Individual Tontine Accounts

• Individual tontine accounts are individualized post-retirement investment vehicles that combine personal control with pooled longevity risk via mortality credits.
• They enable dynamic consumption and portfolio management while redistributing unspent balances from deceased members to enhance pension adequacy.
• Variants include bequest subaccounts, reversible structures, and money-back guarantees, offering flexibility to meet diverse retiree preferences.

Searching arXiv for 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 (Armstrong et al., 2019). In the literature, the term spans several closely related designs: collectivised investment funds with individual accounts (Armstrong et al., 2019), modern retirement-income tontines with predetermined payout rules [(Milevsky et al., 2016); (Milevsky et al., 2013)], account-based tontines with explicit bequest subaccounts (Bernhardt et al., 2019, Dagpunar, 2020), reversible or transaction-cost-aware tontines (He et al., 2022), accumulation-based “money-back” tontines (Milevsky et al., 2024), and recent individual tontine overlays with money-back guarantees under systematic longevity risk (Orozco et al., 18 Feb 2026). 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” (Armstrong et al., 2019), 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 (Armstrong et al., 2019).

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 [(Milevsky et al., 2016); (Milevsky et al., 2013)].

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 (Forsyth et al., 2022, Orozco et al., 18 Feb 2026). 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 (Bernhardt et al., 2019, Dagpunar, 2020).

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” (Dhaene et al., 2024), 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 (Milevsky et al., 2016).

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 (Armstrong et al., 2019), 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 (Armstrong et al., 2019).

Mortality is typically modeled through a deterministic force of mortality or annual death probabilities. In (Armstrong et al., 2019), 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 (Armstrong et al., 2019). 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 (Orozco et al., 18 Feb 2026).

At the individual account level, the core state variable is wealth $X_t$ or $S^1_t$0, with controls given by the consumption or withdrawal rate and the portfolio allocation. A generic continuous-time formulation takes the form

$S^1_t$1

This structure appears explicitly in the collectivised-fund setting (Armstrong et al., 2019) and in various modern tontine models with risky investment (Bernhardt et al., 2019, Dagpunar, 2020, He et al., 2022). 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 $S^1_t$2 evolves between transactions as

$S^1_t$3

so the effective expected return is $S^1_t$4, namely investment return plus longevity credits (He et al., 2022). In the Riccati tontine, the infinite-pool accumulation account satisfies

$S^1_t$5

where the extra drift $S^1_t$6 is the continuous-time mortality-credit term (Milevsky et al., 2024).

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 (Armstrong et al., 2019), 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 $S^1_t$7 is asserted verbatim (Armstrong et al., 2019).

For heterogeneous and finite pools, the allocation problem becomes central. The practical rebalancing algorithm in (Armstrong et al., 2019) proceeds period by period. For each surviving individual $S^1_t$8, provisional post-investment, post-consumption wealth $S^1_t$9 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

$S^2_t$0

where $S^2_t$1 is the probability of surviving the interval. Actual residual wealth of deceased members is aggregated and redistributed among survivors in proportion to these contributions (Armstrong et al., 2019). 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 $S^2_t$2 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

$S^2_t$3

with $S^2_t$4 for almost all individuals (Armstrong et al., 2019). 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 (Dhaene et al., 2024). There, participant $S^2_t$5 contributes $S^2_t$6, has survival probability $S^2_t$7, and receives $S^2_t$8 personalized tontine shares. The time-1 payout is

$S^2_t$9

with $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$0 the survival indicator (Dhaene et al., 2024). The choice of the share-allocation rule $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$1 determines the social character of the pool. The paper studies several explicit rules: the DM rule $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$2, which compensates worse health; the T rule $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$3, which is wealth-proportional and favors longer-lived participants; poor-favoring variants; and the DR uniform-share rule $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$4, which strongly redistributes toward younger and poorer survivors (Dhaene et al., 2024). 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 (Dhaene et al., 2024).

Mixed-cohort equity is treated further in (Milevsky et al., 2016), where cohorts of different ages and contribution sizes receive different participation rates $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$5, so that an investor in cohort $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$6 contributing $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$7 receives $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$8 shares. The expected present value per dollar invested, $$\begin{aligned} d S^1_t &= S^1_t\, r \, d t, \ d S^2_t &= S^2_t \bigl( \mu \, d t + \sigma \, d W_t \bigr). \end{aligned}$$9, is then equalized across cohorts. The paper proves that, for a fixed payout function $r \approx 0.047 - r_{\text{CPI}}$0, any equitable choice of $r \approx 0.047 - r_{\text{CPI}}$1 is unique up to multiplicative scaling if it exists, and provides a necessary and sufficient condition for existence (Milevsky et al., 2016). 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 (Armstrong et al., 2019) 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 $r \approx 0.047 - r_{\text{CPI}}$2 is such that an individual is indifferent between dying at $r \approx 0.047 - r_{\text{CPI}}$3 and living longer while receiving exactly $r \approx 0.047 - r_{\text{CPI}}$4 over the additional period. This anchors welfare evaluation around the distinction between “enough to live on” and merely positive consumption (Armstrong et al., 2019).

Operationally, the paper calibrates a private pension adequacy target using a 70% replacement-rate guideline applied to median pre-retirement earnings $r \approx 0.047 - r_{\text{CPI}}$5, implying total pension $r \approx 0.047 - r_{\text{CPI}}$6, less a deterministic state pension path $r \approx 0.047 - r_{\text{CPI}}$7. The resulting private adequacy target is

$r \approx 0.047 - r_{\text{CPI}}$8

and the present value of a deterministic adequacy-level private pension is

$r \approx 0.047 - r_{\text{CPI}}$9

The baseline investor is assigned initial wealth $r_{\text{CPI}}=0.02$0 (Armstrong et al., 2019).

Preferences are then modeled using exponential Kihlstrom–Mirman preferences with mortality, with satisfaction

$r_{\text{CPI}}=0.02$1

and per-period utility

$r_{\text{CPI}}=0.02$2

with $r_{\text{CPI}}=0.02$3 in the calibration and gain functional

$r_{\text{CPI}}=0.02$4

The paper argues that these preferences capture aversion both to inadequate consumption and to early death (Armstrong et al., 2019).

By contrast, the older optimal-tontine literature adopts CRRA felicity. In (Milevsky et al., 2016), the utility-optimal pool payout function under CRRA is characterized through the Euler–Lagrange condition and the function

$r_{\text{CPI}}=0.02$5

yielding

$r_{\text{CPI}}=0.02$6

The paper proves that the natural tontine, with pool payout proportional to survival probability,

$r_{\text{CPI}}=0.02$7

is exactly optimal for log utility $r_{\text{CPI}}=0.02$8 and near-optimal for realistic $r_{\text{CPI}}=0.02$9 and $\mu \approx 0.082 - r_{\text{CPI}}$0 (Milevsky et al., 2016). Table 8 in that paper gives certainty-equivalent multipliers for using the natural rather than the $\mu \approx 0.082 - r_{\text{CPI}}$1-tailored optimal tontine; for $\mu \approx 0.082 - r_{\text{CPI}}$2, age 60, the multiplier is approximately $\mu \approx 0.082 - r_{\text{CPI}}$3 for $\mu \approx 0.082 - r_{\text{CPI}}$4, which the paper interprets as needing only about 0.34% extra initial capital to match utility (Milevsky et al., 2016).

Several bequest-oriented papers modify the control problem by introducing a bequest subaccount or a time-varying tontine share. In (Bernhardt et al., 2019), a fixed proportion $\mu \approx 0.082 - r_{\text{CPI}}$5 of total wealth is continuously maintained in the tontine account, with the remainder $\mu \approx 0.082 - r_{\text{CPI}}$6 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 (Bernhardt et al., 2019). The closely related closed-form model of (Dagpunar, 2020) allows the bequest proportion to be a control and derives

$\mu \approx 0.082 - r_{\text{CPI}}$7

so the optimal bequest share is the product of the optimal fractional consumption rate and an exponentiated bequest parameter (Dagpunar, 2020).

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 (Bernhardt, 15 Jan 2025). 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 (Armstrong et al., 2019, Forsyth et al., 2022). Several later papers relax this structure.

In the modern tontine with bequest (Bernhardt et al., 2019), total wealth $\mu \approx 0.082 - r_{\text{CPI}}$8 is split into a tontine account $\mu \approx 0.082 - r_{\text{CPI}}$9 and a bequest account $\sigma = 0.15$0, with the key feature that this proportion is continuously rebalanced to remain constant. The tontine account receives direct longevity credits at rate $\sigma = 0.15$1, while the bequest account receives an indirect share of those gains through rebalancing (Bernhardt et al., 2019). This produces a hybrid of pooled longevity insurance and estate preservation.

The reversible modern tontine with transaction costs (He et al., 2022) uses a different architecture: a bequest account $\sigma = 0.15$2 invested risk-free and a tontine account $\sigma = 0.15$3 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 (He et al., 2022).

The Riccati tontine (Milevsky et al., 2024) modifies the basic accumulation tontine by introducing a recovery schedule $\sigma = 0.15$4 such that a representative investor who dies or lapses at time $\sigma = 0.15$5 is expected, though not guaranteed, to receive their money back on average: $\sigma = 0.15$6 In the infinite-pool deterministic-hazard case, the binding schedule satisfies the Riccati ODE

$\sigma = 0.15$7

with explicit solution

$\sigma = 0.15$8

The paper reports, for a 65-year-old with a 20-year horizon, $\sigma = 0.15$9 and expected survivor account value $10^{-5}$0, compared with $10^{-5}$1 without mortality credits (Milevsky et al., 2024). The same paper argues that the underlying assets should ideally have returns negatively correlated with mortality shocks, providing a hedge in high-mortality states (Milevsky et al., 2024).

The most recent extension in the provided corpus adds a money-back guarantee overlay to a dynamic individual tontine account under stochastic mortality and international diversification (Orozco et al., 18 Feb 2026). In that framework, the retiree optimizes expected withdrawals against terminal-wealth CVaR, while a guarantee ensures that the initial premium $10^{-5}$2 is returned either through withdrawals during life or as a death benefit upon early death. The guarantee payout is path-dependent and defined in nominal terms by the shortfall between $10^{-5}$3 and cumulative nominal withdrawals, then converted back to real units (Orozco et al., 18 Feb 2026). The paper prices the guarantee ex post under the induced optimal policy as expected guarantee cost plus a prudential CVaR buffer, concluding that implied loads are driven mainly by tail outcomes and the chosen prudential buffer rather than by mean payouts (Orozco et al., 18 Feb 2026).

6. Quantitative results, comparative performance, and implementation issues

One of the strongest quantitative claims in the literature comes from (Armstrong et al., 2019). Under the calibrated UK-style parameters, with initial budget $10^{-5}$4, annuity-equivalent comparison yields the following results.

Fund type	Annuity Equivalent (£×10^3)	Outperformance
Annuity	126.6	0%
Individual fund	128.7	+1.5%
Collective (tontine)	152.2	+20%

The paper interprets this as showing that an annuity or individual fund would need roughly 20% more initial capital than the collectivised fund to deliver a comparable outcome (Armstrong et al., 2019). It also attributes the advantage to mortality credits, equity risk-premium exposure, and intertemporal substitution (Armstrong et al., 2019).

Earlier work using CRRA preferences reaches a related but distinct conclusion. In (Milevsky et al., 2016), actuarially fair life annuities dominate optimal tontines in utility, but the difference is small, and once realistic annuity loadings are introduced the tontine can be preferred. The paper reports indifference loadings in Table 7; for age 60, $10^{-5}$5, and $10^{-5}$6, the indifference loading is approximately 231.7 bps for $10^{-5}$7, 51.8 bps for $10^{-5}$8, and 5.68 bps for $10^{-5}$9 (Milevsky et al., 2016). This is the basis for the claim that the utility advantage of fair annuities over well-designed tontines is minimal in realistic loaded-annuity markets (Milevsky et al., 2016).

The dynamic-control literature sharpens the comparison against self-directed drawdown. In (Forsyth et al., 2022), the holder of an individual tontine retirement account with annual withdrawal bounds $X_t$0 and $X_t$1, starting from $X_t$2, maximizes expected withdrawals and minimizes expected shortfall at age 95. In the historical market test with tontine overlay and $X_t$3, the paper reports approximately

$X_t$4

while the corresponding no-tontine optimized strategy yields approximately

$X_t$5

and constant 4% withdrawal rules perform worse still (Forsyth et al., 2022). This paper therefore treats the tontine overlay as a quantitatively powerful enhancement to rule-based drawdown.

The recent money-back-guarantee paper (Orozco et al., 18 Feb 2026) extends the efficient-frontier perspective from expected-withdrawal/expected-shortfall to expected-withdrawal/CVaR. It finds that international diversification and longevity pooling jointly deliver the largest improvements in the EW–CVaR trade-off, while stochastic mortality shifts the frontier only modestly in the expected direction (Orozco et al., 18 Feb 2026). It also shows that optimal controls use foreign equity primarily as a state-dependent catch-up instrument (Orozco et al., 18 Feb 2026).

Implementation issues recur across the literature. The collectivised-fund model in (Armstrong et al., 2019) is explicitly linked to Collective Defined Contribution schemes and presented as directly applicable to post-retirement design in CDC settings. The paper notes that implementation requires explicit agreement that no bequests are paid from the pooled account, transparent communication that income is not guaranteed, and clear rules for how mortality credits and investment performance affect payouts (Armstrong et al., 2019). The heterogeneous-pooling literature additionally raises questions about age, sex, health status, anti-discrimination law, and the extent to which explicit health-based share adjustments are socially or legally acceptable (Dhaene et al., 2024, Milevsky et al., 2016).

7. Controversies, misconceptions, and current research directions

A recurrent misconception is that tontines are inherently identical to historical winner-take-all structures. The modern literature rejects this equivalence. Utility-optimal and natural tontines deliberately avoid exploding late-life per-capita payouts and instead target relatively stable lifetime consumption paths [(Milevsky et al., 2013); (Milevsky et al., 2016)]. Another misconception is that there is a uniquely “fair” way to share longevity credits in heterogeneous pools. The recent heterogeneous-pooling literature explicitly disputes this: multiple actuarially coherent allocation rules exist, and rule choice reflects normative judgments about solidarity, redistribution, and individualism rather than a single actuarial truth (Dhaene et al., 2024).

A second controversy concerns bequest. Pure individual tontine accounts generally require full forfeiture at death, which sharpens longevity insurance but makes the product unattractive for bequest-motivated retirees (Armstrong et al., 2019, Forsyth et al., 2022). Modern responses include bequest subaccounts (Bernhardt et al., 2019, Dagpunar, 2020), time-varying bequest preferences (Bernhardt, 15 Jan 2025), money-back-on-average designs (Milevsky et al., 2024), and explicit money-back guarantees (Orozco et al., 18 Feb 2026). This suggests an ongoing convergence between tontine design and insurance-style death-benefit engineering, although these overlays can reduce the raw efficiency of pure mortality pooling.

A third open issue is systematic longevity risk. Much of the earlier literature assumes known future mortality laws and therefore treats longevity risk as purely idiosyncratic and diversifiable (Armstrong et al., 2019, Milevsky et al., 2016, Forsyth et al., 2022). The 2026 neural-network study is notable precisely because it relaxes this assumption and concludes that stochastic mortality shifts the efficient frontier only modestly, though in the expected adverse direction (Orozco et al., 18 Feb 2026). A plausible implication is that systematic longevity risk may matter less for individual-level product ranking than for prudential pricing of guarantees and regulatory capital overlays.

The computational frontier is also shifting. Earlier account-control papers solved HJB or PIDE problems numerically by dynamic programming and Fourier methods (Armstrong et al., 2019, Forsyth et al., 2022), or HJBQVI methods in impulse-control settings (He et al., 2022). More recent work uses neural-network policy approximation to handle high-dimensional state spaces with international assets and stochastic mortality (Orozco et al., 18 Feb 2026). This suggests a methodological migration from low-dimensional stochastic control toward machine-learning approximations in realistic retirement-product design.

Overall, the research literature treats the individual tontine account not as a single contract form but as a design class. Its invariant core is the combination of individualized retirement accounts with mortality-credit redistribution. Around that core, current research varies payout rules, fairness criteria, bequest treatment, reversibility, guarantees, asset universes, and optimization criteria. The resulting body of work frames individual tontine accounts as a technically flexible mechanism for post-retirement risk sharing, with strong capital-efficiency properties but with product design ultimately shaped by normative choices about adequacy, equity, guarantees, and bequest.