Reverse return-volatility asymmetry, and short sale constraints: Evidence from the Chinese markets

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1 Reverse return-volatility asymmetry, and short sale constraints: Evidence from the Chinese markets Liang Wu The School of Economics, Sichuan University, Chengdu,China Abstract Reverse return-volatility asymmetry (i.e. positive returns induce higher price volatilities than negative ones) is documented in the Chinese markets in contrast to the widely accepted negative return-volatility correlation. Using a theoretic model with short sale and borrowing constraints, we show that the asymmetry response of investors to news is one cause of the reverse return-volatility asymmetry. Lifting of short sale constraints can decrease the return-volatility correlation while margin trading increases the correlation. In the context of slow adoption of margin trading and security lending policy, we conduct empirical analysis and verify that the lifting of short sale constraints leads to significantly less reverse return-volatility asymmetry. Keywords: reverse return-volatility asymmetry, anti-leverage, short sale constraints, margin trading 1. Introduction Risk and return are fundamental concepts in finance. For example, the capital asset pricing model (CAPM) provides for an explicit relationship between risk and return by saying that the expected return of an asset is positively related to its undiversified risk. Accordingly, investors taking on additional risks should be compensated with higher expected returns. If the variance and covariance are time-varying, the dynamic nature between risk and volatility can be complex. Theories such as leverage effect (Christie (1982)) and volatility-feeback theory (Pindyck (1983); Campbell address: liangwu@scu.edu.cn (Liang Wu) Preprint submitted to EFMA Annual Meeting 2017 January 15, 2017

2 and Hentschel (1992)) are developed to explain the asymmetric volatility property of individual stock returns in the United States (U.S.) which says that the stock return volatility is negatively correlated with stock returns (see e.g., Bekaert and Wu (2000)). The leverage effect hypothesis suggests that a drop in the value of a stock increases the debt-to-equity ratio, this in turn makes the stock riskier and causes future volatility to rise. Volatility-feedback theory suggests that a volatility increase raises the expected return, leading to an immediate stock price decline after investors revalue the equity. The causalities of the two theories are different: the leverage effect suggests that return shocks lead to the change of future volatility, while volatility-feedback suggests that current or future returns are caused by current or lagged volatility shocks. Since the two effects could happen at intra-day sub-timescales, the observation of contemporaneous return-volatility correlation at a daily level could thus be caused by either effect or by both effects regardless of any differences in order of occurrence of return shocks and volatility shocks. Negative correlation of return and volatility has been observed in most of the financial markets in developed countries (Bollerslev et al. (2008) (U.S. stock markets), Qiu et al. (2006) (German stock markets), Bekaert and Wu (2000) (Japanese stock markets) and Lee (2012) (Korea stock markets)). Other researchers such as Harrison and Zhang (1999); Ghysels et al. (2005); Ludvigson and Ng (2007) find a positive relation. Nelson (1991) and Glosten et al. (1993) argue that either a positive or a negative relationship is possible. While some researchers are discussing whether the correlation of return and volatility is negative or which theory seems to offer the best explanations, other studies focusing upon the Chinese markets reveals the existence of positive return-volatility correlation. Based upon daily raw returns from four market indices of the Shanghai and Shenzhen markets (taking from May, 1992 to August, 1996 for two composite indices and from October, 1992 to August, 1996 for two B-share indices), Yeh and Lee (2001) find that good news has a larger impact upon future volatility than bad news. Menggen (2007) examines the log returns of both Shanghai and Shenzhen Composite Index. He finds a positive and statistically significant risk-return relationship for the daily returns of stock indices composed of stocks traded on the Shenzhen Stock Exchange while the relationship is negative but insignificant for those referencing the Shanghai Stock Exchange. Qiu et al. (2006) discover the positive return-volatility correlation in Chinese stock indices using both daily and 5-min data of Chinese stock indices, compared with the neg- 2

3 ative correlation founded on German DAX Index. They refer to this as the anti-leverage effect, and their result is echoed by Shen and Zheng (2009), who use a retarded volatility model and further argue that leverage effect and the anti-leverage effect are independent of the probability distribution of returns and long-range time correlation of volatilities. Zheng et al. (2014) review this effect together with other features of the Chinese stock markets including the volatility auto-correlation, the return-volatility correlation and the spatial structure at the firm level. Wan et al. (2014) confirm the positive correlation using 5-minute data of 10 individual stocks with high capitalization and high liquidity selected from different industries in the Hushen 300 Index and finds a positive return-volatility correlation for most of the individual stocks in the sample. Possible explanations of the reverse return-volatility asymmetry effect in the Chinese markets given by previous researchers are based on indigenous trading mechanisms. Wan et al. (2014) postulate that the Chinese markets protect investors against daily loss by limiting a stock s daily range to ±10% of the previous day s closing price, which reduces the risk averse of investors. Few or even no dividends cut down their motivations for value investment. The majority of the participants are retail investors who behave like shortterm speculators. They surge for a price rising trend. Klößner et al. (2012) develop a statistical approach to detect intra-day overreaction. They find that Chinese investors over-react to good news with greater emphasis than to bad news, when they are compared with investors in the U.S. and German markets who over-react with greater emphasis to bad news than to good news. However, among all of the empirical studies, few notice the linkage between short sale constraints and the anti-leverage effect observed in Chinese markets. It is worth noting that it is difficult to directly examine the effect of short sales on return-volatility correlation, since short sales have long been practiced in developed markets. To this end, many researchers use proxies such as relative short interest, breadth of ownership, institutional ownership for the level of short sale constraints (see Figlewski (1981); Chen et al. (2002); Nagel (2005)). The 2008 financial crisis, during which short sales were banned to different extents among different countries, provides researchers with the opportunity to test the effects of short sale constraints on pricing efficiency and market stability. Nevertheless, the credence and stability of their results are dampened by the brief periods of the bans and the limited number of affected stocks (e.g. the short ban lasted for only 3 weeks in the U.S.). In this respect, the Chinese stock market is ideal for 3

4 empirical testing: the launches of margin financing and security lending for selected stocks naturally divides the time horizon into two time periods (i.e., before and after the stocks can be sold short). The fact that stocks can be added to the short sale list only if they meet certain requirements provides us with a treatment group as well as a control group composed of shortable and non-shortable stocks, respectively. Based on the margin financing and security lending policy, two previous papers (Chang et al. (2014),Zhao et al. (2014)) studying the effect of short sales focus mainly on market efficiency. The authors find that after the ban is lifted, price efficiency increases and stock return volatility decreases. The first motive of this paper is to test whether return-volatility correlation are positive for daily firm-level returns in the Chinese markets. Most previous studies of the Chinese markets focus on stock indices (see Yeh and Lee (2001); Menggen (2007); Qiu et al. (2006)). One contribution of our study to the literature is that we have a thorough test of the statistics based on raw returns, excess returns and normalized returns of individual stocks. The excess returns retain the idiosyncratic aspects of each stock while the market aspect is removed. The normalized excess returns, on the other hand, remove the effect of time-varying volatility and possible leverage (or anti-leverage) effect through dividing the excess returns by the conditional volatility obtained from the EGARCH model. The second contribution of our study is that we have empirically shown that short sale constraints are responsible for the existence of reverse returnvolatility asymmetry in the Chinese markets, based on a panel regression on a wide range of daily return data covering both shortable and non-shortable stocks spanning from 2006 to 2014 (4 years before and 4 years after the first launch of security lending policy). Some researchers believe that these statistical anomalies are caused by the fact that investors in the Chinese markets tend to be speculators and rush for a price rising (see e.g.qiu et al. (2006); Wan et al. (2014)). It is unlikely that Chinese investors are significantly different from other investors throughout the world. There should be some mechanisms built into the markets which offer the Chinese market investors the motivation to behave differently. In this paper, we present evidence that one possible mechanism to explain the statistical anomalies resides within the policy of the short sale constraints. When the short sales are constrained, investors tend to rush for a price rising hoping to make profits since they can only buy low and sell high. As a result, we can observe that the market reacts more to a positive signal than to a negative signal, and hence the presence 4

5 of reverse return-volatility asymmetry. The remainder of this paper is organized as follows: part 2 gives a theoretic discussion of the effect of short sale constraints on return-volatility correlation especially when margin trading is adopted simultaneously; part 3 gives a brief description of the data set; part 4 presents the data processing method and empirical results; part 4 is the conclusion and discussion. 2. Model In this section, we build a model to study the impact of the short sale and borrowing constraints on return-volatility correlation. Positive returnvolatility correlation means that investors react more intensively to positive returns than their reaction to negative returns. It has been shown in Xu (2007) that the price is convex function of information if the short sales are constrained. One might postulate that the positive return-volatility is also caused by the short sale constraints. The original model framework in Xu (2007) was developed to show that positive skewness is a result from the asymmetric response of the price to information when short sale is strictly constrained. On the other hand, the margin trading policy is adopted simultaneously with the security lending in China. In order to understand how the return-volatility correlation is impacted when both short sale and margin trading are practiced. We extend the theoretical analysis as in Xu (2007) to meet the reality of the Chinese stock markets: the short sale constraints are not strictly imposed; we introduce borrowing constraints to study the effect of margin trading. We will calculate the return-volatility correlation of an asset as we vary the degree of short sales and borrowing constraints. As in Xu (2007), we consider a one period model and assume an economy consisting of two assets, a risky asset with a random payoff of X and a risk-free asset with gross interest rate R. Assume that X follows a normal distribution with mean µ X and precision (the reciprocal of variance) τ X. The public information S = X + ɛ contains a noise ɛ with expected value 0 and precision τ ɛ. All investors can observe the same information S while they hold different opinions towards the precision of noise. There are two representative types of investors who consider τ L and τ H to be the precision of noise, respectively. In addition, τ L < τ H. The two types of investors take proportions of λ and 1 λ respectively in the market. Suppose that information S = s is announced at time 1 and two types of investors trade in the market based on their opinions of error precision. First, 5

6 for θ (θ = L, H) type investors, we calculate the conditional expectation and variance of the risky assets payoff X given the information S = s, E θ [X s] ˆµ θ = τ X µ X + τ θ s (1) τ θ + τ x τ θ + τ X Va θ (X s) ˆτ 1 θ = (τ X + τ θ ) 1 (2) Assume θ type investors have initial wealth of W θ, they maximize their utility function of final wealth W θ by choosing their demand for the risky assets at time 1. The utility function is chosen to be the same constant absolute risk aversion (CARA) utility function U(W θ ) = e aw θ for both types of investors. Since information is a normally distributed random variable, the maximization of the utility function equals the maximization of the mean-variance utility function. max y 1,θ {E θ [ e aw θ ]} max y 1,θ {E θ [W 2,θ s] a 2 Var θ(w θ s)} (3) s.t. W θ = (W θ y 1,θ P 1 )R + y 1,θ X where y θ is the quantity of risky asset demanded by type θ, P 1 is the price of risky asset and a measures absolute risk aversion. Solving the utility maximization, we have the demand function for each type of investors, y θ = (ˆµ θ RP )ˆτ θ a (4) After we set the market clearing condition λy 1,L + (1 λ)y 1,H = 1. This implies the equilibrium price of the risky asset P 1 = 1 R ( µ ā τ ) (5) where τ = λˆτ L + (1 λ)ˆτ H is the average precision, µ = [λˆµ Lˆτ L + (1 λ)ˆµ H ˆτ H ]/ τ is the expected payoff by an average investor (i.e., E[X L, H, λ] after integrating the beliefs from both L and H investors according to their population weights). As seen in Eq. (5), the asset price is a linear function of information s. In the absence of short sale and borrowing constraints, price reacts symmetrically to information. 6

7 2.1. short sale and borrowing constraints If there is a strict short sale constraint (investors are completely forbidden from short selling), the equilibrium solution has been given in Xu (2007). In this paper, we loose the short sale constraints to assume that there is a maximum amount N 0 investors are allowed to short sell the risky asset, i.e., λy L N, (1 λ)y H N. When N = 0, short selling is completely prohibited in the market and we get back to the case discussed in Xu (2007). Note that, as a requirement for the market to clear, two types of investors cannot sell short at the same time. Therefore, at most one type of investors are binding to the short sale constraint, i.e., λy L = N or (1 λ)y H = N. In this paper, we also consider borrowing constraint, which means that there is a limited amount of money investors can borrow. We express the borrowing constraint as the total initial wealth (investors own wealth plus the amount they can borrow) that can be used to invest in the risky asset is less than a limit W. We assume that the borrowing constraints are only applicable to H investors, i.e., W H W, because borrowing constraints on L investors have little impact on skewness based on the following analysis. As shown in Xu (2007), the short constraints lend to asymmetric response of investors because H investors crowd out L investors when the market information s is very bullish. As a consequence, the price only reflects H investors overreaction to market s bullish information. If the initial wealth of H investors is inadequate due to borrowing constraints, the overreaction of H investors to bullish information will largely be damped. At this time, L investors have left the market and has no need for borrowing. Although it is true when s is rather bearish, only L investors will buy the risky asset when H investors overconfidently regard the market signal as bearish and leave the market under short sale constraints, however, at this time, the riksy asset prices is rather cheap, and the amount of capital needed by L investors is much smaller than what is needed by H investors in a bullish market. The following proposition provides the details of the equilibrium solution under short sale and borrowing constraints (see Appendix for proof). Proposition 1. (a) Under condition ˆµ H ˆµ L > a(1+n) (1 λ)ˆτ H + an λˆτ L (or s > µ x + a(1+n)ˆτ L (1 λ)(τ H τ L )τ x + ), the equilibrium price becomes: an ˆτ H λ(τ H τ L )τ x (i) when W (1 + N)P H, P = P H ; (ii) when (1 + N)P s < W < (1 + N)P H, P = W/(1 + N); (iii) when W (1 + N)P s, P = P M ; 7

8 Under (i) and (ii), the demand of L investors is λy L = N, and that of investor H is (1 λ)y H = 1+N; under (iii), λy L = λ(ˆµ L RP M )ˆτ L, (1 a λ)y H = 1 λy L. (b) Under condition ˆµ L ˆµ H > a(1+n) an ˆτ L λˆτ L + an (1 λ)ˆτ H (or s < µ x a(1+n)ˆτ H λ(τ H τ L )τ x λ(τ H τ L )τ x ), the demand of investor H is (1 λ)y H = N while the demand of investor S is λy L = 1 + N; the equilibrium price P = P L ; (c) While neither condition (a) nor (b) is satisfied, short sale constraints no longer affect the equilibrium solution. Then the following two borrowing constraints should be considered: (i) when W (1 λ)p 0 y 0,H, P = P 0 ; (ii) when W < (1 λ)p 0 y 0,H, P = P M ; On condition (i), investor demands are as in Equation (8) and (9), and on condition (ii), investor demands are as in (iii) of (a). Among which P H is the price corresponding to the demand (1 λ)y H = 1 + N of investor H, P L is the price corresponding to the demand λy L = 1 + N of investor L, P M is the equilibrium price when market clears out on condition that investors H are not subject to borrowing constraint W H = W while investors L are not subject to any constraint. Their expressions are given as, P H = 1 R (ˆµ H a(1+n) (1 λ)ˆτ H ) P L = 1 R (ˆµ L a(1+n) λˆτ L ) P M = 1 2λRˆτ L (λˆτ Lˆµ L a + 3. Numerical simulations (a λˆτ Lˆµ L ) 2 + 4aRW λˆτ L ) Under short sale constraint and borrowing constraint, the price of the risky asset is a piecewise function of information s, while s itself is subject to normal distribution and it can be difficult to calculate the return-volatility correlation of the prices, here we adopt the method of Monto Carlo Simulation and generate random samples of x and ɛ. After generating x and ɛ, we calculate the price of risky asset based on each sample and then we calculate return-volatility correlation. As long as the sample space is chosen large enough, for example in this paper we take 1,000,000 samplings, we 8

9 can get a rather smooth curve. In the example of our Monto Carlo Simulation, µ x = 1.5, τ x = 1, R = 1.2, τ L = 0.5, τ H = 1.5, λ = 0.5, a = 0.5, the interval of W is [0.5,5], and N is taken from three dispersed values N = 0, N = 0.1, N = 1. In order to understand the relative magnitudes of different values of W, we need to calculate the market value of the risky asset. Since the total volume of risky asset is 1, the market value is the same as the asset price. When s = µ x, the market provides accurate information of the risky asset and we can obtain P 0 = 1.04, which means the fair value of the asset is We further look at the situation where the information signal is two standard deviations away from its expectation s = µ x + 2 1/τ ɛ + 1/τ x = 4.32, at this time we have P 0 = 2.22, P H = 2.33(N = 0), which means that the market value of risky asset is 2.22 under no short sale constraint and 2.33 when short sale is prohibited. That is to say, with a probability of 97.5%, the market value of risky value will be smaller than From the analysis of the market value we can see that, if the capital of the investor is below 1, the capital is not sufficient for the market to be cleared at its fair value. On the other hand, W = 5 means almost no constraints to the investment capital, since with more than 97.5%, the market value is below For a reasonable range of W to study the effect of borrowing constraints, we use the interval W [0.5, 5]. Under most occasions the interval that would have a real influence on investors borrowing is W [1, 3]. Figure 1 presents the return-volatility correlation under different choices of N and W. We can see that, within the interval W [0.5, 5], the returnvolatility correlation increases as W increases. In addition, if short sale is allowed, the return-volatility correlation would decrease under all levels of borrowing constraints W [0.5, 5]. If some degree of short sale is allowed while the constraint is still strong (for example N = 0.1 which means only 10% of the risky asset can be sold short), the return-volatility correlation of risky asset is still above 0. At this time, the skewness of risky asset is still positively biased. But if the short constraint is very mild (for example N = 1 which means 100% of risky asset can be sold short), while there is borrowing constraint at the same time (for example W < 5), the returnvolatility correlation is negative. 9

10 Correlation N=0,W=+ N=0 N=0.1 N= W Figure 1: Return-volatility correlation of risky asset price under various levels of short sale and borrowing constraint 4. Data We collect Chinese stock data and the designated short sale list from the GuoTaiAn database. We only keep data on A-share stocks traded on the Chinese main board. We exclude the stocks traded in the China Growth Enterprise Market which begin its trading activities on October 23, 2009, significantly after the moment of our earliest data selection. Turnover rate data and the FAMA three-factor data are obtained from the Wind Investor Terminal (WIT). Short sale practices started on March 31, 2010 in China. Only a list of designated securities that meet certain requirements can be sold short. Up to December 31, 2015, there are 756 securities allowed for short sale through the margin trading and security lending policy, including 742 stocks and 14 ETFs listed on the Shanghai and Shenzhen Stock Exchanges. There are a total of 2,674 A-share stocks traded on the Chinese main board up to December 31, Among them, 742 stocks are on the designated list, and 1,932 stocks are not. There are 2,674 A-share stocks in total in our samples as of December 31, Starting from January 1, 2002, four years of data are rolled to estimate the FAMA three-factor model to obtain excess returns. Thus the 10

11 actual effective period is January, 2006 to December, 2015, 10 years in total; four years before the launch of margin financing and security lending, and six years after that. Only those stocks with more than sixty days of data are kept after the calculation of excess returns. Thus, we end up with 2,010 stocks in our samples (865 are on the designated list and 1,145 are not). 5. Empirical study In this section, we want to show empirically that the reverse returnvolatility asymmetry in the Chinese financial markets are all caused at least partially by short sale constraints. First, we present the data processing method Data processing In order to confirm the existence and the robustness of positive risk-return correlation in the Chinese markets, we conduct our experiment on the raw stock returns, excess returns, and normalized excess returns. Excess return is obtained from the FAMA three-factor model for each individual stock. ɛ t = y t (α + β 1 r t1 + β 2 r t2 + β 3 r t3 ) (6) where ɛ t is the excess return (or innovation), r 1, r 2 and r 3 respectively, represent one of the FAMA three factors, namely market return, market capitalization, and price-to-book ratio. The parameters α and βs are estimated from a four-year rolling time window and the excess return is computed for the next one quarter. In this way, the parameters are gradually adapted and do not cause large disruption to ɛ t. We then normalize the excess return by dividing its conditional standard deviation obtained from the EGARCH model. We assume that ɛ t is sampled from a normal distribution N (0, σt 2 ). For the evolution equation of σt 2, as has been documented by many researchers to be suitable for the financial time series in the Chinese markets, the EGARCH (1, 1) model is deployed here to estimate the variances of residuals. log σ 2 t = κ + γ 1 log σ 2 t 1 + η 1 [ ɛ t 1 σ t 1 2 π ] + ξ 1( ɛ t 1 σ t 1 ) (7) By using the conditional volatility estimated from EGARCH model, we can get the normalized excess return ɛ t = ɛ t /σ t. 11

12 The excess return ɛ t keeps the idiosyncratic innovations of each stock while the market part is removed. The normalization of excess return removes the effect of time-varying volatility by dividing the conditional volatility estimated from the EGARCH model. The other advantage of the normalization is that the leverage effect has also been removed after the excess return is normalized by the EGARCH model, which has a leverage term. By the assumption of the EGARCH model, we know that even if a leverage or an anti-leverage effect is present, ɛ t is still symmetrically distributed if it is sampled from N (0, σt 2 ). Therefore, if we can still observe asymmetry in the normalized excess returns, the asymmetry is not due to the leverage or anti-leverage effect described in the EGARCH model Effects of short sales and margin trading on return-volatility relationship The usual way to study the effect of short sale constraints is to take only a few dozen days around the events and thus the time effect and other market events may not be distracting issues. In our study, the time span is rather long (i.e., four years before the first launch of security lending policy and four years after). In order to exclude the possibility that positive return-volatility correlation is caused by factors other than the short sale constraints, we include those stocks which are not yet eligible for short sales for comparison (thanks to the slow adjustment of the security lending policy). Therefore, we divide the stocks into two groups, namely, treatment group which includes stocks that are selected for short sales practice at least once as of March 31, 2014, and control group which includes stocks that are never eligible for short sales. The dependent variables are the three return-volatility correlations measured using the raw returns, excess returns and normalized excess returns. The raw returns is noisy with the market component in it. The excess returns are cleaned and more suitable to examine the return-volatility correlation after we remove the impact from the market. The normalized excess returns are even better after we remove the time-varying volatility. We estimate the variables in non-overlapping time windows spanning from January 1, 2006 to December 31, As a robustness check, we use two time windows to prepare the samples (i.e., quarterly and half-year time windows). If the samples are calculated for every quarter, there are 40 samples for each stock, denoting 40 quarters from January 1, 2006 to December 31, If we prepare the samples for every half-year, then there are 20 samples. 12

13 5.3. Univariate analysis First, we present the average return-volatility correlation for the control group and treatment group. The results are reported in Table.? and Table. 2 for the quarterly samples and semi-annual samples, respectively. We can see that all return-volatility correlations are positive on a 1% significance level. The return-volatility correlations of the excess returns and normalized excess returns are particularly larger than 0 after we remove the market part from the firm-level returns. These results confirm the reverse return-volatility asymmetry in the Chinese markets documented in previous literature. Then, we check the change in the return-volatility correlations before and after the short sale practice. For the control group, we simply divide the samples according to the starting date of the margin trading and security lending policy. The change for correlation measured using the normalized excess return in the control group is nearly 0, while the change of the correlation measured using excess returns is also very small (the statistical significance at 5% level is largely due to the time-varying of the volatility). In contrast, in the treatment group, the return-volatility correlations have significantly reduced when measured using both the excess returns and normalized excess returns. From these comparison, we can see that the short sale policy has substantially reduced the return-volatility correlations. However, the short sale and margin trading are simultaneously implemented in the Chinese markets. We do not know from the univariate analysis which one contributes to the reduction in the return-volatility correlations. In addition, the reduction may be caused by other reasons or by mere time effect. We will conduct a panel regression for a rigorous analysis in the next subsection. 13

14 14 Table 1: Results for univariate analysis conducted on quarterly samples. Raw returns Excess returns Normalized excess returns control group before the policy 0.041*** (15.0) 0.254*** (91.8) 0.31*** (105) control group after the policy 0.022*** (9.06) 0.247*** (104) 0.31*** (122.8) control group change *** (-5.08) ** (-2.06) (-0.027) treatment group before short sale 0.052*** (24.27) 0.24*** (109) 0.29*** (123.5) treatment group after short sale 0.089*** (22.76) 0.21*** (51.8) 0.25*** (57.6) treatment group change 0.037*** (8.25) *** (-7.81) *** (-8.51) Note:Triple asterisks(***)denote significance at 1% critical level, double asterisks(**) at 5% level, and single asterisks(*) at 10% level respectively.

15 15 Table 2: Results for univariate analysis conducted on semi-annual samples. Raw returns Excess returns Normalized excess returns control group before the policy 0.050*** (15.1) 0.28*** (82.1) 0.33*** (91.4) control group after the policy 0.034*** (128.4) 0.27*** (110) 0.33*** (128) control group change *** (-3.95) (-1.17) (1.34) treatment group before short sale 0.064*** (25.6) 0.27*** (106) 0.31*** (117) treatment group after short sale 0.13*** (28.8) 0.23*** (50.6) 0.27*** (55.5) treatment group change 0.062*** (12.3) *** (-6.73) *** (-6.56)

16 5.4. Panal regression results The stocks are different in the treatment group and control group. We use the FAMA three factors as control variables (beta 1, beta 2 and beta 3) to account for the different characteristics between the two groups. We formalize the empirical study in the framework of panel regression. We use the three correlation as explained above as the dependent variables. Explanatory variables include turnover rate (which is the ratio of daily volume to shares outstanding) to proxy for the difference of opinions, return and volatility for each stock, a dummy variable v to represent treatment group vs control group and a dummy variable u to represent short sale practiced. The other two sets of dummy variables account for the fixed effects of industry and time. There are 18 industry categories in total according to the China Securities Regulatory Commission). The designated list is frequently adjusted. Remember that treatment group refers to stocks that are selected for short sales practice at least once as of December 31, There are stocks in treatment group which are not practiced for short sales yet in a time window 1. Since a stock can be allowed for short sale within a time window, we disregard those samples if only a faction of the time a stock is practiced for short sales 2. The regression results are reported in Table 4 and Table 5 for quarterly and semi-annul sampling periods, respectively. The results are similar despite the sampling windows. For each dependent variable, we have two regression equations, Eq. 1 uses the dummy for short sale practice only, and Eq. 2 uses the short sale practice plus the margin rate and short sale rate. As we predict in the theoretic session that margin trading can increase the return-volatility correlation. The impact of short sale and margin trading on return-volatility correlation is opposite. If we observe the reduction of return-volatility correlation, it must be caused by the lifting of short sale constraints. As shown in the Tables, the return-volatility correlation measured using normalized excess returns decreased significantly at a 1% level after short sale is practiced when it is regressed alone. While we include the margin rate and short sale rate in the regression, short sale is still significant less than 0 at 10% level. As predicted, the margin rate has a positive contribution, and the short rate 1 For those stocks in the control group, short sales are never practiced. 2 We also conducted the test by treating a sample with short sale practiced if there are more than half of the days in the given time window the stock is allowed for short selling. The results are not reported in this paper but they do not show big difference. 16

17 Table 3: Descriptive statistics of some of the explanatory and control variables. semi-annual quarterly mean (std.) mean (std.) beta (0.23) (0.24) beta (0.55) (0.571) beta (0.58) (0.60) turnover 3.38 (3.575) 3.36 (3.52) excess return (0.41) (0.29) volatility (0.0298) (0.030) dummy for short sale 0.24 (0.429) 0.26 (0.44) margin rate 1.37% (0.032%) 1.47% (0.033%) short rate % ( %) 0.009% (0.0057%) dummy for shortable (0.499) (0.499) is negative. These results give the direct evidence to support our hypothesis that the short sale constraints are responsible for the reverse return-volatility asymmetry we have observed in the Chinese markets. 17

18 18 Table 4: The table reports the regression conducted on quarterly samples. Dependent variables are the return-volatility correlations measured using raw returns, excess returns, and normalized excess returns, respectively. Explanatory variables include, average daily turnover in a quarter, cumulative return and standard deviation of daily returns in a quarter, a dummy variable for designated list, and a dummy variable for short sale practice, margin rate, and short sale rate. All regressions include dummies (unreported) for each quarter and industry. Raw Returns Excess Returns Normalized Excess Returns Eq. 1 Eq. 2 Eq. 1 Eq. 2 Eq. 1 Eq. 2 beta *** -0.13*** * * (-25.6) (-28.74) (-1.23) (-1.23) (-1.83) (-1.77) beta *** *** *** *** 0.013*** 0.015*** (-19.11) (-20.26) (3.02) (3.22) (4.51) (5.04) beta *** *** * * (-5.27) (-5.14) (0.86) (0.84) (-1.74) (-1.8) turnover *** *** (30.29) (30.01) (0.63) (0.66) (0.82) (0.99) excess return 0.44*** 0.44*** (98.34) (98.62) (1.63) (1.59) (0.42) (0.3) volatility -1.47*** -1.48*** (-27.28) (-27.52) (0.4) (0.43) (-0.049) (0.069) dummy for shortable 0.043*** 0.042*** *** * *** *** (18.02) (17.58) (4.23) (-1.75) (-5.33) (-5.09) dummy for short sale * *** *** *** * *** (-1.85) (-7.03) (-3.72) (-4.99) (-1.76) (-4.665) margin rate -0.38*** *** (-5.29) (0.65) (2.67) shortrate 8.3*** -7.7** -6.9* (3.14) (-2.19) (-1.88) Obj No. of Stocks R-square adjusted F-statistics Time fixed effect YES YES YES YES YES YES Industry fixed effect YES YES YES YES YES YES Note:Triple asterisks(***)denote significance at 1% critical level, double asterisks(**) at 5% level, and single asterisks(*) at 10% level respectively.

19 19 Table 5: The table reports the regression conducted on semi-annual samples. Dependent variables are the return-volatility correlations measured using raw returns, excess returns, and normalized excess returns, respectively. Explanatory variables include, average daily turnover in a half year, cumulative return and standard deviation of daily returns in a half year, a dummy variable for designated list, and a dummy variable for short sale practice, margin rate, and short sale rate. All regressions include dummies (unreported) for each half year and industry. Raw Returns Excess Returns Normalized Excess Returns Eq. 1 Eq. 2 Eq. 1 Eq. 2 Eq. 1 Eq. 2 beta *** -0.12*** (-22.73) (-22.85) (-0.003) (0.12) (-0.4) (-0.31) beta 2-0,045*** *** *** 0.011*** 0.025*** *** (-18.38) (-19.44) (2.69) (3.44) (7.23) (7.64) beta * * (-1.52) (-1.38) (1.129) (1.0) (1.89) (1.82) turnover *** *** (17.38) (17.15) (1.28) (1.52) (-0.35) (-0.22) excess return 0.19*** 0.19*** *** (56.49) (56.72) (-0.021) (-0.39) (1.46) (1.38) volatility (0.49) (1.28) (0.31) (0.51) (0.087) (0.18) dummy for shortable 0.039*** 0.038*** * *** *** (14.66) (14.28) (-1.94) (-1.55) (-3.89) (-3.72) dummy for short sale *** *** -0.04*** * *** (-1.29) (-6.15) (-5.23) (-3.28) (-4.24) (-3.76) margin rate -0.4*** 0.48*** 0.25** (-4.61) (4.26) (2.1) short rate 5.2* (1.71) (-0.25) (1.56) Obj No. of Stocks R-square adjusted F-statistics Time fixed effect YES YES YES YES YES YES Industry fixed effect YES YES YES YES YES YES Note:Triple asterisks(***)denote significance at 1% critical level, double asterisks(**) at 5% level, and single asterisks(*) at 10% level respectively.

20 6. Discussion and conclusion Reverse return-volatility asymmetry (i.e., positive return-volatility correlation) has been documented in the Chinese stock markets in contrast to the widely accepted theory of negative return-volatility correlation. Investors react more intensively on the upside than the downside. One might postulate that the short sale constraints are behind this anomaly since it has shown in Xu (2007) that the price is convex function of information if the short sales are constrained. However, In order to understand how the return-volatility correlation is impacted when both short sale and margin trading are practiced. We extend the theoretical analysis as in Xu (2007) to meet the reality of the Chinese stock markets: the short sale constraints are not strictly imposed; we introduce borrowing constraints to study the effect of margin trading. We calculate the return-volatility correlation of an asset as we vary the degree of short sales and borrowing constraints. The theoretical analysis shows that lifting of short sale constraints can decrease the return-volatility correlation while margin trading increases the correlation. Empirically, we examine the return-volatility correlation of daily firmlevel returns in the Chinese stock market. Three return-volatility correlations are measured using raw returns, excess returns and normalized excess returns. They are all statistically significantly positive. The normalized excess returns are cleaned and more suitable to examine the return-volatility correlation after we remove the impact from the market and time-varying volatility. The return-volatility correlations of the normalized excess returns is significantly positive after the market part is removed. Starting from March, 2010, China adopts the margin finance and security lending policy. The policy provides for a designated list of stocks for short sale practice. The adoption of the security lending policy takes place over time and it is still in process. This affords us sufficient time space from which to collect data to examine the effects of the policy. We divide the Chinese stocks into two groups, namely treatment group and control group. Treatment group includes stocks that are selected for short sales practice at least once as of December 31, 2014; Control group includes stocks that are never eligible for short sales. The univariate analysis indicates that after the margin trading and security lending policy on March 31, 2010, both the return-volatility correlations for the excess returns and normalized excess returns are statistically and economically significantly reduced. However, the 20

21 reduction is nearly zero for the return-volatility correlation of the normalized excess returns for the control group. We then conduct panel regression using daily return data of stocks allowed for short sale after they are added to a designated list and of other stocks never added to the list. Our results show that (i) the lifting of short sale constraints largely decreases the return-volatility correlation of the excess returns and normalized excess returns; Appendix A. Proof of the Proposition 1 Proof. Under condition (a), the market is relatively bullish, H investors overestimate the asset price while L investors are subject to short sale constraints. According to Eq. (4) we have λy 0,L < N, L investors can only sell N thus λy L = N. Market clearing condition requires (1 λ)y H = 1 + N. Under (i), investors H are not subject to the borrowing constraint, then the equilibrium price is determined by investor H s excessive demand alone for risky asset. Substitute y H into Eq. (4) and we get P H ; Under (ii), investors H are subject to the borrowing constraint, then the capital endowment of investor H does not allow the risky asset to be cleared out at price P H which corresponds to its demand, but equilibrium price will still be maintained above P s, the price corresponding to the demand N of investor L. At this time investor L is still affected by short sale constraints and can only sale a maximum amount of N, and the price of risky asset is determined by the initial wealth W instead of the demand 1+N of investor H, consequently the market price under borrowing constraints becomes P = W/(1 + N); Under (iii), the borrowing constraint is so strong that investor H does not have enough capital to support its demand meanwhile even a rather low price P s that corresponds to the negative demand of investor L can not be reached, the asset price would fall until investor L does not need to sort N. Now investors L are not subject to any constraint, and market price P = P M can be solved from the three simultaneous equations as below, (1 λ)y H P = W y L = ˆµ L RP a λy L + (1 λ)y H = 1 (A.1) P M is the non-negative solution to above equations, it can be proved that P M < P s, thus it is true that investor L does not need to short N. 21

22 Under condition (b), the market is relatively bearish, investors H underestimate the asset price due to overconfidence but the amount of asset they can short is limited by short sale constraint, at the same time investor L still has positive demand for risky asset and is not affected by borrowing constraint since the price is already low as explained earlier. According to 4 we have λy 0,H < N, investor H can only short a maximum of N, and the demand of investor L that is not limited by short sale constraint becomes λy L = 1 + N, consequently, the equilibrium price P = P L can be obtained according to the relationship between price and demand as in Eq. (4). Under condition (c), neither of the two investors are subject to short sale constraints, and if W P 0 y 0,H, then investor H will not be affected by borrowing constraints, and the equilibrium solution would be the same as the solution without any constraints i.e. P = P 0 ; if W < P 0 y 0,H then investors H are subject to borrowing constraints, as in (a) P = P M can be obtained by three simultaneous equations mentioned in (a). 22

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