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Long-Run Biological Interest Rate for Pay-As-You-Go Pensions in Advanced and Developing Countries1

Author(s):
Masahiro Nozaki
Published Date:
April 2017
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I. Introduction

Public old-age pensions provide insurance against post-retirement income losses. Broadly speaking, an old-age pension system is run either as a pay-as-you-go scheme, where contributions from the working-age population are used to pay for pension benefits for the retired in the current period, or as a funded scheme, where workers’ contributions are invested in financial assets that pay for their own retirement benefits. The pay-as-you-go scheme is more common in practice,2 while its sustainability hinges on demographic prospects. In fact, in many advanced countries, population ageing has posed challenges in maintaining pay-as-you-go schemes and prompted reforms.

The sustainability of a pay-as-you-go pension system rests ultimately on whether it offers an adequate internal rate of return for current and future generations. If not, working-age individuals opt out and stop contributing to the system. Their exits would leave unfunded liabilities for paying pension benefits to current and future retirees, or defaults of the liabilities would push the pension-dependent elderly into poverty. Facing political pressure, policymakers may be tempted to design pay-as-you-go systems that offer attractive internal rates of return for living generations, at costs with lower internal rate of returns for generations yet to be born. Such Ponzi-like schemes would become unsustainable as soon as it becomes clear that the pension systems could not promise adequate internal rates of return for young generations.

In this context, this paper aims to estimate the internal rate of return that a sustainable pay-as-you-go pension scheme would offer current and future generations. In a simple textbook model of overlapping generations, the internal rate of return for a sustainable pay-as-you-go pension scheme equals the growth rate of the contributions collected from the working-age population. This in turn is decomposed into working-age population growth and productivity growth. Observing this, Samuelson (1958) referred to population growth as the “biological interest rate.”

This paper applies this idea to the real world. It develops an equivalent of Samuelson’s biological interest rate in a general, non-steady-state overlapping generation economy, calling it the Long-Run Biological Interest Rate (LBIR). The LBIR is then estimated for a pay-as-you-go pension scheme that is sustained over 2015–2100 in each of 172 countries, using the United Nation’s age-by-age population projection data (United Nations, 2015). Because the internal rate of return for a sustainable pay-as-you-go pension scheme equals the sum of the LBIR and future productivity growth, a country with a low LBIR would need high productivity growth to provide and maintain an adequate internal rate of return for a pay-as-you-go pension system. The LBIR would serve as a useful benchmark even for countries with funded pension schemes by clarifying what a pay-as-you-go alternative could offer.

Two issues are worth noting in evaluating the LBIR estimates. First, the pay-as-you-go scheme would have to provide pensions for the elderly during its initial years, even though they have never contributed into the scheme. This initial cost is accounted for in the LBIR estimate. Second, it is assumed that the pay-as-you-go pension scheme can accumulate financial reserves to save for rising pension payments in the future, a standard practice in many countries.3 Conditional on a projected demographic path, such a scheme can promise a uniform LBIR for current and future generations. This approach ensures inter-generational equity by construction, an advantage over traditional approaches, such as generational accounting, that tend to be complex and data-demanding. For simplicity, it is assumed that there are no financial reserves at the beginning of the pension scheme.

The key findings of this paper are summarized as follows. First, LBIRs are globally low. The median LBIR for 172 countries is estimated to be as low as 0.9 percent per year, with LBIRs estimated to be negative in 54 countries. Adding historical productivity growth as a proxy for future productivity growth, one can gauge the corresponding internal rate of return for a pay-as-you-go scheme. The median internal rate of return for 172 countries calculated this way is 2.4 percent per year, only marginally exceeding the U.S. risk-free rate of return over the past decades. The internal rates of return are estimated to be negative in 13 countries.

Second, LBIRs vary widely across country groups. In advanced countries, where population ageing has progressed much faster than in developing countries, the median LBIR is -0.6 percent per year, and the LBIRs are estimated to be negative in 31 out of 34 countries. The median of the LBIR plus historical productivity growth is only 1.4 percent per year for this group. Looking at developing countries by region, the median LBIR is slightly negative at -0.1 percent per year in emerging Europe and the Commonwealth of Independent States (CIS), but it is relatively high in sub-Saharan Africa (2.8 percent) and in the Middle East and North Africa (1.6 percent). The LBIRs plus historical productivity growth are relatively high in developing Asia (the median is 3.8 percent) and sub-Saharan Africa (3.4 percent). The former is supported by high historical productivity growth while the latter benefits from the region’s high LBIRs.

Third, raising the retirement age can ratchet up the LBIR. Setting a higher retirement age by five years (from 65 to 70) increases the LBIR by around 40 basis points on average, with the increase being similar across countries, regardless of the LBIR levels. This finding supports raising the retirement age as an effective policy lever for improving the sustainability of a pay-as-you-go pension scheme.

Fourth, fertility prospects have a substantial impact on the LBIR. The LBIRs are calculated under low- and high-fertility scenarios considered by the United Nations’ population projections (United Nations, 2015). These scenarios represent extreme events for most of the countries in the sample, except sub-Saharan African countries. On average, the LBIR would be about 0.5 percentage points lower (higher) under the low-fertility (high-fertility) scenario than under the medium-fertility scenario.

Finally, the LBIR affects the size of pension benefits relative to pension contributions. I define the Replacement-Contribution Ratio (RCR) as the ratio of the pension replacement rate (i.e., pension benefits as a percent of wages) to the contribution rate (i.e., pension contributions as a percent of wages) for a sustainable pay-as-you-go pension scheme. The RCR is positively associated with the LBIR because a higher internal rate of return increases the amount of pension benefits. In advanced countries, the RCRs are particularly low, reflecting their low LBIRs; the median RCR is 2, implying that a replacement rate of 50 percent requires a contribution rate as high as 25 percent. In startling contrast, the median RCR in sub-Saharan Africa is as high as 8.6, thanks to its high LBIR, implying that a replacement rate of 50 percent requires a contribution rate as low as 6 percent. Raising the retirement age improves the RCR not only by lengthening the contribution period and shortening the retirement period but also by ratcheting up the LBIR. The median RCR for all countries improves from 3.5 to 5.5 when the retirement age is raised from 65 to 70. Fertility prospects also affect the RCR—the median RCR decreases (increases) from 3.5 to 3.0 (4.1) under the low-fertility (high-fertility) scenario, as opposed to the medium-fertility scenario.

This paper contributes to the literature by estimating the internal rates of return for pay-as-you-go pensions in most of the countries across the world, based on a generalized framework. Settergren and Mikula (2005) show that the internal rate of return for a pay-as-you-go pension entails not only the biological interest rate but also shifts in income and mortality patterns. Knell (2016) shows that a rise in life expectancy increases the internal rate of return for a pay-as-you-go pension scheme. However, these two papers do not fully take account of non-steady-state population dynamics. Ediev (2014) estimates the internal rate of return for a pay-as-you-go pension scheme in 24 advanced countries and finds the positive relationship between the return and life expectancy. His paper differs from this one in that the estimated internal rate of return is not uniform across generations, but rather age-cohort specific. In other words, the internal rate of returns depends on the year in which the individual is born, because his model assumes that contributions from the working-age population is fully utilized to finance pension payments for retirees in each period.

The rest of the paper is organized as follows. Section II introduces the methodology in detail and defines the LBIR under a generalized overlapping generation model. Section III presents estimates of the LBIR across countries. The RCR is analyzed in Section IV, which discusses the implications of the low- and high-fertility scenarios in Section V. Section VI concludes by discussing policy implications and highlighting the merits of this paper’s analytical framework vis-à-vis traditional methodologies. Country classifications are reported in Appendix I, and country-by-country estimates of the LBIR, together with other demographic and pension indicators, are presented in Appendix II.

II. Setup

The internal rate of return for a public pension scheme refers to the rate that equalizes contributions and pension benefits from individual participants’ perspective. In a standard two-period overlapping generation model, the internal rate of return for a pay-as-you-go pension scheme equals the growth rate of the contribution base, that is, the sum of population growth—Samuelson (1958) called this as the “biological interest rate”—and productivity growth. This is intuitive: contributions from the working-age generation in a given period—which are fully used to pay pension benefits to the retired generation in that period—increase in tandem with the number of workers, as well as the productivity growth that results in higher wages. Note that retirees in the initial period of the scheme receive pensions even though they contributed nothing.4

I build up a framework that applies this idea to the real world. I aim to estimate the highest internal rate of return that a sustainable pay-as-you-go pension can offer equally for current and future generations in a given country. This exercise would be useful not only for countries whose public pension system is primarily pay-as-you-go, but also for those with funded pension schemes by clarifying what a pay-as-you-go alternative could offer. I consider a public pension scheme where workers contribute a portion of wage income to a nationally pooled fund, and retirees receive pensions from that fund. The pension scheme promises a uniform internal rate of return for all current and future generations, assuring inter-generational equity. The pension scheme needs to be sustainable, that is, the present value of contributions over time needs to be at least as large as the present value of pension payments over time. It can borrow to finance a temporary cash shortfall or accumulate reserves for future pension payments (in this sense, the pension scheme differs from a pure pay-as-you-go scheme where worker contributions are fully used to pay for pension payments in each period).

To be specific, consider the following overlapping generation economy. In a given year t, all individuals in the labor force receive the same real wage, W(t), regardless of age. Real wages grow over time at a time-invariant rate of productivity growth denoted by μ, so W(t + 1) = (1 + μ)W(t). A pension scheme is established in year T0. Workers are obliged to contribute at a fixed contribution rate c, starting at age A0 and continuing up until one year before the retirement age at A1. From the perspective of an individual, contributions are invested and accumulated in her notional pension account. The account earns a rate of return denoted by θ, which is time invariant and is the same for all individuals regardless of year of birth. Individuals retire and leave the labor force at age A1, when they receive the balances in their notional accounts as lump-sum pension benefits.

In the early years of the pension scheme, the elderly receive pension benefits, even though they have contributed nothing or have done so for fewer years than the number of years that encompass the regular contribution period (A1A0). To clarify this, I divide the population into four groups as illustrated in Table 1 and Figure 1:

Table 1.Definition of Generation Groups
GroupAge as of year T0Number of years to contribute
Group IA0 (including those not yet born)(A1A0)
Group IIBetween A0 and A1Less than (A1A0)
Group III= A10
Group IV>A10

Figure 1.Population Structure by Generation Group and Age

Source: Author.

Note: This table illustrates how the population is divided into Groups I–IV, with the contribution starting age (A0) set at 20, the retirement age (A1) at 65, and the starting year of the pension scheme (T0) at 2015.

For Group I individuals, the pension scheme exists when they enter the labor force. Their contributions made in year tk become cW(tk)(1 + θ)k in year t in their notional accounts. An individual retiring in year t contributes from year t – (A1A0) to year t − 1. Thus, aggregating contributions over years, she receives in year t a lump-sum pension benefit of

Group II, III, and IV individuals receive pensions despite incomplete contribution histories. I assume that a Group II individual retiring in year t also receives Ψ(t) as if she has contributed fully, although the pension scheme does not exist when she joins the labor force. Similarly, a Group III individual, whose age is A1 in year T0, receives Ψ(T0) even though she has not contributed at all. Group IV individuals receive pension benefits equivalent of those for Group III individuals in annual installments rather than as a lump-sum payment. Starting at year T0 and over her lifetime, a Group IV individual receives annual pensions denoted by Λ(t). For year T0, Λ(t) is set at an amount equal to Ψ(T0) divided by the years of remaining life expectancy for Group III individuals at age A1; so Λ(T0) = Ψ(T0)/B, where B is the remaining life expectancy of individuals of age A1 in year t. This treatment takes into account the prospect that Group IV individuals would have relatively short remaining life expectancy at year T0. For instance, an individual of an age much greater than A1 and a few years of remaining life expectancy would not need to receive Ψ(T0). I assume that Λ(t) increases by productivity growth each year, so Λ(T0+s)=Ψ(T0)B(1+μ)s. This ensures that the replacement rate of the pension benefit, defined by Λ(t)/VK(t), remains constant over time.

The national pension fund collects contributions from workers and pays pension benefits to retirees each year. In year t, the working-age population Nw(t) comprises individuals whose ages fall between A0 and A1 – 1. In each year starting T0, individuals who belong to Groups I, II, and III and reach the age of A1 retire; the number of such individuals in year t is denoted by Nr1(t). The number of Group IV individuals in year t is denoted by Nr2(t). In year t, the fund collects contributions from working-age individuals, denoted by R(t) ≡ cW(t)Nw(t); it pays the lump-sum pension benefits to retiring individuals who belong to Groups I, II, and III, denoted by G1(t) ≡ Ψ(t)Nr1(t); and it also pays annual pension benefits to Group IV individuals, denoted by G2(t) ≡ Λ(t)Nr2(t).

The pension fund can invest in an asset that yields a time-invariant interest rate of r, or can borrow at this interest rate. I assume that r equals the sum of annual productivity growth, average working age population growth (l), and a positive interest-rate-growth-differential (irgd)

The long-run sustainability of the system requires the present value of collected contributions to be larger than or equal to the present value of pension benefit payments:

The highest internal rate of return for the pension scheme, denoted by θ*, is the largest value of θ that satisfies (3) above. Because the left-hand side is independent of θ and the right-hand side is a monotonically increasing function of θ, θ* is uniquely determined as the value that equalizes both sides of (3). Rearranging terms, θ* satisfies

It is convenient to strip off productivity growth from the pension’s internal rate of return. I define ϕ as the internal rate of return obtained after netting out productivity growth from θ*:

Plugging (5) into (4) results in:

I call ϕ as the Long-Run Biological Interest Rate (LBIR). Equation (6) shows that the LBIR can be computed from demographic prospects and an assumed value of the interest-and-growth-rate-differential.

In sum, the highest internal rate of return that a sustainable pay-as-you-go pension can offer equally for all generations is given by the LBIR plus future productivity growth. This approach provides a simple and tractable framework for analyzing the fundamentals of pay-as-you-go pensions in a given country. Note that estimating the LBIR does not require any knowledge of the characteristics of existing pension systems (contribution rate, benefit formula, etc.), which vary significantly by country. In this framework, I assume no financial assets are available in year T0 to finance future pension benefits; if they are available, the internal rate of return would be higher than the LBIR plus productivity growth.

III. Estimating LBIR in Advanced and Developing Countries

In this section, I estimate the LBIR for 172 countries. For each country, the LBIR is calculated by numerically solving equation (6) for ϕ. The pension scheme is introduced in 2015 (so T0 is 2015) with the end year being 2100. Demographic data are taken from the latest United Nations (UN) projection (United Nations, 2015, 2015 Revision of the United Nations World Population Prospects, the medium fertility variant).5 Specifically, (i) Nw(t), Nr1(t), and Nr2(t) are computed from the annual population projection for 2015–2100; (ii) l is calculated as the average rate of increase in the population of ages 20–64 over 2015–2100; and (iii) B is taken from projected remaining life expectancy at age 65 as of 2015. The age at which individuals start contributing (A0) is set at 20 and the retirement age (A1) is set at 65 for all countries to facilitate cross-country comparisons.6 The interest-rate-growth-differential (irgd) is set at 1 percent per year, following the approach by Clements and others (2015).7 I will present sensitivity analyses for alternative values of A1 and irgd. The dataset allows us to compute the LBIRs for as many as 172 countries, consisting of advanced countries and middle- and low-income countries in various regions. See Appendix I for the composition of country groups by income and region.

The UN projection suggests that the population will age in the long run, not only in advanced countries, but also in developing countries. For the countries in this dataset, the average old-age dependency ratio, that is, the ratio of the elderly (age 65 and older) to the working age population (between ages 20 and 64), is projected to increase from 14 percent in 2015 to 50 percent in 2100. Similarly, the average long-run growth rate of the elderly (2 percent per year over 2015–2100) exceeds that of the working age population (0.4 percent). In fact, the population will age faster in developing countries than in advanced countries. For middle- and low-income countries, the old-age dependency ratio more than quadruples between 2015 and 2100. These reflect the global trend of falling fertility and increasing longevity. The question is how this trend translates into the LBIR in each individual country.

The main results of this paper are shown in Table 2, which reports the summary statistics of the estimated LBIR along with demographic indicators (see Appendix II for country-by-country LBIR estimates). Key observations are as follows:

  • Global population ageing translates into a generally low LBIR. The median LBIR for all 172 countries is as low as 0.9 percent per year. The range of the LBIR is somewhat wide, with the standard deviation of 1.5 percent, the minimum of -1.4 percent, and the maximum of 4.1 percent.
  • Comparing the LBIRs across country groups, the median LBIR for advanced countries is negative (-0.6 percent) and lower than that for middle-income countries (0.6 percent) and low-income countries (2.8 percent). Among developing countries, the median LBIR is negative for emerging Europe and CIS countries (-0.4 percent), while it is relatively high for sub-Saharan Africa (3.0 percent) and the Middle East and North Africa (1.7 percent). The median for developing Asia (1 percent) and Latin America and the Caribbean (0.6 percent) lies in the middle.
  • The LBIR is negative in as many as 54 countries, being about 30 percent of the total. Most of the advanced countries have negative LBIRs (31 out of 34 countries), and so do more than half of the countries in emerging Europe and the CIS (16 out of 23 countries). In these countries, the internal rate of return for a pay-as-you-go pension system would be negative unless productivity growth is sufficiently high.
  • The variation of the LBIR across countries reflects the degree of population ageing at present and in the future, as expected. The LBIR is correlated negatively with the old-age dependency ratio in 2015 and 2100, and positively with the projected long-run growth rate of the working-age population (see Figure 2).
Table 2.LBIR Estimates and Demographic Indicators
Long-Run Biological Interest Rate (LBIR) (% per annum)Number of countries with negative LBIRNumber of countriesDependency ratio in 2015 (%) 1/Dependency ratio in 2100 (%) 1/Working-age population growth (%) 2/Old-age population growth (%) 3/
MeanMedianSt. dev.Max.Min.CountCountMeanMeanMeanMean
All countries
All countries1.00.91.54.1−1.45417214.449.70.42.0
By income group
Advanced−0.5−0.60.50.8−1.4313428.763.8−0.20.8
Middle income0.70.61.03.8−1.1228213.254.90.01.8
Low income2.52.81.04.1−0.41567.533.51.43.1
By region
Advanced−0.5−0.60.50.8−1.4313428.763.8−0.20.8
Emerging Europe and CIS−0.1−0.41.02.2−1.1162319.154.1−0.50.8
Developing Asia1.01.00.92.6−0.73199.954.70.22.2
Latin America and the Caribbean0.60.60.61.9−0.533013.359.90.01.8
Middle East and North Africa1.61.70.83.00.10227.146.70.52.9
Sub-Saharan Africa2.83.00.94.1−0.21447.328.91.73.3
Source: Author’s calculations based on the UN Population Projection (United Nations, 2015).

Population aged 65 and over in percent of population of ages 20–64.

Annual percent change in population of ages 20–64, averaged over 2015–2100.

Annual percent change in population of ages 65 and over, averaged over 2015–2100.

Source: Author’s calculations based on the UN Population Projection (United Nations, 2015).

Population aged 65 and over in percent of population of ages 20–64.

Annual percent change in population of ages 20–64, averaged over 2015–2100.

Annual percent change in population of ages 65 and over, averaged over 2015–2100.

Figure 2.LBIR, Old-Age Dependency Ratio, and Working-Age Population Growth

Source: Author’s calculations.

The low levels of LBIRs across the world present challenges for maintaining pay-as-you-go pension schemes over the long run. In a country with a low LBIR, high productivity growth would be necessary to provide future generations with adequate internal rates of return; otherwise, they will have incentive to opt out of the pension scheme and choose alternative options for their retirement savings. In this case, a vicious cycle would emerge: low participation would erode the contribution base, bring down the internal rate of return, and induce further opt-outs.

To explore this point quantitatively, I estimate the LBIR plus productivity growth, along with alternative investment options. Historical productivity growth, measured as the average real per-worker GDP growth rate over 1990–2010, is used as a proxy for future productivity growth. The median productivity growth for all countries in the dataset was 1.5 percent per year during this period. Looking at variations across groups, developing Asia achieved the highest productivity growth (the median is 3.0 percent), followed by emerging Europe and the CIS (1.8 percent) and advanced countries (1.7 percent). The average productivity growth was relatively low in Latin America and the Caribbean (1.0 percent), the Middle East and North America (0.9 percent), and sub-Saharan Africa (0.5 percent). Using the past average as proxy for long-run productivity growth may prove to be optimistic, given that productivity growth is expected to decline in the years ahead in advanced and emerging market economies (IMF, 2015). The LBIR plus historical productivity growth is compared with the risk-free real interest rate, represented by the short-term U.S. real interest rate. It averaged 2.1 percent per year over 1970–2010. Another alternative investment option is represented by the real rate of return for the S&P 500, which averaged 6.8 percent over 1970–2010. This alternative serves an illustrative purpose; needless to say, risky asset returns need to be weighed against risks, and a simple comparison of average returns would be misleading.

The results are reported in Figure 3. The median internal rate of return for 172 countries, measured by the LBIR plus historical productivity growth, is 2.4 percent per year. This is slightly above the risk-free benchmark of 2.1 percent. The median for advanced countries (1.4 percent) and Latin America and the Caribbean (1.8 percent) is below this benchmark, while those for emerging Europe and the CIS (2.1 percent) and the Middle East and North Africa (2.6 percent) marginally exceed the benchmark. In contrast, the estimated internal rate of return is relatively high in developing Asia (median of 3.8 percent) and sub-Saharan Africa (3.4 percent). The former is supported by historically high productivity growth while the latter benefits from its relatively high LBIR, as discussed above. The estimated internal rates of return are negative in 13 countries, and exceed the historical average of the S&P 500 return in only 9 countries.

Figure 3.LBIR Plus Productivity Growth by Region

(In percent per year)

Source: Author’s calculations.

Note: The box chart plots (1) the LBIR plus historical productivity growth, which is measured by the average growth rate of real GDP per working-age population over 1990–2010, (2) the LBIR only, and (3) the historical productivity growth only. The box chart illustrates the distribution of a variable by showing the maximum and minimum (excluding outliers), the 25th and 75th percentiles (as the boundaries of the box), and the median (the line inside the box). The solid and dotted lines exhibit the average returns of the S&P 500 in real terms, and the U.S. one-year interest rate in real terms for 1970–2010, respectively, from Shiller (2015). Figures in brackets indicate the number of countries.

Is the LBIR sensitive to the retirement age and the interest-rate-and-growth differential? The left panel of Figure 4 shows that raising the retirement age from 65 to 70 results in an increase in the LBIR by 40 basis points on average, with the sizes of the increases being similar across countries, regardless of LBIR level. This is substantial, especially for countries with low LBIRs. Nonetheless, the increase in the LBIR (or the internal rate of return) needs to be weighed against longer working years. The right panel of Figure 4 indicates that raising the interest-rate-growth differential does not substantially affect the LBIR; a 200-basis-point increase in the differential, which concomitantly raises the interest rate at which the pension fund can borrow or invest, results in an increase in the LBIR by 20 basis points, on average. The impact tends to be negligible in countries with low LBIRs.

Figure 4.Sensitivity of LBIR to Retirement Age and the Interest Rate

Source: Author’s calculations.

Note: The red line indicates the 45-degree line.

IV. LBIR and Tradeoff Between Pension Contributions and Benefits

In practice, a public pay-as-you-go pension scheme is characterized by the contribution rate and the pension replacement rate (pension benefits divided by pre-retirement wage income). A high replacement rate benefits retirees financially, but it requires a high contribution rate to support the sustainability of the pension system. Nonetheless, a high contribution rate, equivalent to a high tax rate on labor income, discourages labor supply. Therefore, there would be an upper limit for the contribution rate. Raising the retirement age can alleviate this tradeoff by expanding the contribution base and squeezing aggregate pension spending, and is thus deemed effective in preserving the sustainability of a pay-as-you-go pension system (IMF, 2012).

In this section, I analyze the drivers of this tradeoff. A retiree who retires in year t ≥ T0 receives a lump-sum pension benefit Ψ(t) (see equation (1)). Suppose she purchases an annuity in year t in exchange for this. Such an annuity pays her b(t) each year until she dies. Let A2(t) denote remaining life expectancy for individuals retiring in year t (that is, they are expected to live for additional A2(t) years after retirement). Then, b(t) satisfies:

Rearranging terms results in

Equation (8) clarifies the tradeoff between the contribution rate (c) and the replacement rate (b/W) for a sustainable pay-as-you-go pension. In what follows, the ratio of the replacement rate to the contribution rate is defined as the Replacement-Contribution Ratio (RCR). The RCR is a function of the LBIR (ϕ), the interest rate (r), the contribution period (A1A0), and the remaining life expectancy after retirement (A2(t)). A higher LBIR would improve the RCR by boosting the lump sum pension. Similarly, a higher interest rate would improve the RCR by raising the annuity payment as a result of a higher investment return. Raising the retirement age (A1) would improve the RCR through two channels: (i) increasing the contribution period and reducing the retirement period, and (ii) raising the LBIR as discussed in the previous section. Note that productivity growth (μ) would affect the RCR only through the interest rate (reflecting the fixed interest-rate-growth-differential), because both the wage and lump-sum pension benefits grow in tandem with productivity growth.

The left panel of Figure 5 illustrates the RCR as a function of the LBIR. Here, for an illustrative purpose, the interest rate is assumed at 2 percent per annum, the retirement age at 65, and remaining life expectancy at 20 years. For a country with the LBIR of -1 percent per year, the replacement rate can be only around twice as high as the contribution rate. On the other hand, if the LBIR is as high as 4 percent, the replacement rate can be 7 times as large as the replacement rate. The right panel of Figure 5 shows the replacement rate as a function of the contribution rate for given values of the LBIR. For example, for a country with a LBIR of - 1 percent, achieving a 50 percent replacement rate would require a contribution rate of about 25 percent. A labor tax rate as high as this could significantly distort labor supply. In stark contrast, for a country with a LBIR of 4 percent, a 50 percent replacement rate requires a contribution rate as low as about 7 percent.

Figure 5.Ratio of Replacement Rate to Contribution Rate

Source: Author’s calculations.

Now I present estimates of the RCR by country group (see Appendix II for the RCRs of individual countries). The RCR is calculated from the following parameter values: the real interest rate is assumed at 2 percent per year; the retirement age at 65; the remaining life expectancy as of year 2015; and the interest-rate-growth differential at 1 percent. Figure 6 shows box charts of the RCR (left panel), as well as the contribution rate necessary for a 50 percent replacement rate (right panel). Reflecting the low LBIR, the RCR is particularly low for advanced countries, with the median equal to 2.3. With the median being 21.5 percent, the contribution rate required for a 50 percent replacement rate is relatively high compared with other groups. The RCR is also relatively low for emerging Europe and the CIS (median equal to 2.8) and Latin America and the Caribbean (3.2). On the other hand, the replacement-contribution tradeoff is not a concern for most of the sub-Saharan African countries, whose median RCR is 7.5; the contribution rate required for a 50 percent replacement rate is below 10 percent for most of the countries in this group.

Figure 6.Tradeoff Between Replacement Rate and Contribution Rate by Country Group

Source: Author’s calculations.

Note: The box charts illustrate the distribution of a variable by showing the maximum and the minimum (excluding outliers), the 25th and 75th percentiles (as the boundaries of the box), and the median (the line inside the box). Figures in brackets indicate the number of countries.

The retirement age has a significant impact on the RCR (Table 3). As noted earlier, a higher retirement age improves the RCR not only by increasing the contribution period and reducing the retirement period, but also by raising the LBIR. Table 3 shows that the median RCR for all countries increases from 3.6 to 5.4 when the retirement age is raised from 65 to 70. This brings down the contribution rate required for a 50 percent replacement rate by about 5 percentage points. This effect is pronounced in advanced countries, which experience a 6.4 percent reduction in the contribution rate required for a 50 percent replacement rate.

Table 3.Replacement-Contribution Ratio and Required Contribution Rate by Assumption on Retirement Age
Replacement-Contribution Ratio (median)Contribution rate required for 50 percent replacement rate (median, %)
Retirement age65706570
All countries3.65.413.99.2
By country groups
Advanced2.33.321.515.0
Emerging Europe and CIS4.26.611.97.6
Developing Asia2.84.217.911.9
Latin America and the Caribbean3.24.715.610.6
Middle East and North Africa4.77.310.66.9
Sub-Saharan Africa7.513.06.63.8
Source: Author’s calculations.
Source: Author’s calculations.

V. Implications of Demographic Uncertainties

Population projections hinge on long-run demographic assumptions, most notably on fertility, mortality, and migration. Clements and others (2015) highlight substantial uncertainties surrounding these elements, which give rise to substantial fiscal risks connected to age-related public spending on pensions and health care.

In this section, I analyze the implications of demographic uncertainties for the LBIR. I focus on the uncertainties related to fertility. Specifically, I use population projections under low- and high-fertility variants presented by the 2015 UN population projection (United Nations, 2015). The medium-fertility variant of the UN population projection—the basis of the results presented in the previous sections—uses country-specific paths of fertility rates estimated from the latest population survey in each country, as well as the historical evolutions of the fertility rates across countries. Under the low-fertility (high-fertility) variant, the fertility rate is assumed to remain broadly at 0.5 children per woman below (above) the rate envisaged under the medium-fertility variant. I do not analyze uncertainties related to mortality or migration because the UN population projection does not provide alternative variants to treat them in symmetrical settings.8

The low- and high-fertility variants would represent extreme cases for most countries except sub-Saharan Africa. The UN projection produces a probability distribution of the fertility rate for each country over the long run, based on 60,000 trajectories of the fertility rate generated by simulations. For advanced countries, the paths of the fertility rate under the low-and high-fertility variants lie outside the 80th percentile band in 33 out of the 34 countries in the sample, and outside the 95th percentile band in 22 countries. This is illustrated by the example of the United States, as shown in the left panel of Figure 7. Also, in 85 out of 94 developing countries except sub-Saharan African countries, the paths of the fertility rates under the low- and high-fertility variants lie outside the 80th percentile band. Sub-Saharan Africa is an exception, as the paths lie outside the 80th percentile band in only 10 countries out of the 44 countries in the sample. In the right panel of Figure 7, Uganda is shown as an example where the paths under low- and high-fertility variants lie well inside the 80th percentile band.

Figure 7.Fertility Rate Paths Under High- and Low-Fertility Variants

Source: 2015 Revision of the United Nations World Population Prospects (United Nations, 2015).

Note: The UN projection produces a probability distribution of the fertility rate over the long run, based on 60,000 trajectories of fertility rates generated by model simulations. The fan charts exhibits this distribution by showing the 80th percentile and 95th percentile bands. The fertility rate paths assumed under the medium-, high-, and low-variants are also shown.

The results reported in Table 4 and Figure 8 show the implications of low and high fertility rates. Here, the retirement age and the interest-rate-growth-differential are set at 65 and 1 percent, respectively. On average, the LBIR is 0.6 (0.4) percentage points lower (higher) under the low (high) fertility variant than under the medium-fertility variant (the base case). Under the low-fertility variant, the LBIR is negative for most of the advanced countries and three-fourths of the countries in emerging Europe and the CIS, and less than 1 percent for three-fourths of the countries in developing Asia as well as Latin America and the Caribbean. Most of the countries in sub-Saharan Africa have relatively high LBIRs even under the low-fertility variant, with the median being 2.7 percent. Nevertheless, as discussed above, the path of the fertility rate under the low-or the high-fertility variant does not generally represent an extreme event in sub-Saharan Africa.

Table 4.Impact of Fertility Rate Assumptions on LBIR and Tradeoff Between Contribution and Replacement Rates
Long-Run Biological Interest Rate (LBIR) (median, %)Replacement-Contribution Ratio (RCR) (median)Contribution rate for 50% replacement rate (median, %)
Medium fertilityLow fertilityHigh fertilityMedium fertilityLow fertilityHigh fertilityMedium fertilityLow fertilityHigh fertilityNumber of countries
All countries0.9

(1.5)
0.3

(1.5)
1.3

(1.4)
3.6

(2.3)
3.2

(2.1)
4.0

(2.4)
13.9

(6.0)
15.8

(6.6)
12.5

(5.6)
172
By country group
Advanced−0.6

(0.5)
−1.0

(0.5)
−0.3

(0.4)
2.3

(0.3)
2.1

(0.3)
2.5

(0.3)
21.5

(2.6)
23.4

(2.9)
20.1

(2.4)
34
Emerging Europe and CIS−0.4

(1.0)
−1.0

(1.0)
−0.1

(1.0)
2.8

(1.0)
2.5

(0.9)
3.0

(1.1)
17.9

(4.0)
19.9

(4.5)
16.4

(3.8)
23
Developing Asia1.0

(0.9)
0.3

(0.9)
1.4

(0.9)
4.2

(1.3)
3.6

(1.2)
4.6

(1.4)
11.9

(3.7)
13.9

(4.3)
10.8

(3.4)
19
Latin America and the Caribbean0.6

(0.6)
0.1

(0.6)
1.0

(0.6)
3.2

(0.7)
2.8

(0.6)
3.5

(0.8)
15.6

(2.7)
17.6

(3.1)
14.1

(2.5)
30
Middle East and North Africa1.7

(0.8)
1.2

(0.9)
2.0

(0.8)
4.7

(1.4)
4.1

(1.3)
5.2

(1.5)
10.6

(3.0)
12.1

(3.5)
9.6

(2.7)
22
Source: Author’s calculations.Note: Figures in brackets indicate standard deviation.
Source: Author’s calculations.Note: Figures in brackets indicate standard deviation.

Figure 8.Estimates of LBIR Under High- and Low-Fertility Scenarios

(In percent)

Source: Author’s calculations.

Note: The box chart in the right panel illustrates the distribution of a variable by showing the maximum and minimum (excluding outliers), the 25th and 75th percentiles (as the boundaries of the box), and the median (the line inside the box).

Table 4 shows that the median RCR is about 0.4 lower (higher) under the low-fertility (high-fertility) variant than under the medium-fertility variant. The contribution rate required for a 50 percent replacement rate is 1.9 (1.4) percentage points higher under the low-fertility variant than under the medium-fertility variant. The risk of low fertility is substantial in advanced countries, where the median contribution rate needed to achieve a 50 percent replacement rate is 23 percent under the low-fertility variant. On the other hand, the contribution-replacement tradeoff is much less pronounced for sub-Saharan Africa, where the contribution rate of 7.2 percent can still achieve a 50 percent replacement rate under the low-fertility variant.

VI. Concluding Remarks and Policy Implications

In many countries, actuarial studies are conducted periodically to analyze whether the public pension schemes satisfy the long-run budget constraints. If an actuarial study finds that the pension scheme is unsustainable, parametric pension reforms would become necessary. Broadly speaking, the menu of policy options for regaining sustainability consists of raising the contribution rate, cutting pension benefits, and/or raising the retirement age.

Here, note that the long-run budget constraint would not be sufficient to ensure incentive compatibility, that is, the pension scheme offers an internal rate of return adequate for the working-age population to opt for the scheme. The parametric pension reforms may require some generations (typically those who are not born yet) to accommodate an internal rate of return much lower than those offered by alternative investment options. If this is the case, it would be unrealistic to assume in an actuarial study that the pension scheme would achieve the full participation of eligible individuals. In this context, this paper addresses the incentive compatibility problem by computing the internal rate of return for a pay-as-you-go pension scheme created in 2015, to be sustained through 2100.

This paper finds that the internal rate of return for such a pay-as-you-go scheme is the sum of the LBIR and productivity growth. The LBIR estimates across countries indicate that LBIRs are globally low—the median LBIR for 172 countries is only 0.9 percent per year. Demographic risks are significant—a lower-than-envisaged fertility rate in the future would lead to a LBIR that is lower by about 60 basis points on average. This suggests that sustaining the incentive compatibility for a pay-as-you-go pension scheme hinges on ample productivity growth in the future in many countries.

To end on a positive note, raising the retirement age is a viable option to ratchet up the LBIR. Raising the retirement age by five years (from 65 to 70) increases the LBIR in all countries by about 40 basis points on average. Raising the retirement age would improve the tradeoff between the contribution rate and the replacement rate not only by increasing contribution periods and reducing retirement periods, but also by raising the LBIR. This reinforces the argument for raising the retirement age as a more viable option than raising the contribution rate or cutting pension benefits.

Finally, it is worth noting that my approach is much simpler than the traditional approaches that are based on actuarial studies. In particular, calculating the LBIR does not require any knowledge of a country’s pension system, thereby dramatically reducing data and information requirements. Further, by focusing on a uniform internal rate of return for current and future generations, my approach takes account of the long-run budget constraint as well as intergenerational equity. It also presents a simple framework by which to analyze parametric reform options by representing the tradeoff between the contribution rate and the replacement rate as a function of the LBIR and the retirement age. Finally, the LBIR and the RCR augment the list of cross-country indicators to analyze pension sustainability.

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Appendix I. Composition of Country Groups by Income and Region
AdvancedMiddle incomeLow income
AustraliaEmerging Europe and CISEmerging Europe and CIS
AustriaAlbaniaMacedonia, FYRKyrgyz RepublicTajikistan
BelgiumArmeniaMontenegroMoldovaUzbekistan
CanadaAzerbaijan, Rep. ofPoland
CyprusBelarusRomaniaDeveloping Asia
Czech RepublicBosnia & HerzegovinaRussian FederationBangladeshMyanma r
DenmarkBulgariaSerbia, Republic ofBhutanNepal
EstoniaCroatiaTurkeyCambodiaPapua New Guinea
FinlandGeorgiaTurkmenistanLao People’s Dem. Rep.Vietnam
FranceHungaryUkraineMongolia
GermanyKazakhstan
GreeceLatin America and the Caribbean
Hong KongDeveloping AsiaBoliviaHonduras
IcelandBrunei DarussalamMaldivesHaitiNicaragua
IrelandChina, MainlandPhilippines
IsraelIndiaSri LankaMiddle East and North Africa
ItalyIndonesiaThailandAfghanistan, I.R. ofSudan
JapanMalaysiaTimor-LesteDjiboutiYemen, Republic of
Korea, Republic ofMauritania
LatviaLatin America and the Caribbean
LithuaniaAntigua and BarbudaGuatemalaSub-Saharan Africa
LuxembourgArgentinaGuyanaBeninLesotho
NetherlandsBahamas, TheJamaicaBurkina FasoLiberia
New ZealandBarbadosMexicoBurundiMadagascar
NorwayBelizePanamaCameroonMalawi
PortugalBrazilParaguayCentral African Rep.Mali
SingaporeChilePeruChadMozambique
Slovak RepublicColombiaSt. LuciaComorosNiger
SloveniaCosta RicaSt. Vincent & Grens.Congo, Dem. Rep. ofNigeria
SpainDominican RepublicSurinameCongo, Republic ofRwanda
SwedenEcuadorTrinidad and TobagoCote d’IvoireSao Tome & Principe
SwitzerlandEl SalvadorUruguayEritreaSenegal
United KingdomGrenadaVenezuela, Rep. Bol.EthiopiaSierra Leone
United StatesGambia, TheTanzania
Middle East and North AfricaGhanaTogo
AlgeriaMoroccoGuineaUganda
Bahrain, Kingdom ofOmanGuinea-BissauZambia
EgyptPakistanKenyaZimbabwe
Iran, I.R. ofQatar
IraqSaudi Arabia
JordanSyrian Arab Republic
KuwaitTunisia
LebanonUnited Arab Emirates
Libya
Sub-Saharan Africa
AngolaMauritius
BotswanaNamibia
Cape VerdeSeychelles
Equatorial GuineaSouth Africa
GabonSwaziland
Source: Author’s Calculations
Source: Author’s Calculations
Appendix II. Estimated Long-Run Biological Interest Rate and Demographic Indicators by Country
CountryLong-Run Biological Interest Rate (LBIR)Replacement-Contribution Ratio (RCR)Contribution rate required for 50% replacement rateDependency ratio in 2015 (%) 1/Dependency ratio in 2100 (%) 1/Working-age population growth rate (%, age 20–64)
Advanced
Australia0.12.519.625.158.60.5
Austria−0.82.223.230.467.2−0.3
Belgium−0.52.321.630.760.10.0
Canada−0.42.321.826.061.60.1
China, Hong Kong−0.82.024.922.169.1−0.3
Cyprus−0.22.619.020.063.8−0.1
Czech Republic−0.82.321.428.959.5−0.5
Denmark−0.42.420.732.858.10.1
Estonia−0.72.421.130.858.3−0.7
Finland−0.72.222.635.562.6−0.1
France−0.42.222.733.862.10.0
Germany−1.12.024.634.971.0−0.6
Greece−1.12.024.836.172.6−0.7
Iceland−0.22.420.923.168.6−0.1
Ireland−0.12.619.522.159.10.1
Israel0.83.016.721.154.40.8
Italy−1.21.926.437.972.5−0.5
Japan−1.41.828.447.076.1−0.7
Korea, Republic of−0.92.124.319.878.3−0.7
Latvia−0.72.619.631.451.5−0.7
Lithuania−0.62.718.630.950.1−0.6
Luxembourg0.22.718.722.056.60.5
Netherlands−0.62.222.330.763.2−0.2
New Zealand−0.22.420.825.665.00.1
Norway−0.12.520.027.557.40.3
Portugal−1.22.025.234.775.1−0.7
Singapore−0.92.025.017.689.4−0.4
Slovak Republic−0.72.519.721.060.1−0.7
Slovenia−0.92.123.328.663.4−0.5
Spain−1.02.025.130.471.4−0.5
Sweden−0.32.421.034.656.30.3
Switzerland−0.52.222.529.162.80.1
United Kingdom−0.32.420.530.359.20.1
United States−0.12.619.424.753.10.2
Emerging Europe and CIS
Albania−0.52.519.820.680.3−0.9
Armenia−0.42.718.316.765.7−0.9
Azerbaijan, Rep. of0.63.813.38.643.9−0.2
Belarus−0.23.116.321.443.7−0.6
Bosnia & Herzegovina−1.12.321.723.771.0−1.2
Bulgaria−1.12.520.432.654.5−1.1
Croatia−1.02.321.831.367.6−0.8
Georgia−0.52.718.422.461.7−0.8
Hungary−0.72.520.028.557.2−0.7
Kazakhstan1.14.511.011.236.70.3
Kyrgyz Republic1.64.910.27.637.90.5
Macedonia, FYR−0.52.817.919.261.9−0.7
Moldova−0.43.116.314.652.6−1.2
Montenegro−0.52.718.422.461.1−0.6
Poland−0.92.222.224.170.0−1.0
Romania−0.92.420.428.059.3−0.9
Russian Federation−0.13.115.920.439.6−0.4
Serbia, Republic of−0.82.619.128.361.3−0.8
Tajikistan2.25.59.15.834.01.1
Turkey0.43.215.812.964.30.0
Turkmenistan1.34.810.57.137.40.0
Ukraine−0.43.016.923.541.3−0.8
Uzbekistan1.34.411.58.040.20.1
Developing Asia
Bangladesh1.03.912.99.062.20.0
Bhutan0.93.614.08.663.9−0.1
Brunei Darussalam0.33.016.66.970.3−0.1
Cambodia1.44.810.57.653.20.5
China, Mainland−0.52.718.414.270.2−0.8
India1.04.211.910.053.40.2
Indonesia1.04.611.08.941.00.2
Lao People’s Dem. Rep.1.65.29.67.653.70.6
Malaysia0.73.713.79.757.00.2
Maldives0.73.613.98.065.60.0
Mongolia1.54.411.36.750.50.3
Myanmar1.04.511.09.338.00.0
Nepal0.94.211.911.068.60.0
Papua New Guinea2.67.46.86.125.61.2
Philippines1.75.39.48.635.20.7
Sri Lanka−0.12.817.615.969.5−0.6
Thailand−0.72.420.916.071.6−0.9
Timor-Leste2.66.67.613.531.81.6
Vietnam0.53.016.610.757.4−0.1
Latin America and the Caribbean
Antigua and Barbuda0.53.115.911.957.70.1
Argentina0.53.215.819.557.00.2
Bahamas, The0.42.917.013.258.10.1
Barbados−0.52.818.123.659.0−0.3
Belize1.85.49.37.141.00.8
Bolivia1.44.012.512.747.90.7
Brazil0.22.917.212.968.0−0.3
Chile−0.12.321.317.974.8−0.2
Colombia0.33.016.711.765.3−0.3
Costa Rica0.12.619.014.770.3−0.2
Dominican Republic0.93.414.812.359.70.1
Ecuador0.93.315.212.260.70.4
El Salvador0.43.016.515.075.5−0.5
Grenada0.43.514.412.570.8−0.6
Guatemala1.94.511.110.246.61.1
Guyana1.04.710.69.341.9−0.2
Haiti1.64.910.19.142.20.4
Honduras1.23.613.89.361.80.3
Jamaica0.02.818.115.976.8−0.8
Mexico0.63.116.311.470.70.0
Nicaragua0.83.215.49.370.90.0
Panama0.83.116.413.558.10.4
Paraguay1.33.912.911.249.80.3
Peru0.83.414.912.163.70.2
St. Lucia0.22.917.115.266.4−0.3
St. Vincent & Grens.0.23.115.912.464.0−0.6
Suriname0.73.813.112.050.5−0.1
Trinidad and Tobago0.13.315.014.846.5−0.6
Uruguay−0.12.718.525.562.8−0.2
Venezuela, Rep. Bol.0.93.514.311.157.50.2
Middle East and North Africa
Afghanistan, I.R. of2.67.17.15.934.21.1
Algeria0.93.514.310.255.90.4
Bahrain, Kingdom of1.34.211.83.554.9−0.1
Djibouti1.75.19.87.937.30.4
Egypt1.75.29.79.839.51.0
Iran, I.R. of0.33.315.07.960.3−0.4
Iraq3.07.26.96.724.32.1
Jordan1.84.810.47.547.30.8
Kuwait1.85.39.42.843.20.4
Lebanon0.12.818.013.976.6−0.5
Libya1.04.212.07.951.10.2
Mauritania2.87.46.86.921.51.7
Morocco0.63.613.910.660.10.1
Oman1.23.912.93.664.30.0
Pakistan2.05.59.18.935.11.0
Qatar2.04.710.71.553.60.0
Saudi Arabia1.24.311.64.751.40.4
Sudan2.86.77.47.325.01.7
Syrian Arab Republic1.64.710.58.649.91.0
Tunisia0.43.314.912.356.2−0.1
United Arab Emirates1.64.411.31.450.60.0
Yemen, Republic of2.56.87.36.134.11.1
Sub-Saharan Africa
Angola3.89.75.25.919.82.5
Benin3.08.16.16.521.91.8
Botswana1.85.49.36.639.50.6
Burkina Faso3.59.55.35.820.62.2
Burundi3.58.65.85.820.22.4
Cameroon3.17.66.67.426.11.8
Cape Verde1.14.112.18.361.90.2
Central African Rep.2.56.87.38.332.41.4
Chad3.89.85.16.320.42.4
Comoros2.87.36.86.026.71.5
Congo, Dem. Rep. of3.48.36.07.323.12.4
Congo, Republic of3.07.07.18.423.92.2
Côte d’Ivoire3.39.45.37.020.72.1
Equatorial Guinea2.56.57.76.032.41.7
Eritrea3.07.76.55.930.11.6
Ethiopia2.46.08.48.141.41.4
Gabon2.25.68.910.734.51.3
Gambia, The3.79.85.15.719.42.2
Ghana2.77.56.77.123.01.4
Guinea3.08.26.17.026.91.9
Guinea-Bissau2.98.26.17.021.31.6
Kenya2.96.97.36.229.21.8
Lesotho2.77.36.88.631.60.8
Liberia3.08.26.16.823.41.8
Madagascar3.07.46.76.426.82.0
Malawi3.27.26.98.526.42.3
Mali3.69.75.26.423.72.4
Mauritius−0.22.818.015.163.7−0.6
Mozambique3.48.16.28.323.52.2
Namibia2.46.28.17.234.01.2
Niger4.110.64.77.015.53.3
Nigeria3.410.14.96.418.32.1
Rwanda2.35.88.66.142.71.2
Sao Tome & Principe2.96.77.47.126.01.6
Senegal3.17.86.46.828.02.2
Seychelles0.33.315.210.955.4−0.4
Sierra Leone3.09.45.36.023.01.3
South Africa1.75.39.59.038.80.2
Swaziland2.97.56.77.531.30.8
Tanzania3.37.46.77.823.62.4
Togo3.18.46.06.223.91.9
Uganda3.78.85.76.522.72.5
Zambia3.78.45.97.220.32.6
Zimbabwe2.86.77.56.634.21.4
Source: Author’s calculations.

Population aged 65+ as percent of population aged 20–64.

Source: Author’s calculations.

Population aged 65+ as percent of population aged 20–64.

1

The author is grateful to participants in the IMF FAD Seminar in January 2016, Nan Geng, Daehaeng Kim, Giovanni Melina, Mark De Broeck, Hua Chai, and Santiago Acosta-Ormaechea for useful suggestions and comments. Many thanks to Ana Popovich and Liza Prado who provided invaluable administrative assistance. The usual disclaimer applies.

2

Defined-benefit, earnings-linked pensions are publicly provided in 20 out of 34 OECD countries (OECD, 2015).

3

Public pensions hold reserve funds in 16 out of 34 OECD countries (OECD, 2015).

4

In a standard overlapping generation model, the internal rate of return for a funded pension scheme equals the market interest rate, which can be larger than the internal rate of return for a pay-as-you-go scheme (the so-called Aaron (1966) condition). Even if this is the case, the difference in the internal rates of return of the two schemes is fully explained by the initial cost for a pay-as-you-go pension that offers pensions to the elderly with no contribution records. See Sinn (2000), Fenge and Werding (2003), and Blake (2006) for details.

5

Under this variant, net migration is generally assumed to remain constant until around 2050 and decrease gradually decline thereafter, while recent developments such as refugee flows and temporary labor flows are taken into account (United Nations, 2015). See Clements and others (2015) for discussions of these migration assumptions.

6

For OECD countries, the average normal retirement age for defined benefit public pension schemes is 63.3 years (OECD, 2015). The LBIR estimates for alternative values of the retirement age are available from the author upon request.

7

To estimate the net present value of age-related public spending on pension and healthcare in the future, Clements and others (2015) assume that the interest rate-growth differential converges in the long run to 1 percent, the average observed in the advanced economies over the past 25 years.

8

The UN projection only includes variants with constant mortality (the mortality rate kept constant at the 2010–2015 level) and zero migration.

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