Being In The "Zone"

"The optimal strategy might be executing a suboptimal plan at a fast pace. Strategy evolves as lessons are learned—and the person who moves faster, learns faster. Learning is a marathon and perfection is a weighted vest. - James Clear

“It is better to be roughly right than precisely wrong.“ — Carveth Read*

10 years ago I believed, more than I do now, in the grace of specific numbers and precision. Today, not so much. That's because even if I were to have a perfect, optimal retirement model, and if I had successfully tuned it to the infinity of possibilities of whatever reality we know, as of yesterday let's say, then: 1) you and I would still have results different enough today that it would be hard to explain, and 2) for both of us, the output today could be entirely stale as early as tomorrow morning. Agony, right? I used to think so. Instead, I have been thinking how it matters only generally what we spend and how we invest but not necessarily specifically or precisely. Getting into a "close enough zone" and being willing to adapt are stronger and less burdensome concepts than getting it exactly right. At least I am telling myself that so I don't pull out what's left of my hair.

The Model

The model for this exercise has been described before here but a basic schematic should illustrate the method well enough: 

Figure 1. Schematic of an EDULC simulation model

The Setup 

In this post, which was mostly unnecessary but I had some extra covid-time on my hands, I wanted to see if I could illustrate what I mean by "a zone."  To do that I trotted out my consumption utility simulator, plugged in some basic assumptions about investment parameters and risk aversion and then ran it through some paces.  The paces looked roughly like this:

For some initial wealth = W in [500k, 1M 2M 4M]
    For each age 60 -> 100 in increments of 10y (keeps it manageable)
        For each spend rate x->y in .5% increments (say .03 -> .10)
            For each allocation 0->100% in 11 steps in a synthetic portfolio:
                -simulate consumption w utility simulator given the parameters
                -if wealth depletes, snap spending to available income
                -calc the sum = EDULC* as if starting** at that age with W
                -store the matrix of utiles (by spend vs alloc) for age & W
Then...do some blog stuff with the matricies

* Expected discounted utility of lifetime consumption
** e.g., start at each age increment (70 or 80 or 90) with, say, 1M not with what would remain at 70 or 80 or 90 in a standard simulation if starting at 60...but run each age-W pair to infinity (or 120). 

The parameters were set like this

Rp = .035-->.10 N(r,s)
SDp = .04-->.18
Age: as above
Wealth: as above
Consumption: as above
Iterations per matrix cell: 10k
Coefficient of risk aversion: 2
SS: 15k at 70
Life: SOA IAM table set conditionally
Chaotic nudge: on
Recall that this is a model not real life. It reflects the software and sw design only.

Preliminary Output

To give a flavor of the output, here is 1 table of EDULC utiles for age 70 and W = 500k. The shaded zones are: optimal cell (green), within 10% (orangeish), and within 20% (blue). 

There were 20 of these tables in all, one for each age and wealth level.  With 20 tables and 10k iterations per cell and lets say 50 years per iteration that means I had about just under 2 billion sim-years invested in this project. Actually it was probably more like 3B with my errors and re-runs. 

Table 1. EDULC for 500k initial wealth at age 70

In a surface chart Table 1 would look like this where the red zone is analogous to the shaded areas above.

Figure 2. surface of EDULC for variation in spend and allocation

An Interim Conclusion

This is the place where I make my first point on "zone."  While the scale of the z axis in Figure 2 will determine how flat it does or doesn't look to the eye, it's hard to debate that U'' is pretty lean near the top: it is "flat-ish." To me flat-ish means that you have a lot of wiggle room around the optimal point to make choices if you can't get it quite right on the first try. The 10 and 20% thresholds I used in Table 1 are entirely arbitrary but are also my attempt at a policy of: "within 10-20% of lifetime consumption utility" is "close enough"...as long as one keeps an eye on things. 

The Meat of the Output - Spend Rates

With my two billion sim-years what I really wanted to see was the shape of the "spend zone" when charted out by age and for the different wealth levels. In this case I skipped the 2M wealth level just to make it display a little cleaner. Taking -- for each age and wealth level - the max and min spend rates (ignoring allocation for now) within the yellow and blue zones in table 1 gave me this chart: 

Figure 3. "Spend zones" by age and wealth level

Recall that this is not done cohort style. The sim is run at each age with the wealth level indicated. The ROT is a rule of thumb I use for spending that I have described here before as RH40. It is age-based but very conservative. 

Solid line - the optimal solution from the sims at each age increment

Dashed - the 10% off-optimal zone

Dotted - the 20% off-optimal zone

Figure 3 was more or less like looking at the red portion of figure 2 but only along the spend dimension for all the other ages and wealth combos. We'll peek at the asset allocation dimension later.  If we instead only focused on the optimal output for each age and wealth level and the given parameterization we could render it as a surface, thusly:  

Figure 4. Optimal spend rate surface

The Meat of the Output - Asset Allocation

I wasn't sure how to present this. So here I'll just chart the range of allocations that generate Life-Consumption-Utiles within 10-20% of the optimal outcome. I'll do it only for .5 and 4M in W to keep the number of charts down. Recall that I am looking at the highest and lowest yellow and blue shaded allocations in Table 1 (and its relatives) independent of spend rates. 

Figure 5. Optimal Asset Allocation +/- 20% by age for .5M in W

Figure 6. Optimal Asset Allocation +/- 20% by age for 4M in W

As in Figure 4, if in Figure 7 we stick to the optimal outcome for allocation for age and wealth, and not the "zone," we can render it as a surface. Like this:  

Figure 7. optimal allocation choice for this model

Whether this particular output conflicts with other methods for optimizing asset allocation -- say a backward induction and stochastic dynamic programming approach to optimality -- is a concern for another day. Here we are just taking what the model gives. 

Some Observations

I was going to title this section "conclusions" but I have not really concluded anything.  Here, however, is what I observe so far:

• If one is reasonably indifferent to precisely how optimal one want's to be -- something that is elusive over time anyway -- then it looks like there is a lot of room for error. It is maybe a little roomier when it comes to asset allocation, a point I've tried to make before, but also for spending too as we saw in Figure 3. 
  
  
• Mostly in the charts what stands out is not "what to do" but rather "what not to do:" super high or low spend rates and an over-allocation to bonds. Plus high wealth looks like it has more latitude in its ability to prosper with lower risk. Sometimes for low wealth the only way out is the lottery ticket effect of higher risk. This was a point Gordon Irlam made in Portfolio Size Matters. 
  
  
• It looks counter-intuitive but higher wealth tolerates only lower spend rates and vice versa. This is maybe a modeling artifact that may or may not present in real life.  On the other hand, my guess is that what is going on is this: while higher wealth can handle higher spending (obviously) the fall in the convex utility function when wealth fails and spending is forced to snap to available income is from "pretty high" to "pretty low" when spending is high and the penalty in utiles is much harsher than it would be if pre-depletion consumption had been closer to the floor. The model rewards an approach that presents fewer of those death spirals hence the lower spend and longer portfolio lives that go with it. 

* quotes courtesy of David Cantor.