Glossary

10.3 Carhart four-factor model

6 min read•august 21, 2024

The expands on earlier asset pricing theories by adding momentum to market risk, size, and value factors. This model aims to better explain stock returns and mutual fund performance, addressing limitations of simpler approaches like .

By incorporating these four factors, the Carhart model provides a more comprehensive framework for understanding asset pricing and portfolio performance. It offers improved explanatory power for stock returns and serves as a valuable tool for performance evaluation and portfolio construction in finance.

Overview of Carhart model

  • Extends Fama-French three-factor model by adding a fourth factor, momentum
  • Developed by in 1997 to explain stock returns and mutual fund performance
  • Aims to capture additional dimensions of risk and return beyond market

Factors in Carhart model

Market risk factor

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  • Represents the of the market portfolio over the risk-free rate
  • Captures systematic risk associated with overall market movements
  • Calculated as the difference between market return and risk-free rate (Rm−RfR_m - R_f)
  • Measures sensitivity of a stock or portfolio to broad market fluctuations

Size factor

  • Accounts for the historical outperformance of small-cap stocks over large-cap stocks
  • Calculated as the return difference between small and big market capitalization stocks ()
  • Reflects the additional risk premium associated with investing in smaller companies
  • Helps explain why small-cap stocks tend to generate higher returns over long periods

Value factor

  • Captures the tendency of value stocks to outperform growth stocks
  • Computed as the return difference between high and low book-to-market ratio stocks ()
  • Represents the additional risk premium for investing in undervalued companies
  • Addresses the value premium observed in historical stock market data

Momentum factor

  • Measures the excess return of high past returns portfolios over low past returns portfolios
  • Calculated as the average return on high prior return portfolios minus low prior return portfolios ()
  • Accounts for the tendency of stocks with positive momentum to continue performing well
  • Incorporates the empirical observation that past winners tend to outperform past losers

Comparison to CAPM

  • Carhart model expands on CAPM by including three additional risk factors
  • Addresses limitations of CAPM in explaining cross-sectional variation in stock returns
  • Provides a more comprehensive framework for understanding asset pricing and portfolio performance
  • Incorporates size, value, and momentum effects not captured by market beta alone
  • Offers improved explanatory power for stock returns compared to single-factor CAPM

Comparison to Fama-French model

  • Carhart model adds to Fama-French three-factor model
  • Retains market risk, size, and value factors from Fama-French model
  • Addresses the momentum anomaly not explained by Fama-French factors
  • Provides enhanced explanatory power for mutual fund performance
  • Offers a more comprehensive framework for asset pricing and return prediction

Mathematical formulation

Regression equation

  • Expresses expected excess return as a linear combination of factor exposures
  • Ri−Rf=αi+βi(Rm−Rf)+siSMB+hiHML+miUMD+ϵiR_i - R_f = \alpha_i + \beta_i(R_m - R_f) + s_i\text{SMB} + h_i\text{HML} + m_i\text{UMD} + \epsilon_i
  • RiR_i represents the return on asset i, RfR_f is the risk-free rate
  • βi,si,hi,mi\beta_i, s_i, h_i, m_i are for market, size, value, and momentum respectively
  • αi\alpha_i represents the abnormal return not explained by the four factors

Factor calculations

  • Market factor (MKT) calculated as the excess return of market portfolio over risk-free rate
  • (SMB) computed as the average return on small-cap portfolios minus large-cap portfolios
  • (HML) determined by average return on high book-to-market portfolios minus low book-to-market portfolios
  • Momentum factor (UMD) calculated as average return on high prior return portfolios minus low prior return portfolios
  • Factors typically constructed using portfolios formed on size, book-to-market, and prior returns

Applications in finance

Performance evaluation

  • Used to assess mutual fund and portfolio manager performance
  • Helps identify sources of returns beyond market exposure ( generation)
  • Allows for risk-adjusted performance comparisons across different investment strategies
  • Provides insights into factor exposures and style tilts of investment portfolios
  • Facilitates attribution of returns to specific risk factors and manager skill

Portfolio construction

  • Guides factor-based investing strategies and smart beta approaches
  • Assists in creating diversified portfolios with exposure to multiple risk premia
  • Helps identify and target specific factor exposures for enhanced returns
  • Supports risk management by understanding and controlling factor loadings
  • Enables optimization of portfolios based on desired factor exposures and risk-return trade-offs

Strengths of Carhart model

  • Improves explanatory power for stock returns compared to CAPM and Fama-French model
  • Captures , addressing a significant market anomaly
  • Provides a more comprehensive framework for understanding sources of returns
  • Offers enhanced tools for performance evaluation and attribution analysis
  • Supports more sophisticated portfolio construction and risk management techniques
  • Aligns with empirical observations of market behavior and return patterns

Limitations of Carhart model

  • May not fully capture all relevant factors affecting asset returns
  • Assumes linear relationships between factors and returns, which may not always hold
  • Relies on historical data, potentially limiting predictive power for future returns
  • Factor definitions and calculations can be sensitive to methodology choices
  • Does not account for time-varying factor exposures or correlations
  • May suffer from data mining concerns and potential overfitting

Empirical evidence

Historical performance

  • Demonstrates improved explanatory power for mutual fund returns compared to earlier models
  • Shows persistent momentum effect across various markets and time periods
  • Reveals time-varying nature of factor premiums and their relative importance
  • Indicates potential for factor-based strategies to generate excess returns
  • Highlights the importance of considering multiple risk factors in asset pricing

Cross-sectional studies

  • Confirms the relevance of size, value, and momentum factors across different markets
  • Reveals variations in factor exposures and premiums across countries and regions
  • Demonstrates the model's ability to explain cross-sectional differences in stock returns
  • Identifies potential interactions between factors and their impact on asset pricing
  • Supports the use of multi-factor models in international portfolio management

Carhart model vs other models

  • Outperforms CAPM in explaining stock returns and mutual fund performance
  • Provides incremental improvement over Fama-French three-factor model
  • Competes with alternative multi-factor models (Fama-French five-factor model)
  • Offers a balance between simplicity and explanatory power
  • Serves as a benchmark for evaluating more complex asset pricing models
  • Remains widely used in academic research and practical applications

Implementation in practice

Data requirements

  • Necessitates reliable and comprehensive stock market data for factor construction
  • Requires historical returns for individual stocks and market indices
  • Demands accurate and consistent financial statement data for value factor calculation
  • Needs sufficient historical data to estimate factor loadings and perform regressions
  • Involves regular updates to maintain current factor exposures and portfolio characteristics

Statistical considerations

  • Addresses potential multicollinearity among factors through careful model specification
  • Requires attention to outliers and extreme observations in factor construction
  • Involves choice of appropriate estimation periods for factor loadings and regressions
  • Considers the impact of different weighting schemes in portfolio formation
  • Necessitates robust statistical testing to validate factor significance and model fit

Criticisms and debates

  • Questions the stability and persistence of factor premiums over time
  • Debates the economic rationale behind observed factor effects (risk vs. mispricing)
  • Raises concerns about data mining and potential overfitting in factor selection
  • Challenges the assumption of constant factor loadings over extended periods
  • Discusses the impact of trading costs and market frictions on factor-based strategies
  • Examines the potential for factor crowding and diminishing returns as strategies become popular

Extensions of Carhart model

  • Incorporates additional factors to capture other documented anomalies (profitability, investment)
  • Explores time-varying factor exposures and conditional asset pricing models
  • Investigates non-linear relationships between factors and returns
  • Adapts the model for different asset classes (fixed income, commodities, alternatives)
  • Develops industry-specific versions to account for sector-specific risk factors
  • Integrates macroeconomic variables to capture broader economic influences on asset returns

Key Terms to Review (27)

Active management: Active management is an investment strategy where a portfolio manager makes specific investment decisions with the aim of outperforming a benchmark index through ongoing buying and selling of assets. This approach contrasts with passive management, where the goal is to mirror the performance of a specific index. Active management involves in-depth research, market analysis, and adjustments to the portfolio based on market conditions.
Alpha: Alpha is a measure of the active return on an investment compared to a market index or benchmark. It indicates how much more or less an investment has returned relative to its risk, essentially representing the value that a portfolio manager adds beyond the market's performance.
Annualized returns: Annualized returns refer to the geometric average amount of money earned by an investment each year over a given time period, expressed as a percentage. This measure allows investors to compare the profitability of different investments on a standardized basis by projecting the cumulative returns over multiple years into an annual format, which is crucial for evaluating performance in financial models such as the Carhart four-factor model.
Beta: Beta is a measure of a security's or portfolio's sensitivity to market movements, indicating the level of risk in relation to the overall market. A beta greater than 1 means the asset is more volatile than the market, while a beta less than 1 indicates less volatility. Understanding beta helps in assessing investment risk and constructing portfolios that align with an investor's risk tolerance and expected return.
CAPM: The Capital Asset Pricing Model (CAPM) is a financial model that establishes a linear relationship between the expected return of an asset and its systematic risk, as measured by beta. This model is used to determine a theoretically appropriate required rate of return of an asset, factoring in the risk-free rate and the expected market return. It serves as a fundamental concept in modern finance, connecting risk and return to investment decisions and pricing of assets.
Carhart four-factor model: The Carhart four-factor model is an extension of the Fama-French three-factor model that adds a momentum factor to better explain the returns of a portfolio. This model incorporates four key factors: market risk, size, value, and momentum, providing a more comprehensive framework for assessing asset pricing and investment performance. By acknowledging the impact of momentum on stock returns, it helps investors better understand the drivers behind price movements and make informed decisions.
Eugene Fama: Eugene Fama is a prominent economist known as the 'father of modern finance,' who is best recognized for his pioneering work in the fields of asset pricing and market efficiency. His research laid the groundwork for key financial theories, including the efficient market hypothesis, which asserts that asset prices reflect all available information. Fama's contributions extend to factor models, providing insights that have shaped investment strategies and portfolio management.
Excess return: Excess return is the difference between the return of an investment and the return of a benchmark or risk-free rate over a specified period. This concept is crucial as it helps in assessing how well an investment performs relative to a standard, indicating whether the investment has generated a reward for taking on additional risk. In finance, particularly in asset pricing models, excess return serves as a fundamental measure of performance that can guide investors in their decision-making processes.
Factor loadings: Factor loadings are coefficients that represent the relationship between observed variables and their underlying latent factors in a factor model. They indicate the degree to which each observed variable correlates with the factor, providing insight into how much influence each variable has on the factor's behavior. In the context of the Carhart four-factor model, these loadings help investors understand how different factors affect asset returns.
HML: HML, or 'High Minus Low,' is a factor in asset pricing models that represents the difference in returns between high book-to-market (value) stocks and low book-to-market (growth) stocks. This factor reflects the value premium that investors typically receive for investing in undervalued stocks as opposed to overvalued stocks, highlighting an important aspect of market behavior and investor sentiment.
Information Ratio: The information ratio is a measure used to assess the risk-adjusted performance of an investment or portfolio. It calculates the excess return of a portfolio over a benchmark, relative to the volatility of that excess return. A higher information ratio indicates that a portfolio manager is generating more consistent returns compared to its benchmark, making it a critical tool in evaluating portfolio performance and understanding the effectiveness of active management strategies.
Mark Carhart: Mark Carhart is a prominent figure in finance, best known for his contributions to asset pricing models, particularly the Carhart four-factor model. This model builds on the Fama-French three-factor model by adding a momentum factor, which addresses the limitations of previous models by capturing the behavior of stock returns more accurately over time. Carhart's work emphasizes the importance of incorporating multiple risk factors to enhance investment strategies and explain stock performance.
Market risk factor: The market risk factor refers to the inherent risk associated with market fluctuations that can affect the overall value of investments, driven primarily by macroeconomic events or systemic changes. This risk is not specific to a particular asset but rather impacts a wide range of securities, making it crucial for understanding investment performance within models like the Carhart four-factor model. It helps in evaluating how much of an asset's return is influenced by overall market movements versus other risks.
Market risk premium: The market risk premium is the additional return that investors expect to receive from holding a risky market portfolio instead of risk-free assets. This premium is crucial in understanding the compensation investors require for taking on additional risk, reflecting the difference between the expected return on the market and the risk-free rate. It plays a vital role in asset pricing models, influencing investment decisions and portfolio management strategies.
Momentum effect: The momentum effect refers to the tendency of assets that have performed well in the past to continue performing well in the future, while those that have performed poorly tend to continue underperforming. This phenomenon is often attributed to behavioral biases and market inefficiencies, and it plays a significant role in understanding asset pricing and investment strategies.
Momentum factor: The momentum factor refers to the tendency of securities that have performed well in the past to continue to perform well in the future, and those that have performed poorly to continue to perform poorly. This concept is a critical component in financial modeling, particularly in asset pricing, where it is used to capture the persistence of returns over time. It highlights the behavioral aspects of investors who tend to chase trends, leading to a systematic bias in stock returns that can be exploited in investment strategies.
Monthly returns: Monthly returns refer to the percentage change in the value of an investment or portfolio over the course of a month. This metric is essential for assessing the performance of assets, allowing investors to understand how their investments are growing or declining on a monthly basis. By examining monthly returns, investors can also identify trends, assess risk, and make informed decisions about future investments.
Passive Management: Passive management is an investment strategy that aims to replicate the performance of a specific index or benchmark rather than actively selecting individual securities. This approach minimizes trading and typically results in lower costs, as it relies on a buy-and-hold strategy rather than frequent buying and selling. The effectiveness of passive management can be evaluated using various performance measures and models that assess how closely a portfolio tracks its benchmark.
Regression analysis: Regression analysis is a statistical method used to estimate the relationships among variables, often focusing on how the dependent variable changes when one or more independent variables are varied. This technique helps in identifying trends, making predictions, and assessing the strength of predictors in various contexts, including financial modeling and risk assessment.
Risk-adjusted return: Risk-adjusted return is a financial metric that measures the return of an investment in relation to the amount of risk taken to achieve that return. It helps investors understand whether they are being adequately compensated for the level of risk they assume in their investment choices. This concept is crucial in evaluating the performance of portfolios and individual investments, allowing for comparisons that account for varying risk levels across different assets or strategies.
Sharpe Ratio: The Sharpe Ratio is a measure used to assess the risk-adjusted performance of an investment by comparing its excess return to its standard deviation. It provides insights into how much additional return an investor receives for the extra volatility taken on compared to a risk-free asset. This ratio connects various financial concepts, including evaluating probability distributions of returns, optimizing portfolios using mean-variance analysis, and understanding performance measures within models like the Fama-French three-factor and Carhart four-factor models.
Size Factor: The size factor is a concept in finance that refers to the tendency of smaller companies to outperform larger companies on a risk-adjusted basis. This phenomenon is often observed in investment strategies and is a crucial element in asset pricing models, highlighting how company size can influence expected returns. Recognizing the size factor helps investors understand the potential for higher returns from small-cap stocks compared to their large-cap counterparts, contributing to broader investment strategies.
Smb: SMB, which stands for 'Small Minus Big,' is a factor in asset pricing models that captures the historical excess returns of small-cap stocks over large-cap stocks. This concept is integral to understanding how market capitalization influences stock returns, and it plays a significant role in the Carhart four-factor model, which expands on the traditional Fama-French three-factor model by adding momentum as a fourth factor.
Time series analysis: Time series analysis involves statistical techniques used to analyze time-ordered data points to understand underlying patterns, trends, and behaviors over time. This approach is crucial in financial mathematics as it helps in predicting future values based on past observations, revealing volatility and enabling effective modeling. The insights gained from time series analysis can be applied to various financial models, such as evaluating the performance of assets or understanding market dynamics.
Tracking error: Tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked. It quantifies the volatility of the difference between the portfolio returns and the benchmark returns, indicating how much a portfolio deviates from its intended index performance. A low tracking error suggests that the portfolio closely matches the benchmark, while a high tracking error indicates significant divergence, which can be useful for assessing active management strategies and performance consistency.
Umd: UMD, or Upside-Down Market Dynamics, refers to a behavioral finance concept that explains the tendency of investors to overreact to information, leading to mispricing of assets in financial markets. This phenomenon can affect asset pricing models and investment strategies, as it captures the irrational behavior of market participants when responding to news or market signals.
Value Factor: The value factor is a concept in finance that reflects the tendency of undervalued stocks to outperform overvalued stocks over time. This factor is central to various asset pricing models and highlights the importance of price relative to fundamental metrics, such as earnings or book value. It plays a significant role in explaining stock returns and is closely related to risk and expected return in investment strategies.
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