Calculating Multivariate Betas from an EWMA Covariance Matrix
Summary
The document explains how to obtain factor betas for several assets against several indices when only an exponentially weighted covariance matrix is available. It connects the familiar single-factor covariance-over-variance beta to the multiple regression solution: multiply the factor-to-asset covariance block by the inverse of the factor covariance block. This yields a matrix of coefficients, with one row per factor and one column per asset.
The proposed procedure partitions the joint covariance matrix into factor-factor, factor-asset, asset-factor, and asset-asset blocks, then uses the first two relevant blocks in the calculation. A two-index, one-asset expression illustrates the matrix operation, and a regression formulation clarifies the dimensions. A constant can be included in a regression when working from return observations, though the response says it is not needed for the beta calculation discussed. The treatment assumes the factor covariance block can be inverted; it does not address singularity, regularization, or the detailed scaling convention for an EWMA covariance estimate.
Key ideas
- Multivariate betas are obtained by multiplying factor-asset covariances by the inverse factor covariance matrix.
- Partition the joint covariance matrix into factor and asset blocks before computing coefficients.
- The matrix result corresponds to the multiple regression of asset returns on index returns.
- The factor covariance matrix must be invertible for the stated calculation to work directly.
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# How to get Multivariate Betas from an Estimated EWMA co variance Matrix?
# How to get Multivariate Betas from an Estimated EWMA co variance Matrix?
I have a portfolio of 4 assets. I also have returns for 3 indices. I want to get the multivariate betas for these 4 assets-based on these assets. I only have the 7 x 7 covariance matrix estimated by a Exponential Weighted Moving Average Model (EWMA). How would I get these multivariate betas? I know from a univariate percpective I would use the covarience(Asset1Return,Index1Return)/Var(Index1) but this only gives the uni-variate beta.
## Answer by SteinarV (score 1)
https://quant.stackexchange.com/a/16373
With this solution you have to split your covariance matrix somewhat, but it should give you a vector with betas based on you conditional covariances.
Example with two indexes, $x1$ and $x2$, and one asset $y$.
$$[\sigma_{y,x1}, \sigma_{y, x2}]\begin{bmatrix} \sigma_{x1}^2 & \sigma_{x1,x2} \\ \sigma_{x1,x2} & \sigma_{x2}^2 \end{bmatrix}^{-1}$$
## Answer by Kiwiakos (score 1)
https://quant.stackexchange.com/a/17155
Say that you did the calculations in the classic regression way. If you stick the returns of your 4 asset returns in a $(T\times 4)$ matrix $Y$, and your 3 factor returns in a $(T\times 3)$ matrix $X$, then your betas would solve the multiple regressions, collected in a $(3\times 4)$ matrix $$Y = X\cdot \beta + \epsilon$$ You could also add a column of ones in $X$ if you want to also have a constant, but this does not matter. The OLS solution would be $$\beta = (X'X)^{-1}(X'Y)$$
You can see the analogy to the univariate cov/var formula that you describe. The $X'Y$ part corresponds to the $(3\times 4)$ asset-factor covariances, while the $X'X$ parts corresponds to the $(3\times 3)$ factor-factor covariances.
Therefore I would say that your large $(7\times 7)$ EWMA covariance matrix can be partitioned as $$\left( \begin{array}{cc} X'X & X'Y \\ Y'X & Y'Y \end{array} \right) $$ From which you can pick the right elements to calculate the betas.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.