On the Behavior of Long Horizon Adaptive Spending

I always assume that the constant spend assumption -- set spending at the beginning of some interval and then adjust it for inflation -- is well known to be an active risk position because that approach guarantees, in the absence of mortality, that it will someday stop working where "stop working" means zero[1]. But maybe that isn't as obvious as I think since I look at this stuff all the time and others don't.  

On the other hand, I've also heard "% of portfolio" touted often because it kinda-sorta lasts forever. But that is an active risk position as well for a couple reasons:

• Spend volatility becomes high (ignoring that irl that spending is, in fact, even more random than just the portfolio effects and sticky to the down side while loose to the upside). 
• Over long horizons the higher spend rates keeps chipping away at the portfolio and so: while it lasts forever, that high spend also eventually diminishes what one can spend in real dollars over time. 
• Since there is uncertainty, the spending possibilities at some distant horizon are best viewed as a distribution rather than a number if we can even think in distributions anymore. 

But %P is a useful construct because it is at the far end of the spectrum[2] of spend choice relative to "constant" where all sorts of "rules" that try to thread various needles to get a few more miles out of the tires might be considered in the middle somewhere (asserted, not proven).  %P is often referenced as endowment spending because of its tendency to last over long periods of time but even endowments have real world concerns with continuity and stability and lumpy spending and don't really do that %P thing. Whatever. You know what I mean...

We saw in a few past posts here that the literature related to endowment spending seems to be converging on what works in the long run. This idea can also be applied to very early retirements and long-dated trusts. Me? As an early retiree? I've seen various rules and boundaries offered up in this domain of long horizons. For example:

• Ed Thorp (can't remember which book, but let's call it Fortune's Formula -- a worthy read either way) said 2% is as close to perpetual spending as we can get. I have not explored this idea too much other than just asserting it. We'll see more of what I think he means below. 
• Dybvig (2021), Coiner (1990) and Waring (2022), among others[10], suggest that one, in order to foster intergenerational fairness, needs to spend no more than the geometric mean return of a portfolio. I mean, iid and stationarity obtaining and all... This is intellectually appealing and intuitive in a way. 
• Garland (2004) suggests that -- assuming an all equity portfolio -- spending in some intermediate zone between the dividend yield and the earnings yield works to optimize fecundity. This might be similar to the last bullet but I will not be working with Garland in this post. Let's just say you can more or less spend what you earn where what you earn is NOT the arithmetic forthcoming expected return.  

I personally do not have the math skills to replicate Dybvig, Coiner or Waring. All I have is the brute force of simulation, some reasonable guesses about how this works, and some rudimentary ability to code something[3].   So I sat down the other day and started asking myself how this stuff works and whether I could, with my brute force, gather the same insights into long horizon spending that I had read in Thorp, Dybvig, Coiner and Waring. I set it up in my head like this

If: 

• Spending "as a percent" produces a distribution of spends in the future, and 
• We assume "50 years" and "median spend at 50 years" actually mean something, and 
• LIM[med(50ySpend)] = 0 as spend-rate -->0% (portfolio compounds forever), and 
• LIM[med(50ySpend)] = 0 as spend-rate -->100%; (portfolio dies or nearly dies), and
• A provisional goal might be: maximize[med(50y)] without disadvantaging or rewarding future

Then:

• Is there a max[med(50ySpend)] at some spend rate? i.e., the spend rate to max future lifestyle
• Where is that for a set of assumptions N[4/6,12/20] (arbitrary)
• Is there a closed form solution or explanation for max median spend rate?
• What spend rate is the pivot where a net wealth portfolio grows or shrinks?
• Is that spend rate anywhere near the Coiner/Dybvig/Waring assumptions? [4]

To do that I commandeered a piece of code I once got from Milevsky and then nudged for other purposes. In this case I was just looking at a net wealth process[11] unhindered by mortality and what happens to spending (when rendered as a % of Portfolio) over infinite time. Professor M stopped at 100 years. I stop at 50 to borrow an idea from Waring (2022) where we assume 50 is long enough to straddle the zone between successive regimes of trustees of endowments. 100 is too long; 30 is too human. 50 seems more institutional

Here were some of the underlying assumptions 

• Portfolio in four scenarios: ~ N[.04/.06, .12/.20]; this is entirely arbitrary
• Spending is strictly % of portfolio, filtered on state of portfolio in each period
• Spend distrib is examined at year 50. This is entirely arbitrary but “long” or endowmentish
• 50-100k iterations depending on boredom [5]
• Spend rates explored over interval of 1.5 to 5% or so
• I am ignoring stuff like spending vol, consumption utility, human longevity for now
• I assume no jumps or chaos or fat tails and rigid stationarity/iid. I am coming to realize how messed up this assumption is irl but then I am not an academic so idk

So, to summarize what I am doing here:

1. First: frame 4 scenarios 4/12, 4/20, 6/12, 6/20 in return/stand-dev terms
2. Run the portfolio scenarios for 100 years 100k times each for a range of spend rates
3. Examine and chart the distributional density for selected runs at a spend rate
4. Calculate the median[6] spend at year 50 for all scenarios and spend rates
5. Chart the median spend for the interval of spend rates used
6. Look for a local maximum
7. Look for where the portfolio "pivots" from shrink to grow
8. Conclude something, anything, about all of this that might be useful

Here is #3. This is for the 4/12 combo for a selected range of spend rates. This is mostly illustrative to show us what the spend distribution at year 50 looks like where each color is a different spend rate for the 4/12 scenario. I did not repeat this for any other portfolio assumptions, btw. If it is important, paypal me 50k, and I'll make another chart. Heh. What is hard to see here is that while, as spend goes up, the distribution comes in and bunches to the left (a consequence of burning down the portfolio by way of high spending), it also does the same thing at spend rates under 2% where the underspend is by choice (and the portfolio is getting huge). More on that later...

Figure 1. Spending Distribution

Ok so now let's look at the four scenarios -- 4/12, 4/20, 6/12, 6/20 -- and for each one let's examine the median of the equivalent distrib that we'd see in Fig1 and then chart the median as a point for each spend rate over some interval of interest while also looking for what I was asking about in process questions #6 and #7 above: max future spending at year 50, and the pivot spend rate where year 1 and year 50 spending is more or less the same. But let's not check in with reality for a second, though. Suspend your disbelief... Also, be aware of the sim/sampling problem described in note [5]

1. 4% return and 12% std dev

Figure 2. 

2. 6% return and 12% std dev

Figure 3.

3. 4% return and 20% std dev

Figure 4.

4. 6% return and 20% Std Dev

Figure 5.

Discussion

If I were lazy I'd just leave it here because I think the conclusions are pretty much obvious. For starters: Ed Thorp was right about the 2% thing[7] but then I knew he would be. I also got the same result in a mass sim of portfolio longevity in the past, I just didn't have this explicit of a view. Spending less than 2% is a choice, I guess, that will preserve the portfolio and grow it but then so is a 0% spend. But at zero what is the point? Portfolios exist to serve. 2% looks like the bare minimum boundary to serve a purpose of both consumption now and also growing things for better consumption later... ceteris paribus and all, ya know. 

The other conclusion I have here is that Dybvig and Coiner were right, too. But then I knew they would be and I have covered this in the past as well, I think.  The place (in spend terms) where the year 1 and year 50 absolute spending in real terms are at some kind of parity is roughly around the geometric mean return expectation. I mean, we might be skeptical but the intuition of spending what Markowitz (2014) made crystal clear that we earn in the long run makes a lot of sense, something which was hazarded by Coiner in 1990 and codified by Dybvig in 21 and Waring and Seigel in 2022. 

Is there a closed form of this that avoids all the simulation? No idea but I'd start with the reference list below, something I'll not be re-reading anytime soon.  Also, in case it was not clear. Spending less and less down towards 2% looks like it will preserve the portfolio in such a way so that spending can go up in the future (at the median spend at year 50 anyway). Below 2% and you are just flat underspending and giving an unearned gift to the future. Spending more the expected geo mean return[8] basically just means that you are vitiating your ability to spend more in the future and advantaging the present time over the future[9].  

In more prosaic street terms: 

• don't spend less than 2% unless you love the future more than yourself
• don't spend more than the geometric mean expected return unless you hate the future, 
• if you are mortal there is a way to spend even more (not discussed here) 
• if you have life income, you can spend more, too (not discussed here), 
• there may be 100 other ways to frame fecund spending (say Garland 2004 and others), and 
• pray that our world unfolds in the future sorta like it has in the past (ie ignore the Nikkei)

see also

Long Horizon Spending (con't.)

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

[1]. Plenty of ways to evaluate portfolio longevity. Simulation is one. Using a deterministic formula, as Milevsky does in his Seven Equations book using a Fibonacci formula works pretty well, too. Just assuming "30 years" is fraught.

[2] Not strictly true. There is also "perverse spending" where one spends high when markets are down and low when they are high. Ask me and my ex wife about that one...

[3] A footnote testament to Dirk Cotton. When I was 58 I saw an article by Dirk where he said "I coded a simulation in R..." to which I scratched my head and asked myself "what's an R...?" I figured if he could do that I could too and then I was off to the races for a while. Miss that dude. 

[4] I use the convention for geometric mean return = return - Vol/2. But here we have to acknowledge two things: a) that that estimate is at infinity and not really at the N-period or 50 year horizon I use here, and b) there are a crapload of other or better ways to estimate (see Markowitz 2014). I stick with the convention because it is so damn convenient. 

[5] Simulation is often a "sampling from infinity" problem. Too few cycles and your sampling distribution is wide and choppy, too many and it takes an irritatingly long time to get a result that has vanishingly diminishing returns for stuff that no one will read or care about ever again. I tend to keep it as short as makes sense for me. Good thing tenure is not on my Bingo card.  

[6] The spend distribution at year 50 is not normal. I am not equipped to describe its shape. Lognormal? Reciprocal Gamma? No idea. What I do know is that the mean is useless here. Median, to the extent any single number is useful vs looking at the whole distrb, is a pretty good place to start. Mode might be useful too but R makes it hard and I don't want to make the effort. 

[7] all 4 scenarios have a max that hovers around 2%. That it is not exact doesn't bother me. This is amateur simulation, not perfect science or math. See note 5, too. 

[8] I have absolutely not framed a way that one can know how as time passes whether one is inside some safe set of assumptions about whether the expectation set in prior periods still obtain and/or whether one can have any certainty about those assumptions going forward. I'm sure there are whole books and papers on that kind of stuff. 

[9] Some finance/econ papers have various ways to deal with this. One is the time preference discount (not time-value) in some of the models. There are others and they are all over the place but I remember Yaari '65 in particular. I'm sure if I had a more formal ed I'd have a better handle on this.  Then...the OpenYale course I took with Prof Geanakopolos uses another type of nudge but here more in the symmetrical-time-discount realm by way of hyperbolic discounting. I tried it once and get it but idk about the parameters and proper use. The idea here is that projects that produce, say, pollution in the far future under-represent distant future costs thru standard time-symmetrical discounting. It's as if those costs are zero. But that just screws future generations by over-preferencing the present. I think either way it is something to think about in financial and economic analysis. All these abstractions in our spreadsheets and R code can have real consequences for us and/or our kids. 

[10] I'm too tuckered out after writing this but I'd hazard an assertion that this is closely related to Kelly logic too. Markowitz (2016) rips Samuelson on this, btw...a fun read for that alone.  There is a ton of other lit on this too. Frankly, just understanding multiplicativity and how that affects time averages vs ensemble averages is a big leap for some. Ole Peters, when he riffs on ergodicity is often talking about this stuff but note that he did not invent this, it's been circulating for a loooong time.   

[11] In simulation the net wealth process is just a recursive, discrete W(t) = W(t-1)*r' - s where r' is random discrete non-continuous return and s is a spend, in our case here: variable. More formally, we might say something like 

or maybe 

But I do not traffic in SDEs or continuous time much. You get the idea. This not your advisors portfolio tho because there is consumption and a flow through time. They usually work in single periods with no consumption. Strange assumptions, those. 

  

----- 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

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

Garland, J 2004, The Fecundity of Endowments and Long-Duration Trusts, Economics and Portfolio Strategy, Peter L. Bernstein, Inc

Markowitz and Blay 2014, Risk Return Analysis, Volume 1

Markowitz, H. 2016. Risk-Return Analysis, The theory and practice of rational investing [Vol 2]  

Milevsky, M. (2012) The Seven Most Important Equations for Your Retirement. J Wiley

Waring M B, Siegel L B. “Where’s Tobin? Protecting Intergenerational Equity for Endowments, A New Benchmarking Approach.” SSRN 2022