Empirical tests of CAPM: Putting beta on the witness stand

"Theory is elegant. Data is brutally honest." — your econometrics professor, after grading regression papers

You're already familiar with beta estimation techniques (we just learned how to measure beta properly) and the Security Market Line and alpha (we know how to spot mispricing). Now we move from theory rehearsal to courtroom drama: testing whether CAPM actually explains returns in the wild.

Why care? Because if CAPM fails empirically, then the clean intuition that expected return is a simple linear function of beta fails too — and that matters for portfolio construction, performance evaluation, and whether you can still pretend market risk is the only game in town.

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What is an empirical test of CAPM?

An empirical test attempts to check whether the relationship

• Expected return of an asset = riskfree rate + beta * (expected market premium)

holds in observed asset returns. Practically, this means checking whether beta explains cross-sectional differences in average returns and whether estimated alphas are zero.

Think of it as: we estimate betas (remember the beta estimation techniques), we look at realized returns, and we ask: do high-beta assets reliably deliver higher average returns? Or do we get a mess of non-zero alphas and patterns that scream for more factors?

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How do practitioners test CAPM? (The toolbox)

1) Time-series regressions (the market model)

Estimate beta for each asset i via OLS:

R_i,t - R_f,t = alpha_i + beta_i * (R_m,t - R_f,t) + epsilon_i,t

If CAPM holds, alpha_i should be zero on average. This is what you do when you were estimating betas.

2) Cross-sectional regressions (Fama-MacBeth 1973)

Step 1: Estimate betas using time-series regressions.
Step 2: For each period t, run cross-sectional regression of realized returns on betas:

R_i,t = gamma_0,t + gamma_1,t * beta_i + error_i,t

If CAPM holds, the average gamma_1 across t should equal the market premium, and gamma_0 should be the riskfree rate (or zero if we use excess returns). This method gives standard errors that account for cross-sectional dependence.

3) GRS test (Gibbons, Ross, Shanken)

A joint test: are all alpha_i = 0 simultaneously? Uses the matrix of time-series residuals and factors to construct an F-statistic. Powerful for asset portfolios (especially when you test many assets at once).

4) Portfolio sorts and binned tests

Sort assets into portfolios by beta and compare average returns across bins. Simpler, more visual, less prone to measurement error per asset.

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Econometric speed bumps and how we deal with them

Real-world data is not neat. The main issues:

• Errors-in-variables / measurement error in beta: Short estimation windows make betas noisy. Noisy betas attenuate the cross-sectional slope toward zero, making CAPM look better than it is. Fixes: use longer windows, instrument betas, or apply errors-in-variables corrections.

• Time-varying betas: Betas change with market regimes. Using rolling windows or conditional models helps.

• Non-synchronous trading and thin liquidity: This biases beta estimates, especially for small stocks.

• Factor misspecification: If relevant risk factors are omitted, alpha picks up the slack and CAPM fails.

• Sampling and survivorship biases: Watch your dataset.

Practical recipe: estimate stable betas using appropriate windows (balance bias-variance), check with portfolio sorts, use Fama-MacBeth for cross-section, and run a GRS to test joint alpha = 0.

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What do empirical tests usually find? (Short answer: mixed, leaning skeptical)

• Many classic studies find that beta does explain some cross-sectional variation, but not enough. Low beta stocks often outperform what CAPM predicts, and high-beta stocks underperform — the famous low-beta anomaly.

• Size and value patterns (small-cap and high book-to-market stocks earning higher returns) are not explained by CAPM. This motivated multifactor models (hello Fama-French).

• Alpha is often non-zero for many portfolios, and the GRS test frequently rejects the null that all alphas are zero.

• Fama-MacBeth slopes on beta are often statistically weak once you control for other characteristics.

In plain speak: CAPM captures a kernel of truth but misses many predictable patterns.

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Examples and analogies

• Analogy: CAPM is like a single-lens camera that assumes the world is two-dimensional. It captures the broad shapes (market risk) but misses texture and color (size, value, momentum).

• Empirical workflow example:

  1. Estimate asset betas with five years of monthly returns (time-series regression).
  2. Form five portfolios by beta and compute average excess returns.
  3. Run Fama-MacBeth cross-sectional regressions to see whether beta explains returns.
  4. Run GRS to test joint zero-alpha.

Expected outcome: you might see a positive relation, but betas explain only part of the cross-section.

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Table: Quick comparison of tests

Test	What it checks	Strengths	Weaknesses
Time-series regressions	alpha_i = 0 for each asset	Simple, gives betas	Multiple tests, ignores cross-sectional dependence
Fama-MacBeth	Cross-sectional relation of returns on beta	Correct SEs for CS dependence	Requires good beta estimates
GRS	Joint test alpha = 0	Powerful joint inference	Sensitive to model misspecification
Portfolio sorts	Visual, intuitive	Reduces measurement noise	Loses asset-level info

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Why do people keep misunderstanding this?

Because CAPM is elegant and easy to teach. It's the model students fall in love with. But financial markets are messy. If you conflate a useful first-order approximation with a complete law of nature, you'll be disappointed.

Ask yourself: would you bet your career on a one-factor theory when size, value, momentum, liquidity, and behavioral frictions all push returns around? Probably not.

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Key takeaways (punchy, useful)

• CAPM is a baseline, not a gospel. Empirical tests show it captures some cross-sectional variation but leaves persistent anomalies.
• Measurement matters. Noisy betas bias results; careful beta estimation (see previous topic) is critical to fair tests.
• Use multiple tests. Fama-MacBeth, GRS, and portfolio sorts together tell a richer story than any one test.
• If alphas aren’t zero, consider more factors. That’s exactly why multifactor models emerged.

Final thought: CAPM built the stage. Empirical tests reveal the actors. Some days the market behaves like a one-man show; other days it’s a chaotic ensemble cast. Your job is to listen to the data, not the plot summary.

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Want a quick exercise to cement this? Take 50 stocks, estimate betas with 3 years of monthly data, form quintile portfolios by beta, then run a Fama-MacBeth. Report whether the cross-sectional slope on beta is significantly positive and whether alpha is close to zero. Bring snacks; bring code; expect surprises.