisson distribution, but different individuals have different Poisson parameters, that is, the risk is heterogeneous, and these Poisson parameters obey another distribution, then the number of deaths will follow a mixed Poisson distribution. This paper does not fully discuss the models fitting under the assumption of different parameter distributions. Therefore, in the future studies, the mixed Poisson models under various distributions should be fully considered and the corresponding parameter estimation methods should be discussed at the same time. For example, the gamma distribution is a good choice to describe the Poisson parameter distribution, and the mixed Poisson distribution is a negative binomial distribution. The parameter estimation can be easily obtained by iterative calculation with the maximum likelihood method. In the future researches, the conditional maximum likelihood method can be used to avoid the problem that the results in the iterative calculation of parameter estimation of complex model are not easy to converge or converge to local optimum, the multipopulation mortality modelling can also be incorporated into the Bayesian Hierarchical Model Framework, and the MCMC method can be used to avoid the overfitting problem of parameter estimation. The data of population mortality in China come from census and sample survey. The fluctuation of sample data will interfere with the judgment of the trend of mortality. Therefore, the future research should focus on the improvement of the mortality smoothing method and the evaluation of the fitting quality of the mortality data of the elderly population.



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*Source: 2132224-2022-09-12.xml*