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Look in any finance textbook or search about CAPM, it will say that CAPM is a model about how portfolios have systematic risk (risk that can't be diversified away) and idiosyncratic risk (risk that can be diversified away).

But it appears to me that this is impossible to justified based on CAPM alone, which is a model about expected returns ($\mu_i-r_f=\beta_i(\mu_m-r_f)$), and this is actually an implication of the single-index model, which is a model about realized returns ($r_i-r_f= \beta_i(r_m-r_f)+\epsilon_i$).

In fact, the definition of idiosyncratic risk relies critically on the error term $\epsilon_i$. Without the error term, how can idiosyncratic risk and diversification even be defined?

It appears to me that CAPM says absolutely nothing about risk. You can't derive any statement about the variance of assets or portfolio returns just from a relation that is only about the mean of the returns.

My question is this: How is it possible to derive that beta is a measure of systematic risk, and that every asset has systemic risk that can't be diversified away and idiosyncratic risk that can be diversified away, based on CAPM alone and without relying on the single-index model? If it's not possible, it would appear that pretty much everyone is wrongly attributing to CAPM a conclusion that actually comes from the single-index model.

Edit: This observation has also been noted before in this question, but was brushed aside without addressing it.

fe2084
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    You're right. The CAPM only makes a statement about expected returns, not realised or abnormal returns. However, by definition, you can split realised returns into expected returns and "unexpected" returns (both are orthogonal). In the CAPM, expected returns are driven by market beta which is systematic risk because it's the only risk that's priced. Whatever is left over after correcting for beta is then "residual risk" (ie idiosyncratic risk). – Kevin Dec 28 '22 at 12:58
  • @Kevin I don't think that does it. If you split the realized return into expected and unexpected by defining $\epsilon_i = r_i - \mu_i$, where $\mu_i$ is from CAPM, it does not seem to follow that $\epsilon_i$ is uncorrelated with $r_m$, which is a requirement of the single-index model and the decomposition of systematic and idiosyncratic risk. – fe2084 Dec 28 '22 at 22:07
  • The residual is, by definition, independent of your forecasted return (expected return). Otherwise it would have impacted your expectation. You can always decompose a random variable orthogonally into "what you expect" and "how much you screw up". – Kevin Dec 29 '22 at 18:22
  • @Kevin, strictly speaking, a zero-expectation residual is not necessarily independent of the expected value; it is only guaranteed to be linearly independent. – Richard Hardy Dec 29 '22 at 18:46
  • @Kevin No, you can actually work out the covariance of the residual and the predictor and show they are not independent: $Cov(\epsilon_i, r_m) = w_i \sigma_{\epsilon_i}^2\neq 0$, where $w_i$ is the weight of the $i$th asset in the market return. And you do need $Cov(\epsilon_i, r_m)=0$ to get the decomposition of risk into systematic and idiosyncratic risk. The problem with this residual approach is that it's not just $\mu_i$ that needs to replaced with a term that has a residual, but also $\mu_m$. – fe2084 Dec 29 '22 at 23:10

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The beta is just the linear correlation with the "market". This can be interpreted as either a latent factor or an actual market return can be used. Also you've missed the alpha

The model assumes a linear relationship between the above risk-free rate of returns and the market rate of returns:

$$r_j - r_f=\alpha_j + \beta_j (r_m - r_f) + \varepsilon_j$$ where $E[\varepsilon_j]=0$.

$\varepsilon_j$ give the idiosyncratic, diversifiable risk because by investing across a large number of stocks these noise terms even out on average since they're independent. This is basically the same as the central limit theorem.

On the other hand the risk we are exposed to from the market index factor $r_m - r_f$ is not diversifiable because no matter how many stocks we invest in, this term will never be independent across the stocks; it is additive.

Beta is a measure of the systematic risk because it tells you how much you expect your portfolio to change in value in response to the market changing in value.

Bennnn
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  • The model assumes: which model? E.g. the CAPM does not assume that; according to the CAPM, $\alpha_j=0$. noise terms even out on average since they're dependent: did you mean independent? This is basically the same as the central limit theorem. Is it, really? CLT concerns the shape of a distribution (normal) under $\sqrt{n}$ scaling. On the other hand, this might be closer to the Law of Large Numbers: the sample average converges to the population average (which here is zero) as the sample size grows. – Richard Hardy Dec 29 '22 at 18:53
  • To clarify: the analogy to the CLT is for explanation purposes rather than rigorous derivation. By taking an overall portfolio weighting of $\sum_{j=1}^n w_j = 1, w_j > 0$, we can decrease overall idiosyncratic variance by investing fractional amounts in a large number of stocks, hence decreasing variance in a way quite similar to the CLT - just with different variances. If we let the number of available stocks go to infinity then investing over all available stocks will result in convergence of overall idiosyncratic variance to $0$. – Bennnn Dec 29 '22 at 19:11
  • This is the single-index model, not the CAPM, again showing the interpretation comes from the single-index model not CAPM. – fe2084 Dec 29 '22 at 23:06
  • The CAPM just has $\alpha_j = 0$ and utilises the first and second moments of each stock. Even though the assumption of a distribution is not explicitly stated, making decisions based on this is equivalent to assuming a normal distribution since mean, variance and correlation are measures of linearity. Ie. It assumes a model equivalent to what I gave with.

    The CAPM simply has you reduce idiosyncratic risk by reducing overall portfolio variance.

     – Bennnn Dec 30 '22 at 03:37
  • making decisions based on this is equivalent to assuming a normal distribution: it is not exactly equivalent ("if and only if") but is compatible with normality. However, it is also compatible with several other sets of assumptions; see Cochrane "Asset Pricing" Chapter 9. (Also note that there exist other, nonnormal distributions characterized solely by mean and variance, too.) – Richard Hardy Dec 30 '22 at 07:35
  • @Bennn The CAPM is the expectation of that formula for $r_j$ with $\alpha_j=0$ and does not necessarily make an assumption on $r_j$ itself. Furthermore, neither $r_j$ nor $\epsilon_j$ can have a normal distribution since $r_j$ cannot be less than -1 (otherwise the price would be negative), whereas normal distributions are supported on the real line. – fe2084 Jan 03 '23 at 22:14