Whether or not retirement income systems are chaotic is an important issue because chaotic systems are riskier than stochastic (probabilistic) systems. We tend to study retirement income systems with probabilities. If the systems are chaotic, they're riskier than inferential statistics (probabilities) suggests. Bear with me through some background and I will explain the relevance to your retirement plan.
According to the website, FractalFoundation.org, “chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected.” Most retirement income research uses the science of probabilities and statistics that reveal what is unlikely, but not necessarily what is unexpected.
[Tweet this]Are retirement spending models chaotic?
As early as 1887, Henri Poincaré showed that while Newtonian physics could perfectly predict the orbit of two planetary bodies, adding a third body to the mix turned a straightforward problem into one that is virtually unsolvable. A system as simple as the double pendulum simulated below can exhibit chaotic behavior. Its trajectory varies dramatically with small changes in its initial position. Probabilities won't predict the trajectory of the double pendulum because we can't know precisely enough where it will start. As the three body problem and the double pendulum show, systems don't have to be complex to behave chaotically.
When I study the models of retirement income studies, I see a number of characteristics of the models that are also characteristics of chaotic, not probabilistic, systems.
Let’s look at those characteristics as suggested by FractalFoundation.org.
Unpredictability. “Because we can never know all the initial conditions of a complex system in sufficient detail, we cannot hope to predict the ultimate fate of a complex system.” Like the starting point of the double pendulum, we can’t predict precisely enough where we are in the cycle of future market returns at the outset of retirement. Is the market overvalued? Undervalued? Did we pick a fortuitous retirement date for the sequence of future returns? Only time will tell.
The Transition between Order and Disorder. “Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.” Chaotic systems, such as the stock market may be, can remain in a stable state for long periods of time before inexplicably becoming unstable. The 2010 Flash Crash is an example that is still not well-explained. In the example I provided in my previous post, two households went from well-to-do to bankrupt in less than a year when the U.S. economy crashed in late 2007. Everything looked fine for both families just months earlier.
Mixing. “Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Two neighboring water molecules may end up in different parts of the ocean or even in different oceans." Look at the range of outcomes for a retirement portfolio balance simulation in the chart below. One scenario depleted the portfolio in just 19 years while another grew to more than $6M. Each scenario started at the same point (a $1M portfolio balance) under the same initial conditions.
Positive Feedback Loops. “Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.” See my previous post on positive feedback loops for retirement examples.
Here are more characteristics of chaotic systems not included in the Fractal Foundation's list.
Attractors. An attractor is a state toward which a system tends to evolve from a wide variety of starting conditions. System values that get close enough to the attractor tend to remain close to it. A type called a "fixed point attractor", which attracts trajectories to a single point, describes portfolio ruin.
Here's a graphical depiction of a point attractor from Young Scientists Journal. Imagine this as two portfolio balance trajectories that enter a positive feedback loop and spiral downward to ruin.
(Want to see something really cool? Google "images of strange attractors" and you will find some amazing graphics, like this.)
Prediction Horizons. Another characteristic of chaotic systems is a prediction horizon, explained by Professor Jonathan Borwein.
“What at first glance appears to be random behavior is completely deterministic – it only seems random because imperceptible changes are making all the difference. The rate at which these tiny differences stack up provides each chaotic system with a prediction horizon – a length of time beyond which we can no longer accurately forecast its behavior. In the case of the weather, the prediction horizon is nowadays about one week.”
Experts disagree on an exact definition of chaotic systems. They tend to describe their characteristics, instead, much in the way Supreme Court Justice, Potter Stewart once described obscenity – “I know it when I see it.” I'm not an expert in chaos theory, but when I consider the characteristics in common with retirement income models, I think I see it.
My interest in chaos theory is limited to popular books on the subject because the math, differential equations and fractal geometry, is pretty demanding. So, I posed several questions to chaos theory expert, Tom Konrad, who has a doctorate in complex analysis and chaos theory and edits AltEnergyStocks.com. I described spending from a volatile portfolio to Dr. Konrad and asked if he thought it might be a chaotic system.
“It's impossible to ‘prove' that a system is chaotic or is not when we don't completely understand the underlying mechanisms,” he explained.
“It certainly displays chaotic characteristics”, he continued, “but other than acknowledging that, I'm not sure if anything would be accomplished by quantifying them.”
In my interpretation, if it quacks like a duck and tastes like a duck, dinner probably won’t suffer if mathematicians can’t agree to the precise extent of its duck-ness. If the retirement income system displays chaotic characteristics, there may be limited practical negative consequences to treating it as chaotic and it is safer to assume that it is.
Now, why is it important to understand if retirement income systems are chaotic or simply probabilistic? Because stochastic systems are unpredictable but statistically quantifiable, while complex and chaotic systems are even more unpredictable. It was on this point that Dr. Konrad provided my favorite explanation.
“Chaotic systems are less predictable than stochastic systems. Sufficient historical data will eventually allow you to quantify a stochastic system; this is not true for a chaotic system. The stock market seems to be un-quantifiable based on the historic record. That does not necessarily mean that it is chaotic (although there are other reasons, such as positive feedback loops, to believe that it is) but it is clearly harder to quantify than a stochastic system would be.”We debate whether 200 years of stock market returns are enough to characterize the returns of the market's internal processes, or its impact on retirement plans. If the system is chaotic, we will never have enough historical data to make it predictable.
Retirement income studies tend to use probabilities to focus on long-term sustainability of savings as a function of market volatility alone. This approach won't catch many quickly developing expense-related crises, especially since the studies tend to ignore expense uncertainty altogether. When we say a retiree has a 5% risk of outliving her savings, we mean a 5% risk of outliving savings due solely to market volatility. But, there are other risks to those savings that should also be considered.
These studies explain long, slow declines in standard of living, not catastrophic failures, in a world where market returns are normally distributed and mean-reverting and no one ever needs to spend more than their "sustainable withdrawal." Their recommendations – diversification and spending adjustments – provide little help in a spending crisis.
Chaos theory helps explain household finances that veer suddenly from normal equilibrium into a crisis. Debt, divorce or some other expense shock shoves the portfolio balance trajectory into a positive feedback loop and toward the point attractor that is portfolio ruin.
Take another look at the green trajectories in the spiral above and consider the households from my previous post that went from equilibrium to bankruptcy and, in one case divorce, in less than a year. This is not the stuff of 30-year Monte Carlo simulations of normally distributed market returns.
Probabilities and equilibrium are important parts of the story, but they aren't the entire story.
I admit this post is a bit dense, particularly if you have no interest in chaos theory. But, if you take away the following, I think you'll be fine. When a planner tells you that you have a 5% probability of depleting your savings, she typically means a 5% probability of going broke as a result of market volatility. Alas, there are other ways to go broke. If spending systems are chaotic, which I suspect but can't prove mathematically, there are conditions under which their outcomes are unpredictable and probabilities don't help. And lastly, as Dr. Konrad suggests, if they behave chaotically, we might not gain much by proving how chaotic they are.
Unless and until we know that these systems are not chaotic, the safest path for a retiree would be to assume that they are and that the probability of ruin is greater than studies have indicated.
Once again, I note that my posts about market risk shouldn't be taken as an argument against investing in stocks. Retirement income without equities is terribly expensive. But it's important to understand the risks and to be prepared to deal with them. There's more to worry about than a bad sequence of returns and living too long.
Next time, I'll sum up what I learned about retirement finance in 2015.
Looking for some good popular books on chaos theory without the differential equations? Try The Black Swan, Fooled by Randomness, Chaos: Making a New Science, or Dr. Konrad's column in Forbes.
