The Effect of Estimation in Highdimensional Portfolios Luitgard A. - PowerPoint PPT Presentation

The Effect of Estimation in High–dimensional Portfolios Luitgard A. M. Veraart Joint work with Axel Gandy, Imperial College London Analysis, Stochastics, and Applications Vienna University, July 2010 Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 1 / 12

Outline 1 Classical Portfolio Optimisation 2 Plug-In Strategies with Estimated Parameters 3 James-Stein-Shrinkage Applied to Strategies 4 L 1 –Constrained Strategies - LASSO 5 Other Strategies 6 Application to Empirical Data Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 2 / 12

Optimal Portfolio Selection The asset prices: 1 bond S 0 ( t ) = e rt , d risky assets d � dS i ( t ) = S i ( t )[ µ i dt + σ ij dW j ( t )] , S i (0) > 0 , i = 1 , . . . , d , j =1 r > 0 interest rate, µ ∈ R d drift, σ ∈ R d × d volatility matrix of full rank (all constant), W d -variate Brownian motion. Investor has T > 0 fixed time horizon, X 0 > 0 constant initial wealth, chooses π i ( t ) fraction of the wealth invested in the i th asset at time t , resulting in time- t -wealth X t with dX t = � d dS i ( t ) i =0 π i ( t ) X t S i ( t ) . Investor seeks π to maximise V ( π ) := E [log( X T )] . Optimal solution π ∗ = Σ − 1 ( µ − r 1) , where π 0 ( t ) = 1 − � d i =1 π i ( t ), π = ( π 1 , . . . , π d ) T , Σ = σσ T . Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12

Optimal Portfolio Selection The asset prices: 1 bond S 0 ( t ) = e rt , d risky assets d � dS i ( t ) = S i ( t )[ µ i dt + σ ij dW j ( t )] , S i (0) > 0 , i = 1 , . . . , d , j =1 r > 0 interest rate, µ ∈ R d drift, σ ∈ R d × d volatility matrix of full rank (all constant), W d -variate Brownian motion. Investor has T > 0 fixed time horizon, X 0 > 0 constant initial wealth, chooses π i ( t ) fraction of the wealth invested in the i th asset at time t , resulting in time- t -wealth X t with dX t = � d dS i ( t ) i =0 π i ( t ) X t S i ( t ) . Investor seeks π to maximise V ( π ) := E [log( X T )] . Optimal solution π ∗ = Σ − 1 ( µ − r 1) , where π 0 ( t ) = 1 − � d i =1 π i ( t ), π = ( π 1 , . . . , π d ) T , Σ = σσ T . The Problem What if we need to estimate µ ? What if the number of risky assets d → ∞ ? Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12

Plug-in Merton Strategy with Estimated µ General Unbiased Plug-in Estimator π = Σ − 1 (ˆ Estimate µ by ˆ µ and define the plug-in strategy ˆ µ − r 1). π ∼ N (Σ − 1 ( µ − r 1) , V 2 0 ), V 0 ∈ R d × d , then Assume ˆ π ) = V ( π ∗ ) − T 2 trace(Σ V 2 V (ˆ 0 ) . Specific Plug-in Estimator Observation period [ − t est , 0] for t est > 0. Set d � µ i = log( S i (0)) − log( S i ( − t est )) + 1 σ 2 ˆ ij . t est 2 j =1 π ∼ N (Σ − 1 ( µ − r 1) , Σ − 1 / t est ). Then ˆ π ) = V ( π ∗ ) − d T V (ˆ 2 t est . There are realistic scenarios in which even V (ˆ π ) → −∞ as d → ∞ . Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 4 / 12

James-Stein-Type Shrinkage of the Strategy The James-Stein-Strategy µ − r 1), π 0 ∈ R d , a > 0 fixed constants. Consider π = Σ − 1 (ˆ Let ˆ � � π JS ,π 0 = a π − π 0 ) + π 0 . ˆ 1 − (ˆ π − π 0 ) T Σ(ˆ π − π 0 ) (ˆ The Expected Utility for the JS-Strategy µ ∼ N ( µ, Σ / t est ), K ∼ Poisson( λ ), λ = ( π ∗ − π 0 ) T Σ( π ∗ − π 0 ) / 2: Let ˆ � � � � π ) + T 2 d − 2 t est π JS ,π 0 ) = V (ˆ V (ˆ 2 a − a . E t est d − 2 + 2 K π JS ,π 0 dominates ˆ ˆ π for 0 < a < 2( d − 2) / t est ; optimal a = ( d − 2) / t est . Special Choices for π 0 and Optimal a π 0 = π ∗ : V (ˆ π JS ,π 0 ) = V ( π ∗ ) − T t est . π 0 = β π JS ,π 0 ) → ∞ as d → ∞ . d 1, β ∈ R : In some situations V (ˆ Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 5 / 12

L 1 –constrained Strategies - LASSO General Idea Require that π satisfies � π � 1 = � d i =1 | π i | ≤ c for a constant c ≥ 0. � � c max i | µ i − r | + c 2 V ( π ) ≥ log( X 0 ) + rT − T E 2 max i , j | Σ ij | . If max i | µ i − r | , max i , j | Σ ij | bounded, V ( π ) �→ −∞ as d → ∞ . Specific Results For Σ = η 2 ( ρ 11 T + (1 − ρ ) I ), η > 0 , 0 ≤ ρ ≤ 1 analytic results for the optimal L 1 -constrained strategies, if µ known. for the L 1 -constrained plug-in strategy as d → ∞ : ◮ the distribution of # { i : π ∗ i � = 0 } , if ρ = 0, ◮ an upper bound on lim d →∞ P (# { i : π ∗ i � = 0 } > k ), if ρ > 0. Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 6 / 12

Other Strategies and Performance for d → ∞ Other Norm Constraints L 0 -restricted strategies: no degeneration of expected utility as d → ∞ . L 2 -restricted strategies: degeneration possible. Special L 1 -Constraints 1 / d -strategy: Strategy that invests the same amount into all stocks, i.e. π c / d = c d 1 for some c > 0. Equal Weighting of the most Extreme stocks (EWE): = c π EWE β d sign(ˆ a k i ) I ( i ≤ β d ) , i = 1 , . . . , d , k i a i = ˆ µ i − r where ˆ Σ ii , c > 0, β ∈ (0 , 1) constants, k i are such that | ˆ a k 1 | > | ˆ a k 2 | > · · · > | ˆ a k d | . Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 7 / 12

Example - Trading S&P500 Stocks in S&P 500 index on 01/01/2006 having daily returns for all trading days between 2001 and 2008 (373 stocks, n=2011 trading days, daily returns). Specific random ordering of stocks. Allow the strategies to invest in the first d stocks of this ordering. X 0 = 1, r = 0 . 02, roughly T = 1. Use of unbiased estimators based on observed stock prices at time points 0 , ∆ , 2∆ , . . . , ( n − 1)∆: µ data = 1 ξ + 1 ˆ 2 diag( � Σ data ) , ˆ ∆ � � � � n − 2 � 1 � R µ ( i ) − ˆ R ν ( i ) − ˆ Σ data µ,ν = ξ µ ξ ν ∆( n − 2) i =0 � � S µ (( i +1)∆) for µ, ν = 1 , . . . , d , where R µ ( i ) = log , S µ ( i ∆) � n − 2 ˆ 1 ξ µ = i =0 R µ ( i ). n − 1 Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 8 / 12

Analytic and Simulation Results n = 504 n = 1008 n = 2016 Merton−Strategies µ , Σ known 20 µ estimated, Σ known µ , Σ estimated µ ) ) π | µ ( π 10 expected utility V ( 0 −10 −20 James−Stein Strategies JS, π 0 = π * 20 JS, π 0 = π *, Σ est. JS, π 0 = 0 JS, π 0 = 0, Σ est. expected utility V ( π | µ ) 10 0 −10 −20 0.5 L 1 −restricted Strategies expected utility V ( π | µ ) 0.4 L 1 L 1 , µ est. L 1 , µ , Σ Σ est 0.3 1 d EWE, β = 0.1 0.2 0.1 0.0 0 100 200 300 0 100 200 300 0 100 200 300 d d d Expected utility plotted against the number d of available stocks.

Out-of-Sample Performance 2005 2006 2007 2008 1 Strategy 1/d L 1 EWE, β = 0.1 0.5 log ( X T ) 0 −0.5 −1 10 Strategy Merton JS 0 log ( X T ) −10 −20 −∞ 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 d d d d log( X T ) with T = 1 year plotted against the number d of available stocks. Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 10 / 12

Summary Main Contributions Quantification of the effect of estimation in vast portfolios (unknown µ and large d ). Analysis of strategies which are less affected by estimation. Analytic formulae for James-Stein and optimal L 1 -constrained strategies. Specific Conclusions Estimation effects must not be ignored in vast portfolios! Simple plug in strategies have a loss through estimation linear in d . James-Stein shrinkage performs better than simple plug in strategies. L 1 -constrained strategies cannot degenerate. L 1 -constrained strategies and particularly the EWE-strategy and 1 / d strategy perform well also in out-of-sample tests. Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 11 / 12

References Brodie, J., Daubechies, I., De Mol, C., Giannone, D. & Loris, I. (2009). Sparse and stable Markowitz portfolios. P. Natl. Acad. Sci. 106 , 12267–12272. Fan, J., Zhang, J. & Yu, K. (2009). Asset allocation and risk assessment with gross exposure constraints on vast portfolios. Preprint, Princeton University. James, W. & Stein, C. (1961). Estimation with quadratic loss. In Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1 (Univ. of Calif. Press) , 361–379. Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3 , 373–413. Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 , 1135–1151. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. B 58 , 267–288. Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 12 / 12