A 4% withdrawal rate for American retirement spending, derived from a discrete-time model of stochastic returns on assets and their sample moments

ArXiv ID: 2508.10273 “View on arXiv”

Authors: Drew M. Thomas

Abstract

What grounds the rule of thumb that a(n American) retiree can safely withdraw 4% of their initial retirement wealth in their first year of retirement, then increase that rate of consumption with inflation? I address that question with a discrete-time model of returns to a retirement portfolio consumed at a rate that grows by $s$ per period. The model’s key parameter is $γ$, an $s$-adjusted rate of return to wealth, derived from the first 2-4 moments of the portfolio’s probability distribution of returns; for a retirement lasting $t$ periods the model recommends a rate of consumption of $γ/ (1 - (1 - γ)^t)$. Estimation of $γ$ (and hence of the implied rate of spending in retirement) reveals that the 4% rule emerges from adjusting high expected rates of return down for: consumption growth, the variance in (and kurtosis of) returns to wealth, the longevity risk of a retiree potentially underestimating $t$, and the inclusion of bonds in retirement portfolios without leverage. The model supports leverage of retirement portfolios dominated by the S&P 500, with leverage ratios $> 1.6$ having been historically optimal under the model’s approximations. Historical simulations of 30-year retirements suggest that the model proposes withdrawal rates having roughly even odds of success, that leverage greatly improves those odds for stocks-heavy portfolios, and that investing on margin could have allowed safe withdrawal rates $> 6$% per year.

Keywords: 4% rule, Retirement consumption, Risk adjustment, Longevity risk, Leverage optimization, Retirement Portfolio (Equity/Bond mix)

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper uses advanced stochastic calculus and moment-based approximations (Taylor expansions, geometric sums) to model withdrawal rates, reflecting high math complexity. It also grounds its model in historical simulations and historical parameters (e.g., S&P 500 returns, leverage ratios), showing strong empirical backing, though it lacks implementation details or code for true backtesting.

  flowchart TD
    A["Research Goal: What grounds the 4% withdrawal rule?"] --> B["Methodology: Discrete-time consumption model"]
    
    B --> C{"Inputs: Historical S&P 500 Data"}
    C --> D["Compute Sample Moments<br>Mean, Variance, Kurtosis"]
    
    D --> E["Calculate s-adjusted return rate γ<br>γ = f(mean, var, kurtosis)"]
    
    E --> F["Compute Consumption Rate Formula<br>c* = γ / 1 - 1-γ^t"]
    
    F --> G["Key Findings"]
    
    G --> H["4% Rule Explained:<br>High expected returns adjusted<br>down for variance, longevity risk"]
    G --> I["Optimal Leverage > 1.6x<br>for equity-heavy portfolios"]
    G --> J["Historical Simulation:<br>Safe SWR > 6% with leverage"]