A Simple Method for Projecting Pension Deficit Rates and an Illustrative Application

Abstract

In this chapter presented a simple method associated with the ProFamy projection model and software to project the annual pension deficit rate based on (1) The elderly dependency ratio determined by demographic factors of fertility, mortality and migration; (2) The retirement age; and (3) Four (or three) pension program parameters, which can be predicted by trend extrapolation or expert opinions. These input parameters can be derived from commonly available data. The illustrative application to China demonstrates that if the average age at retirement gradually increases from the current very low level to age 65 for both men and women in 2050, the annual pension deficit rate would be largely reduced or eliminated under various possible demographic regimes up to the middle of this century. With everything else being equal, the annual pension deficit rate in the scenario of medium fertility (associated with a two-child policy) would be much lower than that under low fertility (associated with the current fertility policy unchanged) after 2030. The impact of potentially faster mortality decline is likely sizable but relatively moderate; it starts earlier than the effects of fertility change. Note that one may also use the simple method presented in this chapter to explore the magnitude and timing of impacts on future pension deficits due to alternative international migration and/or pension policies by predicting or assuming the size and age/gender structure of international migration and/or the pension program parameters.

Appendix 1: Derivation of the Simple Method

Let W(t) denote the total number of workers in year t; the term “workers” here refers to those who reside/work in rural or urban areas and participate in the DB or DC pension program.

R(t), the total number of retirees in year t; the term “retirees” here refers to those who are retired and receive pension benefits, regardless of whether it is a DB or DC program and rural or urban residence.

d(t), the retiree-worker ratio in year t, namely, the ratio of total number of retirees to total number of workers in year t, $d(t)=\frac{R(t)}{W(t)}$

A(t), the average wage per worker in year t;

P(t) (the contribution rate in year t) and B(t) (the replacement rate in year t) are as defined in the text.

If the total amount of the pension fund premium contributions by workers and employers is equal to the total pension payment to the retirees in year t, the following equation holds: P(t)[A(t)W(t)] = [B(t)A(t)]R(t),

Dividing both sides by A(t) W(t), we get:

$$ P(t)=B(t)\cdot d(t) $$

(7.3)

The above equation is the classic basic equilibrium equation which expresses the annual balanced pension funds (e.g., Becker and Paltsev 2001: 19; Hamayon and Legros 2001; Sin 2005). If d(t) could be reasonably projected into the future years, one could simply use the classic basic equilibrium equation to project the annual pension deficits. However, it is extremely difficult to do so, because d(t), the retiree-worker ratio in year t, mixes the impacts of demographic parameters (such as fertility, mortality, and migration) and changes in the prevalence of pension program coverage among the elderly and participation among workers. Therefore, Zeng (2011) was motivated to decompose d(t) into a couple of demographic parameters and pension program variables that are reasonably predictable.

Let n(t) denote the annual pension deficit rate, which is defined and discussed in the text. The following equation holds in any case of pension fund deficit, balance or surplus:

$$ P(t)\ \left[A(t)\ W(t)\right]+n(t)\ A(t)\ W(t)=\left[B(t)\ A(t)\right]\ R(t) $$

Dividing both sides of the above equation by A(t) W(t), we get:

$$ \begin{array}{l}P(t)+n(t)=B(t)\cdot d(t)\\ {}n(t)=B(t)\ d(t)\hbox{--} P(t)\end{array} $$

(7.4)

We may rewrite Eq. 7.4 as:

$n(t)=B(t){d}_2(t)\frac{d(t)}{d_2(t)}-P(t)$, where d ₂(t) is the dependency ratio of elderly (defined in the text) which is easily predictable as it can be derived from commonly available population forecasting based on parameters of fertility, mortality, and migration.

$$ \begin{array}{l}\frac{d(t)}{d_2(t)}=\\ {}\kern1em \frac{\mathrm{total}\ \mathrm{number}\ \mathrm{of}\ \mathrm{retirees}/\mathrm{total}\ \mathrm{number}\ \mathrm{of}\ \mathrm{workers}}{\mathrm{total}\ \mathrm{number}\ \mathrm{of}\ \mathrm{persons}\ \mathrm{over}\ \mathrm{average}\ \mathrm{age}\ \mathrm{at}\ \mathrm{retirement}/\mathrm{total}\ \mathrm{number}\ \mathrm{of}\ \mathrm{persons}\ \mathrm{of}\ \mathrm{working}\ \mathrm{age}}\end{array} $$

We rearrange the components of $\frac{d(t)}{d_2(t)}$ to make it easier for interpretation,

$$ \begin{array}{l}\frac{d(t)}{d_2(t)}=\\ {}\kern1em \frac{\mathrm{total}\#\mathrm{of}\ \mathrm{retirees}/\mathrm{total}\#\mathrm{of}\ \mathrm{persons}\ \mathrm{over}\ \mathrm{average}\ \mathrm{age}\ \mathrm{at}\ \mathrm{retirement}}{\mathrm{total}\#\mathrm{of}\ \mathrm{workers}\ \mathrm{who}\ \mathrm{participate}\ \mathrm{in}\ \mathrm{the}\ \mathrm{pension}\ \mathrm{program}/\mathrm{total}\#\mathrm{of}\ \mathrm{persons}\kern0.5em \mathrm{of}\ \mathrm{working}\ \mathrm{age}}\end{array} $$

the numerator and denominator of the right side of the above equation is r(t), the retirement rate, and e(t), the pension program participation rate (r(t) and e(t) are defined in the text). Thus,

$$ n(t)=B(t)\cdot {d}_2(t)\frac{r(t)}{e(t)}-P(t) $$

(7.5)

As discussed in the text, when reliable data for estimating/projecting r(t) and e(t) separately are not available, one may simply project (or assume) the c(t) (= r(t)/e(t)). Thus,

$$ n(t)=B(t){d}_2(t)c(t)-P(t) $$

(7.6)

Based on Eq. 7.5 or Eq. 7.6, one may investigate another interesting policy question: How should the contribution rate P(t), replacement rate B(t), and/or average age at retirement be set so that the annual pension deficit rate is zero in future years? In such a policy analysis exercise, one may formulate a set of simultaneous equations: one equation is Eq. 7.5 or Eq. 7.6 with n(t) being set to zero; another one or two equations present the constraints assumed by the analyst, such as simultaneously adjusting the replacement rate and the contribution rate in opposite directions, while ensuring that relative changes in wages for workers and retirement benefits for retirees are the same or different, depending on either DB or DC participants.¹⁶ The estimators for the adjustment indices to adjust the replacement rate and the contribution rate can be obtained by resolving the simultaneous equations, or numerical trailing/simulation.

While n(t) (as percent of the total wages) is a valid indicator of annual pension deficit, one may go one step further to estimate m(t), the index of annual pension deficit as a percentage of GDP. One can compute m(t) by multiplying n(t) by S(t), the total wages as a percentage of GDP, which is relatively easy to predict, and the time series data are usually available.

$$ m(t)=n(t)S(t)=\left[B(t){d}_2(t)\frac{r(t)}{e(t)}-P(t)\right]S(t) $$

(7.7)

Or

$$ m(t)=n(t)S(t)=\left[B(t){d}_2(t)c(t)-P(t)\right]S(t) $$

(7.8)