Market Complexity Magnitudes
A version of this blog first appeared on the Newfound Research blog site.
Did you know that super computers can only simulate out 10-11 moves in a game of chess? Despite a confined board and a fairly “simple” set of rules, every possible move stems a new branch of ways the game can play out. The number of legal chessboard positions is estimated between 1043 (a 10 with 43 zeros after it) and 1047, with a game tree complexity of nearly 10120. This means that looking only three moves ahead requires examining nearly 109 possible board configurations.
Now let us consider the stock market – in fact, only the stocks in the S&P 500® Index – and whether, for any given day, their return is positive or negative. Ignoring move magnitude of correlation considerations, there are 10150 possible return combinations. If we expand our universe to the nearly 5,000 public U.S. equities, there are 101505 possible combinations.
Consider that the known universe only has 1080 atoms in it. That means for every atom, there are 1040 possible chess board configurations. For every possible chessboard configuration, there are 1030 possible label combinations of S&P 500 stocks. For every atom in the universe, there are 1070 possible return combinations for S&P 500 stocks. To put all of that into perspective, since numbers of this magnitude can be difficult to understand, the sun is only 1011 times bigger than an ant.
While fundamental and statistical arbitrage reduce the degrees of freedom in financial markets, the sheer magnitude of the market complexity has several implications. Even if there was a unique return combination every day, it would take ~10148 possible years to achieve every S&P 500 combination. Once again, to put perspective on such a number, it would only take you about 3,500 years to walk to the sun, if such a feat were possible. If we saw a unique combination of returns every second of every day, it would still take 10143 years.
As strategists, this should make us not only question the efficacy of back-testing for predicting live trading results (which pretty much everyone already does), but also the efficacy of live trading results at predicting future results. With such a high degree of complexity, how can we ever know that our strategy will be robust and adaptive to all possible future market environments?
At Newfound, we argue that for models to be successful and robust in uncertain environments, they must be based on simple heuristics tied to a fundamental or economic concept, not based on complex decision rules. We believe that the more complex a model is, the more likely it is to break in untested environments, of which there are many, as we just demonstrated. After all, even though we know the mathematical laws that govern weather patterns down to the molecular level, we still carry an umbrella “just in case.”