2.1 Fairness and self-sustainability

Fairness and self-sustainability are two important properties of a risk-sharing rule in Step 2, as discussed in Denuit et al. (Reference Denuit, Dhaene and Robert2022a), Hieber and Lucas (Reference Hieber and Lucas2022). We equip $(\Omega, \mathcal{F}, P)$ with the filtration $\{\mathcal{F}(t)\}_{t \geq 0}$ , where $\mathcal{F}(t) = \sigma\{(A(s), F(s)) : 0 \leq s \leq t\}$ is the $\sigma$ -algebra generated by $A(t)$ the set of people alive in the pool at time $t$ and $F(t)$ the set of fund balances at time $t$ , where $F(t)=\{F_{1}(t), F_{2}(t), ..., F_{N(t)}(t)\}$ . We denote $E_{t}[\cdot]=E[\cdot|\mathcal{F}(t)]$ . When mortality rates are stochastic, we denote $Q(t)$ as the set of mortality rates $Q(t)=\{q_{1}(t), q_{2}(t), ..., q_{N(t)}(t)\}$ , where $q_{i}(t)$ are random variables of the one-year mortality rates of individual $i$ during the period. We also denote $E_{t}[\cdot|\sigma(Q(t))]=E[\cdot|\mathcal{F}(t)\vee \sigma(Q(t))]$ , where $\sigma(Q(t))$ is the $\sigma$ -algebra generated by $Q(t)$ . Note that the mortality rate $q_{i}(t)$ of any individual $i$ over $[t, t+1]$ is independent to the filtration $\mathcal{F}(t)$ since the mortality rates over time $[t, t+1]$ only rely on random events over that period and are independent to the survivorship and fund balances by time $t$ , so $E_{t}[q_{i}(t)]=E[q_{i}(t)]$ .

Definition 1 (Fairness). A risk-sharing rule is said to be fair if for each member $i$ at time $t+1$ for $t=0, 1, 2, 3, ...$ , the expected fund value after risk sharing conditional on $\mathcal{F}(t)$ is equal to its value before risk sharing:

(2.6) \begin{align}\mathrm{E}_{t}\left[V_{i}(t+1)\right]=s_{i}(t+1).\end{align}

Condition (2.6) promises that members are not taking advantage or disadvantage of other members by joining risk sharing. Note that $s_{i}(t+1)=F_{i}(t)(1+ROR_{i}(t))$ so $s_{i}(t+1)$ is $\mathcal{F}(t)$ -measurable.

Definition 2 (Self-sustainability). A risk-sharing rule is said to be self-sustaining if at any time $t=0, 1, 2, 3, ...$ , the sum of member fund balances before and after risk sharing are equal to each other:

(2.7) \begin{align} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}s_{j}(t+1). \end{align}

When a risk-sharing rule is self-sustaining, it will not pay higher than the total fund balance. Thus, insurance companies do not need to worry about the huge loss in the case of systematic mortality improvement. Therefore, it benefits from a lower capital requirement and thus a lower loading compared with life annuities.

3.1 Proportional rule

First, we define the proportional rule that pays the total mortality credits in proportion to the expected loss when the mortality rates $q_{i}(t)$ are deterministic.

Definition 3. (Deterministic proportional rule). Consider a risk-sharing rule that the weights of the total mortality credits in Equation (2.3) are determined in proportion to the expected loss $s_{i}(t+1)q_{i}(t)$ , which is the product of accumulated fund balance and the one-year probability of death. At the end of a period, the fund value after risk sharing of an individual $i$ initially alive is

(3.1) \begin{align}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+\frac{s_{i}(t+1)q_{i}(t)}{\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)} S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{survives\ this\ period,}\\ \frac{s_{i}(t+1)q_{i}(t)}{\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)} S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{dies\ during\ this\ period.} \end{cases}\end{align}

We call the risk-sharing rule in Equation (3.1) a deterministic proportional rule.

We now extend the proportional risk-sharing rule to the case where mortality rates are stochastic and correlated. That is, we seek a risk-sharing rule of the form

\begin{align*}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+w_{i}(t+1) S(t+1) & \text{if individual $i$ survives this period,} \\[3pt] w_{i}(t+1) S(t+1) & \text{if individual $i$ dies during this period,}\end{cases}\end{align*}

where the weights $w_{i}(t+1)$ are to be determined such that they are proportional to the total risk exposure and the rule is actuarially fair.

Proposition 2. When mortality rates are stochastic and correlated random variables, the risk-sharing rule:

(3.2) \begin{align} V_{i}(t+1) = \begin{cases} s_{i}(t+1)+\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]} S(t+1) & \mathrm{individual}\ i\ \mathrm{survives\ this\ period,} \\[4pt] \frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]} S(t+1) & \mathrm{individual}\ i\ \mathrm{dies\ during\ this\ period,} \end{cases} \end{align}

is fair and self-sustaining.

Proof. Assume the following payout function holds:

\begin{align*}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+w_{i}(t+1) S(t+1) & \text{if individual $i$ survives this period,} \\[3pt] w_{i}(t+1) S(t+1) & \text{if individual $i$ dies during this period.}\end{cases}\end{align*}

Using the law of conditional expectation, we obtain

\begin{align*} E_{t}[V_{i}(t+1)] &= E_{t}\left[E_{t}\left[V_{i}(t+1)|\sigma(Q(t))\right]\right]\\ &= E_{t}\left[s_{i}(t+1) (1-q_{i}(t))+w_{i}(t+1)\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)\right]\\ &= s_{i}(t+1)(1-E[q_{i}(t)])+E_{t}[w_{i}(t+1)]\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)]\\ &= s_{i}(t+1),\end{align*}

which yields

\begin{align*} E_{t}[w_{i}(t+1)]=\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)]}.\end{align*}

Since the weight $w_{i}(t+1)$ is $\mathcal{F}(t)$ -measurable, we can write $w_{i}(t+1)=E_{t}[w_{i}(t+1)]=\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)]}$ . Therefore, the risk-sharing rule in Equation (3.2) is a fair risk-sharing rule.

The risk-sharing rule in Equation (3.2) is self-sustaining because

\begin{align*} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}\frac{s_{j}(t+1)E[q_{j}(t)]}{\sum_{k=1}^{N(t)}s_{k}(t+1)E[q_{k}(t)]} S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j \in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{align*}

The risk-sharing rule satisfying Equation (3.2) is called the stochastic proportional rule. One thing to notice is that $E[q_{i}(t)]$ is assumed to be equal to the $q_{i}(t)$ used in the numerical illustration of the deterministic case. Effectively, this is saying that the results on weighting are not affected when we move from deterministic to stochastic mortality rates.

Lemma 1. The stochastic proportional rule in Equation (3.2) can be rewritten as

\begin{align*} V_{i}(t+1)=\begin{cases} \begin{aligned} &s_{i}(t+1)+s_{i}(t+1)E[q_{i}(t)]\\[4pt] &+w_{i}^{\text{Proportional}}(t+1) (S(t+1)-E_{t}[S(t+1)]) \end{aligned} & \begin{aligned} &\mathrm{if\ individual}\ i \\[4pt] &\mathrm{survives\ this\ period,} \end{aligned} \\[12pt] \begin{aligned} &s_{i}(t+1)E[q_{i}(t)]\\[4pt] &+w_{i}^{\text{Proportional}}(t+1) (S(t+1)-E_{t}[S(t+1)]) \end{aligned} & \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\[4pt] &\mathrm{during\ this\ period,} \end{aligned} \end{cases}\end{align*}

where $w_{i}^{\text{Proportional}}(t+1)=\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]}$ stands for the weighting for individual i at time $t+1$ with the stochastic proportional rule.

Proof. We can rewrite the share of mortality credits as

\begin{align*} &s_{i}(t+1)E[q_{i}(t)]+\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]} (S(t+1)-E_{t}[S(t+1)])\\[4pt] & \quad = s_{i}(t+1)E[q_{i}(t)]+\frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]} S(t+1)-s_{i}(t+1)E[q_{i}(t)]\\[4pt] & \quad = \frac{s_{i}(t+1)E[q_{i}(t)]}{\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{i}(t)]} S(t+1), \end{align*}

which is equal to the form in Equation (3.2).

3.2 Regression rule

Denuit and Robert (Reference Denuit and Robert2021) consider a regression risk-sharing rule, which develops from linear regression and takes the volatility of the risky event into account. The regression risk-sharing rule is also referred to as the covariance risk-sharing rule in Jiao et al. (Reference Jiao, Kou, Liu and Wang2022). Despite the discussion in Hieber and Lucas (Reference Hieber and Lucas2022) under the setting of mortality-sharing products, the factors that affect this risk-sharing rule have not been discussed in detail, nor under stochastic mortality rates.

Definition 4 (Regression rule). Consider a risk-sharing rule that the distribution of the total mortality credits in Equation (2.2) is defined as the following:

(3.3) \begin{align}\begin{aligned}V_{i}(t+1) = \begin{cases}\begin{aligned} &s_{i}(t+1)+E_{t}[X_{i}(t+1)] \\[3pt] &+\frac{Cov_{t}(X_{i}(t+1),S(t+1))}{Var_{t}(S(t+1))}(S(t+1)-E_{t}[S(t+1)])\end{aligned} & \begin{aligned} &\mathrm{if\ individual}\ i \\[3pt] &\mathrm{survives\ this\ period,} \end{aligned} \\[24pt] \begin{aligned} &E_{t}[X_{i}(t+1)]\\[3pt] &+\frac{Cov_{t}(X_{i}(t+1),S(t+1))}{Var_{t}(S(t+1))}(S(t+1)-E_{t}[S(t+1)])\end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\ &\mathrm{during\ this\ period,} \end{aligned}\end{cases}\end{aligned}\end{align}

where $X_{i}(t+1)=1_{i \in D(t+1)}s_{i}(t+1)$ and $S(t+1)=\sum_{j=1}^{N(t)} X_{j}(t+1)$ . We also denote $Cov_{t}(\cdot,\cdot)=Cov(\cdot,\cdot|\mathcal{F}(t))$ and $Var_{t}(\!\cdot\!)=Var(\cdot|\mathcal{F}(t))$ . The risk-sharing rule that satisfies Equation (3.3) is called the regression rule.

Proof. The risk-sharing rule in Equation (3.3) is fair when mortality rates are deterministic because

\begin{align*}E_{t}[V_{i}(t+1)] &= s_{i}(t+1)p_{i}(t)+E_{t}[X_{i}(t+1)]\\ & \quad +\frac{Cov_{t}(X_{i}(t+1),S(t+1))}{Var_{t}(S(t+1))}(E_{t}[S(t+1)]-E_{t}[S(t+1)])\\ &= s_{i}(t+1)p_{i}(t)+s_{i}(t+1)q_{i}(t)\\ &=s_{i}(t+1),\end{align*}

where $p_{i}(t)=1-q_{i}(t)$ is the one-year survival probability of member $i$ between time $t$ and $t+1$ .

The risk-sharing rule in Equation (3.3) is fair when mortality rates are stochastic because

\begin{align*}E_{t}[V_{i}(t+1)] &= E_{t}\left[E_{t}\left[V_{i}(t+1)|\sigma(Q(t))\right]\right]\\ &= E_{t}\Bigr[s_{i}(t+1)p_{i}(t)+E_{t}[X_{i}(t+1)]\\ & \quad + \frac{Cov_{t}(X_{i}(t+1),S(t+1))}{Var_{t}(S(t+1))}(S(t+1)-E_{t}[S(t+1)])\Bigr]\\ &= s_{i}(t+1)E[p_{i}(t)]+s_{i}(t+1)E[q_{i}(t)]\\ &=s_{i}(t+1).\end{align*}

This risk-sharing rule in Equation (3.3) is self-sustaining regardless of whether mortality rates are deterministic or stochastic because

\begin{align*} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}E_{t}[X_{j}(t+1)]\\&+(S(t+1)-E_{t}[S(t+1)])\sum_{j=1}^{N(t)}\frac{Cov_{t}(X_{j}(t+1),S(t+1))}{Var_{t}(S(t+1))}\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+E_{t}[S(t+1)]+(S(t+1)-E_{t}[S(t+1)])\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j \in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{align*}

Proposition 4. When mortality rates are deterministic and assuming independence of future lifetimes of pool members, the regression risk-sharing rule in Equation (3.3) is

(3.4) \begin{align}V_{i}(t+1)=\begin{cases}\begin{aligned} &s_{i}(t+1)+s_{i}(t+1)q_{i}(t)\\ &+w_{i}^{\text{RD}}(t+1)(S(t+1)-\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t))\end{aligned} & \begin{aligned} &\mathrm{if individual}\ {i} \\ &\mathrm{survives\ this\ period,} \end{aligned} \\[24pt] \begin{aligned} &s_{i}(t+1)q_{i}(t)\\ &+w_{i}^{\text{RD}}(t+1)(S(t+1)-\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)) \end{aligned} & \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\ &\mathrm{during\ this\ period,} \end{aligned}\end{cases}\end{align}

where $w_{i}^{\text{RD}}(t+1)=\frac{s_{i}(t+1)^{2} q_{i}(t)(1-q_{i}(t))}{\sum_{j=1}^{N(t)}s_{j}(t+1)^{2} q_{j}(t)(1-q_{j}(t))}$ stands for the weighting for individual i at time $t+1$ with the regression deterministic (RD) risk-sharing rule.

Proof. Using $S(t+1)=\sum_{j=1}^{N(t)} X_{j}(t+1)$ , we have

(3.5) \begin{align}\begin{aligned} Cov_{t}(X_{i}(t+1),S(t+1)) &= Var_{t}(X_{i}(t+1))\\ &= s_{i}(t+1)^{2} q_{i}(t)(1-q_{i}(t)),\end{aligned} \end{align}

and

(3.6) \begin{align}\begin{aligned} Var_{t}(S(t+1)) &= \sum_{j=1}^{N(t)}Var_{t}(X_{j}(t+1))\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1)^{2} q_{j}(t)(1-q_{j}(t)).\end{aligned} \end{align}

Substituting Equations (3.5) and (3.6) into Equation (3.3) completes the proof.

The regression risk-sharing rule with deterministic mortality rates in Equation (3.4) is referred to as the deterministic regression rule in this paper. Since the mortality rate $q_{i}(t)$ in $w_{i}^{\text{RD}}(t+1)$ of Equation (3.4) is deterministic, the source of randomness comes from the uncertainty of survival for given deterministic mortality rates.

Proposition 5. Assuming stochastic mortality rates and independence of future lifetimes of pool members, the fair regression risk-sharing rule in Equation (3.3) becomes

(3.7) \begin{align}\begin{aligned}V_{i}(t+1)=\begin{cases}\begin{aligned} &s_{i}(t+1)+s_{i}(t+1)E[q_{i}(t)]\\ &+w_{i}^{\text{RS}}(t+1)(S(t+1)-\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)])\end{aligned} & \begin{aligned} &\mathrm{if\ individual}\ {i} \\ &\mathrm{survives\ this\ period,} \end{aligned} \\[24pt] \begin{aligned} &s_{i}(t+1)E[q_{i}(t)]\\ &+w_{i}^{\text{RS}}(t+1)(S(t+1)-\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)]) \end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\ &\mathrm{during\ this\ period,} \end{aligned}\end{cases}\end{aligned}\end{align}

where $w_{i}^{\text{RS}}(t+1)=\frac{s_{i}(t+1)^{2}E\left[q_{i}(t)(1-q_{i}(t))\right]+s_{i}(t+1)\sum_{j=1}^{N(t)}s_{j}(t+1)Cov(q_{i}(t),q_{j}(t))}{\sum_{j=1}^{N(t)}s_{j}(t+1)^{2}E[q_{j}(t)(1-q_{j}(t))]+\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}s_{j}(t+1)s_{k}(t+1)Cov(q_{j}(t),q_{k}(t))}$ stands for the weighting for individual i at time $t+1$ with the regression stochastic (RS) risk-sharing rule, and $E[q_{i}(t)(1-q_{i}(t))]=E[q_{i}(t)]-E[q_{i}(t)^{2}]=E[q_{i}(t)]-Var[q_{i}(t)]-E[q_{i}(t)]^{2}.$

Proof. Using the law of conditional expectation, we have

\begin{align*} E_{t}\left[X_{i}(t+1)\right] &= E_{t}\left[E_{t}[X_{i}(t+1)|\sigma(Q(t))]\right]\\ &= E_{t}\left[E_{t}[s_{i}(t+1)1_{i \in D(t+1)}|\sigma(Q(t))]\right]\\ &= E_{t}\left[s_{i}(t+1)q_{i}(t)\right]\\ &= s_{i}(t+1)E\left[q_{i}(t)\right],\end{align*}

and

\begin{align*} E_{t}\left[S(t+1)\right] &= E_{t}\left[E_{t}[S(t+1)|\sigma(Q(t))]\right]\\ &= E_{t}\left[E_{t}[\sum_{j=1}^{N(t)}s_{j}(t+1)1_{j \in D(t+1)}|\sigma(Q(t))]\right]\\ &= E_{t}\left[\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)\right]\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1)E\left[q_{j}(t)\right].\end{align*}

By the law of total covariance, we have

(3.8) \begin{align}\begin{aligned} &Cov_{t}(X_{i}(t+1),S(t+1))\\ & \quad = E_{t}[Cov_{t}(X_{i}(t+1),S(t+1)|\sigma(Q(t)))] +Cov_{t}(E_{t}[X_{i}(t+1)|\sigma(Q(t))],E_{t}[S(t+1)|\sigma(Q(t))])\\ & \quad = E_{t}\left[Cov_{t}\left(s_{i}(t+1)1_{i \in D(t+1)},\sum_{j=1}^{N(t)}s_{j}(t+1)1_{j \in D(t+1)}|\sigma(Q(t))\right)\right]\\ & \quad \quad + Cov_{t}\left(E_{t}\left[s_{i}(t+1)1_{i \in D(t+1)}|\sigma(Q(t))\right],E_{t}\left[\sum_{j=1}^{N(t)}s_{j}(t+1)1_{j \in D(t+1)}|\sigma(Q(t))\right]\right)\\ & \quad = E_{t}\left[Var_{t}(s_{i}(t+1)1_{i \in D(t+1)}|\sigma(Q(t)))\right]+Cov_{t}\left(s_{i}(t+1)q_{i}(t),\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)\right)\\ & \quad = s_{i}(t+1)^{2}E\left[q_{i}(t)(1-q_{i}(t))\right]+s_{i}(t+1)\sum_{j=1}^{N(t)}s_{j}(t+1)Cov(q_{i}(t),q_{j}(t)).\end{aligned}\end{align}

Similarly, we have

(3.9) \begin{align}\begin{aligned} &Var_{t}(S(t+1))\\ &\quad = E_{t}[Var_{t}(S(t+1)|\sigma(Q(t)))]+Var_{t}(E_{t}[S(t+1)|\sigma(Q(t))])\\ &\quad = E_{t}\left[\sum_{j=1}^{N(t)}s_{j}(t+1)^{2}q_{j}(t)(1-q_{j}(t))\right]+Var_{t}\left(\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)\right)\\ &\quad = \sum_{j=1}^{N(t)}s_{j}(t+1)^{2}E[q_{j}(t)(1-q_{j}(t))]+\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}s_{j}(t+1)s_{k}(t+1)Cov(q_{j}(t),q_{k}(t)).\end{aligned}\end{align}

Substituting Equations (3.8) and (3.9) into Equation (3.3) completes the proof.

The regression risk-sharing rule with stochastic mortality rates in Equation (3.7) is referred to as the stochastic regression rule in this paper.

3.3 Alive-only rule

The alive-only rule is a risk-sharing rule proposed in Fullmer and Sabin (Reference Fullmer and Sabin2018) that only distributes the total mortality credits to members alive at the end of each period.

Definition 5 (Alive-only rule). Consider a risk-sharing rule that only the members alive at the end of the period share the total mortality credits as the following:

(3.10) \begin{align}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+\frac{s_{i}(t+1)r_{i}(t)}{\sum_{j\in A(t+1)}s_{j}(t+1)r_{j}(t)} S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{survives\ this\ period,} \\0 & \mathrm{if\ individual}\ i\ \mathrm{dies\ during\ this\ period,}\end{cases}\end{align}

where $r_{i}(t)=\frac{q_{i}(t)}{1-q_{i}(t)}$ , and the major difference is that the dead individual loses everything. The risk-sharing rule in Equation (3.10) is called the alive-only rule.

The weight $\frac{s_{i}(t+1)r_{i}(t)}{\sum_{j\in A(t+1)}s_{j}(t+1)r_{j}(t)}$ is not predetermined, but it depends on the realised survivorship at the end of the period. As the name implies, only members alive receive part of the total mortality credits to compensate, while members who die lose everything.

Proof. The expected value of $V_{i}(t+1)$ is

\begin{align*}\mathrm{E}_{t}\left[V_{i}(t+1)\right] &= E_{t}[E_{t}[V_{i}(t+1)|\mathcal{F}_{i}(t+1)]]\\ &= P(i\in A(t+1))E_{t}[V_{i}(t+1)|i\in A(t+1)]+P(i\notin A(t+1))E_{t}[V_{i}(t+1)|i\notin A(t+1)]\\ &= \left(1-q_{i}(t)\right)\left[s_{i}(t+1)+E_{t}\left[\frac{s_{i}(t+1)r_{i}(t)}{\sum_{j\in A(t+1)}s_{j}(t+1)r_{j}(t)} S(t+1)|i\in A(t+1)\right]\right],\end{align*}

where $\mathcal{F}_{i}(t+1)$ is the filtration representing the survival status of individual $i$ up to time $t+1$ . Fullmer and Sabin (Reference Fullmer and Sabin2018) mention that this is not strictly fair. This is because of the approximation of $E_{t}\left[\frac{s_{i}(t+1)r_{i}(t)}{\sum_{j\in A(t+1)}s_{j}(t+1)r_{j}(t)} S(t+1)|i\in A(t+1)\right]\approx s_{i}(t+1)r_{i}(t)$ , which gives

\begin{align*} \mathrm{E}_{t}\left[V_{i}(t+1)\right] & \approx \left(1-q_{i}(t)\right)s_{i}(t+1)+(1-q_{i}(t))s_{i}(t+1)r_{i}(t)\\ & \approx \left(1-q_{i}(t)\right)s_{i}(t+1)+(1-q_{i}(t))s_{i}(t+1)\frac{q_{i}(t)}{1-q_{i}(t)}\\ & \approx s_{i}(t+1).\end{align*}

This is an approximation because

\begin{align*} E_{t}\left[\frac{S(t+1)}{\sum_{j\in A(t+1)}s_{j}(t+1)r_{j}(t)}|i\in A(t+1)\right] & = E_{t}\left[\frac{\sum_{j=1}^{N(t)}1_{j \in D(t+1)}s_{j}(t+1)}{\sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)\frac{q_{j}(t)}{1-q_{j}(t)}}|i\in A(t+1)\right]\\ & \neq 1\end{align*}

since we cannot move the expectation into the summation in the denominator.

Due to the approximation, the alive-only rule in Equation (3.10) is an almost fair risk-sharing rule. The term almost fair is used since the bias is found to be negligible in Sabin and Forman (Reference Sabin and Forman2016), Fullmer and Sabin (Reference Fullmer and Sabin2018) when the mean amount forfeited by dying members is large compared to the nominal gain of any one member, which is satisfied in the setting of this paper.

Meanwhile, the alive-only rule in Equation (3.10) is still self-sustaining because

\begin{align*} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j \in A(t+1)}\frac{s_{j}(t+1)r_{j}(t)}{\sum_{k\in A(t+1)}s_{k}(t+1)r_{k}(t)} S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j \in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{align*}

Proposition 7. The risk-sharing rule below in Equation (3.11) is a self-sustaining and almost fair alive-only rule under stochastic mortality rates:

(3.11) \begin{align}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+w_{i}^{\text{Alive}}(t+1) S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{survives\ this\ period,} \\0 & \mathrm{if\ individual}\ i\ \mathrm{dies\ during\ this\ period,}\end{cases}\end{align}

where $w_{i}^{\text{Alive}}(t+1)=\frac{s_{i}(t+1)\frac{E[q_{i}(t)]}{1-E[q_{i}(t)]}}{\sum_{j\in A(t+1)}s_{j}(t+1)\frac{E[q_{j}(t)]}{1-E[q_{j}(t)]}}$ stands for the weighting for individual i at time $t+1$ with the alive-only rule under stochastic mortality rates.

Proof. By the law of total expectation, we have

\begin{align*} E_{t}[V_{i}(t+1)] &= E_{t}\left[E_{t}\left[E_{t}\left[V_{i}(t+1)|\sigma(Q(t))\right]|\mathcal{F}_{i}(t+1)\right]\right]\\ &= E_{t}\left[(1-q_{i}(t))\Big(s_{i}(t+1)+\frac{s_{i}(t+1)\frac{E[q_{i}(t)]}{1-E[q_{i}(t)]}}{\sum_{j\in A(t+1)}s_{j}(t+1)\frac{E[q_{j}(t)]}{1-E[q_{j}(t)]}} S(t+1)\Big)|i \in A(t+1)\right]\\ &= s_{i}(t+1)(1-E[q_{i}(t)])+s_{i}(t+1)E[q_{i}(t)]E_{t}\left[\frac{S(t+1)}{\sum_{j\in A(t+1)}s_{j}(t+1)\frac{E[q_{j}(t)]}{1-E[q_{j}(t)]}}|i \in A(t+1) \right].\end{align*}

When we make the approximation $E_{t}\left[\frac{S(t+1)}{\sum_{j\in A(t+1)}s_{j}(t+1)\frac{E[q_{j}(t)]}{1-E[q_{j}(t)]}}|i \in A(t+1) \right]\approx1$ , it gives us

\begin{align*} E_{t}[V_{i}(t+1)]\approx s_{i}(t+1),\end{align*}

which means the risk-sharing rule in Equation (3.11) is almost fair. The risk-sharing rule in Equation (3.11) is self-sustaining because

(3.12) \begin{align}\begin{aligned} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j\in A(t+1)}\frac{s_{j}(t+1)\frac{E[q_{j}(t)]}{1-E[q_{j}(t)]}}{\sum_{k\in A(t+1)}s_{k}(t+1)\frac{E[q_{k}(t)]}{1-E[q_{k}(t)]}} S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{aligned}\end{align}

3.4 A new risk-sharing rule: Joint expectation rule

In this subsection, a new fair and self-sustaining risk-sharing rule is proposed. The weight in the total mortality credits of an individual is higher when the fund balance, the mean of the mortality rate, the standard deviation of the mortality rate, and the correlation between the mortality rates of other fund members are higher. The risk-sharing rule is named the JE rule, and it will reduce to the proportional rule when the mortality rates are no longer stochastic and correlated random variables but deterministic values. In comparison with the regression rule, the weighting of the mortality credits under the JE rule increases monotonically with the mean mortality rate, and the slope of the rate of return from mortality credits is not affected by the fund balance.

Proposition 8. The following risk-sharing rule is a fair and self-sustaining rule incorporating correlations between stochastic mortality rates:

(3.13) \begin{align}V_{i}(t+1)=\begin{cases}\begin{aligned} &s_{i}(t+1) + s_{i}(t+1)E[q_{i}(t)]\\[3pt] &+ w_{i}^{\text{JE}}(t+1) (S(t+1)-E_{t}[S(t+1)])\end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ {i} \\ &\mathrm{survives\ this\ period,} \end{aligned}\\[24pt]\begin{aligned} &s_{i}(t+1)E[q_{i}(t)] \\[3pt] &+ w_{i}^{\text{JE}}(t+1) (S(t+1)-E_{t}[S(t+1)])\end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\ &\mathrm{during\ this\ period,} \end{aligned}\end{cases}\end{align}

where $w_{i}^{\text{JE}}(t+1)=\frac{s_{i}(t+1)\sum_{k=1}^{N(t)}s_{k}(t+1)E[q_{i}(t)q_{k}(t)]}{\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}s_{j}(t+1)s_{k}(t+1)E[q_{j}(t)q_{k}(t)]}$ .

Proof. The risk-sharing rule in Equation (3.13) is fair because

\begin{align*} E_{t}[V_{i}(t+1)] &= E_{t}[E_{t}[V_{i}(t+1)|\sigma(Q(t))]]\\ &= s_{i}(t+1)(1-E[q_{i}(t)])+s_{i}(t+1)E[q_{i}(t)]+w_{i}(t+1)(E_{t}[S(t+1)]-E_{t}[S(t+1)])\\ &= s_{i}(t+1).\end{align*}

The risk-sharing rule in Equation (3.13) is self-sustaining because

\begin{align*} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}s_{j}(t+1)E[q_{j}(t)]+S(t+1)-E_{t}[S(t+1)]\\ &= \sum_{j=1}^{N(t)}1_{j \in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j \in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{align*}

We name the risk-sharing rule in Equation (3.13) as the JE rule because the term $E[q_{i}(t)q_{k}(t)]$ in the numerator is the joint expectation of the one-year mortality rates $q_{i}(t)$ and $q_{k}(t)$ of individual $i$ and individual $k$ at time $t$ .

The joint expectation $E[q_{i}(t)q_{k}(t)]$ in the weight captures not only the expected values of mortality rates but also the volatility of mortality rates and the correlation between mortality rates of different individuals because:

(3.14) \begin{align}\begin{aligned} E[q_{i}(t)q_{k}(t)] & =Cov(q_{i}(t), q_{k}(t))+E[q_{i}(t)]E[q_{k}(t)]\\ & =\rho(q_{i}(t),q_{k}(t)) sd(q_{i}(t)) sd(q_{k}(t))+ E[q_{i}(t)]E[q_{k}(t)],\end{aligned}\end{align}

where $Cov(q_{i}(t),q_{k}(t))$ is the covariance between the one-year mortality rates $q_{i}(t)$ and $q_{k}(t)$ of individuals $i$ and $k$ at time $t$ , $\rho(q_{i}(t),q_{k}(t))$ is the correlation between $q_{i}(t)$ and $q_{k}(t)$ , and $sd(q_{i}(t))$ is the standard deviation of $q_{i}(t)$ .

Lemma 2. When the mortality rates are deterministic, then the risk-sharing rule proposed in Equation (3.13) reduces to the fair proportional rule in Equation (3.1):

\begin{align*}V_{i}(t+1)=\begin{cases}s_{i}(t+1)+\frac{s_{i}(t+1)q_{i}(t)}{\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)} S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{survives\ this\ period,} \\[12pt]\frac{s_{i}(t+1)q_{i}(t)}{\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)} S(t+1) & \mathrm{if\ individual}\ i\ \mathrm{dies\ during\ this\ period.}\end{cases}\end{align*}

Proof. When $E[(q_{j}(t))^{2}]=E[q_{j}(t)]^{2}$ and $E[q_{j}(t)q_{k}(t)]=E[q_{j}(t)]E[q_{k}(t)]$ , the weighting becomes:

\begin{align*} w_{i}^{\text{JE}}(t+1) &= \frac{s_{i}(t+1)\sum_{k=1}^{N(t)}s_{k}(t+1)E[q_{i}(t)q_{k}(t)]}{\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}s_{j}(t+1)s_{k}(t+1)E[q_{j}(t)q_{k}(t)]}\\[4pt] &= \frac{s_{i}(t+1)\sum_{k=1}^{N(t)}s_{k}(t+1)E[q_{i}(t)]E[q_{k}(t)]}{\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}s_{j}(t+1)s_{k}(t+1)E[q_{j}(t)]E[q_{k}(t)]}\\[4pt] &= \frac{s_{i}(t+1)q_{i}(t)\sum_{k=1}^{N(t)}s_{k}(t+1)q_{k}(t)}{\left(\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)\right)\left(\sum_{k=1}^{N(t)}s_{k}(t+1)q_{k}(t)\right)}\\[4pt] &= \frac{s_{i}(t+1)q_{i}(t)}{\sum_{j=1}^{N(t)}s_{j}(t+1)q_{j}(t)}.\end{align*}

Moreover, the proposed JE rule in Equation (3.13) can be extended to include a death benefit so that when a member dies they do not lose all of the accumulated fund balance but get $d_{i}(t+1)$ as the death benefit in the case of death. The inclusion of a death benefit aims to address the loss aversion so that members do not lose all of their fund balances in the event of death. The expected loss thus becomes $s_{i}(t+1)-d_{i}(t+1)$ . The death benefit $d_{i}(t+1)$ can be set as a predetermined proportion of the accumulated fund balance $s_{i}(t+1)$ so that the expected loss is positive. In that sense, $d_{i}(t+1)$ is also $\mathcal{F}(t)$ -measurable.

Proposition 9. With death benefit protection included, the risk-sharing rule in Equation (3.13) can be extended to:

(3.15) \begin{align}V_{i}(t+1)=\begin{cases}\begin{aligned} &s_{i}(t+1) + (s_{i}(t+1)-d_{i}(t+1)) E[q_{i}(t)]\\[3pt] &+ w_{i}(t+1) (S(t+1)-E_{t}[S(t+1)])\end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ {i} \\ &\mathrm{survives\ this\ period,} \end{aligned}\\[12pt] \begin{aligned} &d_{i}(t+1) + (s_{i}(t+1)-d_{i}(t+1)) E[q_{i}(t)]\\ &+w_{i}(t+1) (S(t+1)-E_{t}[S(t+1)]) \end{aligned}& \begin{aligned} &\mathrm{if\ individual}\ i\ \mathrm{dies} \\ &\mathrm{during\ this\ period,} \end{aligned}\end{cases}\end{align}

where $w_{i}^{\text{JE}}(t+1)=\frac{(s_{i}(t+1)-d_{i}(t+1))\sum_{k=1}^{N(t)}(s_{k}(t+1)-d_{k}(t+1))E[q_{i}(t)q_{k}(t)]}{\sum_{j=1}^{N(t)}\sum_{k=1}^{N(t)}(s_{j}(t+1)-d_{j}(t+1))(s_{k}(t+1)-d_{k}(t+1))E[q_{j}(t)q_{k}(t)]}$ , and $S(t+1)= \sum_{j=1}^{N(t)}1_{j\in D(t+1)}(s_{j}(t+1)-d_{j}(t+1))$ . In this case, the risk-sharing rule is still fair and self-sustaining.

Proof. The risk-sharing rule in Equation (3.15) is fair because

\begin{align*} E_{t}[V_{i}(t+1)]=E_{t}[E_{t}[V_{i}(t+1)|\sigma(Q(t))]] &= s_{i}(t+1)(1-E[q_{i}(t)])+d_{i}(t+1)E[q_{i}(t)]\\ & \quad +(s_{i}(t+1)-d_{i}(t+1))E[q_{i}(t)]\\ & \quad + w_{i}(t+1)(E_{t}[S(t+1)]-E_{t}[S(t+1)])\\ &= s_{i}(t+1).\end{align*}

The risk-sharing rule in Equation (3.15) is self-sustaining because

\begin{align*} \sum_{j=1}^{N(t)}V_{j}(t+1) &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}d_{j}(t+1)+\sum_{j=1}^{N(t)}(s_{j}(t+1)-d_{j}(t+1))E[q_{j}(t)]\\&+S(t+1)-E_{t}[S(t+1)]\\ &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}d_{j}(t+1)+S(t+1)\\ &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}d_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}(s_{j}(t+1)-d_{j}(t+1))\\ &= \sum_{j=1}^{N(t)}1_{j\in A(t+1)}s_{j}(t+1)+\sum_{j=1}^{N(t)}1_{j\in D(t+1)}s_{j}(t+1)\\ &= \sum_{j=1}^{N(t)}s_{j}(t+1).\end{align*}

3.5 Summary of risk-sharing rules

We summarise the properties of different risk-sharing rules in Tables 1 and 2. Table 1 compares the weightings in the total mortality credits $S(t+1)$ between different deterministic and stochastic risk-sharing rules. We can see that when extended to stochastic risk-sharing rules, only the JE rule and the stochastic regression rules take the correlation between mortality rates into consideration.

Table 1. Comparison of weighting in $S(t+1)$ of member $i$ between deterministic and stochastic versions of risk-sharing rules.

Table 2. Change in weighting $w_{i}(t+1)$ when the mean, variance of mortality rates, or correlation to mortality rates of other cohorts increase for different risk-sharing rules.

Table 2 further compares how the increment in one statistic (mean, variance, or correlation of mortality rates) affects the weighting in the total mortality credits between different risk-sharing rules, holding the other two statistics the same. We can see that the stochastic regression risk-sharing rule gives a higher weight when the variance and correlation of mortality rates are higher, while the effect of the mean is not monotonic. The proposed JE risk-sharing rule will distribute a higher proportion of total mortality when the mean, variance, or correlation of mortality rates is higher. The proof of the results in Table 2 can be found in the Supplementary Material. Intuitively, it is easy to understand that the weight in mortality credits should increase when the mean mortality rates increase, as a higher mean mortality rate indicates a higher expected loss. Moreover, a higher variance in the mortality rates should also indicate a higher proportion of mortality credits since it means more volatile expected loss, which is associated with more volatile total mortality credits. We note that for the stochastic regression and JE rules, the effect of a change in the variance of mortality rates holds under the assumption that the sum of the covariance-weighted accumulated fund values is positive. Furthermore, higher correlations with the other cohorts mean these cohorts are more likely to have their mortality rates move toward the same direction, leading to more volatile total mortality credits, which is also why these cohorts should take more proportion of the total mortality credits.

To further illustrate how incorporating the volatilities and correlations through the joint expectation would affect risk sharing, we present three scenarios in the Supplementary Material where the JE rule and the proportional rule produce different weights and values after risk sharing. The scenarios are designed around the heterogeneity of risk exposures and probabilities of incurring losses. Scenario 1 assumes that risk exposures are identical, and loss probabilities have the same mean and standard deviation. Scenario 2 assumes that risk exposures are different, and loss probabilities have the same mean and standard deviation. Scenario 3 assumes that risk exposures and the means and standard deviations of the loss probabilities are different. We can see that for a participant of a risk-sharing pool whose probability of loss is negatively correlated with the others, the weight using the JE rule is lower than using the proportional rule. The percentage difference in weight is $-5.26\%$ in Scenario 1, $-16.89\%$ in Scenario 2, and $-5.36\%$ in Scenario 3. Moreover, the percentage difference in the value after risk sharing with respect to the value before risk sharing can be up to $-2.98\%$ in Scenario 1, $-12.79\%$ in Scenario 2, and $-8.04\%$ in Scenario 3. Therefore, there is a need to incorporate the volatilities and correlations through the joint expectation in risk sharing.

4.1 Data and assumptions

We establish a risk-sharing pool that allows mixed-age cohorts with different initial balances and new members to join. The assumptions on the risk-sharing pool are the following:

• • A total of $586$ members in the initial pool at time zero as presented in Table 3.

• • Age range: $60$ – $100$ . Age distribution refers to Australian population exposure in $2020$ (Human Mortality Database, 2022).

• • For each age, half of the members have a high balance and the other half have a low balance, where the high balance is $1.5$ times the low balance. The superannuation balance required for a comfortable retirement in Australia is around $\$600,000$ (ASFA, 2023), so we define high balance at age $60$ as $1.2\times600,000=720,000$ and low balance as $0.8\times600,000=480,000$ .

• • The balance decreases with age to reflect the consumption of retirement balance.

• • New members joining every year have the same size of $586$ members and the same age and balance distributions as in Table 3.

• • Rates of return and the risk-free rate: $3\%$ per annum.

The pool size of $586$ is chosen to demonstrate the pooling effect while maintaining computation efficiency, as we study $50$ times the pool size and perform $30$ years of analysis with new members of the same size joining each year.

The assumptions on mortality rates are

• • Australian male mortality rates.

• • Mortality rates follow a multivariate log-normal distribution since they are nonnegative. The Renshaw and Haberman model (Renshaw and Haberman, Reference Renshaw and Haberman2006) with a log link function and the cohort factor modelled by an ARIMA process (Villegas et al., Reference Villegas, Kaishev and Millossovich2018) implies log-normally distributed mortality rates.

• • The covariance matrix of the multivariate log-normal distribution is estimated by using an $11$ -year bracket with year $2020$ in the middle for each cohort at every point in time and calculating the covariance using these $11$ -element vectors.

Examples of the means, standard deviations, and correlation matrix of mortality rates for a selection of cohorts at ages $60, 70, 80, 90, 100$ in the calendar year $2020$ are displayed in Tables 4 and 5, which are based on Australian male mortality data. We can see that the means of mortality rates increase with age. The standard deviations of mortality rates also increase with age, except for age $100$ . From Table 5, we can see that the correlation is generally higher when the age difference is lower. Age $100$ is also an exception in this case because an improvement in the mortality at younger ages, for example, age $90$ often leads to a worsening mortality at age $100$ since people will die in the end.

Table 4. Mean and standard deviation of mortality rates at different ages in $2020$ .

Table 5. Correlation of mortality rates at different ages in $2020$ .

4.2 Overview of analysis

We compare risk-sharing rules by the rate of return from the distribution of mortality credits in the case of survival: $ROR_{i}^{mc}(t)=(V_{i}(t+1)-s_{i}(t+1))/F_{i}(t).$ We do not directly compare the weightings because they are heavily affected by the pool size and the fund value. The $ROR_{i}^{mc}(t)$ for different risk-sharing rules in the case of survival is shown below:

• • Proportional: $\frac{s_{i}(t+1)E[q_{i}(t)]+ w_{i}^{\text{Proportional}}(t+1) (S(t+1)-E_{t}[S(t+1)])}{F_{i}(t)}$

• • Joint Expectation: $\frac{s_{i}(t+1)E[q_{i}(t)] + w_{i}^{\text{JE}}(t+1)(S(t+1)-E_{t}[S(t+1)])}{F_{i}(t)}$

• • Regression Det: $\frac{s_{i}(t+1)q_{i}(t)+w_{i}^{\text{RD}}(t+1)(S(t+1)-E_{t}[S(t+1)])}{F_{i}(t)}$

• • Regression Sto: $\frac{s_{i}(t+1)E[q_{i}(t)]+w_{i}^{\text{RS}}(t+1)(S(t+1)-E_{t}[S(t+1)])}{F_{i}(t)}$

• • Alive: $\frac{w_{i}^{\text{Alive}}(t+1)S(t+1)}{F_{i}(t)}$

We omit the deterministic proportional and alive-only rules because we assume that the mortality rates $q_{i}(t)$ used in the deterministic case are equal to $E[q_{i}(t)]$ used in the stochastic case. We can see that the difference in risk-sharing rules is their weighting on the difference between the empirical and expected mortality credits $S(t+1)-E_{t}[S(t+1)]$ , except for the alive-only rule which is on $S(t+1)$ . This means different risk-sharing rules have different sensitivities when there is a deviation from the expected mortality credits. We measure this difference in the sensitivity by plotting $ROR^{mc}(t)$ versus the deviation in total mortality credits $S(t+1)-E_{t}[S(t+1)]$ for different (1) risk-sharing rules, (2) ages and thus mortality rates of members, (3) fund balances of members, and (4) pool sizes. The analyses in Sections 4.3–4.5 are between times $0$ and $1$ .

Then, we study the performance of the fund over time with new members joining in Section 4.6. We will assess benefit payments of different cohorts for the next $30$ years since the initial establishment of the pool assuming: (1) no systematic mortality risk and (2) $20\%$ reduction of mortality rates for the first $5$ years.

4.3 Comparison of rates of return from mortality credits against deviation in total mortality credits

We first compare the rates of return from mortality credits $ROR^{mc}(t)$ of different risk-sharing rules against the deviation in total mortality credits for time $t=0$ . As illustrated in Figure 1, $S(t+1)-E_{t}[S(t+1)]=0$ implies zero deviation in mortality credits and this is the point of the expected rate of return from mortality credits $E[ROR^{mc}(t)]$ , where all risk-sharing rules except for the alive-only rule pay the same.

Figure 1. Comparison of $ROR^{mc}(t)$ between risk-sharing rules.

Figure 1(a) shows the comparison of risk-sharing rules for the cohort aged $60$ with a high balance $720,000$ in year $1$ . The slope of the plot implies the sensitivity of $ROR^{mc}(t)$ to the deviation in total mortality credits $S(t+1)-E_{t}[S(t+1)]$ . A higher slope implies a higher sensitivity, and the value of the slope is shown in the figure as $k$ , calculated as the vertical change in $ROR^{mc}(t)$ between the two endpoints. This adjustment controls variations in the scale of $S(t+1)-E_{t}[S(t+1)]$ under different settings in this paper. We can see from Figure 1(a) that the deterministic regression rule and the stochastic regression rule give higher slopes at age $60$ than the other three risk-sharing rules, while the deterministic regression rule has a slightly higher slope than the stochastic regression rule. The differences between the regression rules and other risk-sharing rules are due to the different weightings in mortality credits, as illustrated in Table 1. The weighting of the regression rules has the quadratic of the accumulated fund balance in the numerator, which results in higher slopes than the other risk-sharing rules for cohorts with high fund balances in the pool. In addition to the quadratic accumulated fund balance, this can also be explained by the $E\left[q_{i}(t)(1-q_{i}(t))\right]$ and $q_{i}(t)(1-q_{i}(t))$ terms in the numerators of regression rules, whose relative rates to $E[q_{i}(t)]$ and $q_{i}(t)$ decrease as age increases and thus expected mortality rate increases. Meanwhile, the proportional rule, JE rule, and alive-only rule have minor differences at age 60.

Figure 1(b) shows the comparison at age $100$ with high balance $240,000$ . In contrast to age $60$ , the alive-only rule gives a higher slope than the other four risk-sharing rules with a significant difference. This is because the $\frac{E[q_{i}(t)]}{1-E[q_{i}(t)]}$ term for the alive-only rule in Table 1 is much higher than the expected mortality rates $E[q_{i}(t)]$ at older ages, since the $(1-E[q_{i}(t)])$ term is much smaller than $1$ at older ages with higher expected mortality rates. Meanwhile, the $\frac{E[q_{i}(t)]}{1-E[q_{i}(t)]}$ term at younger ages is close to $E[q_{i}(t)]$ because their expected mortality rates are relatively lower, which explains the minor difference between the alive-only rule with the JE rule and proportional rule at younger ages. The deterministic and stochastic regression rules now have lower slopes than the proportional and JE rules. The expected rate of return from mortality credits at age $100$ is much higher compared with age $60$ because of the much higher expected mortality rate at age $100$ .

Figure 2 compares $5$ cohorts aged $60, 70, 80, 90, 100$ years and with high or low balance. We can see that as age increases, the expected mortality rate $E[q_{i}(t)]$ increases, which leads to higher $E[ROR^{mc}(t)]$ at the point of $S(t+1)-E_{t}[S(t+1)]=0$ . Meanwhile, Table 7 displays the slopes of different risk-sharing rules for members with a high balance in the column ‘Original Size’. It can be seen from Figure 2 and Table 7 that, as age increases, the slopes of the regression rules increase, but not as fast as the proportional rule and JE rule, leading them to be relatively flatter at older ages. This is due to the quadratic accumulated fund balance and the $E\left[q_{i}(t)(1-q_{i}(t))\right]$ term in the regression rules explained earlier. The slope of the alive-only rule increases at the fastest rate with age, and it starts to dominate the other rules for old cohorts. We also find that the slopes of the regression rules are higher with high balance, keeping age the same.

Figure 2. Comparison of $ROR^{mc}(t)$ with different ages, balances, and rules.

4.4 Effect of balance

To further study the effect of balance on the slope, we divide the slope of high balance by the slope of low balance at every age for the period $[0,1]$ and present the results in Table 6. From Table 6, we find that the ratios for the proportional rule, JE rule, and alive-only rule are $1$ , indicating that balance does not affect the sensitivity to deviation in total mortality credits. However, we can see from Table 6 that the ratio for the deterministic regression rule is $1.5$ , which is exactly the ratio of high balance over low balance, indicating that the slope increases proportionally to balance for the deterministic regression rule. Moreover, for the stochastic regression rule, it is above $1$ but not equal to $1.5$ exactly, indicating that the slope still increases with balance, but the ratio is also affected by the mean, variance, and correlation of mortality rates of all fund members.

Table 6. Slope high balance over slope low balance at different ages.

Table 7. Slopes of $ROR^{mc}(t)$ for different risk-sharing rules and for high-balance individuals at ages $60$ , $80$ , and $100$ with different pool sizes.

4.5 Effect of pool size

Table 7 shows the slope values of different risk-sharing rules for different pool sizes and for ages $60$ , $80$ , and $100$ , respectively. The expected rate of return from mortality credits $E_{t}[ROR^{mc}(t)]$ at the point of $S(t+1)-E_{t}[S(t+1)]=0$ does not change with the pool size. We observe from Table 7 that the slopes of the proportional rule, JE rule, alive-only rule, and deterministic regression rule are relatively stable when we increase the pool size to $10$ times and $50$ times the original size. However, we find that the slope for the stochastic regression rule decreases for member age $60$ .

From Table 7, we can see that the slope of the stochastic regression rule increases with pool size at age $80$ . Meanwhile, a decrease in the slope with pool size is observed at age $100$ . When the pool size is $50$ times the original size, a significant difference in slope can be found between the deterministic and stochastic regression rules, as illustrated in Table 7 and Section 2 of the Supplementary Material. In particular, the slope for the cohort age $100$ using the stochastic regression rule reduces from $0.29115$ to $0.04998$ when the pool size is multiplied by $50$ , which is much lower than $0.30555$ using the deterministic regression rule. The decrease in slope at ages $60$ and $100$ can be explained by the low volatility in the mortality rates we assume, and being correlated to fewer members in the pool. Especially at age $100$ , the mortality rates are negatively correlated with the mortality rates of most members at other ages. Meanwhile, the slopes of the JE rule remain the same when the pool size increases. Note that the minor difference for varying pool sizes for the alive-only rule is due to the randomness in the simulation of member survivorship. The comparisons of all cohorts with $10$ and $50$ times the original pool size are displayed in the figures in the Supplementary Material.

To summarise, from the results in Sections 4.3–4.5, individuals can better determine which risk-sharing rule has the characteristics they favour. Since a higher slope indicates more volatility in the rate of return from mortality credits, individuals with higher risk aversion may favor the risk sharing rules with lower slopes, for example, the JE rule at younger ages, and the regression rules at older ages. Moreover, those who do not wish their rates of return from mortality credits to be more volatile with increased fund balances will avoid the regression rules. Meanwhile, those who do not wish their rates of return from mortality credits to change with the pool size will avoid the stochastic regression rule. Furthermore, if individuals believe a longevity shock will happen, they will favor the risk-sharing rule with a lower slope to mitigate the loss.

4.6 Benefit payments over time

The discussions so far are between time $0$ and time $1$ . We now allow new heterogeneous members as shown in Table 3 to enter the pool at the beginning of each year. The ages of existing members increase by one every year, and their balances after benefit payments become the initial balances at the beginning of the next year. Under the dynamic setting, we track the performance of the pool over the next $30$ years since the initial establishment. The means, standard deviations, and correlations of mortality rates at different ages in Tables 4 and 5 are updated over time to reflect the time evolution of mortality rates.

Figure 3 shows the simulated income payments for different cohorts over the next $30$ years. A sudden drop in income payments to zero represents the death of the member. We can see that for younger cohorts, the income payments of all risk-sharing rules stay at a relatively stable level until around age $90$ . The alive-only rule results in higher income payments at older ages, as illustrated in the previous subsections from their higher slope. The income for people who joined at older ages, for example, age $90$ is slightly decreasing because of the very small annuity factor at that age, leading to high-income payments relative to initial contributions and the balance to be consumed very quickly. However, the income keeps increasing for the alive-only rule because of the very high slope at old ages. At age $100$ , the income for risk-sharing rules except for the alive-only rule is decreasing fast, while it is still increasing for the alive-only rule.

Figure 3. Benefit payments over time in a dynamic pool allowing new members to join.

Figure 4 shows the simulated income payments when there is a systematic longevity shock of $20\%$ reduction in the mortality rates of all cohorts for the first $5$ years. We can see that for younger cohorts, their income payments are still relatively stable over their lifetime. However, for older cohorts, the payments for all risk-sharing rules experience a decrease compared with no longevity shock, mainly because the benefits from mortality credits decrease when mortality improves. We can also observe differences between different risk-sharing rules because as illustrated in previous sections, different risk-sharing rules allocate different weightings to the deviation in total mortality credits.

Figure 4. Benefit payments over time in a dynamic pool allowing new members to join, under a $5$ -year systematic longevity shock.

Since the benefit payments are determined by dividing the fund balances over the annuity factors, the differences in the balances between different risk-sharing rules are scaled down and mitigated when transferred into benefit payments. Therefore, to better illustrate the difference between risk-sharing rules, we further compare the fund balances in year $5$ when there is a longevity shock for the first $5$ years in Figure 5. The difference in the way of distributing this deviation leads to the difference in the fund balances, and a lower slope leads to a higher balance. We can see that there is a significant difference when we move from the proportional or JE risk-sharing rule to the deterministic or stochastic regression rule or to the alive-only rule. The two regression rules give lower fund balances at younger ages than the other three rules, but they give higher fund balances at older ages than the proportional rule and JE rule.

Figure 5. Balance at time $5$ , under a $5$ -year systematic longevity shock.

From Figure 5, we can see that the difference in balance is several thousand dollars for ages $60$ , $70$ , and $80$ and can exceed $10,000$ for ages $90$ and $100$ with the initial balances we set in Table 3. An obvious difference is also observed between deterministic and stochastic regression rules. The stochastic regression rule gives higher balances than the deterministic regression rule at age $60$ , $70$ with high balance, $90$ with high balance, and $100$ . In the other cases, they will give lower balances than the deterministic regression rule. This is because when a systematic longevity shock of $20\%$ reduction in mortality rates happens, empirical total mortality credits will tend to be lower than expected. The difference between the JE rule and the proportional rule is small because when we include the variance and correlation for all cohorts, the numerator of every fund member increases, leading to a smaller change in the weighting. Therefore, there is a dilution effect when we move from the proportional rule to the JE rule.

Finally, the difference between the alive-only rule to the other risk-sharing rules is also obvious. We can see from Figure 5 that the difference between the alive-only rule and the JE rule ranges from a few hundred dollars at age $60$ to a few thousand dollars at age $80$ , up until around $50,000$ dollars at age $100$ . We can see that a higher difference between risk-sharing rules normally happens at older ages despite the lower initial contribution at older ages. Therefore, this paper provides a more accurate calculation of risk sharing with different rules under stochastic and correlated mortality rates and can help issuers decide which risk-sharing rule to choose when they establish the product according to their needs. If the issuers do not prefer the distribution of mortality credits to be affected by individual account balances, then they should not choose stochastic or deterministic regression rules. Meanwhile, if the issuers prefer the account balance of younger retirees to be higher than older retirees when a systematic reduction in mortality rates happens, then the proportional, JE, and alive-only rules are preferred. Moreover, if issuers prefer to pay more to surviving members, then they would prefer the alive-only rule. While the difference between different risk-sharing rules exists, we need to emphasise that all risk-sharing rules are still actuarially fair, or almost fair for the alive-only rule.