2.1. NDC Schemes
Our analysis focuses on the effect of COVID-19 on the social adequacy and financial sustainability of the Italian NDC system. Such a system was introduced in Italy by the Dini reform in 1995. In addition to Italy, Sweden, Latvia, and Poland have adopted an NDC system [
19].
The notional rate of return, the fixed contribution rate, the retirement age, the pension indexation, and the conversion coefficient multiplying the final notional account to determine the first pension amount are the principal variables of an NDC system. The system is founded on a Pay-As-You-Go (PAYG) mechanism that uses total current contributions to finance pension benefits. The pension calculation follows a DC approach, where benefits are computed by the amount of contribution paid during the working age and the accumulated earnings through a notional rate of return on contributions. The notional amount accrued at retirement is transformed into a life annuity considering the remaining life expectancy on average. In an NDC scheme, there is no accumulation of financial assets. The word “notional” relates to the PAYG nature that requires the creation of notional accounts used to calculate the amount of pension. In Italy, the notional rate of return is set equal to the nominal GDP five-year moving average.
Despite the establishment of an NDC scheme, the population aging and the reduction of the active population are jeopardizing the financial equilibrium of the Italian NDC pension system. These conditions could compromise the financial sustainability of the system and the adequacy of future pension benefits. The impact of the COVID-19 pandemic on the economic and demographic conditions will probably result in further stress for the pension system.
In order to represent the NDC system dynamics, we follow [
20], which refers to a PAYG pension scheme paying retirement benefits, disregarding survivor benefits, invalidity benefits, and withdrawals. We consider the following states: active (1), pensioner (2), dead (3), and unemployed (4). The transition probabilities between states are a function of age and time. Consequently, the eligibility for pension benefits does not depend on the years of service.
The PAYG scheme is balanced in a generic year
t when annual income from contributions,
, is equal to annual expenditure on pensions,
. The condition
can be expressed as:
where the followingdefinitions apply:
is the total population in the state
i at time
t, and it is given by
, where
is the number of lives in state
i at age
x at time
t. The latter depends on the new entrants in state
i aged
x in year
t,
and the previous-year population who have survived by time
t:
for
, where
is the probability for an individual aged
in year
to remain in state
i for one year (for the equations of new entrants
for
see [
20]. Regarding the unemployed, we assume that
, where
is the relative age distribution of the new unemployed and
is the total new entrants in the unemployed state at time
t. We suppose the same relative age distribution for the new unemployed population and the new actives:
, and
, with
depending on the total active population growth rate
that equally influences contributors).
is the contribution rate of the pension system at time t. Note that in an NDC system, the contribution rate is set constant over time: for all t.
is the average wage at time t, which is given by , where is the total wage at time t, and is the individual wage depending on the growth rate of individual wage from to t, .
is the average pension paid to retirees in year
t. It is given by
, where
is the amount of total pensions paid to retirees at time
t. It is given by
, where
is the total pensions paid to all retirees aged
x at time
t, which depends on the pension indexation rate
and the total benefits paid to the new retirees in the year
t,
(
is a function of the notional rate
that is the rate of return remunerated on the individual notional account, the expected indexation rate
for
, and the expected rate of return,
for
(see [
20] for further details)).
We define the equilibrium contribution rate
as the contribution rate satisfying Equation (
1):
where
is the dependency ratio and
is the average replacement rate of the system in year
t.
The economic dynamics and population structure evolution might threaten the stability of the PAYG pension system. While in a pure Defined Benefit-PAYG system, where benefits are fixed, the equilibrium can be restored by changing the contribution rate (for instance, Ref. [
21] find the optimal contribution rate of a PAYG pension in a Nash equilibrium model to face the shrinking working population), in an NDC pension scheme, in which the contribution rate must be steady, the equilibrium is achieved by modifying (generally decreasing) the replacement rate. It can be attained by changing the notional rate that is related to the economic conditions and changes in the expected survival probabilities of pensioners, reflecting life expectancy evolution (as observed by [
20], there are situations where an NDC system is not able to immediately restore the equilibrium, thus remaining vulnerable to demographic and economic shocks). Modifying the replacement rate on one side might assure the pension scheme is financially sustainable, while on the other side, it might fail to guarantee adequate benefits, lowering the living standard of beneficiaries as well as the attractiveness of the scheme for new members. In this situation, a “social sustainability” question could arise. We quantify it using the replacement rate,
, which is the ratio between the average pension amount and the average wage of active members of the scheme.
The situation described above highlights the need, also in the case of non-funded pension systems such as the PAYG, to constitute a buffer or reserve fund that depends on the difference between income from contributions and expenditure on pensions. The reserve fund
at time
is calculated as follows:
where
represents the unfunded liabilities,
. We assume that
’s rate of return equals the notional rate
and that, at initial time, the reserve fund is null,
.
Evaluating the financial sustainability of the system requires drawing up the actuarial balance, which in PAYG systems is usually compiled by comparing the Net Present Value of the system over a long period,
[
22]:
Equation (
4) can be expressed using the unfunded liabilities’s present value:
Whether was non-negative in the long run (), the pension system would be financially sustainable.
The system’s financial sustainability can be also measured by the annual value of the reserve fund. In this case, the system would be sustainable if in the long run.
The financial sustainability condition may be also explained using the reserve fund. Taking into account Equation (
3), we can express the reserve fund at time
T as follows:
Dividing by
, we obtain:
Consequently, equates , whether .
2.2. Macroeconomic Variables Modeling
To investigate the evolution of an NDC scheme during the COVID-19 pandemic, we have to make assumptions on the evolution of the transition probabilities and the macroeconomic variables relevant to the system. It is extremely complex to forecast the evolution of the main economic variables following the pandemic crisis. COVID-19 has led to high economic uncertainty, and projections are consequently variable, requiring particularly strong assumptions [
23,
24]. In particular, Ref. [
24] highlight that some lessons can be learned from the global financial crisis (GFC) that occurred in 2008 in projecting the GDP values during the COVID-19 crisis and recovery period. They suggest adjusting the original GDP forecasts by an amount similar to the forecast errors made during the GFC. Other scholars have compared the impacts of the COVID-19 pandemic on the economy to those of GFC. For instance, Ref. [
25] studied the efficiency of the US market for each industrial sector, during the GFC and COVID-19 pandemic. During the first, real estate, which caused the crisis, and information technology (IT) were the lowest-efficiency sectors; during the second, the materials sector, which suffered a big reduction in consumption and production, was the lowest-efficiency sector. Ref. [
26] found that the COVID-19 pandemic affected the US economic activity more severely than GFC, while the opposite occurred in the recession probabilities.
However, the aim of the paper is not to provide an exact projection of the evolution of economic variables, which would be hard to reach but to measure the NDC pension system’s resilience to an economic shock produced by the COVID-19 pandemic. Therefore, in the following, we adopt some simplifying assumptions. We assume that all the transition probabilities are deterministic, except for the pensioners’ death probabilities, , which is modeled through a standard stochastic mortality model. While we model the inflation rate , the wage growth rate , and the unemployment rate as stochastic processes, then we introduce two sources of uncertainty in the model: the demographic and the economic risk.
The data considered in the analysis refer to the time period immediately preceding the COVID-19 crisis, while the information, available at the time of writing, about the impact of the pandemic on mortality and the economy is used to determine the magnitude of the shock in the first two years of the pension system projections. Finally, we suppose that the demographic variables are independent from the economic variables. Details on the model selection process are provided in the following.
When dealing with the macroeconomic variables modeling, autoregressive (AR), autoregressive integrated moving average (ARIMA), and vector autoregressive (VAR) models and their variants are, by far, the most popular time series models in the macroeconomic literature. We can mention several contributions relating to modeling and forecasting the macroeconomic variables considered in this paper. For instance, ref. [
27] analyzed the forecasting performance of ARIMA models, bivariate vector autoregressive moving average (VARMA) models, threshold autoregressive (TAR) models, and Markov switching autoregressive (MSA) models applied to the US unemployment rate. Ref. [
28] modeled the real wage growth rate implied in the forecasts of the Social Security trust fund as an AR(1) constrained to a fixed value in the long run. Ref. [
29] developed forecasts for some macroeconomic variables (inflation and unemployment rate included) in the Euro area using AR and VAR models. Ref. [
30] considered both univariate and multivariate linear time series models (random walk, AR, VAR) for forecasting Euro area inflation. Ref. [
31] used VAR models for obtaining forecasts for Swiss inflation, while [
32] analyzed the performance of VAR and ARIMA models (in addition to factor models) for forecasting Austrian inflation. Ref. [
33] used AR and VAR models to forecast a set of US macroeconomic time series. Ref. [
34] provided predictions for macroeconomic variables like the unemployment rate using VAR models and considering data revisions. Ref. [
35] used a time-varying coefficients VAR with stochastic volatility to predict the inflation rate and unemployment rate in the US. Ref. [
36] used ARIMA and VAR models to forecast the Italian youth unemployment rate combining official and Google Trends data.
However, in recent years, advanced non-linear time series methods and artificial neural networks or hybrid approaches combining linear models with autoregressive neural networks have become popular also in the macroeconomic literature. Ref. [
37] studied the forecast accuracy of AR, smooth transition autoregressive, and autoregressive neural network (AR-NN) time series models for a wide set of macroeconomic variables of the G7 economies. Refs. [
38,
39] forecast the unemployment rate, respectively, for some European countries and some Asian countries, using ARIMA models combined with AR-NN and support vector machines. Ref. [
40] forecast inflation of the Euro using Jordan and feedforward neural networks.
Following the relevant literature, we model the inflation rate and wage growth rate through VAR models. The unemployment rate is modeled apart using an ARIMA process as it has a different nature from inflation and wage growth rate. These latter variables represent rates of change between two consecutive years, while the unemployment rate is a ratio between specific groups of people. Each macroeconomic variable is projected using a mean-reversion to an exogenous long-run trend, similar to the approach in [
41] (the authors observed that the structural changes that happened in recent years in some key variables of a social security system, such as fertility, productivity, and interest rate, resulted in mean values that differ from the average of their past values. After much experimentation, they found that satisfactory forecasts were obtained by pre-specifying the long-term means of the series rather than estimating them from the data), where the main reason to include an exogenous long-run trend is to control the long-run simulations to obtain realistic forecasts. This approach meets our scope, which is to obtain reliable forecasts in line with the expected long-term trends and give a robust stochastic structure to our framework to study the impact of COVID-19 on the NDC pension systems. Our approach is also compliant with the literature dealing with the financial sustainability of the pension systems, which generally adopts the hypothesis that the macroeconomic variables are deterministic and consistent with the paths projected by the main financial institutions [
22,
42,
43] or makes a trivial hypothesis [
44].
For our analysis, we consider the following data provided by the Italian National Institute of Statistics (ISTAT):
The unemployment rate of the male population aged 25–75 in the years 1983–2015; the rates for the residual period (2016–2019) are estimated by regression using the unemployment rate of the male population aged 15–64. These data provide values of .
The wage growth rate in the years 1983–2015, and the gross contractual hourly remuneration of employees for the last four years, which are used to estimate .
The consumer price index for blue and white-collar worker households (FOI) in the years 1983–2019, which are used to estimate .
Before choosing the model, for all the macroeconomic variables, we initially inspect their stationarity using the Phillips–Perron (PP) test (null hypothesis: no stationarity) and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test (null hypothesis: stationarity), both at a 5% significance level. The results show that the wage growth rate and inflation rate are stationary, leading to reject the vector error correction model (VECM), which adds to the VAR model an error-correction term, modeling a linear combination of the variables. Then, we analyze the structure of the causal relationships between these variables through the Granger causality test, which is a statistical test for determining whether a one-time series is useful for forecasting another. We perform the Granger causality test at a 10% level. Then, we look for the optimal number of parameters in the VAR model using the R package autoarima and test the validity of the model through a set of well-known goodness of fit measures: the Akaike Information Criterion (AIC), the Hannan-Quinn information criterion (HQ), the Schwarz Criterion (SC) and the Final Prediction Error (FPE). We find VAR(1) as the best model. In the case of unemployment rate, we cannot reject the null hypothesis of PP and KPSS tests, thus questioning the presence of stationarity. Consequently, we consider the ARMA models and not the ARIMA.
The choice of the best model for each macroeconomic variable is made by analyzing the plots of the Auto-Correlation function (ACF) and Partial Auto-Correlation functions (PACF), the Akaike Information Criterium (AIC), and checking the stationarity of residuals and their normality distributions (see
Appendix A.1). As a result, we model the inflation rate and wage growth rate through a VAR(1) with an exogenous long-run trend of 3.5% for the wage growth rate and 2% for the inflation rate, and the unemployment rate through an AR(4) with a 4.7% exogenous long-run trend in line with the long-term values adopted by the Ministry of Finance for the long-term projections of the national pensions expenditure [
45]. Therefore, the unemployment rate considering the exogenous long-run trend
, and denoted as
, is described as follows:
where
are the parameters of the model,
is a constant serving as the intercept of the model, and
is a white noise process. While the inflation rate and the wage growth rate considering the exogenous long-run trends
and
, which are denoted as
and
, respectively, are jointly modeled as a VAR(1) process:
where
,
,
and
are the parameters of the model,
and
are constants serving as the intercept of the model, and
and
are zero-mean white noise processes. Equation (
9) implies that the inflation rate in a certain year is related not only to the previous inflation rate (through parameter
) but also to the previous wage growth rate (through parameter
). A similar consideration can be made for the wage growth rate in Equation (
10).
2.4. The COVID-19 Macroeconomic and Demographic Scenario
We denote
as the COVID-19 shock over the macroeconomic variables here considered, where
represents the shock on the unemployment rate,
represents the shock on the inflation rate, and
represents the shock on the wage growth rate. We set
for all
t with the exception of
and
, where
. Therefore,
,
and
under the COVID-19 scenario are, respectively, modeled as follows:
Concerning the COVID-19 impact on mortality, many proposals have been made in the literature to model adverse mortality jumps. Ref. [
46] suggest to introduce a dummy variable in the time index process to treat the impact of Spanish flu on US mortality. According to this approach, denoting
as the COVID-19 mortality shock,
should be modeled as:
Since then, many models that incorporate adverse jumps in mortality have been proposed in the literature, most of them with the aim to value catastrophic mortality bonds. For instance, Ref. [
47] considered a compound Poisson process to model permanent mortality jumps. Ref. [
48] used independent Bernoulli distribution for transitory jump occurrence and normal distribution for jump severity. Ref. [
49] proposed a stochastic diffusion model with double-exponential jumps. Recently, Ref. [
50] used a Lee–Carter model with a jump diffusion process and a lognormal renewal process for modeling the arrival of mortality jumps.
Most of the proposed models assumed that the sensitivity of each age to the mortality jumps was the same as that of general mortality improvements, which is not supported by the empirical evidence. To avoid this limitation, Ref. [
51] explicitly introduced an age pattern of temporary adverse mortality jumps distinct from that of general mortality improvements, while [
52] proposed two alternative models: in the first, the age pattern of jump effects is represented through a separate constant vector; in the second, they assumed that jump effects for different age groups are not perfectly correlated, allowing the age response to different mortality jumps to be different.
Since in our paper, the objective is not to model the effect of future adverse jumps on mortality but to estimate only the impact of the COVID-19, we propose a simple solution consisting of adding an extra term for the year 2020,
, to the log mortality rates, which allows representing the shift of mortality age profile due to the pandemic:
The increase in mortality due to COVID-19 has affected both the working and retired populations. A comparison of deaths by age group in 2020 and 2019 (see [
53]) shows that only 6.4% of extra deaths in 2020 referred to the 15–64 age group. On the Italian population as a whole, the highest number of deaths recorded in the 15–64 age group in 2020 compared to 2019 is just over 7000. Thus, the effect of extra mortality due to COVID-19 on the total working population is negligible as well as its impact on employment equilibrium. Hence, we can conclude that the number of employees has much more been influenced by the change in the unemployment rate rather than by the change in mortality rates. For this reason, we have neglected the effect of COVID-19 mortality on the working population and measured only the effect on the retired population.