Mortality-weighted stake
ci = qisi / (1 − qi)Fair Tontine Engine — implementing the fair transfer plan (FTP)
How a fair transfer is calculated
The Fair Tontine Engine, which implements the fair transfer plan (FTP), allocates a deceased member’s account among the surviving pool using Sabin’s exact method. Each exact share is a nominal-gain share adjusted once for the recipient and once for the estate.
01 / Worked example
Follow one estate through the pool
Select the deceased member, compare nominal gain with Sabin’s exact allocation and inspect the calculation for any recipient.
Deceased member
Allocation
Display
Estate distributed
C£70,000q = 5.00%Selected transfer
C → BNominal share40.30%×Recipient correction0.9513×Estate correction1.1180=Exact share42.86%
| Member | Death probability | Account | θ | Nominal weight n | Exact weight w | Recipient correction | Estate correction |
|---|---|---|---|---|---|---|---|
| A | 2.00% | £100,000 | 0.1961 | 0.1927 | 0.1667 | 0.8650 | 0.9688 |
| B | 3.00% | £90,000 | 0.2647 | 0.2628 | 0.2500 | 0.9513 | 0.9829 |
| C | 5.00% | £70,000 | 0.3431 | 0.3478 | 0.4167 | 1.1979 | 1.1180 |
| D | 4.00% | £50,000 | 0.1961 | 0.1967 | 0.1667 | 0.8473 | 0.9640 |
Pool feasibilitymax θ = 0.3431Gate: 0.5000
Feasible
02 / Central result
The exact plan keeps the nominal-gain structure
Exact fairness changes the weights, not the form of the allocation. Every exact transfer is a nominal-gain share adjusted once for the recipient and once for the estate being distributed.
Nominal-gain shareANGij
×Recipient correctionwi / ni
×Estate correction(1 − nj) / (1 − wj)
=Exact shareASij
Read the derivation
Nominal weight
ni = ci / ΣckNominal-gain share
ANGij = ni / (1 − nj)Sabin exact share
ASij = wi / (1 − wj)Aij is the fraction of member j’s estate paid to surviving member i. The diagonal is zero and every estate column sums to one.
03 / Methods and benchmarks
Exactness, concentration and speed are separate choices
Nominal-gain iterative proportional fitting (NG-IPF) provides an independent numerical route to the same separable exact matrix as Sabin’s method. Circular exact meets the fairness condition with two recipients per estate and much higher concentration.
Sabin exact
Computes the exact separable weights in O(m); materialising the full transfer matrix remains O(m²).
NG-IPF
A numerical route to the same separable exact matrix. Useful as an independent check.
Circular exact
Fair with two recipients per estate, at the cost of much higher concentration.
Nominal gain
Intuitive and fast, but approximate. Its fairness error remains visible in the benchmark.
Reference synthetic pool · March 2026
Expected pool£255,953
Maximum θ0.214440
FeasibilityPass
| Method | Time | Max fairness error | Recipients / estate | HHI | Distance to NG1 |
|---|---|---|---|---|---|
| Nominal gain | 0.2 ms | £7,722 | 19 | 0.1042 | 0.0000 |
| Sabin exact | 1 ms | £0 | 19 | 0.1174 | 0.1721 |
| NG-IPF | 0.3 ms | £0 | 19 | 0.1174 | 0.1721 |
| Circular exact | 0.5 ms | £0 | 2 | 0.8038 | 3.6939 |
1 Unnormalised Frobenius distance from the nominal-gain transfer matrix: √Σi≠j(Aij − ANGij)². Zero means identical matrices; lower values remain closer to nominal gain.
Indicative runtime
Scaling the exact methods
Indicative runtimes from the March 2026 research run, rounded to avoid false precision. The complete allocation matrix remains O(m²) to materialise.
04 / Implementation
From exact weights to penny-balanced payments
Implementation follows four controls: pool feasibility, exact-weight calculation, transfer attribution and cash reconciliation.
Check the pool
Calculate θ and reject a pool that fails the feasibility condition.
Compute exact weights
Run Sabin’s bisection and verify the weights with NG-IPF.
Explain every transfer
Report the nominal share and both correction factors for every payment.
Pay and reconcile
Round in integer pence, conserve each estate and record any residual adjustment.
The numerical checks require non-negative allocations, no transfer back to the deceased member, and shares that distribute each estate in full. They also measure each member’s fairness residual against the stated tolerance and compare Sabin’s weights with the independent NG-IPF calculation.
Cash settlement is checked separately. Amounts are rounded in integer pence, and any residual adjustment is recorded so that no estate money remains unallocated.
05 / Related research
From one transfer to a pension
This example calculates one transfer. A pension also needs rules for successive deaths, investment and payments. The September 2026 papers and simulations examine those questions under stated assumptions.
Several deaths
The fair-sharing research asks when members’ committed balances can be allocated fairly after a batch of deaths.
Pool continuation
The capacity research examines whether a fair allocation leaves the remaining pool able to share fairly.
Investment and income
The ladder experiment follows investments, survivor allocations and dated payments.
Remaining decisions
The research programme sets out the continuation and payment rules still needed for a complete pension.