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
The calculation uses one declared law for death events. The case in which no members survive 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?
The note separates the proposed allocation from its numerical audit. It sets out admission tests, calculates allocations, and produces independent certificates and settlement instructions. Synthetic populations test the process with members of different ages and account sizes.
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?
The note calculates allocations in death order and checks whether the pool can continue between scheduled payouts. It derives conservative certificates covering specified numbers of deaths and measures their calendar coverage. Paying the same proportion of every account leaves 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
The calculation enumerates the possible events in small pools and covers specified numbers of deaths in the next batch. 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
The paper uses certainty equivalents to work back from nominal payment targets to initial investments. It then tests the ideal survival adjustment against a finite-pool counterexample.
Scope. The analysis uses 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
The experiment prices forty annual targets and compares pooled and unpooled ladders, each funded with £100,000. It follows investments, survivor allocations, releases and payments through three cohorts with different ages and account sizes.
Scope. The experiment runs 1,000 paths for each of three starting pools: 50, 200 and 500 members. It uses synthetic independent mortality, specified investments and ordinary FTP allocations in death order. 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