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May 18

## Efficient market hypothesis is subject to interpretation

The formulation on Investopedia seems to me the best:

The efficient market hypothesis (EMH) is an investment theory that states it is impossible to "beat the market" because stock market efficiency causes existing share prices to always incorporate and reflect all relevant information. According to the EMH, stocks always trade at their fair value on stock exchanges, making it impossible for investors to either purchase undervalued stocks or sell stocks for inflated prices. As such, it should be impossible to outperform the overall market through expert stock selection or market timing, and the only way an investor can possibly obtain higher returns is by purchasing riskier investments.

This is not Math, and the EMH interpretation is subjective. My purpose is not to discuss the advantages and drawbacks of various versions of the EMH but indicate some errors students make on exams.

## Best(?) way to answer questions related to EMH

Since there is a lot of talking, the best is to use the appropriate key words.

Start with "The EMH states that it is impossible to make economic profit".

Then explain why: The stock market is efficient in the sense that stocks trade at their fair value, so that undervalued or overvalued stocks don't exist.

Then specify that "to obtain economic profits, from the revenues we subtract opportunity (hidden) costs, in addition to direct costs, such as transaction fees". What on the surface seems to be a profitable activity may in fact be balancing at break-even.

Next is to address the specification by Malkiel that the EMH depends on the information set $\Omega_t$ available at time $t$.

Weak form of EMH. The information set $\Omega_t^1$ contains only historical values of asset prices, dividends (and possibly volume) up until time $t$. This is basically what an investor sees on a stock price chart. Many students say "historical information" but fail to mention that it is about prices of financial assets. The birthdays of celebrities are also historical information but they are not in this info set.

Semi-strong form of EMH. The info set $\Omega_t^2$ is all publicly available information. Some students don't realize that it includes $\Omega_t^1$. The risk-free rate is in $\Omega_t^2$ but not in $\Omega_t^1$ because 1) it is publicly known and 2) it is not traded (it is fixed by the central bank for extended periods of time).

Strong form of EMH. The info set $\Omega_t^3$ includes all publicly available info plus private company information. Firstly, this info set includes the previous two: $\Omega_t^1\subset\Omega_t^2\subset\Omega_t^3$. Secondly, whether a certain piece of information belongs to $\Omega_t^2$ or $\Omega_t^3$ depends on time. For example, the number of shares Warren Buffett purchased today of the stock is in $\Omega_t^3$ but over time it becomes a part of $\Omega_t^2$ because large holdings must be reported within 45 days of the end of a calendar quarter. If there are nuances like this you have to explain them.

## Implications for time series analysis

Conditional expectation is a relatively complex mathematical construct. The simplest definition is accessible to basic statistics students. The mid-level definition in case of conditioning on a set of positive probability already raises questions about practical calculation. The most general definition is based on Radon-Nikodym  derivatives. Moreover, nobody knows exactly any of those $\Omega_t$. So how do you apply time series models which depend so heavily on conditioning? The answer is simple: since by the EMH the stock price "reflects all relevant information", that price is already conditioned on that information, and you don't need to worry about theoretical complexities of conditioning in applications.

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