Non-ergodicity of Financial Markets

Financial markets are unpredictable and stochastic by nature. They depend on many random events and many unknown unknowns. Nevertheless,  we have an illusion that markets are predictable when we look retrospectively at historical charts, and suddenly everything makes sense. This is known as hindsight bias or I-knew-it-all-along phenomenon. Events like COVID-19 completely change the status quo and prove that we cannot time markets. This brings us down to Earth and we realize that we, as humankind, have a little control of what can happen tomorrow. But even without black swan events, our perception of randomness is distorted, and this is related to non-ergodicity of financial markets. Nassim Taleb wrote: “There is no probability without ergodicity.” Let's have a look at ergodicity and why financial markets are non-ergodic.

There is an excellentpresentation "Time for a Change" by Dr. Ole Peters that's available on YouTube.

Let's have a look at the following game that he talks about on the video. The game has the following rules:

  • Initial investment is $100
  • We are tossing a coin once a minute
  • If the coin shows heads, you win 50%, i.e. $50
  • If it shows tails, you lose 40%, i.e. $40

The game looks attractive because we win more than we lose. It has a positive expectation value. We can calculate the expectation value as 50 x 0.5 - 40 x 0.5 = 5%. 

Here is one sequence where we play for 5 minutes. It is quite simple. On the first 2 tosses, we lost 40%. Then we won and lost again, and we can see trajectory of the wealth developing over time.  But the outcome of the first sequence is negative. 

We will continue tossing the coin for another 55 minutes to play for 60 minutes. Here is the result. 

However, we do not really see much. We lost in the beginning, won in the middle, and lost again at the very end. It looks random. We are going to try again and see how it goes. We will try for 10 times and get different trajectories.

Now it looks even more confusing because it is very difficult to make a conclusion about the outcome of the game. Lines behave very differently, and we don't see our positive expectation value. Let's continue and play 1000 sequences and average all the results to get rid of the noise.

Now this looks much better. We can see a positive trend. Let's increase the number of sequences up to 1 million. 

Now we see a straight line on the logarithmic chart that reflects the positive expectation value. We can make a conclusion that this is a favorable game. This is probably what our intuition told us in the beginning. Let's now make a reality check:

  • We did million sequences and we look at ensemble-average 
  • Is it relevant to me at all?
  • Because we cannot go back in time and access parallel universes
  • We are interested in one sequence only (reality)
  • What happen with time?

As accessing parallel universes is not possible, we are mainly interested in one sequence. But what will happen if we continue playing the same sequence? Let's get rid of the noise through time. Rather than considering many parallel systems, we want to consider what happens if we play for an exceptionally long time. Let's take the original sequence and continue tossing the coin. 

We are tossing the coin for 24 hours. The green part on the chart shows the sequence during the first hour. We can see that we are losing in the game. Let's continue tossing the coin for 1 year. 

Now we can see that the noise diminishes, and the results are clearly devastating. We lost.  

It turns out that we have two different perspectives: 

  • Ensemble perspective that over-emphasizes exceptional success.
  • Time perspective - reflects what happens to me and it is relevant to the individual.

This is a good example of a non-ergodic system because the ensemble perspective does not match with the time perspective. Consequently, a system that has matching ensemble and time perspectives is called ergodic system. 

Why is this important?

  • First, we perceive probability intuitively in ensemble perspective
  • Second, we confuse these perspectives. We use often the ensemble perspective because it looks more appealing. We underestimate time and behave often as if we have access to the parallel worlds. 
  • Finally, life is a single player game and we play the same game over time 

Dr. Ole Peters tells in the presentation that the economic theory has been wrong over the last 80 years. The time perspective has been fully ignored even by many Nobel prize winners - Menger, Samuelson, Markowitz, and Arrow.  Dr. Ole Peters tells that the modern risk management needs to be fully remodeled with the time perspective being the central idea.

Realizing the ensemble and time perspectives is very importing for investing or even opening a new business. For example, investing into a broad-based market index is an ensemble perspective and it has much higher chances for success. Investing into an individual company is a time perspective. Successful entrepreneurs are those who made multiple attempts, failed a lot and succeeded in one of those cases. 

Finally, here are some conclusions on investing:

  • You must make multiple iterations to benefit from the ensemble perspective
  • You must minimize the loss from a single bet
  • You need to use a low leverage to stay longer in the game
  • Each small bet must have probability of an extremely large payout