arxiv: 2604.09342 · v1 · submitted 2026-04-10 · 💱 q-fin.MF

Recognition: 2 theorem links

· Lean Theorem

Optimal Annuitization Time under a Mortality Shock

Matteo Buttarazzi

Pith reviewed 2026-05-10 16:39 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords optimal annuitizationmortality shockoptimal stoppinglife annuityvalue functionclosed-form solutionhealth deteriorationretirement planning

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The pith

Explicit closed-form solutions exist for the value function and optimal annuitization boundaries when retirement wealth faces a sudden permanent mortality shock.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives explicit closed-form solutions for the value of retirement wealth and the best times to convert it irreversibly into a life annuity when the owner can suffer a sudden health deterioration that permanently raises mortality. The decision is framed as an optimal stopping problem with exactly two states: good health before the shock and poor health after it, where the shock arrives at an exponentially distributed random time. A sympathetic reader would care because the solutions reveal how the possibility of worse health shifts the trade-off between keeping wealth in a geometric Brownian motion fund and locking in annuity payments, including the roles of annuity money's worth, investment returns, and bequest motives. Numerical comparisons to the constant-mortality case show that both the likelihood and severity of the shock materially change the optimal boundaries.

Core claim

The paper derives explicit closed-form solutions for the value function and the associated optimal stopping boundaries in an optimal annuitization problem under a mortality shock. An individual's retirement wealth follows geometric Brownian motion and can be converted at any time into a life annuity. The mortality shock is a single permanent jump at an exponentially distributed time, so the problem reduces to optimal stopping across two health states with different mortality intensities. The resulting analytical expressions characterize both the value function in each state and the optimal annuitization thresholds, with the strategy governed by the relative attractiveness of the annuity, the

What carries the argument

The two-state optimal stopping framework with an exponential mortality shock time, where the value function satisfies ordinary differential equations in each health state subject to value-matching and smooth-pasting conditions at the free boundaries.

If this is right

The optimal annuitization boundary in good health differs from the boundary in poor health because of the change in mortality intensity.
Higher money's worth of the annuity pulls the optimal stopping boundary downward in both states.
Stronger bequest motives raise the thresholds, delaying annuitization.
The likelihood of the health shock alters the value of waiting in the initial health state relative to the constant-mortality benchmark.
Explicit formulas allow immediate computation of the regions in which it is optimal to annuitize immediately versus continue investing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework suggests that personalized annuity recommendations should incorporate an individual's estimated probability of future health shocks rather than using average mortality tables.
Similar closed-form techniques could be applied to other irreversible choices under jump risk, such as the timing of long-term care insurance purchases.
If health shocks are correlated across family members, the model implies that joint annuitization decisions would require an expanded state space.

Load-bearing premise

The mortality shock is modeled as a single permanent jump occurring at an exponentially distributed time, with the annuitization decision treated as an optimal stopping problem across exactly two health states.

What would settle it

Empirical records of when individuals actually annuitize after experiencing a documented sudden health decline, compared against the model's predicted boundaries for given annuity money's worth, investment volatility, and bequest strength, would falsify the claim if the observed timings deviate systematically.

Figures

Figures reproduced from arXiv: 2604.09342 by Matteo Buttarazzi.

**Figure 1.** Figure 1: Stopping and continuation regions for Individuals A and B. The green area indicates the stopping region where it is optimal to annuitize, and the grey area represents the continuation region where individuals postpone annuitization. The left panel shows Individual A’s optimal annuitization threshold of x 2 = 68893.49. The right panel shows Individual B’s optimal strategy. Before the shock, annuitization is… view at source ↗

**Figure 2.** Figure 2: (a) The annuity’s money’s worth coefficients δi(∆) in the pre-shock (blue solid line) and post-shock (yellow dashed line) states. (b) The index Mi(∆) in the pre-shock (blue navy line) and post-shock (brown dashed line) states. (c) Optimal annuitization thresholds x 2 l (∆) (red line) and x 2 h (∆) (green line). The coefficients Mi defined in (3.1) provide a useful summary measure of the attractiveness of a… view at source ↗

**Figure 3.** Figure 3: (a) The index Mi(λl) in the pre-shock (navy solid line) and post-shock (brown dashed line) states. (b) Optimal annuitization thresholds x 2 l (λl) (red line) and x 2 h (λl) (green line). 5. Conlusions This study examines the optimal annuitization decision for an individual exposed to the risk of a sudden and permanent health deterioration. We derive explicit analytical characterizations of the value functi… view at source ↗

read the original abstract

In this paper, we derive explicit closed-form solutions for the value function and the associated optimal stopping boundaries in an optimal annuitization problem under a mortality shock. We consider an individual whose retirement wealth is invested in a financial fund following the dynamics of a geometric Brownian motion and has the option at any time to irreversibly convert their wealth into a life annuity. The individual faces a sudden, permanent health deterioration occurring at a random, exponentially distributed time, and the annuitization decision is modelled as an optimal stopping problem across two health states. Our analytical expressions characterise both the value function and the optimal timing of annuitization. The results provide clear economic intuition: the optimal strategy is governed by the critical interplay between the relative attractiveness of the annuity (money's worth), the financial returns from the investment fund, and bequest motives across different health states. A numerical analysis compares the optimal annuitization strategy of an individual facing a health shock against a benchmark case with constant mortality, highlighting how the likelihood and severity of a health shock significantly alter optimal annuitization behaviour.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives closed-form solutions for the optimal annuitization boundary when mortality can jump once at an exponential time, extending standard optimal stopping setups in a clean way.

read the letter

The punchline is that the authors treat annuitization as an optimal stopping problem across two health states and actually produce explicit expressions for the value functions and the free boundaries. Wealth follows GBM, the annuity pays a fixed multiple of wealth at conversion, and the shock arrives at rate lambda and raises the force of mortality permanently. The pre-shock HJB equation is an inhomogeneous ODE whose homogeneous solutions are power functions; after matching value and derivative at the boundary you get a low-dimensional algebraic system that solves explicitly in the parameters. That is the new piece relative to the constant-mortality literature they cite. The derivations follow the usual verification steps for optimal stopping and the stress-test note confirms the ODEs admit the claimed closed forms, so the central claim holds up on its own terms. The economic takeaway they emphasize is intuitive: higher shock probability or severity tilts the boundary, and bequest motives interact with the money's worth ratio in the expected direction. The soft spots are modest but real. The model is deliberately narrow—one irreversible jump, no recovery, no stochastic interest rates or inflation—so the closed forms are tractable but the quantitative importance of the shock depends on how realistic the two-state setup is for actual retirees. The abstract mentions numerical comparisons to the constant-mortality benchmark, yet without seeing the figures or sensitivity tables it is hard to judge how large the differences are or whether they survive reasonable parameter ranges. This is a paper for readers already working on retirement-finance stopping problems or actuarial optimal-control models. A specialist who needs the explicit boundary formulas or wants to extend the framework will find it useful; a general finance reader will not. It is formally grounded enough and the derivations are checkable, so it deserves a serious referee rather than a desk reject. I would send it out, with the usual request that reviewers verify the algebra and ask for clearer numerical robustness checks.

Referee Report

0 major / 2 minor

Summary. This paper models the optimal timing of annuitization for an individual with wealth invested in a geometric Brownian motion fund, facing a sudden permanent health deterioration (mortality shock) occurring at an exponentially distributed time. The annuitization decision is formulated as an optimal stopping problem across two health states (pre- and post-shock), and the authors derive explicit closed-form solutions for the value functions and associated optimal stopping boundaries, along with numerical comparisons to a constant-mortality benchmark.

Significance. The derivation of explicit closed-form solutions for the pre- and post-shock value functions (via the linear second-order ODEs generated by the GBM operator plus the jump-intensity term) and the explicit algebraic system for the free boundaries is a clear technical strength, as is the provision of economic intuition on the interplay between annuity money's worth, investment returns, bequest motives, and shock parameters. If the derivations hold, the results and numerical analysis contribute to the literature by showing how the likelihood and severity of health shocks alter optimal annuitization behavior relative to constant-mortality cases.

minor comments (2)

[Abstract] Abstract: the statement that 'explicit closed-form solutions' are derived would benefit from a brief qualifier noting that these arise from solving the regime-specific linear ODEs and enforcing value-matching/smooth-pasting conditions at the free boundaries.
[Numerical Analysis] Numerical analysis section: the parameter values chosen for the GBM drift/volatility, force of mortality in each state, money's worth, and bequest coefficient should be tabulated or explicitly listed to facilitate replication and interpretation of the reported strategy differences.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation of minor revision. The referee's summary correctly identifies the core modeling framework, the derivation of closed-form value functions and free boundaries, and the numerical comparison to the constant-mortality benchmark. We are pleased that the technical contribution and economic intuition are viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity: standard optimal stopping derivation on two-state Markov model

full rationale

The paper sets up a two-state optimal stopping problem with wealth following GBM and mortality shock as an exponential jump. Pre- and post-shock value functions are defined via the infinitesimal generator (second-order linear ODEs with constant coefficients plus jump intensity term), solved in closed form as linear combinations of power solutions plus particular solution. Optimal boundaries are then fixed by the standard value-matching and smooth-pasting conditions, producing an algebraic system solved explicitly in the parameters. This chain is self-contained mathematical derivation from the stated SDE and payoff structure; no fitted inputs are relabeled as predictions, no self-citations are load-bearing for the closed forms, and no ansatz or uniqueness result is smuggled in. The result is independent of any external fitted quantities or prior author work.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard stochastic-process assumptions and optimal-stopping theory, plus several free parameters that must be chosen or calibrated for the numerical section.

free parameters (4)

money's worth of the annuity
Relative attractiveness parameter governing the annuity payout versus investment returns
drift and volatility of the geometric Brownian motion
Parameters controlling the financial fund dynamics
force of mortality in each health state
Baseline and post-shock mortality intensities
bequest motive coefficient
Strength of bequest preference across health states

axioms (3)

domain assumption Wealth process follows geometric Brownian motion
Standard continuous-time model for the investment fund
domain assumption Mortality shock arrival time is exponentially distributed and permanent
Memoryless property used to create the two-state Markovian structure
standard math Optimal stopping theory applies directly to the two-regime value function
Invoked to characterize the continuation and stopping regions

pith-pipeline@v0.9.0 · 5475 in / 1428 out tokens · 69540 ms · 2026-05-10T16:39:34.241656+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
The problem is solved backwards... explicit solutions of the form A x^gamma + B x^delta plus particular solution... free boundaries fixed by value-matching and smooth-pasting
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
V(x, μ_l) = sup ... e^{-r_l τ} δ_l (X_τ − K) + integral terms with λ_l V(X_t, μ_h)

Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

Agarwal and D

R.P. Agarwal and D. O’Regan.Ordinary and partial differential equations: with special functions, Fourier series, and boundary value problems. Springer Science & Business Media, 2008

2008
[2]

Buttarazzi, T

M. Buttarazzi, T. De Angelis, and G. Stabile. Optimal annuitization with stochastic mortality: Piecewise deter- ministic mortality force.arXiv preprint arXiv:2509.13091, 2025

work page arXiv 2025
[3]

Buttarazzi and G

M. Buttarazzi and G. Stabile. The market value of optimal annuitization and bequest motives. InMathematical and Statistical Methods for Actuarial Sciences and Finance, pages 67–73. Springer, 2024

2024
[4]

Duffie, J

D. Duffie, J. Pan, and K. Singleton. Transform analysis and asset pricing for affine jump-diffusions.Econometrica, 68(6):1343–1376, 2000. M. Buttarazzi: School of Management and Economics, Dept. ESOMAS, University of Torino, Corso Unione Sovietica, 218 Bis, 10134, Torino, Italy. Email address:matteo.buttarazzi@unito.it

2000