Asset Allocation, Certainty Equivalents and Time Horizon

This is a follow on to the last post 

   - Asset Allocation and Risk Aversion in a No Consumption Model

and the only change here is the following 

•  The risk aversion coefficient is pegged at "2," for reasons
•  The horizon is now varied: 10y, 20y and 50 years

With the goal: see what happens to "optimal" certainty equivalent wealth by way of asset allocation at each horizon, keeping in mind my version of "optimal" is not mathematical but a visual shoot-from-the-hip guess. This guess will be poisoned by scaling of the Y axis but so be it. Just winging it here.

If we use the same assumptions and methods as before -- and keep in mind that my approach demands that I admit here that I am not 100% sure I am doing this right ie this is the amateur's anticipatory "mulligan" -- we get the following for a chart of certainty equivalent wealth for three horizons across 11 different portfolio allocations along an EF.

- grey = 10 year horizon

- blue = 20 years

- yellow = 50 years

- red is an eyeball attempt at an optimal "zone" where I define zone as "near the top and not radically different inside the zone vs outside...or the flat area." How's that for science? 

* note: asset (b) is high vol to which the model is quite sensitive

What do I see?

If I ignore the scaling questions -- i.e., were I to chart just the grey line it would look peakier because I'd be using a smaller Y scale -- then the things that seems obvious here are:

• The shorter horizon tolerates a wider approach to risk 20-70% given the fake and arbitrary assets in play and the mid risk aversion. This to me was a tiny bit counter-intuitive but I have to ask myself why I think that and I'm still not sure. My guess, again as an amateur, would have been if I have a 10 year horizon that I'd want to be super careful about adding risk. This is probably true and an examination of the distributions and the relative likelihoods of missing a goal would tell me that a 70% allocation to risk might be dumb. But this is just a quick wing-it post. TBD. 
  
  
• The longer horizon, again ignoring scaling considerations or questions about other portfolios, has a narrower tolerance for risk, say 40-60% allocation to higher risk. Again, I think we should look at the distributions not the utility scores but this is a bit of a utility post...
  
  
• The certainty equiv wealth at all horizons (at RA=2, I should do this again for different RA but I mightn't) appears to be penalized in the extremes of low and high risk towards some kind of weird inter-temporal equivalency. Huh. I hadn't thought about that but I guess it makes sense. Sorta. Maybe. idk.