What happens to consumption utility if I make spend a % of portfolio [amended]

I usually trade in "constant spend" because it is easy to work with. A % of portfolio isn't all that much harder and lasts forever, right? Sorta. The trade off is consumption volatility and the possibility of really low consumption in absolute terms late in the game. In this post I am simulating (50k times) randomized spend rates against a .04/.12 portfolio over 100 years. Lifetime is a survival probability laid over the 100 years, an interval which is arbitrary and used here as a proxy for forever. The utility score is sum of "consumption utility results over all 100 years weighted by the life survival probability in each year." The lifetime parameters are somewhere in a generic middle between average health and annuitant. Risk aversion coefficient is arbitrarily set to 3 in Figure 1.  

[note: fixed an error post-publ]

Figure 1 is the run with grey being all EDULC outcomes and black is the mean for a particular spend rate with the variance of that mean from the granularity of the spend randomization. I think. I think because tho I am pretty sure, my proclivity to carefully test my software goes way down with age. And as Al C reminds me: sometimes with age comes wisdom but sometimes age comes alone. Might be some lonely errors in there as I keep adding stuff to the code.  

Looks like there's an EDULC top somewhere around a 6-7% variable spend for these parameters. Only the top 95% of the vertical range is displayed.  Low spends suck because we could spend more (I quibble in my private life with the philosophy on that whether it is related to bequest or related to maintaining a simple life). High spend will hurt in the variability and the way it can crush a portfolio to leave a very low lifestyle at some point. No other comments since it is Friday and time to be done...

Figure 1. EDULC for parameters

Addendum 4/24/22

I had an error (at least I predicted the possibility, heh) in the code. Basically it was that I have a utility floor that represents the idea that no-one spends zero each year even if broke. Things like SS or family or institutions of govt or association might step in etc. For risk aversion > 1, or "3" in this case, I mis-set the floor too high. In effect it was like having a pension or annuity or other life income that supports higher spending like we'd see in the LCM of LaChance or Yaari. When I correct this Figure 1 looks more like this though the possibility of error still, of course, exists.  Frankly both make sense when viewed through an LCM lens though the amended Figure 1 is what I was expecting which is why I kept scrutinizing the code. 

Figure 1 (amended)

Or this when the expectation operator works on a less granular spend rate of 1 significant digit. The optimal point is closer to 6 now rather than 7. I still wouldn't want to peg my own spending to an exact percent of my assets but that's another post. 

 

 

I did a more formal treatment here, btw:

Spending, Rules and Habit