Closed-form tontine payouts under a trend-adjusted affine longevity model

We study pre-committed payout design for a retirement tontine whose members share idiosyncratic and systematic longevity risk. Mortality is decomposed into a deterministic interacted-Gompertz age–period baseline and a unit-mean stationary affine residual factor. Population death counts are conditionally Poisson, whereas survivors in the closed cohort are conditionally binomial given the common factor path. We prove an exponential-affine transform for finite-cohort survivor moments and use it to obtain an exact backward recursion for the optimal pre-committed payout schedule under representative-member CRRA welfare. In Canadian data calibrated on ages 65–100, longevity pooling is worth 30.9%–32.1% of initial wealth relative to individual drawdown under a zero-bequest benchmark, while an idealized fair annuity adds only 0.11%–0.21% beyond the tontine. A ±10% change in the mortality-improvement rate costs up to 6.71%, disagreement between Lee–Carter and CBD up to 4.58%, and state-dependent adaptation at most 0.16%; by contrast, designing specifically for the fitted residual factor is worth under 0.002%. A bootstrap diagnostic rejects the joint null of the parsimonious baseline and no residual layer, while recovery experiments document substantial finite-sample uncertainty in persistence. Estimates for the United Kingdom, Australia, the United States, and Japan show that persistent residual variation is not specific to Canada.