Assessing Delayed Retirement Policies Linked to Dynamic Life Expectancy with Stochastic Dynamic Mortality

Abstract

The question of how to effectively alleviate the financial pressure on pension insurance due to the increase in life expectancy has become an important issue in the reform of China’s social security system. This paper introduced two life expectancy-related delayed retirement schemes, namely the fixed expected retirement residual life and the fixed life burden ratio. We modeled the financial balance of the employee pension fund and the pension wealth of employees with a dynamic retirement age according to pension policy. Using the population mortality data, the dynamic retirement age under the two schemes was estimated under the stochastic mortality model. Following this, the impact of the two delayed retirement schemes was quantitatively assessed from the perspectives of the financial sustainability of the pension fund and the pension wealth of employees using insurance actuarial methods. This study found that the two life expectancy-related delayed retirement schemes have obvious effects on reducing the gap between the income and expenditure of the pension fund and increasing the pension wealth of employees. Moreover, it found that the fixed expected retirement residual life program contributes more than the fixed life burden ratio program to improve the financial sustainability of the pension fund and the pension wealth benefits of employees.

1. Introduction

Rapid economic growth has led to great improvements in material living and medical and health conditions, and the health of the population has been rapidly rising. The 2021 Statistical Bulletin on the Development of China’s Health Care Program revealed that the life expectancy per capita has reached 78.2 years of age. However, the legal retirement ages for male employees, female workers, and female officials are 60, 50, and 55 years old, respectively. The significant increase in life expectancy has accelerated the rate of population aging and led to more pension liabilities [1]. According to China’s Seventh National Population Census Bulletin (no. 5) on 11 May 2021, there are currently about 264.02 million people over the age of 60 in China, accounting for 18.70% of the total population. Among them, there are approximately 190.64 million people over the age of 65, accounting for 13.50 percent of the total population. China’s basic pension insurance adopts a pay-as-you-go financial system, and its sustainability is heavily dependent on demographic factors. Increasing longevity is changing the actuarial balance of pension funds [2]. The rapid aging of the demographic structure is putting increasing pressure on the pension insurance fund to make payments. In order to alleviate the financial pressure on basic pension insurance, delaying the retirement age has been considered a more effective countermeasure.

As delayed retirement involves the adjustment of the pension interests of workers of different generations, the realistic obstacles facing the policy reform should not be ignored. Survey results from People’s Daily have shown that more than 90% of the respondents have negative feelings towards delayed retirement. The quicker the retirement age is delayed, the more obvious the effect on easing the financial pressure of the pension insurance fund, but it is very likely to cause people’s resistance. Although the slower adjustment of the retirement age can improve public acceptance, the long adjustment cycle will make pension insurance require excessive financial subsidies, and the effect of improving the financial sustainability of the pension insurance fund is relatively limited. In light of the inconsistency of interests, what type of delayed retirement program can be implemented to simultaneously improve the financial situation of the pension insurance fund and increase the level of workers’ pension wealth is an urgent problem to be solved in the reform of the retirement system.

In order to alleviate the financial pressure on public pensions brought about by the increase in life expectancy, a gradual increase in the mandatory retirement age has become a reform policy adopted by the social security systems of OECD countries in response to the impact of population aging. Turner [3] examined the early retirement age policy of social security programs in 23 OECD countries over the years 1949–2035. The automatic adjustment mechanisms linked to life expectancy have been encouraged to be applied when changing key parameters of the pension system, such as retirement age [4]. Under the principle of intergenerational actuarial neutrality, Meneu et al. [5] developed a framework of automatic adjustment mechanisms for defined benefit pension systems that face longevity risk. Taking into account the impact of expected increasing longevity on pension systems, Belloni and Maccheroni [2] analyzed the actuarial characteristics of the Italian pension system throughout its transition from defined benefit (DB) to notional defined contribution (NDC) rules. In fact, some countries have already adjusted their full retirement age according to the evolution of life expectancy, such as The Netherlands. Stevens [6] investigated the effect of five retirement policies linked to survival probabilities on the distribution of future full retirement and longevity risk.

The Institute of Population and Labor Economics of the Chinese Academy of Social Sciences released a research report suggesting that the retirement age for men should be raised by one year every six years and that for women every three years starting in 2018, and that the retirement age for both men and women will reach 65 years old by 2045. However, whether delayed retirement can be effectively implemented also depends on certain practical factors and needs to be justified from multiple perspectives. Zeng et al. [7] compared different options for delaying the retirement age based on the perspectives of pension insurance fund sustainability and fiscal responsibility and argued that no matter what type of retirement delay is implemented, a delay of six months per year is the best or more desirable option. There is no consensus on which types of delaying the retirement age are more suitable for China’s pension fund. From the perspective of employee benefits, how delaying the retirement age affects pension wealth is a key concern. Some scholars believe that delaying retirement will increase the level of employee benefits and act as a gain on pension wealth. Yang et al. [8] constructed a basic pension wealth actuarial model and numerically simulated the personal pension assets corresponding to different retirement ages and found that the retirement age and personal pension wealth present an inverted U-shaped relationship.

Most of the existing relevant studies have discussed the financial impacts arising from the progressive adjustment of the statutory retirement age by targeting a certain age within a specific period of time. Such progressive retirement delay programs are not directly linked to the time-varying pattern of life expectancy. Compared with existing studies, this paper contributes to the existing literature in two aspects: on the one hand, our study introduced the fixed expected retirement residual life scheme and the fixed life burden ratio scheme to reform retirement age in China’s pension fund of employees, and on the other hand, it utilized the stochastic mortality model based on the mortality data of China’s population to estimate future retirement age under two delayed retirement programs linked to life expectancy. The advantage of these delayed retirement schemes is that the adjustment of retirement age depends on the growth of life expectancy. They can effectively reduce the longevity risk that the pension insurance fund faces by realizing the dynamic adjustment of retirement age in accordance with the time-varying law of life expectancy. On the other hand, from the perspective of both government and employees’ interests, we quantitatively assessed the financial impacts of the two delayed retirement schemes on the shortfall of income and expenditure of China’s pension insurance fund of employees and the level of employees’ pension wealth. To a certain extent, the research in this paper provides new ideas for China to formulate a scientific and reasonable progressive delayed retirement scheme.

The remainder of this article is organized as follows: Section 2 first proposes the dynamic equation of delayed retirement schemes linked to life expectancy with stochastic mortality. Then, we constructed the income and expenditure equations of the pension insurance fund and pension wealth of employees using the actuarial method. Section 3 calibrates the parameters of the models. Section 4 estimates the retirement age under the life expectancy-related delaying retirement schemes, and we calculated the financial balance of the pension fund and gain/loss coefficient of pension wealth for employees under thee delayed retirement schemes. Moreover, sensitive analyses were also performed in this section. Section 5 summarizes the conclusions of this article.

2. Mathematical Model

2.1. Stochastic Dynamic Mortality Modeling and Life Expectancy

In the reform of social security systems in developed countries, the life expectancy of the population is an important reference factor in delaying the mandatory retirement age. The life table approach to estimating future life expectancy using historical mortality rates may lead to an underestimation of pension expenditure. In contrast, stochastic dynamic mortality models can portray the time-varying and stochastic nature of population mortality changes and can more accurately predict future life expectancy. Currently, there are a variety of classical stochastic mortality models. Wang and Lu [9] showed that the CBD mortality model [10,11,12] is more suitable for capturing the dynamic and uncertainty of China’s population mortality model than the Lee Carter model and Bayesian hierarchical model. Thus, this paper employed the CBD mortality model to predict the life expectancy of the population. The specific form of the function is as follows:

$$\ln \frac{q_x\left(t\right)}{p_x\left(t\right)}=k_t^1+k_t^2\left(x-\overline{x}\right)+{\epsilon }_x\left(t\right)$$

(1)

where $q_x\left(t\right)$ is the probability of dying within one year for a person $x$ years old living in $t$ period. The survival probability is represented by $p_x\left(t\right)=1-q_x\left(t\right)$. $k_t^1$ and $k_t^2$ denote two stochastic processes, in which $k_t^1$ represents the mortality improvement over time for all ages, and $k_t^2$ represents that the mortality rate improves less in the older age group than in the younger age group. $\overline{x}$ is the sample age mean, derived from the age-weighted average of the ages in the sample, where the weight of each age is the proportion of the population of that age in the total population. ${\epsilon }_x\left(t\right)~\mathrm{N}\left(0,{\sigma }_{\epsilon }^2\right)$. The CBD model does not suffer from variable identification problems and, therefore, does not need to add any constraints. We used Farlle’s method to calculate the probability of death, which corresponds to a life expectancy of:

$$T_x\left(t\right)={\Sigma }_{k=1}^{m-x}k{p}_x\left(t\right)+0.5$$

(2)

$$k{p}_x\left(t\right)={\Pi }_{t=0}^{k-1}\left[1-q_{x+t}\left(t\right)\right]$$

(3)

where $T_x\left(t\right)$ is the remaining lifetime of an individual with age $x$ at time $t$, and $m$ is the maximum lifespan.

2.2. Modeling the Dynamic Delayed Retirement Schemes Linked to Life Expectancy

(1)

The fixed expected residual life in retirement scheme (referred to as Scheme 1).

The adjustment of retirement age under the fixed expected residual life in retirement scheme was determined using the following formula:

$$E_0\left[T_{R{A}_0^{\zeta }}\left(0\right)\right]=E_t\left[T_{R{A}_t^{\zeta ,C}}\left(t\right)\right], \zeta =1, 2$$

(4)

where $R{A}_0^{\zeta }$ is the gender-specific retirement age ($\zeta =1$ for women and $\zeta =2$ for men) at period 0; $R{A}_t^{\zeta ,C}$ is the gender-specific retirement age under the fixed expected residual life in retirement scheme;$E_t$ is the mathematical expectation of the moment conditions; $E_0\left[T_{R{A}_0^{\zeta }}\left(0\right)\right]$ is the gender-specific expected remaining life at retirement for an individual with a retirement age of $R{A}_0^{\zeta }$ at period 0; and $E_t\left[T_{R{A}_t^{\zeta ,C}}\left(t\right)\right]$ is the expected residual life at retirement by gender for an individual with a retirement age of $R{A}_t^{\zeta ,C}$ at period $t$. Delaying the retirement age under this program will not reduce the length of time for which a retired employee can receive a pension.

(2)

The fixed life burden rate scheme (referred to as Scheme 2).

The adjustment of retirement age under the fixed life burden rate scheme was determined using the following formula:

$$\frac{E_0\left[T_{R{A}_0^{\zeta }}\left(0\right)\right]}{R{A}_0^{\zeta }-PA}=\frac{E_t\left[T_{R{A}_t^{\zeta ,R}}\left(t\right)\right]}{R{A}_t^{\zeta ,R}-PA}, \left(\zeta =1, 2\right)$$

(5)

where $R{\mathrm{A}}_0^{\zeta }$ is the current mandatory retirement age; $R{A}_t^{\zeta ,R}$ is the retirement age under the fixed life burden rate scheme; $PA$ is the age of entry into the labor market; and $R{A}_t^{\zeta ,R}-PA$ is the years of work. Under this scheme, the ratio of the length of time that a retired worker will receive a pension to the length of time that he or she will pay pension insurance tax in different periods is constant. This scheme reflects intergenerational equity to some extent.

2.3. Population Growth Modeling

China’s pension insurance is broadly categorized into employees’ pension insurance and residents’ pension insurance. Participants in employees’ pension insurance are all regular employees, requiring the employee’s enterprise to assume the obligation of contributions. The residents’ pension insurance participants are residents without stable employment. China has a typical urban–rural dual economic structure. It is difficult for farmers in China to obtain stable employment, and they have little opportunity to participate in employees’ pension insurance. This paper studied the urban employees’ pension insurance fund, in which the participants are mainly urban employees. Therefore, we needed to calculate the urban population. In this paper, the population growth model by province was used to forecast the urban population of each province in the future, and the model was as follows:

$$\{\begin{array}{c}{N}_{0,y}^{\zeta }\left(t\right)={\omega }^{\zeta }\left(t\right)\rho \left(t\right)N_y\left(t\right) \\ N_y\left(t\right)=\mathrm{TFR}\left(\mathrm{t}\right){\displaystyle \sum _{i=15}^{49}}{N}_{i,y}^1\left(t\right) \\ N_{1,y}^{\zeta }\left(t+1\right)=p_0^{\zeta }\left(t\right)N_{0,y}^{\zeta }\left(t\right)+M_{0,y}^{\zeta }\left(t\right) , \zeta =1, 2\\ \dots \dots \\ N_{i+1,y}^{\zeta }\left(t+1\right)=p_i^{\zeta }\left(t\right)N_{i,y}^{\zeta }\left(t\right)+M_{i,y}^{\zeta }\left(t\right), i=0,1, \cdots , 99\\ CT{N}_{i,y}^{\zeta }\left(t\right)=N_{i,y}^{\zeta }\left(t\right)U_y\left(t\right) \end{array}$$

(6)

where $N_{i,y}^{\zeta }\left(t\right)$ is the number of persons aged $i$ by sex $\zeta$ in province $y$ at time $t$;${\omega }^{\zeta }\left(t\right)$ is the sex ratio at birth in time $t; p_i^{\zeta }\left(t\right)$ is the survival rate of persons aged $i$ at time $t; \rho \left(t\right)$ is the neonatal survival rate; $N_y\left(t\right)$ denotes the number of babies born in province $y$ in time $t; \mathrm{TFR}\left(\mathrm{t}\right)$ is the total fertility rate; $M_{i,y}^{\zeta }\left(t\right)$ is the net in-migration of province $y;$ $CT{N}_{i,y}^{\zeta }\left(t\right)$ is the number of urban population with age $i$ by sex $\zeta$ in province $y$; and $U_y\left(t\right)$ is the urbanization rate in province $y$.

2.4. The Income and Expenditure of the Pension Insurance Fund

The employees’ pension fund in China consists of two parts: the integrated account and the individual account. In terms of ownership, the former is a public pension and the latter is a private pension. In terms of sources of funding, the integrated account is financed by enterprises’ contributions to pension insurance premiums based on a certain percentage of the employee’s salary, while the individual account is financed by a certain percentage of the employee’s salary. In terms of financial systems, the integrated account adopts a pay-as-you-go financial system, while the individual account adopts a fund accumulation system. In terms of liability, the integrated account adopts a defined benefit (DB), while the individual account adopts a defined contribution (DC). Retired workers receive a pension that consists of a basic pension from the integrated account plus a pension from the individual account. However, due to disorganized management, individual accounts have been diverted to the payment of pensions from the integrated account. Individual accounts have essentially become notional accounts.

The income part of the integrated account depends on the statutory contribution base, enterprise contribution rate, number of insured persons, and actual collection rate. The income part of the individual account depends on the statutory contribution base, individual contribution rate, number of insured persons, and actual collection rate. Let $CI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)$ be the income of the integrated account under retirement scheme $\xi$ with retirement age $R{\mathrm{A}}_{\mathrm{t}}^{\xi }$. Let $PI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)$ be the income of an individual account under a retirement scheme with retirement age $R{\mathrm{A}}_{\mathrm{t}}^{\xi }$. The specific formulas are as follows:

$$CI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)={\sigma }_1{\mathsf{\Sigma }}_{\mathsf{\zeta }=1}^2{{\displaystyle \sum }}_{i=PA}^{R{\mathrm{A}}_{\mathrm{t}}^{\zeta ,\xi }-1}CT{N}_i\left(t\right){\alpha }_1{\lambda }_1\gamma Z\overline{w}\left(t\right), \xi =C,R$$

(7)

$$PI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)={\sigma }_2{\mathsf{\Sigma }}_{\mathsf{\zeta }=1}^2{{\displaystyle \sum }}_{i=PA}^{R{\mathrm{A}}_{\mathrm{t}}^{\zeta ,\xi }-1}CT{N}_i\left(t\right){\alpha }_1{\lambda }_1\gamma Z\overline{w}\left(t\right), \xi =C,R$$

(8)

where $\xi =C$ represents the scheme with a fixed retirement residual life, and $\xi =R$ represents the scheme with a fixed life burden rate. ${\sigma }_1$ is the statutory contribution rate for enterprises,${\sigma }_2$ is the statutory contribution rate for individuals,${\alpha }_1$ is the urban employment rate,${\lambda }_1$ is the participation rate of urban active workers,$\gamma$ is the proportion of insured workers in enterprises to insured workers, $Z$ is the actual contribution rate, and $\overline{w}$(t) is the statutory contribution base of a salary.

The total income of the pension fund of employees is:

$$AI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)=CI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)+PI\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\xi }\right)$$

(9)

In 1997, the government carried out a reform of the pension insurance system, establishing a unified basic pension insurance system for enterprise employees. Prior to the reform, employees were not required to pay pension insurance premiums, and the company or the government would pay their pensions after retirement. After the reform, enterprise employees were required to bear pension insurance premiums. Therefore, the government categorizes employees who participate in pension insurance into “old people”, “middle people”, and “new people”. The “old people” are those who retired before 1997. The “new people” are those who joined the workforce after 1997. The “middle people” are those who joined the workforce before 1997 and retired after 1997. Pensions are paid by the pension fund in different ways for different types of retirees. The expenses of an integrated pension insurance account consist of the basic pension for “new people”, the basic pension and transitional pension for “middle people”, and the basic pension for “old people”. Individual account expenses consist of “newcomer” individual account pension expenses and “middle-aged” individual account pension expenses. We established the actuarial model of pension fund expenditure.

The pension benefits for “new people” include a basic pension from the integrated account and a pension from the individual account. Therefore, the expenditure of the pension fund, which is used to pay the pensions of “new people”, also includes two corresponding parts.

The expenditure of “new people” for the integrated account, $P_1\left(t,R{A}_t^{\zeta ,\xi }\right)$, is as follows:

$$\begin{gathered} P_1\left(t,R{A}_t^{\zeta ,\xi }\right)={\displaystyle \sum }_{i=R{A}_t^{\zeta ,\xi }}^{PA+t-1997-1}CT{N}_i^{NM}\left(t\right){\alpha }_2{\lambda }_2\gamma B_i^{NM}\left(t,R{A}_t^{\zeta ,\xi }\right) \\ \times {\prod }_{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)} , \end{gathered}$$

(10)

where $B_i^{NM}\left(t,R{A}_t^{\zeta ,\xi }\right)=\overline{w}\left(\mathrm{t}-\left(\mathrm{i}-R{A}_t^{\zeta ,\xi }\right)-1\right)\left(1+\mathrm{e}\right)\times 0.5\times \left(R{A}_t^{\zeta ,\xi }-\mathrm{PA}\right)\times 1\%, \mathrm{and}$

$$e=\varphi =\frac{1}{t-\left(i-R{A}_t^{\zeta ,\xi }\right)-1997}{\displaystyle \sum }_{j=1}^{t-\left(i-R{A}_t^{\zeta ,\xi }\right)-1997}\frac{w_{R{A}_t^{\zeta ,\xi }-j}\left(t-\left(i-R{A}_t^{\zeta ,\xi }\right)-j\right)}{\overline{w}\left(t-\left(i-R{A}_t^{\zeta ,\xi }\right)-j-1\right)}$$

where ${\alpha }_2$ is the employment rate for “new people”$; {\lambda }_2$ is the participation rate of urban retirees for “new people”$;$ $B_i^{NM}\left(t,R{\mathrm{A}}_{\mathrm{t}}^{\zeta ,\xi }\right)$ is basic pension expenditure for “new people”; $g_2\left(t\right)$ is the annual pension growth rate; $w_i\left(t\right)$ is the actual contribution salary of the insured person;$e$ is the index of the average contribution salary of the participant; $f$ is the index of contributions; and $\varphi$ is the actual average contribution index, and $f=\varphi$.

Let $P{G}_1\left(t,R{A}_t^{\zeta ,\xi }\right)$ be the expenditure on the individual account of the “new people” under retirement age $R{\mathrm{A}}_{\mathrm{t}}^{\zeta ,\xi }$:

$$\begin{gathered} P{G}_1\left(t,R{A}_t^{\zeta ,\xi }\right)={\displaystyle \sum }_{i=R{A}_t^{\zeta ,\xi }}^{PA+t-1997-1}CT{N}_i^{NM}\left(t\right){\alpha }_2{\lambda }_2\gamma \frac{I_i\left(t,R{A}_t^{\zeta ,\xi }\right)}{m^{\zeta }} \\ \times 12\times {\prod }_{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)} , \end{gathered}$$

(11)

where $I_i\left(t,R{A}_t^{\zeta ,\xi }\right)$ is the amount of individual accounts by gender, $I_i\left(t,R{A}_t^{\zeta ,\xi }\right)=\sum _{t=PA}^{R{A}_t^{\zeta ,\xi }-1}{w}_i\left(t\right){\sigma }_2\prod _{t=PA}^{R{A}_t^{\zeta ,\xi }-1}[1+r_p\left(t\right)]$, $r_p\left(t\right)$ is the interest rate of individual account bookings for the year, and $m^{\zeta }$ is the number of months of payment by gender $\zeta$.

The expenditure of “middle people” is more complex, comprising the expenditure on the integrated account and the expenditure on the individual account. Let $P_2\left(t,R{A}_t^{\zeta ,\xi }\right)$ be the expenditure of the integrated account for “middle people”:

$$P_2\left(t,R{A}_t^{\zeta ,\xi }\right)=\{\begin{array}{c}\begin{array}{c}\sum _{i=R{A}_t^{\zeta ,\xi }}^{t-1997+R{A}_t^{\zeta ,\xi }-1}CT{N}_i^{RM}\left(t\right){\alpha }_2{\lambda }_2\gamma (B_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)+T_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)) \\ \times \prod _{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)} ,\\ \mathrm{t}\left[2018, 1997+R{A}_t^{\zeta ,\xi }-\mathrm{PA}+1\right) \\ \end{array}\\ \sum _{i=PA+t-1997}^{R{A}_t^{\zeta ,\xi }+t-1997-1}CT{N}_i^{RM}\left(t\right){\alpha }_2{\lambda }_2\gamma (B_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)+T_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right))\\ \times \prod _{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)},\\ \mathrm{t}\ge 1997+R{A}_t^{\zeta ,\xi }-\mathrm{PA}+1\end{array}$$

(12)

where

$$B_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)=\overline{w}\left(t-\left(i-R{A}_t^{\zeta ,\xi }\right)-1\right)\left(1+e\right)\times 0.5\times (t-\left(i-R{A}_t^{\zeta ,\xi }\right)-1997)\times 1\%$$

$$T_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)=\epsilon \left[i-\left(t-1997\right)-PA\right] \overline{w}\left(t-\left(i-R{A}_t^{\zeta ,\xi }\right)-1\right)f ,$$

$B_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)$ is the basic pension for “middle people”, $T_i^{RM}\left(t,R{A}_t^{\zeta ,\xi }\right)$ is the transitional pension of “middle person”, and $\epsilon$ is the transition coefficient.

Let $P{G}_2\left(t,R{A}_t^{\zeta ,\xi }\right)$ be the individual account expenditure of “middle people”; then:

$$P{G}_2\left(t,R{A}_t^{\zeta ,\xi }\right)=\{\begin{array}{c}\begin{array}{c}\sum _{i=R{A}_t^{\zeta ,\xi }}^{t-1997+R{A}_t^{\zeta ,\xi }-1}CT{N}_i^{RM}\left(t\right){\alpha }_2{\lambda }_2\gamma \frac{I_i\left(t,R{A}_t^{\zeta ,\xi }\right)}{m^{\zeta }}\times 12 \\ *\prod _{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)}, t\left[2018,1997+R{A}_t^{\zeta ,\xi }-PA+1\right)\end{array}\\ \sum _{i=PA+t-1997}^{R{A}_t^{\zeta ,\xi }+t-1997-1}\begin{array}{c}CT{N}_i^{RM}\left(t\right){\alpha }_2{\lambda }_2\gamma \frac{I_i\left(t,R{A}_t^{\zeta ,\xi }\right)}{m^{\zeta }}\times 12 \end{array}\\ *\prod _{j=t-\left(i-R{A}_t^{\zeta ,\xi }\right)}^t\frac{1+g_2\left(j\right)}{1+g_2\left(R{A}_t^{\zeta ,\xi }\right)} , t\ge 1997+R{A}_t^{\zeta ,\xi }-PA+1\end{array}$$

(13)

The pensions of “old people” are paid according to the original policy. Let $P_3\left(t,R{\mathrm{A}}_0^{\zeta }\right)$ be the basic pension expenditure for “old people” with a retirement age of $R{\mathrm{A}}_0^{\zeta }$; then:

$$P_3\left(t,R{\mathrm{A}}_0^{\zeta }\right)={\sum }_{i=t-1997+R{\mathrm{A}}_0^{\zeta }}^{m}CT{N}_i^{OM}\left(t\right){\alpha }_2{\lambda }_2\gamma B_i^{OM}\left(t,R{\mathrm{A}}_0^{\zeta }\right) ,$$

(14)

where $B_i^{OM}\left(t,R{\mathrm{A}}_0^{\zeta }\right)={\left(1+g_2^b\right)}^{i-R{\mathrm{A}}_0^{\zeta }}{B}_{R{\mathrm{A}}_0^{\zeta }}^{OM}\left(t,R{\mathrm{A}}_0^{\zeta }\right)$,

$$B_{R{\mathrm{A}}_0^{\zeta }}^{OM}\left(t,R{\mathrm{A}}_0^{\zeta }\right)=D_t{w}_{R{\mathrm{A}}_0^{\zeta }-1}\left(t-1,R{\mathrm{A}}_0^{\zeta }\right),$$

$$w_{R{\mathrm{A}}_0^{\zeta }-1}\left(t-1,R{\mathrm{A}}_0^{\zeta }\right)=w_{PA}\left(t-1\right){\left(1+g_1^s\right)}^{\left(R{\mathrm{A}}_0^{\zeta }-PA-1\right)},$$

$B_i^{OM}\left(t,R{\mathrm{A}}_0^{\zeta }\right)$ denotes the basic pension expenditure of the “old people”$, g_2^b$ denotes the growth rate of retiree pensions with age,$D_t$ denotes the replacement rate of pensions for urban enterprise workers in 2007, and $g_1^s$ denotes the growth rate of wages before retirement.

The total expenditure of pension fund $AP\left(t,R{A}_t^{\xi }\right)$ is:

$$\begin{gathered} AP\left(t,R{A}_t^{\xi }\right)={\mathsf{\Sigma }}_{\mathsf{\zeta }=1}^2\left(P_1\left(t,R{A}_t^{\zeta ,\xi }\right)+P{G}_1\left(t,R{A}_t^{\zeta ,\xi }\right)+P_2\left(t,R{A}_t^{\zeta ,\xi }\right)\right) \\ +{\mathsf{\Sigma }}_{\mathsf{\zeta }=1}^2\left(P{G}_2\left(t,R{A}_t^{\zeta ,\xi }\right)+P_3\left(t,R{A}_t^{\zeta }\right)\right) \end{gathered}$$

(15)

The balance of the pension insurance fund, $C\left(t,R{A}_t^{\xi }\right)$, is as follows:

$$C\left(t,R{A}_t^{\xi }\right)=AI\left(t,R{A}_t^{\xi }\right)-AP\left(t,R{A}_t^{\xi }\right)$$

(16)

2.5. The Pension Wealth of Employees

Regarding delaying the retirement age, employees are worried about whether delaying the retirement age will lead to a reduction in pension benefits during the whole retirement period. The total pension benefit of retirees comprises the discounted sum of pension benefits received during the whole retirement period, which is defined as pension wealth. According to the policy of pensions, the pension received by retirees is equal to the basic pension from the integrated account plus the individual pension from the individual account. The individual account pension at each month is calculated using the accumulated savings amount of the individual account divided by the number of months of accrual corresponding to the retirement age. To assess the impact of delaying the retirement age on the pension benefits of retirees, we constructed a gain-and-loss index for pension wealth as follows. Let $P_{R{A}_t^{\zeta ,\xi }}$ be the gender-specific pension benefit at the first year of retirement. $pp{w}_{R{A}_t^{\zeta ,\xi }}^P$ denotes the gender-specific pension wealth, which is the sum of the present value of the pension benefits at retirement:

$$\begin{gathered} P_{R{A}_t^{\zeta ,\xi }}=\frac{1+e}{2}{\overline{w}}_{R{A}_t^{\zeta ,\xi }}\left(R{A}_t^{\zeta ,\xi }-PA\right)\times 1\% \\ =\frac{1+e}{2}{\overline{w}}_{R{A}_0^{\zeta }}{\left(1+g_1\right)}^{R{A}_t^{\zeta ,\xi }-R{A}_0^{\zeta }}\left(R{A}_t^{\zeta ,\xi }-PA\right)\times 1\% \end{gathered}$$

(17)

$$\{\begin{array}{c}pp{w}_{R{A}_t^{\zeta ,\xi }}^P=\frac{P_{R{A}_t^{\zeta ,\xi }}\left(1+r\right)}{r-g_2}\left[1-\frac{{\left(1+g_2\right)}^{T_{R{A}^{\zeta ,\xi }\left(t\right)}}}{{\left(1+\mathrm{r}\right)}^{T_{R{A}^{\zeta ,\xi }\left(t\right)}}}\right], \left(r\ne g_2\right)\\ pp{w}_{R{A}_t^{\zeta ,\xi }}^P=P_{R{A}_t^{\zeta ,\xi }}{T}_{R{A}^{\zeta ,\xi }\left(t\right)},\left(r=g_2\right)\end{array}$$

(18)

where $g_1$ is the average growth rate of wages before retirement,$g_2$ denotes the growth rate of pensions, and $r$ is the interest rate.

Let $G_{R{A}_t^{\zeta ,\xi }}$ be the pensions paid by individual accounts in the first year after retirement. $pp{w}_{R{A}_t^{\zeta ,\xi }}^G$ denotes the accumulated wealth from an individual account pension:

$$G_{R{A}_t^{\zeta ,\xi }}=\frac{12\times e{\sigma }_2{\overline{w}}_{R{A}_0^{\zeta }}{\sum }_{i=1}^{R{A}_t^{\zeta ,\xi }-R{A}_0^{\zeta }}{\left(1+g_1\right)}^{R{A}_t^{\zeta ,\xi }-R{A}_0^{\zeta }-i}{\left(1+r_p\left(t\right)\right)}^i}{m^{\zeta }}$$

(19)

$$\begin{gathered} pp{w}_{R{A}_t^{\zeta ,\xi }}^G=G_{R{A}_t^{\zeta ,\xi }}\left(1+\frac{1}{1+r}+{\left(\frac{1}{1+r}\right)}^2+\dots +{\left(\frac{1}{1+r}\right)}^{T_{R{A}^{\zeta ,\xi }\left(t\right)}-1}\right) \\ =G_{R{A}_t^{\zeta ,\xi }}\frac{1}{r}\left(1-\frac{1}{{\left(1+\mathrm{r}\right)}^{T_{R{A}^{\zeta ,\xi }\left(t\right)}-1}}\right) \end{gathered}$$

(20)

Therefore, the pension wealth, $pp{w}_{R{A}_t^{\zeta ,\xi }}$, under delay retirement scheme $\xi$ is:

$$pp{w}_{R{A}_t^{\zeta ,\xi }}=pp{w}_{R{A}_t^{\zeta ,\xi }}^P+pp{w}_{R{A}_t^{\zeta ,\xi }}^G$$

(21)

Then, the gain/loss coefficient of pension wealth, $\beta$, for delaying retirement age schemes is as follows:

$$\beta \left(R{A}_t^{\zeta ,\xi }\right)=\frac{pp{w}_{R{A}_t^{\zeta ,\xi }}}{pp{w}_{R{A}_0^{\zeta }}}, \xi =C,R$$

(22)

where $pp{w}_{R{A}_0^{\zeta }}$ is the pension wealth under the constant retirement age $R{A}_0^{\zeta }$ (present retirement age). When $\beta <1$, it indicates that delayed retirement age leads to a reduction in employee pension wealth, and there is a negative incentive for delayed retirement. When $\beta >1$, it indicates that delayed retirement age increases employee pension wealth and has positive incentives for delayed retirement.

Moreover, delaying the retirement age may have positive and negative effects on employee pension wealth. The employee’s pension wealth mainly depends on the initial pension entitlement (i.e., pension benefit at the first year of retirement) and the time length of the pension receipt. According to China’s policy, the initial pension entitlement has a positive relationship with the time length of the contribution. Delaying the retirement age will increase the time length of the contribution, further increase the initial pension entitlement and have a positive effect on the employee’s pension wealth. At the same time, the change in the time length of receiving pension benefits depends on the speed of delaying the retirement age and the growth rate of life expectancy. When the speed of delaying the retirement age is greater than the growth rate of life expectancy, the length of receiving pension benefits decreases over time, and there is a negative effect of delaying the retirement age on employee pension wealth, and vice versa. If the retirement age is delayed too quickly, it will have a negative impact through a reduction in the time length of receiving pension benefits, and when the intensity of this negative impact is greater than the intensity of the positive impact through an increase in the initial pension entitlement, it will produce a negative net impact on employee pension wealth. It follows that there is a certain threshold value for the speed of delaying the retirement age, below which delaying the retirement age increases the level of pension wealth. This threshold value for the speed of retirement age delay depends on the growth rate of life expectancy.

3. Calibration Parameters of the Model

We first estimated the stochastic dynamic mortality model using the mortality data of China’s population from 1994 to 2021, which were obtained from the population census and population sample survey data published by the National Bureau of Statistics (NBS). Considering the inconsistency of the number of people exposed to mortality risk in the survey samples for different years and the problem of large fluctuations in mortality rates, the data were corrected with reference to the method of Wang and Lu (2019) [9]. Using the estimated population mortality model, we predicted the mortality rate of the population aged 50–89 years old by gender. The mortality rate of the population in the senior age group of 90–100 years old was extrapolated using the CK model; then, we calibrated the parameters of the population growth model, the income and expenditure model of the pension insurance fund, and the pension wealth of employees (the details can be seen in Appendix A).

4. Numerical Analysis

4.1. Estimation of Retirement Age under the Life Expectancy-Related Delayed Retirement Schemes

Under the stochastic mortality model, Equations (4) and (5) were used to estimate the retirement ages of men and women for the period 2023–2043 under the fixed expected retirement residual life and fixed life burden rate. The retirement age will be adjusted to 63.05 years for men and 54.88 years for women in 2033 under the fixed expected retirement residual life scheme (Scheme 1) and will be adjusted to 65.86 years for men and 57.29 years for women in 2043. Under this scheme, the retirement age would be raised (on average) by one year every 3.5 years for men and one year every four years for women. Under the fixed life burden rate scheme (Scheme 2), the retirement age would be adjusted to 60.96 for men and 52.75 for women in 2033, and to 61.88 for men and 53.40 for women in 2043. The average increase under this scenario would be one year every eleven years for men and one year every fifteen years for women. Figure 1 depicts the trend of the retirement age under the two delayed retirement schemes, and it can be seen that the retirement age under the fixed expected retirement residual life scheme is delayed relatively quickly. The faster adjustment of retirement age under the fixed expected retirement residual life scheme can alleviate the problem of financial sustainability of the pension insurance fund in a shorter period of time and reduce the pressure of replenishment of the pension insurance fund in the future, but the faster adjustment may result in damage to the interests of the employees. The slower pace of retirement age adjustment under the fixed life-burden rate scheme had a lower impact on employee pension benefits and is extremely acceptable for the public, but at the same time, a longer adjustment cycle will increase administrative costs and create greater financial pressure on the pension fund to make up for the shortfall.

4.2. Financial Balance of the Pension Fund under Delayed Retirement Schemes

According to the income and expenditure model of the pension fund, we calculated the financial balance of the pension insurance system, which is shown in

Table 1. The financial balance of the pension fund under life expectancy-linked delayed retirement schemes.

Province	2033			2043
	Current Scheme	Scheme 1	Scheme 2	Current Scheme	Scheme 1	Scheme 2
Guangdong	−1643.58	340.39	−1235.58	−9017.12	−5807.75	−8028.39
Jiangxi	−582.17	7.42	−451.83	−2826.75	−1636.11	−2201.91
Liaoning	−1369.25	−500.49	−1003.08	−3753.56	−2433.80	−2981.39
Xinjiang	−751.50	−346.71	−498.71	−2088.43	−1353.85	−1735.62
National average	−1103.60	−356.24	−892.94	−3675.04	−2321.69	−2926.63

Note: The table only shows the current balance of the pension fund for the four provinces representing the east, central, northeast, and west, which have different levels of economic development.

. Under the current fixed retirement age policy, the average balance of the pension insurance fund nationwide will be −110.360 and −367.504 billion yuan in 2033 and 2043, respectively. The deficit of income and expenditure in the pension insurance fund will increase year by year, which will inevitably affect the sustainability of the pension insurance system in the future. Under the fixed retirement residual life scheme in 2033 and 2043, the average financial shortfall of the pension insurance fund in all provinces will be 35.624 and 232.169 billion yuan, respectively. Compared with the current fixed retirement age scenario, the size of the financial shortfall was reduced by 74.736 and 135.335 billion yuan, respectively. This means that the fixed retirement life expectancy scheme (Scheme 1) will reduce the shortfall of the pension fund by 67.72% and 36.83% in the next ten and twenty years, respectively. The average financial shortfall of the pension fund in 2033 and 2043 for all provinces under the fixed life burden rate scheme (Scheme 2) was calculated to be 89.294 and 292.663 billion yuan, respectively. Compared with the current fixed retirement age scenario, the size of the shortfall shrank by 21.066 and 74.841 billion yuan, or 19.09% and 20.36%, respectively. In comparison, it was found that the fixed expected retirement residual life scheme is more effective in mitigating the financial risk of the pension fund. In addition, taking the four provinces of Guangdong, Jiangxi, Liaoning, and Xinjiang as representatives of four regions with different levels of economic development in the east, central, northeast, and west, Table 1 lists the balance of the pension insurance fund in the four provinces under two life expectancy-related delayed retirement programs. It also showed that delayed retirement cannot solve the structural contradiction of the pension insurance fund caused by the differences in development level and demographic structure among provinces.

4.3. The Gain/Loss Coefficient of Pension Wealth with Delayed Retirement Schemes

The gain/loss coefficient of pension wealth for delaying retirement age schemes was used to measure the impact of the two life expectancy-related delayed retirement schemes on the total pension wealth of employees, and the results are shown in

Table 2. The gain/loss coefficients on the pension wealth of employees under two delayed retirement schemes.

Province	2033		2043
	Scheme 1	Scheme 2	Scheme 1	Scheme 2
Hillsides	1.1034	1.0804	1.0617	1.0430
Jiangxi	1.1006	1.0772	1.0590	1.0414
Liaoning	1.1006	1.0771	1.0589	1.0413
Xinjiang	1.1012	1.0779	1.0596	1.0417
National average	1.1020	1.0788	1.0603	1.0422

Note: The table shows only the gain/loss coefficients representing the total pension wealth of employees in the four provinces in the east, central, northeast, and west regions with different levels of economic development.

. It can be seen that the average pension wealth gain/loss coefficient of each province under the two delayed retirement schemes was greater than one, which indicates that the level of employee pension wealth under the two delayed retirement schemes is higher than that under the no-delayed retirement age, and that there is no decrease in employee pension wealth. The level of employee pension wealth mainly depends on the initial pension entitlement and the time length of pension receipt. Under the two delayed retirement schemes, the increase in the number of years of contributions and the increase in the accumulation of personal accounts have led to an increase in the level of initial pension benefits after retirement, which has led to a higher level of pension wealth than that under the current retirement age program.

The average gain/loss coefficients of pension wealth for 2033 and 2043 under Scheme 1 were 1.1020 and 1.0603, respectively, while the average provincial pension wealth gain/loss coefficients under Scheme 1 were 1.0788 and 1.0422, respectively. In comparison, Scheme 1 generated a higher amount of employee pension wealth than Scheme 2. The reason was that although the contribution base was the same under Scheme 1 and Scheme 2, Scheme 1 has a greater deferral, a longer contribution period, and a higher total contribution amount, which resulted in an increase in pension wealth that exceeds the increase in contribution cost. As a result, the positive effect of Scheme 1 on employee pension wealth was deemed to be better. In addition, there was no significant inter-provincial variability in the impact of the two delaying retirement schemes on employee pension wealth.

Employee pension income consists of two parts: the basic pension and the personal account pension.

Table 3. The gain/loss coefficients on the pension wealth of male and female employees.

Groups	2033		2043
	Scheme 1	Scheme 2	Scheme 1	Scheme 2
Gain/loss coefficient for basic pension
Male	0.9323	0.9668	0.9695	0.8460
Female	0.9530	0.9774	0.9781	0.8894
Gain/loss coefficient for personal account pension
Male	1.1309	1.1120	1.0889	1.0584
Female	1.0579	1.0278	1.0150	1.0167
Gain/loss coefficient on total pension wealth
Male	1.1307	1.1119	1.0889	1.0583
Female	1.0577	1.0277	1.0150	1.0165

measures the effects of two delaying retirement schemes on different parts of pension wealth for male and female employees. The gain/loss coefficients of pension wealth from the basic pension and personal account pension were separately calculated. It was found that the gain/loss coefficient of the basic pension part was less than one, but the gain/loss coefficients of the personal account pension were more than one for both male and female employees. This implies that the two delaying retirement schemes reduce pension wealth from the basic pension part, whereas they increase personal account pension wealth compared with the present retirement age policy. In terms of total pension wealth for employees, the gain/loss coefficients for male employees under both delaying retirement schemes were higher than those of female employees, and all of them were more than one. This indicated that the two schemes have a greater impact on the pension wealth of male workers compared with the present retirement age policy. It was further found that Scheme 1 had a greater effect on increasing the total pension wealth of both male and female workers than Scheme 2.

4.4. Sensitivity Analysis

(1)

Sensitivity analysis for the current financial balance of the pension fund.

Table 4. Sensitivity analysis for pension balance with respect to changes in the fertility rate.

Scheme	Fertility Rate Reduced by 30%		Fertility Rate Increased by 30%
	2033	2043	2033	2043
Scheme 1	−1286.43	−4384.86	575.42	−224.35
Scheme 2	−1751.65	−4599.87	−37.97	−1246.74

Note: The values in the table are the national averages by province; the same table below.

,

Table 5. Sensitivity analysis for pension balance with respect to changes in the interest rate.

Scheme	30% Reduction in Interest Rate		30% Increase in Interest Rate
	2033	2043	2033	2043
Scheme 1	−205.83	−1605.43	−549.94	−3203.66
Scheme 2	−741.42	−2328.21	−1103.89	−3791.35

,

Table 6. Sensitivity analysis for pension balance with respect to changes in the wage growth rate.

Scheme	30% Reduction in Wage Growth Rate		30% Increase in Wage Growth Rate
	2033	2043	2033	2043
Scheme 1	−402.30	−2368.22	−299.70	−2139.65
Scheme 2	−942.41	−3016.46	−847.20	−2830.17

and

Table 7. Sensitivity analysis for pension balance with respect to changes in the participation rate.

Scheme	20% Reduction in Participation Rate		20% Increase in Participation Rate
	2033	2043	2033	2043
Scheme 1	−309.13	−1646.14	−463.69	−2469.20
Scheme 2	−761.54	−2315.50	−1142.30	−3473.24

show further analysis of the sensitivity of the pension balance model with respect to changes in the total fertility rate, individual account bookkeeping interest rates, wage growth rates, and participation rates. The results revealed that the financial shortfalls of the pension fund in 2033 and 2043 under Scheme 1 were lower than under Scheme 2.

(2)

Sensitivity analysis for employee pension wealth.

Table 8. Sensitivity analysis for employee pension wealth with respect to changes in the interest rate.

Scheme	20% Reduction in Interest Rate		20% Increase in Interest Rate
	2033	2043	2033	2043
Scheme 1	1.1017	1.0421	1.1023	1.0423
Scheme 2	1.0784	1.0601	1.0791	1.0605

,

Table 9. Sensitivity analysis for employee pension wealth with respect to changes in the pension growth rate.

Scheme	Pension Growth Rate Reduced by 20%		20% Increase in Pension Growth Rate
	2033	2043	2033	2043
Scheme 1	1.0946	1.0344	1.1095	1.0500
Scheme 2	1.0712	1.0522	1.0864	1.0685

and

Table 10. Sensitivity analysis for employee pension wealth with respect to changes in the wage growth rate.

Scheme	20% Reduction in Wage Growth Rate		20% Increase in Wage Growth Rate
	2033	2043	2033	2043
Scheme 1	0.7111	0.6695	1.3192	1.2492
Scheme 2	0.6966	0.6820	1.2910	1.2704

present the analyses of the sensitivity of the pension wealth model with respect to changes in the individual account crediting rates, pension growth rates, and wage growth rate. The results of each table showed that the gain/loss coefficients of employee pension wealth under Scheme 1 was higher than under Scheme 2.

5. Conclusions

With China’s huge population base, the continuous growth of life expectancy has made the social security pressure on the elderly population increasingly prominent. Drawing on the international experience of delayed retirement, this paper introduced two life expectancy-linked delayed retirement schemes (the fixed expected retirement residual life scheme and the fixed life burden rate scheme). Unlike the fixed gradual delayed retirement scheme, the life expectancy-linked delayed retirement scheme is a retirement age adjustment mechanism that relies on the population’s life expectancy growth pattern. From the perspectives of the financial sustainability of the pension insurance fund and employee pension wealth, this paper quantitatively evaluated the economic impacts of the two delayed retirement schemes. The quantitative analysis led to the following specific conclusions: (1) The retirement age under the fixed expected retirement residual life scheme is raised by one year on average every 3.5 years for men, and by one year on average every four years for women. The fixed life burden rate scenario increases by an average of one year every 11 years for men and one year every 15 years for women. (2) Relative to the current retirement age scheme, the fixed expected retirement residual life scheme would result in an average reduction in the financial gap of the pension fund of 67.72% and 36.83% over the next ten and twenty years, respectively. Compared with the fixed life burden ratio scheme, the fixed expected retirement balance life scheme is more effective in mitigating the payment risk of the pension insurance fund. (3) The fixed expected retirement residual life scheme will increase employee pension wealth in the next twenty years, and this effect was found to be better than that achieved with the fixed life burden rate scheme. (4) The relative advantages of the fixed expected retirement life balance program in narrowing the gap between the income and expenditure of the pension fund and increasing the pension wealth of the employees are also reflected in the provinces of different economic levels, among which Guangdong Province is the most significantly affected by the economic impacts of the delayed retirement scheme. Moreover, the fixed expected retirement residual life scheme is more efficient in terms of the financial sustainability of the pension fund and employee pension wealth, but it is also less fair than the fixed life burden rate scheme.

Although our paper constructed the income and expenditure model of the pension insurance fund and the pension wealth model of employees under life expectancy-linked delayed retirement programs, other stochastic aspects have not been definitively addressed to avoid a too complex model. For example, unemployment and surrender risks should be considered in the actuarial models.