Simulating realistic correlation matrices for financial applications: correlation matrices with the Perron–Frobenius property

ABSTRACT

This article is concerned with the simulation of correlation matrices with realistic properties for financial applications. Previous studies found that the major part of observed correlation matrices in financial applications exhibits the Perron-Frobenius property, namely a dominant eigenvector with only positive entries. We present a simulation algorithm for random correlation matrices satisfying this property, which can be augmented to take into account a realistic eigenvalue structure. From the construction principle applied in our algorithm, and the fact that it is able to generate all such correlation matrices, we are further able to prove that the proportion of Perron-Frobenius correlation matrices in the set of all correlation matrices is $2^{1-d}$ in dimension d.

KEYWORDS:

• Random correlation matrix
• Perron–Frobenius property
• Bendel–Mickey algorithm
• eigenvalues

Acknowledgments

The authors thank Matthias Scherer and an anonymous reviewer for helpful comments and suggestions concerning this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 For irreducible non-negative matrices the sharper statements of the original Perron-Frobenius theorem hold.

2 The MST is unique if all edge weights of the original graph are different, i.e. if all entries of the correlation matrix are different, which is typically the case for observed financial correlation matrices.

3 If the rows of X are uniformly distributed on the sphere, the percentage of generated matrices with positive dominant eigenvector vanishes quickly with increasing dimension d, like for the uniform distribution.

4 [6] show that it is not convex in d>3. See also [8,9] for results on the sets of (symmetric) square matrices exhibiting the Perron-Frobenius properties.

5 They show that a necessary and sufficient condition in d=3 for a correlation matrix to have a positive dominant eigenvector is that the sum of any two pairwise correlations is positive.

6 This is always possible as the trace equals d and does not change.

7 Recall that the eigenvalues of any $C\in {\mathcal{C}}_d$ are all non-negative and sum up to d.

8 The eigendecomposition of C is unique up to different signs of columns of V, thus for a fixed C there is only finitely many V such that $V\mathrm{\Lambda }{V}^{\prime }=C$. These V constitute a set of measure zero in ${\mathcal{V}}_{\lambda }^{(d-1)}$.

9 Once the rotation indices i and j are chosen, and it is decided which of the two diagonal entries should be set to 1, we have two possible solutions for the tangent, and two choices for the sign of the cosine, cf. Equations (2(2) $\begin{array}{rl}& (w_{j_k,j_k}^{(k-1)}-1)t_k^2-2t_k{w}_{i_k,j_k}^{(k-1)}+w_{i_k,i_k}^{(k-1)}-1=0,\\ & \tan \left({\theta }_k\right)=t_k=\frac{w_{i_k,j_k}^{(k-1)}\pm \sqrt{(w_{i_k,j_k}^{(k-1)}{)}^2-(w_{i_k,i_k}^{(k-1)}-1\left)\right(w_{j_k,j_k}^{(k-1)}-1)}}{w_{j_k,j_k}^{(k-1)}-1},\\ & c_k=\pm (1+t_k^2{)}^{-1/2},s_k=c_k{t}_k.\end{array}$(2) ) and (3(3) $t_k=\frac{-w_{i_k,j_k}^{(k-1)}\pm \sqrt{(w_{i_k,j_k}^{(k-1)}{)}^2-(w_{j_k,j_k}^{(k-1)}-1\left)\right(w_{i_k,i_k}^{(k-1)}-1)}}{w_{i_k,i_k}^{(k-1)}-1}.$(3) ).

10 cf. [16]: ‘The statistical distribution of the matrices produced by [··· ] is not well understood, [··· ]’ They refer to [18], but the discussion there focuses exclusively on the distribution of eigenvalues and related quantities like the determinant.

11 [2,27]'s power law relies on a lower bound ${\lambda }^{-}$ for the eigenvalues, which is set to ${\lambda }^{-}=0$ here.

12 Extreme cases where all pairwise correlations are larger than 0.9 are possible, but occur very rarely.

13 ${\lambda }_1=0.4\cdot d$ and ${\tilde{\lambda }}_i,i=2,\dots ,d,$ are simulated from density (5(5) $f\left(x\right)=\frac{2}{(x+1{)}^3},$(5) ) and rescaled as described above such that ${\lambda }_2+\cdots +{\lambda }_d=0.6\cdot d$.

14 For comparison, we use a data set of 395 5Y-CDS log return time series of constituents of the four major credit indices (ITRX, ITRX HY, CDX, CDX XO) as described in [1].

15 Although this is not the most reliable method for fitting a power law, we nevertheless stick with this approach for the sake of simplicity, as we only intend to provide an intuition about the behaviour of the degree distribution of the MSTs related to correlation matrices simulated from our Algorithm 1. A thorough fit of a power law to the simulated degree distributions is beyond the scope of this paper. See [28] for more information on the drawbacks of this approach for fitting power-law distributions, and viable alternatives.