10,000 years of a geometric return series - revisited now for time in years to get above a threshold

This is a re-look at a post I did in September. Then, I ran a "return engine" 10,000 years x 1000 times to look at the shape of the annualized geometric return paths that come out of that multiplicative process. The idea then was that there were a ton of differences in the paths that kinda iron themselves out over ~infinite time. The problem now being: none of us have infinite time and the early years -- economists or physicists new to econ or finance notwithstanding -- can be pretty hard or unnerving.  So now, the question today: how bad can it be -- or how long can it take -- over the foreseeable and unforeseeable future? I am sure there are better mathy ways to do this kind of post but I don't know. As I have beat my own drum before, I am an amateur!

Why?

I was reading a paper today by a former physicist that has newly taken-up econ and finance and is enthusiastically trying to correct everybody on what he sees with something called "ergodicity" which is more the physics of constrained gasses than finance. That's fine, but it sometimes grates a bit -- a little like a teenager that has newly discovered sex in a book and wants to tell everyone about the idea. The problem is that he is both right and that it is a story that should be told. It's just that: 1) we already know this stuff*, and 2) it is different for those that actually have to deal with the reality of the consequences on the ground. In this case, I make the boldish claim that retirees are neither physicists nor mathematicians nor economists and don't have time for the "purity of math." In my own case, I have what I want to call a 25 year horizon to get it all right, maybe less. After that horizon I am going to be either decrepit or dead.  The "purity" of the differential equations and Ito calc will not sway me much in maybe as short as 10 or 15 years, Kelly, Markowitz, Hakannson, Latane, or Peters notwithstanding. 

The goal here today is to try to make the case that the  "purity" of the (non)ergodic math of geometric returns, in terms of the years to get above some "threshold," is a harsh taskmaster. In the paper I read, the "ensemble" averages (think arithmetic or linear or single-period returns) are averaged over many lives; the time-average (think geometric, multiplicative return over multi period time) on the other hand, is set up as an N=1 path because, well, because he did it that way...and time is on his side and parallel lives are technically not needed, heh. I think that is a fairly flawed approach because NO ONE knows what path they will follow, they do not have infinite time, and here I am proposing to illustrate "many" paths for the time average i.e., N=1000 not N=1 though it could have been 5000 or 1000000...except I did not want to sit around all day...  

The Setup

• Portfolio 1: N[.07, .25] arbitrary and illustrative only
• Portfolio 2: N[.07, .18] "
• Portfolio 3: N[.07, .12] " 
• Runs (since 1000 iterations is ~short, do a couple of them...): maybe 2/portfolio 
• The Path: 10000 years of multiplicative geometric annualized return 
• 1000 iterations of the path (this takes a long time) 
• Take the minimum annualized geometric return over 10000 years and see how long it takes to get above some threshold level 
• "Threshold" levels wrt annualized geo returns: 0% to ~4 or 5% 
• No leverage, plus or minus, 100% invested into the portfolio 
• No spending; the addition of consumption would CRUSH this analysis but that's another thing

The Goal

• Count and report the years to get, incontrovertibly, above some threshold level over 1000 paths
• We're not following any particular path here but just evaluating the min for all paths
• Goal is not to be exhaustive or thorough, just to see some "year" numbers

The Output - Simulation Charts

Figure 1 - annualized geo returns for Portfolio 1, run 2

Minimum of any annualized geo return path over 10,000 years, Portfolio 1, run 1

The Output - Output Table of Years

The Output - The Table as a Chart

Thoughts

My opinion: the math used by physicists, economists, and sometimes advisors is not really my math. It is not grounded in reality or, as Samuelson said, probably correctly, if I have it right: "rationality principles." Right now I am in a real race against time to get to either death or some future annuitization state at about maybe 80-85. No doubt institutions -- long duration trusts or endowments or abstracted no-spend portfolios that last forever -- can afford the long-dated principles of Kelly or Hakannson or Ergodicity Economics but I'm guessing most 60-ish humans probably can't. I can't. 

Something else needs to be in play here: Talebian robustness and redundancy prep (in theory this might be synonymous with a lower spend rate but that's another post), life income from either inside or outside the model, hedging via some structure or products like trend following, or, as always: other.  Lower vol seems like it pays off but I've made that point before. 

* I've lost track of the people that have written on this with some degree of non-opacity: Markowitz of course, Kelly, Hakannson, Latane, Bernstein, Michaud, Meucci, Collins, Samuelson, Zwecher, Milevsky, Cotton, etc. Whom have I forgotten? A cast of thousands. Me.