Revisiting the Geometric Mean Frontier

I wanted to take a look at the geometric mean frontier again.  This time I wanted to use some real data over a reasonable time frame (looking back) and then sorta project that into a MV frontier going forward and then do two things: 1) look at the four different equations that Mindlin uses to estimate the geo mean return in On the Relationship between Arithmetic and Geometric Returns, Dimitry Mindlin, CDI Advisors 2011, and then 2) take at least one of the equations in #1 and then arbitrarily and unfairly change the risky asset's vol in incremental steps to see how long it takes to get the geo-frontier to have an inflection point.  That last effort in #2 is interesting given my recent foray into estimation problems with standard deviation.  Using a monthly series seems to underestimate annual standard deviation by quite a bit.  So maybe there are some estimation errors when trying to come up with these geo-mean frontier things, too.  If so, maybe the risk of getting to one of Michaud's "critical points," or what I am calling an inflection point, comes up faster and more insidiously than I thought.  Something to think about, which I am.


Let's take a look at AGG and SPY as the low/high risk pair.  AGG in Yahoo goes back to 2003 so we'll estimate from there. I'll use monthly data because that's what Yahoo gives me and now I'm spooked on annualizing stuff so we'll keep all of this in monthly form as long as it make sense.

#1 - See if Mindlin's 4 equations look any different.

In his article, to grossly and quickly summarize, Mr. Mindlin offers 4 different versions of equations to estimate the geometric return using the arithmetic mean and volatility.  Apologies if I got the notation wrong.  In brief they are:

If I understand this correctly, equation 1 is the standard approximation that an awful lot of people use. and equation three is the same one that Michaud recommends in the appendix of his article A Practical Framework For Portfolio Choice, Richard Michaud, Journal of Investment Management 2003.  The Mindlin paper leans towards #4.  Here is the AGG/SPY data with the dotted line being the arithmetic MV frontier and all the pretty colors of other lines being the 4 equations above.  I'd go to the effort of differentiating them but they all pretty much over-lap for this exercise.  I can only see one line.  I'm sure there are cases where they aren't overlapping but today they are so we can leave this one behind for now...

As expected.  The lowest return point is 100% AGG and highest is 100% SPY.  

#2 - See what happens if we dial up the vol of the risky asset one step at a time.

Now let's mess around with SPY and change it's volatility and keep everything else the same (returns, covariance etc) and see what happens.  More precisely, how much change in vol before we get a Michaud point on the geo mean frontier?  My thinking was that the monthly series from 2003 for SPY when annualized says expect a 13.7% std deviation.  I know from experience it is more like 19 or 20%  annual standard deviation so there is some estimation error to be expected.  I thought I'd at least run it up in about 5% increments so we'll go 13.7, 20, 25, 30... Which I did.  It didn't take long; I didn't even bother to go to 30.  Here is what it looks like (It is more than likely I have errors in doing this but let's roll with it). I backed into the monthly standard deviation using excel solver for the annual levels I described. Also, I am using Mindlin #4 from above.

 

So, if you believe the math, there is an inflection point starting at around 6% monthly (20.7% annual portfolio standard deviation if you like multiplying by the square root of 12) when we boldly assert that SPY std dev is 25%.  But I don't think that is totally out of whack.  I remember watching some fun vol in 2009 well north of there.

That's pretty much it.  My take-aways for now? Not much but: 1)  The 4 equations look the same to me but I don't know yet where they might differ in practice and  2) the geo mean frontier looks like it can create some interesting results that one might want to pay attention to a little before I thought it might get interesting before I did this post.