General bounds on functionals of the lifetime under life table constraints in a joint actuarial-financial framework

Introduction to the show: ident: Quantitative Finance. The day in quantitative finance: models of markets, risk and prices.

Ruth: We're looking at the paper: "General bounds on functionals of the lifetime under life table constraints in a joint actuarial-financial framework".

Warren: Here's the gist of the paper. In life insurance, life tables provide survival probabilities at integer ages but leave the distribution of deaths within each year unspecified, which is particularly relevant for modern insurance products such as variable annuities whose payoffs depend jointly on lifetime uncertainty and financial market performance The two most common approaches to mortality modeling in life insurance rely either on fractional age assumptions or mortality rate models <ref:2603.06238#pg2>.

Ruth: Fractional age assumptions, like the uniform distribution of deaths (UDD), constant force of mortality (CFM), and Balducci assumption, produce different continuous-time survival distributions consistent with the same life table at integer ages <ref:2603.06238#pg3>.

Warren: Mortality rate models involve modeling mortality continuously in time through a mortality rate or force of mortality, with examples including the Lee–Carter models and cohort-based extensions <ref:2603.06238#pg4>.

Ruth: Both approaches introduce model risk because they rely on structural assumptions about mortality dynamics <ref:2603.06238#pg2>.

Paper summary: Ruth: We’ve just touched on how this paper addresses the core issue: needing to bridge mortality uncertainty with financial market uncertainty when valuing complex products like variable annuities. The central thesis is that standard approaches relying on fractional age assumptions or specific rate models are too restrictive because the resulting valuations become extremely sensitive to those initial structural choices.

Warren: And the paper proposes a new way of looking at this, moving away from purely actuarial methods into a joint actuarial-financial framework where mortality ambiguity interacts with asset dynamics and embedded guarantees.

Iris: Specifically, they define a complete probability space Ht equals Ft union Gt to handle both uncertainty streams simultaneously, which opens the door to applying stochastic control techniques to the mortality processes in ways that weren't previously explored.

Ruth: The main contribution then is deriving upper and lower bounds for functionals of the lifetime that are compatible with observed life tables at integer ages and the given financial market. These bounds are derived under two distinct assumptions, one strict where mortality paths match survival probabilities almost surely, and another more relaxed setting where it only holds in expectation.

Warren: What matters is that these bounds characterize the extreme contract values—the worst-case and best-case scenarios—over all mortality processes that remain compatible with the observed life table information and the financial market.

Iris: They also show how these bounds behave depending on whether you’re looking at survival-contingent or death-contingent payoffs, noting an asymmetry where lower bounds can be far below baseline for survival liabilities and upper bounds can be far above for death liabilities.

Conclusion: Ruth: Looking at "General bounds on functionals of the lifetime under life table constraints in a joint actuarial-financial framework," Jean-Loup Dupreta and Edouard Mottec have provided a model-free mortality tool. It’s not about finding one single answer, but defining the boundaries of what's possible given the incomplete information we have from just integer ages.

Warren: The implication for us as traders and risk managers is that this framework gives you a robust way to quantify how much uncertainty you are taking on because of that missing death distribution data, especially when dealing with long-term contracts like variable annuities.

Iris: Essentially, the paper moves beyond just comparing different fractional age assumptions and shows how to derive sharp upper and lower bounds for those functionals over the set of admissible mortality intensities compatible with both life tables and arbitrage-free financial valuation.

Ruth: It's a tool that helps incorporate mortality model risk directly into pricing decisions, allowing you to test how adverse mortality dynamics could impact your liabilities while accounting for the interaction with asset price movements.

Warren: So, we’re left with these bounds as a practical way to manage the uncertainty arising from incomplete life table information when pricing these modern financial products.