First some assertions. Some of these are what an old boss called "proof by intimidation" which means your skeptical gene can feel free to kick in.

**Assumptions and Background**

1. I will assume that a 2% consumption rate is a lower bound for

*any*spend program. This is based on common sense, several papers I've read, my own simulations, and the last book that Ed Thorpe wrote. 2% more or less guarantees a perpetuity. That is not a 100% guarantee but close enough for me. Anything less than 2 is more than likely irrational fear and self-denial unless there is some weird condition I have not considered.

2. Where needed, the following assumptions apply: 3% inflation if we are

*not*working in real terms, 7% arithmetic input return if we are

*not*in real terms, 4% arithmetic if we

*are*in real terms, 10% standard deviation which stands in, imperfectly, for a blended 50/50 portfolio.

3. Spending is constant which, of course, we know to be a flawed but convenient modeling assumption

4. We will not be trading in "fail rates" as such here. That's so 2018ish.

5. For the "fecundity" of long term or trust or endowment spending we are leaning entirely on this article here by "the fecundity of Endowments and Long-Duration Trusts" by James Garland. H/T to Andrew Miller. This is relevant because early retirement (like mine) often behaves more like an endowment or a long term trust since the horizons can be really long and the residual uncertainties compound to the point where one might as well treat the problem as a long term trust or endowment...while holding, over the early years, "options" for future spending changes as conditions and longevity permit. In that article, Garland proposes that a spending program that maintains the "fecundity" of a (really) long term portfolio should (if 100% allocated in equities) range from no less than the dividend yield (on, say, the SnP) to the earnings yield, for reasons you will have to read and rationalize on your own.

**The "Ranges"**

**Some Notes**

A. This is based on one of my posts here or here. The range is from my lower 2% bound (in #1 above) to the min(dY/dW) of portfolio longevity for an infinite horizon. That upper bound is hard to defend. Read the post I linked here and decide for yourself.

B. This is based on the same posts as A. The range is from a lower 2% bound to 4.8% based on the same fuzzy dY/dw criterion in A but here it is based on a horizon of 40 years rather than infinity.

C. This was the range we came up with in this post here based on a lifetime consumption utility function for the portfolio parameters described in # 2 above and for a narrow band of risk aversion coefficients. 2% again is asserted as a lower bound even though it looked like "3" in the linked post. My guess is that high risk aversion will drive very low spend rates but I still assert a 2% lower bound for even the worst case. Hide in your closet if you must; I will not go below 2.

D. This is the fecundity model described by Garland. In this case it is the "all equity" model in which case we go from the div yield on the S&P in May 2019 (1.97% coerced to 2) to the E/P ratio of around 4.8%. The rationale is not presented here. Check the link.

E. This is the same as D except we are constructing a 50/50 mix of the S&P and the 10 year. The current 10 year yield this month is ~2.3% which I'll call the lower bound for "E" for today anyway. The upper bound would be a blend of the two yields. Whether this is rational is clearly up for debate.

F. This is the same as B except the range here is now from a 95% cumulative "success rate" for t=0:40 to the same dY/dw boundary asserted in B. Below 95% seems a little too penurious so I threw this scenario F in as a modification to "B."

X. This is the approximate geometric return boundary at infinity for a .04/.10 portfolio. For reference only.

Y. This is "4% rule," just for reference.

**Comments**

You tell me. I am sure there is an infinity of "ranges" one could come up with depending on the tool or framework in question. In general all this looks pretty narrow and centered around 4 (or below). This is not surprising. All bets are off, though, as age expands and horizons come in. All bets are off, too, if portfolio assumptions vary. No doubt that is for a different post.

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