Fuzzy # 3 - optimal spend rates with CRRA Utility > 1 as a cloud of solutions

 This is the third post on fuzzy clouds, the first two being:

• The Challenge of Retirement Finance in a Few Charts, and
• Another Fuzzy Cloud Chart Now with Consumption Utility

Here I am modifying the code a bit by

1. Adding CRRA utility for risk aversion coeff !- 1 where 1 is log utility, and 
2. rounding spend rates -- uniformly random -- to 2 digits from 1
3. Using 100k iterations in the sim vs 25k before (changes little if anything)

When I do this, I still get the look of optimality at some spend rate except now it is more of a cloud within a cloud [1].

Caveat 1 - This is entirely a fake artifact of simulation. One has to trust: a) that I designed it in some meaningful way and that b) I coded it right.

Caveat 2 - The whole point of simulation is to create some variation so some of the variation will come from that simulated wobble. How much? TBD. 

Caveat 3 - This is only run for one set of portfolio assumptions. Take that for what you will.

Caveat 4 - The risk aversion coefficient is necessarily subjective and sometimes controversial I hear. I have read thousands of papers and while I have heard of the RA coeff being tested empirically I have never seen that directly. I put RA in here because other dudes in Ret Fin do CRRA and I'd hate to be a party pooper. My own RA coeff appears to hover between 1.5 and 2 in my experience based on what I spend irl and past models where I have tested it...but ask me tomorrow after my portfolio halves.  ;-)

-----

Except for the changes noted above, we have the same simulation as before where we were calculating the utility of consumption over all 100  years weighted by a conditional survival probability and then summed over the 100 step iteration which is one of 100k in this sim. Each iteration has a different randomly diffuse spend between 1 and 12% here in .01 increments. Some other core assumption in abbrev format:

With the output like follows in Figure 1, where

- Left side is age 63

- Right side is age 75

- Top is risk aversion (RA) coeff = 1 (log U)

- Middle is RA = 1.5

- Bottom is RA = 2

- Red squares are my guess at an acceptable "zone" of spend rates given the fuzz

Figure 1.

Discussion

For any given set of parameters presented here, it looks clear to me that the code wants to find a solution or "best outcome." For example in the upper left panel there is at least a "look of optimality" around 5%.  But given how I coded this and displayed it every outcome between 4 and a little over 6% -- the red box -- looks reasonable to me unless we split hairs... and on what do we have to split hairs?  Obviously 2% and 10% don't evaluate well at that age and risk aversion. I might bias to the left side of the box out of natural conservatism but everything inside the box looks like a family of outcomes. There is zero math or science in that last sentence. 

Addendum

Ok, that's a lot of work. How would this stack up against a rule of thumb, my RH40 say, if we wanted to wing it. RH40 is:  Age / (40 - Age/3)) + n where n is an adjustment for risk. I use n = 1/2% but that is arbitrary. If I take the optimal number in the bottom title of each of the 6 charts -- along with the red box zone -- and then add some points at each age for the FH40 RoT, I'd get a chart like this in Figure 2.  

Figure 2. Optimal zones vs a RoT

Huh. Doesn't look too bad if one squints hard enough. At least they look like they are playing the same age-based game. Caveat 2 might be messing with me a little bit at age 63, idk. 

----------- NOTES --------------------------------

[1] All random spend rates when simulated have a range of utility scores all over the place (the grey dots and the first cloud) and the average of which for a given spend rate is the black dots (the inner or second cloud). I finally figured out the striations, btw: in the sim, we step through the years one year at a time over 100 years. The portfolio will last, say 21 or 22 years, and then drop consumption to a floor. The sum of either outcome has a big impact on the life utility value which is significant only in relative terms and in the small digits to the right of the decimal so the U scores will cluster to either side of that spend cliff toggle difference. Whatever. Probably doesn't matter.