Update on prelim evaluation of real annuitization with stochastic inflation model

I confused a reader in my last post. In a convention that I did not explain, when I ran different spend rates, and used different asset allocations to a stylized efficient frontier, for a parameterization that was unique to me, and therefore not really generalizable, I used a type of shorthand from a prior post. 

For different spend rates for a nominal and for a real annuity, both modeled using stochastic inflation with an autoregressive feature (and compared to a deterministic baseline), I consolidated my moves and took the max utility for each spend rate across all allocations. Easier for me but also easy to misunderstand.  The original chart from (A prelim look at a real annuity and lifetime consumption utility - with a stochastic inflation model) looked like this

Figure 1

Check the past post for why this figure exists. The orange line is the expected discounted lifetime utility (EDULC, using the assumptions in the last post and the underlying model here) for a real annuity with deterministic inflation modeling. The dotted orange is it's analog using a stochastic model.  The orange line, taken in isolation, is really a distilled version of this below in figure 2 where the orange line in figure 1 is the max of all the lines in figure 2, or upper boundary of frontier at each spend rate, i.e., this was a simplification maneuver:

Figure 2

And Figure 2 itself is a type of simplification all by itself.  Shifting a bit*, if we assume that the other lines in figure 1 are constructed just like we did in figure 2, then if we were to take the "figure 2 version" of (from figure 1) the orange dashed (real annuity, stochastic inflation) [upper surface in figure 3]** and the blue dashed line (nominal annuity and stochastic inflation) [lower surface in figure 3] we might actually render the figure 2 versions of those lines as surfaces rather than lines as we did in fig 2.  Fig 2 is easily imagined as a surface I hope and if we are on the right track it'd look like this

I realize this kind of post is tailored to a "reader of 1" and that it is not amenable to generalization since this whole house of cards is sensitive to the parameters as well as model risk but as I mentioned this is a software shake-out not a tutorial. I just thought I'd get myself straight on the basic illustrative concepts first.

Email if you need more than this.  I'm not sure I do.  The basic take-away from this unique parameterization was that the real annuity "wins" over nominal in utility-world and that high spend rates suck in this work and that middle of the road allocations to risk are better than extremes but are still dominated by spend choice. Not shown is that nominal trumps no lifetime income at all...keeping in mind we are still only talking about annuitizing maybe 16% of wealth and not deferring anything. These are perhaps bold and un-explainable assumptions, but we've been here before.  Whether I can make this into a coherent whole in the future is another matter altogether. TBD.  Like I said, a software shakeout, not education.

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* sorry for the shift from orange to orange/blue dashed.  I was not consistent when I was working on this and didn't want to go back.  I got the titles and labeling wrong about 15 times in this post and that was enough...

** just to be clear, the orange dashed line in figure 1 would be the result of going to figure 3, stopping at each spend rate along the spend rate axis for the upper surface, taking the max utility across all 11 different allocations to risk, and then rendering those max values as our dashed orange.