Factor Alignment for Equity Portfolio...

Sebastian Ceria, CEO

Axioma, Inc.

The 19th Annual Workshop on Financial Engineering: Quantitative Asset Management

Columbia University

November 2012

Factor Alignment for Equity Portfolio Management

Factor Alignment Basics

A Decomposition of Expected Returns

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

Why is Misalignment “Bad” in MVO?

Tmax λα −

The optimizer sees no systematic risk in the

orthogonal component of alpha and is hence likely to

load up on it

• 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

Constraint

1 Constraints.t.max T

λα − Qhhhh

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

Alpha vs Implied Alpha Misalignment

Implied Alpha

Optimal Portfolio

With constraint

Constraint

Risk Ellipse

Optimal Portfolio

No longer feasible!

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

How To Align Constraints

Alpha + Constraint Misalignment (approx)

How To Align Constraints AND Proprietary Factors

Alpha Misalignment

Alpha + Constraint Misalignment (approx)

Constraint Misalignment (approx)

Why Do We Have Misalignment,

And The Case For And Against

Alignment

Independent Alpha, Risk and Construction Processes Generate Misalignment

Optimizer

Optimal Portfolio

Alpha Process

Risk Process

Portfolio Construction

Strategy

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

0% 10% 20% 30% 40% 50%

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Fundamental Model

Statistical Model

Correlation between Alpha and Implied Alpha Changes Significantly Over Time, Even For Perfectly Aligned Risk Models

Correlation between alpha and implied alpha

The Opportunity Cost of Misalignment:

Experiments with the USER Model

A Practical Active Strategy: USER Model

Maximize Expected Return

• 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

1.0% 2.0% 3.0% 4.0% 5.0%

Predicted Active Risk

US2AxiomaMH CRM CRM+AAF

Predicted vs Realized Active Risk Significantly Improves with CRM + AAF

Risk Target 3.00%

Base Model 3.81%

CRM 3.38%

CRM + AAF 3.08%

1.0% 2.0% 3.0% 4.0% 5.0% 6.0%

Realized Active Risk

The Realized Risk-Return Frontier Moves Upwards (Annualized Active Returns and 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

1.0% 2.0% 3.0% 4.0% 5.0% 6.0%

CRM CRM+AAF

Opportunity Cost of misalignment from constraints

Relaxing Asset Bounds Further Improves the Frontier Spreads

1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0%

3% 1% 2%

Relaxing the Industry Exposure Bounds Also Improves the Frontier Spreads

1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0%

2.50% 10% 5%

Increasing the Turnover Limit Improves Frontier Spreads

1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0%

16% 8%

Turnover Utilization Improves with CRM + AAF (Portfolios with Similar Realized Active Risk)

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%

Realized Turnover

US2AxiomaMH CRM+AAF

The Opportunity Cost of Misalignment:

Experiments with Client Data*

Dynamically Varying Factor Weights

Maximize Expected Return – Active Variance

• 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

2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0%

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

Latent Voltility (Im

plied Alpha)

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

20% 25% 30% 35% 40% 45% 50% 55% 60%

Latent Volatility (Implied Alpha)

Performance differential vs 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 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

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.

Factor Alignment for Equity Portfolio...

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