There is a book by Benoit B. Mandelbrot called the Misbehavior of Markets describing his discovery and application of fractal geometry to markets.

I’m not going to attempt in a blog post to describe fractal geometry but I will try to describe it’s relevance when it comes to how to think about markets.

Here is how we are trained to project retirement.

We have a 1M nest egg, we suck 40K/yr and the leverage on the nest egg provides the income. This is a simple linear equation and is represented by the black line in the above graph. The linear equation is Y=MX+B where Y = 40K, M= 4% and X= 1M. In this equilibrium you can suck 40K forever.

If you try to suck 60K

The line goes from linear to curve linear aka exponential and you wind up 2M in debt.

Here we increase leverage to 6%

and voila’ we are back to linear but we are now 50% more risky (4% went to 6%). This is the real dilemma of retirement. It’s the relationship between Y and X aka M that determines success.

Here the leverage is constant but the duration is reduced to 29 years. We see it’s the duration that somehow determines success as well. In addition we saw the increase in budget from 40k to 60 has a dramatic effect. So let’s rewrite the equation to Y = M(a)X + B where M is no longer a constant but a variable, a variable of budget, duration, and % change in leverage. Each of these “changes” can be described in terms of rates. 40K to 60K is a 50% rate of change. 4% to 6% is a 50% rate of change. 50 years to 29 years is a 58% rate of change, Each of the parts of (a) have subsidiary rates of change that effects the main variable. Rate of change math is called differential equations and multi-factorial rate of change math is called partial differential equations. The mapping of partial differential equations results in differential surfaces which is a geometric representation of the math.

If you own passive low cost index funds you necessarily are limited and entirely dependent on the economy by definition since index funds by definition return at best market return. If duration goes long and leverage goes long those multiply and cause the expectation to go from linear to curve linear. So it’s the non linear that interferes with the nice clean projection. Mandelbrot enters the picture

This is a picture of smoke (hot gas) above a candle. Notice how for a while the smoke is so well behaved it proceeds up in an entirely Gaussian fashion. If you look REALLY closely you can almost see the Gaussian distribution in the stream of the candle. It’s darker in the center and brighter towards the side. The geometry is entirely predictable. The flow is laminar which means the flow is defined by a Gaussian distribution of an infinite number of parallel plates next to and sliding over each other. The thing that keeps the smoke behaving Gaussian is something called the viscous force. A force which is being dissipated the farther up the laminar column you go. Note just before the chaos you see the smoke start to get wavy, and then you see a little eye form in the center of the smoke this is called cavitation and is the onset of the final extinction of the viscous force that has kept things nice and orderly. You can also see a change in the Gaussian distribution. The color changes and becomes more uniformly dark and the bright edges get very thin. This is where black swans turn while. This is where the tails of the Gaussian distribution get fat and the unexpected begins to happen. The crossing from order to disorder is something called the Reynold’s boundary defined by the Reynold’s number. Beyond the Reynolds boundary is chaos. The viscous force is gone. The smoke above the Reynolds boundary is also predictable using Fractal geometry and so by applying the proper math chaos becomes predictable. The geometry of the chaos is based on a power function (an exponential aka something like X^2 or X^3) as opposed to the nice predictable Gaussian function. What it predicts may however not be desirable, if your life is designed around a Gaussian reality. What is happening along the entire course of the smoke is the smoke is loosing energy, first it’s loosing energy in a way controlled by the viscous force (the Gaussian way) and later by the wild and expansive eddy’s of rotation added to forward motion (the power function way). Note also how the cavitation taken in isolation looks like a hurricane, rotation with a well formed eye. Gives a little perspective on what you are looking at.

This is how markets behave. They behave in an orderly fashion right up till the forces enforcing order dissipate and chaos ensues. Once in chaos it won’t stop until the energy is dissipated (reversion to the mean). The more leverage you have the greater the chaos released when chaos ensues, and that’s the retirement dilemma. How to limit the damage of chaos when chaos finally ensues. You can have a shorter duration (retire on time instead of early, notice the impact of the 29y vs 50 yr retirement. You can have a lower leverage but remember leverage includes things like inflation. You can have a smaller WR. You can have a larger nest egg, You can have diversity across non correlated assets. You can reduce risk dynamically when it hits the fan. You can have a fuse portfolio such that you have something to sell high when your other assets are low. You can own things that tend to grow in the face of a crash like gold and you can allocate that dynamically as the risk of a market crash increases thereby getting off the track before the train hits. If you know how you can go short and become a billionaire instead of a mere mortal, but going short is exercising a power function against a power function so it’s like trying to snuff a nuclear bomb with a nuclear bomb. If you know how to do it you win. If you don’t you are assured to loose.

The main thing is to understand the dynamic nonlinear second order nature of risk. If you look at the graphs above only one line fits a linear reality and that one line has an associated risk. ALL THE OTHER LINES ARE NON LINEAR AND GENERATE AN ERROR COMPARED TO THE LINEAR ASSUMPTION, and errors cost you money.

Pretty often the train blows its whistle before it runs you over and the whistle is reason to get off the tracks. Sometimes whistles are false sometimes true. Buy and hold investing simply and arrogantly throws away the information whistles provide. One thing to understand is once the smoke leaves the candle you can’t put it back in the candle. You have until the smoke leaves to place your bets. Rest assured if you live by the bogglehead mantra you ARE placing a bet.

I may be one of the few, but I do enjoy your mathematical explanations. The problem for most people is that it’s difficult to visualize beyond zero and first order systems. Straight lines are good but curves interacting with curves cause a lot of problems. Unfortunately with multiple variables you’re forced to go beyond algebra and even calculus. As you’ve noted, it is now a partial differential equation. As a former engineer this is now a boundary value problem. In the case of retirement finances, being anywhere on the inside of the boundary functions relatively smoothly and is good (the further away the better). Anything on the outside blows up and is bad, i.e. NW < 0. As always, thanks for a different non-traditional analysis and giving us something to ponder.

It is a boundary problem, but also a boundary optimization problem. You have stated it exactly. Many don’t have the math to quantify, but the picture of multi-factorial differential equations is looking at a rolling country side and imagining how the differential geometry is constructed first a hill is a maximum, next it becomes a minimum only to return to maximum some time later.

Good grief. Your posts have so much info.

All I know is that I am rather paranoid. Thank goodness.

I do not understand all this FIRE stuff any longer. Like you say, it’s all simple until it isn’t.

It all works until it doesn’t.

I understand that concept extremely well.

I might want simple but it is backed up with multiple layers of risk management.

But just watch, I may be the one who takes it on the chin. While these early retirees all relax their way forever.

Oh well, I must admit that I am ready for that scenario too. Because such is life.

It will be what it will be.

If I had your money and your plan I’d throw mine away MB. You’ll do fine. I just like understanding how the gears work.

I am grateful to you for explaining those gears in a way that makes sense. Only calculus I see these days requires a urologist, so even though my skills are tarnished compared to yours and GasFIRE’s, I enjoy the refresher courses immensely.

The best part of retirement is there is always something new to learn or something once learned that can be learned better, and you actually have the time to engage the problem.

Thanks for the post based on one of my favorite books. It is incredible how the financial industry oversimplifies the true nature of markets and severely underestimates the inherent risk.

Oversimplifying is how they make money. “All you have to do is invest monthly in low cost index funds….” “Passive ALWAYS BEATS active….” If I own a low cost index fund, and the world believes this narrative, I’m a trazillionaire. Yet if this was the actual case there would be no active investing. There would be no robots and algorithms trying to shave 0.1bp off trillions of trades. The robots are agnostic and make money on the way up and the way down. The robots are the train speeding down the track and your portfolio is a cow standing on the track. They made cow catchers for that. The cow catcher was a wedge that would launch a cow 100 feet into the air. Easy on the train, hell on the cow. Market was down 30pts on Fri 45 today.

The candle smoke example explanation is excellent. I had 5 semesters of calculus and hit my limit with orthogonal functions and boundary value problems. That was one tough course. The complexity of retirement finance fascinates me. Get math involved and I am in heaven. Keep the posts coming.

Hi Eric

Yes, boundary problems make gray matter leak out of your ears. If there are no obvious functional solutions you resort to actually very cute numeric solutions. I think fractal geometry fits in that bag. Not much except the method of moments was happening on that front before the desktop computer, but now my wrist watch is more powerful than my Commodore. My other favorite was physical chemistry. I took a grad level course and remember after the final going to a bar for a beer to contemplate my fate. It was the hardest test I ever took. I knew I did ok when I looked at the glass and understood why the bubbles form on the bottom of the glass and float to the top. The key is understanding what actually constitutes a bubble. From that knowledge I could infer I understood how the universe worked.

I too spent some time staring at bubble formation in a glass of champagne. My date asked what I was doing, so I explained. I think she was disappointed I was spending time thinking of bubble formation versus conversing with her. Thank goodness I found my wife, someone who understands Engineers.

I think the course that tied it all together for me was Transport Phenomena.

The interesting thing is bubbles are molecules of gas which gather in an envelop of molecules of solvent as the temp of the solvent rises and the envelop is held together by the energy contained in the hydrogen bonds between the local molecules of the solvent. Colder solvents have higher solubility. It’s ultimately a boundary problem based on the most ordered way for a dissolved gas to come out of solution. Turns out bubbles in beer are not sphere’s but hexagonal. That is the lowest energy form of the envelop. The hexagonal nature is why you can tilt a glass beyond a certain critical angle and the bubbles do not form. It’s a different problem than the formation of a single bubble, but clearly a sign of a misspent youth.

I will definitely be experimenting with the critical angle, a good excuse for another beer ! Had no idea of the hexagonal shape, so thank you for the tidbit to look into. I will misspend middle age studying beer bubbles as well as market bubbles. Unfortunately I believe I will experience the bursting of both a stock and bond bubble in the not too distant future.