My Criteria
Let's get right to the criteria. This is some of what I want in a good retirement formula:
1. It is simple. I want a formula that is short, simple, and easy to remember and apply anywhere anytime. It requires no programmable calculators, spreadsheets, R programs, Python programs, or any computer other than a hand calculator or a brain...not even a slide rule. There are no risk gammas, factors for "force of mortality", obscure risk aversion or other coefficients, use of calculus or partial differential equations, alphas or betas, logarithms - natural or otherwise, exponents or square roots, gamma distributions, etc etc. There is no math beyond middle- or even lower-school and not too much in terms of mental gymnastics. Actually none of that is really true. I am supportive of complex math and tools that are necessary for dealing with life-cycle problems, it's just that for certain rules-of-thumb I am quite willing to trade off a little bit of transparency and economic accuracy for simplicity and speed.
2. It is age-adjusted. Rules of thumb like the 4% rule focus on 65 year olds and 30 year retirements. Period. Early retirees have a totally different game to play and their retirements can be longer and much more difficult than it is for someone even in their late 60s. Retirees older than ~80 also have a different game to play. For them, longevity probabilities are on a convex trajectory, bequest motives creep in at a faster rate, and risk aversion is in a confusing flux. A good retirement rule-of-thumb formula respects age differences even if it is extremely reductive.
3. It has "the bend." A good retirement spending formula has to have a good Goldilocks "bend" (not too much not too little) in the math, especially between 60 and median mortality, to reflect what has been observed of either declining risk aversion and/or longevity probabilities that are, counter-intuitively, both declining (longevity expectations are clearly going down) and extending (the median terminal age expectation for survivors gets a little older each survival year) at the same time. A linear formula might respect age in general but it doesn't really have the right bend and more often than not disrespects later ages in ways that are not helpful. See the next point.
4. It has friends in retirement finance math. The formula should conform in its "shape" in some way, even if it is an unexplainable mathematical or statistical artifact, to other well known retirement formulas and rules of thumb. It should also be moderate in the sense that it more or less hangs out in the middle of the pack and is not too extreme at young or old ages. It has friends.
5. It should be more friendly than not to the bequest motive. No small number of retirees have a case of both declining risk aversion and declining consumption utility as they age. For them: the end game approaches, family gets more dear, and mobility and the need to spend on conquering Everest decline which means the utility of consumption may actually go down while legacy planning gathers steam. On the other hand, no small number of ret-fin formulas unrealistically explode in their spending recommendations after age 80. A good formula will not bias towards extreme consumption between 80 and 100 and will have a little respect for the bequest motive instead.
6. It has a plausible link to an economic rationale. Even if it is derivative or indirect or hard to explain, there should be some real link to why the formula exists. An accidental formula might actually be ok just because it works but it's probably better if it works for some kind of explainable reason. On the other hand, "citerion one" above means that I might, in a pinch, actually take something that is simple over something that is rational.
Some Background
Ok, so this formula did not spring Athena-like from the head of Zeus. I lifted this in a way from the "divide by 20" rule devised by Evan Inglis ( see my post here ). If you buy the rationale that underpins that rule, then the new formula will satisfy criterion 6 above. If not, then it might be more of an unsupportable math artifact and most of the rest of this will be a waste of time. For now, I'm a buyer. Note that all attempts at rationalizing my new formula should point back to that link.
The nice thing about the /20 rule, and why it was my starting point, is that it satisfies criteria 1 and 2 in spades. The downside, though, is that it is less helpful when it comes to criterion 3 or 4. After thinking about this problem for a while it dawned on me that if: 1) the meat of most people's retirement problems occur between age 60 and 90, 2) the "divide by 20" paper is acceptable as a rule-of-thumb principle, 3) "/20" is the upper bound for spending conservatism according to Inglis and /10 is his lower bound, and 4) risk aversion declines for at least some cohorts in late retirement but doesn't appear in the /20 rule, then maybe I could modify the /20 rule to make it slide itself from /20 to /10 between the age of 60 and 90 in order to pick up criteria 3, 4, and 5 while still retaining 6, 1 and 2.
To gin up my new rule I simply assumed that /20 is conservative and ok for ages up to about 60 and that then subtracting somewhere around 1 from the rule every three years thereafter is about the right pace to get to the lower risk aversion /10-boundary by 90...after which it probably doesn't matter much. Then it occurred to me that I could instantiate all of this in a formula rather than saying "subtract 1 from divide-by-20 every 3 years after age 60" though that would probably work too. And thus I had a new formula. woo hoo.
The Formula
When using a divide-by-20-at-60-and-then-go-towards-a-divide-by-10-at-90 approach, if you take off the constraint that age 60 and 90 matter a lot and also that 20 and 10 matter precisely then the formula is pretty easy and looks like this:
[RiversHedge RH40 formula 05/12/2017]
But Does it Work?
Let's throw this thing against criteria 3 and 4 to see what happens. To do that I pulled out my other ret-fin formulas plus a Monte Carlo simulator to see how things lined up. Now let's realize that while each formula and tool is fine-tuned to its own assumptions and while I tried to make them all more or less cohere, it is almost impossible to get everything to use the same exact worldview. Each one has its own story and background and usability. All of them are also almost infinitely malleable and so it is entirely possible that I pushed assumptions to unfairly line things up. Even if I did, we are still looking mostly for "shape" and "friends." Actual field-use on all of this is up to the user.
The tools and rules I chose to use are as follows: the 4% rule, the divide by 20 rule, David Blanchett's simple regression formula for complex withdrawal strategies[1], an RMD style formula, the Waring and Seigel ARVA rule using Excel PMT, Gordon Irlam's math formula for retirement spending, Moshe Milevsky's 2005 formula "sustainable spending rate without simulation," a custom-built Monte Carlo Simulator, and, of course, RH40. I think that covers enough ground for now. When one then tries to do a journeyman's job of attempting to make these cats herd, it might look like this:
where green is the Monte Carlo Approach [2], purple is the Blanchett regression formula, red is RH40, and blue is Divide by 20 (these are the formula/tools where I thought there might be a meaningful or useful comparison). Everything else is grey. The 4% rule is the horizontal grey (bad when young, bad when old but maybe it will work out on average...). I particularly like that the RH40 matches Monte Carlo and the Blanchett rule more or less penny for penny up til about 83 (that's because of the assumptions, of course, but isn't that about actuarial mortality for me? One might ask how much it matters after that) and that it bends more than the /20 rule after 65 while also staying relatively conservative to the end.
Conclusion
There are probably a lot of ways to slice and dice and analyse this. But I won't (in a past post I mentioned summer is coming and I get lazy when that happens). After acknowledging that there is a major loss in transparency and believability (where are the return or inflation or allocation assumptions?) my main takeaway is that the rule looks like it has, in terms of the scale and shape of the curve, friends in the ret-fin math world and the friends are all, let's remember, serious solutions offered by serious people. That and: it comes up with similar spend rates and is pretty consistent with the other rules between age 50 and 70+, it bends with the best of them at the right places, it respects age, it respects bequests, it factors in declining risk aversion, it's simple, it is moderate in that it is neither too extreme nor too conservative at older ages, any divergences occur after median mortality anyway, RH40 seems to concur that early retirement is risky and that late retirees can open the gates a bit, and if you believe the rationale for the /20 rule it has some indirect economic support, too. I like it. My new formula.
Postscript
Fwiw, in case anyone was wondering, which I assume to be not true, a more conservative version -- dropping the "divide by" by one every 4 years vs 3 years so that /20 goes to /10 between 60 and 100 -- is:
w = A/(35-A/4).
But at that point you might as well just use the /20 rule (if you believe in it) and maybe change the "divide by" manually as you wish along the way.
Also, if one wanted to do something that has no support in economics or behavioral sciences and is a complete and utter and shameless p-hack that tries to fit the curve to something like Blanchett's formula (given one particular set of assumptions, anyway), one might change the formula to
w = A/(40-A/3) + p
where p is the adjusting p-hack that bends the curve more after about age 80. It is like a late-age "incremental" risk aversion factor. When I tried to curve-fit p in one particular case it looked like this:
p = C*e^(g*A)
where A is age, the constant C turned out to be 1E-13, and g stands in for some type of made-up non-economic risk aversion factor which turned out to be around .284 when fitting to the Blanchett line. Again, at this point we probably lose both criterion #1 and 6 and may want to revert to a legit formula. The other problem is that the hack would have to be custom fit to some "prior" every time you changed your mind about any of the assumptions which is both sketchy and annoying. At least I know it can be done. On the other hand, one really has to wonder about coefficients like .195 .03701 .01255 .04471 and .507 in Blanchett's formula in note 1. I don't recall that those have any particular economic meaning; they just successfully help describe a whole lot of research and simulation done beforehand. At some point I guess you just have to decide if these things work and are helpful.
The question might be partially answered by taking the spending rates implied by the formula and simulating at different ages to see if one can see a constant fail rate. Blanchett's formula, as well as Milevsky's bake a constant risk of ruin assumption right into the math up front. If I could get constant ruin coming out, I might be more comfortable with a hacker formula. That's not fool-proof but it would be interesting to see.
One Last Comment
I hope that the reader can see that there is an awful lot of tongue-in-cheek in all of this. While, for my own odd reasons, I actually did want my own retirement formula, working backwards from that "want" by piggybacking on someone else's simple ROT to make something up that more or less looks like it matches other ret-fin math means that more likely than not the result might be just a little bit thin. I still like it though and think that it has a legitimate purpose (again, as long as you buy the "divide by 20" foundation). For now, all I can say is that I think a post like this and the effort that went into it at least beats Lumosity.com.
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[1] Here for example is what goes for "simple" in the trade lit. This is Blanchett's "simple" formula. Don't get me wrong. He's a really nice guy and was willing and ready to help me with a problem when I emailed him (which I really really appreciate). And a tremendous amount of research and effort and judgement underlie the equation. It's just that while this is rather simple it also isn't. The coefficients are meaningless except as a statistical artifact. Also, while you can probably program it into a programmable hand calculator you probably won't and while you can easily put it in a spreadsheet I guarantee you that you will not get it right on the first try or maybe even the second (that just may say more about me, though). Also see if you can remember the formula and its coefficients tomorrow morning. I don't think so. Compare that to "Divide by 20" or "Divide by (40-A/3)". I seriously doubt you will forget or mis-calculate those. Ever.
[2] the MC assumptions were tuned to try to make things cohere but one thing I wanted to call out is that I assumed all along the way that terminal age was 95. In order to come up with spend rates I then did trial and error at 50, 60, 70, 80 with linear interpolation in between and then trial and error for each age from 80+. The thing is that if one were to be honest with things-actuarial one might push out the terminal age expectation past 95 as one got to age 80 or 85 or 90. The effect would be that the spend rates at 80 or 85 or 90 would be lower than what I modeled. I'm guessing they would look a lot like the red and purple lines. That would just make my point even better so I kinda want to try that some day. Just not today. ...also, note that I tried to be static with all of the formulas' inputs when it came to things like success probability, asset allocation, terminal age, etc. TBD on whether that makes sense in real life especially for very young and very old retirees.
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