Factor models provide a cornerstone for understanding financial asset pricing; however, research on China’s stock market risk premia is still limited. Motivated by this, this paper proposes a four-factor model for China’s stock market that includes a market factor, a size factor, a value factor, and a liquidity factor. We compare our four-factor model with a set of prominent factor models based on newly developed likelihood-ratio tests and Bayesian methods. Along with the comparison, we also find supporting evidence for the alternative t-distribution assumption for empirical asset pricing studies. Our results show the following: (1) distributional tests suggest that the returns of factors and stock return anomalies are fat-tailed and therefore are better captured by t-distributions than by normality; (2) under t-distribution assumptions, our four-factor model outperforms a set of prominent factor models in terms of explaining the factors in each other, pricing a comprehensive list of stock return anomalies, and Bayesian marginal likelihoods; (3) model comparison results vary across normality and t-distribution assumptions, which suggests that distributional assumptions matter for asset pricing studies. This paper contributes to the literature by proposing an effective asset pricing factor model and providing factor model comparison tests under non-normal distributional assumptions in the context of China.
Factor models play a fundamental role in explaining the risk premia on financial assets and serve as the benchmark for constructing investment portfolios. China has the world’s second-largest stock market, but the determinants of risk premia on stocks in China remain largely unknown to both researchers and practitioners. Most research on China’s stock market still follows the tradition in the US stock market by applying the three-factor model of Fama and French (FF3) [
In this paper, we extend the CH3 model of Liu et al. [
When implementing the factor model comparison, we also note that inappropriate distributional assumptions can induce substantial bias to the results [
In the spirit of this, we conduct the model evaluation under the alternative t-distributions. Consistent with tests under normal distribution, we first compare our model with a set of candidate factor models by examining their ability to explain each other’s factors. Utilizing the likelihood-ratio test proposed by Kan and Zhou [
Our contributions to the literature are mainly twofold. First, we achieve a significant improvement on the factor models in China by introducing an effective liquidity factor. Through rigorous model comparison procedures, we show that our four-factor model outperforms the prominent factor models in the literature. In particular, by surveying the most comprehensive list of anomalies for China, we confirm that our proposed factor model has superior explanatory power. Second, we contribute to the factor model comparison literature by providing evidence under alternative distributions instead of normality. Our results suggest that distributional assumptions can cause substantial changes to empirical results and reveal the necessity of considering alternative distributions in factor model studies. To the best of our knowledge, we are the first to evaluate and compare factor models under non-normal distributions for China’s stock market.
Early research on factor models in China mainly replicates the three-factor model (FF3) of Fama and French [
There are researchers who attempt to construct alternative China-specific factor models instead of simply following the US models proposed by Fama and French [
Existing research mainly evaluates factor models based on normality assumption [
Some early studies find no impact of distributional assumptions on empirical conclusions. Harvey and Zhou [
However, Zhou [
Let
This methodology also applies to test whether
Let
If the alphas are not significantly different from zero, then we say
Assuming that
It is noteworthy that the degrees of freedom are assumed to be unknown in the above procedures. In fact, the method also applies when we specify the corresponding degrees of freedom a priori.
Different from the frequentist approaches, Bayesian methods compare competing models in terms of their marginal likelihoods with solid theoretical basis. Higher marginal likelihoods indicate better performances in fitting real data. We adopt a recent Bayesian method of Chib and Zeng [
In the Bayesian method of Chib and Zeng [
Define the stochastic discount factor as
When conducting Bayesian model comparison, we need to sample the posterior distributions of
Specifically, let the prior of
Based on the foregoing setup, we only need to specify the training sample of
Following Chib and Zeng [
The sample period is from January 2000 to December 2019. The trading and financial data in our paper all come from CSMAR database, except for the factors of Liu et al. [
We include all A-share stocks of Shanghai and Shenzhen stock exchanges, whose first two digits of the unique Chinese six-digit stock identifier are 00, 30, and 60. We further impose some filters considering the economic and political background of China’s stock markets following Liu et al. [
The model we propose is built based on the CH3 model of Liu et al. [
We therefore call this new four-factor model CH3 + ILLIQ as an abbreviation.
The finance literature has proposed various factor models while most of them have been ignored in the studies of China’s stock markets. To better examine the performance of our model CH3 + ILLIQ, we compare it with several prominent factor models that are widely recognized along with the China-specific three-factor model (CH3).
Our first choice contains those well-recognized models inspired by economic and finance theory including the five-factor model (FF5) of Fama and French [
Table
Models and factors.
Models | Factors | References |
---|---|---|
Q | MKT, ME, IA, ROE | Hou et al. [ |
FF5 | MKT, SMB, HML, CMA, RMW | Fama and French [ |
FF5CP | MKT, SMB, HML, CMA, RMWCP | Fama and French [ |
FF6 | MKT, SMB, HML, CMA, RMW, UMD | Fama and French [ |
FF6CP | MKT, SMB, HML, CMA, RMWCP, UMD | Fama and French [ |
CH3 | MKT, SMB-CH, VMG, | Liu et al. [ |
CH3 + PMO | MKT, SMB-CH, VMG, PMO | Liu et al. [ |
Literature has extensively explored the anomalies in the US stock market [
Anomalies and definitions.
Acronym | Name | Reference | Details |
---|---|---|---|
AM | Asset-to-market | Fama and French [ | Total assets/fiscal-year-end market capitalization |
DER | Debt-to-equity ratio | Bhandari [ | Total liabilities/fiscal-year-end market capitalization |
LG | Liability growth | Litzenberger and Ramaswamy [ | Annual growth in total liabilities |
OCFP | Operating cash flow-to-price | Desai et al. [ | Operating cash flows/fiscal-year-end market capitalization |
PY | Payout yield | Boudoukh et al. [ | (Income before extraordinary items - the change of book equity)/fiscal-year-end market capitalization |
Rev1 | Reversal | De Bondt and Thaler [ | Cumulative returns from months t-60 to t-13 |
SG | Sustainable growth | Lockwood and Prombutr [ | Annual growth in book equity |
SMI | Sales growth minus inventory growth | Abarbanell and Bushee [ | Annual growth in sales - annual growth in inventory |
TG | Tax growth | Thomas and Zhang [ | Annual growth in taxes payable |
EP | Earnings-to-price | Cakici et al. [ | net profit excluding nonrecurrent gains/losses (in most recent quarterly financial statement)/last month-end market capitalization |
BM | Book-to-market equity | Wang and Xu [ | Book value of equity (in most recent quarterly financial statement)/last month-end market capitalization |
CP | Cash flow-to-price | Cakici et al. [ | net change in cash or cash equivalents (between the two most recent quarterly financial statements)/last month-end market capitalization |
SP | Sales-to-price | Barbee et al. [ | The annual operating revenue divided by fiscal-year-end market capitalization |
ACC | Accruals | Sloan [ | (Income before extraordinary items - operating cash flows)/average total assets |
PACC | Percent accruals | Hafzalla et al. [ | (Total profit - operating cash flow)/net profit |
dBE | Change in shareholders’ equity | Richardson et al. [ | Annual change in book equity/one-year-lagged total assets |
dPIA | Changes in PPE and inventory-to-assets | Lyandres et al. [ | (Annual change in gross property, plant, and equipment + the annual change in inventory)/one-year-lagged total assets |
IA | Investment-to-assets | Cooper et al. [ | Annual change in total assets/one-year-lagged total assets |
IVC | Inventory change | Thomas and Zhang [ | Annual change in inventory/average total assets |
IVG | Inventory growth | Belo and Lin [ | Annual growth in inventory |
NOA | Net operating assets | Hirshleifer et al. [ | (Operating assets - operating liabilities)/total assets |
ATO | Asset turnover | Soliman [ | Sales/net operating assets |
CFOA | Cash flow over assets | Asness et al. [ | Cash flow from operation/total assets |
CTO | Capital turnover | Haugen and Baker [ | Sales/total assets |
EBIT | Earnings before interests and taxes | Greenblatt [ | net profit + income tax expenses + financial expenses |
EY | Earnings yield | Greenblatt [ | EBIT/enterprise value |
GM | Gross margins | Novy-Marx [ | (Operating revenue - operating expenses)/operating revenue |
GP | Gross profitability ratio | Novy-Marx [ | (Operating revenue - operating expenses)/average total assets |
NPOP | Net payout over profits | Asness et al. [ | (net income - changes in book equity)/total profits |
ROIC | Return on invested capital | Greenblatt [ | (EBIT - nonoperating income)/noncash enterprise value |
TBI | Taxable income-to-book income | Green et al. [ | Pretax income/net income |
Z | Z-score | Dichev [ | Z-score = 1.2 × (working capital/total assets) + 1.4 × (retained earnings/total assets) + 3.3 × (EBIT/total assets) + 0.6 × (market value of equity/book value of total liabilities) + (sales/total assets) |
ROE | Return on equity | Guo et al. [ | Net earnings (in most recent quarterly financial statement)/book equity |
CHMOM | Change in 6-month momentum | Gettleman and Marks [ | Cumulative returns from months t-6 to t-1 minus those from months t-12 to t-7 |
MOM6M | 6-month momentum | Jegadeesh and Titman [ | Cumulative returns from months t-6 to t-2 |
MOM12M | 12-month momentum | Jegadeesh and Titman [ | Cumulative returns from months t-12 to t-2 |
MOM36M | 36-month momentum | Jegadeesh and Titman [ | Cumulative returns from months t-36 to t-13 |
VOLT | Volume trend | Haugen and Baker [ | Five-year trend in monthly trading volume/five-year average trading volume |
REV | 1-month reversal | Cakici et al. [ | Cumulative returns over the month t-1 |
PRC | Price | Blume and Husic [ | Share price at month t-1 |
RVOL | RMB trading volume | Chordia et al. [ | Natural log of RMB trading volume from month t-2 |
STD_RVOL | Volatility of RMB trading volume | Chordia et al. [ | Standard deviation of daily RMB trading volume at month t-1 |
STD_TURN | Volatility of turnover | Chordia et al. [ | Standard deviation of daily share turnover at month t-1 |
B_DIM | The Dimson beta | Dimson [ | Estimated using lead, lag, and current market returns |
B_DN | Downside beta | Ang et al. [ | Conditional covariance between a stock’s excess return and market excess return/the conditional variance of market excess return (condition: Market excess return is lower than the average level) |
B_FF | Fama and French beta | Fama and French [ | Calculated by regressing monthly returns on the current and recent lag market return with a five-year rolling window |
B_FP | Frazzini and Pedersen beta | Frazzini and Pedersen [ | Return volatilities for the stock/the market return volatilities × their return correlation |
IVOL | Idiosyncratic return volatility | Ali et al. [ | Standard deviation of residuals of weekly returns on weekly equal-weighted market returns from months t-36 to t-1 |
BETA | Market beta | Fama and MacBeth [ | Estimated market beta from weekly returns and market returns from months t-36 to t-1 |
B_HS | Hong and Sraer beta | Hong and Sraer [ | Summed-coefficients computed by daily returns with a one-year rolling window |
ILLIQ | Illiquidity | Amihud [ | Average of (absolute daily return/daily RMB trading volume) over the past 12 months ending on June 30 |
PRCDEL | Price delay | Hou and Moskowitz [ | The proportion of variation in weekly returns from months t-36 to t-1 explained by 4 lags of weekly market returns incremental to contemporaneous market returns |
TURN3 | 3-month share turnover | Datar et al. [ | Average monthly trading volume for 3 months/number of shares outstanding |
MV | Firm size | Wang and Xu [ | Market value at month t-1 |
STD | One-month volatility | Cakici et al. [ | Standard deviation of daily returns over the month t-1 |
MAX | Maximum daily returns | Carpenter et al. [ | Maximum of daily returns over month t-1 |
TURN | Twelve-month turnover | Liu et al. [ | Average daily share turnover over the past one year |
ABTURN | One-month abnormal turnover | Liu et al. [ | Average daily turnover over the past one month/average daily turnover over the past one year |
AGE | Firm age | Jiang et al. [ | Number of years since IPO year |
CFD | Cash flow-to-debt | Ou and Penman [ | Earnings before depreciation and extraordinary items/average total liabilities |
CR | Current ratio | Ou and Penman [ | Current assets/current liabilities |
CRG | Current ratio growth | Ou and Penman [ | Annual growth in current ratio |
QR | Quick ratio | Ou and Penman [ | (Current assets – inventory)/current liabilities |
QRG | Quick ratio growth | Ou and Penman [ | Annual growth in quick ratio |
SI | Sales-to-inventory | Ou and Penman [ | Sales/total inventory |
It is worth mentioning that not all the anomalies constructed will be used for model comparison. In the spirit of related literature, we only employ the anomalies that cannot be priced by CAPM under proper distributional assumptions. Since we survey the most comprehensive list of anomalies in model comparison studies for China, this also adds to the contribution to related literature. Finally, we derive a list of 15 significant anomalies that fall into four categories according to Jiang et al. [
In this section, we first present distributional tests on both risk factors and return anomalies. Then, under proper distributional assumptions, we report model comparison results using three approaches based on the candidate models: (1) ability to explain the factors in each other, (2) ability to price return anomalies, and (3) Bayesian marginal likelihoods.
First of all, we use the method in Section
We first test the distributional hypothesis of each factor, respectively. The univariate tests of the 14 factors are presented in Table
Univariate distributional tests.
Factor | Skewness | Kurtosis | ||||
---|---|---|---|---|---|---|
MKT | −0.24 | 0.14 | 0.81 | 4.00 | <0.01 | 0.99 |
SMB | −0.39 | 0.02 | 0.70 | 5.90 | <0.01 | 0.81 |
HML | −0.21 | 0.18 | 0.83 | 8.56 | <0.01 | 0.54 |
RMW | −0.54 | <0.01 | 0.60 | 7.05 | <0.01 | 0.68 |
RMWCP | −0.53 | <0.01 | 0.60 | 7.04 | <0.01 | 0.68 |
CMA | 0.16 | 0.32 | 0.87 | 4.84 | <0.01 | 0.93 |
ME | −0.50 | <0.01 | 0.63 | 7.70 | <0.01 | 0.62 |
IA | 0.32 | 0.05 | 0.75 | 4.41 | <0.01 | 0.97 |
ROE | 0.08 | 0.63 | 0.94 | 5.10 | <0.01 | 0.91 |
SMB-CH | 0.06 | 0.71 | 0.95 | 5.06 | <0.01 | 0.91 |
VMG | 0.20 | 0.21 | 0.84 | 4.42 | <0.01 | 0.97 |
PMO | −0.82 | <0.01 | 0.45 | 9.87 | <0.01 | 0.45 |
UMD | −0.02 | 0.91 | 0.98 | 4.41 | <0.01 | 0.97 |
ILLIQ | −0.34 | 0.04 | 0.73 | 8.99 | <0.01 | 0.51 |
Considering that related empirical analysis mainly relies on the joint distributions of factors and anomalies, it is necessary to test the multivariate kurtosis and skewness as well. Note that, following literature, we only employ the anomalies that cannot be priced by CAPM at the 5% significance level for empirical analysis. Based on the results above, we test whether the anomalies can be priced by CAPM under t-distributions, of which the degrees of freedom are assumed to be unknown and estimated based on the methods as illustrated in Section
Table
Multivariate distributional tests.
Skewness | Kurtosis | |||||
---|---|---|---|---|---|---|
Factors | 25.59 | <0.01 | 0.99 | 284.42 | <0.01 | 0.99 |
Anomalies | 134.31 | <0.01 | 0.17 | 605.03 | <0.01 | 0.65 |
In sum, fat tails are a nonnegligible feature of factors and anomalies and deserve serious investigations in related studies. We therefore choose t-distributions as the alternative distributional assumption for normal distributions in the following model comparison analysis.
As discussed earlier, we propose a four-factor model (CH3 + ILLIQ) that combines the three-factor model (CH3) of Liu et al. [
All the empirical tests are carried out under t-distributions and, therefore, we need to specify the d.f. for each test. To ensure that our results are consistent with each other, we use the optimal d.f. of the joint distribution of the 14 candidate factors and 15 anomalies throughout the empirical analysis. Based on the Expectation-Maximum method of Kan and Zhou [
Before any formal comparison, we first need to check the redundancy of the ILLIQ factor in the CH3 + ILLIQ model, i.e., whether ILLIQ can be explained by CH3. To this end, we use the method of Section
Redundancy tests for the factors in CH3 + ILLIQ.
(1) | (2) | (3) | (4) | |
---|---|---|---|---|
ILLIQ | MKT | SMB-CH | VMG | |
LR statistic | 29.57 | 4.06 | 3.85 | 71.83 |
<0.01 | 0.04 | 0.05 | <0.01 |
The redundancy tests above also help justify the validity of CH3 + ILLIQ and confirm that CH3 + ILLIQ outperforms CH3 in the spirit of Fama and French [
We proceed to compare the candidate models based on their ability to explain the factors in each other. Specifically, to test whether model A can explain model B, we regress the exclusive factors in B on A and test whether the intercepts are jointly zero using the LR method in Section
Candidate models’ ability to explain the factors in each other.
Other models’ ability to explain CH3 + ILLIQ | CH3 + ILLIQ’s ability to explain the other models | |||
---|---|---|---|---|
LR statistic | LR statistic | |||
Q | 65.45 | 4.36 | 0.23 | |
FF5 | 75.80 | 6.12 | 0.19 | |
FF5CP | 77.51 | 5.19 | 0.27 | |
FF6 | 86.28 | 8.84 | 0.12 | |
FF6CP | 86.66 | 8.67 | 0.12 | |
CH3 + PMO | 17.18 | 1.40 | 0.24 |
We further evaluate the performances of the factor models in terms of their ability to explain the 15 significant anomalies, which are presented in Section
Candidate models’ ability to explain the anomalies jointly.
t-Distribution | Normal distribution | |||
---|---|---|---|---|
LR statistic | GRS statistic | |||
Q | 69.97 | 4.23 | ||
FF5 | 80.97 | 5.32 | ||
FF5CP | 77.67 | 5.93 | ||
FF6 | 85.36 | 5.64 | ||
FF6CP | 78.02 | 6.09 | ||
CH3 | 36.71 | 0.001 | 1.88 | 0.03 |
CH3 + PMO | 24.85 | 0.05 | 1.43 | 0.13 |
CH3 + ILLIQ | 21.85 | 0.11 | 1.38 | 0.16 |
To further explore the details, we present the results of univariate analysis by regressing each anomaly on the candidate models, respectively. In Table
Candidate models’ ability to explain the anomalies: univariate tests.
Alpha (%) under t-Distribution | Alpha (%) under normal distribution | |||||
---|---|---|---|---|---|---|
CH3 + ILLIQ | CH3 + PMO | CH3 | CH3 + ILLIQ | CH3 + PMO | CH3 | |
ABTURN | 0.36 | 0.33 | 1.00 | 0.41 | 0.30 | 1.14 |
MV | 0.27 | 0.24 | 0.33 | 0.32 | 0.34 | 0.35 |
TURN | 0.08 | −0.25 | −0.69 | 0.05 | −0.20 | −0.62 |
ROE | 0.41 | 0.62 | 0.44 | 0.30 | 0.34 | 0.29 |
STD | −0.24 | −0.52 | 0.08 | 0.03 | −0.19 | 0.33 |
MAX | −0.29 | −0.43 | 0.05 | −0.24 | −0.39 | 0.14 |
SP | 0.44 | 0.14 | 0.25 | 0.76 | 0.57 | 0.48 |
BETA | 0.01 | −0.10 | −0.34 | 0.07 | 0.04 | −0.20 |
CHMOM | 0.27 | 0.23 | 0.03 | −0.12 | −0.02 | −0.40 |
EP | 0.19 | 0.11 | 0.02 | 0.13 | 0.15 | 0.02 |
GP | 0.23 | 0.28 | 0.35 | 0.39 | 0.35 | 0.50 |
IVOL | 0.01 | 0.03 | −0.28 | −0.44 | −0.48 | −0.44 |
PRCDEL | −0.18 | −0.14 | −0.39 | −0.09 | −0.04 | −0.22 |
REV | 0.11 | −0.11 | 0.28 | −0.06 | −0.30 | 0.30 |
Z | 0.09 | 0.24 | 0.29 | 0.06 | 0.15 | 0.38 |
Note.
In sum, the results above show that CH3 + ILLIQ outperforms other candidate models in terms of explaining return anomalies.
Lastly, considering that the above results are based on frequentist approaches, we complement the empirical analysis from a Bayesian perspective using the newly developed method of Chib and Zeng [
We conduct the tests under two alternative distributions, respectively. The left panel of Table
Bayesian marginal likelihoods.
log (marginal likelihood) | ||
---|---|---|
Normal distribution | ||
Q | 7435.64 | 7121.88 |
FF5 | 7426.22 | 7111.24 |
FF5CP | 7422.96 | 7107.92 |
FF6 | 7424.90 | 7108.21 |
FF6CP | 7423.19 | 7105.48 |
CH3 | 7455.77 | 7142.42 |
CH3 + PMO | 7461.53 | 7144.79 |
CH3 + ILLIQ | 7467.62 | 7149.10 |
In sum, our four-factor model CH3 + ILLIQ dominates the other candidate models in the sense of Bayesian marginal likelihoods.
Motivated by the fact that liquidity plays an important role in Chinese stock market, we propose a four-factor model that extends the three-factor model (CH3) of Liu et al. [
Our paper contributes to the literature by proposing a more effective four-factor model for China. Through rigorous model comparisons, we find that our four-factor model outperforms the prominent factor models from the existing literature. In particular, by surveying the most comprehensive list of anomalies for China in related literature, we find that our factor model has a strong explanatory power on return anomalies. Besides, we provide supporting evidence for t-distribution assumption in factor model comparisons. We show that different distribution assumptions can cause substantial changes to empirical results and reveal the necessity of considering alternative distributions in factor model studies.
The data used to support the findings of this study are available upon request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This research was funded by National Natural Science Foundation of China (grant no. 71672079).