The Capital Asset Pricing Model and Fama-French Three-Factor Model
Equilibrium pricing models, asset pricing anomalies, and multi-factor models
1 Introduction
How should investors be compensated for taking on risk? This seemingly simple question lies at the heart of modern finance and has profound implications for how we price assets, construct portfolios, and evaluate investment performance. In this lecture, we develop one of the most influential answers to this question: the Capital Asset Pricing Model (CAPM), derived independently by William Sharpe, John Lintner, and Jan Mossin in the 1960s.
The CAPM provides a surprisingly elegant prediction: the expected return on any asset should depend on just one thing—its sensitivity to market-wide movements, captured by a single number called beta. Assets that move more dramatically with the market should offer higher expected returns to compensate investors for bearing that systematic risk. This insight earned Sharpe a Nobel Prize in 1990 and remains foundational to how practitioners think about risk and return.
Yet financial markets are messy, and reality rarely conforms perfectly to elegant theory. By the early 1990s, researchers had accumulated substantial evidence that the CAPM’s predictions didn’t fully match the data. Certain types of stocks—particularly small companies and those with high book-to-market ratios—seemed to earn returns higher than the CAPM would predict. These anomalies motivated Eugene Fama and Kenneth French to propose an extended model that incorporates additional risk factors beyond market beta.
We begin by examining the assumptions underlying the CAPM, then derive its central predictions and the Security Market Line. After exploring empirical challenges to the model, we introduce the Fama-French three-factor model as an empirically motivated extension. Throughout, we’ll work through numerical examples that illustrate how these models are applied in practice.
2 The Capital Asset Pricing Model (CAPM)
2.1 CAPM Assumptions
The CAPM rests on a set of idealized assumptions about how markets function and how investors behave. While these assumptions are clearly unrealistic—no one believes markets are truly frictionless or that all investors think identically—they serve as a useful starting point for building intuition about how risk should be priced. Think of them as scaffolding that helps us construct a theoretical benchmark, which we can later relax to understand real-world deviations.
The first set of assumptions concerns market structure. Markets are assumed to be perfectly competitive, meaning no single investor is wealthy enough to move prices through their trades. All information relevant to valuing securities is publicly available and costless to obtain. There are no taxes on investment returns and no transaction costs for buying or selling securities. Finally, investors can borrow or lend unlimited amounts at a common risk-free rate of interest.
The second set of assumptions concerns investor behavior. All investors are assumed to plan for a single period—they invest at the beginning and liquidate at the end, caring only about terminal wealth. They are rational mean-variance optimizers, evaluating portfolios solely by their expected return and variance. Crucially, all investors share identical expectations about the means, variances, and covariances of all asset returns. This “homogeneous expectations” assumption implies that every investor perceives the same efficient frontier and identifies the same tangent portfolio.
These assumptions are strong, but they lead to powerful predictions. And importantly, empirical tests of the CAPM don’t test whether the assumptions hold—they test whether the model’s predictions hold. A model can be useful even if its assumptions are unrealistic, as long as its predictions are approximately correct.
2.2 CAPM Predictions
Given these assumptions, the CAPM generates three interconnected predictions about equilibrium in financial markets.
The first and most famous prediction is that the expected excess return on any asset is proportional to the expected excess return on the market portfolio, with the constant of proportionality being the asset’s beta:
\[E[R_i] = R_f + \beta_i \left( E[R_m] - R_f \right)\]
where:
- \(E[R_i]\) is the expected return on asset \(i\)
- \(R_f\) is the risk-free rate
- \(\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}\) is asset \(i\)’s beta, measuring its sensitivity to market movements
- \(E[R_m]\) is the expected return on the market portfolio
- \(E[R_m] - R_f\) is the market risk premium
This equation tells us that the only risk that matters for expected returns is systematic risk—the portion of an asset’s volatility that moves with the market. Idiosyncratic risk, the asset-specific volatility that can be diversified away, earns no compensation. An asset with \(\beta = 1.5\) should earn 50% more excess return than the market, while an asset with \(\beta = 0.5\) should earn half the market’s excess return.
The second prediction is that the market portfolio—the value-weighted portfolio of all risky assets in the economy—is the optimal risky portfolio. In mean-variance terms, the market portfolio lies on the efficient frontier and is precisely the tangent portfolio when we draw a line from the risk-free rate. Every rational investor should hold some combination of the risk-free asset and this market portfolio.
The third prediction follows directly: all investors hold the same risky portfolio (the market portfolio), differing only in how they allocate between this portfolio and the risk-free asset. More risk-averse investors tilt toward the risk-free asset; more risk-tolerant investors tilt toward the market portfolio (or even borrow at the risk-free rate to leverage their market exposure). But the composition of the risky portion is identical for everyone.
Example: Computing Expected Return Using the CAPM
Suppose the risk-free rate is 3%, and the expected return on the market portfolio is 10%. Stock A has a beta of 1.2, while Stock B has a beta of 0.7. What expected returns does the CAPM predict for each stock?
We apply the CAPM equation \(E[R_i] = R_f + \beta_i(E[R_m] - R_f)\) for each stock.
First, calculate the market risk premium: \[E[R_m] - R_f = 10\% - 3\% = 7\%\]
For Stock A with \(\beta_A = 1.2\): \[E[R_A] = 3\% + 1.2 \times 7\% = 3\% + 8.4\% = 11.4\%\]
For Stock B with \(\beta_B = 0.7\): \[E[R_B] = 3\% + 0.7 \times 7\% = 3\% + 4.9\% = 7.9\%\]
Stock A’s higher beta means it is more sensitive to market movements, so the CAPM predicts it should earn a higher expected return (11.4%) compared to Stock B (7.9%) to compensate investors for bearing greater systematic risk.
2.3 A Sketch of the CAPM Derivation
Understanding where the CAPM comes from helps clarify why beta is the relevant measure of risk. The derivation proceeds in several steps, and while a complete proof requires careful mathematics, the core logic is intuitive.
We start from the assumption that all investors are mean-variance optimizers with homogeneous expectations. From our study of portfolio theory, we know that such investors will choose portfolios on the efficient frontier. When we introduce a risk-free asset, the optimal risky portfolio becomes the tangent portfolio—the point on the efficient frontier that, when connected to the risk-free rate, produces the steepest capital allocation line.
Here’s the key insight: if all investors identify the same tangent portfolio and all hold some combination of this portfolio and the risk-free asset, then in equilibrium, the tangent portfolio must be the market portfolio. Why? Because every share of every asset must be held by someone. If all investors hold the same risky portfolio in different amounts, and these holdings must aggregate to equal the total market capitalization of each asset, then that common risky portfolio must be the market portfolio itself.
Now consider adding a small amount of asset \(i\) to the market portfolio. Using calculus, we can show that the marginal impact of asset \(i\) on portfolio variance depends on \(\text{Cov}(R_i, R_m)\), not on \(\text{Var}(R_i)\). This is because within a well-diversified portfolio, an asset’s idiosyncratic risk averages away; only its covariance with the portfolio remains relevant.
For the market portfolio to be optimal, the marginal reward-to-risk ratio must be equal for all assets. Formally, for any asset \(i\):
\[\frac{E[R_i] - R_f}{\text{Cov}(R_i, R_m)} = \frac{E[R_m] - R_f}{\text{Var}(R_m)}\]
The left side is the excess return per unit of covariance with the market; the right side is the same ratio for the market portfolio itself. If these weren’t equal, investors could improve their portfolio by tilting toward assets with higher ratios.
Rearranging this equilibrium condition gives us the CAPM:
\[E[R_i] - R_f = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} \left( E[R_m] - R_f \right) = \beta_i \left( E[R_m] - R_f \right)\]
Beta emerges naturally as the relevant risk measure because it captures exactly what matters for a diversified investor: the asset’s contribution to portfolio variance through its covariance with the market.
2.4 The Security Market Line
The CAPM’s predictions can be visualized using the Security Market Line (SML), which plots the expected excess return of each security against its beta. If the CAPM holds perfectly, all securities should lie exactly on this line:
\[E[R_i] - R_f = \beta_i \left( E[R_m] - R_f \right)\]
The SML passes through two anchor points: the origin (since an asset with \(\beta = 0\) should earn zero excess return) and the market portfolio (which has \(\beta = 1\) by definition). The slope of the SML equals the market risk premium, \(E[R_m] - R_f\).
The SML provides a powerful framework for evaluating investment opportunities. If a security plots above the SML, it offers more expected return than its beta would justify—it has a positive alpha. Such a security is underpriced and represents a good buying opportunity. Conversely, a security below the SML has negative alpha and is overpriced.
It’s important to distinguish the SML from the Capital Market Line (CML). The CML, is the capital allocation line of the market portfolio. So the CML plots expected return against total risk (standard deviation) for efficient portfolios combining the risk-free asset and the market portfolio. The SML plots expected return against beta for all assets, efficient or not. The CML shows how efficient portfolios are priced; the SML shows how individual assets should be priced if the CAPM holds.
Example: Identifying Mispriced Securities
Consider three stocks with the following characteristics when the risk-free rate is 4% and the expected market return is 11%:
| Stock | Beta | Expected Return |
|---|---|---|
| X | 0.8 | 9.5% |
| Y | 1.3 | 12.0% |
| Z | 1.0 | 11.5% |
Which stocks are overpriced, underpriced, or fairly priced according to the CAPM?
First, calculate what the CAPM predicts each stock should earn:
The market risk premium is \(11\% - 4\% = 7\%\).
For Stock X (\(\beta = 0.8\)): \[E[R_X]^{CAPM} = 4\% + 0.8 \times 7\% = 4\% + 5.6\% = 9.6\%\]
For Stock Y (\(\beta = 1.3\)): \[E[R_Y]^{CAPM} = 4\% + 1.3 \times 7\% = 4\% + 9.1\% = 13.1\%\]
For Stock Z (\(\beta = 1.0\)): \[E[R_Z]^{CAPM} = 4\% + 1.0 \times 7\% = 4\% + 7.0\% = 11.0\%\]
Now compare predicted returns to expected returns and compute alpha:
| Stock | CAPM Predicted | Expected | Alpha | Assessment |
|---|---|---|---|---|
| X | 9.6% | 9.5% | -0.1% | Slightly overpriced |
| Y | 13.1% | 12.0% | -1.1% | Overpriced |
| Z | 11.0% | 11.5% | +0.5% | Underpriced |
Stock Z plots above the SML (positive alpha) and is underpriced—it offers more return than its systematic risk warrants. Stock Y has the most negative alpha and is overpriced. Stock X is approximately fairly priced with a negligible alpha.
3 Empirical Challenges: Size and Value Anomalies
While the CAPM provides an elegant theoretical framework, decades of empirical research have revealed systematic patterns in stock returns that the model cannot explain. Two anomalies proved particularly robust and influential in motivating extensions to the CAPM: the size effect and the value effect.
3.1 The Size Anomaly
In 1981, Rolf Banz published a study documenting that small-capitalization stocks earned substantially higher average returns than large-cap stocks, even after adjusting for their higher betas. When stocks are sorted into portfolios by market capitalization (price times shares outstanding), the smallest decile outperforms the largest decile by several percentage points per year—far more than can be explained by differences in market beta.
This finding posed a direct challenge to the CAPM. If beta is the only determinant of expected returns, then once we control for beta, there should be no systematic relationship between firm size and average returns. Yet the data showed otherwise. Small stocks seemed to earn a premium above and beyond what their market risk exposure would predict.
Researchers proposed various explanations. Perhaps small stocks are genuinely riskier in ways that beta doesn’t capture—they may be more exposed to economic downturns, less liquid, or more prone to information asymmetries. Alternatively, the size effect might reflect market inefficiency: investors systematically neglect small stocks, causing them to be underpriced. The debate continues, but the empirical regularity itself was undeniable.
3.2 The Value Anomaly
Around the same time, researchers documented another puzzling pattern: stocks with high book-to-market (B/M) ratios earned higher average returns than those with low B/M ratios, again beyond what beta could explain. The book-to-market ratio compares a firm’s book value of equity (roughly, its accounting net worth) to its market value of equity (stock price times shares outstanding).
Firms with high B/M ratios are called “value” stocks. These are typically companies trading at low prices relative to their fundamentals—perhaps because they’re in declining industries, facing financial difficulties, or simply out of favor with investors. Firms with low B/M ratios are “growth” stocks, commanding premium valuations because investors expect rapid future growth.
The empirical finding was stark: value stocks systematically outperformed growth stocks by a substantial margin. Constructing a portfolio that went long value stocks and short growth stocks generated significant positive returns on average. Like the size effect, this pattern could not be explained by differences in market beta.
3.3 Implications for the CAPM
Together, these anomalies suggested that beta was not the complete story for explaining expected returns. When researchers estimated the CAPM relationship:
\[E[R_i] - R_f = \alpha_i + \beta_i (E[R_m] - R_f)\]
they found that alphas were not zero, as the CAPM predicts. Instead, alphas varied systematically with firm size and book-to-market ratios. Small-cap stocks and value stocks had positive alphas; large-cap stocks and growth stocks had negative alphas.
This evidence could be interpreted two ways. One interpretation is that markets are inefficient—these anomalies represent genuine mispricings that skilled investors could exploit. The other interpretation is that the CAPM is incomplete. Perhaps size and value characteristics proxy for additional risk factors that the market rationally prices but that the single-factor CAPM misses. This second interpretation motivated Fama and French’s influential three-factor model.
4 The Fama-French Three-Factor Model
In their landmark 1993 paper, Eugene Fama and Kenneth French proposed expanding the CAPM by adding two factors designed to capture the size and value effects empirically. Rather than arguing definitively about whether these factors represent risk or mispricing, they took a practical approach: whatever the source, these factors explain substantial variation in stock returns and should be accounted for when evaluating investment performance.
The Fama-French three-factor model expresses an asset’s expected excess return as a function of three systematic factors:
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_{i,m}(R_{m,t} - R_{f,t}) + \beta_{i,smb} \cdot SMB_t + \beta_{i,hml} \cdot HML_t + \epsilon_{i,t}\]
where:
- \(R_{i,t} - R_{f,t}\) is the excess return on asset \(i\) at time \(t\)
- \(\alpha_i\) is the intercept (should be zero if the model fully explains expected returns)
- \(\beta_{i,m}\) is the loading on the market factor, analogous to CAPM beta
- \(R_{m,t} - R_{f,t}\) is the market excess return
- \(\beta_{i,smb}\) is the loading on the size factor
- \(SMB_t\) is the “Small Minus Big” factor return at time \(t\)
- \(\beta_{i,hml}\) is the loading on the value factor
- \(HML_t\) is the “High Minus Low” factor return at time \(t\)
- \(\epsilon_{i,t}\) is the idiosyncratic error term
4.1 Understanding the Factors
The SMB (Small Minus Big) factor captures the size premium. It is constructed as the return on a portfolio of small-capitalization stocks minus the return on a portfolio of large-capitalization stocks. When SMB is positive, small stocks outperformed large stocks that period; when negative, large stocks outperformed.
A stock with a positive \(\beta_{i,smb}\) behaves like small stocks—it tends to do well when small caps outperform. A stock with a negative \(\beta_{i,smb}\) behaves like large stocks. The factor loading tells us how sensitive the stock is to the small-cap versus large-cap spread.
The HML (High Minus Low) factor captures the value premium. It is constructed as the return on a portfolio of high book-to-market (value) stocks minus the return on a portfolio of low book-to-market (growth) stocks. When HML is positive, value stocks outperformed growth stocks; when negative, growth outperformed value.
A stock with a positive \(\beta_{i,hml}\) behaves like value stocks—it tends to do well in periods when value outperforms growth. A stock with a negative \(\beta_{i,hml}\) behaves like growth stocks.
4.2 Interpreting the Model
The Fama-French model dramatically improves our ability to explain the cross-section of stock returns. Many anomalies that generated significant alpha under the CAPM disappear when evaluated against the three-factor model. The model effectively says: expected returns depend not just on market exposure, but also on exposure to size-related and value-related systematic risks.
If we believe the risk interpretation, then SMB and HML represent compensation for genuine economic risks. Perhaps small firms are more vulnerable to recessions, or value firms are more likely to face financial distress. Investors demand higher expected returns from stocks exposed to these risks.
If we believe the mispricing interpretation, the model still serves a useful purpose: it tells us what a “typical” stock with given characteristics should return. Alpha relative to the three-factor model represents performance above or beyond what size and value exposures would predict—a higher bar for demonstrating investment skill.
Example: Estimating Factor Exposures and Expected Return
A portfolio manager estimates the following factor loadings for Portfolio P using regression analysis:
- \(\beta_{P,m} = 1.1\) (market factor loading)
- \(\beta_{P,smb} = 0.3\) (size factor loading)
- \(\beta_{P,hml} = 0.5\) (value factor loading)
Suppose current market conditions suggest the following expected factor returns: the market risk premium is 6%, the expected SMB return is 2%, and the expected HML return is 3%. The risk-free rate is 2%. What is the expected return on Portfolio P according to the Fama-French model?
Using the Fama-French model (assuming alpha equals zero), expected excess return equals:
\[E[R_P] - R_f = \beta_{P,m} \times (E[R_m] - R_f) + \beta_{P,smb} \times E[SMB] + \beta_{P,hml} \times E[HML]\]
Substituting the values:
\[E[R_P] - R_f = 1.1 \times 6\% + 0.3 \times 2\% + 0.5 \times 3\%\] \[E[R_P] - R_f = 6.6\% + 0.6\% + 1.5\% = 8.7\%\]
Adding back the risk-free rate:
\[E[R_P] = 2\% + 8.7\% = 10.7\%\]
The portfolio’s expected return of 10.7% reflects three sources of systematic risk premium: 6.6% from market exposure, 0.6% from small-cap tilt, and 1.5% from value tilt. The positive loadings on SMB and HML indicate this portfolio has characteristics of small-cap value stocks, which historically have earned higher average returns.
Example: Computing Alpha in the Fama-French Model
A mutual fund earned an average monthly excess return of 0.9% over the past five years. Regression analysis against the three Fama-French factors yields the following estimates:
- \(\beta_m = 0.95\)
- \(\beta_{smb} = 0.25\)
- \(\beta_{hml} = -0.15\)
During this period, the average monthly factor returns were: market excess return = 0.7%, SMB = 0.2%, HML = 0.35%. What was the fund’s alpha?
The fund’s expected return based on its factor exposures would be:
\[\text{Expected excess return} = \beta_m \times MKT + \beta_{smb} \times SMB + \beta_{hml} \times HML\] \[= 0.95 \times 0.7\% + 0.25 \times 0.2\% + (-0.15) \times 0.35\%\] \[= 0.665\% + 0.05\% - 0.0525\%\] \[= 0.6625\%\]
The fund’s alpha is the difference between its actual excess return and what the factors predict:
\[\alpha = 0.9\% - 0.6625\% = 0.2375\% \text{ per month}\]
Annualized (multiplying by 12), this is approximately 2.85% per year.
The fund generated positive alpha even after accounting for its factor exposures. Note that the negative HML loading indicates the fund tilts toward growth stocks, which on average had positive returns during this period (HML = 0.35% means value outperformed). The fund’s growth tilt actually reduced its expected return, making its positive alpha more impressive.
5 Key Takeaways
The Capital Asset Pricing Model provides a foundational framework for understanding the relationship between risk and expected return. Built on idealized assumptions about competitive markets and rational mean-variance investors, it delivers a remarkably simple prediction: expected returns depend only on an asset’s systematic risk, measured by beta. The model emerges naturally from the insight that when all investors hold the same tangent portfolio—which must be the market portfolio in equilibrium—the relevant risk measure becomes an asset’s covariance with the market rather than its total variance. The Security Market Line translates this into a practical tool: assets plotting above the line offer positive alpha and are underpriced, while those below offer negative alpha and are overpriced.
Empirical tests, however, revealed persistent patterns that the CAPM could not explain. Small-cap stocks and value stocks (those with high book-to-market ratios) earned higher returns than their betas would predict, generating positive alphas that persisted across time periods and international markets. These anomalies suggested that either markets are systematically inefficient, or beta alone fails to capture all dimensions of risk that investors care about.
The Fama-French three-factor model addresses these shortcomings by supplementing the market factor with two additional factors: SMB, which captures the small-cap premium, and HML, which captures the value premium. This extended model explains a much larger fraction of the cross-sectional variation in stock returns and has become the standard benchmark against which investment performance is evaluated. Understanding both frameworks equips you to think critically about how risk should be priced, to evaluate whether an investment strategy truly generates skill-based alpha, and to appreciate the ongoing tension between elegant theory and messy empirical reality.
6 Key Formulas Summary
| Concept | Formula | When to Use |
|---|---|---|
| CAPM Expected Return | \(E[R_i] = R_f + \beta_i(E[R_m] - R_f)\) | To calculate the required return on an asset based on its market risk |
| Beta | \(\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}\) | To measure an asset’s sensitivity to market movements |
| Security Market Line | \(E[R_i] - R_f = \beta_i(E[R_m] - R_f)\) | To identify mispriced securities by comparing actual returns to predicted returns |
| Alpha (CAPM) | \(\alpha_i = E[R_i] - [R_f + \beta_i(E[R_m] - R_f)]\) | To measure abnormal return not explained by market risk |
| Fama-French 3-Factor Model | \(R_i - R_f = \alpha_i + \beta_m(R_m - R_f) + \beta_{smb} \cdot SMB + \beta_{hml} \cdot HML + \epsilon_i\) | To explain/predict returns using market, size, and value factors |
| SMB Factor | Return on small stocks minus return on big stocks | To capture the size premium in returns |
| HML Factor | Return on high B/M stocks minus return on low B/M stocks | To capture the value premium in returns |
7 Practice Problems
Practice Problem 1: CAPM Expected Returns
The current risk-free rate is 2.5%, and analysts expect the market to return 9% over the coming year. Technology Corp has a beta of 1.4, while Utilities Inc has a beta of 0.6. Calculate the expected return for each stock according to the CAPM.
Apply the CAPM formula: \(E[R_i] = R_f + \beta_i(E[R_m] - R_f)\)
First, find the market risk premium: \[E[R_m] - R_f = 9\% - 2.5\% = 6.5\%\]
For Technology Corp (\(\beta = 1.4\)): \[E[R_{Tech}] = 2.5\% + 1.4 \times 6.5\% = 2.5\% + 9.1\% = 11.6\%\]
For Utilities Inc (\(\beta = 0.6\)): \[E[R_{Util}] = 2.5\% + 0.6 \times 6.5\% = 2.5\% + 3.9\% = 6.4\%\]
Technology Corp’s higher beta reflects its greater sensitivity to market conditions—technology stocks typically amplify market swings. The CAPM predicts a 11.6% expected return to compensate for this risk. Utilities Inc, with its more stable cash flows and lower market sensitivity, warrants only a 6.4% expected return.
Practice Problem 2: Identifying Mispriced Securities
An analyst has gathered the following information about four stocks. The risk-free rate is 3% and the expected market return is 10%.
| Stock | Beta | Expected Return |
|---|---|---|
| A | 1.2 | 12.0% |
| B | 0.9 | 9.5% |
| C | 1.5 | 13.5% |
| D | 0.5 | 7.0% |
Determine which stocks are overpriced, underpriced, or fairly priced according to the CAPM. Rank them by their attractiveness as investments.
Calculate the CAPM-predicted return for each stock and compare to expected return.
Market risk premium: \(10\% - 3\% = 7\%\)
| Stock | Beta | CAPM Prediction | Expected | Alpha | Assessment |
|---|---|---|---|---|---|
| A | 1.2 | \(3\% + 1.2(7\%) = 11.4\%\) | 12.0% | +0.6% | Underpriced |
| B | 0.9 | \(3\% + 0.9(7\%) = 9.3\%\) | 9.5% | +0.2% | Slightly underpriced |
| C | 1.5 | \(3\% + 1.5(7\%) = 13.5\%\) | 13.5% | 0.0% | Fairly priced |
| D | 0.5 | \(3\% + 0.5(7\%) = 6.5\%\) | 7.0% | +0.5% | Underpriced |
Ranking by alpha (most to least attractive): A (+0.6%), D (+0.5%), B (+0.2%), C (0.0%).
All four stocks have non-negative alphas, meaning none are overpriced. Stock A is most attractive because it offers the highest excess return relative to its systematic risk. Stock C is fairly priced—it offers exactly the return warranted by its high beta.
Practice Problem 3: Fama-French Expected Returns
A portfolio has the following estimated factor loadings:
- Market beta (\(\beta_m\)) = 1.0
- SMB loading (\(\beta_{smb}\)) = 0.5
- HML loading (\(\beta_{hml}\)) = -0.3
Expected factor premiums are: Market risk premium = 5.5%, SMB premium = 3%, HML premium = 4%. The risk-free rate is 3.5%. Calculate the expected return according to the Fama-French model.
Using the Fama-French three-factor model (assuming alpha = 0):
\[E[R_P] - R_f = \beta_m(E[R_m] - R_f) + \beta_{smb} \times E[SMB] + \beta_{hml} \times E[HML]\]
Substituting: \[E[R_P] - R_f = 1.0 \times 5.5\% + 0.5 \times 3\% + (-0.3) \times 4\%\] \[= 5.5\% + 1.5\% - 1.2\% = 5.8\%\]
Therefore: \[E[R_P] = R_f + 5.8\% = 3.5\% + 5.8\% = 9.3\%\]
The portfolio’s exposures tell an interesting story. The positive SMB loading (+0.5) indicates a tilt toward smaller stocks, adding 1.5% to expected return. The negative HML loading (-0.3) indicates a growth-stock tilt, which actually reduces expected return by 1.2% since value historically outperforms growth. Net effect: this small-cap growth portfolio is expected to earn 9.3%.
Practice Problem 4: Fama-French Alpha Calculation
A hedge fund reports an average monthly return of 1.2%. Over the same period, the risk-free rate averaged 0.25% per month, and monthly factor returns averaged: Market excess return = 0.8%, SMB = 0.15%, HML = 0.25%. The fund’s estimated factor loadings are: \(\beta_m = 1.15\), \(\beta_{smb} = -0.20\), \(\beta_{hml} = 0.40\). Calculate the fund’s monthly and annualized alpha.
First, calculate the fund’s actual excess return: \[\text{Excess return} = 1.2\% - 0.25\% = 0.95\% \text{ per month}\]
Calculate expected excess return based on factor exposures: \[\text{Expected excess return} = \beta_m \times MKT + \beta_{smb} \times SMB + \beta_{hml} \times HML\] \[= 1.15 \times 0.8\% + (-0.20) \times 0.15\% + 0.40 \times 0.25\%\] \[= 0.92\% - 0.03\% + 0.10\%\] \[= 0.99\%\]
Monthly alpha: \[\alpha = 0.95\% - 0.99\% = -0.04\%\]
Annualized alpha: \[\alpha_{annual} = -0.04\% \times 12 = -0.48\%\]
Despite appearing profitable with a 1.2% monthly return, this hedge fund has a slightly negative alpha of -0.48% annually. Its returns can be largely explained by its factor tilts: high market exposure (1.15), a large-cap tilt (-0.20 on SMB), and a value tilt (0.40 on HML). The fund isn’t generating value beyond what passive exposure to these factors would provide.
8 Ask an LLM
Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:
- The CAPM assumes all investors share identical expectations about asset returns. How would the model’s predictions change if investors had different beliefs about expected returns or risks? What real-world phenomena might this help explain?
- Value stocks have historically outperformed growth stocks (the HML premium), yet many professional investors prefer growth stocks. What behavioral biases or institutional constraints might explain this apparent paradox?
- The Fama-French factors are constructed using specific methodologies for defining “small” versus “big” and “value” versus “growth.” How sensitive are the factor premiums to these construction choices, and what does this suggest about whether these factors represent true risk or data-mining artifacts?
- How would you use the CAPM and Fama-French models together to evaluate whether a mutual fund manager has genuine stock-picking skill versus simply tilting toward small-cap or value stocks?
- Climate risk and ESG considerations are increasingly important to investors. Could these represent additional systematic risk factors that should be added to asset pricing models, similar to how Fama and French added size and value?