Withdrawal Success Optimization in a Pooled Annuity Fund

After a lump sum investment, suppose an investor wishes to complete a pre-determined schedule of withdrawals. In practice, the total amount intended to be withdrawn is usually larger than the lump sum. To overcome this disparity, the lump sum must be invested in various assets having positive expected log-returns, with the goal of maximizing the probability to complete the schedule of withdrawals. Note that this maximization occurs over the time-adapted portfolio weight functions, which control the proportion of available wealth invested in each asset. For the case where rebalancing and withdrawals are made discretely in time, this maximization problem is studied for an individual investor in Brown (2023). The goal here is to extend the results of Brown (2023) to a pool of investors, all hoping to make the same schedule of withdrawals. Of particular interest is the schedule having a constant annual withdrawal until death.

The basic idea of the pooled annuity fund is as follows. At time 0, collect the same lump sum investment from each individual in the pool. Invest the combined funds into a fixed number of assets, rebalancing and withdrawing periodically. When a member of the pool dies, no funds are removed from the pool for a beneficiary. Instead, the dead member’s share of the combined funds is kept in the fund to benefit the remaining living members of the pool. In effect, the members of the pool are mutual beneficiaries, coming together to insure against longevity risk.

These pooled annuity funds can also be called tontines. The word tontine is used to describe a variety of insurance products that pool investors and are structured such that the death of one member benefits the remaining living members. Tontines are ideal for individuals looking to insure against longevity risk, especially those lacking a beneficiary. Obviously, there is concern for such a product to maliciously take advantage of death, ultimately providing a few members with obscene and unfair profits. So tontines must be approached with care. For a brief history of tontines, see McKeever (2009). For some ideas on the modern implementation and regulation of tontines, see Milevsky (2022).

The pooled annuity funds considered here offer investors an increase in the probability to complete a schedule of withdrawals. In other words, the probability available through the pool is larger than what an individual can achieve alone. For the problem of maximizing this probability, only closed pooled annuity funds are considered. In a closed fund, members can only receive the scheduled withdrawals without any exceptions.

Since this lack of liquidity is offputting to many investors, note that if living members are allowed to withdraw the present value of their initial contribution to the fund at any time, a remaining member’s share of the combined funds is at least as large as that present value. This is because a remaining member’s share takes into account the leftover contributions resulting from prior member deaths. A more detailed explanation of this logic, including how this present value is computed, is given in section 5. Allowing a living member to withdraw the present value of their contribution can result in a probability of withdrawal success that differs from the closed pool case. However, this probability will still be larger than if the member had decided to invest alone instead of join the pool, all the while following the same portfolio weights as the pool.

Introduce the filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where $\mathbb{F}:=\{\mathcal{F}(t)\}_{t\in T}$ denotes a filtration of $\mathcal{F}$ and $T\subset[0,\infty)$, $0\in T$. Consider $n$ assets available for investment, each denoted by an index from 1 to $n$. For each $j=1,2,...,n$, let $X_{j}:T\times\Omega\to(0,\infty)$ be an $\mathbb{F}$-adapted process. When convenient, write $X_{j}(t)$ in place of $X_{j}(t,\omega)$, understanding that $X_{j}(t)$ is an $\mathcal{F}(t)$-measurable function on $\Omega$. In this setting, $X_{j}(t)$ denotes the value of asset $j$ at time $t$. Let $\{t_{k}\}_{k=0}$ be an increasing sequence in $T$ with $t_{0}=0$. Require that for each $j$ and $t_{k}$, $\log X_{j}(t_{k+1})-\log X_{j}(t_{k})$ is independent of $\mathcal{F}(t_{k})$.

Introduce a pool of $A_{0}$ annuitants with independent and identically distributed life distributions. By agreeing to be in this pool, each annuitant invests $P>0$ into a closed pooled annuity fund at time $0$. The fund is then invested in the forementioned $n$ assets, rebalancing only at those times $t_{k}$. If an annuitant is alive at time $t_{k}$, where $k=1,2,...$, then the annuitant withdraws $w_{k}\geq 0$ from the fund at time $t_{k}$. No other withdrawals are allowed - not even on a dead annuitants behalf. Note that in this set-up, rebalancing can occur at time $t_{k}$ with $w_{k}=0$.

After accounting for $w_{k}$, denote the combined wealth in the fund at time $t_{k}$ with $W_{k}$. At each time $t_{k}$, rebalance $W_{k}$ according to the $\mathcal{F}(t_{k})$-measurable portfolio weight vector $\boldsymbol{\pi}_{k}:\Omega\to\Pi$, where $\Pi=\{\mathbf{p}\in[0,1]^{n}:\sum_{j=1}^{n}p_{j}=1\}$. When convenient, write $\boldsymbol{\pi}_{k}=(\pi_{k1},\pi_{k2},...,\pi_{kn})$, understanding that $\boldsymbol{\pi}_{k}$ and each $\pi_{kj}$ is an $\mathcal{F}(t_{k})$-measurable function on $\Omega$. In particular, at each time $t_{k}$, invest $\pi_{kj}W_{k}$ in asset $j$ for each $j=1,2,...,n$.

Track the number of annuitants as follows. Suppose an annuitant’s status as alive or dead at time $t_{k+1}$ depends only on the annuitant’s age and status at time $t_{k}$. Let $s$ denote the starting age of all annuitants, and let $d_{k}$ denote the probability of an annuitant aged $s+k$ at time $t_{k}$ dying by time $t_{k+1}$. For $i=1,2,...,A_{0}$ and $k=0,1,...$, define the $\mathcal{F}(t_{k+1})$-measurable random variable $B_{k}^{i}\sim\text{Bernoulli}(d_{k})$. Require that each $B_{k}^{i}$ is independent of $\mathcal{F}(t_{k})$ and $X_{j}(t)$ for every $j=1,2,...,n$ and $t\in T$. Then the number of living annuitants at time $t_{k}$ is given by $A_{k}$, where

To simplify notation, let $X_{jk}=X_{j}(t_{k+1})/X_{j}(t_{k})$ for $j=1,2,...,n$ and $k=0,1,...$. When convenient, write $\mathbf{X}_{k}=(X_{1k},X_{2k},...,X_{nk})$. Let $Y_{k}=\sum_{j=1}^{n}\pi_{kj}X_{jk}$ for $k=0,1,...$. Then the pooled wealth at time step $t_{k}$ is given by $W_{k}$, where the $W_{k}$ are computed recursively via

Again, note that wealth at time step $t_{k}$ is computed after accounting for the withdrawal of $w_{k}$ by all living annuitants at time step $t_{k}$. Furthermore, observe that each $X_{j,k-1}$, $Y_{k-1}$ and $W_{k}$ is an $\mathcal{F}(t_{k})$-measurable function on $\Omega$.

Observe that $W_{k}(\omega)<0$ implies $W_{k+1}(\omega)<0$ for each $\omega\in\Omega$. This guarantees that a failure to execute the scheduled withdrawals up to time $t_{k}$, indicated by $W_{k}(\omega)<0$, will be carried over to time $t_{k+1}$ and indicated by $W_{k+1}(\omega)<0$.

Observe that $W_{k}$ is a function of each $\boldsymbol{\pi}_{i}$ for $i=0,1,...,k-1$. The notation

is used to denote the supremum of $W_{k}$ over all $\mathcal{F}(t_{i})$-measurable portfolio weight vectors $\boldsymbol{\pi}_{i}$, where $i=0,1,...,k-1$. This kind of abbreviation is used in similar situations where there is a $W_{k}$-like function that is constructed using the $\mathcal{F}(t_{i})$-measurable $\boldsymbol{\pi}_{i}$.

Use $\mathbb{E}[\ \cdot\ ]$ to denote the expectation with respect to $(\Omega,\mathcal{F},\mathbb{P})$. Denote the smallest $\sigma$-algebra containing the family of sets $S$ with $\sigma(S)$. Use $\mathbb{R}$ to denote the real numbers, and let $\mathbb{N}_{0}=\{0,1,2,...\}$. Given $u:\mathbb{R}\to\mathbb{R}$ and $Z:\Omega\to\mathbb{R}$, use $u(Z)$ to denote $u\circ Z$. Given sets $\Psi$, $I$ and $\{(f_{i}:\Psi\to\mathbb{R}):i\in I\}$, use $(\sup_{i\in I}f_{i}):\Psi\to\mathbb{R}$ to denote the pointwise supremum of the $f_{i}$, meaning for each $\psi\in\Psi$, $(\sup_{i\in I}f_{i})(\psi)=\sup_{i\in I}(f_{i}(\psi))$. Let $\mathbf{1}=(1,1,...,1)$ denote the $n$-dimensional vector of 1s, and use $\cdot$ to indicate the dot product.

Introduce the recursion starting with

and for $k=1,2,...$,

Here, $I_{k}$ is a flag indicating whether a particular annuitant is alive ($I_{k}=1$) or dead ($I_{k}=0$) at time $t_{k}$. In this setup, $\widehat{W}_{k}=W_{k}$ as long as $I_{k}=1$. If $I_{k}=0$, then $\widehat{W}_{k}\geq 0$ iff $\widehat{W}_{k-1}\geq 0$. In words, $\widehat{W}_{k}$ is non-negative if and only if the combined funds in the pool are not exhausted after executing the scheduled withdrawals for those times, up to and including $t_{k}$, at which the annuitant is alive.

Define $\tau:\Omega\to\mathbb{N}_{0}$ such that

Then $\tau$ indicates the index of the first time step at which the annuitant has expired. The goal is to find

In words, (10) is the sumpremal probability of the annuitant completing the schedule of withdrawals until death, and the supremum is taken over the portfolio vectors $\boldsymbol{\pi}_{i}$, where $i=0,1,...,\tau-1$.

By the law of total probability,

Observe from (3) that if $k\geq\tau(\omega)$, then

It follows that

If $\mathbb{P}(\tau>k)=0$, then (5) and (6) coincide. Alternatively, $k$ can be chosen large enough such that $\mathbb{P}(\tau>k)$ is sufficiently close to $0$, in which case (5) is approximated by (6). It follows that for $k$ sufficiently large,

From here, the goal is to compute the left side of (7).

First define the function $v_{k}:\mathbb{R}\times\mathbb{N}_{0}\to[0,1]$ such that

Let $v_{i}:\mathbb{R}\times\mathbb{N}_{0}\to[0,1]$, $i=0,1,...,k-1$, denote the functions satisfying

For each $i=0,1,...,k-1$ and $a\in\mathbb{N}_{0}$, $v_{i}(x,a)$ is non-decreasing and upper semicontinuous over $x\in\mathbb{R}$. Moreover, the left side of (7) is given by $v_{0}(W_{0},A_{0})$, which can be computed recursively, starting with (8) and then using (9). Proofs of the previous statements are only slight modifications of the proofs given in Brown (2023) for the unpooled case, so they are left out. A simple inductive argument also shows that $v_{i}(x,a)=0$ for $i=0,1,...,k$, $x<0$ and $a\in\mathbb{N}_{0}$. Furthermore, $v_{i}(x,0)=v_{k}(x,0)$ for $i=0,1,...,k$ and $x\in\mathbb{R}$.

In a similar fashion, it is possible to maximize the probability for all annuitants to complete the schedule of withdrawals until death. First define $\widetilde{\tau}:\Omega\to\mathbb{N}_{0}$ such that

Then $\widetilde{\tau}$ indicates the index of the first time step at which all members of the pool have expired. The goal is to find

In words, (10) is the sumpremal probability for every annuitant to complete the schedule of withdrawals until death, and the supremum is taken over the portfolio vectors $\boldsymbol{\pi}_{i}$, where $i=0,1,...,\tau-1$. Like before, a sufficiently large $k$ gives

Using the same kind of logic as before, it follows that the left side of (11) is given by $\widetilde{v}_{0}(W_{0},A_{0})$, where $\widetilde{v}_{k}:\mathbb{R}\times\mathbb{N}_{0}\to[0,1]$ is such that $\widetilde{v}_{k}=v_{k}$, and for $i=0,1,...,k-1$, $\widetilde{v}_{i}:\mathbb{R}\times\mathbb{N}_{0}\to[0,1]$ is such that

Like in Brown (2023), theoretical results are applied in the case where $n=2$. The two assets used are the S&P Composite Index and an inflation-protected bond. The annual real returns of the S&P Composite Index appear to be historically stable in the sense that they are approximately iid (Normal with mean 1.083 and standard deviation .1753) over the past 150 years (Brown, 2023). For $n>2$ the optimization is more difficult to compute, and it is not easy to find other assets like the S&P Composite Index that have such historical stability.

To construct the mortality distribution of annuitants, applications use the 2017 per-age death rates of the US Social Security area population. Section 4.1 describes the S&P Composite Index data and the mortality distribution data. Section 4.2 details the set-up needed to apply theoretical results and describes the algorithms used in applications. Results of applications are given in section 4.3.

Applications show that there is a worthwhile benefit to be gained from joining a pooled annuity fund instead of trying to complete a schedule of withdrawals independently. Moreover, the size of the pool does not have to be very large to reap most of the benefit available. If such a pooled annuity fund ever becomes available to the public, the provider will only need a small number of individuals, say 20, to establish an attractive pool.

Here, only a single contribution, followed by a schedule of withdrawals, is studied. Note that it is also possible to study multiple contributions, occurring at different times, preceding the schedule of withdrawals. For example, each annuitant could make the same annual contribution for 20 years, and then the withdrawal schedule could start after that. This falls more in line with how pensions are structured. However, the heterogeneity of employees contributing to the same pension fund can complicate the kind of analysis conducted here. The emphasis placed here on a single contribution at time 0 is because it appeals directly to those retirement-age individuals looking to insure against longevity risk. The multiple contribution case appeals more to younger individuals planning for retirement.

In applications, the benefit of using optimal instead of constant portfolio weights was demonstrated for a special case. Remaining fully invested in the S&P Composite Index gave a withdrawal success probability that was close to the optimum. On one hand, the difference between the two can be noticeable - as much as .04. On the other hand, this difference is relatively small and may not be considered worth the extra effort that comes with implementing the optimal portfolio weights. Out of curiosity, the author tested the effect of changing the mean and standard deviation of S&P annual returns on this apparent closeness in the success probabilities between the portfolio that remains fully invested in the S&P Composite Index and the optimal portfolio. The closeness remained. So if it is difficult to get annuitants to buy in to this idea of optimal portfolio weights, fund providers can rest somewhat easily knowing that annuitants will not miss out on much by investing in just the S&P Composite Index.

One downside of the pooled annuity funds considered here is that they are closed. This lack of liquidity could easily turn away an interested individual. If living members are allowed the option of withdrawing all of the present value of their initial contribution from the combined funds, at any time, then this can change the probability of withdrawal success for the members who choose to remain in the pool until death. Future research could study the effect of this option on the success probability for those members who choose to remain in the pool until death. Regardless of whether this option is allowed, the success probability for an individual who remains in a pool until death is at least as high as if the individual invests independently, but follows the same portfolio weights as the pool. So if the pool is going all-in on the S&P Composite Index, and this option is allowed, the success probability for an individual who remains in a pool until death is at least as high as if the individual invests independently in just the S&P Composite Index. The logic is as follows.

At time $t_{k}$, the present value of a living member’s contribution to the combined funds is given by $\max\{0,\widetilde{W}_{k}\}$, where

For $t\in(t_{k},t_{k+1})\cap T$, the present value is

Use $\overline{A}_{k}$ to denote the number of living members in the pool at time $t_{k}$, where members can only join the pool at time 0, but they can leave the pool by either dying or exercising the forementioned option. Use $\overline{W}_{k}$ to denote the resulting value of the combined funds at time $t_{k}$. Then since at most $A_{k-1}-A_{k}$ members can exercise the option between times $t_{k-1}$ and $t_{k}$,