Carhart Four Factor Model is one of the most widely used asset pricing frameworks in modern finance, extending the Fama-French Three Factor Model by adding a momentum factor to better explain portfolio returns. Developed by Mark Carhart in 1997, this model helps investors and portfolio managers determine whether a fund's performance is due to genuine skill (alpha) or simply exposure to known risk factors. In this comprehensive guide, you will learn the theory behind each factor, how to calculate them in Excel using MarketXLS, and how to perform portfolio attribution analysis to evaluate investment performance.
The Evolution of Asset Pricing Models
To understand the Carhart Four Factor Model, it helps to see how it evolved from earlier models.
Capital Asset Pricing Model (CAPM)
The CAPM, developed by Sharpe (1964) and Lintner (1965), was the first formal asset pricing model. It proposes that a security's expected return is determined by a single factor: its sensitivity to the overall market.
CAPM Formula: E(Ri) = Rf + βi × (E(Rm) − Rf)
Where:
- E(Ri) = Expected return of asset i
- Rf = Risk-free rate
- βi = Beta of asset i (sensitivity to market)
- E(Rm) = Expected market return
- E(Rm) − Rf = Market risk premium
Limitation: CAPM explains only about 70% of the variation in stock returns. Many anomalies remain unexplained, including the size effect and value premium.
Fama-French Three Factor Model
Eugene Fama and Kenneth French (1993) expanded CAPM by adding two factors that academic research showed consistently affected stock returns:
Fama-French Formula: E(Ri) − Rf = αi + βi × (E(Rm) − Rf) + si × SMB + hi × HML + εi
Where:
- SMB (Small Minus Big): The return difference between small-cap and large-cap stocks. Historically, small-cap stocks have outperformed large-cap stocks.
- HML (High Minus Low): The return difference between high book-to-market (value) stocks and low book-to-market (growth) stocks. Historically, value stocks have outperformed growth stocks.
Improvement: The three-factor model explains roughly 90% of diversified portfolio return variation, a significant improvement over CAPM's 70%.
Limitation: The model still failed to explain the momentum anomaly — the tendency of stocks that have recently performed well to continue performing well.
Carhart Four Factor Model
Mark Carhart (1997) added a fourth factor — momentum — to address this gap:
Carhart Four Factor Formula: E(Ri) − Rf = αi + βi × (Rm − Rf) + si × SMB + hi × HML + wi × WML + εi
Where:
- WML (Winners Minus Losers): Also called MOM or UMD (Up Minus Down). The return difference between stocks with the highest returns over the past 2-12 months and stocks with the lowest returns over the same period.
- αi (Alpha): The portion of returns not explained by the four factors — representing genuine manager skill (or lack thereof).
- εi: Error term
The Four Factors Explained in Detail
Factor 1: Market Risk Premium (Rm − Rf)
This is the same factor from CAPM. It represents the excess return of the overall stock market over the risk-free rate (typically Treasury bills).
In practice:
- When the market rises, high-beta stocks rise more
- When the market falls, high-beta stocks fall more
- A portfolio with β = 1.2 is expected to move 20% more than the market
Using MarketXLS to measure market sensitivity:
=Beta("AAPL") // Individual stock beta
=Last("AAPL") // Current price for position sizing
You can calculate portfolio beta as the weighted average of individual stock betas.
Factor 2: SMB (Size Factor)
The size factor captures the historical tendency of small-cap stocks to outperform large-cap stocks over long periods.
How SMB is constructed:
- Rank all stocks by market capitalization
- Divide into small (bottom 50%) and big (top 50%)
- SMB = Average return of small stocks − Average return of big stocks
Why small caps outperform (theories):
- Higher information asymmetry (less analyst coverage)
- Lower liquidity (investors require compensation)
- Greater business risk (less diversified operations)
- Survivorship bias in some data sets
Using MarketXLS to analyze size:
=MarketCapitalization("AAPL") // Check market cap
=Last("AAPL") // Current price
A positive SMB loading (si > 0) indicates the portfolio tilts toward small-cap stocks.
Factor 3: HML (Value Factor)
The value factor captures the historical tendency of value stocks (high book-to-market ratio) to outperform growth stocks (low book-to-market ratio).
How HML is constructed:
- Rank all stocks by book-to-market ratio
- Divide into three groups: High (top 30%), Neutral (middle 40%), Low (bottom 30%)
- HML = Average return of High B/M stocks − Average return of Low B/M stocks
Why value stocks outperform (theories):
- Value stocks are riskier (distressed companies)
- Behavioral biases (investors overreact to bad news)
- Mean reversion in fundamentals
- Risk premium for uncertainty
Using MarketXLS to assess value characteristics:
=PERatio("AAPL") // Price-to-earnings ratio
=hf_revenue("AAPL", 2024, 2) // Revenue for fundamental analysis
=Revenue("AAPL") // Annual revenue
A positive HML loading (hi > 0) indicates a value tilt; negative indicates a growth tilt.
Factor 4: WML/MOM (Momentum Factor)
The momentum factor — Carhart's key addition — captures the tendency of recent winners to continue winning and recent losers to continue losing.
How WML is constructed:
- Rank all stocks by their cumulative returns over the past 2-12 months (typically skipping the most recent month to avoid short-term reversal effects)
- Divide into Winners (top 30%) and Losers (bottom 30%)
- WML = Average return of Winners − Average return of Losers
Why momentum works (theories):
- Underreaction to information (investors are slow to update beliefs)
- Herding behavior (investors follow trends)
- Confirmation bias (investors seek information supporting their views)
- Gradual information diffusion
Using MarketXLS to analyze momentum:
=GetHistory("AAPL", "2024-01-01", "2025-01-01", "Monthly")
Pull historical prices and calculate trailing returns to assess momentum characteristics.
A positive WML loading (wi > 0) indicates the portfolio tilts toward recent winners.
Calculating the Carhart Four Factor Model in Excel
Here is a step-by-step process to run a four-factor regression in Excel using MarketXLS for data.
Step 1: Gather Historical Returns
Pull historical prices for your portfolio holdings:
=GetHistory("AAPL", "2020-01-01", "2025-01-01", "Monthly")
=GetHistory("MSFT", "2020-01-01", "2025-01-01", "Monthly")
=GetHistory("JNJ", "2020-01-01", "2025-01-01", "Monthly")
Calculate monthly returns from the price data: Return = (Price_t − Price_{t-1}) / Price_{t-1}
Calculate weighted portfolio returns using your allocation weights.
Step 2: Obtain Factor Data
The factor return data (Rm-Rf, SMB, HML, WML) is published monthly by Kenneth French on his data library (Dartmouth). Download the monthly factor returns and align them with your portfolio return dates.
Step 3: Set Up the Regression
Organize your data in columns:
| Month | Portfolio Excess Return (Ri - Rf) | Rm - Rf | SMB | HML | WML |
|---|---|---|---|---|---|
| Jan 2020 | 2.3% | 1.8% | 0.5% | -1.2% | 0.8% |
| Feb 2020 | -3.1% | -2.5% | -0.3% | 0.4% | -1.5% |
| ... | ... | ... | ... | ... | ... |
Step 4: Run the Regression
Use Excel's =LINEST() function for multiple regression:
=LINEST(Y_range, X_range, TRUE, TRUE)
Where:
- Y_range = Column of portfolio excess returns
- X_range = Four columns of factor returns (Rm-Rf, SMB, HML, WML)
This returns the regression coefficients (factor loadings) and statistics.
Alternatively, use the Data Analysis Toolpak → Regression for a more detailed output including R², adjusted R², t-statistics, and p-values.
Step 5: Interpret the Results
The regression output gives you:
| Output | Meaning | Example |
|---|---|---|
| Intercept (α) | Alpha — excess return not explained by factors | 0.15% monthly |
| β (Market) | Sensitivity to market risk | 1.05 |
| s (SMB) | Exposure to size factor | 0.25 |
| h (HML) | Exposure to value factor | -0.10 |
| w (WML) | Exposure to momentum factor | 0.30 |
| R² | Percentage of returns explained by the model | 0.92 |
Interpreting the example:
- The portfolio has slightly higher than market risk (β = 1.05)
- Tilted toward small caps (s = 0.25, positive)
- Slightly tilted toward growth stocks (h = -0.10, negative HML)
- Strong momentum exposure (w = 0.30)
- Alpha of 0.15% monthly ≈ 1.8% annually — genuine outperformance
- The four factors explain 92% of return variation
Portfolio Attribution with the Four Factor Model
One of the most powerful applications of the Carhart model is performance attribution — breaking down where returns come from.
Example Attribution
Suppose your portfolio returned 15% over the past year, and the risk-free rate was 4%.
Portfolio excess return = 15% − 4% = 11%
Using the four-factor model with these loadings and factor returns:
| Factor | Loading | Factor Return | Contribution |
|---|---|---|---|
| Market (Rm - Rf) | 1.05 | 8.0% | 8.40% |
| SMB | 0.25 | 2.0% | 0.50% |
| HML | -0.10 | -1.5% | 0.15% |
| WML | 0.30 | 3.0% | 0.90% |
| Total factor contribution | 9.95% | ||
| Alpha | 1.05% |
Interpretation: Of the 11% excess return, 9.95% came from factor exposures (market risk, size, value, and momentum tilts) and only 1.05% came from genuine stock-picking skill (alpha). This type of analysis is critical for evaluating fund managers and understanding whether their fees are justified.
Using Sharpe Ratio for Risk-Adjusted Evaluation
After decomposing returns with the four-factor model, use the Sharpe Ratio to evaluate risk-adjusted performance:
=SharpeRatio("AAPL")
A high Sharpe Ratio combined with positive four-factor alpha indicates genuinely skilled management. A high return with zero or negative alpha simply means the manager is taking on more factor risk.
Comparison of Asset Pricing Models
| Feature | CAPM | Fama-French 3-Factor | Carhart 4-Factor | Fama-French 5-Factor |
|---|---|---|---|---|
| Number of factors | 1 | 3 | 4 | 5 |
| Factors included | Market | Market, Size, Value | Market, Size, Value, Momentum | Market, Size, Value, Profitability, Investment |
| Year introduced | 1964 | 1993 | 1997 | 2015 |
| R² (typical) | ~70% | ~90% | ~92-95% | ~93-95% |
| Captures momentum | No | No | Yes | No |
| Captures profitability | No | No | No | Yes |
| Primary use | Cost of equity | Academic research | Fund evaluation | Factor investing |
| Key limitation | Too simplistic | Misses momentum | Momentum may be behavioral | Overlaps with value factor |
Practical Applications of the Carhart Model
1. Mutual Fund Evaluation
The Carhart model was originally developed to evaluate mutual fund performance. By controlling for market risk, size, value, and momentum exposures, you can determine whether a fund manager generates genuine alpha or simply loads up on known risk factors.
2. Factor-Based Portfolio Construction
Use the model to build portfolios with specific factor tilts:
- Momentum strategy: Screen for stocks with strong 12-month returns (skip last month)
- Value strategy: Screen for high book-to-market stocks
- Small-cap strategy: Focus on stocks below median market capitalization
- Multi-factor: Combine all factors for diversified factor exposure
3. Risk Budgeting
Allocate risk to each factor based on your investment views:
- If you believe momentum will be strong, increase WML exposure
- If you expect value to underperform, reduce HML loading
- Use the factor decomposition to ensure no single factor dominates portfolio risk
4. Benchmark Selection
The four-factor model helps identify the appropriate benchmark for a portfolio. If your portfolio has high SMB and HML loadings, a large-cap growth index is not an appropriate benchmark — you should use a small-cap value index instead.
The Momentum Factor: A Deeper Dive
Since momentum is the distinguishing feature of the Carhart model, it deserves additional attention.
Constructing Momentum Portfolios
The standard academic approach to constructing momentum portfolios follows these steps:
- Formation period: At the end of each month, rank all stocks by their cumulative return over the previous 12 months, excluding the most recent month (to avoid the short-term reversal effect).
- Portfolio formation: Go long the top 30% (winners) and short the bottom 30% (losers).
- Holding period: Hold the portfolio for one month, then rebalance.
This 12-1-1 approach (12 months lookback, skip 1 month, hold 1 month) has been the standard since Jegadeesh and Titman's seminal 1993 paper.
Momentum Across Asset Classes
Momentum is not limited to individual stocks. Research has documented momentum effects in:
- Country equity indices: Markets that have risen tend to continue rising
- Currencies: Trending currencies tend to continue trending
- Commodities: Strong commodities tend to stay strong
- Fixed income: Bonds with recent price strength tend to outperform
This cross-asset persistence makes momentum one of the most robust anomalies in finance.
Momentum Risk: Crash Risk
The biggest risk of momentum strategies is sudden reversals, known as momentum crashes. The most notable occurred in March 2009, when momentum strategies suffered devastating losses as beaten-down stocks (losers) suddenly surged while former winners declined. During this month, the momentum factor lost roughly 40% in value — more than a decade of accumulated gains.
Understanding this tail risk is critical for anyone using the Carhart model for portfolio construction. The momentum factor has a positive expected return on average but can experience extreme negative outcomes during market regime changes.
Limitations of the Carhart Four Factor Model
-
Static factor loadings: The model assumes factor exposures remain constant over time, but portfolio characteristics change as markets move.
-
Data requirements: You need at least 36-60 months of return data for reliable regression estimates.
-
Factor timing: The model does not account for managers who dynamically adjust factor exposures.
-
Missing factors: Later research has identified additional factors (profitability, investment, low volatility, quality) not captured by the four-factor model.
-
Momentum crashes: The momentum factor can experience sudden, severe reversals (e.g., 2009 momentum crash), which the model does not predict.
-
Transaction costs: Factor returns are calculated without accounting for transaction costs, which can significantly erode momentum strategy returns in particular.
Building a Four-Factor Dashboard in MarketXLS
Create a comprehensive factor analysis dashboard in Excel:
Portfolio Holdings
| Ticker | Shares | Price | Value | Weight |
| AAPL | 50 | =Last("AAPL") | =B2*C2 | =D2/SUM($D$2:$D$10) |
| MSFT | 30 | =Last("MSFT") | =B3*C3 | =D3/SUM($D$2:$D$10) |
Factor Exposures
| Ticker | Beta | Sharpe | Market Cap | P/E |
| AAPL | =Beta("AAPL") | =SharpeRatio("AAPL") | =MarketCapitalization("AAPL") | =PERatio("AAPL") |
Revenue and Fundamental Analysis
=hf_revenue("AAPL", 2024, 2) // Quarterly revenue
=Revenue("AAPL") // Annual revenue
Use these fundamentals to assess whether holdings are value or growth oriented, which directly impacts HML factor exposure.
Historical Data for Returns Analysis
=GetHistory("AAPL", "2020-01-01", "2025-01-01", "Monthly")
Pull monthly prices for all holdings, calculate returns, and run the four-factor regression as described above.
Frequently Asked Questions
What is the Carhart Four Factor Model?
The Carhart Four Factor Model is an asset pricing framework that explains portfolio returns using four systematic risk factors: market risk premium (Rm - Rf), size (SMB — small minus big), value (HML — high minus low book-to-market), and momentum (WML — winners minus losers). Developed by Mark Carhart in 1997, it extends the Fama-French Three Factor Model by adding momentum. The portion of returns not explained by these four factors is called alpha, representing genuine manager skill.
How is the Carhart model different from the Fama-French model?
The primary difference is the addition of the momentum factor (WML). The Fama-French Three Factor Model uses market risk, size (SMB), and value (HML) to explain returns. Carhart added WML to capture the well-documented tendency of recent winning stocks to continue outperforming and recent losing stocks to continue underperforming. This addition typically improves the model's explanatory power from roughly 90% to 92-95% of diversified portfolio return variation.
How do I calculate alpha using the four-factor model?
Alpha is the intercept (constant term) from regressing your portfolio's excess returns against the four factor returns. In Excel, use LINEST or the Data Analysis Toolpak Regression tool. The dependent variable is your portfolio's monthly return minus the risk-free rate. The independent variables are the four monthly factor returns (Rm-Rf, SMB, HML, WML). The intercept represents monthly alpha — multiply by 12 for an approximate annual figure. Positive alpha indicates outperformance beyond factor exposures.
What data do I need for a four-factor analysis?
You need monthly returns for your portfolio (which you can calculate using =GetHistory() in MarketXLS to pull historical prices), and monthly factor return data from Kenneth French's data library at Dartmouth (freely available). You need at least 36 months of data for reliable estimates, with 60 months being preferable. Align the dates carefully — both your portfolio returns and factor returns must use the same monthly periods.
Is the momentum factor still relevant today?
Yes, momentum remains one of the most robust anomalies in financial markets, documented across multiple time periods, asset classes, and geographies. However, momentum strategies can experience sudden crashes (such as in 2009), and transaction costs can significantly reduce net returns. The factor continues to be widely used in academic research and institutional portfolio management. MarketXLS's =GetHistory() function makes it easy to calculate trailing returns for momentum analysis.
Can the Carhart model be applied to ETFs?
Yes. You can apply the four-factor model to any investment that has a return history, including ETFs, mutual funds, hedge funds, and individual stocks. For ETFs, use =GetHistory() to pull historical prices, calculate monthly returns, and run the regression against the four factors. This is particularly useful for evaluating whether an ETF's returns come from factor exposures or from the ETF manager's selection skill.
Getting Started with Factor Analysis in MarketXLS
MarketXLS provides the market data foundation for four-factor analysis directly in Excel. Pull current prices with =Last(), historical data with =GetHistory(), risk metrics with =Beta() and =SharpeRatio(), and fundamental data with =PERatio(), =Revenue(), and =hf_revenue().
Explore MarketXLS pricing and plans to start building factor analysis models. Visit marketxls.com for portfolio management templates and resources.
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