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Capital asset pricing model

The Capital Asset Pricing Model (CAPM) is a foundational financial theory that quantifies the relationship between an asset's expected return and its systematic risk, positing that investors require compensation only for non-diversifiable market risk rather than total risk.^[1] Developed independently in the early 1960s by economists Jack Treynor (1962), William F. Sharpe (1964), John Lintner (1965), and Jan Mossin (1966), the CAPM extends Harry Markowitz's modern portfolio theory by assuming that rational investors hold diversified portfolios and focus on mean-variance efficiency.^[2]^[3]^[4]^[5] Sharpe's formulation earned him the Nobel Prize in Economics in 1990, recognizing CAPM's role in revolutionizing asset pricing and investment decision-making. The model operates under strict assumptions, including that investors are risk-averse and optimize portfolios based solely on expected returns and variance; all investors share identical expectations about asset returns, variances, and covariances; markets are frictionless with no taxes, transaction costs, or short-selling restrictions; and unlimited borrowing and lending occur at a single risk-free rate.^[6] These conditions lead to the derivation of a market equilibrium where the optimal risky portfolio is the market portfolio itself, and all assets lie on the security market line (SML).^[3] At its core, CAPM expresses expected return through the formula E(R_i) = R_f + \beta_i [E(R_m) - R_f], where E(R_i) is the expected return on asset i, R_f is the risk-free rate, \beta_i measures the asset's sensitivity to market movements (covariance with the market divided by market variance), and [E(R_m) - R_f] is the market risk premium.^[6] This implies that assets with higher betas command higher expected returns to compensate for greater systematic risk exposure, while idiosyncratic risk is eliminated through diversification and thus unrewarded.^[3] Widely applied in corporate finance for estimating the cost of equity, performance evaluation, and portfolio management, CAPM remains influential despite empirical challenges revealing deviations such as the flat security market line and multifactor influences on returns.^[6]

History and Development

Inventors and Origins

The Capital Asset Pricing Model (CAPM) emerged as a foundational theory in modern portfolio theory, building directly on Harry Markowitz's pioneering work in mean-variance analysis. In his seminal 1952 paper, Markowitz introduced the framework for portfolio selection that emphasized diversification to minimize risk for a given level of expected return, laying the groundwork for subsequent asset pricing models by quantifying the trade-off between risk and return through variance as a measure of risk.^[7] The initial formalization of CAPM is credited to Jack Treynor, who developed the core ideas in unpublished manuscripts between 1961 and 1962. Treynor's work, including "Market Value, Time, and Risk" (1961) and "Toward a Theory of Market Value of Risky Assets" (1962), proposed a method for valuing securities based on their contribution to overall portfolio risk rather than standalone volatility, marking the first articulation of what would become the CAPM.^[8] These manuscripts circulated privately among academics and influenced later publications, though they remained unpublished until later compilations.^[9] Independently, the model was elaborated and published by several economists in the mid-1960s. William F. Sharpe formalized CAPM in his 1964 paper "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," published in the Journal of Finance, which derived equilibrium pricing for risky assets in a diversified market.^[10] John Lintner extended similar ideas in 1965 with "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," emphasizing investor optimization under risk aversion.^[11] Jan Mossin contributed concurrently in 1966 through "Equilibrium in a Capital Asset Market," published in Econometrica, which analyzed market equilibrium properties for capital assets.^[12] These developments were recognized with the 1990 Nobel Prize in Economic Sciences, awarded jointly to Sharpe, Markowitz, and Merton H. Miller for their contributions to financial economics, particularly the theories of portfolio choice and corporate finance that underpin CAPM.^[13] The early motivations for CAPM stemmed from the need to refine asset pricing by focusing on non-diversifiable, or systematic, risk rather than total variance, addressing limitations in earlier models that treated all risk equally. This shift enabled a more precise assessment of how assets contribute to an investor's overall market exposure.

Key Milestones

In 1965, John Lintner formalized the Capital Asset Pricing Model (CAPM) by deriving an equilibrium relation that integrates the model's risk-return framework into broader capital market pricing, showing that the expected return of a security is linearly related to its systematic risk measured by beta.^[14] This work built on earlier portfolio theory to establish CAPM as a cornerstone of asset pricing equilibrium.^[15] During the late 1960s, empirical methods for estimating beta emerged as researchers began testing CAPM through regressions of individual security returns against market returns, with early tests by Irwin Friend and Marshall Blume in 1970 and adjustments to improve predictive accuracy of historical betas introduced by Blume in 1975 for portfolio risk assessment contexts.^[16]^[17] In 1972, Fischer Black introduced the zero-beta variant of CAPM to relax the assumption of unrestricted risk-free borrowing, proposing instead that asset returns depend on their covariance with the market portfolio and the expected return of a zero-beta portfolio, thus addressing real-world borrowing constraints while preserving the model's core linear risk-return relation.^[18] By the 1970s, CAPM saw early practical applications in U.S. regulatory contexts, particularly for setting rates of return in public utility cases; for instance, the Federal Communications Commission applied it in 1972 to determine a lower cost of equity for AT&T based on an estimated beta of approximately 0.7, and state commissions followed suit, such as the Oregon Public Utility Commissioner in 1977 for Portland General Electric, which set an equity cost of 12.5% using betas between 0.695 and 0.781.^[19] The model's foundational impact was recognized in 1990 when William Sharpe, Harry Markowitz, and Merton Miller received the Nobel Memorial Prize in Economic Sciences, with CAPM specifically cited for explaining financial asset pricing through beta-measured risk; this accolade solidified its role as the standard framework in financial economics textbooks worldwide, including those on corporate finance.^[13]

Core Concepts

Risk and Return Fundamentals

In modern portfolio theory, the risk-return tradeoff represents a foundational principle, positing that investors must balance the potential for higher returns against the uncertainty of those returns, with risk measured primarily through variance or standard deviation of asset returns. Harry Markowitz formalized this concept in his seminal 1952 paper, introducing mean-variance analysis as a framework for portfolio selection, where expected returns serve as the reward for assuming risk, and variance quantifies the dispersion of possible outcomes.^[20] This approach assumes investors are rational and risk-averse, preferring portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given expected return, thereby establishing the efficiency criterion for investment decisions.^[21] Total risk for an individual asset is captured by the standard deviation of its returns, denoted as \sigma, which encompasses all sources of variability in performance, including both diversifiable and non-diversifiable components. In contrast, systematic risk, also known as market or non-diversifiable risk, refers to the portion of total risk that arises from economy-wide factors affecting all assets, such as inflation, interest rate changes, or recessions, and cannot be eliminated through portfolio diversification. Markowitz's work highlighted that while total risk (\sigma) provides a complete measure for undiversified holdings, systematic risk becomes the relevant metric for well-diversified investors, as diversification can substantially mitigate the unsystematic, asset-specific components.^[20] Expected returns function as compensation for bearing risk, with risk-averse investors demanding progressively higher expected returns to justify exposure to greater uncertainty, a direct implication of the utility maximization under risk aversion outlined in Markowitz's framework. The covariance between an asset's returns and the market portfolio's returns, denoted as \operatorname{Cov}(R_i, R_m), plays a critical role in assessing these relationships, as it quantifies the degree to which the asset's performance co-moves with broader market movements—positive covariance indicates amplification of market trends, while negative values suggest hedging potential. In Markowitz's mean-variance optimization, covariances among assets determine the overall portfolio risk, emphasizing that assets with low or negative covariances enhance diversification by reducing the portfolio's total variance without sacrificing expected returns.^[20]

Diversification and Efficient Frontier

Diversification is a fundamental strategy in portfolio theory that reduces overall risk by combining assets whose returns are not perfectly correlated. By holding a portfolio of multiple securities, investors can eliminate unsystematic risk, also known as idiosyncratic or specific risk, which arises from factors unique to individual companies or industries, such as management decisions or product failures.^[7] This type of risk can be substantially mitigated through diversification, with research indicating that a well-diversified portfolio of randomly chosen stocks requires at least 30 stocks for investors who borrow to leverage and 40 stocks for those who lend at the risk-free rate, thereby reducing unsystematic risk to negligible levels. After achieving such diversification, only systematic risk, which affects the entire market and cannot be eliminated, remains.^[7] The concept of the efficient frontier emerges from mean-variance optimization, a framework introduced by Harry Markowitz that identifies the set of portfolios offering the highest expected return for a given level of risk, or equivalently, the lowest risk for a given expected return.^[7] The feasible set of all possible portfolios, plotted in risk-return space where risk is measured by standard deviation or variance, forms a region bounded by a curve known as the efficient frontier, which represents the upper boundary of optimal risky asset combinations.^[7] Portfolios below this frontier are suboptimal, as they provide lower returns for the same risk or higher risk for the same return. Key points on the efficient frontier include the minimum variance portfolio, which achieves the lowest possible portfolio variance among all feasible portfolios, serving as the leftmost point of the frontier and marking the foundation for risk minimization without regard to return constraints.^[7] The tangency portfolio, in the context of mean-variance analysis with solely risky assets, refers to the portfolio on the efficient frontier that maximizes the slope of the line connecting it to the origin in risk-return space, optimizing the risk-return tradeoff for certain investor preferences.^[7] Mathematically, constructing the efficient frontier involves solving a quadratic programming problem to minimize portfolio variance subject to constraints on expected return and asset weights. The portfolio variance is given by

\sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w},

where \mathbf{w} is the vector of portfolio weights, and \Sigma is the covariance matrix of asset returns. This minimization is subject to \mathbf{w}^T \boldsymbol{\mu} = r_p (expected portfolio return equals target r_p) and \mathbf{w}^T \mathbf{1} = 1 (weights sum to unity), solved via quadratic programming techniques that handle the convex nature of the objective function.^[7]

Model Formulation

Formula and Interpretation

The Capital Asset Pricing Model (CAPM) provides a linear equation for estimating the expected return on an asset based on its systematic risk relative to the market. The core formula is

E(R_i) = R_f + \beta_i [E(R_m) - R_f],

where E(R_i) denotes the expected return on asset i, R_f is the risk-free rate of return, \beta_i is the asset's beta coefficient, E(R_m) is the expected return on the market portfolio, and [E(R_m) - R_f] represents the market risk premium.^[22] This formulation, originally derived by Sharpe, posits that the expected return equals the risk-free rate plus compensation for bearing market risk, scaled by the asset's sensitivity to market movements. The beta coefficient, \beta_i, quantifies an asset's non-diversifiable or systematic risk and is defined as the covariance of the asset's return with the market return divided by the variance of the market return: \beta_i = \frac{\Cov(R_i, R_m)}{\Var(R_m)}.^[22] A beta greater than 1 indicates the asset is more volatile than the market, amplifying expected returns (and risks), while a beta less than 1 suggests lower volatility and returns. The risk-free rate serves as the baseline return for zero-risk investments, often proxied by government bond yields, and the market risk premium captures the excess return investors demand for holding the diversified market portfolio over the risk-free asset.^[1] In economic interpretation, the CAPM establishes a linear relationship between expected returns and beta, implying that only systematic risk—uncorrelated with idiosyncratic risks eliminated through diversification—is priced in equilibrium. Deviations from this line are captured by alpha (\alpha_i) in the time-series regression R_i - R_f = \alpha_i + \beta_i (R_m - R_f) + \epsilon_i, where a non-zero alpha signals mispricing or superior performance relative to the model's prediction; in equilibrium, alpha equals zero for all assets. This focus on beta underscores the model's emphasis on market-wide risk as the sole determinant of required returns. Beta is commonly estimated empirically through historical regression analysis, regressing the asset's excess returns (over the risk-free rate) against the market's excess returns using ordinary least squares to obtain the slope coefficient as beta. Typical estimation periods span 3 to 5 years of monthly data to balance statistical reliability with relevance to current conditions.^[23] The derivation of the CAPM stems from market equilibrium conditions under mean-variance optimization, where investors hold combinations of the risk-free asset and the tangency (market) portfolio, equating supply and demand for all risky assets.^[22] In this setting, asset prices adjust such that no investor can improve utility by altering holdings, resulting in the linear pricing relation where expected returns compensate precisely for covariance with the market portfolio. Independent derivations by Lintner and Mossin reinforced this equilibrium outcome, confirming the model's robustness across similar assumptions.^[11]

Security Market Line

The security market line (SML) provides a graphical representation of the capital asset pricing model (CAPM), illustrating the linear relationship between an asset's expected return and its systematic risk, measured by beta (\beta). In equilibrium, the SML is a straight line with the risk-free rate (R_f) as the y-intercept, corresponding to assets with \beta = 0, and a slope equal to the market risk premium (E(R_m) - R_f), which captures the additional return demanded for bearing market risk. The market portfolio lies on the SML at \beta = 1, where its expected return equals E(R_m). This line depicts how asset prices adjust in equilibrium to ensure that expected returns compensate investors solely for non-diversifiable risk.^[22] Assets or portfolios plotting above the SML exhibit a positive alpha, indicating they offer higher expected returns than justified by their beta and thus appear undervalued relative to the market equilibrium. Conversely, those below the SML have a negative alpha, suggesting lower expected returns and overvaluation. Alpha, derived from the intercept in the time-series regression of excess asset returns on market excess returns, quantifies deviations from the SML and serves as a measure of risk-adjusted performance.^[24] The SML differs from the capital market line (CML), which applies exclusively to efficient portfolios formed by combining the risk-free asset and the market portfolio, plotting expected return against total risk (standard deviation). While the CML rewards total risk in diversified contexts, the SML focuses on individual securities or inefficient portfolios, pricing only beta as the relevant risk measure.^[6]

Practical Applications

Asset Valuation

The Capital Asset Pricing Model (CAPM) facilitates asset valuation by providing a framework to estimate the required rate of return for an asset based on its systematic risk, enabling investors to assess whether the asset's market price reflects its intrinsic value. The process begins with estimating the asset's beta, which measures its sensitivity to market movements, alongside the risk-free rate (typically the yield on government securities) and the market risk premium (the expected excess return of the market over the risk-free rate). These inputs yield the required return, which serves as the discount rate for the asset's expected future cash flows, producing a present value that can be compared to the current market price to determine fair valuation.^[25]^[26] In capital budgeting, CAPM-derived required returns are integral to net present value (NPV) analysis, where firms discount projected cash flows from potential investments using this rate to evaluate project viability. For instance, a firm considering a new project estimates the project's beta (often proxied by the firm's or industry's beta if similar), computes the required return, and calculates NPV as the present value of inflows minus outflows; projects with positive NPV are accepted as they enhance shareholder value. This approach ensures that only investments compensating for systematic risk are pursued, aligning with efficient capital allocation.^[26]^[27] Jensen's alpha provides a practical example of CAPM's role in detecting over- or undervaluation, measuring the difference between an asset's actual return and its CAPM-expected return given its beta. A positive alpha indicates the asset has outperformed expectations relative to its risk, suggesting potential undervaluation and superior performance, while a negative alpha signals underperformance and possible overvaluation. For example, in evaluating a portfolio with a beta of 0.73, a risk-free rate of 4%, and a market return of 6%, if the actual return is 8%, the alpha is 2.54%, implying the portfolio delivered excess returns and may be undervalued.^[28] CAPM also plays a key role in mergers and acquisitions by estimating the target's cost of equity, which informs the discount rate in discounted cash flow valuations to gauge acquisition premiums and synergies. Acquirers input the target's beta, the risk-free rate, and market premium into CAPM to derive the cost of equity, often adjusting for leverage to compute the weighted average cost of capital (WACC), ensuring the purchase price aligns with risk-adjusted value creation. For instance, valuing a target with a beta of 1.3, risk-free rate of 4.5%, and market return of 11.5% yields a cost of equity of 13.6%, used to discount equity cash flows and assess deal economics.^[29]

Required Return Calculation

The required return on an asset or firm using the Capital Asset Pricing Model (CAPM) is computed by integrating the model's core formula, which relates expected return to systematic risk, with specific empirical inputs.^[30] The primary inputs for this calculation are the risk-free rate, the market risk premium, and the asset's beta. The risk-free rate is typically proxied by the yield on short-term U.S. Treasury securities, such as three-month Treasury bills, which represent the return on an investment with negligible default and reinvestment risk.^[30] The market risk premium is the expected excess return of the market portfolio over the risk-free rate, often estimated using historical data; long-term averages for the U.S. equity market range from approximately 5% to 8%, depending on the time period and geometric versus arithmetic averaging methods.^[31] Beta measures the asset's sensitivity to market movements and is estimated through ordinary least squares regression of the asset's historical excess returns against the market's excess returns, typically using a broad index like the S&P 500 over a 5-year monthly period.^[23] Adjusted beta models, such as the Blume adjustment, may be applied to correct for regression-induced biases toward unity.^[23] In firm-specific applications, the CAPM-derived required return serves as the cost of equity, a key component in calculating the weighted average cost of capital (WACC), which discounts firm cash flows to reflect the blended cost of debt and equity financing.^[32] The cost of equity is weighted by the proportion of equity in the capital structure, combined with the after-tax cost of debt weighted by the debt proportion, to yield the WACC.^[32] For levered firms, beta must be adjusted to account for financial leverage, as debt amplifies equity risk. Hamada's equation provides this adjustment, relating the levered beta (\beta_L) to the unlevered beta (\beta_U) as follows:

\beta_L = \beta_U \left[1 + (1 - t) \frac{D}{E}\right]

where t is the corporate tax rate, D is the market value of debt, and E is the market value of equity.^[33] The unlevered beta is often derived from comparable firms or industry averages, then relevered using the target capital structure.^[33] Practical challenges in required return calculation arise from the choice between historical and forward-looking estimates for the market risk premium, as historical data may not capture future economic conditions or investor expectations, potentially leading to over- or underestimation of risk compensation.^[34] Forward-looking premiums, derived from implied models like dividend discount valuations, aim to address this but introduce assumptions about growth and dividends.^[34]

Underlying Assumptions

Core Assumptions

The Capital Asset Pricing Model (CAPM) rests on a set of core assumptions that establish an idealized framework for asset pricing under uncertainty. These assumptions, first systematically outlined in the foundational papers by Sharpe (1964) and Lintner (1965), simplify the complexities of real-world markets to derive the model's key predictions about risk and expected returns.^[22]^[35] They encompass investor behavior, expectations, market structure, and portfolio possibilities, enabling the derivation of a linear relationship between expected returns and systematic risk. A primary assumption is that investors are rational, risk-averse agents who maximize expected utility by focusing on the mean and variance of portfolio returns. In this view, investors select portfolios that offer the highest expected return for a given level of risk, as measured by variance, or the lowest variance for a given expected return; this builds directly on Markowitz's mean-variance optimization framework.^[22]^[35] Risk aversion implies that investors require compensation in the form of higher expected returns to hold riskier assets, and they employ diversification to eliminate unsystematic risk. Another key assumption is that all investors share homogeneous expectations regarding the expected returns, variances, and covariances of asset returns. This unanimity ensures that every investor perceives the same joint probability distribution of asset returns, leading to identical assessments of risk and opportunity across the market.^[22]^[35] Without this, divergent views would prevent the convergence to a single market equilibrium portfolio. The model further assumes perfect capital markets, characterized by the absence of taxes, transaction costs, and other frictions that could distort pricing. Information is freely available to all investors at no cost, and markets are fully competitive with no barriers to entry or short-selling restrictions. Additionally, investors can borrow and lend unlimited amounts at a single risk-free rate, which serves as the baseline for pricing risky assets.^[22]^[35] All investable assets are publicly traded and divisible, allowing investors to hold fractional shares and achieve full diversification across the entire market. The investment horizon is a single period, during which returns are realized, and the market portfolio—comprising all risky assets in proportion to their market value—represents the tangency portfolio of efficient risky assets.^[22] These conditions collectively imply that only systematic risk, measured relative to the market portfolio, is priced in equilibrium.

Assumption Implications

The assumptions underlying the Capital Asset Pricing Model (CAPM) yield several key implications for investor behavior and market dynamics. Under these conditions, including the availability of a risk-free asset, unlimited borrowing and lending at the risk-free rate, and investors' mean-variance optimization, the model predicts that rational investors will construct optimal portfolios in a manner that simplifies asset allocation and ensures market equilibrium.^[36] A primary implication is the mutual fund theorem, which states that all investors, regardless of their individual risk tolerances, will hold portfolios consisting solely of combinations of the risk-free asset and the tangency portfolio—known as the market portfolio. This theorem arises because the efficient frontier, when a risk-free asset is introduced, collapses to a straight line (the capital market line), where the tangency point represents the market portfolio of all risky assets weighted by their market values. As a result, in equilibrium, the market portfolio becomes the sole risky fund held by investors, with positions scaled by lending or borrowing at the risk-free rate to match personal risk preferences.^[36] Closely related is the two-fund separation theorem, which posits that the portfolio selection process separates into two independent steps: first, identifying the optimal risky portfolio (the market portfolio), and second, determining the allocation between this portfolio and the risk-free asset based solely on the investor's risk aversion. This separation holds because all investors agree on the composition of the tangency portfolio under homogeneous expectations and mean-variance preferences, rendering the choice of risky assets independent of individual utility functions. Consequently, portfolio construction becomes highly efficient, as no investor needs to hold a diversified set of individual securities beyond the market portfolio itself. In market equilibrium, these implications ensure the absence of arbitrage opportunities, as asset prices adjust such that the expected return on each security is precisely commensurate with its contribution to the market portfolio's risk. The model designates beta—the covariance of an asset's return with the market return divided by the market return's variance—as the sole measure of systematic risk, implying that only non-diversifiable market risk commands a risk premium. Assets with higher betas offer higher expected returns to compensate for their greater sensitivity to market fluctuations, while idiosyncratic risks are eliminated through diversification and thus unrewarded. This equilibrium condition requires that the market portfolio be mean-variance efficient, with supply equaling demand for each asset based on its market capitalization weights.^[36] Finally, the assumption of homogeneous expectations—that all investors share identical estimates of future returns, variances, and covariances—implies strong market efficiency. With unanimous agreement on asset characteristics and full information availability, security prices fully reflect all relevant information, preventing mispricings or opportunities for superior returns through analysis. This leads to an informationally efficient market where the market portfolio's efficiency ensures optimal resource allocation across the economy.^[36]

Criticisms and Evidence

Theoretical Criticisms

One prominent theoretical criticism of the CAPM is Roll's critique, which argues that the model's market portfolio is unobservable in practice, rendering empirical tests of the security market line inherently invalid and the theory untestable.^[37] Specifically, since the true market portfolio must include all assets—such as real estate, human capital, and foreign securities—proxies like stock indices inevitably introduce errors that confound any assessment of the model's predictions.^[38] The CAPM's assumption of unlimited risk-free borrowing at the risk-free rate is also unrealistic, as real-world borrowing constraints faced by investors lead to deviations from the predicted equilibrium, necessitating modifications like the zero-beta CAPM to account for a positive expected return on zero-beta assets.^[18] This critique highlights how the model's reliance on frictionless borrowing distorts the risk-return relationship in leveraged portfolios. Furthermore, the CAPM is formulated as a single-period, static model, which overlooks the multi-period nature of investment decisions and dynamic economic conditions, such as changing investment opportunities over time.^[39] This limitation implies that the model fails to capture intertemporal hedging demands or state variables that influence asset prices in evolving markets. The model further ignores critical market frictions, including taxes, which differentially affect investor returns and alter optimal portfolios, as well as bankruptcy costs that introduce nonlinearities in firm valuation not accounted for in the linear beta-risk framework. Agency issues, arising from conflicts between managers and shareholders, are similarly excluded, potentially leading to suboptimal diversification and risk-taking that the CAPM does not address. Finally, the CAPM's market portfolio excludes non-tradable assets like human capital, which constitutes a significant portion of investors' wealth and introduces unhedgeable risks that the model cannot properly price. This omission biases the beta estimates and undermines the completeness of the efficient frontier.^[37]

Empirical Tests and Anomalies

Early empirical tests of the Capital Asset Pricing Model (CAPM) provided initial support for its predictions. In a seminal study, Black, Jensen, and Scholes (1972) conducted time-series regressions on monthly returns of portfolios formed from New York Stock Exchange (NYSE) stocks between 1931 and 1965, finding that average returns aligned closely with estimated betas, consistent with the model's risk-return relation.^[40] Their analysis also revealed a flatter security market line than predicted, suggesting some deviations but overall validation of beta as a key pricing factor.^[40] Subsequent research highlighted significant empirical shortcomings of the CAPM. Fama and French (1992) examined cross-sectional regressions of average stock returns on NYSE, AMEX, and NASDAQ stocks from 1963 to 1990, demonstrating that beta alone explained little of the variation in returns, while firm size (market equity) and book-to-market equity factors captured much of the cross-sectional differences.^[41] Building on this, Fama and French (2004) reviewed time-series and cross-sectional tests across various periods and markets, confirming that the CAPM's beta premium is weak or insignificant, with size and value factors outperforming beta in explaining returns.^[15] The low-volatility anomaly further challenges the CAPM, as the model implies that only systematic risk (beta) should be priced, while idiosyncratic risk is diversifiable and unrewarded. Ang, Hodrick, Xing, and Zhang (2006) analyzed idiosyncratic volatility in U.S. stocks from 1963 to 2000 using Fama-MacBeth regressions, finding that high idiosyncratic volatility stocks underperform low idiosyncratic volatility ones by approximately 1.17% per month on a risk-adjusted basis.^[42] Post-1990s anomalies extended these critiques. Jegadeesh and Titman (1993) documented the momentum effect through strategies buying past winners and selling past losers in U.S. stocks from 1965 to 1989, generating average monthly returns of 1% that were not explained by CAPM betas.^[43] Similarly, profitability emerged as a predictor; Novy-Marx (2013) showed via portfolio sorts on U.S. stocks from 1963 to 2010 that gross profitability (gross profits over assets) forecasted returns as strongly as book-to-market, with high-profitability firms earning 0.31% excess monthly returns unexplained by beta.^[44] Fama and French (2015) incorporated profitability into their five-factor model, confirming through regressions on U.S. data from 1963 to 2013 that it captures return patterns beyond the CAPM.^[45] Recent evidence from post-2020 studies reinforces the CAPM's limited applicability. In emerging markets, Sun, Tang, and Fang (2022) tested the model using panel regressions on stock returns from 24 emerging economies between 1991 and 2019, finding that beta coefficients were often insignificant and explained less than 10% of return variation, indicating poor fit compared to multi-factor alternatives.^[46] For cryptocurrencies, Liu, Tsyvinski, and Wu (2022) applied CAPM regressions to daily returns of major coins like Bitcoin from 2011 to 2020, estimating betas around 3 but noting their instability and failure to fully price returns.^[47] More recent analyses, such as those examining U.S. stocks during the 2022-2024 high inflation period, continue to show beta's insignificance in explaining returns amid macroeconomic shocks, with R-squared values below 5% in some regressions.^[48]

Extensions and Alternatives

Model Modifications

One prominent modification to the CAPM addresses the assumption of unlimited borrowing at the risk-free rate, which may not hold in practice due to regulatory or market constraints. The zero-beta CAPM, developed by Fischer Black, replaces the risk-free rate with the expected return on a zero-beta portfolio—a portfolio uncorrelated with the market that achieves the minimum variance among all zero-beta assets. This adjustment allows the model to accommodate restricted borrowing while maintaining the security market line, where the intercept is the zero-beta return rather than the risk-free rate, thereby relaxing the perfect capital markets assumption without altering the market portfolio's efficiency.^[18] Another extension, the intertemporal CAPM proposed by Robert Merton, incorporates dynamic investment opportunities over multiple periods, recognizing that investors hedge against changes in future states of the world. Unlike the static single-period CAPM, this version posits that asset returns depend on multiple betas corresponding to different state variables that influence the investment opportunity set, such as interest rates or inflation. By allowing for time-varying risk premia and hedging demands, the intertemporal CAPM relaxes the assumption of constant investment opportunities and provides a framework for multi-factor pricing within the CAPM tradition.^[49] The consumption-based CAPM, introduced by Douglas Breeden, derives from intertemporal utility maximization and links expected asset returns directly to their covariance with aggregate consumption growth, addressing the CAPM's separation of investment and consumption decisions. In this model, the risk premium arises from assets' ability to smooth consumption fluctuations, with the beta measured against consumption rather than the market portfolio, thereby relaxing the mean-variance efficiency assumption by grounding pricing in observable consumption data. This adaptation emphasizes that only consumption risk is systematically priced, as investors care about wealth effects on lifetime utility.^[50] Conditional versions of the CAPM further modify the model by allowing betas and risk premia to vary over time, capturing predictability in returns through conditioning information such as macroeconomic variables or past returns. For instance, using dividend yields or default spreads as instruments, the conditional beta reflects time-dependent systematic risk exposure. Alternatively, GARCH models estimate time-varying covariances between asset returns and the market, enabling dynamic betas that respond to volatility clustering and economic cycles, thus relaxing the CAPM's assumption of constant risk parameters.^[51]^[52] Downside beta models refine the CAPM by focusing exclusively on negative market returns, arguing that investors perceive risk asymmetrically and penalize downside covariance more than symmetric variance. Originating from semivariance-based frameworks, these models replace the standard beta with a downside beta, calculated as the covariance of asset returns with the market conditional on market downturns below a target rate, such as the risk-free rate. This modification addresses the CAPM's reliance on total variance by isolating systematic downside risk, which empirical studies show commands a higher premium, particularly in volatile markets.^[53] Recent extensions include a three-period CAPM incorporating irrational investor behavior, allowing for modeling of behavioral biases over multiple periods (2024), and a functional CAPM tailored for high-frequency financial data, enabling intraday return predictions (2025).^[54]^[55]

Competing Frameworks

The Arbitrage Pricing Theory (APT), proposed by Stephen A. Ross in 1976, serves as a foundational multifactor alternative to the CAPM. Unlike the CAPM, which relies on a single market beta, APT posits that asset returns are determined by multiple systematic risk factors, with expected returns linearly related to the asset's sensitivities (factor loadings) to these factors. This framework does not require the identification of a specific market portfolio, allowing for greater flexibility in factor selection based on empirical evidence of arbitrage opportunities.^[56] Building on multifactor approaches, the Fama-French three-factor model, introduced by Eugene F. Fama and Kenneth R. French in 1993, extends the CAPM by incorporating two additional factors: size (measured by small minus big, SMB) and value (high minus low book-to-market, HML). These factors capture empirical patterns in stock returns, such as the tendency for small-cap stocks to outperform large-cap ones and value stocks to outperform growth stocks, providing a better explanation of cross-sectional return variations than the market factor alone. The model has been widely adopted in academic and practitioner settings for its improved explanatory power over the CAPM.^[57] The Carhart four-factor model, developed by Mark M. Carhart in 1997, further augments the Fama-French framework by adding a momentum factor (winner minus loser, WML), which accounts for the persistence in stock returns where past winners continue to outperform past losers. This extension enhances the model's ability to evaluate mutual fund performance and explain short-term return continuations, making it a standard tool in performance attribution analysis.^[58] More recent developments include the q-factor model by Kewei Hou, Chen Xue, and Lu Zhang in 2015, which combines the market factor with size, investment (conservative minus aggressive, CMA), and profitability (robust minus weak, RMW) factors derived from investment-based asset pricing theory. This model subsumes many anomalies previously unexplained by earlier multifactor specifications, offering superior cross-sectional pricing performance. Advancements in machine learning-based asset pricing, as explored by Shihao Gu, Bryan Kelly, and Dacheng Xiu in 2020, leverage algorithms like neural networks and random forests to identify nonlinear interactions among predictors, achieving higher out-of-sample predictability than linear factor models by flexibly capturing complex return patterns without predefined factors. Recent multifactor models, such as a human capital-based four-factor extension integrating labor market risks (2023), continue to refine pricing by incorporating macroeconomic human capital factors.^[59]^[60]^[61] These competing frameworks have largely supplanted or complemented the CAPM in practice due to their superior empirical fit in addressing well-documented anomalies, such as size, value, and momentum effects, which the single-factor CAPM fails to explain adequately.^[57]

References

[1]
The Capital Asset Pricing Model - American Economic Association
The Capital Asset Pricing Model (CAPM) revolutionized modern finance. Developed in the early 1960s by William Sharpe, Jack Treynor, John Lintner and Jan Mossin.
[2]
[PDF] Capital Asset Prices: A Theory of Market Equilibrium under ... - Finance
Jun 17, 2006 · SEPTEMBER 1964. No. 3. CAPITAL ASSET PRICES: A THEORY OF MARKET. EQUILIBRIUM UNDER CONDITIONS OF RISK*. WILLIAM F. SHARPE†. I. INTRODUCTION. ONE ...
[3]
[PDF] Security Prices, Risk, and Maximal Gains From Diversification
Jun 18, 2006 · Security Prices, Risk, and Maximal Gains From Diversification. John Lintner. The Journal of Finance, Vol. 20, No. 4. (Dec., 1965), pp. 587-615 ...
[4]
None
### Summary of Key Elements of CAPM
[5]
Portfolio Selection - jstor
THE PROCESS OF SELECTING a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with.
[6]
[PDF] THE TREYNOR CAPITAL ASSET PRICING MODEL - Finance
Treynor, J. L. (1962). “Toward a Theory of Market Value of Risky Assets.” Unpublished manuscript. “Rough Draft” dated by Mr. Treynor to the fall of 1962. A ...
[7]
Herding behaviour in energy stock markets during the Global ...
Sep 29, 2020 · Treynor Jack L. 1962. Toward a theory of market value of risky assets. Unpublished manuscript. Subsequently, published as [Chapter 2] of ...
[8]
Capital Asset Prices: A Theory of Market Equilibrium under ... - jstor
WILLIAM F. SHARPEt. I ... The major elements of the theory first appeared in his article "Portfolio Selection," The Journal of Finance, XII (March 1952),.
[9]
risky investments in stock - portfolios and capital budgets - jstor
VALUATION OF RISK ASSETS 37. (6a') d U/l w = U1 [(r - r**) - wg' (w)] +U2 20 ... [I 2] LINTNER, JOHN, "Optimal Dividends and Corpo- rate Growth Under ...
[10]
Equilibrium in a Capital Asset Market - jstor
This paper investigates the properties of a market for risky assets on the basis of a simple model of general equilibrium of exchange, where individual ...
[11]
The Prize in Economics 1990 - Press release - NobelPrize.org
Harry Markowitz is awarded the Prize for having developed the theory of portfolio choice; · William Sharpe, for his contributions to the theory of price ...
[12]
[PDF] Lintner (1965 - Finance
Jun 18, 2006 · The essential purpose of the present paper is to push back the frontiers of our knowledge of the logical structure of these related issues, ...
[13]
The Capital Asset Pricing Model: Theory and Evidence
The capital asset pricing model (CAPM) of William Sharpe (1964) and John Lintner (1965) marks the birth of asset pricing theory.
[14]
[PDF] Capital Market Equilibrium with Restricted Borrowing - Fischer Black
In the postwar period, the estimates obtained by Black, Jensen, and Scholes for the mean of R, were significantly greater than zero. One possible explanation ...Missing: variant | Show results with:variant
[15]
[PDF] Joe Chartoff, * George W . Mayo, Jr. t and Walter A. Smith, Jr. $
In the mid-1960's, the academic community introduced a mathematical formula called the Capital Asset Pricing Model ("CAPM") which, it was con-.
[16]
[PDF] Portfolio Selection Harry Markowitz The Journal of Finance, Vol. 7 ...
Sep 3, 2007 · One type of rule concerning choice of portfolio is that the investor does (or should) maximize the discounted (or capitalized) value of future ...Missing: tradeoff | Show results with:tradeoff
[17]
Modern portfolio theory, 1950 to date - ScienceDirect
Markowitz formulated the portfolio problem as a choice of the mean and variance of a portfolio of assets. He proved the fundamental theorem of mean variance ...
[18]
CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM ...
CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* ; First published: September 1964 ; Citations · 3,840 ; A great many people provided ...Introduction · II. Optimal Investment Policy... · III. Equilibrium in the Capital...
[19]
[PDF] Estimating Beta - NYU Stern
Decide on an estimation period. □ Services use periods ranging from 2 to 5 years for the regression. □ Longer estimation period provides more data, but firms ...
[20]
THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945 ...
The model is illustrated in Section III by an application of it to the evaluation of the performance of 115 open end mutual funds in the period 1945–1964.Introduction · II. The Model · III. The Data and Empirical... · IV. Conclusion
[21]
[PDF] Retirement Income Analysis William F. Sharpe 8. Valuation
and MBA level finance courses, is the Capital Asset Pricing Model described in Chapter 7. ... asset valuation under uncertainty than does the CAPM . That ...
[22]
[PDF] Handout 10: Applying the CAPM to Capital Budgeting
To maximize firm value, the firm should accept the project if NPV> 0, otherwise the firm should reject the project. Result 1 says that the discount rate should ...
[23]
[PDF] The Capital Asset Pricing Model (CAPM) - NYU Stern
The Capital Asset Pricing Model (CAPM) is based on equilibrium analysis, specifying expected returns for capital budgeting, valuation, and regulation.
[24]
[PDF] CHAPTER 4 - NYU Stern
Stock did worse than expected during regression period. The difference between a and Rf (1-β) is called Jensen's alpha, and provides a measure of whether the ...
[25]
[PDF] Mergers & Acquisitions - Restructuring Ownership - NYU Stern
Cost of Equity (from CAPM). 12.50%. Cost of Debt. 5.53%. WACC. 11.38%. Firm Value ... The deadweight costs of making the acquisition (investment banks' fees, etc).
[26]
1. The correct risk free rate to use in the capital asset pricing model
The beta to be used in the CAPM is generated by regressing returns on a stock against returns on a market index. Getting Expected Returns: Disney (9/95).
[27]
[PDF] Estimating Equity Risk Premiums - NYU Stern
We estimate the risk premium by looking at the historical premium earned by stocks over default-free securities over long time periods. These approaches might ...Missing: credible source
[28]
Cost of Equity and Capital (US) - NYU Stern
Cost of Equity and Capital (US) ; Auto & Truck, 34, 1.62 ; Auto Parts, 33, 1.23 ; Bank (Money Center), 15, 0.88 ; Banks (Regional), 591, 0.52 ...
[29]
THE EFFECT OF THE FIRM'S CAPITAL STRUCTURE ON THE ...
There are at least four general procedures that can be used to estimate the effect of the firm's capital structure on the systematic risk of common stocks.Introduction · II. Some Possible Procedures... · IV. Tests of the MM vs...Missing: PDF | Show results with:PDF
[30]
Equity Risk Premiums - NYU Stern
This paper looks at the estimation of an appropriate risk premium to use in risk and return models, in general, and in the capital asset pricing model, in ...
[31]
SECURITY PRICES, RISK, AND MAXIMAL GAINS FROM ...
This paper set forth the simple logic which leads directly to the determination of explicit equilibrium prices of risk assets traded in competitive markets ...II. Investment Choice of an... · IV. The Effects of Correlations...Missing: formula | Show results with:formula<|control11|><|separator|>
[32]
https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/wacc.html
[33]
A critique of the asset pricing theory's tests Part I - ScienceDirect.com
March 1977, Pages 129-176. Journal of Financial Economics. A critique of the asset pricing theory's tests Part I: On past and potential testability of the ...
[34]
[PDF] A Critique of the Asset Pricing Theory's Tests Part I
Roll, Critique of asset pricing theory tests – I. 153 portfolio that were provided in the source papers. The crucial assumption is that the market proxy and its ...
[35]
An Intertemporal Capital Asset Pricing Model - jstor
An intertemporal model for the capital market is deduced from the portfolio selection behavior by an arbitrary number of investors who aot so as to maximize ...
[36]
The Capital Asset Pricing Model: Some Empirical Tests
Jun 13, 2006 · Black , Michael C Fischer , Jensen. Incomplete Measurement of Market Returns and Its Implications for Tests of the Asset Pricing Model ; Fischer ...
[37]
The Cross‐Section of Expected Stock Returns - Wiley Online Library
Two easily measured variables, size and book-to-market equity, combine to capture the cross-sectional variation in average stock returns.Preliminaries · III. Book-to-Market Equity, , and... · Conclusions and Implications
[38]
The Cross-Section of Volatility and Expected Returns
Abstract. We examine how volatility risk, both at the aggregate market and individual stock level, is priced in the cross-section of expected stock returns.
[39]
[PDF] jegadeesh-titman93.pdf - Bauer College of Business
This paper documents that strategies which buy stocks that have performed well in the past and sell stocks that have performed poorly in the past generate ...
[40]
The other side of value: The gross profitability premium - ScienceDirect
Novy-Marx (2013) identifies a proxy for expected profitability that is ... Profitability anomaly and aggregate volatility risk. Journal of Financial ...
[41]
A five-factor asset pricing model - ScienceDirect.com
We test the performance of the five-factor model in two steps. Here we apply the model to portfolios formed on size, B/M, profitability, and investment. As in ...
[42]
An Analysis and Comparison of Multi-Factor Asset Pricing Model ...
This study empirically analyzes and compares return data from developed and emerging market data based on the Fama French five-factor model.Missing: post- | Show results with:post-
[43]
Cryptocurrency returns under empirical asset pricing - ScienceDirect
Using both a CAPM and an intertemporal capital asset pricing framework (ICAPM), we show that both the CAPM's and ICAPM's betas are sizable and significant.
[44]
An Intertemporal Capital Asset Pricing Model
Sep 1, 1973 · An intertemporal model for the capital market is deduced from the portfolio selection behavior by an arbitrary number of investors who act ...
[45]
An intertemporal asset pricing model with stochastic consumption ...
This paper derives a single-beta asset pricing model in a multi-good, continuous-time model with uncertain consumption-goods prices and uncertain investment ...
[46]
[PDF] The Conditional CAPM and the Cross-Section of Expected Returns ...
Apr 2, 2007 · We assume that the CAPM holds in a conditional sense, i.e., betas and the market risk premium vary over time. We include the return on human ...
[47]
[PDF] A Capital Asset Pricing Model with Time-varying Covariances
In this paper the conditional covariance matrix of a set of asset returns is allowed to vary over time following the generalized auto- regressive conditional ...
[48]
Toward the Development of an Equilibrium Capital-Market Model ...
Oct 19, 2009 · The purpose of the present paper is to review and extend some of the implications of an alternative two-parameter portfolio selection model, ...
[49]
The arbitrage theory of capital asset pricing - ScienceDirect.com
December 1976, Pages 341-360. Journal of Economic Theory. The arbitrage theory of capital asset pricing. Author links open overlay panel. Stephen A Ross ∗.
[50]
Common risk factors in the returns on stocks and bonds
This paper identifies five common risk factors in the returns on stocks and bonds. There are three stock-market factors: an overall market factor and factors ...
[51]
On Persistence in Mutual Fund Performance - Carhart - 1997
Apr 18, 2012 · The 4-factor model is consistent with a model of market equilibrium with four risk factors. Alternately, it may be interpreted as a performance ...II. Models of Performance... · IV. Interpreting the... · Longer-Term Persistence in...
[52]
Digesting Anomalies: An Investment Approach - Oxford Academic
Sep 26, 2014 · Abstract. An empirical q-factor model consisting of the market factor, a size factor, an investment factor, and a profitability factor ...Abstract · Conceptual Framework · Factors · Empirical Results
[53]
Empirical Asset Pricing via Machine Learning - Oxford Academic
Abstract. We perform a comparative analysis of machine learning methods for the canonical problem of empirical asset pricing: measuring asset risk premiums.