Friday, December 18, 2015

Retirement Income and Chaos Theory

Are retirement spending models chaotic? In my last post, Positive Feedback Loops: The Other Roads to Ruin, I pointed out the exposure to the risk of these loops in at least three aspects of typical retirement income models: market returns, spending from a volatile portfolio, and the total and rapid collapse of a large real estate portfolio. I noticed these loops because I have an amateur interest in chaos theory, and as I mentioned at the end of that post, positive feedback loops are characteristic of chaotic systems.

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,, “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. 

Are retirement spending models chaotic?
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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
Sensitive dependence on initial conditions. Often colorfully illustrated by "the butterfly effect", this is the characteristic of chaotic systems such that small changes in initial conditions can lead to drastic changes in the outcome. In retirement income studies, someone who retired in 1966 might have exhausted their savings in 30 years while someone retiring in 1967 with identical resources might not have, as a result of unpredictable market returns and unpredictable sequence of returns risk.

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.”
As the Terminal Wealth chart above shows, the prediction horizon for retirement portfolio balances is a less than a year, beyond which the outcomes diverge dramatically and become much more uncertain.

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 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.


  1. Fun topic and timely. I was just trying to build random spending shocks into my personal retirement modeling tools. Also, fwiw, I don't think the history of inflation (i.e., the historically given probabilities) in the US (among other variables) provides a fair representation of what could happen going forward.

    1. I agree. Any estimates of any future financial variables based on past results are highly questionable. Best to not pretend that we know.

      Thanks for writing!

  2. I suppose that a portfolio structure such as Harry Browne's could be looked on as an attempt both to defend yourself against the chaotic slings and arrows, and to admit that you've really no way of knowing what might turn up next. Or turn down, perhaps.

    1. I think if you can admit that, you're way ahead of the game.


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  4. Of note in this great post Dirk, is your point "As the Terminal Wealth chart above shows, the prediction horizon for retirement portfolio balances is a less than a year, beyond which the outcomes diverge dramatically and become much more uncertain."

    The key point ... less than a year ...

    Which is one of the main reasons you, Ken Steiner and I advocate reviewing everything at annual points. First, this removes all of the uncertainty of the prior year (some things are now facts and everything else a moot point). Second, all of the calculations or simulations beyond the 1 year point are simply part and parcel to arriving at the singular answer for the coming year - they're not predictions, simply possible points with even more possible that are not calculated (your point about adverse positive feedback loops veering things way out of considered bounds of estimations; by the way, positive feedback loops may be beneficial as well).

    A great summary of the differences between the two systems. I think the stochastic method gets things close to what is prudent year to year. And well developed decision rules for adverse events help too.

    Chaos may be around the corner for anybody - and that is just how life always has been. Think walking along the savanna 30,000 years ago and the unpredicted lion jumps out. Today, the lion has been replaced by the unseen car when crossing the road, or health events. Our brains interpret scary markets the same way - but as long as rush hour still exists, so will the markets (those companies people are driving to and from).

    Some thought stimulating posts lately Dirk - thanks!

    1. I agree, Larry. I think the stochastic approach works fine when the system is in equilibrium, which is most of the time, but a plan is particularly valuable when the system is not. And while positive feedback loops can be beneficial (bear markets) or harmful (spending crises), in either case they have the potential to upset the equilibrium state and take you somewhere totally unpredictable. And that's what these posts have been about – uncertainty.

      Glad you enjoyed them. They've been fun (though challenging) to write.


  5. It seems a TIPS ladder is expensive from an historical probabilistic perspective. But from a chaos theory perspective a substantial TIPS ladder maybe is a bargain

    1. Interesting point, Derek.

      So, here’s where it gets really messy.

      There are always multiple risks, à la damned if you do, damned if you don’t. Your doctor tells you that there is a risk of dying from the disease you have and recommends a new medication. Then, he tells you there is a small chance that the medication will make your leg will fall off – very rarely happened in trials – and also a small chance that the medication can cause a heart attack, also quite rare. The disease has risk; so does the treatment.

      If you invest in TIPS, then you won’t lose your savings to poor market returns or sequence of returns risk. But, then your investments might not return enough to satisfactorily fund your retirement, so the new risk is inadequate returns. TIPS wouldn’t have helped the households I wrote about in Positive Feedback Loops because their bankruptcies were the result of spending crises and a real estate collapse, while stock losses were merely an aggravating factor.

      I think what chaos theory tells us about retirement spending is that it is ultimately unpredictable, and moreso if you consider expense uncertainty. I’m concerned that a lot of people plan on funding retirement with stocks and bonds and think they have a 95% probability of being successful. It’s a lot more complex than that. You can’t make retirement truly "safe" except with massive wealth. Even if you invest safely in annuities and TIPS bond ladders, you never have total control over the expense side of the equation.

      The correct approach, I believe, is to diversify your bets. Bet on TIPs, bet on life annuities, bet on stocks and bonds, buy insurance, spend conservatively so one or two crises don’t turn into a catastrophe. Chaos theory tells us that may not be enough, but it’s a great start.

      To your point, I think chaos theory does make TIPS ladders a much more attractive alternative for one of those bets.

      Thanks for writing!

  6. As I try to plan for "the other ways to go broke" on the expense side of the equation I find it comforting read of your potential end point for the planning sequence: diversify my bets as best I can (as you suggest) and then give it a rest for a year ... time enough to look into Chaos: Making a New Science! Thank you for the suggestion.

    1. Another book I enjoyed back in the day was The Turbulent Mirror (1989). I was so excited about it that I programmed my IBM PC (running DOS) to generate Mandlebrot diagrams. It took about half an hour to render one.

  7. I'm glad that you've drawn attention to expense uncertainty, Dirk. As noted, it is often overlooked in stochastic retirement modeling. If you did M-C simulations assuming varying degrees of expense uncertainty in addition to the usual market uncertainty, the results would probably look quite different than we're used to seeing, with at least somewhat lower probabilities of not outliving one's assets.

    Expense uncertainty is not just a function of exogenous variables outside of one's control (e.g., the CPI). It can also be due to the evolution of a retiree's core interests. For instance, several years into retirement, I've discovered new hobbies. Some of them require significant spending (e.g., photography and music lessons in my case). I could not have predicted these when I first retired, but they are very important to my well being. And not cheap!

    1. True, but I should respond to two recent comments about "modeling expense uncertainty."

      Expense uncertainty isn't a parametric random variable (or any kind of random variable) and is, therefore, very difficult to model. Sure, there is some degree of volatility around annual expenses, but that probably isn't going to kill your finances. As I have tried to explain in the past few posts, most elder bankruptcies are the results of spending crises, not the slow erosion of savings due to occasional overspending and market volatility.

      Instead of trying to model annual expense volatility, I strongly suggest that you stress-test your finances for major crises. I identified these crises in the past few posts and will write more soon.

      A key point of my recent several posts is that elder bankruptcies aren't predicted by typical Monte Carlo simulation of random variables. They involve spending crises, positive feedback loops and chaos theory. Most retirement research misses this.

      Thanks for writing!

  8. This is fascinating stuff but difficult to apply to life situations. I think for most people the worst case is not a useful model (it leads to living poor and dying rich in too many cases), so we end up living with a model that could fail, and annual rethinking doesn't necessarily help. I saw this most clearly when I was looking at retirement plans against historical data. I noticed a starting point that resulted in being significantly under plan ten years into retirement, but which ended up on plan. The question was whether to count on market recovery or reduce spending. Counting on market recovery would be based on the observation that the 30-40-year performance of the stock market is pretty consistent, while the 10-year performance is not, but counting on this with a depleted portfolio is kind of scary. My mental model of the stock market is chaotic short-term behavior based on emotional positive feedback, but with a long-term attractor based on earnings to keep things from going to zero or infinity. Otherwise the relatively predictable 40-year return would make no sense. The nagging worry is that the whole system that has produced this happy result is not guaranteed. Black swan events within the current system eventually get corrected by stable and increasing earnings, but what about events that change the whole system (war and hyper-inflation come to mind)? I choose to ignore those possibilities because accommodating them would change my lifestyle radically, and I have no way to predict their likelihood. We live with a degree of unavoidable risk, but as a generation we are less comfortable than our parents and grandparents with the idea that the world can change from under us.

    1. Well said, Richard.

      There is a likely outcome and a worst-case outcome and we have to be prepared for the latter. Probabilists rely way too heavily on likely outcomes.

      I think the way to apply the chaos theory insight is simply to recognize that if retirement income systems are stochastic, they involve a great deal of uncertainty. If they are also chaotic, more data on past market returns won't help.

      This leads to the point that a projection of asset values, as Michael Kitces recently noted, is not a retirement plan.

      The reason I wrote my last 4 or 5 posts on this topic is to explain that models are just models. They can't predict the future. What you really need is a plan that addresses a large range of potential outcomes.

      Thanks for writing. Great comment!

    2. Richard, I should add a response to your first statement, "I think for most people the worst case is not a useful model (it leads to living poor and dying rich in too many cases), so we end up living with a model that could fail, and annual rethinking doesn't necessarily help."

      I disagree. I think floor-and-upside strategies are a response to the worst case market scenario and they don't lead to over- or under-spending.

      Second, you shouldn't end up with a "model", at all, you should end up with a plan, and one that covers a wide range of possible outcomes.

      Lastly, "annual rethinking" always helps. When your situation changes, your plan should change. Maybe you planned to drive a straight section of highway, but if you encounter an unexpected curve, you should probably turn.

      Thanks again for your comments!

    3. Bill Bernstein did a short piece on this in 2001:

      In this, he postulates that a probability of portfolio survival greater than 80%-85% is mythical because chaotic societal events such as financial system collapse, etc. do happen as catastrophic fat tail events and there is no way to meaningfully account for them in portfolio construction (e.g. how would a 60 year-old German in 1910 develop a portfolio with a 95% probability of surviving WW I, Versailles Treaty and the Weimar Republic, and WW II? Or a French person?).

      The nice thing about North America is that we have a big moat that does keep armies at bay and we are generally self-sufficient for most requirements like food and resources, so the most likely disasters for us are likely to be self-inflicted financial collapse or nuclear war. That is a lower probability of ruin than Europe or Asia probably have as they seem to have an army crossing various countries at least once every few decades. However, the US financial sector did seem hell-bent on self-immolation in 2005-8.

      So I generally focus on analyses with a 90%-95% success but with a very large grain of salt on the side. Tweaking something to go from 89% to 92% just doesn't seem critical if it would mean a major change.

    4. I linked to that piece in a recent post. It's from a series Bernstein called "The Retirement Calculator from Hell." And while it isn't actually about chaos theory, I completely agree with your approach (and Bill's).

      Thanks for writing!

  9. Even inherently chaotic systems can be controlled with restraining feedback. The thermostat in your home can maintain a constant temperature, depsite the fact that the weather conditions two weeks from now are unknowable. This point of view supports adaptive retirement plans that mitigate the effects of what actually has happened. Simply reducing spending a bit during down markets can greatly reduce the chance of a portfolio going to zero.

    1. Weather is an inherently chaotic system. Can it be controlled by "restraining feedback"?

      The thermostat is a bad example – it's a negative feedback system. "The thermostat in your home can maintain a constant temperature, despite the fact that the weather conditions two weeks from now are unknowable" – if you can afford the energy. If your electric company cuts off service for non-payment, the thermostat is relatively useless. More importantly, chaos stems from positive feedback loops, not negative.

      Adaptive retirement systems, though, make sense for a multitude of reasons, or said differently, there is simply no way to rationalize not adapting. Not adapting means ignoring what you know today in favor of what you thought you knew at some time in the past.

      Reducing spending when a portfolio declines is indeed the rational response and will reduce probability of ruin, as you point out.

      But this post is about families who can't adequately reduce spending. The families I wrote about couldn't, and just a few months earlier had no idea they would need to. I didn't say that reducing spending won't lower your probability of ruin during down markets. It will, but those are times when your finances are in equilibrium. "Reducing spending a little" won't help, however, when your finances crash in a positive feedback loop.

      Thanks for writing!

  10. From what I learned from Michael Kitces, chaotic behaviors were observable only at the daily microscopic level. There has not been a single Black Swan at the yearly level. Better yet, only a bad decade (not a bad year or a bad day) can ruin a well-planned retirement system. By well-planned, I mean diversified and not leveraged. His question was: If an approximate stochastic software tells you that you have about a 5% chance of ruin but a perfect chaotic software tells you that you have exactly 5.5% chance of ruin, what would you do differently? Have you looked at his findings and what do you think?

    I think he has a point.

    1. Isaac, I have read Michael's posts and I completely agree with him. But, we're discussing two totally different issues.

      Michael is referring to Black Swans in the stock market. My post is about bankruptcy due to spending shocks, since no one seems to go broke from sequence risk.

      Second, chaos theory wouldn't tell you that your chance of ruin is 5.5%. It would tell you that you can never have enough historical data to calculate your chance of ruin.

      So, can you ignore chaotic market returns? Sure. You just need to have a plan for losing a large portion of your savings, whether that is gradual or catastrophic.

      But I suggest you don't ignore chaotic spending.

    2. Issac, I should have added this. Of the three families I know locally who went broke and were considered in the post, one was due to the housing crash, one due to prolonged unemployment, and one due to divorce. None owned a substantial stock portfolio.

      It takes a bad decade to ruin a well-planned retirement savings portfolio, not a retirement system. All three of these bankruptcies developed in less than a year.

      Thanks for writing!

    3. Dirk, thanks for correcting me. I indeed fixated on portfolio earlier today. I agree with you that spending shocks are chaotic and warrant taking special measures to lessen their damage. Choatic spending is not to be ignored.

      As for market returns, it looks more normal than chaotic when we look at it on an annual basis, for example,

      Do you suggest that we shouldn't ignore the daily chaotic market returns? Other than rebalancing, what else should we do to respond to the daily chaotic market returns?

    4. Isaac, what you're seeing in Michael's chart there isn't chaos (or, at least, can't be proven to be chaos). Monte Carlo simulations of the stock market employ a spherical cow – an assumption that market returns are normally distributed. The purpose of spherical cows is to make the math easier. It is well documented that market returns are not normal. When these "unexpected" returns occur (which Michael refers to as "black swans", but I don't), it isn't due to chaos, but to the fact that our expectations are wrong. These returns are only "unexpected" if we believe they are normally distributed. As Michael also explained, these unusual returns are easily explained with common statistics (sometimes improbable things happen) and don't require chaos theory.

      The complaint about Monte Carlo simulations that Michael addresses isn't that they don't incorporate chaos (by definition, they can't), but that they use an unrealistic assumption of normal distribution that doesn't capture the extreme outcomes we see far more often than we would if they were, in fact, normally distributed.

      Michael is talking about portfolio ruin due to market volatility (not spending) and the fact that Monte Carlo simulations assuming a normal distribution of market returns are less than ideal statistical approximations. I'm talking about bankruptcy, whether or not you invest in stocks, and chaos theory. Different discussion.

      Your second paragraph should say that market returns look more normal than leptokurtic (has fatter tails than a normal distribution) when looked at annually, not more normal than chaotic.

      I don't think it matters, should you lose a large portion of your retirement savings in the market, whether you lost it catastrophically or over a long period of time due to sequence risk, and that's Michael's point. It's gone either way. You need a plan to address the possibility of that happening, even if the probability is quite small. Rebalancing won't help much, but a lower equity allocation will limit the amount you can lose even if the market goes to zero. A floor is the best solution.

      And, in general, I suggest you ignore anything that is daily.

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