This post falls, like others in the past, into the category of "I wonder what it looks like?" There is no agenda here and all of the input parameters are absolutely 100% arbitrary and are not shaken against any other parameters to see anything else other than just "what happens with the first set?" I hate to say it and my gf would shake her head but this is just a joy ride. The main question here is:
What happens to an efficient frontier when going from a 2 asset portfolio to a 5 asset portfolio but especially when the EF for both is re-rendered using a "realized geometric return at infinity" adjustment?
The portfolio (and don't ask me why these parameters, this just happened to be what I found in an old spreadsheet and is intended to be entirely illustrative) in question will be this:
The 2 asset portfolio is a and b; the 5 asset P is a->e. The covar matrix (again, I don't remember why this but let's roll with it) is this:
Using standard MPT math
the 2-asset EF looks like this
The 5-asset portfolio is harder. Using [(n+k-1)!/(n! (k-1)!)] means there are 1001 combinations of allocations using 10% steps between 0 and 100% allocation to any asset. Using standard MPT, this creates a field like this in red below overlaid on the blue line above...
The blue line is as before and the red field is the set of all 5-asset-allocation combos in 10% increments. The purple line is the outer boundary of the 5-asset portfolio (above ~3.8% return, anyway, where it starts to separate a bit). The striations in red are due to the low resolution of the 10% allocation steps. I wanted to go to smaller steps but 1% steps create 4.6M permutations. The 5A frontier dominates the 2A over part of the interval (estimating prospectively anyway) and there are no critical points on the upper end of either frontier.
Now we add time. For a first cut we'll estimate the geometric return at an infinite horizon using the imperfect but easy estimator: g = r - V/2. We will also focus only on the blue and purple line above (not the rest of the red dots) and I will not attempt to see if there are any alternate geometric frontiers that dominate the purple or blue other than transforming what we already have. We transform like this below where purple (5asset arithmetic frontier) and blue (2 asset arithmetic frontier) become yellow (5 asset geometric) and orange (2 asset geometric). This would look different if we were to look at an N-period-horizon geometric frontier where N < infinity:
Thoughts
This was all a little arbitrary so I can't really make any serious conclusions but here are some thoughts.
- I am not playing the tangency-portfolio and CML game here, fwiw
- Neither the arithmetic nor geometric 2 asset frontier had any critical points. This was true of the 5 asset arithmetic EF as well
- The geometric rendition of the 2A EF (orange) might've had a critical point if: a) we had had a more volatile asset set, or b) we had levered beyond the right end of the EF
- The 5A EF in its geometric rendering does seem to have a critical point near the upper end
- The only comment I think I can make here is just to be cautious with complex portfolios when considering time and volatility but that is nothing new.
- Re-parameterization, as well as the bludgeon of the future, would and will change everything.
- This type of thinking about time is not inconsistent with MPT. Markowitz in 2016, in Risk-Return Analysis Vol 2: "As we noted before, Markowitz (1959) does not recommend that the investor chose the MEL [growth optimal, geometric mean max] portfolio. Rather, it recommends that the investor not select a portfolio that is higher on the efficient frontier, since such a portfolio has greater short run volatility but less long run return--in the Kelly sense--than the MEL portfolio." He dilates a bit more on shorter term horizons but I still am not quite sure what he is saying. Personally a short-ish horizon like mine will more likely than not abhor the volatility of geo mean maximization. TBD
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