اندازه گیری کارایی بازار سرمایه: ساختار همبستگی جهانی و محلی

We introduce a new measure for capital market efficiency. The measure takes into consideration the correlation structure of the returns (long-term and short-term memory) and local herding behavior (fractal dimension). The efficiency measure is taken as a distance from an ideal efficient market situation. The proposed methodology is applied to a portfolio of 41 stock indices. We find that the Japanese NIKKEI is the most efficient market. From a geographical point of view, the more efficient markets are dominated by the European stock indices and the less efficient markets cover mainly Latin America, Asia and Oceania. The inefficiency is mainly driven by a local herding, i.e. a low fractal dimension.

A concept of capital market efficiency is a central notion in financial markets theory [1] and [2]. This notion is generally used for an ideal image of the capital market enabling us to process relevant information to the fundamental price generation. If the relevant information to the fundamental price generation is completely processed by the capital market price mechanism, then such a capital market is said to be efficient. Thus the capital market efficiency accentuates the informational efficiency of capital markets. A notion of the efficient capital market represents such a capital market where prices on traded securities, e.g. stocks, bonds, or property, already reflect all available information and that investors are completely rational. Consequently, the notion of the efficient capital market represents a fair game pattern. No investor can have an advantage in predicting a return on an asset price, since no one has access to information not already available to everyone. It means that investors in the efficient capital market cannot expect to achieve abnormal returns systematically. In other words, the capital market is efficient if the fluctuations of returns are unpredictable [1], [2] and [3]. Paradoxically, an achievement of the ideal efficient capital market, enabling efficient allocation of investments, brings about no activity of investors and no activity of speculators. Because real life experiences with capital markets have shown that there are investors who indeed have been beating the capital markets in the long-term, discrepancies from the above mentioned ideal state are existent and thus worth analyzing. Testing the efficiency of various capital markets in different regions is a popular topic in financial journals (e.g. [4], [5], [6], [7] and [8]). However, the hypothesis of market efficiency is standardly either rejected or not and markets are ranked quite infrequently. Moreover, the researchers majorly focus on a single method and comment on the results. And even more, the whole idea of testing or measuring capital market efficiency has been dealing with the joint-hypothesis problem (i.e. when we reject the efficiency of a specific market, it might be caused by a wrong assumption of the market’s behavior) since its beginnings. This issue was also touched on by Fama himself [2]. In this paper, we try to bypass the problem by defining the efficient market as a martingale. We then analyze the fractal dimension, and long-range and short-range dependence to describe and measure the efficiency of specific markets. The Hurst exponent and a presence of long-term memory have been widely analyzed in recent years—in stock indices [9] and [10], interest rates [11], bonds [12], exchange rates [13] and others. The results vary depending on asset type and on geographical situation as well. Statistically significant long-range dependence was detected in some individual NYSE-listed stocks [14]. Even though the series of developed markets usually possess only short or no memory, emerging markets exhibit a different behavior [15] and [16]. Looking at a different frequency, a significant long memory was found for weekly returns of a large number of Greek stocks [17]. Cajueiro and Tabak [18] rank the markets according to their efficiency and suggest that the Hong Kong stock exchange is the most efficient one followed by Chinese A type shares and Singapore, and finally by Chinese B type shares, which indicates that liquidity and capital restrictions should be taken into consideration in efficiency testing and mainly interpretation. We use the Hurst exponent HH and the fractal dimension DD to construct a new measure of market efficiency based on a deviation from the ideal state (the efficient market) from both local and global perspectives. If the results based on different measures vary, we can further distinguish between local (herding) and global (structure of correlations) effects. We use the fact that the measures are bounded and thus can be used to construct an informative norm representing the said deviation from the ideal state. The measure is estimated for 41 stock indices at different stages of development from the beginning of 2000 till the end of August 2011, i.e. the data set includes the DotCom bubble and its bursting as well as the current Global Financial Crisis. The paper is structured as follows. In Section 2, we define the efficient capital market. Section 3 describes relationships between efficiency and the measures we use. In Section 4, we describe the methods used for the fractal dimension and Hurst exponent estimation. Section 5 covers the results and discusses the implications. Section 6 concludes. The main value of this paper lies in the fact that the proposed methodology bypasses the standard caveats of efficiency testing by building on the martingale definition of efficiency, using different methods and merging them into the efficiency measure. Such a rather bold path leads to very interesting and also meaningful results.

We have introduced a novel approach to measuring capital market efficiency. With a use of bounded measures of dynamic systems connected to a standard martingale definition of capital market efficiency, we constructed a vector containing long-term memory, short-term memory and fractal dimension measures. The efficiency index EIEI is calculated as a simple norm of this vector from its ideal efficient case. Therefore, the distance of a specific market situation from a center of an nn-dimensional cube is taken as a measure of efficiency. The further the market is from the ideal state, the less efficient it is. Such a procedure can be easily generalized to more bounded efficiency quantities. Applying the methodology on a set of 41 stock indices in the period between 2000 and 2011, we found that the Japanese NIKKEI is the most efficient market. From a geographical point of view, the more efficient markets are dominated by European stock indices and the less efficient markets cover mainly the Latin America, Asia and Oceania. More specifically, the least efficient markets are Venezuelan IBC, Malaysian KLSE, Slovakian SAX, CSE of Sri Lanka and Peruvian IGRA (the most inefficient stock market in the analyzed set). We also found that the local characteristics of the series (crowd and herding behavior) partially translate into the global characteristics (correlation structure). Moreover, the local inefficiencies in general dominate the total inefficiency for the strong majority of the indices.