Funding a target-date tontine ladder

All numerical return and survival assumptions below are hypothetical. This note develops a planning calculation; it does not establish the price, delivery or optimality of the proposed heterogeneous tontine using fair transfer plans (FTP). It supplies a funding baseline for comparison with specified allocation and continuation policies.

Working backwards from target payments

A ladder starts with future spending amounts and works backwards to the investments needed today. A payment due soon has little time to earn investment returns. A later payment has more time, but the return on its investment is uncertain. Using the average projected return treats favourable outcomes as if they compensate for disappointing outcomes pound for pound.

A certainty equivalent instead asks: what fixed future amount would the investor regard as equivalent to the uncertain investment outcome, under a stated risk preference? It converts that assessment into a planning growth factor. Dividing the desired future payment by that factor gives an initial investment amount. Greater aversion to risk normally means a lower factor and more money invested today for the same target.

The word “certainty” describes the comparison. It does not make the investment outcome certain. A target funded in this way can still be missed. A spending floor therefore needs its own funding and protection rule.

Longevity sharing adds another calculation. While a rung remains committed, members who survive can receive allocations from accounts whose commitments end on death. An idealised survival adjustment helps explain why less initial capital might support a given conditional payment. The research must then test that adjustment against the actual allocation rules, pool and investment outcomes.

Exact investment-only definition

Let C be one nominal target payment at time T, and G_T > 0 the random gross investment growth of £1 by that date, after any modelled fees. For constant relative risk aversion (CRRA), use

The certainty-equivalent growth factor and planning allocation are

For log utility, K_T = exp(E[log G_T]). CRRA's homogeneity gives CE(A_T G_T)=A_T K_T=C. The annualised continuously compounded planning rate is log(K_T)/T; the annual effective equivalent is K_T^(1/T)−1. These two rate conventions must not be mixed.

These definitions and the lognormal CRRA expression are standard expected-utility results, rather than a novel contribution to tontine theory. Ashwin Rao, Stanford utility notes, pp. 5 and 12.

If log G_T ~ Normal(m_T,v_T), then

For the specific process dG/G = μ dt + σ dW, with constant parameters,

Here μ is the instantaneous proportional drift: E[G_T]=exp(μT). It is neither the annual effective mean return nor the mean log return. Under a deterministic time-varying glidepath and deterministic coefficients, replace the exponent by ∫(μ_t−γσ_t²/2)dt. Under other return models, compute the utility expectation directly; mean and variance alone need not determine it.

Release is earlier than payment

Use two dates. Let τ be the end of mortality sharing and T the payment date. If the investment growth to release is G_τ and the subsequent gross factor is known to be H_(τ,T), then

That assumption requires an asset or cash-flow arrangement that actually fixes the later nominal amount. Moving money into an ordinary fund does not by itself fix its return. If the post-release factor is uncertain, include it in the joint return distribution instead.

A member who survives to release has completed the mortality commitment on that rung. The planning survival probability is therefore s_τ, not s_T. Once released, treatment on a later death follows the ordinary pension/beneficiary contract. Survival to payment still matters for the member's consumption utility. The two dates answer different questions.

Four rungs, four target payments

An annual ladder holds separate investments for separate future payment dates. To isolate the investment calculation, begin with four targets of £1,000, in years 2, 3, 4 and 5. Each rung leaves sharing one year before its payment date. This first illustration applies no survival discount and no mortality credits.

Assume a hypothetical investment following geometric Brownian motion with drift μ=6% and volatility σ=10% a year until each release date. Assume the final year's gross growth is known to be exp(4%). Set CRRA risk aversion to γ=3, with no fees, taxes or inflation. This stylised two-stage investment rule illustrates the calculation; it is not a calibrated glidepath.

The CE planning rate before release is then 6%−3×(10%)²/2=4.5%, continuously compounded. Each initial amount is £1,000/exp(0.045×release_year+0.04).

Totals use unrounded amounts. The different starting amounts reflect the different investment horizons. Each £1,000 remains a CE target under the stated model. It is not the amount that every simulated or realised path will pay. These four rungs illustrate part of a ladder; a retirement design needs a complete schedule and a policy for survival beyond its final rung.

One longer rung, with the funding choices exposed

Target: £10,000 nominal in year 20. Sharing ends in year 19. Use the same hypothetical assumptions: investment drift μ=6% and volatility σ=10% for 19 years, a known gross factor exp(4%) for the final year, and γ=3. Ignore fees, taxes and inflation. These are teaching parameters, not forecasts, current gilt yields or an adopted glidepath.

Thus m=1.085, v=0.19, E[G]=exp(1.18)=3.2543742, and K=exp(0.895)=2.4473358. The pre-release CE planning rate is 4.5% continuously compounded; over the full 20 years it is 4.475% continuously compounded, approximately 4.577% annually effective.

The last row is 0.8 × £4,086, not a separately proven product price. Using expected growth before applying that same survival factor would give £2,458 and would still be an expected-payment calculation.

For the risky CE-funded investment, the model gives a 33.15% probability of a payment below £10,000. Its median is £12,092 and its fifth percentile £5,904. Mean-based funding has a 58.63% probability of falling below £10,000. Funding the same risky strategy to put its fifth percentile at £10,000 instead costs £6,921. That is a 95% model threshold, still not a guarantee. This comparison shows why a CE target and a protected spending floor require separate explanations.

These probabilities exclude finite-pool mortality risk. In the ideal deterministic sharing model, applying the same known uplift leaves the conditional relative payment distribution unchanged. Actual FTP allocations need their own outcome distribution.

Where the survival adjustment stops being a result

Suppose, solely for the initial explanation, every surviving member receives a known gross mortality uplift 1/s_τ, investment risk is independent of mortality, and the same investment distribution applies conditional on survival. Then the conditional payment is A G_T/s_τ, and CRRA homogeneity gives

This is a planning baseline under an idealised deterministic mortality-uplift assumption. It is not a proven funding rule for finite, heterogeneous FTP, and it is not a complete utility valuation including mortality and bequests.

If the model's gross factor already includes survivor allocations, multiplying its CE funding amount by survival again counts the same sharing benefit twice. For an actual scale-invariant policy, define Q as the complete per-pound member payment conditional on the relevant survival event, including investments, charges, sharing and releases, and use A=C/CE(Q | survival). If the policy depends on the member's amount relative to other balances, Q may itself depend on A; the funding equation must then be solved with the actual pool rather than by simple division.

A small homogeneous example demonstrates the limitation without relying on FTP. At one terminal settlement, n equal original accounts are shared among N surviving members. Conditional on a given member surviving, N=1+Binomial(n−1,s). The realised uplift is n/N. For γ=3, its mortality-only CE is

With n=5 and s=0.8, this is 1.16945, below the ideal uplift 1.25. Giving the same conditional CE therefore needs 6.89% more capital than the ideal survival-adjusted calculation. With n=100, the difference falls to 0.373%. These are exact calculations for this toy terminal pool; they are not results for a sequence of heterogeneous FTP settlements. The conditional survivor-count construction follows the standard tontine model in Milevsky and Salisbury, Optimal retirement income tontines, pp. 9–10.

Per-rung CE and lifetime wellbeing

The research objective should distinguish four questions: how much to allocate to each rung; which investments to hold; what payments the allocation policy actually delivers; and how the resulting lifetime income is valued.

Under additive lifetime utility with no bequest motive, a common form is Σ β_T s_T E[u(c_T) | alive at T]. If each rung is the whole of that year's consumption, and its conditional CE equals its target under that same utility function, the ladder has the same expected additive lifetime utility as that deterministic target schedule. This does not establish that the schedule or strategy is optimal.

Background income changes the calculation to u(background income + rung payment). Rung-by-rung CRRA applied only to the payment can then misstate the marginal cost of risk. Bequests, joint-life income, changing spending needs and access to other savings require additional terms. Correlation across years matters for shortfall patterns and adaptive spending; with strictly additive utility and fixed marginal consumption distributions, correlation alone does not change expected utility.

Milevsky and Salisbury optimise lifetime consumption under explicit mortality and budget constraints. Their setup illustrates why a funding identity and a lifetime optimum are separate results. Milevsky and Salisbury, pp. 8–10.

Risk-bearing and guarantee pricing must also stay separate. Chen, Nguyen and Sehner model market-linked tontines with and without guarantees, distinguish pricing from expected-utility assessment, and find preferences depend on investment conditions and risk aversion. Their product is homogeneous and differs from the proposed heterogeneous FTP ladder. Unit-Linked Tontine: Utility-Based Design, Pricing and Performance (2022).

Members may use different beliefs about their own survival and that of peers. These can alter perceived product value even when the operational mortality basis is fixed. A personal preference should not silently change the mortality inputs used to test pool fairness. Chen, Hieber and Rach, Optimal retirement products under subjective mortality beliefs (2021).

Further research

The companion paper, Initial rung funding and simulated FTP income, tests one survival-adjusted allocation using actual heterogeneous FTP paths. The wider question is whether that allocation remains defensible across specified continuation, investment and payment policies. If the adjustment materially understates required capital, the design must alter initial allocations, target payments or the level of protection. The comparison must report both expected utility and the size and frequency of income shortfalls.

What a completed model must show

• A dated schedule of initial investments, target payments, releases and actual payments.
• A traceable separation of investment gains, survivor allocations, charges and distributions.
• The return model and risk preference underlying each CE amount.
• Shortfall measures alongside utility measures, and a separate specification for any protected spending floor.
• The mortality basis, the pool constraints and the treatment of admissions, transfers, large settlement batches and final survivors.

The accompanying calculation source reproduces these illustrations using Python's standard library. It checks the investment CE by numerical integration and the finite-pool expression by summing the binomial outcomes. The inputs can be changed; the resulting numbers remain model outputs until the assumptions and operating policy have been established.