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

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 $\gamma$, 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 $\gamma / (1 - (1 - \gamma)^t)$. Estimation of $\gamma$ (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.