Using an N-period Geometric Mean Return Estimate for Median Horizon-Wealth Outcomes

A reader of my last post (wow, I'm surprised I still have any; one is good, though) pointed out that even with spending set to zero, there is radical uncertainty about future outcomes (of course, because no one can predict the future) and he pointed to page 32 of Michael Zwecher's great book on retirement portfolios where he, Zwecher, starts to introduce the useful concept of income floors. That reader comment in turn reminded me that: 1) straight up terminal wealth sans spending can be estimated by way of the geometric mean without recourse to black box simulators, something I often blather on about here and then point, vaguely, to R. Michaud's work, and 2) I had never actually taken a direct look at the link between the two. Today is the "look."

Geometric Returns vs Net Wealth over Human Horizons

This post won't add much new to what is in a million other papers or posts, just working some personal stuff out. So this is just for me and the nerds.

In a past post I profiled how the annualized geometric return of a (stable) return engine is diffuse even at long horizons but maybe less so at very long horizons. My original point was that -- in terms of human horizons of, say, 20 or 30 or 40 years -- it is really risky to have volatile returns if you have a goal that depends on achieving a particular return (think "locking in a guaranteed lifestyle by purchasing an annuity at age 80"). The individual portfolio return you earn on your one path -- what Zwecher called "one whack at the cat" -- is wildly uncertain. Yes, if you held it to infinity and had some unwarranted conviction that the "return engine" would be stable that long, it would produce a mildly predictable result. This is the basis for the optimization framework of max{E[log(1+r)]} of Kelly, Markowitz, Hakansson, Latane, etc. 

Heat map of the expected time average of a non-ergodic process

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).