In recent years, many countries have introduced pension system reforms to improve their financial sustainability. Many of these are based on the adoption of an automatic link between changes in life expectancy and pensions (Bravo et al., 2021; OECD, 2017).
In this section, we resort to the mechanism proposed by Coppola et al. (2019), who recommended the indexation of retirement age to life expectancy at a given reference age as obtained by projecting mortality tables for different cohorts.
The proposed crosscountry comparative analysis relies on the NRA defined as the age at which an individual could retire in 2018 without any reduction to his/her pension, having had a full working career from age 22 (OECD, 2019). Even if this indicator is sometimes quite different from individuals’ effective retirement age, the broad heterogeneity of pension schemes characterizing the chosen OECD countries makes it necessary to consider a universal and comparable measure.
In the following, we assume that each individual receives a constant monthly payment as long as he/she survives, starting at the NRA. Let ({e}_{mathrm{NRA},mathrm{C}}^{(M)}) be the life expectancy at NRA for a given cohort C, according to the mortality model M.
If we consider the mortality trend and the subsequent longevity risk, we can realistically expect that life expectancy for younger generations will increase with respect to that of a benchmark cohort C^{*}. As suggested by Coppola et al. (2019), we consider a stochastic mortality model M, fit it to data, and then use it for mortality projections for a given cohort C. We then determine the forward shift (({mathrm{lag}}_{mathrm{C}}^{(M)})) applicable to the NRA for cohort C to match the residual expected life span for cohort C^{*}, in the formula:
$${mathrm{lag}}_{C}^{(M)}=mathrm{inf}left{j:{e}_{mathrm{NRA}+mathrm{j},mathrm{C}}^{left(Mright)}le {e}_{mathrm{NRA},{C}^{*}}^{left(Mright)}right}.$$
(1)
By applying that shift to the NRA for cohort C, the pension provider will pay benefits for a number of years not greater than the one for which it will have to pay them to an individual belonging to the reference cohort C*. A similar approach can be found in Denmark’s pension reform, where the statutory retirement age will gradually increase, targeting the age at which the remaining life expectancy is 14.5 years (target retirement age). The target retirement age implies that the remaining life expectancy after retirement should be constant at 14.5 years (Alvarez et al., 2021). For the sake of illustration, we assume C^{*} = 1960 and we consider that the lag is expressed in monthly fractions of a year.
On this basis, the paper presents a comparative analysis across selected OECD countries of the gender differences in life expectancy at NRA and in the lags to be applied to the NRA. The study has an exploratory nature because it is meant to provide guidelines on policy adjustments, and it will be structured in the following phases:

1.
projection of mortality tables by using the Generalized Age–PeriodCohort (GAPC) mortality models shown in Table 1, in order to consider the longevity risk and calculate relative life expectancies;

2.
introduction of an averaged model obtained with an assembling technique that allows us to mitigate the model risk arising when we choose one of the aforementioned models for prediction;

3.
calculation of the shifts to be applied to the NRA for the male and female populations of each OECD country according to the indexing mechanism and the averaged mortality model;

4.
comparative analysis of the gender gap both in life expectancy and in shifts required for NRA.
Mortality projections: GAPC mortality models
In actuarial calculations, agespecific measures of mortality are usually needed, and in a dynamic context, mortality is assumed to be a function of both the age x and the calendar year t (the socalled age–period approach). In our study, we refer to GAPC models to project mortality and we use the GAPC forecasts to obtain projected life table for each specific cohort (calendar year). GAPC models are a class of parametric models that link the force of mortality (({mu }_{x,t}))^{Footnote 1} to a linear or bilinear predictor structure consisting of a series of factors. The reader is referred to Pitacco et al., 2009 for major details on mortality forecasting and to Villegas et al., (2018) for a concise yet comprehensive introduction to model implementation in the free R statistical environment.
The GAPC class includes most of the stochastic mortality models discussed in the demographic and actuarial literature given by the original LC model (Lee & Carter, 1992) and its Poisson version proposed by Brouhns et al., (2002), the extensions of the LC model proposed in Renshaw and Haberman (2006), the age–periodcohort (APC) model (Currie, 2006), the original Cairns–Blake–Dowd (CBD) model (Cairns et al., 2006), the extended CBD (M7) model of Cairns et al., (2009), and the model of Plat (2009). In Table 1,^{Footnote 2} we give a summary definition of the GAPC models we use, pointing out the corresponding systematic component and bibliographic references.
All of these models are derived from the LC model, which is a milestone in the literature related to mortality projections. The LC model has been widely extended to improve model performance and forecasting power. Brouhns et al. (2002) employed this model assuming a Poisson distribution of the number of deaths and using the log link function with respect to the force of mortality ({upmu }_{x,t}). Some other proposals have been introduced in the literature to include components for capturing the cohort effect (({gamma }_{tx})), for instance in Renshaw and Haberman (2006), and a quadratic age effect ({(xoverline{x })}^{2}{widehat{sigma }}_{x}^{2}), as in the M7 model, to obtain the predictor. In the latter, (overline{x }) is the average age in the data, and ({widehat{sigma }}_{x}^{2}) is the average value of ((xoverline{x })). In 2006, Currie introduced the APC model, a substructure of the RH model. Cairns et al. (2006), using the two factor CBD model, propose a predictor structure with two age–period terms ({k}_{t}^{(i)}) (I = 1, 2), with agemodulating terms ({beta }_{x}^{(1)}=1) and ({beta }_{x}^{(2)}=xoverline{x }), no age function ({mathrm{alpha }}_{x}) and no cohort effect. Plat (2009) combines the CBD model with some features of the LC model to obtain a model that is suitable for considering full age ranges and captures the cohort effect.
In order to model and project mortality rates, we follow the steps summarized below:

1.
Fitting the parameter estimates of GAPC stochastic mortality models by maximizing the model loglikelihood;

2.
Inspecting the residuals of the fitted model to analyze mortality models’ goodness of fit;

3.
Modeling the period indexes ({k}_{t}^{(i)}) and the cohort index ({gamma }_{tx}) by using timeseries techniques;

4.
Using the forecasted (simulated) values of the predictors to obtain forecasted (simulated) agespecific central mortality rates or agespecific oneyear death probabilities.
Model averaging
Whatever the criterion we use to select a model to carry out the projections of mortality rates, the model risk arises. As Shang and Haberman (2018) observed, “Model averaging combines forecasts obtained from a range of models, and it often produces more accurate forecasts than a forecast from a single model”. To mitigate this risk, we refer to a modelassembling technique tested by Buckland et al. (1997) and Benchimol et al. (2016).
The assembling technique is based on the use of a weighted average of the forecasts obtained under competing models (see Coppola et al., 2019), with weights that account for the model’s goodness of fit. Referring to the Akaike Information Criterion (AIC), which is widely used for model selection, we calculate the weights for the N projection models using the following formula:
({w}_{i}=frac{{e}^{frac{1}{2}{delta }_{i}}}{sum_{i=1}^{N}{e}^{frac{1}{2}{delta }_{i}}}) where ({delta }_{i}=frac{{mathrm{AIC}}_{i}{mathrm{AIC}}_{mathrm{best}}}{{mathrm{AIC}}_{mathrm{best}}}), i = 1,…,N,
where ({mathrm{AIC}}_{mathrm{best}}) indicates the lowest AIC value among those of the N competing models. Thus, the procedure assigns to projections from the ith model an importance that increases as the goodness of fit increases.
For the purposes of our analysis, we have fitted the selected models on mortality data from 1960 to 2017, which were then projected 30 years onwards. This method has been performed for each country and separately for male and female populations.
Finally, we want to point out that:
– Regarding the fitting time window, we have not considered the period following the cardiovascular revolution. Since the early 2000s, there has been a marked slowdown in progress relating to cardiovascular mortality; consequently, focusing only on the period following the cardiovascular revolution may provide distorted results.
– Regarding the validation of the predictive capacity of the model it could be carried out by splitting the dataset in two subsets: the training set and the test set. We evaluate this approach to be not suitable to our study, as it would result in a test set strongly influenced by the effects of the economic crisis. In the long run, there is an established positive association between economic growth and life expectancy. In the short term, the situation is different; many authors have studied whether the mortality rates are procyclical (for example, CerviniPlá & VallCastelló, 2021). Our approach favors longterm trends in mortality, given that actuarial calculations concerning pensions are based on survival probabilities extending over a longterm horizon.
Lag calculation
For each of the selected countries, we consider the genderspecific NRA given by the OECD in 2019 (see Table 2 in “Data framework” section). We assume C* = 1960 as the benchmark and we calculate the life expectancy at NRA according to the projection derived from the modelassembling (MA) technique, ({e}_{mathrm{NRA},mathrm{C}}^{left(mathrm{MA}right)}), for selected cohorts C. Then we calculate the shift (({mathrm{lag}}_{C}^{left(Mright)})) as defined in (1). This indicator provides the genderspecific adjustment that can be applied to NRA to match ({e}_{mathrm{NRA},{C}^{*}}^{(mathrm{MA})}), thus measuring the extent to which the sustainability of NRA is undermined by the aging of the target population.
Comparative analysis of gender gap
Finally, we carry out a comparative analysis across selected countries with the aim of deriving similarities and differences in the dynamics of longevity risk associated with pension sustainability. To simplify the notation, let ({L}_{M}, {L}_{F}) denote the lags for men and female NRAs for a given cohort, as detailed in (1), “Lag calculation” section. Then, we consider their difference (g={L}_{M}{L}_{F}) as a straightforward measure of the gender gap in sustainability for the NRA. In particular, g = 0 denotes that, with respect to genderspecific NRA, similar adjustments should be made for men and women; g > 0, instead, indicates that NRA is less sustainable for men than it is for women. In this case, stronger adjustments are required for men with respect to the indexation of retirement age needed for women, as a consequence of a higher velocity of growth in life expectancy for men than for women. Conversely, countries for which g < 0 are characterized by a higher velocity in longevity risk for women than for men with respect to NRA. Thus, higher and positive values of this index indicate that longevity risk in NRA programs is more underestimated for men than it is for women. Conversely, the genderspecific evolution of longevity patterns entails that NRA for women exposes National Security Systems to unsustainability to a greater extent than for men if low and negative values occur for the gender gap in retirement age.
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