Estimating geometric returns and wealth over time vs a simulated path

No grand goals here, just looking at an estimator for geometric returns over time (what we really earn) as well as it's correlate - wealth accumulation - and then compare to one very arbitrary simulated path.  Just for fun and to get the formulas into a spreadsheet.  For the estimator I am using R Michaud's estimator for the Nth period geo return and it's variance. Like this:

 

For wealth we compound a dollar using the same math and parameters transformed for the purpose. My assumptions are a normal return distribution based on linear inputs: N[.04,.12]. For the geo returns I'll do it for 10,000 years. For wealth we'll do it over a more human 30 but it is the same process. 

Not to jump to the chase but this is the output. The left is the geo mean return over time, the right is the related wealth accumulations. The solid red line on the left is the geometric N-period return expectation. On the right red is the median wealth. The two are related.  The dotted lines on left and right are the 1 std deviation expectation based on Michaud above.  On the right that means that 68% of the wealth accumulations are between the dotted lines (I believed that but I checked in a side sim). 

The blue lines are the simulated single path. Gasp. One path. Yeah, that's pretty much like the real world where we get "one whack at the cat" a Zwecher phrase. There is no purpose for the blue line other than to scare people. I mean really, this is the thing I have wrestled with for 12 years: what happens if my path sucks?  What do I do? Other than being radically adaptive, that question has not really been answered to my own satisfaction. But then that's life. Ignore here issues like the stability and constancy of the return engine delivering .04,.12 year after year forever without change or fat tails or chaos. Heh. 

Figure 1. Geometric return estimation

Discussion. 

1. Figure 1 makes it easy to see why geometric mean maximization is seductive. This is the point in Latane, Hakansson, Kelly, some Markowitz, Ergodicity economics and some others. Over infinite time, with enough flips of a coin, the probability of heads approaches 50% and the probability of realizing the geometric mean expectation approaches a narrowing but not perfectly certain field of possibility. Maximize that choice, right?! 

2. Ha. The real issue is a combo of the path and the horizon. I am totally a fan of Kelly and his criterion. Samuelson savaged Kelly for reasons that economists know. Markowitz, after S's death got his last word in on this and gave a nod to Kelly and poked S in the eye but even he did not suggest we actually "pick" the geo mean max. He merely and correctly stated that we shouldn't go past it and that we might want something less risky for our own purposes. Figure 1 makes this point.  In real human horizons, taking on the GMM might mean some serious goal-risk no matter how much financial theory we know.

3.  Pure financial concerns love Kelly and GMM. My guess another game is afoot, though. I have 20 years to 82. Pretty good chance that somewhere in there I'll try to lock in lifestyle to manage superannuation risk. To do that I don't want really big down drafts near or before when I want to do that so that GMM is a fools game for me. What I want to do is minimize the risk that I can't lock stuff in. Others have done some papers on this. I'm thinking here of maybe Milevsky and the optimal age to annuitize via evaluation of the real option to do so. Even he admitted once that this is hard stuff and hard to figure out for individuals.  

4. But here's something I always forget. The other side of the superannuation game is early death. My recent experiences with my brother in law and Dirk Cotton tell me I need to live. All this hedging and maximization stuff irk me something big. idk. I have a pretty good life right now. I don't think it's going away. I probably will maximize nothing. But I will keep my eye on it and try to protect my kids from having to carry my sorry a** for longer than they want.   So spend carefully, shy away from excess vol, hedge intelligently, consider life income sooner rather than later. What else is there?   I miss Dirk. He would know.