MS&E348 Portfolio Optimization

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1 Rong Xu Matt LeDuc Ching-Yu Hu

2 Our Methodology Problem Statement Our Data Evaluation of measures of risk Overview of Results Black-Litterman Model

3 Objective To explore pros and cons of various modern portfolio management strategies To implement large-scale portfolio optimizer allowing for long-short positions and leverage ratios Scope S&P5 Universe Monthly data from 12/31/ /1/2

4 In-sample data will be used to calculate Markowitz mean-variance efficient frontier Out-of-sample data will be used to test the validity of the model These data files include mean return, returns, factor-explained returns, index weights of assets, & value of UB on assets

5 Long Only Long-Short Tracking Error Total Risk

9 Use In-Sample data to calculate alphas and covariance matrix Calculate optimal portfolios for 4 cases Vary expected return for each portfolio for.5%, 1%, 5%, 1%, 15% return Fix optimal weights for each portfolio and evaluate on out-of-sample data

10 Store optimized weights and evaluate out-of-sample

11 Comparison Analysis Efficient Frontiers Sharpe Ratio Ex-ante vs. Ex-post Tracking Error

21 Long-Short consistently outperforms Long-Only strategies Predicted risk overestimates out-ofsample risk for long-short strategies and underestimates the risk for long-only strategies Long-Short/Absolute Error emerges as the most robust case

22 The Black-Litterman approach The Black-Litterman approach is a sophisticated portfolio construction method that : overcomes the problem of unintuitive, highlyconcentrated portfolios and input-sensitivity Allows to incorporate investor s view

23 Extreme portfolios Highly sensitive to inputs Unintuitive Cannot incorporate investor s views No way to incorporate confidence level Complete set of expected return is required Market capitalization of assets not taken into account

24 BusnsServs Transport_ Utilities_ Finance OilsEnergy Aerospace Technology MltiSecCng Constructn IndustrlPr BscMaterls AutosTrcks Medical RetailWhol Cons_Discr Cons_Stpls Mean-Var weights New weight Eq. weight

25 The B-L model uses a Bayesian approach that combines the views of an investor on the expected returns of a subset of the assets with the market equilibrium vector of expected returns (the prior) to form a combined expected returns (the posterior). The B-L expected returns are then optimized in a meanvariance way. The size of the bets depends on the confidence levels specified by the user and also on a parameter specifying the weight of the collected investor views

26 w mkt - the weight vector of the market capitalized portfolio, (equilibrium portfolio) lambda - the risk aversion factor Σ - the covariance matrix of asset returns P - a matrix representing the assets in the views. Each row in the matrix contains the weights of assets of one view. Q - a column vector that represents the estimated expected returns in each view. ω i - the variance of view i Ω - a diagonal matrix consisting of the ω i τ - A parameter often referred to as the weight-on-views. Determines the weighting between the view portfolio and the equilibrium portfolio. ECret - this is the B-L modified vector of estimated expected returns Π - The column vector of equilibrium expected excess returns ECret 1 T 1 1 = [( τσ) + P Ω P] [( τσ) 1 Π + P T Ω 1 Q]

27 Π = λσw Market participants collectively reached the equilibrium (supply = demand). So the Markowitz mean-variance optimized weights w mkt are the market capitalization weights yields max{ w' Π 1 λw' Σw} 2 Π = λ Σw mkt the equilibrium returns

28 The risk aversion parameter is ג = (E (r) rf )/σ 2 The Black-Litterman model uses equilibrium returns as a neutral starting point Views will tilt the final weights away from the market portfolio, in a way proportional to level of confidence Views are defined as: Pu=Q+e, where e~φ(,ω) and where u is the future returns, which do not enter directly the model Each row of P represents a set of relative weights on the N assets

29 N ( QΠ, Ω τσ) So the prior is N ( Π, τσ) The view distribution is N ( Q, Ω) Combining with views and performing GLS on this system of equations Π yields to the expression = ECret Q = PECret + ε +η ECret 1 T 1 1 = [( τσ) + P Ω P] [( τσ) 1 Π + P T Ω 1 Q] This expression can then be used as the return in a mean-variance optimization scheme

30 5.25% ε 1 Q + ε =.25% + ε 2 2.% ε 3 Absolute view: View 1: IndustrPr will have an absolute excess return of 5.25% Relative views: View 2: Cons_Stpls will outperform Cons_Discr by 25 bp View 3: US RetailWhol and Medical will outperform AutoTrcks and BscMaterl by 2% = P = P = P

31 BusnsServs Transport_ Utilities_ Finance OilsEnergy Aerospace Technology MltiSecCng Constructn IndustrlPr BscMaterls AutosTrcks Medical RetailWhol Cons_Discr Cons_Stpls Eq. weight New weight Weights 1% Conf.

32 1 T 1 1 ECret ω τ = = p[( Σp τσ) ' + P Ω P] [( τσ) / k k 1 T Π + P Ω 1 Q] A neat characteristic of Black-Litterman framework is possibility to input a confidence level about views Problem: No consensus on how to specify it Parameters τ and Ω relate to the uncertainty in the views. is more or less inversely related to the uncertainty in the views. Lee suggests to set its value between.1 and.5. Satchell and Scowcroft (2) claim it is often set to 1. He and Litterman (1999) calibrate the model so that ω τ = p Σp ' / k k τ

33 New weight Eq. weight Weights 1% Conf. Implied confidence level Cons_Stpls Cons_Discr RetailWhol Medical AutosTrcks BscMaterls IndustrlPr Constructn MltiSecCng Technology Aerospace OilsEnergy Finance Utilities_ Transport_ BusnsServs.2.2.2

34 Idzorek (24) proposes to use the implied confidence level to calibrate model Take advantage of independence of views to incorporate the confidence level separately for each view Neat calibration method that allows user to specify confidence level as a percentage

35 Tilt w ) C ECret k 1% ( w1% mkt k The method consists of determining the values of ω k that results in portfolio tilts of this order Tilt w w ) C k ( 1% 1) For each view, solve ECret for k,1% using BL formula and then obtain corresponding w k,1% 2) Estimate the target weight vector based on the tilt k,% mkt mkt w = w + Tilt k k 3) Find ωk such that it minimizes wk,% wk where wk is obtained from the B-L mean variance optimization.

36 τ τ Note that is held constant throughout and thus is not used in specifying the uncertainty Absolute view: View 1: IndustrPr will have an absolute excess return of 5.25% Relative views: View 2: Cons_Stpls will outperform Cons_Discr by 25 bp View 3: US RetailWhol and Medical will outperform AutoTrcks and BscMaterl by 2%

37 RetailWhol Cons_Discr Cons_Stpls Constructn IndustrlPr BscMaterls AutosTrcks Medical OilsEnergy Aerospace Technology MltiSecCng BusnsServs Transport_ Utilities_ Finance New Weights (8% 8% 8%) Eq. weight New Weights (2% 2% 2%)

38 The views are the source of alpha. The BL framework itself is not a source of alpha. It is interesting to note that the new weights no longer sum up to 1%. This is normal since view #1 is an absolute view with no offsetting position. The portfolio can then be rescaled accordingly When adding constraints to the BL, the resulting weights are less intuitive

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PowerPoint Presentation Decision analysis 量化決策分析方法專論 2011/5/26 1 Problem formulation- states of nature In the decision analysis, decision alternatives are referred to as chance events. The possible outcomes for a chance event