Asset Allocation and Risk Assessment with Gross Exposure Constraints Forrest Zhang Bendheim Center for Finance Princeton University A joint work with Jianqing Fan and Ke Yu, Princeton Princeton University Asset Allocation with Gross Exposure Constraints 1/25
Introduction Princeton University Asset Allocation with Gross Exposure Constraints 2/25
Markowitz’s Mean-variance analysis � Problem: min w w T Σ w , s.t. w T 1 = 1 , and w T µ = r 0 . Solution: w = c 1 Σ − 1 µ + c 2 Σ − 1 1 • Cornerstone of modern finance where CAPM and many portfolio theory is built upon. • Too sensitive on input vectors and their estimation errors. • Can result in extreme short positions (Green and Holdfield, 1992) . • More severe for large portfolio. Princeton University Asset Allocation with Gross Exposure Constraints 3/25
Markowitz’s Mean-variance analysis � Problem: min w w T Σ w , s.t. w T 1 = 1 , and w T µ = r 0 . Solution: w = c 1 Σ − 1 µ + c 2 Σ − 1 1 • Cornerstone of modern finance where CAPM and many portfolio theory is built upon. • Too sensitive on input vectors and their estimation errors. • Can result in extreme short positions (Green and Holdfield, 1992) . • More severe for large portfolio. Princeton University Asset Allocation with Gross Exposure Constraints 3/25
Challenge of High Dimensionality � Estimating high-dim cov-matrices is intrinsically challenging. • Suppose we have 500 ( 2000 ) stocks to be managed. There are 125K ( 2 m ) free parameters! • Yet, 2-year daily returns yield only about sample size n = 500 . Accurately estimating it poses significant challenges. • Impact of dimensionality is large and poorly understood: Risk: w T ˆ Σ − 1 ˆ c 1 ˆ c 2 ˆ Σ − 1 1 + ˆ Σ w . Allocation: ˆ µ . • Accumulating of millions of estimation errors can have a devastating effect. Princeton University Asset Allocation with Gross Exposure Constraints 4/25
Challenge of High Dimensionality � Estimating high-dim cov-matrices is intrinsically challenging. • Suppose we have 500 ( 2000 ) stocks to be managed. There are 125K ( 2 m ) free parameters! • Yet, 2-year daily returns yield only about sample size n = 500 . Accurately estimating it poses significant challenges. • Impact of dimensionality is large and poorly understood: Risk: w T ˆ Σ − 1 ˆ c 1 ˆ c 2 ˆ Σ − 1 1 + ˆ Σ w . Allocation: ˆ µ . • Accumulating of millions of estimation errors can have a devastating effect. Princeton University Asset Allocation with Gross Exposure Constraints 4/25
Efforts in Remedy � Reduce sensitivity of estimation. • Shrinkage and Bayesian: — Expected return (Klein and Bawa, 76; Chopra and Ziemba, 93; ) — Cov. matrix (Ledoit & Wolf, 03, 04) • Factor-model based estimation (Fan, Fan and Lv , 2008; Pesaran and Zaffaroni, 2008) � Robust portfolio allocation (Goldfarb and Iyengar, 2003) � No-short-sale portfolio (De Roon et al., 2001; Jagannathan and Ma, 2003; DeMiguel et al., 2008; Bordie et al., 2008) � None of them are far enough; no theory. Princeton University Asset Allocation with Gross Exposure Constraints 5/25
Efforts in Remedy � Reduce sensitivity of estimation. • Shrinkage and Bayesian: — Expected return (Klein and Bawa, 76; Chopra and Ziemba, 93; ) — Cov. matrix (Ledoit & Wolf, 03, 04) • Factor-model based estimation (Fan, Fan and Lv , 2008; Pesaran and Zaffaroni, 2008) � Robust portfolio allocation (Goldfarb and Iyengar, 2003) � No-short-sale portfolio (De Roon et al., 2001; Jagannathan and Ma, 2003; DeMiguel et al., 2008; Bordie et al., 2008) � None of them are far enough; no theory. Princeton University Asset Allocation with Gross Exposure Constraints 5/25
About this talk � Propose utility maximization with gross-sale constraint. It bridges no-short-sale constraint to no-constraint on allocation. � Oracle (Theoretical), actual and empirical risks are very close. No error accumulation effect. Elements in covariance can be estimated separately; facilitates the use of non-synchronized high-frequency data. Provide theoretical understanding why wrong constraint can even beat Markowitz’s portfolio (Jagannathan and Ma, 2003) . � Portfolio selection and tracking. Select or track a portfolio with limited number of stocks. Improve any given portfolio with modifications of weights on limited number of stocks. Princeton University Asset Allocation with Gross Exposure Constraints 6/25
About this talk � Propose utility maximization with gross-sale constraint. It bridges no-short-sale constraint to no-constraint on allocation. � Oracle (Theoretical), actual and empirical risks are very close. No error accumulation effect. Elements in covariance can be estimated separately; facilitates the use of non-synchronized high-frequency data. Provide theoretical understanding why wrong constraint can even beat Markowitz’s portfolio (Jagannathan and Ma, 2003) . � Portfolio selection and tracking. Select or track a portfolio with limited number of stocks. Improve any given portfolio with modifications of weights on limited number of stocks. Princeton University Asset Allocation with Gross Exposure Constraints 6/25
Outline Portfolio optimization with gross-exposure constraint. 1 Portfolio selection and tracking. 2 Simulation studies 3 Empirical studies: 4 Princeton University Asset Allocation with Gross Exposure Constraints 7/25
Short-constrained portfolio selection E [ U ( w T R )] max w w T 1 = 1 , � w � 1 ≤ c , Aw = a . s.t. Equality Constraint : • A = µ = ⇒ expected portfolio return. • A can be chosen so that we put constraint on sectors. Short-sale constraint : When c = 1 , no short-sale allowed. When c = ∞ , problem becomes Markowitz’s. • Portfolio selection: solution is usually sparse. Princeton University Asset Allocation with Gross Exposure Constraints 8/25
Short-constrained portfolio selection E [ U ( w T R )] max w w T 1 = 1 , � w � 1 ≤ c , Aw = a . s.t. Equality Constraint : • A = µ = ⇒ expected portfolio return. • A can be chosen so that we put constraint on sectors. Short-sale constraint : When c = 1 , no short-sale allowed. When c = ∞ , problem becomes Markowitz’s. • Portfolio selection: solution is usually sparse. Princeton University Asset Allocation with Gross Exposure Constraints 8/25
Risk optimization Theory Actual and Empirical risks : R ( w ) = w T Σ w , R n ( w ) = w T ˆ Σ w . ˆ w opt = argmin R ( w ) , w opt = argmin R n ( w ) || w || 1 ≤ c || w || 1 ≤ c � � R n (ˆ • Risks: R ( w opt ) — oracle, w opt ) — empirical; � R (ˆ w opt ) — actual risk of a selected portfolio. Theorem 1 : Let a n = � ˆ Σ − Σ � ∞ . Then, we have 2 a n c 2 | R (ˆ w opt ) − R ( w opt ) | ≤ | R (ˆ w opt ) − R n (ˆ a n c 2 w opt ) | ≤ a n c 2 . | R ( w opt ) − R n (ˆ w opt ) | ≤ Princeton University Asset Allocation with Gross Exposure Constraints 9/25
Risk optimization Theory Actual and Empirical risks : R ( w ) = w T Σ w , R n ( w ) = w T ˆ Σ w . ˆ w opt = argmin R ( w ) , w opt = argmin R n ( w ) || w || 1 ≤ c || w || 1 ≤ c � � R n (ˆ • Risks: R ( w opt ) — oracle, w opt ) — empirical; � R (ˆ w opt ) — actual risk of a selected portfolio. Theorem 1 : Let a n = � ˆ Σ − Σ � ∞ . Then, we have 2 a n c 2 | R (ˆ w opt ) − R ( w opt ) | ≤ | R (ˆ w opt ) − R n (ˆ a n c 2 w opt ) | ≤ a n c 2 . | R ( w opt ) − R n (ˆ w opt ) | ≤ Princeton University Asset Allocation with Gross Exposure Constraints 9/25
Accuracy of Covariance: I Theorem 2 : If for a sufficiently large x , i,j P {√ n | σ ij − ˆ σ ij | > x } < exp( − Cx 1 /a ) , max for some two positive constants a and C , then � (log p ) a � � Σ − ˆ Σ � ∞ = O P √ n . • Impact of dimensionality is limited. Princeton University Asset Allocation with Gross Exposure Constraints 10/25
Accuracy of Covariance: I Theorem 2 : If for a sufficiently large x , i,j P {√ n | σ ij − ˆ σ ij | > x } < exp( − Cx 1 /a ) , max for some two positive constants a and C , then � (log p ) a � � Σ − ˆ Σ � ∞ = O P √ n . • Impact of dimensionality is limited. Princeton University Asset Allocation with Gross Exposure Constraints 10/25
Algorithms w T Σ w . min w T 1 =1 , � w � 1 ≤ c Quadratic programming for each given c ( Exact ). 1 Coordinatewise minimization. 2 LARS approximation. 3 Princeton University Asset Allocation with Gross Exposure Constraints 11/25
Connections with penalized regression Regression problem : Letting Y = R p and X j = R p − R j , var( w T R ) E ( w T R − b ) 2 = min b E ( Y − w 1 X 1 − · · · − w p − 1 X p − 1 − b ) 2 , = min b Gross exposure : � w � 1 = � w ∗ � 1 + | 1 − 1 T w ∗ | ≤ c , not equivalent to � w ∗ � 1 ≤ d . • d = 0 picks X p , but c = 1 picks multiple stocks. Princeton University Asset Allocation with Gross Exposure Constraints 12/25
Connections with penalized regression Regression problem : Letting Y = R p and X j = R p − R j , var( w T R ) E ( w T R − b ) 2 = min b E ( Y − w 1 X 1 − · · · − w p − 1 X p − 1 − b ) 2 , = min b Gross exposure : � w � 1 = � w ∗ � 1 + | 1 − 1 T w ∗ | ≤ c , not equivalent to � w ∗ � 1 ≤ d . • d = 0 picks X p , but c = 1 picks multiple stocks. Princeton University Asset Allocation with Gross Exposure Constraints 12/25
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