Factor Alignment for Equity Portfolio Management Sebastian Ceria, CEO Axioma, Inc. The 19th Annual Workshop on Financial Engineering: Quantitative Asset Management Columbia University November 2012
Factor Alignment Basics
A Decomposition of Expected Returns max h T α h λ 2 h T Qh The portion of alpha explained by the risk factors is referred to as the spanned component Q = XΩ X T + Δ α α + = α X If alpha and risk factors are aligned, then α = 0, or, in other words, there is misalignment if and only if α 0 The residual obtained by regressing the alphas against the factors in the risk model is referred to as the orthogonal component of alpha
Why is Misalignment Bad in MVO? max h T α h λ 2 h T Qh The optimizer sees no systematic risk in the orthogonal component of alpha and is hence likely to load up on it α = α + No Factor Risk, Only Specific Risk α X Contains Factor Risk and Specific Risk In MVO, we are aiming to create portfolios that have an optimal risk-adjusted expected return If a portion of systematic risk is not accounted for then the resulting riskadjusted expected return cannot be optimal
Alignment is Also About Constraints Implied Alpha max h s.t. T λ T λ T α h h Qh max α h h Qh 2 Constraint1 Constraint m Optimal Portfolio h* h * T 2 Implied alpha acts as the de facto alpha in the case of constrained MVO problems Optimizer sees no systematic risk in the orthogonal component of implied alpha and is hence likely to load up on it Implied alpha is a dynamic entity determined by the interaction of alpha, risk factors and constraints
Alpha vs Implied Alpha Misalignment Implied Alpha Alpha Optimal Portfolio With constraint Risk Ellipse Optimal Portfolio No longer feasible! Constraint
Misalignment Problems and Two Ways Out Problem: Misalignment due to proprietary factors not being represented in the risk model Having non-zero orthogonal components Solution: Custom Risk Models Add the proprietary factors to the risk models and completely regenerate them Problem: Misalignment due to the usage of constraints The difference between alpha and implied alpha (even with Custom Risk Models) Solution: Alpha Alignment Factor Methodology Add to the risk model the orthogonal component of implied alpha (Axioma proprietary and patented)
From Misalignment To Alignment Base Model Custom Risk Model Base Model + AAF Custom Risk Model + AAF
How To Align Proprietary Factors With Custom Risk Models Alpha Misalignment Base Model Custom Risk Model Base Model + AAF Custom Risk Model + AAF
How To Align Constraints Base Model Custom Risk Model Alpha + Constraint Misalignment (approx) Base Model + AAF Custom Risk Model + AAF
How To Align Constraints AND Proprietary Factors Alpha Misalignment Base Model Custom Risk Model Alpha + Constraint Misalignment (approx) Constraint Misalignment (approx) Base Model + AAF Custom Risk Model + AAF
Why Do We Have Misalignment, And The Case For And Against Alignment
Independent Alpha, Risk and Construction Processes Generate Misalignment Alpha Process Risk Process Portfolio Construction Strategy Optimizer Optimal Portfolio
To Align or Not To Align? (Proprietary Factors & Constraints) Proprietary Factors: Against The Free Lunch Theory: I don t want my factors in the risk model, otherwise the risk model will not let me bet on them (Never mind the systematic risk) Constraints: Indifferent The presence of constraints is ignored in most of the literature that concerns alignment issues
Empirical Evidence of Why Alignment Matters The USER Model * and Client Data * Guerard et. al
Proprietary Factors in the USER Model have Orthogonal Components with Realized Systematic Risk Comparable with Other Axioma Factors Axioma Style Factors, 25-75% Range of Systematic Risk
Proprietary Factors Have Statistically Significant Orthogonal Components Percentage of statistically significant periods (90% cf) CTEF RSP RCP RBP REP SP CP BP EP 0% 10% 20% 30% 40% 50% Axioma Style Factors, 25-75% Range % of SSP
Average Correlation Between Alpha and Implied Alpha Is Low for Most Clients 1.0 0.9 0.8 Fundamental Model Statistical Model 0.7 Correlation 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Correlation between Alpha and Implied Alpha Changes Significantly Over Time, Even For Perfectly Aligned Risk Models 80% Correlation between alpha and implied alpha 60% 40% 20% 0% Mar-99 Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02 Mar-03 Sep-03 Mar-04 Sep-04 Mar-05 Sep-05 Mar-06 Sep-06 Mar-07 Sep-07 Mar-08 Sep-08
The Opportunity Cost of Misalignment: Experiments with the USER Model
A Practical Active Strategy: USER Model Maximize Expected Return s.t. Fully invested long only portfolio GICS Sector exposure constraints (20%) GICS Industry exposure constraints (10%) Active asset bounds constraint (2%) Turnover Constraint (16% two-way) Active Risk Constraint (3%) Base Model = US2AxiomaMH (Axioma Fundamental Model) Benchmark = Russell 3000 Monthly backtest, 1999-2009 time period Expected Return = USER.BP + US2AxiomaMH.Medium-Term- Momentum
Predicted vs Realized Active Risk Significantly Improves with CRM + AAF US2AxiomaMH CRM CRM+AAF Realized Active Risk 6.0% 5.0% 4.0% 3.0% 2.0% Risk Target 3.00% Base Model 3.81% CRM 3.38% CRM + AAF 3.08% 1.0% 1.0% 2.0% 3.0% 4.0% 5.0% Predicted Active Risk
The Realized Risk-Return Frontier Moves Upwards (Annualized Active Returns and Risk) 4.0% US2AxiomaMH CRM CRM+AAF Realized Active Return 3.0% 2.0% 1.0% 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% Realized Active Risk Frontier Spreads: CRM: 1% Additional Realized Active Annualized Return CRM + AAF: 1.75% Additional Realized Active Annualized Return
Frontier Spreads Should be Interpreted as Opportunity Costs for Misalignment CRM CRM+AAF 2.0% Frontier Spread 1.6% 1.2% 0.8% 0.4% Opportunity Cost of misalignment from constraints 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% Realized Active Risk
Relaxing Asset Bounds Further Improves the Frontier Spreads 2.00% 3% 1% 2% Frontier Spread 1.50% 1.00% 0.50% 0.00% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Realized Active Risk
Relaxing the Industry Exposure Bounds Also Improves the Frontier Spreads 2.00% 2.50% 10% 5% Frontier Spread 1.50% 1.00% 0.50% 0.00% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Realized Active Risk
Increasing the Turnover Limit Improves Frontier Spreads 2.00% 16% 8% Frontier Spread 1.50% 1.00% 0.50% 0.00% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Realized Active Risk
Turnover Utilization Improves with CRM + AAF (Portfolios with Similar Realized Active Risk) US2AxiomaMH CRM+AAF Realized IR 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% Realized Turnover
The Opportunity Cost of Misalignment: Experiments with Client Data * * With permission from Madison Square Investors
Dynamically Varying Factor Weights Maximize Expected Return Active Variance s.t. 130/30 long short portfolio GICS Sector exposure constraints (20%) GICS Industry exposure constraints (10%) Active asset bounds constraint (2%) Turnover Constraint (30% two-way) Maximum shorting constraint Base Model = US2AxiomaMH (Axioma Fundamental Model) Benchmark = Russell 1000 Monthly backtest, 2002-2011 time period Expected Return = Dynamically varying combination of proprietary factors
Improvements From Alignment Are Also Significant for More Complex Alpha Models 11.0% US2AxiomaMH CRM CRM+AAF Realized Active Return 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Realized Active Risk
When are FAP solutions most valuable?
The Outperformance of CRM+AAF Is Very Strongly Correlated With the Latent Volatility of the Orthogonal Component of Implied Alpha Performance Differential Latent Volatility 10% 8% 6% 4% 2% 0% -2% -4% 70% 60% 50% 40% 30% 20% Latent Voltility (Implied Alpha) Feb-01 Jun-01 Oct-01 Feb-02 Jun-02 Oct-02 Feb-03 Jun-03 Oct-03 Feb-04 Jun-04 Oct-04 Feb-05 Jun-05 Oct-05 Feb-06 Jun-06 Oct-06 Feb-07 Jun-07 Oct-07 Feb-08 Jun-08 Oct-08 Feb-09 Performance Differential
The Performance Differential Using AAF in Different Market Regimes Has 60% Correlation With Latent Volatility of the Orthogonal Part of Implied Alpha 10% Performance differential vs Latent Volatility (Implied Alpha) Performance Differential 8% 6% 4% 2% 0% -2% -4% 20% 25% 30% 35% 40% 45% 50% 55% 60% Latent Volatility (Implied Alpha)
Theoretical Foundations: Pushing Frontier Theorem (Saxena and Stubbs) Theorem The increment in the utility function that results when FAP solutions such as AAF or CRM are employed increases as a function of systematic risk associated with hidden systematic risk factors Periods of high cross sectional correlations are often accompanied by rising factor volatilities of both common and hidden systematic risk factors (Renshaw and Saxena, 2011) The incremental value of FAP solutions tends to be highest during periods of high cross sectional correlations as we are currently witnessing
Lessons Learned From Client Implementations
Include individual components of alpha as distinct custom factors It increases the flexibility of the optimizer in finding better risk/return trade-offs (example: a medium-return, high-volatility component combined with a low-volatility, medium-return component)
Generate frontiers to analyze riskadjusted performance Portfolios with similar levels of realized risk should be compared If a risk constraint rarely binds, addressing mis-alignment will have limited impact on portfolio construction
Re-examine your strategy and consider loosening constraints Many constraints are typically used to compensate for risk under-prediction in traditional MVO. CRMs provide greatly improved risk estimates, which may obviate the need for tight constraints
Observations and Conclusions Q1. What are the sources of factor alignment problems (FAP)? A. Independent alpha, risk, and strategy design processes Q2. What is the opportunity cost of FAP? A. Pushing realized frontier upwards Q3. When are FAP solutions most valuable? A. During periods of high cross sectional correlations
References S. Ceria, A. Saxena and Robert A. Stubbs, Factor Alignment Problems and Quantitative Investing, Journal of Portfolio Management, 2012. A. Saxena and R. A. Stubbs, Alpha alignment factor: A solution to the underestimation of risk for optimized active portfolios. Journal of Risk, To Appear. A. Saxena and R. A. Stubbs, An empirical case study of factor alignment problems using the USER model, Journal of Investing, 2011. A. Saxena and R. A. Stubbs, Pushing the Frontier (literally) with the Alpha Alignment Factor. Technical report, Axioma, Inc. Research Report #022, September 2010. A. Saxena, C. Martin and R. A. Stubbs, Aligning alpha and risk factors, a panacea to factor alignment prolems? Technical report, Axioma, Inc. Research Report #028, September 2010. A. Renshaw and A. Saxena, Using Axioma s Risk Models to Explain the Recent Surge in Equity Correlation. Technical report, Axioma, Inc. Research Report #026, 2011.