An empirical study of CAPM in Chinese market

Ruiyang Zheng (10389881) Supervisor: Liang Zou

University of Amsterdam

Faculty of Economics and Business

Bachelor Thesis Economics and Finance

Abstract

Capital Asset Pricing Model (CAPM) is one of the most important asset pricing models in . However, most financial economists focus on testing CAPM in well-developed financial markets and only a few financial economists try to test CAPM in emerging financial markets. In this article, some classic literatures of empirical study of CAPM are discussed. And then I use the multivariate test method to perform the empirical test of CAPM in Chinese . I find that the null hypothesis that the market proxy is on the mean-variance frontier is rejected. This result is different from that of a former empirical test of CAPM in Chinese stock market. Several possible explanations of this difference are listed in the article.

1. Introduction

In November, 2014, the Hugangtong Mechanism was put into operation after a period of preparation. This mechanism facilitates the capital flows between the Chinese stock market and the Hong Kong stock market. The qualified Chinese can invest in the Hong

Kong stock market while the qualified Hong Kong investors can invest in the Chinese stock market. However, compared to the well-developed stock market in Hong Kong, the Chinese stock market still has many features of developing country stock markets. First, restrictions of transaction exist in the Chinese stock market. T+0 transactions are not allowed, and selling is limited. Second, the daily fluctuation of stock price is restricted in the region of

10% of the opening price. Third, the behavior of Chinese investors may be different from the investors in the developed countries. For example, is wildly used by the Chinese investors, but this technique is proven to be useless in a weak-form efficient market. Forth, individual investors account for a large proportion of the Chinese investors.

These features make the Chinese stock market different from those in developed countries.

The Chinese stock market, as well as the stock markets in other developing countries, has a growing influence on the world economy. However, in the field of empirical study of

Capital Asset Pricing Model (CAPM), researchers mainly focus on the stock markets in the developed countries. CAPM is one of the most important parts of the modern investment theory. As Cochrane (2005) claims, “The CAPM, credited to Sharpe (1964) and Lintner

(1965a, b), is the first, most famous, and (so far) most widely used model in asset pricing.”

And since Sharp and Lintner established the CAPM, many financial economists have made effort to test the model in reality. Black, Jensen and Scholes (1972) and Fama and MacBeth

(1973) can represent the early works of the empirical test of CAPM. And Gibbons (1982) established a new method, namely the multivariate test, to test the model. And these works mainly test the CAPM in the US stock market, which is well-developed before the 1980s.

In the past two decades, some Chinese scholars performed the empirical test of CAPM on Chinese stock market, but most of empirical tests performed by these Chinese scholars use the method of Black, Jensen and Scholes (1972), which may have some problems such as measurement error (Jin & Liu, 2001). In addition, most of those tests were performed more than ten years ago, but Chinese stock market has many developments in the 21th century. 2

These developments may make the test result different from that of the former study.

As a result, the central research question of my thesis is whether the CAPM is applicable in Chinese stock market. Because it is hard to find the real , the goal of this study is to test whether the market proxy of the Chinese stock market is on the mean-variance frontier. Monthly returns of the Shanghai A share from January, 2004 to January,

2014 will be collected to perform the multivariate test, which is a test about the mean-variance efficiency of the market proxy.

The rest of the paper is organized as follows: Section 2 will introduce the former theoretical and empirical studies in CAPM. Some background information that is relevant to the test will also be explained. Section 3 will provide the methodology, which will be applied in Section 4. Finally, I will draw a conclusion in Section 5.

2. Theoretical Framework

A. Basic knowledge of CAPM

According to Fama and French (2004), the CAPM builds on the model developed by

Markowitz (1959). In his model, an selects a portfolio at time t-1 that produces a stochastic return at t. And the investors will select a portfolio that is mean-variance-efficient.

A mean-variance-efficient portfolio means that the portfolio minimizes the variance of returns given expected returns and maximizes the expected returns given the variance of returns

(Fama & French, 2004). The authors also claim that Sharp (1964) and Lintner (1965) added two important assumptions to Markowitz‟s mean-variance model. One of the assumptions they added is the unlimited borrowing and lending, which means that all investors can unlimitedly borrow and lend any amount of money at the risk-free rate (Fama & French,

2004). And the other assumption is complete agreement. It means that the investors agree on the joint distribution of asset returns, and that distribution is the true one. In other words, the investors have a homogeneous expectation and they analyze securities in the same way

(Bodie, Kane & Marcus, 2011).

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(Fama & French, 2004)

With these three assumptions, the CAPM can be described as Figure 1. In Figure 1, the horizontal axis is the standard deviation of an asset and the vertical axis is the expected return of an asset. Investors can group the risky assets into a portfolio. Because the individual risk of each risky asset in the portfolio can be diversified, investors can minimize the risk

(standard deviation) of the portfolio given a level of expected return. So investors can get a minimum variance frontier for risky assets, represented by curve abc in Figure 1. And then the risk-free asset is introduced and it is on the vertical axis. To combine the risk-free asset and risky asset portfolio, we can draw a line which passes the expected return of risk-free portfolio and tangents the minimum variance frontier at point T. And this line is so called the line (CPL). Portfolios on this line are mean-variance-efficient. According to the first assumption of CAPM, every investor should choose a portfolio on this line according to his risk preference. And because of the complete agreement assumption, all investors in the market are facing the same CPL. Thus all investors hold the same portfolio

T for risky assets (Fama & French, 2004). So portfolio T must be the value-weight market portfolio of risky assets and must be on the mean variance frontier if the market is to be clear.

According to the authors, this implies the following mathematical relation:

E(ri)=E(rzm)+[E(rm)-E(rzm)]βim, i=1,….N (1)

ri: Expected return of asset i

rzm: Expected return of risk-free asset

rm: Expected return of market portfolio

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Bim: of asset i

And this is the formula of Sharp-Lintner CAPM.

The Sharp-Lintner CAPM is built on the three assumptions that mentioned above, but those assumptions may be unrealistic. For example, the assumption of unrestricted risk-free borrowing and lending is unrealistic (Fama & French, 2004). Black (1972) shows that market portfolio is mean-variance-efficient when unrestricted short selling is allowed, even if the unlimited risk-free borrowing and lending is not allowed (Fama & French, 2004). But the Fama and French (2004) also claim that the assumption of unrestricted short selling is also unrealistic. Huang and Litzenberger (1988) claims that the empirical test are more consistent with the Black CAPM. But in this article, I will mainly focus on the

Sharp-Lintner CAPM because in China the deposit rate of state-own banks, which is secured by the Chinese government, can be regarded as the risk-free rate. And the unlimited short selling assumption of Black CAPM is even more unrealistic in Chinese stock market.

B. OLS Time-series method to test CAPM

According to Jensen et.al (1972), the formula of the time-series OLS test can be described as follows:

Rit=αi+βiRmt+eit (i=1……N, t=1……T) (2)

The dependent variable Rit is the excess return of asset i at time t, and the independent variable Rmt is the market excess return at time t. Excess return is the difference between the return of an asset (or a portfolio) and the risk free asset return. The Treasury bond rate is often regarded as the risk-free rate in the empirical research of Sharp-Lintner CAPM. The term is introduced by Jensen (1968) to measure the difference between E[Rit] and

E[βiRmt]. In other words, Jensen‟s alpha indicates the abnormal return of a security according to CAPM. Theoretically, if the market portfolio is mean-variance-efficient, the alphas of the assets should equal to zero (Jensen, Black & Scholes, 1972). Different from the betas of the cross-sectional test, which are fixed independent variables, the betas of the 5 time-series test are parameters to be estimated (Huang & Litzenberger, 1988).

Jensen et.al (1972) claim that using the OLS time-series method to test CAPM may have the problem that the residuals of individual stocks are correlated (E(ejt, eit)≠0 for i≠j). This problem is proved by King (1966). Jensen et.al (1972) claim that this problem can be solved by grouping the data. It means that one can form the securities into portfolios and estimates the portfolio betas and alphas. The portfolio alphas will appropriately incorporate the non-independent errors problem because the residual variance from the regression will incorporate the effects of any cross-sectional interdependencies in the errors among the securities in each portfolio (Jensen et.al, 1972). In order to obtain the maximum possible dispersion of the risk coefficients, the authors construct the portfolios by suing the ranked values of betas of individual stocks. However, they also remind that this method may cause the problem of selection bias, since those securities entering the high-beta portfolio would tend to have positive measurement errors in their betas. And this will lead to a positive bias in the estimated portfolio beta and a negative bias in the estimated portfolio alpha. For the low-beta portfolio, the selection bias will thus lead to a negative bias in the estimated portfolio beta and a positive bias in the estimated portfolio alpha.

The approach of Jensen et.al (1972) to solve the selection bias problem is the rolling regression. The authors intend to use an instrumental variable, the independent estimate of beta of the individual security in the past, to perform the time-series test. It means that they estimate the betas of the stocks by the data of the first period and then rank these observed betas to form the portfolios. And then, they use the data of these portfolios in the subsequent period to perform the time-series test. After this, they repeat this process until the end of the research time period.

C. Cross-sectional method to test CAPM

The cross-sectional method can be descripted as follows: The first step is the same as the time-series regression using model (2). And then, another regression is performed using the following model:

푅̅i=γ0+γ1훽̂ i+μi (i=1……N) (3) 6

E(Ri ) is the expected excess return of security i (average realized excess returns), and 훽̂ i is the beta of security i that is estimated in the time-series test. According to the Jensen et.al

(1972), the main result of CAPM is the relation between the expected risk premium on individual assets and their systematic risk. This relation can be descripted as:

E(Ri)= βiE(Rm) (4)

As a result, the expected value of γ0 should be equal to zero, and the expected value of γ1 should be the expected market excess return (Jensen et.al, 1972). One can test the CAPM by testing whether the test results are significantly different from the expected value.

Jensen et.al (1972) use both of time-series method and cross-sectional method to test the

Sharp-Lintner CAPM on the stocks listed on New York between 1926 and

1966. The cross-sectional model they use is the same as (3). The authors claim that the main task of the research is to verify whether γ0 equals to zero. Although the authors find a linear relation between the excess stock returns and betas, the observed value of γ0 is significantly different from zero. In addition, the intercepts and the slopes of the cross-sectional relation vary in different sub-period. So the authors conclude that the

Sharp-Lintner form of CAPM is inconsistent with the data.

Fama and MacBeth (1973) use the similar method to test the CAPM on common stocks listed on NYSE between 1926 and 1968. But the authors‟ cross-sectional model, which is showed below, is different from that of Jensen et al.

Rit=γ0t+γ1tβi+γ2tβi^2+γ3tsi+ηit （5）

Fama and MacBeth claim that the term βi^2 is added to test the linearity. The term si is the standard error of the excess returns of stock i. And it can be used to test whether the risk of stock i is deterministically related to the beta of stock i.

The authors claim that their goal is to test the three testable implications of CAPM: (C1)

The linear relation between the excess returns and betas. (C2) No other measure of risk other 7 than betas. (C3) Higher risk is associated with higher return. The results of the authors‟ empirical test cannot reject those three implications of CAPM.

One of the common concerns of Fama and MacBeth (1973) and Jensen et.al (1972) is the measurement error problem. The betas that one estimates in the time-series test may contain measurement errors, making the test result biased and inconsistent. If one uses these estimated betas to perform the cross-section test, the measurement error will make the result of the regression inconsistent. This can be descripted as follows:

훽̃i=βi+ξi

E(Ri)= γ0+γ1(훽̃i -ξi)+ei=γ0+γ1훽̃i+(ei-γ1ξi)

cov((ei-γ1ξi), 훽̃i)=cov((ei-γ1ξi), (βi+ξi))= γ1*var(ξi),where

E(ξi )=0, E(ξi, βi)=0,

2 E(ξi, ξj)= σ (ξ) if i=j

E(ξi, ξj)=0 if i≠j (Jensen et.al, 1972)

The estimator of beta is correlated with the error term if var(ξi) is not equal to 0 , violating the

OLS assumption (Stock & Watson, 2007). If one performs the cross-sectional test using model (3), the probability limit of the estimator of γ1 that is subjected to the measurement error can be expressed as:

2 2 plim 훾̃1 =γ1/[1+σ (ξ)/S (βi)], (Jensen et.al, 1972)

2 where S (βi) is the cross-sectional sample variance of the true βi. As a result, the a positive

2 σ (ξ) may lead to a biased and inconsistent estimation of γ1. (Jensen et.al, 1972)

Jensen et.al (1972) and Fama and MacBeth (1973) try to deal with the problem of measurement error by grouping the data into portfolios. Jensen et.al (1972) show that the grouping procedure can significantly reduce the sampling error in the estimated risk measures and thus can virtually eliminate the measurement error problem. This can be described as follows:

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2 2 2 var(훽̃i∣βi)=σ (ξ)= σ (ejt)/Φ=σ (e)/Φ

where ejt are the errors in time-series model as (2). The ejt are assumed to be independently distributed and have constant variance for all j and t (Jensen et.al, 1972). And

Φ=Σ(Rmt-Rm)2

2 2 plim 훾̃1 =γ1/[1+σ (e)/ ΦS (βi)]

Given N securities, one can group them into M equal-size contiguous subgroups. The average return of each group is:

Rk=(1/L)ΣRkit, where L=N/M

So the time-series model becomes:

Rkt=αk+βkRmt+ekt (k=1……M, t=1……T)

and the estimate of βk is 훽̃k=βk+ξk. As a result,

2 2 var(훽̃k∣βk)=σ (ξk)= σ (e)/LΦ

Similar to (3), the cross-section test model of group data is

E(Rk)=γ0+γ1훽̃k+ek

And for large sample, the probability limit of estimate of γ1 is:

2 2 plim 훾̃1=γ1/[1+plim((1/L)σ (e)) / ΦS (βi)]= γ1, as long as L→∞ as N→∞

D. Roll’s critique 9

In 1976, Roll published his famous critique on empirical CAPM test. He claims that there is only one testable hypothesis associated with the CAPM, namely „the market portfolio is mean-variance efficient‟. All the other implications of the model, such as the linear relation between stock excess returns and betas, are not independently testable. That is because if the betas are calculated against a mean-variance efficient portfolio, the linear relation between stock excess returns and betas will be automatically satisfied. This is so called the mean-variance tautology.

As a result, the author claims that CAPM is not testable unless the exact composition of the true market portfolio is known and used in the tests. Using the market proxy to test

CAPM may have two difficulties: First, the market proxy might be mean-variance efficient while the true market portfolio is not. Second, the market proxy might be mean-variance inefficient while the true market portfolio is mean-variance efficient. Since the composition of market portfolio is impossible to be known, the author claims that there is not possible to test CAPM.

The author also reviews the research of Fama and MacBeth (1973). He claims that (C1),

(C2) and (C3) are simply implications of the fact that the market portfolio is assumed to be ex-ante efficient. So the test of Fama and MacBeth (1973) is a tautological test.

E. Six difficulties in testing the CAPM

Huang and Litzenberger (1988) summarize three conceptual problems that are related to the empirical test of CAPM. The first conceptual problem is that we are using the ex post stock price data to test the ex ante relationship between risk premiums and betas. This problem can be solved by the assumption of rational expectation, which implies that the realized returns of assets in a given time period are drawings from the ex ante probability distributions of returns of those assets.

The second conceptual problem is the nonstationarity of risk premiums and betas on individual assets (Huang & Litzenberger, 1988). Because the CAPM is a two period model, it is implicitly assumed that the CAPM holds period by period. But in the empirical test, it is unlikely that the risk premium and beta of an individual asset are stationary over time. The authors claim that there are two ways to solve this problem. The first approach is to form 10 portfolios that are constructed to have stationary betas and to assume that the risk premiums of these portfolios and of the market portfolio are stationary over time. The second approach is “to interpret the tests in terms of the distributions of asset returns conditional on a coarser information set and assume that these distributions are time-stationary.” (Huang &

Litzenberger, 1988) The authors prove that even when the risk premiums and betas conditional on information sets available to investors over time are nonstationary, they can be stationary conditional on a coarser information set.

And the third conceptual problem is the unobservability of the true market portfolio, which contains all kinds of asset including financial assets, real estates, human capital, etc.

And it is, apparently, impossible to observe such a portfolio. So in the empirical test researchers often use a market proxy to stimulate the function of the true market portfolio.

But according to Roll (1976), if we assume that the market proxy is on the mean-variance frontier, the empirical test of the CAPM will become a tautological test, since the only testable implication of CAPM is that the market portfolio is on the mean-variance frontier.

Huang and Litzenberger (1988) provide three approaches to deal with this problem. The first approach is to ignore the problem by assuming that the disturbance terms form the regressing the asset returns on the market proxy returns are uncorrelated with the true market portfolio and that the market proxy has a unit beta. The authors prove that under these assumptions, the betas can still be estimated when an appropriate market proxy is used. And the second approach is to interpret the test as a test of whether the market proxy is on the portfolio frontier. The third approach is to view the test as a test of a single factor APT with a pre-specified factor. The authors claim that it is empirically indistinguishable from the second approach.

In addition to these three conceptual problems, Huang and Liztenberger (1988) also claim that there are three econometrics problems related to the empirical test of CAPM. The first problem is that the disturbance terms of the cross-sectional test are heteroscedastic and correlated across assets, because variances of rates of return differ across assets and asset returns are correlated. The authors claim that the problem can be solved by using OLS estimators and to calculate the correct variances of the coefficients. This solution requires the estimation of the variance-covariance matrix. 11

The second econometric problem is the measurement problem which is mentioned above

(Huang & Litzenberger, 1988). The first solution of this problem is to group the data as the methods in Fama and MacBeth (1973) and Jensen et.al (1972). The second solution is to use an instrumental variable and the third solution is to use an adjusted GLS approach that takes account of the variances of the measurement errors in betas.

The third econometrics problem is that the CAPM implies a non-linear constraint on the return generating process in the time-series test (Huang & Litzenberger, 1988). The author claims that in order to estimate the parameters in formula (6), constrains as follows should be considered:

αj=E[Rzc(m)](1-βjm)

ejt=Rjt-E[Rjt∣Rmt]

where Rzc(m) is the excess on the minimum variance zero covariance portfolio with respect to the market proxy. But the constraints itself depend upon the parameters that are to be estimated (βjm and Rjt). This problem can be solved by using a maximum likelihood estimation that takes these interactions into account (Huang & Litzenberger, 1988).

F. The Chinese stock market

In the past two decades, the Chinese economy develops rapidly since the market economy reform. And nowadays, the Chinese stock market is playing an important role in the world capital market. On 27/11/2014, the total capitalization of Chinese stock market reached

44,880 trillion dollars, making the Chinese stock market the second largest stock market in the world. However, the Chinese stock market still has many features of a developing country stock market.

One of the main differences between the Chinese stock market and the stock markets in developed countries is the restriction imposed by the trading regulation. First, the Chinese stock market has the limitation of daily fluctuation of individual stocks. According to Su and Fleisher (1998), the Chinese stock regulatory authorities imposed a 10% daily price change limit on any individual stock to reduce the . Second, the T+0 transaction 12

(buy and sell the shares of a stock in the same trading day) is forbidden in the Chinese stock market since 1995. But T+0 transactions are allowed in the Chinese Treasury bond market and future market. Third, short selling a stock is not allowed in the Chinese stock market, while short selling the index future is allowed.

The other difference between Chinese stock market and the developed country stock market is the investor behavior. According to Chen and Li (2006), technical analysis is very popular among Chinese investors. Security analysts, as well as many other stock commentators and newsletter writers, base their recommendations on technical analysis.

However, technical analysis is proved to be useless in the weak-form efficient market. In addition, the individual investors play such an important role in Chinese stock market that their transactions account for more than 90% of the total trading volume (Ng & Wu, 2007).

In the last two decades, some Chinese financial economists try to perform the empirical test of CAPM on the Chinese stock market. However, most of these researches test the

CAPM by the method of Jensen et al (Jin & Liu, 2001). Jin and Liu (2001) test

Sharp-Lintner CAPM on the 496 stocks listed on Shanghai Stock Exchange and Shenzhen

Stock Exchange between 1996 and 2001 by many different methods, including the methods of

Fama and MacBeth (1973) and multivariate test. (The method of multivariate test will be explained in the Methodology section) They show that when using the method of multivariate test, the null hypothesis that alphas are all equal to zero cannot be rejected at the significance level of 1% and 5%. It implies that the market portfolio is mean-variance efficient. However, using the method of Fama and MacBeth, the author claim that only γ2 in

(5) is significant at the level of 10%. It implies that there is a relation between the average excess returns of the portfolios and the betas, but this relation is not linear.

One problem in Jin & Liu (2001) is that the authors use the daily excess returns instead of monthly excess returns in the multivariate test. And the daily excess returns have substantial departures from normality that are not observed with monthly data (Brown &

Warner, 1985). So using the daily excess returns violates the multivariate normality return assumption of multivariate test.

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3. Methodology

A. The multivariate test

The Roll‟s critique puts forward a new problem to the financial economists who want to test

CAPM. Since the true market portfolio is not observable and the only testable implication of CAPM is whether market portfolio is mean-variance efficient, many financial economists interpret the test as a test of whether the market proxy is mean-variance efficient. However, both the OLS time series method and the cross-sectional method are not the best method to test the efficiency of the market proxy. That is because the OLS time series method cannot test the efficiency of the market proxy directly and the cross-sectional method is subject to the measurement error problem.

Multivariate test is a method to test the efficiency of the market proxy (Huang &

Litzenberger, 1988). The first multivariate test in the literature is finished by Macbeth

(1975), but Gibbons (1980, 1982) presents the first extensive treatment (MacKinlay, 1987).

According to Gibbons (1982), this method can avoid the measurement error problem and increase the precision of parameters estimates for the risk premiums. This method is developed further by MacKinlay (1987) and Gibbons, Ross and Shanken (1989).

The multivariate test of CAPM can be described as follows:

First, assume that the asset returns are multivariate normal distributed and the excess returns are independently and identically distributed though time (MacKinlay, 1987). To test whether the market proxy is on the mean-variance frontier, one has the null hypothesis that

αi=0, ∀i (Huang & Litzenberger, 1988). And apparently the alternative hypothesis is that

αi≠0, ∀i.

The second step is to obtain an estimate of beta for each of N assets by an OLS regression (Huang & Litzenberger, 1988). But the intercept term is not included in the model. The model can be descripted as:

c c Rit=β iRmt+e it (i=1……N, t=1……T) (6)

And then, the time-stationary cross-sectional variance-covariance matrix of the disturbance term of (6) can be estimated as: 14

c c c ┬ Ve =(∑e te t )/T (7)

c where e t is the N-vector of time t residuals for the N assets in the sample.

The third step of the test is to use the same method as the second step to estimate Ve (the time-stationary cross-sectional variance-covariance matrix of the disturbance term), βi (betas) and et (vector of residuals) of the unconstrained model (2).

The log of the likelihood function of Rit using unconstrained estimates of betas, variances, and covariance is:

┬ -1 ln L= -(NT/2)ln2π - (T/2)ln|Ve| - (1/2)∑et Ve et (8)

The log of the likelihood function of Rit using constrained estimates of betas, variances, and covariance is:

c c c ┬ -1 c ln L = -(NT/2)ln2π - (T/2)ln|V e| - (1/2)∑e t Ve e t (9)

According to Huang and Litzenberger (1988), because (8) and (9) are maximized,

┬ -1 c ┬ -1 c (1/2) ∑et Ve et = (1/2) ∑e t Ve e t

Since Lc is the maximized and is with an additional constraint, ln Lc – ln L <= 0. So the test statistic for the null hypothesis that the market proxy is on the ex ante portfolio frontier is:

c c 2 -2λ=-2 (ln L – ln L) = T(ln |V e| - ln |Ve|) ~ χ N-1 when T → ∞ (Gibbons, 1982) (10)

The test is actually testing whether the sample data of the market proxy fits the constrained model in which alpha equals to zero and which indicates the mean-variance efficiency. If the sample data of the market proxy fits the unconstrained model, the estimates of the time-stationary cross-sectional variance-covariance matrix of the disturbance 15 terms of these two models will not be significantly different from zero.

As a test about the mean-variance efficiency of the market portfolio, the multivariate test solved the three econometrics problems described by Huang and Litzenberger (1988). By using the estimates of the time-stationary cross-sectional variance-covariance matrix of the disturbance terms in the test statistics, it solved the problem that the disturbance terms of the cross-sectional test are heteroscedastic and correlated across assets. And the multivariate test eliminates the measurement error problem of betas in the OLS process. In addition, by using the maximum likelihood estimates, this method solved the problem of the non-linear constraint on the return generating process.

B. Data

The main disadvantage of the multivariate test is the lack of power. MacKinlay (1987) proves that the multivariate test lack the power to detect the plausible deviation from CAPM.

The author claims that when the deviations from the model are randomly introduced without regard to the covariance structure of the residuals, the test has reasonable power. However, the test is very weak when the same sorts of deviations are introduced using a factor model.

As a result, the selection of data and time scope is important in this study to ensure that the test has a reasonable power.

First, the N (number of portfolios) should be no larger than T (periods) to ensure that the estimates of the covariance matrices nonsingular (Gibbons et al., 1989). Second, an appropriate combination of N and T should be selected. The authors claim that it is normally assumed that a research period that is less than five years can match the stationary assumption of the betas. And in the multivariate test, researchers mainly used the monthly returns which reasonably meet the normality requirement (Fama, 1976). And according to

Gibbons et al. (1989), to get a satisfying statistics power for the multivariate test, the N should be a third to one half of the T.

As a result, if we selected a research period of five years, the N should be 20 to 30

(Gibbons et al., 1989). However, it is hard to selected 20 to 30 assets to represent many assets in the market and at the mean time maximized the statistics power of the test. As a result, the common approach is to group the assets into portfolios. There are many ways to 16 group the data but the most common approach is to use beta-sorted portfolios (Gibbons et al.,

1989).

According to the analysis above, in this article I choose a research period of five years

(from Jan, 2009 to Jan, 2014). The Shanghai A share Index will be used as the market proxy.

The average deposit rate of Chinese state-owned banks in the research period will be used as the risk-free rate, which can be collected on the website of World Bank. The average deposit rate from Jan, 2009 to Jan, 2014 is 2.92%.

The grouping procedure can be described as follows: First, the betas of each stock will be estimated by OLS method using the data from Jan, 2004 to Jan, 2009. And then I will form

21 portfolios according to the values of betas of these stocks. And then I can use these 21 portfolios to perform the multivariate test in the research period (Jan, 2009 to Jan, 2014). So the T in the test is 60 and the N in the test is 21. This grouping procedure is made to avoid the selection bias. But it requires that only 734 Shanghai A share stocks with complete data from Jan, 2004 to Jan, 2014 will be considered in this article. The stock price data will be collected from the Thomson Reuters Datastream.

4. Results

After estimating the betas of 734 stocks from Jan, 2004 to Jan, 2009, I sort 21 portfolios according to the ranking of the betas (from small to large). So the first portfolio has the smallest beta from Jan, 2004 to Jan, 2009 and the 21st portfolio has the largest beta in that period. There are 36 stocks in each portfolio except the 21st portfolio, which contains only

14 stocks. The betas (model with a constant) of these 21 portfolios from Jan, 2009 to Jan,

2014 are shown below (ranking from small to large):

Portfolios Betas Portfolios Betas P17 0.074243 P9 1.126487 P2 0.8225 P8 1.142305 P3 0.922193 P18 1.189022 P1 0.973911 P16 1.189499 P7 1.018415 P11 1.198147 P4 1.041387 P10 1.205572 P6 1.048969 P15 1.215222 P13 1.087523 P19 1.289356 P14 1.104546 P20 1.302696 P12 1.108224 P21 1.408917 P5 1.112331 17

Then the regression of each portfolio‟s excess return on the excess return of Shanghai A share Index is performed, using the constrained model and the unconstrained model respectively. And after that I calculate the constrained time-stationary cross-sectional

c variance-covariance matrix (V e) and the unconstrained time-stationary cross-sectional variance-covariance matrix (Ve) using STATA. Finally, the test statistics is calculated:

c where CCC is V e and NNN is Ve. (Specific description of CCC and NNN is in the

Appendix). When the degree of freedom is 20, the (two-tail) 5% critical value of χ2 distribution is 34.170, which is smaller than the test statistics. As a result, the null hypothesis that αi=0, ∀i is rejected. So the Shanghai A share Index, as a market proxy, is not on the mean-variance frontier.

Although some researchers criticize that the power of the multivariate test is weak, in this article the null hypothesis is rejected. This result is different from that of Jin and Liu

(2001). In my opinion, there are some possible reasons of this difference of test result.

The first possible reason is the problem in Jin and Liu (2001). As mentioned in Section

2 Part F, the main problem in Jin and Liu (2001) is that they use the daily excess returns to perform the multivariate test. The daily data can hardly meet the normality requirement of the test so it may influence the result of the test. The other problem of the article is that the authors form the portfolios according to the ranking of betas in the test period. As mentioned in Section 2 Part B, this may lead to the selection bias problem, which influences the result of the test.

The second possible reason is that the restrictions on the Chinese stock market make the market inefficient. For example, the daily fluctuation of each stock in the Shanghai Stock

Exchange is at most by 10%, so the stock prices cannot immediately reflect the information that may lead to a fluctuation of stock prices by more than 10%. The other evidence of the inefficiency of Chinese stock market is the popularity of technical analysis, which is proved to be useless in the weak-form efficient market.

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The third possible reason is that the Shanghai A share Index might not be a good market proxy. In the research period of the test of this paper, the Chinese real estate market also had a fast development, especially in the period from 2008 to 2013 (Gyourko & Deng, 2012).

As a result, investors may invest a large proportion of their fund into the real estate market.

So in this period, the Shanghai A share Index, which only contains stocks, is not a good proxy to represent all the accessible assets on the .

The forth possible reason is the behavior of the Chinese investor. In developed stock markets, investors are mainly institutional investors who are well trained and use similar ways to analyze the stock market. In contrast, individual investors account for a large proportion of transactions in the Chinese stock market. And these investors have many different backgrounds and may analyze the market in completely different ways. As a result, the homogeneous expectations assumption of CAPM is violated. The research period of Jin and

Liu (2001) is from 1997 to 2000. And the research period of this article is from 2009 to

2014. China experiences a huge development since 2000 and Chinese people have more available fund to invest compared to the late 1990s. Such a phenomenon might increase the number of Chinese individual investors, resulting in a larger deviation from the CAPM assumption.

5. Conclusion

In this article some classic literatures about the empirical study of CAPM are introduced, and then the multivariate test is selected to be the most appropriate method to solve the research question of this article, namely the efficiency of Shanghai A share Index as a market proxy.

The test result shows that the null hypothesis is rejected, implying that the Shanghai A share

Index is not on the mean-variance frontier.

There are many possible reasons to explain why the Shanghai A share Index is not on the mean-variance frontier, but the specific explanation of this result is beyond the scope of this article. I can only draw a conclusion that the Shanghai A share Index, as a market proxy, in the research period of this article is inefficient. One of the main limitations of this kind of empirical study is that the result of efficiency (or inefficiency) is hard to extend to different research period. For example, the test result between this article and that of Jin and Liu 19

(2001) is different, but it is hard to distinguish the main reason(s) of this difference.

However, I can still put forward some suggestions for further research. CAPM is a classic asset pricing model, but in recent years, financial economics have modified the model and put forward many different forms of CAPM such as the Fama and French three factor model or Fama and French four factor model. These models improve the CAPM by putting more relevant variables into the model. Another type of improvement is to modify the assumption of CAPM. For example, Zou (2006) introduce the BCAPM which modify the mean-variance preference into that investor perceives risk as the second moment of portfolio returns around a personal target return. It results in an improvement of the CAPM pricing error. Testing these models in the Chinese stock market will be an interesting research topic since CAPM does not have a good empirical performance in the Chinese stock market, as shown in this article.

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Appendix

c 1. The constrained time-stationary cross-sectional variance-covariance matrix (V e)

2. The unconstrained time-stationary cross-sectional variance-covariance matrix (Ve)

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3. GDP Data of China and Its Trend

(Retrieved from the website of Shanghai Stock Exchange)

4. Total and number of listed companies on the Shanghai Stock

Exchange and the Shenzhen Stock Exchange, 2003-2012

(Retrieved from the website of Shanghai Stock Exchange)

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