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**1.0 Introduction**

One of the most important quantification model in modern financial economics is the Sharpe-Lintner capital asset pricing model (CAPM). The model allows economists to study differences in returns based on risks in the market. It demonstrates that cross-section excepted excess returns are linked to the market betas in a linear relation. However, it is noticeable that there are several assumptions underlying the CAPM model and thus lead to inherent limitations of the model. To combat the limitations and make the explanatory power stronger, a series of study have concentrate on developing multi-factor models to address the limitation revolved around CAPM. Despite a lot of efforts, the improved models cannot fully address all the limitations of CAPM but help to alleviate the problems to some modest extent.

**2.0 A review of CAPM 1**

In this section, a literature review regarding CAPM will be conducted. Based on the previous literature and empirical tests, some limitations relate to CAPM will be identified as summarized.

*2.1 CAPM empirical tests *

There are three widely cited empirical tests on CAPM, including Black, Jensen, and Scholes (Jensen et al., 1972), Fama and MacBeth (Fama and MacBeth, 1973), and Blume and Friend (Blume and Friend, 1973).

Black, Jensen, and Scholes (Jensen et al., 1972)fail to talke the efficient of the market portfolio into consideration and do not claim that they have conduced tests regarding any other hypothesis and no linearity test is provided. By using the same set of sample, Banz proposed different findings .

Banz in his paper studies the relation between the the value of the stock of a company and the return with a sample of 25 portfolios and suggests the size effect (Banz, 1981). He argues that smaller firms have common stocks with higher adjusted returns than larger firms have. As a result, the study provides evidence that goes against CAPM that states market risk and expected return are linearly related. The unexplained returns of small firms do not proportionate with the beta and other explanatory factors need to be added into the model such as the size of the firm, gamma. Accordingly, Banz believes that the CAPM is not well specified and shows that the there would be divergence between the theoretical beta of risk free assets and the beta of the portfolios predicted by the model (Banz, 1981). An additional residual analysis also demonstrates the misspecification of the linear model. Consequently, using size coefficient to estimate the size premium also has its limitations but the coefficient is still significate, especially for firms in small sizes. For small firms, there is a strong correlation between return and market share in a non-linear form, but it does not work for firms with medium to large market cap. The possible explanation might rely on the diversification effect in the portfolio.

The study of Fama and MacBeth has three testable hypotheses. First, it assumes investors believe mean-variance efficient portfolios are optimal portfolios. In addition, given homogeneous expectations of investors, the market portfolio is ex-anti efficient, which requires three strong assumptions, including perfect capital market, homogeneous investor expectations, and returns that follow two parameters probability distribution. Last, it assumes that there is portfolio that is ex-post efficient to have the same linearity between sample mean returns and betas that is the same as the market.

*2.2 Limitations of CAPM *

According to Roll, CAPM lacks support from unambiguous test based on previous studies and literatures and the test is still far from possible given its practicability (Roll, 1977). In terms of hypothesis that is linked with the two parameters CAPM (Jensen et al., 1972), there is only one testable hypothesis, the market portfolio has the the mean-variance feature. However, there would always be ex-post mean-variance efficient portfolios given a sample of individual returns observed. Since the beta is calculated by the observations of a portfolio, the result is sure to fit exactly into the linearity relation regardless the existence of true mean-variance efficient in the market portfolio. In fact, the CAPM bears little testability since the exact composition of the portfolio in a real-setting market remains unknown and it is not used in the tests. As a result, unless the sample covers all individual securities, the theory falls short of testability.

In addition, while an alternative is to make use of proxies for the market portfolio, there are two difficulties to employ a chosen proxy for the true market portfolio. Firstly, it might be possible that while the true market portfolio is not mean-variance efficient, the proxy of the market portfolio is. In this case, the sample meet the assumptions underlying the model whereas the portfolio in the real market fail to meet all these implications. In addition, the proxy might eventually prove to be inefficient. There are high correlations among these chosen proxies and correlations between proxies and the market portfolio regardless of the efficiency of the mean variance. As a result, the composition of the portfolio might be underestimated but the inferences will differ accordingly.

Based on discussions above, the two-parameter theory has severe testability limitations unless following issues can be solved. First, due to the unknown sample distribution and difficulty in computation of the full sample covariance matrix, it is hard to test the mean-variance efficiency of the proxy directly. Second, the test based on the linearity relation between return and beta also caused difficulties since the theory only predicts the cross-sectional relation in terms of form rather than parameter values and the individual deviation in securities can setoff each other in portfolio formation. Consequently, the testability of CAPM is still questionable.

*2.3. Limitation Summary*

The model also has limitations regarding its data. CAPM relies on historical data to predict future returns. As a result, the calculated beta might not be able to yield appropriate predictions for the future variability and returns. Accordingly, the model is not a deterministic model and the results should be regarded as estimations. Additionally, there are some practical problems regarding data measurement. It difficult to measure risk free rate, beta, and the market risk premium.

In addition, there are several unrealistic assumptions regarding investors. For example, it assumes that investors have homogeneous expectations and anticipations regarding future returns. However, if investors have different expectations, the security market line will vary. It is next to impossible that all investors will have same expectations. Moreover, it assumes that the capital market is perfect and efficient, while the it is a rare case. It relies on the fact that all the securities and assets are correctly priced and the consistent outperformance is impossible (Fama and French, 1996). It assumes that all investors are rational and risk averse and have the same set of information to make investment decisions. It simply ignores all the transaction costs, taxes, and limitation on accessibility to investment. Furthermore, it also believes that investors are highly diversified in their portfolio but in fact, small investors might not be able to hold a portfolio that is fully diversified.

Furthermore, the model fails to consider a multiple period scenario. It simply assumes investors only adopts a single step action and narrows the horizon into one period of time. However, different time horizons might result in different security market lines. Moreover, the model solely relies on market portfolio to make conclusion and both the beta and risk premium are correlated with the market portfolio.

**3.0 A review of Multifactor models **

Unrealistic assumptions and limitations of CAPM lead to multi-factor models that extend the original CAPM pricing model. Various factors have been added in to the original model or replaced the variables in the CAPM model. This study will examine several most important models, including Arbitrage Pricing Theory (APT), Fama‐French 3 factor model, and Fama-French-Carhart (FFC) model.

*3.1 Arbitrage Pricing Theory (APT) *

In order to address the shortcomings of CAPM, Ross in 1976 proposed an alternative pricing theory named arbitrage pricing theory (APT) (Ross, 1976). APT has a few distinguished differences with traditional CAPM model. It determines asset values y law of one price and on arbitrage and is originated from statistical model, compared to CAPM an equilibrium model. First, the new APT is a model consists multiple factors compared with the single-factor nature of CAPM theory. Second, APT does not require the market portfolio to have the feature of mean-variance efficiency and to provide the optimal risk to return combination. Alternatively, it assumes that there are no arbitrage profits in a market equilibrium situation. As a result, it is not necessary to assume that every investor is optimizing the profits and return.

APT assumes that there is no arbitrary in market equilibrium and investors cannot earn arbitrage profits. As a result, the expected return of an asset is linearly correlated with its sensitivity to other involved independent factors. Investors are believed to construct zero risk portfolio without any investment outlay but still earn positive returns. All securities have finite values and variances. APT also takes other factors into consideration. It assumes that besides market risks premium, a portfolio can also reach the expected rate of returns consisting of other risk premium. The sensitivity to multiple unexpected changes in other economic factors also contribute to the return. In 1980 ad 1984, Roll and Ross identified four general factors that can be included in the model to get asset returns, including unanticipated changes in inflation, default risk premium of bonds, industrial production, and the term structure of interest rate.

*3.2 Fama**‐**French 3 factor model *

To describe returns of securities and stocks, Eugene Fama and Kenneth French proposed a 3 factor model in the paper the cross section of expected stock return (Fama and French, 1992). The three factors include company size, the price to book ratio of the particular company, and the market risk. In contrast to conventional CAPM that only uses one variable, the Fama-French model takes three factors into consideration. The starting point is that two types of stocks tended to outperform the market when they have small caps and the stocks have low price-to-book rations. SMB means small minus big, indicating the difference in market capitalization and measuring the historic excess returns of small caps over large caps. HML means high minus low, indicating the difference in book to market ratio and measuring the value stocks over growth stocks. The SMB is divided into two groups, with the first group including small stocks and the other including large stocks. HML is divided into three groups, including low, medium, and high level. In addition, they found that the book value to market value ratio are more powerful in explanting the stock return compared with the size of the company, so the factor is divided into three instead of two.

In their study, Fama and French also tested the leverage and the earnings to price ratio, which is less relevant and then is discarded later on. The three factor model is an extension of CAPM with other two beta coefficients. As a result, unlike the single beta formula, the current model has three coefficient of betas. Accordingly, the Fama–French three-factor model can explain more than 90 percent of the returns in a diversified portfolios compared with 70 percent under a CAPM model.

Later Griffin finds that the Fama and French factors are specific to each country and believes that some local variables can be used to improve the explanation of variations in returns instead of global factors. As a result, the updated the model with some countries specific factors in both emerging and developed markets. Liews and Vassalous shows that the style portfolios that are based on the size and book to market value are closely related to business risk proxies and future growth in a macro economic level. Petkova an ZHnag in 2005 also found that the higher that average return of portfolios with higher book to market value is associated with the risk premiums.

*3.3 Fama-French-Carhart (FFC) model *

As a tool for valuating mutual funds, a four-factor model is structured based on the three factor model discussion above. The extension of the Fama-French three factor model is the Fama-French-Carhart (FFC) model, which takes into account the factor momentum facto, which is the MOM factor monthly momentum (Carhart, 1997). The factor of momentum is based on study of Jegadeesh and Titman (Jegadeesh and Titman, 2002), claiming that good and bad performance of stocks can persist during a certain period of time, which is the so called momentum effect (WML). According to study by Bello (Bello, 2008), he found out that the three-factor model outperformed the CAPM and the four factor model excelled the three factor model. The Fama French Carhart model makes use of firm characteristics and specifies four factor portfolios.

**4.0 Discussion **

Limitations of CAPM lead to multi-factor models that extend the original CAPM pricing model and aim to design a better proxy for the market portfolio that can resemble the constituents of systematic risks. The updated models address the problems and limitations by CAPM in one of the other ways, but neither of them has perfectly addressed all the problems and limitations encountered by using CAPM model.

In APT model, the overall risk is decomposed into two parts, including systematic part and unsystematic part, whereas the return on assets only involves the essential risks. In CAPM model, the systematic components are divided into two parts, including essential part and inessential part. Since the APT include other economic factors, it is considered superior to CAPM. In addition, the assumptions of APT model are less restricted that assumptions required by the CAPM theory. For example, while CAPM assumes investors consider portfolio by required rate of return and variance, the APT model does not require the market portfolio to be mean-variance efficient. In addition, the CAPM only focuses on one single time horizon, the APT can be used for multiple steps and take the dynamics of market into consideration by incorporating factors such as inflation rate.

However, theoretically, the APT model is more complex than the traditional CAPM theory thus both the quantity and nature of the risk factors remain largely unknown. Besides, since there are multiple risk factors included in this model, it is more difficult to establish the coefficient relationship for each risk premium and the sensitivity. As a result, the model is still relatively less used and less practical compared with CAPM. To formulate the systematic risks, a finite factor is used to formulate the model. In addition, the expected return of a security is closely associated with the exposure to each factor and then concluded in a vector of factors. By considering portfolio of a large number of securities, the reward to the unsystematic or idiosyncratic risk in the return can be make small (Khan and Sun, 1997). Consequently, the CAPM neglects unsystematic risks and only emphasizes on efficient diversification, whereas APT emphasizes on law of large numbers and naïve diversification.

The three factor model is considered to improve the explanatory power of the returns on portfolios better in comparison to CAPM model. Value firms more tangible capital and thus firms are at higher risk when an economic downturn happened while growth firms possess few fixed assets thus can withdraw their investment easily. Consequently, as indicated by the Fama-French model, value firms should earn higher average returns considering the risks they bear. Beta of the HML will be negative in good times and positive in recessing period. As a result, beta in CAPM of value stocks would be less than that of growth ones when the economy is in good condition, but higher in recessing situation. Therefore, in the improved model, the beta of HML is an adjustment to the CAPM and betas are allowed to differ based on the business cycle (Jagannathan and Wang, 1996).

As a result, the multi-factor models help to address the limitations in using CAPM to a modest extent by importing more relevant variables. In addition, other factors, such as irrational pricing and data problems might also contribute to the explanation of returns of assets. A three-factor model can fit in to the pattern of returns and risks but fail to explain the irrational behaviors of high premium for distress. The acid test shows that three factor model is a useful both theoretically and practically. However, the underlying reasoning is still unclear. It is difficult to trace the economic state variables such as SMB and HML, as well as explaining their reasonableness and soundness. The other multi-factors models, just like the three factor model, are just models and cannot fully explain the anticipated returns on securities and portfolios, such as continuation the short-term returns (Jegadeesh and Titman, 1993).

**5.0 Conclusion **

The Sharpe-Lintner capital asset pricing model (CAPM) is one of the most important quantification model in modern financial economics. The single factor model can be used to study the differences in returns of portfolios and securities in the financial market. It assumes that there is mean-variance efficiency. However, it is hard to test the mean-variance efficiency of the proxy directly considering the unknown sample distribution and difficulty in computation of the full sample covariance matrix. In addition, the test based on the linearity relation between return and beta also caused difficulties. Thus, the testability of CAPM is still questionable. To address the limitations and make the explanatory power stronger, a series of study have concentrate on developing multi-factor models to address the limitation revolved around CAPM. However, theoretically, the APT model IS less restricted than assumptions required by the CAPM theory and does not require the market portfolio to be mean-variance efficiencies. However, it is more complex than the traditional CAPM theory thus both the quantity and nature of the risk factors remain largely unknown. The three factor model is considered to improve the explanatory power of the returns on portfolios better in comparison to CAPM model, but it still fails fully explain the anticipated returns on securities and portfolios, such as continuation the short-term returns. In conclusion, the multi-factors can address limitations of the CAPM to some modest extent.

**Reference**

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Fama, E. F. & French, K. R. 1996. Multifactor explanations of asset pricing anomalies. *The journal of finance,*51**,**55-84.

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Jagannathan, R. & Wang, Z. 1996. The conditional CAPM and the cross‐section of expected returns. *The Journal of finance,*51**,**3-53.

Jegadeesh, N. & Titman, S. 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. *The Journal of finance,*48**,**65-91.

Jegadeesh, N. & Titman, S. 2002. Cross-sectional and time-series determinants of momentum returns. *Review of Financial studies,*15**,**143-157.

Jensen, M. C., Black, F. & Scholes, M. S. 1972. The capital asset pricing model: Some empirical tests.

Khan, M. A. & Sun, Y. 1997. The capital-asset-pricing model and arbitrage pricing theory: a unification. *Proceedings of the National Academy of Sciences,*94**,**4229-4232.

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