A 4% Withdrawal Rate (WR) works (under Trinity study rules - 30 year retirement) with 1.22% real Compound Annual Growth Rate (CAGR).

A 4.5% real CAGR would sustain (again under Trinity study rules - 30 year retirement) a 6% WR.

Change 30 years to 55 years and the above numbers change to

3.35% CAGR

4.9% WR

All assuming that early returns are not bad enough to drive one to a negative balance, regardless of what the CAGR for the asset allocation ends up being at the end of the time period.

For now I'll have to take your word for it because the 4% rule always seemed close enough that I didn't bother getting into debates over semantics. What is your source for these numbers?

This does assume constant (or at least not-too-bad-in-the-first-few-years) CAGR.

Given that, it's simply a calculation. See cells F16:G22 on the 'Misc. calcs' tab of the case study spreadsheet, or just use Excel's RATE and PMT functions respectively to get the CAGR and WR numbers.

Constant returns are probably not a good assumption for this case. I used the monte carlo simulator at

https://www.portfoliovisualizer.com/monte-carlo-simulation#analysisResults Using an annualized 4.5% return (assumed real), 15% annualized standard deviation, and fat tailed distribution, a 4% constant withdrawal rate succeeded 75% of the time at 30 years, and about 50% for 50 years. A 3% withdrawal rate succeeded 89% of the time at 30 years and 71% of the time at 50 years.

Using real annualized returns of 1.22% in the above, the 30 year success rate is about 34%. Using real annualized returns of 3.35% for a 50 year withdrawal results in a 34% success rate. Not 100% safe.

To check the simulation, 6.7% returns at 4% constant withdrawal succeeded about 90% of the time at 30 years and 79% at 50 years. I think that is a little lower success rate than actually happened so the simulation might be slightly biased.

The 4% rule was true over a period with about 6.7% annualized returns. Doing some simple math, 4/6.7, it seems like an appropriate safe withdrawal rate is very approximately 60% of the annualized returns of the period. If we assume 5% real returns then 3% is probably an appropriate constant safe withdrawal rate. In general volatility seems to be more damaging to withdrawal rates than most people assume. It is also more beneficial to accumulation than most people assume.