Improved Risk Reporting with Factor-Based Diversification Measures

Improved Risk Reporting with Factor-Based Diversification Measures — February 2014

Appendices

orthogonal Procrustes problem solved in Schönemann (1966). Note that in our case, Q is not orthogonal so we cannot apply directly the results obtained by P. Schönemann.

In order to do so, we compute the singular value decomposition of Λ P'D , leading to: Λ P'D = USV' , (B.4) where U and V are two orthogonal matrices, and S is a diagonal matrix containing the singular values of Λ P'D . This leads to:

,

where Z = V'Q'U satisfies ZZ' = I N . Hence, we obtain:

(B.5)

From Equation (B.5), it is straightforward to see that the maximum is attained for z kk = 1 for all k = 1, ..,N, leading to Z = I N . Finally, we obtain Q' = UV' , which gives us:

(B.6)

Remark 1 In the simple case where we impose D = I N , i.e. the factors have a unit variance, then the solution A * is simpler since we are left with the singular value decomposition of Λ P' . Noticing that Λ is a diagonal matrix and P an orthogonal matrix simply leads to: U = I N , S = Λ and V = P . Therefore, Q * is simply given by UV' = P' , and matrix A * takes the following expression: A * = P Λ −1 P' , (B.7)

which coincides with the Riccati decomposition of Meucci (2009b).

Remark 2 The problem of maximising tr( Q'M ) over orthogonal matrices Q is also known as a particular case of the

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