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Factor Alignment for Equity Portfolio...
Transcript of Factor Alignment for Equity Portfolio...
Sebastian Ceria, CEO
Axioma, Inc.
The 19th Annual Workshop on Financial Engineering: Quantitative Asset Management
Columbia University
November 2012
Copyright © 2012 Axioma
Factor Alignment for Equity Portfolio Management
Factor Alignment Basics
Copyright © 2012 Axioma
A Decomposition of Expected Returns
Qhhhh
T2
Tmax λα −
Q = XΩ XT + Δ
The residual obtained by regressing the alphas against the factors in the
risk model is referred to as the orthogonal component of alpha
Xααα += ⊥
The portion of alpha explained by the risk factors is referred to as the spanned component
If alpha and risk factors are aligned,
then α⊥ = 0, or, in other words, there is misalignment
if and only if α⊥ ≠ 0
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Why is Misalignment “Bad” in MVO?
Qhhhh
T2
Tmax λα −
The optimizer sees no systematic risk in the
orthogonal component of alpha and is hence likely to
load up on it
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• 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 risk-adjusted expected return cannot be “optimal”
Risk Specific and Risk Factor Contains
Risk Specific Only Risk, Factor No
X α α α + = ⊥
Alignment is Also About Constraints
Optimal Portfolio h*
Implied Alpha
m
Qhhhh
Constraint
1 Constraints.t.max T
2T
λα − Qhhhh
T2
T*max λα −
• 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
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Alpha vs Implied Alpha Misalignment
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Alpha
Implied Alpha
Optimal Portfolio
With constraint
Constraint
Risk Ellipse
Optimal Portfolio
No longer feasible!
Misalignment Problems and Two Ways Out
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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
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How To Align Proprietary Factors With Custom Risk Models
Base Model Custom Risk Model
Base Model + AAF Custom Risk Model + AAF
Alpha Misalignment
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How To Align Constraints
Base Model Custom Risk Model
Base Model + AAF Custom Risk Model + AAF
Alpha + Constraint Misalignment (approx)
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How To Align Constraints AND Proprietary Factors
Base Model Custom Risk Model
Base Model + AAF Custom Risk Model + AAF
Alpha Misalignment
Alpha + Constraint Misalignment (approx)
Constraint Misalignment (approx)
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Why Do We Have Misalignment,
And The Case For And Against
Alignment
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Independent Alpha, Risk and Construction Processes Generate Misalignment
Optimizer
Optimal Portfolio
Alpha Process
Risk Process
Portfolio Construction
Strategy
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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
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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
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Axioma Style Factors, 25-75% Range of Systematic Risk
Proprietary Factors Have Statistically Significant Orthogonal Components
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RCP
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CTEF
Percentage of statistically significant periods (90% cf)
Axioma Style Factors, 25-75% Range % of SSP
Average Correlation Between Alpha and Implied Alpha Is Low for Most Clients
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Fundamental Model
Statistical Model
Correlation between Alpha and Implied Alpha Changes Significantly Over Time, Even For Perfectly Aligned Risk Models
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Correlation between alpha and implied alpha
The Opportunity Cost of Misalignment:
Experiments with the USER Model
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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
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Predicted Active Risk
US2AxiomaMH CRM CRM+AAF
Predicted vs Realized Active Risk Significantly Improves with CRM + AAF
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Risk Target 3.00%
Base Model 3.81%
CRM 3.38%
CRM + AAF 3.08%
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US2AxiomaMH CRM CRM+AAF
The Realized Risk-Return Frontier Moves Upwards (Annualized Active Returns and Risk)
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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
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Realized Active Risk
CRM CRM+AAF
Opportunity Cost of misalignment from constraints
Relaxing Asset Bounds Further Improves the Frontier Spreads
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Relaxing the Industry Exposure Bounds Also Improves the Frontier Spreads
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Increasing the Turnover Limit Improves Frontier Spreads
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Realized Active Risk
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Turnover Utilization Improves with CRM + AAF (Portfolios with Similar Realized Active Risk)
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The Opportunity Cost of Misalignment:
Experiments with Client Data*
Copyright © 2012 Axioma * 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
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Improvements From Alignment Are Also Significant for More Complex Alpha Models
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Realized Active Risk
US2AxiomaMH CRM CRM+AAF
When are FAP solutions most
valuable?
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The Outperformance of CRM+AAF Is Very Strongly Correlated With the Latent Volatility of the Orthogonal Component of Implied Alpha
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Perf
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Performance Differential Latent Volatility
The Performance Differential Using AAF in Different Market Regimes Has 60% Correlation With Latent Volatility of the Orthogonal Part of Implied Alpha
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Performance differential vs Latent Volatility (Implied Alpha)
Theoretical Foundations: Pushing Frontier Theorem (Saxena and Stubbs)
Theorem
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• 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
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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)
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Generate frontiers to analyze risk-
adjusted 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
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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
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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
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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.
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