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Security market line

The Security Market Line (SML) is a graphical representation of the Capital Asset Pricing Model (CAPM), depicting the linear relationship between the expected return of a security or portfolio and its systematic risk, as measured by beta.^[1]^[2] Developed in the 1960s by William Sharpe, John Lintner, and Jan Mossin in the context of modern portfolio theory, the SML originates from the equilibrium conditions where investors hold diversified portfolios and price assets based on non-diversifiable risk.^[1]^[3] It serves as a benchmark for evaluating whether securities offer appropriate compensation for their risk exposure in efficient markets. The SML is defined by the CAPM equation:
E(R_i) = R_f + \beta_i [E(R_m) - R_f]
where E(R_i) represents the expected return on security i, R_f is the risk-free rate, \beta_i is the security's beta (the covariance of its returns with the market divided by the market's variance), and [E(R_m) - R_f] is the market risk premium.^[2] In this framework, the line's intercept is the risk-free rate, and its slope reflects the additional return demanded for bearing market risk.^[4] Securities plotting above the SML are considered undervalued (offering higher returns for their beta), while those below are overvalued, guiding investment decisions toward mispriced assets.^[4] The model assumes beta captures all relevant systematic risk, ignoring idiosyncratic risk that can be diversified away.^[2] In practice, the SML finds applications in corporate finance and investment management, such as estimating a firm's weighted average cost of capital (WACC) for capital budgeting, valuing equity securities, and assessing portfolio performance relative to benchmarks.^[5] For instance, analysts use it to determine required returns on projects or stocks by inputting estimated betas and prevailing market premiums.^[5] However, the SML's validity depends on CAPM assumptions, including rational and risk-averse investors, homogeneous expectations about asset returns, perfect capital markets without taxes or transaction costs, and the ability to borrow and lend at the risk-free rate.^[6] Empirical tests have shown deviations, such as a flatter SML than predicted, prompting extensions like multifactor models.

Theoretical Foundations

Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is a theoretical framework that establishes a linear relationship between the expected return on an asset and its systematic risk relative to the overall market portfolio. It posits that investors are compensated only for bearing non-diversifiable market risk, as idiosyncratic risks can be eliminated through diversification. Developed independently in the mid-1960s, the model builds on Harry Markowitz's mean-variance portfolio theory by incorporating equilibrium pricing in competitive capital markets.^[7] The CAPM was first formalized by William F. Sharpe in 1964, followed by John Lintner in 1965 and Jan Mossin in 1966.^[8]^[9] Sharpe's seminal work outlined the conditions under which asset prices adjust to ensure market equilibrium, emphasizing the role of risk aversion in portfolio selection.^[8] Lintner extended this by integrating corporate finance decisions, such as capital budgeting, into the pricing framework.^[9] Mossin provided a general equilibrium analysis, demonstrating how investor preferences lead to a unique efficient frontier in asset markets. Central to the CAPM are several key assumptions about investor behavior and market structure. Investors are assumed to be rational and risk-averse mean-variance optimizers, seeking to maximize utility based on expected returns and variances over a single-period horizon. Markets are perfect, implying no taxes or transaction costs, unlimited borrowing and lending at a risk-free rate, homogeneous expectations among investors, and free access to all information.^[7] Under these conditions, all investors hold a combination of the risk-free asset and the tangency portfolio, which is the market portfolio of all risky assets.^[8] In equilibrium, the CAPM implies that asset prices adjust such that the supply of securities equals demand, resulting in a linear tradeoff between expected return and systematic risk for all assets. This equilibrium ensures no arbitrage opportunities exist, as any mispriced asset would be exploited until prices realign. The model's core equation captures this relationship:

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 risk, and E(R_m) is the expected market return.^[7] This formulation serves as the theoretical foundation for the security market line, with \beta_i quantifying the contribution of systematic risk.^[9]

Systematic Risk and Beta

In finance, risk is broadly categorized into systematic risk and unsystematic risk. Systematic risk, also known as market risk or non-diversifiable risk, arises from factors that affect the entire market or economy, such as inflation, interest rate changes, recessions, or geopolitical events, and cannot be eliminated through diversification.^[10] In contrast, unsystematic risk, or specific risk, is unique to individual assets or companies, stemming from factors like management decisions, product failures, or industry-specific issues, and can be mitigated by holding a diversified portfolio.^[11] The Capital Asset Pricing Model (CAPM) posits that only systematic risk is compensated with higher expected returns, as investors can diversify away unsystematic risk in a well-balanced portfolio. Beta (β) serves as the primary measure of an asset's systematic risk within the CAPM framework, quantifying the sensitivity of an asset's returns to market returns. Formally, beta for asset i is defined as:

\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}

where R_i is the return on the asset, R_m is the return on the market portfolio, Cov denotes covariance, and Var denotes variance.^[12] This measure captures the asset's contribution to the overall portfolio's non-diversifiable risk, as introduced in the foundational CAPM theory.^[13] The value of beta provides insight into an asset's volatility relative to the market. A beta greater than 1 indicates that the asset is more volatile than the market, amplifying both upward and downward movements; for example, technology stocks often exhibit betas above 1 due to their sensitivity to economic cycles. A beta of 1 signifies that the asset's returns move in line with the market, as seen in broad market indices like the S&P 500. A beta less than 1, such as those common in utility stocks, implies lower volatility and reduced sensitivity to market fluctuations. Negative betas, though rare, suggest an inverse relationship with the market, where the asset tends to perform well when the market declines, exemplified by assets like gold or certain hedging instruments.^[11]^[10] Beta is typically calculated using historical regression analysis, where the asset's excess returns are regressed against the market's excess returns over a period, often 3 to 5 years of monthly data, to estimate the slope coefficient, which is beta.^[14] For instance, consider hypothetical monthly returns for a stock and the market over five periods: stock returns of 2%, -1%, 3%, 1%, -2% and market returns of 1%, -0.5%, 2%, 0.5%, -1%. Running a linear regression yields a beta of approximately 1.7, indicating the stock is 70% more volatile than the market. This method relies on past data to infer future systematic risk exposure, though adjustments like the Vasicek Bayesian approach may be applied to account for estimation error and mean reversion toward 1.^[15] In the CAPM, beta's central role underscores that expected returns are determined solely by systematic risk, as unsystematic risk is diversified away in equilibrium, ensuring that investors are rewarded only for bearing undiversifiable market exposure.^[12] This principle, derived from the model's assumptions of rational investors and efficient markets, positions beta as essential for assessing an asset's placement relative to the security market line.

Definition and Formula

Mathematical Expression

The security market line (SML) is expressed mathematically as

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

where E(R_i) denotes the expected return on security i, R_f is the risk-free rate of return, \beta_i is the beta coefficient of security i, and E(R_m) is the expected return on the market portfolio. This equation, central to the Capital Asset Pricing Model (CAPM), posits that the expected return on any security is linearly related to its systematic risk as measured by beta.^[12]^[16] In the formula, R_f acts as the y-intercept, representing the baseline return available from a theoretically risk-free investment, such as the yield on short-term U.S. Treasury bills. The market risk premium, [E(R_m) - R_f], forms the slope of the line, quantifying the compensation per unit of market risk borne by investors. Beta \beta_i, which briefly references the covariance of the security's returns with market returns divided by the market's variance, captures the security's sensitivity to market movements.^[12]^[16]^[17] The notation aligns directly with CAPM conventions, where returns may be expressed as decimals (e.g., 0.03 for 3%) or percentages, ensuring computational consistency regardless of the unit chosen. For illustration, consider a security with \beta_i = 1.2, a risk-free rate R_f = 3\%, and a market risk premium [E(R_m) - R_f] = 5\%; the expected return is then E(R_i) = 3\% + 1.2 \times 5\% = 9\%. These parameter values reflect typical historical estimates used in CAPM applications.^[17]

Derivation from CAPM

The Capital Asset Pricing Model (CAPM) derives the security market line (SML) through its core principle known as the separation theorem, which posits that under equilibrium conditions, all investors will hold portfolios consisting solely of combinations of the risk-free asset and the tangency portfolio—identified as the market portfolio—regardless of their individual risk preferences.^[12] This theorem emerges from mean-variance optimization, where investors seek to maximize expected utility by balancing expected returns against portfolio variance, assuming homogeneous expectations about asset returns and the ability to borrow or lend at a single risk-free rate.^[12] The derivation begins with the expected return of the market portfolio itself. For the market portfolio M, its beta \beta_M equals 1 by definition, as it represents the benchmark against which all assets are measured. Thus, the CAPM equation applied to the market yields E(R_M) = R_f + \beta_M [E(R_M) - R_f], simplifying to E(R_M) = R_f + [E(R_M) - R_f], which holds tautologically and confirms the market's position on the line.^[12] For any individual asset i, its expected return must align with the portfolio's overall efficiency in equilibrium. The asset's contribution to the portfolio's risk is captured through its covariance with the market return, \Cov(R_i, R_M), rather than its total variance, because diversification eliminates idiosyncratic risk.^[12] Portfolio theory underpins this by decomposing an asset's total variance into systematic and idiosyncratic components: \Var(R_i) = \beta_i^2 \Var(R_M) + \Var(\epsilon_i), where \beta_i = \Cov(R_i, R_M) / \Var(R_M) measures systematic risk, and \epsilon_i is the unsystematic portion uncorrelated with the market.^[12] In market equilibrium, only systematic risk is priced, as investors hold the diversified market portfolio and demand compensation proportional to non-diversifiable risk. This leads to the condition that the marginal contribution of asset i to portfolio utility—via its covariance—must equal that of other assets, implying a linear pricing relationship.^[12] The mathematical proof outline proceeds as follows. Consider a portfolio P with weights w_j summing to 1, where the expected return is E(R_P) = \sum w_j E(R_j) and variance is \Var(R_P) = \sum \sum w_i w_j \Cov(R_i, R_j). For the efficient market portfolio, the first-order condition for optimization requires that the partial derivative of utility with respect to each weight equals zero, leading to E(R_i) - R_f = \lambda \Cov(R_i, R_M), where \lambda is the market price of risk.^[12] Normalizing by the market's own covariance gives \lambda = [E(R_M) - R_f] / \Var(R_M), yielding the SML formula:

E(R_i) = R_f + \left[ \frac{\Cov(R_i, R_M)}{\Var(R_M)} \right] [E(R_M) - R_f]

This equation demonstrates the linear risk-return tradeoff central to the SML, with beta as the slope coefficient.^[12]

Interpretation and Visualization

Risk-Return Relationship

The Security Market Line (SML) embodies the core principle of the Capital Asset Pricing Model (CAPM) by delineating the fair expected return that investors should demand for bearing a given level of systematic risk, measured by beta (β). In equilibrium, asset prices adjust such that only non-diversifiable market risk is compensated, ensuring that expected returns align precisely with this risk exposure; securities plotting above the SML offer higher returns than warranted by their beta and are thus undervalued, while those below provide insufficient returns and are overvalued. This framework posits that investors achieve optimal diversification by holding the market portfolio, with the SML serving as the benchmark for equilibrium pricing under rational expectations.^[1] The SML illustrates a linear tradeoff between systematic risk and expected return, where the line's intercept represents the risk-free rate (R_f) and its slope captures the market risk premium (E[R_m] - R_f), quantifying the additional compensation per unit of beta. A steeper slope indicates a higher risk premium, often reflecting heightened market uncertainty or growth prospects, which elevates the required returns across all beta levels. For instance, low-beta assets, such as utility stocks with β ≈ 0.5, earn returns close to R_f since they contribute minimally to portfolio volatility, whereas high-beta assets, like technology firms with β ≈ 1.5, demand a substantial premium to justify their amplified market sensitivity. Investors leverage this relationship to evaluate whether an asset's projected return adequately recompenses its beta-driven risk, guiding allocation decisions toward mispriced opportunities.^[1] The position of the SML is influenced by macroeconomic factors, particularly shifts in the risk-free rate due to central bank policies or interest rate fluctuations, which cause parallel upward or downward movements in the line. Similarly, variations in the market risk premium—driven by expectations of economic growth, such as projected GDP expansions—alter the slope, as stronger growth anticipates higher market returns and thus steeper compensation for risk. These dynamics underscore the SML's sensitivity to broader economic conditions, enabling investors to anticipate adjustments in required returns for beta exposure during periods of monetary tightening or optimistic forecasts.^[18]

Graphical Representation

The security market line (SML) is graphically represented as a straight line in a two-dimensional plot, with the y-axis denoting the expected return of an asset E(R_i) and the x-axis representing the asset's beta \beta_i, a measure of its systematic risk relative to the market. The line originates at the point (0, R_f), where R_f is the risk-free rate, and exhibits a slope equal to the market risk premium [E(R_m) - R_f], where E(R_m) is the expected return on the market portfolio. This setup visually illustrates the linear relationship posited by the Capital Asset Pricing Model (CAPM), linking higher systematic risk to proportionally higher expected returns. Key features of the SML graph include its y-intercept at the risk-free rate R_f, reflecting the return available to investors without exposure to market risk, and its passage through the market portfolio point (\beta = 1, E(R_m)), as the market by definition has a beta of unity. These elements make the graph a fundamental tool for visualizing the equilibrium pricing of assets in efficient markets. Individual securities or portfolios are plotted as points on this graph based on their estimated beta and expected return; assets lying on the line are considered fairly priced, while those above the line indicate undervaluation (offering higher returns for the given risk) and those below suggest overvaluation. Deviations from the SML thus highlight potential mispricings, guiding investment decisions toward assets that plot above the line for superior risk-adjusted performance. The SML's position and orientation can shift dynamically in response to changes in market conditions: an increase in the risk-free rate R_f causes a parallel upward shift of the entire line, raising required returns across all beta levels without altering the slope, while variations in the market risk premium—such as during periods of heightened economic uncertainty—adjust the slope, steepening it for greater compensation per unit of risk or flattening it otherwise. These shifts underscore the graph's utility in assessing how macroeconomic factors influence asset pricing. In a hypothetical diagram, the SML might depict the risk-free rate at 3% on the y-axis intersecting the origin, rising with a slope determined by a 5% market premium to reach the market return of 8% at \beta = 1; asset A, with \beta = 1.2 and an expected return of 10%, would plot above the line (indicating undervaluation), whereas asset B, at \beta = 0.8 and 5% return, would lie below (suggesting overvaluation), providing a clear visual cue for portfolio adjustments.

Applications in Portfolio Management

Treynor Ratio

The Treynor ratio, introduced by Jack Treynor in 1965, serves as a performance metric within the Capital Asset Pricing Model (CAPM) framework to assess a portfolio's excess return relative to its systematic risk.^[19] It quantifies the reward obtained for bearing market risk, measured by beta, making it a key tool for evaluating how well a portfolio aligns with the Security Market Line (SML).^[19] The formula for the Treynor ratio is given by:

T_p = \frac{E(R_p) - R_f}{\beta_p}

where E(R_p) is the expected return of the portfolio, R_f is the risk-free rate, and \beta_p is the portfolio's beta, representing its sensitivity to market movements.^[19] A higher Treynor ratio signifies superior risk-adjusted performance, as it indicates greater excess return per unit of systematic risk assumed.^[20] The benchmark for comparison is the market portfolio's Treynor ratio, which equals the market risk premium [E(R_m) - R_f]/1, since the market beta is 1.^[19] Portfolios plotting on the SML exhibit a Treynor ratio identical to this market premium, reflecting fair compensation for systematic risk under CAPM assumptions.^[21] Those above the SML demonstrate outperformance, with Treynor ratios exceeding the market premium, while those below underperform.^[21] For illustration, consider a portfolio with an expected return of 12%, a beta of 1.1, and a risk-free rate of 4%; its Treynor ratio is (12\% - 4\%)/1.1 = 7.27\%. If the market risk premium is 8%, this portfolio underperforms the SML benchmark.^[20] In practice, the Treynor ratio is particularly applicable to well-diversified portfolios, where unsystematic risk is minimized and beta effectively captures the relevant exposure to market fluctuations.^[19]^[20]

Jensen's Alpha

Jensen's alpha, denoted as \alpha, measures the excess return of a portfolio or security relative to the return predicted by the Security Market Line (SML) based on its systematic risk. It quantifies the difference between the actual expected return and the SML-expected return, serving as an indicator of the portfolio manager's ability to generate returns above or below what would be anticipated given the asset's beta. This metric was developed by Michael C. Jensen in his 1968 study on mutual fund performance, where it was introduced as a risk-adjusted performance measure to assess whether fund managers could consistently outperform the market after accounting for risk.^[22] The formula for Jensen's alpha is given by:

\alpha_p = E(R_p) - \left[ R_f + \beta_p \left( E(R_m) - R_f \right) \right]

where E(R_p) is the expected return of the portfolio, R_f is the risk-free rate, \beta_p is the portfolio's beta, and E(R_m) - R_f is the expected market risk premium. This expression directly subtracts the SML-predicted return from the portfolio's actual or expected return, capturing any vertical deviation from the SML.^[22] A positive \alpha suggests that the portfolio has outperformed the SML benchmark, potentially indicating superior stock selection or market timing by the manager, though it may also arise from luck or unaccounted risks; a negative \alpha indicates underperformance relative to the expected risk-return tradeoff; and an \alpha of zero implies the portfolio lies exactly on the SML, delivering returns commensurate with its systematic risk. In practice, alpha is often estimated using a time-series regression model:

R_{p,t} - R_{f,t} = \alpha_p + \beta_p (R_{m,t} - R_{f,t}) + \epsilon_t

where the intercept term \alpha_p represents the average excess return not explained by the market, and the regression is run over historical return data to derive point estimates of \alpha and \beta. This estimation method allows for empirical evaluation of performance over time, with statistical significance tests applied to assess whether the alpha is meaningfully different from zero.^[22] For illustration, consider a portfolio with an actual annual return of 15%, a beta of 0.8, a risk-free rate of 3%, and a market risk premium of 7%. The SML-predicted return is $3\% + 0.8 \times 7\% = 8.6\%, yielding an alpha of $15\% - 8.6\% = 6.4\%, which would suggest strong outperformance relative to the benchmark. Unlike the Treynor ratio, which evaluates efficiency per unit of systematic risk, Jensen's alpha focuses on the absolute magnitude of excess return as a deviation from the SML.^[22]

Empirical Evidence and Criticisms

Testing the SML

Empirical tests of the security market line (SML) primarily assess whether the cross-section of asset returns aligns with the predictions of the Capital Asset Pricing Model (CAPM), using historical data to evaluate the linear relationship between beta and expected returns.^[23] Key methodologies include cross-sectional regressions, which estimate the relation between average returns and betas across assets, as pioneered by Fama and MacBeth (1973), who ran month-by-month regressions on U.S. stocks from 1927 to 1968 and found a positive but flatter-than-expected slope.^[24] Time-series tests examine the significance of Jensen's alpha, the intercept in regressions of excess returns on market excess returns, to detect deviations from the SML; for instance, Jensen (1968) applied this to mutual funds from 1955 to 1964, revealing non-zero alphas inconsistent with CAPM. Portfolio sorts by beta quintiles further test linearity by grouping stocks into portfolios based on estimated betas and comparing realized returns to SML predictions, often using data from the Center for Research in Security Prices (CRSP) for U.S. equities. Generalized Method of Moments (GMM) estimation has also been employed for joint tests of CAPM restrictions, incorporating multiple moments to evaluate pricing errors across assets.^[23] Early empirical evidence from the 1960s and 1970s provided initial support for the SML's linear risk-return relation in U.S. data. Black, Jensen, and Scholes (1972) conducted cross-sectional tests on NYSE stocks from 1931 to 1965, forming equal- and value-weighted portfolios sorted by beta, and found a positive relation between average returns and betas, though the line was flatter than the Sharpe-Lintner version, with low-beta stocks earning higher-than-predicted returns and high-beta stocks lower.^[25] Fama and MacBeth (1973) corroborated this rough linearity but noted the intercept exceeded the risk-free rate and the slope was below the market risk premium, using CRSP data to estimate betas via 60-month rolling regressions.^[24] These studies, drawing on post-1926 U.S. market data, suggested the CAPM held approximately, attributing deviations to the Black version allowing a non-zero intercept.^[23] Subsequent research in the 1990s revealed significant challenges to the SML's predictive power, particularly a flat relation for low-beta stocks and the emergence of anomalies. Fama and French (1992) analyzed NYSE, AMEX, and NASDAQ stocks from 1963 to 1990 via cross-sectional regressions and portfolio sorts, finding that beta alone explained little of the cross-section of average returns, with a nearly flat SML; instead, size and book-to-market factors captured much of the variation, as low-beta small-value stocks outperformed CAPM predictions.^[26] Their time-series tests on 25 size- and book-to-market-sorted portfolios confirmed significant alphas, rejecting the CAPM.^[23] Beta-sorted quintile portfolios from CRSP data showed low-beta groups yielding returns approximately 0.24% per month above predictions, while high-beta groups underperformed by about 0.17% per month, highlighting nonlinearity.^[26] Post-2000 studies have affirmed partial validity of the SML in international contexts while underscoring persistent anomalies like size and value factors that challenge its linearity. Fama and French (1998) extended tests to twelve major non-U.S. markets from 1975 to 1995 using global indices and local data, finding the CAPM's beta-return relation weak internationally, with scaled-price variables (e.g., book-to-market) explaining returns better than beta alone, though a multifactor extension partially restored pricing power.^[27] Leverage effects have been invoked to explain the low-beta anomaly, where constraints on borrowing prevent investors from leveraging low-beta assets to match high-beta risk; Baker, Bradley, and Wurgler (2011) documented this in U.S. and global data from 1968 to 2008 via a "betting against beta" factor, showing leveraged low-beta portfolios outperforming high-beta ones by 0.55% monthly after risk adjustment, using CRSP for U.S. and Datastream for international stocks.^[28] GMM-based international tests, such as those in Fama and French (2012), confirm the SML's slope is positive but too flat, with anomalies persisting across developed markets. Recent studies as of 2025, such as applications in emerging markets like Indonesia, suggest CAPM retains utility for certain contexts, though anomalies continue to challenge its universality in developed economies.^[29]^[30]^[31] Overall, while early U.S. evidence supported the SML, later and global findings indicate it captures systematic risk partially but fails to fully explain return cross-sections due to multifactor influences.^[23]

Limitations and Alternatives

The Security Market Line (SML), derived from the Capital Asset Pricing Model (CAPM), relies on several assumptions that are often criticized for their lack of realism in actual markets. These include the absence of taxes and transaction costs, unlimited borrowing and lending at the risk-free rate, homogeneous expectations among investors, and perfect information availability, all of which simplify theoretical modeling but fail to reflect real-world frictions such as varying investor beliefs and regulatory constraints.^[32] Additionally, the SML's focus on systematic risk via beta overlooks other systematic factors beyond market movements and assumes diversification eliminates non-systematic risk entirely, which is impractical for many investors with concentrated holdings.^[33] Beta estimates themselves exhibit instability over time due to changes in firm leverage, business models, or market regimes, leading to unreliable predictions of future risk exposure.^[34]^[35] Empirically, the SML demonstrates limited explanatory power, with cross-sectional regressions of stock returns on beta typically yielding low R² values, often below 10%, indicating that market beta accounts for only a modest portion of return variation. The betting-against-beta anomaly further undermines the model, as low-beta assets often deliver higher risk-adjusted returns than predicted, while high-beta assets underperform, suggesting a flatter SML than CAPM implies.^[36]^[28] To address these shortcomings, alternatives extend the single-factor framework of the SML. The Fama-French three-factor model incorporates size (small minus big) and value (high minus low book-to-market) factors alongside market risk, capturing additional dimensions of systematic risk that better explain return cross-sections. The Carhart four-factor model builds on this by adding a momentum factor, accounting for persistence in stock performance and improving out-of-sample predictions. The Arbitrage Pricing Theory (APT) offers a more flexible multifactor approach, positing that returns are driven by multiple unspecified macroeconomic or fundamental risks rather than a single market factor, allowing for empirical customization without rigid assumptions.^[37] The SML remains useful for quick benchmarking of expected returns in stable, diversified portfolios, but multifactor models are preferred for sophisticated analysis involving heterogeneous risks or non-U.S. markets.^[38] In the 2020s, evolving perspectives integrate behavioral finance elements, such as investor biases affecting risk perceptions, and ESG factors, which increasingly influence pricing through sustainability risks and stakeholder preferences, prompting hybrid models that blend traditional factors with these non-financial drivers.^[39]^[40]

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A betting against beta (BAB) factor, which is long leveraged low-beta assets and short high-beta assets, produces significant positive risk-adjusted returns.
[27]
Understanding the CAPM: Key Formula, Assumptions, and ...
CAPM and the Security Market Line (SML) The efficient frontier assumes the same things as the CAPM and can only be calculated in theory. If a portfolio existed ...
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The capital asset pricing model – part 3 - ACCA Global
Perfect capital market. This assumption means that all securities are valued correctly and that their returns will plot on to the SML. A perfect capital market ...
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Capital Asset Pricing Model (CAPM): Definition & Formula
Oct 18, 2024 · 3. Instability of beta over time. CAPM uses beta to measure a stock's sensitivity to market movements, but beta can be unstable. It is based on ...Overview Of The Capm Formula · Capm Assumptions · Capm And The Efficient...
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The Capital Asset Pricing Model - Tidy Finance
The Capital Market Line. Now, note the following three simple derivations: The expected excess return of the efficient tangency portfolio, is given by E ( ω ...Portfolio Return & Variance · The Capital Market Line · Asset Pricing And The Market...<|control11|><|separator|>
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Stock regression - RPubs
Sep 17, 2021 · The CAPM, Fama-French 5 factor model and Carhart four-factor model will be illustrated to evaluate whether the models are statistically significant.Missing: typical R2
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Tutorial: Stock Price Returns and Beta Coefficients - edgarstat.com
Feb 16, 2023 · The R2 = 25.45%, showing that the CAPM linear model is a bad fit for Chevron. Example 2: The CAPM Coefficients of ConocoPhillips (GVKEY 8549, ...Missing: typical | Show results with:typical
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[PDF] Betting Against Beta - NYU Stern
Baker, M., B. Bradley, and J. Wurgler (2011), “Benchmarks as Limits to Arbitrage: Understanding the Low Volatility Anomaly,” Financial Analysts Journal 67(1) ...
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How to Calculate and Interpret the Arbitrage Pricing Theory (APT)
Apr 9, 2025 · This line is similar to the Security Market Line (SML) from CAPM. However, the APT allows for multiple risk factors, while the SML considers ...
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[PDF] Multi-Factor Models and the Arbitrage Pricing Theory (APT)
▻ HML: high minus low book-to-market. The Fama-French-Carhart 4-factor model adds an additional momentum factor. Econ 471/571, F19 - Bollerslev. APT. 54. Page ...
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ESG Investing and the Popularity Asset Pricing Model (PAPM)
Feb 1, 2024 · Under PAPM, individual investors may all have unique views on how ESG characteristics or sub-ESG characteristics influence expected risk and ...
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The asset pricing effects of ESG investing
In this note, we discuss the asset pricing implications of the increasing consideration of Environmental, Social and Governance (ESG) issues in investing.