--- title: Individual Tontine Accounts url: https://www.emergentmind.com/topics/individual-tontine-account type: topic --- # Individual Tontine Accounts Searching arXiv for recent papers on individual tontine accounts, modern tontines, and collectivised post-retirement investment. An **individual tontine account** is a retirement decumulation arrangement in which each retiree maintains an individualized post-retirement account and individualized consumption and investment choices, while participating in a mortality-pooling mechanism under which residual balances of deceased members are redistributed to survivors as **mortality credits** rather than bequeathed [1909.12730]. In the literature, the term spans several closely related designs: collectivised investment funds with individual accounts [1909.12730], modern retirement-income tontines with predetermined payout rules [1610.10078; 1307.2824], account-based tontines with explicit bequest subaccounts [1903.05990; 2005.00715], reversible or transaction-cost-aware tontines [2209.09709], accumulation-based “money-back” tontines [2402.14555], and recent individual tontine overlays with money-back guarantees under systematic longevity risk [2602.16212]. Across these variants, the defining feature is the combination of individual account ownership and control with pooled sharing of idiosyncratic longevity experience. ## 1. Conceptual definition and product taxonomy In the terminology of “Collectivised Post-Retirement Investment” [1909.12730], an individual tontine account is an **individual post-retirement investment account that participates in a collectivised fund**. Each retiree has their own account value and their own optimal consumption path; on death, any remaining funds in that account are not bequeathed, but are shared among surviving members. The resulting mortality credits compensate survivors for the risk of long life and make higher sustainable withdrawals possible [1909.12730]. This account-based formulation differs from the classical image of the tontine as a pure winner-take-all survivor lottery. Modern work treats the tontine as a decumulation technology for lifetime spending rather than as a gambling contract. In the retirement-income formulations of Milevsky and Salisbury, the pool pays a prescribed payout schedule \(d(t)\) and each survivor receives a pro rata share of that scheduled pool payout; the sponsor bears no longevity risk because the payout stream is financed from the pooled assets rather than guaranteed by insurer capital [1610.10078; 1307.2824]. A second branch of the literature embeds the tontine mechanism inside a personal investment account. In this account-centric view, the retiree chooses withdrawals and asset allocation dynamically, while mortality pooling appears as an additional return term or annual mortality-credit top-up [2211.10509; 2602.16212]. A third branch further splits wealth into a tontine portion and a bequest-preserving portion, yielding “modern tontine with bequest” designs in which the retiree participates in longevity pooling only on a selected fraction of wealth [1903.05990; 2005.00715]. The literature also distinguishes between **actuarial fairness**, **collective actuarial fairness**, and **equity**. In the one-period heterogeneous-pool framework of “Egalitarian pooling and sharing of longevity risk” [2402.00855], actuarial fairness for an individual means expected payout equals accumulated initial investment, while collective actuarial fairness means aggregate expected payouts match aggregate capital plus return. In mixed-cohort tontines, “equity” is distinct from full fairness and refers to equal expected present value per dollar invested across cohorts, even though residual end-of-pool value prevents strict actuarial fairness in closed tontines [1610.09384]. ## 2. Financial market structure, mortality modeling, and account dynamics A standard account-based specification uses a Black–Scholes–Merton post-retirement market with one real risk-free asset \(S^1_t\) and one real risky asset \(S^2_t\), with dynamics \[ \begin{aligned} d S^1_t &= S^1_t\, r \, d t, \\ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned} \] In the calibration used in [1909.12730], the setting is UK-oriented, with real gilt return \(r \approx 0.047 - r_{\text{CPI}}\), \(r_{\text{CPI}}=0.02\), real equity return \(\mu \approx 0.082 - r_{\text{CPI}}\), and volatility \(\sigma = 0.15\). Consumption is modeled in discrete annual steps, while investment is continuous-time [1909.12730]. Mortality is typically modeled through a deterministic force of mortality or annual death probabilities. In [1909.12730], the main numerical example uses a representative UK woman aged 65 in 2019, with mortality following the CMI 2018 model and survival truncated where probability falls below \(10^{-5}\). A central modeling assumption there is **no systematic longevity risk**: future mortality distributions are treated as known, so longevity uncertainty is purely idiosyncratic and diversifiable through pooling [1909.12730]. By contrast, recent work explicitly incorporates systematic longevity risk through Lee–Carter or Cairns–Blake–Dowd type generalized age–period–cohort models, which generate path-dependent annual death probabilities and hence path-dependent mortality-credit rates [2602.16212]. At the individual account level, the core state variable is wealth \(X_t\) or \(W_t\), with controls given by the consumption or withdrawal rate and the portfolio allocation. A generic continuous-time formulation takes the form \[ dX_t = X_t\bigl[r + \pi_t(\mu-r)\bigr]dt + X_t \pi_t \sigma dW_t - \gamma_t dt + \text{(mortality credits)}. \] This structure appears explicitly in the collectivised-fund setting [1909.12730] and in various modern tontine models with risky investment [1903.05990; 2005.00715; 2209.09709]. In the infinite-pool limit, mortality credits often collapse into an additional deterministic drift term proportional to the current tontine balance and the hazard rate. For example, in the reversible two-account modern tontine with transaction costs, the tontine account \(Y(t)\) evolves between transactions as \[ dY(t) = (\mu + \lambda(t)) Y(t)\,dt + \sigma Y(t)\,dW_t, \] so the effective expected return is \(\mu+\lambda(t)\), namely investment return plus longevity credits [2209.09709]. In the Riccati tontine, the infinite-pool accumulation account satisfies \[ dZ_t = \mu Z_t\,dt + \sigma Z_t\,dB_t + (1-k_t)\lambda_t Z_t\,dt, \] where the extra drift \((1-k_t)\lambda_t Z_t\,dt\) is the continuous-time mortality-credit term [2402.14555]. ## 3. Mortality-credit allocation and fairness mechanisms The simplest mortality-sharing rule arises in a homogeneous infinite pool, where individual deaths average out and each survivor effectively receives a deterministic mortality-credit rate. In [1909.12730], the evolution of a single account in the collectivised fund can be modeled by an SDE with an extra deterministic drift term due to mortality credits, although the paper emphasizes that no simple closed form such as \(q_x \times \text{account value}\) is asserted verbatim [1909.12730]. For heterogeneous and finite pools, the allocation problem becomes central. The practical rebalancing algorithm in [1909.12730] proceeds period by period. For each surviving individual \(i\), provisional post-investment, post-consumption wealth \(\mathring{X}^i_{t+\delta t}\) is computed using the optimal homogeneous strategy for that member’s own type. The member’s contribution to the mortality pool is then defined as \[ \Gamma^i_{t+\delta t} = (1 - s^i_t)\,\mathring{X}^i_{t+\delta t}, \] where \(s^i_t\) is the probability of surviving the interval. Actual residual wealth of deceased members is aggregated and redistributed among survivors in proportion to these contributions [1909.12730]. This rule is intended to be actuarially fair across heterogeneous members because it reflects both mortality probability and wealth at risk. The same paper reports that with only \(n=100\) heterogeneous members, the resulting allocation yields for almost all members at least **98% of the maximum possible benefit** they would obtain in their own infinite homogeneous pool, formalized through the optimality ratio \[ \text{OR}_i := \frac{u^i_S - u^i_1}{u^i_\infty - u^i_1}, \] with \(\text{OR}_i \ge 0.98\) for almost all individuals [1909.12730]. This is one of the clearest finite-pool efficiency results in the literature. A different but related heterogeneous-pool framework is provided by the one-period share-allocation approach of [2402.00855]. There, participant \(i\) contributes \(T_i\), has survival probability \(p_i\), and receives \(f_i\) personalized tontine shares. The time-1 payout is \[ W_i = (1+R)\,\Big(\sum_{j=1}^{n+1} T_j\Big)\, \frac{f_i I_i}{\sum_{j=1}^{n+1} f_j I_j}, \] with \(I_i\) the survival indicator [2402.00855]. The choice of the share-allocation rule \(f_i(T,p)=T_i g(p_i)\) determines the social character of the pool. The paper studies several explicit rules: the DM rule \(f_i=T_i/p_i\), which compensates worse health; the T rule \(f_i=T_i\), which is wealth-proportional and favors longer-lived participants; poor-favoring variants; and the DR uniform-share rule \(f_i=1\), which strongly redistributes toward younger and poorer survivors [2402.00855]. The central conclusion is methodological: actuarial science cannot guarantee a unique sharing rule; the chosen rule depends on the extent of solidarity, altruism, and individualism within the pool [2402.00855]. Mixed-cohort equity is treated further in [1610.09384], where cohorts of different ages and contribution sizes receive different participation rates \(\pi_i\), so that an investor in cohort \(i\) contributing \(w_i\) receives \(u_i=\pi_i w_i\) shares. The expected present value per dollar invested, \(F_i(\pi)\), is then equalized across cohorts. The paper proves that, for a fixed payout function \(d(t)\), any equitable choice of \(\pi\) is unique up to multiplicative scaling if it exists, and provides a necessary and sufficient condition for existence [1610.09384]. This result is directly relevant to the pricing of individual tontine shares in heterogeneous retail or pension pools. ## 4. Preference theory, pension adequacy, and optimal control A major contribution of [1909.12730] is the claim that pension design cannot be fully understood using homogeneous expected utility alone. The paper introduces **pension adequacy** as a primitive preference concept: an adequacy level \(a_t\) is such that an individual is indifferent between dying at \(\tau\) and living longer while receiving exactly \(a_t\) over the additional period. This anchors welfare evaluation around the distinction between “enough to live on” and merely positive consumption [1909.12730]. Operationally, the paper calibrates a private pension adequacy target using a 70% replacement-rate guideline applied to median pre-retirement earnings \(£24{,}100\), implying total pension \(£16{,}800\), less a deterministic state pension path \(SP_t\). The resulting private adequacy target is \[ AL_t = \pounds 16800 - SP_t, \] and the present value of a deterministic adequacy-level private pension is \[ X_{AL} \approx £126{,}636. \] The baseline investor is assigned initial wealth \(X_0=X_{AL}\) [1909.12730]. Preferences are then modeled using **exponential Kihlstrom–Mirman preferences with mortality**, with satisfaction \[ s_{\gamma,\tau} = \sum_{t \in {\cal T}, \, t \le \tau} u(\gamma_t, t)\,\delta t, \] and per-period utility \[ u(\gamma_t, t)= \begin{cases} a\,(\gamma_t + SP_t)^\rho - a\,(AL_t + SP_t)^\rho & \gamma_t \ge 0 \ \forall t, \\ -\infty & \text{otherwise}, \end{cases} \] with \(\rho=-1\) in the calibration and gain functional \[ {\cal J}(\gamma,\tau) = \mathbb{E}\Bigl[-\exp\bigl(-s_{\gamma,\tau}\bigr)\Bigr]. \] The paper argues that these preferences capture aversion both to inadequate consumption and to early death [1909.12730]. By contrast, the older optimal-tontine literature adopts CRRA felicity. In [1610.10078], the utility-optimal pool payout function under CRRA is characterized through the Euler–Lagrange condition and the function \[ \beta_{n,\gamma}(p)=p\,\theta_{n,\gamma}(p), \] yielding \[ D_{n,\gamma}^{\mathrm{OT}}(p)=D_{n,\gamma}^{\mathrm{OT}}(1)\,\beta_{n,\gamma}(p)^{1/\gamma}. \] The paper proves that the **natural tontine**, with pool payout proportional to survival probability, \[ d(t)=c_0\,{}_tp_x, \] is exactly optimal for log utility \((\gamma=1)\) and near-optimal for realistic \(n\) and \(\gamma\) [1610.10078]. Table 8 in that paper gives certainty-equivalent multipliers for using the natural rather than the \(\gamma\)-tailored optimal tontine; for \(n=100\), age 60, the multiplier is approximately \(1.003388\) for \(\gamma=2\), which the paper interprets as needing only about 0.34% extra initial capital to match utility [1610.10078]. Several bequest-oriented papers modify the control problem by introducing a bequest subaccount or a time-varying tontine share. In [1903.05990], a fixed proportion \(\alpha\) of total wealth is continuously maintained in the tontine account, with the remainder \((1-\alpha)\) in a bequest account. Under power utility, the paper reports that more risk-averse retirees allocate a fairly stable proportion of pension savings to the tontine account regardless of the strength of the bequest motive, and that this optimal proportion is often around **80–90%** in the numerical study, while very low-risk-aversion cases can generate a non-monotonic response because of extremely large tail bequests at advanced ages [1903.05990]. The closely related closed-form model of [2005.00715] allows the bequest proportion to be a control and derives \[ 1-a^*(t)=c^*(t)b^{1-\gamma}, \] so the optimal bequest share is the product of the optimal fractional consumption rate and an exponentiated bequest parameter [2005.00715]. A more recent note argues that constant bequest preferences yield unrealistic behavior and proposes time-dependent bequest preferences calibrated to “100% payback upon death at the start that vanishes over time,” producing an almost linearly increasing allocation in the tontine from 0% to 100% over time [2501.08972]. This suggests a design interpretation in which explicit refund or money-back features are used to reconcile early bequest concerns with later-life full pooling. ## 5. Product variants: bequest, reversibility, money-back protection, and transaction costs The simplest individual tontine account is a no-bequest arrangement: all residual wealth is forfeited at death and redistributed to survivors [1909.12730; 2211.10509]. Several later papers relax this structure. In the **modern tontine with bequest** [1903.05990], total wealth \(X(t)\) is split into a tontine account \(Y(t)=\alpha X(t)\) and a bequest account \(Z(t)=(1-\alpha)X(t)\), with the key feature that this proportion is continuously rebalanced to remain constant. The tontine account receives direct longevity credits at rate \(\lambda(t)Y(t)\), while the bequest account receives an indirect share of those gains through rebalancing [1903.05990]. This produces a hybrid of pooled longevity insurance and estate preservation. The reversible modern tontine with transaction costs [2209.09709] uses a different architecture: a bequest account \(X(t)\) invested risk-free and a tontine account \(Y(t)\) fully invested in a risky asset, with all consumption drawn from the bequest account. Transfers between the two accounts are allowed in both directions but incur fixed and proportional transaction costs. The resulting optimization is a combined stochastic and impulse-control problem with infinite horizon and HJBQVI characterization. Numerically, the optimal allocation region is **V-shaped** and exhibits two phases: initially the retiree decreases tontine exposure to smooth wealth as longevity credits build up, but at very advanced ages increases tontine exposure to gamble for large longevity credits [2209.09709]. The **Riccati tontine** [2402.14555] modifies the basic accumulation tontine by introducing a recovery schedule \(k_t\) such that a representative investor who dies or lapses at time \(t