The long-run time-series properties of equity prices (among other variables) is, of course, of particular interest to financial analysts. There is a strong interest in determining whether stock prices can be characterised as random walk or mean reverting processes because this has an important effect on an asset's value. A stock price follows a mean reverting process if it has a tendency to return to its trend path over time, which means that investors may be able to forecast future returns better by using information on past returns to determine the level of reversion to the long-term trend path. A random walk has no memory, which means that any shock to a stock price following a random walk process is permanent and there is no tendency for the price level to return to a trend path over time. The random walk property also implies that the volatility of stock price can grow without bound in the long run: increased volatility lowers a stock's value, so a reduction in volatility due to mean reversion would increase a stock's value.

The most popular tests for the random walk hypothesis are the augmented Dickey and Fuller (1979, 1981) tests and the Phillips and Perron (1988) tests. An important drawback of these tests is that they do not have the statistical power to differentiate a random walk from a slow-speed mean reversion if the data set is small.

For a stock price following a Lognormal random walk, we have:

In ~ (S_{t+1}+d_{t+1}) = Normal ~(In~(S_t)+\bar{r},\sigma) Equation 1

where

St is the stock price at time t;

d_{t} is the dividend at time t;

\bar{r} (or 'rbar') is the mean long-run log return; and

s is the volatility.

Introducing mean reversion with a 'reversion speed' b, we have:

In ~(S_{t+1}+d_{t+1}) = Normal (In ~(S_t) + \bar{r} - b (In~(S_t+d_t)-In~(S_{t-1})-\bar{r}),\sigma) Equation 2

In other words the log stock value is pulled back a fraction b of how far it had deviated in the previous period from its expected value.

Viewed in terms of log returns, the Equation 2 becomes clearer:

r_{t+1} = Normal ~ (\bar{r}-b(r_t-\bar{r}),\sigma) Equation 3

where

r_{t+1} = In ~ \bigg(\frac{S_{t+1}+d_{t+1}}{S_t}\bigg) |

When b = 0, this is the Lognormal random walk of Equation 1:

r_{t+1} = Normal(\bar{r},\sigma) |

And if b=1:

r_{t+1} = Normal(2\bar{r}-r_t,\sigma) |

So, the expected return in period t+1 is:

\bar{r}_{t+1} = 2\bar{r}-r_t |

and the expected return in period (t+2) is:

\bar{r}_{t+2} = 2\bar{r}-\bar{r}_{t+1} = r_t |

which average to equal the long-term mean return, fluctuating alternatively above or below in a zig-zag pattern.

From Equation 3, you can see that if you performed a least squares linear regression in Excel of historical rt+1 against rt:

The gradient would be (-b), i.e. b=-SLOPE({r_{t+1}},{r_{t}});

The intercept would be rbar(1+b), i.e. rbar=INTERCEPT({r_{t+1}},{r_{t}})/(1-SLOPE({r_{t+1}},{r_{t}}));

Syx would be s, i.e. s=STEYX({r_{t+1}},{r_{t}}).

It is therefore a simple matter to estimate the mean reversion parameters from historic data, assuming of course that the model is a good representation of the variable's true stochastic behavior.

#### Example

The model Mean Reversion illustrates how one can evaluate historical returns of a stock (price and dividend), estimate the mean reversion parameters, and make a time series forecast with those projected values. The problem with making such a projection out further than one period is that one needs to distribute a change in the fortunes of the company between a change in the stock price and the dividend payout. In this model we have simply taken a random historic percentage of the stock+dividend that is released as dividend, but you could create more focused models based on an understanding of the company's management, etc.

The links to the Mean Reversion software specific models are provided here:

#### Discussion and uses

Stock prices are most often quoted as an example of mean reversion, perhaps because this is an area where a lot of mathematically minded people are trying to find a way of getting ahead of the others. But is it a good example?

The principle of mean reversion is that we have an extra predictive ability, so there is an additional expected return to be made on the investment. However, in an efficient market (freely available information, no transaction charges, similar utility, many small investors, etc.) there are no expected excess returns. Going back to Equation 3, the mean return of a stock exhibiting mean reversion is:

\bar{r}-b(r_t-\bar{r}) |

i.e. the stocks long term return is a function of where the current return is compared to the long term average return (rbar). If rt exceeds the mean return rbar, then the stock's expected return in the future is less than the mean return so everyone would sell. But then the stock price would go down until r_{t} = rbar.

So mean reversion should not exist in an efficient market - but of course an efficient market does not exist, perhaps mostly because investors do not analyze new information rationally, but act initially over-zealously (e.g. dot-com boom, moving out of IT shares in general because one IT company has a problem, etc.) and then correct. This is the domain of an area known as behavioural finance.

**Other uses**

Many commodities may exhibit mean reversion towards their marginal cost, or towards their real market value. For example, the oil price goes up and down like a yo-yo with wars, politics and resultant supply shortages, but reverts to a rational market value when stability returns.