Reversion to the mean? Still tough to profit from it
#decisionMaking #riskManagement #quantitativeIntuition
When things do not go as planned, we often make decisions on the subtle belief that events will soon “revert to the mean”. Say, something in our professional or personal lives is performing extremely poorly, so badly we had never believed possible. Say also, despite the recent results, we have decided to stick to our initial positions because we think things will soon change course.
Are we sure we are making the right decision by waiting for better times to come?
The problem is often different than we think. It is not only about if, but it is also about how and when. In general, thinking that things will revert toward the mean is probably correct, however, thinking we can easily exploit that knowledge is probably wrong. In general, our fortunes are not functions of state, therefore depending only on the initial and final states. On the contrary, our results are in general extremely dependent on the path.
While some of these arguments are studied in applied mathematics through puzzles like the Gambler’s Ruin Problem and Chasing the Losses, this post wants to give readers possible general intuitive tools. The objective is to detach intuition from game-specific situations where the chosen rules could play important roles (e.g. the type of possible reward at each step). This is also why no specific simulation is shown here. To gain effective intuition for decision making, below, three main factors making paths relevant and reversion to the mean difficult to exploit. In the final takeaway at the end of the post, possible practical principles will be suggested.
1) Limited resources
Normally, our resources are limited; however, the point to stress is that with finite quantities their allocation becomes crucial. In general, while trying to catch an opportunity requiring funding, we usually make excessive allocations – despite them appearing to us conservative. Let us consider a rigged game where we are given probability to win 55% of the time. That game would often sound like a good game to play, being something that “on average” would pay out. Unfortunately, even a couple of rounds in excel could show that if we allocated something like 25% of our wealth on it, we could end up in ruin 80% of the time. In terms of probability of success, the optimal bet for that game would be closer to 10%. That is maybe not that intuitive; people would probably believe 20-25% to be an optimal starting point, even not too “aggressive”. So, sometimes we worry about reversion to the mean and other probable future factors, while we have positioned ourselves up for failure right from the beginning.
2) Big swings
We might already agree on our tendency to under-estimate potential swings and volatility of events. The problem is to be found in the wrong idea about the underlying probability distributions. Remaining on an intuitive level, applications with milder swings described by random-walks and Gaussian distributions – more common in physics for example – have potential swings that can be intuitively predicted through past data. On the contrary, potential swings in common real-life applications could be completely missed by any indicator calculated over past data. Moreover, even though these distributions do have quantitative indicators that could tell us something, they usually completely change in time because of new and unexpected outcomes altering the statistics. For example, the standard deviation computed over distributions of the same class describing the wealth of people across the world (i.e. Pareto or 80-20% distribution) has no real meaning when the parameters of those distributions are in particular ranges. So, these considerations make the reversion toward the mean something completely unpredictable; the mean becomes often just an unstable mathematical middle point between numbers quite different in magnitude and probability.
3) Long-term dependence of events
We often do not reconcile properly our minds with the dependence in time of events. Often, hidden processes determine and characterize long streaks of good and bad events – longer than processes with independent events. In general, if something has performed badly lately, there is no supreme justice ensuring us that an offsetting positive outcome is soon to be due. On the contrary, similarly to what chaos-theory would suggest, underlying unknown processes are likely to be characterizing and sustaining our adverse experience. Eventually, an abrupt movement in the opposite direction is likely to come, but that is something completely unpredictable and, to some extend, related to point-2 above – surprising events of a magnitude not seen before among past data. So, chaotic processes have underlying dynamics possibly playing against the reversion toward the mean of extreme events. Eventually an adjusting round is likely to occur, but that could take much longer than expected. The offsetting event, while bigger in magnitude (point-2), it could be rarer than normally predicted by independent random occurrences.
Takeaway
Before giving possible practical principles derived from the arguments above, here is a possible educated question somebody might ask at this point: “so, even assuming finite resources, what if I find an opportunity with limited swings and independent events? They would be kind-of governed by the normal [bell] distribution, should I be then more comfortable in trying to exploit concepts like reversion to the mean?”. That would probably be a good point and a generally good way to check the reasoning above. As we mentioned in the first part of the post, the answer could depend in part on the specific conditions of the game we play (e.g. are we changing the bet depending on the result at each step as if we were chasing the losses?). However, something could be inferred: in general, phenomena governed by the bell distribution will yield more predictable and smaller potential losses. Extreme outcomes could have their magnitude related to the square root of the number of rounds played – thinking about the usual standard deviation. That milder character, coupled with the independence of the events, could indeed yield better results in case we decided to adopt strategies based on ideas like the reversion toward the mean. Obviously, the way we play (e.g. percentage allocation, point-1 above) and the metric we choose for the comparison, still affect the conclusion.
As readers might have perceived by now, this post implies that in some ways good decisions go beyond the reversion to the mean and other similar specific situations. Good decision-making is based more on effective principles adopted right from the beginning; here are possible ones from the discussion above:
Allocate an initial percentage of the total resources smaller than intuitively deemed correct – even in the single digits range
Expect swings much bigger than the biggest one ever occurred in the past – doubling and above is not inconceivable
Be ready to wait for better times longer and longer as time passes by – exhausting waiting times could be revealing us we are getting further from the end rather than closer
The interested reader can research arguments leveraged in this post regarding topics like fractals, fat-tails and similar arguments approached and discussed by professionals in different fields such as mathematics, risk management, and more - not citing anyone on purpose.
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