Two Equivalent Ways to Measure Basket Realized Variance
Summary
The document compares realized variance computed from a basket’s own time series of returns with variance reconstructed from its constituents’ volatilities, weights, and pairwise correlations. The first method squares and sums basket returns over time; the second applies the covariance expansion across assets. Under consistent inputs, the two are mathematically equivalent: expanding each basket return into weighted constituent returns and aggregating over the same observation period produces the same result.
The equivalence depends on using matching return periods, weights, and estimates. The discussion notes that practical differences can arise when the traded basket does not track its constituents exactly, such as an ETF trading away from net asset value because of trading costs or market mechanics. The source offers no numerical comparison or empirical study, and its comments about such discrepancies being small are not substantiated. The zero-mean assumption also matters to the variance interpretation when squared returns are used directly.
Key ideas
- Portfolio-level squared returns and constituent covariance calculations can yield the same basket variance.
- The equivalence requires consistent weights, return observations, and correlation estimates over the same period.
- The constituent method expands variance into weighted individual variances and pairwise covariance terms.
- Tracking gaps between a traded basket and its underlying assets can create practical differences.
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# Difference between volatility measures of a basket of assets
# 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.
## Answer by XYQ (score 1, accepted)
https://quant.stackexchange.com/a/43478
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.
## Answer by Ezy (score -1)
https://quant.stackexchange.com/a/43477
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).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.