Why CAPM Does Not Imply a Perfect Return Regression
Summary
This discussion clarifies the difference between the CAPM relation among expected returns and beta, and a regression of realized returns through time. CAPM says that expected excess return is related to market beta; it does not assert that each period’s realized security return equals beta times the market return plus the risk-free rate. Idiosyncratic variation and random outcomes can therefore leave observations away from a fitted line even when the expected-return relation holds.
The answer also notes that expected returns and betas are estimated from finite samples, so their estimated values need not lie exactly on the theoretical Security Market Line. A perfect fit in realized-return regression is not a CAPM requirement, and an R-squared of one would indicate a perfect sample linear relationship instead. The explanation is conceptual and does not assess empirical CAPM performance or specify a regression design.
Key ideas
- CAPM constrains expected returns rather than every realized return.
- A realized-return regression can contain residual variation even if the expected-return relation holds.
- Finite-sample estimates of expected returns and beta need not lie exactly on the theoretical Security Market Line.
- An R-squared of one describes a perfect sample fit, not a necessary consequence of CAPM.
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# Error term, R-square and perfectly holding CAPM?
# Error term, R-square and perfectly holding CAPM?
It is known that if the CAPM holds then the E(R) = CAPM predicted return and all securities lie on the SML. However, in each period, there is an error term that is non-zero in every single observation but that is 0 on average (as in any regression) In my view, if the CAPM perfectly holds and we run a regression between the market portfolio and a security, the R square should be 1, however, we would still see deviations from the line of best fit (because of the idiosyncratic risk represented by the error term). However, examples show that if the R square is 100%, then all points are on the line of best fit: .
Is my logic flawed? How would it look graphically if security returns are regressed on the mean-variance efficient market portfolio (theoretically, not empirically)? Wouldn't we see points deviating from the line of best fit (with Beta as slope)?
## Answer by Richard Hardy (score 2)
https://quant.stackexchange.com/a/74847
The CAPM states that $$ \mathrm{E}(R_i)-r_f=\beta_i[\mathrm{E}(R_m)-r_f]. $$ None of these quantities are observable, excerpt for $r_f$. The data that is typically used in relation to the CAPM are realized returns $r_{i,t}$ and $r_{m,t}$ for $t=1,\dots,T$ from which we can estimate the three unknown quantities. We do not expect the estimated version of the CAPM, $$ \hat{\mathrm{E}}(R_i)-r_f=\hat\beta_i[\hat{\mathrm{E}}(R_m)-r_f], $$ to hold precisely because of the estimation imprecision. Therefore, we do not expect the estimates $\hat{\mathrm{E}}(R_i)$ and $\hat\beta_i$ to lie precisely on a straight line, regardless of whether $\mathrm{E}(R_i)$ and $\beta_i$ do (i.e. regardless of whether the CAPM actually holds or not).
> if the CAPM perfectly holds and we run a regression between the market portfolio and a security, the R square should be 1
This is even further from the truth, because the CAPM does not say $$ r_{i,t}-r_f=\beta_i[r_{m,t}-r_f] $$ for some or all $t=1,\dots,T$. Even if the hypothesized relationship between the expected values and the beta, $\mathrm{E}(R_i)-r_f=\beta_i[\mathrm{E}(R_m)-r_f]$, holds, this does not imply a perfect linear relationship for the realizations $r_{i,t}$ and $r_{m,t}$ of the random variables $R_i$ and $R_m$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.