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."

The Assumptions

- a return engine with a distribution defined as N[.05,.18]. This is arbitrary and used for illustration only. We assume iid. Normal just makes the programming easy but could be nudged if we wanted to. 

- 100 periods of time. I could have gone far but, really, why? 100 is still within reasonable human scale, sorta. It also allows me to make a point about intermediate horizon return estimates when we are not dealing with infinity.

- 5000 iterations. Don't need too many to make the point.  

- Initial wealth is in units, here set to 25. Ordinarily I'd set the spend to 1 unit but here it is set to zero.

A Simulation.  

Here is the simulated outcomes using a rudimentary stub of what can be fairly called Monte Carlo simulation...but only in it's very basic sense. The net wealth process is defined here as 

W(t+1) = W(t)*r' - (a spend unit)

where r' is the randomized return and in this post the spend unit is set to zero. When we take the assumptions and plug it into the sim and then boxplot the results we get this (in Zwecher terms I might have drawn a horizontal line at 1 unit to represent a floor; Note that I limited the horizon by accident to 50 periods)

Figure 1. Net wealth process for spend = 0

where the black line is the median wealth outcome at period N and the tan boxes are the 25th and 75th percentiles. The rest is what we might call conjecture or magical thinking or something like that. 

Testing the Geometric Mean Estimates against the Simulation

In a "A Practical Framework for Portfolio Choice" R Michaud reminds us that we don't really need a simulator to estimate the median wealth outcome (median because the average gets whacked by the asymmetry of the "unfolding" and is therefore a better estimate than the mean) we can say that median wealth can be estimated by way of (1 + G(m))^N where G(m) is the geometric mean estimate and N is the number of periods. Since this post is working with finite periods, not infinity, we'll borrow Michaud's N-period estimator rather than the common estimator: 

Then we will compare that estimator -- using initial wealth and the progression of N periods -- to the median horizon-wealth black line from Figure 1.  In theory -- even though simulation is imprecise and the estimator itself is not perfect -- they should be pretty close.  Here are the two charted together:

Figure 2. Comparing The Sim to The Estimate

So, yeah, this maybe means you don't really need to pay an advisor to run a Monte Carlo sim. I mean there are other issues like taxes and complex estates and weird spending and all, but at it's core, the simulation is just chaining returns multiplicatively which can be done at home. The other issue is the extreme (but invisible in Figure 2) growth in the dispersion of possible outcomes -- for which the median is only a proxy and for which just looking at the median might distract from the issue of real risk or uncertainty. But there are ways around that too that I have not illustrated here. Maybe just knowing that the outcomes: a) go wide, and b) go asymmetric to the upside might be enough for now. Anyway, Dirk Cotton once made the point that in the presence of chaotic systems (i.e., real world not my fake blog world) the prediction horizon is really really short so that saying anything past a year or two is a pretty fraught enterprise.