Calculating Beta with Variance and Covariance
The total risk of a security can be divided into systematic risk and unsystematic risk.
This means that the total risk of a security, measured by standard deviation of returns, is only relevant in isolation. But in a well-diversified portfolio, the only relevant risk measure is beta, which represents systematic risk (or market risk).
The calculation of beta as a measure of market risk begins with the single-index model of the return premium for a security:
$$\displaystyle R_i - R_f = \beta_i(R_M - R_f) + e_i $$
This equation relates the returns of the market and of a given security in each period. If you just solved this algebraically for beta, you would get something that's not too useful, since it would just be the beta for each and every observation. What do you think should be next?
Yes!
Incorrect. Again, that wouldn't be too useful, as it would work for just a single observation, and using just one random observation discards most of the information you have in the data available. The best plan is to use a calculation which looks at all of the observations at once.
What's really nice about algebra is that you can always do anything to both sides of an equation. Here, the plan is to add the risk-free rate to both sides to isolate the asset return, and then take the covariance of both sides.
$$\displaystyle Cov(R_i,R_M) = Cov(\beta_iR_M - \beta_iR_f + e_i + R_f,R_M) $$
The very next simplifying step will get rid of all of the $$R_f$$ terms entirely. Why do you think that is?
No, one of them has a beta attached; they can't cancel each other exactly.
That's right!
Risk-free indicates zero variance, so it would have to have zero covariance with everything. It's kind of like measuring how your movements throughout the day covary with your couch, or something. It doesn't move, so no connection.
No, the risk-free rate is assumed to be positive in nearly every case.
Now you're getting close. Break this long right-hand side term into individual covariances and see what's left:
$$\displaystyle Cov(R_i,R_M) = Cov(\beta_iR_M,R_M) + Cov(e_i,R_M) $$
Again, are two more terms that don't make this cut, because they are just the risk-free rate. And in fact, consider the remaining terms here. The first term is the market covarying with itself, and the second term is the market covarying with an independent error term.
Can you identify a logical next form of this equation with that in mind?
No, the error term isn't covarying with itself, but with the market return, and remember that these are independent.
No, the covariance of something with itself isn't one. The correlation of something with itself is one, but that's different.
To summarize this discussion:
[[summary]]
Exactly!
The covariance of something with itself is its variance, and the covariance of independent variables should be zero. So then with a little more algebra, beta can now be isolated as:
$$\displaystyle \beta_i = \frac{Cov(R_i,R_M)}{\sigma_M^2} $$
The beta of an asset is its covariance with the market, divided by the market variance. So a covariance of 0.032 with the market of asset i, while the market has a variance of 0.04, indicates a beta of asset i of $$0.032/0.04 = 0.8$$, for example.
Solve explicitly for beta and use a random observation
Use some calculation that looks at all of the observations at once
The risk-free rate terms cancel each other
The covariance of a risk-free rate with anything is zero
The $$R_f$$ terms are always assumed to be zero
$$ Cov(R_i,R_M) = \beta_i\sigma_M^2 + 0 $$
$$ Cov(R_i,R_M) = \beta_i \times 1 + 0 $$
$$ Cov(R_i,R_M) = \beta_i\sigma_M^2 + \sigma_e^2 $$
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