Therandom walk hypothesis is afinancial theory which states that theprices of financial assets, particularly those in thestock market, follow arandom walk. According to this hypothesis, price variations occur in an essentially random manner, which implies that they cannot be systematically predicted or consistently exploited to achieve returns above those of the overall market.

The concept can be traced to French brokerJules Regnault who published a book in 1863, and then to French mathematicianLouis Bachelier whose Ph.D. dissertation titled "The Theory of Speculation" (1900) included some remarkable insights and commentary. The same ideas were later developed byMIT Sloan School of Management professorPaul Cootner in his 1964 bookThe Random Character of Stock Market Prices.^[1] The term was popularized by the 1973 bookA Random Walk Down Wall Street byBurton Malkiel, a professor of economics atPrinceton University,^[2] and was used earlier inEugene Fama's 1965 article "Random Walks In Stock Market Prices",^[3] which was a less technical version of his Ph.D. thesis. The theory that stock prices move randomly was earlier proposed byMaurice Kendall in his 1953 paper,The Analysis of Economic Time Series, Part 1: Prices.^[4] In 1993 in theJournal of Econometrics,K. Victor Chow and Karen C. Denning published a statistical tool (known as the Chow–Denning test) for checking whether a market follows the random walk hypothesis.^[5]

Whether financial data can be considered a random walk is a venerable and challenging question. One of two possible results are obtained, the data does fall under random walk or the data does not. To investigate whether observed data follows a random walk, some methods or approaches have been proposed, for example, the variance ratio (VR) tests,^[6] theHurst exponent^[7] andsurrogate data testing.^[8]

Burton G. Malkiel, an economics professor at Princeton University and author ofA Random Walk Down Wall Street, performed a test where his students were given a hypotheticalstock that was initially worth fifty dollars. The closing stock price for each day was determined by a coin flip. If the result was heads, the price would close a half point higher, but if the result was tails, it would close a half point lower. Thus, each time, the price had a fifty-fifty chance of closing higher or lower than the previous day. Cycles or trends were determined from the tests. Malkiel then took the results in chart and graph form to achartist, a person who "seeks to predict future movements by seeking to interpret past patterns on the assumption that 'history tends to repeat itself'."^[9] The chartist told Malkiel that they needed to immediately buy the stock. Since the coin flips were random, the fictitious stock had no overall trend. Malkiel argued that this indicates that the market and stocks could be just as random as flipping a coin.

Modelling asset prices with a random walk takes the form:

${\displaystyle S_{t+1}=S_t+\mu \mathrm{\Delta }t{S}_t+\sigma \sqrt{\mathrm{\Delta }t}{S}_t{Y}_i}$

where

${\displaystyle \mu }$ is a drift constant

${\displaystyle \sigma }$ is thestandard deviation of the returns

${\displaystyle \mathrm{\Delta }t}$ is the change in time

${\displaystyle Y_i}$ is ani.i.d. random variable satisfying${\displaystyle Y_i\sim N(0,1)}$.

There are other economists, professors, and investors^[who?] who believe that the market is predictable to some degree. These people believe that prices may move intrends and that the study of past prices can be used to forecast future price direction.^{[clarification needed ConfusingRandom andIndependence?]} There have been some economic studies that support this view, and a book has been written by two professors of economics that tries to prove the random walk hypothesis wrong.^[10]

Martin Weber, a leading researcher in behavioural finance, has performed many tests and studies on finding trends in the stock market. In one of his key studies, he observed the stock market for ten years. Throughout that period, he looked at the market prices for noticeable trends and found that stocks with high price increases in the first five years tended to become under-performers in the following five years. Weber and other believers in the non-random walk hypothesis cite this as a key contributor and contradictor to the random walk hypothesis.^[11]

Another test that Weber ran that contradicts the random walk hypothesis, was finding stocks that have had an upward revision forearnings outperform other stocks in the following six months. With this knowledge, investors can have an edge in predicting what stocks to pull out of the market and which stocks — the stocks with the upward revision — to leave in. Martin Weber’s studies detract from the random walk hypothesis, because according to Weber, there are trends and other tips to predicting the stock market.

ProfessorsAndrew W. Lo and Archie Craig MacKinlay, professors of Finance at theMIT Sloan School of Management and the University of Pennsylvania, respectively, have also presented evidence that they believe shows the random walk hypothesis to be wrong. Their bookA Non-Random Walk Down Wall Street, presents a number of tests and studies that reportedly support the view that there are trends in the stock market and that the stock market is somewhat predictable.^[12]

One element of their evidence is the simple volatility-based specification test, which has a null hypothesis that states:

${\displaystyle X_t=\mu +X_{t-1}+{ϵ}_t}$

where

${\displaystyle X_t}$ is the log of the price of the asset at time${\displaystyle t}$

${\displaystyle \mu }$ is a drift constant

${\displaystyle {ϵ}_t}$ is a random disturbance term where${\displaystyle \mathbb{E}[{ϵ}_t]=0}$ and${\displaystyle \mathbb{E}[{ϵ}_t{ϵ}_{\tau }]=0}$ for${\displaystyle \tau \ne t}$ (this implies that${\displaystyle \tau }$ and${\displaystyle t}$ are independent since${\displaystyle \mathbb{E}[{ϵ}_t{ϵ}_{\tau }]=\mathbb{E}[{ϵ}_t]\mathbb{E}[{ϵ}_{\tau }]}$).

To refute the hypothesis, they compare the variance of${\displaystyle (X_t-X_{t+\tau })}$ for different${\displaystyle \tau }$ and compare the results to what would be expected for uncorrelated${\displaystyle {ϵ}_t}$.^[12] Lo and MacKinlay have authored a paper, theadaptive market hypothesis, which puts forth another way of looking at the predictability of price changes.^[13]

Peter Lynch, a mutual fund manager atFidelity Investments, has argued that the random walk hypothesis contradicts theefficient-market hypothesis (EMH). In Lynch's view, if prices are rational and reflect all available information as EMH proposes, then their fluctuations should not be "random"; conversely, if prices do follow a random walk, then they cannot be rational.^[14]According tomodern portfolio theory, this is imprecise.Samuelson (1965) showed that in an informationally efficient market the (properly discounted) price follows a martingale, implying that price changes are unpredictable given available information; the "random" in "random walk hypothesis" meansunforecastable rather thanirrational.^[15]

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