2.1. GSA scheme

We assume that a community of $l_x$ workers aged x retire at time $t=0$ and are free to decide the decumulation strategy to follow. They evaluate the competing strategies according to their expected present values at time $t=0$ . As a benchmark decumulation strategy, we consider an immediate annuity product with annual benefit $b_{A}$ paid at the beginning of each year. In the following and in the remainder of the paper, we will be assuming that there is a reference population whose survival probabilities are used by the insurance company to price a lifetime annuity. Assuming that

(2.1) \begin{equation}\{{}_{t}{p}_{x}^r\}_{t=0, \dots,\omega-x-1},\end{equation}

is the vector that collects all the survival probabilities for a head aged x over t years for the reference population, and ignoring commission expenses and safety loadings, the single premium of the unitary immediate lifetime annuity paid in advance sold to a policyholder aged x is

\begin{equation*}\ddot{a}_{x} = \sum_{t=0}^{\omega-x-1}\ {}_{t}{p}^r_{x}v^{t},\end{equation*}

where $v^{t}=(1+i)^{-t}$ is the t-years financial discount factor and $\omega$ denotes the limiting age (i.e. $p_{\omega}=0$ ). Then, the expected present value at time $t=0$ of the benefits paid by the annuity with periodic payment $b_{A}$ for a policyholder aged x whose survival probabilities are $\{{}_{t}{p}_{x}\}_{t=0, \dots,\omega-x-1}$ is

\begin{equation*}EPV_A(0) = \sum_{t=0}^{\omega-x-1}\ {}_{t}{p}_{x}v^{t}b_{A}.\end{equation*}

The alternative we focus on in this paper is a collective self-insurance scheme, namely the group self-annuitization scheme proposed by Piggott et al. (Reference Piggott, Valdez and Detzel2005). The scheme works as follows. When it is set up, the scheme pools together into a fund the resources collected by the $l_x$ policyholders. To make the comparison with the annuity fair, we assume that each individual contributes $ b_A \ddot{a}_{x}$ to the fund. Hence, the total fund at time $t=0$ is

\begin{equation*}F(0)=l_{x}b_{A}\ddot{a}_{x}.\end{equation*}

At time 0, each individual receives a benefit $b_{A}$ , equal to the annuity benefit, therefore

(2.2) \begin{equation} b_{GSA}(0)=b_A.\end{equation}

Assuming investment at the risk-free interest rate level i, at time $t=1$ the fund is valued

\begin{equation*}F(1)=(F(0)-l_{x}b_{A})(1+i)=l_{x}b_{A}(\ddot{a}_{x}-1)(1+i),\end{equation*}

where the annuity price $\ddot{a}_{x}$ (as well as all the other annuity prices $\ddot{a}_{x+t}$ for all $t\geq 1$ ) is computed using the survival probabilities (2.1) of the reference population. Given the actual number of survivors at age $x+1$ , $l_{x+1}^{*}$ , the benefit at time $t=1$ received by each survivor in the scheme, $b_{GSA}(1)$ , obtained sharing among the survivors the annuitized value of the fund at time 1, is

\begin{align*}b_{GSA}(1)= \frac{1}{l_{x+1}^{*}}\left(\frac{F(1)}{\ddot{a}_{x+1}}\right)=\frac{1}{l_{x+1}^{*}}\left(\frac{l_{x}b_{A}(\ddot{a}_{x}-1)(1+i)}{\ddot{a}_{x+1}}\right).\end{align*}

Considering that (i) the recursive relationship for the annuity prices is

\begin{equation*}\ddot{a}_{x+1}= (\ddot{a}_{x}-1)(1+i)/{p}^r_{x},\end{equation*}

and that (ii) the number of survivors at time 1, $l^*_{x+1}$ , is given by $l_x p^*_x, p^*_x$ being the realized 1-year survival probability at age x, $b_{GSA}(1)$ can be rewritten as

\begin{align*}b_{GSA}(1)=\frac{1}{l_{x+1}^{*}}\left(\frac{l_{x}b_{A}(\ddot{a}_{x}-1)(1+i)}{(\ddot{a}_{x}-1)(1+i)/{p}^r_{x}}\right)=b_{A}\left(\frac{p^r_{x}}{p_{x}^{*}}\right).\end{align*}

Observing that at a generic time $t\geq 1$

(2.3) \begin{equation} l^*_{x+t}=l^*_{x+t-1}p^*_{x+t-1},\end{equation}

the benefit of the GSA scheme at time t is

\begin{eqnarray*}b_{GSA}(t)&=&\frac{F(t)}{l_{x+t}^{*}\ddot{a}_{x+t}}=\frac{l_{x+t-1}^{*}b_{GSA}(t-1)\left(\ddot{a}_{x+t-1}-1\right)\left(1+i\right)}{l_{x+t}^{*}\ddot{a}_{x+t}}=\\&=& b_{GSA}(t-1)\frac{l_{x+t-1}^{*}}{l_{x+t}^{*}}\frac{\left(\ddot{a}_{x+t-1}-1\right)\left(1+i\right)}{\left(\ddot{a}_{x+t-1}-1\right)\left(1+i\right)/{p}^r_{x+t-1}}=\\&=&b_{GSA}(t-1)\left(\frac{p^r_{x+t-1}}{p_{x+t-1}^{*}}\right).\end{eqnarray*}

Hence,

(2.4) \begin{equation}b_{GSA}(t)=b_{GSA}(t-1) \cdot MEA_{t},\end{equation}

where

(2.5) \begin{equation}MEA_{t}=\frac{p^r_{x+t-1}}{p_{x+t-1}^{*}}.\end{equation}

The factor $MEA_t$ , $t=1,2,...,\omega-x-1$ is the mortality experience adjustment, that is the ratio of the expected to the actual survival rates in year $[t-1,t]$ . Notice that in Piggott et al. (Reference Piggott, Valdez and Detzel2005) the interest rate risk is considered too, and the benefit at time t turns out to be equal to the benefit at time $t-1$ times the adjustment for the mortality risk ( $MEA_t$ ) and the adjustment for the interest rate risk ( $IRA_t$ ). In this paper, we ignore interest rate risk by assuming a constant interest rate i over time.Footnote ¹

The realized survival probabilities $p^*_{x+t}, t=0,1,...$ are found via simulation of the number of deaths in the pool, and therefore simulation of the realized number of survivors $l^*_{x+t}$ . Piggott et al. (Reference Piggott, Valdez and Detzel2005) assume the randomness in mortality to increase linearly with age. Differently from Piggott et al. (Reference Piggott, Valdez and Detzel2005), we model the number of survivors in each time period by simulating the evolution of the stochastic mortality intensity process $\lambda(t)$ and the time of death of each individual in the pool as the first jump time of a doubly stochastic process with intensity $\lambda$ (see Section 3).

The time-0 expected present value for an individual enrolled in the GSA scheme aged x and whose survival probabilities are $\{{}_{t}{p}_{x}\}_{t=0, \dots,\omega-x-1}$ is

\begin{equation*}EPV_{GSA}(0) = \sum_{t=0}^{\omega-x-1}v^{t}{}_{t}{p}_{x}b_{GSA}(t),\end{equation*}

where the GSA benefits $b_{GSA}(t)$ are as in (2.4).

2.2. The redistributive GSA scheme

In the description of the GSA scheme, we have neglected the fact that the scheme can pool together individuals belonging to different sub-populations, which can display different mortality patterns. A simplified description of this situation is to consider three different categories of individuals: high-risk (HR), medium-risk (MR) and low-risk (LR) individuals, with survival probabilities $\{{}_{t}{p}^{HR}_{x}\}_{t=0, \dots,\omega-x-1}$ , $\{{}_{t}{p}^{MR}_{x}\}_{t=0, \dots,\omega-x-1}$ and $\{{}_{t}{p}^{LR}_{x}\}_{t=0, \dots,\omega-x-1}$ , respectively. In this context, by “high (low) risk individual” we refer to an individual with lower (higher) survival probabilities with respect to the medium ones at every time horizon:

\begin{equation*} {}_{t}{p}_{x}^{HR} \lt{}_{t}{p}_{x}^{MR}\lt{}_{t}{p}_{x}^{LR} \qquad \text{ for all } t=0,\dots, \omega-x-1. \end{equation*}

If we calculate the expected present values at $t=0$ of the GSA benefits for the three categories, $EPV_{GSA}^j(0)$ , $j=HR, MR, LR$ , using their own respective survival probabilities, we obtain that

\begin{equation*} \sum_{t=0}^{\omega-x-1} {}_{t}{p}_{x}^{HR}v^{t}b_{GSA}(t)\lt \sum_{t=0}^{\omega-x-1} {}_{t}{p}_{x}^{MR}v^{t}b_{GSA}(t)\lt \sum_{t=0}^{\omega-x-1} {}_{t}{p}_{x}^{LR}v^{t}b_{GSA}(t), \end{equation*}

that is

(2.6) \begin{equation} EPV_{GSA}^{HR}(0)\lt EPV_{GSA}^{ MR}(0) \lt EPV_{GSA}^{LR}(0). \end{equation}

It is clear that, due to the different survival probabilities of each sub-population, distributing the same benefit to each sub-group leads to a solidarity transfer across individuals, in particular from high-risk to low-risk individuals. If, as usual, high-risk individuals belong to lower socio-economic classes, such transfer exhacerbates inequality. It is thus interesting to study redistribution mechanisms within the scheme to reduce those inequalities.

We consider a redistribution mechanism obtained with a simulative approach. In particular, we simulate 10,000 scenarios of mortality patterns for each sub-population and implement an optimal redistribution policy by minimizing for each simulation k $(k=1, \ \dots, 10,000 )$ the squared distance between a re-scaling of $EPV_{GSA}^{j,k}(0)$ ( $j=HR, MR, LR$ ) through the use of redistributive shares $\alpha^j$ and the baseline $EPV_{GSA}(0)$ level of the medium policyholder in the absence of other sub-populations in the scheme.

The redistributive scheme works as follows. We assume that each retiree pays $b_A \ddot{a}_x=1000$ in the pooled fund at time $t=0$ . The unique pool has value $N_{0}b_A \ddot{a}_x=1000 N_0$ , where $N_{0}$ is the total number of retirees aged x given by the sum of the number $N_{0}^{j}$ of retirees in each sub-population $j \in \{HR,MR,LR\}$ :

\begin{equation*} N_{0}=N_{0}^{HR}+N_{0}^{MR}+N_{0}^{LR}. \end{equation*}

The optimal redistributive shares, $\alpha_{*}^{j}$ , are found via a simulative approach by solving the following problem:

(2.7) \begin{align} {\min}_{\{\alpha^{j}\}_{j \in \{LR,MR,HR\}}}&\left\{\frac{1}{10,000}\sum_{k=1}^{10,000}\frac{1}{3}\sum_{j\in \{LR,MR,HR\}}\Big(1000-\alpha^{j}EPV_{GSA}^{j,k}(0)\Big)^2\right\} \end{align}

(2.8) \begin{align} &\qquad \qquad \qquad s.t. \sum_{j\in \{LR,MR,HR\}}\alpha^{j}=3. \\[8pt]\nonumber \end{align}

where $EPV_{GSA}^{j,k}(0)$ is the expected present value of GSA benefits for sub-population j in simulation k. At time $t=0$ , the optimal redistributive share $\alpha_{*}^{j }$ defines the benefits paid to sub-group j in the redistributive scheme:

(2.9) \begin{equation} b_{RE}^{j}(0)= \alpha_{*}^{j}b_{GSA}^{j}(0)=\alpha_{*}^{j}b_A. \end{equation}

Due to the recursive mechanism (2.4), for each $t\gt 0$ , it holds

(2.10) \begin{equation} b^{j}_{RE}(t)= b^{j}_{RE}(t-1) \cdot MEA_{t}, \end{equation}

with $MEA_{t}$ defined as in Equation (2.5).

The approach results to be financially sustainable since the constraint in (2.8) ensures that the funds awarded to each individual are a fraction of the total funds available at each time t.

The effect of the redistribution can be measured evaluating the EPV of the benefits obtained by the groups after the redistribution:

(2.11) \begin{align} EPV_{RE}^{j}(0)= \sum_{t=0}^{\omega-x-1}{}_{t}{p}_{x}^{j}v^{t}b_{RE}^{j}(t)=\alpha_{*}^{j}EPV^j_{GSA}(0). \end{align}

The goal after the redistribution is to obtain a reduced gap among the $EPV_{RE}(0)$ of the different sub-populations.

3.1. The theoretical framework

We consider a complete filtered probability space $(\Omega, \mathcal{F},\mathbb{P})$ and a filtration $\{\mathcal{G}_{t}: t \geq 0\}$ of sub- $\sigma$ -algebras of $\mathcal{F}$ . We introduce a nonexplosive counting process $M_{t}$ with intensity $\lambda(t)$ and another filtration $\{\mathcal{F}_{t}: t \geq 0\}$ such that $\mathcal{F}_{t}\subset \mathcal{G}_{t}$ .

The process $M_{t}$ is said to be doubly stochastic driven by $\{\mathcal{F}_{t}: t \geq 0\}$ , if $\lambda(t)$ is $(\mathcal{F}_{t})$ -predictable and for all t,s with $t\lt s$ , conditional on the $\sigma$ -algebra $\mathcal{G}_{t}\vee \mathcal{F}_{s}$ generated by $\mathcal{G}_{t}\cup \mathcal{F}_{s}$ , the process $M_{s}-M_{t}$ is Poisson distributed with parameter

\begin{equation*}\int_{t}^{s}\lambda(u)du.\end{equation*}

Conditional on the knowledge of a particular trajectory $t \rightarrow \lambda(t, \widetilde{\omega})=\widetilde{\lambda}(t)$ for fixed $\widetilde{\omega} \in \Omega$ , the counting process M becomes a Poisson process with (conditionally deterministic) time varying intensity $\widetilde{\lambda}(t)$ .

A stopping time $\tau$ is said to be doubly stochastic with intensity $\lambda$ if the underlying counting process whose first jump time is $\tau$ is doubly stochastic with intensity $\lambda$ . The specification of the sub-filtration $\mathbb{F} = (\mathcal{F}_{t})_{t \geq 0}$ is meant to indicate that the first jump time of M is a stopping time with respect to $(\mathcal{G}_{t})_{t\geq0}$ , but outside the span of $(\mathcal{F}_{t})_{t\geq0}$ , which carries sufficient information to reveal the intensity $\widetilde{\lambda}(t)$ (i.e. the likelihood that the jump will happen) but not enough to predict the occurrence of the jump. Doubly stochastic processes with a stopping time are typically exploited to model the survival process of individuals. In particular, the time of death is typically modelled as a doubly stochastic stopping time with intensity given by the mortality intensity $\lambda$ that is a stochastic force of mortality. The mortality intensity is typically modelled as an affine process in order to exploit well-known analytical results for the computation of the survival function (see Duffie et al. Reference Duffie, Pan and Singleton2000). For the specification of the affine stochastic mortality intensity, we consider non-mean reverting processes among those introduced by Luciano and Vigna (Reference Luciano and Vigna2008). In particular, we model the mortality intensity $\lambda_x(t)$ of an individual aged $x+t$ of a specific cohort with initial age x with a Feller process with dynamics given by

(3.1) \begin{equation}d\lambda_{x}(t) = a\lambda_{x}(t)dt + \sigma \sqrt{\lambda_{x}(t)} dW_{x}(t),\end{equation}

where $a\gt 0$ and $\sigma\geq0$ represent the drift and diffusions parameters associated to the processes, W_x (t) being a Brownian motion. Following Duffie et al. (Reference Duffie, Pan and Singleton2000), given an affine mortality intensity $\lambda_{x}(t)$ , the survival probability $S_{x}(t)$ describing the probability of an individual aged x to survive t years is:

(3.2) \begin{equation}S_{x}(t) = {}_{t}{p}_{x} =\mathbb{E}[e^{-\int_{0}^{t}\lambda_{x}(u)du}|\mathcal{G}_{0}]=e^{\alpha(t)+\beta(t)\lambda_{x}(0)},\end{equation}

where $\alpha(\cdot)$ and $\beta(\cdot)$ solve appropriate ODEs and the initial observed intensity $\lambda_{x}(0)$ is taken to be equal to $-ln(\hat{{p}}_{x})$ , where $\hat{{p}}_{x}$ is the observed survival rate at age x.

In the case of the Feller process, we have

(3.3) \begin{equation}\begin{cases}\alpha(t)=0 \\\beta(t)=\frac{1-e^{bt}}{c+de^{bt}}\end{cases}\end{equation}

with $b=-\sqrt{a^{2}+2\sigma^2}$ , $c=\frac{b+a}{2}$ , $d=\frac{b-a}{2}$ .

The inequality constraint

(3.4) \begin{equation}e^{bt}\left(\sigma^{2}+2d^{2}\right)\gt \sigma^{2}-2dc\end{equation}

must hold in order for the survival process $S_{x}(t)$ to be a decreasing function of t.

3.2. Methodology for the simulation of the number of deaths in one simulated scenario

Let $\tau$ be the time of death of the individual modelled as a doubly stochastic stopping time with intensity $\lambda$ . For the death time of each individual in the group self-annuitization pool it holds

(3.5) \begin{equation}\mathbb{P}(\tau\gt t) = \mathbb{P}({ M_{t}}=0) = \mathbb{E}[e^{-\int_{0}^{t}\widetilde{\lambda}(u)du}|\mathcal{F}_{0}],\end{equation}

where the knowledge of the parameter $\int_{0}^{t}\widetilde{\lambda}(u)du$ is conditional to the filtration $\mathcal{F}_{t}$ , $M_{t}$ being the nonexplosive counting process with intensity $\lambda(t)$ described in Section 3.1. It can also be proven that, conditional on the particular trajectory of the intensity process $\widetilde{\lambda}(t)$ , $\tau$ satisfies

(3.6) \begin{equation}\tau = \inf\limits_{t\geq 0}\left\lbrace t: E_{1}\leq \int_{0}^{t}\widetilde{\lambda}(u)du\right\rbrace,\end{equation}

where $E_{1}\sim Exp(1)$ .

In order to obtain the number of deaths in the GSA pool and the $p_{x+t-1}^{*}$ probabilities in one simulated scenario, we first simulate for each sub-population $j\in\{HR, MR, LR\}$  one trajectory of the intensity process $\tilde{\lambda}(t)$ for $t\in[0,\omega-x]$ ; then, assuming the future remaining lifetime of pool members to be independent, we simulate, for $j \in \{HR, MR, LR\},$ $N^j_0 = 1000$ independent realizations from an Exp(1) random variable; finally, thanks to (3.6) every single extraction leads to the death time of the different $N_{0}^{j}=1000$ individuals. Thus, for a given j, the exponential random vector

\begin{equation*}\{E^j_{n} \}=\{E_1^j,E_2^j,...{,} E_{1000}^j \},\end{equation*}

for $n=1,...,N_{0}^j=1000$ , leads through Equatrion (3.6) to the death time random vector for the simulated scenario

\begin{equation*}\{\tau_{n}^j\}=\{\tau_1^j,\tau_2^j,...{,} \tau_{1000}^j\}.\end{equation*}

It follows that the number of individuals of sub-population j dying between ages $x+t-1$ and $x+t$ in the simulated scenario is given by

(3.7) \begin{equation}d_{x+t}^j = \mathbb{E}\left[\sum_{n=1}^{N_{0}^{j}}{\unicode{x1D7D9}}_{\left\lbrace t-1\lt\tau_{n}^j\leq t\right\rbrace}\right] .\end{equation}

This leads to the computation of the survivors $l^{*j}_{x+t}$ at age $x+t$ belonging to sub-population j in the GSA pool as

(3.8) \begin{equation}l_{x+t}^{*j} = l_{x+t-1}^{*j}-d_{x+t}^j,\end{equation}

and for each t the total number of survivors in the GSA pool is given by

(3.9) \begin{equation}l_{x+t}^{*}= \sum_{j \in \{HR,MR,LR\}}l_{x+t}^{*j}.\end{equation}

From the sequence $\{l_{x+t}^{*}\}$ provided by Equation (3.9), one can compute the realized survival probabilities $\{{}_{t}{p}_{x}^{*}\}$ as in (2.3) in the simulated scenario. Finally, using (2.4), it is possible to compute in the simulated scenario the benefits $\{b_{GSA}(t)\}_{t=0,1,...,\omega-x-1}$ for all survivors in the GSA scheme.

4.1. The dataset

Individuals belonging to distant socio-economic classes bear a different longevity risk. More disadvantaged social classes are more exposed to riskier environments, riskier habits and to access barriers to healthcare. According to Ardito et al. (Reference Ardito, Leombruni and Costa2019), the longevity gap is an increasing function of the degree of deprivation, and it is more pronounced for male individuals than for females. In addition, the gap shrinks with the policyholder’s age.

To model the mortality differences in socio-economic classes, we use a dataset which groups the English population in ten deciles, according to the English Indices of Deprivation (version of 2015).Footnote ² The deciles are the result of the subdivision of the 32,844 lower-layer super output areas in ten equal-sized groups such that the first decile corresponds to the most deprived areas and the tenth decile corresponds to the least deprived ones. The ranking is based on income deprivation, employment deprivation, education, skills and training deprivation, health deprivation and disability, crime, barriers to housing and services, living environment deprivation.

The dataset has been used in other papers on mortality differentials by socio-economic classes such as Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Jones and Rowney2022), Wen et al. (Reference Wen, Cairns and Kleinow2021), Haberman et al. (Reference Haberman, Kaishev, Millossovich, Villegas, Baxter, Gaches, Gunnlaugsson and Sison2014). As stressed by Wen et al. (Reference Wen, Cairns and Kleinow2021), “the health deprivation and disability” factor does not perfectly suit the purposes of a research analysis based on socio-economic mortality differentials. It concerns the quality of the health status rather than a mere economic wealth based discriminant. Still, considering the correlation among the health status and all the remaining factors, the English Indices of Deprivation dataset is a useful and proficient tool to spot differences in mortality experience according to the wealth level.

The dataset provides figures for the number of deaths and risk exposures (mid-year estimates) in each population decile area by gender and single year of age for calendar years 2001–2017. Therefore, our calibration relies on seventeen observations for each sub-population. We select three sub-populations out of the ten deciles of the dataset: we consider as low socio-economic status representatives the first decile sub-population individuals, as middle class representatives the fifth decile sub-population individuals and as high socio-economic status representatives the tenth decile sub-population individuals. Therefore, given the differences in the survival probability levels, we consider three categories of policyholders by assuming first decile retirees as high-risk individuals $(j=HR)$ , fifth decile retirees as medium-risk individuals $(j=MR)$ and tenth decile retirees as low risk individuals $(j=LR)$ . Notice that, as stressed in Remark 1, in making this risk classification, we are ignoring the fact that the risk profile of each individual can change over time. Considering the construction of this dataset, a change of risk class can occur not only due to a sudden change in the health status, but also due to the members’ mobility or the re-classification of the geographical districts used to construct the dataset.

Although policyholders contribute identically to the scheme funding, high socio-economic status individuals (i.e. LR individuals) will receive on average benefits for a longer period with respect to lower socio-economic groups retirees (i.e. HR individuals). Thus, lower socio-economic classes will bear the longevity risk associated to the richest.

4.2. Calibration

For the calibration procedure, we focus on the cohort of male individuals born in 1936, entering the scheme in 2001 at the age of $x=65$ Footnote ³ and, as mentioned in Section 4.1, we consider the three sub-populations of HR, MR and LR individuals belonging to the first, fifth and tenth percentile, respectively. The mortality intensity of each sub-population follows the Feller process in $\left(3.1\right)$ :

(4.1) \begin{equation}d\lambda_{65}^j(t)=a^j \lambda_{65}^j(t) dt+ \sigma^j \sqrt{\lambda_{65}^j(t)} dW_{65}^j(t) \qquad j=HR, MR, LR.\end{equation}

We calibrate the processes (4.1) on seventeen observations, from 2001 to 2017, over the age span 65 to 81 and we obtain closed-form expressions for the remaining survival probabilities up to the limiting age $\omega=110.$

In detail, we first derive from the dataset described in Section 4.1 three vectors of observed empirical survival probabilities $\{{}_{t}{\hat{p}}_{65}^{j}\}$ for $j\in\{HR,MR,LR\}$ relative to males born in 1936 and observed yearly from 2001 to 2017. Then, we calibrate the parameters of the processes $\{\lambda_{65}^{HR}, \lambda_{65}^{MR}, \lambda_{65}^{LR}\}$ by minimizing the mean squared error (MSE), that is the average squared difference between the model implied and the observed survival probabilities:

(4.2) \begin{equation}\min_{a^j, \sigma^j} \sum_{t=1}^{17}\frac{1}{17}\left({}_{t}{\hat{p}}_{65}^{j}-{}_t p_{65}^j\right)^2,\end{equation}

for $j \in \{HR,MR,LR\}$ , where ${}_tp_{65}^{j}$ is defined as in (3.2)–(3.3) and applied to the j-th sub-population. From the calibration procedure we get the minimizing parameters $a^j, \sigma^j$ that are reported in Table 1 together with $\lambda_{65}^j(0)=-\log \hat{p}^j_{65}$ and the calibration error. The observed mortality intensity at age 65, $\lambda_{65}^j\left(0\right)$ , ranges from 0.011 to 0.025. The value of the long term mean parameter $a^j$ ranges between 0.073 and 0.075 while the volatility $\sigma^j$ ranges from 0.00025 to 0.00048. The low volatility levels are associated to a low calibration error ranging from 0.004 to 0.0049. The Supplementary Material reports for each $j \in \{HR,MR,LR\}$ the calibrated model implied survival function ${}_tp_{65}^j$ vs. the observed survival curve and shows that the two overlap quite well, their differences being below 1% in absolute value for all ages and risk classes.

Table 1. Feller model’s calibration.

Table 1 reports also the fitted life expectancies at age 65, $e_{65}^j$ , which can be computed from the model-implied survival curves. We see that $e_{65}^{HR}=15.0129$ for the high-risk population, $e_{65}^{MR}=19.0285$ for the medium-risk population, $e_{65}^{LR}=22.3383$ for the low risk population. These figures are informative, because they highlight a remarkable difference in life expectancy among the different sub-populations.

4.3. Simulation of the benefits under the GSA scheme

We have run 10,000 Monte Carlo simulations of the processes (4.1). In order to compute the GSA and the annuity benefits, we need a reference population (see Section 2). We, therefore, set the reference population r to be the fifth decile male policyholders, that is $r=MR$ , that implies

(4.3) \begin{equation} \{{}_{t}{p}^r_x\}_{t=0,...,\omega-x-1}=\{{}_{t}{p}^{MR}_x\}_{t=0,...,\omega-x-1}. \end{equation}

Accordingly, given that the premium paid by the policyholders is 1000, the annuity level benefit is

(4.4) \begin{equation}b_{A}=\frac{1000}{\ddot{a}_{x}^{r}}=\frac{1000}{\ddot{a}_{x}^{MR}}.\end{equation}

In the calculation of the annuity, we set the interest rate $i=2\%$ . In the case of the GSA scheme, the starting point for the GSA benefits as in Equation (2.2) is given by

\begin{equation*}b_{GSA}(0) = \frac{1000}{\ddot{a}_{x}^{MR}}=b_A=62.565.\end{equation*}

Considering a cohort of three equally sized sub-populations of $N_{0}^{j}=1000$ males, $j \in \{HR,MR,LR\}$ (so that $N_{0}=3000$ ) coming from the first, the fifth and the tenth deciles of the English population, the simulation process is based on the following algorithm:

• • For each simulated scenario, we simulate the trajectory of $\lambda^j_{65}(t)$ for $t=1, \dots, 45$ for each sub-population j. We run 10,000 simulated scenarios.

• • For each simulated scenario we use the procedure illustrated in Section 3.2 to simulate the number of deaths and obtain the realized survivors $\{l^*_{x+t}\}$ and the realized survival probabilities ${}_{t}{p}_{x}^{*}$ for that scenario.

• • For each of the 10,000 scenarios we compute the benefits for the GSA scheme $b_{GSA}(t)$ for $t =1, \dots, 45$ using the $\{{}_{t}{p}_{x}^{*}\}_{t=1,...,45}$ as illustrated in Section 2.

• • Therefore, we get the distribution of the 10,000 paths for the GSA benefits $b_{GSA}(t)$ for $t=1, \dots, 45$ .

Figure 1 shows the level annuity benefit $b_{A}$ and the 10th, 30th, 50th, 70th and 90th percentiles of the distribution of benefits $b_{GSA}(t)$ over ages 65–110 obtained from the 10,000 simulations. The GSA and annuity benefits are unique across individuals; therefore, Figure 1 holds indiscriminately for each sub-population $j=\{HR,MR,LR\}$ .

Figure 1. Cash flow streams: GSA versus annuity.

Figure 2 reports the scheme benefits over ages 65–95, while Figure 3 reports the expectation and the standard deviation of the GSA benefits for all ages 65–110.

Figure 2. Cash flow streams: GSA versus annuity. Detail, age 65–95.

Figure 3. $\mathbb{E}(b_{GSA}(t))$ , $\sigma(b_{GSA}(t))$ .

At time $t=0,$ we clearly have $b_{GSA}(0)=b_{A}$ . In most cases, the $b_{GSA}(t)$ distribution outperforms the level annuity benefit until about the age $x=90$ , when it bends towards a minimum of approximately 20 (see Figure 1). In line with this behaviour, we see from Figure 3 that the expectation of the GSA benefit $\mathbb{E}(b_{GSA}(t))$ in the first years slightly increases with age and is decreasing after age 90; the standard deviation of the GSA benefit $\sigma(b_{GSA}(t))$ is an increasing function of age. From Figures 1 and 2, we observe a systematic rise and decline of GSA benefits before and after an approximate age of 90. In the comparison with the annuity, which provides flat benefits, we see that the GSA benefit is larger than the annuity one from age 65 to age 95 (the age when the GSA benefit equals the annuity one depends on the scenario, with the median value being 95 years). This rise–decline pattern can be due to the coexistence of different sub-populations with different mortality in the same pool. This creates a number of deaths larger in the first years and lower in later years with respect to the reference population, leading to an immediate rise and a subsequent fall in the benefits. Sensitivity analyses (see the Supplementary Material) support this intuition: by considering three sub-populations that are more similar to the reference one we obtain a benefit behaviour that is more similar to that of the annuity.

Regardless of the causes, this pattern could be considered a negative feature for retirees. There is indeed evidence that for elderly people liquidity needs increase with age, for persons older than 85 being six times higher than for persons below 65 (see Weinert and Gründl Reference Weinert and Gründl2021). Also, Weinert and Gründl (Reference Weinert and Gründl2021) talk about a retirement smile: high consumption levels in the early retirement years followed by a consumption decline as people become more and more home-bound, before a consumption peak at very old ages due to costly nursing care.

However, many considerations are in order. First, most of the members of the GSA scheme would never experience such decline in benefits because they die before age 95 (the probability of dying between ages 65 and 95 from the dataset is ${}_{30}{q}_{65}^{HR}=0.9387$ for HR, ${}_{30}{q}_{65}^{MR}=0.8277$ for MR, ${}_{30}{q}_{65}^{LR}=0.7115$ for LR). Thus, this problem would be suffered only by a minority of individuals, while most of them would in any case enjoy the extra benefit of GSA in the first years after retirement.

Second, it is true that the annuity guarantees a level benefit that does not decrease with age. However, the annuity benefits reported here do not consider commission expenses (that are higher for annuities than for self-insurance schemes): if expenses were taken into account, the annuity benefit would be reduced and the positive gap between GSA and annuity benefits in the first years would be larger. Therefore, a possible way to address the decreasing GSA benefits in old age could be to invest in the first decades the extra money exceeding the annuity benefits in financial instruments and use the accumulated money after age 95. A natural way to do it and transfer longevity risk, would be to buy a deferred lifetime annuity that starts to pay periodic benefits at age 95: given the low probability of reaching age 95, such deferred annuity would likely be affordable with the extra benefit from the GSA scheme.

Last but not least, the retirees actually surviving to age 95 and over would mainly belong to the less deprived socio-economic class (LR), which would probably be less affected by the decline of benefits in elderly age.

4.4. Calculation of EPV(0) for each socio-economic class

The benefit from the GSA scheme is the same for all individuals. However, different socio-economic classes experience different lifetime durations. Therefore, the value of the benefit’s stream will be different. A natural way to measure such value is the expected present value at time $t=0$ of the benefits using the class-specific survival function.

Therefore, for each of the 10,000 simulated scenarios of $b_{GSA}(t)$ we calculate $EPV_{GSA}^{j}(0)$ , where

\begin{align*}EPV_{GSA}^{j}(0)= \sum_{t=0}^{\omega-x-1}v^{t} {}_{t}{p}_{x}^{j}b_{GSA}(t),\end{align*}

where in ${}_tp_{x}^{j}$ we plug the calibrated parameters $a^{j}$ , $\sigma^{j}$ $(j \in \{HR,MR,LR\})$ of Tab. 1. Thus, for each socio-economic class $j \in \{HR,MR,LR\}$ , we obtain a distribution of the 10,000 $EPV_{GSA}^{j}(0)$ .

Similarly, we calculate the expected present value of the benefits for the annuity, $EPV_{A}^{j}(0)$ , where

\begin{align*}EPV_{A}^{j}(0) = \sum_{t=0}^{\omega-x-1}v^{t} {}_{t}{p}_{x}^{j}b_{A}.\end{align*}

Because of (4.3) and (4.4) when $j=MR$ we have $EPV_{A}^{MR}(0)=1000$ .

Table 2 collects $EPV_{A}^{j}(0)$ and some percentiles of the distribution of $EPV_{GSA}^{j}(0)$ .

Table 2. $EPV_{A}^{j}(0)$ and some percentiles of the distribution of $EPV_{GSA}^{j}(0)$ , $j \in \{HR,MR,LR\}$ .

Table 2 shows the effect of ignoring mortality differentials across socio-economic classes: if the same benefit is awarded to all individuals regardless of their heterogeneous contribution to the overall risk, the expected present value of cash flow streams at time $t=0$ is considerably different across sub-groups when the actualization process is weighted by the specific mortality experience of each sub-population individually. Without redistribution, the more deprived socio-economic individuals (HR) enjoy a value about 15–17% lower than the reference individuals (MR) who, in turn, enjoy a value about 12–13% lower than the one of the least deprived individuals (LR). More importantly, on average, the transfer from HR to LR individuals is about 30% of the benefit enjoyed by the reference individuals. This strong solidarity from the high-risk retiree to the low-risk retiree can be dealt with by means of the redistributive scheme illustrated in Section 2.2.

4.5. Simulation of redistributive GSA benefits

We implement the redistributive mechanism as described in Section 2.2. In particular, using the simulated $EPV_{GSA}^{j,k}(0)$ for every j and every k, obtained in Sections 4.3–4.4, we find numerically the optimal shares $\alpha_{*}^j, j \in \{HR, MR, LR\}$ that minimize the performance criterion in (2.7) satisfying restriction (2.8), using the library “Rsolnp” of the R software environment.

Table 3 reports the optimal redistributive shares $\alpha_{*}^j$ , $j\in \{HR,MR,LR\}$ .

Table 3. $\alpha_{*}^{j}$ , $b_{RE}^{j}(0)$ , $b_{A}$ for $j \in\{HR,MR,LR\}$ .

From Table 3, we see that the redistributive mechanism ensures that retirees enjoying a more favourable mortality experience are penalized by a lower initial benefit:

(4.5) \begin{equation}b_{RE}^{HR}\left(0\right)\gt b_{A}\gt b_{RE}^{MR}\left(0\right) \gt b_{RE}^{LR}\left(0\right) .\end{equation}

Figure 4, top, medium and bottom panels report the benefits $b_{RE}(t)$ paid by the redistributive scheme to HR, MR and LR members, respectively. Figures 5 and 6 report, respectively, the expectation and the standard deviation of $b_{RE}^j(t)$ for all $j \in \{HR,MR,LR\}$ together with the expectation and standard deviation of the GSA benefit. Table 4 reports the optimal shares and some percentiles of the distribution of $EPV_{RE}^j(0).$

Figure 4. Annuity benefits and relevant percentiles of the GSA scheme benefits distribution. Top panel: HR individuals; Medium panel: MR individuals; Bottom panel: LR individuals.

Figure 5. $\mathbb{E}(b_{GSA}(t))$ and $\mathbb{E}(b_{RE}^j(t)), j \in \{HR,MR,LR\}$ .

Figure 6. $\sigma(b_{GSA}(t))$ and $\sigma(b_{RE}^j(t)), j \in \{HR,MR,LR\}$ .

We observe the following:

• • Remarkably and as expected, the effect of redistribution becomes evident in Table 4. Thanks to the redistributive mechanism, the EPV(0) gaps across socio-economic groups have been significantly shrunk. The $EPV_{RE}(0)$ of HR individuals is still lower than that of LR individuals, but now the transfer from HR to LR individuals is on average less than 0.5–1%.

• • For high-risk policyholders, redistribution triggers a wider dispersion of the $b_{RE}(t)$ distribution at each time t with respect to the case with no redistribution, as it emerges by observing Figure 6.

• • When analysed over time, the dispersion is an increasing function of the policyholder’s riskiness. We compute two types of ranges: the minimum and the maximum values reached by $b_{GSA}(t)$ over time and over paths, $[{\min}_{t}b_{GSA}(t), {\max}_{t}b_{GSA}(t)]$ , and the minimum value reached by $b_{GSA}(t)$ over time over the 10th percentile and the maximum value reached by $b_{GSA}(t)$ over time over the 90th percentile, $[{\min}_{t}b_{GSA}(t)10{\text{th}}, {\max}_{t}b_{GSA}(t)90{\text{th}}]$ . We conclude the following:

  1. 1. Before redistribution, $[{\min}_{t}b_{GSA}(t), {\max}_{t}b_{GSA}(t)]$ ranges in $[16.96,81.70]$ . After redistribution, for high-risk individuals $b_{RE}^{HR}(t)$ ranges in $[19.74,95.12]$ , for medium-risk individuals it ranges in $[16.48,79.39]$ , finally for low-risk individuals it ranges in $[14.65,70.59]$ .

  2. 2. Before redistribution $\ [{\min}_{t}b_{GSA}(t)10{\text{th}}, {\max}_{t}b_{GSA}(t)90{\text{th}}\ ]$ ranges in $\left[24.29,66.73\right]$ , while after redistribution for high-risk individuals $b_{RE}(t)$ ranges in $\left[28.28,77.70\right]$ , for medium-risk individuals it ranges in $\left[23.60,64.84\right]$ , while for low-risk individuals it ranges in $\left[20.99,57.66\right]$ .

• • For all the sub-populations $j \in \{HR,MR,LR\}$ the expectation of the redistributive GSA benefit $\mathbb{E}(b_{RE}^j(t))$ is slightly increasing for approximately 25 years and then it is decreasing with age, while the standard deviation of the redistributive GSA benefit $\sigma(b_{RE}^j(t))$ is an increasing function of age (Figures 5 and 6).

Table 4. $EPV_{A}^{j}(0)$ and some percentiles of the distribution of $EPV_{RE}^{j}(0)$ , $j \in \{HR,MR,LR\}$ for the redistributive GSA scheme.