Brownian Bridge™

1. In the CAPM derivation via the mean-variance efficient frontier, the Security Market Line is derived by constructing a portfolio of asset $i$ and the market portfolio $M$ , then requiring tangency at $\alpha = 0$ . What is the economic rationale for evaluating at $\alpha = 0$ ?

Because investors hold zero weight in individual assets in equilibriumBecause asset

i

is already contained in

M

, so the efficient frontier in the

(\sigma, E[R])

space must be tangent to the CML at this point — adding more of asset

i

would be a redundant deviation from the market portfolioBecause

\alpha = 0

minimises tracking error to the market portfolioBecause the drift cancels at

\alpha = 0

by Girsanov's theorem

2. The APT (Ross 1976) derives that $\mathbb{E}[R_i] = \lambda_0 + \sum_k \beta_{i,k} \lambda_k$ without specifying which factors to use. What is the minimal assumption required to derive this result — and what does the APT explicitly NOT require that CAPM does?

APT requires normally distributed returns; CAPM does notAPT requires a risk-free asset; CAPM does notAPT requires only no-arbitrage and a factor structure for returns; CAPM additionally requires mean-variance optimisation by all investors and identical beliefsAPT requires that factor risk premia are positive; CAPM does not

3. In the Fama-MacBeth two-pass regression, the second-pass cross-sectional regression at time $t$ regresses $R_{i,t}$ on estimated betas $\hat{\beta}_{i,k}$ . What is the standard error of the estimated factor premium $\hat{\lambda}_k$ , and why does Shanken (1992) correction inflate it?

\text{SE}(\hat{\lambda}_k) = \text{std}(\hat{\gamma}_{k,t})/\sqrt{T}

, uncorrected; Shanken adjusts for cross-sectional heteroskedasticity

\text{SE}(\hat{\lambda}_k) = \text{std}(\hat{\gamma}_{k,t})/\sqrt{T}

, uncorrected; Shanken adjusts for the fact that

\hat{\beta}

are estimated with error, creating an errors-in-variables bias that OLS ignores

\text{SE}(\hat{\lambda}_k) = 1/\sqrt{N}

where

N

is number of assets; Shanken adjusts for time-series autocorrelation

\text{SE}(\hat{\lambda}_k) = \text{std}(R_{i,t})/\sqrt{T}

; Shanken adjusts for the factor correlation structure

4. Roll's critique (1977) states that the CAPM is untestable because the true market portfolio — which must include all investable assets (human capital, real estate, private equity, foreign assets) — is unobservable. Therefore, any test of CAPM using a proxy portfolio (e.g., the S&P 500) is a joint test of the CAPM and the adequacy of the proxy, making individual rejection ambiguous.

truefalse

5. Fama and French (1992) find that size (SMB) and value (HML) effects are not explained by CAPM beta. Two competing interpretations exist: (1) SMB and HML are risk factors — small-cap and value stocks are riskier in ways beta doesn't capture; (2) SMB and HML are mispricing — investors systematically overprice large-cap growth stocks and underprice small-cap value. Which argument is falsifiable, and what evidence discriminates between them?

Both are equally falsifiable; evidence from international markets discriminates between themThe risk story is falsifiable via consumption-CAPM tests; the mispricing story cannot be tested. International evidence supports the risk story since the value premium appears globallyNeither story is testable since asset pricing models are always observationally equivalentThe mispricing story is falsifiable; the risk story is definitionally true since any return difference must reflect some risk

6. You have estimated a Fama-French three-factor model for a long/short equity fund and find $\hat{\alpha} = 0.4\%$ per month with $t$ -statistic 1.8. The fund has been operating for 4 years (48 months). Is this alpha statistically significant, and what is the minimum track record (in months) needed for significance at the 5% one-sided level, assuming the alpha estimate is correct?

Significant at 5%; minimum track record is 36 monthsNot significant (t < 1.96 for two-sided); minimum track record ≈ 67 months using the asymptotic Sharpe formula for monthly dataSignificant at 10% but not 5%; minimum track record depends on the SR, approximately 58 monthsNot significant; minimum track record requires the information ratio to exceed 2.0

Quiz: Factor Models: CAPM, APT, and Fama-French

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