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EigenvaluesofLévy CovariationMatrices

Random matrix models for datasets with fixed time horizons

Gregory Zitelli

SIAM-ALA18, Hong Kong May, 2018

Laloux et al. (1999)

Which is real? Which is fake?

Contents

• 1. Motivation

Markowitz portfolio theory, Bai’s Theorem

• 2. M–P Law

Universality, eigenvalue outliers

• 3. Sample Lévy Covariance Ensemble (SLCE)

Lévy processes, GGC/EGGC processes

• 4. Limiting Spectral Distributions

Stieltjes transform algorithm, approximate densities

Motivation

Markowitz Portfolio Theory

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p risky assets with return vector r = [r1 . . . rp]. µ µ µ = E[r] is the fixed 1 × p vector of expected returns Σ Σ Σ is the symmetric p × p covariance matrix. w is the 1 × p vector of weights on each asset, w1† ≤ 1.

For fixed 0 < σ2 < ∞, consider the optimization maximize wµ µ µ† (expected return on portfolio) subject to w ∈ R1×p, w1† ≤ 1, wΣ Σ Σw† ≤ σ2 (acceptable volatility) Let R(σ2,µ µ µ,Σ Σ Σ) denote the optimal return µ = wµ µ µ†.

Markowitz Portfolio Theory

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maximize wµ µ µ† = R(σ2,µ µ µ,Σ Σ Σ) subject to w ∈ R1×p, w1† ≤ 1, wΣ Σ Σw† ≤ σ2

R(σ2,µ µ µ,Σ Σ Σ) and the optimal w have closed-form expressions in terms of σ2, µ µ µ, and Σ Σ Σ. In practice, µ µ µ and Σ Σ Σ are unknown. How accurate is the portfolio construction when µ µ µ and Σ Σ Σ are estimated by independent sampling?

Estimating the Efgicient Frontier

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Markowitz Portfolio Theory

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Rectangular N × p data matrix. Let λ = p/N ∈ (0, 1), N samples, p assets. Bouchaud and Potters, mid 2000s, out-of-sample risk underestimated by √ 1 − λ, unproven. 2006, Bai Zhidong, Wong Wing-Keung, A Note on the Mean-Variance Analysis of Self-Financing Portfolios (unpublished), full proof published in 2009. Overestimation of returns, underestimation of risk. Relies on Marčenko–Pastur law.

Bai’s Theorem

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p assets, N observations, p/N → λ(0, 1). Σ Σ ΣN sequence of p × p covariance matrices µ µ µN sequence of 1 × p expected return vectors. Model: XN = 1 1 1†µ µ µN + YNΣ Σ Σ1/2

N .

Under mild assumptions (finite fourth moment), lim

N→∞

R(σ2, µ µ µN, Σ Σ ΣN) R(σ2,µ µ µN,Σ Σ ΣN) = √ 1 1 − λ

Proof is built on the Marčenko–Pastur (M–P) law applied to the sample data matrix YNΣ Σ Σ1/2

N .

M–PLaw

Marčenko–Pastur Law

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Y =       y1,1 · · · y1,p . . . ... . . . . . . . . . yN,1 · · · yN,p                 

• p

N, yj,k i.i.d. ∼ Y for all N

Rows are independent samples of p i.i.d. random variables with true covariance Ip, sample covariance matrix is S = 1 NY†Y If p is fixed and N → ∞, S → Ip, S has p eigenvalues converging to 1. What if p, N → ∞, p/N → λ ∈ (0, 1)?

Marčenko–Pastur Densities

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Example

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N = 2000, p = 500, λ = 0.25, i.i.d. Gaussian entries.

Laloux et al. (1999)

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Outliers?

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N = 2000, p = 500, λ = 0.25, i.i.d. Lognormal entries (normalized).

SampleLévyCovariance Ensemble(SLCE)

N → ∞

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Marčenko–Pastur: N [y1,k y2,k y3,k . . . yN−1,k yN,k]† 

• ↘

N′ [y1,k y2,k y3,k . . . yN−1,k yN,k yN+1,k . . . yN′−1,k yN′,k]† New approach, fixed horizon [0, T]: N [y1,k y2,k y3,k . . . yN−1,k yN,k]† 

• 
• 
• 
• N′

[y1,k y2,k y3,k . . . yN−1,k yN,k yN+1,k . . . yN′−1,k yN′,k]†

Marčenko–Pastur

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Marčenko–Pastur, N increases

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Fixed time horizon

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Fixed time horizon, N increases

13/29

Lévy Processes

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Correspondence between Lévy processes Xt ⇐ ⇒ infinitely divisible (ID) distributions X through X1 ∼ X Lévy–Khintchine: Decomposition of characteristic function in terms

• f Brownian motion (continuous part) and an additional measure

Π.

Lévy Processes

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1 t log φXt(ϑ) = iµϑ − 1 2σ2ϑ2 + ∫

R

[ eiϑx − 1 − iϑx 1 + x2 ] dΠ(x) Π positive Borel measure, Π({0}) = 0 dΠ(x) integrable near ±∞ x2dΠ(x) integrable near zero behavior of dΠ(x) near 0 ⇐ ⇒ path properties of Xt tail behavior of dΠ(x) ⇐ ⇒ tail behavior of fX

GGCs

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Thorin (1977): Nonnegative Lévy process such that dΠ(x) = x−1g+(x) dx, x > 0 where g+ : (0, ∞) → R is completely monotone (CM). Extended GGC (EGGC): Lefu and right tails, dΠ(x) = { x−1g+(x) dx, x > 0 |x|−1g−(|x|) dx, x < 0 g± both CM.

EGGCs

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Good properties: infinite activity, continuous densities. Lots of common distributions are GGC or EGGC.

α-stable, Student’s t, lognormal, gamma, Laplace, Pareto, generalized skew hyperbolic, generalized inverse Gaussian.

Popular models of asset returns are EGGC.

variance-gamma (VG) model of Madan and Seneta (1990), normal-inverse Gaussian (NIG) model of Barndorfg-Nielsen (1997), CGMY model of Carr et al. (2002).

Symmetric EGGCs are precisely the processes created by Brownian motion time-changed by a GGC!

EGGCs

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Sample Lévy Covariance Ensemble

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X =       x1,1 · · · x1,p . . . ... . . . . . . . . . xN,1 · · · xN,p                 

• p

N, xj,k i.i.d. ∼ XT/N for each N

Fix an EGGC process Xt. Fix a time horizon T > 0. Fix a shape parameter λ ∈ (0, 1), p : N → N with p(N)/N → λ. Let X = X(N) be a sequence of independent random matrices with i.i.d. entries, such that [X(N)]jk ∼ XT/N.

Sample Lévy Covariance Ensemble

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X =       x1,1 · · · x1,p . . . ... . . . . . . . . . xN,1 · · · xN,p                 

• p

N, xj,k i.i.d. ∼ XT/N for each N

Now consider the sample covariance S = 1 TX†X and its eigenvalues. Are these distinct from M–P?

Variance-Gamma Process

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Normal-Inverse-Gaussian Process

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LimitingSpectralDistributions

Variance of columns

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MP law relies on the convergence of the sample variance of the columns to converge to something non-random.       x1,1 · · · x1,p . . . ... . . . . . . . . . xN,1 · · · xN,p       1 N

N

∑

j=1

x2

j,k

If xj,k ∼ Y (mean zero) are i.i.d. then these converge to Var(Y). If xj,k ∼ XT/N then these converge to [X]T, the (random) quadratic variation process corresponding to Xt at time t = T.

Random column variance

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      x1,1 · · · x1,p . . . ... . . . . . . . . . xN,1 · · · xN,p       ∼       z1,1 · · · z1,p . . . ... . . . . . . . . . zN,1 · · · zN,p             √ν1 . . . √ν2 . . . . . . ... · · · √νp      

νj ∼ [X]T are i.i.d. copies of the quadratic variation process. This type of matrix has been well-studied, limiting spectral densities can be approximated by a modification

• f the algorithm introduced in Yao et al. (2015).

Algorithm

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Let s(z) = ∫

R 1 x−zdL(x) be the Stieltjes transform of the

limiting spectral distribution L for z ∈ C+. The modified Stieltjes transform is s(z) = − 1−λ

z

+ λs(z). If ν ∼ [X]T is the measure of the quadratic variation process at time T > 0, then s(z) is given by s(z) = 1 −z + λ ∫

R w 1+ws(z)dν(w)

Iterations of the map for z = x + iϵ can be used with Stieltjes inversion to approximate the density for L. Density is approximately f(x) ≈ s(x + iϵ).

Estimating the Density

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Empirical deviations from M–P

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Comparison of S&P 500 (lefu, 476 assets) and Nikkei 225 (middle, 221 assets). Daily data (top) June 2013-May 2017 (approx. 900 datapoints) and minute-by-minute (bottom) January 2017-May 2017 (approx. 40000 [SPX] and 30000 [NKY] datapoints). Right column: SLCE generated with i.i.d. NIG entries, T = 0.05 × 900 (top) and T = 0.05 × 100 (bottom).

Empirical deviations from M–P

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Conclusions

Conclusions

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Empirical evidence shows limiting distributions are independent of odd cumulants (skewness). Sensitive to higher moments, kurtosis not enough to determine shape (VG and NIG were normalized to have same excess kurtosis). Behavior of eigenvectors and top eigenvalue need to be investigated, efgects accuracy of weights in portfolio selection.

Thank you!

References

Bai, Z., Liu, H., Wong, W.K., 2009: Enhancement of the applicability of Markowitz’s portfolio optimization by utilizing random matrix theory. Mathematical Finance,

• Vol. 19.

Laloux, L., Pierre, C., Bouchaud, J.P., Potters, M., 1999: Noise dressing of financial correlation matrices. Phys. Review Letters, Vol. 83, No. 7. Marčenko, V.A., Pastur, L.A. 1967: Distribution of eigenvalues for some sets of random matrices. Journal of Multivariate Analysis, Vol. 20. Yao, J., Zheng, S., Bai, Z., 2015: Large Sample Covariance Matrices and High-Dimensional Data Analysis. Cambridge University Press, New York.

Convergence of a histogram

Y is N × p, p/N ≈ λ ∈ (0, 1), i.i.d. ∼ Y (fixed). S = 1

NY†Y, p × p symmetric matrix.

The histogram measure of S is µS = 1 p ∑

σ∈Eig[Y]

δσ M–P Law: As N, p → ∞, µS converges weakly to the M–P distribution with parameter λ (almost surely).

Markowitz Portfolio Theory

Model: XN = 1 1 1†µ µ µN + YNΣ Σ Σ1/2

N .

p : N → N such that p(N)/N = p/N → λ ∈ (0, 1) µ µ µN ∈ R1×p, Σ Σ Σ is a positive-definite p × p symmetric matrix, s.t. the following limits exist 1Σ Σ Σ−1

N 1†

N → a1, 1Σ Σ Σ−1

N µ

µ µ†

N

N → a2, µ µ µNΣ Σ Σ−1

N µ

µ µ†

N

N → a3 YN is an N × p matrix whose entries are i.i.d. ∼ Y, where Y is some probability distribution which has mean zero, unit variance, finite fourth moment.

Bai’s Theorem

Given the previous model, almost surely we have 1 Σ Σ Σ

−1 N 1†

N → 1 1 − λa1, 1 Σ Σ Σ

−1 N

µ µ µ†

N

N → 1 1 − λa2,

• µ

µ µN Σ Σ Σ

−1 N

µ µ µ†

N

N → 1 1 − λa3 and in particular for any σ2 > 0 we have lim

N→∞

R(σ2, µ µ µN, Σ Σ ΣN) R(σ2,µ µ µN,Σ Σ ΣN) = √ 1 1 − λ where µ µ µN and Σ Σ ΣN = Σ Σ Σ1/2

N Y† NYNΣ

Σ Σ1/2

N

are the sample mean and covariance. Proof is built on the Marčenko–Pastur (M–P) law applied to the sample data matrix YNΣ Σ Σ1/2

N .