10,000 years of a geometric return series done 1000 times

I keep saying I'm done here on RH but that does not eliminate my curiosity. I was wondering what 10,000 years of simulated geometric returns would look like. That's way way outside any reasonable lifetime but it is closer to infinity than not in practical terms. Let's see what it looks like...  

I took the (arbitrary) annualized return for N(.07,.25) and ran it through this:

 

for 10,000 years and I did that 1000 times. 1000 is an ok but not great amount of times to do it but I did not have all afternoon to wait for the PC to run it all. Then, cuz graphing is hard, I graphed the above for only the first 100 out of 1000 iterations over 10000 years. When I do that, in grey, and I add the mean, in blue, it'd look like this: 

Of course as we do this some core assumptions come to mind:

1. there is a stable return engine that kicks out N(.07,.25) forever with no change...gasp. This is a pretty weak assumption.

2. the return distribution of the "return engine" is in fact normal over all those years. This is also a very weak assumption. 

3. we can examine the G distributions at [n = 5, 18, 10000]  and assume they are normal-ish. This is also weak, and not considered very carefully.

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If we were to create a table of the various outcomes, it might look like this:

Then, if, for some kinda fun, we were to sample the density of the distribution of G at 5(red), 20(grey), and 10,000(blue) years, it'd probably look like this which is consistent with the intuition of the first chart and maybe makes us question the normally distributed assumption:

Some Thoughts:

- the assumptions are pretty important. See 1, 2, 3 above. 

- In the near years (0-20+) there is wide variance of outcomes with reference to the closed form math[1] as well as just wide variance in general. This is a difficult zone to deal with in practical terms for those that tend to perseverate on max G or max e^[E(log(1+r))] at infinity. The wonks have no skin in either your game or, sometimes, even their own game. 

- in the simulation that uses only 1000 alternative lives there is pretty wide difference between what we might expect via the closed form math (and also via the ability to sim more than 1000 times) [see table].  But this also accentuates the point that we have only one life, so which of the many paths on the chart, or other unseen charts, will one's life be? No idea, 

- The expectation math is pretty far from what the sim kicked out. I have to assume that this is from too few iterations in the sim,

- the "20 year" interval comes from the idea that I personally have only about 20 years before I might, might!, annuitize all assets. So that means while I like the idea of things like consumption utility and/or N-period wealth maximization, ie the zone where academics and some practitioners focus on U(c) or N=infinity, I am managing my "wealth" to some end-goal year, say, 2040 AND I am also thinking about consumption utility. This horizon and its split objective are an area that the geo-return maximizers will never deal with. 

- at 10,000 years we still see some variance about where we will end up. Expectation math is fun but not certain at even those long time-frames.

- Look at the first chart. there are paths there that would really hurt me for a really long time, not to mention the 20 year horizon thing. 

- if one were to have an advisor, or professor, that focuses on geo return: 1) omg he/she understands multi period math!, and 2) it's not enough; ask them "for how long and what path?" 

- where tf did our 7% return assumption go, eh?

No one knows this kind of stuff except the soothsayers and they usually have unaligned incentives. 

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1. The estimation method using Michaud 2003