Empirical Tests of CAPM and Their Limits

Why Testing CAPM Requires More Than Comparing Returns

The CAPM makes a statement about expected returns: in the model, an asset's expected excess return is proportional to its beta relative to the market portfolio. Researchers, however, observe realized returns over a finite sample, not the expected returns themselves. They must estimate beta, choose an observable stand-in for the market portfolio, and decide whether the observed deviations from the model are larger than sampling variation could plausibly produce.

These choices matter to an actuarial analyst using CAPM estimates as well as to a researcher evaluating the model. A regression result is evidence conditional on its return data, proxy, and statistical assumptions; it is not a direct observation of the CAPM's unobservable expected returns. This tutorial is aligned with the SOA VEE Accounting and Finance syllabus.

The Time-Series Regression

A common first test uses observations over time for one asset or portfolio. For period \(t\), let \(R_{it}\) be the asset's realized return, \(R_{ft}\) the risk-free return, and \(R_{Mt}\) the return on the chosen market proxy. The regression is

Here, \(\beta_i\) is the estimated sensitivity to the proxy's excess return, \(\alpha_i\) is the regression intercept, and \(\varepsilon_{it}\) is the period-specific residual. Under the CAPM, with the true market portfolio and suitable conditions on the return-generating process, the population intercept is zero. A sample intercept need not equal zero: realized returns fluctuate, so a nonzero estimate can arise by chance.

The regression beta is the slope describing how the asset's excess return co-moves with the proxy's excess return. With an intercept, the ordinary least-squares estimate is the sample covariance of those two series divided by the sample variance of the proxy series. It is not the correlation: correlation is standardized and has no return-per-return scale.

Why the OLS Slope Has This Form

The slope formula follows directly from least squares. Write \(x_t=R_{Mt}-R_{ft}\) and \(y_t=R_{it}-R_{ft}\). Least squares chooses \(\alpha\) and \(\beta\) to minimize the sum of squared residuals, \(\sum_t(y_t-\alpha-\beta x_t)^2\). Differentiating with respect to the intercept and setting the derivative to zero gives \(\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}\). Substituting this intercept into the slope condition gives the covariance-over-variance result:

The numerator measures whether the asset and market-proxy excess returns tend to be above or below their respective sample means together. The denominator scales that co-movement by variation in the proxy. If the proxy has no variation, the denominator is zero and its slope cannot be estimated. This calculation estimates a historical linear association; interpreting it as a forward-looking required-return beta requires the additional judgment that the relationship is informative for the future.

Testing Whether Alpha Is Zero

A common time-series test asks whether the population intercept is zero. Under the regression conditions stated above, calculate the estimated intercept's standard error and form a \(t\)-statistic. For \(n\) observations and one market-proxy regressor plus an intercept, the conventional residual degrees of freedom are \(n-2\). For a two-sided test at significance level \(\gamma\), reject the zero-alpha null if the absolute \(t\)-statistic exceeds the corresponding critical value, or equivalently if its \(p\)-value is below \(\gamma\).

A rejection means that zero alpha is inconsistent with the data under the specified regression and inference assumptions at the chosen significance level. It does not, by itself, identify which assumption or model component is responsible. A failure to reject is also not proof that alpha is exactly zero; it may reflect limited information or a noisy estimate.

Cross-Sectional Tests of the Security Market Line

A second approach tests whether average returns across assets line up with their betas. A standard two-pass procedure first estimates each asset's beta from time-series returns. In the second pass, researchers relate the assets' average excess returns to those estimated betas. In the CAPM, the cross-sectional relation has zero intercept and a slope equal to the expected market risk premium. In practice, researchers must account for uncertainty in both the average returns and the first-pass betas when assessing whether the estimates fit that prediction.

Why Empirical Conclusions Are Limited

An empirical CAPM test is a joint-hypothesis test. A rejection may indicate that the CAPM prediction fails, but it may instead—or also—reflect a poor market proxy, errors in the return data, an unsuitable sample period, or incorrect assumptions used for statistical inference. The true market portfolio includes all relevant risky assets and is not directly observed. A broad equity index may be a practical proxy, but it is not the same object as the theoretical market portfolio. Consequently, a test using that index evaluates the CAPM together with the adequacy of the proxy.

Several sources of uncertainty deserve attention:

• Sampling noise. Returns vary substantially from period to period. A finite sample produces uncertain estimates of alpha, beta, and average returns. A nonzero estimate need not be statistically significant, and a statistically insignificant estimate need not be exactly zero.
• Estimated-beta error. First-pass betas are estimates, not fixed known quantities. In a second-pass regression, treating noisy betas as exact can distort the estimated risk-return slope and its standard errors. Limited variation in betas makes the comparison especially imprecise.
• Proxy choice. Different plausible market proxies can produce different beta estimates and different alpha estimates. A changed result may reflect proxy construction rather than a change in the asset's underlying economics.
• Return measurement. Frequency, timing, dividend treatment, and the alignment of asset, market, and risk-free returns must be consistent. Data errors or selective inclusion of assets can also affect conclusions.
• Error dependence. Returns may exhibit changing variance or serial dependence. Conventional standard errors can then misstate uncertainty. Robust inference can address specified forms of dependence, but it cannot make the proxy identical to the true market portfolio.

Common Exam Trap: Treating a Regression Result as a Direct Verdict

A common mistake is to say, “The estimated alpha is positive, so CAPM is false.” The reasoning skips two steps: whether the estimate is statistically distinguishable from zero, and whether the tested regression uses a suitable proxy and valid inference assumptions. A positive point estimate can arise from sampling noise. Even a statistically significant intercept is evidence against the specified empirical regression, not a diagnosis that isolates the theoretical CAPM as the sole source of failure.

The reverse error is equally important. A small or statistically insignificant alpha does not prove CAPM. The test may have low power, the beta estimate may be noisy, or the proxy may obscure a relation that would be clearer with the true market portfolio. State the conclusion at the level supported by the analysis: describe the estimate, its uncertainty, the proxy and period, and the limits of the test.