Multi-period efficient frontier contextualized on a surface

 A few posts back I created a geometric return surface based on combinations of arithmetic return (x), standard deviation (y) and realized long horizon geometric return (z).  That was a relatively empty exercise since there is no context as in "why wouldn't we just pick the highest z?" Well, because you can't. You are limited to what is investable along the effects of diversification that are implied in the efficient frontier.

So in this post I'll use the same surface but now take a hypothetical 2-asset efficient frontier -- where the arithmetic single period returns are re-rendered as multi-period long horizon geometric returns -- and place it on the surface just to see what it looks like. Since Excel was not up to the task and I didn't want to figure out how to do it in R, I had to hand-place the points on the chart (2nd one below) which was a pain and not all that precise.  

First, the hypothetical portfolio - hypothetical because this is just an illustration - is like this: N(.035,.04) N(.07,.25) ρ = -.10 and the geometric return is at N = infinity with G=a-V/2 as an approximation.  The efficient frontiers in R^2 would look like this:

Orange is the single-period arithmetic, blue the multi-period long-horizon geometric. The green dot is the critical point max. Now since R^2 is not cool enough we'll take the blue dots and place them (and change the color to red just to confuse you; green is still the critical point) into the R^3 surface from the past post. Like this: 

Discussion

- Is there any real purpose to this? No, not really, I just wanted to see what it looked like.

- I noticed that the critical point (green) is not dramatically different than it's neighbors along z. I'd say something smart about gradients and their second derivatives here but that's way above where I have gotten so far in my calc classes. Basically by this I mean that it almost looks like it doesn't matter too much how one allocates in the broad middle of the interval [0,100%] (for these parameters only). This is a point I've tried to make before in different ways,

- The "up and to the left" of single period finance becomes "climb the hill" hence the gradient comment.  The go-left-first of my "retirement opinion" would, in this context, become "climb the hill while leaning a little more left" (when consuming a portfolio, volatility has a big impact on portfolio longevity).  The climb should probably start at the green dot but the last bullet means it probably doesn't matter as long as one is reasonably close (not defined) to green.

- Using leverage would suck as would no risk assets at all. 

- The geo return is horizon dependent and this illustration uses a not so practical assumption of infinity. Even if we were immortal, assuming that  N(.035,.04) N(.07,.025) ρ = -.10  is stable over large N is a weak assumption. This would all be a very different post if N were, say, 20-30 years and spending were to be added.  

- Other parameterizations might not have a critical point or it would be past the 100% risk allocation.