Stock markets & the allure of the Gaussian Inference

Current bad is future better, but not true for cash as it is true for assets that money can buy now. But let us be careful of the GAUSSIAN Inference with a data that is following Power Laws.

When we look at the sharp uptick in stock returns for the stock indices of the world from S&P 500, the Nasdaq or the Dow, we are overawed by the steep rise in equity values and our Gaussian preference of expecting a stock return from this index would prompt us to make decisions on individual stocks as well. But we must be careful that these distributions actually follow power laws, expecting anything wildly close to the Gaussian would be fundamentally wrong.

A part of this rise in stock value is because cash has lost its value and real returns on cash are hugely negative throughout the world in every economy, so it makes huge sense to put money in any other asset that is most likely to inflate in the future given the current situation of the pandemic.

The other part of the rise is the constant diminishing size of the outstanding stocks in the market. So as the buybacks increase, the number of shares outstanding diminishes raising the income per share, one of the major parameters for evaluating stocks as an asset class.

But more importantly as more money or cash moves into a market that is constantly reducing the number of stocks on offer, the result has to be a rising value or price, provided majority of investors believe in it. 

Do not make a mistake that every asset within an asset class is same. Each could be on its own and clubbing them together could be a wrong way of thinking and expecting about them. Take the stock market and you will find the concentration of preferences and therefore of risk in just a handful of equities. Conventional thinking would lead one to think that as one class (say Technology stocks) rises, the others could fall and could act as a hedge, but not really as there is no correlation between the two as the Gaussian worldview would suggest.

In layman’s analysis only 5 stocks in the S&P 500 (Apple, Amazon, Google, Microsoft & Facebook) are worth $7 Trillion, the market cap delta that Apple created in the last three months is more than the combined reverse delta of the bottom most 30% in the S&P 500. 

Our minds are hardwired into constant paradigms of Gaussian distributions that are built around data that are largely structured around mean and standard deviation that can represent the data well; if you think of heights of individuals and the sample mean is 170 cm and standard deviation of 6 cm we know what kind of population data we are dealing with, we can almost plot the data in our minds. If you select a random sample and calculate the mean, you can reasonably estimate the population mean. This is the typical Gaussian expectation. 

The problem starts with sets of data that do not follow the Gaussian distribution and such data abound while our hardwired minds keep pondering in the same manner trying to interpret the data with a Gaussian expectation. 

Think of a power law distribution and you will find yourself in trouble thinking that you know the sample mean you can estimate population mean for the whole data set but unfortunately that mean will be meaningless as the next single data point collected from the next sample which is about to come would sway your expectation away; existing sample data has no connection to the future sample data in a non Gaussian power law distribution.

This will be easier to understand with our everyday view of stock markets which follow power laws. 

The S&P 500 for example, take the equity returns and they never follow Gaussian distributions but power laws. So if you think that there is a mean return or a median or a standard deviation the way we understand distributions, we will be totally wrong. Naively if you would have calculated these statistic, they would have mis-represented the entire data in more ways than you can possibly imagine. 

If we randomly select two securities from S&P 500 and the combined return of the two is $36 million, what is the most likely distribution of wealth, it would follow the power law and the answer would be $35,999,000 and $ 1,000.

It is like measuring the depths closer to the banks to estimate the average depth between two lands about 1000 feet apart that are connected by a sea; as you move away from the shore the depths become so large that all your earlier samples become meaningless. If you randomly select two samples of depth data, the most likely distribution of depths would be 0.2 feet and 3500 feet. If you did this for two land masses divided by a river the two would have remained closer to each other. 

The stock return distributions wander away from the Gaussian (Normal) to the Power Laws, the probability of an event staying within one standard deviation of the mean rises to between 75 and 95 percent. It will be extremely rare to find a 6-sigma deviation in a Gaussian distribution, the probability of finding that outcome would be infinitesimally small, 0.00034%.

The entire Gaussian world of statistics built around mean and standard deviation does not apply in these distributions, neither does mean square regression rules. We would have to wait for all data to be captured to be able to say anything about the distribution, which could be difficult to interpret and take action on.

This is the fundamental problem with analyzing stock return trends that it is never amply clear what the next set of data should look like; a single presence of a large deviation that has not shown in the recent data could sway the entire inference; statistical significance becomes reduced to a rhetoric.

Stock market analysis cannot be based on the Gaussian worldview of things, clubbing two completely dissimilar entities within the same basket may not reduce risk, it could end up increasing risk, we must delve into the underlying data and decipher it for statistical significance.