A note on bequest preferences in utility maximisation for modern tontines

Paper: arXiv 2501.08972

Authors: Thomas Bernhardt

Abstract

In this short note, we address two issues in the literature about modern tontines with bequest and utility maximisation: how to verify optimal controls and the decreasing allocation of funds in the tontine. We want to raise awareness in the actuarial community about the dual approach to solve optimal control problems when working with power utilities. Additionally, we point out that bequest preferences should be time-dependent or otherwise yield unrealistic investment strategies. We base our attempt at modelling bequest preferences on common sense rules like 100% payback upon death at the start that vanishes over time. Our modelling shows that the resulting investment strategy almost linearly adjusts the allocation in the tontine from 0% to 100% over time.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of utility maximisation in modern tontines using a dual approach to resolve verification issues with power utility. However, it is purely theoretical and lacks empirical backtesting or real-world dataset validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Tontine Utility Maximisation & Address Bequest Issues] --> B[Input: Power Utility & Time-Dependent Bequest Rules]
    B --> C[Methodology: Apply Dual Approach to Verification of Optimal Controls]
    C --> D[Modeling: Simulate Bequest Preference Vanishing Over Time]
    D --> E[Outcome 1: Validated Dual Approach Framework for Actuarial Models]
    D --> F[Outcome 2: Investment Strategy Adjusts Tontine Allocation 0% to 100% Linearly]