Papers and calculations
The research papers
From the first allocation question to funding, payments and pool continuation.
Start with the question you want to examine. The fair-sharing explanation introduces FTP; the notes below contain the derivations, examples and checks. The funding and simulation papers connect the allocation rules to the investment ladder.
Fair tontine transfers for arbitrary death batches
When can the deceased committed balances be shared fairly?
The death-set formulation makes fairness and conservation a transportation problem. It gives existence conditions for individual members and groups, the common survivor-weight rule, and the precise entropy interpretation of ‘closest fair’.
Assumptions and verification
One declared event law. The all-dead event needs explicit treatment. Under fixed margins, different positive member-only proportional references give the same entropy minimiser.
Fairness and feasibility: §§1–2. Nominal-gain interpretation: §3. Count conditioning: §4.
Auditable FTP for heterogeneous pools
How is an allocation proposed and then checked separately?
Admission gates, compressed allocation calculations, independent numerical certificates and settlement instructions, with synthetic heterogeneous population tests.
Assumptions and verification
The certificate checks a stated mathematical rule under declared inputs. It does not validate mortality estimates, death records or a pension product.
Read ‘The operating contract’, ‘A separate certificate for numerical fairness’ and the mortality-calibrated scenarios.
Keeping a tontine fair as the pool changes
Can a fair transfer create an infeasible successor?
Exact examples separate current fairness from the ability to continue. The note connects continuation constraints with death ordering and the mortality ledger.
Assumptions and verification
The event convention and horizon matter. A chronological-death result is not an unrestricted calendar-batch guarantee, and a closed capital pool differs from a distributing pension.
The opening examples explain the mechanism; the technical appendix records the mathematical conditions.
FTP between scheduled payouts
What can we check between distributions, and what do payouts change?
Constrained chronological allocation, a fast ordinary-policy check, conservative death-budget certificates and calendar-coverage calculations. Common proportional payouts leave relative feasibility unchanged.
Assumptions and verification
Large-pool replay results cover audited states and their immediate successors. Coverage of a conservative certificate is not a failure probability. Actual member-specific distributions still have to be specified.
See ‘What the next payout date changes’ and ‘A guarantee covering more than the sampled histories’.
Keeping the next death batch fair
Can the next pair or triple remain fair after this batch is allocated?
A constrained batch calculation repairs explicit failures of ordinary FTP. Nearby rational policies satisfy current fairness exactly and leave every covered successor feasible. A separate countercase proves that some states cannot be repaired by reallocation.
Assumptions and verification
Small-pool, enumerated current events and specified next-count layers. Exact feasibility of the corrected policies does not establish their exact entropy optimality or fairness throughout the ladder.
See the pair/triple examples, ‘Verification and computational scope’ and ‘Some pools cannot be repaired by reallocating credits’.
Funding a target-date tontine ladder
Defines the certainty-equivalent growth factor, works back from nominal payment targets to initial investments, and tests the ideal survival adjustment against a finite-pool counterexample.
Scope. Established expected-utility mathematics and hypothetical teaching examples. The ladder experiment applies the funding baseline to three heterogeneous cohorts. Preferences, policy calibration and complete continuation are further research questions.
Initial rung funding and simulated FTP income
Prices forty annual targets, compares separately funded pooled and unpooled ladders from the same £100,000, and follows investments, survivor allocations, releases and payments through three fixed heterogeneous cohorts.
Scope. 1,000 paths each for 50, 200 and 500 members; synthetic independent mortality, specified investments and ordinary chronological FTP. Numerical audits and coverage records accompany the outcomes. Closure rights, batch continuation and reliable extreme-age estimates remain open.
The next research
The research programme connects the remaining funding, distribution and continuation questions.
Capacity after credits and the proposed portability principle