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Would be very grateful for some help in comparing the single index model with other multi-index models in computing the variance-covariance matrix.

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In a single factor model the covariance matrix of the returns is

$\Omega=\sigma_M^2\beta\beta^T+D$

where $\sigma_M^2$is the market variance, $\beta$ is an N vector containing the Betas of the N securities, and $D$ is a diagonal matrix containing the residual variances of the N securities. This kind of matrix is fairly simple and far from general.

In a multiple factor model with K factors this generalizes to

$\Omega= B\Omega_f B^T+D$

now $B$ is an $K \times N$ matrix of factor loadings and $\Omega_f$ is the $K \times K$ covariance of the factors. Again $D$ is a diagonal matrix of idiosyncratic variances for the N securities. This kind of model can produce matrices more complex (of higher rank) than the prior method. However, it still cannot match any arbitrary covariance matrix unless $K=N$. With $K \lt N$ it involves some approximation or simplification compared to an arbitrary covariance matrix.

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