High Dimensional Minimum Risk Portfolio Optimization

Transcription

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

2 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-Supélec June 26, / 35

3 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-Supélec June 26, / 35

4 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 Portfolio return Efficient frontier Minimum variance point Optimal solution specifies an efficient frontier Portfolio risk Figure: Efficient frontier Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

5 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 Portfolio return Efficient frontier Minimum variance point Optimal solution specifies an efficient frontier Portfolio risk 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-Supélec June 26, / 35

6 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 Portfolio return Efficient frontier Minimum variance point Optimal solution specifies an efficient frontier Portfolio risk 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-Supélec June 26, / 35

7 Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ 2 (h) = h T C N h s.t. h T 1 N = 1 h: portfolio allocation vector C N : covariance matrix of asset returns Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

8 Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ 2 (h) = h T C N h s.t. h T 1 N = 1 h: portfolio allocation vector C N : covariance matrix of asset returns Optimal allocation h GMVP = C 1 N 1 N 1 T N C 1 N 1 N Minimum risk σ 2 (h GMVP) = 1 1 T N C 1 N 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

9 Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ 2 (h) = h T C N h s.t. h T 1 N = 1 h: portfolio allocation vector C N : covariance matrix of asset returns Optimal allocation h GMVP = C 1 N 1 N 1 T N C 1 N 1 N Problem: C N unknown Minimum risk σ 2 (h GMVP) = 1 1 T N C 1 N 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

10 Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ 2 (h) = h T C N h s.t. h T 1 N = 1 h: portfolio allocation vector C N : covariance matrix of asset returns Optimal allocation h GMVP = C 1 N 1 N 1 T N C 1 N 1 N Minimum risk σ 2 (h GMVP) = 1 1 T N C 1 N 1 N Problem: C N unknown construct estimator ĈN Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

11 Motivation and Problem Statement Problem Statement Asset allocation problem in GMVP framework min h σ 2 (h) = h T C N h s.t. h T 1 N = 1 h: portfolio allocation vector C N : covariance matrix of asset returns Optimal allocation h GMVP = C 1 N 1 N 1 T N C 1 N 1 N Minimum risk σ 2 (h GMVP) = 1 1 T N C 1 N 1 N Problem: C N unknown construct estimator ĈN Thus, in practice, GMVP selection is ĥ GMVP = Ĉ 1 N 1 N 1 T N Ĉ 1 N 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

12 Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ 2 (ĥgmvp) = ĥt GMVPC N ĥ GMVP = 1TNĈ 1 N C N Ĉ 1 N 1 N ) 2 ( 1 T NĈ 1 N 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

13 Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ 2 (ĥgmvp) = ĥt GMVPC N ĥ GMVP = 1TNĈ 1 N C N Ĉ 1 N 1 N ) 2 ( 1 T NĈ 1 N 1 N Traditional sample covariance matrix (SCM): Ĉ SCM = 1 n n t=1 x t x T t with x t = x t 1 n n i=1 x i 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-Supélec June 26, / 35

14 Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ 2 (ĥgmvp) = ĥt GMVPC N ĥ GMVP = 1TNĈ 1 N C N Ĉ 1 N 1 N ) 2 ( 1 T NĈ 1 N 1 N Traditional sample covariance matrix (SCM): Ĉ SCM = 1 n n t=1 x t x T t with x t = x t 1 n n i=1 x i 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 C N [Yang et al., 2015] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

15 Motivation and Problem Statement Problem Statement The portfolio risk (out-of-sample or realized risk) : σ 2 (ĥgmvp) = ĥt GMVPC N ĥ GMVP = 1TNĈ 1 N C N Ĉ 1 N 1 N ) 2 ( 1 T NĈ 1 N 1 N Traditional sample covariance matrix (SCM): Ĉ SCM = 1 n n t=1 x t x T t with x t = x t 1 n n i=1 x i 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 C N [Yang et al., 2015] Optimal shrinkage of eigenvalues in the spiked model Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

16 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-Supélec June 26, / 35

17 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-Supélec June 26, / 35

18 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-Supélec June 26, / 35

19 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-Supélec June 26, / 35

20 A Robust Approach to Minimum Risk Portfolio Optimization System Model Data samples: a time series of independent and identically distributed observations x t R N (returns of N assets) x t = µ + τ t C 1/2 N y t, t = 1, 2,..., n µ: mean vector of returns C N : covariance matrix of returns τ t: real, positive scalar random variable, independent of y t y t: zero mean unitarily invariant random vector with y t 2 = N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

21 A Robust Approach to Minimum Risk Portfolio Optimization System Model Data samples: a time series of independent and identically distributed observations x t R N (returns of N assets) x t = µ + τ t C 1/2 N y t, t = 1, 2,..., n µ: mean vector of returns C N : covariance matrix of returns τ t: real, positive scalar random variable, independent of y t y t: zero mean unitarily invariant random vector with y t 2 = N Embraces a class of elliptical distributions Multivariate normal Exponential Multivariate Student-t... Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

22 A Robust Approach to Minimum Risk Portfolio Optimization Tyler s Robust M-estimator with Linear Shrinkage Shrinkage Tyler s estimator ĈST(ρ) For ρ (max{0, 1 n N }, 1], the unique solution to Ĉ ST (ρ) = (1 ρ) 1 n x t x T t n 1 t=1 N xt t Ĉ 1 ST (ρ) x + ρi N t with x t = x t 1 n n i=1 x i Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

23 A Robust Approach to Minimum Risk Portfolio Optimization Tyler s Robust M-estimator with Linear Shrinkage Shrinkage Tyler s estimator ĈST(ρ) For ρ (max{0, 1 n N }, 1], the unique solution to Ĉ ST (ρ) = (1 ρ) 1 n x t x T t n 1 t=1 N xt t Ĉ 1 ST (ρ) x + ρi N t with x t = x t 1 n n i=1 x i The realized portfolio risk σ 2 (ĥst(ρ)) = 1T N Ĉ 1 ST (ρ)c N Ĉ 1 ST (ρ)1 N (1 T NĈ 1 ST (ρ)1 N) 2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

24 A Robust Approach to Minimum Risk Portfolio Optimization Tyler s Robust M-estimator with Linear Shrinkage Shrinkage Tyler s estimator ĈST(ρ) For ρ (max{0, 1 n N }, 1], the unique solution to Ĉ ST (ρ) = (1 ρ) 1 n x t x T t n 1 t=1 N xt t Ĉ 1 ST (ρ) x + ρi N t with x t = x t 1 n n i=1 x i The realized portfolio risk σ 2 (ĥst(ρ)) = 1T N Ĉ 1 ST (ρ)c N Ĉ 1 ST (ρ)1 N (1 T NĈ 1 ST (ρ)1 N) 2 Goal: find the optimal ρ to minimize σ 2 (ĥst(ρ)) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

25 A Robust Approach to Minimum Risk Portfolio Optimization Tyler s Robust M-estimator with Linear Shrinkage Shrinkage Tyler s estimator ĈST(ρ) For ρ (max{0, 1 n N }, 1], the unique solution to Ĉ ST (ρ) = (1 ρ) 1 n x t x T t n 1 t=1 N xt t Ĉ 1 ST (ρ) x + ρi N t with x t = x t 1 n n i=1 x i The realized portfolio risk σ 2 (ĥst(ρ)) = 1T N Ĉ 1 ST (ρ)c N Ĉ 1 ST (ρ)1 N (1 T NĈ 1 ST (ρ)1 N) 2 Goal: find the optimal ρ to minimize σ 2 (ĥst(ρ)) Main difficulty: unknown C N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

26 A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ĈST(ρ o ) Step 1: Find the deterministic equivalent N σ 2 (ρ) of Nσ 2 (ĥst(ρ)) under double limits 1 Non-random function of C N and ρ Gives deterministic approximation for the true portfolio risk 1 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

27 A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ĈST(ρ o ) Step 1: Find the deterministic equivalent N σ 2 (ρ) of Nσ 2 (ĥst(ρ)) under double limits 1 Non-random function of C N and ρ Gives deterministic approximation for the true portfolio risk Step 2: Provide a (scaled) consistent estimator N ˆσ 2 sc(ρ) of N σ 2 (ρ) under double limits which depends only on observable quantities 1 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

28 A Robust Approach to Minimum Risk Portfolio Optimization Method of Developing Risk-minimizing ĈST(ρ o ) Step 1: Find the deterministic equivalent N σ 2 (ρ) of Nσ 2 (ĥst(ρ)) under double limits 1 Non-random function of C N and ρ Gives deterministic approximation for the true portfolio risk Step 2: Provide a (scaled) consistent estimator N ˆσ 2 sc(ρ) of N σ 2 (ρ) under double limits which depends only on observable quantities Step 3: Find the optimal ρ o that minimizes N ˆσ 2 sc(ρ), and construct optimized portfolio based on this ĥ o ST = Ĉo 1 ST (ρo )1 N 1 T N Ĉo 1 ST (ρo )1 N 1 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

29 A Robust Approach to Minimum Risk Portfolio Optimization Deterministic Equivalent of Realized Portfolio Risk Recall realized portfolio risk: σ 2 (ĥst(ρ)) = 1T N Ĉ 1 ST (ρ)c N Ĉ 1 ST (ρ)1 N (1 T N Ĉ 1 ST (ρ)1 N ) 2 Theorem 1 Under mild assumptions, Nσ sup 2 (ĥst(ρ)) N σ2 (ρ) a.s. 0 ρ R ε where σ 2 (ρ) = 1 1 βk2 (γ+αk) 2 1 T N ( ) 1 ( k (γ+αk) CN + ρin CN ( 1 T N ( k (γ+αk) CN + ρin ) 1 1N ) ) 1 2 k (γ+αk) CN + ρin 1N and where, for ε (0, min{1, c 1 }), R ε := [ε + max{0, 1 c 1 }, 1]. κ, α, β, γ are functions of ρ and C N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

30 A Robust Approach to Minimum Risk Portfolio Optimization Consistent Estimation of the Realized Portfolio Risk Recall realized portfolio risk: σ 2 (ĥst(ρ)) = 1T N Ĉ 1 ST (ρ)c N Ĉ 1 ST (ρ)1 N (1 T N Ĉ 1 ST (ρ)1 N ) 2 Theorem 2 1 Denote κ = lim N tr[cn ]. Under the settings of Theorem 1, N sup N ˆσ2 sc(ρ) 1 κ Nσ2 (ĥst(ρ)) a.s. 0, ρ R ε where ˆσ 2 sc(ρ) = ) T 2 (ˆγsc + ˆαscˆk) 1N Ĉ 1 ST (ĈST(ρ) (ρ) ρi N Ĉ 1 ST (ρ)1n. ˆkˆγ sc (1 T N Ĉ 1 ST (ρ)1n )2 ˆk, ˆγ sc, ˆα sc are (observable) consistent estimators of k, γ/κ, α/κ ˆσ 2 sc(ρ) only depends on observable quantities Since κ does not depend on ρ, minimizing σ 2 (ĥst(ρ)) over ρ is approximated by minimizing ˆσ 2 sc(ρ) over ρ Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

31 A Robust Approach to Minimum Risk Portfolio Optimization Consistent Estimation of the Realized Portfolio Risk Corollary 3 Denote ρ o and ρ the minimizers of ˆσ sc(ρ) 2 2 and σ (ĥst(ρ)) over Rε respectively. Under the settings of Theorem 1 and Theorem 2, Nσ 2 (ĥst(ρo )) Nσ 2 (ĥst(ρ )) a.s. 0. In words... Choosing ρ to minimize ˆσ 2 sc(ρ) is as good (asymptotically) as minimizing the unobservable σ 2 (ĥst(ρ)) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

32 A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρ o = arg min ρ [ε+max{0,1 c 1 N },1] ˆσ 2 sc(ρ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

33 A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρ o = arg min ρ [ε+max{0,1 c 1 N },1] ˆσ 2 sc(ρ) Obtain Ĉo ST, the unique solution to: Ĉ o ST = (1 ρ o ) 1 n n x t x T t 1 N xt t Ĉo 1 ST x + ρ o I N t t=1 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

34 A Robust Approach to Minimum Risk Portfolio Optimization Portfolio Design for Minimizing Risk Find the optimal ρ via a numerical search ρ o = arg min ρ [ε+max{0,1 c 1 N },1] ˆσ 2 sc(ρ) Obtain Ĉo ST, the unique solution to: Ĉ o ST = (1 ρ o ) 1 n Construct the optimized portfolio: ĥ o ST = n x t x T t 1 N xt t Ĉo 1 ST x + ρ o I N t t=1 Ĉo 1 ST 1 N 1 T NĈo 1 ST 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

35 A Robust Approach to Minimum Risk Portfolio Optimization Synthetic Data Simulation Realized Risk 5.5 x n (N=200) Ĉ o ST ĈP ĈC ĈC2 ĈLW ĈR 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 Ĉ P Abramovich-Pascal Estimate [Couillet and McKay, 2014] Ĉ C Chen Estimate [Couillet and McKay, 2014] Ĉ C2 [Chen et al., 2011] Ĉ LW [Ledoit and Wolf, 2004] Ĉ F [Rubio et al., 2012] Ĉ o ST achieves the smallest realized risk both when n N and n > N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

36 A Robust Approach to Minimum Risk Portfolio Optimization Real Data Simulation Ĉ o ST ĈP Annualized Standard Deviation n (N=45) Figure: Realized portfolio risks achieved by different covariance estimators using HSI data set ĈC ĈC2 ĈLW ĈR 736 days of HSI daily returns (from Jan. 3, 2011 to Dec. 31, 2013) Rolling window method Ĉ o ST outperforms over the entire span of estimation windows Lack of stationarity when n > 300 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

37 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-Supélec June 26, / 35

38 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 Density 15 Density Eigenvalues (a) All eigenvalues included Eigenvalues (b) The largest eigenvalue excluded Figure: Histogram of eigenvalues of SCM of S&P100 data set. Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

39 Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors x t i.i.d. N(µ, C N ), t = 1,..., n C N = I N + t 1 v 1 v T 1 + t 2 v 2 v T t r v r v T r r: the number of spikes t 1... t r 0 are fixed independently of N and n v 1,..., v r are population eigenvectors Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

40 Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors x t i.i.d. N(µ, C N ), t = 1,..., n C N = I N + t 1 v 1 v T 1 + t 2 v 2 v T t r v r v T r r: the number of spikes t 1... t r 0 are fixed independently of N and n v 1,..., v r are population eigenvectors Ĉ 1 N = I N + w 1 u 1 u T 1 + w 2 u 2 u T w r u r u T r u 1,..., u r: sample eigenvectors Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

41 Spiked Covariance Model in the Minimum Risk Portfolio Design System Model and Problem Formulation N-dimensional vectors x t i.i.d. N(µ, C N ), t = 1,..., n C N = I N + t 1 v 1 v T 1 + t 2 v 2 v T t r v r v T r r: the number of spikes t 1... t r 0 are fixed independently of N and n v 1,..., v r are population eigenvectors Ĉ 1 N = I N + w 1 u 1 u T 1 + w 2 u 2 u T w r u r u T r u 1,..., u r: sample eigenvectors Find the optimal (w 1,..., w r ) that minimize the portfolio risk: C N Ĉ 11 N where σ 2 = 1T N Ĉ 1 N N (1 T N Ĉ 1 N 1 N) 2 arg min σ 2, w 1,...,w r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

42 Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing Ĉ 1 N Step 1: Find the deterministic equivalent N σ 2 of Nσ 2 under double limits 2 Non-random function of C N and (w 1,..., w r) Gives deterministic approximation for the true portfolio risk 2 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

43 Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing Ĉ 1 N Step 1: Find the deterministic equivalent N σ 2 of Nσ 2 under double limits 2 Non-random function of C N and (w 1,..., w r) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w 1,..., w r) that minimize N σ 2 2 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

44 Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing Ĉ 1 N Step 1: Find the deterministic equivalent N σ 2 of Nσ 2 under double limits 2 Non-random function of C N and (w 1,..., w r) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w 1,..., w r) that minimize N σ 2 Step 3: Provide a consistent estimator (ŵ 1,..., ŵ r) which depends only on observable quantities 2 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

45 Spiked Covariance Model in the Minimum Risk Portfolio Design Method of Developing Risk-minimizing Ĉ 1 N Step 1: Find the deterministic equivalent N σ 2 of Nσ 2 under double limits 2 Non-random function of C N and (w 1,..., w r) Gives deterministic approximation for the true portfolio risk Step 2: Find the optimal (w 1,..., w r) that minimize N σ 2 Step 3: Provide a consistent estimator (ŵ 1,..., ŵ r) which depends only on observable quantities Step 4: Construct the optimized Ĉ 1 risk and the corresponding portfolio selection ĥ = Ĉ 1 risk 1 N 1 T N Ĉ 1 risk 1 N 2 i.e., N, n, N/n = c N c (0, ) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

46 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ 2 = 1 N 1T N Ĉ 1 N C N Ĉ 1 N 1 N ( 1 N 1T N Ĉ 1 N 1 N) 2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

47 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ 2 = 1 N 1T N Ĉ 1 N C N Ĉ 1 N 1 N ( 1 N 1T N Ĉ 1 N 1 N) 2 The numerator of Nσ 2 : 1 N C N Ĉ 1 N 1 N = 1 N 1T N (I N + w 1u 1u T w ru ru T r ) N 1T N Ĉ 1 (I N + t 1v 1v T t rv rv T r )(I N + w 1u 1u T w ru ru T r )1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

48 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ 2 = 1 N 1T N Ĉ 1 N C N Ĉ 1 N 1 N ( 1 N 1T N Ĉ 1 N 1 N) 2 The numerator of Nσ 2 : 1 N C N Ĉ 1 N 1 N = 1 N 1T N (I N + w 1u 1u T w ru ru T r ) N 1T N Ĉ 1 (I N + t 1v 1v T t rv rv T r )(I N + w 1u 1u T w ru ru T r )1 N The denominator of Nσ 2 : ( ) 2 ( ) N 1T N Ĉ 1 N 1 N = N 1T N (I N + w 1u 1u T w ru ru T r )1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

49 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Recall that Nσ 2 = 1 N 1T N Ĉ 1 N C N Ĉ 1 N 1 N ( 1 N 1T N Ĉ 1 N 1 N) 2 The numerator of Nσ 2 : 1 N C N Ĉ 1 N 1 N = 1 N 1T N (I N + w 1u 1u T w ru ru T r ) N 1T N Ĉ 1 (I N + t 1v 1v T t rv rv T r )(I N + w 1u 1u T w ru ru T r )1 N The denominator of Nσ 2 : ( ) 2 ( ) N 1T N Ĉ 1 N 1 N = N 1T N (I N + w 1u 1u T w ru ru T r )1 N Useful results: When N, n, 1 N 1T N u iu T i 1 N 1 N 1T N u iu T i v ivi T u iu T i 1 N a.s. 1 s i N 1T N v ivi T 1 N, a.s. s 2 i where s i = 1 c/(ti)2 1 + c/t i, i = 1,..., r. 1 N 1T N u iu T i v iv T i 1 N 1 N 1T N v iv T i 1 N a.s. 1 s i N 1T N v ivi T 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

50 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Denote k i = 1 N 1T N v iv T i 1 N. The deterministic equivalent of the numerator of Nσ 2 is (t 1s 2 1k 1 + s 1k 1)w (s 1k 1 + t 1s 1k 1)w (t rs 2 rk r + s rk r)w 2 r + 2(s rk r + t rs rk r)w r t 1k t rk r The deterministic equivalent of the denominator of Nσ 2 is (s 1k 1w 1 + s 2k 2w s rk rw r + 1) 2 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

51 Spiked Covariance Model in the Minimum Risk Portfolio Design Deterministic Equivalent of Realized Portfolio Risk Denote k i = 1 N 1T N v iv T i 1 N. The deterministic equivalent of the numerator of Nσ 2 is (t 1s 2 1k 1 + s 1k 1)w (s 1k 1 + t 1s 1k 1)w (t rs 2 rk r + s rk r)w 2 r + 2(s rk r + t rs rk r)w r t 1k t rk r The deterministic equivalent of the denominator of Nσ 2 is (s 1k 1w 1 + s 2k 2w s rk rw r + 1) 2 Observe that N σ 2 takes the form as a 1x a 2x a rx 2 r + f (b 1x 1 + b 2x b rx r + d) 2 (1) We can use the Cauchy-Schwarz inequality to find (x 1, x 2,..., x r ) that minimize (1) Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

52 Spiked Covariance Model in the Minimum Risk Portfolio Design Optimal Shrinkage of Eigenvalues Since (b 1x 1 + b 2x b rx r + d) 2 (a 1x a 2x a rx 2 r + f)( b2 1 a 1 + b2 2 a b2 r a r + d2 f ) where = can be reached when x 1 = b1f a 1d In our case,, x2 = b2f a 2d brf,...,xr =, we obtain that a rd a 1x a 2x a rx 2 r + f (b 1x 1 + b 2x b rx r + d) 1 2 b 2 1 a 1 + b b2 r a r + d2 f f = 1 + t 1k t rk r d = 1 t1s1k1 + s1k1 t 1s a 1 = t 1s 2 1k 1 + s 1k 1 b 1 = s 1k 1 x 1 = w 1 + t1 + 1 t 1s (s1k1 + t1s1k1)2 t 1s 2 1k1 + s1k1 trsrkr + srkr t rs r + 1 a 2... (srkr + trsrkr)2 t rs 2 rk r + s rk r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

53 Spiked Covariance Model in the Minimum Risk Portfolio Design Precision Matrix Estimation The optimal w1(t i, s i, k i ) = b 1f a 1 d t t 1 s 1 + 1, and similar results for w 2,..., wr The estimators of t i, s i and k i, i = 1,..., r: ˆt i = λ i + 1 c + (λ i + 1 c) 2 4λ i 2 ŝ i = 1 c/(ˆt i ) c/ˆt i ˆk i = 1 1 ŝ i N 1T N u i u T i 1 N where λ 1,..., λ r are the top r sample eigenvalues 1 Ĉ 1 risk = I N + ŵ 1(ˆt i, ŝ i, ˆk i )u 1 u T ŵ r(ˆt i, ŝ i, ˆk i )u r u T r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

54 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 20 risktrue 18 riskest riskdet 16 N*Risk n (N=n/2) C N = I N + 14v 1v1 T + 9v 2v2 T + 4v 3v3 T v 1 = 3/N[1 N/3 ; 0 2N/3 ] v 2 = 3/N[1 N/3 ; 0 N/3 ; 1 N/3 ] v 3 = 3/N[0 2N/3 ; 1 N/3 ] Figure: The deterministic equivalent and the estimator of the true risk Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

55 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations Ĉrisk Ĉclip N*Risk ĈFro ĈFroinv n (N=n/2) Figure: The average realized portfolio risk of different covariance estimators in the GMVP framework using synthetic data Benchmarks Ĉ clip : Eigenvalue Clipping [Laloux et al., 2000] Ĉ Fro: Frobenius norm minimization [Donoho et al., 2013] Ĉ Froinv: Frobenius norm minimization of precision matrix [Donoho et al., 2013] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

56 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.7 x 10 3 N*Risk Ĉrisk Ĉclip ĈFro ĈFroinv 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 r (N=95) 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-Supélec June 26, / 35

57 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.7 x 10 3 N*Risk Ĉrisk Ĉclip ĈFro ĈFroinv r (N=95) 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 Ĉrisk perform the best? 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-Supélec June 26, / 35

58 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.7 x 10 3 N*Risk Ĉrisk Ĉclip ĈFro ĈFroinv r (N=95) 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 Ĉrisk perform the best? Time correlation unconsidered Impulsiveness of the data Too large estimation error Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

59 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.7 x 10 3 N*Risk Ĉrisk Ĉclip ĈFro ĈFroinv r (N=95) 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 Ĉrisk perform the best? Time correlation unconsidered Impulsiveness of the data Too large estimation error Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

60 Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model y t = x t T 1/2 R N, t = 1,..., n i.i.d. N-dimensional vectors x t N(µ, C N ), t = 1,..., n C N = I N + t 1v 1v1 T + t 2v 2v2 T t rv rvr T T R N is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

61 Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model y t = x t T 1/2 R N, t = 1,..., n i.i.d. N-dimensional vectors x t N(µ, C N ), t = 1,..., n C N = I N + t 1v 1v1 T + t 2v 2v2 T t rv rvr T T R N is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] New estimators ˆt i,ŝ i and ˆk i, i = 1,..., r [Vinogradova et al., 2013] Define Y N = [y 1,..., y n] Denote λ 1... λ N the eigenvalues of 1 n YN YT N ˆm(x) = 1 N r N j=r+1 ˆt i = 1 λ j x, ĝ(x) = ˆm(x)(xc ˆm(x) + c 1) ( ĝ(λ 1 i) n tr[ 1 ) 1 N YT N Y N ], ŝ i = ˆm(λi)ĝ(λi), ĝ ˆki = 1 1 (λ i) ŝ i N 1T N u iu T i 1 N Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

62 Spiked Covariance Model in the Minimum Risk Portfolio Design Time Correlated Data Model y t = x t T 1/2 R N, t = 1,..., n i.i.d. N-dimensional vectors x t N(µ, C N ), t = 1,..., n C N = I N + t 1v 1v1 T + t 2v 2v2 T t rv rvr T T R N is Hermitian nonnegative with constraints detailed in [Vinogradova et al., 2013] New estimators ˆt i,ŝ i and ˆk i, i = 1,..., r [Vinogradova et al., 2013] Define Y N = [y 1,..., y n] Denote λ 1... λ N the eigenvalues of 1 n YN YT N ˆm(x) = 1 N r N j=r+1 ˆt i = 1 λ j x, ĝ(x) = ˆm(x)(xc ˆm(x) + c 1) ( ĝ(λ 1 i) n tr[ 1 ) 1 N YT N Y N ], ŝ i = ˆm(λi)ĝ(λi), ĝ ˆki = 1 1 (λ i) ŝ i N 1T N u iu T i 1 N Ĉ 1 risk = I N + ŵ 1(ˆt i, ŝ i, ˆk i )u 1 u T ŵ r(ˆt i, ŝ i, ˆk i )u r u T r Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

63 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 4 x 10 3 Ĉrisk 3.8 Ĉclip ĈFro 3.6 ĈFroinv N*Risk r (N=95, n=250) 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-Supélec June 26, / 35

64 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 4 x 10 3 Ĉrisk 3.8 Ĉclip ĈFro 3.6 ĈFroinv N*Risk 3.4 Ĉ risk performs the best r (N=95, n=250) 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-Supélec June 26, / 35

65 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 4 x 10 3 Ĉrisk 3.8 Ĉclip ĈFro 3.6 ĈFroinv N*Risk 3.4 Ĉ risk performs the best r (N=95, n=250) The risk realized by Ĉrisk doesn t change much when r 15. Why? Haven t figured out yet... 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-Supélec June 26, / 35

66 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.25 x Ĉrisk Ĉclip ĈFro N*Risk ĈFroinv r = n (N=95) 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 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

67 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.25 x Ĉrisk Ĉclip ĈFro N*Risk ĈFroinv r = 11 Lack of stationarity when n grows too big n (N=95) 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 Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

68 Spiked Covariance Model in the Minimum Risk Portfolio Design Numerical Simulations 3.25 x Ĉrisk Ĉclip ĈFro N*Risk ĈFroinv r = 11 Lack of stationarity when n grows too big n (N=95) 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 Problem unsolved: the determination of the number of spikes Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

69 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-Supélec June 26, / 35

70 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-Supélec June 26, / 35

71 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-Supélec June 26, / 35

72 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-Supélec June 26, / 35

73 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-Supélec June 26, / 35

74 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-Supélec June 26, / 35

75 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-Supélec June 26, / 35

76 References Reference I Harry Markowitz. Portfolio selection. J. Finance, 7(1):77 91, Mar Liusha Yang, Romain Couillet, and Matthew R McKay. A robust approach to minimum variance portfolio optimization. arxiv preprint arxiv: , David E Tyler. A distribution-free M-estimator of multivariate scatter. Ann. Statist., 15(1): , Ricardo Antonio Maronna. Robust M-estimators of multivariate location and scatter. Ann. Statist., 4(1):51 67, 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 , Honolulu, HI, Apr F. Pascal, Y. Chitour, and Y. Quek. Generalized robust shrinkage estimator and its pplication to STAP detection problem. Submitted for publication, URL Yilun Chen, Ami Wiesel, and Alfred O. Hero. Robust shrinkage estimation of high-dimensional covariance matrices. IEEE Trans. Signal Process., 59(9): , Sept Romain Couillet and Matthew R McKay. Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators. J. Mult. Anal., 131:99 120, Olivier Ledoit and Michael Wolf. A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal., 88(2): , Feb Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35

77 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): , Aug 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): , David L Donoho, Matan Gavish, and Iain M Johnstone. Optimal shrinkage of eigenvalues in the spiked covariance model. arxiv preprint arxiv: , Julia Vinogradova, Romain Couillet, and Walid Hachem. Statistical inference in large antenna arrays under unknown noise pattern. Signal Processing, IEEE Transactions on, 61(22): , Yang (HKUST) Minimum Risk Portfolio Optimization Centrale-Supélec June 26, / 35