The Efficient Market Hypothesis is the proposition that financial security markets are informationally efficient; i.e., the current price incorporates all information known currently concerning factors in the future which may affect the price of a stock. The origins of the Efficient Market Hypothesis (EMH) can be traced to Louis Bachelier in terms of theory and to Alfred Cowles' discovery, in practice, that apparently no one can predict the changes in stock prices. The preliminary version of the observation noted that the price of a stock over any given period was just as likely to go up as to go down. A more refined version of the observation led to the random walk model of stock prices that said the net change in a stock price is, on average, equal to zero. In mathematical terms:

if Δp = p_t+1-p_t
then
E{Δp} = 0,
 

where E{} stands for "expected value of." To state this idea more precisely it should be expressed as:

E{Δp:I_t} = 0,
 

where

E{___ :I_t}
stands for
 
"the expected value of ____,
given the information I
that is available at time t".
 

An immediate implication of the random walk model is that:

E{p_t+1:I_t} = p_t
or, more generally,
E{p_t+s:I_t} = p_t.
 

In words, the expected value of the price of the stock s time periods in the future given the information that is available at time t is just the price at time t.

The random walk model can be stated in more complete form as follows:

p_t+1 = p_t + u_t,
 

where u_t is a random variable with normal distribution and expected value of zero. Further u_t is uncorrelated with any previous values, u_t-1, u_t-2, ...; i.e.,

E{u_tu_t-k} = 0 for k=1,2,....
 

The above version of the random walk model was the accepted version for a long time, but analysts worried about a theoretical flaw in its formulation. If p_t+1 = p_t + u_t and u_t takes on a negative value that is larger in magnitude than p_t the model implies that the market price of a stock could be negative. But that is impossible. (This version of the model can be called the additive model.)

The problem was resolved by treating the rate of return on holding the stock one time period as the random variable. The rate of return is given by:

r_t = (d_t + (p_t+1-p_t)/p_t
 

where d_t is the dividend paid in period t. The future price can then be expressed as:

p_t = (1+r_t)p_t - d_t
 

Consequently, the expected price is:

E{p_t} = (1+E{r_t})p_t - E{d_t}
 

Let E{r_t}=r and E{d_t}=0 (since dividends are not paid in most time periods). Then

E{p_t+1} = (1+r)p_t
and
E{p_t+s} = (1+r)^sp_t
 

This is a random walk with drift. (This can be called the multiplicative model.)

The Efficient Market Hypothesis says, in effect, that because the market is informationally efficient there is a simple forecast of the future price of a stock based upon its current price. Furthermore the EMH implies there is no better forecast of price possible.

Another implication of the EMH is that the expected net present value of any transaction in financial markets, such as buying a stock, is zero.

E{NPV} =
E{(p_t+s/(1+r)^s - p_t):I_t} = 0