2. Problem setting

In order to be consistent with practitioner literature, we will consider the scenario set out in Bengen (Reference Bengen1994). This scenario posits that the retiree desires fixed (real) minimum cash flows and that the cash flows are desired over a fixed planning horizon. The investor’s primary concern is that of exhausting savings during the planning horizon. We conjecture that the reason that the Bengen rule continues to be very popular in practice is that it directly addresses the typical concerns of retirees (see, e.g., Ameriks et al., Reference Ameriks, Veres and Warshawsky2001; Scott et al., Reference Scott, Sharpe and Watson2009; Pfau, Reference Pfau2015; Ruthbah, Reference Ruthbah2022; Daily et al., Reference Daily, Palmer and Mizell2023).

It may seem counterintuitive to use a fixed, relatively long-term planning horizon (usually 30 years for a 65 year old retiree). Based on current mortality tables, the probability of a 65-year-old Canadian male reaching the age of 95 is about $0.13$ . A 95-year-old male has only a one in six chance of reaching his $100{\text{th}}$ birthday.

However, surveys show that retirees fear exhausting their savings more than death (Hill, Reference Hill2016). Along these lines, Pfau (Reference Pfau2018) writes:

1. Play the long game. A retirement income plan should be based on planning to live, not planning to die. A long life will be expensive to support, and it should take precedence over death planning.

Therefore, use of a long, fixed planning horizon has become the default test of the risk of running out of funds.

It is also of interest to not only estimate the probability of ruin but also the size of the shortfall. To this end, we allow the retiree to continue to withdraw the minimum desired cash flows under a stochastic scenario where savings are exhausted. This debt accumulates at the borrowing rate. This allows us to measure the size of the shortfall for this scenario. We use the mean of the worst 5% of the outcomes as a quantitative measure of shortfall. A negative shortfall signals that the retiree has run out of cash and has an accumulated debt. An accumulated debt of one dollar at age 95 is certainly less concerning than a debt of $100,000 at that time. Although we are measuring the shortfall at the same point in time, the case with a more negative shortfall is likely due to having run out of money earlier and having debt accumulate. This can occur due to the forced minimum annual withdrawals. Effectively, the accumulated debt in this case penalizes strategies which run out of cash early during the planning horizon.

How would this work in practice? Basically, we assume that the investor divides his wealth into mental accounts, containing funds intended for different purposes (e.g., current spending or future needs). The standard life cycle model assumes that all wealth is completely fungible. In contrast, the behavioral approach posits that all wealth is not fungible and that mental bucketing is commonplace (Shefrin and Thaler, Reference Shefrin and Thaler1988). In particular, we will assume that the investor has mortgage-free residential real estate, which is in a separate mental account. This real estate is to be considered a hedge of last resort, if needed.Footnote 8 If the investments work out well, or if the retiree passes away, this real estate can be considered as a bequest.

It is commonplace in actuarial applications to mortality-weight cash flows. While this is clearly appropriate for annuity providers, it does not seem to be very informative for an individual retiree. Consider the perspective of a 65-year-old male with median life expectancy of about 87. The standard mortality-weighting approach would weight the minimum cash flows at 22 years after retirement by one half. However, if the retiree is planning to live rather than planning to die, he needs the entire minimum cash flow at age 87, not half of it.

As noted above, we assume that the investor trades off the reward of total real withdrawals over the 30-year horizon with the risk of expected shortfall at the end of the horizon. This risk/reward tradeoff is reminiscent of the tradeoff between expected return and standard deviation from traditional portfolio theory, but with different measures of risk and reward. An obvious alternative would be to specify a utility function. Most prior academic studies involving various forms of tontines have done so, typically assuming constant relative risk aversion (CRRA) utility (see, e.g., Milevsky and Salisbury, Reference Milevsky and Salisbury2015, Reference Milevsky and Salisbury2016; Bernhardt and Donnelly, Reference Bernhardt and Donnelly2019; Chen et al., Reference Chen, Hieber and Klein2019, Reference Chen, Hieber and Rach2021). However, we believe it is useful to consider an objective function based on the expected withdrawal, expected shortfall criteria. First, utility functions in principle should include all of the investor’s wealth, but this would be incompatible with the mental accounts framework discussed above. Second, related to the inclusion of the entire amount of the investor’s wealth, utility functions often have infinite marginal utility of wealth at zero. This means that the investor would always avoid reducing wealth to zero.Footnote 9 However, this is incompatible with a minimum withdrawal constraint: if the investor must withdraw some funds each year, there is inevitably some chance of insolvency if the investor survives long enough. Third, we believe that in practice it is easier to communicate with retired clients if the discussion is framed in terms of monetary amounts, which can be directly compared to the value of a residential real estate hedge.

3. Overview of individual tontine accounts

3.1 Intuition

We give a brief overview of modern tontines in this section. We restrict attention to the case of an individual tontine account (Fullmer, Reference Fullmer2019), which is a constituent of a perpetual tontine pool. Consider a pool of m investors, who are alive at time $t_{i-1}$ . Let $v_i^j$ be the balance in the portfolio of investor j at time $t_i$ . We assume that $v_i^j \geq 0$ . In a tontine, if investor j participates in a tontine pool in time interval $(t_{i-1}, t_i)$ , and investor j dies in that interval, then her portfolio $v_i^j$ is forfeited and given to the surviving members of the pool in the form of mortality credits (gains). Suppose that the probability that j dies in $(t_{i-1}, t_i)$ is $q_{i-1}^j$ .

Consider tontine members $j=1, \dots, m$ who are alive at $t_{i-1}$ . Let

(3.1) \begin{align} {\bf{1}}^j_i &= \begin{cases} 1 & \text{ Investor }\textit{j} \text{ is alive at } t_{i-1} \text{ and alive at } t_i \\ 0 & \text{ Investor $\textit{j}$ is alive at $t_{i-1}$ and dead at } t_i \\ \end{cases} \nonumber \\ E_{i-1} \big[{\bf{1}}^j_i\big] &= 1 -q_{i-1}^j, \end{align}

where $E_{i-1}[\cdot]$ denotes an expectation operator conditional on mortality information known at $t_{i-1}$ .

Let the tontine gain (mortality credit) for investor j, conditional on ${\bf{1}}_i^j = 1$ , for the period $(t_{i-1}, t_i)$ , paid out at time $t_i$ , be denoted by $c_i^j$ . The tontine will be a fair game if, for each player j, the expected gain from participating in the tontine is zero,Footnote 10

(3.2) \begin{equation} - v_i^j E_{i-1}[1 - {\bf{1}}^j_i] + E_{i-1}\big[{\bf{1}}^j_i\big] \cdot \mathbb{E}_{i-1}^j \big[c_i^j\big] = 0,\end{equation}

where

(3.3) \begin{equation}\mathbb{E}_{i-1}^j [\cdot] \equiv E_{i-1} \Big[\cdot \big \vert {\bf{1}}_i^j = 1;\, \big\{v_i^j\big\}_{j=1}^m\Big]\end{equation}

that is conditional on member j being alive at $t_i$ , with known (realized) values of investment accounts $\{v_i^j\}_{j=1}^m$ . This is because the member enters into the pool at $t_{i-1}$ , with unknown mortality states of the other members of the pool at $t_i$ . However, the mortality credits are divided up based on the realized investment accounts at $t_i$ . From Equation (3.2), this gives

(3.4) \begin{equation}\mathbb{E}_{i-1}^j\big[c_i^j\big] = \left(\overbrace{\frac{q_{i-1}^j}{1-q_{i-1}^j}}^{\textit{Gain rate}} \right) v_i^j .\end{equation}

We emphasize that $v_i^j$ are unknown at $t_{i-1}$ since we allow investment in risky assets. However, in terms of fairness, we only require that the realized investment proceeds are reallocated fairly, taking into account random mortality.

Note that the right-hand side of Equation (3.4) is independent of the other members’ accounts in the pool, their investment strategies, and their mortality status. This is surprising and counterintuitive, as the expectation operator on the left-hand side as defined in Equation (3.3) does depend on the values of other members’ accounts and implicitly their investment strategies, since these determine the account values at the end of the period. In fact, it is easy to construct somewhat pathological cases where Equation (3.4) will fail. For example, suppose that every investor other than j invests their entire account value in losing lottery tickets, driving their account values to zero. If every other investor’s account value is zero, that would include the accounts of investors who pass away during the period and there would be no tontine gains to be distributed to investor j. Underlying the apparent independence of the right-hand side of Equation (3.4) from the values of other members’ accounts are implicit assumptions that any member’s gain is small compared to the overall expected amount of mortality credits available and that the pool is sufficiently large so that the realized amount of mortality credits is approximately equal to the expected amount. This small bias condition is stated more precisely below (see Condition 3.1) with additional discussion in Remark 3.4.

We also have the budget rule that the total of the tontine gains distributed is equal to the total amounts forfeited

(3.5) \begin{equation}\sum_{j=1}^m {\bf{1}}^j_i c_i^j = \sum_{j=1}^m \left(1 -{\bf{1}}_i^j\right) v_i^j .\end{equation}

Let

(3.6) \begin{equation} \Omega_i = \left\{ \left\{{\bf{1}}^j_i\right\}_{j=1}^{m}, \left\{v_i^j\right\}_{j=1}^{m} \right\}.\end{equation}

In general $c_i^j = c_i^j(\Omega_i \big\vert {\bf{1}}_i^j = 1)$ .

Remark 3.1 (No tontine gains to members who have died). Note that Equation (3.2) assumes that no tontine gains accrue to members who have just died. This will maximize tontine gains of survivors, which is the focus of the current study. Several previous studies have specified payments to the estates of members who die in $(t_{i-1},t_i)$ (see, e.g., Bernhardt and Donnelly, Reference Bernhardt and Donnelly2018; Hieber and Lucas, Reference Hieber and Lucas2022; Denuit et al., Reference Denuit, Hieber and Robert2022).

We can always write $c_i^j $ as

(3.7) \begin{equation}c_i^j = \left(\frac{q_{i-1}^j}{1-q_{i-1}^j}\right)\, v_i^j \, H_i^j\left( \Omega_i \big\vert {\bf{1}}_i^j=1\right), \end{equation}

for some function $H_i^j( \Omega_i \big\vert {\bf{1}}_i^j = 1)$ . Then, the fairness condition (3.4) becomes

(3.8) \begin{equation}\mathbb{E}_{i-1}^j \left[H_i^j\right] = 1 .\end{equation}

It is also desirable to impose the condition that in each period, a surviving investor is never made worse off by participating in a tontine pool, that is the tontine gain is non-negative

(3.9) \begin{equation}H_i^j\left(\Omega_i \big\vert {\bf{1}}_i^j = 1\right) \geq 0 .\end{equation}

It is convenient to summarize the essential tontine properties. These properties are largely the same as in Hieber and Lucas (Reference Hieber and Lucas2022), with the exception that there are no payments to the estates of deceased members.

Property 3.1 (Tontine Properties). The desirable properties of a tontine are

1. (i) Fairness: $\mathbb{E}_{i-1}^j [H_i^j] = 1\, ;\, j=1,\dots,m$ (see Equation (3.8)).

2. (ii) Budget constraint (3.5).

3. (iii) Tontine gain non-negativity: $H_i^j \geq 0\, ;\, j=1,\dots,m$ .

Sharing rules which satisfy Property 3.1 are discussed in Sabin (Reference Sabin2010, Reference Sabin2011), and sharing rules for the case where payments are made to just deceased members are described in, for example, Bernhardt and Donnelly (Reference Bernhardt and Donnelly2018), Hieber and Lucas (Reference Hieber and Lucas2022), and Denuit et al. (Reference Denuit, Hieber and Robert2022), and the references therein.

3.2 A simplified approach

One of the downsides of sharing rules which exactly satisfy Property 3.1 for finite-sized pools is that these rules are somewhat complex. Many of these exact rules require processing deaths one at a time. It is argued in Sabin and Forman (Reference Sabin and Forman2016) and Fullmer and Sabin (Reference Fullmer and Sabin2019) that complex sharing rules can impede consumer acceptance. These authors argue that it is sufficient to have simple rules which satisfy Property 3.1(i)–(iii) in the limit of large, perpetual pools.

3.2.1 Group gain

In the terminology of Sabin and Forman (Reference Sabin and Forman2016), we define the group gain $G_i$ as

(3.10) \begin{equation} G_i = \frac{\sum_k \left(1 - {\bf{1}}_i^k\right) v_i^k } {\sum_k {\bf{1}}_i^k \left(\frac{ q_{i-1}^k}{ 1- q_{i-1}^k } \right) v_i^k}.\end{equation}

Note that the $G_i$ is the same for all members j. $G_i$ has the convenient interpretation as being the ratio of the total realized mortality credits to the total expected credits for survivors.

Sabin and Forman (Reference Sabin and Forman2016) suggest the following simplified sharing rule

(3.11) \begin{equation}c_i^j = \left(\frac{q_{i-1}^j}{1 - q_{i-1}^j} \right) v_i^j \, G_i,\end{equation}

which uses the group gain $G_i$ in place of the function $H_i^j$ in Equation (3.7). By assumption, $v_i^j \geq 0$ , and hence, Property 3.1(iii) is satisfied.

The total tontine gains are

(3.12) \begin{align}\text{Total Tontine Gains} &= \sum_{j=1}^m {\bf{1}}^j_i c_i^j = G_i \sum_{j=1}^m {\bf{1}}^j_i \left(\frac{ q_{i-1}^j}{1 - q_{i-1}^j} \right) v_i^j \nonumber \\ &= \sum_{j=1}^m \left(1 - {\bf{1}}_i^j\right) v_i^j\, = \text{ Total Forfeited} \, ;\end{align}

hence, Property 3.1(ii) is satisfied.

In essence, the group gain $G_i$ in Equation (3.10) is a scaling factor to adjust for actual deaths compared to expected deaths, as suggested in Piggott et al. (Reference Piggott, Valdez and Detzel2005) and Qiao and Sherris (Reference Qiao and Sherris2013).

Remark 3.2 $\left(\sum_{j=1}^m {\bf{1}}^j_i =0\right)$ Note that Equation (3.10) is undefined if all members die in $(t_{i-1},t_i)$ . We assume that the tontine is large (in terms of members) and perpetual (i.e., open to new members), so that the probability of all members dying in a period is negligible. For mathematical completeness, we can suppose that if all members die in $(t_{i-1},t_i)$ , we collapse the tontine and distribute all remaining account values $v_i^j$ to the estates of members j.

However, in general Property 3.1(i) will not be satisfied for a finite-sized pool, that is

(3.13) \begin{equation}\exists j\, s.t.\, \mathbb{E}_{i-1}^j \left[ G_i \big\vert {\bf{1}}_i^j = 1\right] < 1,\, \, j \in \{1,\cdots,m \}.\end{equation}

This means that there is a bias that favors some members over others; that is, some members have a negative expected gain, implying that other members must have a positive expected gain. This is illustrated in Winter and Planchet (Reference Winter and Planchet2022), using an example with a pool consisting of a large number of young investors (with small individual portfolios), and a single elderly member with a large portfolio. The elderly member effectively subsidizes the younger members. Sabin and Forman (Reference Sabin and Forman2016) show that the bias is negligible under the following conditions:

Condition 3.1 (Small bias condition). Suppose that:

1. (a) the pool of participants in the tontine is sufficiently large, and

2. (b) the expected amount forfeited by all members is large compared to any member’s nominal gain, that is

   (3.14) \begin{equation}\left(v_i^j \frac{q_{i-1}^j}{1-q_{i-1}^j}\right) \ll \sum_k q_{i-1}^k v_i^k\, \, ;\, \, j=1,\dots, m .\end{equation}

Then the bias is negligibly small (Sabin and Forman, Reference Sabin and Forman2016).

Equation (3.14) is essentially a diversification requirement: no member of the pool has an abnormally large share of the total pool capital. In addition, of course, if the pool is sufficiently large, then the actual number of deaths in $(t_{i-1}, t_i)$ will converge to the expected number of deaths (ignoring systematic mortality risk).

In Fullmer and Sabin (Reference Fullmer and Sabin2019), simulations were carried out to determine the magnitude of the volatility of $G_i$ under practical sizes of tontine pools. Given a tontine pool of 15,000 members, with varying ages, initial capital, and randomly assigned investment policies (i.e., the bond/stock split), the simulations showed that $E[G_i] \simeq 1$ and that the standard deviation was about $0.1$ . This standard deviation at each $t_i$ actually resulted in a smaller effect over a long term (assuming that the tontine member lived long enough). This is simply because everybody dies eventually, so that if fewer deaths than expected are observed in a year, then more deaths will be observed in later years, and vice versa. More detailed analysis of the probability density of $G_i$ is given in Denuit and Vernic (Reference Denuit and Vernic2018), with a slightly different use of the factor $q_i^j$ .

Henceforth, we will assume that the pool is sufficiently large and that it satisfies the diversity condition (3.14), so that there is no significant error in assuming that $G_i \equiv 1$ . More precisely, our optimal control problem will be formulated assuming that

(3.15) \begin{equation}c_i^j = \left(\frac{q_{i-1}^j}{1-q_{i-1}^j}\right) v_i^j .\end{equation}

As a sanity check, we also carry out a test where we simulate the effect of randomly varying G, based on the statistics of the simulations in Fullmer and Sabin (Reference Fullmer and Sabin2019). Our results show that the effect of randomness in G can be safely ignored for a reasonably sized tontine pool. To be more precise, we will modify Equation (3.15) so that

(3.16) \begin{equation}c_i^j = \left(\frac{q_{i-1}^j}{1-q_{i-1}^j}\right) v_i^j \hat{G}_i .\end{equation}

where $\hat{G}_i$ is a random variable. We will give a numerical example showing the effects of randomness of $\hat{G}_i$ in Monte Carlo simulations. However, our computation of the optimal strategy will always assume $G_i \equiv 1$ .

For notational convenience, we define the tontine gain rate at $t_i$ for investor j as

(3.17) \begin{equation}\left(\mathbb{T}_i^g \right)^j = \left(\frac{q_{i-1}^j}{1-q_{i-1}^j}\right) .\end{equation}

In our optimal control formulation, we will typically drop the superscript j from Equation (3.17),

(3.18) \begin{equation}\mathbb{T}_i^g = \left(\frac{q_{i-1}}{1-q_{i-1}}\right),\end{equation}

since we will consider a given investor j with conditional mortality probability of $q_{i-1}$ in $(t_{i-1}, t_i)$ . Using this notation, Equation (3.15) becomes (for a fixed investor j)

(3.19) \begin{equation}c_i = \mathbb{T}_i^g v_i .\end{equation}

Remark 3.3 (Other sharing rules which satisfy Property 3.1(i)--(iii)). We have used sharing rule (3.11) as an example of a practical scheme, which only approximately satisfies Property 3.1(i)–(iii). However, our optimal control formulation will also apply to any sharing rule which satisfies Property 3.1(i)–(iii), provided that the pool is large enough so that $\text{var}[ H_i^j]$ is small, where $\text{var}[\cdot]$ denotes variance.

Remark 3.4 (Effect of investment decisions of other members in the pool). Provided the small bias condition (Condition 3.1) holds, it does not matter what investment strategy is followed by any given investor in period $(t_{i-1}, t_i)$ . Each investor can choose whatever policy they like, since only the observed final portfolio value at $t_i$ matters.Footnote 11 At first glance, the idea that the investment strategies of other pool members do not affect the strategy of the individual investor seems counterintuitive. However, it is important to realize that it is the reallocation of the realized forfeiture amounts that matters. We provide some brief intuition here; for further discussion, see Fullmer and Sabin (Reference Fullmer2019) and Winter and Planchet (Reference Winter and Planchet2022) and references cited therein.

Assuming that Equation (3.8) holds with sufficient accuracy (and with the group gain $G_i$ in place of the function $H_i^j$ ), then each member’s expected gain will be given by Equation (3.4). Consider an aggressive investor in a pool dominated by conservative investors. Assume that the aggressive investor has a larger account balance than the conservative investors. Since the aggressive investor’s stake is larger, she will get a larger share of the (smaller) forfeitures. In addition, if $\text{var}(G_i)$ is small, then the realized mortality credits will be close to the expected value (3.4). Moreover, we will show that the optimal strategy for an individual investor (computed assuming $G_i \equiv 1$ ) is robust to volatile $G_i$ .

3.6 Survivor benefits

Many DB plans have survivor benefits which are received by a surviving spouse. A typical case would involve the surviving spouse receiving 60–75% of the yearly pension after the DB plan holder dies.

Consider the following case of a male, same-sex couple, both of whom are exactly the same age. As an extreme case, suppose the survivor benefit is 100% of the tontine cash flows, which continue until the survivor dies. From the CPM2014 table from the Canadian Institute of Actuaries,Footnote 13 the probability that an 85-year old Canadian male dies before reaching the age of 86 is about $0.076$ . Assuming that the mortality probabilities are independent for both spouses, then the probability that both 85-year old spouses die before reaching age 86, conditional on both living to age 85 is $(0.076)^2 \simeq 0.0053$ . From Equation (3.17), the tontine gain rate per year is

(3.20) \begin{equation} \text{ tontine gain rate } = \frac{0.0053}{1-0.0053} \simeq 0.0053 .\end{equation}

We will assume in our numerical examples that the base case fee charged for managing the tontine is 50 bps per year. This means that net of fees, there are essentially no tontine gains for our hypothetical couple for the first 20 years of retirement, which is surely undesirable. Once one of the partners passes away, the tontine gain rate will, of course, take a jump in value.

As another extreme case, suppose that the surviving spouse receives 50% of the tontine cash flows. In this case, the total cash flows accruing to this couple are exactly the same as those that would have resulted from dividing the original total wealth in half and then having each spouse invest in their own individual tontine.

It is possible to determine the distribution of the cash flows for a survivor benefit which is intermediate to these edge cases. However, this requires additional state variables in our optimal control problem and is probably best tackled using a machine learning approach (Li and Forsyth, Reference Li and Forsyth2019; Ni et al., Reference Ni, Li, Forsyth and Caroll2022). We will leave this case for future work and focus attention on the individual tontine case with no survivor benefit. Note that in the tontine context, survivor benefits are typically provided by a separate insurance overlay.Footnote 14

4. Formulation

We assume that the investor has access to two funds: a broad market stock index fund and a constant maturity bond index fund. The investment horizon is T. Let $S_t$ and $B_t$ respectively denote the real (inflation-adjusted) amounts invested in the stock index and the bond index, respectively. In general, these amounts will depend on the investor’s strategy over time, as well as changes in the real unit prices of the assets. In the absence of an investor determined control (i.e., cash withdrawals or rebalancing), all changes in $S_t$ and $B_t$ result from changes in asset prices. We model the stock index as following a jump diffusion.

In addition, we follow the usual practitioner approach and directly model the returns of the constant maturity bond index as a stochastic process (see, e.g., MacMinn et al., Reference MacMinn, Brockett, Wang, Lin, Tian, Mitchell, Maurer and Hammond2014; Lin et al., Reference Lin, MacMinn and Tian2015). Consistent with the stock index, we will assume that the constant maturity bond index also follows a jump diffusion. Empirical justification for this can be found in Forsyth et al. (Reference Forsyth, Vetzal and Westmacott2022), Appendix A. This will also be discussed in Section 9.

Let $ S_{t^-} = S(t - \epsilon), \epsilon \rightarrow 0^+$ , that is $t^-$ is the instant of time before t, and let $\xi^s$ be a random number representing a jump multiplier. When a jump occurs, $S_t = \xi^sS_{t^-}$ . Allowing for jumps permits modeling of non-normal asset returns. We assume that $\log(\xi^s)$ follows a double exponential distribution (Kou, Reference Kou2002; Kou and Wang, Reference Kou and Wang2004). If a jump occurs, $u^s$ is the probability of an upward jump, while $1-u^s$ is the chance of a downward jump. The density function for $y = \log (\xi^s)$ is

(4.1) \begin{equation}f^s(y) = u^s \eta_1^s e^{-\eta_1^s y} {\bf{1}}_{y \geq 0} + (1-u^s) \eta_2^s e^{\eta_2^s y} {\bf{1}}_{y < 0}.\end{equation}

We also define

(4.2) \begin{equation}\gamma^s_{\xi} = E[ \xi^s -1 ] = \frac{u^s \eta_1^s}{\eta_1^s - 1} + \frac{(1 - u^s)\eta_2^s}{\eta_2^s + 1} - 1 .\end{equation}

In the absence of control, $S_t$ evolves according to

(4.3) \begin{equation}\frac{dS_t}{S_{t^-}} = \left(\mu^s -\lambda_\xi^s \gamma_{\xi}^s \right) \, dt + \sigma^s \, d Z^s + d\left( \displaystyle \sum_{i=1}^{\pi_t^s} \left(\xi_i^s -1\right) \right), \end{equation}

where $\mu^s$ is the (uncompensated) drift rate, $\sigma^s$ is the volatility, $d Z^s$ is the increment of a Wiener process, $\pi_t^s$ is a Poisson process with positive intensity parameter $\lambda_\xi^s$ , and $\xi_i^s$ are i.i.d. positive random variables having distribution (4.1). Moreover, $\xi_i^s$ , $\pi_t^s$ , and $Z^s$ are assumed to all be mutually independent.

Similarly, let the amount in the bond index be $B_{t^-} = B(t -\epsilon), \epsilon \rightarrow 0^+$ . In the absence of control, $B_t$ evolves as

(4.4) \begin{equation}\frac{dB_t}{B_{t^-}} = \left(\mu^b -\lambda_\xi^b \gamma_{\xi}^b + \mu_c^b {\bf{1}}_{\left\{B_{t^-} < 0\right\}} \right) \, dt + \sigma^b \, d Z^b + d\left( \displaystyle \sum_{i=1}^{\pi_t^b} \left(\xi_i^b -1\right) \right), \end{equation}

where the terms in Equation (4.4) are defined analogously to Equation (4.3). In particular, $\pi_t^b$ is a Poisson process with positive intensity parameter $\lambda_\xi^b$ , and $\xi_i^b$ has distribution

(4.5) \begin{equation}f^b\left( y= \log \xi^b\right) = u^b \eta_1^b e^{-\eta_1^b y} {\bf{1}}_{y \geq 0} + \left(1-u^b\right) \eta_2^b e^{\eta_2^b y} {\bf{1}}_{y < 0},\end{equation}

and $\gamma_{\xi}^b = E[ \xi^b -1 ]$ . $\xi_i^b$ , $\pi_t^b$ , and $Z^b$ are assumed to all be mutually independent. The term $\mu_c^b{\bf{1}}_{\{B_{t^-} < 0\}}$ in Equation (4.4) represents the extra cost of borrowing (the spread).

The diffusion processes are correlated, that is $d Z^s \cdot d Z^b =\rho_{sb}\, dt$ . The stock and bond jump processes are assumed mutually independent. See Forsyth (Reference Forsyth2020b) for justification of the assumption of stock-bond jump independence.

We define the investor’s total wealth at time t as

(4.6) \begin{equation}\text{Total wealth } \equiv W_t = S_t + B_t.\end{equation}

The term “total wealth” refers to the sum of the values of the investor’s tontine account plus any accumulated debt arising from insolvency due to the minimum required withdrawals. This is perhaps a bit misleading since it excludes the value of any additional assets that the investor has such as real estate or the value of government benefits, but it simplifies our exposition. We impose the constraints that (assuming solvency) shorting stock and using leverage (i.e., borrowing) are not permitted. As noted above, in case of insolvency, the portfolio is liquidated, trading stops and debt accumulates at the borrowing rate.

5. Notational conventions

Consider a set of discrete withdrawal/rebalancing times $\mathcal{T}$

(5.1) \begin{equation} \mathcal{T} = \{t_0=0 <t_1 <t_2< \ldots <t_M=T\} \end{equation}

where we assume that $t_i - t_{i-1} = \Delta t =T/M$ is constant for simplicity. To avoid subscript clutter, in the following, we will occasionally use the notation $S_t \equiv S(t), B_t \equiv B(t)$ and $W_t \equiv W(t)$ . Let the inception time of the investment be $t_0 =0$ . We let $\mathcal{T} $ be the set of withdrawal/rebalancing times, as defined in Equation (5.1). At each rebalancing time $t_i$ , $i =0, 1, \ldots, M-1$ , the investor (i) withdraws an amount of cash $\mathfrak{q}_i$ from the portfolio and then (ii) rebalances the portfolio. At $t_M =T$ , the portfolio is liquidated and no cash flow occurs. For notational completeness, this is enforced by specifying $\mathfrak{q}_M = 0$ .

In the following, given a time dependent function f(t), we will use the shorthand notation

(5.2) \begin{equation}f\left(t_i^+\right) \equiv \displaystyle \lim_{\epsilon \rightarrow 0^+} f(t_i + \epsilon) \, \, ; \, \, f\left(t_i^-\right) \equiv \displaystyle \lim_{\epsilon \rightarrow 0^+} f(t_i - \epsilon) \, .\end{equation}

Let

(5.3) \begin{equation} (\Delta t)_i = \begin{cases} \Delta t & i = 1, \ldots M, \\[5pt] 0 & i=0 \, {\text{ or }} \, W\left(t_i^-\right) \leq 0 \end{cases} \, .\end{equation}

We assume that a tontine fee of $\mathbb{T}^f$ per unit time is charged at $t \in \mathcal{T}$ , based on the total portfolio value at $t_i^-$ , after tontine gains but before withdrawals.Footnote 15 Recalling the definition of tontine gain rate $\mathbb{T}^g_i$ in Equation (3.18), we modify this definition to enforce no tontine gain at $t=0$ ,

(5.4) \begin{equation} \mathbb{T}_i^g = \begin{cases} \left(\dfrac{ q_{i-1}}{ 1 - q_{i-1}} \right) & i=1, \ldots, M \\[5pt] 0 & i = 0 \, \, {\text{ or }} \, W\left(t_i^-\right) \leq 0\\ \end{cases} .\end{equation}

Then $W\left(t_i^+\right)$ is given by

(5.5) \begin{equation} W\left(t_i^+\right) = \left(S(t^-_i)+B\left(t_i^-\right)\right)\left(1 + \mathbb{T}^g_i \right) \exp\left({-}(\Delta t)_i \mathbb{T}^f\right) -\mathfrak{q}_i \, ;\, i \in \mathcal{T}, \end{equation}

where we recall that $\mathfrak{q}_M \equiv 0$ and $(\Delta t)_0 \equiv0$ . With some abuse of notation, we define

(5.6) \begin{equation} W\left(t_i^-\right) = \left( S(t^-_i) + B\left(t_i^-\right) \right) \left( 1 + \mathbb{T}^g_i \right) \exp\left({-}(\Delta t)_i \mathbb{T}^f \right)\end{equation}

as the total portfolio value, after tontine gains and tontine fees, the instant before withdrawals and rebalancing at $t_i$ .

Typically, DC plan savings are held in a tax-advantaged account, with no taxes triggered by rebalancing. With infrequent (e.g., yearly) rebalancing, we also expect other transaction costs, apart from the tontine fees, to be small, and hence can be ignored. It is possible to include transaction costs, but at the expense of increased computational cost (van Staden et al., Reference van Staden, Dang and Forsyth2018).

We denote the multi-dimensional controlled underlying process by $X\left(t\right)= \left( S \left( t \right), B\left( t \right) \right)$ , $t\in\left[0,T\right]$ and the realized state of the system by $x = (s,b)$ . Let the rebalancing control $\mathfrak{p}_i({\cdot})$ be the fraction invested in the stock index at the rebalancing date $t_i$ , that is

(5.7) \begin{equation} \mathfrak{p}_i \left(X\left(t_i^-\right)\right) = \mathfrak{p} \left(X\left(t_i^-\right), t_i \right) = \frac{S\left(t_i^+\right)}{S\left(t_i^+\right) + B\left(t_i^+\right)}.\end{equation}

Let the withdrawal control $\mathfrak{q}_i({\cdot})$ be the amount withdrawn at time $t_i$ , that is $\mathfrak{q}_i \left( X\left(t_i^-\right) \right) = \mathfrak{q} \left( X\left(t_i^-\right),t_i \right)$ . Formally, the controls depend on the state of the investment portfolio, before the rebalancing occurs, that is $\mathfrak{p}_i({\cdot}) = \mathfrak{p}\left(X\left(t_i^-\right), t_i \right)= \mathfrak{p}\left(X_i^-, t_i \right)$ , and $\mathfrak{q}_i({\cdot}) = \mathfrak{q}\left(X\left(t_i^-\right), t_i \right)= \mathfrak{q}\left(X_i^-, t_i \right)$ , $t_i \in \mathcal{T}$ , where $\mathcal{T}$ is the set of rebalancing times.

However, it will be convenient to note that in our case, we find the optimal control $\mathfrak{p}_i({\cdot})$ among all strategies with constant wealth (after withdrawal of cash). Hence, with some abuse of notation, we will now consider $\mathfrak{p}_i({\cdot})$ to be function of wealth after withdrawal of cash

(5.8) \begin{align} W\left(t_i^-\right) &= \begin{cases} \left(S\left(t_i^-\right) + B\left(t_i^-\right)\right) (1 + \mathbb{T}^g_i) \exp\bigl({-}(\Delta t)_i \mathbb{T}^f\bigr) \text{ if $ \left(S\left(t_i^-\right) + B\left(t_i^-\right)\right) > 0$} \\ \left(S\left(t_i^-\right) + B\left(t_i^-\right)\right) \text{ otherwise} \\ \end{cases} \nonumber \\[3pt] W\left(t_i^+\right) &= W\left(t_i^-\right) - \mathfrak{q}_i({\cdot}) \nonumber \\[3pt] \mathfrak{p}_i({\cdot}) &= \mathfrak{p}(W\left(t_i^+\right), t_i) \nonumber \\[3pt] S\left(t_i^+\right) &= S_i^+ = \mathfrak{p}_i\left(W_i^+\right)\, W_i^+ \nonumber \\[3pt] B\left(t_i^+\right) &= B_i^+ = \left(1 -\mathfrak{p}_i\left(W_i^+\right)\right) \, W_i^+ .\end{align}

Note that the control for $\mathfrak{p}_i({\cdot})$ depends only $W_i^+$ . Since $\mathfrak{p}_i({\cdot}) = \mathfrak{p}_i(W_i^- - \mathfrak{q}_i)$ , then it follows that

(5.9) \begin{equation} \mathfrak{q}_i({\cdot}) = \mathfrak{q}_i\left(W_i^-\right),\end{equation}

which we discuss further in Section 8.

A control at time $t_i$ , is then given by the pair $(\mathfrak{q}_i({\cdot}),\mathfrak{p}_i({\cdot}))$ where the notation $({\cdot})$ denotes that the control is a function of the state. Let $\mathcal{Z}$ represent the set of admissible values of the controls $(\mathfrak{q}_i({\cdot}), \mathfrak{p}_i({\cdot}))$ . We impose no-shorting, no-leverage constraints (assuming solvency). We also impose maximum and minimum values for the withdrawals. We apply the constraint that in the event of insolvency due to withdrawals ( $W\left(t_i^+\right)< 0$ ), trading ceases and debt (negative wealth) accumulates at the appropriate borrowing rate of return (i.e., a spread over the bond rate). We also specify that the stock assets are liquidated at $t=t_M$ .

More precisely, let $W_i^+$ be the wealth after withdrawal of cash, and $W_i^-$ be the total wealth before withdrawals (but after fees and tontine cash flows), then define

(5.10) \begin{align} \mathcal{Z}_{\mathfrak{q}}\left(W_i^-, t_i\right) &= \left\{\begin{array}{l@{\quad}l} [\mathfrak{q}_{\min},\mathfrak{q}_{\max}] & t_i \in \mathcal{T} \, ;\, t_i \neq t_M\, ;\, W_i^- \geq \mathfrak{q}_{\max} \\[4pt] \,\left[\mathfrak{q}_{\min},\max\left(\mathfrak{q}_{\min},W_i^{-}\right)\,\right] & t_i \in \mathcal{T} \, ;\, t \neq t_M \, ;\, W_i^- \lt \mathfrak{q}_{\max} \\[4pt] \{0\} & t_i = t_M \end{array}\right., \end{align}

(5.11) \begin{align} \mathcal{Z}_{\mathfrak{p}} \left(W_i^+,t_i\right) &= \left\{\begin{array}{l@{\quad}l} \,[0,1\,] & W_i^+ \gt 0 \, ;\, t_i \in \mathcal{T}\, ;\, t_i \neq t_M \\[4pt] \,\{0\,\} & W_i^+ \leq 0 \, ;\, t_i \in \mathcal{T}\, ;\, t_i \neq t_M \\[4pt] \,\{0\,\} & t_i=t_M \end{array}\right. .\qquad\qquad\qquad\end{align}

The rather complicated expression in Equation (5.10) imposes the assumption that as wealth becomes small, the retiree (i) tries to avoid insolvency as much as possible and (ii) in the event of insolvency, withdraws only $\mathfrak{q}_{\min}$ .

The set of admissible values for $(\mathfrak{q}_i,\mathfrak{p}_i), t_i \in \mathcal{T}$ , can then be written as

(5.12) \begin{equation} (\mathfrak{q}_i,\mathfrak{p}_i) \in \mathcal{Z}\left(W_i^-, W_i^+,t_i\right) = \mathcal{Z}_{\mathfrak{q}}\left(W_i^-, t_i\right) \times \mathcal{Z}_{\mathfrak{p}} \left(W_i^+,t_i\right).\end{equation}

For implementation purposes, we have written Equation (5.12) in terms of the wealth after withdrawal of cash. However, we remind the reader that since $W_i^+ = W_i^- -\mathfrak{q}_i$ , the controls are formally a function of the state $X\left(t_i^-\right)$ before the control is applied.

The admissible control set $\mathcal{A}$ can then be written as

(5.13) \begin{equation} \mathcal{A} = \left\{ (\mathfrak{q}_i, \mathfrak{p}_i)_{0 \leq i \leq M} \,:\, ( \mathfrak{q}_i, \mathfrak{p}_i) \in \mathcal{Z}\left(W_i^-, W_i^+,t_i\right) \right\}.\end{equation}

An admissible control $\mathcal{P} \in \mathcal{A}$ can be written as

(5.14) \begin{equation} \mathcal{P} = \{(\mathfrak{q}_i({\cdot}), \mathfrak{p}_i({\cdot})) \, :\, i=0, \ldots, M \} .\end{equation}

We also define $\mathcal{P}_n \equiv \mathcal{P}_{t_n} \subset\mathcal{P}$ as the tail of the set of controls in $[t_n, t_{n+1},\ldots, t_{M}]$ , that is

(5.15) \begin{equation} \mathcal{P}_n =\{(\mathfrak{q}_n({\cdot}), \mathfrak{p}_n({\cdot})), \ldots, (\mathfrak{q}_{M}({\cdot}), \mathfrak{p}_{M}({\cdot})) \} .\end{equation}

For notational completeness, we also define the tail of the admissible control set $\mathcal{A}_n$ as

(5.16) \begin{equation} \mathcal{A}_n = \left\{ (\mathfrak{q}_i, \mathfrak{p}_i)_{n \leq i \leq M} \,:\, (\mathfrak{q}_i, \mathfrak{p}_i) \in \mathcal{Z}\left(W_i^-, W_i^+,t_i\right) \right\},\end{equation}

so that $\mathcal{P}_n \in \mathcal{A}_n$ .

6. Risk and reward

6.1 Risk: Definition of Expected Shortfall (ES)

Let $g(W_T)$ be the probability density function of wealth $W_T$ at $t=T$ . Suppose

(6.1) \begin{equation}\int_{-\infty}^{W^*_{\alpha}} g(W_T) \, dW_T = \alpha, \end{equation}

that is $\textit{Pr}[W_T > W^*_{\alpha}] = 1 - \alpha$ . We can interpret $W^*_{\alpha}$ as the Value at Risk (VAR) at level $\alpha$ . For example, if $\alpha = 0.05$ , then 95% of the outcomes have $W_T >W^*_{\alpha}$ . If $W^*_{\alpha}$ is sufficiently large and positive, this suggests very low risk of running out of savings.Footnote 16 The Expected Shortfall (ES) at level $\alpha$ is then

(6.2) \begin{equation} {\text{ES}}_{\alpha} = \frac{\int_{-\infty}^{W^*_{\alpha}} W_T \, g(W_T)\, dW_T}{\alpha},\end{equation}

which is the mean of the worst $\alpha$ fraction of outcomes. Typically, $\alpha \in \{ 0.01, 0.05 \}$ . The definition of ES in Equation (6.2) uses the probability density of the final wealth distribution, not the density of loss. Hence, in our case, a larger value of ES (i.e., a larger value of average worst case terminal wealth) is desired. The negative of ES is commonly referred to as Conditional Value at Risk (CVAR).

Define $X_0^+ = X(t_0^+), X_0^- = X\left(t_0^-\right)$ . Given an expectation under control $\mathcal{P}$ , $E_{\mathcal{P}} [\cdot]$ , as noted by Rockafellar and Uryasev (Reference Rockafellar and Uryasev2000), $ \text{ES}_{\alpha}$ can be alternatively written as

(6.3) \begin{equation} {\text{ES}}_{\alpha}\left( X_0^-, t_0^-\right) = \sup_{W^*} E_{\mathcal{P}_0}^{X_0^+, t_0^+} \left[W^* + \frac{1}{\alpha} \min( W_T - W^*, 0 ) \right] .\end{equation}

The admissible set for $W^*$ in Equation (6.3) is over the set of possible values for $W_T$ .

The notation ${\text{ES}}_{\alpha}\left( X_0^-, t_0^-\right)$ emphasizes that ${\text{ES}}_{\alpha}$ is as seen at $(X_0^-, t_0^-)$ . In other words, this is the pre-commitment ${\text{ES}}_{\alpha}$ . A strategy based purely on optimizing the pre-commitment value of ${\text{ES}}_{\alpha}$ at time zero is time-inconsistent and hence has been termed by many as non-implementable, since the investor has an incentive to deviate from the time zero pre-commitment strategy at $t >0$ . However, in the following, we will consider the pre-commitment strategy merely as a device to determine an appropriate level of $W^*$ in Equation (6.3). If we fix $W^*$ $\forall t >0$ , then this strategy is the induced time-consistent strategy (Strub et al., Reference Strub, Li, Cui and Gao2019; Forsyth, Reference Forsyth2020a; Cui et al., Reference Cui, Li, Qiao and Strub2022) and hence is implementable. We delay further discussion of this subtle point to Appendix A.

An alternative measure of risk could be based on variability of withdrawals (Forsyth et al., Reference Forsyth, Vetzal and Westmacott2020). However, we note that we have constraints on the minimum and maximum withdrawals, so that variability is mitigated. We also assume that given these constraints, the retiree is primarily concerned with the risk of depleting savings, which is well measured by ES.

6.2 A measure of reward: Expected Total Withdrawals (EW)

We will use expected total withdrawals as a measure of reward in the following. More precisely, we define EW (expected withdrawals) as

(6.4) \begin{equation} {\text{EW}}\left( X_0^-, t_0^-\right) = E_{\mathcal{P}_0}^{X_0^+, t_0^+} \left[\sum_{i=0}^{M} \mathfrak{q}_i \right],\end{equation}

where we assume that the investor survives for the entire decumulation period, consistent with the Bengen (Reference Bengen1994) scenario.

Note that there is no discounting term in Equation (6.4) (recall that all quantities are real, i.e., inflation-adjusted). It is straightforward to introduce discounting, but we view setting the real discount rate to zero to be a reasonable and conservative choice. See Forsyth (Reference Forsyth2022b) for further comments.

7. Problem EW-ES

Since expected withdrawals (EW) and expected shortfall (ES) are conflicting measures, we use a scalarization technique to find the Pareto points for this multi-objective optimization problem. Informally, for a given scalarization parameter $\kappa > 0$ , we seek to find the control $\mathcal{P}_0$ that maximizes

(7.1) \begin{equation} \text{EW}\left( X_0^-, t_0^-\right) + \kappa\, \text{ES}_{\alpha}\left( X_0^-, t_0^-\right).\end{equation}

More precisely, we define the pre-commitment EW-ES problem $(PCES_{t_0}(\kappa))$ problem in terms of the value function $J(s,b,t_0^-)$

(7.2) \begin{align}\left({PCES_{t_0}}\left(\kappa \right)\right)\,:\, J\left(s,b, t_{0}^-\right) &= \sup_{\mathcal{P}_{0}\in\mathcal{A}} \sup_{W^*} \left\{E_{\mathcal{P}_{0}}^{X_0^+,t_{0}^+} \left[\sum_{i=0}^{M} \mathfrak{q}_i + \kappa\left(W^* + \frac{1}{\alpha} \min \left(W_T -W^*, 0\right) \right)\right.\right. \nonumber \\ & \quad + \epsilon W_T \vert X\left(t_0^-\right) = (s,b) \, \Bigg] \Bigg\} \end{align}

(7.3) \begin{align}\text{ subject to } &\begin{cases} (S_t, B_t) \text{ follow processes (4.3) and (4.4)}; \, \, t \notin \mathcal{T} \\[4pt] W_{\ell}^+ = W_{\ell}^{-} - \mathfrak{q}_\ell \,; \, X_\ell^+ = \left(S_\ell^+, B_\ell^+\right) \\[4pt] W_{\ell}^- = \left(S(t^-_i) + B\left(t_i^-\right) \right) \left(1 + \mathbb{T}^g_i \right) \exp\left({-}(\Delta t)_i \mathbb{T}^f\right)\\[4pt] S_\ell^+ = \mathfrak{p}_\ell({\cdot}) W_\ell^+ \,; \, B_\ell^+ = (1 - \mathfrak{p}_\ell({\cdot}) ) W_\ell^+ \, \\[4pt] \left(\mathfrak{q}_\ell({\cdot}), \mathfrak{p}_\ell({\cdot})\right) \in \mathcal{Z}\left(W_\ell^-, W_\ell^+,t_\ell\right) \\[4pt] \ell = 0, \ldots, M \, ;\, t_\ell \in \mathcal{T} \\\end{cases}.\end{align}

Note that we have added the extra term $E_{\mathcal{P}_{0}}^{X_0^+,t_{0}^+}[ \epsilon W_T ]$ to Equation (7.2). If we have a maximum withdrawal constraint, and if $W_t\gg W^*$ as $t \rightarrow T$ , the controls become ill-posed. In this fortunate state for the investor, we can break investment policy ties either by setting $\epsilon < 0$ , which will force investments in bonds, or by setting $\epsilon > 0$ , which will force investments into stocks. Choosing $ |\epsilon| \ll 1$ ensures that this term only has an effect if $W_t \gg W^*$ and $t \rightarrow T$ . See Forsyth (Reference Forsyth2022b) for more discussion of this.

Interchange the $\sup \sup ({\cdot}) $ in Equation (7.2), so that value function $ J\left(s,b, t_{0}^-\right)$ can be written as

(7.4) \begin{align}\qquad J\left(s,b, t_{0}^-\right) &= \sup_{W^*} \sup_{\mathcal{P}_{0}\in\mathcal{A}} \Biggl\{ E_{\mathcal{P}_{0}}^{X_0^+,t_{0}^+} \Biggl[\sum_{i=0}^{M} \mathfrak{q}_i + \kappa \left( W^* + \frac{1}{\alpha} \min \left(W_T -W^*, 0\right) \right) \Biggr. \Biggr. \nonumber \\ & \Biggl. \Biggl. \quad +\, \epsilon W_T \bigg\vert X\left(t_0^-\right) = (s,b) \Biggr] \Biggr\} .\end{align}

Noting that the inner supremum in Equation (7.4) is a continuous function of $W^*$ and that the optimal value of $W^*$ in Equation (7.4) is bounded,Footnote 17 then define

(7.5) \begin{align}\mathcal{W}^*(s,b) &= \displaystyle \mathop{\textrm{arg max}}_{W^*} \sup_{\mathcal{P}_{0}\in\mathcal{A}} \Biggl\{ E_{\mathcal{P}_{0}}^{X_0^+,t_{0}^+} \Biggl[ \, \sum_{i=0}^{M} \mathfrak{q}_i \, + \, \kappa \left( W^* + \frac{1}{\alpha} \min \left(W_T -W^*, 0\right) \right) \nonumber \\ & \quad +\, \epsilon W_T \bigg\vert X\left(t_0^-\right) = (s,b)\Biggr]\Biggr\} .\end{align}

See Forsyth (Reference Forsyth2020a) for an extensive discussion concerning pre-commitment and time-consistent ES strategies. We summarize the relevant results from that discussion in Appendix A.

8. Formulation as a dynamic program

We use the method in Forsyth (Reference Forsyth2020a) to to solve problem We (7.4). write Equation (7.4) as

(8.1) \begin{equation} J\left(s,b,t_0^-\right) = \sup_{W^*} V\left(s,b,W^*, 0^-\right),\end{equation}

where the auxiliary function $V(s, b, W^*, t)$ is defined as

(8.2) \begin{align} V\left(s,b, W^*, t_n^-\right) &= \sup_{\mathcal{P}_{n}\in\mathcal{A}_n} \Biggl\{ E_{\mathcal{P}_{n}}^{\hat{X}_n^+,t_{n}^+} \Biggl[\sum_{i=n}^{M} \mathfrak{q}_i + \kappa \left( W^* + \frac{1}{\alpha} \min\left(\left(W_T -W^*\right),0\right) \right) \Biggr. \Biggr. \nonumber \\ & \quad +\, \epsilon W_T \bigg\vert \hat{X}\left(t_n^-\right) = \left(s,b, W^*\right) \Biggr] \Biggr\}.\end{align}

(8.3) \begin{align}\text{ subject to } &\begin{cases} (S_t, B_t) \text{ follow processes (4.3) and (4.4)}; \, \, t \notin \mathcal{T} \\[3pt] W_{\ell}^+ = W_{\ell}^- - \mathfrak{q}_\ell \,; \, X_\ell^+ = \left(S_\ell^+, B_\ell^+, W^*\right) \\[3pt] W_{\ell}^- = \left(S(t^-_i) + B\left(t_i^-\right)\right) \left(1 + \mathbb{T}^g_i \right) \exp\left({-}(\Delta t)_i \mathbb{T}^f\right) \\[3pt] S_\ell^+ = \mathfrak{p}_\ell({\cdot}) W_\ell^+ \,; \, B_\ell^+ = \left(1 - \mathfrak{p}_\ell({\cdot}) \right) W_\ell^+ \, \\[3pt] (\mathfrak{q}_\ell({\cdot}), \mathfrak{p}_\ell({\cdot})) \in \mathcal{Z}\left(W_\ell^-, W_\ell^+, t_\ell\right) \\[3pt] \ell = n, \ldots, M \, ;\, t_\ell \in \mathcal{T}\end{cases} .\end{align}

We have now decomposed the original problem (7.4) into two steps:

• • For given initial cash $W_0$ , and a fixed value of $W^*$ , solve problem (8.2) using dynamic programming to determine $V(0,W_0, W^*, 0^-)$ .

• • Solve problem (7.4) by maximizing over $W^*$

  (8.4) \begin{equation} J\left(0,W_0, 0^-\right) = \sup_{W^*} V\left(0,W_0, W^*, 0^-\right) .\end{equation}

8.1 Dynamic programming solution of problem (8.2)

We give a brief overview of the method described in detail in Forsyth (Reference Forsyth2022b). Apply the dynamic programming principle to $t_n\in \mathcal{T}$

(8.5) \begin{align} V\left(s,b,W^*, t_n^-\right) &= \sup_{\mathfrak{q} \in \mathcal{Z}_{\mathfrak{q}}(w^-,t_n)} \left\{ \sup_{\mathfrak{p} \in \mathcal{Z}_{\mathfrak{p}}(w^- - q, t_n)} \left[\mathfrak{q}+ V\left((w^- -\mathfrak{q})\mathfrak{p}, (w^- - \mathfrak{q}) (1-\mathfrak{p}), W^*, t_n^+\right) \right] \right\} \nonumber \\[8pt] &= \sup_{\mathfrak{q} \in Z_{\mathfrak{q}}\left(w^-, t_n\right)} \left\{\mathfrak{q} + \left[\sup_{\mathfrak{p} \in \mathcal{Z}_{\mathfrak{p}}\left(w^- - q, t_n\right)} V\left((w^- -\mathfrak{q})\mathfrak{p}, (w^- - \mathfrak{q}) (1-\mathfrak{p}), W^*, t_n^+\right) \right] \right\} \nonumber \\[8pt] & \quad w^- = (s+b) \left(1 + \mathbb{T}^g_i \right) \exp\left({-}(\Delta t)_i \mathbb{T}^f\right).\end{align}

For computational purposes, we define

(8.6) \begin{equation}\tilde{V}\left(w, t_n, W^*\right) = \left[\sup_{\mathfrak{p} \in \mathcal{Z}_{\mathfrak{p}}( w, t_n)} V\left(w \mathfrak{p}, w(1-\mathfrak{p}), W^*, t_n^+\right) \right] .\end{equation}

Equation (8.5) now becomes

(8.7) \begin{align}V\left(s,b,W^*, t_n^-\right) &= \sup_{\mathfrak{q} \in \mathcal{Z}_{\mathfrak{q}}(w^-,t_n)} \left\{ \mathfrak{q} + \left[\tilde{V}\left( (w^- -\mathfrak{q}), W^*, t_n^+\right)\right] \right\} \nonumber \\ & \quad w^- = (s+b) (1 + \mathbb{T}^g_i ) \exp\left({-}(\Delta t)_i \mathbb{T}^f\right).\end{align}

This approach effectively replaces a two dimensional optimization for $(\mathfrak{q}_n, \mathfrak{p}_n)$ , to two sequential one-dimensional optimizations. From Equations (8.6)–(8.7), it is clear that the optimal pair $(\mathfrak{q}_n, \mathfrak{p}_n)$ is such that

(8.8) \begin{align}\mathfrak{q}_n &= \mathfrak{q}_n(w^-, W^*) \nonumber \\ & \quad w^- = (s+b) ( 1 + \mathbb{T}^g_i ) \exp( -(\Delta t)_i \mathbb{T}^f ) \nonumber \\\mathfrak{p}_n &= \mathfrak{p}_n( w, W^*) \nonumber \\ & \quad w = w^- - \mathfrak{q}_n. \end{align}

In other words, the optimal withdrawal control $ \mathfrak{q}_n$ is only a function of total wealth (after tontine gains and fees) before withdrawals. The optimal control $\mathfrak{p}_n$ is a function only of total wealth after withdrawals, tontine gains, and fees.

At $t=T$ , we have

(8.9) \begin{equation}V(s, b, W^*,T^+) = \kappa ( W^* + \frac{\min( (s+b -W^*), 0 )}{\alpha} ) + \epsilon (s+b).\end{equation}

At points in between rebalancing times, that is $t\notin \mathcal{T}$ , the usual arguments (from SDEs (4.3–4.4), and Forsyth, Reference Forsyth2022b) give

(8.10) \begin{align}V_t &+ \frac{(\sigma^s)^2 s^2}{2} V_{ss} +\left(\mu^s - \lambda_{\xi}^s \gamma_{\xi}^s\right) s V_s + \lambda_{\xi}^s \int_{-\infty}^{+\infty} V( e^ys, b, t) f^s(y)\, dy + \frac{\left(\sigma^b\right)^2 b^2}{2} V_{bb} \nonumber \\ &+ \left(\mu^b + \mu_c^b {\bf{1}}_{\{ b < 0 \}} - \lambda_{\xi}^b \gamma_{\xi}^b\right) b V_b + \lambda_{\xi}^b \int_{-\infty}^{+\infty} V\left( s, e^yb, t\right) f^b(y)\, dy -\left(\lambda_{\xi}^s + \lambda_{\xi}^b \right) V + \rho_{sb} \sigma^s \sigma^b s b V_{sb} = 0, \nonumber \\ & \qquad \qquad \qquad s \ge 0 \, ;\, b \ge 0.\end{align}

In case of insolvencyFootnote 18 $s = 0, b < 0$ and then

(8.11) \begin{align}V_t &+ \frac{\left(\sigma^b\right)^2 b^2}{2} V_{bb} + \left(\mu^b + \mu_c^b {\bf{1}}_{\{ b < 0 \}} - \lambda_{\xi}^b \gamma_{\xi}^b\right) b V_b + \lambda_{\xi}^b \int_{-\infty}^{+\infty} V( 0, e^yb, t) f^b(y)\, dy - \lambda_{\xi}^b V = 0, \nonumber \\ & \qquad \qquad \qquad s = 0;\, b < 0.\end{align}

It will be convenient for computational purposes to re-write Equation (8.11) in terms of debt $\hat{b} = -b$ when $b<0$ . Now let $\hat{V} (\hat{b}, t) = V(0, b,t), b < 0, b = -\hat{b}$ in Equation (8.11) to give

(8.12) \begin{align}\hat{V}_t &+ \frac{\left(\sigma^b\right)^2 \hat{b}^2}{2}\hat{V}_{\hat{b}\hat{b}} + (\mu^b + \mu_c^b - \lambda_{\xi}^b \gamma_{\xi}^b)\hat{b}\hat{V}_{\hat{b}} + \lambda_{\xi}^b \int_{-\infty}^{+\infty} \hat{V}(e^y \hat{b},t)f^b(y)\, dy - \lambda_{\xi}^b \hat{V} = 0,\nonumber \\ & \qquad \qquad \qquad s = 0 \, ;\, b < 0\, ;\, \hat{b} = -b.\end{align}

Note that Equation (8.12) is now amenable to a transformation of the form $\hat{x} = \log \hat{b}$ since $\hat{b} > 0$ , which is required when using a Fourier method (Forsyth and Labahn, Reference Forsyth and Labahn2019; Forsyth, Reference Forsyth2022b) to solve Equation (8.12).

After rebalancing, if $b \ge 0$ , then b cannot become negative, since $b=0$ is a barrier in Equation (8.11). However, b can become negative after withdrawals, in which case b remains in the state $b<0$ , where Equation (8.12) applies, unless there is an injection of cash to move to a state with $b>0$ . The terminal condition for Equation (8.12) is

(8.13) \begin{equation}\hat{V}\left(\hat{b}, W^*,T^+\right) = \kappa \left( W^* + \frac{\min( ({-}\hat{b} -W^*), 0 )}{\alpha} \right) + \epsilon ({-}\hat{b}) \, ;\, \hat{b} > 0.\end{equation}

A brief overview of the numerical algorithms is given in Appendix B, along with a numerical convergence verification.

9. Data

We use data from the Center for Research in Security Prices (CRSP) on a monthly basis over the 1926: 1–2020:12 period.Footnote 19 Our base case tests use the CRSP US 30 day T-bill for the bond asset and the CRSP value-weighted total return index for the stock asset. This latter index includes all distributions for all domestic stocks trading on major U.S. exchanges. All of these various indexes are in nominal terms, so we adjust them for inflation by using the U.S. CPI index, also supplied by CRSP. We use real indexes since investors funding retirement spending should be focused on real (not nominal) wealth goals.

We use the threshold technique (Mancini, Reference Mancini2009; Cont and Mancini, Reference Cont and Mancini2011; Dang and Forsyth, Reference Dang and Forsyth2016) to estimate the parameters for the parametric stochastic process models. Since the index data is in real terms, all parameters reflect real returns. Table 1 shows the results of calibrating the models to the historical data. The correlation $\rho_{sb}$ is computed by removing any returns which occur at times corresponding to jumps in either series, and then using the sample covariance. Further discussion of the validity of assuming that the stock and bond jumps are independent is given in Forsyth (Reference Forsyth2020b).

Table 1. Estimated annualized parameters for double exponential jump diffusion model. Value-weighted CRSP index, 30-day T-bill index deflated by the CPI. Sample period 1926:1–2020:12.

10. Historical market

We compute and store the optimal controls based on the parametric model (4.3)–(4.4) as for the synthetic market case. However, we compute statistical quantities using the stored controls, but using bootstrapped historical return data directly. In this case, we make no assumptions concerning the stochastic processes followed by the stock and bond indices. We remind the reader that all returns are inflation-adjusted. We use the stationary block bootstrap method (Politis and Romano, Reference Politis and Romano1994; Politis and White, Reference Politis and White2004; Patton et al., Reference Patton, Politis and White2009; Cogneau and Zakalmouline, Reference Cogneau and Zakalmouline2013; Dichtl et al., Reference Dichtl, Drobetz and Wambach2016; Cavaglia et al., Reference Cavaglia, Scott, Blay and Hixon2022; Simonian and Martirosyan, Reference Simonian and Martirosyan2022; Anarkulova et al., Reference Anarkulova, Cederburg and O’Doherty2022). A key parameter is the expected blocksize. Sampling the data in blocks accounts for serial correlation in the data series. We use the algorithm in Patton et al. (Reference Patton, Politis and White2009) to determine the optimal blocksize for the bond and stock returns separately, see Table 2. We use a paired sampling approach to simultaneously draw returns from both time series. In this case, a reasonable estimate for the blocksize for the paired resampling algorithm would be about $2.0$ years. We will give results for a range of blocksizes as a check on the robustness of the bootstrap results. Detailed pseudo-code for block bootstrap resampling is given in Forsyth and Vetzal (Reference Forsyth and Vetzal2019).

Table 2. Optimal expected blocksize $\hat{b}=1/v$ when the blocksize follows a geometric distribution $Pr(b = k) = (1-v)^{k-1} v$ . The algorithm in Patton et al. (Reference Patton, Politis and White2009) is used to determine $\hat{b}$ . Historical data range 1926:1–2020:12.

11. Investment scenario

Table 2 shows our base case investment scenario. We use thousands of dollars as our units of wealth. For example, a withdrawal of 40 per year corresponds to $\$40,\!000$ per year (all values are real, i.e., inflation-adjusted), with an initial wealth of 1000 (i.e., $\$1,\!000,\!000$ ). This would correspond to the use of the four per cent rule (Bengen, Reference Bengen1994). Our base case scenario assumes a fee of 50 bps per year for the tontine overlay. See Chen et al. (Reference Chen, Hieber and Rach2021) for a discussion of tontine fees.

As a motivating example, we consider a 65-year old Canadian retiree with a pre-retirement salary of $100,000 per year, with $1,000,000 in a DC savings account. Government benefits are assumed to amount to about $20,000 per year (real). The retiree needs the DC plan to generate at least $40,000 per year (real), so that the DC plan and government benefits together replace 60% of pre-retirement income. We assume that the retiree owns mortgage-free real estate worth about $400,000. In an act of mental accounting, the retiree plans to use the real estate as a longevity hedge, which could be monetized using a reverse mortgage. In the event that the longevity hedge is not needed, the real estate will be a bequest. Of course, the retiree would like to withdraw more than $40,000 per year from the DC plan, but has no use for withdrawals greater than $80,000 per year. We further assume that the real estate holdings can generate $200,000 through a reverse mortgage. Hence, as a rough rule of thumb any expected shortfall at $T=30$ years greater than $-\$200,\!000$ is an acceptable level of risk.

Our view that personal real estate is not fungible with investment assets (unless investment assets are depleted) is consistent with the behavioral life cycle approach originally described in Shefrin and Thaler (Reference Shefrin and Thaler1988) and Thaler (Reference Thaler1990). In this framework, investors divide their wealth into separate “mental accounts” containing funds intended for different purposes such as current spending or future need.

We take the view of a 65-year-old retiree, who wants to maximize her total withdrawals and minimize the risk of running out of savings, assuming that she lives to the age of 95. We also assume that the retiree has no bequest motive.

Recall that Bengen (Reference Bengen1994) attempted to determine a safe real withdrawal rate, and constant allocation strategy, such that the probability of running out of cash after 30 years of retirement was small. In other words, Bengen (Reference Bengen1994) maximized total withdrawals over a 30 year period, assuming that the retiree survived for the entire 30 years. This is, of course, a conservative assumption.

In our case, we are essentially answering the same question. The key difference here is that we allow for (i) dynamic asset allocation, (ii) variable withdrawals (within limits) and (iii) a possible tontine overlay.

12. Constant withdrawal, constant equity fraction

As a preliminary example, in this section we present results for the scenario in Table 3, except that a constant withdrawal of 40 per year (recall that units are thousands so that this is $40,000) is specified, along with a constant weight in stocks at each rebalancing date.

Table 3. Input data for examples. Monetary units: thousands of dollars. CPM2014 is the mortality table from the Canadian Institute of Actuaries.

Table 4 gives the results for various values of the constant weight equity fraction in the synthetic market. The best resultFootnote 20 for ES (the largest value) occurs at the rather low constant equity weight of $p=0.1$ , with $\text{ES} = -239$ . Table 5 gives similar results, this time using bootstrap resampling of the historical data (the historical market). Here the best value of $\text{ES} = -305$ occurs for a constant equity fraction of $p=0.4$ . Consequently, in both the historical and synthetic market, the constant weight, constant withdrawal strategy fails to meet our risk criteria of $\text{ES} > -200$ .

Table 4. Constant weight, constant withdrawals, synthetic market results. No tontine gains. Stock index: real capitalization weighted CRSP stocks; bond index: real 30-day T-bills. Parameters from Table 1. Scenario in Table 3. Units: thousands of dollars. Statistics based on $2.56 \times 10^6$ Monte Carlo simulation runs. $T = 30 $ years.

Table 5. Constant weight, constant withdrawals, historical market. No tontine gains. Historical data range 1926:1–2020:12. Constant withdrawals are 40 per year. Stock index: real capitalization weighted CRSP stocks; bond index: real 30-day T-bills. Scenario in Table 3. Units: thousands of dollars. Statistics based on $10^6$ bootstrap simulation runs. Expected blocksize $=2$ years. $T =30$ years.

These simulations indicate that there is significant depletion risk for the constant withdrawal, constant weight strategy suggested in Bengen (Reference Bengen1994).

13. Synthetic market efficient frontiers

Figure 1 shows the efficient EW-ES frontiers computed in the synthetic market for the following cases:

1. 1. Tontine: the case in Table 3. The control is computed using the algorithm in Section 8 and then stored and used in Monte Carlo simulations. The detailed frontier is given in Table D.1.

2. 2. No Tontine: the case in Table 3, but without any tontine gains. The control is computed and stored and then used in Monte Carlo simulations The detailed frontier is given in Table D.2.

3. 3. Const q=40, Const p: The best single point from Table 4, based on Monte Carlo simulations.

Figure 1. Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 1). Tontine case is as in Table 3. The No Tontine case uses the same scenario, but with no tontine gains, and no fees. The Const q, Const p case has $q=40$ , $p=0.10$ , with no tontine gains, which is the best result from Table 4, assuming no tontine gains, and no fees. Units: thousands of dollars.

Note that all these strategies produce a minimum withdrawal of 40 per year (i.e., 4% real of the initial investment) for 30 years. However, the best result for the constant weight strategies was $(\text{EW},\text{ES}) = (40, -239)$ This can be improved significantly by optimizing over withdrawals and asset allocation, but with no tontine gains. For example, from Table D.2, the nearest point with roughly the same level of risk is $(\text{EW},\text{ES}) =(58,-242)$ . However, the improvement with optimal controls and tontine gains is remarkable. For example, it seems reasonable to target a value of $ES \simeq 0$ . From Table D.1, we note the point $(\text{EW},\text{ES}) = (69,47)$ , which is dramatically better than any No Tontine Pareto point. This can also be seen from the large outperformance in the EW-ES frontier compared to the No Tontine case in Figure 1.

13.1 Effect of fees

Figure 2 shows the effect of varying the annual fee in the synthetic market, for the scenario in Table 3. Recall that the base case specified a fee of 50 bps per year. Assuming a shortfall target of ES $\simeq 0$ , then the effect of fees in the range 0–100 bps is quite modest. Even with annual fees of 100 bps, the Tontine case still significantly outperforms the No Tontine case (which is assumed to have no fees).

Figure 2. Effect of varying fees charged for the Tontine, basis points (bps) per year. Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 1). Base case Tontine is as in Table 3 (fees 50 bps per year). The No Tontine case uses the same scenario, but with no tontine gains, and no fees. Units: thousands of dollars.

13.2 Effect of Random G

Recall the definition of the group gain $G_i$ at time $t_i$ in Equation (3.10). Basically, the group gain is used to ensure that the total amount of mortality credits disbursed is exactly equal to the total amount forfeited by tontine participants who have died in $(t_{i-1}, t_i)$ .

If Condition 3.1 holds, then we expect that randomly varying $G_i$ will have a small cumulative effect. In Fullmer and Sabin (Reference Fullmer and Sabin2019) and Winter and Planchet (Reference Winter and Planchet2022), the authors create synthetic tontine pools where the investors have different initial wealth, ages, genders, and investment strategies. These pools are perpetual, that is new members join as original members die. It is assumed that the investors can only select an asset allocation strategy from a stock index and a bond index, both of which follow a geometric Brownian motion (GBM).

The payout rules are different from those suggested in this paper, but it is instructive to observe the following. In Fullmer and Sabin (Reference Fullmer and Sabin2019), the perpetual tontine pool has 15,000 investors in steady state. After the initial start-up period, the expected value of the group gain $G_i$ at each rebalancing time is close to unity, with a standard deviation of about $0.1$ . Fullmer and Sabin (Reference Fullmer and Sabin2019) also note that there is essentially no correlation between investment returns and the group gain.

Further simulations were carried out in Winter and Planchet (Reference Winter and Planchet2022), using the same sharing rule as in Fullmer and Sabin (Reference Fullmer and Sabin2019) for a single period (one year), with 500–1000–5000 participants. The investors had different allocation amounts with ages from $40-70$ , but all participants had the same investment strategy. The variance of $G_i$ was negligible for the pool with 5000 initial investors. The Fullmer and Sabin (Reference Fullmer and Sabin2019) study, with the additional variability of random asset allocation to risky assets, had a low standard deviation for $G_i$ at around 15,000 participants. Consequently, it would appear that the number of participants required to be reasonably sure that the assumption that $var(G_i)$ is small is in the range of 5000–15,000, depending on restrictions for individual asset allocation.

Figure 3 shows the effect of randomly varying $G_i$ . The curve labeled $G=1.0$ is the base case EW-ES curve from the scenario in Table 3, in the synthetic market (parameters in Table 1). The controls from this base case are stored and then used in Monte Carlo simulations, where G is assumed to have a normal distribution with mean one, and standard deviation of $0.1$ . The EW-ES curves for both cases essentially overlap, except for very large values of ES, which are not of any practical interest. We get essentially the same result if we use a uniform distribution for G with $E[G] = 1$ , with the same standard deviation. This is not surprising, since, assuming that the value function is smooth, then a simple Taylor series argument shows that, for any assumed distribution of G with mean one, the effect of randomness of G is a second order effect in the standard deviation.

Figure 3. Effect of randomly varying group gain G (Section 3.2.1). Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 1). Base case Tontine ( $G=1.0)$ is as in Table 3. Random G case uses the control computed for the base case, but in the Monte Carlo simulation, G is normally distributed with mean one and standard deviation $0.1$ . Units: thousands of dollars.

Of course, we cannot determine the actual distribution of G without a detailed knowledge of the characteristics of the tontine pool. In fact, if we knew the distribution, we could include it in the formulation of the optimal control problem. However, knowledge of the distribution of G is unlikely to be available to pool participants in practice.

Nevertheless, the simulations in Fullmer and Sabin (Reference Fullmer and Sabin2019) and Winter and Planchet (Reference Winter and Planchet2022), coupled with our results as shown in Figure 3, suggest that for a sufficiently large, diversified pool of investors the effects of randomly varying G are negligible. Note that we are only considering idiosyncratic mortality risk here, not systematic risk, for example unexpected mortality improvement.

14. Bootstrapped results

As discussed in Section 10, a key parameter in the stationary block bootstrap technique is the expected blocksize. In Figure 4(a), we show the results of the following test. We compute and store the optimal controls, based on the synthetic market. Then, we use these controls, but carry out tests on bootstrapped historical data. The efficient frontiers in Figure 4(a), for $\text{ES} <1000 $ essentially overlap for all expected blocksizes in the range 0.5–5.0 years. Since it is probably not of interest to aim for an ES of 1000 (which is one million dollars) at age 95, this indicates that the computed strategy is robust to parameter uncertainty.

Figure 4. Optimal strategy determined by solving Problem 7.2 in the synthetic market, parameters in Table 1. Control stored and then tested in bootstrapped historical market. Inflation-adjusted data, 1926:1–2020:12. Non-Pareto points eliminated. Expected blocksize (Blk, years) used in the bootstrap resampling method also shown. Units: thousands of dollars. The const q, const p case had $(p,q) = (0.4, 40)$ (no tontine gains). This is the best result for the constant (p,q) case, shown in Table 5.

Figure 4(b) compares the efficient frontier tested in the historical market (expected blocksize 2 years), with the efficient frontier in the synthetic market. We observe that the synthetic and historical curves overlap for $\text{ES} < 1000$ , which again verifies that the controls are robust to data uncertainty. The efficient frontiers/points for the No Tontine case and the constant weight and constant withdrawal strategy (computed in the historical market) are also shown. The Tontine overlay continues to outperform the No Tontine case by a wide margin.

15. Detailed historical market results: EW-ES controls

In this section, we examine some detailed characteristics of the optimal EW-ES strategy, tested in the historical market for the scenario in Table 3. Figure 5 shows the percentiles of fraction in stocks, wealth, and withdrawals versus time for the EW-ES control with $\kappa = 0.18$ , with $(\text{EW},\text{ES})= (69, 204)$ . To put this in perspective, recall that this strategy never withdraws less than 40 per year. Compare this to the best case for a constant withdrawal, constant weight strategy (no tontine) from Table 5, which has $(\text{EW},\text{ES}) = (40, -306) $ , or to the optimal EW-ES strategy, but with no tontine, from Table E.2, which has $(\text{EW},\text{ES}) = (70,-806)$ .

Figure 5. Scenario in Table 3. EW-ES control computed from problem EW-ES Problem (7.2). Parameters based on the real CRSP index, and real 30-day T-bills (see Table 1). Control computed and stored from the Problem (7.2) in the synthetic market. Control used in the historical market, $10^6$ bootstrap samples. $q_{\min} = 40, q_{\max} = 80$ (per year), ${\kappa}= 0.18$ . $W^* = 385$ . Units: thousands of dollars.

Figure 5(a) shows that the median optimal fraction in stocks starts at about $0.60$ and then drops to about $0.20$ at 15 years, finally ending up at zero in year 26. Figure 5(b) indicates that for the years in the span of 20–30, the median and fifth percentiles of wealth are fairly tightly clustered, with the fifth percentile being well above zero at all times. The optimal withdrawal percentiles are shown in Figure 5(c). The median withdrawal starts at 40 per year and then increases to the maximum withdrawal of 80 in years 3–4 and remains at 80 for the remainder of the 30-year time horizon.

Figure 6 shows the optimal control heat maps for the fraction in stocks and withdrawal amounts, for the scenario in Table 3. Figure 6(a) shows a smooth behavior of the optimal fraction in stocks as a function of (W,t). This can be compared with the equivalent heat map for the EW-ES control in Forsyth (Reference Forsyth2022b) (no tontine gains), which is much more aggressive at changing the asset allocation in response to changing wealth amounts. The smoothness of the controls in Figure 6(a) appears to be due to the rapid de-risking of a strategy which includes tontine gains, which provides a natural protection against sudden stock index drops. The upper blue zone in Figure 6(a) is de-risking due to the fact that, with sufficiently large wealth, there is essentially no probability of running out of cash even at the maximum withdrawal amount. The use of the stabilization factor $\epsilon = -10^{-4}$ forces the strategy to increase the weight in bonds for large values of wealth (see Equation (7.2)).Footnote 21 The lower red zone is in response to extremely poor wealth outcomes, which means that the optimal strategy is to invest 100% in stocks and hope for the best. However, this is an extremely unlikely outcome, as can be verified from Figure 5(b).

Figure 6. Optimal EW-ES. Heat map of controls: fraction in stocks and withdrawals, computed from Problem EW-ES (7.1). Real capitalization weighted CRSP index, and real 30-day T-bills. Scenario given in Table 3. Control computed and stored from the Problem 7.2 in the synthetic market. $q_{\min} = 40, q_{\max} =80$ (per year). ${\kappa} = 0.18$ . $W^* = 385$ . $\epsilon = -10^{-4}$ . Normalized withdrawal $(q - q_{\min})/(q_{\max} - q_{\min})$ . Units: thousands of dollars.

From Figure 6(b), we can observe that the optimal withdrawal strategy is essentially a bang-bang control, that is withdraw at either the maximum or minimum amount per year. This is not unexpected, as discussed in Appendix C. We also note that this type of strategy has been suggested previously, based on heuristic reasoning.Footnote 22