Adding incremental uncertainty to a consumption utility model

I once joked on Twitter that extreme "risk aversion" (in terms of the coefficient and the convex CRRA model iteself) was less of an "economic" topic and more one of psychotherapy. Heh. I got some pushback on that from a retirement quant but I think in the far-extreme it makes sense.  If one were to be so risk averse in the non-financial sense that one couldn't leave the house, that is not the domain of models and math but of getting mental health help to reduce the aversion.  

Another way to look at it is that the risk-averse almost don't trust anything for a variety of reasons. I have no idea what the economists say since am an autodidact amateur. The advisors to the risk averse and the black box models they lay on the table, plus the intuitive uncertainty humans reasonably have about the future, makes risk aversion of some kind make sense to me. I mean if one had a perfect predictive model it'd be deterministic and we could make rational choices. At that point, residual risk aversion really would be pathological. I think. Maybe.  So it might be worth exploring some phenomena that represent some of the uncertainty inherent in what others call might risk aversion...though I accept the idea that "all else equal" some are more fearful than others given the same information. But there we are back to the counseling idea I think.  

In this post I don't want to be too rigorous. No tables or graphs; certainly no R3 surfaces (I promised David I'd pull back on that). BUT, before and during happy-hour today, another covid afternoon of 2020, I had a question that was a blogging itch to scratch.  Went kinda like this.

If I ran my lifetime consumption utility simulator, with assumptions set to my own self, using basic stuff, then: a) what is the baseline spend rate and allocation from that imperfect piece of software (no triangulating anything else here), b) what happens, incrementally, to the spend rate if I nudge inflation only from deterministic to stochastic (using US history) with an AR(1) autoregressive feature laid on top, and c) what happens again if I add an odd pulse of chaos theory to "b?"

That last (c) is a hit to net wealth so it can be viewed as maybe adding a fat tail (that we don't understand) to returns or maybe adding another level of uncertainty to the spend rate or maybe it's adding a chance for a post-antifa reparations tax. No idea. Whatever it is, what I am trying to do is add some layers of atypical "uncertainty" to a model that are not often seen in the first place (except maybe MaxFi?) and then adding some more uncertainty on top of that. What happens? I can guess: spending has to come in (gasp) a bit in the near term ... but short of the alchemy of turning lead into gold, what else is there? I am not pinning my hopes on portfolio engineering...yet. Going back to work is the closest to alchemy I can conjure in this world. Maybe the risk pools for annuities is another but that's another post. 

Scenarios

Using the assumptions I came up with[1] I ran these scenarios: 

1) a baseline which attempts to select an optimal spend and allocation from a set of many iterations and scenarios run and which uses the "expected discounted utility of lifetime consumption" as a max critierion

2) Use #1 + add some stochastic, autoregressive inflation. The basic idea is to change inflation from static to being a random draw from a table of historical data but also to ensure that it mean reverts a bit if it trends too far. I used a coefficient of reversion from some econ luminary I can't remember so I'm comfortable that I am not a total moron. It's just a way of getting off the no-randomness train which seems reasonable.

3) Use #2 + add an unlikely chaotic hit to net wealth at some time t from some shock. See note [2] for where and how I've done this before. The chaos magnitude is intended to replicate something the likelihoods that might be seen in phenomena like earthquakes, forest fires, and sand-pile avalanches. These are power-law magnitudes: usually small but sometimes - rarely - big, big enough to hurt. Whether I modeled it right I have no idea. Doesn't matter. It's existence in the model is enough for now and the model I validated in the past against earthquakes. Or was it fires? Idk.  

Output by Scenario

Scenario 1. spend rate = .040, allocation to risk = .40 [3]

Scenario 2. spend rate = .035, allocation to risk = .60

Scenario 3. spend rate = .029, allocation to risk = .60

Discussion

I'm not really in the mood to discuss since this sorta stands on its own and I weary of charts. But here is a wee bit of commentary. For most of the last decade, at ages well under the one assumed in this run, I tried to keep my spend rate around or under, if possible, 3%. BUT, I was explicitly -- and sometimes very unprofessionally -- shamed for that choice by: advisors, ex-girlfriends, and even my ex-wife in a moment of unnecessary post-divorce pique. For 7 years now I have been trying to understand the reasonableness of my conservatism which I knew intuitively to be right but could not explain.  This post is pretty close to what I saw in my head in my past without really knowing it at the time. This post also shows why I'd rather spend less time ramping up "coefficients of risk aversion" in econ models that I don't understand and spend more time trying to find some additional intuition on why I should or shouldn't do something. I mean, short of being god or having the future revealed to me, I will not ever have enough intuition to really bridge the gap of uncertainty. But...It's a worthy effort and it makes more sense than nudging my coefficient from an arbitrary 2 to an arbitrary 3. Quick, tell me, what does a 3 mean. See?

Also, as an aside, I hadn't really analyzed my own data in a serious way yet like this (omg, really? you've been a retirement blogger for years). Yes, but I tend to focus on other important things like an actuarial balance sheet (think Ken Steiner), income statement, and a spending control chart. This post here, though, I guess, goes a long way towards validating the statistical control boundaries of my spend control chart...so that's useful. But kind of a yawn though since nothing here will cause me to change my current control boundaries. It's just confirmation.  Next year is another matter, though. TBD 

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[1] basic assumptions 

• Inflation is either deterministic at 1.03 or stochastic using a table of US history to bootstrap with an AR(1) feature. with a coefficient of .64
• RV<-c(.0350, .0415, .0480, .0545, .0610, .0675, .0740, .0799, .0870, .0935, .10)  
• SDV<-c(0.04, 0.0386, 0.0457, 0.0583, 0.0736, 0.0902, 0.1076, 0.1254, 0.1434, 0.1616, 0.18)
• Our current low-return expectation environment helps none of this...
• Risk is defined in the context of a 2 asset portfolio so we go with that...
• age 62
• Starting W (redacted)
• spend rates (independent variable); Note that in-model spend is constant which is a fairly flawed assumption. Or, at least, it's constant until it drops to some either moderate or horrible floor when wealth depletes. 
• Allocation (independent variable)
• iterations 10000 worlds x n spend rates x 11 allocations x T life-years
• coefficient of risk aversion = 2 CRRA relative risk utility
• subjective discount = .005
• 20k of SS at 70, No other pension or annuity
• Normally distributed returns
• No return regime overrides
• Chaos instability like this post (figure 2) or this maybe
• Might be some others
• Model for evaluation is here

[2] The chance of a chaos shock in any one life is low. The chance of seeing it over 10000 iterations x a factor = f(spend rates tested, allocations tested, scenarios tested) is high but this approach still mutes the implications for a particular life when chaos hits which, no matter what this post looks like, could be catastrophic. Such is modeling. 

[3] Hey! 4% just like Bengen. Don't be fooled though. These are different methods and different worlds. This was coincidental.