A Tax-Efficient Model Predictive Control Policy for Retirement Funding

Kasper Johansson; Stephen Boyd

REVIEW 3 major objections 5 minor 47 references

A Tax-Efficient Model Predictive Control Policy for Retirement Funding

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Solving a convex optimization problem each year to re-plan withdrawals and Roth conversions funds a fixed inflation-adjusted consumption while leaving a larger bequest than the 4% rule in about 70% of simulated retirements.

desk verdict Solid convex-MPC formulation for retirement funding, but the claimed edge over the 4% rule rests on a weak proportional-withdrawal benchmark, not a fair tax-aware comparison. read the letter →

arxiv 2507.10603 v1 pith:GPPUACJF submitted 2025-07-12 math.OC cs.CE

Kasper Johansson , Stephen Boyd This is my paper · ORCID

classification math.OCcs.CE MSC 90C2591G10

keywords retirementfundingmodelpredictivecontrolconvexoptimizationtax-efficientwithdrawalsRothconversion4%ruleMonteCarlosimulationbequestmaximization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a retiree can reliably fund a constant, inflation-adjusted consumption and still leave a larger inheritance by re-solving a modest convex optimization problem once a year, rather than following a fixed withdrawal rule. The yearly plan picks withdrawals, deposits, and Roth conversions across brokerage, traditional IRA, and Roth accounts to maximize the eventual bequest subject to meeting a consumption target, with taxes and account rules modeled explicitly. Because the plan is recomputed each year from current account values, life expectancy, and updated forecasts, the policy absorbs shocks from investment returns, inflation, and longevity. In Monte Carlo simulation the policy's consumption matches the 4%-rule benchmark in almost every scenario, while its bequest is larger in around 70% of cases with a median increase of about 6%.

What carries the argument

The engine of the paper is the retirement funding planning (RFP) problem, a convex optimization problem with roughly $13T$ variables for a $T$-year horizon that selects withdrawals, deposits, and Roth conversions across brokerage, IRA, and Roth accounts. Its convexity comes from relaxing the tax constraint $\tau_t = \phi(\omega_t) + \xi(b_t)_+(1-\delta_0)_+$ to an inequality $\tau_t \ge \phi(\omega_t) + \xi(b_t)_+(1-\delta_0)$, since overpaying taxes can never improve the objective and the relaxation is tight at the optimum. The second moving part is the annual update-plan-act loop: each year the retiree solves the RFP with current account values, a planning horizon set to 150% of remaining life expectancy, and current forecasts, then executes only the first year's actions. In the simulations the policy's signature behavior is aggressive Roth conversion in the early retirement years, while the retiree is in a lower tax bracket, followed by tax-free Roth growth that funds the larger bequest.

What would settle it

Run the same MPC-versus-benchmark comparison out-of-sample: fit the Gaussian mixture and VAR models only on data through 1990, generate trajectories with the realized 1991-2023 returns and inflation, and check whether the relative bequest still centers near 1.06 with the MPC policy better in about 70% of cases. A cleaner version is a pure historical backtest, replaying both policies on the actual 1962-2023 sequence and on block-bootstrap reorderings of it; if the bequest advantage shrinks toward zero or reverses, the claimed superiority is an artifact of the chosen statistical models rather than a property of the policy.

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Extended reading notes

Core claim

The paper's central claim is that the retirement funding planning problem admits a convex formulation, and that wrapping this planner in a model predictive control loop yields a policy that dominates the standard 4% rule. The planner maximizes the bequest subject to a constant real consumption target; the key move is relaxing the tax-bill equality constraint to an inequality, justified because a retiree would never voluntarily pay more tax than required, so the relaxation is tight at any optimum. Solved annually with updated balances, life expectancy, and return forecasts, the planner's first-year actions become the policy. Across 1,000 simulated lifetimes for two typical US retirees, the policy delivers the target consumption in over 98% of scenarios and a bequest larger than the benchmark in about two-thirds to 70% of scenarios, with a median relative bequest of 1.06; when it beats the benchmark, the median increase is 12-14%.

Load-bearing premise

The evaluation treats statistical models fitted to historical data, a three-component Gaussian mixture for stock returns and a vector autoregression with a hand-chosen piecewise-linear transform for Treasury rates and inflation, as a faithful stand-in for future market behavior, so the simulated bequest advantage may not transfer to real retirement outcomes if the future deviates from the fitted distributions.

Editorial extensions

If this is right

If the paper's simulations are representative, replacing a fixed withdrawal rule with the yearly re-planning policy preserves consumption in nearly all scenarios and increases the expected inheritance left to heirs, with a median bequest gain of about 6%.
The policy's advantage traces to a specific mechanism, front-loaded Roth conversions, so retirees looking to copy the result should expect to convert IRA funds to Roth early, before taxable income rises at age 70.
Because each planning problem solves in about 0.01 seconds, the policy is cheap enough to run on a laptop and to evaluate over thousands of scenarios before adoption.
The same convex planner extends to 401(k) and Roth 401(k) accounts, five-year Roth rules, MAGI-dependent contribution limits, and collar-protected portfolios without changing the architecture.
If the retiree uses collar options or conservative return forecasts in the planner, worst-case bequest outcomes tighten while the median outcome is largely unchanged, so the policy can be tuned for risk aversion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: a natural stress test the authors do not run is to fit the return and inflation models on a pre-1990 window and evaluate the policy on the post-1990 holdout, to see whether the roughly 70% bequest advantage is a property of the policy or of the fitted distributions. This is my inference, not a paper claim.
Beyond the paper: the tight-relaxation trick for taxes is general, since any convex cost that the optimizer would never voluntarily exceed can be relaxed the same way, which suggests the formulation carries over to progressive capital-gains taxes, state taxes, and carryforward losses. This is my inference, not a paper claim.
Beyond the paper: the authors use historical-average return forecasts and report that fancier VAR forecasts do not change results; a further test would be deliberately wrong forecasts, such as planning on a 2% real return when returns average 5%, to map how much safety margin the 150% horizon and re-optimization provide. This is my suggestion, not a paper result.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Two things to know about arXiv:2507.10603. First, the convex optimization and MPC machinery for retirement funding is genuinely new in its specific package: multiple account types with RMDs, Roth conversion limits, a bequest objective with a soft consumption target, and annual re-planning. The reformulation is correct, the tightness argument for the tax relaxation is standard and fine, and the open-source implementation makes the simulations reproducible. Second, the headline claim—median bequest up about 6%, better in roughly 70% of cases—is plausible but rests on a benchmark that is not the 4% rule as normally practiced. The benchmark withdraws proportionally from brokerage, IRA, and Roth, does no Roth conversions, and ignores the standard advice to spend taxable assets first. So the measured advantage is likely an artifact of that strawman comparison.

What the paper does well: the planning problem is well-posed, the MPC loop with cash discrepancy carry-forward is sensible, and the simulation uses exact progressive capital-gains taxes even though the planner uses a flat rate. The withdrawal and conversion patterns are intuitively explained. The authors are transparent about simplifications and provide code and data.

The main soft spot is the benchmark. To claim outperformance over "the 4% rule" you need to compare against a reasonable tax-aware strategy, e.g., spend taxable first, then tax-deferred, then Roth, or at least a 4% rule variant with sensible ordering. The conclusion's generalization to "traditional withdrawal strategies" is not supported by the single hand-chosen benchmark. Also minor: Eq. (2) writes ξ(bt)+(1−δ0) without the positive part, while the original tax bill has (1−δ0)+; the intended meaning is clear, but the notation should be fixed. The Monte Carlo evaluation covers only two retiree profiles and reports no confidence intervals. The statistical models for returns and inflation are reasonable historical proxies, but the advantage over MPC could shrink under different distributions.

Who this is for: readers interested in applied convex optimization, MPC for personal finance, or tax-aware retirement planning. The method is a solid engineering contribution; the comparative claim needs strengthening. I would send it to peer review. The optimization contribution deserves referee time, but the authors should be asked to benchmark against a fairer comparison, add uncertainty quantification, and soften the conclusion to match what the evidence actually supports.

Referee Report

3 major / 5 minor

Summary. This paper proposes a model predictive control (MPC) policy for retirement funding. The planning layer is a finite-horizon convex optimization problem that maximizes bequest subject to maintaining consumption near a target, with linear models for brokerage, traditional IRA, and Roth IRA balances, piecewise-linear progressive income taxes, flat capital-gains taxes, RMDs, Roth conversion limits, external income, and liabilities. The MPC layer re-solves this problem each year using updated balances, life expectancy, and forecasts, executes only the first-year action, and carries forward cash-balance discrepancies. The policy is evaluated by Monte Carlo simulation using a Gaussian mixture model for stock returns and a VAR model for Treasury rates and inflation, compared with a benchmark described as a 4%-rule analogue. The paper reports roughly equal consumption and bequest outcomes better in about 70% of simulations, with a median increase around 6%.

Significance. The optimization and MPC formulation is a solid methodological contribution: the convex relaxation of the tax constraints is standard and tight for this problem, the model incorporates the major U.S. retirement account rules, and the authors provide an open-source implementation and reproducible numerical experiments. The simulation pipeline also has strengths: evaluation uses exact progressive capital-gains taxes even though planning uses a flat rate, and the cash-balance discrepancy is handled in a sensible way. The main weakness is that the headline 'better than the 4% rule' claim rests on a benchmark that is not the 4% rule and is constructed to be tax-inefficient; this undermines the external conclusion, though not the internal validity of the optimization method.

major comments (3)

[Section 5.2, Tables 7 and 8, and the Conclusion] The benchmark policy is not the 4% rule as normally understood. It withdraws 3.75% of initial balances plus projected Social Security until age 85, takes only the RMD from the IRA, and splits all other withdrawals across brokerage, IRA, and Roth in proportion to remaining balances. By construction it spends Roth assets at the same rate as taxable assets, performs no Roth conversions, and ignores the standard advice to exhaust taxable accounts before tax-deferred accounts before tax-free accounts. Figures 10 and 15 show that the MPC policy instead spends the brokerage account first and executes large early Roth conversions. The reported bequest advantage may therefore be an artifact of a deliberately tax-inefficient benchmark rather than a benefit of MPC. A fair comparison should include at least one conventional tax-aware withdrawal heuristic, for example taxable-first spending with RMDs and no conversions, or a simple conversion rule, calibrated to the same consumption target.
[Sections 5.3-5.4 and the Conclusion] The claim that MPC 'outperforms traditional withdrawal strategies, such as the 4% rule' generalizes from only two retiree profiles and one benchmark parameterization. The two examples differ in wealth, Social Security, and capital-gains tax rate, but they do not constitute a broad test; the 'around 70%' and 'median around 6%' statistics in Tables 7 and 8 are specific to these two cases and this benchmark. The conclusion should be restricted to the tested comparison or the experiments should be extended to additional profiles and benchmark variants.
[Sections 5.1 and 5.3] Several parameters of the MPC policy are chosen 'to give good performance' (gamma = 500, the 150% horizon extension factor, and the capital-gains tax rate xi), and no sensitivity analysis is reported. Because the comparison is between two policies and the MPC policy is tuned on the same simulation setup, it would be useful to show that the bequest advantage is robust to reasonable variations in gamma, the horizon factor, and xi. This is not required for the optimization contribution, but it is needed to support the empirical 'outperforms' claim.

minor comments (5)

[Section 5.2] 'required mininum amount' should read 'required minimum amount'.
[Appendix A.3] 'CDSs' should read 'CDFs'.
[Section 4.2] The VAR forecast formula writes 'At−τ' with τ > t, which appears to be a typo for A^(τ−t); as written it suggests a negative exponent.
[Section 5.2] The statement that 3.75% of initial balances 'roughly speaking corresponds to a 4% pre-tax withdrawal rate' is not immediate, because the benchmark also includes projected Social Security to age 85; the calculation should be stated explicitly.
[Table 7] 'There is no maximum relative bequest' is confusing; please state that the ratio is undefined or infinite when the benchmark bequest is zero, and report finite quantiles with that caveat.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the MPC bequest advantage is a simulation output, not a fitted or self-citational identity.

full rationale

The claimed chain is: (i) formulate the retirement funding planning problem as a convex problem (Eq. 1 with the relaxed tax constraints Eq. 2); (ii) wrap it in an annually re-solving MPC policy; and (iii) evaluate via Monte Carlo over statistical models for returns, Treasury rates, inflation, and lifetime. The headline bequest comparison is an output of that simulation, not a parameter fitted into the model. The GMM and VAR models are fitted to historical data, but the MPC plans use simple historical-average forecasts (Sections 4.3 and 5.1), and the simulated advantage is computed from realized account dynamics and exact tax calculations, so it is not an algebraic identity with the fitted inputs. Self-citations (BV04, DB16, BBD+17, MKBA21, MB21) supply background methods and are not load-bearing; no uniqueness theorem is imported from prior work by the authors. Two non-circularity caveats remain: the benchmark is a deliberately simple proportional-withdrawal rule that skips Roth conversions and tax-aware ordering, so the comparison supports 'MPC beats this benchmark' more strongly than the conclusion's broader 'such as the 4% rule'; and gamma (and the 150% horizon multiplier) are chosen using the same simulation environment used for scoring. These are external-validity and tuning concerns, not reductions of the result to its inputs by construction.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central performance claim depends on (i) statistical models of returns and inflation fitted to historical data, (ii) policy parameters (gamma, horizon multiplier, capital-gains rate) chosen by hand or by simulation, and (iii) the benchmark construction. None of these are derived from first principles; the paper is transparent about them.

free parameters (7)

gamma (consumption shortfall penalty) = 500
Objective parameter U(c,q)=q-gamma*(c_tar-c)_+ chosen by simulation to give good performance with small risk of running out of money (Sections 5.3 and 5.4).
Planning horizon extension factor = 1.5 (or until age 120)
Planning horizon set to 150% of remaining life expectancy, capped at age 120, as a conservative choice (Section 5.1).
Capital gains tax rate xi (planning) = 0.15 (upper-middle), 0.0 (lower-middle)
Fixed long-term capital-gains tax rate used in the planning problem, chosen as typical for each income class (Sections 5.3 and 5.4).
GMM market return model parameters = means 28%, -11%, 11%; stds 11%, 16%, 12%; weights 0.38, 0.25, 0.38
Three-component Gaussian mixture fitted to historical annual market returns from 1927 to 2023 (Section 4.1).
VAR Treasury/inflation model parameters = mu=(0.058,0.029), A=[[0.80,0.24],[-0.04,0.88]], Sigma_eps=1e-4*[[0.72,0.48],[0.48,1.47]]
First-order VAR with mean fitted to 1962-2023 Treasury and transformed inflation data (Section 4.2).
PWL inflation transform parameters = k=2.9%, s-=2.5, s+=0.75
Slopes chosen by hand to make the transformed inflation rates approximately Gaussian (Section 4.2).
Portfolio return forecasts for planning = rho_B=1.032, rho_I=1.055, rho_R=1.055
Historical average inflation-adjusted returns used as constant forecasts in the planning problem (Section 5.1).

assumptions (7)

domain assumption The retiree is a US tax resident aged 60 or older at retirement, so withdrawal penalties and Roth five-year rules are mostly avoided.
Section 2: 'We assume the retiree retires at age 60 or older, which simplifies some of the rules and eliminates some penalties.' Restricts applicability.
domain assumption Planning under the unrealistic assumption of known future returns, inflation, and lifetime is adequate when the plan is re-solved annually in an MPC loop.
Sections 1.3 and 3: strong assumptions in the planning problem are tolerated because each year a new plan is formed; the paper cites MPC robustness literature (MR22, GMTT07, KR24).
ad hoc to paper The simplified tax model (fixed capital gains rate, no loss benefits, full taxation of additional income) is conservative and sufficient for planning.
Section 2.4 lists these simplifications; simulations use exact progressive capital gains tax, so mismatch is handled through the cash-balance discrepancy in Section 3.3.
ad hoc to paper The brokerage basis-to-value ratio stays constant at the last observed value over the planning horizon.
Section 2.4: 'our approximation is that the ratio of basis to value in the brokerage account remains constant'.
domain assumption The Gaussian mixture and VAR models fitted to historical data adequately represent future stock, Treasury, and inflation dynamics for simulation and forecasting.
Section 4 builds the models from market data (1927-2023) and Treasury/inflation data (1962-2023); the evaluation of the MPC policy is only as good as these models.
domain assumption A cash-balance discrepancy between planned and realized taxes and cash flows can be rolled into next year's liabilities.
Section 3.3: 'This discrepancy is carried over to the next year's liabilities by adjusting l_{t+1}.'
domain assumption The benchmark policy, withdrawing 3.75% of initial balance (including projected future earnings to age 85) and depleting accounts proportionally, is a fair analog of the standard 4% rule.
Section 5.2 constructs the benchmark 'in analogy to the popular 4% rule'; the headline comparison depends on this baseline.

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Cite this review

Pith. "Pith review of A Tax-Efficient Model Predictive Control Policy for Retirement Funding." pith.science (2026). https://pith.science/paper/GPPUACJF

@misc{pith2026250710603,
  author       = {Pith},
  title        = {Pith review of: A Tax-Efficient Model Predictive Control Policy for Retirement Funding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPPUACJF}},
  note         = {Machine review of arXiv:2507.10603}
}

read the original abstract

The retirement funding problem addresses the question of how to manage a retiree's savings to provide her with a constant post-tax inflation adjusted consumption throughout her lifetime. This consists of choosing withdrawals and transfers from and between several accounts with different tax treatments, taking into account basic rules such as required minimum distributions and limits on Roth conversions, additional income, liabilities, taxes, and the bequest when the retiree dies. We develop a retirement funding policy in two steps. In the first step, we consider a simplified planning problem in which various future quantities, such as the retiree's remaining lifetime, future investment returns, and future inflation, are known. Using a simplified model of taxes, we pose this planning problem as a convex optimization problem, where we maximize the bequest subject to providing a constant inflation adjusted consumption target. Since this problem is convex, it can be solved quickly and reliably. We leverage this planning method to form a retirement funding policy that determines the actions to take each year, based on information known at that time. Each year the retiree forms a new plan for the future years, using the current account values and life expectancy, and optionally, updated information such as changes in tax rates or rules. The retiree then carries out the actions from the first year of the current plan. This update-plan-act cycle is repeated each year, a general policy called model predictive control (MPC). The MPC retirement policy reacts to the effects of uncertain investment returns and inflation, changes in the retiree's expected lifetime or external income and liabilities, and changes in tax rules and rates. We demonstrate the effectiveness of the MPC retirement policy using Monte Carlo simulation.

Figures

Figures reproduced from arXiv: 2507.10603 by the authors.

**Figure 1.** Annual market returns from 1927 to 2023. these models to generate realistic data for use in Monte Carlo simulations, and also to create simple forecasts of future values for use in MPC. We focus here on traditional stock/bond investment portfolios since they are simple and commonly used. More sophisticated portfolios that limit the left and right tails of the return distribution using collars, Treasury inflation sec… view at source ↗

**Figure 2.** CDF of historical and simulated market returns. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Historical 10-year Treasury and inflation rates. where µ ∈ R 2 is the mean, A ∈ R 2×2 is the VAR (matrix) coefficient, and ϵt ∼ N (0, Σ ϵ ), where Σϵ is the covariance matrix of the residuals. We take µ to be the empirical mean of xt , and A is chosen to minimize the sum of squared errors on the historical data. We take Σ ϵ to be the empirical covariance of the residuals of the VAR model. The fitted parameter values… view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: CDF of historical and simulated Treasury and inflation rates. Forecast. We can use the VAR model to give an elementary forecast of future Treasury and inflation rates. Denoting our estimate of xτ in time period t, with τ > t, as ˆxτ|t , we have xˆτ|t = µ + A t−τ (xt − …

**Figure 5.** Figure 5: Two simulated trajectories of Treasury and inflation rates, initialized from the 1962 values. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: Forecasted Treasury and inflation rates. The shaded and solid curves represent forecasted and realized rates, respectively [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: Inflation adjusted portfolio returns. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

**Figure 8.** Figure 8: CDF of inflation adjusted portfolio returns. Percentiles Portfolio Mean Volatility 25th 50th 75th Historical 3.2% 4.0% 0.6% 3.7% 6.0% Simulated 3.1% 4.4% 0.3% 3.5% 6.3% [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

**Figure 9.** Figure 9: Empirical CDF of the relative bequest for the upper-middle-class retiree. bequest for the MPC policy divided by that of the benchmark policy. When these numbers are larger than one, the MPC policy delivered more to the retiree (or her heirs) than the benchmark. The rel…

**Figure 10.** Figure 10: Realized withdrawals from the brokerage, IRA, and Roth accounts for the upper-middle-class retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th perc…

**Figure 11.** Figure 11: Realized Roth conversions with the MPC policy for the upper-middleclass retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th percentiles. sions are…

**Figure 12.** Figure 12: Realized balances of the brokerage, IRA, and Roth accounts for the upper-middle-class retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th percentil…

**Figure 13.** Figure 13: Average realized taxes with the MPC and benchmark policies for the upper-middle-class retiree. 5.4 Example: Lower-middle-class male We now consider a lower-middle-class male retiree, age 65, who begins receiving Social Security payments of $2,013 monthly at age 70, es…

**Figure 14.** Figure 14: Empirical CDF of the relative bequest for the lower-middle-class retiree. higher potential for a large bequest. This is further illustrated in the empirical CDF of the relative bequest in figure 14. Withdrawals. The realized withdrawals from the brokerage, IRA, and Ro…

**Figure 15.** Figure 15: Realized withdrawals from the brokerage, IRA, and Roth accounts for the lower-middle-class retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th perc…

**Figure 16.** Figure 16: Realized Roth conversions with the MPC policy for the lower-middleclass retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th percentiles. Once Soci…

**Figure 17.** Figure 17: Realized balances of the brokerage, IRA, and Roth accounts for the lower-middle-class retiree. The dark lines represent realized trajectories, the red line represents the median over all trajectories at each age, and the blue lines represent the 5th and 95th percentil…

**Figure 18.** Figure 18: Average realized taxes with the MPC and benchmark policies for the lower-middle-class retiree. While this simplification introduces some conservatism, incorporating more precise tax rules into the model is relatively straightforward. Five-year rules. There are two key…

**Figure 19.** Figure 19: Histograms of simulated inflation adjusted returns of the simple stock- /bond portfolios and the collar portfolios. • The cap C is the strike price of the call option with the same price as the put option with floor F. A.2 Experimental setup We consider the two retire…

**Figure 20.** Figure 20: Empirical CDFs of the bequest under the MPC policy, with and without a collar, for the two retirees. provides a meaningful improvement over a simple stock/bond portfolio. The exact benefit depends on factors such as the risk level of the underlying portfolio, as well …

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