Pricing Ability of Carhart Four-Factor and Fama–French Three-Factor Models: Empirical Evidence from Morocco

Abstract

In this study, the reliability of the Fama–French Three-Factor model (FF3F) and the Carhart Four-Factor model (C4F) is examined thoroughly. In order to determine which of the asset pricing models is the best to explain portfolio returns on the Moroccan share market, these two models are indeed evaluated in the Moroccan market. Additionally, it is worth mentioning that five years of monthly data from the firms that listed on the Casablanca Stock Exchange are used in this research, as well over the period of nine years. The results of this inquiry show that these models barely have a partial hold on the Casablanca Stock Exchange (CSE), which limits their ability to predict the cross-sections of returns. In accordance with this, the C4F model has somewhat greater explanatory power than the FF3F Model. Moreover, our research adds to the body of knowledge by inserting two learned material asset pricing theories to the proof in the market, which is still evolving, and where distinctive anomalistic traits still exist (the CSE).

1. Introduction

Generally speaking, some of the professionals in both fields of finance and academics have, for a long time, disagreed on how to estimate the cross-sectional setting returns. Additionally, research has been conducted in this area to create a reliable asset pricing model for assessing the forecasted stock returns. The very first sincere endeavor was the Capital Asset Pricing Model by (Sharpe 1964), which later served as a framework for more intricate asset pricing models such as the (FF3FM) (Fama and French 1993). However, due to the fact that the aforementioned prototype performed better than its forerunner (CAPM) in terms of explanatory power, it was discovered that it was inefficient in capturing a number of significant market anomalies, which include accruals (Sloan 1996), net share issues, (Ikenberry et al. 1995; Loughran and Ritter 1995), volatility (Ang et al. 2006) and, especially, momentum. As a result, Carhart (1997) introduced the model of Carhart Four Factor, which added a risk factor that resembles the momentum of the FF Three-Factor formula. Furthermore, as a means for defining the connection that correlates the risk and the returns, the Fama–French Three-Factor model (1993) can assist the investors in developing reliable investing strategies and constructing an investment portfolio. While many researchers look into asset values in developing countries, just a few of them are interested in African emerging markets in particular. Especially, the Moroccan stock market has received minimal attention. Additionally, as far as scholars know, Aguenaou et al. (2011), Tazi et al. (2022) have only presented studies that address the possibility of the application of the FF3F model and the CFF model in the Moroccan setting. Moreover, our work incorporates a comparison with Carhart’s momentum-driven model and updates the conclusions of the previous studies using more up-to-date data. Additionally, a fundamental aspect of financial markets also empirically confirms the existence of momentum patterns, in which the aim is to show shareholders that the financial sector is beneficial but not unduly risky. They might also benefit from market anomalies by utilizing the momentum and winner–loser interaction. Determining the sensitivity of momentum returns will also enable researchers to explore the relationship between this momentum and risk. Additionally, financial advisors will also receive the data they require to advise their customers on the profitability of a range of equities featured on the CSE from this report. Therefore, the major objective of this thesis is to assess the validity of the FF3F and the four-factor prototype in the Moroccan context using the OLS method, as well as the approaches proposed by prominent scholars, namely Fama and French (1993) and Carhart (1997). Thus, the key inquiries of the study are:

Firstly, is the FF3F framework superior to that of the Carhart Four-Factor model at predicting portfolio results on the Moroccan stock market?

Secondly, which model should Moroccan stock market portfolio managers use?

Therefore, the sections of the study are further broken down as the following. An overview of the entire body of research on predicting the cross-sectional stock returns is provided in Section 2 of this paper. Section 3 contains the data that pertain to cross-sectional stocks. Additionally, the empirical techniques for using the OLS regression method to assess the pricing capabilities of the FF3F framework and the Fama–French paradigm are also covered in Section 4, while the results and conclusions are generally covered in the fourth chapter.

2. Theoretical Review

Generally speaking, Markowitz (1952) was the first person who created the Modern Portfolio theory, which is used to determine the connection between asset risk and returns. It is also used to develop the Capital Asset Pricing Model (CAPM) according to Sharpe (1964), who identify the market return as the only risk factor affecting the expected returns on the stock. However, this model has been widely condemned by several scholars, namely Black et al. (1972), Merton (1973), Banz (1981), Reinganum (1981), Fama and French (1992), and Campbell and Vuolteenaho (2004), with the most well-known criticism being that the single-factor model has failed in real-world applications because of its irrational assumptions. Due to their underwhelming findings and weak explanatory power, researchers have moved away from CAPM investigations and are now searching for alternative variables. For instance, Stattman (1980) and Davis et al. (2000) found that the B/M ratio has a favorable effect on US equities. Accordingly, Banz (1981) also identified the size impact, which showed a strong and unfavorable relationship between average return and company size. Additionally, Basu (1983) found a favorable correlation between average return and earnings to price. Furthermore, Bhandari (1988) demonstrated that average return is connected to leverage positively. All of these studies, on the other hand, are unable to identify strong particular characteristics that explain excess returns. Likewise, Fama and French (1992) found that size and B/M are two easily quantifiable parameters that give a clear-cut representation of a cross-section of typical share prices. Furthermore, by 1993, they offered a three-factor framework that shows the cross-section of average results using the excessive market return, size component, and B/M parameters. The momentum approach was later improved by Jegadeesh and Titman (1993), and it was revisited by Carhart (1997), who added the short-term impetus as one of the three elements in the Fama–French framework. They also projected that the stocks that had previously done well would do so in the future. Thus, the stocks that have performed poorly in the past would continue to do so. In addition, Carhart (1997) examined the mutual fund’s and found that their returns were remarkably consistent. He also found that the momentum impacts are more pronounced for funds in the top and bottom performing deciles, which may help to explain mutual fund persistence. Furthermore, purchasing investments in the top momentum decile and selling investments in the bottom momentum decile can result in an average return of 8%, according to Carhart (1997). This theory has generally undergone extensive empirical testing in both established and developing markets. For instance, Connor and Sehgal (2001) evaluated the FF3F model systematically in India and discovered compelling evidence supporting the effects of the market, size, and book-to-market ratios on stock returns. Additionally, Karp and Van Vuuren (2017) asserts that a number of variables, such as market volatility, inadequate market proxy measurements, market liquidity constraints, and unpriced risk factors, contributed to the model’s poor performance. However, they also assert that the model’s performance was subpar. Nevertheless, Pusuwanaratana and Tachasermsukkul (2017) found that the CAPM framework had greater explanatory power than the FF3F model in the Thai market when it came to explaining profits. By using a survey of 832 French-listed companies over an 18-year timeframe (1995–2012), Boubaker et al. (2018) investigated the risk indicators that properly represent the default risk. They created 12 sizes and used book-to-market, leverage-sorted portfolios, and the portfolio of distressed companies not only to test out whether the distress risk is systematic but also to determine whether adding a leverage variable to the three-factor framework would improve its ability to describe the sample’s returns. Therefore, the findings of this investigation demonstrate the applicability of the Fama–French three-factor approach with a leverage risk premium in the French setting, indicating the need for additional variables to account for the default risk. Additionally, Pojanavatee and Khuppakun (2019) demonstrated it using the three-factor model of Fama and French where the size, value, and market beta aspects influence the formation of the gain rate on Property and Construction stocks over 61 equities from July 2015 to June 2018 in Thailand (1993). In line with this, Aguenaou et al. (2011) investigated the validity of this model on the Casablanca Stock Exchange (CSE), finding confirmation of ubiquitous market and value risk variables. None of the measurements, nevertheless, indicate that the model is not totally valid in the Moroccan financial market. Furthermore, Chowdhury (2017) found that the FF3F model has a poor equity returns explanation when using the Chittagong Stock Exchange in Bangladesh as an illustration. Chen and Fang (2009) found the same in markets around the Pacific Basin, which include: Japan, Singapore, South Korea, Indonesia, Thailand, Malaysia, and Hong Kong. Despite their findings, they could not find any evidence in favor of Carhart’s Four-Factor model’s influence on momentums. Accordingly, the Polish stock market’s asset pricing model has also been studied by Czapkiewicz and Skalna (2010), Urbański (2012), Waszczuk (2013), and Zaremba (2014). They discovered that when stocks are resolved by size and value rather than it is restored by momentum, the three-factor model does not describe the variance in portfolio returns. Additionally, Fama and French (2011) examined the stock markets of North America, Asia Pacific, Japan, and the European region using the C4F model, and all except Japan showed statistically substantial worth and momentum premiums. Likewise, Cakici et al. (2013) also looked at the four-factor model’s utility by also looking at the size, value, and momentum effects of 18 growing European countries. They also claimed that all emerging European stock markets have a strong value impact, but there is no momentum effect in Eastern European countries, where stock markets are only getting started and most of their structures are still developing. Bretschger and Lechthaler (2012), however, investigated the C4F utility in the Japanese stock market and discovered that it has performed well in predicting stock returns. In the same line, the asset-pricing framework was also adapted to the emerging markets by Cakici et al. (2013), who found that all areas with the exception of Eastern Europe experienced significant value and momentum impacts. Additionally, they have looked into the effectiveness of the C4F model in developing markets around the world. For instance, they comprise more than 800 stocks from several Asia countries namely China, Thailand, Malaysia, Indonesia, Philippines, South Korea, Taiwan, and India using data from 1990 to 2011. As a consequence, they found that the value factor has a negative correlation with the momentum element and that this element can explain the stock returns in Asia. Furthermore, Nwani (2015) discovered that, except for the size effect, the FF3F and C4F models both had significant evolutionary power on the London stock exchange. Similar to this, Kholkin and Haug (2016) predicted the returns of Norwegian equity mutual funds using monthly measurements. The findings show that 14 funds of the momentum parameter proposed by Jegadeesh and Titman (1993) and Carhart (1997) accurately predicted the variation in returns with a precision of 97%. Using data from the Colombo Stock Exchange (CSE), Abeysekera and Nimal (2017) tested the C4F model’s performance and compared it with the CAPM and FF3F paradigms. Accordingly, this study, which includes the C4F, adequately accounts for the diversity in the cross-section of the average stock returns in the CSE. Furthermore, the four-factor framework has also been proven to outperform the CAPM and the three-factor approach in terms of performance. Additionally, on the Amman Stock Exchange (ASE) equities industry, Momani (2021) investigated the viability of the FF and Carhart asset pricing frameworks. According to the research that has been found in this inquiry, the methods are unable to accurately represent the cross-section of average gains to portfolios which are organized by size/book-to-market and size/momentum. In addition, the Fama–French approach and the Carhart prototype are both capable of describing profits. In their study of the FF3F and the C4F models’ applicability in Morocco over a more than five-year period, Tazi et al. (2022) discovered that both of these models only partially remain true for the Casablanca Stock Exchange, making it impossible to fully rely on them to anticipate cross-sections of the returns. Peillex et al. (2021) evaluated the MSCI Japan Empowering Women Index’s financial performance over eight years (2010–18). Additionally, they examined precisely whether the stock index, which symbolizes the investments in gender diversity (WIN), performs differently from its traditional parental indices (i.e., the IMI). Moreover, this performance is assessed on a variety of subperiods utilizing conventional risk-adjusted returns markers, including the Treynor ratio, the Sharpe ratio, Jensen’s alpha, the Fama–French three-factor alpha, the Carhart alpha, and finally the Fama–French five-factor alpha. Hence, their findings demonstrate that WIN displays remarkably identical risk-adjusted profits to those of its traditional peers, behindhand the performance framework that has been utilized or the time frame analyzed. Thus, based on these findings, we draw the conclusion that there are no economic disadvantages to investing in the WIN equities index as opposed to its parental index.

3. Data

Within this research, a selection of 59 out of the 75 firms on the CSE are examined. Additionally, we use data from each month from January 2008 to December 2017. The time frame in question is quite brief in comparison with other asset pricing model studies, notably those addressing the US stock market. Moreover, along with monthly pricing at the end of each year, the dataset also contains accounting information from the Decypha database, such as Total Assets, Total Liabilities, Shares Outstanding, and Operating Income. We substituted the risk-free level with the 13-weak Moroccan Treasury Bill; the statistics are available on the Bank Al Maghreb website.

4. Methodology

4.1. Statistical Model

The C4F model, which is an expansion of the FF3F model, is generally used in the investigation. Additionally, the evaluation of the preceding regressions, which were calculated utilizing the Ordinary Least Squares (OLS) method, is necessary for the FF3F and Carhart’s Four-Factor models. In order to evaluate the effectiveness of the FFM and the Carhart framework, which are used to illustrate the variation in cross-sectional returns in the Casablanca stock market, F-tests along with their p-values and adjusted R-squared were also examined.

4.1.1. Fama–French Three-Factor Model

The linear regression model is as follows, according to Fama and French (1993):

$$R_t^p=a_i+b_i\left(R_{Mt}-R_{Ft}\right)+s_i{SMB}_t+h_i{HML}_t+{ei}_t$$

where $R_t^p$ is the monthly returns on test portfolios, where: p, p = S/L, S/M, S/H, B/L, B/M, and B/H. Furthermore, $R_{Mt}-R_{Ft}$ is the excess return on the market index at time t, ${SMB}_t$ (Small minus big) is the portfolio return on the size factor, and ${HML}_t$ (High minus low) refers to the portfolio return on the value factor at time t. The sensitivity of the asset to each of the factors is represented by the coefficients $b_i$, $s_i$, and $h_i$. Finally, $a_i$ denotes the intercept, and ${ei}_t$ denotes the error term at time.

4.1.2. Carhart Four-Factor Model

For the C4F model, the following formula is used:

$$R_t^p=a_i+b_i\left(R_{Mt}-R_{Ft}\right)+s_i{SMB}_t+h_i{HML}_t+w_i{WML}_t+{ei}_t$$

The monthly returns on test portfolios are denoted by $R_t^p$. Additionally, p, p = {S/L, S/M, S/H, B/L, B/M, B/H}, and the $R_{Mt}-R_{Ft}$ formula represents the excess returns on the market index at the time t, ${SMB}_t$ present a portfolio’s returns on the size factor, and ${HML}_t$ also refers to the returns on portfolio on the value factor at the period t. In line with this, ${WML}_t$ (Winners minus losers) is the return on portfolio that is based on historical stock performance, likewise at time; the sensitivity of the fortune to each of the factors is represented by the coefficients $b_i$, $s_i$, $h_i$, and $w_i$ at time t, where $a_i$ is the intercepted and ${ei}_t$ is the error term.

Thus, in order to undertake empirical testing of the asset pricing model, the risk factors that act as explanatory variables (right-hand-side (RHS) portfolios) and portfolio assets that operate as dependent variables (left-hand-side (LHS) portfolios) must be specified (LHS).

4.2. Dependent Variables

According to Aguenaou et al. (2011), Tazi et al. (2022) and Nwani’s (2015) methodology. The monthly returns of the S/L, S/M, S/H, B/L, B/M, and B/H portfolios are the dependent variables for The Fama–French Three-Factor model, as well as for the Carhart Four-Factor model. Additionally, the goal of the LHS portfolios is to gather a representative prototype of the equity returns across the Moroccan market, where “test portfolios” were built, as opposed to testing the model on specific equities. Hence, the portfolios must be well-diversified as a whole. Likewise, the dependent variables are the six test portfolios, which are: “B/L, B/M, B/H, S/L, S/M, and S/H”. Additionally, the three BE/ME classifications and the two size categories are combined to form these portfolios. Based on their market capitalization during each year of the study period, the stocks are divided into two categories: big (B) and small (S). The sample size is also divided into three groups according to their BE/ME ratio, where high (H) is the top 30%, moderate (M) is the middle 40%, and low (L) is the lowest 30%. The resulting portfolios can be described as follows:

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S/L portfolio: Contains small capitalization (S) stocks that are also low-value (L) ones.

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S/M portfolio: Contains small capitalization (S) stocks that are also medium-value (M) ones.

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S/H portfolio: Contains small capitalization (S) stocks that are also high-value (H) ones.

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B/L portfolio: Contains big capitalization (B) stocks that are also low-value (L) ones.

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B/M portfolio: Contains big capitalization (B) stocks that are also medium-value (M) ones.

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B/H portfolio: Contains big capitalization (B) stocks that are also high-value (H) ones.

4.3. Explanatory Variables

4.3.1. Fama–French Three-Factor Model

The risk factors are the explanatory variables, and they were developed according to the Fama–French guidelines (1993).

Market Factor (Rm-Rf)

The difference between the weighted monthly return of all stocks and the 13-week Moroccan Treasury Bill is used to compute the market portfolio factor. In fact, the 13-week Moroccan Treasury Bill is used as a risk-free rate proxy in the current paper, while the Moroccan All Shares Index (MASI) is used as a market portfolio surrogate.

Size Factor (SMB)

In this component, it has been said that Small minus big is the mathematical expression of the discrepancy between the average returns on small-cap stocks (S/L, S/M, and S/H) and the average returns on big-cap stocks (B/L, B/M, and B/H).

Thus:

$${SMB}_t=\frac{1}{3}\left(R_t^{SL}+R_t^{SM}+R_t^{SH}\right)-\frac{1}{3}\left(R_T^{BL}+R_t^{BM}+R_t^{BH}\right)$$

where $R_t^{SL},R_t^{SM},R_t^{SH}$,$R_T^{BL},R_t^{BM},\mathrm{and}{R}_t^{BH}$ represent the monthly returns on the following portfolios SL, SM, SH, BL, BM, and BH at period t.

Value Factor (HML)

This element demonstrates the difference in average returns between portfolios imitating the high B/M ratio shares (B/H, S/H) and portfolios matching a low B/M ratio equity (B/L, S/L); thus:

$${HML}_t=\frac{1}{2}\left(R_t^{SH}+R_t^{BH}\right)-\frac{1}{2}\left(R_t^{SL}+R_t^{BL}\right)$$

where $R_t^{SH},R_t^{BH},$$R_t^{SL}$, and $R_t^{BL}$ represent the monthly returns on the stock portfolios SH, BH, SL, and BL at period t.

4.3.2. Carhart Four-Factor Model

The very first three factors of the structure, which are Rm-Rf, SMB, and HML, are developed identically to the three-factor framework. What contributes to make this distinction is the insertion of a fourth element, which is WML (Winners minus loser).

Momentum Factor (WML)

The momentum element depicts the difference in average returns on winners’ portfolios (B/W and S/W) and losers’ portfolios (B/Losers and S/Losers). Moreover, six additional stocks are formed to establish this aspect: S/Lc, S/M, S/W, B/Lc, B/M, and B/W, where the companies are sorted depending on their average returns record for a year (t − 1) and market capitalization.

Furthermore, the stocks are divided into two main categories based on their market capitalization in each year of the period of study: big (B) and small (S). Furthermore, the firms are divided into three main groups, which are based on their average return over the previous 12 months. These groups are Losers (from 0% to 30%), Medium (from 40% to 70%), and Winners (from 70% to 100%). The resulting portfolios can be described as follows:

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S/W portfolio: Contains small capitalization (S) stocks that are also Winner (W) ones.

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S/M portfolio: Contains small capitalization (S) stocks that are also Medium (M) ones.

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S/Lc portfolio: Contains smalls capitalization (S) stocks that are also Loser (Lc) ones.

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B/W portfolio: Contains big capitalization (B) stocks that are also Winner (W) ones.

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B/M portfolio: Contains big capitalization (B) stocks that are also Medium (M) ones.

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B/Lc portfolio: Contains big capitalization (B) stocks that are also Loser (Lc) ones.

Thus:

$${WML}_t=\frac{1}{2}\left(R_t^{SW}+R_t^{BW}\right)-\frac{1}{2}\left(R_t^{SLc}+R_t^{BLc}\right)$$

where $R_t^{SW},R_t^{BW}$$,R_t^{SLc}$, and $R_t^{BLc}$ signify the monthly return on stock portfolios SW, BW, SLc, and BLc at time t, correspondingly.

5. Empirical Results

5.1. Descriptive Statistics of the Pricing Factors

Table 1. Summary statistics of pricing factors.

	MKT	SMB	HML	WML
Mean	−0.0333583	−0.00139	−0.00935	0.004147
Median	−0.0329897	0.000692	−0.00749	0.006773
Maximum	0.00417692	0.105536	0.108485	0.124916
Minimum	−0.073314	−0.08141	−0.14635	−0.09586
St deviation	0.0110824	0.032528	0.046112	0.039159
Skewness	−0.2298151	0.107978	−0.18886	−0.24984
Kurtosis	1.48404341	0.332785	0.930628	0.65318
Observation	120	120	120	120
JB test stat	12.0682238	0.786914	5.043693	3.381646 1
p-value	0.00239562	0.67472	0.080311	0.184368

¹ Note: The Jarque–Bera statistic is distributed as a ${\chi }^2$ with two levels of freedom and evaluates the null hypothesis of a normal distribution.

summarizes the descriptive statistics of the four asset pricing parameters MKT, SMB, HML, and WML.

Additionally, during the course of our sample period, the market component, value element, and its size had negative monthly average profits of −0.0333583, −0.00139, and −0.00935. On the other side, the impetus element has generated positive monthly data on an average of 0.004147. The theory of Fama and French states that the value stocks outperform growth stocks, while small firms outperformed big firms.

In line with this, the SMB has a negative value which suggests that, from 2008 to 2017, the firms that have large market capitalization value outperformed enterprises with small market capitalization. Furthermore, the HML factor has a negative value, which indicates that throughout the period of 2008–2017, companies with a low BE/ME ratio outperformed those with a high BE/ME ratio. In addition to this, it is also worth noting that the WML variable is positive. This indicates that there is a momentum effect throughout this time span, and the companies with the best returns over the past year continue to beat stocks with the worst returns through the coequal time period.

Furthermore, Table 1 shows that none of the portfolios with a minimum return are positive. Additionally, the winner’s portfolio is second in terms of minimal value, and the value’s portfolio (HML) has the lowest minimum value. The size of the SMB portfolio and the market portfolio both had a minimal value, which is about zero. Likewise, the portfolio of the winner has the highest maximum value. The market portfolio, on the other hand, has the lowest standard deviation/risk. The value portfolio, on the other hand, has the biggest standard deviation. Furthermore, the size and value portfolios are the riskiest. The same thing goes with the exception of market and value portfolios (HML).The JB statistics in each of the portfolio variables are insignificant, meaning that the portfolio returns are non-Gaussian. Additionally, for independent and dependent variables, the normality condition is not required. The prediction error is the sole variable that must have a normal distribution.

Figure 1 displays the four portfolio returns’ patterns as diagrams. It is obvious that over the study period, the returns of the winner portfolio increases more quickly than the returns of the other portfolios. Additionally, the size portfolio beats both the value and market portfolios in comparison.

Thus, concerning all of the factors that appear to be inactive for some months, the market return has fallen below 0%. This is just proof that the market portfolio has a low-returning portfolio, and investors can diversify to the winner portfolio, value portfolio, or size portfolio, which all guarantee higher returns.

5.2. Correlation between Factor and Market Portfolios

Before running the cross-sectional regression, the correlation coefficients between the factor portfolios were computed to check for multicollinearity, as indicated in the

Table 2. Correlation coefficient among pricing factors.

	SMB	HML	WML	MKT
SMB	1
HML	−0.47937392	1
WML	0.25184097	−0.28470258	1
MKT	−0.01071568	−0.15971367	0.00644869	1

Note: The period covered by the survey is from January 2008 to December 2017. The four-component model proposed by Carhart (1997) includes the pricing components MKT, SMB, HML, and WML.

.

Additionally, the correlation matrix demonstrates that the market component and the value factor have a weak and negative correlation (−0.15971367). There is also a very weakly negative correlation with the size factor (−0.01071568) and a very minor positive correlation with the momentum factor (0.00644869). Similarly, the momentum factor exhibits a moderately weak negative correlation with the value factor (−0.28470258) and a weak positive correlation with the size factor (0.25184097). However, the size factor and the value factor have the only reported high negative correlation (−0.47937392).

Overall, the findings reveal that all risk factors are only weakly correlated. This suggests that the model assumptions of Fama and French (1993) are supported, implying that there is no proof for a possible multicollinearity problem.

5.3. Patterns of Average Returns

Effects are commonly used to describe patterns in average returns. As a result, we talk about the size effect, momentum effect, and the value effect in this section. In addition,

Table 3. The average excess returns per month for portfolios built by using size and BE/BE.

			B/M Equity
	Mean Excess Return				st Deviation
	B	C	Average		B	C	Average
H	−0.00558	−0.00436	−0.497%	H	0.043051	0.087962	6.551%
M	0.004583	0.001248	0.292%	M	0.038154	0.033369	3.576%
L	0.002982	0.007516	0.525%	L	0.049912	0.031453	4.068%
Average	0.066%	0.147%		Average	4.371%	5.093%

summarizes the study’s findings.

5.3.1. Size–B/M Portfolios

The LHS portfolio typically applies 2 × 3 levels based on size and B/M ratio, and six more portfolios subsequently appear as a result of this. Thus, Table 3 also includes the standard deviation and average excess returns for these six size–B/M portfolios.

Table 3 shows that, overall, the growth stocks (low B/M) outperform value stocks (high B/M) in terms of average return (−0.497 percent vs. 0.525 percent). Furthermore, Fama and French (2015) published research in which they discovered that companies with a high BE/ME ratio create higher profits on average than firms with a low BE/ME ratio when size is held constant. In our sample, however, the effect is the polar opposite. If we then concentrate on the size effect while keeping the value relatively constant, we find that, on average, the returns in our sample increase as size increases by 0.066 percent and 0.147 percent, respectively. As a result, previous empirical research suggests that small firms should have larger excess returns to make up for the risk taken by investors. The growth stocks have a lower standard deviation in excess return than value stocks 4.068 percent and 6.551 percent.

5.3.2. Size–Momentum Portfolios

The LHS portfolio employs 2 × 3 sorting based on size and momentum. Additionally, six portfolios subsequently appear as a result of this. The mean excess returns and standard deviation for these six size–momentum portfolios are displayed in

Table 4. Average excess returns per month for portfolios built using size and momentum.

			Momentum
	Mean Excess Return				st Deviation
	B	C	Average		B	C	Average
W	−0.00186	0.008654	0.340%	W	0.043371	0.036675	4.002%
M	0.003243	0.001939	0.259%	M	0.033463	0.02894	3.120%
L	−0.0045	0.003004	−0.075%	L	0.04959	0.048613	4.910%
Average	−0.104%	0.453%		Average	4.214%	3.808%

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When compared with Loser stocks, Winners have a larger return and a lower risk. This backs up Carhart’s assertion that stock returns are persistent. To sum up, the large firms outperform small companies, where the growth stocks surpass value stocks, and the momentum effect is present, indicating that prior Winners are still performing well. Additionally, it should be noted that the dataset only includes companies that are indicated in the Casablanca Stock Exchange (CSE), and that this effect is present throughout the research term (2008–2017).

5.4. Regression Results

5.4.1. The Fama–French Three-Factor Model

Table 5. The FF3F Model Regression Outputs.

Portfolios		SL	SM	SH	BL	BM	BH
Coefficients
Intercept		0.0199733	0.0196058	0.01983	0.0195778	0.0201103	0.0292661
MKT		0.5546107	0.4224234	0.5444236	0.4550916	0.6010875	0.8951256
SMB		0.5260314	0.4240155	0.8146344	−0.3987554	−0.149205	−0.8116451
HML		−0.2395995	0.0367429	0.6545962	−0.2745184	−0.1050278	1.032028
WML		0	0	0	0	0	0
t-Stats
Intercept		1.53	1.81	2.03	2.32	2.05	1.65
MKT		1.52	1.39	1.99	1.92	2.19	1.83
SMB		3.75	3.64	7.76	−4.39	−1.42	−3.87
HML		−2.39	0.44	8.72	−4.23	−1.40	7.25
WML		0	0	0	0	0	0
Regression Statistics
Multiple R
R Square		0.2622	0.1296	0.4431	0.2207	0.0717	0.7316
Adjusted R Square		0.2432	0.1071	0.4287	0.2005	0.0477	0.7197
Observations		120	120	120	120	120	72
p-values
Intercept		0.129	0.073	0.045	0.022	0.042	0.103
MKT		0.132	0.167	0.049 **	0.057 *	0.030 **	0.071 *
SMB		0.000 ***	0.000 ***	0.000 ***	0.000 ***	0.158	0.000 ***
HML		0.018 **	0.659	0.000 ***	0.000 ***	0.165	0.000 ***
WML		0	0	0	0	0	0

Note: * p < 0.1; ** p < 0.05; *** p < 0.01. The sample is from January 2008 to December 2017. The table above displays the outcomes of the OLS.

displays the OLS regression assessment utilized to evaluate the FF3F model framework. It also shows the excess returns for six portfolios, which are dependent variables: B/H, B/M, B/L, S/H, S/M, and S/L.

In all of these six portfolios that indicate positive exposure to the size factor, there is just one relevant output from the FF3F model. Additionally, the market’s coefficients are all positive. Furthermore, SL and SM have significant market betas at the 5% level, whereas the market betas of BL and BH are significant at the 10% level.

In addition to this, in the smallest size portfolios, SL, SM, and SH have negative SMB coefficients. Similarly, in terms of statistical significance, SMB seems significant in all portfolios at the 1% degree, except the exception of the portfolio BM, demonstrating the existence of the size effect.

However, the HML coefficient is negative in the lowest B/M portfolios, which are SL, BL, and BM. Furthermore, in terms of statistical significance, the HML factor is not strong for midsize stocks. This HML has further lost relevance in the SM and BM portfolios, although the SL, SH, BL, and BH portfolios are outstanding at the scale of 1%. Therefore, the presence of the value effect is confirmed.

Furthermore, the values vary significantly between 5 percent and 72 percent, according to the Adjusted R-Square. For the largest size portfolios (32 percent on average), the FF3F model has a higher predictive potential than that of the portfolios of the smallest size (26 percent on average). It also demonstrates a lower predictive power (22 % on average) for the lowest B/M portfolios than the one which has the highest B/M portfolio (57 percent on average).

The Fama–French three-factor analysis assumes significantly greater predictability overall in each of the aforementioned six junction portfolios but to a variable degree. The results show that the three-factor model explains a substantial portion of the multiple in the dependent variables. The outcomes of the three variables, MKT, SMB, and HML are valid for all sample portfolios.

To summarize, the size impact appears to be entirely intact in the Moroccan stock market from 2008 to 2017, whereas the value effect appears to be partially intact. Based on the findings, we can deduce the pricing ability of the FF3F analysis by utilizing monthly data, which is only and partially significant for Moroccan companies’ data.

5.4.2. Carhart (1997) Four-Factor Model

The regression analysis, which makes use of the OLS technique, is used to assess the pricing potential of Carhart’s (1997) asset pricing model.

Table 6. Carhart Four-Factor Analysis Regression Outputs.

Portfolios		SL	SM	SH	BL	BM	BH
Coefficients
Intercept		0.0199165	0.0195537	0.0197423	0.0195079	0.0200224	0.0297627
MKT		0.5441255	0.4127957	0.5282116	0.4421862	0.5848465	0.866516
SMB		0.5455164	0.441907	0.8447618	−0.3747729	−0.1190237	−0.7883288
HML		−0.2600818	0.0179357	0.6229267	−0.2997285	−0.1367539	1.001826
WML		−0.1103045	−0.1012836	−0.1705515	−0.135765	−0.1708564	−0.1681573
t-Stats
Intercept		1.53	1.81	2.05	2.34	2.08	1.69
MKT		1.49	1.36	1.96	1.89	2.17	1.78
SMB		3.86	3.77	8.10	−4.14	−1.14	−3.76
HML		−2.55	0.21	8.27	−4.59	−1.81	6.95
WML		−1.03	−1.14	−2.16	−1.98	−2.16	−1.19
Regression Statistics
Multiple R
R Square		0.2690	0.1394	0.4648	0.2464	0.1079	0.7372
Adjusted R Square		0.2436	0.1094	0.4462	0.2202	0.0769	0.7215
Observations		120	120	120	120	120	72
p-values
Intercept		0.130	0.073	0.043	0.021	0.040	0.096
MKT		0.140	0.176	0.053	0.061	0.032 **	0.080 *
SMB		0.000 ***	0.000 ***	0.000 ***	0.000 ***	0.257	0.000 ***
HML		0.012 **	0.833	0.000 ***	0.000 ***	0.072 *	0.000 ***
WML		0.305	0.256	0.033 **	0.050 **	0.033 **	0.237

Note: * p < 0.1; ** p< 0.05; *** p< 0.01. The sample is from January 2008 to December 2017. The table above displays the outcomes of the OLS.

shows the excess return for six portfolios, with the following variables B/H, B/M, B/L, S/H, S/M, and S/L as the dependent variables:

From the above figure, except for the SL portfolio, which has an insignificant alpha, all the portfolios have a positive significant alpha, with SM and BH having a significant value at the 10% level, and SH and BM have an outstanding scale at the level of 5%, which signifies that these portfolios have transmitted the excess returns compared with their benchmark, and it is named the MASI. Furthermore, except for the BM and BH portfolios, which are significant at 10% and 5%, the market’s b coefficients are all positive. However, if b > 1, then this portfolio is more volatile than the MASI; conversely, if b < 1, then this portfolio is less volatile than the MASI.

In addition, all portfolios have a b coefficient MKT that is far from one, indicating that they are less volatile than the benchmark. Likewise, in the smallest-sized portfolios, the SMB beta coefficients are positive, and all sorts of portfolios have a robust significance. At the level of 1%, they are much more significant, with the exception of one portfolio, BM, which has an insignificant beta. Therefore, these findings support the presence of a size effect in our sample. On one hand, the HML beta coefficients are significant in all portfolios, with the SL portfolio having the highest BE/ME ratio and the lowest BE/ME ratio at the 1% and 5% levels, respectively. Portfolios with a Medium BE/ME ratio, on the other hand, exhibit insignificant value at the 10% level for the BM portfolio. Hence, the presence of the value impact is, thus, confirmed. In addition to this, in SH, BL, and BM, the WML beta coefficients are all significant at the 5% level. The fact that all of the portfolios have a negative coefficient is intriguing; the WML effect for our sample only marginally holds.

Additionally, speaking of values, the Adjusted R-Square indicates that they differ wildly between 8% and 72%. Furthermore, the FF3F model has a higher average prediction performance for the biggest size assets (34%) than for the smallest portfolios (27%) but a lower average statistical ability for the lowest B/M portfolios (23%) compared with the highest B/M portfolios (58 percent on average). In addition to this, the retail price capabilities of Carhart’s (1997) Four-Factor model were demonstrated by utilizing six test portfolios as dependent variables. As a result, the MKT, SMB, and HML variables produce consistent results across all sample portfolios. The WML factor, on the other hand, lost its costing capability for the conditional on three main variables, which are SH, SM, and BH.

In summary, this suggests that the size and value effects remained in the Moroccan environment during the aforementioned period, whereas the momentum effect partially held. Additionally, we may infer from the results that the four-factor model’s valuing abilities using monthly data would only be marginally significant for Moroccan financial records.

5.5. Fama–French Three-Factor Model versus Carhart Four-Factor Model

The p-value, Adjusted R-Square, and significance F values of the FF3F framework as well as the Carhart Four-Factor model were investigated using the OLS regression outcome shown in

Table 7. Fama-French Three-Factor Regression Model.

Portfolios		SL	SM	SH	BL	BM	BH
F-test		13.74	5.76	30.76	10.95	2.99	61.78
p-value		0.0000	0.0010	0.0000	0.0000	0.0340	0.0000
Significance		YES	YES	YES	YES	YES	YES
Adjusted R-Square		0.2432	0.1071	0.4287	0.2005	0.0477	0.7197

Note: The OLS method is employed to calculate the variables.

and

Table 8. Carhart Four-Factor Regression Model.

Portfolios		SL	SM	SH	BL	BM	BH
F-test		10.58	4.66	24.97	9.40	3.48	46.98
p-value		0.0000	0.0016	0.0000	0.0000	0.0101	0.0000
Significance		YES	YES	YES	YES	YES	YES
Adjusted R Square		0.2436	0.1094	0.4462	0.2202	0.0769	0.7215

Note: The OLS method is employed to calculate the variables.

. Additionally, it displays the excess return of six portfolios, designated as the dependent variables, which are B/H, B/M, B/L, S/H, S/M, and S/L.

Additionally, the F-tests and p-values found indicate that all six portfolios of the FF3F and C4F regression models are statistically significant. Generally speaking, both the FF3F model and the Carhart model regressions have a comparable adjusted coefficient of determination (Adjusted R-square), with some differences. Moreover, although the difference in Carhart is minor, it cannot be underestimated. Furthermore, the C4F model outperforms the FF3F model in four out of six portfolios. In line with this, it is important to note that when the WML factor was added to the FF3F model, the factor HML in the portfolio BM became significant. In addition, it is safe to presume that, with reference to the Casablanca Stock Exchange (CSE), the C4F model contributes only somewhat over the course of the research process to the explanatory capacity of the FF3F model. Furthermore, while the size and value impacts are somewhat real, the momentum effect is hardly noticeable over the course of the investigation. These findings suggest that it is not possible to totally rely on the FF3F model and the C4F model to predict cross-sections of return on the Casablanca Stock Exchange (CSE) across the four selected years (2013 to 2017). As a result, Aguenaou et al. (2011), Tazi et al. (2022) found during the study period that, when compared with other research conducted in the Moroccan context, the C4F model did not significantly improve the explanatory power of the FF3F model.

6. Conclusions

In summary of evertything that was said, the goal of this inquiry is to evaluate the accuracy and efficiency of the Carhart Four-Factor model for estimating excess return in particular CSE securities (Casablanca Stock Exchange). Additionally, this study looks into how the market, value, size, and momentum affect the required excess return, and a comparison of the higher explanatory power of the FF3F paradigm and the Carhart analysis is provided. Additionally, a 9-year period span was adopted as the study’s dataset. Based on the explanatory variables, the regression model was examined to estimate excess stock return (market, size, value, and momentum factors). Furthermore, in this study, the four effects were found in the Casablanca stock exchange. Thus, we conclude that market, size, value, and momentum are somehow held but do not explain the excess stock return in CSE based on regression analysis of the FF3F model and the C4F model. In a nutshell, our analysis advises CSE purchasers to trade in large-cap firms with low B/M rates and firms that have performed well in the previous years. Consequently, the value, size, and momentum influences should indeed be considered when assessing the effectiveness of portfolios in the CSE. Elliptically, the C4F model seems to have a higher Adjusted R-Square than that of the FF3F model whenever it comes to describing the excess return in the CSE. These findings demonstrate that it is impossible to reliably anticipate the cross-sections of return on the Casablanca Stock Exchange (CSE) between 2008 and 2017 using the FF3F model and the C4F model.