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Definition of Beta

It is generally understood that the beta of an asset $i$ is given by coefficient of the linear regression of the asset returns on market ($m$) returns, i.e. $$\beta_i = \frac{\rho\sigma_i\sigma_m }{\sigma_m^2}=\frac {\rho \sigma_i}{\sigma_m}$$ where $\sigma$ indicates standard deviation of returns.This is based on the linear model where $$r_i=\alpha_i+\beta_i \ r_m$$ which when rearranged gives $$\beta_i=\frac {r_i-\alpha_i}{r_m}\tag{1}$$

See wiki article here.

Use in CAPM

This beta is then used in the well-known formula CAPM formula for expected return or cost of equity, i.e. $$r_i=r_f+\beta_i(r_m-r_f)$$ ($r_f$ being the risk-free rate) which when rearranged gives $$\beta_i=\frac {r_i-r_f}{r_m-r_f}\tag{2}$$

Question

How can we reconcile $(1)$ and $(2)$?

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    $\begingroup$ The CAPM is actually $r_{it}=r_f+\beta_i(r_{mt}−r_f)+\tilde{e}_t$ for $t=1,\cdots,T$ which does not give equation (2) at all. The only way to find $\beta_i$ is via regression, which gives $\beta_i=\frac{\rho \sigma_i}{\sigma_m}$, your first equation. $\endgroup$
    – Alex C
    Oct 7, 2019 at 1:22
  • $\begingroup$ The typo in the CAPM formulas has been corrected. Equation $(2)$ remains unchanged. The question is not how to find beta, but why the two equations do not appear consistent. $\endgroup$ Oct 7, 2019 at 2:16
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    $\begingroup$ For starters, your capm formula is true for expectations only, right? $\endgroup$
    – LazyCat
    Oct 7, 2019 at 17:58

5 Answers 5

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I slightly disagree with Alex’s comment. The CAPM does not read as \begin{align*} r_{i,t} = r_{f,t}+ \beta_{i} (r_{m,t}-r_{f,t}) + \varepsilon_{i,t}. \end{align*} There is an important difference between the single index model (aka market model) (SIM) which reads as \begin{align*} r_{i,t}-r_{f,t}= \alpha_{i} + \beta_{i}(r_{m,t}-r_{f,t}) + \varepsilon_{i,t} \end{align*} and the (conditional) capital asset pricing model (CAPM) which reads as \begin{align*} \mathbb{E}_t[r_{i,t+1}] - r_{f,t} &= \frac{\mathbb{C}\text{ov}_t(r_{i,t+1},r_{m,t+1})}{\mathbb{V}\text{ar}_t[r_{i,t+1}]} \cdot (\mathbb{E}_t[r_{m,t+1}]-r_{f,t}) \\ &= \beta_{i,t} \cdot (\mathbb{E}_t[r_{m,t+1}]-r_{f,t}). \end{align*}

(Subscripts indicate that the conditional expectation/variance/covariance is meant). The CAPM is an equilibrium asset pricing model about expected returns which can be, for instance, derived from a stochastic discount factor (SDF) framework assuming the SDF is linear in the market return. In particular, there is no idiosyncratic risk component $\varepsilon_{i,t}$ and the CAPM makes no statement about realised returns, variances of returns or anything of the sort. It is only concerned with expected returns.

You can immediately see how the two CAPM equations agree with the equations you quoted: The CAPM implies that the expected excess return of any asset is proportional to the expected excess return of the market portfolio (value weighted portfolio of all assets), i.e. $\alpha_{i,t}=0$.

The SIM is purely and merely a statistical (econometric) model which regresses historical returns against the returns of some factor (this may be a portfolio mimicking the market portfolio but may be any other factor which is believed to drive the returns). As in a standard OLS regression, the slope coefficient in the SIM model, $\beta_{i}$, is estimated to be $\frac{\mathbb{C}\text{ov}(r_{i,t}^e,r_{m,t}^e)}{\mathbb{V}\text{ar}[r_{m,t}^e]}$. So, the SIM may be used to test the CAPM empirically (i.e. if the CAPM was true, we would find $\alpha$ to be not statistically different from zero etc.) but the SIM may not be confused with the CAPM. As a matter of fact, empirical tests raise indeed strong doubts that the CAPM is a good model for average stock returns.

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  • $\begingroup$ Your third equation could be expressed as a regression model just like the first two. That might make it easier to see how it differs from the first two equations. Accordingly, I wonder if the following actually holds: In particular, there is no idiosyncratic risk component $\varepsilon_{i,t}$. A term $\varepsilon_{i,t}$ can be easily introduced as you switch from the expectation to a realization on the left hand side (see e.g. my answer on this thread). Also, is it correct that $\alpha_{i,t}=r_{f,t}$? I get $\alpha_{i,t}=(1-\beta_i)r_{f,t}$ instead. $\endgroup$ Feb 5, 2023 at 16:09
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    $\begingroup$ And then in your last paragraph, in the expression of $\beta_{i,t}$, should the variance not be of the market return rather the asset's return? Also, if we use SIM to test the CAPM empirically, we suffer from measurement error ($\mathbb{E}(r_m)$ gets replaced by $r_m$); that might be worth mentioning to prevent naive applications of SIM as a test of the CAPM. $\endgroup$ Feb 5, 2023 at 16:13
  • $\begingroup$ @RichardHardy Firstly, I love all the questions you been posting recently. Very insightful! Regarding this answer: I corrected the typo in the formula for $\beta$. I agree that measuring the return on the market portfolio is difficult and poses a problem. Frankly, that's probably not the biggest problem for the CAPM and using the CRSP vw average is so common, I don't think this is the big challenge for testing the theory. $\endgroup$
    – Kevin
    Feb 5, 2023 at 18:39
  • $\begingroup$ @RichardHardy Regarding the idiosyncratic component: yes, you can add a mean zero variable and go to realisations, but there's more. For example, you could introduce a second return factor that affects stocks but is not priced. Adding this factor would add a more complex factor structure in realised returns and variances but still keep expected excess returns proportional to the market return (CAPM still holds). Both are possible paths how to go from CAPM (expected returns) to realised returns. The CAPM simply makes no statement whatsoever about realised returns. $\endgroup$
    – Kevin
    Feb 5, 2023 at 18:42
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    $\begingroup$ Thank you, Kevin! I appreciate your encouragement very much! $\endgroup$ Feb 7, 2023 at 16:43
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Consult any econometrics textbook (if you're able to take a stiff non-pharmaceutical sedative). You'll get hundreds of pages littered with equations with betas: some with hats, some with stars, some with stars and hats. "Beta" is just the conventional shorthand for a regression co-efficient.

So if a 1% change in say bond yields is associated with an X% change in stock prices, that's one beta.

The CAPM is just a special case, arguing that the fair excess returns (ie returns less cash) for any asset should be proportional to its undiversifiable risk. It's an intuitive argument, that ticks a bunch of economic theory boxes. But whether it bears any resemblance to market reality and outcomes is a different matter.

Assume it does; and it's just one application of the beta family that the theory says should be relevant. But it's just one of the many betas out there that the theory says should be relevant to asset valuations.

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One thing to note is that the CAPM can be rewritten as

$$r_i=r_f(1-\beta_i)+\beta_i r_m$$

Thus, one can identify $\alpha_i$ with $(1-\beta_i)r_f$.

Then, we we have, for the CAPM,

$$\beta_i=\frac {r_i-r_f}{r_m-r_f}\tag{2}$$

and for the regression

$$\beta_i=\frac {r_i-\alpha_i}{r_m} =\frac {r_i-(1-\beta_i)r_f}{r_m}, \tag{1}$$ so $$ \beta_i - \beta_i r_f / r_m =\frac {r_i-r_f}{r_m} $$ or $$ \beta_i =\frac {r_i-r_f}{r_m(1- r_f / r_m)} =\frac {r_i-r_f}{r_m - r_f } \tag{1'}$$

So (1) = (2), after all, with suitable $\alpha_i$.

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The two betas are different. Beta in (2) is CAPM beta, beta in (1) is a coefficient of the linear regression, not a CAPM beta.

For more, see CAPM is neither a cross sectional model, nor a time series model

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Notation

  1. Latin capital letters denote random variables. Latin lowercase letters denote their realizations.
  2. Asterisks denote returns in excess of the risk-free rate.
  3. Greek letters denote parameters.

Definition of beta

The definition of $\beta_i$ is $$ \beta_i\equiv\frac{\text{Cov}(R_i)}{\text{Var}(R_m)}=\frac{\text{Cov}(R^*_i)}{\text{Var}(R^*_m)} $$ where $R^*_i\equiv R_i-r_f$ and $R^*_m\equiv R_m-r_f$. Equivalently, $\beta$ corresponds to the slope coefficient in a linear model of $R_i$ (or $R^*_i$) on $R_m$ (or $R^*_m$), $$ \mathbb{E}(R_i\mid R_m=r_m)=\alpha_i+\beta_i r_m \quad\quad \text{or} \quad\quad \mathbb{E}(R^*_i\mid R^*_m=r^*_m)=\alpha_i'+\beta_i r^*_m \tag{i} $$ or equivalently $$ r_i=\alpha_i+\beta_i r_m+\varepsilon \quad\quad \text{or} \quad\quad r^*_i=\alpha_i'+\beta_i r^*_m+\varepsilon'. $$ Thus we do not have $(1)$ but rather $$ \beta_i=\frac{\mathbb{E}(R_i\mid R_m=r_m)-\alpha_i}{r_m} \quad\quad \text{or} \quad\quad \beta_i=\frac{\mathbb{E}(R^*_i\mid R^*_m=r^*_m)-\alpha_i'}{r^*_m} \tag{1'}. $$

Use in the CAPM

The CAPM does not state that $$ r_i=r_f+\beta_i(r_m-r_f) $$ but rather $$ \mathbb{E}(R_i)=r_f+\beta_i(\mathbb{E}(R_m)-r_f) \quad\quad \text{or} \quad\quad \mathbb{E}(R^*_i)=\beta_i\mathbb{E}(R^*_m) \tag{ii} $$ or equivalently $$ r_i=r_f+\beta_i(\mathbb{E}(R_m)-r_f)+\varepsilon_i'' \quad\quad \text{or} \quad\quad r^*_i=\beta_i\mathbb{E}(R^*_m)+\varepsilon_i'''. $$ Thus we do not have $(2)$ but rather $$ \beta_i=\frac{\mathbb{E}(R_i)-r_f}{\mathbb{E}(R_m)-r_f} \quad\quad \text{or} \quad\quad \beta_i=\frac{\mathbb{E}(R^*_i)}{\mathbb{E}(R^*_m)} \tag{2'}. $$

Reconciling the two

It will be easier to consider $(i)$ vs. $(ii)$ than $(1')$ vs. $(2')$. Since $(i)$ is equivalent to $(1)$ and $(ii)$ is equivalent to $(2)$, we can do that. Now, $(i)$ and $(ii)$ are not directly comparable due to the former involving a conditional expectation on the left hand side while the latter only involving an unconditional expectation there. To take care of that, take the expectation of $(i)$ to obtain $$ \mathbb{E}[\mathbb{E}(R_i\mid R_m=r_m)]=\mathbb{E}[R_i]=\alpha_i+\beta_i \mathbb{E}[R_m] \quad\quad \text{or} \quad\quad \mathbb{E}[\mathbb{E}(R^*_i\mid R^*_m=r^*_m)]=\mathbb{E}[R^*_i]=\alpha_i'+\beta_i \mathbb{E}(R^*_m). $$ Now we see that the CAPM implies $\alpha_i=(1-\beta_i)r_f$ or $\alpha_i'=0$.

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