Why Beta Depends on Return Definitions and Model Assumptions

# Beta using only price returns?


# Beta using only price returns?












It is my understanding that one can use both excess returns and price returns to compute a beta coefficient. In the former way, beta would be interpreted in the standard way (a 1 unit change in market excess returns is associated with a beta unit change in share excess returns). In the latter way, you have the same interpretation but only considering the actual returns of the market and the shares. In this sense, using both excess returns and price returns are both valid ways to compute beta. Correct?

## Answer by Dave Harris (score 0)

https://quant.stackexchange.com/a/66549

The short answer is no. The problem is with what you mean by beta.

There are two hidden assumptions in your posting as well. The first is that $\beta$ exists in the data generating function. The second is that both representations are part of the data generation process.

Let us imagine a simple case where $y=5+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma^2).$ If you would build a model, such as $y=\beta{x}+\alpha+\varepsilon,\varepsilon\sim\mathcal{N}(0,\sigma^2)$, then your problem is that it does not match nature. Excluding false positives, your model should accept the null that $\beta=0$, which is the same as saying that $\beta$ does not exist because $x\perp{y}$.

Since the standard way has been extensively falsified, it is also not a valid representation of reality, so you cannot use that either. Because it has been falsified so many times in so many markets in so many different ways, you can treat the $\beta$ from models like the CAPM, APT, or Fama-French which is also not validated, as though they do not exist. It is beyond reasonable to assume that $\beta\equiv{0}$ though the finance textbooks do not say that.

The models are misspecified models, at best.

That does not imply that assets do not move together, it just implies that $$\beta=\frac{\sigma_{i,j}^2}{\sigma_{i,i}^2}$$ is not a useful mathematical construction. That is not the only way nor unique in any sense to view assets as moving together. It is insanely inconvenient, however, which is why that discussion is avoided. That construction is inconsistent with heavy tails.

Do note that if $x_{t+1}=\beta{x}_t+\epsilon_{t+1},\beta>1,\epsilon\sim{f}(0,\sigma^2),0<\sigma^2<\infty$, where $f$ is any distribution with finite variance centered on zero,then no solution for $\beta$ exists in parametric frequentist methodologies. If $\beta=1$ that should be currency. If $\beta<1$ then $x_{t+1}$ should converge to zero, an undesirable property for an investment. If $x_t$ is your amount invested, you do not want $\beta\le{1}$. Once you drop parametric methods though, you find yourself without a mean or a variance anymore.

## Answer by Michael Williamson (score 0)

https://quant.stackexchange.com/a/69529

The beta will be the same regardless if you use total return or excess return because the same risk free rate is applied to both sides of the equation. Beta is the slope coefficient, and applying a linear shift on both series does not affect the slope.

The beta would only changed if you use excess returns for the security vs total returns for the market.