Early investigators of stock prices perceived that stock prices seemed to be as likely to go up as to go down. This led to the formulation of random walk models of stock prices. This means that if P_t represents the stock price at time t then

P_t+1 = P_t + u

If this model is valid then the expected value of P_t+1, given the information that is available at time t, is equal to the price at time t, P_t.

This same relationship prevails for not just one period ahead but for any number of time periods ahead. That is to say, the expected value of the the price of the stock at time t+s, given the information that is available at time t, is just the price of the stock at time t; i.e.,

E{P_t+s:_t} = P_t.

The forecasts of future stock prices based upon the Additive Random Walk model are easy and available to everyone. Of course, such forecasts are just the expected values of the future prices. We would also like to know the variability of the future prices, the margin of error or confidence range of the forecasts. The variability of the forecast can be expressed in terms of the standard deviation. Thus the standard deviation of the forecast one period ahead is just the standard deviation of the random variable u, . More generally

P_t+s = P_t + u_t+1 + ... + u_t+s
so
Var(P_t+s) = P_t + Var(u_t+1) + ... + Var(u_t+s)

Var(P_t+s) = sV
and
C = s^1/2

If the random variable have a normal, bell-shaped distribution then the confidence range such that there is a 95 percent probability that the actual future value will fall into that range is just the range

the expected value - 1.96 to
the expected value + 1.96

The Multiplicative Random Walk Model of Stock Prices

Although the Additive Random Walk model passed all of the statistical tests applied to it there was a troublesome element. If the change in stock price is a normal variable then according to the Additive Random Walk model there is a nonzero probability that the future stock price will be negative, whereas that is an impossibility. Analyst tried to eliminate this possibility by means of a reformulation of the Random Walk model. Consider first the notion of the rate of return for holding a share of stock. The net gain from holding a stock during some period of time is the sum of the dividend paid during that period of time and the net price change during that period. If this net gain is divided by the price of the stock at the beginning of the period the result is the rate of return. In symbols this is:

r_t+1 = (D_t+1 + (P_t+1 - P_t)/P_t

The above definition of the rate of return may be used to find the future stock price associated with each rate of return; i.e.,

P_t+1 = (1+r)P_t - D_t+1.

E{P_t+1:_t} =
(1+E{r:_t})P_t - E{D_t+1:_t}.

E{P_t+s:_t} =
(1+E{r:_t})^sP_t
- {Future value of the
expected dividends over the period from t to t+s}
where this future value is
E{D_t+1:_t}(1+r_f)^s-1 +
E{D_t+2:_t}(1+r_f)^s-2 +
E{D_t+s:_t}

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