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. 

But in my presenting this type of "funnel" chart

a friend reminded me that Zvi Bodie said charts like this make it look like stocks get safer in the long run. That is a warranted concern. That is because not only is the annualized geometric return: a) an expectation not a path, and b) very misunderstood in terms of risk for short run horizons -- but we also ignore the inversion of the chart above. By inversion I mean that there is a flip side to the funnel that can be rendered in terms of wealth outcomes where the dispersion of wealth is related to the geometric mean return at the end of the funnel horizon by way of the expected median wealth. Those widening wealth outcomes, btw, get really really wide over time and show how the game really works in real terms when thinking about "safety."

So, for this post I'll do that wealth dispersion thing but just to make this an unjustified apples to oranges thing (heh) I will convert the wealth process to a "net wealth" process so that we can savor some sense of how it works for a retiree spending real money. See bottom section for my assumptions. I have not attempted to play with different parameterizations here, I'm just looking at the flow or process for one set. The following chart is made of two parts:

- the net wealth process related to the funnel above -- with the exception that we are spending from the portfolio now -- rendered here in box plot form* (left axis)

- an overlay of mortality probability (red, right axis) conditional on being 62 with M (mode) and b (dispersion) from a Gompertz model and parameterized as below

These two (boxplot + red) don't make perfect sense to chart together but I need both to make sense of real risk. What, if anything, can we say in looking at this?

• It's hard to see but there is both growing dispersion and growing, lognormalish, asymmetry (to the upside) in the distributions of terminal wealth
• median wealth, a metric of interest, declines convexly and monotonically towards or beyond zero
• There is negative wealth here but that is not a modeling conviction, it just makes the coding easier. Think of it as a type of fail magnitude for now or maybe it represents the possibility of frictionless moral-hazardless borrowing. Hopefully this does not subvert my point. I'll do better next time ;-)
• I don't know how R determines "outliers" but we can see here that whomever coded "boxplot" thinks that the out-year wealth outcomes on the upside are outliers and they probably should be considered so. 
• The probability of being dead (red) peaks before the worst of the downside net wealth kicks in, I think.
• Being in volatile assets over a long horizon appears to be very risky when looked at through a lens of wealth dispersion, especially in the presence of consumption, ..."predictability" of returns at infinite horizons notwithstanding. 
• The original point in the last post -- annualized geometric returns over the very short term in one's own unique path in life can seriously suck -- still stands, something that cannot really be seen in the wealth dispersion chart but is real nonetheless. I guess we need both charts to intuit the personality of risk. 

   

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Footnotes

* best way I could find to visualize it. The R heat map did not work as well as I wanted.

Assumptions

• Periods: max = 100 but showing 50 in a chart or two
• iterations - 5000
• returns - .05 (this is arbitrary)
• standard dev - .18 (this is arbitrary)
• wealth units - 25
• spend unit -  1   (4%) (this is arbitrary)
• Mode of longevity - 90 
• dispersion of longevity - 8.5
• age - 62
• Net wealth is W(t+1) = W(t)*r -1 recursively, random r

Aux Chart

This is the zoom in on the median wealth line in the second figure. Don't forget the uncertainty above and below....