2.1 Notation

Forman and Sabin (Reference Forman and Sabin2016) and Donnelly and Young (Reference Donnelly and Young2017) introduced a pooling mechanism where (surviving) participants can increase their income by sharing their mortality risk and agreeing about a mutual inheritance scheme. Precisely, the pool operates over one period, starting at time $t=0$ to end at time $t=1$ , say. At time 0, participant i contributes to the fund an amount $c_{i}$ , $i=1,2,\ldots ,n$ . This amount is invested and accumulates to $a_{i}$ at time 1 (with the same deterministic accumulation rate for all participants). Contrarily to regular investments, the terminal amount $\sum_{i=1}^{n}a_{i}$ is divided among participants or beneficiaries at time $1$ , according to some agreed mutual inheritance rule.

The respective death probabilities before time 1 are denoted as $q_{x_{1}},q_{x_{2}},$ $\ldots ,q_{x_{n}}$ where the sequence of probabilities $q_{x_i}$ is contained in a closed subinterval of [0,1] and $x_{i}$ refers to age for participant i. These values are assumed to account for the effects of age, gender, socio-economic profile, etc. and all participants agree about them. Notice that no guarantee is offered with respect to the life table. Probabilities $q_{x_i}$ are only used to distribute mortality credits among participants, their values entering the sharing formula (see e.g. formula (2.2) below). The mechanisms discussed in this paper thus account for heterogeneity among participants as different probabilities $q_{x_i}$ and initial contributions $c_i$ are permitted, under mild conditions. The terminal payout depends on participants’ actual mortality experience.

Let $I_{i}$ denote the survival indicator for individual i, that is

\begin{equation*}I_{i}=\left\{\begin{array}{l}1\,,\text{ if individual i survives up to time }1, \\ \\0\,,\text{ otherwise}.\end{array}\right.\end{equation*}

Thus, $\textrm{P}[I_{i}=0]=q_{x_{i}}=1-\textrm{P}[I_{i}=1]$ . Throughout the paper, we assume that the random variables $I_1,\ldots,I_n$ are mutually independent.

2.2 Mutual inheritance scheme

Participants agree about a mutual inheritance scheme defining the final payout to each of them or to their designated beneficiaries in case of death. The latter can be members of participant’s family or a charity, for instance. Survivors get back at least their accumulated assets $a_{i}$ while the mutual inheritance agreement embedded in the survivor fund provides them with an extra return above the purely financial one. Let us denote as

\begin{equation*}X_{i}=(1-I_{i})a_{i}\end{equation*}

the accumulated contribution $a_{i}$ lost in case participant i dies. The amount

\begin{equation*}S_{n}=\sum_{i=1}^{n}X_{i}\end{equation*}

is the total amount of mortality credits to be distributed among the n participants (or their beneficiaries) according to a predefined rule.

Denuit (Reference Denuit2019) proposed to adopt the conditional mean risk sharing rule to define the mutual inheritance scheme so that participant i receives an amount $\textrm{E}[X_{i}|S_{n}]$ corresponding to his or her expected share in mortality credits. Precisely, the terminal cash flow $W_{i,n}$ for participant i (or his or her beneficiaries) at time $t=1$ is equal to

\begin{equation*}W_{i,n}=\left\{\begin{array}{ll}a_{i}+\textrm{E}[X_{i}|S_{n}], & \text { if participant i survives} \\ \\ \textrm{E}[X_{i}|S_{n}], & \text{ if participant i dies}.\end{array}\right.\end{equation*}

This can be rewritten as

\begin{equation*}W_{i,n}=a_{i}I_{i}+\textrm{E}[X_{i}|S_{n}]\,.\end{equation*}

Clearly,

\begin{equation*}\sum_{i=1}^{n}W_{i,n}=\sum_{i=1}^{n}\textrm{E}[X_{i}|S_{n}]+\sum_{i=1}^{n}a_{i}I_{i}=\sum_{i=1}^{n}(1-I_{i})a_{i}+\sum_{i=1}^{n}a_{i}I_{i}=\sum_{i=1}^{n}a_{i}\end{equation*}

so that the entire resources are pooled within the group. Moreover,

\begin{equation*}\textrm{E}[W_{i,n}]=(1-q_{x_{i}})a_{i}+\textrm{E}[X_{i}]=(1-q_{x_{i}})a_{i}+q_{x_{i}}a_{i}=a_{i}\end{equation*}

so that the gain is zero, on average, for each participant. The game is thus fair and does not transfer money from some participants to other ones, on average (ex ante). There is thus no donation embedded in the survivor fund as long as the death probabilities truly reflect participants’ mortality.

Since

(2.1) \begin{equation} X_1,X_2,\ldots,X_n\text{ identically distributed}\Rightarrow \textrm{E}[X_i|S_n]=\frac{1}{n}\sum_{i=1}^nX_i\end{equation}

we recover the uniform allocation of mortality credits as a particular case when the group is homogeneous with respect to mortality as well as contributed amounts. The conditional mean risk sharing rule thus extends the uniform allocation applying in the homogeneous case to heterogeneous pools.

Let

\begin{equation*}\mathcal{S}_{n}=\left\{ \sum_{i=1}^{n}j_{i}a_{i},j_{i}\in\{0,1\},i=1,\ldots ,n\right\}\end{equation*}

be the set of values of the random variable $S_{n}$ . The conditional expectation $\textrm{E}[X_{i}|S_{n}=s]$ is defined for $s\in \mathcal{S}_{n}$ , but not for $s\notin \mathcal{S}_{n}$ , so that their properties are only discussed within $\mathcal{S}_{n}$ . Clearly, $0\in \mathcal{S}_{n}$ and $\textrm{E}[X_{i}|S_{n}=0]=0$ while for all positive $s\in \mathcal{S}_{n}$ , we exploit the independence together with the discreteness of $X_i$ to obtain

(2.2) \begin{align} \textrm{E}[X_i|S_n=s] & = \frac{ \textrm{E}[ X_i{\unicode[Times]{x1D7D9}}_{S_n=s}]}{\textrm{P}[S_n=s]} \nonumber\\[5pt] & = \frac{1}{ \textrm{P}[S_n=s] } \left( \int_{\mathbb{R}}\textrm{E}\left[x{\unicode[Times]{x1D7D9}}_{x+\sum_{j\ne i}X_j = s }\right]\, \textrm{d}\textrm{P}_{X_i}(x) \right) \nonumber \\[5pt] & = \frac{1}{ \textrm{P}[S_n=s] } \left( \int_{\mathbb{R}} x\cdot \textrm{P}\left[x+\sum_{j\ne i}X_j = s \right]\, \textrm{d}\textrm{P}_{X_i}(x) \right) \nonumber \\[5pt]& = \frac{ 1 } { \textrm{P}[S_n=s] } a_i\cdot \textrm{P}\left[ a_i +\sum_{j\ne i}X_j = s \right] \cdot q_{x_i}\,.\end{align}

Formula (2.2) provides the actuary with a practical way to compute conditional expectations $\textrm{E}[X_{i}|S_{n}=s]$ . Computation time nevertheless grows with the number n of participants. Henceforth, we assume that $a_i = a + k_i\cdot h$ , $i=1,2,\ldots,n$ for $k_i\in\mathbb{N}$ and $a,h\in\mathbb{R}^+$ .

2.3 Comonotonic mortality credits

2.3.1 Definition

A common property for a mortality risk sharing scheme is that the extra amount paid to participant i if he or she survives increases with the amount $S_{n}$ of mortality credits to be distributed at maturity. This property is referred to as the comonotonicity property and means for the conditional mean risk sharing rule that

(2.3) \begin{equation}s\mapsto \textrm{E}[X_{i}|S_{n}=s]\text{ is nondecreasing over }\mathcal{S}_{n}\text{ for every }i\,. \end{equation}

If (2.3) holds true then mortality credits all move in the same direction with $S_n$ , increasing when $S_n$ increases and decreasing when $S_n$ decreases, and the random vector $(\textrm{E}[X_1|S_{n}=s],\ldots,\textrm{E}[X_n|S_n])$ is said to be comonotonic. We refer the reader to Dhaene et al. (Reference Dhaene, Denuit, Goovaerts, Kaas and Vyncke2002a,b) for a general presentation of comonotonicity. It is important to stress that (2.3) is not in general valid (counterexamples are easy to build, as shown next), but some important cases for which the property holds have been studied in Denuit and Robert (Reference Denuit and Robert2021b).

2.3.2 Comonotonicity and heterogeneity in invested amounts

It is easy to see that the dispersion of the accumulated distributions $a_{1},a_{2},\ldots ,a_{n}$ plays a crucial role in that respect. To see why this is the case, assume that all $a_{1},a_{2},\ldots ,a_{n}$ are different. If $a_{i}$ cannot be written as a sum of some other $a_{j}$ ’s, then the event $\{S_{n}=a_{i}\}$ is equivalent to the event $\{I_{i}=0,I_{j}=1$ for $j\neq i\}$ and it follows that $\textrm{E}[X_{i}|S_{n}=a_{i}]=a_{i}$ . If all participants die, then we have

\begin{equation*}\textrm{E}\left[X_{i}\,\Big|\,S_{n}=\sum_{j=1}^{n}a_{j}\right]=a_{i}\,.\end{equation*}

Therefore, for a finite pool size n and $a_{1},a_{2},\ldots ,a_{n}$ as described before, comonotonicity cannot hold unless conditional expectations are actually constant. Heterogeneity in invested amounts thus impacts on survivor funds. Sections 3 and 4 study the case where all contributed amounts $a_i$ are equal but participants are subject to different death probabilities, and the case where different contributions are allowed but selected from a predefined menu (limited heterogeneity). Before proceeding with this analysis, let us explain why comonotonic mortality credits ease the management of survivor funds by considering possible guarantees that may be offered to participants.

2.3.3 Minimum guarantees

Participants to survivor funds are exposed to randomness in mortality credits. This is generally the case with tontine-like arrangements. The need for protection has been addressed in several papers and resulted for instance in hybrid schemes such as the tonuity proposed by Chen et al. (Reference Chen, Hieber and Klein2019) and the tontine with minimum guarantee in Chen and Rach (Reference Chen and Rach2019).

In order to ensure that participants get a minimum payoff, it seems to be desirable to reinsure the lower layer of $S_n$ : if $S_n$ falls below a given threshold w then the (re-)insurer covers the shortfall $(w-S_n)_+ = \max\{w-S_n,\,0\}$ . A natural candidate for w consists in a given percentage of $\textrm{E}[S_n]=\sum_{i=1}^nq_{x_i}a_i$ . For instance, one could set $w=0.9\cdot \textrm{E}[S_n]$ if pool members are ready to accept mortality credits that are at most 10% below their expected value for the entire pool.

When mortality credits are comonotonic, the global threshold w on the aggregate mortality credits $S_n$ to be shared among participants can easily be split into individual amounts $w_i$ . Here, $w_i$ is the minimal guarantee on the share of mortality credits allocated to participant i, as shown next. Under assumption (2.3), there exist $w_1,\ldots,w_n$ such that $\sum_{i=1}^nw_i=w$ and the identities

(2.4) \begin{equation} (w-S_n)_+=\sum_{i=1}^n(w_i-\textrm{E}[X_i|S_n])_+\text{ and }(S_n-w)_+=\sum_{i=1}^n(\textrm{E}[X_i|S_n]-w_i)_+\end{equation}

both hold true. Formula (2.4) can be found in the proof of Theorem 6 in Kaas et al. (Reference Kaas, Dhaene, Vyncke, Goovaerts and Denuit2002); see the first formula on p. 78. The decomposition of $(w-S_n)_+$ appearing in (2.4) allows us to charge each participant with his or her own contribution to the (re-)insurance premium, based on the respective $(w_i-\textrm{E}[X_i|S_n])_+$ entering the decomposition of the lower layer $(w-S_n)_+$ . The price of the (re-)insurance treaty can thus be split according to (2.4), participant i paying $(1+\theta)\textrm{E}[(w_i-\textrm{E}[X_i|S_n])_+]$ summing up to $(1+\theta)\textrm{E}[(w-S_n)_+]$ where $\theta$ is the loading applied by the reinsurer. Although proportional loadings are common in practice, notably when expenses are referred to, a safety loading could rely on diversification arguments, then being based on some risk measure. The results of this section still apply in that case provided the selected risk measure is additive for comonotonic risks. All spectral risk measures enjoy this property, including Value-at-Risk or Tail-Value-at-Risk.

Identities (2.4) also provide the analyst with the right way to allocate $\max\{S_n,w\}$ among participants:

(2.5) \begin{eqnarray}\max\{S_n,w\}&=&w+(S_n-w)_+ \notag \\[5pt]&=&\sum_{i=1}^n\left(w_i+(\textrm{E}[X_i|S_n]-w_i)_+\right) \notag \\[5pt]&=&\sum_{i=1}^n\max\{\textrm{E}[X_i|S_n],w_i\}. \end{eqnarray}

Then, $\max\{S_n,w\}$ is distributed among participants according to formula (2.5). Precisely, participant i gets mortality credits $\max\{\textrm{E}[X_i|S_n],w_i\}$ when the pool buys protection ensuring mortality credits $\max\{S_n,w\}$ .

3.1 Risk elimination

We know from Denuit and Robert (Reference Denuit and Robert2021c) that the income $\textrm{E}[X_{i}|S_{n}]$ decreases in convex order with the size n of the pool. This ensures that the variance of $\textrm{E}[X_{i}|S_{n}]$ is nonincreasing when n gets larger. In that respect, conditional mean risk sharing offers a way to distribute the total mortality credits $S_{n}$ among the n participants so that enlarging the pool is always beneficial and participants thus prefer joining the larger pool. It is therefore interesting to study mortality credits when n becomes large. The following result shows that diversification operates within survivor funds, under the conditional mean risk sharing rule.

Proposition 3.1. If $a_{i}=a$ for all i and the sequence of probabilities $q_{x_i}$ is contained in a closed subinterval of [0,1], then we have

\begin{equation*}\textrm{Var}\left[\textrm{E}[X_{i}|S_{n+1}]\right]\leq \textrm{Var}\left[\textrm{E}[X_{i}|S_{n}]\right]{\;for\;all\;}n\;{ and }\;\lim_{n\to\infty}\textrm{Var}\left[\textrm{E}[X_{i}|S_{n}]\right]=0\,.\end{equation*}

Proof. The inequality for the variance of mortality credits directly follows from a general result in Denuit and Robert (Reference Denuit and Robert2021c). Proposition 2.1 in that paper shows that $\textrm{E}[X_{i}|S_{n}]$ decreases in the convex order for any independent random variables $X_i$ , which implies the announced ranking for variances. The mean-squared convergence of $\textrm{E}[X_{i}|S_{n}]$ to $\textrm{E}[X_{i}]$ can be deduced from Strasser (Reference Strasser2012) who derived large-pool approximations for $\textrm{E}[X_{i}|S_{n}]$ in the particular case when $a_{i}=1$ for all i. In Theorem 2.1 of this paper, it is indeed proved that

\begin{equation*}\textrm{E}[X_{i}|S_{n}]=q_{x_{i}}+\frac{q_{x_{i}}(1-q_{x_{i}})}{\sqrt{\textrm{Var}[S_{n}]}}Z_{n}+\frac{\tau _{i,n}}{\textrm{Var}[S_{n}]}{(Z_{n}^{2}-1)}+r_{ni}\left( Z_{n}\right)\end{equation*}

where

\begin{equation*}Z_{n}=\frac{S_{n}-\textrm{E}[S_{n}]}{\sqrt{\textrm{Var}[S_{n}]}},\qquad \tau_{i,n}=q_{x_{i}}(1-q_{x_{i}})\left(q_{x_{i}}-\frac{1}{\textrm{Var}[S_{n}]}\sum_{j=1}^{n}q_{x_{j}}^{2}(1-q_{x_{j}})\right),\end{equation*}

and $\textrm{E}[|r_{ni}\left( Z_{n}\right) |]=O\left( n^{-3/2}\right) $ . Using Corollary 5.2 and Lemma 5.3 of the same paper, it is also true that $\textrm{E}[|r_{ni}\left( Z_{n}\right) |^{2}]=O\left( n^{-3/2}\right) $ . Since

\begin{align*}\left( \textrm{E}[X_{i}|S_{n}]-q_{x_{i}}\right) ^{2} & \leq 4\left( \frac{q_{x_{i}}(1-q_{x_{i}})}{\sqrt{\textrm{Var}[S_{n}]}}Z_{n}\right) ^{2}+4\left(\frac{\tau _{i,n}}{\textrm{Var}[S_{n}]}{(Z_{n}^{2}-1)}\right) ^{2}\\[5pt] & +4\left(r_{ni}\left( Z_{n}\right) \right) ^{2},\end{align*}

we deduce that the variance of $\textrm{E}[X_{i}|S_{n}]$ tends to 0 as $n\rightarrow \infty $ . This ends the proof.

The limit in Proposition 3.1 extends the classical weak law of large numbers for averages of independent and identically distributed $X_i$ corresponding to mortality credits allocations within homogeneous pools as shown in (2.1). Proposition 3.1 shows that individual risk can generally be fully eliminated in an infinitely large pool.

The following numerical example illustrates the results derived in Proposition 3.1. All calculations are carried out with the distr package available in R, see distr.r-forge.r-project.org/.

Example 3.2. Consider n participants partitioned into two groups: 60% of them are subject to a death probability equal to 0.1 and the remaining 40% of them to a death probability equal to 0.2. All values $a_i$ are set to 1. The left panel in Figure 1 shows the variance of $\textrm{E}[X_i|S_n]/a_i$ for $n\in\{10,20,30,\ldots,1000\}$ . The decreasing trend to 0 established in Proposition 3.1 is clearly visible there. The lower quartile, the median and the upper quartile (that is, quantiles associated to probability levels 0.25, 0.5 and 0.75) of $\textrm{E}[X_i|S_n]/a_i$ are displayed in the other two panels of Figure 1 for $n\in\{100,200,300,\ldots,1000\}$ for a participant in group 1 in the middle panel and for a participant in group 2 in the right panel. We can see there that the interquartile range narrows at speed $n^{-1/2}$ .

Figure 1. Variance of $\textrm{E}[X_i|S_n]/a_i$ as a function of $n\in\{10,20,30,\ldots,1000\}$ (left panel) for a participant in group 1 (continuous line) and in group 2 (broken line), median and interquartile range (IQR) of $\textrm{E}[X_i|S_n]/a_i$ for $n\in\{100,200,300,\ldots,1000\}$ in group 1 (middle panel) and in group 2 (right panel) for the pool described in Example 3.2. Horizontal lines in the central and right panels correspond to expected mortality credits.

3.2 Comonotonic mortality credits

If all the contributions are equal, then the comonotonicity property (2.3) always holds true, as formally shown next.

Proof. We can assume without loss of generality that $a=1$ . Considering (2.2) with $a_i=1$ for all i, we can see that

\begin{equation*}\textrm{E}[X_{i}|S_{n}=s] =q_{x_{i}}\frac{\textrm{P}\left[ \sum_{j\neq i}X_{j}+1=s\right]} {\textrm{P}\left[ S_n=s\right]}.\end{equation*}

Hence, comonotonicity holds provided the ratio of probabilities appearing in the latter expression is nondecreasing. This means that $S_n$ is smaller than $\sum_{j\neq i}X_j+1$ in the sense of the likelihood ratio order. The announced result then follows from stochastic inequality (1.5) in Xu and Balakrishnan (Reference Xu and Balakrishnan2011). Precisely, these authors established that given two sets of independent Bernoulli random variables $\{J_1,J_2,\ldots,J_n\}$ and $\{K_1,K_2,\ldots,K_n\}$ with respective means $q_{J,1},q_{J,2},\ldots,q_{J,n}$ and $q_{K,1},q_{K,2},\ldots,q_{K,n}$ such that $q_{J,1}\leq q_{J,2}\leq\ldots\leq q_{J,n}$ and $q_{K,1}\leq q_{K,2}\leq \ldots \leq q_{K,n}$ ,

\begin{equation*}\sum_{l=1}^m\frac{1}{q_{J,l}}\leq \sum_{l=1}^m\frac{1}{q_{K,l}}\text{ for }m=1,\ldots,n\Rightarrow s\mapsto\frac{\textrm{P}\left[\sum_{l=1}^nJ_l=s\right]}{\textrm{P}\left[\sum_{l=1}^nK_l=s\right]} \text{ is nondecreasing}.\end{equation*}

It suffices to apply this result here with $q_{J,l}=q_{K,l}=q_{x_l}$ for $l\neq i $ , $q_{J,i}=1$ and $q_{K,i}=q_{x_i}$ (with appropriate re-indexing). This ends the proof.

The following numerical example illustrates the comonotonicity property established in Proposition 3.3.

Example 3.4. Let us consider the same pool as in Example 3.2 with participants partitioned into two groups: 60% of them are subject to a death probability equal to 0.1 and the remaining 40% of them to a death probability equal to 0.2, with $a_i=1$ for all i. The functions $s\mapsto\textrm{E}[X_i|S_n=s]$ are displayed in Figure 2 for $n\in\{10,100\}$ for a participant in group 1 or in group 2. The increasingness established in Proposition 3.3 is clearly visible there.

Figure 2. Functions $s\mapsto\textrm{E}[X_i|S_n=s]$ for $n=10$ (left panel) and $n=100$ (right panel) for a participant in group 1 (continuous line) and in group 2 (broken line) in the pool described in Example 3.2.

4.1 Risk elimination

When contributions vary among participants, some structure must be imposed on $a_1,\ldots,a_n$ to get diversification within the survivor fund, as shown by the following example which appears to be similar to developments in Section 2.3.2.

The difference between the present paper and the previous works devoted to conditional mean risk sharing rule appears to be the knowledge of amounts that could be lost and can identify the participant who died in some extreme cases. To solve this issue, we assume in this section that only a limited choice is offered to participants for the amounts of contribution. Precisely, we assume that participants choose their level of contributions within a predefined menu $b_j = a + k_j \cdot h$ , for $j=1,2,\ldots,p$ , for consecutive $k_j\in\mathbb{N}$ and $p\in\mathbb{N}$ . The number of possible contributions p in this predefined menu $\{b_{1},\ldots,b_p\}$ does not vary with n. Theoretical results are valid for any p but diversification is better and the proposed approximations are more accurate when p is smaller. The predefined menu still contains Example 4.1 as special case. However, participant 1 is now part of a pool of others with the same accumulated distribution $a_1$ .

Let $\mathcal{G}_{j}=\{i|a_{i}=b_{j}\}$ be the group of participants with accumulated contribution equal to $b_{j}$ , $j=1,2,\ldots,p$ . Then,

(4.1) \begin{equation}S_{j,n}=\sum_{i\in \mathcal{G}_{j}}X_{i}=b_{j}\sum_{i\in \mathcal{G}_{j}}\left( 1-I_{i}\right)\end{equation}

is the total amount of mortality credits generated by group $\mathcal{G}_{j}$ . Henceforth, we denote as $n_j=\#\mathcal{G}_{j}$ the number of participants in group j, with accumulated contribution $a_i=b_j$ .

We are now in a position to assess diversification effects within survivor funds, under conditional mean risk sharing rule when contributions are selected from a predefined menu (limited heterogeneity) by keeping p fixed when n increases. In this way, we can show that full risk elimination is still possible provided all group sizes $n_j$ tend to infinity. This is precisely stated next.

Proposition 4.2. Let p be a fixed integer and the sequence of probabilities $q_{x_i}$ is contained in a closed subinterval of [0,1]. If $a_{i}\in \{b_{1},\ldots,b_p\}$ for all i, then we have

\begin{equation*}\textrm{Var}\left[\textrm{E}[X_{i}|S_{n+1}]\right]\leq \textrm{Var}\left[\textrm{E}[X_{i}|S_{n}]\right]\;{ for\;all\;}n\;{ and }\;\lim_{n_1,n_2,\ldots,n_p\to\infty}\textrm{Var}\left[\textrm{E}[X_{i}|S_{n}]\right]=0.\end{equation*}

Proof. The inequality for the variance of mortality credits again follows from Denuit and Robert (Reference Denuit and Robert2021c). The mean-squared convergence of $\textrm{E}[X_{i}|S_{n}]$ to $\textrm{E}[X_{i}]$ can be deduced from the following property established by Denuit and Robert (Reference Denuit and Robert2021f): for any independent random variables U, V and W, $\textrm{E}[U|U+V+W]$ is smaller or equal than $\textrm{E}[U|U+V]$ in the sense of the convex order so that

\begin{equation*}\textrm{Var}\left[\textrm{E}[U|U+V+W]\right]\leq \textrm{Var}\left[\textrm{E}[U|U+V]\right].\end{equation*}

Without the loss of generality, we can set $i=1$ and assume that $a_1=b_1$ . Applying this result to $U=X_1$ , $V=S_{1,n}$ and $W=\sum_{j=2}^pS_{j,n}$ shows that

\begin{equation*} \textrm{Var}\left[\textrm{E}[X_1|S_{n}]\right]\leq \textrm{Var}\left[\textrm{E}[X_1|S_{1,n}]\right].\end{equation*}

Since the variance appearing on the right-hand side tends to 0 when $n_1$ tends to infinity by Proposition 3.1, this ends the proof.

Example 4.3 illustrates the convergence established in Proposition 4.2.

Figure 3. Variance of $\textrm{E}[X_i|S_n]/a_i$ as a function of $n\in\{10,20,30,\ldots,1000\}$ (left panel) for a participant in group 1 (continuous line) and in group 2 (broken line), median and interquartile range (IQR) of $\textrm{E}[X_i|S_n]/a_i$ for $n\in\{100,200,300,\ldots,1000\}$ in group 1 (middle panel) and in group 2 (right panel) for the pool described in Example 4.3. Horizontal lines in the central and right panels correspond to expected mortality credits.

Figure 4 compares the variance of the case where groups 1 and 2 are pooled together (continuous line) as a function of the pool size n to the case where group 1 (pool size $40\%\cdot n$ ) and group 2 (pool size $60\%\cdot n$ ) are treated separately. We observe that pooling the two heterogeneous groups reduces the variance and is beneficial for both groups.

Figure 4. Variance of $\textrm{E}[X_i|S_n]/a_i$ as a function of $n\in\{10,20,30,\ldots,500\}$ for a participant in group 1 (black) and in group 2 (grey). We compare the case where the two groups are treated separately (broken line) to the case where both groups are pooled together (continuous line).

4.2 Comonotonicity

In the general case of heterogeneous $a_i$ , comonotonicity property (2.3) does not necessarily hold as shown by the following example.

Example 4.4. Consider again the pool described in Example 4.3. The functions $s\mapsto\textrm{E}[X_i|S_n=s]$ are displayed in Figure 5 for $n\in\{10,20,50,100\}$ for a participant in group 1 or in group 2. We can see there that (2.3) does not hold for every i when n is small. Interestingly, when the size of the pool increases while keeping the same structure, the functions $s\mapsto \textrm{E}[X_{i}|S_{n}=s] $ ultimately become monotonic.

Figure 5. Functions $s\mapsto\textrm{E}[X_i|S_n=s]$ for $n=10$ (upper left panel), $n=20$ (upper right panel), $n=50$ (lower left panel), and $n=100$ (lower right panel), for a participant in group 1 (continuous line) and in group 2 (broken line) in the pool described in Example 7.

The next section derives accurate approximations for mortality credits within large pools.

4.3 Large-pool comonotonic approximations to mortality credits

Within the groups $\mathcal{G}_{j}$ , we can apply the results of Section 3 since $a_{i}=b_{j}$ for all i. The idea is thus to distribute the mortality credits among the different groups $\mathcal{G}_{j}$ and then within each group. The amount $S_{j,n}$ of mortality credits generated by $\mathcal{G}_{j}$ has been defined in (4.1). The first two moments of this random variable are given by

(4.2) \begin{equation}\textrm{E}[S_{j,n}]=b_{j}\sum_{i\in \mathcal{G}_{j}}q_{x_{i}}\text{ and }\textrm{Var}[S_{j,n}]=b_{j}^{2}\sum_{i\in \mathcal{G}_{j}}\left( 1-q_{x_{i}}\right)q_{x_{i}}.\end{equation}

The total amount of mortality credits to be distributed among the n participants can be decomposed into $S_{n}=\sum_{j=1}^{p}S_{j,n}$ .

We propose a hierarchical mortality credit sharing to distribute $S_{n}$ that proceeds in two steps. First, the total amount of mortality credits is shared between the groups $\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_p$ with the help of the linear regression sharing rule $h_{j,n}^{\textrm{lin}}$ . The latter only uses the two first moments of $S_{j,n}$ and attributes to group $\mathcal{G}_{j}$ the amount of mortality credits

\begin{equation*}\textrm{E}[S_{j,n}|S_{n}=s]\approx h_{j,n}^{\textrm{lin}}(s)=\textrm{E}[S_{j,n}]+\frac{\textrm{Var}[S_{j,n}]}{\sum_{j=1}^p\textrm{Var}[S_{j,n}]}\left( s-\textrm{E}[S_{n}]\right) .\end{equation*}

This approximation can be justified by the fact it is asymptotically equivalent to the conditional mean risk sharing rule as n and the numbers $n_{j}$ of participants in groups $\mathcal{G}_{j}$ are large. In such a case, the amounts $S_{j,n}$ are approximately normally distributed with expected values $\textrm{E}[S_{j,n}]$ and variance $\textrm{Var}[S_{j,n}]$ given in (4.2). The proposed approximation $h_{j,n}^{\textrm{lin}}(s)$ corresponds to the conditional expectation of $S_{j,n}$ given $S_{n}=s$ in that limiting case. Notice that the functions $s\mapsto h_{j,n}^{\textrm{lin}}(s)$ are also increasing in s so that the allocation of mortality credits between groups results in comonotonic shares.

For $s\in \mathcal{S}_{n}$ , let us define the vector $\left(n_1(s),n_2(s),\ldots,n_p(s) \right)\in\mathbb{N}^p$ as

\begin{align*}&\left(n_1( s),n_2 ( s),\ldots,n_p( s) \right)\\[5pt] &=\arg\min_{(k_1,\ldots,k_p)\in\mathbb{N}^p}\sum_{j=1}^p\big\vert k_{j}b_{j}-h_{j,n}^{\textrm{lin}}(s)\big\vert \text{ such that }\sum_{j=1}^pk_{j}b_{j}\le s\,.\end{align*}

For a realization s of $S_n$ , $n_{j}(s) $ provides a number of participants such that $n_{j}(s)\cdot b_{j}$ is very close to the mortality credits $h_{j,n}^{\textrm{lin}}(s)$ allocated to group j. This allocation cannot distribute more than the realized mortality credits, that is it needs to satisfy the budget constraint $\sum_{j=1}^pk_{j}b_{j}\le s$ . Notice that the functions $s\mapsto n_{j}( s) $ are nondecreasing in s. The mortality credits allocated to participant i in group j are then defined as $\textrm{E}\left[ X_{i}\,|\, S_{j,n}=n_{j}( S_{n})\cdot b_{j}\right]$ . This procedure is in fact similar to the approach developed in Denuit and Robert (Reference Denuit and Robert2021e) when pools are partitioned into teams and we end up with a comonotonic mortality credit allocation.

The conditional expectation involved in the second step can be computed directly or the following approximation can be applied. Using Theorem 2.1 in Strasser (Reference Strasser2012), we obtainFootnote ¹

\begin{equation*}\textrm{E}[X_{i}|S_{j,n}]=q_{x_{i}}b_{j}+\frac{q_{x_{i}}(1-q_{x_{i}})b_{j}^{2}}{\sqrt{\textrm{Var}[S_{j,n}]}}Z_{j,n}+\tau_{i,j,n}{(Z_{j,n}^{2}-1)}+O_{\textrm{P}}(n_{j}^{-3/2})\,,\end{equation*}

where

\begin{eqnarray*}Z_{j,n}&=&\frac{S_{j,n}-\textrm{E}[S_{j,n}]}{\sqrt{\textrm{Var}[S_{j,n}]}}\,,\\[5pt]\tau _{i,j,n}&=&{\frac{q_{x_{i}}(1-q_{x_{i}})b_{j}^{3}}{\textrm{Var}[S_{j,n}]}}\left(q_{x_{i}}-\frac{b_{j}^{2}}{\textrm{Var}[S_{j,n}]}\sum_{l\in \mathcal{G}_{j}}q_{x_{l}}^{2}(1-q_{x_{l}})\right)\,.\end{eqnarray*}

Letting

\begin{equation*}H_{j,n}:=\frac{h_{j,n}^{\textrm{lin}}(S_{n})-\textrm{E}[S_{j,n}]}{\sqrt{\textrm{Var}[S_{j,n}]}}=\frac{\sqrt{\textrm{Var}[S_{j,n}]}}{\sum_{j=1}^p\textrm{Var}[S_{j,n}]}\left( S_{n}-\textrm{E}[S_{n}]\right)\,,\end{equation*}

we finally obtain that

\begin{eqnarray*}&&\textrm{E}\left[ X_{i}|S_{j,n}=n_{j}( S_{n})\cdot b_{j}\right] \\[5pt]&=&q_{x_{i}}b_{j}+\frac{q_{x_{i}}(1-q_{x_{i}})b_{j}^{2}}{\sqrt{\textrm{Var}[S_{j,n}]}}H_{j,n}+\tau _{i,j,n}{(H_{j,n}^{2}-1)}+O_{\textrm{P}}(n_{j}^{-3/2}) \\[5pt]&=&q_{x_{i}}b_{j}+\frac{q_{x_{i}}(1-q_{x_{i}})b_{j}^{2}}{\sum_{i=1}^{n}q_{x_{i}}(1-q_{x_{i}})a_{i}^{2}}\left( S_{n}-\textrm{E}[S_{n}]\right) \\[5pt]&&+q_{x_{i}}(1-q_{x_{i}})b_{j}^{3}\left(q_{x_{i}}-\frac{b_{j}^{2}}{\textrm{Var}[S_{j,n}]}\sum_{l\in \mathcal{G}_{j}}q_{x_{l}}^{2}(1-q_{x_{l}})\right)\, \\[5pt] &&\times \left(\left( \frac{S_{n}-\textrm{E}[S_{n}]}{\sum_{i=1}^{n}q_{x_{i}}(1-q_{x_{i}})a_{i}^{2}}\right) ^{2}-\frac{1}{\textrm{Var}[S_{j,n}]}\right) \\[5pt]&&+O_{\textrm{P}}(n_{j}^{-3/2}).\end{eqnarray*}

Example 4.5 illustrates the accuracy of the proposed approximation.

Example 4.5. Consider the pool described in Example 4.3 consisting of two groups: group 1 (60% of the pool) with $a_i=1$ and death probability 0.1 and group 2 (40% of the pool) with $a_i=3$ and death probability 0.2. Figure 6 shows the functions $s\mapsto\textrm{E}[X_i|S_n=s]$ for $n\in\{100,500,1000\}$ together with their large-pool approximations. For $n=100$ , we can see there that the approximation is accurate in the central region but deteriorates in the tail. For larger pool sizes, the quality of the approximation seems to be excellent in this example.

Figure 6. Functions $s\mapsto\textrm{E}[X_i|S_n=s]$ (continuous line) and their large-pool approximation (broken line) for $n=100$ (left panels), $n=500$ (middle panels), and $n=1000$ (right panels), and a participant in group 1 (upper panels) and in group 2 (lower panels) for the pool described in Example 4.3.