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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 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. $\endgroup$ – hypergeometric Oct 7 at 2:16
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    $\begingroup$ For starters, your capm formula is true for expectations only, right? $\endgroup$ – LazyCat Oct 7 at 17:58
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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,t} (r_{m,t}-r_f) + \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} = \alpha_{i,t} + \beta_{i,t}(r_{m,t}-r_{f,t}) + \varepsilon_{i,t} \end{align*} and the capital asset pricing model (CAPM) which reads as \begin{align*} \mathbb{E}_t[r_{i,t}] - 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). So, 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 $\varepsilon_{i,t}$.

You can immediately see how the two CAPM equations agree with the equations you quoted: The CAPM assumes 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. $\alpha_{i,t}=r_{f,t}$.

The SIM is purely and merely a statistical 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). By standard OLS regression of the SIM model, the coefficient $\beta_{i,t}$ is estimated to be $\frac{\mathbb{C}\text{ov}(r_{i,t},r_{m,t})}{\mathbb{V}\text{ar}[r_{i,t}]}$. So, the SIM may be used to test the CAPM empirically (i.e. if the CAPM was true, we wound find $\alpha$ to be not statistically different from zero etc.) but 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.

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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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