**2. Literature Review**

The multi-factor risk model has undergone a series of developments which can be divided into four major steps. At the beginning, in order to figure out the leading factors for security return, several researchers have contributed to the development of an asset pricing model. Initially, Markowitz (1952) proposed a mean-variance model to illustrate the statistical relationship between security risk and return in terms of standard deviation and expected rerun. It has established the foundation of modern finance theory. However, he had not specified the factors that explain expected security returns.

Based on the modern portfolio theory, the capital asset pricing model (CAPM) was developed by Sharpe (1964) and Lintner (1965) to illustrate the linear relationship between expected return and market risk premium.

$$\text{Ri} = \text{Rf} + \beta\_{\text{market}}(\text{Rm} - \text{Rf}) \tag{1}$$

With the empirical tests in the stock market, they managed to find out the pattern of stock returns in line with the general stock market index. To be specific, the model estimates the relationships between the return on market index (the explanatory variable) and the return on the stock (the dependent variable). The regression coe fficient of the single index model is referred to as beta which is a measure of the sensitivity of a stock to general movement in the market index. This empirical research symbolized the transition from qualitative analysis to quantitative analysis which also laid the foundation for the subsequent asset pricing models.

Since then, the CAPM model has been widely applied in research and empirical testing. However, with increasing abnormal returns which cannot be explained by existing factors, the market beta was no longer su fficient to describe expected return (Fama 1996). Therefore, Fama and French proposed a multifactor model consisting of three factors for market risk (Rm − Rf), market value, and book-to-market ratio (Fama and French 1993). In this model, Ri, Rf, and Rm stands for security expected return, risk free rate, and market return. SMB and HML are the risk premium factors. They illustrated that small stocks can generate higher returns than large stocks while value stocks can generate higher returns than growth stocks.

$$R\_i - R\_f = \mathbf{a} + \beta\_{markxt} \left( R\_m - R\_f \right) + \beta\_{size} \text{SMB} + \beta\_{BM} \text{HML} \tag{2}$$

It is also a supplement of arbitrage pricing theory (Ross 1976) which emphasizes that the expected return is not only a ffected by market risk but also a series of other factors. The generalized model (Bodie et al. 2014) illustrated the linear relationship of expected return and di fferent factors (Bodie et al. 2017). In Equation (3), *Fj* represents the factors and *b* represents the coe fficients.

$$r\_i = a\_i + \sum\_{j=1}^k b\_{ij} F\_j + \epsilon\_{i\prime} \ i = 1, \ 2 \dots, N \tag{3}$$

Moreover, Banz (1981) illustrated that return of securities are a ffected by several index including B/M ratio and E/P ratio which represent a series of risk premium. These articles largely contribute to the development of the multi-factor model.

In latter decades, the three-factor model was faced with a series of challenges. As the three-factor model was applied to stock trading and empirical testing, it appeared that some of the phenomenon cannot be explained by the model which can lead to unpredictable abnormal return. Therefore, more specific factors need to be added into the model to improve the accuracy. Novy-marx (2013) proved that profitability, measured by gross profits-to-assets, has roughly the same power as book-to-market value in explaining the average return. The following equation is based on the dividend discount model, *Yt*<sup>+</sup>τ represents the earning for period *t* + τ, d*Bt*<sup>+</sup>τ is the change in book equity, *r* represents the expected return. It implied that higher market value leads to lower expected return, while higher earnings imply higher expected return.

$$M\_t = \sum\_{\tau=1}^{\infty} E(Y\_{t+\tau} - dB\_{t+\tau}) / (1+r)^{\tau} \tag{4}$$

Aharoni et al. (2013) recorded an insignificant but statistically reliable relationship between investment pattern and average return. Other evidence also illustrated that the profitability and investment factors can explain some of the variation in average return.

Carhart (1997) conducted a study on the common factors in stock return and investment expense by adding momentum factor in three-factor model. It measures the tendency of price changes with a portfolio of long previous-12-month return winners and short previous-12-month loser stocks, which had an 8% accumulated yield. The momentum factor can explain 6.4% of the excess return.

Therefore, Fama and French (2015) added the two new factors, investment (CMA) and profitability (RMW), to build the five-factor model for measuring the effects of company size, valuation, profitability, and investment pattern in average stock returns. In this equation, they divide both sides by book value at time *t* to create book-to-market ratio and present the relationship between return *r* and valuation factor.

$$R\_i - R\_f = \mathbf{a} + \beta\_{\text{market}} \left( R\_m - R\_f \right) + \beta\_{\text{six}} \text{SMB} + \beta\_{\text{RM}} \text{HML} + \beta\_{\text{profitality}} \text{RMW} + \beta\_{\text{incormutum}} \text{CMA} + \varepsilon \tag{5}$$

Moreover, they use the RMW factor to represent the profitability and the CMA factor to represent investment. According to empirical study, the five-factor model has a higher effectiveness than the three-factor model, which can explain 71–94% of the variation of average return. In the test, researchers divided all the stocks with three sets of factors. The result also implied that book-to-market factor is redundant for describing average return using the American stock data from 1963 to 2013. Also, the model fails to capture the low average return for small firms with low profitability and high investment.

Sehgal and Vasishth (2015) tested the model in various emerging markets and discovered that the change of price and trading volume are partly risk based and partly behavioral. The research indicated that behavioral factors are necessary in the study of the multi-factor model. Except for the existing risk premium factors in Fama and French model, Peng et al. (2014) studied the effect of different investment sentiments on the market return from which they found out customer satisfaction is a significant factor for abnormal return. Moreover, a human capital factor was considered in terms of compensation level (Moinak and Balakrishnan 2018). Zahedi and Rounaghi (2015) applied an artificial neural network to assess the components of a multi-factor model and predict future stock prices.

Considering the features of data and the time-varying factors, a range of studies applied various methods to address problems via multi-factor models. Akter and Nobi (2018) examined the distribution and frequency distribution for both daily stock returns and volatility. Furthermore, Chen and Kawaguchi (2018) distinguish two significant regimes (a persistent bear market and a bull market) to examine market time-varying risk factors to achieve Markov regime-switching. These studies examined the model specifically in certain periods which enables the researchers to compare the model performances in different situations which can improve the applicability of the model.

Some researchers conductedmodification on the basicmodelwith specific structure. Ronzani et al. (2017) suggested that β (systematic risk) evolves over time and the model with time-varying β provide less

conservative VaR measures than the static β. While Cisse et al. (2019) examined the dynamics of the model with Kalman filter and Markov switching (MS) model and proved that the former method fits better in the model. Bhattacharjee and Roy (2019) proposed a social network dependence structure to address such misspecifications. For the investment aspect, Frazzini et al. (2013) suggesting that Buffett's returns are more due to stock selection than to his effect on management.
