Playing with Gaussian Mixes and "jumps" again

There is no real hard science or rigor past this sentence so begone if you need that sort of thing.  

The reason to fling blog crud today was that I was coerced into reading a paper[1] on jump-diffusion processes by the inimitable David Cantor. I know nothing, really, of those processes but I came to the conclusion that my amateur attempts at doing a Gaussian mix in the past was kinda close. I mean, my vol was not really stochastic but both the return and vol "jump" within a random process and so we can, over time, with either jump modeling or mixes - close enough, right? - more or less mimic the fat tails of real world distributions, which in modeling-for-retirement terms is desirable. I think. 

The underlying issue is that returns in real life are not normally distributed depending on the interval. That doesn't matter much if you are an amateur or don't care. It also means nothing if you lean into deterministic models which can be very useful. It also means little if your discrete time is in years rather than days or months -- the issue of fat tails starts to go away a bit with longer intervals. I was once told this was a central limit theorem thing. Fine. Also, I was told that this process of fat tails fading as one extends the horizon is called "aggregative Gaussianity." Erp; that's a heck of a phrase.... Also that phrase in its essence depends on horizon, so the term structure of one's analysis probably matters. I usually work in years so it matters less. TBD. If I happened to manage a 5B pension fund, though, with monthly or daily payouts, my interest in fancy models might be a little higher. No way to know. That is not my world.

A quick side note. Here is my opinion. I've read a ____ ton of papers over the years. While I admire the impulse to complexity and modeling stuff towards more and more real world processes, I am not sure where all this goes. If one is attempting mightily to model how the past works I have no doubt that the past will successfully be modeled. If you weren't able to do that you might be fired from your advisory job or not get tenure at university. So I get the impulse, it's just not mine.  On the other hand, the future is wide open so modeling the past will nearly always miss in the future. Think 1987 or LTCM or Covid or my divorce. Or death. If I based my predictive model on the last 62 years, then technically speaking I might model immortality. Heh; I know better.  So, some guy modeling autoregressive this or jump that is cool and super smart. Me? I might actually respect a modeler that throws in really weird unnatural things to widen the universe of possibilities. Idk, maybe an alien invasion or something. Maybe an asteroid strike. Or at least spending shocks like wealth taxes or divorce or war. Widen the future, rather than fit the curve I say...

That being said, I started to play with Gaussian mixes again today just for fun and because they are a very very simple version of a jump process though the volatility is not strictly stochastic though I guess it is in a binomial way as a product of the jump to some other state. TBD, I am not an actuarial hero. In this case I wanted to figure out the following:

- If I had some annual SnP return data (Stern/Damodaran)

- If I had a fabricated normal distribution based on the annual S&P return data 

- If I had a fake fat tailed distribution that very very vaguely matched the first two moments of the SnP

- If I feared a 50% drop in the index in any given year

- If I measured things only annually 

then

- how likely is a 50% drop in a normal model

- how likely is a 50% drop in a very very fake-ly and not well fit sorta-matched fat-tailed model

I realize others and statisticians pour over stuff like this all the time. I generally will see papers and articles that talk about how such an such event is an X standard deviation event so hyper unlikely in a normal distribution. And then they go on discuss fancy fat tailed distribution this or that and say it is not all that unlikely. Taleb is famous for this. Me? I don't have the skills so here I'll just play a bit with what I have and then you can go read the other papers.  

If I have not sandbagged enough I'll lay it on even thicker...  In constructing the mix below, note that I failed at constructing a proper mix in terms of the output of the Expectations Maximization routines I typically try in R. Not sure why it didn't work. I think because maybe working at an annual interval the differences are too small. idk.  That means I did my amateur thing here -- I just made stuff up.  I fit a mixed distribution based on the first two moments - sorta - and also by eye and instinct. Science, right? Yeah, whatever. I'll roll with it. Frankly I would love to figure it out but at 62 I'm not going back to school. I'd take instruction though, but that has been rare in this 8 year blog process. Instruct away if you have the goods. I'm game. 

So here is the setup

1. SnP data 1928-2018 from Damodaran/Stern, annual 

2. A normal distribution based on the first 2 moments of #1 - N[.11356, .1958]

3. A really really fake "mix" based on 75% N[.18,.17] and 25% N[-.08, .60] ... note[2]. This fit only by eye and wasn't really super close in the first two moments. It'll make my point though. At least there are two processes, one with a good high return and some random vol number and then a second wildly dispersed "negative" process. Whether this is a good dynamic is unknown-to-unlikely. Works for now.   

The question, if I have not undermined myself yet, is the following

A. What is the probability of a 50% decline in #2 and #3 (integrating the pdf (pmf) for each to -.50)?

B. Is this meaningful or useful?

The Output

Here are the distributions from the setup

• Blue is the normal distribution ie #2
• Red is the SnP stern data, so #1
• Purple is the mix so #3

• blue and red mean 0.1132927
• blue and red sd      0.1959537
• purple mean           0.1153863
• purple sd                0.1879278

So not a perfect match but close-ish... I'll work on precision later...

If we believe in this world, and since we are mid-post I guess we must, then the question, if we take the programmatic distributions for #2 and #3, is what is the likelihood of a -50% in #2 and then #3? 

- for #2 the likelihood of -.50 is .0009

- for #3 the likelihood of -.50 is .01098

Is this meaningful or useful?

This would no doubt be a bigger difference in a monthly series but here it is at least a ~12x greater chance in #3 vs #2. Real world enough for now or, per my point above about modeling and realism, at least the mix model allows me to open up the gates of hell on returns when coding. And this is before we even get to spending, btw. If I have not been clear, note that I have not attempted to do a crisp statistical analysis of the difference between normal and historical market returns. My only goal is to build tools, whether fat tails or chaotic processes, that can give me some humility in coding forward expectations for returns and/or spending.  This is easily achieved I think without hyper sophistication. 

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[1] Do Jumps Matter in the Long Term? A Tale of Two Horizons, Jean-Francois Begin and Mathieu Boudreault 2021, North American Actuarial Journal, 0(0), 1–20, 2021 # 2021 Society of Actuaries ISSN: 1092-0277 print / 2325-0453 online

[2] please remember my sandbagging. This is entirely fabricated just for the fun of the post and it sorta "fit" only by way of two moments and my eye. Go back and read the first sentence of the post...