The CML, Perfect Correlation, and the Limits of a CAPM Derivation

# Deriving the CAPM from the CML


# Deriving the CAPM from the CML












In the paper "A Simple Derivation of the Capital Asset Pricing Model from the Capital Market Line" the authors reason:

Given the CML

$$R_p = R_f +\frac{R_m - R_f}{\sigma_m}\sigma_p$$

where:

- $R_p$ is the return on an efficient portfolio

- $R_f$ is the risk-free rate

- $R_m$ is the return on the market portfolio

- $\sigma_m$ is the standard deviation of returns on the market portfolio

- $\sigma_p$ is the standard deviation of returns on efficient portfolio p.

They claim that:

> Efficient portfolios along the CML and are perfectly correlated with the market portfolio.

Based on this statement they can extend the CML-equation by multiplication with $1 = \rho_{pm}$, i.e. the correlation between the efficient portfolio P and the market-portfolio, to define the return on any portfolio as a function of its total risk:

$$ R_p = R_f +\frac{R_m - R_f}{\sigma_m}\sigma_p\rho_{pm} $$

From which the CAPM-formula immediately follows.

Now I have two questions:

- The statement that all efficient portfolios are perfectly correlated seems wrong, as only holding the risk-free asset is an efficient portfolio. The risk-free asset is not correlated to the market-protfolio at all. So this statement cannot be correct? Is there really in error in the paper? Where is my misunderstanding here?

- If the simple approach above does not work, is the idea in general salvagable, and can the CAPM be derived from the CML?

## Answer by Fab (score 2, accepted)

https://quant.stackexchange.com/a/70862

This is a terrible paper.

$\rho_{pm}=1$ is trivially true, in a CAPM with risk free asset, for all efficient portfolios on the CML (combinations of risk free asset and the market portfolio) (except, as you point out, for the risk-free portfolio itself, but that corner case can be dispatched with trivially.)

So, sure, you can multiply with $\rho_{pm}=1$ for portfolios that are on that line.

However, the author then proceeds to claim (implicitly) that the equation holds also for other, non-efficient portfolios, thus begging the question. Yes, the CAPM equation does indeed hold for those other, non-efficient portfolios, but only if you include the $\rho_{pm}$, which is now not $=1$ anymore. So the original justification for including it doesn't hold.

This is like saying: I will now derive Ohm's law. We do know that $V=I$ in the special case that R equals $1$ (ignore units..). Thus, I can multiply the right hand side with R, since R equals 1. Therefore, we have $V=RI$. Now I perform some complicated algebraic operations, $V/I = R$, invert, $I/V=1/R$, $I=V/R$, and boom, hey presto, I've derived Ohm's law.

## Answer by phdstudent (score 3)

https://quant.stackexchange.com/a/69426

> The statement that all efficient portfolios are perfectly correlated seems wrong, as only holding the risk-free asset is an efficient portfolio. The risk-free asset is not correlated to the market-protfolio at all. So this statement cannot be correct? Is there really in error in the paper? Where is my misunderstanding here?

The statement is correct. All efficient portfolios are perfectly correlated. As all efficient portfolios are a combination of risk-free and tangency/market portfolio. You example is an extreme example (where one only holds the risk free). Under standard Mean-Variance preferences, you would only be on such a portfolio if your risk-aversion is infinite. So ignoring that corner solution, all efficient portfolios are indeed perfectly correlated. Think about a portfolio that has $100\% - \epsilon$ invested on the risk-free and $\epsilon$ invested on the market with $\epsilon$ being very small. Even such portfolio would be perfectly correlated with the market.

> If the simple approach above does not work, is the idea in general salvagable, and can the CAPM be derived from the CML?

It does work!

## Answer by demully (score 2)

https://quant.stackexchange.com/a/69417

Urghh! CAPM is rubbish; but none of the claims above are an inconsistent question about the consistent rubbish of the world in which they choose to reside ;-)

All portfolios on the CML are indeed 100% correlated to market, mixing cash with the market, meeting at the tangency portfolio. Very different from saying that all portfolios along the efficient frontier are correlated thus, let alone at all correlated. Apples-vs-oranges, here. Stright-line CML vs buxom-curved tangency...

CAPM can indeed be derived from the CML... IF, a very big IF, with a cherry on top and an atom bomb in the icing underneath, the CML has been pre-defined for you. Absent that, you're making guesses on random data using a model that is more often wrong that right ;-)

[Google "AQR- betting against beta versus betting against correlation"]

best wishes, DEM