Asset risk relative to market portfolio risk - derivation problem

Question

I am currently studying the CAPM, and at the moment I am focusing on beta. I am using the following book:

Danthine, J-P and J. B. Donaldson (2014): Intermediate Financial Theory (3rd Edition) http://www.sciencedirect.com.ez.statsbiblioteket.dk:2048/science/book/9780123865496

page. 211-212

'- and my problem is with a derivation they make in the book, that i just don't follow, the books states the following:

For efficient portfolios, we have the simple linear relationship in Eq. (8.1).

(8.1) $E \tilde{r}_{p}=r_{\mathrm{f}}+\frac{E \tilde{r}_{M}-r_{\mathrm{f}}}{\sigma_{M}} \sigma_{p}$

The CML applies only to efficient portfolios. What can be said of an arbitrary asset j that does not belong to the efficient frontier? To discuss this essential part of the CAPM, we first rely on Eq. (8.2), and limit our discussion to its intuitive implications:

(8.2) $E \tilde{r}_{j}=r_{\mathrm{f}}+\left(E \tilde{r}_{M}-r_{\mathrm{f}}\right) \frac{\sigma_{j M}}{\sigma_{M}^{2}}$

$\beta_{j}=\sigma_{j M} / \sigma_{M}^{2}$ i.e., the ratio of the covariance between the returns on asset j and the returns on the market portfolio divided by the variance of the market returns. We can thus rewrite Eq. (8.2) as Eq. (8.3).

(8.3) $E \tilde{r}_{j}=r_{\mathrm{f}}+\left(\frac{E \tilde{r}_{M}-r_{\mathrm{f}}}{\sigma_{M}}\right) \beta_{j} \sigma_{M}=r_{\mathrm{f}}+\left(\frac{E \tilde{r}_{M}-r_{\mathrm{f}}}{\sigma_{M}}\right) \rho_{j M} \sigma_{j}$

Comparing Eqs. (8.1) and (8.3), we obtain one of the major lessons of the CAPM: only the fraction $\rho_{j M}$ of the total risk of an asset $j, \sigma_{j}$, is remunerated by the market.

So i understand the general idea that the market portfolio is a efficient portfolio that contain all risky assets, and that the risk added by asset j, only will be systematic risk. I also understand that we are trying to find the relative risk that asset j are adding to the market portfolio, but i just don't get the derivation.

I hope you can help me!

Answer 1

Comparing 8.1 and 8.3 we see that $\sigma_p$ is replaced by $\rho_{jM}\sigma_j$. Importantly it is NOT replaced by $σ_j$ as hasty thinking might have led us to guess: "just replace $p$ by $j$ in the subscripts of 8.1 and it would still work, right?" No, that would be wrong.

8.3 shows that what matters for expected return is not $σ_j$, the volatility of stock $j$, but only the component of that volatility which is parallel to market fluctuations, whereas the component orthogonal to those fluctuations does not matter.

So only a portion (since $0≤\rho_{jM}≤1$) of $\sigma_j$ matters for expected return estimation. Not the whole thing.

Answer 2

The idea and method in Danthine and Donaldson (2014) are obsolete.

The CAPM formula holds under the partial equilibrium of purely risky assets, which is equivalent to the condition that the market portfolio is the tangency portfolio. The risk-return characteristics is a false impression from the partial equilibrium of purely risky assets. Thus it is illusory, for the assets are priced as a whole and the prices are endogenous, the return of market portfolio is not exogenous but endogenous.

Only when market return is given in advance, the CAPM formula could at best be considered a relative pricing formula. In this case it can only be used to price the portfolio of primitive securities. As relative pricing, it does not make sense to discuss risk in the CAPM formula. Because the relative pricing is based on the equilibrium prices of primitive securities and is realized through an arbitrage mechanism (replication), whereas arbitrage is not affected by risk preference. When the CAPM equilibrium prices are free of arbitrage, the CAPM formula must be a risk-neutral pricing formula. For more discussion, see The CAPM formula holds under the partial equilibrium of purely risky assets