# Asset risk relative to market portfolio risk - derivation problem

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!

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.