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Money-Back Tontines for Retirement Decumulation: Neural-Network Optimization under Systematic Longevity Risk

German Nova Orozco, Duy-Minh Dang, Peter A. Forsyth

Abstract

Money-back guarantees (MBGs) are features of pooled retirement income products that address bequest concerns by ensuring the initial premium is returned through lifetime payments or, upon early death, as a death benefit to the estate. This paper studies optimal retirement decumulation in an individual tontine account with an MBG overlay under international diversification and systematic longevity risk. The retiree chooses withdrawals and asset allocation dynamically to trade off expected total withdrawals (EW) against the Conditional Value-at-Risk (CVaR) of terminal wealth, subject to realistic investment constraints. The optimization is solved under a plan-to-live convention, while stochastic mortality affects outcomes through its impact on mortality credits at the pool level. We develop a neural-network based computational approach for the resulting high-dimensional, constrained control problem. The MBG is priced ex post under the induced EW--CVaR optimal policy via a simulation-based actuarial rule that combines expected guarantee costs with a prudential tail buffer. Using long-horizon historical return data expressed in real domestic-currency terms, we find that international diversification and longevity pooling jointly deliver the largest improvements in the EW--CVaR trade-off, while stochastic mortality shifts the frontier modestly in the expected direction. The optimal controls use foreign equity primarily as a state-dependent catch-up instrument, and implied MBG loads are driven mainly by tail outcomes (and the chosen prudential buffer) rather than by mean payouts.

Money-Back Tontines for Retirement Decumulation: Neural-Network Optimization under Systematic Longevity Risk

Abstract

Money-back guarantees (MBGs) are features of pooled retirement income products that address bequest concerns by ensuring the initial premium is returned through lifetime payments or, upon early death, as a death benefit to the estate. This paper studies optimal retirement decumulation in an individual tontine account with an MBG overlay under international diversification and systematic longevity risk. The retiree chooses withdrawals and asset allocation dynamically to trade off expected total withdrawals (EW) against the Conditional Value-at-Risk (CVaR) of terminal wealth, subject to realistic investment constraints. The optimization is solved under a plan-to-live convention, while stochastic mortality affects outcomes through its impact on mortality credits at the pool level. We develop a neural-network based computational approach for the resulting high-dimensional, constrained control problem. The MBG is priced ex post under the induced EW--CVaR optimal policy via a simulation-based actuarial rule that combines expected guarantee costs with a prudential tail buffer. Using long-horizon historical return data expressed in real domestic-currency terms, we find that international diversification and longevity pooling jointly deliver the largest improvements in the EW--CVaR trade-off, while stochastic mortality shifts the frontier modestly in the expected direction. The optimal controls use foreign equity primarily as a state-dependent catch-up instrument, and implied MBG loads are driven mainly by tail outcomes (and the chosen prudential buffer) rather than by mean payouts.
Paper Structure (48 sections, 74 equations, 12 figures, 6 tables, 1 algorithm)

This paper contains 48 sections, 74 equations, 12 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Tontine modeling
  3. Modeling of individual tontine accounts
  4. Individual fairness condition
  5. Pool-level budget constraint
  6. Large-pool approximation
  7. Mortality models
  8. Deterministic mortality
  9. Stochastic mortality
  10. From stochastic mortality models to tontine gain rate.
  11. Money-back guarantee overlay
  12. Description
  13. MBG pricing load
  14. Stochastic control framework
  15. Index dynamics
  16. ...and 33 more sections

Figures (12)

  • Figure 8.1: Validation of the EW--CVaR efficient frontiers in the synthetic market. Reference frontiers with and without a tontine overlay, labelled respectively as "No Tontine (PDE Control)" and "Tontine (PDE Control)", are compared with the corresponding NN frontiers, together with the constant withdrawal/constant allocation benchmark. Units: thousands of dollars.
  • Figure 9.1: EW--CVaR efficient frontiers for domestic--only (two--asset) and internationally diversified (four--asset) portfolios, with and without a tontine overlay. Constant--weight benchmarks are shown as single points. Units: thousands of real AUD.
  • Figure 9.2: EW--CVaR efficient frontiers for the four--asset tontine portfolio under deterministic table mortality and stochastic mortality (LC and CBD models). Units: thousands of real AUD.
  • Figure 9.4: Optimal fraction in domestic equities in the two--asset (domestic--only) tontine with stochastic mortality (LC model) for a representative point on the efficient frontier ($\gamma=1.5$). Colours show the fraction of wealth invested in domestic equities as a function of time and real wealth. Units: thousands of real AUD.
  • Figure : (a) Australian equity index (domestic)
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 2.1: Mortality inputs in simulation
  • Remark 3.1: Timing convention
  • Remark 4.1: Minimum withdrawals under insolvency
  • Remark 5.1: Discounting and mortality weighting
  • Remark 5.2: Pre-commitment and induced time-consistent interpretation
  • Remark 7.1: Interpreting $f_g$ in starting--rate terms
  • Remark 7.2
  • Remark 7.3: Inflation treatment in MBG pricing
  • Remark 9.1