I know very, very little about geometric means and returns
but I'm starting to think that I should know more. I have tried to read a little bit about this but
it can be a tough slog through some of this stuff.
That means that this post is not a demonstration of any kind of
comprehensive knowledge. It is, rather, more of an attempt to convince myself
that I know anything at all from my recent journey into geo land and an attempt to
try to consolidate some of it "on paper" if I do. As always, let me know if there are any major
errors or omissions.
I was too lazy to do anything other than grab a dataset I already
had handy from a few months ago when I calculated for myself a MV frontier for random
allocations of 5 ETFs over about 35 months somewhere in the 2014-2016 range. That is not the best place to start for this
particular task but at least I had a place to start. I cheated a little because back then I had
calculated geometric returns rather than arithmetic. For this post I transformed them back to
arithmetic by means described in the referenced articles. Let's pretend,
though, that they were arithmetic from the start though calculating them from
scratch would probably take less time than writing this paragraph about why I
didn't. The thing to do, then, I gather,
is to transform, for the array of MV risk levels, the arithmetic returns to geometric
by way of whatever of the many approximations seems appropriate[1]. In this case we will use Bernstein and
Wilkinson's equation[2] for approximating a geometric return and ignore it's exact
appropriateness or variance behavior from real historical returns:
In addition we will use, for comparison and what I'll
loosely call "fun," Richard Michaud's equation 6a [3] for
approximating geometric returns over some set of time intervals when the
intervals are finite ( I use 2 periods below) rather than large or infinite:
When one does this, the geometric mean frontier, relative to
the traditional MV frontier, can look like this below…or at least it does for
the data I hacked out for what I was trying to do earlier this year. The lines probably look a little choppy
because my sampling of the possible asset allocation choices was a little choppy. That, and I couldn't figure out how to best transform
the 5000 MV allocation sample-pairs of return & standard deviation that I
used for 5 assets into one MV "line."
For today I just roughly binned the data, looked for the highest and least
volatile points in the bin, and then did the best I could. This is what I got. Choppy. Close enough.
So, the geometric returns along the geometric mean frontier
-- as expected since geometric returns are always (correct me if I don't have this right but
I think they are equal when variance is zero) below arithmetic -- are below the
MV line and, this is important to look for in portfolio analysis, as risk rises
to the right the curves look like they start bending a bit lower in relative
terms. The blue line is Bernstein equation. The red line is the Michaud equation for geometric returns when the
number of N periods is finite (2 periods in this case). Red becomes more or less blue as N gets big.
The problem for me in this illustration, as lovely as it is, is that most of the charts of geometric frontiers I have seen lately in my reading seem to bend down a bit more than mine does. But then again the time period I happened to pick was relatively low vol and it's volatility (why don't these guys ever talk about spending…that's the real killer) that does funky stuff to expected (and realized) geometric returns. In fact, I was so disappointed that I did not get the "bend," or what Michaud calls "critical points," that I decided I had to create some fake data to get it and so to complete this post. The "critical point," according to Michaud, is an optimum along the curve of the GMF (not always present or maybe it's beyond the right end of the MV frontier) where the return is the highest and after which one might start to get lower (geo) return for higher risk, something that no one I know wants. While you can kind of see it start to bend at 11-12% volatility, and I could have maybe imported charts from others websites as well to illustrate this concept, I wanted to exaggerate the bend in this post with my fake data to make a point. Here, for example, is a fake MV frontier made with two assets. The return/standard_deviation for asset 1 and 2 are 3%/7% and 9%/20% respectively. The correlation coeff is -.5. I don't think this setup is wildly unrealistic but it is fake and the lower risk asset is maybe a little too high risk but it works for my point. Given those assumptions, then, the arithmetic and geometric mean frontier and the MV capital market line might look a little like this (ignore debate about risk-free rate levels while we do this since this is hypothetical):
The problem for me in this illustration, as lovely as it is, is that most of the charts of geometric frontiers I have seen lately in my reading seem to bend down a bit more than mine does. But then again the time period I happened to pick was relatively low vol and it's volatility (why don't these guys ever talk about spending…that's the real killer) that does funky stuff to expected (and realized) geometric returns. In fact, I was so disappointed that I did not get the "bend," or what Michaud calls "critical points," that I decided I had to create some fake data to get it and so to complete this post. The "critical point," according to Michaud, is an optimum along the curve of the GMF (not always present or maybe it's beyond the right end of the MV frontier) where the return is the highest and after which one might start to get lower (geo) return for higher risk, something that no one I know wants. While you can kind of see it start to bend at 11-12% volatility, and I could have maybe imported charts from others websites as well to illustrate this concept, I wanted to exaggerate the bend in this post with my fake data to make a point. Here, for example, is a fake MV frontier made with two assets. The return/standard_deviation for asset 1 and 2 are 3%/7% and 9%/20% respectively. The correlation coeff is -.5. I don't think this setup is wildly unrealistic but it is fake and the lower risk asset is maybe a little too high risk but it works for my point. Given those assumptions, then, the arithmetic and geometric mean frontier and the MV capital market line might look a little like this (ignore debate about risk-free rate levels while we do this since this is hypothetical):
MV(A) is the standard Markowitz linear frontier. The
tangency point is right about 40% allocated to the higher risk asset. MV(G) is the Bernstein-converted (from above)
geometric frontier given the parameters I set up. The MV(G) "high vol" is an alternative completely fake setup with the same assets and returns and covariance as MV(G)
except that asset 2 now has a 35% standard deviation rather than 20%. I hope I got the math right (my chart looks a
little extreme to me but who knows).
What it does show, whether I got it right or not, is that there is now a
"critical point" somewhere around the 15% risk level (~75% allocated
to the risk asset at that point) which is what I wanted and most articles on this subject try
to show (usually at higher risk levels?).
The critical point (pun intended) here, and the end of this post, is
that the lesson might be that sometimes when one thinks one is signing on for a
single-period-mean-variance-optimal-or-maybe-riskier-but-higher-return
portfolio, it is entirely possible that one might not be as efficiently well
compensated for taking the additional risk [5] as one thinks when looking at it
in geometric return terms…the returns that are the real true long term "effective" ones that one really does receive in life. If
that is all I learned from this (if I got it right) I guess it was worth the effort.
Notes ---------------------------------------
[1] there are quite a few approximations in quite a few
papers. Their derivation is beyond me.
The approximations all seem to have both pros and cons as well as a ton of
similarities and relationships between themselves. This post is not a survey, though. Mindlin, in the "Links" below has
a good math-heavy survey of this topic.
There is, of course a lot of cross over and equivalency between the discussions of Mindlin, Michaud,
Bernstein etc.
[2] http://www.efficientfrontier.com/ef/198/gmf.pdf
[3] from "A Practical Framework For Portfolio Choice" Note that in the end he gives preference to the
following equation which more or less converges with Bernstein's equation and
Michauds 1/N formula as N gets large:
but see note 1 and Mindlin for a broader discussion of geometric approximations.
[4] fwiw, the "textbook" definition of geometric
average return:
[5] the "critical points" may or may not exist at all in any given portfolio analysis or may be beyond the relevant MV universe in question. I guess you gotta look. It seems like someone with a portfolio made up of unusually volatile components who also has a long planning horizon is a good candidate for thinking this kind of thing through.
Links ------------------------------------
Diversification, Rebalancing, And The Geometric Mean Frontier,
William J. Bernstein and David Wilkinson 1997.
A Practical Framework For Portfolio Choice, Richard Michaud,
Journal of Investment Management 2003.
On the Relationship between Arithmetic and Geometric Returns, Dimitry Mindlin, CDI Advisors 2011.
Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management, A. Meucci 2010.
Geometric Mean, Wikipedia
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