On some futility in thinking about consumption smoothing rules

I spent the better part of an afternoon trying to excelify some math on consumption smoothing, succeeded, and then gave up after the fact for reasons below.

Let's say, as a convenient-for-me strawman, that there are four broad categories of spending in Ret-fin models:

• Constant inflation-adjusted spend
• Percent of portfolio
• Honorable attempts to be somewhere in the middle of the last two for reasons, and
• Irrational or non-mathematically necessary rules or heuristics that might/not accidentally work

1. The First is a fav in Ret-Fin for reasons now lost to me. Habit? Ease of use and coding? Some connection to observed behaviors? Economic LCM theory? Maybe all of these. It has its pros and cons. It is easy and kinda makes sense and economists like it. On the other hand it is almost an explicit bet that death will beat the wealth-depletion cliff. 

2. The Second has the beneficence of lasting forever (sorta) but in exchange one (academically speaking?) gives up lifestyle stability and risks a severely diminished lifestyle later. For example.  

3. The Third has a long history of which I know very little. I have never really done spend rules here like "snake in a tunnel" or "Guyton and Klinger." They might or might not work -- probably better than nothing -- but get ding-ed sometimes for being neither "optimal" nor mathematically necessary  though I have some quibbles with "optimal" solutions applied to real life but that is another post. In addition I have seen a number of papers lately (eg Mirrlees (2020) and Dybvig (2021)) that talk about consumption smoothing that is not constant and cover some of the issues and trade offs. For Mirrlees, I covered a bit of that here. Dybvig is clearly rigorous in the mathematics of the trade off between the future and the present. I had a hard time reading it but I liked it and am guessing it could be important if not necessarily for retirees

4. The fourth is just there as a placeholder because we know that dumb or mis-applied rules must exist. 

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This post is mostly, in a very very limited way, about #3 and so about Mirrlees and Dybvig. And that only because that was what I was working on this afternoon. So, this is not a comprehensive cover of "rules." It is not even a good cover of the references. I didn't even finish my thoughts on my afternoon. I mostly just threw up my hands and stopped.  Also, both of those works are more likely to be relevant to endowments and perpetuities rather than retirees but they are also not totally unapplicable to us I think. 

In both works the idea was to create a math rule for smoothing consumption over time given that an endowment value is going to vary quite a bit over time. In retirement we don't usually model it exactly that way because: a) we are not a perpetuity, we are eventually going to die, and b ) the portfolio often is assumed, or the interval is assumed to make it so, to last "long enough." Convenient, right?  We also focus less on intergenerational equity (ex bequest) and focus more on stuff like risk of ruin or consumption utility over finite human lives. And some other stuff. 

The rules in the two works, if I recall, help institutions deal with lumpy spending and lumpy portfolios. The problem, identified in Dybvig (and Coiner (1990)) is that: a) a pure pct of portfolio rule that uses the arithmetic return expectation as a spend rate will encounter "death spirals" because of that optimistic spend assumption and b) it implicitly biases the present over the future and so becomes a fairness issue. Not un-retiree or bequest related tho I think. Dybvig's solution, along with Michael Coiner (1990), is to focus more on a spend rule that uses, more or less, the geometric return input assumption[1]. The rule that "smooths" this geo return spend assumption is still problematic because the smoothed spending that is too high -- by way of smoothing that doesn't catch up to declining wealth fast enough -- effectively can burn the portfolio too hard too soon and still create a death spiral that kills the process early. He innovates a quant solution that adjusts the spend-smoothing to deal with the problem I just described but it only works under certain assumptions not listed here. Whatever. Nothing works perfectly. 

Here are the three formulations I was playing around with. I am sure there are others before we even get into "rules" which we won't:

eq 1. Mirrlees Habit function

eq 2. Dybvig simple mean reversion adjustment.
k is the speed of adjustment, T a target spend

eq 3. Dybvig trying to smooth without biasing
the present over the future. Sorta works but read the paper

Ok, so this post is not going to go through any of these three. After chugging through excel for about 2 hours I did finally get the last two to work and the first I'd already done. Basically they all look kind of the same, depending on the parameters AND the particular run which is always going to be unique. Any random run is going to sorta like this although you have to imagine a blue and grey line going down vs up and with grey trailing blue and grey tracking blue tighter or more loosely for different adjustment factors...

Figure 1. Smoothed spending (y,grey) over time (x) where
dashed is a constant spend, blue is %P and grey
is some unspecified smoothing rule tho in this case it was eq3.

I'm not going to run these equations through any paces here or suss out the outcome-impacts of choosing one or the other, because I concluded, as I stared at another pointless spreadsheet for the millionth time, the following:

1. I'm not an endowment. I only have a 25 year horizon or maybe 37 at the very very max. So, not forever. [2]
   
   
2. It is almost always in the nature of retirement finance, done with real people, to bias the near-future over the far-future given finite resources and limited time and the grind of mortality. This is one reason why the Bengen 4% rule worked back whenever. It assumed 30 years (only) using probably-outlier mid-20th C US (only) data and it worked close enough most of the time, especially for 65 year olds or older...less so for 50 year olds or non-US retirees...or futures that don't or won't comply with the 20th C data. This is not really good enough, imo, for a perpetuity which is playing an entirely different game (sorta, except maybe for the early retiree[2]). Even in the Life Cycle Model with its life-consumption utility there is: a) a time preference discount (not time value discount though there can be that) that shifts some utility forward, and b) a hard weighting for conditional survival probability that effectively moots the out-years past about 30-40 years. 
   
   
3. Me? There are limits to how much I can budget and/or adjust spending. In effect there is only so much downward movement I can do in my spending or planning. I have structural and psychological limits, for now I think, on how much I can adjust downward. I mean I could adjust up to infinity, yes, but down? Not so much. Anyway, I already have. In 2012 or so I cut out 50% of lifestyle which is quite a bit harder than it sounds. What's next? A shelter? ;-)  I spend as I do and there are both limits and lumps I cannot control. This last year saw a new roof and AC and fridge and water heater and some major tree work. Only so much smoothing I can do.[3]

So, no elegant endowment smoothing rules for me. I'll just look around and adapt every year until I hand all of it over to either an advisor, my kids, a warden, or a personal estate representative and the courts. That's why I gave up this afternoon. 

----- Notes -------------------------------

[1] better be monitored. The minute the rule and expectation is created the world will change and more than just planned "vol." 

[2] I have theory that if one retires at 49 the problem space is probably not too different than an endowment. One might treat it as perpetual even though we know longevity will catch up to us. Then, when longevity does start to catch up - say mid 60s or so - we phase-shift into a more normal retirement problem. I am probably over stating it but it felt like this as a proceeded from 50 to 63. 

[3] Technically these hits were already spent since I had booked recurring deferred maintenance to my balance sheet along the way so the 2021 and 2022 spending lumps disappeared a bit. But this post is not about personal financial statements though that would be a better topic. 

----- References -------------------------

Coiner, Michael. (1990) The Lognormality of University Endowment in the Far Future and its Implications. Economics of Education Review Vol 9 No, 2 157-161

Diamond, Peter and Mirrlees, James (2000), “Adjusting One’s Standard of Living: Two-Period Models,” in Incentives, Organization, and Public Economies, Papers in Honor of Sir James Mirrlees.

Dybvig, Philip H. and Qin, Zhenjiang, How to Squander Your Endowment: Pitfalls and Remedies (October 11, 2021). Available at SSRN:https://ssrn.com/abstract=3939984