My portfolio longevity modeling artifacts vs economic utility theory on one set of parameters

See the correction of errors that I made in this post here.

I guess I have to confess to being a geek here.  In the last post (One more obscure artifact of Portfolio Longevity) I was looking at the second derivative of Portfolio Longevity (L) percent at L = infinity (ok 100 years) or an artifact of which I was labeling "Y." In that look-see, Y ramps up the fastest at ~3.4% spend (w) for the given, and entirely illustrative, portfolio assumptions of arithmetic input 4% return and a 10% standard deviation.

The basic math model for this was like the following where M is endowment, w is spending and V(tilde) is random, volatile returns:

Net wealth process...recursive

The universe of L and w for that parameterization and that model looked like this when run

Figure 1. Portfolio Longevity over infinite (100 years here) time for given parameters

The first and second derivatives of Y -- strictly modeling artifacts from an overly simplified model of L - looked like this. This, by the way, is the curve of the z axis of figure 1 at PL = 100 over all w in the interval:

Figure 2, First and second deriv of portfolio longevity at infinity

Figure 2 happened to give me an "area of interest" at around 3.4% for the particular parameters I used. In the last post I tried to interpret that point but I wasn't sure since I was pretty deep into abstract modeling over infinite time, which looks nothing like human lifetime and also looks nothing like economic consumption utility.  Recall that my interpretation of 3.4 in this narrow universe was this

But even if it is not coincidence, is this even interpretable? My guess is "no" in any practical sense though in the context of the model it might make sense. I'll have to ask professor M some day. In this model with the r = ~N[.04,.10], a 3.4% spend rate is about where the portfolios along w(i=2:12) are tipping over to infinity at the highest rate. The closest I can come to meaning here is that below 3.4 consumption portfolios are probably robust (but self-denying in lifestyle?) and above 3.4 they are increasingly less so.

Ok, so far so good. But then I got to thinking (and this is where my type of thinking can be somewhere between distracting and relationship-ending): "what if I ran the same parameters for a portfolio and spending in an economic consumption utility context? Would that give any context to the 3.4?"  hmmm. Gotta go...

So, I did it. I just happened to have an existing model for consumption utility over a random lifetime. I call it my "wealth depletion time" model. The basic schematic looked like this and is described in more detail here: 

Figure 3. Consumption Utility Model over random lifetime

And into this model I plugged the same assumptions as my last post, plus a few others[1] with a critical note that there is no social security or other annuity or pension income, which is really important to consider. The output of this model is the "expected discounted utility of lifetime consumption" or EDULC. When I run it for constant spends between 3 and 4% this is what I get for EDULC for the parameters that match as closely as I can the last post I did:

Figure 4. Expected Discounted Utility of Lifetime Consumption 

Huh? whattya know? There is a flat-ish lifetime utility area below 3.4 and a strongly declining utility of lifetime consumption above 3.4.  Weird.  Part of me wants to say that this proves my "artifact." The other, much bigger part of me, admits I have no idea and that this might all be different for different parameters.  Who knows?  My take away though is that there is, again, no such thing as a 4% rule, but there is more than likely a "tipping point" in whatever (portfolio longevity at infinity, economic consumption utility, other...) somewhere in the mid 3s to 4 and above which you are playing an increasingly difficult and dangerous game (changes for older ages, btw). If I had an advisor when I was 50 who said "go ahead and spend 7%" I would immediately offer him or her to play a game of Russian Roulette with as many chambers under 30 or 40 that they would like.

  

Other random thoughts
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I know I hinted at this above but let me emphasize that the presence, in a lifetime consumption utility model, of lifetime income (social security, annuities, pensions, side gigs, etc) is a huge factor.  Having income or buying income is going to have a dramatic impact in lifetime consumption utility and how one depletes wealth over a life-cycle.  This is probably a much bigger deal than returns and vol. None of this is modeled here.

I have not totally shaken out my WDT model. It's been a long while since I coded it and, as usual, I can't remember exactly how I structured it.  There are some funky assumptions and interactions that I need to figure out. So...take all of this with a grain of salt.  Inflation in particular needs some review in what I did.  It would change the inflection point but not the basic idea.  TBD.  Terribly embarrassing if I am way off but whatever...  (just confirmed that spending is strictly rendered in real terms...)

Notes
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[1]

- age = 60 and we are using SOA IAM mortality data to construct random lifetime

- inflation of .03 if it is needed

- endowment of 1M

- spend is varied between 3 and 4%

- sim is run 100k times

- risk aversion is a low value of gamma =1 

- Portf r and sd are .04 and .1 to sync to my other illustration. No fat tails but I could

- there is an income floor of 1000 to reflect family, social, and govt resources at WDT

- SS income is zero here, no annuites or pensions

- subjective discount is .005