Comparing my naive complexity model to earthquakes

In my last post (My baby steps into "critical states" in a decumulation model) I cooked up a simulation that would hit a retirement plan with some chaotic negative strikes -- like the ones we see in the physical world: earthquakes, forest fires, and sand pile avalanches (in terms of how often and how big).  It was a first pass effort so I was winging it for fun and not paying attention to anything real because there is no real underlying stressor-process in retirement that is coherent. That I know of. Yet.

Then, after the post in question, I started to wonder: "huh, I wonder if this machine I cooked up is even close to any real world complexity-dynamic...in at least the way it looks and in terms of prevalence?" In this case I also said "let's try earthquakes first."


Disclaimer: I realize my "magnitude idea" does not really map onto the 10x changes in destructiveness of the Richter scale. But work with me here for a minute. On the other hand I'd argue that a 50% loss of wealth is more than 5 x 10% and that a 90% hit is likely to be a fatal blow to a lifestyle. I guess I'd have to look at the utility of wealth, maybe with a CRRA model. Later...

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To do this post I borrowed the Gutenberg-Richter formula after a 7.3 minute self-education using a random PDF I found somewhere on the internet. Trust me yet? Heh.  The formula looks like this:

Log(10)[N] = a - bM

where N is the number of earthquake per year of magnitude
M is the earthquake magnitude
a is called "the" productivity (a constant, let's use 5 for fun)
b is called the b-value in the range of .8-1.1 (let's bracket .7 to 1.3 here) 
M is magnitude (see my disclaimer, eh?) [(.10-->.90)*10...so 1-->9 ]

If I plug all that in and calculate N (over M) and then N/N(tot) for M=1:9 and chart it against the N/N(tot) I had for my simulation in the last post, I get something that looks like this:

Figure 1. my sim v Earthquake model
upper grey is b=.7, lower is b=1.3

Figure 1 in Log Y scale

Huh.  I'm surprised it's even close. I mean I figured I'd wandered into an angry ocean from a wading pool, but this isn't too bad. It (my propensity sim, last post) looks like it exists within a reference range for real earthquakes. Cool.  I could try to interpret the divergences but really, I have no idea and it doesn't matter. If I were to do this sim again I might just use the GR formula to work backwards to get the fake retirement finance quakes I was looking for in the first place. But it looks like I am playing the same game either way (tho It looks like I am intra-plate; not sure how that translates). It also means I don't really have to go back and re-code the last post. I was worried that I had something that wasn't even close to real chaotic/complex processes. Now I can skip it. I feel more calibrated. Plus I'm getting lazy...