The EMH vs MPT redux: Does Distribution Matter?

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). 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." Basically, a random variable is one that we do not know the outcome but we rely on probabilities to estimate it.

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 "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.

Theories of Investing

...or why every theory is a greater fool theory.

Imagine, dear reader, that you are currently thinking about purchasing a company's stock. What would your rationale for buying might be? Usual answers would be great potential for future growth, low P/E ratio, excellent management, stable growth over a period of time or a strong company record. Others may say that they have received a tip from a "good source" that the price might rise or that their charts tell them that there will be a "head-and-shoulders" recovery or their MACD combined with their 7- or 30-day average has indicated that they should do so. No matter what the approach might be, there really is just one simple reason people buy stock: they believe that the price will rise.

Some have longer horizons than others, some bet for a 1% return, some bet with high leverage; it does not really matter. Their purpose is simple and can be fully described in one word: profit. Why would any us bother to trade a stock if their was no potential of gaining something for it? There is no-one out there (I think!) who would be willing to buy a stock if he believed that the price would remain stable and he wouldn't be able to make a single cent of profit. Keeping his money in the bank, earning his meager but stable return would be a better alternative. The same holds if the stock-picker was certain he would face a loss in his investment.

I doubt that there is anyone who would disagree that people are into stocks for profits. There might also be other reasons for it, such us obsessive gambling behaviour, but this just an outcome. The only real reason everyone got involved with the stock market was profit. Note that before, my statement was "if one believed that there was no profit potential" he wouldn't bet in the market. Not if I (or any other) told John, Jack or Jill that the stock was going down and they had a different opinion; only if they believed it themselves, although someone might be able to influence them one way or another. Yet, any other person cannot trade with their money (unless they give it to him) which means that if he cannot influence them, they will continue to buy; at the same time I would prefer to sell. This is essentially what makes stock markets move up and down: differences of opinion. 

It does not really matter who is right and who is wrong when the market moves. You cannot even know who is. Just because the market has moved with you for a day does not make you right just as it does not make you wrong if the market moved against you for a day. What makes you right or wrong is whether the market moves with you or against you in the time-frame you have personally selected, no matter how big or small that might be. If your horizon is infinite, then you have less to worry about day-to-day movements, yet more to worry about longer-term ones. If it's a couple of days or hours then the opposite holds.

Then the question becomes: when do you sell? Is it just when the investment horizon hits or when the price hits a certain limit? Let's think about it. People tend to have some sort of price or return in their minds if their horizon isn't infinite (which usually isn't). Returning to the previous arguments, people always buy for a profit. But they want the profit to be substantial enough to cover their costs and gain some return on top of that. Thus, they have to at least know about the minimum price at which they will sell. How about the maximum then? If one uses more sophisticated techniques (value investing is an example that springs to mind) they can even calculate the exact price they expect the stock reach; after reaching that plateau, they will almost certainly sell it. If ones does not use any sophisticated technique then he will just sell it when his gut tells him to do so (this does not make the "gut" technique better or worse than the others).

In any case, deep down, there is only one reason for selling a security and that it that the investor believes that it has run its course, i.e. that it cannot provide him with any more profit at the time being. If an investor is asked, explanations offered may appear to be different but at the heart of every argument about the sale of a particular security is the simple notion that the investor does not expect it to provide any more profits at the time being. The stock may continue to rise or it may fall, yet this is of no meaning to the investor. He may purchase it again later if his forecast changes, but at the time of sale he is almost certain that the winning strike will no longer continue.*

Consequently, when the investor's beliefs are that the stock must be sold, it means that anyone suggesting that it will continue its course will not matter. At the time of selling, his belief dictates that doing anything else but selling would be irrational, given his knowledge and the interpretation of that knowledge at the time. Thus, for the investor, any other person acting against what he believes is nothing but a "fool". This might appear to be a rather broad generalization given that others might have better information or better understanding of the situation. Yet, the idea behind this is subjectivity. If I know something and keep on postponing my actions because I do not know what others know then I would be indefinitely postponing closing the position. This will mean that I will never capitalize on any gains which will mean that there is no point in ever opening the position. Under my subjective decision, I can only say that I am doing the best of what I can understand and have information of, which means that my decision is optimal based on what I know and understand at the time.

Thus, returning to the original argument, if an investor decides, using whatever technique or idea, that the position must be closed, any other way of acting would be foolish, including the person who buys the stocks he sells. Once again: the decision may be proven right or wrong in the future, but it does not matter. All that matter is that the investor believes it is the correct one at the time. Subsequently, when we are selling we are depending on finding a fool to buy the stock. We might have been foolish in buying the stock or we might have been clever, it does not matter. All that matters is that, when we decide to sell it, a greater fool than us will be willing to buy it. Maybe the buyer is not a fool and in the end proves to have better understanding or information. Yet, from our subjective point of view, the buyer is a greater fool since we are giving stock at a price we would consider ourselves a fool for buying.

Most critics will say that approaches like value investing are not like that. Let's think again: we have a firm with 1 million of profits selling for 9 million when we believe that the proper price would be 10 million. Thus, we buy at 9 and wait until it reaches 10 to sell. Would we have bought it at 10? No profit potential so I would assume the answer would be a clear no. Thus, for the value investor, anyone buying the stock at 10 or higher is a fool since he cannot possible get any profit from the transaction (unless information changes that is).

Summing up, it does not really matter why a person buys a stock; he wouldn't be doing it if there was no hope of profit. And if he does buy it then we assume he has to sell it at some point, whatever that point may be and no matter how many times the point is revised. When finally reaching it, the investor will sell; and from the investor's point of view, anyone else buying what he is selling, is a fool.

*It is straightforward to say that if the market turns against the investor and is forced to close his position then all of the above do not matter; leverage, willingness to hold the position despite worsening of the situation are what will define his ability to remain there.

Inside Information and Trading

The US has been the prominent pursuant of investors who, after having undisclosed information about a corporation try to exploit it by either buying or selling according it. The Wikipedia definition is "any individual who trades shares based on material non-public information in violation of some duty of trust", not limited to people who are part of the specific company but to all others who use such information to their benefit. The rationale for going after people who trade as such is simply that they are taking advantage of people with less information than their own.

The other side of the coin promotes that people with inside information should not be persecuted but instead, in the words of Milton Friedman "You want more insider trading, not less. You want to give the people most likely to have knowledge about deficiencies of the company an incentive to make the public aware of that." Others, argument that insider trading is a victimless act, meaning that those who were going to sell were going to sell anyhow, thus the fact that their counterpart has more information does not affect them. Nevertheless, this rather simplistic argument does not take into account that if a person is presented with more information he might change his mind from selling his current position. Thus, the seller is prevented from reaching a decision under full information. Obviously, those who argue that this kind of asymmetric information is legal in other practices such as real estate, are making the argument that if something bad is happening somewhere else then we should let it happen elsewhere even if we can control for it.

Friedman's argument brings to mind the theses of the Efficient Market Hypothesis (EMH) and the Noise Trader approach. The former, as already discussed, suggests that all available information (either past, present or future, public or private) has already been reflected in the stock price, thus making the market efficient and inappropriate for gaining from them. The latter states that in a stock market, there exist two types of agents: noise traders, who trade erratically and irrationally and rational ones. The noise traders, who even under erroneous beliefs can dominate the market and earn significantly higher returns than rational investors, are responsible for the stock market deviating from fundamental values and they are keeping arbitrageurs from correcting those values (the interested reader may have a look here, here and here for more details).

Returning to the argument, Friedman implicitly assumes that the markets are not efficient but if insiders with more information come to dominate the market then noise traders would diminish thus making markets fully efficient. If someone has information which is useful in estimating a company's future prospects (regardless of the way the estimation is done) then his knowledge would alter the company's stock price, even slightly, to his favour. No trader with new information is ever small, no matter how large the market is. The only difference is that if a trader has private information about a company and that information will never become public, then he stands no chance of ever winning in the stock market no matter how important his information is.

Although this is a rather strange thought, it is easy to understand: if I have information that a company faces trouble but no-one else has that information then I would sell the stock short in order to make a profit from its fall. However, this fall will not occur until the rest of the crowd finds out about it as well. The crowd has no incentive to sell the company if they do not have that information: for them, the company is doing as great as always and no news to the opposite have come to shutter that belief. In addition, the ends do not justify the means either: the public would be aware of that but the only way that insiders would ever disclose such information would be if they stand a chance of winning from it. Thus, they would not be doing it for the public's benefit and neither would the public be benefited from this new information. In fact, new information would increase the duration of the effect as it would start with the insiders' trading and finish when the stock market had assimilated the effect (to be fair it might decrease volatility in the market since prices would not have to adjust so rapidly; still that would depend on the state of the economy, the current trend in the market and so on).

Thus, any inside information trading is bound to disrupt the current market state; although readers may assume that this happens every time new information presents itself, I would like to remind you that this is not a collective action: the market shifts because of the action of some individuals who are in fact determining the price of a stock based on their beliefs and information. This is essentially the same as stock manipulation: the motive to earn a greater return than others, exploiting information others do not have (in the original case the public had no idea that someone was manipulating the stock while in this case they have no idea that new information exists).

The above, make the simple case that if private information was obtained by a person and used in such a manner as to secure a profit, it would only be illegal if that information was to reach the public at a point in the future; if the information was to remain private then no such case would hold. In fact, this also brings forth an issue of importance: it is not the information per se that matters but the timing of that information. As a recent experiment (whose link I was unfortunately unable to find) has shown, those investors who (on purpose) received information on the CPI publication minutes before the actual were published gained on average more than those who received it later. The information on the CPI would have been practically useless if they had received it 3 months prior to the actual publication as the stock market would have demolished their potential profits. In common parlance the investor would have been "too smart for his own good". (The same holds in experimental game theory as participants who understood the game before everyone else faced the problem of being right too soon, thus losing their bets.)

The main idea of all the above arguments can be put simply in just a sentence: insider trading is knowing information sooner than the public does but not too soon as volatility would destroy any potential for gain. In addition, you have to be certain that the public will find out that information and that it will subsequently act on it. Thus, instead on focusing on information which could potentially be used by insiders we should focus on the people who can get that information just minutes before everyone else does and act on it. Inside information is nothing more than timing and not new information.

The Real Money Multiplier

Conventional economic theory states that banks are bounded by their reserve requirements when it comes to creating new money. In the conventional way, if just one bank exists in the economy, the money multiplier roughly equals the inverse of the reserve ratio. For example, if banks are required to hold a 10% reserve ratio, then the value of the multiplier is roughly 10. Thus if money introduced in the economy via the government is M then with the banks' assistance they will reach 10M. (for a more detailed analysis and some examples, Wikipedia provides an excellent introduction)

Nevertheless, the idea that the money multiplier is a myth, in its conventional estimation at least, has been promoted by a number of academics and non-academics. Notably, a 2010 paper by the Washington Federal Reserve questions its existence while a recent article by Scott Fullwiler does an excellent work in explaining the endogenous money theory (which basically states that banks are not constrained by deposits but by regulatory capital requirements) as well as the interactions between the economy and monetary and fiscal policies. What has been left unanswered, however, has been whether a money multiplier exists in the endogenous money theory and whether this may be calculated up to any degree. It is this author's belief that both of the above questions can be solved and the answer is affirmative in both case.

Model:

Consider an economy with just one bank. The bank chooses to invest the money it receives in one of the following types of assets, depending on its risk appetite and the risk-return relationship. Each asset has its own specific risk weight which is as defined by the Basel II capital adequacy rules (in the case where the asset types could have more than just 1 risk-weight value, I have assumed for simplicity the lowest)

1. Government bonds (0%)

2. Loans to corporates or securities companies (20%)

3. Retail Loans (75%)

4. Secured residential property loans (35%)

5. Secured commercial real estate loans (100%)

6. Other loans (100%)

7. Cash (0%)

Now suppose that at time 0 we witness an influx of  €X of new money in the economy (again for simplicity we assume that no cash in circulation exist. The reader may find it easier to understand this as an economy where every transaction is carried out via a credit/debit card and subsequently when an account is debited another is credited). The money, (which as we have seen here is basically a transfer from the government to the people) is subsequently deposited in a bank. Then, it is up to the bank's discretion to select one of the 7 asset choices mentioned above. Through the analysis, the assumption is that the bank holds €0.1X in equity and the minimum capital requirement is 10%.

If, for example, the bank chooses to invest in option 7, then the currency available in the economy does not change at all. Using the broad definition of M2 (can be found here) if the money deposited in the bank stays in the bank as cash, then the money stock is exactly equal to €X. In this case, it is more than obvious that the money multiplier value is exactly 0.

Now suppose that the bank decides to invest fully in loans secured by residential property. Then, the first time these funds are lent out the bank has a risk weight of 0.35*€X. Thus, the bank's capital ratio will be:

Thus, the bank after producing the aforementioned loans, still has a margin of 18% which is to its best interest to use; which means that the bank will choose to lend again and again until the ratio equals 10%.

The value of the multiplier here is essentially based on the amount of money lent out. Thus, since the money supply in the economy has been increased by 285%, the money multiplier is 2.85, or exactly 1/0.35. In other words, if the initial money supply was €1bn, then the bank was able to raise the amount of money in the economy to €3.85bn with just €100m of equity. 

Money Multiplier Using Reserves

While in the previous example we have not used the notion of bank reserves at all, we will now show that these do not matter at all to the money multiplier (i.e. they do not affect the multiplier value). Suppose now that the bank has to save 10% of its existing deposits each time, or in other words it can only lend 90% of its existing deposits. Now, the capital requirement ratio for the first round of lending will become 

Continuing with our substitutions, we find that the multiplier value which will again make the regulatory capital equal 10% will be 2.85, yet it will be reach in an additional round of lending

Thus, any change in the reserves ratio does not have any affect whatsoever on the multiplier value. All it does is make the rounds of lending until the final value is reached more. In addition, what can also be inferred from the above analysis is that a Y% increase in equity will mean a similar Y% increase in the multiplier value. The following graph indicates the multiplier values for 0.1X equity (solid line) and the effect a 20% increase on the equity value (dashed line).

What is considered as a fact (regardless of the approach) is that approximately 90% of the money supply in the economy comes as a result of the banking sector. Nevertheless, as we have seen above, the multiplier appears to be very small to accommodate such an increase in the money supply. In fact, the banks in this model represent about 75% of the money supply under the 35% risk weight assumption; if the weight is at the low of 20% then the banks represent up to 83% of the supply of money. What the reader should note is that these values are much larger than real-life ones, as it is customary that a bank would choose a combination of the above assets and not just focus on a specific asset category. 

An issue which would further increase the amount of money in the economy would be the introduction of a new bank. If a new bank enters the market then the maximum value of the multiplier given the asset selection has been reached, then money supply may be increased, again up to the point where capital regulations allow it. If, for example, bank 2 has an equity value of 0.05X then the money supply in the economy can be increased by 1.425, making bank-created money supply reach 82% of the total money in the economy. Thus, after many more banks are considered in the economy, with each bank contributing to the total money supply (yet, not all banks having the same equity values) the bank-created money would reach approximately 90% (or even higher) of the total money stock. The same would occur if 2 (or more) banks were in the economy at time 0.

The above model presents what the author understands to be a much better representation of the money multiplier values. As we have seen, these values are heavily determined by the bank's equity and (most importantly) the investment decisions each bank chooses to make. Although we cannot pinpoint the exact value of the money multiplier, we can be sure that this has no relation to the reserve ratio. In addition, the number of banks in the economy plays an important role to the total money stock, although their individual multiplier values are constrained just by their equity.

The policy implications of the money multiplier will be duly discussed in a follow-up post.