High Dimensional Minimum Risk Portfolio Optimization Liusha Yang Department of Electronic and Computer Engineering Hong Kong University of Science and Technology Centrale-Sup´ elec June 26, 2015 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 1 / 35

Outline 1 Motivation and Problem Statement 2 A Robust Approach to Minimum Risk Portfolio Optimization 3 Spiked Covariance Model in the Minimum Risk Portfolio Design 4 Concluding Remarks Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 2 / 35

Motivation and Problem Statement 1 Motivation and Problem Statement 2 A Robust Approach to Minimum Risk Portfolio Optimization 3 Spiked Covariance Model in the Minimum Risk Portfolio Design 4 Concluding Remarks Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 3 / 35

Motivation and Problem Statement Background Markowitz’s Mean-Variance Portfolio Optimization Framework [Markowitz, 1952] Asset allocation: spread bets across multiple financial assets to minimize risk for given expected return, or maximize expected return for given risk Optimal solution specifies an “efficient frontier” Portfolio return Portfolio risk Minimum variance point Efficient frontier Figure: Efficient frontier Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 4 / 35

Motivation and Problem Statement Background Markowitz’s Mean-Variance Portfolio Optimization Framework [Markowitz, 1952] Asset allocation: spread bets across multiple financial assets to minimize risk for given expected return, or maximize expected return for given risk Optimal solution specifies an “efficient frontier” Portfolio return Portfolio risk Minimum variance point Efficient frontier Figure: Efficient frontier Global Minimum Variance Portfolio Framework (GMVP) Target: find the portfolio (lying on frontier) with minimal risk Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 4 / 35

Motivation and Problem Statement Background Markowitz’s Mean-Variance Portfolio Optimization Framework [Markowitz, 1952] Asset allocation: spread bets across multiple financial assets to minimize risk for given expected return, or maximize expected return for given risk Optimal solution specifies an “efficient frontier” Portfolio return Portfolio risk Minimum variance point Efficient frontier Figure: Efficient frontier Global Minimum Variance Portfolio Framework (GMVP) Target: find the portfolio (lying on frontier) with minimal risk Technical problem: require accurate covariance estimation Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 4 / 35

Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ2(h) = hT CN h s.t. hT 1N = 1 h: portfolio allocation vector CN : covariance matrix of asset returns Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 5 / 35

Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ2(h) = hT CN h s.t. hT 1N = 1 h: portfolio allocation vector CN : covariance matrix of asset returns Optimal allocation hGMVP = C−1 N 1N 1T N C−1 N 1N Minimum risk σ2(hGMVP ) = 1 1T N C−1 N 1N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 5 / 35

Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ2(h) = hT CN h s.t. hT 1N = 1 h: portfolio allocation vector CN : covariance matrix of asset returns Optimal allocation hGMVP = C−1 N 1N 1T N C−1 N 1N Minimum risk σ2(hGMVP ) = 1 1T N C−1 N 1N Problem: CN unknown Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 5 / 35

Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ2(h) = hT CN h s.t. hT 1N = 1 h: portfolio allocation vector CN : covariance matrix of asset returns Optimal allocation hGMVP = C−1 N 1N 1T N C−1 N 1N Minimum risk σ2(hGMVP ) = 1 1T N C−1 N 1N Problem: CN unknown construct estimator ˆ CN Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 5 / 35

Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ2(h) = hT CN h s.t. hT 1N = 1 h: portfolio allocation vector CN : covariance matrix of asset returns Optimal allocation hGMVP = C−1 N 1N 1T N C−1 N 1N Minimum risk σ2(hGMVP ) = 1 1T N C−1 N 1N Problem: CN unknown construct estimator ˆ CN Thus, in practice, GMVP selection is ˆ hGMVP = ˆ C−1 N 1N 1T N ˆ C−1 N 1N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 5 / 35

Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ2(ˆ hGMVP ) = ˆ hT GMVP CN ˆ hGMVP = 1T N ˆ C−1 N CN ˆ C−1 N 1N 1T N ˆ C−1 N 1N 2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 6 / 35

Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ2(ˆ hGMVP ) = ˆ hT GMVP CN ˆ hGMVP = 1T N ˆ C−1 N CN ˆ C−1 N 1N 1T N ˆ C−1 N 1N 2 Traditional sample covariance matrix (SCM): ˆ CSCM = 1 n n t=1 ˜ xt ˜ xT t with ˜ xt = xt − 1 n n i=1 xi Known to yield poor performance (high portfolio risk) when: n not N Data is non-Gaussian or contains outliers Existing time correlation Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 6 / 35

Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ2(ˆ hGMVP ) = ˆ hT GMVP CN ˆ hGMVP = 1T N ˆ C−1 N CN ˆ C−1 N 1N 1T N ˆ C−1 N 1N 2 Traditional sample covariance matrix (SCM): ˆ CSCM = 1 n n t=1 ˜ xt ˜ xT t with ˜ xt = xt − 1 n n i=1 xi Known to yield poor performance (high portfolio risk) when: n not N Data is non-Gaussian or contains outliers Existing time correlation Robust shrinkage estimator of CN [Yang et al., 2015] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 6 / 35

Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ2(ˆ hGMVP ) = ˆ hT GMVP CN ˆ hGMVP = 1T N ˆ C−1 N CN ˆ C−1 N 1N 1T N ˆ C−1 N 1N 2 Traditional sample covariance matrix (SCM): ˆ CSCM = 1 n n t=1 ˜ xt ˜ xT t with ˜ xt = xt − 1 n n i=1 xi Known to yield poor performance (high portfolio risk) when: n not N Data is non-Gaussian or contains outliers Existing time correlation Robust shrinkage estimator of CN [Yang et al., 2015] Optimal shrinkage of eigenvalues in the spiked model Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 6 / 35

A Robust Approach to Minimum Risk Portfolio Optimization 1 Motivation and Problem Statement 2 A Robust Approach to Minimum Risk Portfolio Optimization 3 Spiked Covariance Model in the Minimum Risk Portfolio Design 4 Concluding Remarks Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 7 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Motivation and Objectives Robust covariance estimators [Tyler, 1987, Maronna, 1976] For n N, good performance for non-Gaussian samples; robust to outliers For n ≈ O(N) performance degraded due to finite sampling For n < N, estimators do not exist Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 8 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Motivation and Objectives Robust covariance estimators [Tyler, 1987, Maronna, 1976] For n N, good performance for non-Gaussian samples; robust to outliers For n ≈ O(N) performance degraded due to finite sampling For n < N, estimators do not exist Shrinkage (regularized) robust estimators [Abramovich and Spencer, 2007, Pascal et al., 2013, Chen et al., 2011, Couillet and McKay, 2014] Joint robustness and resilience to finite sampling Key challenge: Design optimal shrinkage parameter for specified objective function Mean squared error minimization in [Couillet and McKay, 2014] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 8 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Motivation and Objectives Robust covariance estimators [Tyler, 1987, Maronna, 1976] For n N, good performance for non-Gaussian samples; robust to outliers For n ≈ O(N) performance degraded due to finite sampling For n < N, estimators do not exist Shrinkage (regularized) robust estimators [Abramovich and Spencer, 2007, Pascal et al., 2013, Chen et al., 2011, Couillet and McKay, 2014] Joint robustness and resilience to finite sampling Key challenge: Design optimal shrinkage parameter for specified objective function Mean squared error minimization in [Couillet and McKay, 2014] Objective: Design shrinkage robust estimator for minimizing risk under the GMVP framework Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 8 / 35

A Robust Approach to Minimum Risk Portfolio Optimization System Model Data samples: a time series of independent and identically distributed observations xt ∈ RN (returns of N assets) xt = µ µ µ + √ τtC1/2 N yt , t = 1, 2, ..., n µ µ µ: mean vector of returns CN : covariance matrix of returns τt : real, positive scalar random variable, independent of yt yt : zero mean unitarily invariant random vector with yt 2 = N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 9 / 35

A Robust Approach to Minimum Risk Portfolio Optimization System Model Data samples: a time series of independent and identically distributed observations xt ∈ RN (returns of N assets) xt = µ µ µ + √ τtC1/2 N yt , t = 1, 2, ..., n µ µ µ: mean vector of returns CN : covariance matrix of returns τt : real, positive scalar random variable, independent of yt yt : zero mean unitarily invariant random vector with yt 2 = N Embraces a class of elliptical distributions Multivariate normal Exponential Multivariate Student-t . . . Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 9 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Tyler’s Robust M-estimator with Linear Shrinkage Shrinkage Tyler’s estimator ˆ CST (ρ) For ρ ∈ (max{0, 1 − n N }, 1], the unique solution to ˆ CST (ρ) = (1 − ρ) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ C−1 ST (ρ)˜ xt + ρIN with ˜ xt = xt − 1 n n i=1 xi Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 10 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Tyler’s Robust M-estimator with Linear Shrinkage Shrinkage Tyler’s estimator ˆ CST (ρ) For ρ ∈ (max{0, 1 − n N }, 1], the unique solution to ˆ CST (ρ) = (1 − ρ) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ C−1 ST (ρ)˜ xt + ρIN with ˜ xt = xt − 1 n n i=1 xi The realized portfolio risk σ2(ˆ hST (ρ)) = 1T N ˆ C−1 ST (ρ)CN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 10 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Tyler’s Robust M-estimator with Linear Shrinkage Shrinkage Tyler’s estimator ˆ CST (ρ) For ρ ∈ (max{0, 1 − n N }, 1], the unique solution to ˆ CST (ρ) = (1 − ρ) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ C−1 ST (ρ)˜ xt + ρIN with ˜ xt = xt − 1 n n i=1 xi The realized portfolio risk σ2(ˆ hST (ρ)) = 1T N ˆ C−1 ST (ρ)CN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 Goal: find the optimal ρ to minimize σ2(ˆ hST (ρ)) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 10 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Tyler’s Robust M-estimator with Linear Shrinkage Shrinkage Tyler’s estimator ˆ CST (ρ) For ρ ∈ (max{0, 1 − n N }, 1], the unique solution to ˆ CST (ρ) = (1 − ρ) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ C−1 ST (ρ)˜ xt + ρIN with ˜ xt = xt − 1 n n i=1 xi The realized portfolio risk σ2(ˆ hST (ρ)) = 1T N ˆ C−1 ST (ρ)CN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 Goal: find the optimal ρ to minimize σ2(ˆ hST (ρ)) Main difficulty: unknown CN Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 10 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ˆ CST(ρo) Step 1: Find the deterministic equivalent N ¯ σ2(ρ) of Nσ2(ˆ hST (ρ)) under double limits1 Non-random function of CN and ρ Gives deterministic approximation for the true portfolio risk 1i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 11 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ˆ CST(ρo) Step 1: Find the deterministic equivalent N ¯ σ2(ρ) of Nσ2(ˆ hST (ρ)) under double limits1 Non-random function of CN and ρ Gives deterministic approximation for the true portfolio risk Step 2: Provide a (scaled) consistent estimator N ˆ σ2 sc (ρ) of N ¯ σ2(ρ) under double limits which depends only on observable quantities 1i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 11 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ˆ CST(ρo) Step 1: Find the deterministic equivalent N ¯ σ2(ρ) of Nσ2(ˆ hST (ρ)) under double limits1 Non-random function of CN and ρ Gives deterministic approximation for the true portfolio risk Step 2: Provide a (scaled) consistent estimator N ˆ σ2 sc (ρ) of N ¯ σ2(ρ) under double limits which depends only on observable quantities Step 3: Find the optimal ρo that minimizes N ˆ σ2 sc (ρ), and construct optimized portfolio based on this ˆ ho ST = ˆ Co−1 ST (ρo)1N 1T N ˆ Co−1 ST (ρo)1N 1i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 11 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Deterministic Equivalent of Realized Portfolio Risk Recall realized portfolio risk: σ2(ˆ hST (ρ)) = 1T N ˆ C−1 ST (ρ)CN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 Theorem 1 Under mild assumptions, sup ρ∈Rε Nσ2(ˆ hST (ρ)) − N ¯ σ2(ρ) a.s. −→ 0 where ¯ σ2(ρ) = 1 1 − βk2 (γ+αk)2 1T N k (γ+αk) CN + ρIN −1 CN k (γ+αk) CN + ρIN −1 1N 1T N k (γ+αk) CN + ρIN −1 1N 2 and where, for ε ∈ (0, min{1, c−1}), Rε := [ε + max{0, 1 − c−1}, 1] . κ, α, β, γ are functions of ρ and CN Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 12 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Consistent Estimation of the Realized Portfolio Risk Recall realized portfolio risk: σ2(ˆ hST (ρ)) = 1T N ˆ C−1 ST (ρ)CN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 Theorem 2 Denote κ = limN 1 N tr[CN ]. Under the settings of Theorem 1, sup ρ∈Rε N ˆ σ2 sc (ρ) − 1 κ Nσ2(ˆ hST (ρ)) a.s. −→ 0, where ˆ σ2 sc (ρ) = (ˆ γsc + ˆ αsc ˆ k)2 ˆ kˆ γsc 1T N ˆ C−1 ST (ρ) ˆ CST (ρ) − ρIN ˆ C−1 ST (ρ)1N (1T N ˆ C−1 ST (ρ)1N )2 . ˆ k, ˆ γsc , ˆ αsc are (observable) consistent estimators of k, γ/κ, α/κ ˆ σ2 sc (ρ) only depends on observable quantities Since κ does not depend on ρ, minimizing σ2(ˆ hST (ρ)) over ρ is approximated by minimizing ˆ σ2 sc (ρ) over ρ Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 13 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Consistent Estimation of the Realized Portfolio Risk Corollary 3 Denote ρo and ρ∗ the minimizers of ˆ σ2 sc (ρ) and σ2(ˆ hST (ρ)) over Rε respectively. Under the settings of Theorem 1 and Theorem 2, |Nσ2(ˆ hST (ρo)) − Nσ2(ˆ hST (ρ∗))| a.s. −→ 0. In words.... Choosing ρ to minimize ˆ σ2 sc (ρ) is as good (asymptotically) as minimizing the unobservable σ2(ˆ hST (ρ)) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 14 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρo = arg min ρ∈[ε+max{0,1−c−1 N },1] ˆ σ2 sc (ρ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 15 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρo = arg min ρ∈[ε+max{0,1−c−1 N },1] ˆ σ2 sc (ρ) Obtain ˆ Co ST , the unique solution to: ˆ Co ST = (1 − ρo) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ Co−1 ST ˜ xt + ρoIN Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 15 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρo = arg min ρ∈[ε+max{0,1−c−1 N },1] ˆ σ2 sc (ρ) Obtain ˆ Co ST , the unique solution to: ˆ Co ST = (1 − ρo) 1 n n t=1 ˜ xt ˜ xT t 1 N ˜ xT t ˆ Co−1 ST ˜ xt + ρoIN Construct the optimized portfolio: ˆ ho ST = ˆ Co−1 ST 1N 1T N ˆ Co−1 ST 1N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 15 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Synthetic Data Simulation 160 180 200 220 240 260 280 300 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 10−3 n (N=200) Realized Risk ˆ Co ST ˆ CP ˆ CC ˆ CC2 ˆ CLW ˆ CR bound Figure: The average realized portfolio risk of different covariance estimators in the GMVP framework using synthetic data. Distribution of the asset returns: Student-T distribution (DoF=3) Benchmarks ˆ CP Abramovich-Pascal Estimate [Couillet and McKay, 2014] ˆ CC Chen Estimate [Couillet and McKay, 2014] ˆ CC2 [Chen et al., 2011] ˆ CLW [Ledoit and Wolf, 2004] ˆ CF [Rubio et al., 2012] ˆ Co ST achieves the smallest realized risk both when n ≤ N and n > N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 16 / 35

A Robust Approach to Minimum Risk Portfolio Optimization Real Data Simulation 50 100 150 200 250 300 350 400 450 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 n (N=45) Annualized Standard Deviation ˆ Co ST ˆ CP ˆ CC ˆ CC2 ˆ CLW ˆ CR Figure: Realized portfolio risks achieved by different covariance estimators using HSI data set 736 days of HSI daily returns (from Jan. 3, 2011 to Dec. 31, 2013) Rolling window method ˆ Co ST outperforms over the entire span of estimation windows Lack of stationarity when n > 300 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 17 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design 1 Motivation and Problem Statement 2 A Robust Approach to Minimum Risk Portfolio Optimization 3 Spiked Covariance Model in the Minimum Risk Portfolio Design 4 Concluding Remarks Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 18 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Histogram of the Eigenvalue Distribution in Financial Data Observe ”spikes” in the spectrum distribution of real data −10 0 10 20 30 40 50 60 0 5 10 15 20 25 30 Eigenvalues Density (a) All eigenvalues included −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 Eigenvalues Density (b) The largest eigenvalue excluded Figure: Histogram of eigenvalues of SCM of S&P100 data set. Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 19 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1v1vT 1 + t2v2vT 2 + ... + trvrvT r r: the number of spikes t1 ≥ ... ≥ tr ≥ 0 are fixed independently of N and n v1 , ..., vr are population eigenvectors Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 20 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1v1vT 1 + t2v2vT 2 + ... + trvrvT r r: the number of spikes t1 ≥ ... ≥ tr ≥ 0 are fixed independently of N and n v1 , ..., vr are population eigenvectors ˆ C−1 N = IN + w1u1uT 1 + w2u2uT 2 + ... + wruruT r u1 , ..., ur : sample eigenvectors Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 20 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1v1vT 1 + t2v2vT 2 + ... + trvrvT r r: the number of spikes t1 ≥ ... ≥ tr ≥ 0 are fixed independently of N and n v1 , ..., vr are population eigenvectors ˆ C−1 N = IN + w1u1uT 1 + w2u2uT 2 + ... + wruruT r u1 , ..., ur : sample eigenvectors Find the optimal (w1 , ..., wr ) that minimize the portfolio risk: arg min w1 ,...,wr σ2, where σ2 = 1T N ˆ C−1 N CN ˆ C−1 N 1N (1T N ˆ C−1 N 1N )2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 20 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing ˆ C−1 N Step 1: Find the deterministic equivalent N ¯ σ2 of Nσ2 under double limits2 Non-random function of CN and (w1 , ..., wr ) Gives deterministic approximation for the true portfolio risk 2i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 21 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing ˆ C−1 N Step 1: Find the deterministic equivalent N ¯ σ2 of Nσ2 under double limits2 Non-random function of CN and (w1 , ..., wr ) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w∗ 1 , ..., w∗ r ) that minimize N ¯ σ2 2i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 21 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing ˆ C−1 N Step 1: Find the deterministic equivalent N ¯ σ2 of Nσ2 under double limits2 Non-random function of CN and (w1 , ..., wr ) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w∗ 1 , ..., w∗ r ) that minimize N ¯ σ2 Step 3: Provide a consistent estimator ( ˆ w∗ 1 , ..., ˆ w∗ r ) which depends only on observable quantities 2i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 21 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing ˆ C−1 N Step 1: Find the deterministic equivalent N ¯ σ2 of Nσ2 under double limits2 Non-random function of CN and (w1 , ..., wr ) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w∗ 1 , ..., w∗ r ) that minimize N ¯ σ2 Step 3: Provide a consistent estimator ( ˆ w∗ 1 , ..., ˆ w∗ r ) which depends only on observable quantities Step 4: Construct the optimized ˆ C−1 risk and the corresponding portfolio selection ˆ h∗ = ˆ C−1 risk 1N 1T N ˆ C−1 risk 1N 2i.e., N, n → ∞, N/n = cN → c ∈ (0, ∞) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 21 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ2 = 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N ( 1 N 1T N ˆ C−1 N 1N )2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 22 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ2 = 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N ( 1 N 1T N ˆ C−1 N 1N )2 The numerator of Nσ2: 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N = 1 N 1T N (IN + w1 u1 uT 1 + ... + wr ur uT r )× (IN + t1 v1 vT 1 + ... + tr vr vT r )(IN + w1 u1 uT 1 + ... + wr ur uT r )1N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 22 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ2 = 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N ( 1 N 1T N ˆ C−1 N 1N )2 The numerator of Nσ2: 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N = 1 N 1T N (IN + w1 u1 uT 1 + ... + wr ur uT r )× (IN + t1 v1 vT 1 + ... + tr vr vT r )(IN + w1 u1 uT 1 + ... + wr ur uT r )1N The denominator of Nσ2: 1 N 1T N ˆ C−1 N 1N 2 = 1 N 1T N (IN + w1 u1 uT 1 + ... + wr ur uT r )1N 2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 22 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ2 = 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N ( 1 N 1T N ˆ C−1 N 1N )2 The numerator of Nσ2: 1 N 1T N ˆ C−1 N CN ˆ C−1 N 1N = 1 N 1T N (IN + w1 u1 uT 1 + ... + wr ur uT r )× (IN + t1 v1 vT 1 + ... + tr vr vT r )(IN + w1 u1 uT 1 + ... + wr ur uT r )1N The denominator of Nσ2: 1 N 1T N ˆ C−1 N 1N 2 = 1 N 1T N (IN + w1 u1 uT 1 + ... + wr ur uT r )1N 2 Useful results: When N, n → ∞, 1 N 1T N ui uT i 1N a.s. −→ si 1 N 1T N vi vT i 1N , 1 N 1T N ui uT i vi vT i 1N a.s. −→ si 1 N 1T N vi vT i 1N 1 N 1T N ui uT i vi vT i ui uT i 1N a.s. −→ s2 i 1 N 1T N vi vT i 1N where si = 1 − c/(ti )2 1 + c/ti , i = 1, ..., r. Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 22 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Denote ki = 1 N 1T N vivT i 1N . The deterministic equivalent of the numerator of Nσ2 is (t1 s2 1 k1 + s1 k1 )w2 1 + 2(s1 k1 + t1 s1 k1 )w1 + ... + (tr s2 r kr + sr kr )w2 r + 2(sr kr + tr sr kr )wr + 1 + t1 k1 + ... + tr kr The deterministic equivalent of the denominator of Nσ2 is (s1 k1 w1 + s2 k2 w2 + ... + sr kr wr + 1)2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 23 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Denote ki = 1 N 1T N vivT i 1N . The deterministic equivalent of the numerator of Nσ2 is (t1 s2 1 k1 + s1 k1 )w2 1 + 2(s1 k1 + t1 s1 k1 )w1 + ... + (tr s2 r kr + sr kr )w2 r + 2(sr kr + tr sr kr )wr + 1 + t1 k1 + ... + tr kr The deterministic equivalent of the denominator of Nσ2 is (s1 k1 w1 + s2 k2 w2 + ... + sr kr wr + 1)2 Observe that N ¯ σ2 takes the form as a1 x2 1 + a2 x2 2 + ... + ar x2 r + f (b1 x1 + b2 x2 + ... + br xr + d)2 (1) We can use the Cauchy-Schwarz inequality to find (x1 , x2 , ..., xr ) that minimize (1) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 23 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Optimal Shrinkage of Eigenvalues Since (b1 x1 + b2 x2 + ... + br xr + d)2 ≤ (a1 x2 1 + a2 x2 2 + ... + ar x2 r + f)( b2 1 a1 + b2 2 a2 + ... + b2 r ar + d2 f ) where “=” can be reached when x1 = b1 f a1 d , x2 = b2 f a2 d ,...,xr = br f ar d , we obtain that a1 x2 1 + a2 x2 2 + ... + ar x2 r + f (b1 x1 + b2 x2 + ... + br xr + d)2 ≥ 1 b2 1 a1 + b2 2 a2 + ... + b2 r ar + d2 f In our case, f = 1 + t1 k1 + ... + tr kr − (s1 k1 + t1 s1 k1 )2 t1 s2 1 k1 + s1 k1 − ... − (sr kr + tr sr kr )2 tr s2 r kr + sr kr d = 1 − t1 s1 k1 + s1 k1 t1 s1 + 1 − ... − tr sr kr + sr kr tr sr + 1 a1 = t1 s2 1 k1 + s1 k1 b1 = s1 k1 x1 = w1 + t1 + 1 t1 s1 + 1 . Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 24 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Precision Matrix Estimation The optimal w∗ 1 (ti , si , ki ) = b1 f a1 d − t1 + 1 t1 s1 + 1 , and similar results for w∗ 2 , ..., w∗ r The estimators of ti, si and ki, i = 1, ..., r: ˆ ti = λi + 1 − c + (λi + 1 − c)2 − 4λi 2 − 1 ˆ si = 1 − c/(ˆ ti )2 1 + c/ˆ ti ˆ ki = 1 ˆ si 1 N 1T N uiuT i 1N where λ1 , ..., λr are the top r sample eigenvalues ˆ C−1 risk = IN + ˆ w∗ 1 (ˆ ti , ˆ si , ˆ ki )u1uT 1 + ... + ˆ w∗ r (ˆ ti , ˆ si , ˆ ki )uruT r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 25 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 200 250 300 350 400 450 8 10 12 14 16 18 20 n (N=n/2) N*Risk risktrue riskest riskdet Figure: The deterministic equivalent and the estimator of the true risk CN = IN + 14v1 vT 1 + 9v2 vT 2 + 4v3 vT 3 v1 = 3/N[1N/3 ; 02N/3 ] v2 = 3/N[1N/3 ; 0N/3 ; 1N/3 ] v3 = 3/N[02N/3 ; 1N/3 ] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 26 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 200 250 300 350 400 450 8 10 12 14 16 18 20 22 24 n (N=n/2) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: The average realized portfolio risk of different covariance estimators in the GMVP framework using synthetic data Benchmarks ˆ Cclip : Eigenvalue Clipping [Laloux et al., 2000] ˆ CFro : Frobenius norm minimization [Donoho et al., 2013] ˆ CFroinv : Frobenius norm minimization of precision matrix [Donoho et al., 2013] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 27 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 x 10−3 r (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes 1005 days of daily returns of 95 stocks from S&P500 (from 2011 to 2014) Realized risk under different assumed number of spikes U-shape curve Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 28 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 x 10−3 r (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes 1005 days of daily returns of 95 stocks from S&P500 (from 2011 to 2014) Realized risk under different assumed number of spikes U-shape curve Why doesn’t ˆ Crisk perform the best? Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 28 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 x 10−3 r (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes 1005 days of daily returns of 95 stocks from S&P500 (from 2011 to 2014) Realized risk under different assumed number of spikes U-shape curve Why doesn’t ˆ Crisk perform the best? Time correlation unconsidered Impulsiveness of the data Too large estimation error Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 28 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 x 10−3 r (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes 1005 days of daily returns of 95 stocks from S&P500 (from 2011 to 2014) Realized risk under different assumed number of spikes U-shape curve Why doesn’t ˆ Crisk perform the best? Time correlation unconsidered Impulsiveness of the data Too large estimation error Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 29 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model yt = xtT1/2∈ RN , t = 1, ..., n N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1 v1 vT 1 + t2 v2 vT 2 + ... + tr vr vT r T ∈ RN is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 30 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model yt = xtT1/2∈ RN , t = 1, ..., n N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1 v1 vT 1 + t2 v2 vT 2 + ... + tr vr vT r T ∈ RN is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] New estimators ˆ ti,ˆ si and ˆ ki, i = 1, ..., r [Vinogradova et al., 2013] Define YN = [y1 , ...., yn ] Denote λ1 ≥ ... ≥ λN the eigenvalues of 1 n YN YT N ˆ m(x) = 1 N−r N j=r+1 1 λj −x , ˆ g(x) = ˆ m(x)(xc ˆ m(x) + c − 1) ˆ ti = ˆ g(λi ) 1 n tr[ 1 N YT N YN ] −1 , ˆ si = ˆ m(λi )ˆ g(λi ) ˆ g (λi ) , ˆ ki = 1 ˆ si 1 N 1T N ui uT i 1N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 30 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model yt = xtT1/2∈ RN , t = 1, ..., n N-dimensional vectors xt i.i.d. ∼ N(µ, CN ), t = 1, ..., n CN = IN + t1 v1 vT 1 + t2 v2 vT 2 + ... + tr vr vT r T ∈ RN is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] New estimators ˆ ti,ˆ si and ˆ ki, i = 1, ..., r [Vinogradova et al., 2013] Define YN = [y1 , ...., yn ] Denote λ1 ≥ ... ≥ λN the eigenvalues of 1 n YN YT N ˆ m(x) = 1 N−r N j=r+1 1 λj −x , ˆ g(x) = ˆ m(x)(xc ˆ m(x) + c − 1) ˆ ti = ˆ g(λi ) 1 n tr[ 1 N YT N YN ] −1 , ˆ si = ˆ m(λi )ˆ g(λi ) ˆ g (λi ) , ˆ ki = 1 ˆ si 1 N 1T N ui uT i 1N ˆ C−1 risk = IN + ˆ w∗ 1 (ˆ ti , ˆ si , ˆ ki )u1uT 1 + ... + ˆ w∗ r (ˆ ti , ˆ si , ˆ ki )uruT r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 30 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.8 3 3.2 3.4 3.6 3.8 4 x 10−3 r (N=95, n=250) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 31 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.8 3 3.2 3.4 3.6 3.8 4 x 10−3 r (N=95, n=250) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes ˆ Crisk performs the best Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 31 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 0 5 10 15 20 25 30 2.8 3 3.2 3.4 3.6 3.8 4 x 10−3 r (N=95, n=250) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of assumed spikes ˆ Crisk performs the best The risk realized by ˆ Crisk doesn’t change much when r ≥ 15. Why? Haven’t figured out yet... Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 31 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 100 200 300 400 500 600 700 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 x 10−3 n (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of samples r = 11 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 32 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 100 200 300 400 500 600 700 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 x 10−3 n (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of samples r = 11 Lack of stationarity when n grows too big Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 32 / 35

Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 100 200 300 400 500 600 700 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 x 10−3 n (N=95) N*Risk ˆ Crisk ˆ Cclip ˆ CFro ˆ CFroinv Figure: Realized portofio risks achieved out-of-sample over 1005 days of S&P100 real market data (from 2011 to 2014) under different number of samples r = 11 Lack of stationarity when n grows too big Problem unsolved: the determination of the number of spikes Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 32 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Robust approach Robust to outliers and local nonstationary effects Robust to finite-sampling effects Minimization of the portfolio risk Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Robust approach Robust to outliers and local nonstationary effects Robust to finite-sampling effects Minimization of the portfolio risk Spiked model Spiked covariance structure exploited Time correlation considered Minimization of the portfolio risk Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Robust approach Robust to outliers and local nonstationary effects Robust to finite-sampling effects Minimization of the portfolio risk Spiked model Spiked covariance structure exploited Time correlation considered Minimization of the portfolio risk Future work Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Robust approach Robust to outliers and local nonstationary effects Robust to finite-sampling effects Minimization of the portfolio risk Spiked model Spiked covariance structure exploited Time correlation considered Minimization of the portfolio risk Future work Combine spiked model with robust estimation Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

Concluding Remarks Concluding Remarks Two novel minimum risk portfolio optimization strategies Shrinkage Tyler’s robust M-estimator & Spiked covariance model Deterministic characterization and consistent estimation of the portfolio risk Parameter calibration to minimize risk Key advantages Robust approach Robust to outliers and local nonstationary effects Robust to finite-sampling effects Minimization of the portfolio risk Spiked model Spiked covariance structure exploited Time correlation considered Minimization of the portfolio risk Future work Combine spiked model with robust estimation Exploit time correlation structure more finely Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 33 / 35

References Reference I Harry Markowitz. Portfolio selection. J. Finance, 7(1):77–91, Mar. 1952. Liusha Yang, Romain Couillet, and Matthew R McKay. A robust approach to minimum variance portfolio optimization. arXiv preprint arXiv:1503.08013, 2015. David E Tyler. A distribution-free M-estimator of multivariate scatter. Ann. Statist., 15(1): 234–251, 1987. Ricardo Antonio Maronna. Robust M-estimators of multivariate location and scatter. Ann. Statist., 4(1):51–67, 1976. YI Abramovich and Nicholas K Spencer. Diagonally loaded normalised sample matrix inversion (LNSMI) for outlier-resistant adaptive filtering. In Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), volume 3, pages 1105–1108, Honolulu, HI, Apr. 2007. F. Pascal, Y. Chitour, and Y. Quek. Generalized robust shrinkage estimator and its pplication to STAP detection problem. Submitted for publication, 2013. URL http://arxiv.org/abs/1311.6567. Yilun Chen, Ami Wiesel, and Alfred O. Hero. Robust shrinkage estimation of high-dimensional covariance matrices. IEEE Trans. Signal Process., 59(9):4097–4107, Sept. 2011. Romain Couillet and Matthew R McKay. Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators. J. Mult. Anal., 131:99–120, 2014. Olivier Ledoit and Michael Wolf. A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal., 88(2):365–411, Feb. 2004. Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 34 / 35

References Reference II F. Rubio, X. Mestre, and D. P. Palomar. Performance analysis and optimal selection of large minimum variance portfolios under estimation risk. IEEE J. Sel. Topics Signal Process., 6(4): 337–350, Aug. 2012. Laurent Laloux, Pierre Cizeau, Marc Potters, and Jean-Philippe Bouchaud. Random matrix theory and financial correlations. International Journal of Theoretical and Applied Finance, 3 (03):391–397, 2000. David L Donoho, Matan Gavish, and Iain M Johnstone. Optimal shrinkage of eigenvalues in the spiked covariance model. arXiv preprint arXiv:1311.0851, 2013. Julia Vinogradova, Romain Couillet, and Walid Hachem. Statistical inference in large antenna arrays under unknown noise pattern. Signal Processing, IEEE Transactions on, 61(22): 5633–5645, 2013. Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Sup´ elec June 26, 2015 35 / 35