Stochastic Efficiency is Real and It’s Spectacular.

Have you ever read something, and instantly thought, that just blew my mind.  Well this post is about one of those moments for me, and hopefully it will be the same for you as well.  I’m also going to introduce you to the brilliance of Ole Peters and Alexander Ademou and their work on Ergodicity. In the process, I will show one of their insightful predictions is 100% correct.

What is Stochastic Efficiency

In my post on the equity premium puzzle, I added a reference to Ole Peters’ and Alex Adamou’s work on Ergodicity Economics in the footnotes a week after I published the post.  I discovered their work soon after writing that post, and realized their solution to the puzzle was extremely novel and paired exceptionally well with my own.  They used a concept called stochastic efficiency in their solution, and it’s brilliant.

Stochastic efficiency begins with an understanding of how to mix stocks in a portfolio, similar to what I described here.  (If you haven’t read the post, it will help your understanding). I came to understand the optimal way to mix a portfolio through a graphical understanding of the problem shown below.  Essentially, cash and a risky asset (i.e. stocks) have a preferred mixing rate for optimal long term wealth.  The formula for the optimal portfolio is:

%  Stocks in Portfolio = (Expected Return of Stocks – Risk Free Rate) /  Standard Deviation of Stocks²

The above parameters create returns in various portfolios as shown in the chart below.  You can see there is a clear maximum – a clear optimal portfolio – for the long term growth. It makes sense than an investor would prefer to hold this optimal portfolio in his or her own investing account. 

Peters and Adamou figured this out mathematically about a decade ago. Others in finance similarly discovered this formula decades before.  But Peters and Adamou brilliant insight took it one step further and asked what this meant for an investor?  If there is an optimal portfolio, shouldn’t the market trend towards this portfolio? Obviously, yes it should.

More specifically, would the “market” prefer one kind of optimal portfolio, which they called optimal leverage, over others? They answered (section 5.2.3):

Imagine that [optimal leverage] > 1 in our model market. This would mean that the simple strategy of borrowing money to buy stock will achieve faster long-run growth than buying stock only with our own money. If we associate putting all our money in stock, [optimal leverage]= 1, with an investment in the market, then it would be a trivial matter for us to beat the market (by doing nothing more sophisticated than investing borrowed money).  Similarly, imagine that [optimal leverage] < 1. In this scenario, the market could again be beaten very easily by leaving some money in the bank (and, if [optimal leverage] < 0, by short selling).

It would strain language to consider our market efficient if consistent out-performance were so straightforward to achieve. This suggests a different, fluctuations-based notion of market efficiency, which we call stochastic market efficiency: it is impossible for a market participant without privileged information to beat a stochastically efficient market simply by choosing the amount he invests in stock, i.e. by choosing his leverage.

Here they’re saying, any optimal leverage above 100% or less than 0% should not last for long, as market forces should quickly act to correct this position. People don’t turn down easy money, it’s what separates us from animals.  A portfolio between 0% and 100% of stocks vs cash should be “stable”.  But even here, they believe market forces (efficiency) should still push the portfolio toward an optimal leverage of 1.

1 = (Expected Return of Stocks – Risk Free Rate) /  Standard Deviation of Stocks ²

Therefore :

Expected Return of Stocks – Risk Free Rate =  Standard Deviation of Stocks²

Graphically, this means the market should try to “balance” in a position where the risk-free rate matches the bottom of the mixing range in chart 1, shown below.

Testing the Theory.

Peters and Ademou went about trying to test this theory through three different methods.  The simplest example involves using historical data from markets as inputs to the optimal portfolio construction formula.  If they were correct, the optimal leverage of the market using historical data would lie between zero and one.  A pure proof of their theory would show the market organized itself toward an optimal portfolio of nearly one.

I don’t feel the published test supported their claim very well.  Using inputs from market indexes (S&P 500, DAX, etc), they didn’t find a historical “leverage” matching their predictions.  The results were directionally correct, but from a scientific point of view, not very precise. 

However, I quickly recognized the insight and predictions were nearly perfect. They just made the same mistake everyone is the financial community makes – using a stock market index to represent the stock market as a whole. 

A Market Index is not the Market.

The stock market index is an investment strategy, not a representation of the market.  Their properties of return and standard deviation are not the same.  Every single investing and economic study mistakenly uses a stock market index as a proxy for the stock market leading to many false conclusions. 

A better representation of the market is an average of stock returns.  It is subtle, but this view produces very different qualities than a market index.  The adjacent tables represent the return of Dow COMPONENTS, from 1940 to 2010 as well as the returns from the Dow itself.¹ Clearly, different.  The long term geometric returns of the Dow components experienced significant shrinkage compared to the Dow itself. The average of index components represents the “market” far better than a market index.  You may recognize this data from my prior post on market index investing.  Here I added the dividends to the return streams to show the total return.

Proving Stochastic Efficiency and Optimal Leverage.

So what happens if you use the average properties of stocks instead of the market index in the optimal leverage formula? ¹

 (Expected Return of Stocks – Risk Free Rate) /  Standard Deviation of Stocks ²  =  % stock in portfolio

(12.01% – 4.5%)      /     27.45% ² = .997

You don’t get much closer to 1.

Over a 70 year sample, Peters’ and Adamou’s prediction, mixed with my realization that a market index is not the market, works absolutely perfectly.  Our philosophies focusing on the geometric mean (which they call Ergonomic Economics) explain the way the world actually works unbelievably well.

Is This Just a Coincidence?

Let’s look deeper.

Hypothesis Then Test

First, this conclusion was not data mined. Peters and Adamou proposed this idea for how the market should work, without great supporting data. It was a pure hypothesis based on their work to date. I knew how to support the theory with data I had previously analyzed. That’s how science is supposed to work: hypothesis, and then discovery of proof.

Works with Gold Too

Second, Ole’s hypothesis works with more than just stocks. I ran the numbers with gold and found an optimal leverage of 1.009 from August 1971 (when Nixon closed the gold window) to 2018 (9.7% return, 21% standard deviation, 5.2% risk free rate).

Ole proposed a second way to evaluate the theory: backtest the asset’s returns with varying leverage. The “optimal leverage” will provide the highest return.²

Its clear the peak is just past a leverage of 1. Keep some cash, and you will end up with less wealth. Leverage the asset more than 1%, and you also end up with less wealth. An optimal leverage of 1, just as predicted.

Can it be a coincidence gold also precisely follows the same organizing principle?

Now the Bad News

The last nine years don’t work as expected for stocks. I showed a sample of stocks stopping at 2010, only because that’s were the raw data I found on the internet stopped(I analyzed this data years ago to study internal correlations in market indexes). I don’t have access to expensive historical data, or a research team to collect and analyze it, so I use what I can find.

Knowing the data I had on hand was a bit old, I have tried to approximate the last 9 years, and, yada yada yada….. it’s way off. The optimal leverage is around 2.5 (approximately 10% return, 20% vol, 0% risk free rate).

Was this caused by Quantitative Easing (QE)? Maybe. Was it something else, or is optimal leverage just not a real concept? It may be that nine years isn’t enough time for the data to converge and settle out. If that’s true, we could be in for a major crash.

Personally I lean towards QE. It’s very hard to overlook seventy years of evidence in stocks and fifty years of evidence in gold perfectly aligning with the prediction. The proof will come in how well it explains the future.

Ergodicity and the Coming Revolution in Economics

I encourage anyone reading this blog to learn more about Peters’ and Adamou’s work at the London Mathematics Laboratory (LML).  They have been developing these ideas for years now, with few other academics joining them.  They truly are masters of their domain.  While there are many similarities between their work and my own, the LML does a far better job than I at showing the pure math behind these concepts. My blog is focused on the applying similar principles to investing.  I’ve already applied these and other concepts to:

• Create a spectacular investments strategy
• Show the superiority of the geometric average
• Solve the Equity Premium Puzzle
• Explain through rebalancing why index investing works
• Show that all investments ultimately fail
• Explain why most investment research misses the mark
• Show that mean reversion and momentum in stocks may be a statistical illusion
• Demonstrated how to balance a simple portfolio
• Re-defined the efficient market theory geometrically

More to come. Please keep exploring this subject. Much of the world that once seemed confusing and puzzling will begin to make sense.  It’s truly an eye-opening experience.     

A new economics is out there, and I’m loving every minute of it.

1- Data of returns from down components from Quantifying the Behavior of Stock Correlations Under Market Stress: Tobias Preis, Dror Y. Kenett, H. Eugene Stanley. Dirk Helbing, & Eshel Ben-Jacob

Fed fund rate from FRED through 1954. Approximated with 3-month t-bills from 1940-1954.

2- I really love this idea. It shows very clearly how leverage often hurts your long term returns. And it also shows very clearly that each asset is chasing its own optimal leverage.