An Advanced Time-Varying Capital Asset Pricing Model via Heterogeneous Autoregressive Framework: Evidence from the Chinese Stock Market

Abstract

The capital asset pricing model (CAPM) is a foundational asset pricing model that is widely applied and holds particular significance in the globally influential Chinese stock market. This study focuses on the banking sector, enhancing the performance of the CAPM and further assessing its applicability within the Chinese stock market context. This study incorporates a heterogeneous autoregressive (HAR) component into the CAPM framework, developing a CAPM-HAR model with time-varying beta coefficients. Empirical analysis based on high-frequency data demonstrates that the CAPM-HAR model not only enhances the capability of capturing market fluctuations but also significantly improves its applicability and predictive accuracy for stocks in the Chinese banking sector.

1. Introduction

The capital asset pricing model (CAPM), initially introduced by Sharpe in 1964, is a fundamental technique in financial research for assessing systematic risk exposure in equity markets [1]. The conventional CAPM assumes a constant variance in market returns and models asset pricing statically. It overlooks the time-varying nature of asset return volatility and the dynamic correlations between assets, leading to a clear disconnect from real financial market behavior [2]. Consequently, it is unable to capture the persistence and volatility inherent in real-world asset returns.

The concept of time-varying betas has been widely explored as an extension of the CAPM in several academic investigations [3,4,5]. However, the majority of dynamic CAPM relies on simple linear regression using low-frequency data to model time-varying betas, neglecting the significant advances in volatility estimation models, such as the heterogeneous autoregressive (HAR) model, in capturing the complex dynamics of financial time series. Low-frequency data, being overly discrete, leads to intraday information loss and significantly weakens the model’s ability to accurately represent financial risk and volatility. The application of CAPM holds significant value in the Chinese stock market, the world’s second-largest [6]. By the end of 2022, the total market capitalization of the Chinese stock market reached $12.5 trillion (approximately RMB 88 trillion), further solidifying its position as the second-largest market globally. The distinctive characteristics and vast scale of the Chinese stock market make it a focal point for academic and industry research on emerging markets [7]. Extensive empirical and theoretical research on the effectiveness of CAPM in the Chinese stock market has significantly enriched the existing literature and contributed to the sustained development of China’s financial markets [8,9]. The financial sector occupies a critical strategic role in China, especially the banking industry, which is intricately linked to the nation’s economic progress. In this context, examining the effectiveness of the CAPM in publicly listed banks in China holds significant academic and practical relevance.

The contributions of this study in comparison to the existing literature are as follows. Firstly, this study employs high-frequency data to construct HAR models, effectively predicting realized covariance and realized volatility. High-frequency data capture intraday trading information at extremely short intervals, providing a more detailed view of market dynamics compared to low-frequency data. This greatly improves the accuracy of volatility analysis. Secondly, this study introduces an innovative approach by utilizing the HAR model to predict the covariance between asset and market returns, as well as the variance in market returns in the CAPM betas, thereby constructing the CAPM-HAR model with time-varying betas. Supported by high-frequency data, the CAPM-HAR model offers more accurate dynamic predictions of asset returns through time-varying betas. This extension creates new opportunities for the empirical application of time-varying CAPM and holds considerable theoretical and practical significance. Finally, by conducting an empirical analysis of the stocks of listed banks on the Shanghai Stock Exchange, this study validates the applicability and effectiveness of the CAPM-HAR model in the context of China’s stock market, thereby providing valuable insights into China’s financial markets.

The remainder of this paper is structured as follows: Section 2 provides a concise review of the existing literature on the time-varying CAPM and the recent developments in volatility modeling. Section 3 describes the data utilized in the empirical analysis. Section 4 outlines the methodological approach adopted in this study. Section 5 reports and discusses the empirical findings. Finally, Section 6 presents the conclusions of the study, summarizing key insights and potential directions for our future research.

2. Literature Review

The traditional CAPM assumes that the coefficients are constant, employing a static approach for asset pricing analysis, which often fails to capture the complexities of financial markets. Empirical studies have consistently highlighted that dynamic models with time-varying betas offer more accurate predictions of asset returns. Local linear regression was employed to estimate volatility and betas, and empirical analysis strongly rejected the assumption of constant volatility [10]. Subsequently, time-varying beta models have been proposed, incorporating methodologies such as state-space models (SSM), dynamic conditional beta (DCB) models, and autoregressive conditional beta (ACB) models, with several studies leveraging high-frequency data [11]. These dynamic models have demonstrated superior predictive performance and practical applicability compared to traditional static approaches, particularly in forecasting risk variations within the Real Estate Investment Trust (REIT) market. Moreover, realized betas, derived from daily returns, have been shown to outperform the Fama–MacBeth beta estimates based on monthly returns in explaining momentum profits, thereby significantly enhancing predictive accuracy [12].

In the existing literature on dynamic CAPM, the majority of studies rely on simple regression methods to estimate time-varying betas, overlooking the significant advances in volatility estimation models that are more adept at capturing the complex dynamic characteristics of financial time series. In contrast, innovative volatility estimation methodologies, such as Generalized Autoregressive Conditional Heteroskedasticity (GARCH), HAR models on high-frequency data, and multifactor dynamic betas model, have substantially improved the understanding of asset volatility and risk dynamics [13]. A multivariate GARCH(1,1) model has been employed in prior research to estimate the conditional covariance matrix, incorporating economic fundamentals and investor sentiment as informational variables to dynamically assess market risk premiums, thereby emphasizing the significance of investor sentiment in market risk pricing [14]. The HAR model has been shown to outperform GARCH models in forecasting short-term and medium-term volatility [15,16], effectively capturing volatility characteristics across different time scales [17,18].

A novel statistical methodology utilizing multivariate realized kernels has been introduced to estimate covariance and correlation from high-frequency data, ensuring that the covariance estimates remain positive and semi-definite even in the presence of noise and asynchronous observations [19,20]. On this basis, realized covariance matrices have been computed using high-frequency data, with the HAR model incorporated into correlation forecasting to ensure computational feasibility and significantly improve predictive precision when analyzing high-dimensional asset return distributions [21]. However, the application of high-frequency data is not without limitations. High-frequency financial data, distinguished by their large volume and complexity, present substantial challenges to storage and computational capacities [22]. In addition, issues related to data quality, such as erroneous records, disordered time sequences, and price jumps induced by macroeconomic events, further exacerbate the difficulties associated with data preprocessing and modeling [23]. These challenges compromise the stability and reliability of models, underscoring the limitations of existing statistical methods in effectively capturing the dynamic characteristics of high-frequency data [24]. In this study, data with fixed time intervals are selected, and the high overall liquidity of the banking sector helped to partially overcome the limitations mentioned above through data cleaning and pre-processing techniques. Future research could further improve the applicability and predictive performance of the model by integrating liquidity adjustment methods with online forecasting techniques.

Although the HAR model has been widely applied in modeling asset return volatility, it has not yet been integrated into the CAPM framework. This paper utilizes non-parametric methods to compute realized covariance and realized volatility, independently constructs HAR models for each, and integrates the HAR model into the CAPM framework to better capture the time-varying characteristics of liquidity risk across multiple time scales.

The Chinese stock market exhibits a unique structure, marked by high volatility and a diverse investor base comprising both institutional and individual participants. This uniqueness limits the effectiveness of conventional CAPM models, necessitating more flexible approaches to account for the complexities inherent in the market [25]. This paper integrates HAR volatility models into the dynamic CAPM framework to improve the model’s applicability under the high-volatility conditions characteristic of the Chinese stock market. This approach provides a foundation for future research to systematically investigate the role of volatility in asset pricing, particularly how the integration of multi-time-scale dynamic volatility models can enhance the performance of the CAPM under diverse market conditions. This not only broadens the applicability of the CAPM but also offers new tools for volatility forecasting in financial markets.

3. Data

This study examines the 36 publicly listed banks on the Shanghai Stock Exchange between 1 January 2020 and 31 December 2022, excluding six companies with substantial missing data. Using high-frequency data from the RESSET database, daily and 5 min interval trading records for the remaining 30 banks were collected for empirical analysis. These banks exhibit differences in size and nature, which ensures that the sample characteristics meet the model’s requirement for sample heterogeneity. The codes and names of the 30 listed banks are presented in

Table 1. Codes and names of public companies.

Company Code 1	Company Name 1	Company Code 2	Company Name 2
600000	Shanghai Pudong Development Bank Co., Ltd.	601577	Bank of Changsha Co., Ltd.
600015	Huaxia Bank Co., Ltd.	601658	Postal Savings Bank of China Co., Ltd.
600016	China Minsheng Banking Corp., Ltd.	600318	Anhui Xinli Finance Co., Ltd.
600036	China Merchants Bank Co., Ltd.	601818	China Everbright Bank Co., Ltd.
600705	AVIC Capital Co., Ltd.	601838	Bank of Chengdu Co., Ltd.
600901	Jiangsu Financial Leasing Co., Ltd.	601860	Bank of Jiangsu Co., Ltd.
600908	Wuxi Rural Commercial Bank Co., Ltd.	601916	Zhejiang Chouzhou Commercial Bank Co., Ltd.
600919	Bank of Jiangsu Co., Ltd.	601939	China Construction Bank Corporation
600926	Bank of Hangzhou Co., Ltd.	601328	Bank of Communications Co., Ltd.
600928	Xi’an Bank Co., Ltd.	601988	Bank of China Limited
601009	Bank of Nanjing Co., Ltd.	601997	Bank of Guiyang Co., Ltd.
601077	Chongqing Rural Commercial Bank Co., Ltd.	601998	China CITIC Bank Corporation Limited
601128	Jiangsu Changshu Rural Commercial Bank Co., Ltd.	603323	Jiangsu Suzhou Rural Commercial Bank Co., Ltd.
601166	Industrial Bank Co., Ltd.	601229	Bank of Shanghai Co., Ltd.
601169	Bank of Beijing Co., Ltd.	601288	Agricultural Bank of China

.

The selection of the banking sector as the research focus is motivated by several considerations. First, the banking sector, characterized by its high representativeness and stability, occupies a strategically vital position within China’s financial market and is closely interconnected with the nation’s macroeconomic growth. Second, publicly listed banks offer comprehensive and reliable data, bolstered by high market liquidity, which facilitates the robust validation of the model’s effectiveness. Additionally, the banking industry is a perfect place to research the CAPM because of its strong systematic risk. The selection of the Chinese stock market is motivated by its status as the world’s second-largest equity market, offering vast research potential and abundant empirical data. Moreover, previous studies have highlighted the limited effectiveness of the CAPM in the Chinese market, particularly in capturing dynamic market characteristics and accurately pricing risk [26,27]. Therefore, this study aims to overcome these limitations by incorporating multi-scale volatility modeling and proposing the CAPM-HAR model to enhance the effectiveness of asset pricing significantly. This approach offers new insights and tools for financial research and practical applications in the Chinese market.

The one-year Chinese government bond yields are selected as the risk-free rates, and the annualized risk-free interest rates are converted to daily data using a compounding method, with data sourced from the China Bond Information Network. The one-year government bond yields for 2020–2022 were recorded at $1.5\%,1.5\%,1.65\%$, translating to daily rates of $0.0041\%,0.0041\%,0.0045\%$. The Shanghai Composite Index meets the structural requirements of the CAPM for the market, accurately reflecting market trends and exhibiting good market representation. Additionally, the 30 selected sample stocks are all traded on the Shanghai Stock Exchange. Therefore, this study uses the returns of the Shanghai Composite Index as a proxy for market returns, with data sourced from the RESSET high-frequency database. Figure 1 illustrates the time series of returns for selected stocks and the Shanghai Composite Index from 1 January 2020 to 31 December 2022.

By conducting Augmented Dickey–Fuller (ADF) unit root tests [28] on the return data of individual stocks and the Shanghai Composite Index, we assess whether these time series are stationary. The test results indicate that the p-values for all series are close to zero, strongly rejecting the null hypothesis of a unit root. This suggests that the return series are stationary, confirming that the data are suitable for further development of the CAPM.

4. Methodology

4.1. CAPM

In the context of the CAPM, various components are defined as follows: $R_j$ represents the expected return of asset j; ${\mu }_f$ denotes the risk-free rate, typically approximated by the government bond yield, which reflects the return investors can achieve without taking on risk. The ${\beta }_j$ corresponds to the beta coefficient of asset j, measuring its systematic risk relative to the overall market. $R_M$ is the expected return of the market, often proxied by the return on a broad market index. The term ${\alpha }_j$ represents the intercept, capturing the deviation between the asset and the market pricing. Theoretically, if the model is correctly specified, ${\alpha }_j$ should be zero.

Based on these definitions, the conventional CAPM is expressed as follows:

$$R_j={\mu }_f+{\alpha }_j+{\beta }_j\left(R_M-{\mu }_f\right)+u_j$$

(1)

where, $u_j$ represents the error term. Let $R_j^{*}=R_j-{\mu }_f$ and $R_M^{*}=R_M-{\mu }_f$ represent the excess returns of asset j and the market, respectively.

$$R_j^{*}={\alpha }_j+{\beta }_j{R}_M^{*}+u_j$$

(2)

For the CAPM represented in Equation (2), the beta is calculated as follows [1]:

$${\beta }_j={\displaystyle \frac{COV(R_j,R_M)}{VAR\left(R_M\right)}}$$

(3)

Here, $COV(R_j,R_M)$ represents the covariance between asset j and the market, while $VAR\left(R_M\right)$ denotes the variance of the market. This study will leverage high-frequency data to establish HAR models for the covariance $COV(R_j,R_M)$ and variance $VAR\left(R_M\right)$ separately, thereby obtaining a time-varying beta.

4.2. HAR

This section constructs HAR-RC and HAR-RV models for the estimators $\widehat{CO{V}_t}$ and $\widehat{R{V}_t}$ of $COV(R_j,R_M)$ and $VAR\left(R_M\right)$, respectively, in order to forecast the realized covariance among assets returns and the market returns, as well as the realized variance of the market. In this paper, $VAR\left(R_M\right)$ and $COV(R_j,R_M)$ in the CAPM are considered time-varying variables, and are estimated on a daily basis to capture their dynamic behavior over time. In CAPM, $VAR\left(R_M\right)$ represents the variance of the market return, while volatility is the square root of the return variance. Therefore, the square of the realized volatility $\widehat{R{V}_t}$ can be used as a proxy for the market return variance. Similarly, $COV(R_j,R_M)$ represents the covariance between the returns of asset j and the market, which can be approximated by the corresponding component of the realized covariance matrix $\widehat{CO{V}_t}$. This paper predicts the next-period beta by constructing HAR models for realized volatility and realized covariance, respectively. The HAR model is a linear autoregressive model designed for time series data, frequently used for modeling and forecasting realized volatility. These predictions will subsequently inform the estimation of the next period’s beta.

4.2.1. HAR-RV

In the context of realized volatility, let $p_{t,i}$ represent the price vector at time i on day t, and $r_{t,i}=ln\left(p_{t,i}\right)-ln\left(p_{t,i-1}\right)$ denote the log return vector of the market during the i-th sampling interval on trading day t. The sampling frequency within the trading day is defined as $1/N$, where N is the total number of sampling intervals.

Based on these definitions, the realized volatility on trading day t is calculated as follows:

$$R{V}_t=\sqrt{\sum _{i=1}^N{\left(r_{t,i}\right)}^2}$$

(4)

Typically, a 5 min sampling frequency is employed to construct realized volatility [29]. The HAR-RV model is defined as follows [17]:

$$\widehat{R{V}_t}={\beta }_0+{\beta }_{d}R{V}_{t-1}+{\beta }_{w}R{V}_w+{\beta }_{m}R{V}_m+u_t.$$

(5)

To capture the long-memory characteristics of realized volatility, this paper selects lags of 1 day, 1 week, and 1 month, using 5 days and 22 days to compute the weekly and monthly realized volatility, respectively:

$$\begin{array}{cc}\hfill R{V}_w& ={\displaystyle \frac{1}{5}}\sum _{i=1}^{5}R{V}_{t-i},\hfill \end{array}$$

(6)

$$\begin{array}{c}\\ \hfill R{V}_m& ={\displaystyle \frac{1}{22}}\sum _{i=1}^{22}R{V}_{t-i}\hfill \end{array}$$

(7)

4.2.2. HAR-RC

In this paper, a weighted kernel method [30,31] is employed to estimate the covariance matrix among asset returns and market returns. Let $\widehat{{COV}_t}$ denote the realized covariance matrix on day t, where the kernel function $k(·)$ is the Parzen kernel, and the optimal bandwidth $H_t$ is selected through cross-validation. The specific formula for the covariance estimation is given as follows:

$$\begin{array}{c}\hfill \widehat{CO{V}_t}=\sum _{h=-n}^{n}k\left({\displaystyle \frac{h}{H_t+1}}\right)\sum _{i=\left|h\right|+1}^n\left(r_{t,i}{r}_{t,i-h}^{\prime }{\mathbf{1}}_{\{h\ge 0\}}+r_{t,i-h}{r}_{t,i}^{\prime }{\mathbf{1}}_{\{h<0\}}\right)\end{array}$$

(8)

Here, $r_{t,i}$ represents the log-return series, and h denotes the lag length. Let ${RV}_t=diag\left({COV}_t\right)$ be the diagonal matrix of the covariance matrix, which contains the realized volatilities of each asset. The covariance matrix ${COV}_t$ can be further expressed [32] as follows:

$$\begin{array}{c}\hfill {COV}_t=\sqrt{{RV}_t}·{RCorr}_t·\sqrt{{RV}_t}\end{array}$$

(9)

By decomposing ${COV}_t$, the correlation matrix ${RCorr}_t$ is obtained. A dynamic model is then constructed for the correlation matrix, specifically using the following HAR-RC model [21]:

$$\begin{array}{cc}\hfill \widehat{vech\left({RCorr}_t\right)}& =vech\left({RCorr}_{t-1}\right)(1-{\beta }_1-{\beta }_2-{\beta }_3)\hfill \\ \hfill & +{\beta }_1·vech\left({RCorr}_t\right)+{\beta }_2·{\displaystyle \frac{1}{4}}\sum _{k=2}^{5}vech\left({RCorr}_{t-k}\right)\hfill \\ \hfill & +{\beta }_3·{\displaystyle \frac{1}{15}}\sum _{k=6}^{20}vech\left({RCorr}_{t-k}\right)+u_t\hfill \end{array}$$

(10)

In the above equation $\widehat{vech\left({RCorr}_t\right)}$ represents the vectorization operation applied to the matrix ${RCorr}_t$, and $u_t$ denotes the random error term.

This study constructs HAR-RC and HAR-RV models for the estimators $COV(R_j,R_M)$ and $VAR\left(R_M\right)$, respectively, which will be used subsequently to compute time-varying betas for the development of the dynamic CAPM-HAR model.

4.3. Time-Varying CAPM-HAR

In practice, ${\beta }_j$ varies dynamically over time, reflecting the time-varying relationship between an individual asset’s expected return and market systemic risk. This variability allows a more precise capture of an asset’s risk exposure under different market conditions. Therefore, this paper utilizes high-frequency data to establish a dynamic CAPM-HAR model:

$$R_{j,t}^{*}={\alpha }_{j,t}+{\beta }_{j,t}{R}_{M,t}^{*}+u_{j,t}$$

(11)

In the above equation $R_{j,t}^{*}=R_{j,t}-{\mu }_{f,t}$ denotes the expected excess return of asset j at time t. Here, ${\mu }_{f,t}$ represents the risk-free rate at time t, which is still approximated by the government bond yield. ${\beta }_{j,t}$ refers to the beta coefficient of asset j at time t. $R_{M,t}^{*}=R_{M,t}-{\mu }_{f,t}$ denotes the expected excess return of the market at time t, with the Shanghai Stock Exchange Composite Index selected in this study as the proxy. Finally, ${\alpha }_{j,t}$ is introduced to capture the asset’s deviation from the market:

For the CAPM-HAR represented in Equation (11), this study establishes a time-varying $\beta$ estimation model:

$${\beta }_{j,t}={\displaystyle \frac{COV(R_{j,t},R_{M,t})}{VAR\left(R_{M,t}\right)}}\approx {\displaystyle \frac{\widehat{CO{V}_{jM,t}}}{{\widehat{R{V}_t}}^2}}$$

(12)

In this equation, $COV(R_{j,t},R_{M,t})$ and $VAR\left(R_{M,t}\right)$ represent the covariance between the returns of asset j and the market on day t, and the variance of market returns on day t, respectively. $\widehat{CO{V}_{jM,t}}$ refers to a component of the covariance matrix $\widehat{CO{V}_t}$, representing the predicted value of $COV(R_{j,t},R_{M,t})$. ${\widehat{R{V}_t}}^2$ denotes the realized variance of the market returns, serving as the predicted value for $VAR(R_M,t)$. Both $CO{V}_t$ and $R{V}_t$, derived from the HAR model established in the previous section using high-frequency data are used to obtain a time-varying beta according to Equation (12). Based on these results, a dynamic CAPM-HAR model is established in this section to capture the time-varying relationships among assets and the market.

4.4. Evaluation Metrics

Jensen’s Alpha (commonly denoted as $\alpha =R_{j,t}-{\widehat{R}}_{j,t}$) was introduced by the American economist Michael C. Jensen in 1968 as a performance evaluation metric [33]. This measure is based on the CAPM and is employed to assess the excess return of a portfolio or asset after accounting for systematic risk.

Specifically, $\alpha$ quantifies the difference between the actual return of an asset and the expected return predicted by CAPM, representing the unexplained excess return of the model, i.e., the non-systematic risk or pricing error of the asset. If $\alpha$ is not significantly different from zero, it indicates that the model adequately explains the asset return. Conversely, if $\alpha$ significantly deviates from zero, it suggests that part of the asset return remains unexplained by the model and may be driven by other factors. As a singular performance metric, $\alpha$ effectively highlights the portion of the return not accounted for by the model, offering supplementary insights for investment decisions.

This study applies the t-test for statistical significance by formulating the null hypothesis ($H_0:\alpha =0$) and calculating the t-statistic and corresponding p-value to assess whether $\alpha$ significantly deviates from zero. If the t-test fails to reject the null hypothesis, it indicates that the model sufficiently explains the asset excess return.

5. Results

This section provides a detailed analysis of the predictive performance of the CAPM. To evaluate the validity of the CAPM and the enhanced CAPM-HAR, this study conducts an empirical analysis using daily and 5 min frequency trading data from the Shanghai Stock Exchange banking sector, covering the period from 1 January 2020 to 31 December 2022. In out-of-sample forecasting, a rolling window approach is employed to compare the two models, the key differences between the models lie in the data frequency and the complexity of their structures. The rolling forecast approach flexibly adapts to the time-varying characteristics of financial markets, effectively capturing the dynamic relationship between asset returns and risk factors, and enhancing the model’s responsiveness to market volatility.

First, in the classical CAPM framework, a linear model is dynamically fitted to daily data from the previous 132 trading days to estimate the Beta coefficient for the next trading day. This coefficient measures the systematic risk of assets relative to the market. Using the Beta coefficients, the relationship between each asset’s expected return and the market’s excess return can be determined. The CAPM-HAR model offers a more refined estimation using high-frequency data. The HAR-RV and HAR-RC models are trained on 5 min high-frequency data from the past 132 trading days. Specifically, the HAR-RV model captures the market index’s volatility characteristics across different time scales, with the parameters ${\beta }_d$, ${\beta }_w$, and ${\beta }_m$ representing the contributions of short-term (daily), medium-term (weekly), and long-term (monthly) volatilities, respectively. Meanwhile, the HAR-RC model captures the dynamic changes in the realized covariance matrix through the parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$. Figure 2 compares the out-of-sample realized volatility predictions of the HAR-RV model with the actual observed volatility of the SSE Composite Index. In this figure, Realized Volatility represents the actual volatility data, while Realized Volatility Forecasts represent the predictions made by the HAR-RV model.

In this study, by comparing the predicted excess returns of each asset with the actual excess returns, the $\alpha$ values for both models were calculated. Figure 3 compares the out-of-sample predicted returns with the actual observed returns for a selection of stocks, using both the CAPM-HAR and CAPM models. The figure clearly demonstrates that the predicted values from the CAPM-HAR model align more closely with the actual observations, highlighting its superior capability in capturing the dynamic behavior of asset returns. In contrast, the CAPM model exhibits significant deviations from actual values across multiple time points, particularly during periods of heightened market volatility, indicating its limitations in reflecting the dynamic characteristics of the market. The results presented in the figure further validate the enhanced predictive accuracy of the CAPM-HAR model, achieved through the integration of high-frequency data and multi-scale volatility modeling. This improvement underscores the model’s potential to provide more reliable support for asset pricing and risk evaluation.

To further evaluate the effectiveness of the CAPM and CAPM-HAR models, a t-test was conducted for all assets to assess whether the $\alpha$ values significantly deviated from zero at the 0.01 significance level. The results, presented in

Table 2. T-test results for CAPM with p-values marked with * if greater than 0.01.

Stock Code	Mean	T-Statistic	p-Value	Stock Code	Mean	T-Statistic	p-Value
600000	−0.0017	−29.1015	5.8062 × ${10}^{-89}$	600705	−0.0001	−1.3978	0.1632 *
600015	−0.0018	−46.7679	5.1373 × ${10}^{-139}$	600901	−0.0008	−9.6676	2.1181 × ${10}^{-19}$
600016	−0.0024	−54.2205	4.3341 × ${10}^{-156}$	600908	0.0005	6.3343	8.8071 × ${10}^{-10}$
600036	0.0023	26.2298	2.6379 × ${10}^{-79}$	600919	0.0005	6.9911	1.8046 × ${10}^{-11}$
600318	0.0011	5.5964	4.9759 × ${10}^{-8}$	600926	0.0021	19.5863	2.0250 × ${10}^{-55}$
600705	−0.0001	−1.3978	0.1632 *	600928	−0.0012	−17.7064	2.1723 × ${10}^{-48}$
601009	0.0002	2.2867	0.0229 *	601166	0.0011	11.6167	5.7192 × ${10}^{-26}$
601077	−0.0018	−37.7062	3.0265 × ${10}^{-115}$	601169	−0.0019	−45.9838	4.2458 × ${10}^{-137}$
601128	−0.0004	−5.1355	5.1016 × ${10}^{-7}$	603323	0.0003	4.1141	5.0378 × ${10}^{-5}$
601166	0.0011	11.6167	5.7192 × ${10}^{-26}$	601229	−0.0022	−57.3362	1.1395 × ${10}^{-162}$
601288	−0.0031	−80.6201	6.0797 × ${10}^{-204}$	601328	−0.0022	−45.9075	6.5418 × ${10}^{-137}$
601577	−0.0002	−3.2557	0.0013	601658	−0.0011	−22.5011	3.9319 × ${10}^{-66}$
601818	−0.0013	−16.3157	3.6044 × ${10}^{-43}$	601838	0.0019	20.0498	3.8392 × ${10}^{-57}$
601860	−0.0009	−12.7629	4.8764 × ${10}^{-30}$	601916	−0.0019	−25.4701	1.1210 × ${10}^{-76}$
601939	−0.0020	−31.8985	6.1371 × ${10}^{-98}$	601988	−0.0029	−84.1285	3.3050 × ${10}^{-209}$
601997	−0.0013	−21.0410	8.3765 × ${10}^{-61}$	601998	−0.0016	−29.8956	1.4866 × ${10}^{-91}$

and

Table 3. T-test results for CAPM-HAR with p-values marked with * if greater than 0.01.

Stock Code	Mean	T-Statistic	p-Value	Stock Code	Mean	T-Statistic	p-Value
600000	−0.003911	−5.2876	2.4064 × ${10}^{-7}$	600705	0.003495	1.3307	0.1843 *
600015	−0.003100	−3.0811	2.2557 × ${10}^{-3}$	600901	−0.002470	−1.7076	8.8761 × ${10}^{-2}$ *
600016	−0.002138	−2.4536	1.4717 × ${10}^{-2}$ *	600908	−0.000367	−0.2764	0.7824 *
600036	−0.001812	−0.9196	3.5852 × ${10}^{-1}$ *	600919	−0.004906	−3.7464	2.1546 × ${10}^{-4}$
600318	−0.001462	−0.3260	7.4463 × ${10}^{-1}$ *	600926	−0.002157	−1.1638	2.4545 × ${10}^{-1}$ *
601009	−0.002473	−1.7047	8.9299 × ${10}^{-2}$ *	600928	−0.001694	−1.4344	1.5252 × ${10}^{-1}$ *
601077	−0.007823	−2.3695	1.8449 × ${10}^{-2}$ *	601166	−0.001352	−0.8742	3.8272 × ${10}^{-1}$ *
601128	−0.003383	−0.6444	5.1982 × ${10}^{-1}$ *	601169	−0.003836	−2.6535	8.3951 × ${10}^{-3}$
601229	−0.004580	−5.8430	1.3463 × ${10}^{-8}$	603323	−0.003047	−2.7046	7.2324 × ${10}^{-3}$
601288	0.006497	1.6553	9.8914 × ${10}^{-2}$ *	601328	−0.004880	−3.8815	1.2794 × ${10}^{-4}$
601577	−0.002299	−1.3757	1.6994 × ${10}^{-1}$ *	601658	0.001939	0.4149	6.7848 × ${10}^{-1}$ *
601818	−0.003080	−2.5603	1.0953 × ${10}^{-2}$	601838	−0.001659	−0.8083	4.1954 × ${10}^{-1}$ *
601860	−0.002204	−1.2371	2.1703 × ${10}^{-1}$ *	601916	−0.000785	−0.5226	6.0163 × ${10}^{-1}$ *
601939	−0.007591	−6.4566	4.3598 × ${10}^{-10}$	601988	−0.006349	−5.7645	2.0504 × ${10}^{-8}$
601997	−0.000529	−0.4697	6.3894 × ${10}^{-1}$ *	601998	−0.006895	−5.5574	6.0949 × ${10}^{-8}$

, reveal that the CAPM-HAR model produced 20 $\alpha$ values near zero, indicating that the excess returns of these assets were well explained by the model and demonstrating a high level of fit. In contrast, the CAPM model produced significantly fewer $\alpha$ values close to zero, with many exhibiting substantial deviations, highlighting its limitations in capturing the dynamic relationships between assets and the market. Additionally, in the CAPM-HAR model, only a small number of assets showed $\alpha$ values significantly deviating from zero, suggesting that these returns may be driven by unsystematic risks or other factors not included in the model. This underscores the improvements achieved by the CAPM-HAR model in explaining excess returns by incorporating high-frequency data and multi-scale volatility dynamics. Notably, assets with $\alpha$ values close to zero demonstrate the model’s ability to effectively capture systematic risk, offering a more reliable tool for risk assessment. Compared to the traditional CAPM, the CAPM-HAR model exhibits clear advantages in explanatory power and predictive accuracy, providing stronger theoretical support and practical utility for asset pricing and risk management in financial markets.

In summary, the CAPM model relies on low-frequency data for simplified risk estimation, while the CAPM-HAR model, by incorporating high-frequency data and multi-scale volatility, captures more complex market dynamics. The two models differ significantly in terms of data usage and model complexity, with the CAPM-HAR model providing a more detailed depiction of market volatility, leading to more accurate estimates of the risk-return relationship for assets. Future research could consider extending the model into a multi-factor framework, incorporating additional risk factors and firm-specific information to account for unsystematic risks and other influential factors, thereby enhancing the depth and breadth of risk assessment.

By comparing the characteristics and advantages of these two models, this study provides investors and decision-makers in financial markets with a more precise tool for risk evaluation. Particularly, in the face of rapidly changing market conditions, the CAPM-HAR model offers more reliable predictions for investment decisions, helping investors make more informed choices in complex financial markets. Future research could not only extend the applicability of the model but also introduce online estimation techniques to dynamically process real-time data. This would enable the model to better adapt to immediate market changes and information flows, enhancing its applicability and real-time responsiveness in practical investment scenarios, and providing more flexible and timely support for risk management and capital allocation strategies.

6. Conclusions

This study conducts an in-depth analysis of high-frequency data, modeling the covariance between asset returns and market returns, as well as the variance in market returns. Based on this, the HAR model is incorporated into the CAPM framework to derive a time-varying beta, effectively extending the conventional CAPM to better capture dynamic market relationships. This process leads to the development of a time-varying CAPM-HAR model based on high-frequency data, which effectively captures the dynamic relationships and volatility characteristics between assets.

By capturing dynamic changes in the covariance between asset returns and market returns, as well as the time-varying characteristics of market return variance, the model demonstrates outstanding performance in analyzing volatility and risk premiums. For investors and asset managers, the model aids in optimizing asset allocation and risk management strategies, thereby enhancing the scientific rigor and precision of investment decisions. For risk management institutions, the model facilitates real-time monitoring of market fluctuations and provides quantitative tools for risk hedging and dynamic adjustments. For regulators, the model offers a systematic framework for risk assessment, contributing to improved market efficiency and the maintenance of financial system stability. This approach efficiently integrates high-frequency data, improving the depiction of market dynamics and providing a more accurate and practical tool for asset pricing and risk management.

The application of the CAPM-HAR model to banking sector stocks listed on the Shanghai Stock Exchange has significantly enhanced its explanatory power in the Chinese stock market. Although this study is based on empirical data from the Chinese banking sector, its methodology and conclusions demonstrate a degree of general applicability. By leveraging high-frequency data to dynamically analyze the relationship between asset returns and systematic market risk, the CAPM-HAR model can be applied to markets characterized by high volatility or rich datasets. Specifically, international markets with similar high-frequency trading data characteristics, such as those in the United States, Europe, and the Asia-Pacific region, could adopt this model to analyze time-varying risk premiums and asset pricing dynamics. Moreover, differences in market structures, investor behavior, and regulatory environments across countries may influence the parameterization and predictive performance of the model. Future research should validate the robustness and generalizability of the CAPM-HAR model using broader international datasets to address these variations effectively.

The application of this model lays a strong foundation for further research, encouraging deeper exploration of the time-varying beta effect across various industries and market contexts. Expanding the model into a multi-factor framework that incorporates additional risk factors represents one promising direction. Integrating streaming data and online prediction methods could also enhance its adaptability to real-time market dynamics.