# Random Walk vs. Efficient Market Hypothesis

"A random walk is one in which future steps or directions cannot be predicted on the basis of past actions."

Malkiel

"One of the earliest and most enduring models of the behavior of security prices is the random walk hypothesis, an idea that was conceived in the 16^{th} century as a model of games of chance."

COOTNER, Paul H. (Edited by), The Random Character of Stock Market Prices, 1964.

"But with the benefit of hindsight and the theoretical insights of LeRoy (1973) and Lucas (1973) [should be Lucas (1978)], it is now clear that efficient markets and the random walk hypothesis are two distinct ideas, neither one necessary nor sufficient for the other. The reason for this distinction comes from one of the central ideas of modern financial economics mentioned above: the necessity of some trade-off between risk and expected return. If a security's expected price change is positive, it may be just the reward needed to attract investors to hold the asset and bear the corresponding risks. Indeed, if an investor is sufficiently risk averse, he or she might gladly pay to avoid holding a security which has unforecastable returns. In such a world, prices do not need to be perfectly random, even if markets are operating efficiently and rationally."

Andrew W. LO (2000) In: COOTNER, Paul H. (Edited by), The Random Character of Stock Market Prices, page ix, 1964.

The EMH and the RWH are two distinct ideas, neither one necessary nor sufficient for the other, due to the necessity of some trade off between risk and expected return, see LeRoy (1973) and Lucas (1978).

"Thanks for reading my book, and for your question. The answer is simple: if fundamental changes in economic conditions cause expected returns to vary through time (e.g., due to business cycle fluctuations), then stock returns should contain a predictable component to reflect the time-varying risks associated with those fundamental changes. In such an environment, the EMH requires that stock returns be predictable, i.e., a non-random walk, so if they are random, that would be inefficient because it would not properly reflect changes in the underlying fundamentals." Andrew Lo e-mail [cootner] But with the benefits of hindsight and the theoretical insights of LeRoy (1973) and Lucas (1978), it is now clear that efficient markets and the random walk hypothesis are two distinct ideas, neither one necessary nor sufficient for the other. The reason for the distinction comes from one of the central ideas of modern financial economics mentioned above: the necessity of some trade off between risk and expected return. If a security's expected price change is positive, it may be just the reward needed to attract investors to hold the asset and bear the corresponding risks. Indeed, if an investor is sufficiently risk averse, he or she might gladly pay to avoid holding a security which has unforecastable returns. In such a world, prices do not need to be perfectly random, even if markets are operating efficiently and rationally.Under very special circumstances, e.g. risk neutrality, the two are equivalent. However, LeRoy (1973), Lucas (1978), and many others have shown in many ways and in many contexts that the Random Walk Hypothesis is neither a necessary nor a sufficient condition for rationally determined security prices.

============== [ME] NB: Strictly, we mean a martingale when we speak of a random walk. The EMH and the RWH are two distinct ideas, neither one necessary nor sufficient for the other, due to the necessity of some trade off between risk and expected return, see LeRoy (1973) and Lucas (1978). (Assume that the risk-free interest rate is zero.) EMH ^ ~RWH Suppose that a security has a positive expected return with a given standard deviation. Let us suppose that the return is precisely the reward needed to attract investors to hold the asset and bear the corresponding risks. There is no compelling reason to either buy or sell this security. Reveal this information to all market participants and the market would not move. ~EMH ^ RWH Suppose that there are no rational participants in the market. Suppose that a security has zero expected return with a given standard deviation. Reveal this information to the irrational traders and the market would move.

A Martingale has increments that are unpredictable, i.e., the expected value of the increment is unpredictable. A random walk has independent increments&emdash;none of the moments are predictable. The popular "ARCH" class models in finance violate random walks because their variance is predictable.

Random walk processes with zero drift are martingale processes but not all martingale processes are random walks. EMH vs RW

- Cox, J. and S. Ross, 1976, "The Valuation of Options for Alternative Stochastic Processes,"
*Journal of Financial Economics*, 3, 145-166. - Harrison, M., and D. Kreps, 1979, "Martingales and Arbitrage in Multiperiod Securities Markets,"
*Journal of Economic Theory*, 20, 381-408. - Leroy, S., 1973, "Risk Aversion and the Martingale Property of Stock Returns",
*International Economic Review*14, pp. 436-46. - Lucas, R., 1978, "Asset Prices in an Exchange Economy",
*Econometrica*46, pp. 1429-46.