In 2013 the Nobel Prize committee gave the economics prize to two academics with fundamentally opposite beliefs.1 On one side they gave Eugene Fama the award for his work developing and expanding the efficient market theory. And on the other side, Robert Shiller won the award for effectively disproving the efficient market theory. If you ever needed proof of the deep confusion within economics, this should be exhibit 1A.
In many ways Dr. Fama and Dr. Shiller created the two leading financial economic theories. One theory believes the markets are efficient. They incorporate all available information into their prices, and in doing so, behave rationally. The other believes investors act irrationally, causing inefficiency in market prices. Available information can not possibly justify stock’s volatile price action (think fall 1987 or 2008), leading to the creation of behavioral finance to explain investors’ irrational behavior. No one has ever unified these two theories, until now.
This post will explain, rationally, why markets behave as they do. They are efficient, but not in the way the efficient market camp believes; they are geometrically efficient. Geometric efficiency explains why markets shoot up to into “bubbles”, and why they “crash”. And it explains, with indisputable mathematical concepts, that these “bubbles and crashes” are totally rational, actually proving market efficiency.
What are Efficient Markets?
The efficient market theory says that stocks always trade at their fair value, that all information available to investors is incorporated into the price of the stocks immediately. It’s not possible to find mispriced securities unless one has inside information. Stocks are never under priced or over priced – they are always correctly priced. And changes in security prices are always justified by known information. As such, an investor can’t pick stocks and outperform the “market” because there isn’t any inefficiency to exploit.2
The leading theory from Fama’s day explained a market gets “efficient” across all assets by equalizing risk adjusted returns. The more risky an investment, the more the investment should expect to return. Investments that don’t rest on the “security market line”, are in the process of quickly moving towards the line.
Furthermore, under this framework an investor can’t use leverage alone to create a better investment. Leveraging up a lower returning asset just creates the same expected return as a riskier asset. The market is therefore “efficient” as no asset provides a risk adjusted return superior to another.3
Shiller’s Insight on Efficient Markets.
Robert Shiller didn’t believe in efficient markets. Dr. Shiller in 1981 wrote a seminal paper showing that markets do not behave rationally and efficiently. The variation in stock fundamentals, aka the “available information”, do not vary enough to justify the amount of price volatility actually seen in the market.
He analyzed the belief that asset prices should reflect the value of all future cash payments, and concluded that if prices acted efficiently they would fluctuate far less than they do in the real world. Bubble and Crashes would not exist.
Put bluntly, the stock market information available to investors does not explain the wild swings in stock prices. Therefore markets continuously overshoot and undershoot fair value. This realization lead to the creation of behavioral finance to explain the clear discrepancy between efficient markets and actual investor behavior.
Based on the framework given for the efficient market hypothesis, Dr. Schiller was mostly right.4 But his ultimate conclusion that markets are inefficient and investors irrational is misguided. Instead, his work disproved the current framework describing the markets construction, not their efficiency or rationality.
The Challenge of Disproving EMT
Fama understood the unique challenge with trying to disprove the efficient market theory. Each test of the “theory” is actually a test of the efficient market theory AND a test of a specific asset pricing model of the theory. It is a “joint hypothesis problem”. You can’t isolate one from the other. He described this problem in his Nobel prize speech.
The Problem With the Efficient Market Theory
Let’s think again about the the working framework of asset price falling along uniform risk/return security market line. Which return becomes efficient along this line, the arithmetic or the geometric? They both can’t be efficient can they?
Most efficient market theories assume markets get efficient across the the arithmetic return. Researchers, such as Markowitz pay lip service to the geometric return, but never in the core parts of their theories. CAPM is entirely based around the arithmetic return, and so are most other asset pricing models. Over very short time frames that are not repeated, this model of the markets works.
But as readers of this blog have come to learn, everything in investing repeats. The arithmetic return is meaningless. The geometric return is all that matters.
Efficient Theory of Geometric Markets
Calculating the geometric return of assets on the theoretical “security market line” gives a much different picture. The long term returns become curved. There is a peak. The end of the security market line is no longer efficient at all. The whole concept breaks.
The proper framework states markets aim to maximize their long term geometric return. I showed the foundation of this concept in the optimal way to create a portfolio. Geometrically, there is an optimal portfolio for all assets, and the market will try and efficiently move toward this portfolio at all times.
We’re going to stick with a very simple two asset stocks/cash portfolio I introduced in the prior post to walk through the philosophy. A portfolio of two assets (stocks and cash) will have a curved long term expected return.
There is a clear maximum point showing there is a singular superior portfolio composition. The equation for determining the maximum portfolio is:
% Stocks in Portfolio = (Asset Return – Risk Free Rate) / (Asset Variance)
The rational investor in this market should try and invest at this maximum point. A truly efficient market doesn’t organize itself around a securities line, it organizes itself around the maximum return point on a geometric return curve. This is the correct framework for testing the efficient market theory.
Lets Evaluate the Model for Expected Volatility.
So lets walk through a simple thought experiment, and you check for any irrational decisions. We’re going to use the 7% risk free rate, 10% risky asset rate, and 20% volatility (uncertainty) for the thought experiment from this post.
- I realize the geometric return is what matters to my future wealth.
- Therefore I decide to structure my portfolio to maximize my geometric return through this method and hold 75% of my portfolio in stocks and 25% cash.
- All is well in the market, nothing new changes. I keep the portfolio the same.
- Suddenly a major financial company goes bankrupt. I believe the uncertainty in the market increases due to this event. Lets say the uncertainty (volatility) is now estimated at 25%.
- My portfolio needs to change because the properties of the market have changed. What is the right portfolio now?
- Based on the math of maximizing the geometric return, my new portfolio should be:
(10%-7%)/ 25%^2 = 48% Stocks
Wait a second. You’re telling me a 5% change in uncertainty (volatility) should cause the percentage portfolio to drop by 36%? Yes I am.
And what if Everyone did this?
Mathematically, the value of the risky asset in your portfolio should fall from 75% to 48%, a 36% drop in allocation to the risky asset. It is perfectly rational for your own value of stocks to fall by 36%. So, by extension, doesn’t this also mean it’s perfectly rational for the market value to fall by 36% as well? If everyone is being rational, then the market (everyone) should also act to reduce their exposure to stocks until the percentage of stocks in their portfolio also falls to 48%, correct?
Since there is a buyer of a stock for ever seller, the only way for the average investor (the market) to reduce their stock exposure by 36% is for the PRICE TO FALL by 36%5. Therefore, rationally, a 5% increase in expected volatility of the market, should cause the price of the market to fall by 36%. A geometric view of the markets explains why prices crash, efficiently.
Large Fluctuations in Price are PERFECTLY Rational.
This rational six step thought experiment was loosely based on the 2008 crash and Lehman Brothers failure. And I ask you again, did any action described seem irrational? Isn’t that exactly how an intelligent investor should invest? Mathematically, it is. If we are going to say markets are efficient, incorporate all information, and act rationally, the crash is the only logical outcome. Markets organize themselves around maximizing their geometric return. When viewed through a geometric lens the crash is actual proof the markets are rational and efficient.
In future posts, I’ll walk you through the various applications of the theory in history. The 1973-4 crash, the 1987 crash, the 1999 bubble, 2008 in more detail, the 2017-2018 bubble and winter crash, and the downturn from last December. All are perfectly rational, and easily explained6 when market behavior is viewed through the geometric return.
1-Lars Peter Hansen also won that year.
2- I believe this is true. Individual assets are mostly efficient and can’t be predicted. But there is a difference between the individual assets being efficient and a group of asset, like a market index, being efficient. This leads to the conclusion that you can’t beat the market by picking individual stocks, but you can beat the market by strategically combining the individual assets together to exploit the differences in the geometric and arithmetic returns.
3- Another way of stating this assumption is all assets have the same sharp ratio (risk adjusted return). Risk Parity funds usually make this assumption when they construct their portfolios.
4-I say mostly because for he assumed a constant discount rate, which was a mistake. There is no logical reason to believe the discount rate is constant. I mean, look at the history of the risk free rate. Its always moving, so why wouldn’t the discount rate be moving too?.
5- Its a strange concept that price movements in the market actually indicate changes in market portfolio compositions. The best explanation of this phenomenon was written by an anonymous non-investment manager like myself, Jesse Livermore, in “The Single Greatest Predictor of Future Stock Market Returns”.
6-Take the first derivative of the different input variables in the portfolio construction formula to see how variations in interest rates, volatility, and return ultimately change the portfolio construction, and by extension prices.
Wow, very profound. Surprised no one else has got this yet.
With your work and Ole Peters and crew, seems like economics has a lot of catching up to do with reality.
Amazing!
On the other hand, I can almost hear Dr. Shiller saying, “yes but even market variance does not change that quickly”. It really just needs to change a little.
Average return expectation held constant, volatility changing from %20 to %21 would explain a ten percent drop in the markets. So some development that pushes the consensus estimation error up by half a percent (e.g. stdev from %20 to %20.1) should result in a one percent market drop. Seems reasonable to me on a daily basis.
Correction to above: As the market drops, the expected return should go up, given the future value of dividend/payoff has not changed much but price has gotten cheap. Therefore half of the effect would be absorbed by the nominator. In that case, there’d be one to one correspondence between volatility/stdev and price level. %1 vol increase would result in %1 market drop. Still reasonable.
The 1% change in vol’s effect depends on it current state. A 1% change from 10 is different than a 1% change from 30%. You are correct that after the price falls the expected return could be greater, but what’s first, the price falling or the volatility increase?
Why is your risk free rate 7%? Is it because numbers become way off when it’s realistic 1-2% or there’s something behind that number?
It was just and example. Rates have been that high in the past (1987) and if you allow leverage in the example, as long as the risk free rate doesn’t match the geometric return, the effect is the same.
Great explanation, thanks for sharing all of this! Just one thing I didn’t understand: You write that “since there is a buyer of a stock for ever seller, the only way for the average investor (the market) to reduce their stock exposure by 36% is for the price to fall by 36%”. However, someone always has to hold all stocks, so the “average” investor cannot actually reduce their exposure. Given this, let’s say the dominant players in the market following this volatility adjustment reduce their exposure by 36%. This would certainly add selling pressure and lower prices, but the decline could be (much) different from 36% (?) Would love to hear your thoughts about that. Also, I think Mustafa is correct regarding the change in expected return, and his conclusion of 1% vol increase to result in 1% market drop seems both intuitive and close to reality.
Investors that don’t trade regularly, which is most of them, have thier exposure reduced for them by the price. If they are 50/50 and one of thier assets falls by 36%, then they are now 61/39 in thier holdings and they didn’t do anything. The price changed thier allocations for them.
The price change from 1% increase in vol is not linear. A change from 10% vol to 11% vol is not the same as a change from 40% vol to 41% because the vol need to be squared. There is a point though where 1% should equal 1%.
This nonlinearity between vol change and price change is why the market downturns often look like waterfalls and accelerate.
He is right that the numerator should be going up as the price falls. But that’s a more complicated topic.
One thing I’m not convinced of is that every rational investor targets maximizing their geometric return. Aren’t there many alternative optimizations such as limiting sequence of returns risk (in the case of family offices, endowments and early retirees)? Couldn’t multiple different investors with opposing goals prevent the market from having a single converging optimal allocation?
I agree most individuals don’t maximize their geometric return. We all have our own quirks and biases and goals. I’m saying in aggregate they should. Kind of the wisdom of the crowds idea.
Ok, as a fellow engineer with a mathematics background, let me try to follow your stock re-balance algorithm more generically.
% Stocks in Portfolio = (Asset Return – Risk Free Rate) / (Asset Variance)
So:
%of Asset_A = (Asset_A_averageReturn – Asset_B_averageReturn) / (Asset_A_volatility ^2)
with “Asset Variance” being (“volatility” ^2)
You lost me in that last part.
I feel like there is a general equation for % of an asset within a set of assets relative to the variance of the asset compared to “average” returns.
I am working on a similar strategy, and trying to understand your algorithm equation and the sources for the variables.
If your looking for two risky assets see this, and for an exact formula see the footnotes. https://breakingthemarket.com/optimal-portfolios-for-two-assets/
Is CAGR (which usually is quoted) arithmetic or geometric? I thought the way we usually provide average returns is you take the total stock return (X) over a period of n years and then raise X^(1/n) to get the “average” rate of return. Isn’t that a geometric rate?
CAGR is Geometric. Depends on the document as to what’s normally quoted, or focused on I think.