Mortality-weighted stake
ci = qisi / (1 − qi)Fair Tontine Engine — implementing the Fair Transfer Plan (FTP)
Exact fairness, explained transfer by transfer
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.
No negative off-diagonal allocation
Zero diagonal
Every estate sums to one
Fairness residual within tolerance
Infeasible pools rejected
Sabin and NG-IPF agree
No unallocated penny
Deterministic preset results
05 / Research frontier
The atomic transfer rule is the foundation
Current research extends the single-death FTP into simultaneous events, delayed notification, proof-of-life settlement and target-date retirement ladders.
01Single-death FTP
02Pair and count-specific layers
03Hall feasibility gates
04Chronological event replay
05Shadow mortality-credit ledger
06Proof-of-life settlement
07Adaptive target-date ladder