Improved Risk Reporting with Factor-Based Diversification Measures

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

Appendices

B. Technical Details

Matrix D 2 can be anything, e.g. one may want to impose that the factors have the same variance as the initial asset, and pick D 2 = diag( Σ ), or one may want to impose that the factors have the same unit variance, in which case we pick: D 2 = I N .

B.1 Proof of Lemma 1 Since the weights of a portfolio sum up to 1, then the mean of the weights is simply equal to 1/ N , and the expression of the sample variance applied to the weights takes to the following form:

If we compute the sum of squared tracking errors, we have:

8 - Note that we could also use the Cholesky decomposition as well, leading to: Σ = LL'

Then, we conclude the proof by recalling the expression of the ENC 2 measure:

where a k is the k th column of matrix A , and e k the k th elementary vector. Since Σ and D are given, so are their traces, then minimising the sum of the squared tracking errors is equivalent to maximising tr( A' Σ ). Now, if we use the PCA decomposition 8 of Σ = P Λ 2 P' , we can rewrite tr( A' Σ ) as:

B.2 Derivation of the Minimal Linear Transformation We look for matrix A such that the returns on the factors r F defined as: r F = A' r , (B.1) are uncorrelated with a variance matrix A' Σ A equal to a diagonal matrix D 2 , and also such that the basis of the factors is the smallest linear torsion of the basis of the original assets in the sense that the sum of squared tracking errors between the new uncorrelated factors’ returns r F and the original correlated assets’ returns r is minimal:

where Q satisfies the following property: Q'Q = I N . Therefore we are now solving the following problem: (B.3)

(B.2)

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