Asset Allocation and Risk Aversion in a No Consumption Model

I swear I've done this before but idk. I did a quick spreadsheet the other day on asset growth and asset allocation for a conversation with a friend and I thought I'd flesh out the idea a little more [3]. The basic idea is what is the expected CRRA utility of wealth, in certainty equivalent terms, at T=20 (arbitrary) for a given simple 2 asset portfolio with return and std dev of .01/.04 and .07/.25 corr coeff = -.20 (really really arbitrary) in real terms? For my conversation I had used coefficient of risk aversion of 2 (another arbitrary but I've always thought of 2 as "mine" but whatever). Here the goal is to push the RA coeff up and down to see what happens to asset allocation. 

So here, the basic idea is 

• Asset A = N[.01,.04]
• Asset B = N[.07,.25] * note: this is high vol to which the model is quite sensitive
• Correlation = -.2
• Portfolios are x/y% of A and B stepping from 0% B to 100% B=high risk in 11 steps
• The r/sdv of the 11 steps are along a MV efficient frontier using the usual math
• 3-N asset portfolios are ignored but we can imagine a different EF up/leftish
• Starting wealth w = 100k
• Iterations = 10k
• Spending = 0
• Periods = 20y
• No leverage allowed
• Shorter and longer horizons are ignored, for now
• Evaluation = E[U(w, t=20)] ...or the certainty equivalent CE of w if E[U] is inverted
• Using CRRA in form of U(w) = [w^(1-ra) - 1] / (1 - ra)
• For ra = 1 -- I cheated and set the coeff to 1.000010 cuz I didn't want to code an "if" [1]
• Used RA coeff = 1 to 3.  Higher order risk ignored [2]
• No fat tails, no chaos, no taxes, no spend, and literally no other nods to reality
• Let's call this whole thing "illustrative" or "for fun"

And if we follow the above, we get this: 

Figure 1

Where 

• Green is RA = 1
• Blue is  RA = 2
• Grey is  RA = 3
• Red overlay is a weak attempt to guess at the optimal "zone" by me with no real math other than my eyes.
• X is asset allocation to asset (b)
• Y is the certainty equivalent wealth for the given risk aversion at t=20

What do I notice? 

• For higher (2-3 here) risk aversion the range of asset allocation is maybe 30-50/70% which is consistent with past work of mine that had spending in it that showed middle ranges of risk to be relatively fecund.
  
  
• Risk aversion, if we believe such things, matters in this model and "lower" means one can take more risk but again, this is a "duh" and anyway we do not have any sense here of a mid-portfolio levered up...which might work differently
  
  
• The area of optimality in red is really flat which means, to my naïve eye, that the exact dialing in of asset allocation probably doesn't matter much within some range, Just gotta stay away from the extremes I guess.
   
• The intuition on the CRRA "curve" in Figure 1 is that the distributions of outcomes at T=20 are going from narrow to wide as the risk increases. This is obvious, right?. But at the same time, the left tails are moving out and this, via the convexity of CRRA, punishes that leftward tendency. But more importantly the center mass, via the mode, is moving first right and then left even while the right tail is extending to absurd levels. This whole movement thing benefits and then crushes the evaluation of the mean Utility. Kinda makes sense if one believes utility theory, right?.  I could probably have come to similar conclusions just using the mode but CRRA is that thing that academics use so I had to go there. Heh. 
  

Figure 2. Watch the modes and left tails - Growth of $1 at t=20

• I'm thinking there is a related analysis with Kelly math and blah blah Samuelson etc. Higher risk, via geometric mean analysis shows we can make money but sometimes the vol just doesn't pay past some risk-point or at or beyond some horizon. I'm not doing any of that today, no way. Markowitz [2014] admitted all this, btw: agreed with Kelly, poked Paul, and concluded that the portfolio on the EF that meets the Kelly Max E[log(1+r)] criterion is literally as far as one need go in MPT and that for many normies, based on psychology and behaviors, they are likely further left or down on the EF ... which is where I find myself today. Tomorrow? TBD. Again, I call my own RA coeff (not explained here but I should be more careful here) somewhere around 2. Figure 1 explains in a general sense my approach to my own allocation (ex an allocation to trend following, which is a more interesting topic). 
  
  
• Certainty equivalent wealth in real terms below the start wealth is an unexplored and unexplained phenomenon. Maybe a reader can help me out on this. Probably doesn't matter except as a "don't go there" piece of advice. 

----- Notes -------------------------------

[1] Typically we'd see U(w) = ln(w) for RA=1. I could have coded this but got lazy. It's harder to do if/then in Excel than R and I am winging an Excel back-of-the napkin thing here. I figured that 1.000010 (or was it 1.00010?) was kind of a limit ->1 kind of thing. Let me know if this is bogus. 

[2] I'm an amateur dork so in the past I have found risk aversion of higher order, idk 6 or 8 or 16, to be weird. The math is odd and the significant digits to the right get odd. I know it's all relative not absolute but still. Check it out. Do CRRA on 100k with RA = 16 and see how it works. Also I got whipped once for saying it -- but I'll stick with it -- that super high RA like 8 or 16 doesn't need math, what it needs is therapy.  At that level one is probably afraid to leave the house. 

[3] My careful readers, all three of you, are probably tired of seeing me say "omg, I'm quitting this horrible blog" and then dropping another post. Whatever. Self expression has its oddities. At least one of you corresponds privately with great consistency. That's the real way to nudge my behaviors.