Why Beta Can Stay Similar Across Daily and Weekly Returns

Summary

The document explains why a security’s beta can be similar when estimated from daily or weekly returns. Beta is defined as the covariance between security and market returns divided by market-return variance. Under an assumption that daily security and market log returns are independent and identically distributed draws from a joint normal distribution, weekly returns are sums of daily returns. Both covariance and market variance scale by the same factor when the sampling interval is lengthened, so their ratio—and thus beta—stays unchanged.

This is a theoretical result under specific assumptions, not a guarantee about observed estimates. If returns are not independent, identically distributed, or jointly normal, weekly covariance may not scale proportionally to daily covariance. The discussion also notes that investors may care about forward-looking beta calculated using arithmetic returns; converting from log returns makes that calculation more involved. No empirical comparison or estimation procedure beyond the covariance ratio is provided.

Key ideas

• Beta is the covariance of an asset’s returns with market returns divided by market-return variance.
• When weekly log returns are sums of independent daily returns, covariance and variance scale together.
• Under those assumptions, changing from daily to weekly data leaves beta unchanged.
• Dependence or departures from the assumed return distribution can make beta vary across sampling frequencies.
• Forward-looking beta on arithmetic returns requires additional care when converting from log returns.

Tags

# Beta and Frequency of Data


# Beta and Frequency of Data












Why are the betas of individual securities essentially the same whether we use daily or weekly data when calculating?

## Answer by John (score 3)

https://quant.stackexchange.com/a/10915

Suppose you have $$X\equiv\left(x_{1},\: x_{2}\right) $$ where $x_{1}$ are the daily log returns of the security and $x_{2}$ are the daily log returns of the market. Assume further that $X$ is iid multivariate normal $$X\sim N\left(\mu,\Sigma\right) $$ People frequently calculate beta as $$\beta_{1,2}\equiv\frac{\Sigma_{1,2}}{\Sigma_{2,2}} $$ If you convert $X$ from a daily series to a weekly series, you could say that the weekly variables are just the sum of the daily variables. Due to the properties of a normal distribution, this means you could write $$X_{weekly}\sim N\left(5\mu,5\Sigma\right) $$ This implies a weekly beta of $$\beta_{1,2}^{weekly}\equiv\frac{5\Sigma_{1,2}}{5\Sigma_{2,2}}=\frac{\Sigma_{1,2}}{\Sigma_{2,2}}$$ or that the beta is the same as the daily version.

There are a few wrinkles to this argument. First, returns may not be iid normally distributed, which would mean that the covariance of the weekly data may not be proportional to the covariance of the daily data. Second, the beta that really matters to an investor is the forward-looking beta on arithmetic returns. That beta is more complicated to calculate since it involves the conversion between the log returns and the arithmetic returns.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.