Bounds on Market Beta from Volatility and Market Weights

# How high can Beta be in CAPM?


# How high can Beta be in CAPM?












I recently got an interview question for a junior analyst role asking if risk could be infinite in CAPM, and I wasn't sure how to answer it. I don't see how an asset could be infinitely more volatile than the market. I understand a theoretical 0-beta asset, but not infinity. So how high can beta really be? In what case would it be infinite (if any)?

I apologize if this question is very elementary and I'm missing a key point here.

## Answer by Matthew Gunn (score 8, accepted)

https://quant.stackexchange.com/a/36043

Infinity is rather non-sensical. A better question perhaps is whether you can put some theoretical bounds on an asset's market beta.

#### An asset's volatility bounds its market beta

Let $R_i$ be the return of security $i$ and $R_m$ be the return of the market. Market beta would be given by:

$$ \beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$$ Let $\rho \in [-1, 1]$ be the correlation coefficient, $\sigma_i$ the standard deviation of return $i$, and $\sigma_m$ the standard deviation of the market return. Since $\operatorname{Cov}(R_i, R_m) = \rho_{im} \sigma_i \sigma_m$ we can rewrite the above expression as:

$$ \beta_i = \rho_{im} \frac{\sigma_i}{\sigma_m} $$

$\rho_{im} \in [-1, 1]$, hence if we know $\sigma_i$ and $\sigma_m$, we can put an upper bound on the market beta: $\beta_i \in [-\frac{\sigma_i}{\sigma_m}, \frac{\sigma_i}{\sigma_m}]$. To have a high market beta, you need high volatility. This is perhaps rather obvious.

#### Another constraint: value weighted average beta must be 1

Let $w_i$ be security $i$'s share of the market portfolio. The market portfolio return is then:

$$R_m = \sum_i w_i R_i $$

Take covariance of both sides and divide by variance of the market:

$$ \frac{\operatorname{Cov}(R_m, R_m)}{\operatorname{Var}(R_m)} = \sum_i w_i \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$$

Observe that the first side is 1 and the second side are market betas:

$$ 1 = \sum_i w_i \beta_i$$

The value weight mean market beta of all securities in the market portfolio must be 1! Speaking loosely, the larger $i$'s weight is in the market portfolio, the more market beta is pulled toward 1.

On the other hand, there's no theoretical reason the security must have positive net supply (eg. derivatives don't have positive net supply).

#### Last comment

Whenever people talk about market betas, there's some tendency to say "CAPM." Resist the urge. You can estimate market betas and run the following regression whether the CAPM is true or not.

$$R_t - R^f_t = \alpha + \beta (R^m_t - R^f_t) + \epsilon_t$$

The CAPM is an economic theory that implies that $\alpha_i$ in the above regression is zero. The CAPM, while simple and beautiful, is an empirical failure. It doesn't work. That said, you can still estimate market betas and use them in sensible ways. Just don't use them as a sufficient statistic to forecast returns.