Random Walk
What is Random Walk?
A random walk is a price series whose next change cannot be predicted from its past, so today's price is the best forecast of tomorrow's price.
The logic is competitive, not statistical. If past prices contained a usable pattern, traders would buy or sell now to capture it, and their own trades would move the price immediately and erase the pattern. What survives is movement driven by genuinely new information, and new information cannot be anticipated by definition, so successive price changes end up close to uncorrelated. Two testable predictions follow: trading rules built on price charts should not beat buying and holding once costs are counted, and because independent shocks add in variance rather than in size, the standard deviation of returns should grow with the square root of the holding period. The model is an approximation that frays at the edges: real returns show volatility clustering, extreme moves more often than a normal bell curve allows, mild short-run momentum and slow long-horizon reversal, so a strict random walk is best treated as a close first description rather than a law.
Random Walk: a worked example
Start a stock at $100 and let each day's move be plus or minus $1 with equal probability. The expected price tomorrow is 0.5 x $101 + 0.5 x $99 = $100, exactly today's price, which is what makes today's price the best forecast. Each step has variance of 1, and independent steps add their variances, so after 100 days the variance is 100 and the standard deviation is the square root of 100, or $10. The expected level is still $100, but the likely range has widened to roughly $90 to $110. Ten rising days in a row look like a trend and are not: their probability is 0.5 raised to the tenth power, under 0.1 percent, which still means many stocks in a large market will produce such a run every year.
The mistake students make with random walk
Students hear random walk and conclude that stock prices should go nowhere on average, which would make long-run investing pointless. That describes a walk with zero drift; a random walk with positive drift trends upward while every deviation from the trend stays unpredictable, and equity prices historically have drift. The other frequent error is the gambler's fallacy applied to markets, expecting a rise because the price has fallen five days running. Independence means the past run carries no information about the next step, so nothing is due.
Random Walk questions
How is a random walk related to the efficient market hypothesis?
The random walk is the price behavior that weak-form market efficiency predicts. If past prices held any usable information, traders would act on it and the pattern would disappear into the current price, leaving changes unpredictable. The hypothesis is the statement about information and the random walk is the observable consequence, which is why tests of the weak form are mostly tests for predictable patterns in past returns.
Does a random walk mean stock prices are meaningless?
No, it says the changes are unpredictable from past prices, not that the level is arbitrary. The price still reflects expected earnings, interest rates and risk, and it moves when those expectations change. Because expectations are already priced in, only surprises move prices, and surprises by definition arrive without warning.
Do real stock prices actually follow a random walk?
Approximately, and closely enough that chart-based trading rules rarely beat buy and hold after transaction costs. The measured departures are real, though: volatility arrives in clusters of calm and turbulent stretches, very large moves happen more often than a normal distribution implies, and studies find mild momentum over months with some reversal over several years. These effects are small relative to trading costs, so knowing about them is easier than profiting from them.
Formula / Example
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