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.

• Edited:Sharpe ratio is the tangent line to the parabola, max sharpe ratio is a single point. It likely intersects with the risk free rate. If the roll critique were restricted to assets on this two fund theorem line (as opposed to any on the parabola)then I think I understand. Rerrading the quote it seems to refer to max sharpe ratio which is the tangency 2 fund portfolio only, then the answer is simple. Apr 3, 2023 at 13:50
• To be perfectly honest, the following statement is quite confused, "In short, it is saying that any asset on the parabola will solve the CAPM exactly because the CAPM represents the best possible trade-off between risk and return, which is the same concept as mean-variance efficiency." Apr 3, 2023 at 14:24
• Thanks you are right, I have removed that line. Apr 3, 2023 at 16:11
• For future reference: 1.from the CAPM, RFR+B*(R-RFR), B*(R-RFR) is the premium for that stock. Therefore B*(R-RFR)/stdev is the sharpe ratio for a stock, 2.the CML is the optimal tangent line with a risk free asset, the slope represents the maximum sharpe ratio. Therefore any asset on the CML will return the same sharpe ratio and will reproduce the same Beta, representing the optimal risk/return relationship. Apr 11, 2023 at 16:55
• Keep in mind that the Roll Critique and the Black Jensen and Scholes (1972) Critique (or "Beta Anomaly" as you rightly call it) are separate and distinct. The former is more theoretical the latter more empirical. Feb 22 at 19:58

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.