Discrete Dynamics in Nature and Society

Volume 2015, Article ID 980768, 17 pages

http://dx.doi.org/10.1155/2015/980768

## The Risk of Individual Stocks’ Tail Dependence with the Market and Its Effect on Stock Returns

^{1}School of Statistics & Collaborative Innovative Center of Financial Security, Southwest University of Finance and Economics, Chengdu 611130, China^{2}Aix-Marseille School of Economics, Aix-Marseille University, CNRS & EHESS, 2 rue de la Charité, 13002 Marseille, France^{3}Cardiff Business School, Cardiff University, C26 Aberconway Building, Colum Drive, Cardiff CF10 3EU, UK^{4}School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, China

Received 28 May 2015; Accepted 22 October 2015

Academic Editor: Gian I. Bischi

Copyright © 2015 Guobin Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traditional beta is only a linear measure of overall market risk and places equal emphasis on upside and downside risks, but actually the latter is always much stronger probably due to the trading mechanism like short-sale constraints. Therefore, this paper employs the nonlinear measure, tail dependence, to measure the extreme downside risks that individual stocks crash together with the whole market and investigates whether such tail dependence risks will affect stock returns. Our empirical evidence based on Shanghai A shares confirms that most stocks display nonnegligible tail dependence with the whole market, and, more importantly, such tail dependence risks can indeed provide additional information beyond beta and other factors for asset pricing. In cross-sectional regression, it is proved that this tail dependence does help to explain monthly returns on Shanghai A shares, whereas the time-series regression further indicates that mimicking portfolio returns for tail dependence can capture strong common variation of Shanghai A stock returns.

#### 1. Introduction

Based on the classical Capital Asset Pricing Model (CAPM) developed by Sharpe [1], Lintner [2], and Black [3], it is the comovements of individual stocks with the whole markets that determine stocks’ expected returns. The stock prices, especially, depend on market betas, which are defined as the covariance of the stock and market returns divided by the variance of market returns. As a linear measure, beta provides an overall description of market risk and cannot distinguish upside from downside risk. However, it is well-documented that the degree of downside market risk is usually stronger than upside market risk’s; see Longin and Solnik [4] and Ang and Chen [5]. As advocated by Hong and Stein [6], due to the short-sale constraints existing in many stock markets, the trade of bearish investors who want to sell stocks short is always prohibited and hence the adverse information held by these investors could not be released to the market. If such adverse information is pent-up for a long time, it may accumulate to a very large amount. As long as such adverse information is finally flushing out to the market, it would provoke a heavy crash and bring the extreme downside risk which is much stronger than the upside counterpart.

Therefore, in this paper, we want to turn our focus onto this extreme downside market risk. We employ tail dependence, which is a flexible measure of extreme comovements and can be easily calculated using the sound Copula theory, to capture the risk that individual stocks crash together with the whole market and further explore the effect of such tail dependence risk on stock returns. The main purpose of our investigation is to compare the role of this new tail dependence in the asset pricing framework with that of classical beta. Such a comparison is motivated by two aspects. The first motivation is from the institutional perspective. As suggested by the theory of Hong and Stein [6], the extreme downside risk generated by short-sale constraints is stronger than upside risk and thus demands greater compensation, so we expect that the tail dependence risk defined in our paper would play a more significant role than linear beta in affecting the stock returns. The second motivation is from the statistical perspective: beta is a linear measure which describes the overall degree of dependence, whereas tail dependence is a nonlinear measure of extreme comovements. Intuitively, tail dependence can measure the probability that both the returns of individual stocks and the market returns are extremely negative (or extremely positive). Beta is calculated based on all the observations but tail dependence only looks at the tails of the distribution, and thus those stocks sharing the same betas might have different tail dependence with the market; see Hu [7]. We believe that the tail dependence should provide additional important information beyond beta for asset valuations.

Before 2010^{1}, China’s stock market was one of the few markets completely prohibiting the short selling of stocks and lacking other financial derivatives like stock index futures which can be sold short; see Comerton-Forde and Rydge [8] and Bris et al. [9]. According to our discussion above, such a specific feature would make heavy extreme downside market risk a strong feature of the Chinese stock market. Considering this, we choose to carry out our empirical analysis based on the data of Shanghai A shares. Besides, there are also many other special features in Shanghai’s stock market. For example, the fact that many firms in China have far lower free float may make the size effect weaker or even disappear, and, because of the dubious accounting practice, the ability of book-to-market variables to explain stock returns might become questionable too. These unique features in Shanghai’s stock market are quite different from those in mature stock markets, so this study can also allow us to diagnose whether those factors which proved to be useful in explaining the stock returns on mature markets would be important factors in pricing Chinese stocks, in addition to verifying the existence of tail dependence and examining its effects on stock returns.

As early as 1970s, Bawa and Lindenberg [10] proposed that the CAPM should be extended by taking into account the asymmetry of downside and upside market risks. And, recently, a study which is very close to ours, Ang et al. [11], defined a “downside beta” to measure the downside market risk and confirmed that the stock returns are significantly affected by downside betas. The downside beta in their paper was calculated as the covariation of individual stocks and the market when the market return falls below its average. But, differently, our focus is the* extreme* downside market risk during market crisis (i.e., the market returns are extremely negative, not just below the average). Such extreme downside market risks cannot be captured by just using downside beta, so we employ the tail dependence to measure the extreme comovements of individual stocks and the market. Besides, by introducing the sound Copula theory^{2} into the calculation of tail dependence, we are able to provide a more explicit representation of the dependence structure between individual stocks and the market and capture their tail behaviours without the discretionary choice of a threshold to define “downside beta” as in Ang et al. [11]. Huang et al. [12] also defined a measure of extreme downside risk and explored its effects on expected stock returns, but it is also different than our extreme downside* market* risk in as much as individual stocks crash down with the whole market, measured by the tail dependence between individual stocks and the market. Huang’s measure of extreme downside risk is constructed by the left tail index, only based on the information of each stock’s marginal distribution^{3}.

Our empirical evidence based on A shares of Shanghai’s stock market confirms that remarkable tail dependence with the whole market does exist for most stocks in this market. More importantly, we find that this tail dependence plays a nonnegligible role in explaining the cross-sectional stock returns in Shanghai’s stock market; the stocks with stronger tail dependence tend to have higher average monthly returns. Even after controlling the effects of linear beta and other factors, the tail dependence still shows significant relation with stock returns. In contrast, the coefficients of linear beta are consistently insignificant in our cross-sectional regressions. Furthermore, we also employ the time-series regression approach of Black et al. [13] to analyze the role of tail dependence in asset pricing, and the results suggest that a portfolio constructed as proxy for risk factor related to tail dependence can capture strong common variations in returns of Shanghai A shares. Therefore, we advocate that the tail dependence of individual stocks with the market may contain additional information beyond linear beta and other factors; thus tail dependence risk should be taken into account in asset pricing.

Our investigation can provide two contributions to the existing literature. Firstly, we define a “tail dependence” index to represent a new dimension for market risk, “extreme downside market risk,” that is, the risk of individual stocks crashing together with the whole market. More importantly, we recommend by providing supportive evidence that this “tail dependence” index has the potential to be a new pricing factor for stock returns. Secondly, our analysis may provide a possible explanation for the inconspicuous relation of betas with stock returns in previous studies like Fama and French [14] and Easley et al. [15]. These studies found that the cross section of returns on common stocks shows little relation to the market betas, but our results show that although no significant relation with stock returns could be found for betas, tail dependence will significantly affect stock returns. The insignificant relation of beta with stock returns found before is probably only due to the fact that beta is an overall measure for market risk. We insist that the pricing ability of market risk should not be doubted, and thus the extreme downside market risk measured by tail dependence is still a necessary factor for explaining stock returns.

The rest of this paper is organized as follows: Section 2 first introduces the calculation of tail dependence based on Copula theory and explains its difference with linear market beta; Section 3 then provides a brief description of the data used, and some important preliminaries on methodology; Section 4 outlines the empirical analysis for the existence of tail dependence and its effects in asset pricing; finally, we conclude our main results in Section 5.

#### 2. Methodology

##### 2.1. Tail Dependence and Copula Theory

As discussed above, in the stock markets like China’s, a large amount of adverse information may be sidelined due to the existence of short-sale constraints first and then flushing out during market crashes. As a result, a majority of stocks tend to crash down with the whole market and hence the extreme downside market risk exists for most stocks. In our paper, we employ the measure of tail dependence to describe such a risk of individual stocks’ crashing together with the market, which cannot be measured by classical betas or even downside beta.

Tail dependence is a nonlinear measure of extreme comovements and thus could be a perfect candidate to describe the relations between individual stocks and the market during crisis. For two variables and with the cumulative distribution functions (CDFs) of and , respectively, their tail dependence can be defined as follows:where and represent upper and lower tail dependences, respectively. Loosely said, the bivariate tail dependence looks at the concordance in the tail, that is, the relation between the extreme values of and . Geometrically, it measures the dependence in the upper-right or lower-left quadrant tail of a bivariate distribution. The value of tail dependence is essentially calculated as a limit, so we can avoid the discretionary choice of a threshold as in the definition of downside beta. Besides, the tail dependence can be very easily calculated using the sound Copula functions.

Copula is a function that incorporates marginal distributions into a joint distribution. Defined precisely, it is a joint distribution function of standard uniform random variables with a probability integral transformation applied to marginals. For more details, see Nelsen [16] and Cherubini et al. [17]. The linkage between the joint distribution and its marginal is demonstrated by Sklar’s theorem. Let be a vector of univariate variables with the marginal distributions denoted by ; Sklar’s theorem states that there exists a Copula function which could link the joint -dimensional distribution function to its marginals as follows:This relation can be expressed in terms of densities by deriving both sides of (2), and we getwhere represents the joint density function and the marginal density function. And the Copula density function is defined by . Hence, the joint density can be defined as the product of the Copula density and the univariate marginal densities^{4}.

Therefore, the information of marginal distributions is contained in , while the information of dependence structure is completely captured by Copula functions. Copula can capture the nonlinear dependence structure and allows the dependence degree in the tails to be different from that in the middle of distribution. One important feature of Copula functions is that it can always be easily related to the tail dependence we focus on in this paper; there exists a formula relating Copula functions to tail dependence as follows [18]:If the formula in (4) exists finitely, is said to have upper tail dependence if , no upper tail dependence if . The value , called the “upper tail dependence coefficient,” represents the limit of the conditional probability that the distribution function of exceeds the threshold , given that the corresponding function for does, when tends to one, and analogously for the lower tail dependence coefficient . Through this formula, we can introduce the sound Copula theory into the calculation of tail dependence between individual stocks and the market. Especially for some well-known parametric families of Copula functions, their parameters could be directly related to tail dependence; see Table 1.