In the early 1950s Harry Markowitz developed a theory of portfolio selection which has resulted in a revolution in the theory of finance leading to the development of modern capital market theory (1952, 1959). He formulated a theory of investor investment selection as a problem of utility maximization under conditions of uncertainty. Markowitz discusses mainly the special case in which investors’ preferences are assumed to be defined over the mean and variance of the probability distribution of single-period portfolio returns, but he also treated most issues developed more fully in the subsequent literature.

J. Tobin (1958) utilized the foundations of portfolio theory to draw implications with regard to the demand for cash balances. He also demonstrated that given the possibility of an investment in a risk-free asset as well as in a risky asset (or portfolio), an investor can construct a combined portfolio of the two assets to achieve any desired combination of risk and return. Subsequently, W. F. Sharpe, using one of the efficient methods for constructing portfolios discussed in the appendices to the Markowitz book (1959), developed what he called the ‘diagonal model’ in his dissertation under the direction of Markowitz, the results of which were later summarized in an article (1963). This represented another step towards general equilibrium models of asset prices developed almost simultaneously by Treynor (1965), Sharpe (1964, 1970), Lintner (1965a, b), and Mossin (1966, 1969). Important contributions were made by Fama (1971, 1976) and by Fama and Miller (1972).

These works resulted in the development of the relationship between return and risk summarized in what has been called the Security Market Line of the Capital Asset Pricing Model (CAPM).

$$ E\left({R}_j\right)={R}_F+\left[\frac{E\left({R}_M\right)-{R}_F}{\sigma_M^2}\right]\mathrm{COV}\left({R}_j,{R}_M\right). $$
(1)

This equation says that the return required (ex ante) by investors on any asset is equal to the return, RF, on a risk-free asset plus an adjustment for risk. Alternatively, the risk adjustment can be defined as the market risk premium weighted by the risk of the individual asset normalized by the variance of market returns. This latter measure has been referred to as the beta measure (β) of the risk of an individual asset or security \( \left[\beta =\mathrm{COV}\left({R}_j,{R}_M\right)/{\sigma}_M^2\right]. \)Leading synthesis papers on the CAPM are by Jensen (1972) and Rubinstein (1973).

The CAPM model assumes that the market functions in a reasonably perfect way in the sense that: all individuals act as if they are price-takers of all relevant prices; all securities are perfectly divisible and can be sold both long and short without margin and/or escrow requirements; there are no transaction costs or taxes; and, as in nearly all useful economic theory, arbitrage opportunities are absent so that an appropriate one price law obtains. Individuals are assumed to be risk averse, expected utility maximizers. In that differential assessment of probabilities generally explains too much, it is usual (although not necessary for all purposes) to require that probability beliefs are homogeneous (Krouse 1986). Subsequent work established that the main principles of the CAPM held up with the successive relaxation of the above assumptions (Black 1972; Brennan 1971; Lintner 1969; Mayers 1972, 1973; Merton 1973).

Roll’s critique (1977) has had a major impact. His major conclusions are: (1) The only legitimate test of the CAPM is whether or not the market portfolio (which includes all assets) is mean-variance efficient; (2) If performance is measured relative to an index which is ex post efficient, then from the mathematics of the efficient set, no security will have abnormal performance when measured as a departure from the Security Market Line; (3) If performance is measured relative to an ex post inefficient index, then any ranking of portfolio performance is possible depending on which inefficient index has been chosen. The Roll critique does not imply that the CAPM is invalid, but that tests of the CAPM are joint tests with market efficiency and that its uses must be implemented with due care.

Three basic types of models of asset pricing have been most frequently employed. The simplest, called the market model, is based on the fact that returns on security j can be linearly related to returns on a ‘market’ portfolio, namely:

$$ {R}_{jt}={a}_j+{b}_j{R}_{Mt}+{\varepsilon}_{jt} $$
(2)

where εjt is the mean zero classical normally distributed error term. The market model assumes that the slope and intercept terms are constant over the time period during which the model is fit to the available data, a strong assumption.

The second model is the capital asset pricing theory. It requires the intercept term to be equal to the risk-free rate, or the rate of return on the minimum variance zero-beta portfolio, both of which may change over time. In its simplest form, the CAPM is written

$$ {R}_{jt}-{R}_{Ft}=\left[{R}_{Mt}-{R}_{Ft}\right]{\beta}_{jt}+{\varepsilon}_{jt}. $$
(3)

Systematic risk, βjt, is generally assumed to remain constant over the interval of estimation.

The third model is the empirical counterpart to the CAPM, referred to as the empirical market line

$$ {R}_{jt}={\widehat{\gamma}}_{0t}+{\widehat{\gamma}}_{1t}{\beta}_{jt}+{\varepsilon}_{jt}. $$
(4)

This formulation does not require that the intercept term equal the risk-free rate. No parameters are assumed to be constant over time. In contrast to the market model, which is a time series expression, both the intercept, \( {\widehat{\gamma}}_{0t} \), and the slope, \( {\widehat{\gamma}}_{1t}=\left({R}_{Mt}-{R}_{Ft}\right) \), are the estimates taken from cross-section data each time period (typically each month). The betas in Eq. 4 are (following Fama and MacBeth 1973) calculated from the market model (Eq. 2). (See Copeland and Weston, 1983, Chaps. 7 and 10).

Empirical tests of the CAPM were conducted by Miller and Scholes (1972), Fama and MacBeth (1973), and Reinganum (1981), among others. Most of the studies use monthly total returns (dividends are reinvested) on listed common stocks.

Asset pricing models have been used to measure portfolio performance by mutual funds, pension fund advisers, etc., and in residual analysis of the impact of accounting reports, stock splits, mergers, etc. Some studies have used the market model to measure the error terms or residuals-positive or negative performance. However, the generally accepted procedure is first to calculate the β’s from the market line (Eq. 2). Portfolios ranked by β’s provide groupings to minimize errors in the measurement of variables problem. These portfolio betas are used to develop the parameters (intercept and slope terms) in Eq. 4 which is the empirical market line used to estimate the CAPM of Eq. 3. With estimates of the γ terms, the empirical market line can then be used to calculate ‘abnormal’ returns or residuals from predicted security returns.

The empirical tests of CAPM typically are conducted in excess return form. The equation in this form should have an intercept term not significantly different from zero, with a slope equal to the excess market portfolio return. The empirical tests have found an intercept term significantly above zero with a slope less than predicted. Thus the empirical securities market line is tilted clockwise implying that low beta securities earn more than the CAPM would predict and high beta securities earn less. But the main predictions of the CAPM of a positive market price for risk and a model linear in beta are supported.

The recognition that the market return alone might not explain all of the variation in the return on an asset or a portfolio gave rise to a multiple factor analysis of capital asset pricing. This more general approach formulated by Ross (1976b) was called the Arbitrage Pricing Theory (APT). Requiring only that individuals be risk averse, the APT has multiple factors and in equilibrium all assets must fall on the arbitrage pricing line. Thus the CAPM is viewed as a special case of the APT in which the return on the market portfolio is the single applicable factor.

Empirical work on the APT was performed by Gehr (1975), Roll and Ross (1980), Reinganum (1981), and Chen et al. (1984). These studies use data on equity daily rates of return for the New York and American Stock Exchange listed stocks. The initial studies establish that other factors contribute to an explanation of required returns but did not identify them. Later studies suggest that economic influences such as unexpected changes in inflation rates, default premia (measured by the difference between high- and low-grade bond yields), and the term premium in interest rates (measured by the difference between yields on short- and long-term bonds) correlate highly with the identified explanatory factors.

The CAPM and APT have provided useful conceptual frameworks for business finance applications such as capital budgeting analysis and for measurement of the cost of capital. Although the CAPM has not been perfectly validated by empirical tests, its main implications are upheld: systematic risk (beta) is a valid measure of risk, the model is linear in beta, and the tradeoff between return and risk is positive. The earliest empirical tests of the APT have shown that asset returns are explained by three or possibly four factors and have ruled out the variance of an asset’s own returns as one of the factors.

See Also