Testing CAPM Beta Predictions with Future Stock Returns

Summary

The document asks how the CAPM’s prediction about expected returns can be tested using realized returns. It outlines a two-stage empirical setup: estimate each stock’s market beta from a time-series regression of excess returns on the market excess return, then regress next-period stock returns across stocks on those estimated betas. The theoretical prediction is that higher market beta corresponds to higher expected return, with the slope representing the market premium.

The central issue is that expected returns are not directly observed, whereas future realized returns are observable but noisy outcomes. The document frames this distinction as a question and does not include the answer, estimation details, statistical evidence, or discussion of issues such as sampling error and repeated testing. It therefore introduces an asset-pricing test rather than establishing whether the CAPM relation holds in data.

Key ideas

• The CAPM links expected stock returns to market beta.
• Betas are estimated from time-series regressions of excess returns on market excess returns.
• A cross-sectional test relates next-period realized returns to estimated betas.
• Realized future returns serve as noisy observations of expected returns.
• The document poses the empirical interpretation question but does not resolve it.

Tags

# Cross section of expected returns vs cross section of returns


# Cross section of expected returns vs cross section of returns












Typically, one estimates the CAPM beta of stock $i$ of a time series regression of stock excess returns $R_t$ on the market excess return $MRP$:

$R_{t} = \alpha + \beta MRP_t + \epsilon_t$,

where $t$ denotes a period (i.e. month hereafter). The sample for this regression spans from $T-k$ up to $T$.

Repeated for different stocks $i$, one gets a cross section of betas $\beta_i$. According to e.g. Bali et al. (2016), p.131, the CAPM predicts that "there is a positive relation between market beta and expected stock returns, and the slope defining this relation represents the market premium". This relation is then, inter alia, tested by performing "a cross-sectional regression of one-month-ahead future excess stock ($R_{T+1}$) on the given measure" (Bali et al. (2016), p.140), i.e. the CAPM beta:

$R_{i,T+1} = \gamma_1 + \gamma_2 \beta_{i} + \epsilon_{i,T+1}$.

Now my question is, how does one get from a theoretical model that involves expected returns (the first quote), to an empirical model that involves future realized returns (the second quote)? Those two concepts are completely different in my opinion.

Thank you in advance for clarifying.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.