Mean-variance portfolio optimization when means and covariances are - PowerPoint PPT Presentation

Introduction and review A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Conclusion Mean-variance portfolio optimization when means and covariances are estimated Zehao Chen June 1, 2007 Joint work with Tze Leung Lai (Stanford Univ.) and Haipeng Xing (Columbia Univ.) Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Conclusion Outline Introduction and review 1 The Markowitz framework Different efficient frontier definitions under stochastic setting A high dimensional plug-in covariance matrix estimator 2 The L 2 boosting estimator Simulation and empirical study A modified Markowitz framework 3 The framework An example and simulation Conclusion 4 Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion The basic formulation We denote the returns of p risky assets (e.g. stock returns) by a p × 1 vector R , and an unobserved future return by r , E ( R ) = µ, Cov ( R ) = Σ The mean-variance optimization solves for an asset allocation w , which minimizes the portfolio risk σ 2 w , while achieving a certain target return µ ∗ , i.e, w σ 2 w w T Σ w = min min w subject to w T µ ≥ µ ∗ , w 1 + w 2 + ... + w p = 1 Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion The basic formulation This formulation has a closed form solution for w , w ∗ = f ( µ, µ ∗ , Σ − 1 ) The weights sometimes have additional constraints, i.e. l i ≤ w i ≤ u i , i = 1 , ..., p If l i ≥ 0, the constraint is also called no short selling constraint. The solution under this additional constraint requires quadratic programming. Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Efficient frontier Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion ”Plug-in” efficient frontier For w = w ( X , µ ∗ ) and reasonable µ and Σ estimates ˆ µ ( X ) and ˆ Σ( X ), w w T ˆ min Σ w subject to w T ˆ µ ≥ µ ∗ Define the ”plug-in” efficient frontier as parametrized by µ ∗ C ( µ ∗ ) = ( w T Σ w , w T µ ) Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion ”Plug-in” covariance estimates Factor model (with domain knowledge) R = α + BF + ǫ Σ = W 1 + W 2 = B Ω B T + Cov ( ǫ ) Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion ”Plug-in” covariance estimates Factor model (with domain knowledge) R = α + BF + ǫ Σ = W 1 + W 2 = B Ω B T + Cov ( ǫ ) Shrinkage estimator α ˆ F + (1 − α )ˆ Σ Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Random efficient frontier? −3 −3 x 10 x 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.04 0.06 0.08 0.1 0.12 0.14 0.16 −3 −3 x 10 x 10 9 8 7.5 8 7 7 6.5 6 6 5 5.5 4 5 3 4.5 2 4 1 3.5 0 3 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Figure: n = 50 , p = 5 i.i.d multivariate normal Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Random efficient frontier? The data dependent efficient frontier is not well defined, and is a random curve. Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Random efficient frontier? The data dependent efficient frontier is not well defined, and is a random curve. By plugging the estimates of mean ˆ µ , it’s essentially constraining on w T ˆ µ ≥ µ ∗ Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Random efficient frontier? The data dependent efficient frontier is not well defined, and is a random curve. By plugging the estimates of mean ˆ µ , it’s essentially constraining on w T ˆ µ ≥ µ ∗ Conceptually, one should constrain on E ( w T r ) ≥ µ ∗ Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Resampled efficient frontier For bootstrapped samples X ∗ 1 ,..., X ∗ B of X , w M ( X , µ ∗ ) = 1 B i =1 w ( X ∗ i , µ ∗ ) Σ B Define the resampled efficient frontier as C M ( µ ∗ ) = ( w T M Σ w M , w T M µ ) Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review A high dimensional plug-in covariance matrix estimator The Markowitz framework A modified Markowitz framework Different efficient frontier definitions under stochastic setting Conclusion Previous definitions of efficient frontiers ”Plug-in” efficient frontier w w T ˆ Σ w , w T ˆ min µ ≥ µ ∗ C ( µ ∗ ) = ( w T Σ w , w T µ ) Resampled efficient frontier w M ( X , µ ∗ ) = 1 B i =1 w ( X ∗ i , µ ∗ ) Σ B C M ( µ ∗ ) = ( w T M Σ w M , w T M µ ) Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review The L 2 boosting estimator A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Simulation and empirical study Conclusion If one really wants to use the plug-in Estimate Σ or Σ − 1 ? Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review The L 2 boosting estimator A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Simulation and empirical study Conclusion If one really wants to use the plug-in Estimate Σ or Σ − 1 ? Employ a sparsity assumption (in practice, residuals from some factor model): reduce the number of parameters to estimate. Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review The L 2 boosting estimator A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Simulation and empirical study Conclusion If one really wants to use the plug-in Estimate Σ or Σ − 1 ? Employ a sparsity assumption (in practice, residuals from some factor model): reduce the number of parameters to estimate. Impose proper weight constraints. Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review The L 2 boosting estimator A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Simulation and empirical study Conclusion A high dimensional covariance estimator Use Modified Cholesky decomposition Σ − 1 = T ′ D − 1 T to reparametrize covariance matrix, and enforce sparsity on T . Zehao Chen M.V. optimization when means and covariances are estimated

Introduction and review The L 2 boosting estimator A high dimensional plug-in covariance matrix estimator A modified Markowitz framework Simulation and empirical study Conclusion A high dimensional covariance estimator Use Modified Cholesky decomposition Σ − 1 = T ′ D − 1 T to reparametrize covariance matrix, and enforce sparsity on T . Use coordinate-wise greedy (component-wise L 2 boosting) with a modified BIC stopping criterion. Can be shown to be a consistent estimator. Zehao Chen M.V. optimization when means and covariances are estimated