On June 30, 2011, an Empirical Finance post made some interesting points:

In 1926, if you invested in the market starting with $2,709 and if it made a constant return of 40% annually year in year out, you would have owned the entire U. S. equity market by 1989. If the return was lower, it would take longer to own the market. But, if the return was less than 31%, you might never be able to own the entire market. He then went on to conclude that “nobody can continuously outperform the market over long periods of time, because eventually (you) will own the market! And by definition, if you own the market, you can’t outperform the market.

This article was quoted by Cullen Roche on July 4 on Seeking Alpha, where he concluded:

It’s not impossible to beat the market, but it’s important for investors to maintain reasonable and rational expectations. Beating the market takes an extraordinary effort, intellect, approach and a little luck never hurt.

Here I am going to recast EF’s view with some simple algebraic formulas and, by injecting some realism into them, show why it is impossible to beat the market over long periods of time or to own the entire market. Let me start with an overly simplified expression for the expanding U. S. equity market. The U.S. market capitalization has been increasing steadily for the past century and this expansion may be expressed by a simple exponential (or logarithmic) formula, like

Log (An /A0) = n Log (1 + X)

A0 is the total capitalization of the U. S. equity market at some point in time (for example, about $27 billion in 1926); n is the number of years since 1926; and An is the market capitalization at the nth year. The important number in this formula is X --the annual growth rate of the equity market. I do not have access to CRSP data but I could infer from EF’s presentation that the U. S. market had been growing at an average annual rate of about 8.5%, or X= 0.085. By this formula, the current U. S. market capitalization is about $28 trillion.

This formula is exactly the same one commonly used to calculate the effect of compound interest, where X is the annual interest, n is the number of year, A0 is the starting deposit, and An the total deposit at the end of the nth year.

An investment fund with a constant annual return as posed by wes also performs like an instrument with compound interest, and as such can be described by a similar formula, as follows:

Log (Bn/B0) = n Log (1 + Y)

Here, B0 is the initial capitalization (e.g., $2,709 in the EF example) and Bn is the value of the fund at the end of the nth year after growing at a constant rate of Y per year.

Mathematically, as long as Y is greater than X, the investment fund will eventually own the whole market. At that time,

An = Bn

The question is when. You can get the answer by solving the three formulas for n:

n = Log [A0/B0]/ Log [(1+Y)/(1+X)]

The following chart shows the effect of the growth rate of fund Y on the year 1926+n, when the fund may own the whole market.

[Click all to enlarge]

Here I show the results of my calculation as well as some of EF’s results. They are remarkably close to each other.

**Modified Fund Growth**

Next comes the question: Can you really “own” the whole market? The answer is a resounding “no.” It is because in the real world, Y -- the annual return of a fund -- is affected by many factors including the size of the fund Bn relative to the total market capitalization An.

As the EF piece said: “When you own the entire market, the growth rate of your fund is that of the entire market. “ Therefore, at the start of the fund, say 1926, the growth rate of the fund could be Y. However, when you own the entire market, the growth rate of the fund is no longer Y but is equal to X, the growth rate of the market itself. We can devise many different relationships between the growth rate of the fund Yn at the nth year and the size of the fund relative to the size of the market of the same year, Bn /An. The simplest relationship is a linear one. This may not be realistic but is easy to handle mathematically, and can be expressed as follows.

Yn = X + (Y – X)(1 – Bn/An)

Here, Yn is the fund’s growth rate in the nth year. It can be seen that at the start of the fund, Bn/An is close to 0 and Yn=Y, and, when the fund size approaches that of the market, Bn/An is almost 1 and Yn=X.

With the value of Y varying from one year to another, fund size Bn in the nth year becomes

Bn = B0 (1 + Y1) (1 + Y2) (1 + Y3) … (1 + Yn)

Here, the value of Y is Y1 when n=1, Y2 when n=2, and so forth.

The last two formulas and the very first one in this article can be solved to see how the fund size Bn increases over the years. The result is shown in the figure below (the red curve). The initial fund growth rate used here is Y=40% per year. Shown also in the same figure are the total capitalization of the U. S. equity market as it increases over time (the blue line), and the fund size based on the second formula in this article with a constant annual return of Y (the green line).

You can see that the original formula predicts the fund size to zoom right through the total market capitalization by approximately 1989. The figure below shows the details around where the three lines come together.

At the beginning, the growth based on the modified formula is essentially the same as that of the original formula with a constant annual return (the green line). They are almost identical up to about 1980 when the fund size has grown to about 10% of the total market capitalization. The effect of fund size on the growth of the fund becomes significant after 1980. At this point, the fund growth based on the modified formula starts to bend beneath the total market capitalization; it then approaches that of the total market capitalization but it (the red line) would never catch up with the total market capitalization (the blue line).

The assumption above that the growth rate of a fund is linearly related to the fund size relative to the market is overly simplistic. In actuality, it appears that the growth rate is affected more severely by the fund size. In other words, the red curve in the last chart above should be, realistically, way below what it is shown there. If you are mathematically inclined, you can set up your own relationship between the growth rate of the fund and its size to arrive at a more realistic simulation.

**Summary**

A simple fund growth model that assumes a fund can grow at a constant rate year in and year out and can one day outgrow beyond the total market capitalization is not realistic. An injection of some realism into the growth rate of the fund, in the form of modulating the growth rate with the size of the fund relative to the total market could make the predicted fund growth somewhat more accurate. With this modulation the fund would never grow beyond the total market capitalization. In reality, when a fund size becomes large enough to be a significant portion of the total capitalization of the equity market, there would be many practical limitations restricting the growth of the fund. These limitations include the effectiveness of the management, the execution of stock trades in a gigantic volume, a market that may no longer function “normally” with one fund (or a small number of funds) that owns almost everything, to name a few. When all of these real world situations are figured into the estimation of the growth of a fund, you may find that the growth of a fund may peter out long before its size reaches 1/100th or even 1/1,000th of the market.

Exxon Mobil (NYSE:XOM) or Apple (NASDAQ:AAPL), the two largest corporations in the U.S., each has a market capitalization of about $400 billion which is about 1/30th of the current total market valuation. Suppose you have been very successful and one day you own the entire Exxon Mobil or Apple. Now, think of how you can trade that $400 billion worth of stock with stocks of other corporations so that you would make more money per year than sticking with the Exxon Mobil or Apple you already own?

A steady high return looks good on paper, but in reality there are many theoretical and practical limitations that would stop such a growth after only a few glorious years. Roche’s point that “it is not impossible to beat the market” is valid up to a point. I don’t want to put words in his mouth but I guess what he meant by maintaining “reasonable and rational expectations” implies that there is a limit to how long such a good performance can last.

**Disclosure: **I have no positions in any stocks mentioned, but may initiate a long position in XOM, AAPL over the next 72 hours.