Why Correlation Matrices Are Not Annualized
Summary
The document explains why a daily correlation matrix should not be multiplied by the number of trading days when annualizing covariance. Correlation is dimensionless: it is covariance divided by the product of the two assets’ standard deviations. If covariance and variance are both scaled by the same time factor under the usual aggregation assumptions, those factors cancel in the correlation ratio.
The answers emphasize that multiplying correlations by an annualization factor could produce values outside the valid range or distort perfect relationships. They also note that covariance scaling depends on assumptions about return dependence and time aggregation. In particular, the stated equivalence between daily and longer-period correlations is not universal when returns exhibit serial or cross-asset dependence across time; the excerpt does not provide an empirical example.
Key ideas
- Correlation is a unitless ratio of covariance to the product of standard deviations.
- Scaling covariance and variance by the same factor leaves correlation unchanged under the stated assumptions.
- Multiplying a correlation matrix by an annualization factor can create invalid or misleading values.
- The scaling argument depends on assumptions about return dependence across time.
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Full text
# How to annualize the correlation matrix?
# How to annualize the correlation matrix?
If asset returns are daily, and the asset return covariance matrix, $\Sigma$, is annualized by $\Sigma \times 252$, do I also multiply the correlation matrix by 252 to annualize it?
## Answer by siou0107 (score 8, accepted)
https://quant.stackexchange.com/a/58005
No, because correlation is a unitless quantity. As you use volatilities to do the scaling, the $\sqrt{252}$ factor should already be taken into account in them.
If you take a correlation of 1 between two assets, multiplying your correlation matrix by a factor $C \neq 1$ risks either to underestimate correlations (by hiding perfect (anti)correlations) or have your matrix not making any sense (correlation greater than 1).
## Answer by Rodolfo Oviedo (score 0)
https://quant.stackexchange.com/a/77743
No.
By definition,
```
Corr[x, y] = Cov[x, y] / Sqrt( Var[x] Var[y] )
```
where `x` and `y` are the daily returns of two assets.
If annualizing the (variance and) covariance matrix of daily returns requires multiplication by 252, then the correlation of annual annual returns `X` and `Y` is
```
Corr[X, Y] = Cov[X, Y] / Sqrt( Var[X] Var[X] ) = 252 Cov[x, y] / Sqrt( 252 Var[x] 252 Var[y] )
```
because variances and covariances are the elements of the covariance matrix.
Canceling 252^2 inside que square root and 252 in the numerator yields makes the last member to satisfy the definition of the correlation of daily returns. Therfore
```
Corr[X, Y] = Corr[x, y]
```
Becouse 252 could be substituded by any other constant, the correlations in invariant with the period of returns.
Notes:
- The assumption that the covariance matrix of log returns scales with the period of such returns holds if the return of an asset is not correlated with past returns of the same or other assets, and, given an arbitrary period to measure returns, the covariances are the same whatever the start of such period.
- Variance is a particular case of a covariance of a variable with itself: Cov(X, X) = Var(X). Therefore, if the covariance scales with the period of the returns, so must the variance.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.