In financial theory study, researchers give strength on how to measure the reasonable price of capital assets. Capital asset pricing, in particular the return of capital assets, is one of key issues in the whole process of financial study. Well-known Capital Asset Pricing Model (CAPM) was given as a possible answer.
The CAPM was introduced by Sharp (1965) and Linter (1965), which base on mean-variance theory. The theory is built upon several strict assumptions. The CAPM states that the expected return is affected by only one factor systematic risk and there is a linear relationship between return and systematic risk. The expected return equivalent risk-free rate add risk premium. Generally speaking, high systematic risk is relevant to great yield. Huge amounts of financial studies were finished to test the right of this theory. Some earlier studies were support the CAPM while later more and more researchers doubted it. Roll (1977) thought it is un-testable. Ross (1976) who proposed Arbitrage Pricing Theory (APT) believed that more factors should be imported to the model. Over the past thirty years, due to plenty of anomalies, the CAPM was denied. The workers found "size effect" (Banz 1981), the Value Line enigma(Copeland and Mayers 1982), exchange rate and so on also can explain the return.
Shanghai Stock Exchange is the largest stock exchange in China. It was established on November 26, 1990 and was in operation on December 19 in the same year. It is a non-profit organization and directly administered by the China Securities Regulatory Commission (CSRC). SSE indices, the authoritative indices, are published by the SSE. It is widely useful to measure the China's stocks market performance. It is a great important benchmark to analyze portfolio for investors. Since the Shanghai Stock Exchange play an key role in the national economy, it is very important to study the applicability of CAPM in the SSE. Hence, this paper concentrates on make sure the efficiency of the CAPM in the SSE.
The remaining chapters are arranged as follow:
Chapter two is the literature review. It will state the background of the CAPM from the beginning with Markowitz's (1952) a mean-variance theory to the end with the recent study situation. Introduce the theory in detail, including all the assumptions, the two-fund separation theorem, three type stocks with different Beta values. After that display the situation of the previous empirical works, like Lintner (1969), Sharp and Cooper(1972), Black,Jensen and Scholes(1972), Miller and Scholes(1972), Fama and MacBeth(1973) and so on. At last the criticism about the CAPM is disclosed.
In chapter three, after reviewing the earlier study, the methodology will be explained. 50 stocks, one-year deposit rate and SSE A Share Index with weekly data from 2004 to 2009 are employed. Stocks and portfolios will be tested respectively to exam the CAPM by the time series test and cross-sectional test. To be more precise, other relevant tests also will be applied, for instance unit root test, significant test and residual tests. Lastly, the limitations in this paper will be presented.
In Chapter four, empirical results and analysis for stocks and portfolios will be addressed. By comparing the results of these two cases, the finding will be shown. The reasons for the finding will be analyzed.
Chapter five is the conclusion of this paper. It will be review the methodology and the result of empirical test. Give a brief conclude of this paper.
2 Literature Review
2.1 Background of the CAPM theory
In an efficient capital market, the prices of asset which are exact signals for capital allocation will fully and immediately reflect all available related information. Asset pricing plays an important role in financial field. As a core of modern financial theories, the Capital Asset Pricing Model (CAPM) theory has been widely discussed and applied in finance literature and investment decision.
Markowitz (1952) described a mean-variance portfolio optimization model in the study "Portfolio Selected" which is regarded the outset of the modern investment. With the mean-variance model analysis, he concluded that the portfolio can effectively reduce the risk. He found an effective portfolio boundary that is with a given risk level maximizing expected return or for a given benefit level minimizing the risk of portfolio. According to his study, under uncertain conditions investors tend to select portfolios to optimal their utilities. Suppose that all investors are risk-averse individuals who pursue portfolios with higher expected returns and lower variances. The variance of portfolio is applied to measure its risk. It relate to the individual variances of expected returns and the covariance of all assets.
On that basis, Sharp(1964) and Lintner (1965) proposed the Capital Asset Pricing Model (CAPM). After that CAPM has developed rapidly in past few years but also experienced huge difficulties. In the 1970s, many scholars have done a great number of empirical tests on the validity of the CAPM. Earlier some researchers agreed this theory. But Roll (1977) who believed CAPM is un-testable published a critique of asset pricing theories, which initiate people criticize the classical theory of the CAPM.
At the same time, Ross (1976) illustrated Arbitrage Pricing Theory(APT) that is also an equilibrium asset pricing model and very similar to the CAPM. Any risky asset return is seen to be a linear combination of various common factors that affect returns. Compare with the CAPM, the assumptions of APT are simpler. The CAPM predicts that the return will be linearly related to a single-factor that the rate of return on the market portfolio, while the APT is based on similar intuition but is much more general. The APT is more reasonable to explain the asset pricing issue.
Over the past three decades, a growing number of researchers contradicted the CAPM. In that a large number of anomalies that can be used to explain stock yields were found in the CAPM studies. The famous anomalies contain the "size effect"(Banz,1981);the Value Line enigma(Copeland and Mayers,1982); the relation between price/earnings ratios and expected returns that low ratio portfolios tend to have higher returns(Basu,1977);the relation between book-value/market-value ratios and expected returns(Fama and French,1992) and so on. The power of interpretation of the return on Beta coefficient becomes weaker even disappear.
2.2 The CAPM theory
The CAPM is a ratiocinative outcome of mean-variance portfolio theory. It is developed in a hypothetical world with some stringent assumptions about investors and the opportunity set. Tobin(1958), Markowitz(1959), and Fama(1965b) suppose that the capital market is perfect or frictionless that there are neither transactions costs nor information costs. Moreover, investors are price takers, which mean their trades do not impact on market prices. Sharp(1964) assumed that all individual investors views the expected returns that have a probability distribution over a single-period time horizon and focuses on only two parameters of this distribution that is its expected value and standard deviation. Investors are risk-averse individuals who choose an investment offering lower risk and greater benefits from a set of investment opportunities. In other words, investors will select the one that maximizes their utilities. Fama and MacBeth(1973) supposed that all investors achieve the available information costless and simultaneously and they have "homogeneous expectations". Investors can borrow and lend unlimited at a risk-free rate. Copeland, Weston and Shastri (2005) gave a fully and systematical description of the CAPM assumptions. Including the mention earlier, they stated that the number of assets is settled and all assets are traded availably in the market. In addition, all assets are perfectly and infinitely divisible implies that slavery is permitted in the model. For example, investors can sell several portions of human capital like writing ability to others at market prices. Finally, there is no market weakness for instance taxes, regulations, or restrictions on short selling. Bailey (2005) condensed the assumptions into three sets of conditions:(1)the capital market is in equilibrium;(2)an isolated individual investor behave depend on a mean-variance standard;(3) "homogeneous beliefs".
The equation of the CAPM for the asset return is: , ,where as the expected return on an individual asset i, as the risk-free rate, as the expected return on the market portfolio, as the covariance between the asset I and the market portfolio, as the variance of the market portfolio. It indicates that in equilibrium the expected excess return on a single risky asset i () is equal to the difference between the expected return on the market portfolio and the risk-free rate times a constant of proportionality given by the beta. It is important to remember that this expression not only be used for individual assets but also for all portfolios. It is obvious that it is linear between beta and expected return. In the CAPM framework, beta is used to measure of risk that implies higher value of beta require higher returns. Elton et al(2003) noted that although stocks with high beat are looked forward to a greater return, it does not mean that they always produce a superior yield. Due to more risky, they sometimes give lower returns.
Cuthbertson(2004) states that the two-fund separation theorem is essential to the CAPM. Under the two-fund separation principle, all investors will hold a combination of the risk-free asset and the market portfolio. Within a mean-variance portfolio framework, all investors will hold the same proportion of risky assets. Only the level of holdings will differ across investors. The relationship can of course breakdown when investors have risky income streams that are correlated with asset returns. Furthermore, the separation principle can apply even without a risky asset. Indeed, a risky portfolio can be formed as a linear combination of two different portfolios on the frontier, one being a zero beta portfolio.
Cuthbertson and Nitzsche(2001) confirmed that the influential market portfolio variance factors attribute to the co-variances between all the asset returns in the market. The main contribution of an asset to the market portfolio variance actually relate to the covariance of this asset with all other assets in the market. is the variance of the market portfolio that measures the overall risk of the market portfolio. Hence, the beta shows the 'contribution of an asset to market portfolio risk'. According to the equation of the CAPM, they also states that in general investors tend to hold the stocks with positive beta that will have a high expected return. If a stock is uncorrelated with the market return that means =0, the stock only require to produce the return at the risk-free rate. Facing a multitude of investments, investors can rank the stocks by risk-factor β. In case of β=1 a stock is called neutral stock, its expected return changes the same to the market portfolio. When β>1 a stock is named aggressive stock, its expected return increases or falls more than that of the market portfolio. In this situation, due to its greater risk than the market portfolio, the return of this stock is required greater than the average market return. Risk premium compensations exceed risk. On the contrary, a stock termed defensive stock have β<1.
2.3 Empirical tests of the CAPM
After the CAPM theory was proposed by Sharp(1964) and Lintner (1965), a large number of financial economists focus their attention on the test of the validity and efficiency of the CAPM that is ensure whether expected return can be explained by Beta. Most of them implement the empirical tests with three propositions: first of all, check the intercept whether equal to zero that abnormal return exists or not. Secondly, make sure the Beta that is the only single variable can explain the expected return. Thirdly, ensure the market risk premium is differing to zero. Empirical appraisal of the CAPM has two major aims: firstly, test whether or not reject the theorem; secondly, provide some necessary information which could service portfolio selection, i.e., evaluate the riskiness of different investments according to the beta-coefficients of the CAPM.
In the most of previous studies, investigators estimated Betas by using a time series regression and tested the hypotheses by using a cross-sectional regression. Lintner(1969) first finished the empirical work of the CAPM. He used the data of 301 common stocks and calculated each stock's annual return and the market return as the average return for all stocks from 1954 to 1963. He evaluated the Beta by regressing each stock return against the market return. The form of the time series regression is where the coefficient is the estimation of the real Beta for stock i. The form of Lintner's cross-sectional regression is where is the variance of the residual from the time series regression. All parameters should be hypothetical value. On the basis of the CAPM theory, is expected to be zero. and are expected to be and respectively. The results he gained were that =0.108, =0.063,=0.237. It shows that it is break the CAPM.
It seems worthwhile reviewing the testing results of the CAPM to see whether greater return related to higher risk. Sharp and Cooper(1972) tested if equally weighted portfolios of stocks traded in New York Stock Exchange with respect to risk over long periods of time will generate consistent returns for each year from 1931to 1967. They computed average annual returns and divided all New York Stock Exchange stocks into ten groups according to their rank by Beta measured with 5 year of previous data for each year. Then they who used the slope of the regression line examined the relationship between expected return and Beta. The examination provides confidence that there is a positive relationship which is both strong and linear between expected return and risk (Beta).
Black,Jensen and Scholes(1972) who first recommended the in-depth time series test used the sample that were all the stocks traded on New York Stock Exchange between 1926 to 1965. When test the validity of the CAPM, it is necessary to apply a huge amount of stocks. The most excellent method is that test the CAPM equation with the data of each stock in the market and then test the intercept. But, it is obviously improper for that the assumption of the intercept is the independence of the residuals, and in fact it is not. In order to avoid this situation, it is good choice to launch the regression with portfolios. While Black, Jensen and Scholes composed portfolios, they employed an influential variable that is also greatly associated with the Beta but independent. They utilized first 60 monthly data from 1926-1930 to exam Betas and rank the stocks into ten portfolios from highest beta to lowest beta and then computed the returns of ten portfolios for the next year. To achieve the 35 years of monthly returns of ten portfolios, the whole process was operated for 1933, 1934, and so on, until 1965. At last, ten portfolios were regressed. They noted that the excess return was explained well by the CAPM. They also found that while β>1 the intercept be likely to be negative and while β<1 the intercept be likely to be positive. It showed that the stock with low value of beta exceed the expected return in theory and the stock with high value of beta under the expected return.
Miller and Scholes(1972) submitted a authoritative paper that focus on discussing the statistical and theoretical troubles in all empirical appraisal of the CAPM. In accordance with their article, misspecification of the estimate equations is response for the bias result of previous studies. Firstly, on condition that returns are actually produced by the CAPM, the equation for estimating Beta will be consistent. Secondly, equation misspecification given an explanation for observing an high intercept and a low slope when the expected return is not in proportion to the Beta. Thirdly, it is the existence of heteroscedasticity. They also thought the influence of the mistakes. The incorrect measuring Beta in the first-pass regression should lead to that the beta downward and the intercept upward in the cross-sectional regression. Finally, the distributions of expected returns showing positive skewness implied that there is some connection during residual variance and return, although there is not.
Fama and MacBeth(1973) completed the empirical evaluation of the CAPM by using the data of monthly returns for all common stock traded on New York Exchange from January 1926 to June 1968. They acted three tests and divided time into nine stages that every 3 stages were created for each test: a four years for estimate beta of every stock and make up the portfolio, a five years for estimate portfolio's beta and a five years for test the model. They formed the portfolios by the same method as Black et al. However, they tested one of the equation that was for each month. They considered that whether residual risk influences return; whether there are linearity in the stock market; whether there is risk premium in the market. The results proved that over the whole time is statistical significant non-zero and is usually bigger than . The results also concluded that the expected returns are positively linear with Beta. It also showed that Beta squared has no effect on expected return. Furthermore the conclusion of Fama and MacBeth are different to that of Lintner and Douglas in residual risk. They believed that residual risk that has no effect on expected return is not important. The study of Fama and MacBeth is better than that of Black et al for the reason that Fama and MacBeth lauched coefficient betas which were used to forecast the sub-period return and calculated the average return between the different periods.
Fama and French(1992) presented a classic study that an empirical test of the CAPM used the data of daily individual stock returns from 1963-1990 in NYSE and AMEX and of that in NASDAQ from 1973 to 1990. The consequences obviously illustrated that while the portfolios were ranked by Beta, the relationship between return and Beta is unsubstantiated. That means the CAPM does not explain the nearly 50 years of the average stock returns. Roll and Ross(1994) evidenced that there are no relationship during Beta and average cross-sectional returns.
Cuthbertson(1999) described the concept of the single index model(SIM) that is not a really model but it is only a statistical assumption. It is that any stock return for all time periods can be effectively embodied as follow: , where as a single economic variable (e.g. GNP) , as white noise. When the unexplained element for any two stock are independent, Cov()=0 i. When and are independent, Cov()=0 for all I and t. Under the above conditions, unbiased estimations of () for any stock or portfolio can be attained by an OLS regression using on time series data for and . But, it is important to remember that the SIM is a poor expression of expected returns and especially the assumption of independence, Cov()=0, hardly ever holds in reality.
There are also many researches about the CAPM in China. In Donghui's (1996) analysis, he found that there is a negative relationship between expected return and systematic risk. In addition, unsystematic risk has an important influence to expected return. Chaojun and Jing (1998) first systematic studied the CAPM in China. In their opinions, the CAPM theory cannot be applied in Shanghai Stock Exchange (SSE) and there are some other variables that impact on the return. Tao and Shaogong (2000) searched out the same result by collecting 40 stocks with the data during 1996 to 1998 in SSE. Yang and Yuan (2007) chosen 100 stocks in SSE that the sample for the period January 1,2004 to December 31,2006. They did time series regression and cross-sectional regression for four models by ordinary least squares(OLS). The statistic displayed that firstly, the linear relation of the average return and unsystematic risk do not exist; secondly, the Beta is not the only factor to determine the return; finally, the pricing of Shanghai Stock Exchange do not conform the traditional CAPM theory. Shangli (2009) studied the three year data of Shengzhen Stock Market. The empirical assessment was similar to that of Black, Jensen and Scholes (1972), included time series regression and cross-sectional regression. He realized there is positive relation that but is not linear between systematic risk and return. Returns are influenced by some other risk factors. The atmosphere of speculation outweigh in the market. That means the market is a immature market. Therefore, the stock market of Shenzhen cannot conform with the CAPM. In brief, many empirical studies show that the CAPM cannot be applied to the Chinese stock market.
Cuthbertson (2004) concluded some points about the empirical works. Firstly, it's difficult to devise the model properly, especially in evaluating variances accurately. Secondly, the hypothesis that the Betas are constant over time is probably incorrect. Thirdly, the econometric techniques is advancing thereby the previous researches might offer wrong results.
2.4 Criticism of the CAPM
At one time, people did empirical tests to support the CAPM that the Beta is the only factor with explanatory power. But, due to the Roll's criticism, investigators began to call the CAPM in question. More factors were found to explain the stock returns, like firm size, leverage, price earnings ratio, and so on.
Roll (1977)proposed an famous critique in the paper "A critique of asset pricing theory's Test: Part I. On the past and potential testability of the theory". In respect to some principles of the CAPM test, he insisted that it is incorrect and unambiguous. Moreover, it is not possible to complete the tests in practice. In his powerful criticism, he advocated that only one hypothesis that is 'the market portfolio is mean-variance efficient' can be testable in the CAPM model. He argued that all exam of the CAPM means nothing unless prove this hypothesis. The joint hypotheses that the test of the CAPM is conditional on the efficiency of the market portfolio are nearly not possible to exam because the true market portfolio is so hard to measure. Firstly, the stock market maybe mean-variance efficient, however, the true market portfolio is not. Secondly, the chosen proxy tends to be inefficient. It is also very difficult to test the proxy's mean-variance efficiency directly. According his critique, although market is perfect and the CAPM is legitimate, the cross-sectional return line cannot be applied without additional discussion. He pointed that the test result has high sensitivity to the chosen market proxy. Roll and Ross represented this point in 1994. They advised that the inefficient market proxy lead to the incorrect consequence. However, Stambaugh(1982) demonstrated that the cross-sectional test of the CAPM is insensitive to the market proxy. The reason is that whatever use market proxy, stock or bond, even the fixed asset based on stock and bond, the statistical results are similar.
Lo and MacKinlay (1999) designated that the possible explanations of evident violations of the CAPM can be separated into two types: in the first place, the risk-based alternatives contains multifactor asset pricing models and perfect capital markets; secondly, the non-based alternatives contains biases launched in the empirical methodology, the subsistence of market frictions, or the existence of irrational investors. The empirical result that the intercepts of the CAPM is not zero lead to the empirical test of multifactor asset pricing models includes the arbitrage pricing theory by Ross(1976) and the intertemporal capital asset pricing model by Merton(1973). The main difference from the CAPM is that additional factors are introduced in these models. Fama and French(1993) recorded that the estimations of the CAPM intercepts are not zero both for portfolios grouped due to the value of equity and grouped due to market capitalization. But they found it is interesting that a three-factor model induce a near zero intercept for the same portfolios. Lo and MacKinlay (1990) presented data-snooping that is difficult to control is one of explanations. Furthermore, due to the CAPM is assumed in a perfect market, the effects of market frictions can led to that the test result of the CAPM intercept is not zero. Amihud and Mendelson(1986) provided some evidence showing that market frictions and demands for liquidity could affect the expected returns.
In addition to the above mentioned, Rosenberg and Marathe(1977) affirmed that it is greater to join some variables that for instance dividend yield, trading volume and firm size in the model. Banz(1981) first discovered the "size effect" that is one of the most famous anomalies. Rozeff &Kinney (1976),Keim (1983),and Roll(1983) reported that in general small firms which are likely to achieve great unusual returns outperform large portfolio-sized firms. Fama(1981) claimed that stock returns relate to several economic factors, for instance interest rate, inflation and exchange rate. Basu (1982) taken a point that some other factors must be taken into account. Take the price earnings ratio for example, low ratio portfolios tend to have higher returns. Rosenberg, Reid and Lansten (1985) approved of this point. They believed book-to-market has attribution to the return.
3 Methodology
In this section, firstly describe the data collection. After that, the empirical tests of the CAPM are introduced at length. Lastly, the limitation of this empirical research will be displayed.
3.1 Data Collection
In this paper, the data begins on January 1, 2004 and ends on December 31, 2009. Usually, the estimated Beta value will be more accurate if the period is longer so that there are the more the samples used in the regression. But on the other hand, if sample period is excessively long, it will actually neglect the constitutive changes in the company's achievement and financial characteristic during this period of time. Hence, the period of six years is selected. Moreover, the time is close to recent days for more convincing.
I collect 50 stocks from Shanghai Stock Exchange (SSE) at random. Take short-term noise effects into account, I choose the stock weekly closing price and calculate the returns by continuously compounded (ln(/)). In previous studies, most financial economists employed the rate of short term government bond, for example one-month or one-year Treasury bill rate, as the risk-free rate. However, only medium and long-term government bond is supplied in the Chinese market. Therefore, in this study, one-year deposit rate represent the risk-free rate. Obviously, the rate will be computed as weekly rate. Concerning the proxy of market, I decide to account the market return by using the SSE A Share Index. The method is same as to the stocks by continuously compounded ((ln(/)).All the data is obtained via the database DataStream.
3.2 Tests of the CAPM
There are so numerous empirical tests of the CAPM in previously research, although some of them are fruitless. With the purpose of the best exams, it is extremely important to consider some severe and complex econometric troubles. Hence, in this paper, the tests summarize as follow.
3.2.1 Two main approaches
3.2.1.1 Time series test
There are two main approaches to check the validity of the CAPM in my paper. The first one is time series test. This test has two major aims: firstly, evaluate beta for every stock; secondly, examine how suitably the data has. It regress the excess stock returns against the excess market return utilizing observations for a sequence of dates by ordinary least squares(OLS). The form of the regression for beta estimation as follows:
Where,
is the return on stock i,
is the risk-free rate,
is the rate of return on the market.
Attempt to use excess return notation, new variables are defined: and . Then the regression equation becomes
Where,
is the excess return of stock i,
is the average risk premium.
The betas estimation can be launched by regression. The null hypothesis of the test is that regression intercepts for all stocks are jointly equal to zero and the alternative hypothesis is that the intercepts differ from zero. The intercept, ,is expected to be zero for each stock or portfolio in the CAPM theorem.
3.2.1.2 Cross-sectional test
The second one is cross-sectional test that is dependent on average excess returns and the value of estimated beta. The purpose is to catch the Security Market Line (SML) that graphically displays the expected return as a linear function of systematic risk (beta). In this test, the average excess return of stock or portfolio will be computed over certain time period. The estimates of beta for stocks or portfolios are attained from the time series tests. Each pair data, the average excess return and beta, corresponds to one stock or portfolio. The regression formula by OLS is :
Where,
is the average excess return of stock i or portfolio i,
is the estimate of beta for stocks i or portfolio i.
The null hypothesis of this test are that both and are equivalent to zero respectively and the alternative hypothesis is that they statistically differ from zero. There are some predictions from this test of the CAPM. In the first place, the intercept, , is expected to be zero by the CAPM. It indicates there have a risk-free rate. When the intercept term significantly differ from zero, it proves that something deviate the CAPM theory. Secondly, there have one and only variable, beta, to explain the expected return. Other variables for instance residual variance, firm size, dividend yield and so on should not have any explanatory power. Thirdly, it should be a linear relationship between the expected return and beta. That means the results should show that high beta lead great yield and vice versa. Finally, the slope of beta, , is expected to equal to the excess return on the market().
