Using approximations to intuit the output of simple non-spend MC simulation

My first foray into retirement finance was in 2012 with Milevsky's "7 Equations..." book. Those seven equations took me pretty far and I still think that simple deterministic equations (ok, well maybe not the Komogorov equation in that book, but even that can be handled by amateurs) embedded in an adaptive triangulation methodology along with some common sense can go a really long way into teasing out some intuition about the consequences of the portfolio and spend choices that one needs to make over a lifetime.

Obviously people, myself included, complicate this mission greatly. Sometimes it is for good reasons or there is "necessary" complexity (unique asset mixes, oddly shaped spend paths, unusual tax situations, etc) but sometimes it seems unnecessary. How much "lift" exactly in decision making quality comes from autoregressive stochastic inflation models or 20 independent asset classes modeled separately or a highly-fit-to-history stochastic yield curves or a super duper utility model? No one ever -- or less often than I would have guessed -- says anything about that. I mean there is, of course, a simplicity on the far side of complexity (Holmes) but I'd say few get there and the rest leave behind only the opacity of a modeler-biased black box that might or might not help with the decision at hand and still gives me limited confidence in facing the unknowable-ness of a future that will subvert our modeling efforts at almost every step...especially if we are set-and-forgeters. Dirk Cotton had a great take on this once when he was still with us.

For the rest of this post I will flog my favorite over-linked paper -- A Practical Framework for Portfolio Choice by R Michaud [RM], 2003 -- because it sticks to principles and provides tools that are mathematically sound and easy to deploy while also being (radically?) transparent. The basic question or goal is to see if -- in one narrow example that proves little -- approximations look reasonable at an "eyeball check" level for getting insight into the multiperiod consequences of portfolio choice (and here I won't even be choosing between portfolios ... which subverts my point a bit). And here when I say approximation I mean things like:

a) using a normal distribution vs other (say lognormal) is probably fine (not really proved)

b) using deterministic equations for the geometric mean return of a choice gives insight

c) geo mean analysis can help us make conclusions similar to MC sim but quickly and transparently

For "a" I was going to do some coding but got lazy. I'll lean on (RM): 

"It is well known that the geometric mean is asymptotically lognormally distributed. However, it is also true that it can be approximated asymptotically by a normal distribution. This second result turns out to have very useful applications. Asymptotic normality implies that the mean and variance of the geometric mean can be convenient for describing the geometric mean distribution in many cases of practical interest. The normal distribution can also be convenient for computing geometric mean return probabilities for MV efficient portfolios. ...

Because of asymptotic normality, the expected geometric mean is asymptotically equal to the median, and, consequently, the expected geometric mean is a consistent and convenient estimate of median terminal wealth ... Since the multiperiod terminal wealth distribution is typically highly right-skewed, the median of terminal wealth, rather than the mean, represents the more practical investment criterion for many institutional asset managers, trustees of financial institutions, and sophisticated investors. As a consequence, the expected geometric mean is a useful and convenient tool for understanding the multiperiod consequences of single-period investment decisions on the median of terminal wealth"

For "b" this is the quote I keep looking for every time I forget it. Pay attention to footnote 21 since that is right at the center  ; -)  of my point about approximating and the relationship between a quick and dirty estimation formula and the median of modeled wealth over some horizon:

For "c" I will lean on this quote and then do a little estimator-calc vs. MC-sim thing: 

"4.3 Monte Carlo Versus Geometric Mean Financial Planning

The advantage of Monte Carlo simulation [in] financial planning is its extreme flexibility. Monte Carlo simulation can include return distribution assumptions and decision rules that vary by period or are contingent on previous results or forecasts of future events However, path dependency is prone to unrealistic or unreliable assumptions. In addition, Monte Carlo financial planning without an analytical framework is a trial and error process for finding satisfactory portfolios. Monte Carlo methods are also necessarily distribution specific, often the lognormal distribution.28

Geometric mean analysis is an analytical framework that is easier to understand, computationally efficient, always convergent, statistically rigorous, and less error prone. It also provides an analytical framework for Monte Carlo studies. An analyst armed with geometric mean formulas will be able to approximate the conclusions of many Monte Carlo studies.

For many financial planning situations, geometric mean analysis is the method of choice. A knowledgeable advisor with suitable geometric mean analysis software may be able to assess an appropriate risk level for an investor from an efficient set in a regular office visit. However, in cases involving reliably forecastable path-dependent conditions, or for whatif planning exercises, supplementing geometric mean analysis with Monte Carlo methods may be required. 29

For portfolio choice and many common cash flow planning applications, geometric mean analysis may often be the method of choice. With a knowledgeable advisor, experience suggests that an engaged investor may often be able to choose an appropriate resampled portfolio from an efficient set with suitable geometric mean analysis software within a half hour. However, in cases involving reliably forecastable path-dependent conditions, or for what-if planning exercises, supplementing geometric mean analysis with Monte Carlo methods may be desirable."

For the sim vs calc thing I am going to keep it simple. For the geo mean I will use the equations in section b (RM 6a and 6b) to calculate the n-period geometric mean estimate over 100 years (blue dashed lines). I will also use 6b (variance but w sqrt(V) to get standard dev) to infer the 68% of geo returns between the upper and lower estimate on either side of the mean (I have to do a little work to transform that into wealth outcomes but it is pretty straightforward). 

For the simulation, we want to look at median terminal wealth by way of footnote 21 above. Therefore I will simulate and then extract the median wealth over each of the N sim periods 0->100 (black line) to which the geometric mean (chained in a 1+g^n chain) should relate, as in the footnote. Around that median I will simply use percentiles from the simulation output at .1587, .50 (median) and .8413 (i.e., the interval = .6827) and then eyeball all of it in an unscientific way to see if the quick and dirty deterministic estimator is playing the same game as the more "robust" stochastic processes of the sim. Some assumptions:

• the portfolio is r= N[.05,.12] which is arbitrary and uses the simplifying normal distrb for r
• initial wealth is = 1
• 50,000 iterations
• spending is set to zero, something I rarely do in the blog
• blue dashed lines (Fig 2) are the Michaud estimates from 6a and 6b: the mean r(g) +/- 1 std dev
• black (Fig 2) is the median wealth from the simulation
• green lines (Fig2) are the 15.9th  and 84.1th percentiles from the simulation
• grey lines are the simulation squiggle paths, and any one of which we could be on in our 1 life 

The sim vs estimator output

Figure 1. the simulated, chained geometric return over 100 periods
blue is the geometric mean which converges towards  ~.0428

Figure 2. Wealth outcomes: sim vs geo mean estimator.
Median sim wealth (black) and the geo mean return
estimate for the median (blue center dashed)

Discussion

What is to discuss? ;-) The dirty ugly dreaded deterministic formulas that are usually outshone by their fancy pants simulator cousins seem to give some insight into multiperiod consequences of choosing different portfolios (haven't shown that difference here, mostly just showing the flow of "one") without much fuss...and they look like they more or less exactly overlap each other. The hidden issue or decision vector mentioned in some of my other posts is that lower return portfolios can outperform higher depending on the parameters and the estimators can help us in the quick selection of mean-variance efficient portfolios in the context of multi-period time, all without a black box.  The point here is that one can intuit the portfolio selection issues based on basic finance principles and heuristics alone without the need to code an MC sim. I mean, except for weird tax issues or odd spending "shapes" or other non-return things (see my post on chaotic strikes to a plan). That's really where simulation shines. But take Michaud's warning on unrealistic or unreliable assumptions seriously, especially the more customization and design is involved in the MC tool used to influence your decision making.