A market is efficient with respect to information when price variations cannot be forecast and when prices completely incorporate the expectations and the available information that investors are supposed to have. Fama (1970) describes this concept as "efficient", assuming that in a market prices always and completely reflect the available information. The satisfactory conditions to define a market as "efficient" are very stringent:
- the existence of a majority of investors who act in a rational way (attempting to attain the maximum gain) and not connected between themselves (they are price takers);
- the set of information (I) available for agents without cost in various moments of time t;
- the investors have homogenous expectations, they have the same opinions about the effect that information could have on the present and expected asset price;
- there are no transaction costs or taxes.
Maikiel (1992) states that efficiency based on a certain set of information I implies the impossibility to gain profits through trading based on the same information. Basically, if prices do not vary after the divulgation of specific information, the market can be considered efficient with respect to the aforesaid information. Naturally, considering the previous affirmations, a crucial point of modern economic theory emerges; that is, the question of information asymmetry.
The classical rate structure in terms of market efficiency requires the following division:
- weak efficiency: the set of information only considers the prices’ "past history";
- strong efficiency: the set of information includes all the information publically available (for instance: gain forecasts given by analysts, periodical communications compulsory for the specific firm, and obviously the prices’ "past history", etc.);
- semi-strong efficiency: represents a level of efficiency higher than the previous and implies that the set of information I contains all the information which every market investor knows. Obviously, this last context is very unlikely in real financial markets.
Returning to the previously introduced concept of the impossibility to forecast asset returns, their "random nature" deserves some explanation in order to avoid confusion.
Indeed, individuals generally believe that in an efficient context, the asset price has to follow a "smooth" route and not a random one. The assumption of "randomness" entails the improbable supposition of a small increasing (or decreasing) set of variations of the asset price.
When the price rises, this increase will occur all at once and not through a series of small variations. In this manner, it is difficult to get gains since the price changes quickly once the information is available for market participants.
The result is that the price of subsequent transactions is a random process and not a continuous one.
Therefore, in contrast with the general opinion, efficiency does not require the market price to systematically coincide with the true value, but implies unbiased error in the market-price determination.
To sum up, prices could increase or decrease but these deviations from the true value should be random, that is, the stock over/under value probability is assumed as identical each time (future values are not predictable). Furthermore, these variables need to be uncorrelated with any other variable. In consequence of the previous reasoning where market-price deviations from the true value are random, no investors can pick under/over-valued assets by adopting well known investment strategies.
It is also possible that several markets may be efficient while others are not, or that a specific market is efficient with respect to a certain group of investors and not with respect to another one: this is the direct consequence of differences in rates and transaction costs which can give advantages to particular categories while hindering others.
The market efficient hypothesis could be formalised as follows:
a.
b. ,
where:
is the set of information available in ; this set is decisive in determining the stock quotation at time and the expected quotation at time t;
is the set of information used by investors;
P_tis the stock price at time t;
and are the conditional probability distribution functions of the stock quotation at time t.
The first of the previous equations (hypothesis a) states that all the information available at time is used immediately by the market. Therefore, the second one states that the function is formulated by investors considering the set of information available at . The same equation implies the following:
. Briefly, it is possible to state that the efficient market hypothesis suggests the following price formation process:
1. The set , liberally available, is used by investors () to forecast the distribution probability and the expected value
of the price at time t ;
2. Once the expectations have been established, the market, on the basis of a specific equilibrium model (for instance the Security Market Line notation), determines the expected return equilibrium .
3. The expected return of a period can be defined in the following way:

Therefore, the price will be:

4. Subsequently, the asset price at time should be the following

Where is not coherent with the expected price and the expected return, the quotation at time could vary. In order to clarify where:
, considering and the price at time would rapidly increase in the presence of instantaneous market activity: if such a situation should occur (i.e. ), an effective return which coincides with that of equilibrium will be registered (this return comprehends the information acquiring costs). With respect to the equality , this suggests the exact and complete incorporation of the set of information in the price; however, in order for this phenomenon to occur, that set of information should be reflected in and . According to this situation, it is not possible to gain extra-returns by simultaneously carrying out more than one transaction at time t-1 based on the knowledge of the set. The correction of to this equilibrium value occurs quickly as long as the set of information that is available to all the investors at the same instant is correctly interpreted by the operators.

Bibliography
FAMA  E. F., 1970, Efficient Capital Markets: A Review of Theory and Empirical Work, in Journal of Finance, Vol. 25(2), pp. 383-417;
FAMA  E. F., 1991, Efficient Capital market: II, in Journal of Finance, Vol. 46, No. 5 (dicembre), pp. 1575-1617;
GROSSMAN  S. J., 1976, On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information, in Journal of Finance, Vol. 31 (maggio), pp. 573-585.
GROSSMAN  S. J. and STIGLITZ J. E., 1980, On the Impossibility of Informationally Efficient Markets, in American Economic Review, Jun 1980, Vol. 70(3), pp. 393-408.

Editor: Rocco CICIRETTI
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