Fallacies in the Arguments of Efficient Market Critics: Part Two

By fundamentally analyzing a company one can determine the intrinsic value and see if the market has mispriced the security.

This statement is the bread and butter of fundamental analysts, and is also blatantly incorrect from what the majority of research has came to conclude.  It has been proposed with extremely strong evidence that although profitability is mean reverting, and therefore somewhat predictable, things such as revenue or its growth, are not.  Popular methods such as the forecasting of cash flows are usually involve some kind of simple extrapolation, they are highly susceptible to subjectivity, and usually revolve around some sort of mean reverting assumption that has been shown to be without basis.  Knowing that the foundation of fundamental analysis for the purpose of stock selection is the ability to correctly forecast what boils down to be random numbers with some degree of consistency, we can unequivocally say that the process is frivolous.

How have investor greats like Warren Buffett and Peter Lynch beaten the market year after year; don’t you say this is impossible?

That’s a valid question, but easily answered.  Out of any large number of investors exposed to a small sub-group of overall market assets, some, albeit an extremely tiny percentage, will perform above the market with consistency.  How many more can you name?  Two, maybe three who have performed on a similar scale to those listed above?  Now think of how many investors there are total.  That’s an incredibly tiny percentage, as would be expected.

Let me ask you this to counter the question.  Why haven’t those who duplicated the methods of those listed performed the same in an identical time period?

What about behavioral finance?

Again, this is a valid question.  Behavioral finance is still a new and rather undeveloped field.  Its research has yet to fully combat the efficient market theory.  Until it can provide the burgeoning amount of high quality empirical evidence like EMH has, it can never be put on the same level.

Estimating Non-Linear Functions with OLS Regression

In empirical research, you’ll often come across variables that are not related in a linear way.  There are methods to estimate non-linear functions, but many involve things far more complicated than OLS regression.  Luckily we have a method of estimating non-linear functions using OLS!

A prime example of a non-linear function in economics is the Cobb-Douglas production function (CDPF).  Below you’ll see the CDPF and subsequently the resulting form once when take the log of both sides.

Notice how the log-log form now highly resembles our OLS regression equation?  Below is another image showing how these results can be interpreted in our OLS regression.

Sometimes this way of estimating non-linear functions will not work very well.  There are other methods to estimate non-linear functions, but this OLS-based method should be sufficient to describe most non-linear relationships.

(More information on the Cobb-Douglas Production Function can be found here and some good working examples not related to regression can be found here)

Read up! Previous Research You’ll Need to Know

Here’s a few papers hosted on the SSRN network you’ll find helpful for background purposes while engaging in financial research.  The links are to the abstracts, but you can usually download the paper from the sources further down the page.

The Equity Premium

Forecasting Profitability and Earnings

The Adjustment of Stock Prices to New Information

Profitability, Growth, and Average Returns

SSRN is a massive collection of academic working papers.  It’s an excellent source of information if you don’t have journal subscriptions, as the working-paper of the journal article is usually still available and viewed for free!  Registration has many benefits in terms of what you can gain access to and it’s completely free!

Identifying and Correcting for Heteroscedasticity

Heteroscedasticity is where regression results produce error terms that are of significantly varying degrees across different observations.  Another way of saying this is basically that as your independent variable gets larger, so does the distribution of error terms around it.  The picture below may help significantly if your having trouble what we’re describing.  (Sometimes it’s also spelled Heteroskedasticity, from what I can tell, either way is correct)

Do you see how the terms are more “spread out” on the right hand side of the chart when compared to the left hand side of the chart?  That’s heteroscedasticity!

There is a more formal, and more correct way to detect heteroscedasticity than just looking at the chart though.  It’s called the Goldfeld-Quandt Test.

In the Goldfeld-Quandt Test we’ll first need to sort the variable we think is causing this heteroscedasticity in order of it’s magnitude.  Divide this sample size, T, into two equal sections (T/2) and take the variance of each independently so that we get two measures (Sigma squared one and sigma squared two, representing each section of T/2 respectively).  Now the higher variance divided by our lesser variance will become our test-statistic, and well test it against an F-distribution with a comparative F-statistic at some specificied level of significance.  The statistic we calculate from our sample we’ll call GQ.

The hypothesis we’re testing is that the variance is equal to the variance multiplied by the x at time t.  Comparatively, our null hypothesis will be that our variance is equal to our variance.  A rejection of the null means we have heteroscedasticity.

The F we’re testing GQ against (The F sub C) has (T/2 – K) degrees of freedom in both its numerator and its denominator at the significance level of alpha.  If GQ is larger than our F statistic of (T/2 – K, T/2 – K) degrees of freedom, we can reject the null else, we accept it.

It’s very easy to correct for heteroscedasticity though.  Just divide through the square root of x at time t for each term in your sample at each time t. The other option is much more painstaking.  It requires one to perform an entirely new least-squares estimate to derive new coefficients using the weighted least-squares method.  Blah, who wants to do that when we can correct for it dividing through the square of the independent variable.

You may be asking why do I need to care about heteroscedasticity anyways?  Well, when doing your research, you’ll be questioned about it as our coefficients are still linear unbiased estimators, but they now they are not however the best linear unbiased estimators (BLUE)!  Also hypothesis testing and confidence intervals based off the standard errors will not be correct as their assumptions are violated.  Correcting for heteroscedasticity alleviates this violation and returns the residuals back to a normal distribution satisfying all the assumptions of our original OLS model.  Just remember, with our OLS assumptions violated, our results are meaningless and invalid.

Fallacies in the Arguments of Efficient Market Critics: Part One

I will attempt to counter the flawed arguments and wrongly placed assumptions often used in attempt to invalidate the presence of market efficiency.  Rarely is it argued that the financial markets are perfectly efficient, but the overwhelming consensus of research has provided tremendous amounts of significant evidence that more than hint at its existence.

The out-performance of small-cap stocks and value stocks throughout time show that market isn’t efficient.

The reason some small cap equities, and distressed equities (a.k.a. “value” stocks) produce returns of much larger magnitude than an overall market index is due to their higher than average risk.  Equity risk premia vary throughout economic cycles and move randomly over time depending on the aggregate investors willingness to take on risk.  Risky firms, such as firms who are small and firms who are in financial distress, have a much higher risk premia.  Due to this, when payoff is realized, the magnitude of the return is inherently larger compensating the investor for the amount of risk taken.  These larger returns realized from small cap and distressed equities is merely a reflection of risk!

With the presence of irrational market participants (“technical” analysts and such) trading stocks based on non-valuation related information, there are going to be inefficiently priced stocks.

The statement above is one that grossly misinterprets the theory.  The assumption is that the aggregate investor, the market as a whole, is rational.  Stated nowhere is the assumption that every market participant be rational, but rather that the end result, the culmination of all market participants decisions be rational.

It isn’t a question of the number of categorized rational agents, but rather the result coming from the aggregate agent.  Rationality here should be thought of as a variable which is continuous over some ranked interval.  Expanding the thought process to degrees of rationality rather than a binary view allows us to look at market rationality more correctly.  It would make sense that the higher ranked degrees of rational agents also participate in the markets with a larger amounts of capital and that the amount of capital controlled correlates exponentially with rationality as rationality approaches infinity.  With this known, we see that the actions of the financial markets are overwhelmingly dominated by rational decisions.

Looking at a chart of the S&P 500 since 1920 till the present, it’s obvious that it differs from a random walk.

Thanks for pointing out the obvious!  The S&P 500 is a proxy for the level of economic well-being of firms as a whole reflected through their valuation.  The occurrence and content of information relevant to a firm is indeed random, but is the information influenced randomly?  The news that affects the fundamentals of firms are aggregate factors that are based on decisions intended to maximize our economic standard of living and the value of a firm through policies implemented by government and decisions made by management of the firm, respectively.

Acute readers have already made the connection, but some may be asking what this has to do with market efficiency?  The obvious difference from a random walk the S&P 500 portrays is due to the decisions implemented by government and aggregate management of firms to foster an environment of economic growth which positively effects the value of firms on an aggregate level.  The upward path of the S&P 500 index actually shows market efficiency rather than refuting it.  As our economic vitality increased, henceforth so did the value of firms as those goals were achieved, compensating the investor by the amount of the risk premium, and the index reflects that.

Here in the US, looking at the 20th century, we succeeded in reaching these goals.  There’s no guarantee this will happen in the future.  The news is random because it is unpredictable, not because it is randomly generated.  The price of a stock today reflects its value today, and changes in the price today are independent of historical news and prices.

The Fama-French Three Factor Model

Research of stock prices, fund performance, and other related topics has been long studied by academics.  In 1993 Eugene Fama and Kenneth French developed a three factor model that has significant explanatory power when describing returns.  These three factors are the market premium, HML, & SMB.

This post is not meant to explain the model, that would be an overburdening task.  Rather this post is meant to describe when to use the model.  I also include a few tips on how to refine the model to your needs if desired.

If you’re looking for a comprehensive explanation of the model itself, please see this paper hosted at IFA.

When looking at how to describe returns, this is the model you want to use.  Individual stocks, mutual funds, and other miscellaneous portfolios that produce returns are appropriate dependent variables for this model.

Additional factors that may influence returns in a particular portfolio are appropriate to be added if they do not get captured by the original three factors.  An example would be the exposure to credit risk and that can be measured as the change in the corporate credit spread or some other similar proxy.

The market premium can be manipulated directly as well if necessary.  If a fund has its choices limited to a narrow universe of stocks, a special index should be used.  An example would be if we were measuring the returns of a large cap growth oriented mutual fund.  We would not use the S&P 500 index as our proxy for market returns.  Instead we would use the Russell Large Cap Growth Index.  This would provide a better measure for the return in the universe of stocks this fund is limited to.

Again if you’re looking for a great explanation of the model, please see this wonderful publication hosted by the IFA published by the Tuck School of Business.

Choosing a Functional Form

So you have your data, independents, and some particular dependent variable.  You run your regression and bam, you get some pretty bad numbers.  Your r-squared it low, your ANOVA F test is not significant, and your variables have some pretty low t-statistics.  It all made sense, the variables have viable economic impact on your dependent variable, what went wrong?

You probably have the wrong functional form!  Data can be mysterious.  Perfectly descriptive data can hide its usefulness unless you have chosen the correct functional form.  The mathematical notation is a little beyond this script-editors capabilities, but I did find this nice pdf file on another website that may be of use suggesting some different forms.  You can find this handy document here.

One very popular form is the log-log form.  Given that your data allows for this form to be applied to your model (remember the domain of natural log!) .  It can simply be put:

Your original model:  Y = A + BX + CZ; given Y is your dependent variable, A is your intercept, and B & C are the slope coefficients to X and Z, respectively.

The resulting Log-Log Model:  Ln(Y) = A + B*Ln(X) + C*Ln(Z)

Once you have manipulated the data, you must re-run the regression using this manipulated data to get the new model intercept and coefficients.

Keep in mind through, all this manipulation of your independent variables will dramatically much change the interpretation of your regression coefficients.

The following document that shows the use of functional forms as the relate specifically to economic cost-profit functions.  It can be found here on the Cornell website.

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