A fantasy of exogeneity

The Setup

Past this sentence there is no modeling of real financial phenomena. This is just playing around with an idea just to see what it looks like.  This is also the second whack at an idea about modeling "critical states" like forest fires, sand pile avalanches and earthquakes.  Here is the idea: most research papers I read perseverate on returns and return distributions.  The normal distribution is the flawed baseline but usually close enough. There are others. T-distributions have usable fat tails but need to be fit. Gaussian mixes (GM) are often very usable but also need to be fit. I like GM since there is a "high note" of a relatively regular, probabilistic, narrow variance return process and a "low note" of much lower and/or very wide variance returns. This is easy to model but conceiving of the low note as a stochastic process might be "fittable" in the end but also wrong. What if the world had darker forces -- sometimes related to returns -- that are not a regular random process and not always a function of returns.  What if the earthquakes that hit us financially come from things other than returns (or regular spending).  Here we can take a stab at some ideas for what I mean:

• The 87 crash was embedded in returns but the underlying source was not normal variance but a self-organized cascade related to everyone leaning into portfolio insurance and selling hard into a sell-off.  That is something embedded in the long term lookback variance, of course, but not something I would plan on every month or year. Also the annual return in 87 does not include the fear and loss of mind we saw in the fall.  
• Divorce. Trust me on this. This is basically a wealth tax or capital haircut. Sometimes it is foreseeable but usually not. The consequences for everyone are severe. 
• A new roof.  Speaking of which. I just put one on.  That bill would have paid for a year at an elite college.
• Long term care
• Comet strikes
• Capital haircuts like Cyprus
• Revolution and war
• Uninsured health expense
• A child or three moving back home
• Uninsured liability claims 
• Levered real estate bankruptcy cascades (this was Dirk Cotton's go-to example from a real person he worked with. Classic case of problems starting slowly and then fast to the point of end-game in moment.) 
• Some weird and unexpected phase-shift in spending

The Model

I'm sure there are others than what I bulleted but most of these have nothing really to do with return variance and can't be modeled with a T distribution or a Gaussian mix on returns. The way I did it in the past post "My baby steps into "critical states" in a decumulation model" was to build an earthquake-like model that throws an exogenous power law curve-ball at a plan. My first attempt was a custom model that created build-up stressors, threshold points, and uncertainty about scale and timing thereafter. That was close enough to an earthquake model that I then borrowed Gutenberg Richter (GR) for doing the same thing since it was easier. That is the model I'll use here. Recall the first sentence though. I am not really modeling a real world data set, I'm just goofing around. 

GR looks like this: 

Log10 N = a - bM

where M is the magnitude, N is the number of events greater than or = mag M,  a and b are constants that depend on the geo-area being modeled. For example b is framed like this: 

• 1.8-1.0 oceanic ridge   - big
• 1.0-0.7 interplate - med
• 0.7-0.4 intraplate - small

while "a" usually ranges from 2 to 8 in real data sets and depends on things I don't know. For this look-see I am using an alpha of 5 and a beta of .6. These, of course, are entirely arbitrary. Also for magnitude I am using a 9 point scale where the lowest loss (strike against net wealth in time t) is nothing and the highest is a 90% loss. Again this is arbitrary and models nothing real in any data I've seen. This is entirely fiction. I don't know if this is a legit way to do it but it's what I'm going with today. Here is the probability mass embedded in my code where we can see the power law look. Did I mention fake yet? Oh, and I did not model aftershocks. 

Figure 1. my fake financial earthquake model

Some other core assumptions I used for anything below include:

• 50k iterations
• return and standard dev = N[05, .12]
• spend is set to 0 or at least "baseline" recurring spending anyway
• horizon is 40 periods, long but not impossible, total scope of run was 100
• interval for Figure 1 was .01 from 0 to .99 in the program, 0->.9 x .1 in the figure
• initial wealth = 1

The net wealth model is a very simple MC sim or recursive net wealth chain where 

W(t) = [W(t-1)*r(t)]*(1-trouble) - s(t) + i(t) 

s and i are set to zero here and "trouble" is sampled from something like figure 1.  I hope I have all this right for my flight of fancy. 

The Output

1. Net wealth and geometric returns - no chaos.  On the left is the net wealth process without chaos over 100 periods. On the right is the chained, compound, simulated returns over 100 periods and xx,000 iterations

left

   

right

2. Net wealth and geometric returns - earthquake model. On the left is the net wealth process with chaos over 100 periods. On the right is the chained, compound, simulated returns over 100 periods and xx,000 iterations

left

right

3. Comparison of the density of net wealth at t=40 using R density and bw="SJ." X axis is net wealth. Black is the original. Blue includes the earthquake model. 

4. Comparison of compound annualized return at t = 40. compound r = (Wt/W0)^(1/t)-1. Black is the original. Blue includes the earthquake model. 

Discussion

I'm a little reluctant to do a classic discussion because of how fictional this is.  There are no real world conclusions I am comfortable in making given the fantasy.  However, if we are literally "through the looking glass" and accept that, then what can we say?

1. Compound interest is called the 5th force. And it probably is if it is unimpeded by reality. We see that in the no-chaos scenario

2. If reality does intrude - I mean, in this fake world anyway - then we maybe need to be a little more circumspect

3. The introduction of a "small quake" model seems to clip all of the run-away upside scenarios for wealth

4. The chaos in this post's model appears to destroy wealth over the 40 period horizon

5. All of this was kind of predictable. Take a normal model and then throw really bad stuff at it and things will get worse. Duh. Everything will shift left, which they do.

6. If we were to add regular consumption, it'd even be worse so let's not.

7. The underlying return process, the thing we see in 10 million finance papers, hums along pretty much like we'd expect as if there were no storm outside and plunging temperatures. The gravity of regular finance has not changed.

8. The real lesson is more likely than not as it always is: save more, spend less, retire later, buy or get income - pensions, annuities, SS, work. 

9. Pretty sure I have proven nothing other than that one can code a program that "shows some fake stuff." Otoh, I guess the message -- if we could infer anything from this exercise -- is maybe to be a little circumspect about overly tuned or fitted models and advice. My divorce, my kids, my life experience, 40 years of markets, and my household budget all tell me all of this.