3.1 Pool structure

We consider a GSA scheme that provides post-retirement benefits to its members. Individuals pay an initial capital when joining the scheme; upon death, any residual amount is retained by the pool (i.e., no death benefit is included). In return for their initial contribution, members receive an annual lifetime income, whose amount is differentiated depending on the specific mortality profile of the individual. In fact, the GSA is structured with multiple levels of benefits, with higher benefit amounts provided to those with a shorter life expectancy. The individual’s mortality profile is first assessed at entry and then reviewed annually. Upon a change of the mortality profile, for example, following the onset of a disease, the individual will be assigned the level of benefit consistent with his(her) new status. However, similarly to a traditional GSA, annual amounts for the various benefit levels are not guaranteed but are subject to annual adjustment based on the investment and longevity experience of the pool. In the following, we focus on longevity only, and we propose an approach to define such adjustments, which is motivated by actuarial arguments.

Mortality is modelled adopting the Markov ageing setting described in Section 2. We first assume that it is possible to classify individuals according to the ageing states and to differentiate benefits based on such states. However, as already pointed out in Section 1, biological age (which is at the root of ageing states) is not directly measurable. In the second step of the discussion (in particular, in Section 3.5), we assume that individuals are classified according to proxy risk factors. It is unlikely that risk classes identified this way correspond exactly to the ageing states of the Markov model. The implications are discussed in Section 3.5; for the moment, we assume that both at entry and at any time, a risk classification based on the ageing states of the Markov model can be performed. In particular, we assume that it is possible to describe the dynamics of each individual’s mortality profile in terms of the transition to higher ageing states.

We consider a single cohort consisting of $L_x$ individuals who join the scheme at time $t=0$ , at chronological age x, with $L_x^i$ individuals placed in state i, $i\in E$ (clearly, $\sum_{i=1}^m L_x^i=L_x$ ). The maximum attainable age is $\omega$ for each individual; however, we assume that, to avoid the impact of major random fluctuations at later ages (when the cohort will shrink to low size), the scheme is committed to pay benefits up to time $t_{\max}$ (i.e., up to chronological age $x_{\max}=x+t_{\max}\lt\omega$ ). At time $t_{\max}$ , the available funds will be transferred to an insurer willing to take charge of the surviving participants and their funds and guarantee them a lifetime annuity. The price and optimal time of such a transfer (which undoubtedly constitute interesting research questions) are not addressed in this paper, as our aim for now is to define acceptable benefit adjustment rules when multiple levels of benefits are involved; addressing also the transfer at time $t_{\max}$ would excessively broaden the scope of the present study. Thus, in this paper, we simply assume that the scheme will pay an income to survivors at times $t=0,1,\ldots,t_{\max}$ , for a chosen (reasonable) $t_{\max}$ , and the scheme’s payout rate will not account for annuity costs concerning times $t\gt t_{\max}$ .

Let $B_0^{i}$ be the benefit set at time $t=0$ for state $i \in E$ . An individual entering the scheme in state i will initially get the benefit $B_0^i$ ; however, upon transition to a state $j\gt i$ , $j\in E$ , (s)he is entitled to cash the benefit $B_0^{j}$ . Based on the usual actuarial arguments, the initial capital to be paid by an individual in state i at time $t=0$ is then assessed as follows:

(3.1) \begin{equation} {} {}V_0^{i} = \sum_{j\geq i}^{m} B_0^{j}\, \ddot{a}^{i,j}_{x}, {}\end{equation}

where $\ddot{a}^{i,j}_{x} = \sum_{h=0}^{t_{\max}}(1+r)^{-h} {_hp_x^{i,j}}$ is the actuarial value of the annuity paid when in state j, computed using the transition probabilities $_hp_x^{i,j}=\exp(\boldsymbol{\Lambda}h)_{i,j}$ predicted by the mortality model and a given (deterministic) discount rate r. We point out that no loading is embedded in the choice of parameters of (3.1), as it is natural for a GSA scheme. We note also that the differentiation by state concerns not only the benefits but, consistently, the initial capital too.

3.2 Pool funds

A GSA is grounded on the principle of funded liabilities. In other words, it is required that a balance be maintained at all times between available resources and the value of obligations. This is achieved by updating the benefit amount, upwards or downwards, depending on the experience with longevity and investments. We assume that the adjustment is applied annually, right before the benefit is paid at time t, $t\geq 1$ . Let $B_0^i$ be the benefit set at time $t=0$ for state $i\in E$ ; reasonably, $B_0^1\lt B_0^2\lt\ldots\lt B_0^m$ , since higher-indexed states correspond to lower life expectancy. Taking $B_0^1$ as the baseline benefit amount, it may be convenient to express the initial benefit amount for each state $i, i\gt1$ , as a proportion of $B_0^1$ , as follows: $B_0^i=c^i B_0^1$ , where the $c^i$ ’s ( $1\lt c^2\lt c^3\lt\ldots\lt c^m$ ) are chosen according to some target. We let $B_t^{i}$ denote the updated benefit amount at time t for state i, according to an adjustment rule which is described in Section 3.4. In the following, we understand $i\in E$ and $t=0,1,\ldots, t_{\max}$ , unless otherwise stated (the adjustment of the benefit at time $t=0$ is, of course, not under consideration; $B_0^i$ is simply the starting chosen amount).

Following the adjustment of the benefit, the value of the obligation to an individual in state i at time t can be assessed as follows:

(3.2) \begin{equation} {} {}V_{t}^i= \sum_{\substack{j\geq i}}^m B_{t}^j\, \ddot{a}_{x+t}^{i,j}, {}\end{equation}

with an obvious meaning of $\ddot{a}_{x+t}^{i,j}$ for $t\gt0$ . The total value of the obligations of the pool at time t is then

(3.3) \begin{equation} {} {}F_t^{R} =\sum_{i=1}^m L_{x+t}^{i}V_t^{i}, {}\end{equation}

where $L_{x+t}^{i}$ is the (observed) number of individuals in state i at time t. We refer to $F_t^{R}$ as the required pool fund at time t (in line with this, we will refer to $V_t^i$ as the individual fund required in state i at time t).

The resources available to a GSA, which we call the available pool fund and denote as $F_t^{A}$ at time t, are the money contributed by members, net of the payments made so far, plus interest. We focus on longevity risk only and assume a deterministic financial setting; we take r as the annual deterministic return on investments. Starting from $F_0^{A}=\sum_{i=1}^m L_x^{i} V_0^{i}$ , the dynamics of the available pool fund can be described recursively for $t\gt0$ as follows:

(3.4) \begin{equation} {} {}F_t^{A} = \left(F_{t-1}^{A}-\sum_{i=1}^m L_{x+t-1}^{i} B_{t-1}^{i}\right) (1+r). {}\end{equation}

The basic requirement for running the GSA can now be written as:

(3.5) \begin{equation} {} {} {} {}F_t^{R} = F_t^{A}, \quad t\geq0. {}\end{equation}

If benefits are undifferentiated (i.e., the same benefit amount is provided in all states), Equation (3.5) univocally defines the amount admitted for the benefit at time t (since, in such a case, there is only one unknown). When benefits are differentiated by state, Equation (3.5) admits an infinite number of solutions, as the number of unknowns is now m. The introduction of additional requirements is necessary to identify workable solutions. We point out that the benefit adjustment not necessarily will preserve the initial ratio between benefits, that is, not necessarily we find $B_t^i=c^i B_t^1$ . If preserving such a ratio is a priority, then we need to impose $B_t^i=c^i B_t^1$ at all times t; in this case, Equation (3.5) has only one unknown, namely $B_t^1$ . It would follow that, at any time $t\gt0$ , benefits would be differentiated by state, but they would all be adjusted in the same proportion.

At this point, it is convenient to examine the causes that may result in a mismatch between available and required pool funds. Disregarding financial risks, we may consider the following circumstances: mortality rates in the various states higher or lower than expected; ageing process (i.e., transition rates) higher or lower than expected. It should be taken into account that deviations between expected and observed rates may differ across states. Instead of adopting a common adjustment proportion, we may find it more appropriate to differentiate by state not only the benefits but also the adjustment coefficients. In the following, we investigate this solution. To this end, we split the available and required pool funds by state.

3.3 Splitting the pool funds by state

It is rather natural to identify

(3.6) \begin{equation} {} {}F_t^{i,R}=L_{x+t}^{i}\,V_t^{i} {}\end{equation}

as the pool fund required at time t for state i. Clearly, $\sum_{i=1}^m F_t^{i,R}=F_t^R$ .

Let $F_t^{i,A}$ denote the pool fund made available to state i at time t, such that $\sum_{i=1}^m F_t^{i,A}=F_t^A$ .

We now require that the principle of funded liabilities is applied by state, that is, we impose

(3.7) \begin{equation} {} {} {} {}F_t^{i,R} = F_t^{i,A}, \quad t \geq 0,\quad i\in E. {}\end{equation}

Such a requirement clearly also satisfies (3.5). To proceed further with (3.7), it is useful to note that the amount of resources available per state is affected by cross-subsidy effects between states. Part of these cross-subsidies are embedded in the individual fund required (and available) in state i at time $t-1$ . To check this, we start by considering that

\begin{equation*}\ddot{a}_{x+t-1}^{i,i}=1+(1+r)^{-1}\, p_{x+t-1}^{i,i}\,\ddot{a}_{x+t}^{i,i}, {}\end{equation*}

and

\begin{equation*} {} {}\ddot{a}_{x+t-1}^{i,j}=(1+r)^{-1} \sum_{i\leq k\leq j}\,p_{x+t-1}^{i,k}\, \ddot{a}_{x+t}^{k,j}, \quad j\gt i. {}\end{equation*}

Then, we can write the following recursion for the individual fund in state i:

(3.8) \begin{equation} {} {}V_{t-1}^i=B_{t-1}^i+\sum_{j\geq i}p_{x+t-1}^{i,j}\, V_{t^-}^j\, (1+r)^{-1}, {}\end{equation}

where

\begin{equation*} {} {}V_{t^-}^j= \sum_{\substack{k\geq j}}^m B_{t-1}^k\, \ddot{a}_{x+t}^{j,k} {}\end{equation*}

represents the amount of individual fund required in state j at time t, before the adjustment of the benefits at that time.

Let

(3.9) \begin{equation} {} {}F_{t^-}^{i,A} = \left(F_{t-1}^{i,A}-L_{x+t-1}^i B_{t-1}^i\right)(1+r) {}\end{equation}

denote the pool fund dragged to state i at time t, $t\gt0$ , by the pool fund available in such a state at time $t-1$ , net of the benefits paid at time $t-1$ and credited with interest. Comparing Equation (3.9) with (3.4) it can be easily seen that $\sum_{i=1}^m F_{t^-}^{i,A}=F_t^A$ .

Since because of (3.7) we have $F_{t-1}^{i,A}=F_{t-1}^{i,R}$ , using (3.8) we can rearrange (3.9) as:

(3.10) \begin{eqnarray} {} {}F_{t^-}^{i,A} {} {}& = & L_{x+t-1}^{i}\, p_{x+t-1}^{i,i}\,V_{t^-}^i+ L_{x+t-1}^{i}\sum_{j\gt i}p_{x+t-1}^{i,j}\,V_{t^-}^j , {}\end{eqnarray}

that we can interpret as follows. Given $L_{x+t-1}^{i} $ , the quantity $L_{x+t-1}^{i}\, p_{x+t-1}^{i,i}$ represents the number of individuals expected to stay in state i during the year, while $L_{x+t-1}^{i}\, p_{x+t-1}^{i,j}$ represents the number of individuals expected to be in state j at the end of the year. Equation (3.10) then shows that $F_{t^-}^{i, A}$ , that is, $F_{t-1}^{i, A}$ net of the benefits paid at the beginning of the year in state i and credited with interest, consists of the resources required for those expected to remain alive, either in state i or in a higher state $j, j\gt i$ (we point out that for $j=m+1$ , that is, for the dead state, $V_{t^-}^j=0$ ).

Since $p_{x+t-1}^{i,i}=1-\sum_{j\gt i}^{m+1} p_{x+t-1}^{i,j}$ , Equation (3.10) can be further rearranged as follows:

(3.11) \begin{equation} {} {}F_{t^-}^{i, A}=L_{x+t-1}^{i}\,\left(1-p_{x+t-1}^{i,m+1}\right)\,V_{t^-}^i+ L_{x+t-1}^{i}\sum_{j\gt i}^m \,p_{x+t-1}^{i,j}\,\left(V_{t^-}^j-V_{t^-}^i\right). {}\end{equation}

Equation (3.11) allows us to give a further useful interpretation to $F_{t^-}^{i,A}$ . Since $L_{x+t-1}^{i}\, p_{x+t-1}^{i,m+1}$ is the expected number of deaths reported by those in state i at time $t-1$ , the amount $L_{x+t-1}^{i}\, p_{x+t-1}^{i,m+1}\, V_{t^-}^i$ can be related to the expected mortality credits in such a state. Conversely, $L_{x+t-1}^{i}\sum_{j\gt i}^m\, p_{x+t-1}^{i,j}$ is the number of individuals expected to move from state i to a living state $j, j\gt i,$ and $L_{x+t-1}^{i}\sum_{j\gt i} \,p_{x+t-1}^{i,j}\,\left(V_{t^-}^i-V_{t^-}^j\right)$ is a measure of the (total) net cross-subsidy ‘cost’ charged to state i for those expected to be alive, but in higher ageing states.

The annual dynamics of the population of state i can be described for $t\gt0$ as follows:

(3.12) \begin{equation} {} {} {} {}L_{x+t}^{i} = L_{x+t-1}^{i} + \sum_{j\lt i} L_{x+t-1}^{j,i} - \sum_{j\gt i}^{m+1} L_{x+t-1}^{i,j}, {}\end{equation}

where we use the notation $L_{x+t-1}^{j,i}$ (or $L_{x+t-1}^{i,j}$ ) to denote the observed number of individuals in state i (j) at time t being in state j (i) at time $t-1$ . Equation (3.12) makes it clear that the number of individuals in state i at time t is the result of those who left state i (moving to a living state $j, \, i\lt j\leq m$ , or due to death, $j=m+1$ ) and those who have entered it from a state $j\lt i$ . Clearly, $L_{x+t-1}^{j,i}=0$ for $i=1$ . Note that situations such as $L_{x+t-1}^i=0$ and $L_{x+t}^i\gt0$ or $L_{x+t-1}^i\gt0$ and $L_{x+t}^i=0$ are possible, as a result of transitions into and out of state i. In the following, we use the notation $\widetilde{p}$ to denote the realised (observed) transition or permanence rates; for instance, $\widetilde{p}_{x+t-1}^{\ i,j}=\frac{L_{x+t-1}^{i,j}}{L_{x+t-1}^{i}}$ . Such rates are obviously defined for states that are non-empty at time $t-1$ , that is, for states i such that $L_{x+t-1}^{i}\gt0$ . Using this notation, Equation (3.12) can be rewritten as:

(3.13) \begin{equation} {} {}L_{x+t}^{i}= L_{x+t-1}^i \left(1-\widetilde{p}_{x+t-1}^{\ i,m+1}\right) +\sum_{j\lt i}L_{x+t-1}^{j}\, {}\widetilde{p}_{x+t-1}^{\ j,i}-\sum_{j\gt i}^{m}L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,j} . {}\end{equation}

Let us now compare $F_{t^-}^{i,A}$ to the pool fund required at time t for state i. As stated by (3.6), such a fund is proportional to $L_{x+t}^{i} $ . Its amount before the adjustment of the benefits is represented by:

(3.14) \begin{equation} {} {}F_{t^-}^{i, R}=L_{x+t}^i\, V_{t^{-}}^i. {}\end{equation}

If we plug (3.13) into (3.14), we obtain the expression:

(3.15) \begin{equation} {} {}F_{t^-}^{i, R}= L_{x+t-1}^i \left(1-\widetilde{p}_{x+t-1}^{\ i,m+1}\right)\, V_{t^{-}}^i+\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,V_{t^{-}}^{i}-\sum_{j\gt i}^{m}L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,j}\, V_{t^{-}}^{i}, {}\end{equation}

that we can compare with (3.11) for an interpretation. It is unlikely that $F_{t^-}^{i,A}=F_{t^-}^{i,R}$ , for a number of reasons:

• • The realised number of deaths, $L_{x+t-1}^i\, \widetilde{p}_{x+t-1}^{\ i,m+1}$ , can be other than what expected, $L_{x+t-1}^{i} \,p_{x+t-1}^{i,m+1}$ , and then the realised mortality credits in state i can be higher or lower than expected. We may consider it appropriate for any discrepancy to be absorbed by state i, given that this mismatch arises exclusively from mortality in state i.

• • In face of individuals entering state i during the year, it is required to hold in this state an additional amount $\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,V_{t^{-}}^{i}$ , which is not already included in $F_{t^-}^{i,A}$ . This is an amount that should be cross-subsidised by the previous states $j,j\lt i$ .

• • Individual funds $V_{t^-}^j$ for those moving to a living state $j,j\gt i$ are no longer required in state i and should be taken out from $F_{t^-}^{i,A}$ , to cross-subsidise the next states. This is in line with what was discussed with regard to those entering state i (previous point).

• • Finally, we note that when $L_{x+t}^i=0$ , then $F_{t^-}^{i,R}=0$ ; it can happen, instead, that $F_{t^-}^{i,A}\gt0$ , given that this latter fund is assessed based on expected number of transitions, in contrast to $F_{t^-}^{i,R}$ which considers realised transitions. In such a circumstance, $F_{t^-}^{i,A}$ should be released to non-empty states.

Based on the above considerations, we define the pool fund available for state i at time t as follows:

(3.16) \begin{equation} {} {}F_{t}^{i, A}= \begin{cases} {} {} {}F_{t^-}^{i, A} +\displaystyle\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,V_{t^{-}}^{i}-\sum_{j\gt i}^{m}L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,j}\, V_{t^{-}}^{j} +X_t^i & \text{if }L_{x+t}^i\gt0;\\ {} {} {}0 & \text{if }L_{x+t}^i=0. {} {}\end{cases} {}\end{equation}

With $X_t^i$ we denote the amount of resources left at time t by empty states, released to state i, so to ensure that $\sum_{i}F_t^{i, A}=F_t^A$ . We assume a redistribution proportional to the required funds before the adjustment of the benefits; more precisely, we assume (for non-empty states i):

\begin{equation*} {} {}X_t^i=\left(F_t^A-\sum_{i:\,L^i_{x+t}\,\gt0} F_{t^-}^{i,A}\right)\,\frac{F_{t^-}^{i,R}}{F_{t^-}^R}. {}\end{equation*}

As already noted, definition (3.16) ensures $\sum_{i}F_t^{i,A}=F_t^A$ . The amounts $\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,V_{t^{-}}^{i}$ and $\sum_{j\gt i}^{m}L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,j}\, V_{t^{-}}^{j}$ measure the cross-subsidy, respectively, in favour of and released by state i. These amounts are also present in $F_t^A$ because, when we deal with multiple states, necessarily cross-subsidy effects between states are involved; however, such cross-subsidies are not explicit in (3.4) because they sum up to zero at the pool level.

3.4 Adjustment of the benefits by state

Once the pool fund available for state i at time t, that is $F_t^{i, A}$ , is defined, we proceed to the adjustment of benefits imposing (3.7). We point out that this is a set of equations that must be solved backwards, from state m down to state 1. In particular, for $i=m$ , we have

\begin{equation*} {} {} {}L_{x+t}^m B_{t}^m \ddot{a}_{x+t}^{m,m}=F_t^{m, A}, {} {}\end{equation*}

and the unknown $B_t^m$ straightforwardly follows:

\begin{equation*} {} {} {}B_t^m=\frac{F_{t}^{m, A}}{L_{x+t}^{m}}\frac{1}{\ddot{a}^{m,m}_{x+t}}. {} {}\end{equation*}

For $i=m-1$ , by imposing

\begin{equation*} {} {} {}L_{x+t}^m \left( B_{t}^{m-1} \ddot{a}_{x+t}^{m-1,m-1}+ B_{t}^m \ddot{a}_{x+t}^{m-1,m}\right)=F_t^{m-1, A} {} {}\end{equation*}

the unknown is $B_t^{m-1}$ , being $B_t^{m}$ previously determined, and it holds that:

\begin{equation*} {} {} {}B_t^{m-1}=\left(\frac{F_{t}^{m-1, A}}{L_{x+t}^{m-1}}-B_{t}^m \ddot{a}_{x+t}^{m-1,m}\right)\frac{1}{\ddot{a}^{m-1,m-1}_{x+t}}. {} {}\end{equation*}

Proceeding iteratively, for the non-empty states, we find

(3.17) \begin{equation} {} {} {}B_t^i=\left(\frac{F_{t}^{i, A}}{L_{x+t}^{i}}-\sum_{j\gt i}B_t^j\,{\ddot{a}^{i,j}_{x+t}} \right)\frac{1}{\ddot{a}^{i,j}_{x+t}}, {} {}\end{equation}

while we assume that if $L_{x+t}^i=0$ , then $B_t^i=B_{t-1}^i$ (i.e., no adjustment is applied to benefits relating to empty states).

An interesting interpretation of the adjustment can be found in terms of individual fund $V_t^i$ . For brevity, we develop it under the assumption $L_{x+t-1}^i\gt0$ and $X_t^i=0$ for all states i.

If we equate (3.6)–(3.16), and rearrange, we get

(3.18) \begin{equation} {} {}\begin{aligned} {} {} {}V_t^{i} \left(L_{x+t}^i-\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\right) & = L_{x+t-1}^{i}\,p_{x+t-1}^{i,i}\, V_{t^-}^{i}+ \sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,(V_{t^-}^{i}-V_{t}^{i})\\& \quad- L_{x+t-1}^{i}\sum_{j\gt i}^m\left(\widetilde{p}_{x+t-1}^{\ i,j}-p_{x+t-1}^{i,j}\right)\,V_{t^-}^j. {} {}\end{aligned} {}\end{equation}

Since (3.13) can also be written as

\begin{equation*} {} {}L_{x+t}^i= L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,i}+\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}, {}\end{equation*}

from (3.18) we obtain

(3.19) \begin{equation} {} {}V_t^{i} = \frac{p_{x+t-1}^{i,i}}{\widetilde{p}_{x+t-1}^{\, i,i}}\, V_{t^-}^{i}-\frac{\displaystyle\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}}{L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,i}}\,(V_{t}^{i}-V_{t^-}^{i})-\frac{\displaystyle \sum_{j\gt i}^m\left(\widetilde{p}_{x+t-1}^{\, i,j}-p_{x+t-1}^{i,j}\right)}{\widetilde{p}_{x+t-1}^{\, i,i}}\, V_{t^{-}}^{j}. {}\end{equation}

When adjusting the benefits, clearly, we also need to adjust the individual required funds. Equation (3.19) suggests the following interpretation. The individual fund required at time t for state i must be adjusted because:

• • The proportion of individuals remaining in state i differs from what was expected (see the first term of Equation (3.19));

• • The cross-subsidy received for individuals entering state i from lower states $j,j\lt i$ is based on $V_{t^-}^i$ , but it must be updated to $V_t^{i}$ after the adjustment (see the second term of Equation (3.19));

• • The cross-subsidy transferred to higher states $j,j\gt i$ is based on the realised transition rates, whereas the amount set aside in $F_{t^-}^{i, A}$ is based on the expected transition rates (see the third term of Equation (3.19)).

We would like to emphasise that the first term recalls the mortality experience adjustment (MEA) factor introduced in Piggott et al. (Reference Piggott, Valdez and Detzel2005). It expresses the benefit update implied by the discrepancy between expected and realised mortality. In our multi-state framework, such an adjustment applies state-specific, accounting for ageing and mortality deviations. The second and third terms in Equation (3.19) correct the state-specific adjustment by accounting for, respectively, cross-subsidy inflows and outflows. Naturally, cross-subsidy cash flows can be defined in alternative ways; for instance, their amounts may be shaped based on the expected transition rates rather than realised values. Beyond the possible choices and consequent algebraic expressions, individual funds and benefits must be updated when experience differs from expectations, that is, realised cross-subsidies differ from those expected.

3.5 Smoothing benefits across states

The reasons that may lead an individual to prefer participation in a GSA scheme over the purchase of a life annuity are extensively discussed in the literature (see also Section 1); above all, cost savings, albeit at the expense of guarantees. When dealing with a multi-state GSA, one should additionally wonder why an individual should choose it, rather than one offering identical benefits across all states.

Similarly to a standard annuity, an individual who is aware of being in substandard health or maintaining unhealthy lifestyle patterns might feel disadvantaged by the use of undifferentiated annuity rates (see, e.g., Olivieri and Tabakova Reference Olivieri and Tabakova2024 for the case of annuities and Kabuche et al. Reference Kabuche, Sherris, Villegas and Ziveyi2024 for the case of a GSA). On the other hand, an individual currently in good health may feel better protected by a scheme that guarantees an upgrade in the benefit in the event of a deterioration in their health status. In this preference, one may recognise an interest akin to that observed for long-term care insurance products or disability income annuities.

Although GSA members are expected to accept benefit fluctuations across states depending on experience, they may reasonably expect a consistent and coherent ordering of those benefits, that is, $B_t^1\lt B_t^2\lt\ldots\lt B_t^m$ at all times $t\geq0$ . At time $t=0$ , this is ensured by the initial choice of the benefit amounts; however, Equation (3.17) does not necessarily maintain such an ordering at later times ( $t\gt0$ ), as it does not explicitly consider the relative differences among benefit levels. We also note that, even when the ordering of benefits is preserved, the distance between them may fall below desirable thresholds. These considerations suggest the introduction of corrective mechanisms to the adjustment rule described in Section 3.4.

There is a further circumstance that may require modifications to the adjustment rule. As we have discussed earlier (see Sections 1, 2 and 3.1), ageing states are usually not observable. The current mortality profile of an individual can reasonably be detected based on proxy risk factors, which may lead to a classification of individuals that does not precisely replicate the intended ageing states. Moreover, either by design or in connection with the available proxy risk factors, it could be considered more appropriate to structure benefits across $m^*$ classes, not necessarily matching the number of ageing states, that is, $m^*\leq m$ . In settings where m is large, adopting fewer benefit classes, $m^*\lt m$ , may offer a more practicable implementation. Illustrative proxy risk factors include medical events like the development of diabetes or a heart attack, as well as milder but indicative conditions such as hypertension, elevated glucose or cholesterol levels, or macular disease. They may also encompass functional or cognitive impairments, such as difficulties in performing activities of daily living or the onset of dementia. It should be noted that, given the absence of guarantees, the scheme may allow for a classification based on medical conditions or diseases that are typically excluded from standard insurance coverage. However, since more severe conditions result in access to a higher benefit class, safeguards must be anyhow in place to avoid potential misuse or fraud.

In the following, we still refer to E as the set of ageing states and to $E^*=\{1,2,\ldots, m^*\}$ as the set of benefit classes. Benefits $B_t^i$ assessed as in Equation (3.17) ensure the actuarial equivalence between benefits and available resources by ageing state and for the whole pool, as discussed in Section 3.3. In the following, we refer to them as the notional benefits (or benefits by ageing state). We note that

(3.20) \begin{equation} {} {}B_t^{TOT}=\sum_{i\in E} L_{x+t}^i\,B_t^i {}\end{equation}

measures the total payout admitted by the actuarial equivalence condition (3.7) at time $t,t\geq0$ . We denote with $\overline{B}_t^k$ the benefit for class $k\in E^*$ , and to keep the actuarial equivalence at the pool level, we require

(3.21) \begin{equation} {} {}\sum_{k\in E^*} l_{x+t}^k\,\overline{B}_t^k = B_t^{TOT} , {}\end{equation}

where $l_{x+t}^k$ is the number of survivors placed in benefit class k at time t (note that $\sum_{i\in E} L_{x+t}^i=\sum_{k\in E^*}l_{x+t}^k$ ). We refer to $\overline{B}_t^k$ also as the actual benefits. Based on the motivations mentioned above, we introduce the further requirement:

(3.22) \begin{equation} {} {}\overline{B}_t^k = u^k\,\overline{B}_t^1 , {}\end{equation}

where $\overline{B}_t^1$ is the benefit amount for the lowest benefit class, and $u^k$ , $k=2,3,\ldots,m^*$ , is a coefficient chosen to achieve the desired differential between the benefit in class k and class 1. Reasonably, $1\lt u^2\lt u^3\lt\ldots \lt u^{m^*}$ . To avoid potential misunderstandings when comparing the coefficients $c^i$ ’s introduced in Section 3.2 and the coefficients $u^k$ ’s, we stress that, by design, $u^k$ measures the ratio at any time t between the actual benefit in class k in respect of class 1. The coefficient $c^i$ , instead, describes the initial level of differentiation across ageing-state benefits at time 0; at the following times $t,t=1,2,\ldots$ , it may turn out $B_t^i\gtreqless c^i B_t^1$ , as a result of the different benefit dynamics by ageing state. Based on (3.22), Equation (3.21) admits only one solution. After noting that requirement (3.22) introduces a kind of smoothing of benefits, we point out that benefits subjected to (3.22) bring us back to the solution discussed in Section 3.2 after Equation (3.5). Indeed, they may differ from that solution because of the number $m^*$ of benefit classes (and the specific choice of the proportion among benefits), but basically, they follow the same idea. Yet, the analysis conducted in Sections 3.3 and 3.4 is not redundant, as it enables us to assess the extent of solidarity entailed by this choice.

As already noted, while actual benefits ensure actuarial equivalence at the pool level, the actuarial equivalence at the ageing state level will clearly be lost. This means that by adopting benefits $\overline{B}_t^k$ ’s instead of $B_t^i$ ’s, solidarity mechanisms are introduced in the benefit design. With reference to an individual in ageing state i, placed in benefit class k, the size and direction of solidarity can be understood by comparing the actual benefit $\overline{B}_t^k$ with the notional benefit $B_t^i$ . Clearly, for some individuals $\overline{B}_t^k\lt B_t^i$ , meaning that they are contributing solidarity, whereas for others $\overline{B}_t^k\gt B_t^i$ , that is, they are receiving solidarity. We underline that this constitutes a design choice, albeit one that, in specific contexts, is imposed by the limited information available on individual mortality patterns. As pointed out in Section 1, understanding the solidarity effects introduced enables a rational justification and communication of design decisions, which is particularly important in arrangements where participants are exposed to the full range of shared risks. A numerical assessment is discussed in Section 4.2, which also clarifies why the introduction of solidarity effects may prove to be recommended.

4.1 Model fitting and simulation

To fit the Markov ageing model, both the initial distribution $\boldsymbol{\pi}$ and the intensity matrix $\boldsymbol{\Lambda}$ of the ageing process must be estimated, and the choice of fitting method may be affected by the data availability. At first glance, the maximum-likelihood estimation (MLE) approach represents a natural candidate for fitting the Markov ageing model. To perform the MLE procedure, independent trajectories of the ageing process must be completely observed up to the time of death. Each path should consist of the sequence $\left\{\left(i_k,s_k\right), \, k=1,2,\ldots,n\right\}$ , where $i_1,i_2,\ldots, i_n$ are the ageing states visited before death, and $s_1,s_2,\ldots,s_n$ are the corresponding sojourn times. However, due to the latent nature of the ageing states, trajectories of the ageing process are not detectable, and the likelihood function maximisation cannot be directly executed. Nonetheless, if realisations of i.i.d. times until death are accessible, the MLE can be carried out via the expectation-maximisation (EM) algorithm. Assuming that an underlying ageing process generates such absorption times, the conditional expected value of the log-likelihood function can be computed and then maximised w.r.t. model parameters. In case no information on the times until death is available, they could be randomly drawn from a valid PH distribution. The simulation exercise requires that several combinations of the initial probability vector and the intensity matrix must be tried, as well as different numbers of EM iterations should be considered. It could be computationally expensive, and there is no guarantee that the maximum of the likelihood function is achieved. Indeed, while the EM algorithm always converges, it does not necessarily converge to the ML estimator as it can be trapped in a local maximum. The EM algorithm can be exploited to its full effectiveness, for instance, when survival data are observable; however, this is not the case for the present analysis because only cohort mortality rates are available to us.

Given the above considerations, we opt for an alternative fitting method to MLE. For the sake of completeness, we refer the reader to the Section 1 of the Supplementary Material, where technical details on the EM-based fitting of the Markov ageing model are provided.

We proceed to fit the Markov ageing model by adopting a target approach, that is, a general procedure allowing for dealing directly with the observed mortality or survival curve. Let $\boldsymbol{\theta}_{c}=(\boldsymbol{\pi}_{c},\boldsymbol{\Lambda}_{c})$ be the parameter characterising the Markov ageing model for the cohort c. For each cohort, the deviations between the observed mortality data (the target) and the corresponding value provided by the ageing model are measured through a proper loss function $\mathcal{L}$ . We choose the mean squared error as the loss function, while our target is the t-year survival probability at age 65. Then, we have

(4.1) \begin{equation} {} {} {} {} {} {}\mathcal{L}(\boldsymbol{\theta}_c) = \dfrac{1}{\left\vert \mathcal{X}_{c} \right\vert} \sum_{t=1}^{\left\vert \mathcal{X}_{c} \right\vert} \left({}_t p_{65,c}-S_{65}\left(t;\;\boldsymbol{\theta}_{c}\right)\right)^2, {} {}\end{equation}

where ${}_t p_{65,c}$ is the t-year survival probability at age 65 for the cohort c and computed as ${}_t p_{65,c} = \prod_{h=0}^{t-1} \left(1-\frac{m_{65+h, c}}{1+0.5 \,m_{65+h, c}}\right)$ , while $S_{65}\left(t;\;\boldsymbol{\theta}_{c}\right)$ is the survival function for the same cohort and its expression is given in Equation (2.7). We find the estimate $\hat{\boldsymbol{\theta}}_{c}$ as the solution of the following constrained mean squared error minimisation problem:

(4.2) \begin{equation} {} {} {} {} {} {}\min_{\boldsymbol{\theta}_{c}}\mathcal{L}(\boldsymbol{\theta}_c), \quad \text{s.t.} \quad \pi_{i, c} \in [0,1] \ \forall \, i \in E, \ \sum_{i \in E} \pi_{i, c} = 1, {} {}\end{equation}

which we solve numerically for each cohort and gender.

As mentioned in Section 2, two other aspects deserve consideration in fitting the Markov ageing model: the number of ageing phases, m, and possible non-uniqueness problems in parameter specifications. Concerning the latter, along the lines of Su and Sherris (Reference Su and Sherris2012), we impose the following specification for intensity matrix elements in (2.3):

(4.3) \begin{equation} {} {} {} {} {} {}\lambda_i = \lambda \quad \text{and} \quad q_i = \gamma + \alpha\,\text{e}^{\beta \, i}, \quad \forall \, i \in E, \, \gamma, \alpha, \beta \gt0. {} {}\end{equation}

A constant transition intensity between states means that, on average, a linear decline of various physiological functions is acknowledged. The transition towards the death state accounts for a background rate, namely $\gamma$ , and an ageing-dependent component, $\alpha\,\text{e}^{\beta \, i}$ . The former allows for the representation of ageing termination due to accidental death, while the latter describes an exponential increase in mortality as ageing proceeds. Regarding the number of ageing phases, an accurate fit of the entire mortality schedule could be attained for high values of m (see, e.g., Lin and Liu Reference Lin and Liu2007; Cheng et al. Reference Cheng, Jones, Liu and Ren2020); however, if only part of the mortality schedule is of interest, a low-dimensional PH distribution furnishes a satisfactory fit (see, e.g., Liu and Lin Reference Liu and Lin2012). Therefore, we select the number of phases empirically, starting with low m values and increasing them until the loss function reduces. In Table 2, mean squared error values are reported for different numbers of ageing phases, distinguishing by cohorts and genders. For the 1920 and 1930 Australian cohorts, we find that $m=15$ ageing phases grant the best model fitting, for both genders; the same occurs for the 1940 cohorts with $m=7$ ageing phases. Figure 2 furnishes a graphical visualisation for the comparison between the observed t-years survival probabilities and the corresponding fitted values of the PH’s survival function. The latter are determined according to the m value providing the lowest mean squared error for each cohort and gender. We observe a satisfactory fit across all cohorts and both genders. The greater fitting accuracy is achieved for the younger cohorts, since their survival curve consists of a few observations outlining a non-complex trend over time.

Table 2.

Mean squared error values by increasing the number of ageing phases, m, and distinguishing by cohorts and genders. The lowest mean squared error values are in bold.

Figure 2.

Comparison between the observed t-years survival probabilities (red dots) and the corresponding fitted values of the PH’s survival function (blue triangles). The latter are determined according to the m value providing the lowest mean squared error for each cohort and gender (see Table 2).

For the sake of simplicity, from now on, we focus on the 5-state Markov ageing model for the 1920 Australian cohort, considering both genders. Using a low-dimensional Markov ageing model facilitates the readability of both the estimated parameters and the results of the Markov ageing GSA presented in Section 4.2. Comments we draw in the following discussions can be easily extended to high-dimensional Markov ageing models, as well as to the other cohorts. We refer the reader to Section 2 of the Supplementary Material for the parameters estimate concerning the befitting Markov ageing models.

Table 3 collects the 5-state Markov ageing mode’s parameters estimate concerning the 1920 Australian cohort. We note that, for both genders, the initial distribution of the process is mainly concentrated in the first ageing state, and it is residually located in the other states, in particular the last one. Then, 65-year-old individuals cannot be attributed to a unique ageing status; individuals differ in physiological state, even though they have the same chronological age. For the male gender, the probability of being in the first ageing state is lower than for the female gender; the opposite occurs if we consider the fifth state. Therefore, the physiological status of females chronologically aged 65 should be better than that of males. Such evidence is accompanied by the gender-based estimates of $\lambda$ . Females present a $\lambda$ value lower than that for males, indicating a slower decline in their physiological functions. Considering the parameters embedded in the transition intensity to the death state, the background rate estimate appears negligible for both genders. Indeed, the background rate $\gamma$ measures the effects of accidental deaths, which mainly occur at adult ages, while we are referring to older ages. The $\beta$ parameter represents the rate at which the transition intensity to the death state varies as the ageing phases progress. The larger the value of $\beta$ , the more likely the transition to the death state will be. As shown in Table 3, the female gender has a higher $\beta$ than that for the male gender, but at the same time, there is a compensation with the respective $\alpha$ parameter values. On the whole, we find that the female gender tends to experience a slower ageing process and a longer lifespan, compared to the male gender.

Table 3.

Estimate of the 5-state Markov ageing model’s parameters for the 1920 Australian cohort, both genders. We recall that: $\alpha$ is an ageing-related scale parameter, $\beta$ denotes the rate of change of $q_i$ as ageing progresses, $\gamma$ is the rate measuring the ageing termination due to accidental deaths, $\lambda$ is the transition intensity among states, and $\pi_i$ is the probability that the process starts in the state i, for $i=1,\ldots, 5$ .

Following the model fitting, the evolution over time and by state of the initial population can be simulated. Similarly to Kabuche et al. (Reference Kabuche, Sherris, Villegas and Ziveyi2024), the realised number of individuals transitioning from state i to state j in the interval $(t-1,t]$ is randomly drawn from a multinomial distribution, that is,

(4.4) \begin{equation} {} {} {} {}L_{x+t-1}^{i,j} \sim \text{Multinom}\left(L_{x+t-1}^i,p_{x+t-1}^{i,j}\right), \quad i \in E,\, j \in \mathcal{S}. {}\end{equation}

In (4.4), $L_{x+t-1}^i$ refers to the actual number of individuals belonging to the ageing state i at time $t-1$ , while $p_{x+t-1}^{i,j}$ is the expected 1-year transition probability, from state i to state j, stemming from the calibrated Markov ageing model. At each time step, we run 10,000 simulations to generate population transitions among states. The total number of individuals attaining state j at time t is used for generating population transitions at time $t+1$ . In addition, for each random draw, the simulated number $L_{x+t-1}^{i,j}$ is used to compute the realised 1-year transition probabilities as $\widetilde{p}_{x+t-1}^{\, i,j}=\frac{L_{x+t-1}^{i,j}}{L_{x+t-1}^i}$ .

4.2 Implementation of the Markov ageing GSA

4.2.1 Pool composition and population dynamics

We assume that at time $t=0$ the GSA scheme is joined by a cohort of $L_x=10,000$ individuals (either females or males), chronological age $x=65$ ; we further assume that the scheme closes upon reaching time $t_{\max}=30$ . Consistent with the Markov ageing model calibrated as described in Section 4.1, individuals are distributed in five ageing states, with initial composition as in Table 4. We point out that such a composition (for both genders) naturally follows from the estimated initial distribution of the Markov process; see Table 3.

Table 4.

Initial pool composition, by gender.

Figure 3 plots the average number of survivors in the five ageing states across time, separately for females and males; this way, we have a representation of the population dynamics. The average paths of the five ageing states are similar for both genders: the decreasing pattern of state 1 is explained by the fact that individuals can only exit this state (either by jumping to a higher living state or because of death). The non-monotonic trajectories of the other states are driven by the net balance of transitions in and out of each state. States representing higher ageing levels reach their maximum size later than the others, and the peak population tends to decrease with increasing state severity. Nevertheless, at older chronological ages, and especially at the final time $t_{\max}$ , the higher ageing states are larger than the lower ones. In particular, state 1 exhibits a population size that almost disappears at later times, although for both genders, it does not die out by time $t_{\max}$ . When comparing these patterns between females and males, the greater average longevity of the former results in higher average sizes of the lower-index ageing states for females, at any given time.

Figure 3.

Population dynamics (average value) by time (annual basis) and state for both genders. We refer the reader to the Section 3 of the Supplementary Material for the graphical representation of the population transitions among states, for both genders.

4.2.2 Available pool fund by state

The GSA scheme pays annual benefits to survivors at the beginning of the year, from time $t=0$ until time $t_{\max}=30$ . We set $B_0^1=1.00$ account unit and $B_0^i=1.2^{i-1}\,B_0^1$ ; Table 5 lists the initial benefit amounts by ageing state and the corresponding initial capital, assessed according to Equation (3.1). The discount rate is set to $r=0.02$ . For each gender, the higher initial capital required from members in ageing state 1 reflects their longer life expectancy, notwithstanding their lower initial benefit. Similarly, for a given state, higher initial capitals are charged to females as a consequence of their higher expected lifetime.

Table 5.

Initial notional benefit amount and capital, by gender.

Figure 4 shows the time profile of the average values of the pool fund available by state for females. The time profiles for males are similar and, for brevity, are not shown here, but they can be found in the Section 3 of the Supplementary Material.

Figure 4.

Available fund (average value), and its components, by time (annual basis) and state for the female gender. y-axis values are on $10^{-4}$ scale. Since the fund value ranges differ significantly across the various states, we adopted separate y-axis scales to enhance the readability of the fund patterns for states $i \gt 1$ . For the male gender, refer to the Section 3 of the Supplementary Material.

The time profiles can be interpreted, in particular, considering the several components of the available pool fund by state, as defined in Equation (3.16). For each state, these components are included in the picture. In this regard, we point out that all quantities are plotted in absolute value; however, cross-subsidies may be either incoming or outgoing, depending on the states j considered in relation to state i.

In general, the time profile of each $F_t^{i, A}$ resembles that of survivors in the ageing state, that is, $L_{65+t}^i$ , apart from scaling factors due to the benefit amounts. As the population ages, resources are shifted towards the more severe ageing states. In particular, cross-subsidy is prevalent between consecutive states. For instance, ageing state 2 receives people from state 1 increasing its heap of individual funds, that is, $L_{65+t}^{1,2}V_{t^-}^2$ , and transfers part of its individuals (and the relative funds) primarily to state 3, that is, $L_{65+t}^{2,3}V_{t^-}^3$ . In turn, ageing state 3 acquires people and funds from state 2, and part of its population is mainly relocated to ageing state 4, and so on. Both cross-subsidy inflows and outflows by state are reported in Table 6. It is worth noting that the cross-subsidy obtained by state 1 is null for every $t\gt0$ , as is the cross-subsidy offered by state 5. Since only forward ageing transitions are allowed in the Markov ageing model, the lowest ageing state subsidises all the next ones; conversely, the highest ageing state solely receives from the others. Given the initial population composition (see Table 4), the ageing state 1 brings individuals and related funds to subsequent states during the first year. Its cross-subsidy outflow at $t=1$ is fully allocated among states $j\gt1$ , corresponding to their respective inflows. As individuals age (or die), state 1 tends to empty, and fewer funds will be transferred. Meanwhile, ageing states $j\gt1$ experience an initial increase in incoming cross-subsidies, which then gives way to greater outflows as ageing progresses.

Table 6.

Ingoing and outgoing cross-subsidies (average value) by state. Values refer to the female gender and, for conciseness, are reported for selected years. Note that, by means of Equation (3.16), the ingoing cross-subsidy for the state i at time $t\gt0$ is given by $\sum_{j\lt i}L_{x+t-1}^{j}\, \widetilde{p}_{x+t-1}^{\ j,i}\,V_{t^{-}}^{i}+X_t^i$ , while the outgoing cross-subsidy is quantified as $\sum_{j\gt i}^{m}L_{x+t-1}^{i}\, \widetilde{p}_{x+t-1}^{\ i,j}\, V_{t^{-}}^{j}$ . For the male gender, we refer the reader to the Section 3 of the Supplementary Material.

Finally, we notice that $X_t^i$ is null over time and for all states. Such a result arises from state populations never becoming extinct; therefore, no state leaves resources to other states due to population depletion.

4.2.3 Notional benefits

Figure 5 plots the time profile of the average value of the notional benefits by gender. For all states, benefits on average are quite stable; however, at higher chronological ages, they show either an increasing or decreasing path. Such a behaviour can be interpreted in view of cross-subsidy flows, arising from deaths, entries into the state and transitions to higher living states. We note, in particular, an increasing trend in state 5 (which can be exited only upon death) and a decreasing trend for the other states (which are likely to become thinnier, as a result of deaths and ageing). The effect of cross-subsidies for the various states can also be understood in light of the dynamics of the available pool funds by state (see Figure 4). We note that the male population in state 1 is the one that shrinks the most (due to deaths and ageing), see Figure 3; then, the average time profile of benefits in state 1 for males may suggest that cross-subsidies rooted in actuarial equivalence lead to undesirable benefit patterns when the number of participants becomes very low.

Figure 5.

Individual notional benefits (average value) by time (annual basis) and ageing state, for both genders. For each ageing state, the black dashed line represents the initial benefit level, that is, $B_0^i$ for $i=1,\ldots,5$ , outlined in Table 5.

Indeed, the benefit dynamics in ageing state 1 deserves careful consideration. While it is true that individuals in this state are exposed to the risk of jumping to a higher state (where they would receive a higher benefit amount), it is also the case that those who remain in state 1 often do so as a result of healthier lifestyles and well-being-oriented behaviours (that may also prove to be more expensive than more passive, sedentary or unhealthy lifestyles). A reduction in benefits for state 1 (especially when, at the same time, benefits in the other states are increasing) may be perceived as a form of penalisation and a disincentive for behaviours that should instead be encouraged due to their social value. This is one of the reasons why it may be appropriate to introduce an adjustment to actuarially equivalent cross-subsidies, by admitting some kind of solidarity among states (see Section 3.5). Numerical results, in particular, suggest that the introduction of solidarity should be considered in favour of those in ageing state 1, that is, individuals with longer life expectancy, who spend their lives in better ageing conditions. This conclusion may appear counterintuitive, as we are naturally inclined to believe that financial support should be directed from healthier individuals to those in poorer health. Thanks to our modelling framework, we are able to gain deeper insight into this issue, so as to enable more informed decision-making with respect to the GSA design.

4.2.4 Actual benefits

Notional benefits can be repackaged into actual benefits in a number of ways, each of which implies some degree of solidarity that individuals may perceive as either acceptable or not. For instance, if we consider a plain GSA, that is, a scheme with undifferentiated benefits, it is not difficult to conclude from Figure 5 that the common benefit amount falls between the lowest and the highest notional benefit amounts. Individuals in lower ageing states would implicitly receive solidarity from those in higher ageing states. The difference between notional and actual benefits in this case can be significant enough to lead certain categories of people to opt out of the scheme (provided that participation is not mandatory).

We examine numerically some benefit smoothing solutions that may be seen as suitable by various population groups. We consider four cases:

• • $m^*=1$ , meaning that a unique benefit class is acknowledged. In other words, we look at a plain GSA wherein all the ageing states are grouped since the scheme provides undifferentiated benefits;

• • $m^*=2$ , that is, two classes of benefit, assuming that $u^2=1.5$ , $l_{x+t}^1 = L_{x+t}^1 + L_{x+t}^2$ and $l_{x+t}^2 = L_{x+t}^3 + L_{x+t}^4 + L_{x+t}^5$ . This way, ageing states 1 and 2, on the one side, and ageing states 3, 4, and 5, on the other, are grouped. These groupings are based on the assumption that individuals in adjacent ageing states are unlikely to clearly distinguish between their respective mortality characteristics;

• • $m^*=3$ , that is, three classes of benefit, assuming that $u^2=1.5$ , $u^3=2$ , $l_{x+t}^1 = L_{x+t}^1 + L_{x+t}^2$ , $l_{x+t}^2 = L_{x+t}^3 + L_{x+t}^4$ and $l_{x+t}^3 = L_{x+t}^5$ . In this case, ageing class 5 is kept by itself;

• • $m^*=5$ , that is, five classes of benefit, assuming that $u^2=1.25$ , $u^3=1.5$ , $u^4=1.75$ , $u^5=2$ and $l_{x+t}^k = L_{x+t}^i$ for every $k,i =1,\ldots,5$ , with $k=i$ . We point out that this case corresponds to the solution described in Section 3.2, after Equation (3.5).

The numerical results are plotted in Figure 6, for females (for males, see Section 3 of the Supplementary Material). The smoothing effect is apparent in all cases (as it can be easily seen when comparing Figure 5 with Figure 6); in particular, as a consequence of the imposed linking among the various benefit classes, the benefit trajectory is identical across all classes, apart from the chosen scaling factor. It should be noted that, in any case, the total payout of the pool remains the same (i.e., $B_t^{TOT}$ ) at all times and is determined by requiring funded liabilities by ageing state (i.e., imposing (3.7)). Solidarity contributions can be understood by comparing the notional with the actual benefit for each ageing state. In Table 7, both notional and actual benefits are presented by considering different numbers of benefit classes. In general, higher ageing states offer solidarity to lower ones. The size of such contributions depends on the specific choice of the benefit classes and the chosen proportions among benefit amounts. For instance, if a unique benefit class is acknowledged (i.e., $m^\star=1$ ), then lower ageing states receive solidarity from higher ageing states along the entire scheme duration. In particular, it holds that $B_t^i\lt\overline{B}_t^1$ , for $i=1,2$ and for every $t\gt0$ , while ageing states 3 and 4 offer solidarity, respectively, until the end of the 10th and 25th year. State 5, instead, only contributes to solidarity each year. Now, let us consider $m^\star=2$ benefit classes. We recall that individuals belonging to the first and second ageing states, that is, $L_{x+t}^1$ and $L_{x+t}^2$ , are relocated into the first benefit class since $l_{x+t}^1= L_{x+t}^1+L_{x+t}^2$ . As emerges in Table 7, individuals in the first ageing state will earn solidarity, being $B_t^i\lt\overline{B}_t^1$ for every $t\gt0$ ; conversely, individuals in the ageing state 2 obtain an actual benefit lower than the corresponding notional ones until the twentieth year. The second benefit class pays the actual benefits $\overline{B}_t^2$ to individuals residing in the states 3, 4 and 5. Comparing the actual benefits with the notional benefits $B_t^3$ , $B_t^4$ and $B_t^5$ , it is easy to see that states 4 and 5 give solidarity to state 3. By also inspecting $m^\star=3$ and $m^\star=5$ cases, we observe that as the number of benefit classes increases, the magnitude of the solidarity from higher ageing states to lower ones shrinks. Then, if the scheme aims to provide many benefit classes, a constraint to the solidarity mechanism between ageing states is implicitly introduced.

Figure 6.

Actual benefits (average value) by time (annual basis) for the female gender, by considering different numbers $m^\star$ of benefit classes. For the male gender, refer to Section 3 of the Supplementary Material.

Table 7.

Notional benefits (average value), $B_t^i$ , $i=1,\ldots,5$ , and their related actual benefits, $\overline{B}_t^k$ , $k=1,\ldots,m^\star$ , under different numbers $m^\star$ of benefit classes. For conciseness, values are reported for every 5 years up to scheme maturity, and they refer to the female gender. For males, refer to Section 3 of the Supplementary Material.

Once again, we note that the analysis of notional benefits allows us to assess both the magnitude and direction of solidarity effects, which would otherwise remain unspecified. Comparing actual with notional benefits can help identify the most workable smoothing rules, especially in light of the implied solidarity mechanisms.

Finally, we emphasise that in the case of a traditional GSA (i.e., $m^\star=1$ ), the actual benefits are increasing over time, but they are undifferentiated. Then, individuals belonging to greater ageing phases perceive the same benefit as individuals in better health statuses, and in the meantime, they offer solidarity. However, if differentiated benefits are provided, more aged individuals continue to contribute solidarity, but they can access greater benefit levels.