I know I've done the kind of charts in this post before, but whatever. David C pointed out to me I've been kinda re-hashing my past stuff lately but sometimes that's necessary to pound it into one's own head. Here (Figure 1) I was running a "portfolio longevity" calc for a .04/.12 consumption portfolio (different spend rates) 4 million times (uh, there is a reason for that big nbr) to see how many portfolios "tip over" into portfolios that last to infinity (or in this case 100 years which is a convenient proxy for forever but not really).
This portfolio longevity concept is important if one were to happen to perseverate on the preservation of capital over long horizons. Some recent papers I've read (Waring ('22), Coiner ('90), Dybvig ('21))* about spending by endowments make the point that if one is not careful that one can bias spending (by way of either innumerate spend rules or by way of behavioral smoothing rules) to the present over the future and that that is not fair -- unfair to the point of failing but there is zero discussion of fail here -- to the future and may overcontribute consumption to the present. For mere mortals like me this may sometimes not always matter [1] but for immortals like endowments and long-dated trusts it might actually matter quite a bit down the road. And here I am not yet discussing the "smoothing" of spending which can really screw up inter-temporal-equity even more but that is for another post...
In really really long retirements -- say me at 50 (I'm now 63) with a possible 50 or 55 year horizon -- this care about inter-year equity (inter-generational equity in an endowment model) matters, especially if one cares about one's older self (or one's beneficiaries for that matter). My ex girlfriends never understood this idea of future me...or my kids; they always wanted me to spend on them now ;-) heh
My take-away from the papers above* and some others is that most endowments seem set their spend rules, colloquially speaking, and over simplifying, to:
Spend <= portfolio return expectations - inflation
Which sounds good but if you know finance reality it probably isn't so good. This spend rate will burn a portfolio over the very long haul for reasons in the papers referenced and that I try to touch on below. I mean, let's forget about ex-post f'kery of the universe that can screw with things like iid and heteroskedasticity and so forth -- and also let's forget deeply flawed assumptions like the ones Japanese retirees might have made in 1989 -- and let's just say there are at least fees and inefficiencies and stuff. But that is not really what the papers were mentioning.
In finance, if we read any source that deals with muti-period time, we know that the return expectation -- my comment above about universe-f'kery notwithstanding -- is not really the arithmetic input expectation. The geometric return is what we earn in the long run[2]. So a pension or endowment that were to set their spending policy to the linear/arithmetic return would probably be shooting themselves in the foot over a long horizon and be disadvantaging the people depending on the endowment in the future. Since the median of the wealth distribution is a better metric for long-haul results where N years = "a lot" and since that median is tied at the hip to the geometric return expectation, that means that there is kind of a 50/50 chance that we will end up at about the median wealth or geo return over time and that spending more than that return will almost certainly destroy wealth over time and spending less will maybe have an unbounded upside (we are ignoring human longevity for a second here). Dybvig in particular has some nuance on this but I have not totally digested that paper yet. Basically the general idea that I take away is more or less like this: for a spend that will not destroy wealth over very long horizons[3]:
Spend <= N-period geometric real return +/- some kind of other considerations
where the other considerations here might include whether or not we are actually smoothing spending...which apparently can be destructive in the sense that if one's wealth crashes, the smoothing rule might not respond fast enough and may or may not result in a wealth crash or depletion before the time where it would be convenient for future beneficiaries, either self or others... for reasons not articulated here.
So, to test this idea, I went back to something I did back in 2018 or so. If I run an unbounded (by human longevity, anyway) portfolio longevity calc -- how long do portfolios last in simulation where 100 years is a proxy for infinity for convenience -- for a portfolio with .04/.12 assumptions (arbitrary, real r, discrete) What happens? How often do portfolios "tip over" to 100 years (infinity). And what, btw, is the spend rate at which the portfolios that last forever seem to accumulate the fastest. Is it "r" or something else? The point here is that we are working with a supposition and guessing that at the point that they accumulate to infinity the fastest -- not r -- we also think that there is a (50/50?) chance that maybe they will and maybe they won't be on either side. It's just that at that spend rate (spoiler: the geometric return rate), it starts looking like we might have more of a perpetuity than not. Ed Thorpe said once that anything below a 2% spend rate is probably going guarantee a perpetuity but: a) maybe not even then, and b) that is not really our question which is what is the highest rate of spend over, say 100 years, that does not destroy capital in some sense.
In Figure 1, we have a chart of the frequency of the 911,000 portfolios of the 4M run that lasted forever under this particular parameterization. For each spend rate (on the X axis or "a" in Fig 1) we have the frequency of the subset for that spend rate of the 911k lasting forever (on the Y axis (or "b") or the "frequency of X"). Un-shown is the dY/dX calc I did that shows the rate at which that frequency changes. The dotted line, fwiw, is the 4% discrete return input assumption (say the naïve pension/endowment spend we talked about above) and the other one, the solid line, is both (coincidentally? probably not) the point on the curve with the highest slope (the accumulation of portfolios that are tipping over to infinity the fastest or max(dY/dX)) AND the point that represents the geometric return given the parameters I posited. I calculated that geometric mean return, btw, 6 different ways using, among others, Markowitz 2014 Risk Return Vol 1 for the estimators. I have not adjusted them for continuous math though, because I am not all that conversant in continuous math. The exact slope max (max(dY/dx)) is ever so slightly a tiny bit off from the geo mean estimates, a little lower by 1/10th of a percent of a spend rate, but I will naively assume that that is from variability in the simulation but I really don't know yet. TBD. It's very close.
One other point here. The geometric mean return also represents the median wealth at N="a long time" That means that it is a marker and only that. Half the portfolios would be smaller than expected, half bigger at that N-time point given things like iid and stability. So, a question: have we destroyed wealth under our spend rate of 3.2% not= 4%? Meh, a 50/50 chance I guess. It is still a "bet" but probably better over a very very long horizon than spending at the average arithmetic return expectation of 4% which is almost certainly guaranteed to destroy wealth over very long horizons. Read Waring or Coiner for that whole story.
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| Figure 1. Freq of portfolios going "infinite" at the 100y mark |
For some more context, here below is the heat scatter of Portfolio Longevity in Years for different spend rates. The various dots represent the number of years (X axis) a particular portfolio lasted under the assumptions for some given spend rate (Y axis). The color is basically a representation of frequency or probability depending on how you look at it. The very right side of Figure 2 is really Figure 1 if you flip it around...and as Figure 1 goes to the left the very right side of Fig 2 goes down and gets more orange as figure 1 goes up. Idk, I hope you know what I mean. The "lines" are the 4% input return (dotted) and the geometric output expectation (solid). Basically at about 3.2-3.3% spend rate, or around the geo return expectation, portfolios start lasting a very long time, longer than a human scale by the way, but that is a different post/story.
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| Figure 2. Portfolio Longevity Heat Scatter For different spend rates given params |
----- Notes ---------------------
[1] This retirement finance stuff, if we don't literally go back to the Bible or Fibonacci, at least starts with Bengen. He probably didn't realize it but his 4% rule, based on a 30 year fixed term, depended very very heavily on an implicit bet about death. Mitchell et al wrote about death and capital markets once, from which I borrow that idea. But a bet it is. Somewhere in-between a fixed bet on a 30 year death and an endowment for immortals is the loooooong horizon retiree. This is something I have struggled with until just recently. At 63, my horizon probably looks more normal than it did at 50 but I still have to be careful since people with long horizons -- say annuitant or people with good health expectations -- might still have really long horizons. So, I might still be playing a long game which, while not exactly an "endowment," is still pretty long. I have some ideas on this. TBD...
[2] The literature is super long on this. I am thinking about people like Michaud, Meucci, Markowitz, Milevsky, Mindlin (lotta Ms there), Kelly, textbooks, others, blah blah blah. Not gonna rehash it here...
[3] we are very close to the Kelly criterion and log utility here. Dybvig '21 explains why this approach here and in his paper more or less moots the Samuelson critique. Otoh, I have a personal critique of Kelly/Hakannson that is consistent with Samuelson and others. Basically if one were to have a short human horizon -- eg I have say 10-15 years before it might be optimal to annuitize some wealth -- then the volatility of a max-log approach could be absolutely soul and goal crushing. Trust me on this.
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