One of the posts that have stirred most comments and provoked interesting discussions on this blog is the one on the contradiction between the EMH and the MPT. In a nutshell, in it I claim that if the EMH holds and prices are random then no prediction can be made about the future of a portfolio based on previous prices. A common criticism I usually get is that the EMH does not say that prices are fully random but that they follow a distribution which (in the long run) does not allow for excess returns since prices are anticipated perfectly and thus are random (as per Samuelson 1965).

The first question we then have to answer is what randomness is. The Wikipedia definition of a random variable is

*"a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).*Basically, a random variable is one that we do not know the outcome but we rely on probabilities to estimate it.^{}As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability."
The most common and widely used distribution in statistics is the normal. Basically the normal encompasses 99.7% of all possible outcomes if one calculates its mean plus, minus 3 variances. Probability-wise, it says that the most efficient estimate would be the mean, which, for example has an 8% chance of occurring if the game is toss a coin for 100 repetitions. Thus, proponents may add, although we cannot forecast the outcome with accuracy it is quite possible that we can forecast the long-term distribution of returns.

Let's think about this for a second. When we say that we know the distribution it means that we can be certain that any outcome will be part of it. The probability of the range of the outcome may be large or small but still it would have to be part of our distribution. Such a distribution is not hard to create. We know for example that a stock cannot fall to less than 0 (i.e. 100%) in any time-frame thus we may use that as the lower limit and build a higher limit from there. We would not even care about the skewness of the distribution as a 200% increase is possible but a 200% decrease is not. Thus, such a distribution, either based on past data or constructed by the analyst is not a difficult task. Up to this point the MPT and the EMH appear quite complementary: the first defines that risk-adjusted returns may be obtained by utilizing past price data and the latter confirms that no excess returns may occur from that.

Yet, the magic word here is the long-term. What is the long-term? Although there is no standard definition I would say that the short-term is the time when we do not have enough data to "significantly affect" our distribution. If I have 200 data points then 10 points make a difference; if I have 200,000 points then 10 points do not matter at all. Thus, in the latter we have a short-run. The MPT basically says nothing about short-run dynamics. It doesn't know what will happen in the short run since the assumption of a normal distribution does not allow it to know; remember that a distribution is useful in predicting large samples but terrible in single observations. In other words, the probability of finding the exact return is zero.

Thus, in the short run the MPT allows the model, via its statistical distribution not to abide by its rules. In fact, it does not care if it abides by its rules. All it cares is that in the long-run, whatever that may be, the average return is the one predicted by the data. Yet, this means that in the short-run, for example a daily return or an hourly return may be higher than the mean. In other words, excess returns may be gained from actions. This is not against statistics as returns will move in the distribution, both up and down. Yet, in the long-term they should not be in excess of the expected return.

So far, I have not moved away from anything that any reader with a basic understanding of statistics does not know and neither have I proposed anything that the student of the MPT or the EMH is unaware of. Yet the consequences of accepting that in the short-run an investor could earn excess returns is something odd in the EMH literature. The EMH states that

*"one cannot consistently achieve returns in excess of average market returns on a risk-adjusted basis"*which is rather in disagreement with what we have suggested before. Basically, the MPT does not really deny the possibility of excess returns. It merely wishes to either maximize return given risk or minimize risk given return. This does not, in any sense, mean that excess returns are "forbidden" or impossible. They may be hard to be achieved or they may not be achieved by many but they are allowed to happen.
In contrast the EMH does not accept that any such possibility exists (even in its weak form the long-run consistency of excess returns is not allowed). In the long-run, nobody will outperform the market. No matter how short or long the definition is, people are not "allowed" to win excess returns. As already said, not only does the MPT allow for such deviations and it does not even care about them. If excess returns can consistently be earned, then the EMH does not hold as it does not allow for any such events. If they cannot be consistently earned, then the EMH holds and the risk-adjusted returns of MPT are the norm. Yet, the MPT allows for excess returns as we have already seen but the EMH does not. Here, if the EMH holds, the MPT will not.

As we already said, a random variable is one whose values are not known but they are picks form a distribution. Is this distribution known in the stock market? Perhaps the analyst could make up one as stated before. Yet, as values are added to the distribution, the probability of each band of outcomes is reduced. If I take 260 observations of daily returns from the market (i.e. a year) then it will probably not fit my estimated distribution (it might but it's highly unlikely). But if the MPT holds, then very little risk (i.e. very little variance) would give me back something close to my expected return. This expected return is basically the mean of a distribution comprised of past observations. Yet, if I have a great approximation of the return, I have a great approximation of the price. But if I have a great approximation of the price a year ago, then the price was not random for me. As we have seen, the probability of getting a correct price under the EMH is 8% for a coin toss of 100 times. If the price is very close to the estimation then my probability was much higher than 0.08, probably 0.80 or even more. Then, unless we will have to change the definition of a random variable to "a value which has a very large probability of happening but we are not 100% certain of" then the EMH, even in distribution, does not occur. Even in risk-adjusted returns, if one can forecast will accuracy, then prices are not "random variables"Thus, if the MPT holds, then the EMH will not hold.

As Paul Samuelson commented, the EMH is

As we already said, a random variable is one whose values are not known but they are picks form a distribution. Is this distribution known in the stock market? Perhaps the analyst could make up one as stated before. Yet, as values are added to the distribution, the probability of each band of outcomes is reduced. If I take 260 observations of daily returns from the market (i.e. a year) then it will probably not fit my estimated distribution (it might but it's highly unlikely). But if the MPT holds, then very little risk (i.e. very little variance) would give me back something close to my expected return. This expected return is basically the mean of a distribution comprised of past observations. Yet, if I have a great approximation of the return, I have a great approximation of the price. But if I have a great approximation of the price a year ago, then the price was not random for me. As we have seen, the probability of getting a correct price under the EMH is 8% for a coin toss of 100 times. If the price is very close to the estimation then my probability was much higher than 0.08, probably 0.80 or even more. Then, unless we will have to change the definition of a random variable to "a value which has a very large probability of happening but we are not 100% certain of" then the EMH, even in distribution, does not occur. Even in risk-adjusted returns, if one can forecast will accuracy, then prices are not "random variables"Thus, if the MPT holds, then the EMH will not hold.

As Paul Samuelson commented, the EMH is

*"micro efficient but macro inefficient"*meaning that it is better applied in specific stocks and not in the aggregate market. In essence, sometimes, prices move randomly (Robert Shiller has already shown that in much of his work) but this holds more of a single daily (or even of higher frequency) return and yet is more often than not focused on some specific piece of information. Prices do not move without news, yet the effect these news have is not ex ante known to any participant. Although I have serious doubts whether markets are efficient, I agree that they reflect information, or better perceived information. Yet, I also think that forecasts can and are made. The EMH and the MPT is their original forms are not complementary; in fact they are quite contradictory. This does not make one correct the other erroneous. If anything, it should just make us understand that no theory holds for everything.
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