# Difference between volatility measures of a basket of assets

I am trying to understand intuitively the difference between two different measures of realized variance of a basket of assets.

The first measure I am aware of is when you take the realized variance to be the sum of squared log returns of the basket. I.e.

$$\sigma_B^2 = \sum_{i=1}^T{r_i}^2$$

where $$\sigma_B^2$$ is the realized variance of the basket and $$r_i$$ is the log returns of the basket ($$r_i = \sum_{j=1}^n\omega_jr_j$$ for weights $$\omega_j$$ and single stock returns $$r_j$$)

The second measure is obtained via consideration of basket components standard deviations and correlations:

$$\sigma_B^2 = \sum_{i=1}^n \omega^2_i\sigma^2_i + 2\sum_{i=1}^n\sum_{j=i+1}^n\omega_i\omega_j\sigma_i\sigma_j\rho_{ij}$$

where $$\sigma_B^2$$ is the basket variance, $$\omega_i, \sigma_i$$ is the weight and volatility corresponding to the $$i^{th}$$ basket component respectively and $$\rho_{ij}$$ is the pairwise correlation between the $$i^{th}$$ and $$j^{th}$$ components.

I know the former of these two measures assumes that the expected return of the basket is $$0$$, but this is potentially assumed in the second as well if each $$\sigma_i$$ is found by summing the $$i^{th}$$ components squared log returns and taking the answers square root. Is there a fundamental intuitive difference between these two measures of realized variance?

Thanks.

If the correlation estimate $$\rho_{ij}$$ is calculated using the same period $$[1,T]$$ then 2 expressions are identical. They just differ in the order in which you are performing the 2 summations (one in time direction and the other in the cross-sectional direction).