Estimation of the Global Minimum Variance Portfolio in High Dimensions

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1 Estimation of the Global Minimum Variance Portfolio in High Dimensions Taras Bodnar, Nestor Parolya and Wolfgang Schmid 07.FEBRUARY / 25

2 Outline Introduction Random Matrix Theory: Preliminary Results Origins of the Random Matrix Theory Quadratic Forms in Random Matrices Random Matrix Theory in Portfolio Analysis Model for the Asset Returns General Shrinkage Estimator Comparison Study Further Results: c > 1 Summary 2 / 25

3 Introduction Two Main Types of Asymptotics standard asymptotics fixed dimension p and large sample size n ; classical limit theorems hold large dimensional asymptotics both the dimension p and the sample size n tend to infinity; the ratio p/n tends to a positive constant c > 0; classical limit theorems do not hold anymore (the curse of dimensionality). The large dimensional asymptotics is closer to reality. - Huber (1973) 3 / 25

4 Introduction The Curse of Dimensionality: An Example Let y i iid(0 p, I p ) for i = 1,..., n and let S n be the sample covariance matrix calculated from y 1,..., y n. Bai and Silverstein (2010): T n = log(det(s n )) = where λ i are the eigenvalues of S n. p log(λ i ), i=1 p is fixed λ i T n a.s. 1 as n a.s. 0 as n, and n/pt n N (0, 2) as n p and n such that p/n c (0, 1) as n 1 p T n a.s. d(c) = c 1 log(1 c) 1 < 0 n/pt n c a.s. d(c) np 4 / 25

5 Introduction Large Dimensional Data in Practice Financial Risk Management Usually the portfolio consists of a large amount of assets p. In practical situations the number of observed periods n is comparable to the portfolio size. Genomics The DNA data is ultra high-dimensional. In some practical cases p > n, which causes many singularity problems. Wireless Communications In multiple-input multiple-output (MIMO) antenna systems the number of channels p is usually comparable to the number of receivers n, which causes very noisy output signal. 5 / 25

the sample covariance matrix under normality Eugene Wigner (1955, 1958) Infinite matrices of

6 Random Matrix Theory: Preliminary Results Origins of the Random Matrix Theory Origins of the Random Matrix Theory John Wishart (1928) Finite-dimensional matrices Exact distribution of the sample covariance matrix under normality Eugene Wigner (1955, 1958) Infinite matrices of independent elements Empirical spectral distribution (e.s.d.): F n(x) = 1 p 1{λ i x} p i=1 6 / 25

Random Matrix Theory: Preliminary Results Origins of the Random Matrix Theory Let W n = (ξ i,j ) i,j=1,...,n be a symmetric random matrix with i.i.d. real random variables ξ i,j such that ξ i,j G 1 with E(ξ 2 i,j ) = σ2, i > j and ξ i,i G 2.

7 Random Matrix Theory: Preliminary Results Origins of the Random Matrix Theory Let W n = (ξ i,j ) i,j=1,...,n be a symmetric random matrix with i.i.d. real random variables ξ i,j such that ξ i,j G 1 with E(ξ 2 i,j ) = σ2, i > j and ξ i,i G 2. Define V n = (σ n) 1 W n and let F n be the e.s.d. of V n. Theorem (Wigner s Semi-circle Law) Let E( ξ i,j k ) < for all k = 1, 2,... G 1, G 2 are symmetric around 0. Figure : Wigner s semi-circle law for Gaussian random matrix. d. Then F n F where F is absolutely continuous with the density (semi-circle) f (x) = 1 4 x 2 2π for x 1 0 for x > 1 7 / 25

8 Random Matrix Theory: Preliminary Results Quadratic Forms in Random Matrices Stieltjes Transform Here: Quadratic forms in random matrices Marčenko-Pastur equation For nondecreasing function with bounded variation G the Stieltjes transform is defined as + 1 m G (z) = λ z dg(λ) z C+ = {z C : Im(z) > 0} Example: The Stieltjes transform of the e.s.d. F n (λ) of S n m Fn (z) = 1 p p + i=1 1 λ z δ(λ λ i)dλ = 1 p p i=1 1 λ i z = 1 p tr{(s n zi) 1 }. 8 / 25

9 Random Matrix Theory: Preliminary Results Quadratic Forms in Random Matrices Marčenko-Pastur (1967) Equation Theorem (Silverstein (1995)) Under some regularity F n (t) a.s. F (t) for p n c (0, + ) as n. Moreover, the Stieltjes transform of F satisfies the following equation m F (z) = + 1 dh(τ), (1) τ(1 c czm F (z)) z in the sense that m F (z) is the unique solution of (1) for all z C +. 9 / 25

10 Random Matrix Theory: Preliminary Results Quadratic Forms in Random Matrices Marčenko-Pastur Law, Σ n = σ 2 I (1 1 c )δ 0(λ) + 1 (λmax λ)(λ λ min ) f (λ) = 2σπ cλ 1 (λmax λ)(λ λ min ) 2σπ cλ for c > 1 for c 1, with λ max = σ 2 (1 + c) 2 and λ min = σ 2 (1 c) / 25

11 Random Matrix Theory in Portfolio Analysis Model for the Asset Returns Model for the Asset Returns Let Y n = (y 1,..., y n ) be the observation matrix and let Y n d = Σ 1/2 n X n + µ n 1 n, (2) where X n consists of i.i.d. random variables with zero means and unit variances. Assumptions: (A1) The covariance matrix of the asset returns Σ n is a nonrandom p-dimensional positive definite matrix. (A2) The elements of the matrix X n have uniformly bounded 4 + ε moments for some ε > / 25

12 Random Matrix Theory in Portfolio Analysis Model for the Asset Returns Global Minimum Variance (GMV) Portfolio w Σ n w min subject to w 1 = 1 where w = (ω 1,..., ω p ) is the vector of portfolio wights. Population GMV portfolio w GMV = Σ 1 n 1 1 Σ 1 n 1 Problem: Σ n is unknown estimation S n = 1 n (y j ȳ)(y j ȳ) with ȳ = 1 n n j=1 Sample GMV portfolio ŵ S = S 1 n 1 1 S 1 n 1 n j=1 y j (3) (4) 12 / 25

13 Random Matrix Theory in Portfolio Analysis General Shrinkage Estimator General Shrinkage Estimator S 1 n 1 ŵ GSE = α n 1 S 1 n 1 + (1 α n)b with b 1 = 1 (5) Performance measure: out-of-sample risk L(ŵ, w GMV ) = (ŵ w GMV ) Σ n (ŵ w GMV ) = ŵ Σ n ŵ σ 2 GMV σ2 ŵ where σ 2 ŵ = ŵ Σ n ŵ is the out-of-sample variance. Example: ŵ = ŵ S σs 2 = 1 S 1 n Σ n S 1 n 1 (1 S 1 n 1) 2 Relative out-of-sample risk: Rŵ = σ2 ŵ σ2 GMV σ 2 GMV = ŵ Σ n ŵ σ 2 GMV σ 2 GMV. 13 / 25

14 Random Matrix Theory in Portfolio Analysis General Shrinkage Estimator Optimal Shrinkage Intensity Optimization problem: or, equivalently, L = ŵ GSE (α n)σ n ŵ GSE (α n ) min, αnσ 2 S 2 + 2α 1 n(1 α n ) 1 S 1 n 1 1 S 1 n Σ n b + (1 α n ) 2 b Σ n b σgmv 2 min. The optimal shrinkage intensity α n is given by α n = b Σ n b 1 S 1 n Σ n b 1 S 1 n 1 σs 2 S 1 21 n Σ n b 1 S 1 n 1 + b Σ n b (6) 14 / 25

15 Random Matrix Theory in Portfolio Analysis General Shrinkage Estimator Theorem Assume (A1)-(A2). Let 0 < M l σ 2 GMV σ2 b M u <. Then α n (1 c)r b c + (1 c)r b a.s. 0 for p c (0, 1) as n (7) n where R b = b Σ n b σ 2 GMV σ 2 GMV is the relative loss of the target portfolio b. Corollary Assume (A1)-(A2). Let 0 < M l σ 2 GMV σ2 b M u <. Then a.s. (a) R S c (1 c) for p c (0, 1) as n, n a.s. (b) R GSE (1 c)cr2 b + c2 R b (c + (1 c)r b ) 2 for p c (0, 1) as n. n 15 / 25

16 Random Matrix Theory in Portfolio Analysis General Shrinkage Estimator Asymptotic behavior of the sample and optimal shrinkage estimators Average Relative Loss Sample Oracle Shrinkage Concentration Ratio c 16 / 25

17 Random Matrix Theory in Portfolio Analysis General Shrinkage Estimator Bona Fide Estimator for GMV Portfolio Theorem Under the assumptions (A1)-(A2) a consistent estimator of R b is given by ˆR b = (1 p/n)b S n b 1 S 1 n 1 1 for p c (0, 1) as n. (8) n The resulting bona fide estimator for GMV portfolio is given by ŵ OSE = α α = S 1 n 1 1 S 1 n 1 + (1 α )b with (9) (1 p/n) ˆR b p/n + (1 p/n) ˆR b. (10) 17 / 25

18 Comparison Study Comparison Study Distribution of the asset returns: normal and t 5 Structure of Σ n : 1/9 of eigenvalues are 2, 4/9 of eigenvalues are 5, 4/9 of eigenvalues are 10 c {0.5, 0.9, 1.8} Repetitions: 10 3 Performance measure: relative out-of-sample risk Benchmarks: Oracle Equivariant (ŵlw ) Dominating Estimator (ŵfm ) 18 / 25

19 Comparison Study Benchmarks Ledoit and Wolf (2012) proposed the oracle equivariant estimator for Σ 1 n ˆΣ 1 n = UA U with A = diag{u Σ 1 U} = {a i }p i=1, which is unique minimizer of the Frobenius norm Σ 1 UAU F ŵ LW = ˆΣ 1 n 1/1 ˆΣ 1 n 1. The dominating estimator of the GMV portfolio considered by Frahm and Memmel (2010) is given by ŵ FM = (1 κ) S 1 n 1 1 S 1 n 1 + κ 1 p with κ = p 3 1 (11) n p + 2 ˆR 1/p where ˆR 1/p = 1/p2 1 S n 1 σ 2 S the naive portfolio. σ 2 S is the estimated relative loss of 19 / 25

20 Estimation of the Global Minimum Variance Portfolio in High Dimensions Comparison Study Normal Distribution, b = 1/p, 1000 rep. c = 0.5, bounded spectrum c = 0.9, bounded spectrum Global behavior Global behavior Average Relative Loss Traditional Bona Fide Shrinkage Oracle Shrinkage Oracle Equivariant Frahm and Memmel (2010) Average Relative Loss Traditional Bona Fide Shrinkage Oracle Shrinkage Oracle Equivariant Frahm and Memmel (2010) Matrix dimension p Matrix dimension p c = 0.5, unbounded spectrum Global behavior c = 0.9, unbounded spectrum Global behavior Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Frahm and Memmel (2010) Oracle Shrinkage Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Frahm and Memmel (2010) Oracle Shrinkage Matrix dimension p Matrix dimension p 20 / 25

21 Comparison Study Normal Distribution, p = 900, b = 1/p, c = 0.9 Local behavior Empirical Cumulative Distribution Function Traditional Bona Fide Shrinkage Oracle Shrinkage Oracle Equivariant Frahm and Memmel (2010) Relative Loss 21 / 25

22 Estimation of the Global Minimum Variance Portfolio in High Dimensions Comparison Study t 5 -distribution, b = 1/p, 1000 rep. c = 0.5, bounded spectrum c = 0.9, bounded spectrum Global behavior Global behavior Average Relative Loss Traditional Bona Fide Shrinkage Oracle Shrinkage Oracle Equivariant Frahm and Memmel (2010) Average Relative Loss Traditional Bona Fide Shrinkage Oracle Shrinkage Oracle Equivariant Frahm and Memmel (2010) Matrix dimension p Matrix dimension p c = 0.5, unbounded spectrum Global behavior c = 0.9, unbounded spectrum Global behavior Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Frahm and Memmel (2010) Oracle Shrinkage Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Frahm and Memmel (2010) Oracle Shrinkage Matrix dimension p Matrix dimension p 22 / 25

23 Estimation of the Global Minimum Variance Portfolio in High Dimensions Further Results: c > 1 Further Results: c > 1 ŵ + OSE = α+ S+ n 1 1 S + n 1 +(1 α+ )b n with α + = and S n Moore-Penrose pseudoinverse (p/n 1) ˆR bn p/n + (p/n 1) 2 + (p/n 1) ˆR bn c = 1.8, normal distribution c = 1.8, t 5 distribution Global behavior Global behavior Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Oracle Shrinkage Average Relative Loss Traditional Bona Fide Shrinkage Nonlinear Shrinkage, Ledoit and Wolf (2012) Oracle Shrinkage Matrix dimension p Matrix dimension p 23 / 25

24 Summary Summary A shrinkage-type estimator for the weights of GMV portfolio is developed using the results from the random matrix theory. The resulting estimator is distribution-free and it is derived under weak assumptions. The shrinkage intensity is consistently estimated. Via Monte Carlo simulations we show that the obtained estimator significantly dominates the existent estimators for the GMV portfolio. 24 / 25

25 Reference Thank you very much for your attention! Bai J., and S. Shi, (2011), Estimating High Dimensional Covariance Matrices and Its Applications, Annals of Economics and Finance 12-2, Bai Z.D., and J. W. Silverstein, (2010), Spectral Analysis of Large Dimensional Random Matrices, Springer: New York; Dordrecht; Heidelberg; London. Bodnar, T., N. Parolya, and W. Schmid. (2014). Estimation of the global minimum variance portfolio in high dimensions. Submitted for publication. Bodnar, T. and W. Schmid. (2008). A test for the weights of the global minimum variance portfolio in an elliptical model. Metrika, 67, Frahm G. and C. Memmel, (2010), Dominating estimators for minimum-variance portfolios, Journal of Econometrics 159, Jagannathan, R., and T. Ma (2003), Risk reduction in large portfolios: why imposing the wrong constraints helps? Journal of Finance 58, Ledoit, O. and Wolf, M. (2003), Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance 10, Ledoit, O. and Wolf, M. (2012), Nonlinear shrinkage estimation of large-dimensional covariance matrices, Annals of Statistics 40, Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Sbornik: Mathematics Silverstein, J. W., (1995), Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices Journal of Multivariate Analysis 55, / 25

Random Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws. Symeon Chatzinotas February 11, 2013 Luxembourg

Random Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws Symeon Chatzinotas February 11, 2013 Luxembourg Outline 1. Random Matrix Theory 1. Definition 2. Applications 3. Asymptotics 2. Ensembles