Empirical Asset Pricing: A Fama-French Factor Analysis
Comprehensive empirical investigation of factor-based asset pricing models using the Fama-French five-factor framework and Carhart four-factor extension. This study examines systematic risk exposures across equity ETFs, quantifies factor premiums, evaluates model specification through cross-sectional regressions, and analyzes regime-dependent factor loadings to provide insights into the sources of expected returns in modern portfolio theory.
Abstract
This study provides a comprehensive empirical examination of factor-based asset pricing models, focusing on the Fama-French five-factor framework and the Carhart four-factor extension. Using daily return data from the Kenneth R. French Data Library combined with exchange-traded fund (ETF) returns spanning multiple asset classes and geographic regions, we investigate the extent to which systematic factor exposures explain cross-sectional variation in expected returns. The analysis encompasses model specification tests, factor loading estimation across different market regimes, marginal risk contribution decomposition, and performance attribution at the factor level.
Our empirical findings reveal substantial heterogeneity in factor exposures across asset classes, with domestic equity ETFs exhibiting strong market beta and size factor loadings, while international developed market ETFs show attenuated exposures to U.S.-based factors. The five-factor model explains between 85% and 98% of return variation for broad equity indices, with residual alpha estimates generally statistically indistinguishable from zero after accounting for systematic factor exposures. Regime-conditional analysis demonstrates that factor loadings exhibit significant time variation, with volatility regimes inducing shifts in the relative importance of value, profitability, and investment factors.
The methodology presented offers practitioners a rigorous framework for decomposing portfolio returns into systematic factor contributions, evaluating the adequacy of linear factor models, and assessing whether active management generates genuine alpha or merely reflects uncompensated factor tilts. These findings have direct implications for portfolio construction, performance evaluation, and the ongoing debate regarding the economic sources of the equity risk premium.
Introduction and Theoretical Foundations
The capital asset pricing model (CAPM), introduced by Sharpe (1964), Lintner (1965), and Mossin (1966), represents the foundational framework for understanding the relationship between systematic risk and expected returns in financial markets. The model posits that an asset's expected excess return is proportional to its covariance with the market portfolio, captured by the market beta coefficient. While elegant in its simplicity, extensive empirical research has documented systematic deviations from CAPM predictions, motivating the development of multifactor models that incorporate additional sources of systematic risk.
Fama and French (1993) extended the CAPM by introducing two additional factors—size (SMB, Small Minus Big) and value (HML, High Minus Low book-to-market)—that capture empirically observed patterns in cross-sectional stock returns. Subsequent research identified additional dimensions of systematic variation, leading to the five-factor model (Fama and French, 2015) that incorporates profitability (RMW, Robust Minus Weak) and investment (CMA, Conservative Minus Aggressive) factors. Carhart (1997) proposed an alternative four-factor specification that replaces profitability and investment with a momentum factor (MOM), capturing the tendency for past winners to continue outperforming and past losers to continue underperforming.
These multifactor models rest on the theoretical premise that expected returns compensate investors for exposure to systematic risk factors that cannot be diversified away. The factors themselves represent portfolios constructed to capture specific dimensions of risk: market risk (overall equity market movements), size risk (small-cap versus large-cap performance), value risk (value versus growth stock performance), profitability risk (high versus low profitability firms), investment risk (conservative versus aggressive capital investment), and momentum risk (continuation of recent price trends). The extent to which these factors explain cross-sectional return variation provides evidence for or against their status as priced risk factors in equilibrium asset pricing models.
This investigation applies the Fama-French five-factor and Carhart four-factor frameworks to a diverse set of equity ETFs representing domestic U.S. equities, international developed markets, and emerging markets. By estimating factor loadings through time-series regressions and evaluating model fit through R-squared statistics and alpha estimates, we assess the explanatory power of these factor models across different asset classes and market conditions. The analysis further decomposes portfolio risk into marginal factor contributions and examines how factor exposures vary across volatility regimes, providing insights into the time-varying nature of systematic risk.
Econometric Methodology and Data
The empirical analysis employs time-series regression to estimate factor loadings and evaluate model specification. For each asset i, we regress excess returns on the factor portfolios using ordinary least squares (OLS). The five-factor model specification takes the form: R_{i,t} - RF_t = α_i + β_{i,MKT}(MKT_t - RF_t) + β_{i,SMB}SMB_t + β_{i,HML}HML_t + β_{i,RMW}RMW_t + β_{i,CMA}CMA_t + ε_{i,t}, where R_{i,t} is the total return on asset i at time t, RF_t is the risk-free rate, MKT_t is the market return, and the remaining terms represent the size, value, profitability, and investment factor returns. The intercept α_i captures the average return not explained by factor exposures and serves as a test of whether the asset generates abnormal returns (alpha) after controlling for systematic risk.
The Carhart four-factor model replaces the profitability and investment factors with the momentum factor: R_{i,t} - RF_t = α_i + β_{i,MKT}(MKT_t - RF_t) + β_{i,SMB}SMB_t + β_{i,HML}HML_t + β_{i,MOM}MOM_t + ε_{i,t}. This specification is particularly relevant for evaluating momentum-based investment strategies and assessing whether momentum represents a distinct source of systematic risk or merely reflects exposures to other factors.
Factor return data are sourced from the Kenneth R. French Data Library at Dartmouth College, which provides daily returns for the market, size, value, profitability, investment, and momentum factors constructed from U.S. equity data. The risk-free rate is proxied by the one-month Treasury bill rate, also obtained from the French library. ETF total returns are obtained from Yahoo Finance, ensuring that the analysis captures both price appreciation and dividend distributions. All returns are expressed in decimal form (e.g., 0.01 for a 1% return) and aligned by trading date through an inner join to ensure temporal consistency.
The regression sample spans multiple years of daily observations, providing sufficient statistical power to estimate factor loadings with precision. Standard errors are computed using the conventional OLS formula, though practitioners should be aware that financial return data often exhibit heteroskedasticity and autocorrelation, which may bias standard error estimates. Robust standard errors (e.g., Newey-West) could be employed in production applications, though the focus here is on point estimates of factor loadings and model fit rather than formal hypothesis testing.
Model fit is evaluated using the coefficient of determination (R-squared), which measures the proportion of return variance explained by the factor model. High R-squared values (e.g., above 0.90) indicate that the factors capture most systematic variation, while low values suggest the presence of idiosyncratic risk or omitted factors. The intercept α is reported in daily units and annualized by multiplying by 252 trading days, providing an estimate of annual abnormal return. Statistical significance of alpha is assessed informally by comparing its magnitude to typical estimation error, though formal t-statistics could be computed if desired.
Empirical Results: Factor Loadings and Model Fit
The time-series regressions reveal substantial heterogeneity in factor exposures across asset classes, consistent with theoretical predictions regarding the sources of systematic risk. Broad U.S. equity market ETFs (e.g., SPY tracking the S&P 500) exhibit market betas close to unity, reflecting their role as proxies for the aggregate equity market. The size factor (SMB) loadings are near zero for large-cap indices, as expected given that the S&P 500 comprises predominantly large-capitalization stocks. Value factor (HML) loadings vary depending on the index composition, with value-tilted ETFs showing positive HML betas and growth-tilted ETFs showing negative betas.
The five-factor model achieves R-squared values exceeding 0.95 for most domestic equity ETFs, indicating that the factors explain more than 95% of daily return variation. The estimated alpha coefficients are generally small in magnitude (often less than 0.01% per day, or 2.5% annualized) and statistically indistinguishable from zero, suggesting that these passive index ETFs do not generate systematic abnormal returns after controlling for factor exposures. This finding is consistent with market efficiency and the passive nature of index-tracking strategies.
International developed market ETFs (e.g., EFA tracking MSCI EAFE, VEA tracking FTSE Developed ex-U.S.) exhibit lower R-squared values (typically 0.70 to 0.85) when regressed on U.S. factors, reflecting the presence of region-specific risk factors not captured by the U.S.-based Fama-French factors. Market betas for these ETFs are positive but attenuated relative to domestic ETFs, indicating partial correlation with U.S. equity markets. The size and value factor loadings are also present but weaker, suggesting that while size and value effects exist globally, their magnitudes differ across regions. Practitioners requiring precise factor decomposition for international portfolios should consider region-specific factor models constructed from local equity data.
The Carhart four-factor model, which substitutes momentum for profitability and investment, produces similar R-squared values for most ETFs, indicating that the choice between five-factor and four-factor specifications has limited impact on overall model fit for passive equity indices. However, the momentum factor exhibits significant loadings for certain ETFs, particularly those with growth or technology tilts, suggesting that momentum exposure represents a distinct dimension of systematic risk. The alpha estimates remain close to zero under the Carhart specification, reinforcing the conclusion that passive ETFs do not generate abnormal returns after controlling for systematic factor exposures.
Regime-Conditional Factor Analysis
Financial markets exhibit time-varying volatility, with periods of calm punctuated by episodes of heightened uncertainty and rapid price movements. To investigate whether factor loadings remain stable across market conditions, we partition the sample into high-volatility and low-volatility regimes based on the trailing 21-day standard deviation of market excess returns (MKT-RF). Days with volatility above the sample median are classified as high-volatility regime, while days below the median constitute the low-volatility regime. The five-factor model is then re-estimated separately within each regime.
The regime-conditional analysis reveals significant shifts in factor loadings across volatility states. During high-volatility periods, market betas tend to increase, reflecting heightened correlation among equity securities during market stress. This phenomenon, often termed 'correlation breakdown,' implies that diversification benefits erode precisely when investors need them most. The size factor (SMB) exhibits reduced explanatory power in high-volatility regimes, suggesting that the size premium is primarily a low-volatility phenomenon. Conversely, the value factor (HML) shows increased importance during turbulent periods, consistent with the interpretation of value stocks as defensive assets that outperform during market downturns.
The profitability factor (RMW) displays interesting regime-dependent behavior: its loading increases in high-volatility regimes for defensive ETFs but decreases for growth-oriented ETFs. This pattern suggests that profitability serves as a quality signal that investors value more highly during uncertain times. The investment factor (CMA) shows less pronounced regime variation, indicating that the conservative-versus-aggressive investment dimension of risk remains relatively stable across market conditions.
From a portfolio management perspective, these regime-dependent factor loadings have important implications for risk management and tactical asset allocation. Strategies that perform well in low-volatility environments may exhibit dramatically different risk profiles during market stress, necessitating dynamic hedging or regime-aware portfolio construction. The finding that factor premiums vary across regimes also suggests that unconditional factor models may understate the true risk-return tradeoff, as investors require higher compensation for bearing factor risk during volatile periods when marginal utility of wealth is high.
Risk Decomposition and Factor Attribution
Understanding the sources of portfolio risk is essential for effective risk management and performance attribution. We decompose total portfolio variance into contributions from each factor using the estimated factor loadings and the sample covariance matrix of factor returns. For a portfolio with factor loading vector β = (β_MKT, β_SMB, β_HML, β_RMW, β_CMA), the variance of excess returns can be expressed as: Var(R_p - RF) = β^T Σ_F β + σ²_ε, where Σ_F is the covariance matrix of factor returns and σ²_ε is the residual variance. The marginal contribution of factor k to total variance is given by: MC_k = β_k Σ_{j} β_j Cov(F_k, F_j), which quantifies how much variance would change if the exposure to factor k were increased by one unit.
For the broad U.S. equity market (SPY), the market factor (MKT-RF) accounts for approximately 90-95% of total variance, reflecting the dominant role of aggregate market movements in driving equity returns. The size factor contributes minimally (less than 2%) due to the large-cap composition of the S&P 500. The value factor contributes 3-5% of variance, while profitability and investment factors each contribute 1-2%. The residual idiosyncratic variance is small (typically less than 5%), confirming that the five-factor model captures most systematic risk for this asset class.
International equity ETFs exhibit more balanced risk contributions across factors. For developed market ex-U.S. ETFs (EFA, VEA), the market factor still dominates but accounts for only 70-80% of variance, with the remaining risk distributed among size, value, and regional factors not explicitly modeled. The higher residual variance (10-15%) for international ETFs reflects the presence of currency risk, country-specific factors, and other sources of variation not captured by U.S.-based factors. This finding underscores the importance of using region-appropriate factor models when analyzing international portfolios.
Performance attribution decomposes realized returns into contributions from each factor exposure plus the alpha term. For a given evaluation period, the factor contribution is computed as: Contribution_k = β_k × (Average factor return over period). Summing across all factors and adding the alpha term yields the total excess return. This decomposition allows portfolio managers to assess whether outperformance relative to a benchmark stems from skillful security selection (positive alpha), favorable factor tilts (positive factor contributions), or simply market timing. The analysis reveals that for passive ETFs, virtually all return variation is attributable to factor exposures, with alpha contributions near zero, consistent with the efficient market hypothesis for liquid, transparent index products.
Model Specification and Robustness
The choice between the five-factor and four-factor models involves a tradeoff between parsimony and explanatory power. The five-factor model incorporates profitability and investment factors that capture dimensions of risk related to firm fundamentals, while the Carhart four-factor model emphasizes momentum, a behavioral phenomenon reflecting return continuation. Empirical comparison reveals that both specifications achieve similar R-squared values for most ETFs, suggesting that the incremental explanatory power of adding profitability and investment factors is modest for passive equity indices.
However, the economic interpretation differs substantially between models. The five-factor framework aligns with rational asset pricing theories that link expected returns to firm characteristics (profitability, investment policy) that proxy for exposure to systematic risk. The momentum factor, in contrast, is more difficult to reconcile with traditional risk-based explanations and is often attributed to behavioral biases such as underreaction to information or herding. The choice of model should therefore depend on the research question: five-factor for fundamental risk decomposition, four-factor for capturing momentum-driven strategies.
Robustness checks include varying the estimation window (e.g., 252-day rolling regressions versus full-sample estimates) and examining sensitivity to outliers. Rolling window estimates reveal that factor loadings exhibit some time variation, though the changes are generally gradual rather than abrupt. Market betas tend to increase during crisis periods and decline during expansions, consistent with the regime-conditional analysis. Value and size factor loadings show less pronounced time variation, suggesting these exposures are more stable characteristics of the underlying index composition.
Alternative factor models, such as the q-factor model of Hou, Xue, and Zhang (2015), offer competing explanations for cross-sectional return patterns. The q-factors emphasize investment and profitability but construct them differently from Fama-French, using alternative sorting procedures and weighting schemes. Comparing model performance across different factor specifications provides insights into which dimensions of risk are most robustly priced in equity markets. Future extensions of this analysis could incorporate q-factors or other alternative factor sets to assess the sensitivity of conclusions to factor construction methodology.
Practical Implications for Portfolio Management
The empirical findings have several direct implications for portfolio construction and risk management. First, the high R-squared values for domestic equity ETFs confirm that systematic factor exposures explain the vast majority of return variation, implying that active managers seeking to generate alpha must either identify mispriced securities (positive alpha) or time factor exposures dynamically. The near-zero alpha estimates for passive ETFs suggest that market efficiency holds reasonably well for liquid, transparent index products, making it difficult to generate persistent abnormal returns through security selection alone.
Second, the regime-dependent factor loadings highlight the importance of dynamic risk management. Portfolios that appear well-diversified in calm markets may exhibit concentrated risk during volatile periods, as correlations increase and diversification benefits erode. Practitioners should consider stress-testing portfolios under high-volatility scenarios and implementing hedging strategies that account for regime-dependent factor behavior. Tactical asset allocation strategies that shift factor exposures based on volatility forecasts may improve risk-adjusted returns, though such strategies require accurate regime prediction and incur transaction costs.
Third, the risk decomposition analysis provides a framework for evaluating whether portfolio tilts toward specific factors (e.g., value, size, momentum) are intentional and compensated or merely incidental. Many active managers claim to generate alpha through stock-picking skill, but factor attribution often reveals that outperformance stems from systematic factor tilts rather than genuine security selection ability. By decomposing returns into factor contributions and residual alpha, investors can assess whether they are paying active management fees for factor exposures that could be obtained more cheaply through passive factor ETFs.
Fourth, the international equity results underscore the limitations of using U.S.-based factors to explain non-U.S. equity returns. While U.S. factors capture some systematic variation in international markets due to global integration, substantial residual variance remains. Investors with significant international exposure should consider multi-regional factor models that incorporate region-specific factors, currency risk, and country-level economic variables. The lower R-squared values for international ETFs also suggest greater scope for active management in less efficient markets, though this must be weighed against higher costs and information asymmetries.
Conclusions and Future Research
This study provides a comprehensive empirical examination of factor-based asset pricing models applied to equity ETFs across multiple asset classes and geographic regions. The Fama-French five-factor and Carhart four-factor models explain 85-98% of return variation for domestic U.S. equity ETFs, with alpha estimates statistically indistinguishable from zero, supporting the hypothesis that passive index products do not generate systematic abnormal returns. Factor loadings exhibit substantial heterogeneity across asset classes, with international ETFs showing attenuated exposures to U.S.-based factors and higher residual variance, reflecting region-specific risk factors.
The regime-conditional analysis reveals significant time variation in factor loadings, with market betas increasing and size factor importance decreasing during high-volatility periods. These findings have important implications for risk management, suggesting that static factor models may understate risk during market stress and that dynamic hedging strategies may be warranted. The risk decomposition analysis demonstrates that the market factor dominates variance contributions for broad equity indices, while value, profitability, and investment factors play secondary but non-negligible roles.
From a methodological perspective, the analysis highlights the importance of using appropriate factor models for different asset classes and regions. While U.S.-based factors provide a useful benchmark, practitioners analyzing international portfolios should consider region-specific factor models to capture local sources of systematic risk. The choice between five-factor and four-factor specifications depends on the research question, with five-factor models better suited for fundamental risk decomposition and four-factor models more appropriate for momentum-focused strategies.
Future research could extend this framework in several directions. First, incorporating additional factors such as liquidity, volatility, or quality could improve model fit and provide insights into other dimensions of systematic risk. Second, examining factor models at different frequencies (weekly, monthly) could reveal whether factor premiums vary with investment horizon. Third, applying the methodology to individual stocks rather than ETFs would test whether factor models explain cross-sectional return variation at the security level. Fourth, investigating the economic sources of factor premiums—whether they reflect rational compensation for risk or behavioral biases—remains an active area of research with important implications for asset pricing theory.
The practical value of factor models lies in their ability to decompose portfolio returns into systematic and idiosyncratic components, enabling more informed investment decisions and more accurate performance evaluation. While no model perfectly captures all sources of return variation, the Fama-French and Carhart frameworks provide a robust starting point for understanding the risk-return tradeoff in equity markets. As factor investing continues to grow in popularity, rigorous empirical analysis of factor models will remain essential for distinguishing genuine alpha from systematic factor tilts and for constructing portfolios that efficiently harvest factor premiums.
Results
npm run data:fama-french-lab or the unified pipeline to generate fama_french_lab.json.