Obscure Artifacts

In my last two posts on portfolio longevity (see content and assumptions here) I ran a simulation that randomized both returns and withdrawal rates (uniform) to create a 3D picture of portfolio longevity with respect to withdrawal rates.  I was interested in one of the shapes in that chart. In particular I was interested in the nonlinear change in the percent of portfolios that lasted to infinity in that simulation (right side of figure 1).  The number of portfolios that do that start slow and then ramp up.  It seemed to confirm, in a way, in my own head anyway, my contention that 4% (given those assumptions only) looks like a "tipping point," rather than a "rule."

Here is another way to look at it. First here is the original chart with the region of interest pointed out by the red arrows...

Figure 1. 3D PL vs W

Now let's focus on this red curve exclusively. Let's also shift here from raw frequencies to the percent, for each withdrawal rate in .10 chunks, of portfolios that survive to infinity (or 100 years in this case) which, for convenience, we'll call Y though it happens to be the z axis in figure 1 if it were to be re-conceived as a probability mass along each w.  Then, in addition, let's now also look at the first derivative of Y with respect to w and see for fun where the rate is highest. Humor me by imagining for a moment that the curve is actually smoothly continuous.

The wall of probability implied by what happens at PL = 100 in figure 1 looks like this below in figure 2:

Figure 2. Percent of portfolios in sim that go infinite at diff withdrawals

Now, I'm guessing that this has more to do with the real growth rate being around the withdrawal rate (I'd have to check other parameterizations) but I thought that it was interesting that the rate at which these things go infinite reaches a peak or inflection point when changing from 3% to 5% or around 4.  This is another example of why in my 5-process posts I said I thought second derivatives were as insightful as anything else in retirement finance.  Note also that the inflection point is roughly where the median number of portfolios go into orbit.  hmmm.  Is this useful info? No idea. Just fun to see.


random other thoughts
----------------------------------------
- I've never really considered 4% "safe" for anyone younger than 60. Even here in this post, where age is not part of the game, somewhere around 4%, half of this parametization has failed before infinity and half not. That begs the question on horizon which is typically discussed as "30 years" or "random human life." I recall a paper I read that showed that there is a direct link between consumption rates and mortality or hazard rates which dominate the spending choice. Probably true. I'll try to find the link.

- You can see in figure 1 that an opinion about whether a portfolio will last is, in fact, sensitive to choice about horizon. You can also see why spend rates can go up with age as the horizon comes in and why for early retirees, discretion might be the order of the day.

- The 4% rule, a lucky outcome from what might be called a lucky back-test looks here like one of an infinite array of possible rules.  In trading terms one might say the 4% rule was curve-fit to the data (and a fixed lifetime) and might suffer under an out of sample test (and random lifetime).

- My portfolio choice of .04/.10 was illustrative only and maybe arbitrary but, as in a prev post, it's not too far off from a blended portfolio after taxes and fees depending on what era the data model is based.