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?



The firs is at portfolio level while the second is from estimated from the constituents. Mathematically they should be the same assume the mean return is zero for your first formula. But in reality they may not in certain context like ETF, because of the trading cost and trading mechanism, there might be a gap between ETF price and the NAV, but the difference caused by this should be small.


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).


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