When to Annualize Covariance, Beta, and R²
Summary
The discussion explains when monthly covariance can be scaled to an annual horizon. Under zero covariance between different months and identical monthly distributions, annual covariance is twelve times monthly covariance, just as annual variance is twelve times monthly variance. The independence assumption is stronger than necessary; zero interperiod covariance is sufficient for the variance aggregation step.
Beta and R² are dimensionless regression measures and do not need annualization: consistently scaling both return series leaves the regression slope and fit unchanged. One response also recommends correlation as a unitless alternative to covariance. These scaling conclusions depend on the stated assumptions for aggregating returns, and serial dependence can invalidate simple annualization of covariance or variance.
Key ideas
- Annual covariance is twelve times monthly covariance when monthly observations have identical distributions and zero covariance across periods.
- Independence is sufficient for annualization, but zero interperiod covariance is the weaker condition needed for variance aggregation.
- Beta and R² do not change when both return series are scaled consistently.
- Correlation is unitless, while covariance retains the units of the two variables.
Tags
Full text
# Annualized Covariance
# Annualized Covariance
I have two time series. One with monthly returns on an asset and one with monthly returns on a benchmark index. I have calculated the covariance using the `=COVARIANCE.P()` formula in excel. Further I want to look at the Beta and $R^2$ and therefore wonder:
Are there any reasons to annualize the covariance? And if, how to do so?
Beta is the:
> "... slope of the regression equation"
so does it make any sense to annualize it?
What about $R^2$?
## Answer by betonviews (score 1, accepted)
https://quant.stackexchange.com/a/4758
If you assume that your monthly returns are independent from each other, then the annualized variance of each series, and the covariance can be annualized. This assumption allows you to use V(x1+X2+...+x12) = V(x1) + V(x2) + ... + V(x12) where xi is the return for the month "i". Actually, for this to happen you only need a weaker assumption: that is that interperiod returns correlation or covariance be zero since V(x1 + .. + x12) = Sum(i=1..12,j=1..12,Cov(xi,xj)).
Then if you add the "identically distributed" assumption which means that x1, ... , x12 are just the repetition of the same experiment and follows the same probabilistic law: you get in particular E(x1) = ... = E(x12) (same expected returns) V(x1) = V(x2) = ... = V(x12) (same variance)
Finally, V(x1 + .. X12) = V(x1) + ... + V(x12) = 12 * V(x1) That is: V(annual returns) = 12 times the variance of monthly returns.
Beta and R² are already "normalized" so no need to "annualize" them. Under the same assumptions, you are trying to explain one series of returns with the other using a linear model. Whatever the relation of monthly returns, you will have the same on annual returns.
Final remark: not that assuming independent expected returns means that your monthly returns have no memory. But sometimes, they do: the returns from month i and returns from month i+1 are correlated (see Markov chains for example).
## Answer by ash (score 0)
https://quant.stackexchange.com/a/4754
Can you Annualize Covariance , Beta and R^2 ? Mathematically yes you can. For example you can multiple monthly returns by 12 for asset and benchmark. But Beta is slope regression equation as you pointed out so it will come back as exact same as non-annualized beta. I cannot come up with any reason why you would like to annualize Covariance or R^2 as well.
## Answer by Akavall (score 0)
https://quant.stackexchange.com/a/4756
Why not use correlation coefficient instead of covariance?
$ \rho_{x,y} = \frac{Cov_{x,y}}{std.dev_x * std.dev_y} $
An issue with covariance is that it has units; correlation coefficient is unitless.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.