Closed-form Optimal Tontine Payouts in an Affine GARCH Model

We consider a discrete-time tontine with proportional payouts, a bequest motive, and stochastic mortality, where a representative participant maximizes expected constant relative risk aversion (CRRA) utility of consumption and terminal wealth. To ensure a stable and operationally viable consumption path, we solve for the optimal pre-commitment payout rate (π t), leading to closed-form backward recursion based on the market portfolio's cumulative return moment generating function (m.g.f.). We show that the difference of such pre-commitment to the global optimal is economically negligible. An empirical illustration for a T = 35 year retirement horizon compares optimal payout profiles under an affine GARCH(1,1) model, a matched lognormal benchmark and an optimal risk-free cash account. The analysis demonstrates that ignoring dynamic volatility yields relative payout deviations of up to 4.5% at inception, with modest Wealth Equivalent Gain (WEG) from model mis-specification. The WEG could be up to 11% if acting optimal but avoiding risky investments. We also detect a WEG of up to 6.0% due to mismatch frequency, i.e., the operational rigidity of weekly rather than hourly administrative rebalancing. These findings, supported by robustness checks across major equity indices, underscore the necessity of integrating discrete-time GARCH dynamics with the proper rebalancing frequency into modern actuarial frameworks.