2.1 Utility

We model the retirement phase of a representative individual in discrete time from age 66 ( $t=1$ ) to age 105 ( $T=40$ ). Our subject’s utility drawn from consuming a single, non-durable consumption good is described by a time-separable CRRA utility function. The individual seeks to maximise lifetime utility, which – in recursive form – is specified as

(1) \begin{equation}V_t=u(C_t)+\beta E_t\left[\,p_tV_{t+1}+\left(1-p_t\right)b\cdot u(D_{t+1})\right]\end{equation}

with

\begin{equation*}u({*})=\begin{cases}log({*}) & \text{for } \rho = 1 \\[3pt] \frac{({*})^{1-\rho}}{1-\rho} & \text{for } \rho \neq 1\end{cases}\end{equation*}

$C_t$ denotes consumption at time t, $D_t$ is the level of bequest at time t, $\rho$ is the coefficient of relative risk aversion, b is the strength of bequest motive, and $\beta> 0$ is the time preference discount factor. $p_t$ is the survival probability from time t to $t+1$ , which we assume to be known at time t, but which evolves stochastically over time as described below. In the final period of life at the maximum attainable age T, $p_T=0$ and therefore, $V_T=u(C_T)+\beta E_T\left[b\cdot u(D_{T+1})\right]$ .

2.2 Capital market

The individual has access to capital markets through investing in risk-free bonds, risky stocks (denoted by $\bullet$ ), and risky tontines (denoted by $\circ$ ). The one-period real gross bond return is constant over time and described by the accumulation factor $R_f$ . The stochastic real gross stock return from time $t-1$ to time t is denoted by the accumulation factor $R_t^{\bullet}$ , which is assumed to be serially independent and identically log-normally distributed with an expected value of $E(R_t^{\bullet})=(1+\mu^{\bullet})$ and a volatility (i.e. standard deviation) of $\sigma^{\bullet}$ .

In addition to bonds and stocks, the individual can – at each point in time – invest into revolving 1-year tontines, which allow members to decide every year anew how much money they want to commit over the next period. This contrasts with other tontine designs where funds are committed for life. The pool of tontinists, for which we assume that our investor is a representative member, is closed and exists until there is only one survivor remaining. Pool members are assumed to be homogeneous in terms of their personal and economic characteristics, that is, they have the same age, sex, preferences, income, and wealth profiles. Assuming that our investor is representative for all members of the tontine pool has two implications. First, if our representative individual decides to commit funds to the tontine in a given period, all members of the tontine pool will do so. Moreover, in every period, each member of the tontine pool commits an equal amount of funds to the tontine.

The initial 1-year tontine is set up at retirement in $t=1$ and consists of $N_1$ members. Over the first year, the funds invested into the tontine accumulate with the tontine’s annual gross investment return $R_2$ . At the end of the first year, the accumulated tontine funds are paid out in full, evenly distributed amongst the $N_2$ surviving participants, and the 1-year tontine renews for those $N_2$ survivors. This process continues as long as $N_{t}\geq2$ . In case $N_t=1$ , there is no further opportunity to pool mortality risks, which is a defining characteristic of the tontine. Instead, the tontine collapses into a purely financial product with returns equal to those of the underlying asset. In that case, we consider the tontine to have ceased to exist.

Formally, the tontine’s total gross return from time $t-1$ to time t is

(2) \begin{equation}R_t^{\circ}=\begin{cases}\frac{N_{t-1}}{N_t}R_t&\text{if alive in }\textit{t}\\[3pt]0&\text{if dead in }\textit{t}\\[3pt] \end{cases}\end{equation}

where $R_t \in \{R_f,R_t^{\bullet}\}$ ^{Footnote 1}. As such, the tontine return is risky by nature, since it depends on the random number of surviving tontinists in the pool. If the tontine funds are invested in bonds, the tontinists’ mortality risk will be the only source of risk in the tontine return. If the underlying investment is risky, the tontine return is subject to mortality risk as well as to the underlying market risk.

For CRRA investors, Milevsky & Salisbury (Reference Milevsky and Salisbury2015) show that the optimal/close to optimal (depending on the level of risk aversion) tontine design is one that provides individual tontinists with a payout structure that resembles that of a constant annuity. This can be achieved through a natural tontine, which they define as a tontine with total payouts that decrease proportionally to the survival rates, such that the decrease in the number of participants does not result in an exponential increase in individual payouts.

If, however, the tontine fractions in the investors’ portfolios could be adjusted flexibly over time, it might not be necessary to adjust the tontine payout scheme itself. Instead, investors could hold shares in a more traditional tontine design, such as the one described in equation (2) and studied here, and adapt their portfolios in order to smooth consumption, for example, by reducing the tontine fraction over time. This could provide investors with more flexibility, for example, in case of increasing liquidity needs in old age due to health expenditures.

2.3 Retirement income and health state

At the beginning of every period, the household receives an exogenous, health-state-dependent retirement income $Y_t$ . This income is assumed to be constant ( $Y_t=\bar{Y}$ ) as long as the individual remains healthy. In contrast, if the individual becomes frail, income is assumed to decrease over time (and to potentially become negative) to reflect an age-increasing liquidity need $Y^{-}_t$ associated with frailty. The individual’s health state is described by a two-state Markov chain, for which we assume that frailty is an absorbing state. With $\pi_{f}$ representing the probability of a healthy individual becoming frail, the transition matrix from state 1 (healthy) to state 2 (frail) is given by $P=\begin{pmatrix} 1-\pi_{f} & \pi_{f} \\ 0 & 1\\ \end{pmatrix}$ . Therefore, the health-state-dependent retirement income is $Y_t=\bar{Y}-\mathds{1}_P\cdot Y^{-}_t$ , where $\mathds{1}_P$ is one if the household is frail and zero otherwise.

Naturally, the most obvious approach to considering frailty-related health expenses would be to directly model an increased consumption need. The reason we refrain from doing so and rather model decreasing retirement income is to reduce computational effort in determining the optimal policies. The net result of these two approaches is the same. One could argue that – since securing one’s health and standard of living in case of frailty is of fundamental importance – health expenses in case of frailty are prioritised over the consumption of other goods and services. Hence, health expenses are immediately deducted from retirement income and decisions with respect to consumption and portfolio choice are only made subsequently based on the remaining budget.

2.4 Wealth accumulation

At the beginning of every period, the individual can spread the wealth on hand $W_t$ across bonds $B_t$ , stocks $S_t$ , tontines $\varUpsilon_t$ , and consumption $C_t$ . The budget constraint is

(3) \begin{equation}W_t=B_t+S_t+\varUpsilon_t+C_t\end{equation}

Individual disposable wealth on hand in $t+1$ is

(4) \begin{equation}W_{t+1}=B_tR_f+S_tR_{t+1}^{\bullet}+\varUpsilon_{t}R_{t+1}^{\circ}+Y_{t+1}\end{equation}

$B_tR_f+S_tR_{t+1}^{\bullet}+\varUpsilon_tR_{t+1}^{\circ}$ describes the value of financial wealth in $t+1$ and $Y_{t+1}$ is the retirement income. Short selling is not allowed, thus

(5) \begin{equation}B_t,S_t,\varUpsilon_t\geq 0\end{equation}

Since, in case of death, the tontine investment goes to the tontine pool and not to the heirs, the bequest in $t+1$ is

(6) \begin{equation}D_{t+1}=B_tR_f+S_tR_{t+1}^{\bullet}\end{equation}

2.5 Mortality dynamics

To incorporate systematic mortality risks, we rely on the two-factor approach of Cairns et al. (Reference Cairns, Blake and Dowd2006). Specifically, we model the stochastic dynamics of the logits of the time-varying conditional 1-year mortality rates ( $q^t_x$ ) for an individual aged x in year t according to

(7) \begin{equation}\text{logit}\!\left(q^t_x\right) = \log\left(\frac{q^t_x}{1-q^t_x}\right) = \kappa^t_1 + \kappa^t_2 \left(x-\bar{x}\right)\end{equation}

where $\bar{x}=\left(x_u-x_l+1\right)^{-1}\sum_{x=x_l}^{x_u}x$ is the mean of the range of ages considered to be fitted, $\kappa^t_1$ is the “level” of mortality, which usually has a downward trend reflecting generally improving mortality rates over time, and $\kappa^t_2$ is the “slope” coefficient, which has a gradual upward drift reflecting the fact that, historically, mortality at high ages has improved at a slower rate than at younger ages. The column vector $\kappa^t=(\kappa_1^t, \kappa_2^t)'$ is assumed to follow a two-dimensional random walk with drift $\kappa^{t+1}= \kappa^t+\tau +\chi\cdot Z_{t+1}$ , where $\tau$ is the drift of $\kappa^t$ , $\chi$ is the lower triangular Cholesky matrix of the $\kappa^t$ -covariance matrix $\Sigma$ , and $Z_{t+1}$ is a bivariate standard normal shock. Furthermore, we assume that the mortality rate at the end of our projection horizon, that is at age 105, is $q^T_{105}=1$ .

2.6 Calibration

For the base case calibration of the model, we set the risk aversion parameter to $\rho=1$ (Aït-Sahalia & Hurd, Reference Aït-Sahalia and Hurd2016). Following Weinert & Gründl (Reference Weinert and Gründl2016), we set the subjective discount factor to $\beta=0.98$ . We refrain from an accumulation phase and begin the analysis at retirement age 66 ( $t=1$ ) with an initial wealth endowment of $W_1=150,000$ EUR. Using Ordinary Least Squares (OLS) regression, we calibrate the described Cairns et al. (Reference Cairns, Blake and Dowd2006) mortality model to mortality data for US males aged 65–105 over the period 1950–2014 provided by the Human Mortality Database (HMD). We obtain the following point estimates:

\begin{equation*}\hat{\kappa}^0 =\begin{pmatrix}-2.4106763\\[3pt] 0.1050414\end{pmatrix}\!,\hat{\tau} =\begin{pmatrix}-0.0090878\\[3pt] 0.0003335\end{pmatrix}\!, \text{and}\ \hat{\Sigma} =\begin{pmatrix}0.0005446 & 0.0000162\\[3pt] 0.0000162 & 0.0000008\end{pmatrix}\\\end{equation*}

We assume a maximum attainable age of 105 ( $T=40$ ). The individual receives a constant yearly retirement income of $\bar{Y}=24,000$ EUR. However, with probability $\pi_f=0.03$ , the individual becomes frail and faces an increased liquidity need, which depends on the age of the individual. In this case, the liquidity need is 12, 000 EUR at age 66 ( $t=1$ ) and linearly increases up to 30, 000 EUR at age of 105 ( $T=40)$ . That is, $Y^{-}_t =\frac{15}{26}t+12,000$ . Once the individual faces a higher liquidity need, this state will remain until death. Therefore, the health-state-dependent retirement income $Y_t=\bar{Y}-\mathds{1}_P\cdot Y^{-}_t = 24,000 - \mathds{1}_P\cdot (\frac{15}{26}t+12,000)$ . The frailty probability, the resulting age-dependent liquidity needs, as well as the initial wealth endowment are estimated based on the German Socio-Economic Panel (SOEP) in the period 1984–2013. The risk-free gross bond return is $R_f=1.02$ , the expected stock return is $\mu^{\bullet}=0.06$ , and the volatility of stock return is $\sigma^{\bullet}=0.18$ as in Horneff et al. (Reference Horneff, Maurer and Rogalla2010). In the base case scenario, we assume that the tontine funds are invested into the risk-free asset, that is, $R_t=R_f=1.02$ . The initial size of the tontine is $N_1=10,000$ . Furthermore, we set the bequest parameter to $b=1$ . We assume no correlation between stock returns and income shocks. We use 10, 000 simulated life-cycle trajectories of stock and tontine returns, health state, mortality model parameters, and tontine pool size to determine expected lifetime utilities and optimal policies (see the description of the numerical strategy below).

In subsequent variations of the base case, we consider a scenario in which the investor does not have a bequest motive ( $b=0$ ) and one in which tontine funds are invested in stocks (i.e. $R_t=R^{\bullet}_t$ ). Further, we analyse the impact of alternative mortality dynamics by disabling the random mortality shocks (trending mortality; $\chi=0$ ) and then by additionally disabling the mortality drift (constant mortality; $\chi,\tau=0$ ). Finally, we study the impact of alternative initial tontine pool sizes ( $N_1=100$ and $N_1=1,000$ ). Table 1 summarises the parameters for our model calibrations.

Table 1. Model calibration

2.7 Numerical strategy

We seek to maximise equation (1) with respect to consumption $C_t$ subject to equations (3)–(6). Traditionally, the workhorse of solving such discrete-time portfolio choice models is dynamic optimisation via backward induction over a discretised, potentially high-dimensional, state space comprising all state variables. While this approach is straightforward, it suffers from the curse of dimensionality, as the computational effort to find optimal solutions increases exponentially with the number of state variables. To ease the computational burden related to a greater number of state variables, alternative solution strategies have been proposed more recently. Among those are various simulation/regression approaches (Brandt et al., Reference Brandt, Goyal, Santa-Clara and Stroud2005; Koijen et al., Reference Koijen, Nijman and Werker2007, Reference Koijen, Nijman and Werker2010). Given the high dimensionality of our optimisation problem with six state variables (age, wealth, health, remaining tontine population, and two mortality factors), we also employ a simulation/regression approach, specifically the one discussed in Denault & Simonato (Reference Denault and Simonato2017) and Denault et al. (Reference Denault, Delage and Simonato2017). In what follows, we briefly discuss the main aspects of this solution technique. For a detailed step-by-step description of the algorithm, including examples, we refer to Denault et al. (Reference Denault, Delage and Simonato2017, p. 380 ff).

Generally, the simulation/regression approach employs standard dynamic optimisation techniques. Key element in reducing the computational effort associated with solving the model, however, is the differential treatment of endogenous and exogenous state variables. Endogenous state variables, in our case wealth, are discretised following the traditional approach. The interrelation between the value function and the exogenous state variables, on the other hand, is captured by specifying future utility as a parametric function of policy and exogenous state variables. To this end, I (in our case 10, 000) life-cycle trajectories of our exogenous state variables are simulated using the stochastic dynamics described above, and a sample set of M policies (in our case all asset portfolios with weight steps of 10%, excluding those that contain any asset weights of zero) is defined. Then, for each (discretised) value of the endogenous variable at time t, the utility at time $t+1$ for each of the $I \cdot M$ combinations of sample policies and exogenous state variable realisations is determined. These future utilities are then regressed onto a basis of policy and exogenous state variables (in our case, a constant, the 3 asset weights, their squares and cross-products, the 4 exogenous state variables, and the 12 cross-products between asset weights and exogenous state variables) to derive the parametric function. Using this functional relation between policies, (exogenous) state variables, and future utility, we then determine for each state variable trajectory the utility maximising asset allocation^{Footnote 2}.

Figure 1 Simulated development of the number of remaining tontine members over time. Initial number of tontine members at age 66: $N_1=100$ , $N_1=1,000$ , and $N_1=10,000$ , respectively. The graphs show the 0.1–99.9 percentiles simulated paths. Darker areas represent higher probability mass.