The competitions of time-varying and constant loadings in asset pricing models: empirical evidence and agent-based simulations

Abstract

Under two frameworks of cross-section and time-series factors, we implement asset pricing models to dissect the abnormal returns in the Chinese stock market. Our findings indicate that the model using the earnings-to-price factor outperforms the model using the book-to-market factor in the framework of cross-section factors. Moreover, we further compare the time-varying loadings with constant loadings in the asset pricing models. Existing research has implied the outperformance of time-varying loadings in the US market. However, we consider the effects of backdoor listings in the Chinese stock market. Our evidence documents that the time-varying loading factor model cannot perfectly surpass the constant loading model. Our agent-based simulations indicate that such a finding originates from the static collective behaviors and stable beliefs of the Chinese traders.

Appendices

Appendix A: More details of the agent-based model

In Sect. 2, we have mentioned that there are two crowds of agents in the agent-based model. Therefore, in this appendix, we present the details of heterogeneous agents’ beliefs, as well as the determinations of asset prices and returns.

1.1 A.1: Heterogeneous agents and their beliefs

Within the crowd of fundamental traders, the agents believe the actual prices of assets vary around the fundamental prices. Hence, the fundamental traders long the underpriced assets and short the overpriced assets. As for the crowd of trend-chasing traders, the agents buy assets when the assets experience price increases in the past and sell assets due to the past decreases in prices. Consequently, the beliefs regarding expectations about returns for these crowds of traders proceed as follows:

$${E}_{f,t}\left({\mathbf{r}}_{t+1}\right)={\varvec{\omega}}+m\left(1-{\mathbf{P}}_{t}^{*-1}{\mathbf{p}}_{t-1}\right)$$

(A1)

$${E}_{c,t}\left({\mathbf{r}}_{t+1}\right)={\mathbf{g}}_{t-1}$$

(A2)

where ${E}_{f,t}\left({\mathbf{r}}_{t+1}\right)$ is the beliefs regarding expectations about returns for the crowd of fundamental traders, ${E}_{c,t}\left({\mathbf{r}}_{t+1}\right)$ is the beliefs regarding expectations about returns for the crowd of trend-chasing traders, ${\varvec{\omega}}={\left({\omega }_{1},{\omega }_{2},\cdots ,{\omega }_{N}\right)}^{T}$ denotes the time-series average of the fundamental dividend yields of each asset, ${\mathbf{p}}_{t}^{*}={\left({p}_{1,t}^{*},{p}_{2,t}^{*},\cdots ,{p}_{N,t}^{*}\right)}^{T}$ denotes the fundamental price of each asset, ${\mathbf{P}}_{t}^{*}:=diag\left[{\mathbf{p}}_{t}^{*}\right]$, ${\mathbf{p}}_{t}={\left({p}_{1,t},{p}_{2,t},\cdots ,{p}_{N,t}\right)}^{T}$ denotes the actual price of each asset, ${\mathbf{g}}_{t-1}$ is a vector containing the averages of past returns, and ${\mathbf{g}}_{t-1}=h{\mathbf{g}}_{t-2}+\left(1-h\right){\mathbf{r}}_{t-1}$.

In Chiarella et al. (2013), the fundamental traders stabilize the market, while trend-chasing traders boost the market risks. Therefore, the variance–covariance matrices for the returns of these traders are defined in equations (A3) and (A4):

$${\Sigma }_{f}=diag\left({\sigma }_{1}^{2},{\sigma }_{2}^{2},\cdots ,{\sigma }_{N}^{2}\right)$$

(A3)

$${\Sigma }_{c,t}={\Sigma }_{0}+\zeta {\mathbf{V}}_{t-1}$$

(A4)

where ${\Sigma }_{f}$ is the beliefs regarding the variance–covariance matrix for the returns for the crowd of fundamental traders, ${\Sigma }_{c,t}$ is the beliefs regarding the variance–covariance matrix for the returns for the crowd of trend-chasing traders, ${\Sigma }_{0}$ denotes the initial variance–covariance matrix, and ${\mathbf{V}}_{t-1}$ is calculated as follows:

$${\mathbf{V}}_{t-1}=h{\mathbf{V}}_{t-2}+h\left(1-h\right)\left({\mathbf{r}}_{t-1}-{\mathbf{g}}_{t-2}\right){\left({\mathbf{r}}_{t-1}-{\mathbf{g}}_{t-2}\right)}^{T}$$

(A5)

1.2 A.2: The determinations of asset prices and returns

To perform simulations for constant and time-varying betas, we should calculate asset prices and returns in accordance with Chiarella et al. (2013). Equations (A6) and (A7) show the procedures used to obtain the prices and returns.

$${\mathbf{p}}_{t}={\mathbf{S}}^{-1}\left\{\frac{{n}_{f}}{{\varrho }_{f}}{\Sigma }_{f}^{-1}\left[{E}_{f,t}\left({\mathbf{r}}_{t+1}\right)\right]+\frac{{n}_{c}}{{\varrho }_{c}}{\Sigma }_{c,t}^{-1}\left[{E}_{c,t}\left({\mathbf{r}}_{t+1}\right)\right]-\left(\frac{{n}_{f}}{{\varrho }_{f}}{\Sigma }_{f}^{-1}+\frac{{n}_{c}}{{\varrho }_{c}}{\Sigma }_{c,t}^{-1}\right){r}_{f}1\right\}+{\mathbf{S}}^{-1}{\boldsymbol{\vartheta }}_{t}$$

(A6)

$${\mathbf{r}}_{t}=\left({\mathbf{P}}_{t-1}^{-1}{\mathbf{p}}_{t}-1\right)+{\mathbf{P}}_{t-1}^{-1}{\mathbf{P}}_{t-1}^{*}{{\varvec{\rho}}}_{t}$$

(A7)

where ${{\varvec{\rho}}}_{t}={\left({\rho }_{1,t},{\rho }_{2,t},\cdots ,{\rho }_{N,t}\right)}^{T}$ contains the fundamental dividend yield of each asset, ${{\varvec{\rho}}}_{t}$ is randomly drawn from a normal distribution, ${\mathbf{P}}_{t}:=diag\left[{\mathbf{p}}_{t}\right]$, $\mathbf{S}=diag\left({s}_{1},{s}_{2},\cdots ,{s}_{N}\right)$ is a matrix of average asset supply per trader, and ${\boldsymbol{\vartheta }}_{t}$ denotes a vector of demand shocks for each asset and is randomly drawn from a normal distribution.

Appendix B: Variable definitions

First of all, firm size is measured by market capitalization, which is calculated by the stock price times the number of shares. Four ratios related to accounting metrics are applied in this paper, including earnings-to-price, book-to-market, cash-flow-to-price, and the return on equity. We divide the earnings by firm size to acquire the earnings-to-price ratio. Earnings is net profit in excess of a nonrecurring gain or loss. The ratio of shareholder equity to firm size produces the book-to-market ratio. To obtain the cash-flow-to-price ratio, we divide the changes in cash flow between the two most recent periods by firm size. The return on equity is the ratio of earnings to shareholder equity. Since the accounting information is revealed semi-annually or quarterly in China, the most recent data in month t-1 are used to generate these ratios.

Two volatility measures regarding the 1-month volatility and MAX are used in this paper. We compute the time-series standard deviation of daily returns for the past 20 trading days to acquire the 1-month volatility, while MAX is the maximum daily return for the same horizon. We use the average daily turnover for the past 250 trading days as the 12-month turnover. Besides, we divide the average daily turnover for the past 20 trading days by the 12-month turnover to obtain the 1-month abnormal turnover. The first day of the past T-trading-day period is the last trading day in month t-1.

Appendix C: The construction of the anomaly portfolios and factor portfolios

Following the concepts of Fama and French (2020), the portfolios targeted by model factors and anomaly portfolios as well as related excess returns are created. In each month, the stocks are sorted into 5 groups simultaneously based on firm size, earnings-to-price, and book-to-market for the prior month. The intersections between the sorts based on firm size and earnings-to-price produce 25 value-weighted portfolios besides the intersections between the sorts on firm size and book-to-market. Hence, we obtain 50 value-weighted portfolios targeted by model factors.

The anomaly variables in the Chinese stock market are cash-flow-to-price, the return on equity, 1-month volatility, MAX, 12-month turnover, and 1-month abnormal turnover in accordance with Liu et al. (2019). To be more specific, in each month, the stocks are simultaneously sorted into 5 groups based on firm size, cash-flow-to-price, the return on equity, 12-month turnover, and 1-month abnormal turnover for the last month.

From the intersections between size sorts and the sorts, respectively, on cash-flow-to-price, the return on equity, 12-month turnover, and 1-month abnormal turnover, there are 100 value-weighted portfolios. The above sorts are simultaneous, with two exceptions. It is rare for the stocks with large firm size to have high volatility, and so the portfolios related to 1-month volatility and MAX are obtained by conditional sorts to avoid empty or tiny portfolios (Fama and French 2020). In each month, the stocks are classified one by one based on predetermined firm sizes into 5 groups. Within each size group, the stocks are sorted into 5 groups in accordance with 1-month volatility and MAX for the last month, again resulting in 50 value-weighted portfolios. Consequently, we obtain 150 value-weighted anomaly portfolios.

Appendix D: The construction of the long-short portfolios

Unconditional sorts and size-neutral sorts are executed to generate long-minus-short portfolios following Liu et al. (2019). The unconditional long-minus-short portfolios include those based on firm size, earnings-to-price, book-to-market, cash-flow-to-price, the return on equity, 1-month volatility, MAX, 12-month turnover, and 1-month abnormal turnover. In each month, the stocks are sorted into 10 groups based on firm size for the last month. The group with the lowest firm size is the long leg and the group with the highest firm size is the short leg.

As for the long-minus-short portfolios related to earnings-to-price, book-to-market, cash-flow-to-price, and the return on equity, these variables for the previous month are, respectively, used to classify the stocks into 10 groups in each month. The group with the highest ratio is the long leg and the group with the lowest ratio is the short leg. The long-minus-short portfolios related to 1-month volatility and MAX are engendered as well. The stocks are one by one sorted into 10 groups for each month based on 1-month volatility and MAX for the last month. The group with the highest 1-month volatility or MAX is the short leg, whereas the group with the lowest 1-month volatility or MAX constitutes the long leg.

In each month, the stocks are classified into 10 groups based on the previous 12-month turnover and 1-month abnormal turnover to obtain the long-minus-short portfolios related to 12-month turnover and 1-month abnormal turnover. What forms the long leg is the group with the lowest 12-month turnover or 1-month abnormal turnover. By contrast, the short leg is the group with the highest 12-month turnover or 1-month abnormal turnover.

The size-neutral long-minus-short portfolios are also based on the above variables, but the stocks are sorted based on firm size before selecting the long and short legs. To be specific, the stocks are first sorted into 10 groups based on firm size for the last month for each month. Within every group from the sorts on firm size, the long and short legs are screened out. All long and short legs across size groups are pooled together to construct the total long and short legs. We calculate the value-weighted returns of the long and short legs for each month as well as the long-minus-short return spreads for these long-minus-short portfolios.

Appendix E: Comparing the LSY and FF models through pricing errors

This appendix presents the pricing errors and loadings in dissecting return spreads using the cross-section factors. Table 10 shows the results of the LSY cross-section factors and the outcomes of the FF cross-section factors are presented in Table 11. Since the asset pricing model using the LSY cross-section factors captures more anomalies than the model using the FF cross-section factors, the LSY model is superior in dissecting return spreads.

From Panel A of Table 10, the asset pricing model using the LSY cross-section factors just fails to fully dissect the anomalies based on the return on equity, 1-month volatility, and MAX since the absolute t-statistics of the alphas range from 2.41955 to 23.75747. In addition, the model fully dissects the remaining six anomalies as the absolute t-statistics of the alphas are all lower than 1.96. Hence, the asset pricing model using the LSY cross-section factors fully captures six anomalies out of a total of nine anomalies from unconditional sorts.

When we obtain the return spreads of anomalies from size-neutral sorts, the results just slightly deteriorate in Panel B. The asset pricing model using the LSY cross-section factors fails to fully capture the anomalies related to earnings-to-price, cash-flow-to-price, the return on equity, and 1-month abnormal turnover since the absolute t-statistics of the alphas lie between 2.19063 and 5.00453. The model fully dissects the remaining four anomalies because the absolute t-statistics of the alphas are below 1.96. In short, the asset pricing model using the LSY cross-section factors captures six unconditional anomalies and four size-neutral anomalies.

In Panel A of Table 11, the results of unconditional anomalies show that the asset pricing model using the FF cross-section factors fails to capture the anomalies related to earnings-to-price, the return on equity, 1-month volatility, and MAX because the absolute t-statistics of the alphas range from 4.71424 to 24.53758. The model captures the remaining five unconditional anomalies due to the insignificant absolute t-statistics of the alphas being below 1.96. The results from size-neutral sorts in Panel B indicate that the asset pricing model using the FF cross-section factors is characterized by poor performance. There are eight anomalies from size-neutral sorts, while the asset pricing model using the FF cross-section factors only captures two anomalies based on book-to-market and cash-flow-to-price (the absolute t-statistics of the alphas are below 1.5). Moreover, the model fails when facing the remaining six anomalies as the absolute t-statistics of the alphas exceed 2 or even 7.

Table 10 Dissecting return spreads using the LSY cross-section factors

Table 11 Dissecting return spreads using the FF cross-section factors

Appendix F: The construction of the momentum portfolios

In accordance with Jegadeesh and Titman (1993), the constructions of the momentum portfolios require formation and holding periods. Based on a rolling procedure, for each month, we sort the stocks based on the past returns for the formation period to generate the winner and loser portfolios. The winner portfolio consists of the stocks with the highest past returns, while the loser portfolio contains the stocks with the lowest past returns. In the holding period, we calculate the future returns of the winner and loser portfolios. The momentum profit is the difference in future returns between the winner and loser portfolios (winner minus loser).

Appendix G: The strength of collective behavior

In compliance with Feng et al. (2012), the strength of collective behavior (c) in the context of this paper can be conceptually measured as:

$$c=\left(\frac{a}{\left|\sum_{i=1}^{a}{D}_{i}\right|}\right)$$

$${D}_{i}=\left\{\begin{array}{c}1,\mathrm{with }P\Rightarrow \mathrm{fundamental}\\ -1,\mathrm{with} 1-P\Rightarrow \mathrm{trend chasing}\end{array}\right.$$

where a is the number of agents in a crowd, ${D}_{i}$ is each agent’s decisions, and P is the probability to be a fundamental trader.

When all agents are fundamental traders or trend-chasing traders, $\left|\sum_{i=1}^{a}{D}_{i}\right|$ arrives at its maximum value, i.e., $\left|\sum_{i=1}^{a}{D}_{i}\right|=a$ and $c=1$, indicating the strongest collective behaviors. In other words, collective behaviors are the strongest once all the crowds of agents have the same strategies. When c is higher, the collective behavior is weaker. Using the agent-based model of this paper as an example, if ${n}_{f}=0.1$, then $c=1.25$.