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arxiv: 2606.03158 · v1 · pith:ZC4MH75Ynew · submitted 2026-06-02 · 💱 q-fin.PM

Portfolio Choice with Competing Precautionary and Accumulation Goals

Steven Campbell , Agostino Capponi , Ananya Parashar This is my paper

Pith reviewed 2026-06-28 07:41 UTC · model grok-4.3

classification 💱 q-fin.PM
keywords portfolio choiceprecautionary goalsretirement planningforced fundingnon-monotonic value functiondual goalsBlack-Scholes marketoption value
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The pith

The value function for a household with both a random emergency goal and a fixed retirement goal need not increase with wealth under forced funding.

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

This paper examines optimal portfolio choice for a household that must simultaneously pursue a random-deadline precautionary goal and a fixed-deadline accumulation goal in a Black-Scholes market. It establishes that a forced funding rule, requiring full payment of each goal whenever wealth suffices, creates a growth crowding-out effect and a deadline pressure effect while also producing non-monotonicity in the value function. A sympathetic reader cares because this non-monotonicity means a household just above the random-goal threshold can end up worse off than a slightly poorer one, since paying the random goal depletes resources needed for the fixed goal. The analysis further shows that an optional funding variant yields the greatest value of flexibility at intermediate wealth levels.

Core claim

Under forced funding of both a random-deadline goal and a fixed-deadline goal, the household maximizes a weighted sum of the probabilities of fully funding each goal. The resulting value function is not monotone in wealth: a household just above the random-goal threshold is forced to pay it when the shock arrives, depleting its wealth for the fixed goal and ending up worse off than a slightly poorer household that missed the random goal. This non-monotonicity is absent from single-goal benchmarks and arises purely from the interaction between the two goal types. The model also identifies a growth crowding-out effect in which precautionary saving distorts investment and a deadline pressure ef

What carries the argument

The forced funding rule, under which each goal is paid in full whenever the household's wealth meets or exceeds the required amount at shock arrival or deadline.

If this is right

  • The value function exhibits non-monotonicity in wealth near the random-goal threshold.
  • Precautionary saving for the random goal crowds out investment suited to the fixed goal.
  • A shorter effective saving horizon forces households to accept higher risk.
  • The option to decline the fixed goal at the terminal date is most valuable at intermediate wealth levels.

Where Pith is reading between the lines

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

  • Allowing households to choose the timing or size of goal payments could eliminate the non-monotonicity.
  • Empirical tests of household portfolios should separate single-goal from dual-goal cases to detect crowding-out patterns.
  • Policy designs that provide flexibility in goal funding may raise welfare by avoiding forced depletion effects.

Load-bearing premise

The forced funding rule is imposed exogenously rather than chosen by the household itself.

What would settle it

A numerical solution or controlled simulation that tracks the value function across wealth levels straddling the random-goal threshold and checks whether it decreases when crossing the threshold under forced funding.

Figures

Figures reproduced from arXiv: 2606.03158 by Agostino Capponi, Ananya Parashar, Steven Campbell.

Figure 1
Figure 1. Figure 1: Growth crowding-out effect under forced funding. Sensitivity of the value function [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Deadline pressure effect. Sensitivity of the value function [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sensitivity of the value function V R,D(0, w) (left panel) and the initial optimal risky position π ∗ (0, w) (right panel) to the random-deadline intensity λ under the public-college baseline [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity of the value function V R,D(0, w) (left panel) and the initial optimal risky position π ∗ (0, w) (right panel) to the preference weights under the public-college baseline calibration. We impose αD + αR = 1 and vary αD ∈ {0.1, 0.3, 0.5, 0.7, 0.9}. In Appendix C, we show that our sensitivity analysis remains qualitatively the same under plausible variations of the calibrated parameters. 6.4 Optio… view at source ↗
Figure 5
Figure 5. Figure 5: Ex ante option value (left panel) and terminal option value (right panel) as a function of wealth. [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Terminal option value ∆T (w) = EG[∆(w, G)] for different dispersions σG of the fixed-deadline goal distribution. Wealth w is fixed on the horizontal axis and the average is taken only over G, with future random-deadline uncertainty already embedded in V R,K [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ex ante option value (left panel) and optimal policy difference (right panel) under baseline [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Fixed-deadline single-goal benchmark. The clipped fixed-deadline HJB solution [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Random-deadline single-goal benchmark. The clipped random-deadline HJB solution [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Option value as fixed-goal priority per dollar varies. The horizontal axis is ( [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sensitivity of the initial two-goal value function [PITH_FULL_IMAGE:figures/full_fig_p054_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity to the specification of the random-deadline goal for the public-college baseline. [PITH_FULL_IMAGE:figures/full_fig_p055_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sensitivity to the fixed-deadline goal dispersion [PITH_FULL_IMAGE:figures/full_fig_p055_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sensitivity of the initial two-goal value function [PITH_FULL_IMAGE:figures/full_fig_p056_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Sensitivity of the initial two-goal value function [PITH_FULL_IMAGE:figures/full_fig_p056_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Sensitivity of the ex ante option value ∆ [PITH_FULL_IMAGE:figures/full_fig_p057_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Sensitivity of the terminal option value ∆ [PITH_FULL_IMAGE:figures/full_fig_p057_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Ex ante option value and terminal option value as a function of the fixed-deadline goal priority [PITH_FULL_IMAGE:figures/full_fig_p058_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Sensitivity of the ex ante option value ∆ [PITH_FULL_IMAGE:figures/full_fig_p058_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Ex ante option value and optimal policy difference under a lower arrival intensity [PITH_FULL_IMAGE:figures/full_fig_p059_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Value function convergence in T. Panel (a) plots the time-0 value function V (0, w; T) for different deterministic deadlines T, together with the terminal operator T K(w). Panel (b) plots the average absolute error over log wealth MAElog w(V, T K) = 1 log wmax−log wmin R log wmax log wmin |V (e x ) − T K(e x )| dx. As T → 0, the value converges to the terminal operator because there is no time to trade be… view at source ↗
Figure 22
Figure 22. Figure 22: Option value as the deterministic deadline varies. Panel (a) plots the ex ante option value [PITH_FULL_IMAGE:figures/full_fig_p060_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Small-λ limit. As the random-arrival intensity decreases, the two-goal value becomes increasingly Browne-shaped. The plotted reference is αR + αDV D Browne(0, w; G). At the opposite extreme, as λ → ∞, the random-deadline goal is resolved immediately. The limiting value is therefore the instant-arrival operator applied at time zero: JR[V D](0, w) = ER [PITH_FULL_IMAGE:figures/full_fig_p061_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Random-arrival asymptotics. Panel (a) plots [PITH_FULL_IMAGE:figures/full_fig_p062_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Ex ante option value as a function of the random-deadline survival probability Pr( [PITH_FULL_IMAGE:figures/full_fig_p062_25.png] view at source ↗
read the original abstract

We study optimal portfolio choice for a household simultaneously managing a random-deadline goal, such as a medical emergency or job loss, and a fixed-deadline goal such as retirement or college tuition. Under a forced funding rule, in which each goal is paid in full whenever affordable, the household maximizes a weighted sum of the probabilities of fully funding both goals in a Black--Scholes market. We identify two novel effects absent from single-goal models: a growth crowding-out effect, in which precautionary saving for the random goal distorts investment toward the fixed goal, and a deadline pressure effect, in which a compressed saving horizon forces excess risk-taking. A striking implication is that the value function need not be monotone in wealth: a household just above the random-goal threshold is forced to pay it when the shock arrives, depleting its wealth for the fixed goal, and ends up worse off than a slightly poorer household that missed the random goal but kept its wealth intact. This non-monotonicity is absent from all single-goal benchmarks and arises purely from the interaction between the two goal types under forced funding. We further study an optional funding variant in which the household may decline the fixed-deadline goal at time $T$ rather than being required to fund it. We characterize the ex ante option value, i.e., the full time-$0$ value of this flexibility and the terminal option value, i.e., its value at the funding decision node. We find that both options are most valuable at intermediate wealth levels where paying the fixed-deadline goal would substantially reduce the continuation value of the random-deadline problem.

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.

Referee Report

2 major / 1 minor

Summary. The paper studies optimal portfolio choice in a Black-Scholes market for a household simultaneously pursuing a random-deadline goal (e.g., medical emergency) and a fixed-deadline goal (e.g., retirement), maximizing a weighted sum of funding probabilities under an exogenous forced full-funding rule. It identifies a growth crowding-out effect and a deadline pressure effect, shows that the value function is non-monotone in wealth due to the interaction of the two goals, and analyzes the ex ante and terminal option values of an optional-funding variant for the fixed goal.

Significance. If the modeling assumptions hold, the work contributes to multi-goal portfolio choice by isolating interaction effects absent from single-goal benchmarks, including the novel non-monotonicity result and the characterization of option values being highest at intermediate wealth levels. The first-principles stochastic-control derivation in a standard market setting is a strength.

major comments (2)
  1. [§2] §2: The forced full-funding rule for the random goal is imposed exogenously rather than derived from the household's optimization problem. This rule is load-bearing for the central non-monotonicity claim (a household just above the threshold ends up worse off than one below), yet the paper provides no robustness analysis under endogenous choice, delay, or partial funding for the random goal (the optional variant in §5 applies only to the fixed goal at T).
  2. [Abstract, §5] Abstract and §5: The non-monotonicity and crowding-out results are presented as arising purely from the goal interaction under forced funding, but without reported verification that the non-monotonicity survives reasonable perturbations to parameters such as goal sizes, arrival intensities, or risk aversion (as would be needed to confirm it is not an artifact of specific modeling choices).
minor comments (1)
  1. [§2] Notation for the two value functions and the weighting parameter could be introduced more explicitly at the start of §2 to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address the two major comments point by point below. We agree that additional discussion and robustness checks would strengthen the paper and will revise accordingly.

read point-by-point responses

  1. Referee: §2: The forced full-funding rule for the random goal is imposed exogenously rather than derived from the household's optimization problem. This rule is load-bearing for the central non-monotonicity claim (a household just above the threshold ends up worse off than one below), yet the paper provides no robustness analysis under endogenous choice, delay, or partial funding for the random goal (the optional variant in §5 applies only to the fixed goal at T).

    Authors: The forced full-funding rule is an exogenous modeling assumption that captures situations where goals like medical emergencies require immediate full funding if affordable. This assumption is central to the non-monotonicity because it triggers wealth depletion upon goal arrival. The paper's focus is on analyzing the portfolio choice and value function under this rule, which is a standard approach in goal-based portfolio optimization. An endogenous funding model would be a valuable extension but would require a different setup. We will revise the manuscript to include a discussion of this limitation and potential implications of relaxing the assumption in the concluding section. revision: partial

  2. Referee: Abstract and §5: The non-monotonicity and crowding-out results are presented as arising purely from the goal interaction under forced funding, but without reported verification that the non-monotonicity survives reasonable perturbations to parameters such as goal sizes, arrival intensities, or risk aversion (as would be needed to confirm it is not an artifact of specific modeling choices).

    Authors: While the non-monotonicity is a structural consequence of the forced funding interaction—specifically, the forced payment depleting resources for the fixed goal—we acknowledge the value of explicit robustness checks. The analytical derivation shows the effect holds generally, but to address the referee's concern, we will add numerical sensitivity analysis in the revised version, varying key parameters like goal sizes, arrival rates, and risk aversion to confirm the persistence of non-monotonicity and crowding-out effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity; first-principles stochastic-control derivation

full rationale

The paper sets up a stochastic control problem in Black-Scholes with an exogenous forced-funding rule as a modeling assumption, then derives the value function, crowding-out and deadline-pressure effects, and non-monotonicity as consequences of the interaction between the two goal types. No fitted parameters are renamed as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The non-monotonicity result follows directly from the state jumps induced by the rule and is not equivalent to the inputs by construction. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on continuous-time stochastic control in a complete Black-Scholes market with exogenous forced funding; no free parameters, ad-hoc axioms, or invented entities are described in the abstract.

axioms (2)
  • domain assumption Black-Scholes market dynamics with constant coefficients
    Standard complete-market assumption invoked for the portfolio problem.
  • domain assumption Forced funding rule is imposed exogenously
    Central modeling choice that generates the interaction effects.

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Reference graph

Works this paper leans on

34 extracted references · 2 canonical work pages

  1. [1]

    Brookings Papers on Economic Activity , year =

    Lusardi, Annamaria and Schneider, Daniel and Tufano, Peter , title =. Brookings Papers on Economic Activity , year =

  2. [2]

    , title =

    Fulford, Scott L. , title =. Journal of Monetary Economics , year =

  3. [3]

    and Natvik, Gisle J

    Fagereng, Andreas and Holm, Martin B. and Natvik, Gisle J. , title =. American Economic Journal: Macroeconomics , year =

  4. [4]

    2025 , url =

    Real-Life Data: An Innovative Approach to Calculating Income Replacement Rates , institution =. 2025 , url =

    2025

  5. [5]

    arXiv preprint arXiv:2506.06654 , year=

    Goal-based portfolio selection with mental accounting , author=. arXiv preprint arXiv:2506.06654 , year=

    Pith/arXiv arXiv

  6. [6]

    Advances in Applied Probability , year =

    Sid Browne , title =. Advances in Applied Probability , year =

  7. [7]

    Jean L. P. Brunel , title =

  8. [8]

    Management Science , year =

    Agostino Capponi and Yuchong Zhang , title =. Management Science , year =

  9. [9]

    Chhabra , title =

    Ashvin B. Chhabra , title =. The Journal of Wealth Management , year =

  10. [10]

    Rossi , title =

    Francesco D'Acunto and Alberto G. Rossi , title =. Annual Review of Financial Economics , year =

  11. [11]

    Journal of Financial and Quantitative Analysis , year =

    Sanjiv Das and Harry Markowitz and Jonathan Scheid and Meir Statman , title =. Journal of Financial and Quantitative Analysis , year =

  12. [12]

    Das and Daniel Ostrov and Anand Radhakrishnan and Deep Srivastav , title =

    Sanjiv R. Das and Daniel Ostrov and Anand Radhakrishnan and Deep Srivastav , title =. Journal of Banking and Finance , year =

  13. [13]

    Rossi , title =

    Francesco D'Acunto and Nagpurnanand Prabhala and Alberto G. Rossi , title =. Review of Financial Studies , year =

  14. [14]

    The Journal of Finance , year =

    Harry Markowitz , title =. The Journal of Finance , year =

  15. [15]

    Journal of Financial Economics , year =

    Michael Reher and Stanislav Sokolinski , title =. Journal of Financial Economics , year =

  16. [16]

    Journal of Financial and Quantitative Analysis , year =

    Hersh Shefrin and Meir Statman , title =. Journal of Financial and Quantitative Analysis , year =

  17. [17]

    The Journal of Finance , year =

    Lettau, Martin and Ludvigson, Sydney , title =. The Journal of Finance , year =

  18. [18]

    2025 , month =

    Ma, Jennifer and Pender, Matea and Hu, Xiaowen , title =. 2025 , month =

    2025

  19. [19]

    , title =

    Lusardi, Annamaria and Mitchell, Olivia S. , title =. Journal of Monetary Economics , year =

  20. [20]

    Consumer Expenditure Survey Anthology, 2003 , publisher =

    Duly, Abby , title =. Consumer Expenditure Survey Anthology, 2003 , publisher =. 2003 , pages =

    2003

  21. [21]

    Journal of Political Economy , year =

    Scholz, John Karl and Seshadri, Ananth and Khitatrakun, Surachai , title =. Journal of Political Economy , year =

  22. [22]

    , title =

    Mehra, Rajnish and Prescott, Edward C. , title =. Journal of Monetary Economics , year =

  23. [23]

    The Real Deal: 2012 Retirement Income Adequacy at Large Companies , institution =

    2012

  24. [24]

    Foundations of modern probability , author=

  25. [25]

    2017 , publisher=

    Enlargement of filtration with finance in view , author=. 2017 , publisher=

    2017

  26. [26]

    2016 , issn =

    Lifetime ruin under ambiguous hazard rate , journal =. 2016 , issn =. doi:https://doi.org/10.1016/j.insmatheco.2016.06.007 , url =

    work page doi:10.1016/j.insmatheco.2016.06.007 2016

  27. [27]

    , title =

    D'Acunto, Francesco and Rossi, Alberto G. , title =. Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices , editor =

  28. [28]

    North American Actuarial Journal , year=2004, volume=

    Virginia Young , title=. North American Actuarial Journal , year=2004, volume=. doi:10.1080/10920277.2004.10596174 , url=

    work page doi:10.1080/10920277.2004.10596174 2004

  29. [29]

    Insurance: Mathematics and Economics , volume=

    Optimally investing to reach a bequest goal , author=. Insurance: Mathematics and Economics , volume=. 2016 , publisher=

    2016

  30. [30]

    Bulletin of the American mathematical society , volume=

    User’s guide to viscosity solutions of second order partial differential equations , author=. Bulletin of the American mathematical society , volume=

  31. [31]

    Mathematics of Operations Research , volume=

    Survival and growth with a liability: Optimal portfolio strategies in continuous time , author=. Mathematics of Operations Research , volume=. 1997 , publisher=

    1997

  32. [32]

    2009 , publisher=

    Continuous-time stochastic control and optimization with financial applications , author=. 2009 , publisher=

    2009

  33. [33]

    Optimal stochastic control, stochastic target problems, and backward

    Touzi, Nizar , volume=. Optimal stochastic control, stochastic target problems, and backward. 2013 , publisher=

    2013

  34. [34]

    and Border, Kim C

    Aliprantis, Charalambos D. and Border, Kim C. , title =