Monday, January 25, 2010

The Rowe Conjecture and the Efficient Market Hypothesis

Nick Rowe of Toronto presented a conjecture concerning two variables as relates to the efficient market hypothesis, EMH. (Note: We have written this in some detail on the Telmarc web site for reference).

Rowe defined them as follows:

Recall that the EMH simply stated is the assumption that the market value of a stock is a reflection of all the information available regarding the stock. Now we know two things. First, that there is a herd mentality in the market that make many people to believe that the stock has or does not have value independent of whatever information is available. In fact some people may have information not available to others. Second, the herd mentality is driven by a percent of those who believe the EMH whereas when the herd develops the true existence of the EMH may actually disappear.

Thus the two variables, the one being the belief in the EMH and the second being the actual operation of the EMH are related. If the true existence of the EMH is say 100% then we have an efficient market and herd mentality is at a minimal because everyone distrusts the herd and does their own analysis, assuming equality to information and equality of access to trading.
We now develop a dynamic model based on the Rowe conjectures. We have changed these variables slightly from what Rowe had stated so that they are probabilities and that they are time dependent. Now Rowe sets the problem up as a supply and demand model wherein he disregards temporal dynamics and further looks at the people percent as the quantity and the probability of validity as the price variables.

We disregard the supply demand paradigm and look at them as interlinked temporal variables. Rowe has presented a compelling model of market behavior. We build upon it and do so in a dynamic fashion.

We assume a generalized model of the following type:

Now this is a generalized model which we will add some structure to. We will do so by applying a discrete time version and then go back to the continuous time version to analyze the results in a phase plane methodology.

Let us now write:

This is a linear model. We will expand this shortly but this is a good place to commence the analysis. This simple model states the following:

1. At some time k+1, the percent of people who now believe that the EMH is true is some multiple of the percent who believe before, and this may be greater or less than one, and some percent of the probability that it is actually in force.

2. The EMH is often true if those in the market are of the belief that it is not and that the market is not reflecting the true value and that they must do their own work to seek the truth.

3. The EMH is often false, namely its presence has a low probability, if there is a herd mentality. Namely the greater the belief in an EMH the smaller the probability that an EMH is true.

4. Market Bubbles occur when the herd approaches 100% and this also means that the truth that EMH exists is reduced to zero. When a market Bubble occurs the market then is subject to collapse, and the belief in the EMH drops precipitously.

5. Thus the model should reflect the dynamic as follows:

a. when the belief is low then the truth is high
b. when the belief is high, it grows the level of belief to a point and then collapses the level of belief
c. when the belief is high the truth is low
d. truth is dependent only upon the belief, and it is the belief that solely drives the Bubble

Thus we can create a model which can be written as follows. First for truth we have:

This is the Clock equation of Andronov et al and it describes an oscillatory system in the space of x, and dx/dt. Namely we have a phase space of the two variables, orthogonal to one another and there is a oscillatory behavior in the box as shown below. This can be simulated in discrete time by the following:

The typical solution for this may look as follows:

The above is a time plot of the two variable over time and they cycle back and forth. This shows the following:

1. There can be a model for the EMH that demonstrates the relationship between the two variables. It is not a model using a supply demand model.

2. The model demonstrates market cycles as expectations and reality cycle with each other.

3. The model can be tested against real data to ascertain its validity.

It would be interesting to see how this compares with reality.