This not really a dig on Ergodicity Economics. I did dig in the past but my point here today is to continue to look at the reality on the ground for human retirees when it comes to finance. EE makes the proper point that the time average matters more than the ensemble average and then they make maybe a teeny tiny bit of a "stretched point" that there is only one (ie log) utility function that matters. So, I quibble, but only on the edges. And as before, EE was not the first to the world on time averages in finance. Others were into this point well before EE. I don't know the proper list but let's say Kelly, Hakansson, Latane, Markowitz, Thorp, and a host of others.
I mean, the geometric mean in finance matters and is also, notably, also a reasonable proxy for Monte Carlo simulation in the right hands (not 25 year old advisors, btw) because the geometric time-averaged mean is also representative of the distribution of terminal wealth outcomes. It is, in that sense, directly correlated to median terminal wealth (cuz of course the average is meaningless due to extreme wealth outcomes on the upside).
The Setup
To run through this exercise again (search "10000 years" on my search bar) I took an entirely arbitrary (people ask me about my numbers but they don't matter really, we are just illustrating a point) normal return engine (N[.05,.18]) and I ran it out for 100 years and I did that 10,000 times. I did this kind of thing in the past for 10,000 years but this time I wanted to stay closer to a human scale, plus it runs faster at 100. The results below are not dollar outcomes, btw, but annualized geometric returns over the path for each of the 10,000 parallel universes.
Then, with the output in hand, I created a heat map of the dispersion of annualized geometric return outcomes along the way. Why a heat map? Idk, just cuz. I wanted to see it. Plus, it is easier to characterize what is happening with a heat map rather than just plotting a junk pile of 10,000 paths, each one on top of the other. The point of the heat map is to be able to intuit return risk given a stable return engine where "stability" itself is kind of a stupid assumption undiscussed here. So, if I happen to NEED a particular return to survive through and to some horizon, how risky is that expectation (for fixed parameters here)? Also I like to look at returns rather than terminal wealth because there is a convergence rather than a dispersion and -- just for today only -- I like that.
The Chart
Discussion
1. The expectation
On the right side of the chart is the expected time average formula. Typically this is represented with an estimate using the formula g ~ A - V/2. But, three things:
a. that estimate is for infinity not intermediate time horizons. On the right I show the N period estimator I picked up from R Michaud.
b. while g ~ A - V/2 is fair for normally distributed stochastic GBM processes it is less so for real markets. See Markowitz 2014 for a deep dive on this.
c. It is only an expectation. Even at very very long horizons there is still some dispersion around the expectation. At human horizons there is quite a bit more dispersion.
2. The risk
The main point for me and this post is my need to land my retirement 747 on the ground safely at age 90 or so, with some deep consideration as well for my liabilities in the interval that occurs before age 90. That means that the big black circle above is of keen interest to me. Kelly and Hakansson and Latane and Markowitz and EE and others all seem to like to propose the Max[E(log(1+r))] criterion. This is lovely and mathematically pure and I respect it soooo soooo immensely. But man. None of those dudes will be there to take the other side of my risk in a pinch in 30 years. So, obviously I don't really do either N[.05,.18] or Max[E(log(1+r))]. I hedge and diversify and monitor obsessively.
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