2.1 Coverage measures

With minor variations, ‘coverage’ stands for either (A) passive coverage: the proportion of elderly individuals who receive a pension or (B) active coverage: the proportion of economically active individuals that contribute to any pension system. The extensive study of Latin American Countries by Arenas (Reference Arenas de Mesa2019), for example, applies these measures. ‘Coverage’ is a useful SM for pension systems because it is easy to compute and understand, it is nationally representative, and it provides an international ranking since most people will agree on the social convenience of a high coverage. Indeed, coverage might currently be the principal SM of pensions in international comparisons.

However, passive coverage is insensitive to the size of pension allowances, and, to that extent, to adequacy, sufficiency, sustainability, and other dimensions commonly used to analyze the performance of pension systems. Passive coverage, in other words, could be high even within a pension system that consistently provides little consumption smoothing and little redistribution. The passive coverage measure, then, is not very close to the fundamental goals of pension systems. Active coverage, on the other hand, depends heavily on labor market characteristics like formalization, and is insensitive to the size of contributions, thus again as a measure it is not necessarily close to the fundamental goals (active coverage could be high even if smoothing and redistribution are low).Footnote ⁴

It is then hard for current coverage measures to rise as a leading SM of the pension situation.

2.2 RR measures

The RR concept stands for the proportion of labor income that is replaced by pension income. The RR takes the form of a ratio between some measure of pensions in the numerator and some measure of labor income in the denominator.

The RR is always among the top pension indicators. It has tremendous appeal for the general public, academia, and policymakers. Furthermore, most defined-benefit pension systems establish the size of pension allowances based on the idea of RRs,Footnote ⁵ and individual capitalization systems are probably set keeping a desired RR in mind. Another example is ILO Convention N°102 for pension and unemployment insurance, currently ratified by about 65 countries, which establishes minimum benefits using RRs.

Indeed, the RR concept points to the heart of one of the fundamental roles of pension systems: the consumption smoothing role. From the perspective of a public pension system, the RR from mandatory deposits can be interpreted as a ranking: a higher RR is better than a lower one, at least up to some advisable standard.

All leading pension publications give the RR concept a central place. Adequacy (2021) uses two measures of the RR:

• • The empirical aggregate-level replacement ratio: the empirical ratio of the gross median individual pension income of the population aged 65–74, relative to gross median individual earnings from work of the population aged 50–59 excluding other social benefits.

• • The theoretical individual-level RRs: the RR of hypothetical workers that follow certain careers, most of them being a lifetime full-length and uninterrupted stream of pension contributions from a certain average or minimum wage.

Ageing (2021) uses one:

• • The empirical aggregate-level replacement ratio among new pensioners: the empirical average new pension allowance over the average yearly gross wage at retirement.

OECD (2021) uses theoretical individual-level RRs along the lines of Adequacy (2021). ILO (2021) reproduces the ones in Ageing (2021).

2.3 Other pension SMs

There are a handful of other measures in the current literature that perform international comparisons. Actuarial valuations, for instance, provide measures for sustainability that can be used as international rankings, and sustainability is certainly a very important dimension. All too relevant since many (if not the majority) of public pension systems in the world constantly operate below the standard sustainability line. However, some big problems arise: (A) Sustainability does not have a sufficiently close link with the fundamental roles of pension systems; a very small system, with very small pensions, could be perfectly sustainable without having a substantive impact on the population. (B) What is the sustainability standard for public systems? It is not clear, and many will dispute the idea that they are just the same as the one for private institutions. (C) Actuarial valuations are complex and costly, and require expensive and painful actuarial analysis and assumptions that are not easy to perform continuously across countries. (D) Actuarial valuations rarely focus on more than one subsystem in any given country.

The financial returns of pension funds are another important dimension of comparison. Each pension system has a different role for financial returns, though, and has its own particular set of preferences over the risk-return profile too.

Another international comparison refers to pension indexations. The underlying idea is that pensions should be indexed by either inflation or costs of living (usually captured by a wage growth index) or another index, rather than no indexing at all. However, this dimension captures only a small fraction of the pension situation.

Finally, another measure is the ‘benefit ratio’, defined in Ageing (2021) as the average pension as a share of the economy-wide average wage (gross wages and salaries divided by employees).Footnote ¹⁰ While very interesting as a measure, it shares most of the weaknesses of the current RR measures discussed above.

3.1 An actuarial formula for the RR

The next formulas assume a constant wage and density of contributions, and then it is shown that using wage or density trajectories does not alter the formula for the self-financeable RR.

As shown in the actuarial or financial fields, it is relatively easy to derive the following formula for the final account balance of a person that saves a contribution rate ‘c’ of a constant monthly wage ‘w’ with a density of contributions ‘d’, under constant monthly interest rate ‘i’, from age at entry (say, 25) until age at exit (say, 65)Footnote ¹¹:

(1)$${\rm Final\ savings} = \underbrace{{\displaystyle{{\left[{{( {1 + i} ) }^{12\left({\matrix{ {{\rm exit}} \cr {{\rm age}} \cr } -\matrix{ {{\rm entry}} \cr {{\rm age}} \cr } } \right)}-1} \right]} \over i}}}_{{\matrix{ {{\rm Final\ savings\ from\ a\ monthly\ }} \cr {{\rm deposit\ of}\; 1} \cr } }} \ \ \ \ \ \cdot \underbrace{ { w\cdot d\cdot c}} _{{\matrix{ { {\rm Average\ monthly\ }} \cr {{\rm deposit}} \cr } }}$$

(2)$${\rm Pension\ allowance} = \displaystyle{{{\rm Final\ savings}} \over {{\rm Actuarial\ factor}}}$$

Then, to transform those final savings into a self-financed pension, the only required operation is to divide the final savings by an ‘actuarial factor’, as shown in equation (2). The actuarial factor is standard in the actuarial literature, and it represents precisely the neccesary ratio of dollars of savings for each dollar of self-financed pension allowance. More precisely, the actuarial factor can be defined as the present value of a monthly pension of $1 dollar payable until death for an individual currently of age ‘age at retirement’; it depends only on life expectancy and the expected interest rate of low-risk assets which are the typical investment of savings during retirement.Footnote ¹² A typical actuarial factor for a 65-year-old individual may lie around 220, meaning that at 65 years old, $220 of savings are necessary to, on average, finance a pension of $1 until death. So, savings of, say, $100,000 would translate into a monthly pension of around $100,000/220 = $450. This self-financed pension could also be called the actuarially fair pension.

The self-financeable RR, finally, is the ratio between the pension allowance and labor income, as shown in equation (3). The labor income here is w × d, because, as mentioned in Section 2.2, the year average is what counts as labor income. The self-financeable RR formula is:

(3)$${\rm Self}-{\rm financeable\ RR} = \displaystyle{{{\rm pension\ allowance}} \over {w \times d}} = \underbrace{{\displaystyle{{\left[{{( {1 + i} ) }^{12\left({\matrix{ {{\rm exit}} \cr {{\rm age}} \cr } -\matrix{ {{\rm entry}} \cr {{\rm age}} \cr } } \right)}-1} \right]} \over {ac \times i}}}}_{{\matrix{ {{\rm Replacement\ rate\ of\ the\ dollar\ }} \cr {{\rm that\ is\ deposited\ monthly}} \cr } }} \ \ \ \ \cdot \underbrace{c}_{{\matrix{ {\% \;{\rm of\ the\ dollar\ }} \cr {{\rm that\ is\ deposited\ monthly}} \cr } }}$$

where ac stands for actuarial factor. Equation (3) shows many important issues. First, it shows the key and simple linear role of the contribution rate: twice the contribution rate maps into twice the RR. Clearly, this accumulation stage variable, the contribution rate, is a prime determinant of the potential consumption smoothing of a pension system. Furthermore, across countries of similar levels of development there is a similar retirement age, entry age, and actuarial factor, because of which the contribution rate is probably the principal source of international heterogeneity in the self-financeable RR.

Second, from equation (3) it is easy to propose a general rule of thumb for how much RR buys each 1% of contribution rate. Plugging in the common 25/65 for age at entry/exit, 220 for the actuarial factor, and a constant 4% for the real annual interest rate, we get that each 1% of contribution rate buys about 5% of RR. If the target for the RR is 70%, for example, then the required contribution rate is 70/5 = 14%.

Third, it is very important to note from equation (3) that the self-financeable RR is not a function of labor income (wage or density of contributions); thus, the self-financeable RR is comparable across populations of different wages and densities.

If the wage or density is not constant across the life cycle, the reference labor income in the denominator of the RR should be the lifetime average labor income, rather than measures of recent labor income, as discussed in Section 2.2. Financially, the natural choice for the lifetime average is the permanent income that would attain the same final savings; in this case, by construction the RR in equation (3) does not depend on the permanent labor income (trajectories of wage and density of contributions), because this parameter is present in the numerator and the denominator of the RR and therefore it is canceled out in the same way that the constant labor income got canceled from equation (3). In other words, the RR with respect to the permanent labor income is comparable not only across different constant wages and densities, but also across different trajectories of wages and densities of contributions.

Thus, it is possible to generalize that with a real annual interest of 4% and a 25/65 life career, each 1% of contribution rate buys about a 5% RR with respect to the permanent income, and therefore a 14% contribution rate buys about a 70% RR, all this not depending on the particular trajectories of wages or densities of contributions. The spreadsheet with Formula 3 is available in the online Appendix.

3.2 An SM of the RR

We propose to compute the RR detailed in equation (3) using country-level aggregates. The resulting measure captures the potential consumption-smoothing of the pension system. It is not a prediction, but a projection of the current situation, which makes things simpler and avoids somewhat arbitrary assumptions for the future. The country-level aggregates to be plugged into equation (3) can be estimated from cross-sectional observations across recent years.

Before discussing how the proposed measure deals with the challenges described in Section 2.2, we discuss each country-level aggregate.

The contribution rate is an essential component of the RR detailed in equation (3). We base the country-level aggregate for this rate on the labor share of the GDP.Footnote ¹³ For the proposed measure, we define a macro contribution rate as the ratio between (A) the summation of all the working-age individuals' mandatory deposits into the pension dynamics as a percentage of GDP, and (B) the macroeconomic labor share as a percentage of GDP. The spirit of this macro contribution rate, then, is the percentage of national labor income that is mandatorily delivered by working-age individuals into the pension dynamics. The mandatory deposit stands here for the mandatory contributions and taxes for contributory pensions plus the working-age individuals' attributable share of the financing of non-contributory pensions.

The attributable share of the financing of non-contributory pensions is added because it represents additional economic resources that working-age individuals put into the pension dynamics.Footnote ¹⁴ We simply approximate the attributable share of working-age individuals by the proportion of the population below the legal retirement age.

For the remaining parameters in equation (3), we use the same ‘25’ for the average age of entry into the labor market across countries, and the average legal retirement age as ‘age at retirement’. Both are well known and understood parameters. For the real annual interest rate during working age, we propose a 4%, in the idea that although each country may prefer a particular risk-return profile, it is still a good idea to use the same conservative position when comparing them. For each country, we compute the actuarial factor using country-level data on life expectancy (mortality rates, to be precise) plus a conservative real annual interest rate of 1%.

The proposed measure, therefore, can be called the country-level mandatory self-financeable RR of the permanent income (‘CRR’). It measures what fraction of their middle-career labor income could replace a working-age population mandatorily saving the current macro contribution rate, given their age of entry into the labor market and their legal retirement age. The CRR has the same structure as equation (3):

(4)$$ \underbrace{{{\rm CRR}}}_{{\matrix{ {{\rm Country}-{\rm level}, \;} \cr {{\rm mandatory}, \;} \cr {{\rm self}-{\rm financeable\ }} \cr {{\rm replacement\ rate}} \cr } }} \!\! = \!\! {\rm \;}\underbrace{{\displaystyle{{\left[ \! {{( {1 + i} ) }^{12\left({\matrix{ {{\rm legal\ retirement}} \cr {{\rm age}} \cr } -\matrix{ {{\rm average\ entry}} \cr {{\rm age}} \cr } } \right)} \!-1} \! \right]} \over {af \times i}}}}_{{\matrix{ {{\rm Replacement\ rate\ of\ the\ dollar\ }} \cr {{\rm that\ is\ monthly\ deposited}} \cr } }}\cdot \underbrace{ {\displaystyle{{\matrix{ {{\rm working}-{\rm ag}{\rm e}^{\prime}} \cr {{\rm mandatory\ contributions\;}} \cr {{\rm and \ share\ of\ financing\ }} \cr {{\rm of \ non}\hbox{-}{\rm contributory}, \;\;} \cr {{\rm pensions \ as}\;\% \;{\rm of\ GDP}} \cr } } \over {{\rm labor\ share\ of\ GDP}}}}}_{{\displaystyle{\rm Macro\ contribution\ rate}}}$$

The CRR has several advantages as an SM, which makes it comparable as a ranking across countries and times. Unlike current RR measures:

• First, the CRR focuses clearly on one fundamental role of pension systems, the consumption smoothing potential of current pension systems. Current measures of the RR are focused on actual current pensions being paid, where it is usually hard to, empirically, disentangle consumption smoothing from redistribution.

Second, the CRR presents the replaceability of the permanent labor income, which implies that the CRR does not depend on either the level or the volatility of wages or density of contributions. Applied to the country-level aggregates, therefore, the CRR is comparable across countries with different levels and trajectories of GDP's labor share. Current measures of RRs focus on the replaceability of pre-retirement labor income without a clear conceptual and empirical framework, as discussed in Section 2.2.

Third, the CRR does not depend on sustainability issues (since it is a self-financeable RR), thus simplifying its use as an SM.

Fourth, the CRR does not depend on redistributive dynamics (since it is a macro level and self-financeable RR), again simplifying its use as a SM.

Fifth, the CRR does not depend on the heterogeneity of benefits rules (other than the legal retirement age), because all countries and times are measured by the same one: the self-financeable benefit at retirement age. To be precise, the CRR assumes the same ‘likelihood of a survivorship’ and ‘survivorship allowance equals 60% of the original pension’ configuration across countries, but the survivorship configuration is far from being a protagonist in pension debates.

Sixth, the CRR includes all working-age individuals and all labor income in the economy. All individuals are involved in the labor share of GDP, in the financing of non-contributory pensions, and in the computation of the average age of entry into the labor market; besides, the focal point of a legal retirement age is proposed as legitimate for everyone. Current measures of RRs, on the other hand, are focused on some forms of labor income, and some subgroups of real or hypothetical individuals, probably far from national representativeness.

Additionally, the CRR depends only mildly on the evolution of the system. The macro contribution rate might change if the public spending on non-contributory changes, but there is considerable risk in taking a stance about the future configurations of the pension system; the CRR takes the conservative position of projecting the situation with the current value of the macro contribution rate. The age of entry in the labor market and the interest rate in the CRR formula would probably remain stable as the system evolves. Finally, the life expectancy and the retirement age might evolve in a highly predictable way, in which case their expected value in the near future can be used; the results in the next section are based on the projected values for 2030.