Yes, the CAPM is dead

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    Yes, the CAPM Is Dead


    Tsong-Yue Lai

    Professor of Finance

    California State University Fullerton

    Fullerton, CA 92634


*Please do not quote or copy without permission from the author

    All comments are welcome

First version: October, 2008

    Current version: February, 2011

I am grateful to James Ang, John Erickson, Sharon Lai for their helpful comments and

    suggestions. All remaining errors remain my own.

    Yes, the CAPM Is Dead


    This paper proves the CAPM derived by Sharpe, Lintner, and Mossin is just a tautology rather than an asset pricing model. In statistics, the expectation of an unconditional random variable is a constant parameter and is decided by the density function and should not be affected by other factors. However, the CAPM asserts the expected excess rate of return on an asset depends on the beta, which depends on the covariance between the asset’s return and the market portfolio’s rate of return. In fact, the expected rate of return

    must be given before the covariance can be calculated in statistics. Thus if the CAPM were held then the beta in the CAPM would have depended on the expected rate of return on an asset not vice versa. This paper also addresses the validity of the market index model being used in empirical studies in financial economics. Since the market index is constructed by its components, the returns on the components of the market index affect

    not vice versa. Consequently, using the return on the the return on the market index, and

    market index as an explanatory variable to explain the return on the security violates the assumptions for the regression model in statistics. Hence, all previous findings which used the market index as the explanatory variable in the regression model should be closely re-examined, and their findings suspect.

    Keywords: CAPM, Alpha, Beta, Tautology, Mean-Variance Efficient Portfolio, Market Portfolio, Market Risk Premium, Systematic Risk, Unsystematic Risk, and Market Index.

    Yes, the CAPM Is Dead

I. Introduction and Background of the CAPM

    The capital asset pricing model (CAPM) was developed by Sharpe (1964), Lintner (1965), and Mossin (1966) to explain the relationship between the expected rate of return and the risk on the capital assets. Since then the CAPM has become the foundation of modern financial economics. For example, the required rate of return on a risky asset based upon the CAPM is used as the discount rate to value an asset in the investment decision. In corporate finance, the CAPM also has been used to calculate the cost of equity and cost of debt, and applied in capital budgeting as well. Empirically, the CAPM has been used as the basic model to test (1) the expected rate of return on a stock, (2) market efficiency,

    (3) the performance of the mutual funds among other things in the finance literature. The premise of these applications is that the CAPM must be valid theoretically. However, what if the CAPM is not valid and not an asset pricing model, would this render subsequent applications and findings still valid?

    Recently, the validity of the CAPM has been extensively discussed in numerous empirical studies. For example, Fama and French (1992) claim the CAPM is useless for

    precisely what it was developed to do based on their empirical results. Fama and French

    (1996) even challenged the validity of the CAPM in their empirical paper entitled “The

    CAPM is Wanted, Dead or Alive”. Unfortunately, other researchers appeared resistant and

    skeptical of their empirical conclusions. For instance, Ross et. al (2008) and Kothari et. al (2001) argues that the findings of Fama and French are due to data mining. Jagannathan and Wang (1994, 1996) demonstrate that if the assumptions of the S&P market index used as the


    proxy of the market portfolio and constant beta are relaxed, then the empirical support for CAPM is very well and the CAPM is alive. Campbell and Vuolteenaho (2004) break the beta of a stock with the market portfolio into two; the cash-flow beta and discount- rate beta, and found empirically that higher average returns compensate investors for higher average cash-flow beta. Cremers (2001) attributes the poor performance of the CAPM to the measurement problems of the market portfolio and its beta and concludes the CAPM may still be alive with the unobservable nature of the true market portfolio. Hur and Kumar (2007) claim that the beta measurement errors and portfolio grouping procedures cause the biases in cross sectional tests of the CAPM, hence the results and conclusions of the Fama and French (1992, 1996) may not be valid and the beta may not be dead after all.

    Fama and French (2004) argue the theoretical failing of the CAPM may be attributed to many simplifying assumptions and the difficulties in using the market portfolio in the tests of the model. Indeed, the fact that market portfolio is a vital factor in the CAPM and that its implementation is difficult is the consensus in finance. However, to attribute the failing of the CAPM to the many simplifying assumptions may not be convincing for both practitioners and academicians. For example, the supporters of the CAPM embrace the notion that how well the CAPM predicts or describes the rate of return on a risky security in the market is all that matters. They argue that if the CAPM can provide a good explanation of the expected rate of return, then unrealistic assumptions are of less importance. In addition, simplifying assumptions is inevitable for a theoretical model like the CAPM.


    However, if the premise is not held, then the conclusions based on that premise are meaningless in logics. Similarly, any conclusions drawn regarding the validity of the CAPM based on previous empirical findings may not be plausible if any one of the assumptions used in the derivation of the CAPM is violated, or shown to be false. For example, the market portfolio must be the mean-variance efficient in the CAPM, while previous empirical studies substitute a proxy index for the market portfolio. Using a mean-variance inefficient market proxy index in empirical studies violates the conditions of the CAPM and therefore, the conclusions from empirical findings, which claim that the CAPM is useless, may not be justified using this logic.

    The testability of the CAPM has focused on a vital variable; the market portfolio. Previous studies of testability presume the CAPM is valid theoretically but cannot be tested due to the implementation of the market portfolio or for various other reasons. For example, Roll (1977) criticized the testability of the CAPM. Roll contends that the

    1CAPM is not testable unless the exact composition of the market portfolio is known and

    used in the test. Kandel (1984) analyzes the mean-variance efficiency of the market index and concludes that his results do not support the notion that the mean variance efficiency is testable on a subset of the assets. Roll and Ross (1994) explain that the lack of the exact linear relation between the expected returns and the betas could due to the mean-variance inefficiency of the market portfolio proxies. Interestingly, as shown in the Proposition 5 in this paper, the linearity between the expected rate of return and its beta does not exist because the beta in the CAPM depends on the expected excess rate of return and not vice versa. Besides the market portfolio problem, as pointed out by Jensen


    (1972), the assumptions such as the Markowitz (1952) mean-variance criterion for investors’ portfolio selection, a perfect market, and a single period required to derive the CAPM, have also been criticized in the finance literature.

    Empirically, Fama and French (1992) use a cross-sectional regression to confirm the firm size, the price-earning, debt-equity, and the book-to-market ratios to explain the expected return on assets. Fama and French (1996) use a time-series regression to reach the same conclusion. Further, Fama and French (1993, 1996) use a three factor model and

    found that the beta factor matters less than other two factors; the firm’s price-earnings

    ratio and the firm’s market-to-book ratio, to the average rate of return on assets. Their empirical evidences contradict the assertion of CAPM that the expected rate of return on a security should depend solely on its beta and not other factors. Their seminal work has attracted a great deal of attention and stoked controversy regarding the validity of the CAPM. For example, Kothari, Shanken and Sloan (1995, 1998) examine the cross-section of expected returns on assets. They argue that the selection bias in the data of Fama and French could exaggerate the effect of the market-to-book ratio.

    An application of the relationship between the individual asset’s expected rate of

    2return and the market risk premium presented in the CAPM is the market index model,

    in which the rate of return on the market index is used as an explanatory variable in the regression model in different studies such as the event studies, the prediction of the asset rate of return, mutual funds performance, and others. The market index model shows that the rate of return on the market index affects the rate of return on the individual security


    or mutual fund. If the individual stock price moves before the market index does then how can the ex-post market index rate of return be used as the explanatory variable to explain the ex-ante dependent variable of the rate of return on security or on mutual funds? This paper will address this issue later.

    The purpose of this paper is to prove that the CAPM is simply a tautological model, and demonstrates the futility of continued use of the CAPM, as claimed by Fama and French (1992). However, the difference between this approach and previous studies is that this paper shows the characteristic of the market portfolio, the exact compositions of the beta, and the market risk premium in the CAPM and then proves that the CAPM is dead in theory. This paper is different from previous studies by conclusively showing that the CAPM is not useful as a model rather than reliance on empirical findings conducted

    3by other researchers such as by Fama and French (1992, 1994) and by others.

    The rest of this paper is organized as follows. Section I provides the background of the problem presented. Section II explores the meaning and definition of the market

    portfolioin the CAPM. The next section delves into analysis of the greater the expected rate of return on an asset, the greater the beta in the CAPM. Section IV shows the market risk premium in the CAPM is decided by the expected excess rate of return on assets and

    rtible covariance matrix among the assets’ rate of return. Section V uses algebra the inve

    to prove that the CAPM is not an asset pricing model because the product of the beta and risk premium is rewritten based on the expected excess rate of return. Section VI explores some statistical problems of the market index model in empirical studies. Section VII


presents the significance of these results, and conclusions that can be validly reached

    based on the mathematical results.

    II. What is Exactly the Market Portfolio in the CAPM?

    ~RAssume there are n risky securities and one is the risk-free asset. The notation of is the i

    risky asset i and its expected rate of return is R, for i=1,2..,n, r stochastic rate of return on i

    4is the risk-free rate of return on the risk-free asset, the nx1 vector of the expected excess

    5rate of return is R-r, and the nxn non-singular covariance matrix of the risky asset’s rate

    of return is Ω. The non-singular covariance matrix Ω and the nx1 vector R-r imply there exists an unique nx1 vector λ such that the expected excess rate of return vector R-r can

    6be expressed by

     R-r = Ωλ = cΩω , (1) ω

    -1-1where, λ = Ω(R-r) and can be rewritten as λ = cω, ω is a nx1 vector ω = Ω(R-ω

    T-1T-1TTTr)/[eΩ(R-r)], c = eΩ(R-r) = ω(R-r)/ωΩω is a constant scalar, e is 1xn vector ω

    Tof one. Since eω =1, ω is interpreted as a portfolio in this paper.

     The R-r = cΩω in equation (1) is the necessary and sufficient condition for the ω

    7optimal portfolio decision ω within the mean-variance framework. Thus, the unique ω in

    equation (1) must be the optimal mean-variance efficient portfolio. The following

    proposition presents this result.


Proposition 1: Given the expected excess rate of return vector R-r on n risky securities

    and the non-singular covariance matrix Ω between n risky securities rate of returns, the portfolio ω in equation (1) must be the unique optimal mean-variance efficient within the

    -1T-1mean-variance framework if and only if ω = Ω(R-r)/[eΩ(R-r)] .

    On the other hand, the CAPM is presented as the following:

     R = r +β (R-r), for all i =1,2,…,n (2) iim

    ~~2RRwhere, R r is the market risk premium, β= Cov(,)/σ, Cov(.,.) is the mi mim

    covariance operator, ω is the market portfolio, R is the expected rate of return on the mm

    ~~T2RRmarket portfolio = ω, and σ is the variance of the market portfolio rate of mmm


    ~~2RRIn equation (2), the β is defined by β = E[(-R)( R)] /σ, where, E[..] is iimmiim

    the expectation operator in statistics. The definition of β implies that the expected rate of i

    return R and R on the asset i and on the market portfolio, respectively, must be given im

    before the β can be calculated. That is, using systematic risk β to explain the R in the iii

    CAPM is implausible in statistics because the R exists already before the beta appears. i

     In terms of vector algebraic, equation (2) can be rewritten as

    2 R- r = β(R r) =(R r)Ωω/σ = cΩω = Ωλ (3) mmmmmmm,


    22Where, β =Ωω, c = (R-r)/σ, λ = cω, β is the nx1 vector of beta, Ωω is mmmmmmmmm

    ~the nx1 vector of covariance between the rate of return on i-th risky asset i=1,2,…,n Ri

    ~2Tand the market portfolio rate of return, σ = ωΩω. Rmmm m

    -18Equations (1) and (3) imply that λ = Ω(R-r)= λ and thus the market portfolio m

    -1T-1ω in equation (3) must be identical to the unique portfolio Ω(R-r)/[eΩ(R-r)] = ω in m

    equation (1). That is, equation (3) must be identical to equation (1). Since equation (1) holds for any positive integer n, thus, equation (3) must hold for any number of assets as well once the expected rate of return and their covariance existent or being given.

    Equation (1) is simply an algebraic result and is irrelevant to the demand and supply of the risky assets. Therefore, equation (1) must be irrelevant to market equilibrium. Since equation (1) is identical to equation (3), the market equilibrium must be irrelevant to the CAPM. Hence, the required rate of return on an asset based on the market equilibrium in the CAPM must be identical to the assumed expected rate of return on an asset in equation (1).

    In addition, the matter of equation (3) is the optimal mean-variance efficient portfolio for n securities rather than the market equilibrium. In other words, previous studies substitute the unobservable mean-variance efficient portfolio by the market portfolio under the equilibrium condition in the CAPM is not justifiable. Unfortunately, Elton et. al(2010), Fama and French (2004) argue that if all investors select the same


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