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The asset pricing theory aims to provide explanations for asset prices associated with uncer- tainty. Consequently, implicitly high rates of return should follow low prices. More generally, the theory explains why certain assets earn more profits than others. In this context, the valua- tion of assets must account for two dimensions: time effects and the risk of its underlying pay- ments. At this juncture, the price of time is reflected by the pure interest rate, usually repre- sented by the risk-free interest rate. The price of risk translates into the additional expected return per unit risk born (Sharpe, 1964).

There are two competing views on the role of asset pricing: the normative and the positive. From the normative perspective, asset pricing theory determines the true value of an asset so that investors can determine which assets might be mispriced. This practically creates oppor- tunities for the investor to trade and earn adequate risk-adjusted profits. The positive perspec- tive, on the other hand, sees the world as it is so that asset prices are assumed to be accurate. Deviations of asset pricing models must, therefore, be erroneous so that corrections of the mod- els need to be applied (Cochrane, 2001). The normative perspective has prevailed especially for practitioners since this view allows for derivations of asset pricing theory-based investment strategies.

The beginning of asset pricing theory is most commonly credited to Sharpe (1964) and Lintner (1965) summarized with the Sharpe-Lintner-CAPM. As of now, the application of the CAPM is widely-spread, such as for the estimation of a firm’s cost of capital and the perfor- mance evaluation of a managed portfolio. The CAPM signifies that a firm’s risk must be meas- ured relative to an efficient market portfolio, which as a matter of principle must not only con- sist of financial assets but also consumer durables or real estate (Fama and French, 2004). The CAPM builds upon the portfolio theory credited to Markowitz (1952). In his model of portfolio choice, the investor selects a “mean-variance-efficient” portfolio so that either the portfolio 1) minimizes the variance of the portfolio return at an expected return or 2) maximizes the ex- pected return at a given variance (Markowitz, 1959). The Sharpe-Lintner-CAPM rests on two additional assumptions to the mean-variance-efficient portfolio selection. First, investors in a capital market fully agree on the statistical distribution of asset returns in the future. Secondly, investors have unrestricted opportunities to borrow or lend at a risk-free rate (Fama and French, 2004). The Sharpe-Lintner-CAPM finally describes the linear relationship between an ex- pected return and risks born by the security. Thus, an investor expecting high returns must, therefore, accept higher risks expressed as the volatility of a security. The CAPM can be formally expressed as follows:

𝐸(𝑅_𝑖) = 𝑅_𝑓+ 𝛽_𝑖,𝑀(𝐸(𝑅𝑀) − 𝑅𝑓) (1)

where E(Ri) is the expected return for security i, Rf is the risk-free interest rate, E(RM) is the expected market return, and βi,M is the market beta of security i. The market beta of a security can be determined with the covariance of the return of security i divided by the variance of the market return as shown in the following equation:

𝛽𝑖.𝑀 =

𝑐𝑜𝑣(𝑅_𝑖,𝑅_𝑀)

𝜎_𝑀2 (2)

Hence, the market beta measures the sensitivity of a security’s return to the variation in market returns. In other words, the market beta reflects the covariance risk of a security relative to the covariance risk of all securities of the market portfolio, which equals the variance of the market return. Referring back to the two dimensions of asset pricing, the price of time and risk, Rf, therefore, reflects the price of time whereas (E(RM) - Rf) denotes the price (or premium) per unit of market beta risk (Fama and French, 2004).

The unrealistic assumption of free borrowing and lending in the CAPM prompted Black (1972) to develop an extended CAPM model with limitations on risk-free borrowing and lend- ing. He further assumes unrestricted short selling opportunities for investors of risky assets and that the market portfolio results from the weighted portfolio chosen by each investor. The baseline of his results implies that his extended model only differs to the CAPM regarding the treatment of the risk-free rate. The Black-CAPM requires the risk-free rate to be smaller than the expected market return, which is necessary for a positive premium for the market beta.

One of the major critiques on the CAPM is the empirically appropriate consideration of the market portfolio, which must not be limited to financial assets. Roll (1977) and Roll and Ross (1980) demonstrate the sensitivity of CAPM to financial securities but also all other individual assets. Roll and Ross (1980) formulate the following equation as the central conclusion of the so-called Arbitrage Pricing Theory (APT):

𝐸_𝑖 = 𝜆₀+ 𝜆₁𝛽_𝑖,1+ ⋯ + 𝜆_𝑘𝛽_{𝑖,𝑘 (3)}

where Ei is the expected return of asset i, λ0 is the return for a riskless asset and βi,j (j = 1, …,

k) is the coefficient vector for the risk factor premium λj. Roll and Ross (1980) argue that fun- damental economic variables, such as the Gross National Product, must be a component of

systematic risk and as such part of the model (3). However, the authors do not finally resolve the issue which economic variables must finally be included in such an asset pricing model to fully capture the systematic risk required to explain the expected returns of an asset.

An enduring challenge of asset pricing models is to consistently explain returns independent from time regimes. The validity of asset pricing models is prone to price fluctuations in the short run. Some researcher argue that returns indeed can be deducted from past behavior. If historical prices repeat themselves in patterns, they also should occur in the future, contradict- ing the random walk theory and the associated independence of successive price changes (Fama, 1965). Hence, one must differentiate between historical patterns which might predict future returns as propagated by behavioral finance theory or risk premiums which compensate the investor's risks born with the investment. The following section provides an overview of return patterns observed in the past which potentially contradict the understanding of the effi- cient market hypothesis.