or in his extended version it looks like this where the second term on the right side reflects future income and the third term is, as above (FV), there for legacy goals.
This is elegant but non-trivial stuff. It was hard for me to figure out. It is more or less the stringing together the results of a set of returns in complex ways over time so it becomes one of those multi-period geometric return things. Not for the faint of heart I think. But reasonable. The outcome is a withdrawal rate for a given set of assumptions and for a random set of returns[2] over some period of time. Since the RH40 rule of thumb comes up with a withdrawal rate as well I wanted to see how these two approaches stacked up just for fun. In a past post (I think this is the one) I did the same thing with RH40 vs other math like Blanchett's simple formula, excel pmt function and some others. The trick here in this post was to figure out 1) what return distribution do I use, 2) what other assumptions do I make, and 3) how do I execute the comparison?
1. What returns? Let's start by asserting that something like 14% is unrealistic since US large cap over the long run looks more like 7 or 8%. Let's also say that 7 or 8% isn't right either since I personally am not 100% allocated to equities; so maybe 5% sounds about right?? Thomas Pikkety says 5% is about the general return on "capital" if the lookback period is a couple thousand years. But most if not all of that period did not have the degree of inflation and taxes we face today. So let's throw in a dampener and say the real rate of return is .05-.03 or .02. Buzz kill but gotta do it. For volatility I figure not 16 or 18% like equities might play out. Let's call it 10% with no analysis to back that up.
2a. In the equation in Note 1 let's set FV to zero so no legacy plan. Also, since he extends the equation to reflect future income like SS let's assume that away to zero, too. Let's generate returns randomly in a normal distribution[2] because I don't have the skills to do otherwise which I should. Note that for RH40 I use a modified version where the withdrawal rate is subject to a minimum of 2% which is close to perpetuity range (short of some Armageddon): i.e., w = max[ age / (40-age/3)/100, 2% ]
2b. Speaking of assumptions, the thing to notice here is that -- since returns in the ERN math are random and the n-period geometric return can be random when they are all strung together -- the withdrawal rates when you set this up will be a distribution[2]. That means there is a question of what withdrawal rate result we look at. The mean? the mode? median? Min? Max? How about this? Let's look at mean, median, and the very, very conservative 25th percentile.
3. As far as execution, I just happen to have had a program in R that already instantiated this math and was documented here. It is a simple Rscript simulation using the note 1 math and a fake return generator. The hard part is getting one's head around the "sum" term in the denominator used for calculating "w" which is done by working backwards from time T. A happy hour cocktail after that effort was necessary! I figured 10k runs was enough to get a reasonable result but who knows. I did the 10k for each of 7 different (annual, though it could or should be done monthly??) durations from 20 to 50 years which, if one were to assume age 95 is a planning expectation, might mean retirement starts from ages 45 to 75.
When I do this it looks like this in withdrawal rate terms:
The distribution of simulated withdrawal rates (for 50 and 20 periods) might look like this if I got it right. Note the pronounced skew in the longer duration assumption:
I don't really have an objective way of coming to any conclusion here. My takeaway is that when using really really conservative assumptions (and yes, these are pretty darn conservative but I was doing that on purpose) my RH40 formula, which started out as a kind of self-aware inside joke, stands up pretty well. It shows particularly well against that ERN 25th percentile level. Since the ERN withdrawal rate distribution is quite skewed for long durations ( no idea which withdrawal rate would have worked so best be conservative, eh? ) and since young retirees like me have other reasons to be conservative when starting out other than merely managing uncertain returns and uncertain longevity (a big deal of course) it would be easy to construct a narrative that makes RH40's early conservatism pretty reasonable.
I guess my personal test is not really a comparison but the "bar" test. If I'm sitting in a bar and I have to pull a withdrawal rate out of thin air based on someone's age, and it has to be extremely conservative, and it also has to be easy to remember and requires no spreadsheets or programming, then I can probably stand by RH40. In practice I'd probably say something like "RH40 says ___ , but if you are feeling lucky, add a point."
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[1] From ERN: deriving SWRs analytically.
[2] Or at least that's the way I'm doing it here. It might be easier to do in non-sim mode and just use 1 deterministic return assumption. On the other hand I already had the code and also I think the stringing together of the random returns into a geometric sequence to get a distribution of possible withdrawal rates that might work is more realistic. It's basically simulation at that point which begs the question of why not go the rest of the way and just do a full simulation. Well, I guess because that is not the game we are playing today. The purpose of the re-used code was originally to look at how geometric returns with spending play out over short horizons. In that case, random returns were necessary to get the effect on expected value that made the analysis work. That approach might be overkill here but it still works.
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