Why Mean-Variance Efficiency Implies the CAPM
Summary
The note explains the Roll critique: for a portfolio on the mean-variance efficient frontier, asset excess returns satisfy a CAPM-style relation using that portfolio as the benchmark. It gives the efficient portfolio’s risky-asset weights in terms of the covariance matrix and expected excess returns, then derives the portfolio variance and each asset’s covariance with the portfolio. Their ratio yields the CAPM beta relation when the portfolio has a nonzero expected excess return.
The result is an algebraic identity for an efficient portfolio, rather than evidence that the market portfolio is efficient or that the CAPM holds empirically for a particular market. The answer also states the converse: if the relation holds with the portfolio as benchmark, that portfolio must be efficient. The discussion does not address estimation error, market frictions, or how to identify the true efficient portfolio from data.
Key ideas
- A mean-variance efficient portfolio has weights proportional to the inverse covariance matrix applied to expected excess returns.
- The portfolio’s variance and its covariance with each asset imply the CAPM relation using that portfolio as the benchmark.
- The relation requires a portfolio with nonzero expected excess return in the derivation shown.
- The converse stated in the answer is that a portfolio satisfying this CAPM relation is mean-variance efficient.
- The identity does not establish that the actual market portfolio is efficient or validate an empirical CAPM.
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# Roll Critique - CAPM and mean variance tautology?
# Roll Critique - CAPM and mean variance tautology?
Wikipedia introduces the Roll Critique mean-variance tautology:
Any mean-variance efficient portfolio $R_p$ satisfies the CAPM equation exactly: $$ E(R_i) = R_f + \beta_{ip}[E(R_p) - R_f] $$ A portfolio is mean-variance efficient if there is no portfolio that has a higher return and lower risk than those for the efficient portfolio. Mean-variance efficiency of the market portfolio is equivalent to the CAPM equation holding. This statement is a mathematical fact, requiring ''no'' model assumptions."
Does anyone have a simple proof or intuition of this. The mean variance frontier is a parabola with expected return on the left - the tangent line (sharpe ratio) is different for each point in the parabola,if you used any portfolio, you would get a different sharpe ratio.
I know the answer is very near but am not smart enough to see it.
Maybe my question is: in what way does the CAPM represent optimal risk and return - is there a relationship to the Sharpe Ratio?
$$ \beta_{\text{CAPM}}= \mathrm{Cov}(x,m)/\mathrm{Var}(x) \\ \text{Sharpe}=\mathrm{E}(x)/\mathrm{Stdev}(x). $$ Also it is described as a tautology - for example in the Beta anomaly, Betas do not line up with Returns (too flat), but the Roll Critique wording is very strong that mean variance efficiemcy and CAPM are exactly the same,not approximately.
## Answer by Alphie (score 3)
https://quant.stackexchange.com/a/78436
Let $R$ denote the vector of risky asset returns, $\Sigma:=\text{Cov}[R]$ the covariance matrix of returns, $\mu:=E[R]$ the vector of expected returns, and $r:=R_f$ the risk-free rate.
Recall that the mean-variance efficient portfolio $R_p$ with mean $p:=E[R_p]$ has weights
$$w_{p}:=\frac{\Sigma^{-1}(\mu-1r)}{(\mu-1r)'\Sigma^{-1}(\mu-1r)} (p-r)$$
in the risky assets and $1-1'w_{p}$ in the risk-free asset.
Now, it is easy to check that
$$\text{V}[R_{p}]=\frac{(p-r)^2}{(\mu-1r)'\Sigma^{-1}(\mu-1r)} $$
$$\text{Cov}[R,R_p]=\frac{\mu-1r}{(\mu-1r)'\Sigma^{-1}(\mu-1r)} (p-r)$$
so that $\mu-1r=\frac{\text{Cov}(R,R_{p})}{\text{V}(R_{p})}[p-r]$ if $p\neq r$. Therefore the CAPM holds for $R_p$.
Note that the converse also holds, i.e. if $R_p$ satisfies the CAPM, then $R_p$ must be mean-variance efficient.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.