Optimal covariances in risk model aggregation

Alec N. Kercheval

Portfolio risk forecasts are often made by estimating an asset or factor covariance matrix. Practitioners commonly want to adjust a global covariance matrix encompassing several sub-markets by individually correcting the sub-market diagonal blocks. Since this is likely to result in the loss of positive semi-definiteness of the overall matrix, the off-diagonal blocks must then be adjusted to restore that property. Since there are many ways to do this adjustment, this leads to an optimization problem of Procrustes type. We discuss two solutions: a closed form solution using an adapted norm, and a fast iterative approach due to Koschat and Swayne.