# Can Limits to Arbitrage Explain Stock Price Idiosyncratic Volatility Premium Puzzle in China’s A-Share Market?

Authors: Xiaohui Chen; Jianhua Ye
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Investigating the existence and causes of idiosyncratic volatility premium puzzle in developing stock market can enrich the research on this asset pricing puzzle. To investigate the existence and whether limits to arbitrage in China’s A-share market can explain the idiosyncratic volatility premium puzzle, this paper uses listed stocks in China’s A-share market from 2002 to 2019 as a sample. We calculate three individual measures and one comprehensive measure of limits to arbitrage based on Chinese specific regulations. After that, we conduct univariate portfolios analysis, regression analysis, and bivariate portfolios analysis to obtain evidence. We prove that idiosyncratic volatility premium puzzle exists in China’s A-share market and is robust and that limits to arbitrage in this market can partly explain this asset pricing puzzle. This paper enriches research on asset pricing anomaly and can help us evaluate the effect of China’s A-share market reform.


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## Body

## 1. Introduction

The CAPM predicts that investors can only require compensation for bearing systematic risk because idiosyncratic risk is completely diversified away when the financial market is completely efficient. Prior studies show that investors in some markets cannot fully diversify their portfolios in practice [1]. Thus, idiosyncratic volatility (IV) will be expected to play a role in the risk-return relationship. One of the important tenets of modern portfolios theory is that systematic risk is the sole driver of returns, and thus idiosyncratic volatility cannot explain why returns of some stocks or portfolios are higher or lower than others [2]. Nevertheless, Merton [3] posited that investors cannot diversify their portfolios and require a positive risk premium for bearing idiosyncratic volatility. As a result, the relationship between idiosyncratic volatility and stock returns is positive.However, the empirical asset pricing literature showed that the sign of the relationship between idiosyncratic volatility and stock returns is one of the most controversial topics. First, most of these empirical studies argue that idiosyncratic volatility is not priced [4]. Second, other literature concludes that stocks with higher idiosyncratic volatility command higher risk premium [5]. Third, many works argue that the relationship between idiosyncratic volatility and stock returns is negative [6], and this implies that the average return of stocks with low (high) idiosyncratic volatility is higher (lower) than others. The lack of consensus on the sign of the idiosyncratic volatility premium is called idiosyncratic volatility premium puzzle.Some papers investigate the existence and causes of the idiosyncratic volatility premium puzzle, but it is also necessary for us to further investigate the causes of the idiosyncratic volatility premium puzzle in China’s A-share market. Gu et al. [7] use a sample containing all listed companies in Shanghai Stock Exchange and Shenzhen Stock Exchange, and the sample period spans from January 2002 to December 2012. They find that negative idiosyncratic volatility premium exists in China’s A-share market and this asset pricing effect is much stronger and more persistent in stocks with high limits of arbitrage. However, many new regulation changes have taken place in China’s A-share market after 2012. One of the most important reforms is the enforcement of securities margin trading regulation. On March 30, 2010, securities margin trading regulation was enforced in some stocks in China’s A-share market. During the whole year 2010, the purchasing amount of margin trading accounts for about 0.16 percent of the total trading amount in China’s A-share market, and the selling amount of margin trading is zero. The number of stocks that are available for margin trading and the number of total listed companies in China’s A-share market are 96 and 2041 at the end of year 2010, respectively. In addition, the margin trading activities increase quickly after 2010. During the whole year 2019, purchasing (selling) amount of margin trading accounts for about 8.94% (0.23%) of the total trading amount in China’s A-share market. The number of stocks that are available for margin trading and the number of total listed companies in China’s A-share market are 1761 and 3857, respectively, at the end of year 2019. During the period from 2010 to 2019, the average percentage of purchasing (selling) amount of margin trading accounts for a total trading amount of about 7.14% (0.52%). Securities margin trading is one important regulation that may have an important impact on China’s A-share market. First, securities margin trading is one important factor affecting arbitraging activities, which are the main mechanism that corrects mispricing and improves capital market efficiency.Against this background, we want to investigate two issues. First, we want to know whether the idiosyncratic volatility premium puzzle still exists in China’s A-share market as that found by Gu et al. [7] using the sample during 2002 and 2012. Second, can limits to arbitrage measures related to this new background explain this asset pricing puzzle? This paper uses listed stocks in China’s A-share market during 1999 and 2019 as sample to investigate the existence of idiosyncratic volatility premium puzzle in China’s A-share market and whether limits to arbitrage in the new background can explain this kind of asset pricing puzzle.


## 2. Materials and Methods

### 2.1. Sample

This paper uses a panel data set consisting of all listed companies in China’s A-share market. Original accounting data and stock trading data are all obtained from RESET data provider, and the variablesRi,t, MKTd, SMBd, and HMLd are used directly. The sample period is from January 2002 to December 2019. Financial sector companies, special treated companies, and companies with missing value are excluded from the original sample.


### 2.2. Data and Descriptive Statistics

Following Ang et al. [6], stock i’s idiosyncratic volatility in month t equals to stock i’s standard deviation of daily abnormal returns within that month to proxy for stock i’s idiosyncratic volatility in month t. Stock i’s daily abnormal returns are calculated based on Fama and French’s [8] three-factor model in month t. Equation (1) is the three-factor model used to calculate individual stock’s daily abnormal return.(1)Ri,d−rfd=αi+βiMKTd+siSMBd+hiHMLd+εi,d.Equation (1) is used to calculate daily abnormal return on stock i in day d, where Ri,d is the return on stock i in day d, rfd is the risk free return in day d, and MKTd, SMBd, and HMLd are Fama and French’s [8] three factors in day d. For each stock i, we estimate equation (1) every month and calculate daily abnormal return on stock i in day d.The main variables used in this paper include monthly stock return (Ri,t), equal to price change of stock i in month t divided by beginning price of stock i in month t; monthly abnormal stock return (ARi,t), equal to monthly return of stock i adjusted by Fama and French’s [8] three-factor model; ratio of equity divided by market value (BMi,t), equal to equity of company i at the beginning of the year divided by total market value of its common stock at the beginning of month t; turnover ratio (TURNi,t), equal to turnover ratio of tradable common stock during the period from month t − 5 to month t; mean of top 5 maximum daily returns of stock i in month t (MAXi,t), equal to mean of stock i’s top 5 highest stock returns within month t; and firm size lnMVi,t, equal to log of total market value of common stock i at the end of month t.Based on China’s A-share market background and regulations, we design the following four variables used to measure limits to arbitrage, three of which are individual measures and the last one is a comprehensive measure which is constructed on the first three measures.The first one is Amihud [9] illiquidity measure (AMHi,t). This measure reflects the effect of trading dollar amount on stock price. The higher value of this measure reflects less liquidity of the stock, and it is more difficult and costly for arbitragers to exploit the mispricing opportunity, and the idiosyncratic volatility premium puzzle may be more evident.The second one is the number of daily closing prices reaching price limits in the last six months (PLNi,t). China’s A-share market introduced the price limit regulation which allows for 10 percent price change in the current trading day compared with last trading day’s closing price. Chen et al. [10] argue that price limit regulation hinders price discovery. By intuition, large PLNi,t indicates low price efficiency; moreover, it is more difficult for arbitrageurs to correct mispricing, and the idiosyncratic volatility premium puzzle may be more significant.The third one is the dummy variable proxy for whether margin stock trading is allowed (MSTi,t). China’s A-share market introduced marginal security trading regulation in 2010, but this regulation is only applied in some of the trading securities. Diamond and Verrecchia [11] believe that short selling constraints hinder price discovery; the price is more efficient in market without short selling constraints. We deduce that idiosyncratic volatility premium puzzle is more significant in stocks with short selling constraints.The last one is comprehensive measure of limits to arbitrage which is constructed on the above three individual limits to arbitrage measures. We first construct three equal portfolios sorted by AMHi,t and PLNi,t each month, respectively. Second, we use discrete variables D1i,t and D2i,t, values of which are 0, 1, and 3, representing low, middle, and high level of limits to arbitrage stocks measured by AMHi,t and PLNi,t, respectively. After that, we use variable CMi,t proxies for comprehensive level of limits to arbitrage. The value of CMi,t equals to [(3 − D1i,t)/2 + D2i,t/2 + MSTi,t].Table1 reports the descriptive statistics and correlation coefficients for key variables. Panel A shows the descriptive statistics. Mean, median, and skewness of Ri,t+1 (ARi,t+1) are 0.01 (−0.00), −0.00 (−0.01), and 0.78 (0.82), respectively. This evidence indicates that distribution of stock return in China’s A-share market is right skewed. The mean values of IVOLi,t−1, BMi,t, and TURNi,t are all larger than the corresponding median values of these variables, and the corresponding skewness values of these variables are all positive. The mean values of lnMVi,t and MAXi,t are both smaller than the corresponding median values of these variables, and the corresponding skewness values of these variables are both negative. This evidence indicates that the distributions of variables IVOLi,t−1, BMi,t, and TURNi,t are all right skewed, and the distributions of variables lnMVi,t and MAXi,t are both left skewed. Evidence from panel A tells us that the number of lower returns is less than the number of higher returns.Table 1
Descriptive statistics and correlation coefficients of main characteristic variables.
Panel A: descriptive statistics of main characteristics of our sampleVarMeanMediStdQ1Q3MinMaxSkewKurtARi,t+1−0.00−0.010.10−0.060.05−1.681.220.824.82Ri,t+10.01−0.000.13−0.070.08−0.531.420.783.48IVOLi,t0.020.020.010.010.030.000.081.061.40BMi,t0.390.330.270.210.51−6.365.430.6418.82lnMVi,t21.9922.041.2621.2322.7817.4628.17−0.050.25TURNi,t45.4934.2238.3318.8359.890.41397.902.096.90MAXi,t−0.16−0.140.09−0.20−0.10−0.510.50−1.233.56Panel B: correlation coefficients of main variables (up-right is Spearman, and down-left is Pearson)VarARi,t+1Ri,t+1IVOLi,tBMi,tlnMVi,tTURNi,tMAXi,tARi,t+11.000.58∗∗∗−0.09∗∗∗0.04∗∗∗−0.05∗∗∗−0.08∗∗∗0.03∗∗∗Ri,t+10.63∗∗∗1.00−0.01∗∗∗0.05∗∗∗−0.04∗∗∗0.01∗∗0.01∗∗∗IVOLi,t−0.07∗∗∗−0.01∗∗∗1.00−0.29∗∗∗0.07∗∗∗0.46∗∗∗−0.53∗∗∗BMi,t0.02∗∗∗0.03∗∗∗−0.25∗∗∗1.00−0.07∗∗∗−0.27∗∗∗0.06∗∗∗lnMVi,t−0.03∗∗∗−0.04∗∗∗0.07∗∗∗−0.02∗∗∗1.000.000.02∗∗∗TURNi,t−0.05∗∗∗0.01∗∗0.41∗∗∗−0.24∗∗∗0.01∗∗1.00−0.41∗∗∗MAXi,t0.00∗∗0.02∗∗∗−0.52∗∗∗0.06∗∗∗0.01∗∗−0.37∗∗∗1.00Note.∗∗, ∗∗∗Statistical significance at 5% and 1% levels, respectively.Panel B shows us the Spearman and Pearson correlation coefficients of main variables. The up-right part is Spearman correlation coefficients and the down-left part is Pearson correlation coefficients. The Pearson correlation coefficients of IVOLi,t−1 with ARi,t+1 and Ri,t+1 are −0.09 and −0.01, respectively, and both are statistically significant at 1% level. The corresponding Spearman correlation coefficients are −0.07 and −0.01, and both are statistically significant at 1% level. All these pieces of evidence primarily indicate that firms with higher idiosyncratic volatility are always followed by lower subsequent stock return. The evidence from panel B also tells us that next month’s return of firms with higher book-to-market ratio and firms with higher maximum daily return is higher than that of other firms. Furthermore, returns of firms with bigger size and higher turnover are always lower than those of other firms in the nearest future month.The analysis above tells us the basic characteristics of our sample, and the correlation analysis primarily tells us that stock price idiosyncratic volatility is negatively correlated with next month’s stock return.


### 2.3. Methodology

In Section3.1, we conduct univariate portfolios analysis and regression analysis to obtain more evidence on the existence of idiosyncratic volatility premium puzzle. We construct 5 and 10 portfolios based on IVOLi,t−1 and calculate equally weighted and market value-weighted mean of each portfolio’s monthly return. In the regression section, we first construct one asset pricing factor IVOLFt−1 based on IVOLi,t−1, introduce this factor into Fama and French’s [8] regression, and conduct Fama and MacBeth’s [12] stepwise regression. The results of these two tests are reported in Tables 2 and 3 , respectively.Table 2
Mean of equally and market value-weighted portfolios returns when sorted by IVOLi,t−1.
Panel A: equally and market value-weighted return of five equal portfolios sorted by IVOLi,t−1PortfoliosQ1Q2Q3Q4Q5Q1–Q5t-statEqually weightedRt1.23∗∗∗1.28∗∗∗1.18∗∗0.90∗∗∗−0.18∗∗1.4217.09ARt0.32∗∗0.33∗∗0.21∗∗−0.09∗−1.19∗∗∗1.5124.67Market value weightedRt1.87∗∗∗1.97∗∗∗2.22∗∗∗2.164∗∗∗1.51∗∗∗0.374.30ARt0.97∗∗∗0.78∗∗0.93∗∗∗0.66∗∗−0.42∗∗∗1.3920.32Panel B: equally and market value-weighted returns of ten equal portfolios sorted by IVOLi,t−1PortfoliosD1D2D3D4D5D6D7D8D9D10D10–D1t-statEqually weightedRt1.19∗∗∗1.27∗∗∗1.28∗∗∗1.291.25∗∗1.11∗∗1.060.73∗0.35∗∗−0.72∗∗∗1.9116.20ARt0.31∗∗∗0.33∗∗∗0.33∗∗∗0.330.26∗∗∗0.16∗∗0.10−0.27−0.65∗∗∗−1.73∗∗∗2.0423.17Market value weightedRt1.69∗∗∗2.07∗∗∗1.60∗∗∗2.332.44∗∗∗1.99∗∗∗2.202.12∗∗∗2.04∗∗∗0.91∗∗0.786.41ARt0.92∗∗∗1.03∗∗∗0.50∗1.071.14∗∗∗0.71∗∗0.730.60∗0.32∗−1.24∗∗∗2.1621.95Note.∗, ∗∗, ∗∗∗Statistical significance at 10%, 5%, and 1% levels, respectively.Table 3
Fama and MacBeth [12] regression results.
Equal weightMarket value weightedIntercept0.003∗∗∗ (6.83)0.001 (1.38)0.007∗∗∗ (16.03)0.003∗∗∗ (3.98)IVOLF−0.013∗∗∗ (−4.50)−0.306∗∗∗ (−4.67)MKT0.985∗∗∗ (98.24)0.972∗∗∗ (90.32)0.977∗∗∗ (88.22)0.957∗∗∗ (79.51)SMB0.777∗∗∗ (54.30)0.748∗∗∗ (40.66)0.717∗∗∗ (45.48)0.683∗∗∗ (33.84)HML−0.257∗∗∗ (−14.60)−0.262∗∗∗ (−14.76)−0.351∗∗∗ (−17.11)−0.353∗∗∗ (−17.24)Adj_R249.23050.79147.95049.931Note.∗∗∗Statistical significance at 1% level.In Section3.2, we conduct two tests to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market. First, we conduct bivariate portfolio analysis based on IVOLi,t−1 and comprehensive measure of limits to arbitrage CMi,t. Second, we conduct bivariate portfolio analysis based on IVOLi,t−1 and individual measures of limits to arbitrage AMHi,t, PLNi,t, and MSTi,t. The results of these two tests are reported in Tables 4 and 5 , respectively.Table 4
Mean returns of each portfolio sorted by CMi,t and IVOLi,t−1.
CMReturnD1D2D3D4D5D6D7D8D9D10D1–D10TEqually weightedL-CMi,tRi,t1.581.701.731.661.821.781.761.611.541.010.572.97ARi,t0.630.680.750.680.820.800.750.630.51−0.060.684.97M-CMi,tRi,t0.911.101.211.211.111.140.860.680.61−0.020.934.62ARi,t0.100.280.380.280.210.19−0.09−0.28−0.31−1.041.157.69H-CMi,tRi,t0.660.580.720.710.580.40−0.04−0.08−0.67−1.512.1710.27ARi,t−0.23−0.40−0.30−0.34−0.44−0.54−1.04−1.10−1.67−2.552.3114.23Market value weightedL-CMi,tRi,t1.881.892.832.212.452.692.732.332.592.40−0.53−0.66ARi,t0.730.791.630.981.251.351.371.121.210.590.140.96M-CMi,tRi,t1.671.682.042.372.022.131.711.731.951.753−0.08−0.39ARi,t1.230.930.991.081.060.770.200.220.19−0.461.6910.37H-CMi,tRi,t1.521.482.682.291.861.851.962.061.460.211.315.91ARi,t0.610.341.190.790.640.490.290.21−0.12−2.192.8115.33Table 5
Return differences between extreme IVOLi,t−1 portfolios within different portfolios sorts by individual limits to arbitrage measure.
Panel A: equally weighted extreme portfolios’ mean return differencesIndividual limits to arbitrage measuresAMHi,tPLNi,tMSTi,tLevel of limits to arbitrageReturnsP1–P10t-statisticsP1–P10t-statisticsP1–P10t-statisticsLowestRi,t0.854.140.512.971.455.58ARi,t1.057.090.685.411.748.14MiddleRi,t1.447.121.346.42ARi,t1.469.831.318.42HighestRi,t2.3711.772.4310.632.605.26ARi,t2.5416.112.3914.053.048.18Panel B: market value equally weighted extreme portfolios’ mean return differencesIndividual limits to arbitrage measuresAMHi,tPLNi,tMSTi,tLevel of limits to arbitrageReturnsP1–P10t-statisticsP1–P10t-statisticsP1–P10t-statisticsLowestRi,t−1.30−5.77−0.44−2.76−0.15−0.52ARi,t0.261.560.292.221.094.41MiddleRi,t−0.60−2.800.903.96ARi,t0.714.330.784.29HighestRi,t1.497.052.6810.570.510.89ARi,t3.2518.252.5312.775.1211.53


## 2.1. Sample

This paper uses a panel data set consisting of all listed companies in China’s A-share market. Original accounting data and stock trading data are all obtained from RESET data provider, and the variablesRi,t, MKTd, SMBd, and HMLd are used directly. The sample period is from January 2002 to December 2019. Financial sector companies, special treated companies, and companies with missing value are excluded from the original sample.


## 2.2. Data and Descriptive Statistics

Following Ang et al. [6], stock i’s idiosyncratic volatility in month t equals to stock i’s standard deviation of daily abnormal returns within that month to proxy for stock i’s idiosyncratic volatility in month t. Stock i’s daily abnormal returns are calculated based on Fama and French’s [8] three-factor model in month t. Equation (1) is the three-factor model used to calculate individual stock’s daily abnormal return.(1)Ri,d−rfd=αi+βiMKTd+siSMBd+hiHMLd+εi,d.Equation (1) is used to calculate daily abnormal return on stock i in day d, where Ri,d is the return on stock i in day d, rfd is the risk free return in day d, and MKTd, SMBd, and HMLd are Fama and French’s [8] three factors in day d. For each stock i, we estimate equation (1) every month and calculate daily abnormal return on stock i in day d.The main variables used in this paper include monthly stock return (Ri,t), equal to price change of stock i in month t divided by beginning price of stock i in month t; monthly abnormal stock return (ARi,t), equal to monthly return of stock i adjusted by Fama and French’s [8] three-factor model; ratio of equity divided by market value (BMi,t), equal to equity of company i at the beginning of the year divided by total market value of its common stock at the beginning of month t; turnover ratio (TURNi,t), equal to turnover ratio of tradable common stock during the period from month t − 5 to month t; mean of top 5 maximum daily returns of stock i in month t (MAXi,t), equal to mean of stock i’s top 5 highest stock returns within month t; and firm size lnMVi,t, equal to log of total market value of common stock i at the end of month t.Based on China’s A-share market background and regulations, we design the following four variables used to measure limits to arbitrage, three of which are individual measures and the last one is a comprehensive measure which is constructed on the first three measures.The first one is Amihud [9] illiquidity measure (AMHi,t). This measure reflects the effect of trading dollar amount on stock price. The higher value of this measure reflects less liquidity of the stock, and it is more difficult and costly for arbitragers to exploit the mispricing opportunity, and the idiosyncratic volatility premium puzzle may be more evident.The second one is the number of daily closing prices reaching price limits in the last six months (PLNi,t). China’s A-share market introduced the price limit regulation which allows for 10 percent price change in the current trading day compared with last trading day’s closing price. Chen et al. [10] argue that price limit regulation hinders price discovery. By intuition, large PLNi,t indicates low price efficiency; moreover, it is more difficult for arbitrageurs to correct mispricing, and the idiosyncratic volatility premium puzzle may be more significant.The third one is the dummy variable proxy for whether margin stock trading is allowed (MSTi,t). China’s A-share market introduced marginal security trading regulation in 2010, but this regulation is only applied in some of the trading securities. Diamond and Verrecchia [11] believe that short selling constraints hinder price discovery; the price is more efficient in market without short selling constraints. We deduce that idiosyncratic volatility premium puzzle is more significant in stocks with short selling constraints.The last one is comprehensive measure of limits to arbitrage which is constructed on the above three individual limits to arbitrage measures. We first construct three equal portfolios sorted by AMHi,t and PLNi,t each month, respectively. Second, we use discrete variables D1i,t and D2i,t, values of which are 0, 1, and 3, representing low, middle, and high level of limits to arbitrage stocks measured by AMHi,t and PLNi,t, respectively. After that, we use variable CMi,t proxies for comprehensive level of limits to arbitrage. The value of CMi,t equals to [(3 − D1i,t)/2 + D2i,t/2 + MSTi,t].Table1 reports the descriptive statistics and correlation coefficients for key variables. Panel A shows the descriptive statistics. Mean, median, and skewness of Ri,t+1 (ARi,t+1) are 0.01 (−0.00), −0.00 (−0.01), and 0.78 (0.82), respectively. This evidence indicates that distribution of stock return in China’s A-share market is right skewed. The mean values of IVOLi,t−1, BMi,t, and TURNi,t are all larger than the corresponding median values of these variables, and the corresponding skewness values of these variables are all positive. The mean values of lnMVi,t and MAXi,t are both smaller than the corresponding median values of these variables, and the corresponding skewness values of these variables are both negative. This evidence indicates that the distributions of variables IVOLi,t−1, BMi,t, and TURNi,t are all right skewed, and the distributions of variables lnMVi,t and MAXi,t are both left skewed. Evidence from panel A tells us that the number of lower returns is less than the number of higher returns.Table 1
Descriptive statistics and correlation coefficients of main characteristic variables.
Panel A: descriptive statistics of main characteristics of our sampleVarMeanMediStdQ1Q3MinMaxSkewKurtARi,t+1−0.00−0.010.10−0.060.05−1.681.220.824.82Ri,t+10.01−0.000.13−0.070.08−0.531.420.783.48IVOLi,t0.020.020.010.010.030.000.081.061.40BMi,t0.390.330.270.210.51−6.365.430.6418.82lnMVi,t21.9922.041.2621.2322.7817.4628.17−0.050.25TURNi,t45.4934.2238.3318.8359.890.41397.902.096.90MAXi,t−0.16−0.140.09−0.20−0.10−0.510.50−1.233.56Panel B: correlation coefficients of main variables (up-right is Spearman, and down-left is Pearson)VarARi,t+1Ri,t+1IVOLi,tBMi,tlnMVi,tTURNi,tMAXi,tARi,t+11.000.58∗∗∗−0.09∗∗∗0.04∗∗∗−0.05∗∗∗−0.08∗∗∗0.03∗∗∗Ri,t+10.63∗∗∗1.00−0.01∗∗∗0.05∗∗∗−0.04∗∗∗0.01∗∗0.01∗∗∗IVOLi,t−0.07∗∗∗−0.01∗∗∗1.00−0.29∗∗∗0.07∗∗∗0.46∗∗∗−0.53∗∗∗BMi,t0.02∗∗∗0.03∗∗∗−0.25∗∗∗1.00−0.07∗∗∗−0.27∗∗∗0.06∗∗∗lnMVi,t−0.03∗∗∗−0.04∗∗∗0.07∗∗∗−0.02∗∗∗1.000.000.02∗∗∗TURNi,t−0.05∗∗∗0.01∗∗0.41∗∗∗−0.24∗∗∗0.01∗∗1.00−0.41∗∗∗MAXi,t0.00∗∗0.02∗∗∗−0.52∗∗∗0.06∗∗∗0.01∗∗−0.37∗∗∗1.00Note.∗∗, ∗∗∗Statistical significance at 5% and 1% levels, respectively.Panel B shows us the Spearman and Pearson correlation coefficients of main variables. The up-right part is Spearman correlation coefficients and the down-left part is Pearson correlation coefficients. The Pearson correlation coefficients of IVOLi,t−1 with ARi,t+1 and Ri,t+1 are −0.09 and −0.01, respectively, and both are statistically significant at 1% level. The corresponding Spearman correlation coefficients are −0.07 and −0.01, and both are statistically significant at 1% level. All these pieces of evidence primarily indicate that firms with higher idiosyncratic volatility are always followed by lower subsequent stock return. The evidence from panel B also tells us that next month’s return of firms with higher book-to-market ratio and firms with higher maximum daily return is higher than that of other firms. Furthermore, returns of firms with bigger size and higher turnover are always lower than those of other firms in the nearest future month.The analysis above tells us the basic characteristics of our sample, and the correlation analysis primarily tells us that stock price idiosyncratic volatility is negatively correlated with next month’s stock return.


## 2.3. Methodology

In Section3.1, we conduct univariate portfolios analysis and regression analysis to obtain more evidence on the existence of idiosyncratic volatility premium puzzle. We construct 5 and 10 portfolios based on IVOLi,t−1 and calculate equally weighted and market value-weighted mean of each portfolio’s monthly return. In the regression section, we first construct one asset pricing factor IVOLFt−1 based on IVOLi,t−1, introduce this factor into Fama and French’s [8] regression, and conduct Fama and MacBeth’s [12] stepwise regression. The results of these two tests are reported in Tables 2 and 3 , respectively.Table 2
Mean of equally and market value-weighted portfolios returns when sorted by IVOLi,t−1.
Panel A: equally and market value-weighted return of five equal portfolios sorted by IVOLi,t−1PortfoliosQ1Q2Q3Q4Q5Q1–Q5t-statEqually weightedRt1.23∗∗∗1.28∗∗∗1.18∗∗0.90∗∗∗−0.18∗∗1.4217.09ARt0.32∗∗0.33∗∗0.21∗∗−0.09∗−1.19∗∗∗1.5124.67Market value weightedRt1.87∗∗∗1.97∗∗∗2.22∗∗∗2.164∗∗∗1.51∗∗∗0.374.30ARt0.97∗∗∗0.78∗∗0.93∗∗∗0.66∗∗−0.42∗∗∗1.3920.32Panel B: equally and market value-weighted returns of ten equal portfolios sorted by IVOLi,t−1PortfoliosD1D2D3D4D5D6D7D8D9D10D10–D1t-statEqually weightedRt1.19∗∗∗1.27∗∗∗1.28∗∗∗1.291.25∗∗1.11∗∗1.060.73∗0.35∗∗−0.72∗∗∗1.9116.20ARt0.31∗∗∗0.33∗∗∗0.33∗∗∗0.330.26∗∗∗0.16∗∗0.10−0.27−0.65∗∗∗−1.73∗∗∗2.0423.17Market value weightedRt1.69∗∗∗2.07∗∗∗1.60∗∗∗2.332.44∗∗∗1.99∗∗∗2.202.12∗∗∗2.04∗∗∗0.91∗∗0.786.41ARt0.92∗∗∗1.03∗∗∗0.50∗1.071.14∗∗∗0.71∗∗0.730.60∗0.32∗−1.24∗∗∗2.1621.95Note.∗, ∗∗, ∗∗∗Statistical significance at 10%, 5%, and 1% levels, respectively.Table 3
Fama and MacBeth [12] regression results.
Equal weightMarket value weightedIntercept0.003∗∗∗ (6.83)0.001 (1.38)0.007∗∗∗ (16.03)0.003∗∗∗ (3.98)IVOLF−0.013∗∗∗ (−4.50)−0.306∗∗∗ (−4.67)MKT0.985∗∗∗ (98.24)0.972∗∗∗ (90.32)0.977∗∗∗ (88.22)0.957∗∗∗ (79.51)SMB0.777∗∗∗ (54.30)0.748∗∗∗ (40.66)0.717∗∗∗ (45.48)0.683∗∗∗ (33.84)HML−0.257∗∗∗ (−14.60)−0.262∗∗∗ (−14.76)−0.351∗∗∗ (−17.11)−0.353∗∗∗ (−17.24)Adj_R249.23050.79147.95049.931Note.∗∗∗Statistical significance at 1% level.In Section3.2, we conduct two tests to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market. First, we conduct bivariate portfolio analysis based on IVOLi,t−1 and comprehensive measure of limits to arbitrage CMi,t. Second, we conduct bivariate portfolio analysis based on IVOLi,t−1 and individual measures of limits to arbitrage AMHi,t, PLNi,t, and MSTi,t. The results of these two tests are reported in Tables 4 and 5 , respectively.Table 4
Mean returns of each portfolio sorted by CMi,t and IVOLi,t−1.
CMReturnD1D2D3D4D5D6D7D8D9D10D1–D10TEqually weightedL-CMi,tRi,t1.581.701.731.661.821.781.761.611.541.010.572.97ARi,t0.630.680.750.680.820.800.750.630.51−0.060.684.97M-CMi,tRi,t0.911.101.211.211.111.140.860.680.61−0.020.934.62ARi,t0.100.280.380.280.210.19−0.09−0.28−0.31−1.041.157.69H-CMi,tRi,t0.660.580.720.710.580.40−0.04−0.08−0.67−1.512.1710.27ARi,t−0.23−0.40−0.30−0.34−0.44−0.54−1.04−1.10−1.67−2.552.3114.23Market value weightedL-CMi,tRi,t1.881.892.832.212.452.692.732.332.592.40−0.53−0.66ARi,t0.730.791.630.981.251.351.371.121.210.590.140.96M-CMi,tRi,t1.671.682.042.372.022.131.711.731.951.753−0.08−0.39ARi,t1.230.930.991.081.060.770.200.220.19−0.461.6910.37H-CMi,tRi,t1.521.482.682.291.861.851.962.061.460.211.315.91ARi,t0.610.341.190.790.640.490.290.21−0.12−2.192.8115.33Table 5
Return differences between extreme IVOLi,t−1 portfolios within different portfolios sorts by individual limits to arbitrage measure.
Panel A: equally weighted extreme portfolios’ mean return differencesIndividual limits to arbitrage measuresAMHi,tPLNi,tMSTi,tLevel of limits to arbitrageReturnsP1–P10t-statisticsP1–P10t-statisticsP1–P10t-statisticsLowestRi,t0.854.140.512.971.455.58ARi,t1.057.090.685.411.748.14MiddleRi,t1.447.121.346.42ARi,t1.469.831.318.42HighestRi,t2.3711.772.4310.632.605.26ARi,t2.5416.112.3914.053.048.18Panel B: market value equally weighted extreme portfolios’ mean return differencesIndividual limits to arbitrage measuresAMHi,tPLNi,tMSTi,tLevel of limits to arbitrageReturnsP1–P10t-statisticsP1–P10t-statisticsP1–P10t-statisticsLowestRi,t−1.30−5.77−0.44−2.76−0.15−0.52ARi,t0.261.560.292.221.094.41MiddleRi,t−0.60−2.800.903.96ARi,t0.714.330.784.29HighestRi,t1.497.052.6810.570.510.89ARi,t3.2518.252.5312.775.1211.53


## 3. Results and Discussion

### 3.1. Empirical Evidence on the Existence of Idiosyncratic Volatility Premium Puzzle in China’s A-Share Market

We conduct univariate portfolios analysis to obtain evidence on the existence of idiosyncratic volatility premium puzzle in China’s A-share market. Each month, all the stocks are sorted into 5 and 10 equal portfolios based on IVOLi,t−1, respectively. Then, we calculate equally weighted and tradable common stock market value-weighted return of each portfolio. After that, we calculate time series mean of each portfolio return and the corresponding mean return differences between extreme portfolios. Q1 (Q5) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing five portfolios. D1 (D10) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing 10 portfolios.Table2 reports equally weighted and market value-weighted time series means of each portfolio’s returns, sorted by IVOLi,t−1. Panel A reports returns of five portfolios sorted by IVOLi,t−1. These results show that the higher (lower) the IVOLi,t−1, the lower (higher) the returns of the portfolios. Equally weighted Rt (ARt) difference between portfolio Q1 and Q5 is 1.42 (1.51), and the corresponding t-statistics is 17.09 (24.67). The tradable common stock market value-weighted Rt (ARt) difference between portfolio Q1 and Q5 is 0.37 (1.39), and the corresponding t-statistics is 4.30 (20.32).Panel B reports returns of ten portfolios sorted by IVOLi,t−1. These results show that the higher (lower) the IVOLi,t−1, the lower (higher) the returns of the portfolios. Equally weighted Rt (ARt) difference between portfolio D1 and D10 is 1.91 (2.04), and the corresponding t-statistics is 16.20 (23.17). Tradable common stock market value-weighted Rt (ARt) difference between portfolio D1 and D10 is 0.78 (2.16), and the corresponding t-statistics is 6.41 (21.95).In summary, evidence from univariate portfolios analysis tells us that idiosyncratic volatility can negatively predict future stock return; the higher the IVOLi,t−1, the lower the future stock return. The following regression analysis will provide us with more related evidence.In the Section2, we construct idiosyncratic volatility pricing factor (IVOLFt).The calculation process used to calculate VOLFi,t asset pricing factor consists of the following three steps. First, we sort and construct five portfolios based on IVOLi,t each month. Second, we calculate market value-weighted mean return of stocks in each portfolio as the return of that portfolio in that month. Third, we calculate extreme portfolios return difference (IVOLFt), equal to return of the largest IVOLi,t portfolio minus return of the lowest IVOLi,t portfolio in each month. Furthermore, we define IVOLFt as the idiosyncratic volatility asset pricing factor. We introduce this IVOLF into Fama and French’s [8] three-factor asset pricing model and conduct Fama and MacBeth’s [12] regression to test the asset pricing effect of the IVOLF. The corresponding stepwise regression results are presented in Table 3.Table3 reports the Fama and MacBeth [12] stepwise regression results. We can see that the estimated regression coefficients of IVOLF in equally weighted and market value-weighted regression results are −0.01 and −0.31, respectively, and both are significant at 1% level. What is more, changes of intercept (Adj_R2) before and after the introduction of VOLF in equally weighted and market value-weighted regression results are 0.00 (1.56%) and 0.00 (1.98%), respectively. All these pieces of evidence above show us that idiosyncratic volatility factor has a significant asset pricing effect.In summary, the evidence from the descriptive, correlation, and regression analysis proves that stock price idiosyncratic volatility can negatively predict future stock return in China’s A-share market. In the following section, we will conduct bivariate analysis to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market.


### 3.2. Empirical Evidence on Whether Limits to Arbitrage Can Explain Idiosyncratic Volatility Premium Puzzle in China’s A-Share Market

We perform bivariate sorts on CMi,t and IVOLi,t−1 to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market. In each month, we first constructed low, middle, and high level of limits to arbitrage portfolios sorted by CMi,t, and the three portfolios are indicated by L-CMi,t, M-CMi,t, and H-CMi,t, respectively. Second, we constructed ten portfolios sorted by IVOLi,t−1 within each CMi,t portfolio every month. D1 (D10) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing 10 portfolios. After this, we constructed thirty portfolios each month. Third, we calculate equally weighted and market value-weighted mean of Ri,t and ARi,t of the thirty portfolios each month. At last, we calculate the time series mean of equally weighted and market value-weighted return of each portfolio and the time series mean of equally weighted and market value-weighted return difference between extreme portfolios. The results are presented in Table 4.We can see that the equally weighted mean differences ofRi,t (ARi,t) between extreme portfolios within low (L-CMi,t), middle (M-CMi,t), and high (H-CMi,t) level of limits to arbitrage portfolios are 0.57 (0.68), 0.93 (1.15), and 2.17 (2.31), and all the differences are statistically significant. What is more, means of Ri,t and ARi,t are inclined to decrease with the increase of IVOLi,t−1, in general. This tendency is more evident in portfolios with the highest level of limits to arbitrage. For example, in high level of limits to arbitrage portfolios, the means of ARi,t from portfolio D1 to portfolio D10 are −0.23, −0.40, −0.30, −0.34, −0.44, −0.54, −1.04, −1.10, −1.67, −2.55, respectively.We can also find that the market value-weighted mean differences ofRi,t (ARi,t) between extreme portfolios within low, middle, and high level of limits to arbitrage portfolios are −0.53 (0.14), −0.08 (1.69), and 1.31 (2.81). Except −0.53 and −0.08, all the other differences are statistically significant. This evidence indicates that means of Ri,t and ARi,t are inclined to decrease with the increase of IVOLi,t−1, but this trend is only prevalent in stocks with high level of limits to arbitrage.In summary, evidence from Table4 confirms that stock price idiosyncratic volatility in the past few months can negatively predict future month stock return. This predictability is more prevalent in stocks with high level of limits to arbitrage.To confirm the robustness of the above conclusions, we replace CMi,t with each individual measure of limits to arbitrage and conduct bivariate portfolios analysis. The extreme IVOLi,t−1 portfolios return differences are reported in Table 5.When limits to arbitrage are measured by AMHi,t, PLNi,t, and MSTi,t, respectively, the equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t−1 portfolios are 0.85 (1.05), 0.51 (0.68), and 1.45 (1.74) in the lowest level of limits to arbitrage portfolios, and the values of equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t−1 portfolios are 2.37 (2.54), 2.43 (2.39), and 2.60 (3.04) in the highest level of limits to arbitrage portfolios.When limits to arbitrage are measured by AMHi,t, PLNi,t, and MSTi,t, respectively, the market value-weighted Ri,t (ARi,t) differences between extreme IVOLi,t portfolios are −1.30 (0.26), −0.43 (0.29), and −0.15 (1.09) in the lowest level of limits to arbitrage portfolios, and the values of equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t portfolios are 1.49 (3.25), 2.68 (2.53), and 0.51 (5.12) in the highest level of limits to arbitrage portfolios.The evidence from Tables4 and 5 tells us that returns of the lowest IVOLi,t−1 portfolios minus return of the highest IVOLi,t−1 portfolio are always negative, especially in stocks with high level of limits to arbitrage, and this result is robust. This indicates that idiosyncratic volatility premium puzzle exists in Chinese A stock market, and this effect is more evident in stocks confronting high level of limits to arbitrage.


### 3.3. Robust Test: Extended Return Period

Table6 shows us cumulative return (CRi,t+n) and cumulative abnormal return (CARi,t+n) difference between extreme IVOLi,t−1 portfolios within low, middle, and high level of limits to arbitrage portfolios when limits to arbitrage measure are CMi,t and the return periods covered are extended from two to seven months after the portfolios are constructed (n equals 2 to 7 and indicates the number of months after the portfolios are constructed).Table 6
Extended periods’ mean return difference of each extreme IVOL portfolio within different portfolios sorts by individual limits to arbitrage measure.
ReturnsCRi,t+2CRi,t+3CRi,t+4CRi,t+5CRi,t+6CRi,t+7L-CM−0.86∗−1.00∗−1.49−1.06−0.45−0.55M-CM0.56∗∗0.89∗1.19∗2.08∗∗3.26∗∗∗4.27∗∗∗H-CM3.03∗∗∗3.51∗∗∗4.99∗∗∗6.92∗∗∗8.06∗∗∗9.97∗∗∗H-L3.89∗∗∗4.51∗∗∗6.48∗∗∗7.97∗∗∗8.51∗∗∗10.51∗∗∗ReturnsCARi,t+2CARi,t+3CARi,t+4CARi,t+5CARi,t+6CARi,t+7L-CM0.230.14−0.19−0.0500.229∗−0.107M-CM2.76∗∗∗3.59∗∗∗4.46∗∗∗4.98∗∗∗6.272∗∗∗7.184∗∗∗H-CM4.79∗∗∗5.23∗∗∗6.88∗∗∗8.44∗∗∗10.04∗∗∗11.09∗∗∗H-L4.57∗∗∗5.09∗∗∗7.07∗∗∗8.49∗∗∗9.82∗∗∗11.19∗∗∗Note.∗, ∗∗, ∗∗∗Statistical significance at 10%, 5%, and 1% levels, respectively.When the time period covered is from 2 months to 7 months, the extreme IVOLi,t−1 portfolios CRi,t differences are −0.86, −1.00, −1.49, −1.06, −0.45, and −0.5 in low level of limits to arbitrage portfolios, and the corresponding CARi,t differences are 3.03, 3.51, 4.99, 6.92, 8.06, and 9.97 in high level of limits to arbitrage portfolios. When the time period covered is from 2 months to 7 months, the extreme IVOLi,t−1 portfolios CARi,t differences are 0.23, 0.14, −0.19, −0.05, 0.23, and −0.11 in low level of limits to arbitrage portfolios, and the corresponding CARi,t differences are 4.79, 5.23, 6.88, 8.44, 10.04, and 11.09 in portfolios with the highest level of limits to arbitrage.In summary, the evidence from Table6 confirms that the idiosyncratic volatility premium puzzle is robust when extending return periods, and it is likely that the longer the return periods, the more prevalent the idiosyncratic volatility premium puzzle; this tendency is most evident in stocks confronting highest level of limits to arbitrage.


## 3.1. Empirical Evidence on the Existence of Idiosyncratic Volatility Premium Puzzle in China’s A-Share Market

We conduct univariate portfolios analysis to obtain evidence on the existence of idiosyncratic volatility premium puzzle in China’s A-share market. Each month, all the stocks are sorted into 5 and 10 equal portfolios based on IVOLi,t−1, respectively. Then, we calculate equally weighted and tradable common stock market value-weighted return of each portfolio. After that, we calculate time series mean of each portfolio return and the corresponding mean return differences between extreme portfolios. Q1 (Q5) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing five portfolios. D1 (D10) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing 10 portfolios.Table2 reports equally weighted and market value-weighted time series means of each portfolio’s returns, sorted by IVOLi,t−1. Panel A reports returns of five portfolios sorted by IVOLi,t−1. These results show that the higher (lower) the IVOLi,t−1, the lower (higher) the returns of the portfolios. Equally weighted Rt (ARt) difference between portfolio Q1 and Q5 is 1.42 (1.51), and the corresponding t-statistics is 17.09 (24.67). The tradable common stock market value-weighted Rt (ARt) difference between portfolio Q1 and Q5 is 0.37 (1.39), and the corresponding t-statistics is 4.30 (20.32).Panel B reports returns of ten portfolios sorted by IVOLi,t−1. These results show that the higher (lower) the IVOLi,t−1, the lower (higher) the returns of the portfolios. Equally weighted Rt (ARt) difference between portfolio D1 and D10 is 1.91 (2.04), and the corresponding t-statistics is 16.20 (23.17). Tradable common stock market value-weighted Rt (ARt) difference between portfolio D1 and D10 is 0.78 (2.16), and the corresponding t-statistics is 6.41 (21.95).In summary, evidence from univariate portfolios analysis tells us that idiosyncratic volatility can negatively predict future stock return; the higher the IVOLi,t−1, the lower the future stock return. The following regression analysis will provide us with more related evidence.In the Section2, we construct idiosyncratic volatility pricing factor (IVOLFt).The calculation process used to calculate VOLFi,t asset pricing factor consists of the following three steps. First, we sort and construct five portfolios based on IVOLi,t each month. Second, we calculate market value-weighted mean return of stocks in each portfolio as the return of that portfolio in that month. Third, we calculate extreme portfolios return difference (IVOLFt), equal to return of the largest IVOLi,t portfolio minus return of the lowest IVOLi,t portfolio in each month. Furthermore, we define IVOLFt as the idiosyncratic volatility asset pricing factor. We introduce this IVOLF into Fama and French’s [8] three-factor asset pricing model and conduct Fama and MacBeth’s [12] regression to test the asset pricing effect of the IVOLF. The corresponding stepwise regression results are presented in Table 3.Table3 reports the Fama and MacBeth [12] stepwise regression results. We can see that the estimated regression coefficients of IVOLF in equally weighted and market value-weighted regression results are −0.01 and −0.31, respectively, and both are significant at 1% level. What is more, changes of intercept (Adj_R2) before and after the introduction of VOLF in equally weighted and market value-weighted regression results are 0.00 (1.56%) and 0.00 (1.98%), respectively. All these pieces of evidence above show us that idiosyncratic volatility factor has a significant asset pricing effect.In summary, the evidence from the descriptive, correlation, and regression analysis proves that stock price idiosyncratic volatility can negatively predict future stock return in China’s A-share market. In the following section, we will conduct bivariate analysis to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market.


## 3.2. Empirical Evidence on Whether Limits to Arbitrage Can Explain Idiosyncratic Volatility Premium Puzzle in China’s A-Share Market

We perform bivariate sorts on CMi,t and IVOLi,t−1 to obtain evidence on whether limits to arbitrage can explain idiosyncratic volatility premium puzzle in China’s A-share market. In each month, we first constructed low, middle, and high level of limits to arbitrage portfolios sorted by CMi,t, and the three portfolios are indicated by L-CMi,t, M-CMi,t, and H-CMi,t, respectively. Second, we constructed ten portfolios sorted by IVOLi,t−1 within each CMi,t portfolio every month. D1 (D10) stands for portfolios with lowest (highest) IVOLi,t−1 when constructing 10 portfolios. After this, we constructed thirty portfolios each month. Third, we calculate equally weighted and market value-weighted mean of Ri,t and ARi,t of the thirty portfolios each month. At last, we calculate the time series mean of equally weighted and market value-weighted return of each portfolio and the time series mean of equally weighted and market value-weighted return difference between extreme portfolios. The results are presented in Table 4.We can see that the equally weighted mean differences ofRi,t (ARi,t) between extreme portfolios within low (L-CMi,t), middle (M-CMi,t), and high (H-CMi,t) level of limits to arbitrage portfolios are 0.57 (0.68), 0.93 (1.15), and 2.17 (2.31), and all the differences are statistically significant. What is more, means of Ri,t and ARi,t are inclined to decrease with the increase of IVOLi,t−1, in general. This tendency is more evident in portfolios with the highest level of limits to arbitrage. For example, in high level of limits to arbitrage portfolios, the means of ARi,t from portfolio D1 to portfolio D10 are −0.23, −0.40, −0.30, −0.34, −0.44, −0.54, −1.04, −1.10, −1.67, −2.55, respectively.We can also find that the market value-weighted mean differences ofRi,t (ARi,t) between extreme portfolios within low, middle, and high level of limits to arbitrage portfolios are −0.53 (0.14), −0.08 (1.69), and 1.31 (2.81). Except −0.53 and −0.08, all the other differences are statistically significant. This evidence indicates that means of Ri,t and ARi,t are inclined to decrease with the increase of IVOLi,t−1, but this trend is only prevalent in stocks with high level of limits to arbitrage.In summary, evidence from Table4 confirms that stock price idiosyncratic volatility in the past few months can negatively predict future month stock return. This predictability is more prevalent in stocks with high level of limits to arbitrage.To confirm the robustness of the above conclusions, we replace CMi,t with each individual measure of limits to arbitrage and conduct bivariate portfolios analysis. The extreme IVOLi,t−1 portfolios return differences are reported in Table 5.When limits to arbitrage are measured by AMHi,t, PLNi,t, and MSTi,t, respectively, the equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t−1 portfolios are 0.85 (1.05), 0.51 (0.68), and 1.45 (1.74) in the lowest level of limits to arbitrage portfolios, and the values of equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t−1 portfolios are 2.37 (2.54), 2.43 (2.39), and 2.60 (3.04) in the highest level of limits to arbitrage portfolios.When limits to arbitrage are measured by AMHi,t, PLNi,t, and MSTi,t, respectively, the market value-weighted Ri,t (ARi,t) differences between extreme IVOLi,t portfolios are −1.30 (0.26), −0.43 (0.29), and −0.15 (1.09) in the lowest level of limits to arbitrage portfolios, and the values of equally weighted Ri,t (ARi,t) differences between extreme IVOLi,t portfolios are 1.49 (3.25), 2.68 (2.53), and 0.51 (5.12) in the highest level of limits to arbitrage portfolios.The evidence from Tables4 and 5 tells us that returns of the lowest IVOLi,t−1 portfolios minus return of the highest IVOLi,t−1 portfolio are always negative, especially in stocks with high level of limits to arbitrage, and this result is robust. This indicates that idiosyncratic volatility premium puzzle exists in Chinese A stock market, and this effect is more evident in stocks confronting high level of limits to arbitrage.


## 3.3. Robust Test: Extended Return Period

Table6 shows us cumulative return (CRi,t+n) and cumulative abnormal return (CARi,t+n) difference between extreme IVOLi,t−1 portfolios within low, middle, and high level of limits to arbitrage portfolios when limits to arbitrage measure are CMi,t and the return periods covered are extended from two to seven months after the portfolios are constructed (n equals 2 to 7 and indicates the number of months after the portfolios are constructed).Table 6
Extended periods’ mean return difference of each extreme IVOL portfolio within different portfolios sorts by individual limits to arbitrage measure.
ReturnsCRi,t+2CRi,t+3CRi,t+4CRi,t+5CRi,t+6CRi,t+7L-CM−0.86∗−1.00∗−1.49−1.06−0.45−0.55M-CM0.56∗∗0.89∗1.19∗2.08∗∗3.26∗∗∗4.27∗∗∗H-CM3.03∗∗∗3.51∗∗∗4.99∗∗∗6.92∗∗∗8.06∗∗∗9.97∗∗∗H-L3.89∗∗∗4.51∗∗∗6.48∗∗∗7.97∗∗∗8.51∗∗∗10.51∗∗∗ReturnsCARi,t+2CARi,t+3CARi,t+4CARi,t+5CARi,t+6CARi,t+7L-CM0.230.14−0.19−0.0500.229∗−0.107M-CM2.76∗∗∗3.59∗∗∗4.46∗∗∗4.98∗∗∗6.272∗∗∗7.184∗∗∗H-CM4.79∗∗∗5.23∗∗∗6.88∗∗∗8.44∗∗∗10.04∗∗∗11.09∗∗∗H-L4.57∗∗∗5.09∗∗∗7.07∗∗∗8.49∗∗∗9.82∗∗∗11.19∗∗∗Note.∗, ∗∗, ∗∗∗Statistical significance at 10%, 5%, and 1% levels, respectively.When the time period covered is from 2 months to 7 months, the extreme IVOLi,t−1 portfolios CRi,t differences are −0.86, −1.00, −1.49, −1.06, −0.45, and −0.5 in low level of limits to arbitrage portfolios, and the corresponding CARi,t differences are 3.03, 3.51, 4.99, 6.92, 8.06, and 9.97 in high level of limits to arbitrage portfolios. When the time period covered is from 2 months to 7 months, the extreme IVOLi,t−1 portfolios CARi,t differences are 0.23, 0.14, −0.19, −0.05, 0.23, and −0.11 in low level of limits to arbitrage portfolios, and the corresponding CARi,t differences are 4.79, 5.23, 6.88, 8.44, 10.04, and 11.09 in portfolios with the highest level of limits to arbitrage.In summary, the evidence from Table6 confirms that the idiosyncratic volatility premium puzzle is robust when extending return periods, and it is likely that the longer the return periods, the more prevalent the idiosyncratic volatility premium puzzle; this tendency is most evident in stocks confronting highest level of limits to arbitrage.


## 4. Conclusions

This paper proves that idiosyncratic volatility premium puzzle exists during the periods from 2002 to 2019 in the context of China’s A-share market after controlling for size, book-to-market, and systematic risk factors. This effect is robust when we extend the return periods from two months to seven months. When we construct proxies for limits to arbitrage based on Chinese special stock market regulation, we also prove that this idiosyncratic volatility premium puzzle is more prevalent in stocks that are more difficult to arbitrage, which is consistent with the viewpoint that asset pricing puzzle can be explained by limits to arbitrage.



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