Journal of Risk
ISSN:
14651211 (print)
17552842 (online)
Editorinchief: Farid AitSahlia
Research on the premium for the joint lowertail risk of liquidity and investor sentiment
Need to know
 The authors construct a measure of joint lower tail risk between liquidity and investor sentiment (LISR) based on the bivariate extreme value theory and copulas.
 We demonstrate that the premiums for LISR measures exist regardless of the sentiment at the market level or firm level.
 LISR premiums are robust to various doublesort portfolios based on a series of firm characteristics and LISR measures, and not affected by the risk factors in the pricing model and influencing factors of crosssectional returns.
Abstract
Considering the dual risks of extreme downside liquidity and extreme negative sentiment, we introduce the concept of the joint lowertail risk of liquidity and investor sentiment (LISR) and construct measures to study the issue of lowertail risk premiums in the Chinese stock market. Our findings provide convincing evidence that the premiums for LISR measures are significant regardless of the sentiment at the market or firm level. Downside liquidity risk and extreme negative sentiment cannot explain the LISR premiums separately, which means that an extreme downside change in liquidity and extreme negative sentiment have a joint effect on future stock returns. In addition, LISR premiums are robust to various portfolio double sorts, hold for various asset pricing factor models and remain significant when controlling for an extensive list of firm characteristics. Our conclusions have obvious value for improving and enriching the theoretical research on investor sentiment and liquidity risk premiums, and they provide a valuable reference for investors aiming to construct portfolios matching their own risk preferences, and for regulators supervising the market.
Introduction
1 Introduction
Since the start of the twentyfirst century, crisis events have periodically broken out around the world, such as the 2007–9 global financial crisis, the 2015 stock market crash and the Covid19 pandemic. During a crisis, social instability and global economic uncertainty rise (Yang et al 2020) while investor sentiment becomes extremely negative and liquidity continuously declines or even evaporates (Chiu et al 2018). These hazards highlight the huge tail risk that crisis events bring to the market. The tail behavior of risks has gained much attention in the literature in recent years (Santolino et al 2021). An accurate assessment of tail behavior among financial assets is essential for risk minimization and portfolio optimization, and for monitoring the stability of financial markets.
Defined as the tail of the liquidity distribution for all stocks, extreme downside liquidity is a direct measure of extreme liquidity risk. This extreme risk arises from the simultaneous drying up of liquidity across assets and can lead to a market freeze that seriously damages investors’ interests (Wu 2019). Liquidity has always been one of the important bases on which investors make decisions (Amihud and Mendelson 1986). However, extreme downside changes in liquidity due to crisis events have challenged the traditional liquidity premium theory. Recent research on liquidity has shown the importance of extreme downside liquidity risk in asset pricing. Menkveld and Wang (2012) define the “liquileak risk” as the probability of a security remaining illiquid for more than a week, and they find a significant premium on this risk. Wu (2019) focuses on the effect of a marketwide extreme liquidity event in the US stock market and finds that the tail distribution of liquidity risk is significantly related to expected returns. Jeong et al (2018), also looking at the US stock market, take the liquidity skewness to reflect extreme downside changes in liquidity and prove that it has a positive premium. Anthonisz and Putnins (2017) develop a (liquidityadjusted) downside capital asset pricing model (LDCAPM) and find that downside liquidity risk has a significant premium. Ruenzi et al (2020) propose the concept of extreme downside liquidity risk, which reflects both downside return risk and downside liquidity risk, and they show that it has provided a very large premium in recent years.
Pastor and Stambaugh (2003) believe that liquidity can be seen to behave differently in good and bad market states. Investor sentiment may be the main reason for this difference (Kyle 1985; De Long et al 1990). Kuhnen and Knutson (2011) find that negative sentiment is related to a higher level of risk aversion, especially in extreme conditions, and that investors under extreme negative sentiment can change their beliefs and avoid risks regardless of their own information. Thus, stocks that suddenly become very illiquid are unattractive for such investors. However, investors probably care less about an extreme downside change in liquidity during periods of stable or positive sentiment. In sum, extreme downside liquidity is a risk for investors only during a period of extreme negative sentiment. In addition, during such periods, returns typically become increasingly unstable, while stock illiquidity and investors’ funding illiquidity reinforce each other (Brunnermeier and Pedersen 2008). As a result, the tail dependence between liquidity and investor sentiment will increase, and the market will see the dual risks of extreme downside liquidity and extreme negative sentiment. However, the existing research on the liquidity risk premium has ignored the important role of investor sentiment. Investor sentiment is linked to various stock market anomalies (Yu and Yuan 2011) and closely related to stock valuation (Verma and Verma 2020), which will affect investors’ investment decisions. It is therefore necessary to incorporate extreme negative sentiment into the study of extreme downside liquidity premiums so as to potentially enhance the understanding of liquidity pricing and provide an important supplement to the research on investor sentiment and liquidity premiums.
In considering the dual risks of extreme downside liquidity and extreme negative sentiment, we use bivariate extreme value theory and copulas to construct measures of the joint lowertail risk of liquidity and investor sentiment (LISR) in order to study the joint lowertail risk premium in the Chinese stock market. The main work and contributions are as follows. First, we apply bivariate extreme value theory and copulas to the research fields of liquidity and investor sentiment. For multivariate financial time series, most of the existing research focuses on an overall dependence structure, such as correlation (Engle 2002; Pakel et al 2020). However, correlation structures cannot help reveal joint tail behavior (Zhao 2021). In contrast, the LISR measures constructed based on bivariate extreme value theory and copulas can well reflect the extreme dependence structure between the two variables, providing theoretical support for the study of the joint lowertail risk premium. Second, we generalize the existing literature on tail risk premiums from the univariate dimension to the joint dimension. Under extreme negative sentiment, liquidity and investor sentiment interact with one another, but the existing literature ignores the investor sentiment dimension when studying the extreme downside liquidity risk premium. Our paper incorporates extreme negative sentiment into the study of liquidity risk premiums. We test the dual effects of extreme downside liquidity and extreme negative sentiment on future stock returns through univariate sorts, bivariate sorts, factor analysis and Fama–MacBeth regression. The conclusions have clear theoretical value for improving and enriching the research on investor sentiment and liquidity risk premiums, and they provide a valuable reference for investors aiming to construct portfolios matching their own risk preferences, and for regulators supervising the market.
2 Data and variable construction
2.1 Data source and sample
We select Ashare listed companies from January 2003 through September 2022 as the sample. All data are from the RESSET financial research database (RESSET/DB) and the China Stock Market and Accounting Research (CSMAR) database. We filter the original data by excluding “special treatment” stocks (tagged “ST”), stocks at risk of special treatment (tagged “${}^{*}$ST”), “particular transfer” stocks (tagged “PT”), financial stocks and stocks with fewer than 10 trading days within a month. To ensure there are enough stocks in the sample to make the conclusions more widely applicable, we adopt the rolling elimination method to eliminate stocks month by month. By the end of September 2022 the number of stocks stood at 3922.
2.2 Variable construction
2.2.1 Measuring return
We employ four types of return measures: excess returns ($R$); returns adjusted by the CAPM (${\alpha}^{\mathrm{CAPM}}$); returns adjusted by the Fama–French threefactor model (${\alpha}^{\mathrm{FF3}}$); and returns adjusted by the Carhart fourfactor model (${\alpha}^{\mathrm{Car}}$). The excess return $R$ is the difference between the monthly return and the riskfree rate, and the monthly return is the difference between the logarithm of the closing price of the current month and that of the previous month.
We select a 12month rolling window to regress each model and obtain the following regression parameters for the CAPM, FF3 and Carhart models, respectively:
${R}_{i,t}$  $={\alpha}_{i}^{\mathrm{CAPM}}+{\beta}_{i}{\mathrm{MKT}}_{t}+{\epsilon}_{i,t}^{\mathrm{CAPM}},$  (2.1)  
${R}_{i,t}$  $={\alpha}_{i}^{\mathrm{FF3}}+{\beta}_{i}^{\mathrm{FF3},1}{\mathrm{MKT}}_{t}+{\beta}_{i}^{\mathrm{FF3},2}{\mathrm{SMB}}_{t}+{\beta}_{i}^{\mathrm{FF3},3}{\mathrm{HML}}_{t}+{\epsilon}_{i,t}^{\mathrm{FF3}},$  (2.2)  
${R}_{i,t}$  $={\alpha}_{i}^{\mathrm{Car}}+{\beta}_{i}^{\mathrm{Car},1}{\mathrm{MKT}}_{t}+{\beta}_{i}^{\mathrm{Car},2}{\mathrm{SMB}}_{t}+{\beta}_{i}^{\mathrm{Car},3}{\mathrm{HML}}_{t}+{\beta}_{i}^{\mathrm{Car},4}{\mathrm{MOM}}_{t}+{\epsilon}_{i,t}^{\mathrm{Car}},$  (2.3) 
where ${R}_{i,t}$ denotes the excess return of stock $i$ in month $t$; ${\mathrm{MKT}}_{t}$, ${\mathrm{SMB}}_{t}$, ${\mathrm{HML}}_{t}$ and ${\mathrm{MOM}}_{t}$ represent the market factor, size factor, value factor and momentum factor in month $t$; ${\beta}_{i}$ denotes the sensitivity coefficient of the excess return to various factors; ${\epsilon}_{i,t}$ is the residual term; and ${\alpha}_{i}$ in the three models are the riskadjusted returns, denoted elsewhere in the paper by ${\alpha}^{\mathrm{CAPM}}$, ${\alpha}^{\mathrm{FF3}}$ and ${\alpha}^{\mathrm{Car}}$, respectively.
2.2.2 Measuring liquidity
We use the Amihud (2002) illiquidity ratio (ILLIQ) as our main liquidity measure. The ILLIQ can measure liquidity from both the price and the trading volume, reflecting the change in price caused by a unit change in trading volume.
The ILLIQ of stock $i$ in month $t$ is defined as
$${\mathrm{ILLIQ}}_{i,t}=\frac{1}{{N}_{i,t}}\sum _{d=1}^{{N}_{i,t}}\frac{{r}_{i,t,d}}{{\mathrm{vol}}_{i,t,d}},$$  (2.4) 
where ${N}_{i,t}$ indicates the number of trading days of stock $i$ in month $t$, ${r}_{i,t,d}$ denotes the return of stock $i$ on day $d$ of month $t$ and ${\mathrm{vol}}_{i,t,d}$ is the daily volume (in millions) of stock $i$ on day $d$ of month $t$.
The illiquidity ratio is an inverse measure of liquidity; that is, the larger the value of ILLIQ, the lower the liquidity. To facilitate the description of liquidity levels in the following and to simplify the estimation of joint lowertail risk measures, we define LIQ as the opposite of ILLIQ:
$${\mathrm{LIQ}}_{i,t}={\mathrm{ILLIQ}}_{i,t},$$  (2.5) 
where ${\mathrm{LIQ}}_{i,t}$ is the liquidity of stock $i$ in month $t$. A higher value of LIQ indicates a higher level of liquidity.
2.2.3 Measuring investor sentiment
Market level.
Following Yi and Mao (2009), we select for principal component analysis the indicators “discount of closedend fund” (DCEF), “trading volume” (TURN), “initial public offering (IPO) number” (IPON), “IPO firstday returns” (IPOR), “consumer confidence index” (CCI) and “new investor account” (NIA), and we construct a comprehensive investor sentiment index (CISI) to measure investor sentiment at the market level. The specific steps are as follows.
 (1)
Construct the investor sentiment index (ISI) with 12 variables. Since there may be forwardlooking or lagged relationships between the above six indicators and investor sentiment, we also perform principal component analysis on the forwardlooking and lagged variables of the six indicators (which results in 12 variables in total).^{1}^{1} 1 Yi and Mao (2009) believe that investor sentiment may be influenced by one lag of DCEF, TURN, IPON, IPOR, CCI and NIA. This paper therefore considers one lag only, instead of other lags. To eliminate the influence of unit difference, we standardize the variables and then compute the weighted average of the principal components whose cumulative variance interpretation rates exceed 85%.^{2}^{2} 2 Compared with the Baker–Wurgler index, which retains only the first principal component value, our method can retain more information.
 (2)
Select the source indicators of the CISI. The correlation analysis shows that the correlations between the ISI and ${\mathrm{DCEF}}_{t}$, ${\mathrm{TURN}}_{t1}$, ${\mathrm{IPON}}_{t}$, ${\mathrm{IPOR}}_{t1}$, ${\mathrm{CCI}}_{t1}$ and ${\mathrm{NIA}}_{t}$ are relatively high. Therefore, these six variables are selected as the source indicators of the CISI.
 (3)
Construct the CISI. We perform principal component analysis on the six source indicators to construct the CISI. The final result is^{3}^{3} 3 Because the uniqueness value of ${\mathrm{NIA}}_{t}$ is greater than 0.6, this variable is removed.
${\mathrm{CISI}}_{t}$ $=0.7894{\mathrm{DCEF}}_{t}+0.0843{\mathrm{TURN}}_{t1}$ $\mathrm{\hspace{1em}\hspace{1em}}+0.7848{\mathrm{IPON}}_{t}+0.5414{\mathrm{IPOR}}_{t1}+0.8218{\mathrm{CCI}}_{t1},$ (2.6) where ${\mathrm{CISI}}_{t}$ denotes the market sentiment in month $t$, giving a time series. In the same month, all stocks in the sample have the same CISI value.
Firm level.
Referring to Coulton et al (2015), we take firm sentiment sensitivity to represent investor sentiment at the firm level. We estimate the following model to generate firm sentiment sensitivity:
${R}_{i,t}$  $={\alpha}_{i}+{\beta}_{\mathrm{MKT},i}{\mathrm{MKT}}_{t}+{\beta}_{\mathrm{SMB},i}{\mathrm{SMB}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\beta}_{\mathrm{HML},i}{\mathrm{HML}}_{t}+{\beta}_{\mathrm{MOM},i}{\mathrm{MOM}}_{t}+{\beta}_{\mathrm{SENT},i}\mathrm{\Delta}{\mathrm{CISI}}_{t}+{\epsilon}_{i,t}.$  (2.7) 
This is the Carhart fourfactor model with the change in market sentiment ($\mathrm{\Delta}\mathrm{CISI}$) as an additional variable. The monthly excess returns of stock $i$ in month $t$ are denoted by ${R}_{i,t}$; ${\mathrm{MKT}}_{t}$, ${\mathrm{SMB}}_{t}$, ${\mathrm{HML}}_{t}$ and ${\mathrm{MOM}}_{t}$ are the four Carhart factors, representing the market factor, size factor, value factor and momentum factor in month $t$, respectively; ${\beta}_{\mathrm{MKT},i}$, ${\beta}_{\mathrm{SMB},i}$, ${\beta}_{\mathrm{HML},i}$ and ${\beta}_{\mathrm{MOM},i}$ are the coefficients of the sensitivity of the excess return to the four Carhart factors; and ${\beta}_{\mathrm{SENT},i}$, a firmlevel measure, is the coefficient of the sensitivity of the excess return to the change in market sentiment, also known as the sentiment sensitivity. In month $t$ we use the data of the four Carhart factors, the change in market sentiment over the previous 12 months and the excess return data of stock $i$ to estimate (2.7), and we obtain the sentiment sensitivity of stock $i$ in month $t$ (${\beta}_{\mathrm{SENT},i}$). With the same method, we calculate the monthly ${\beta}_{\mathrm{SENT}}$ of each stock in the sample to compose a ${\beta}_{\mathrm{SENT}}$ panel estimation matrix.
2.2.4 Measuring the LISR
The lowertail dependence coefficient of the between liquidity and investor sentiment is taken as the proxy variable of the LISR. The LISR is a panel data matrix, because we calculate the lowertail dependence coefficient between liquidity and investor sentiment for each stock in each month to measure the joint lowertail risk at the stock–month level.
The lowertail dependence coefficient between two random variables reflects the likelihood that a realization of one random variable is in the extreme lower tail of its distribution conditional on the realization of the other random variable also being in the extreme lower tail of its distribution (Ruenzi et al 2020). Given two random variables ${X}_{1}$ and ${X}_{2}$, lowertail dependence ${\lambda}_{\mathrm{L}}$ is formally defined as
$${\lambda}_{\mathrm{L}}({X}_{1},{X}_{2})=\underset{u\to {0}^{+}}{lim}P({X}_{1}\le {F}_{1}^{1}(u)\mid {X}_{2}\le {F}_{2}^{1}(u)),$$  (2.8) 
where $u\in (0,1)$ denotes the value of the distribution function and ${lim}_{u\to {0}^{+}}$ indicates the limit if we approach the lower tail of the distribution from above.
Based on (2.8), the LISR measures are defined as
$\mathrm{LISR1}$  $={\lambda}_{\mathrm{L}}(\mathrm{LIQ},\mathrm{CISI})=\underset{u\to {0}^{+}}{lim}P(\mathrm{LIQ}\le {F}_{\mathrm{LIQ}}^{1}(u)\mid \mathrm{CISI}\le {F}_{\mathrm{CISI}}^{1}(u)),$  (2.9)  
$\mathrm{LISR2}$  $={\lambda}_{\mathrm{L}}(\mathrm{LIQ},{\beta}_{\mathrm{SENT}})=\underset{u\to {0}^{+}}{lim}P(\mathrm{LIQ}\le {F}_{\mathrm{LIQ}}^{1}(u)\mid {\beta}_{\mathrm{SENT}}\le {F}_{{\beta}_{\mathrm{SENT}}}^{1}(u)),$  (2.10) 
where we capture the joint lowertail risk of stock liquidity and investor sentiment at the market level (LISR1) by (2.9), and the joint lowertail risk of stock liquidity and investor sentiment at the firm level^{4}^{4} 4 A value of ${\beta}_{\mathrm{SENT}}>0$ indicates that $\mathrm{\Delta}\mathrm{CISI}$ positively affects excess returns, while $$ indicates that $\mathrm{\Delta}\mathrm{CISI}$ negatively affects excess returns. A greater degree of either positive or negative impact means a higher sentiment sensitivity. We therefore use the absolute value of ${\beta}_{\mathrm{SENT}}$ to construct the LISR2 measure. (LISR2) by (2.10).
The lowertail dependence coefficient between two variables can be expressed in terms of a copula function, $C:{[0,1]}^{2}\mapsto [0,1]$. McNeil et al (2005) show that a simple expression for ${\lambda}_{\mathrm{L}}$ in terms of the copula $C$ of the bivariate distribution can be derived based on
$${\lambda}_{\mathrm{L}}=\underset{u\to {0}^{+}}{lim}\frac{C(u,u)}{u}.$$  (2.11) 
Following the methods of ChabiYo et al (2018) and Weigert (2016), we form 64 convex combinations of the 12 basic copulas. To determine which convex copula combination delivers the best fit for the data, we fit all 64 combinations to the bivariate distributions of each stock’s liquidity and CISI or ${\beta}_{\mathrm{SENT}}$ in 36month rolling windows. Then, we select a specific copula combination for each stock in each month based on the estimated loglikelihood values for the 64 different copulas. Finally, we use the copula that best fits the respective stock, together with the two LISR measures over the previous 36 months, to estimate the tail dependence coefficients by (2.11). As this procedure is repeated for each stock $i$ and month $t$, we end up with a panel of tail dependence coefficients, ${\mathrm{LISR1}}_{i,t}$ and ${\mathrm{LISR2}}_{i,t}$.
2.2.5 Control variables
In addition to the measures outlined above, we control for a battery of return predictors, referring to Jeong et al (2018) and Cakici and Zaremba (2021). The firm size (SIZE) is proxied by the logarithm of the stock market capitalization at the end of the previous month. The booktomarket ratio (BM) is the reciprocal of the pricetobook ratio. The market Beta ($\beta $) is the slope coefficient; specifically, to derive $\beta $ for month $t$, we regress the excess stock returns over month $t12$ to month $t1$ on the market return. The turnover (TURN) can be obtained directly from the database. The momentum (MOM) for month $t$ is represented by the total stock return in months $t12$ to $t2$. The shortterm reversal (REV) for month $t$ is the total stock return in month $t1$. The idiosyncratic volatilities (IVOL3 or IVOL4) are computed as the residual term derived from, respectively, the Fama–French threefactor model or the Carhart fourfactor model based on data of the previous 12 months.
3 Analysis of premiums for the joint lowertail risk of liquidity and investor sentiment
3.1 Descriptive statistics
Type  Variable  Definition  Mean  SD  CV  Min.  Max. 
Returns  $R$  Excess return  $$0.007  0.150  21.429  $$1.978  2.667 
${\alpha}^{\text{CAPM}}$  Returns adjusted by the CAPM  0.005  0.037  7.400  $$0.453  1.383  
${\alpha}^{\text{FF3}}$  Returns adjusted by the Fama–French  $$0.015  0.042  2.800  $$0.691  0.441  
threefactor model  
${\alpha}^{\text{Car}}$  Returns adjusted by the Carhart  $$0.015  0.048  3.200  $$0.830  0.665  
fourfactor model  
Joint lowertail risk  LISR1  Joint lowertail risk of liquidity and  0.069  0.239  3.464  0.000  0.101 
sentiment at the market level  
LISR2  Joint lowertail risk of liquidity and  0.066  0.223  3.379  0.000  0.111  
sentiment at the firm level  
Control variables  SIZE  Firm size  22.310  1.200  0.054  17.040  28.660 
BM  Booktomarket ratio  0.419  0.281  0.671  0.000  5.556  
$\beta $  Market Beta  1.057  0.766  0.725  $$28.370  29.980  
TURN  Turnover  46.300  47.700  1.030  0.022  1064.000  
MOM  Momentum  $$0.051  0.485  9.510  $$4.002  5.641  
REV  Shortterm reversal  $$0.004  0.151  37.750  $$1.976  2.668  
IVOL3  Idiosyncratic volatility derived from  0.000  0.093  —  $$1.504  0.998  
the Fama–French threefactor model  
IVOL4  Idiosyncratic volatility derived from  $$0.001  0.085  85.000  $$1.355  0.946  
the Carhart fourfactor model 
We calculate each variable according to Section 2.2 and then report their means, standard deviations, coefficients of variation,^{5}^{5} 5 The coefficient of variation is the absolute value of the ratio of the standard deviation to the mean. minimums and maximums in Table 1. From the table we can see the following.
 •
The mean of $R$ is $$0.007, indicating that the average excess return of individual stocks during the sample period is negative. The coefficients of variation of ${\alpha}^{\mathrm{CAPM}}$, ${\alpha}^{\mathrm{FF3}}$ and ${\alpha}^{\mathrm{Car}}$ are smaller than that of $R$, indicating that the returns after riskadjustment are more stable.
 •
The means of LISR1 and LISR2 are 0.069 and 0.066, respectively, and the coefficients of variation are 3.464 and 3.379, respectively, indicating that the averages and fluctuations of the two joint lowertail risk measures are similar.
 •
The statistics of control variables are also reported in detail in the table for reference.
In addition, to ensure the accuracy and rationality of the variable construction, we test whether the sentiment level of the Chinese stock market can be measured by the CISI, which is calculated by principal component analysis and used to construct the ${\beta}_{\mathrm{SENT}}$ and LISR measures. The time series of the CISI is shown in Figure 1.
In the figure there are three periods in which the CISI both fluctuates violently and drops significantly, corresponding respectively to the global financial crisis in 2007–9, the stock market crash in 2015 and the Covid19 pandemic in 2020. This shows that the CISI can capture well the change in marketwide investor sentiment during crisis events – a finding that can guarantee the accurate and reasonable construction of the ${\beta}_{\mathrm{SENT}}$ and LISR measures.
3.2 Correlation analysis
$\bm{R}$  LISR1  LISR2  SIZE  BM  $\bm{\beta}$  TURN  MOM  REV  IVOL3  IVOL4  

$\bm{R}$  1${}^{***}$  
LISR1  0.067${}^{***}$  1${}^{***}$  
LISR2  0.102${}^{***}$  0.497  1${}^{***}$  
SIZE  0.052${}^{***}$  0.642${}^{***}$  0.509${}^{***}$  1${}^{***}$  
BM  0.023${}^{***}$  $$0.051${}^{***}$  $$0.033${}^{***}$  $$0.040${}^{***}$  1${}^{***}$  
$\bm{\beta}$  0.001  $$0.052${}^{***}$  $$0.009${}^{***}$  $$0.029${}^{***}$  $$0.048${}^{***}$  1${}^{***}$  
TURN  0.186${}^{***}$  0.178${}^{***}$  0.140${}^{***}$  $$0.056${}^{***}$  $$0.200${}^{***}$  0.055${}^{***}$  1${}^{***}$  
MOM  $$0.018${}^{***}$  0.253${}^{***}$  0.159${}^{***}$  0.206${}^{***}$  $$0.233${}^{***}$  $$0.014${}^{***}$  0.135${}^{***}$  1${}^{***}$  
REV  $$0.010${}^{***}$  0.148${}^{***}$  0.114${}^{***}$  0.056${}^{***}$  $$0.022${}^{***}$  $$0.001  0.216${}^{***}$  $$0.025${}^{***}$  1${}^{***}$  
IVOL3  0.643${}^{***}$  $$0.002  0.020${}^{***}$  0.000  0.040${}^{***}$  $$0.008${}^{***}$  0.084${}^{***}$  $$0.149${}^{***}$  $$0.101${}^{***}$  1${}^{***}$  
IVOL4  0.585${}^{***}$  $$0.000  0.023${}^{***}$  $$0.002  0.039${}^{***}$  $$0.011${}^{***}$  0.082${}^{***}$  $$0.142${}^{***}$  $$0.078${}^{***}$  0.910${}^{***}$  1${}^{***}$ 
To preliminarily understand the correlations between important variables and provide a basis for further analysis of the LISR premiums, we calculate Pearson coefficients, as reported in Table 2. We can see from the table that the correlation coefficient of $R$ and LISR1 is 0.067, and that of $R$ and LISR2 is 0.102; both are significant at the 1% level, indicating that the excess return is significantly positively correlated with joint lowertail risks. The results preliminarily show that there may be a risk premium phenomenon. The relationship between LISR1 and LISR2 is not significant, which indicates that the content captured by the two risk measures is different. In addition, there are different degrees of correlation between excess returns and most control variables, so these variables should be controlled during the regression analysis.
3.3 Univariate portfolio sorts
To study whether there are LISR premiums in the Chinese stock market, we start our empirical analysis with univariate portfolio sorts. The specific steps are as follows.
 (1)
Build five stock portfolios. For each month $t$ we sort stocks into five quintiles based on their LISR (ie, LISR1 or LISR2). Portfolio 1 consists of stocks with the lowest values of LISR1 or LISR2, respectively, while portfolio 5 consists of stocks with the highest values of LISR1 or LISR2, respectively.
 (2)
Calculate the stock portfolio returns. For all four types of returns, we compute the equalweighted portfolio return and the valueweighted portfolio return, which is the average of returns weighted by the market capitalization of stocks.
 (3)
Test the return differences between portfolio 5 and portfolio 1 (“return differences” for short) using the Newey–West $t$test.
(a) LISR1 portfolios  
Equalweighted portfolio returns  Valueweighted portfolio returns  
Stock  
portfolio  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
1 (low LISR1) ($L$)  $$1.628  $$0.409  $$2.271  $$2.094  $$1.482  $$0.357  $$2.021  $$1.863 
2  $$1.123  $$0.109  $$2.025  $$1.886  $$1.392  $$0.116  $$1.881  $$1.755 
3  $$0.695  0.327  $$1.684  $$1.636  $$1.342  0.318  $$1.564  $$1.528 
4  $$0.103  0.878  $$1.248  $$1.299  $$0.971  0.867  $$1.128  $$1.176 
5 (high LISR1) ($H$)  0.453  1.737  $$0.453  $$0.604  $$0.773  1.752  $$0.381  $$0.503 
$HL$  2.081${}^{***}$  2.146${}^{***}$  1.818${}^{***}$  1.490${}^{***}$  0.709${}^{***}$  2.109${}^{***}$  1.640${}^{**}$  1.360${}^{***}$ 
(b) LISR2 portfolios  
Equalweighted portfolio returns  Valueweighted portfolio returns  
Stock  
portfolio  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
1 (low LISR2) ($L$)  $$1.556  0.141  $$1.826  $$1.732  $$1.658  0.805  $$1.157  $$1.183 
2  $$0.716  0.236  $$1.727  $$1.670  $$1.019  0.978  $$0.938  $$0.955 
3  $$0.274  0.342  $$1.668  $$1.611  $$0.730  1.013  $$1.076  $$1.061 
4  $$0.296  0.622  $$1.420  $$1.409  $$0.902  1.270  $$0.794  $$0.889 
5 (high LISR2) ($H$)  $$0.070  1.269  $$0.883  $$0.967  $$0.730  1.615  $$0.501  $$0.620 
$HL$  1.486${}^{*}$  1.128${}^{***}$  0.943${}^{**}$  0.765${}^{**}$  0.928${}^{*}$  0.810${}^{***}$  0.656${}^{***}$  0.563${}^{*}$ 
The results for the stock portfolio returns and the return differences ($HL$) are reported in Table 3. We begin with univariate portfolio sorts based on LISR1 in part (a) of the table. The equalweighted and valueweighted excess returns of the five stock portfolios show a clear pattern of increasing, on the whole, with LISR1. The return differences are 2.081 for equalweighted returns and 0.709 for valueweighted returns, with statistical significance for both at the 1% level, indicating that there is a positive premium for LISR1. After we riskadjust the returns using the three models, all the return differences are between 1.360 and 2.146, and significant at at least the 5% level, indicating that the premiums are still significantly positive after considering the impact of other risk factors.
We proceed to analyze univariate portfolio sorts based on LISR2 in part (b) of the table. Here, the equalweighted and valueweighted excess returns of the five stock portfolios again show a trend of increasing with LISR2. The return differences are 1.486 for equalweighted returns and 0.928 for valueweighted returns, and significant at the 10% level, indicating that there is a positive premium for LISR2. We find qualitatively similar results for ${\alpha}^{\mathrm{CAPM}}$, ${\alpha}^{\mathrm{FF3}}$ and ${\alpha}^{\mathrm{Car}}$, which means that the LISR2 premium again remains significantly positive after considering the impact of other risk factors.
In summary, the results from Table 3 provide evidence that the LISR measures have substantial return premiums, regardless of the sentiment at the market or firm level. The results show that investors demand a return premium to compensate for low liquidity under extreme negative sentiment. In addition, we find that the return differences calculated by the valueweight method are lower than those calculated by the equalweight method, so firm size may have an impact on the LISR premiums.
3.4 Bivariate portfolio sorts
Since the LISR measures are proxied by the lowertail dependence coefficients between liquidity and investor sentiment, we conduct bivariate equalweighted portfolio sorts based on the two LISR measures and either downside liquidity risk or extreme negative sentiment to test whether either of those two single variables affects the LISR premiums. In addition, because the results in Table 2 document that the two LISR measures are correlated with control variables, we then go on to investigate bivariate equalweighted portfolio sorts based on the two measures and each control variable to test whether the LISR premiums are affected by firm characteristics.
3.4.1 LISR and downside liquidity risk
To test whether the LISR premiums are affected by downside liquidity risk, we first construct a downside liquidity risk variable and form equalweighted bivariate portfolios based on it and the LISR measures, and we then analyze the levels of premiums in different downside liquidity risk groups.
Following Jeong et al (2018), we use the nonparametric skewness measure (SKILLIQ) to measure downside liquidity risk, which is defined as
$${\mathrm{SKILLIQ}}_{i,t}=\frac{{\mu}_{i,t}{\mathrm{median}}_{i,t}}{{\sigma}_{i,t}},$$  (3.1) 
where ${\mu}_{i,t}$, ${\sigma}_{i,t}$ and ${\mathrm{median}}_{i,t}$ respectively represent the mean, standard deviation and median of the daily price impact of stock $i$ during month $t$. A higher value of SKILLIQ indicates a higher level of downside liquidity risk.
The bivariate portfolio returns and return differences are reported in Table 4. The excess return difference is denoted by $R(HL)$, and ${\alpha}^{\mathrm{CAPM}}(HL)$, ${\alpha}^{\mathrm{FF3}}(HL)$ and ${\alpha}^{\mathrm{Car}}(HL)$ indicate the return difference after being adjusted by the CAPM, Fama–French threefactor model and Carhart fourfactor model.
From Table 4 we can see the following.
 •
In part (a), among the five SKILLIQ portfolios, excess returns show monotonic patterns, and the highLISR1 portfolio appears to be characterized by high returns. All groups show positive excess return differences and are statistically significant at at least the 5% level. These results indicate that the LISR1 premium exists under all conditions of SKILLIQ. After riskadjusting the returns, the conclusion remains unchanged.
 •
In part (b), among the five SKILLIQ portfolios, all groups show significantly positive $R(HL)$, ${\alpha}^{\mathrm{CAPM}}(HL)$ and ${\alpha}^{\mathrm{FF3}}(HL)$, and four of them are statistically significant at at least the 5% level after controlling for the four Carhart factors. These results indicate that the LISR2 premium exists under all conditions of SKILLIQ.
 •
According to the results of parts (a) and (b), in all the downside liquidity risk groups, the two LISR measures have significant premiums regardless of the sentiment at the market or firm level. Thus, downside liquidity risk cannot account for the LISR premiums, which also means that the LISR measures contain risk information that cannot be captured by downside liquidity risk.
3.4.2 LISR and extreme negative investor sentiment
To test whether the LISR premiums are affected by extreme negative sentiment, we first define an extreme negative sentiment variable and construct equalweighted bivariate portfolios based on it and the LISR measures, and we then analyze the levels of premiums in different sentiment groups.
(a) LISR1 and SKILLIQ  

SKILLIQ  
Stock  
portfolio  1 (low)  2  3  4  5 (high) 
1 (low LISR1) ($L$)  $$0.926  $$0.851  $$0.807  $$0.658  $$0.629 
2  $$0.809  $$0.733  $$0.655  $$0.589  $$0.554 
3  $$0.787  $$0.605  $$0.589  $$0.565  $$0.550 
4  $$0.586  $$0.571  $$0.564  $$0.545  $$0.524 
5 (high LISR1) ($H$)  $$0.492  $$0.487  $$0.470  $$0.437  $$0.427 
$R(HL)$  0.434${}^{***}$  0.364${}^{**}$  0.337${}^{***}$  0.221${}^{**}$  0.202${}^{***}$ 
${\alpha}^{\text{CAPM}}(HL)$  0.069${}^{*}$  0.081${}^{*}$  0.126${}^{**}$  0.101${}^{***}$  0.056${}^{**}$ 
${\alpha}^{\text{FF3}}(HL)$  0.059${}^{**}$  0.057${}^{**}$  0.106${}^{**}$  0.084${}^{*}$  0.027${}^{**}$ 
${\alpha}^{\text{Car}}(HL)$  0.055${}^{*}$  0.076${}^{**}$  0.083${}^{***}$  0.075${}^{*}$  0.044${}^{*}$ 
(b) LISR2 and SKILLIQ  
SKILLIQ  
Stock  
portfolio  1 (low)  2  3  4  5 (high) 
1 (low LISR2) ($L$)  $$0.903  $$0.819  $$0.783  $$0.637  $$0.550 
2  $$0.799  $$0.743  $$0.610  $$0.595  $$0.548 
3  $$0.65  $$0.592  $$0.582  $$0.566  $$0.517 
4  $$0.623  $$0.513  $$0.490  $$0.489  $$0.405 
5 (high LISR2) ($H$)  $$0.522  $$0.462  $$0.394  $$0.385  $$0.384 
$R(HL)$  0.381${}^{***}$  0.357${}^{**}$  0.389${}^{*}$  0.252${}^{**}$  0.166${}^{***}$ 
${\alpha}^{\text{CAPM}}(HL)$  0.112${}^{***}$  0.019${}^{*}$  0.059${}^{*}$  0.053${}^{*}$  0.073${}^{**}$ 
${\alpha}^{\text{FF3}}(HL)$  0.160${}^{**}$  0.086${}^{*}$  0.125${}^{**}$  0.082${}^{**}$  0.062${}^{**}$ 
${\alpha}^{\text{Car}}$ ($HL$)  0.145${}^{**}$  0.062  0.148${}^{***}$  0.137${}^{***}$  0.069${}^{***}$ 
Referring to Jia et al (2022), we define the extreme negative sentiment condition as the lower 5% tail of the investor sentiment proxies. We construct a dummy variable $D$ that equals 1 when the CISI or ${\beta}_{\mathrm{SENT}}$ falls within the lower 5% tail of their distributions and 0 otherwise.
(a) LISR1 and $D$  
$\bm{D}\mathbf{=}\text{\U0001d7ce}$  $\bm{D}\mathbf{=}\text{\U0001d7cf}$  
Stock  
portfolio  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
1 (low LISR1)  $$1.617  $$0.511  $$2.315  $$2.135  $$1.610  $$0.377  $$2.246  $$2.074 
2  $$1.438  $$0.289  $$2.112  $$1.962  $$1.106  $$0.078  $$2.006  $$1.87 
3  $$1.41  $$0.058  $$1.872  $$1.77  $$0.646  0.354  $$1.66  $$1.614 
4  $$1.367  0.343  $$1.712  $$1.675  $$0.049  0.902  $$1.233  $$1.287 
5 (high LISR1)  $$1.275  0.765  $$1.272  $$1.307  0.523  1.738  $$0.453  $$0.603 
$HL$  0.342${}^{*}$  1.276${}^{**}$  1.043${}^{**}$  0.828${}^{***}$  2.133${}^{**}$  2.115${}^{**}$  1.793${}^{***}$  1.471${}^{***}$ 
(b) LISR2 and $D$  
$\bm{D}\mathbf{=}\text{\U0001d7ce}$  $\bm{D}\mathbf{=}\text{\U0001d7cf}$  
Stock  
portfolio  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
1 (low LISR2)  $$1.812  $$0.568  $$1.944  $$1.868  $$1.512  0.323  $$1.761  $$1.670 
2  $$1.621  $$0.370  $$1.596  $$1.534  $$0.605  0.002  $$1.369  $$1.250 
3  $$1.543  $$0.112  $$1.479  $$1.446  $$0.256  0.075  $$1.091  $$0.993 
4  $$1.278  0.154  $$1.294  $$1.282  $$0.201  0.446  $$1.006  $$0.955 
5 (high LISR2)  $$1.208  0.572  $$0.716  $$0.766  $$0.008  1.546  $$0.899  $$0.895 
$HL$  0.604${}^{***}$  1.140${}^{***}$  1.228${}^{***}$  1.102${}^{***}$  1.504${}^{***}$  1.223${}^{*}$  0.862${}^{***}$  0.775${}^{***}$ 
We first sort all the firms into two groups based on their $D$values. Subsequently, within the two groups, we sort the stocks into five quintiles based on LISR1 or LISR2. The bivariate portfolio returns and return differences are reported in Table 5.
From the table we can see the following.
 •
In part (a), considering investor sentiment at the market level, portfolio excess returns increase with the increase of LISR1. The return differences are 0.342 and 2.133, respectively, and significant at at least the 10% level. The results indicate that the positive LISR1 premium exists whether or not the current market sentiment is in the extreme negative situation. We obtain similar conclusions in the results of ${\alpha}^{\mathrm{CAPM}}$, ${\alpha}^{\mathrm{FF3}}$ and ${\alpha}^{\mathrm{Car}}$.
 •
In part (b), considering investor sentiment at the firm level, we reach the same conclusion as for part (a): the excess returns and three riskadjusted returns of the bivariate portfolios have a clear trend of increasing with LISR2. All return differences are significantly positive at at least the 10% level. The results indicate that the positive LISR2 premium exists whether or not the firm sentiment is in the lowertail of its distribution.
 •
According to the results in parts (a) and (b), we conclude that the LISR premiums are significant in both groups (ie, $D=0$ and $D=1$). Neither extreme negative sentiment at the market level nor that at the firm level can explain the LISR premium. This also shows that the LISR measures contain risk information that cannot be captured by extreme negative sentiment.
The empirical results in this subsection and Section 3.4.1 show that downside liquidity risk and extreme negative sentiment cannot explain the joint lowertail risk premium separately. Thus, extreme negative sentiment and extreme downside liquidity have a joint effect on future stock returns, and considering only a single indicator may lead to an underestimation of risk and therefore to errors in investment decisionmaking.
3.4.3 LISR and firm characteristics
To test whether the LISR premiums are affected by a series of firm characteristics, we form equalweighted portfolios from bivariate sorts on LISR measures and SIZE, BM, $\beta $, TURN, MOM, REV, IVOL3 and IVOL4, respectively, and analyze the levels of premiums in different firm characteristic groups.
First, based on LISR measures and SIZE, we construct bivariate portfolios and calculate the portfolio excess returns and return differences. The results are reported in Table 6.
(a) LISR1 and SIZE  

Stock portfolio  Small SIZE  2  3  4  Large SIZE 
Low LISR1 ($L$)  $$0.051  $$0.708  $$1.062  $$1.215  $$1.317 
2  0.132  $$0.600  $$0.940  $$1.196  $$1.312 
3  0.564  $$0.544  $$0.940  $$1.179  $$1.268 
4  0.767  $$0.284  $$0.916  $$1.038  $$1.069 
High LISR1 ($H$)  1.391  $$0.244  $$0.691  $$1.003  $$0.807 
$R(HL)$  1.442${}^{***}$  0.464${}^{***}$  0.371${}^{**}$  0.212${}^{**}$  0.510${}^{***}$ 
${\alpha}^{\text{CAPM}}(HL)$  0.256${}^{***}$  $$0.214  0.268${}^{**}$  0.468${}^{***}$  0.396${}^{**}$ 
${\alpha}^{\text{FF3}}(HL)$  0.266${}^{***}$  0.219${}^{**}$  0.173${}^{**}$  0.376${}^{**}$  0.418${}^{**}$ 
${\alpha}^{\text{Car}}$ ($HL$)  0.248${}^{***}$  0.191${}^{*}$  0.134${}^{*}$  0.328${}^{***}$  0.346${}^{**}$ 
(b) LISR2 and SIZE  
Stock portfolio  Small SIZE  2  3  4  Large SIZE 
Low LISR2 ($L$)  $$0.065  $$0.584  $$0.948  $$1.206  $$1.260 
2  0.212  $$0.485  $$0.928  $$1.172  $$1.178 
3  0.537  $$0.392  $$0.922  $$1.096  $$1.160 
4  0.931  $$0.341  $$0.910  $$1.044  $$1.106 
High LISR2 ($H$)  1.382  $$0.119  $$0.665  $$0.971  $$1.087 
$R(HL)$  1.447${}^{***}$  0.465${}^{***}$  0.283${}^{**}$  0.235${}^{***}$  0.173${}^{**}$ 
${\alpha}^{\text{CAPM}}(HL)$  0.077  0.051${}^{*}$  0.122${}^{**}$  0.206${}^{**}$  0.226${}^{**}$ 
${\alpha}^{\text{FF3}}(HL)$  0.129${}^{**}$  0.107${}^{**}$  0.063  0.081${}^{*}$  0.175${}^{**}$ 
${\alpha}^{\text{Car}}$ ($HL$)  0.137${}^{**}$  0.135${}^{**}$  0.053${}^{*}$  0.053${}^{*}$  0.155${}^{***}$ 
We can see the following from the table.
 •
In part (a) the positive relation between LISR1 and the excess returns is significant after controlling for SIZE in general. Among the five SIZE groups, all groups show positive excess return differences and are significant at at least the 5% level. These results indicate that there is a significantly positive LISR1 premium across firms of different sizes. We obtain similar conclusions in the results of the riskadjusted return differences.
 •
In part (b) we find that LISR2 is also significantly associated with the excess returns after controlling for SIZE. The excess returns of the LISR2based quintile portfolios show monotonically increasing patterns within each SIZE group. Among the five SIZE groups, all four types of return differences are significantly positive at at least the 10% level. These results also mean that the premium for LISR2 exists independently of firm size.
 •
From the results of parts (a) and (b), we can see that the LISR premiums exist in all size groups, regardless of sentiment at the market or firm level. The results show that firm size has no effect on the LISR premium.
 •
In addition, we observe that the excess return difference in portfolio 1 is significantly higher than in other groups. This indicates that investors bear a higher premium for holding micro stocks due to high uncertainty, which is consistent with the findings of An et al (2019).
Next, we form equalweighted portfolios from bivariate sorts on LISR measures and the seven other control variables, respectively. We find that the return differences are significantly positive in all control variable groups, indicating that the LISR premium has nothing to do with these seven control variables.^{6}^{6} 6 Due to space limitations, the specific test results are not listed in this paper. They are available from the corresponding author on request.
3.5 Factor model analysis
In Section 3.4 the impact of some risk characteristics is indirectly controlled through the bivariate sorts. To further control other risk factors, we study whether the LISR premiums can be explained by multifactor models.
We combine the LISR measures with the risk factors in the CAPM, Fama–French threefactor model and Carhart fourfactor model to form new regression models:
${R}_{i,t+1}$  $={\alpha}_{i}+{\beta}_{\mathrm{MKT}}{\mathrm{MKT}}_{t}+{\beta}_{\mathrm{LISR}}{\mathrm{LISR}}_{i,t}+{\epsilon}_{i,t},$  (3.2)  
${R}_{i,t+1}$  $={\alpha}_{i}+{\beta}_{\mathrm{MKT}}{\mathrm{MKT}}_{t}+{\beta}_{\mathrm{SMB}}{\mathrm{SMB}}_{t}+{\beta}_{\mathrm{HML}}{\mathrm{HML}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\beta}_{\mathrm{LISR}}{\mathrm{LISR}}_{i,t}+{\epsilon}_{i,t},$  (3.3)  
${R}_{i,t+1}$  $={\alpha}_{i}+{\beta}_{\mathrm{MKT}}{\mathrm{MKT}}_{t}+{\beta}_{\mathrm{SMB}}{\mathrm{SMB}}_{t}+{\beta}_{\mathrm{HML}}{\mathrm{HML}}_{t}$  
$\mathrm{\hspace{1em}\hspace{1em}}+{\beta}_{\mathrm{MOM}}{\mathrm{MOM}}_{t}+{\beta}_{\mathrm{LISR}}{\mathrm{LISR}}_{i,t}+{\epsilon}_{i,t},$  (3.4) 
where ${R}_{i,t+1}$ denotes the monthly excess returns of stock $i$ in month $t+1$; ${\mathrm{MKT}}_{t}$, ${\mathrm{SMB}}_{t}$, ${\mathrm{HML}}_{t}$ and ${\mathrm{MOM}}_{t}$ represent the market factor, size factor, value factor and momentum factor in month $t$, respectively; ${\mathrm{LISR}}_{i,t}$ denotes the joint lowertail risk of liquidity and investor sentiment of stock $i$ in month $t$ (ie, ${\mathrm{LISR1}}_{i,t}$ or ${\mathrm{LISR2}}_{i,t}$); ${\beta}_{\mathrm{MKT}}$, ${\beta}_{\mathrm{SMB}}$, ${\beta}_{\mathrm{HML}}$, ${\beta}_{\mathrm{MOM}}$ and ${\beta}_{\mathrm{LISR}}$ are the coefficients of independent variables; and ${\epsilon}_{i,t}$ is the error term.
CAPM  Fama–French  Carhart  
(1)  (2)  (3)  (4)  (5)  (6)  
LISR1  0.051${}^{***}$  0.051${}^{***}$  0.050${}^{***}$  
LISR2  0.028${}^{**}$  0.028${}^{**}$  0.028${}^{**}$  
MKT  0.185${}^{***}$  0.134${}^{***}$  0.176${}^{***}$  0.116${}^{***}$  0.173${}^{***}$  0.110${}^{***}$ 
SMB  $$0.212${}^{***}$  $$0.218${}^{***}$  $$0.226${}^{***}$  $$0.237${}^{***}$  
HML  $$0.315${}^{***}$  $$0.383${}^{***}$  $$0.334${}^{***}$  $$0.416${}^{***}$  
MOM  $$0.053  $$0.079${}^{*}$ 
We put ${\mathrm{LISR1}}_{i,t}$ and ${\mathrm{LISR2}}_{i,t}$, in turn, into (3.2)–(3.4) and perform regression tests. The regression results are shown in Table 7.
We can see from the table that the three coefficients of LISR1 are significantly positive at the 1% level, and the three coefficients of LISR2 are significantly positive at the 5% level. These results indicate that, after controlling the risk factors in the factor models, the joint lowertail risks still have significantly positive impacts on the future excess returns. That is to say, the LISR premiums are not affected by other risk factors, and the LISR measures have independent explanatory power for future excess returns.
3.6 Fama–MacBeth regression analysis
To control the impact of multiple return prediction variables and to further analyze the independent impact of joint lowertail risk on stock returns, we examine the relationship between the two LISR measures and returns by using Fama–MacBeth regressions.
Following Fama and MacBeth (1973), we construct a crosssectional regression model:
$${r}_{i,t+1}={\gamma}_{0}+{\gamma}_{\mathrm{LISR}}{\mathrm{LISR}}_{i,t}+\sum _{j=1}^{n}{\gamma}_{K}{K}_{i,t}^{j}+{\epsilon}_{i,t},$$  (3.5) 
where ${r}_{i,t+1}$ is the return of stock $i$ in month $t+1$, in the form of ${R}_{i,t+1}$, ${\alpha}_{{\mathrm{CAPM}}_{i,t+1}}$, ${\alpha}_{{\mathrm{FF3}}_{i,t+1}}$ or ${\alpha}_{{\mathrm{Car}}_{i,t+1}}$; ${\mathrm{LISR}}_{i,t}$ denotes the joint lowertail risk of liquidity and investor sentiment of stock $i$ in month $t$ (ie, ${\mathrm{LISR1}}_{i,t}$ or ${\mathrm{LISR2}}_{i,t}$); ${\epsilon}_{i,t}$ is the error term; ${\gamma}_{0}$, ${\gamma}_{\mathrm{LISR}}$ and ${\gamma}_{K}$ are monthly regression parameters; and ${K}_{i,t}^{j}$ refers to the set of additional control variables listed in Section 2.2.5.^{7}^{7} 7 If the dependent variables are excess returns, or returns adjusted by the CAPM or Fama–French threefactor model, we employ IVOL3, and if they are returns adjusted by the Carhart fourfactor model, then we employ IVOL4. For robustness, we perform both univariate regression with no control variables included and multivariate regression accounting for the role of other stock return predictors. The results are reported in Table 8.
(a) Univariate regression  

LISR1related results  LISR2related results  
Variable  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
LISR  0.017${}^{*}$  0.040${}^{***}$  0.030${}^{***}$  0.029${}^{***}$  0.005${}^{**}$  0.008${}^{*}$  0.020${}^{***}$  0.017${}^{***}$ 
${R}^{\text{2}}$ (%)  0.92  2.45  1.43  1.40  2.01  1.60  1.71  1.89 
(b) Multivariate regression  
LISR1related results  LISR2related results  
Variable  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$  $\bm{R}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{A}\mathbf{P}\mathbf{M}}}$  ${\bm{\alpha}}^{\text{\mathbf{F}\mathbf{F}\U0001d7d1}}$  ${\bm{\alpha}}^{\text{\mathbf{C}\mathbf{a}\mathbf{r}}}$ 
LISR  0.010${}^{**}$  0.041${}^{***}$  0.026${}^{***}$  0.024${}^{***}$  0.360${}^{**}$  0.224${}^{***}$  0.172${}^{***}$  0.122${}^{***}$ 
SIZE  0.031${}^{**}$  $$0.133${}^{***}$  $$0.135${}^{***}$  $$0.130${}^{***}$  $$0.002  $$0.134${}^{***}$  $$0.114${}^{***}$  $$0.090${}^{***}$ 
BM  $$0.007${}^{**}$  $$0.015${}^{***}$  $$0.016${}^{***}$  $$0.015${}^{***}$  $$0.005  $$0.007${}^{*}$  $$0.010${}^{**}$  $$0.011${}^{***}$ 
$\beta $  $$0.011  $$0.037${}^{**}$  0.007  0.014  $$0.018  0.010  0.004  0.009 
TURN  2.862${}^{*}$  1.716  2.630${}^{**}$  2.780${}^{**}$  1.378  1.470  1.467  1.940${}^{*}$ 
MOM  0.049${}^{***}$  0.000  0.006  0.004  0.035${}^{**}$  0.006  0.006  0.002 
REV  0.011  $$0.004  0.024  0.016  0.003  $$0.005  $$0.004  $$0.001 
IVOL3  0.001  $$0.002  0.002  —  0.001  0.004  $$0.001  — 
IVOL4  —  —  —  0.005${}^{***}$  —  —  —  0.000 
${R}^{\text{2}}$ (%)  3.10  2.19  2.05  2.01  2.88  2.00  1.93  1.98 
From the table we can see the following.
 •
When the dependent variables are excess returns, the coefficients of LISR1 and LISR2 in the univariate regression are significantly positive at at least the 10% level. In the multivariate regressions, the coefficients are significantly positive at at least the 10% level, indicating that the two LISR measures can significantly and independently predict the crosssectional returns.
 •
When the dependent variables are riskadjusted returns, the coefficients of LISR1 and LISR2 in the univariate regressions and multivariate regressions are still significantly positive, indicating that the ability of the joint lowertail risk to predict the cross section of stock returns is unaffected by risk factors.
 •
In summary, we provide strong evidence that the LISR measures are priced in the cross section of expected stock returns regardless of investor sentiment at the market or at firm level.
Through the empirical analysis in this section we find that the LISR premiums are significant and that they are robust to various portfolio double sorts, hold for various asset pricing factor models and remain significant when controlling for an extensive list of firm characteristics. We document that the LISR premiums are the result of the joint effect that extreme downside liquidity and extreme negative sentiment have on future stock returns. At the market level, under extreme negative sentiment, investors have higher levels of risk aversion (Kuhnen and Knutson 2011). Stocks that suddenly become very illiquid during such a period are unattractive; thus, investors demand additional compensation for holding them. At the firm level, lower sentiment sensitivity means lesstimely price discovery (Coulton et al 2015). Therefore, the stocks contain less idiosyncratic information and investors have insufficient understanding of the firm information that they do contain. In that case, the risk costs of lowliquidity stocks increase due to high uncertainty, and investors demand higher expected returns as compensation. For stocks with high sentiment sensitivity, the rapid incorporation of information into the stock price can improve the decisionmaking environment and reduce the risk cost caused by uncertainty, so investors pay less attention to liquidity.
4 Robustness tests
To ensure the robustness of the empirical results, we perform three more analyses.^{8}^{8} 8 Due to space limitations, the specific test results are not listed. They are available from the authors on request.
 Alternative liquidity measures.

The empirical analysis in Section 3 is performed using LISR measures constructed with the Amihud illiquidity ratio, which defines liquidity in terms of the priceimpact dimension. One potential concern is that our main findings are driven by this single dimension used to proxy liquidity. To ensure the stability of our findings, we now test whether our results are robust to using alternative liquidity measures in different dimensions. We use the Datar et al (1998) measure (turnover ratio), the Chung and Zhang (2014) measure (bid–ask spread) and the Lesmond et al (1999) measure (zeroreturn days) to replace LIQ, and we perform the relevant tests in Section 3 again.
 Sample period splitting.

To ensure that our results are robust over time, we split the whole sample period into two subsamples. Specifically, the first subperiod spans January 2003 to December 2012 and the second spans January 2013 to September 2022. We perform the relevant tests in Section 3 for the two subsamples.
 Different estimation windows.

In Section 2.2.4 we use a 36month rolling window to fit the constructed copula combination to the bivariate distribution of liquidity and investor sentiment when calculating the lowertail dependence coefficient of the two variables. To ensure that our conclusions are robust to different lengths of estimation window, we select a shorter window (24 months) and a longer window (48 months) to reconstruct the lowertail dependence coefficient, and we then perform the relevant tests in Section 3 again.
The three robustness tests show that there are significantly positive LISR premiums, which is consistent with our previous results, indicating that the conclusions are robust.
5 Conclusion
Liquidity and investor sentiment are vulnerable to crisis events and can show extreme downside changes. Against this background, we consider the dual risks of extreme downside liquidity and extreme negative sentiment, and we construct LISR measures based on bivariate extreme value theory and copulas to study the joint lowertail risk premium in the Chinese stock market.
Our empirical results show the following.
 •
The LISR premiums are significant. The cross section of stock returns reflects a premium if a stock’s liquidity is extremely low at the same time as the sentiment at the market or firm level is extremely low.
 •
Downside liquidity risk and extreme negative sentiment cannot explain the joint lowertail risk premium separately, which means that an extreme downside change in liquidity and extreme negative sentiment have joint effects on future stock returns. The LISR measures contain risk information that cannot be captured by the two single indicators (ie, extreme negative sentiment and downside liquidity risk).
 •
The tested firm characteristics have no impact on the joint lowertail risk premium. Among firms of different sizes, micro firms have a higher joint lowertail risk premium, due to high uncertainty.
 •
The LISR measures can significantly and independently predict future stock returns, an ability that is unaffected by the risk factors in the pricing models and the influencing factors of crosssectional returns.
 •
Our results are stable across LISR measures constructed with alternative liquidity measures, sample period splitting and different estimation windows.
This study can help further improve and enrich the research on the link between investor sentiment and liquidity and premiums, and it provides a valuable reference for investors aiming to accurately identify the joint tail risk and construct portfolios matching their own risk preferences, and for regulators supervising the market.
Declaration of interest
The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.
Acknowledgements
The authors are grateful for financial support from the National Natural Science Foundation of China (grants 71571041 and 72171039).
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