I'm dumber than I look

In My portfolio longevity modeling artifacts vs economic utility theory on one set of parameters I tried to cheat a little and moved too fast. So fast that I made two errors: a sin of commission and one of omission.  I won't revise the last post, I'll just keep rolling here. But I don't think I'll tweet this one. It's hard enough to admit to error, harder still to do it in public.

1. The Sin of Commission

I forgot that I had written my econ-utility sim in nominal terms.  Oops.  All I had to do is look at the code. That's pretty dumb. I am embarrassed. I also conveniently suppressed the idea that the output is sensitive to risk aversion, which is one of the bigger points of that simulator.  Also it was an apples to oranges comparison between PL and WDT utility.  The portfolio longevity model was purposefully run to a proxy for infinity while the wealth depletion time (WDT) consumption utility sim is weighted by conditional survival probabilities using individual annuitant mortality...so scaled to human life.

That means my "aha" about the expected discounted utility of lifetime consumption (EDULC) conveniently matching up with the max rate at which portfolios, unconstrained by human life, tip over to infinity, was more of an "uh-oh," a fake result intemperately presented.  The real version, if we say that a arithmetic nominal return of 7% is more or less equivalent to a 4% real (is it?) then the actual results of the WDT sim would look like this for coefficients of risk aversion of 1, 1.5 and 2, where a lot of papers I've read tend to point to "3" as something that might be observed in the field. The line at 3.4 spend was the "tipping point" in the last post for reference:

So, is "3.4," for the parameterization I was using, meaningful in a utility context? No. Maybe. Probably not. Nothing to see here....move along. <pan to men putting up the yellow crime scene tape, news at 11...>

2. The Sin of Omission. 

Given that the portfolio longevity model at infinity that I used in the past post is just plain math and sim, I forgot entirely to talk about geometric returns over multi-period time. For my parameter set, 4% real return and 10% sd, it now makes a little bit more sense.  If, over an infinite number of periods, a geometric realized return g approaches a limit that can (not exactly correctly) be estimated from the arithmetic input a, given volatility V=sd^2, by 

g = a - V/2

then .04/.10 would approach ~3.5% expected realized return at infinity.  It makes intuitive sense now, to me anyway, that a consumption portfolio would probably be at its peak rate of fecundity (over an infinite interval) when consumption is just at or below the long term expected realized return. Zero consumption would, of course, be of no value to the living.  A high spend rate, on the other hand, one that is well above what the portfolio produces in real expected terms, just whittles away the nest egg over infinite time towards zero at some point, probably sooner rather than later. In this type of context, a max rate of portfolio longevity "leap to infinity" at around 3.4 spend now makes a lot of sense.  The formal finance investigation is unexplored and is probably already out in the world somewhere.