2. Hybrid convex pension systems

In a pay-as-you-go (PAYG) public pension system, a representative of the working population receives at each t ≥ 0 a salary S _t > 0 and pays to the system a fraction of this salary as a contribution rate Π_t > 0 in order to cover an average present pension P _t > 0. If N _t and M _t are respectively the number of workers and pensioners at time t it must hold

(1)$$M_tP_t = N_t\Pi _tS_t.$$

We define further the dependency ratio D _t > 0 to be the ratio of the retired population to the active workforce:

(2)$$D_t = \displaystyle{{M_t} \over {N_t}}.$$

For all t ≥ 0 we can thus write the following from of the budget equation

(3)$$D_tP_t = \Pi _tS_t, \;$$

The individual average replacement rate Δ_t > 0 is defined as the ratio of the average pension paid to the salary i.e.,

(4)$$\Delta _t = \displaystyle{{P_t} \over {S_t}}.$$

Substituting in (3) we attain the equilibrium equation

(5)$$D_t\Delta _t = \Pi _t.$$

Clearly, the process D _t does not completely define the pension plan unless a second equation determining an allocation rule of the cashflows coming from the workers and to be transferred to the pensioners is defined.

In national PAYG pension systems essentially two allocation rules are considered, DB and DC. A DB system assumes that the replacement rate is constant to the current value at all times t ≥ 0, i.e.,

(6)$$\matrix{ {\Delta _t = \Delta _0, \;} & {\Pi _t = \displaystyle{{D_t} \over {\Delta _0}}} \cr } $$

Conversely, a DC system assumes that the contribution rate is kept constant in time, i.e., for all t ≥ 0:

(7)$$\matrix{ {\Pi _t = \Pi _0, \;} & {\Delta _t = \displaystyle{{\Pi _0} \over {D_t}}.} \cr } $$

From these equations we deduce that the ageing of a country, reflected in an increasing value of D _t, in a DB (respectively DC) system is a cost borne by the workers (resp. retirees) alone in the form of an increased contribution rate (resp. a decreased replacement rate). It is self-evident (and will be later more rigorously justified) that neither any of these two attain inter-generational social fairness in that the burden (or advantage) of a demographic shock is carried (or enjoyed) by either one of the cohorts of pensioners or workers.

In a recent paper, Devolder and de Valeriola (Reference Devolder and de Valeriola2019) have introduced the idea of a new allocation rule for the public resources available through (5), based on the idea of hybridizing DB and DC using a convex combination of the two systems.

For a fixed α ∈ [0, 1] the Defined Convex Combination (DCC) scheme is the pension system that gathers contributions and pays pensions in such a way to keep the α-convex combination between Δ_t and Π_t constant, regardless of the variations in D _t. That is we require that for all t ≥ 0 the condition:

(8)$$\left\{{\matrix{ {\alpha \Delta_t + ( {1-\alpha } ) \Pi_t = C_\alpha } \cr {C_\alpha = \alpha \Delta_0 + ( {1-\alpha } ) \Pi_0} \cr } } \right.$$

is in force alongside (3). Conditions (3) and (8) fully describe the pension system. It is clear that DC and DB are naturally embedded in this framework, and correspond respectively to the values α = 0 and α = 1.

The intuition behind the DCC systems is that even neither the DB of DC scheme attain social fairness, equitability could still be achieved by some kind of co-movement in the contribution or replacement rate, able to resolve the shocks in D _t in a ‘socially optimal’ way. Naturally, in order to do so, the national security system must be statutorily empowered with the ability of dynamically adjusting the rates as to maintain the relationships above.

As we shall see further on, in a DCC scheme such optimality could for example be achieved solving a minimization problem on α of some kind of loss functional applied to the contribution and replacement rates.

Although a novel concept, a non-trivial instance of the DCC pension scheme class was already present in prior literature, and that is the one associated to the Musgrave rule (Musgrave, Reference Musgrave1981). The Musgrave rule states that social fairness is achieved if the system leaves the ratio of the replacement rate to the salaries net of contribution constant. In the formula

(9)$$\displaystyle{{\Delta _t} \over {1-\Pi _t}} = \displaystyle{{\Delta _0} \over {1-\Pi _0}}: = C_M$$

with C _M > 0. We call this system the Defined Musgrave scheme (DM). To see that this is of DCC type one just rearranges the above and divides by 1 + C _M, obtaining

(10)$$\displaystyle{{\Delta _t} \over {1 + C_M}} + \Pi _t\displaystyle{{C_M} \over {1 + C_M}} = \displaystyle{{C_M} \over {1 + C_M}}$$

which is a DCC with α = (1 + C _M)⁻¹ and $C_\alpha = 1-\alpha$.

As it turns out this correspondence can be generalized. Consider a pension system that keeps constant the ratio of the replacement rate to the salary net of a multiple γΠ_t of the contribution rate, with 0 ≤ γ < 1/Π₀, leading to:

(11)$$\matrix{ {\displaystyle{{\Delta _t} \over {1-\gamma \Pi _t}} = \displaystyle{{\Delta _0} \over {1-\gamma \Pi _0}}: = C} \cr } $$

for some C > 0 and all t ≥ 0. Note that when γ = 0, 1 we have respectively a DB and a DM pension system. We can rearrange (11) to

(12)$$\Delta _t + \gamma C\Pi _t = C$$

which after dividing by 1 + γC can be seen to be a convex combination of the form

(13)$$\alpha \Delta _t + ( {1-\alpha } ) \Pi _t = C_\alpha $$

where

(14)$$\matrix{ {\alpha = \displaystyle{{1-\gamma \Pi _0} \over {1 + \gamma ( {\Delta_0-\Pi_0} ) }}, \;} & {C_\alpha = \displaystyle{{\Delta _0} \over {1 + \gamma ( {\Delta_0-\Pi_0} ) }}.} \cr } $$

Observe 0 < α < 1. Since these transformations are invertible, implementing the constant replacement rate to income net-of-a-multiple-of-contribution ratio rule is fully equivalent to the design of a DCC scheme. In other words, we could say that the DCC plans are effectively a generalization of pension systems based on the Musgrave rule. The role of γ is that of setting the balance of the adjustments made in reaction to D _t more favourably to either pensioners or workers. As we shall see γ is strictly linked to the maximum value that Π_t can assume.

3. The instantaneous dependency rate and explicit solutions

It is evident from the foregoing discussion that once the dependency ratio D _t is specified the DCC architecture is fully determined by the equilibrium and convexity conditions (5)–(8). An appropriate choice of the dependency ratio model is thus the critical element to the design of a DCC scheme that adequately describes a realistic pension system.

Naturally, the positive process D _t is better modelled as an exponential evolution, i.e., modelling is equivalently effected on E _t = log D _t. In Devolder and de Valeriola (Reference Devolder and de Valeriola2019) the authors assume two alternative versions of D _t. The first is a Brownian motion with drift, i.e., the stochastic differential equation (SDE)

(15)$$dE_t = \mu dt + \sigma dW_t$$

for some Brownian motion W _t and constant drift and volatility μ, σ > 0. The second is a mean-reverting OU process:

(16)$$dE_t = \kappa ( {\theta -E_t} ) dt + \sigma dW_t$$

for a long run mean θ and a mean reversion speed coefficient κ > 0. According to the authors, these alternative choices are rooted in opposing views regarding mean-reversion in mortality rates expressed in the literature (Luciano and Vigna, Reference Luciano and Vigna2005; Zeddouk and Devolder, Reference Zeddouk and Devolder2018), whose absence (resp. presence) supports model (15) (resp. (16)).

The view of this paper is that models (15) and (16) should not be used to model the dependency ratio. With regards to the theoretical motivation offered for these models, we observe that D _t is not a pure mortality process, but it is impacted by at least three socio-economic and demographic factors: mortality, fertility and employment rates. So a model selection with reference to literature related to only one of such factors is debatable. Second, and most importantly, models (15) and (16) are rejected by the statistical evidence in a large number of countries. We will detail and discuss this thoroughly in Section 4.

This provides motivation for a change of paradigm, and what we propose is the introduction of an instantaneous stochastic rate r _t driving the variation of the dependency ratio. Under such a specification D _t does not suffer from the limitations exposed above. The choice we operate is well-grounded in classic demographic theory, produces tractable solutions for Π_t and Δ_t providing insight on the role of the DCC parameter α, and is adequately supported by the empirical data.

We assume D _t follows a classic exponential population growth model (e.g., Kendall, Reference Kendall1949) with a stochastic growth rate r _t, that is, D _t solves for any possible realizations of a stochastic process r _t the equation

(17)$$\matrix{ {\displaystyle{{dD_t} \over {dt}} = r_tD_t, \;} & {D_0 > 0} \cr } $$

that is

(18)$$D_t = D_0\exp \left({\mathop \int \nolimits_0^t r_udu} \right).$$

The process r _t, which we shall leave unspecified at the moment, is the instantaneous dependency rate, encoding the time evolution of the demographic, economic and social determinants combining to form the dependency ratio. Note that the dependency ratio is increasing in t if and only if r _t > 0. This means that a positive process for the rate r _t would be excessively binding for modelling of D _t, since it would not allow its (at least) temporary downturn.

Henceforth we also assume that Π_t and Δ_t are pathwise differentiable. Without loss of generality we can thus set, for Π₀, Δ₀ > 0:

(19)$$\Pi _t = \Pi _0\exp ( {\pi_t} ) $$

(20)$$\Delta _t = \Delta _0\exp ( {\delta_t} ) $$

for some differentiable processes π _t, δ _t, with δ ₀ = π ₀ = 0, chosen such that the equilibrium equation (5) is fulfilled, which together with (18) implicates

(21)$$r_t = {\pi }^{\prime}_t-{\delta }^{\prime}_t.$$

Furthermore, (8) must be in force, and we know from the previous section that this is equivalent to the expression in (11) being constant. Therefore, differentiating and using (21) we have that

(22)$$\matrix{ {\displaystyle{d \over {dt}}\displaystyle{{\Delta _t} \over {1-\gamma \Pi _t}} = \displaystyle{{\Delta _t( {\delta_t^{\prime} ( {1-\gamma \Pi_t} ) + \pi_t^{\prime} \gamma \Pi_t} ) } \over {{( {1-\gamma \Pi_t} ) }^2}} = \displaystyle{{\Delta _t( {\delta_t^{\prime} + \gamma r_t\Pi_t} ) } \over {{( {1-\gamma \Pi_t} ) }^2}} = 0} \hfill & \qquad \hfill & {} \hfill \cr } $$

holds if and only if

(23)$$\matrix{ {\delta _t = {-}\mathop \int \nolimits_0^t \gamma \Pi _ur_udu.} \hfill & \qquad \hfill & {} \hfill \cr } $$

Replacing (23) in (20) and using again (21), we have that π _t must satisfy pathwise the ODE

(24)$$\left\{{\matrix{ {\pi_t^{\prime} = ( {1-\gamma \Pi_0e^{\pi_t}} ) r_t, \;} \hfill \cr {\pi_0 = 0.} \hfill \cr } } \right.$$

Remembering π _t = log (Π_t/Π₀) we arrive at the relation:

(25)$$\matrix{ {\Pi _t^{\prime} = \Pi _t( {1-\gamma \Pi_t} ) r_t, \;} & {\Pi _0 > 0.} \cr } $$

Equation (25) is a familiar logistic equation which naturally arises in population evolution models, and nicely complements (17). Its solution is:

(26)$$\Pi _t = \displaystyle{{D_t\Pi _0} \over {\gamma D_t\Pi _0 + D_0( {1-\gamma \Pi_0} ) }}$$

for 0 ≤ γ ≤ 1/Π₀, as it can be shown by a standard separation of variables argument, which for the reader's convenience we recall in the Appendix. Therefore:

(27)$$\Delta _t = \displaystyle{{\Pi _0} \over {\gamma D_t\Pi _0 + D_0( {1-\gamma \Pi_0} ) }}.$$

For γ = 0, we have Δ_t = Δ₀ from (27) which leads to the DB scheme. For γ = 1/Π₀ it is Π_t = Π₀ from (26) and thus we recover a DC scheme, something which cannot be obtained from (11) or (13). For γ = 1 the two equations above also clearly imply the DM scheme.

We see that the stochastic intensity rate r _t is also the stochastic growth factor of the contribution rate: to an increase in the dependency ratio, that is, ageing of the population, must correspond to an increase in the contribution rate to maintain the equilibrium. The faster the population ages, the steeper this revision must be.

Also the role of γ is of interest. In a logistic equation the quantity γ ⁻¹ is the capacity of the dynamical system, that is, the maximum valueFootnote ² of the solution. In other words, for any chosen γ, its inverse γ ⁻¹ is the maximum value of contribution which is needed to maintain the system in equilibrium: remarkably this value is <1 with certainty (implying <100% of the salary required to be paid as a pension contribution) if and only if 1 ≤ γ ≤ 1/Π₀. This means that if instead 0 < γ < 1 then we have that Π_t > 1 with positive probability, that is with positive probability the pension system could potentially face collapse,Footnote ³ which is something one might want to remember. We do observe that this property is not a byproduct of the specification (18), but rather a characteristic of the DCC system as a whole, as a numerical example readily shows.Footnote ⁴

The conclusion is that the parameter γ – which in the constructive definition of a DCC scheme is the multiple of the contribution rate such that the ratio of the dependency rate to the salary netted of such a multiple is constant – retains the following socio-economic interpretation. It is the inverse of the maximum contribution rate that workers will ever need to pay into the pension system. When γ ⁻¹ = 100% = γ, we are in a DM system, and the maximum contribution is the whole salary. In the case γ ⁻¹ = Π₀ we are in a DC scheme and such a maximum is the given fixed rate of contribution. As another example, if γ = 2 then γ ⁻¹ = 50%, and the workers will never have to contribute to pensions with more than the half of their salaries. The parameter γ could be explicitly or implicitly set by the national authorities in accordance with the country's economic outlook.

In the next section we provide statistical backing for DCC plans with an instantaneous dependency rate.

4. Statistical evidence of an instantaneous dependency rate

In this section we conduct a brief statistical study motivating the reasons for going beyond the modelling of the dependency ratio as the logarithm of an Ito diffusion, and providing supporting evidence for the choice of an instantaneous dependency rate. When looking at the annual time series of the dependency ratio, if either one of (15) or (16) were a plausible model, the time series analysis would support either some sort of stationarity, or a drifting random walk behaviour. As we shall see both of these hypotheses are strongly rejected by the statistical analysis. In contrast, assuming a mean reverting SDE for the rate r _t in the model (18) finds empirical support.

Let $\{ {d_t^C } \} _{t_{min} \le t \le t_{max}}$ be the available data set for the generic country C and set $x_t^C = \log d_t^C$. We begin by assuming the true generating data process of $d_t^C$ to be of the form (18): since the data visualization shows curves with very low convexity/concavity at the given annual frequency, we can use backward differences with Δt = 1 and write

(28)$$x_t^C -x_{t-1}^C \sim \displaystyle{d \over {dt}}\log D_t = r_t.$$

Hence, the realizations of the process r _t can be thought as being approximated by the series

(29)$$y_t^C = \Delta x_t^C : = x_t^C -x_{t-1}^C .$$

Therefore, in order to justify the introduction of a mean-reverting instantaneous dependency rate r _t to model D _t we must provide adequate evidence of stationarity of the series $y_t^C$, or, equivalently, show that the series $x_t^C$ are integrated of order one.

In order to test this hypothesis we first visualize the data, study the correlations and select reasonable models. We then check for stationarity combining two popular statistical tests: the augmented Fuller–Dickey (ADF) test (Dickey and Fuller, Reference Dickey and Fuller1979) and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test (Kwiatkowski et al., Reference Kwiatkowski, Phillips, Schmidt and Shin1992). These tests are complementary to each other, in the sense that the null hypothesis of the former is the presence of a unit root in the regression model, whereas the null of the second is stationarity.

We used the OECD official datasetFootnote ⁵ for the old-age dependency ratio,Footnote ⁶ comprising a set of 36 countries, with records ranging from 1950 to 2014. Figures 1 and 2 present the time series $x_t^C$ and $y_t^C$ for selected countries in the sample. The series $x_t^C$ exhibits a marked growth trend, and the drifting random walk model or a stationary process reverting to a constant are already suspect. In contrast, it does seem plausible to postulate an autoregressive model for $y_t^C$.

Figure 1. Time series of the logarithmic dependency ratios $x_t^C$ for various countries.

Figure 2. Time series of the differenced logarithmic dependency ratios $y_t^C$.

The correlograms of the two set of series (two examples are shown in Figures 3 and 4) suggest strong, slowly decaying autocorrelation function (ACF) function for the series $x_t^C$ for all the countries considered, which is at odds with stationarity, and ultimately model (16). After differencing, we notice that in most cases the slow decay of the ACF is resolved, while the partial autocorrelation function (PACF) is still non-negligible for the first one or two lags (see table in Appendix B); in some cases the correlations change sign so that a moving average component can be conjectured. Non-zero PACF for $y_t^C$ at lag one for all C is in contrast to the random walk with drift hypotheses for $x_t^C$, that is, with model (15). We reject such model on this basis. Overall, the correlogram plots suggest an autoregressive modelFootnote ⁷ with dependency up to the first two lags for $y_t^C$, and therefore an ARIMA(p, 1, q) model for $x_t^C$, p = 1, 2, q ≥ 0.

Figure 3. Correlograms of $x_t^{EL}$ and $y_t^{EL}$.

Figure 4. Correlograms of $x_t^{HU}$ and $y_t^{HU}$.

The ADF test for a time series x _t is concerned with finding the statistical significance of δ in the autoregression model

(30)$$\Delta x_t = \alpha + \beta t + \delta x_{t-1} + \mathop {\mathop \sum \nolimits }\limits_{i = 1}^k \delta _i\Delta x_{t-i} + \epsilon _t$$

where ε _t is white noise and k is an optional lag applied to account for possible higher order correlations (when k = 0 the sum is absent). The null hypothesis is δ = 0 which is equivalent to a unit root being present in (30). The regressors α and β may or may not be equal to zero, which must be assessed from the qualitative character of the series under inspection: the ADF test thus tests the random walk hypotheses, with optional drift and trend. The ADF statistics $DF_\tau ^k$ is a negative number, and its critical values at a given level of confidence depend on the sample size. From the study of the correlograms of the OECD data we argue that an AR(1) model is adequate for $y_t^C$ in all but ten countries, for which an AR(2) model is better suited. Therefore we set k = 1 for this latter group, and k = 0 in all the others.

The KPSS test for stationarity decomposes the process x _t as a sum of a deterministic part, a random walk u _t and white noise ε _t:

(31)$$x_t = \beta t + u_t + \epsilon _t$$

with u _t = u _t−1 + η _t, $\eta _t\sim {\cal N}( {0, \;\sigma^2} )$; the null hypothesis is thus σ ² = 0, and the case β ≠ 0 incorporates a trend. An additional parameter k appears in the formulation of the estimator of the test, whose recommended choice in Kwiatkowski et al. (Reference Kwiatkowski, Phillips, Schmidt and Shin1992) –which we follow– is fixed as the entire part of 4(N/100)^1/4 where N is the length of the series. To our knowledge exact p-values for the test statistic $K_\tau$ are not easily obtained, but bounds are available.

Both the ADF and KPSS tests are known to have low power, so the failure to reject the null by either test must be accompanied by confirmatory evidence (rejection) from the other one in order to have a good degree of certainty that the null can be indeed considered true.

Preliminarily, we analyse model (16); in order to do so we apply the ADF and KPSS tests with α ≠ 0, β = 0 to the series $x_t^C$. In the KPSS test the null hypothesis of stationarity is rejected at a 5% significance level for 35 of the 36 countries (97.2%); for the ADF test the presence of a unit-root is rejected at the same confidence level for only 5 out of 36 countries (13.8%). This is overwhelming evidence that model (16) is not supported by the data. Insisting in this direction, we then asked ourselves whether Figure 1 could be seen as the realization of a trend-stationary processes, so that a deterministic linear trend could be incorporated in (16) to produce a reliable model, e.g., by replacing the constant θ with a linearly growing mean reversion level θ(t) = tθ. This tantamount to testing the KPSS and ADF regressions with α, β ≠ 0: the result is that H ₀ is rejected at a 5% significance for 32 out of the 36 countries (88.8%) in the KPSS test, and for 4 out of 36 countries (11.1%) for the ADF test. This indicates that there also appears to be no obvious way of extending model (16) as to include the observed linear drift in the log-dependency ratio.

We then run the ADF and KPSS tests on the set $y_t^C$ using the specification α > 0 and β = 0 suggested by the data visualization. The frequencies of the p-values of the ADF test and the statistic $K_\tau$ of the KPSS test are reported respectively in Figures 5 and 6. The country-by-country summary is given in Appendix B. For the ADF p-values we observe a skewed frequency with a clustering around small values (the median is 0.0899), from which we can conclude that the observed data are consistently in contrast to the null hypotheses of non-stationarity, which is thus unlikely to be true. By the same token, there is a high frequency of low values of the KPSS test statistics, the median being 0.2768; exactly 66.6% of the values are lower than the critical value 0.347 of $K_\tau$ for a 10% significance level, which is no basis for the rejection of H ₀. Taken jointly, the two test results show that the hypothesis of stationarity of the true data generating process of $y_t^C$ is compatible with the historically observed data.

Figure 5. Frequency histogram of the p-values of the ADF test.

Figure 6. Frequency histogram of the KPSS test statistics $K_\tau$.

In conclusion, this short cross-country study finds evidence that the differenced log-dependency ratio can be modelled with a simple autoregressive process. This provides a first statistical basis for an instantaneous dependency rate based on a mean-reverting continuous time process (essentially of OU type, see the next section).

5. Model specification and calibration

In this section we instantiate our model and fit it to the Italian data as made available by the OECD in the period 1955–2014. The residuals analyses of the times series $y_t^{IT}$ fitted using ordinary least squares on an AR(1) model is rather satisfactory, as shown in Figure 7, which means a simple autoregression of order one is a reasonable model for the data. We therefore can assume r _t to be the continuous counterpart of an AR(1) regression, i.e., a stationary mean-reverting OU process driven by the SDE:

(32)$$dr_t = \kappa ( {\theta -r_t} ) + \eta W_t$$

for some standard one-dimensional Brownian motion W _t. The parameters retain the following interpretation: the mean reversion parameter θ is the drift at which the dependency ratio is expected to grow; κ is the speed of mean reversion towards this drift, and the volatility of the rate η expresses the degree of reliability attached to the estimate of the future dependency ratio.

Figure 7. Residual time series, ACF and distribution, from the fit of model (36) to $y_t^{IT}$. The p-value of the Ljung–Box test is 0.6481.

The solution of r _t started at s < t is

(33)$$r_t = r_se^{-\kappa ( {t-s} ) } + \theta ( {1-e^{-\kappa ( {t-s} ) }} ) + \eta \mathop \int \nolimits_s^t e^{-\kappa ( {t-u} ) }dW_u$$

where the last term indicates the stochastic integral with respect to the Brownian motion W _t.

We recall that r _t is normally distributed with time s < t conditional mean and variance:

(34)$${\rm {\opf E}}_s[ r_t] = r_se^{-\kappa ( t-s) } + \theta ( 1-e^{-\kappa ( t-s) }) $$

(35)$${\rm Va}{\rm r}_s( r_t) = \displaystyle{{\eta ^2} \over {2\kappa }}( 1-e^{{-}2\kappa ( t-s) }) $$

where r ₀ ~ N(θ, σ ²/2κ) since we start the process from the stationary distribution. As observed, since r _t can be either positive or negative, D _t can both increase or decrease: when θ > 0 (ageing countries) D _t will have a tendency to rise, whereas if θ < 0 (rejuvenating countries) will have a tendency to decrease.

We can calibrate r _t by matching the coefficients of the three terms in (33) with the corresponding constant, autoregressive and noise coefficients in the AR(1) model

(36)$$y_t = c + \phi y_{t-1} + \epsilon _t$$

with $\epsilon _t\sim {\cal N}( {0, \;\sigma^2} )$. The coefficients of the fit of (36) to the differentiated series $y_t^{IT}$ are

(37)$$\phi = 0.72456$$

(38)$$c{\rm} = 0 .00424$$

(39)$$\sigma ^2 = 4.838 \times 10^{{-}5}.$$

Coefficient matching of (36) and (33) when s = t − 1 and solving for the OU parameters yields:

(40)$$\theta = \displaystyle{c \over {1-\phi }}$$

(41)$$\kappa = {-}\log \phi $$

(42)$$\eta = \sigma \sqrt {\displaystyle{{-2\log \phi } \over {1-\phi ^2}}} .$$

Substituting the numerical values in the above we finally have

(43)$$\theta = 0.01536$$

(44)$$\kappa = 0.32214$$

(45)$$\eta = 0.00810.$$

We simulate D _t, Π_t and Δ_t for the DM pension system (γ = 1) in the calibrated model based on a simple Euler scheme for r _t combined with equations (26) and (27). The time horizon is 40 years. We use as initial data the 2019 dependency ratio D ₀ = 35.6% in the website dati.istat.it by the Italian Institute of Statistics (ISTAT), and a replacement rate Δ₀ = 79.8% as estimated by the Italian Accountant General.Footnote ⁸ The two imply an initial contribution rate Π₀ of 28.4%.

Sample trajectories of the three processes are shown in Figure 8. The positive mean reversion level θ is what generates the clearly visible upward pull of the dependency ratio. As the plots show, as the population gets older and the dependency ratio increases, in order to maintain constant the ratio of the replacement rate to salaries-net-of-contribution, the national pension system revises the contribution rates upwards and lets the replacement rate decrease. Finally we observe that consistently with what already remarked, the paths of D _t, although upward trending, are not monotonically increasing. This is a realistic feature: for example, the Italian dependency ratio shows a clear downturn in the 1980s and in the early 2000s.

Figure 8. Sample paths of r _t (top left panel), D _t (top right), Π_t (bottom left) and Δ_t (bottom right). The D _t paths trend upwards (although not monotonically) because the population gets older: contributions must increase and replacement rates drop.

6. Social equitability in DCC schemes. DB or DC?

In Devolder and de Valeriola (Reference Devolder and de Valeriola2019) the authors suggest that intergenerational fairness is attained when the present (or target) relationship between the living standards of pensioners and workers is maintained through time. The living standards of the two groups can be for instance taken to be respectively the replacement rate and the salary net of contribution. The key idea is thus that the variability of some kind of ‘spread’ between the two should be minimized in order to attain social equitability. This idea is consistent with the original approach in Musgrave (Reference Musgrave1981). As explained in Section 2 the formulation of one such minimization problem is made possible precisely by the introduction of the DCC pension systems and it is one of the main motivation for considering hybrid DB and DC type schemes.

Two natural variables quantifying the divergence (or similarity) of the living standards are given by the difference and the ratio of the replacement rate to the salary net of contributions, i.e.,:

(46)$$d_t = \Delta _t-( {1-\Pi_t} ) = \Delta _t + \Pi _t-1$$

and

(47)$$m_t = \displaystyle{{\Delta _t} \over {1-\Pi _t}}.$$

As suggested, the second is exactly the quantity first analysed by Musgrave.

We introduce then six different measures of social fairness by applying three different dispersion statistics to both of the variables above: namely variance, interquartile range and median absolute variation. We denoted these respectively SD^x, IQR^x, MAD^x, where x = m, d according to which one of (47) and (46) is used. Social fairness is thus achieved, for each of the measures under consideration, at the minimizer:

(48)$$\gamma _M^x = \mathop {{\rm argmin}}\limits_{\gamma \in [ {0, 1/\Pi_0} ] } M^x( \gamma ) $$

where x = m, d and M ∈ {SD, IQR, MAD}.

Let us observe that such minimizers can be found analytically, and the minimum in all cases is zero. Indeed in a DCC scheme for all M we have

(49)$$M^d = M^d( {\Delta_t + \Pi_t} ) = 0$$

if and only if α = 1/2, since by definition of DCC scheme 2(Δ_t/2 + Π_t/2) must be constant in t. Inverting the first equation in (14) one gets the equivalent value of $\gamma _M^d$:

(50)$$\gamma _M^d = \displaystyle{1 \over {\Delta _0 + \Pi _0}}.$$

In the case when the ratio of the living standards is considered, the minimizers of all the three measures is trivially $\gamma _M^m = 1$ because this choice generates the DM scheme which is defined by the property that the ratio of the replacement rate to the salary net of contribution is constant.

Using the calibrated model and the numerical methods of Section 5, we ran a Monte Carlo simulation of 10,000 sample paths of Π_t and Δ_t at an horizon of t = 40 years, for values of γ ranging from γ = 0 (DB) to γ = 1/Π₀ (DC). We computed all of the six social fairness measures: the results are the blue lines shown in Figure 9. The shapes are those of a pasting between two concave functions, one decreasing and one increasing, with a kink in the minima at the points $\gamma _M^d$ and $\gamma _M^m$.

Figure 9. Social fairness measures in the calibration to the Italian data. Note the concave shapes and the minima $\gamma _M^d = 0.9241$ and $\gamma _M^m = 1$. The blue line is the result of the full calibration, the red line is the simulation outcome when altering the mean reversion level to θ ^′ = 0. The endpoints relative positions, corresponding to the DB scheme (left endpoint) and the DC scheme (right endpoint) are inverted in the two cases.

The general outlook from the results of the time series calibration presented confirms the consideration expressed in Devolder and de Valeriola (Reference Devolder and de Valeriola2019) stating that social fairness is best attained using an allocation rule of DCC type rather than relying on pure DB or DC architectures. As highlighted above, which value of γ is optimal depends on the chosen living standard spread measure.

Another important element of analysis, having policy implications, is the relative intergenerational fairness of DB and DC pension systems. When comparing among them the two systems we notice that in all six cases the values of the equitability measures are lower for DC, implicating that such a system is socially fairer. In other words, the calibration to the Italian data of a DCC structure seem to suggest that the choice of a DC public pension system (as it has been recently put in place in the country), for such a country is not only preferable to DB in terms of financial sustainability, but also in terms of social equitability.

However, we should not be tempted to draw undue general conclusion on the social equitability of various DCC schemes from this specific case study. Of course, different parameter estimation procedures may give raise to different results, even when they refer to the same country. For example, our analysis is fully based on the historically observed data: however the model could be also calibrated to the projections of the National Institute of Statistics, which predicts – as in many other European countries – the dependency ratio increase to slow down once the baby boom generation effect on pensions ceases.

To illustrate how the scenario may change under different model parameter estimations, we re-run the simulations of the Italian case using the same random numbers drawing and κ, η, but we assume a different future evolution of the dependency ratio is estimated, lower than historical one, which implies θ ^′ = 0. The new sample paths for the rates are shown in Figure 10: r _t is now oscillating around a lower mean reversion and thus becomes negative with higher probability, determining less steep paths for D _t and thus less extreme scenarios for Π_t and Δ_t. The corresponding results for the measures M ^x are the red lines in Figure 9: what we obtain now is that among DB and DC it is DB to show the lower values of M ^x.

Figure 10. Sample paths of r _t, D _t, Π_t and Δ_t when θ = 0. There is equal probability of r _t assuming positive or negative values: the dependency ratio has no tendency to increase, and thus contribution and replacement rate fluctuate freely.

The mean reversion level θ is therefore a critical parameter: it is the annualized expected value of the dependency ratio growth. When such a value is low, the dependency ratio rises, reflecting a high longevity risk, low fertility, and/or a shrinking economy. In such a case DB contributors are inevitably overburdened with such risk factors being transferred to them from the older generations, and this tips the balance of the living standards in favour of the current pensioners. In this scenario, a DC system may then be preferred.

However, when θ is low or negative, implicating a low or decreasing dependency ratio, we might be observing a period of low longevity risk or booming economy. In such a case, workers could expect to be able to adequately support with their contributions current and future retirees living standards within a DB architecture, without excessively penalizing their own. In fact, they must do so, in order to maintain the intergenerational risk sharing equilibrium.

A DCC system, through its automatic adjustment mechanisms, is precisely one that permits to address a dynamic macroeconomic evolution between the two scenarios described above, such as the one that has already historically materialized during the 1960s boom (demographic and economic) and the 1980s recession.