**1. Introduction**

The e fficient-market hypothesis (EMH) is one of the most important economic and financial hypotheses that have been tested over the past century. Traditional finance theory supporting EMH is based on some important financial theories, including the arbitrage principle (Modigliani and Miller 1959, 1963; Miller and Modigliani 1961), portfolio principle (Markowitz 1952a), Capital Asset Pricing Model (Treynor 1961, 1962; Sharpe 1964; Lintner 1965; Mossin 1966), arbitrage pricing theory (Ross 1976), and option pricing theory (Black and Scholes 1973). In addition, Adam Smith (Smith 1776) commented that the rational economic man will chase the maximum personal profit. When a rational economic individual comes to stock markets, they become a rational economic investor who aims to maximize their profits in stock markets.

However, an investor's rationality requires some strict assumptions. When not every investor in the stock market looks rational enough, the assumptions could be relaxed to include some "irrational" investors who could trade randomly and independently, resulting in o ffsetting the e ffects from each other so that there is no impact on asset prices (Fama 1965a).

What if those "irrational" investors do not trade randomly and independently? In this situation, Fama (1965a) and others have commented that rational arbitrageurs will buy low and sell high to eliminate the e ffect on asset prices caused by "irrational" investors. Fama and French (2008) pointed out that the financial literature is full of evidence of anomalies. Another school (see, for example, Guo and Wong (2016) and the references therein for more information) believes that Behavioral Finance is not caused by "irrational" investors but is caused by the existence of many di fferent types of investors in the market.

In this paper, we review the theory and the literature on market e fficiency and market anomalies. We give a brief review on market e fficiency and define clearly the concept of market e fficiency and the e fficient-market hypothesis (EMH). We discuss some e fforts that challenge the EMH. For example, we document that investors may not carry out the dynamic optimization problems required by the tenets of classical finance theory, or follow the Vulcan-like logic of the economic individual, but use rules of thumb (heuristic) to deal with a deluge of information and adopt psychological traits to replace the rationality assumption, as suggested by Montier (2004). We then review di fferent market anomalies, including the Winner–Loser E ffect, reversal e ffect; Momentum E ffect; calendar anomalies that include January e ffect, weekend e ffect, and reverse weekend e ffect; book-to-market effect; value anomaly; size e ffect; Disposition E ffect; Equity Premium Puzzle; herd e ffect and ostrich effect; bubbles; and di fferent trading rules and technical analysis.

Thereafter, we review di fferent theories of Behavioral Finance that might be used to explain market anomalies. This review is useful to academics for developing cutting-edge treatments of financial theory that EMH, anomalies, and Behavioral Finance underlie. The review is also beneficial to investors for making choices of investment products and strategies that suit their risk preferences and behavioral traits predicted from behavioral models. Finally, when EMH, anomalies, and Behavioral Finance are used to explain the impacts of investor behavior on stock price movements, it is invaluable to policy makers in reviewing their policies to avoid excessive fluctuations in stock markets.

The plan of the remainder of the paper is as follows. In Section 2, we define the concept of market efficiency clearly, review the literature on market e fficiency, and discuss several models to explain market e fficiency. We discuss some market anomalies in Section 3 and evaluate Behavioral Finance in Section 4. Section 5 gives some concluding remarks.

#### **2. Market E** ffi**ciency**

The concept of market e fficiency is used to describe a market in which relevant information is rapidly impounded into the asset prices so that investors cannot expect to earn superior profits from their investment strategies. In this section, we define the concept of market e fficiency clearly, review the literature of market e fficiency, and discuss several models to explain market e fficiency.

#### *2.1. Definition of Market E*ffi*ciency*

The definition of market e fficiency was first anticipated in a book written by Gibson (1889), entitled *The Stock Markets of London, Paris and New York*, in which he wrote that, when "shares become publicly known in an open market, the value which they acquire may be regarded as the judgment of the best intelligence concerning them".

In 1900, a French mathematician named Louis Bachelier published his PhD thesis, *Théorie de la Spéculation* (Theory of Speculation) (Bachelier 1900). He recognized that "past, present and even discounted future events are reflected in market price, but often show no apparent relation to price changes". Hence, the market does not predict fluctuations of asset prices. Moreover, he deduced that "The mathematical expectation of the speculator is zero", which is a statement that is in line with Samuelson (1965), who explained e fficient markets in terms of a martingale. The empirical implication is that asset prices fluctuate randomly, and then their movements are unpredictable. Bachelier's contribution to the origin of market e fficiency was discovered when his work was published in English by Cootner (1964) and discussed in Fama (1965a, 1970).

#### *2.2. Early Development in EMH*

Pearson (1905) introduced the term *random walk* to describe the path taken by a drunk, who staggers in an unpredictable and random pattern. Kendall and Hill (1953) examined weekly data on stock prices and finds that they essentially move in a random-walk pattern with near-zero autocorrelation of price changes. Working (1934) and Roberts (1959) found that the movements of stock returns look like a random walk. Osborne (1959) showed that the logarithm of common stock prices follows Brownian motion and finds evidence of the square root of time rule.

If prices follow a random walk, then it is di fficult to predict the future path of asset prices. Cowles (1933, 1944) and Working (1949) documented that professional forecasters cannot successfully forecast, and professional investors cannot beat the market. Granger and Morgenstern (1963) found that short-run movements of the price series obey the random-walk hypothesis, using spectral analysis, but that long-run movements do not. There is evidence of serial correlation in stock prices (Cowles and Jones 1937) which, however, could be induced by averaging (Working 1960, and Alexander 1961). Cowles (1960) reexamined the results in Cowles and Jones (1937) and still found mixed evidence of serial correlation even after correcting an error caused by averaging.

#### *2.3. Recent Developments in Market E*ffi*ciency*

The 2013 Nobel laureate, Eugene Fama, provided influential contributions to theoretical and empirical investigation for the recent development of market e fficiency. According to Fama (1965a), an e fficient market is defined as a market in which there are many rational, profit-maximizing, actively competing traders who try to predict future asset values with current available information. In an efficient market, competition among many sophisticated traders leads to a situation where actual asset prices, at any point in time, reflect the e ffects of all available information, and therefore, they will be good estimates of their intrinsic values.

The intrinsic value of an asset depends upon the earnings prospects of the company under study, which is not known exactly in an uncertain world, so that its actual price is expected to be above or below its intrinsic value. If the number of the competing traders is large enough, their actions should cause the actual asset price to wander randomly about its intrinsic value through o ffsetting mechanisms in the markets, and then the resulting successive price changes will be independent. Independent successive price changes are then consistent with the existence of an e fficient market.

A market in which the prices of securities change independently of each other is defined as a random-walk market (Fama 1965a). Fama (1965b) linked the random-walk theory to the empirical study on market e fficiency. The theory of random walk requires successive prices changes to be independent and to follow some probability distribution.

When the flow of news coming into the market is random and unpredictable, current price changes will reflect only current news and will be independent of past price changes. Hence, independence of successive price changes implies that the history of an asset price cannot be used to predict its future prices and increase expected profits. It is then consistent with the existence of an e fficient market. Using serial correlation tests, run tests, and Alexander's (1961) filter technique, Fama (1965b) concluded that the independence of successive price changes cannot be rejected. Then, there are no mechanical trading rules based solely on the history of price changes that would make the expected profits of the market traders higher than buy-and-hold.

The random-walk theory does not specify the shape of the probability distribution of price changes, which needs to be examined empirically. Fama (1965b) found that a Paretian distribution with characteristic exponents less than 2 fit the stock market data better than the Gaussian distribution; this finding is in line with the findings of Mandelbrot (1963). Hence, the empirical distributions have more relative frequency in their extreme tails than would be expected under a Gaussian distribution while the intrinsic values change by large amounts during a very short period of time.
