Correlation Matrices and the Perron-Frobenius Theorem

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1 Wilfrid Laurier University July

2 Acknowledgements Thanks to David Melkuev, Johnew Zhang, Bill Zhang, Ronnnie Feng Jeyan Thangaraj for research assistance.

3 Outline The - theorem extensions to negative elements

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5 This is preliminary work. Comments welcome. Markowitz efficient portfolios selected by investors These portfolios have desirable properties Mean variance efficient Sharpe used these ideas to develop the CAPM Equilibrium model relating expected return to risk Market portfolio is mean variance efficient

6 Compatible Σ, Ü (m),µ Σ covariance matrix; µ return vector, Ü (m) market weights These three entities have to be compatible since Ü (m) on frontier Best Grauer (1985) Assume Σ known ΣÜ (m) = γ 1 µ+γ 2 (1) Assume- for now- we have a way to find Ü (m) Task is to find compatible µ

7 Picking the market portfolio Origins of idea from Sharpe Ross (APT) Dominant common factor that influences stock returns PCA used to identify this factor Principal eigenvector of the correlation matrix Trzcinka (1986), Laloux et al(1999), Avellaneda Lee (2010) & Allez Bouchaud (2011) Market portfolio should have positive weights When will principal eigenvector have positive weights?

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9 The - (-) A real n n matrix,, with positive entries has a unique largest real eigenvalue the corresponding eigenvector has strictly positive components. Provides sufficient conditions Result can be weakened Matrix can have some negative elements retain the - (PF) property. There are sometimes negative correlations between stock returns. Hence we are interested in correlation matrices which have the PF property

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11 Empirical experiment We obtain CRSP daily returns for S&P1500 components from Divide data to five-year periods Select 10,000 rom samples of 50 stocks, compute C All matrices positive-definite For non-positive matrices, we study the distribution of elements Test whether the type can be determined based on simple rules for the elements

12 Changes in correlation through time Breakdown of Matrices by Type Non-PF PF w negative Positive Proportion (%) Period

13 Visualizing changes in correlation through time Daily Return sp1500_90_ Daily Return sp1500_94_ Daily Return sp1500_99_ Daily Return sp1500_04_

14 Distribution of elements Fatter left tails for Non-PF correlation matrices. E.g : Non-PF Density of elements PF

15 Count of negative elements Estimated density for the count of negative elements. E.g : Non-PF PF Count of negative elements

16 Characteristics of stock correlations Which stocks are most prevalent in Non-PF matrices? - rank stocks by average correlation with the rest (ascending) - take the top three; these are low correlation stocks Low correlation stocks appear as (almost) full rows of negative elements When these stocks are selected we end up with Non-PF matrices - Consistent with proposition on strictly negative rows (discussed later)

17 Element distribution: empirical vs simulated Statistics of negative elements Empirical Simulated Stat PF Non-PF PF Non-PF Count* N/A 612 Mean N/A Std dev N/A Min N/A % N/A % N/A % N/A Max N/A * average number per matrix

18 Visualizing negative rows Daily Return sp1500_94_ Figure: matrix visualization. 100 stocks daily returns,

19 Visualizing negative rows Different sampling frequency (e.g. weekly) can sometimes reduce the number of negative correlation Daily Return sp1500_94_ Weekly Return sp1500_94_ Figure: matrix visualization. Daily (left) returns vs weekly (right) returns

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21 Notation PF(n) is the set of n n correlation matrices possessing the strong - property PF + (n) is the set of matrices in PF(n) that have only positive elements PF (n) is the set of matrices in PF(n) that have at least one negative element PD(n) is the set of n n positive-definite correlation matrices PF PD (n) is the set PF(n) PD(n)

22 Analytic result for three by three case Consider the 3 3 correlation matrix 1 ρ 12 ρ 13 C = ρ 12 1 ρ 23 ρ 13 ρ 23 1 C PF(3) when the following condition holds ρ 12 +ρ 13 > 0 ρ 12 +ρ 23 > 0 ρ 13 +ρ 23 > 0 When A,B PF PD (3), then convex combination always preserve the above condition PF PD (3) is convex

23 Negative row Assume C PF(n), λ υ are dominant eigenpair. Then λ > 1 υ i > 0,i = 1,...,n. For each row i, υ i + j i υ j ρ i,j = λυ i (2) υ 1 (λ 1) = j i υ j ρ i,j (3) The LHS of (3) is positive. Thus, ρ i,j cannot be all negative. In other words, PF matrix cannot have rows of only negative off diagonals!

24 Constant correlation matrices Suppose P is a correlation matrix. P has all off-diagonal entries equal to ρ Whenever ρ > 1 n 1, P PD(n) Furthermore, P PF PD (n) if only if ρ > 0

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26 Simulate 10,000 rom positive-definite correlation matrices using Harry Joe s method (2006) Study the distribution of Non-PF PF matrices among various dimensions Within the set of PF matrices, we test convexity properties Better underst eventually positive condition

27 Simulated proportions by type Dimension PF + (n) PF (n) Non-PF % 10.04% 75.18% % 9.31% 87.40% % 5.59% 93.96% % 3.02% 96.90% % 1.55% 98.44% % 0.79% 99.21% Table: Proportion of sample correlation matrices by type from dimension 3 to 8 The set of PF matrices shrinks while matrix dimension increases The decreasing portion of positive matrices proposes a limitation of simulation method - low likelihood of getting positive matrices

28 Eventually positive matrices Definition An n n matrix A is said to be eventually positive if there exists a positive integer k 0 such that A k > 0 for all k > k 0. (Noutsos) For any symmetric n n matrix A the following properties are equivalent. 1 A possesses the strong - property. 2 A is eventually positive

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30 Summary Interest in finding compatible Ü (m) µ Market portfolio has positive weights Proxied by dominant eigenvector of correlation matrix When does dominant eigenvector have positive weights - property Explored this question in three ways 1 Using 2 analysis 3 simulation

31 Conclusions Empirical Negative correlation has declined during last 20 years Negative correlations tend to occur in rows Has implications for PF property Analytical A row of negative correlations destroys the PF property Obtained a simple characterization of 3 3 matrices Constant correlation matrices have simple classification Simulation PF matrices rare in high dimensions for rom matrices Failure of PF related to negative elements Related to number, size position of negative elements