Factor Alignment for Equity Portfolio...

41
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

Transcript of Factor Alignment for Equity Portfolio...

Page 1: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Page 2: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Factor Alignment Basics

Copyright © 2012 Axioma

Page 3: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 4: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

• 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 + = ⊥

Page 5: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 6: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Alpha vs Implied Alpha Misalignment

Copyright © 2012 Axioma

Alpha

Implied Alpha

Optimal Portfolio

With constraint

Constraint

Risk Ellipse

Optimal Portfolio

No longer feasible!

Page 7: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Misalignment Problems and Two Ways Out

Copyright © 2012 Axioma

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)

Page 8: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

From Misalignment To Alignment

Base Model Custom Risk Model

Base Model + AAF Custom Risk Model + AAF

Copyright © 2012 Axioma

Page 9: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

How To Align Proprietary Factors With Custom Risk Models

Base Model Custom Risk Model

Base Model + AAF Custom Risk Model + AAF

Alpha Misalignment

Copyright © 2012 Axioma

Page 10: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

How To Align Constraints

Base Model Custom Risk Model

Base Model + AAF Custom Risk Model + AAF

Alpha + Constraint Misalignment (approx)

Copyright © 2012 Axioma

Page 11: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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)

Copyright © 2012 Axioma

Page 12: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Why Do We Have Misalignment,

And The Case For And Against

Alignment

Copyright © 2012 Axioma

Page 13: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Independent Alpha, Risk and Construction Processes Generate Misalignment

Optimizer

Optimal Portfolio

Alpha Process

Risk Process

Portfolio Construction

Strategy

Copyright © 2012 Axioma

Page 14: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 15: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Copyright © 2012 Axioma

Empirical Evidence of Why

Alignment Matters

The USER Model* and Client Data

* Guerard et. al

Page 16: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Proprietary Factors in the USER Model have Orthogonal Components with Realized Systematic Risk Comparable with Other Axioma Factors

Copyright © 2012 Axioma

Axioma Style Factors, 25-75% Range of Systematic Risk

Page 17: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Proprietary Factors Have Statistically Significant Orthogonal Components

Copyright © 2012 Axioma

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

EP

BP

CP

SP

REP

RBP

RCP

RSP

CTEF

Percentage of statistically significant periods (90% cf)

Axioma Style Factors, 25-75% Range % of SSP

Page 18: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Average Correlation Between Alpha and Implied Alpha Is Low for Most Clients

Copyright © 2012 Axioma

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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

Co

rrela

tio

n

Fundamental Model

Statistical Model

Page 19: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

0%

20%

40%

60%

80%

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

Correlation between alpha and implied alpha

Page 20: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

The Opportunity Cost of Misalignment:

Experiments with the USER Model

Copyright © 2012 Axioma

Page 21: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 22: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

1.0% 2.0% 3.0% 4.0% 5.0%

Rea

lized

Act

ive

Ris

k

Predicted Active Risk

US2AxiomaMH CRM CRM+AAF

Predicted vs Realized Active Risk Significantly Improves with CRM + AAF

Copyright © 2012 Axioma

Risk Target 3.00%

Base Model 3.81%

CRM 3.38%

CRM + AAF 3.08%

Page 23: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

0.0%

1.0%

2.0%

3.0%

4.0%

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

Rea

lized

Act

ive

Ret

urn

Realized Active Risk

US2AxiomaMH CRM CRM+AAF

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

Copyright © 2012 Axioma

Frontier Spreads: CRM: 1% Additional

Realized Active Annualized Return CRM + AAF: 1.75% Additional Realized Active Annualized

Return

Page 24: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Frontier Spreads Should be Interpreted as Opportunity Costs for Misalignment

Copyright © 2012 Axioma

0.0%

0.4%

0.8%

1.2%

1.6%

2.0%

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

Fron

tier

Spr

ead

Realized Active Risk

CRM CRM+AAF

Opportunity Cost of misalignment from constraints

Page 25: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Relaxing Asset Bounds Further Improves the Frontier Spreads

Copyright © 2012 Axioma

0.00%

0.50%

1.00%

1.50%

2.00%

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

Fron

tier

Spr

ead

Realized Active Risk

3% 1% 2%

Page 26: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Relaxing the Industry Exposure Bounds Also Improves the Frontier Spreads

Copyright © 2012 Axioma

0.00%

0.50%

1.00%

1.50%

2.00%

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

Fron

tier

Spr

ead

Realized Active Risk

2.50% 10% 5%

Page 27: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Increasing the Turnover Limit Improves Frontier Spreads

Copyright © 2012 Axioma

0.00%

0.50%

1.00%

1.50%

2.00%

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

Fron

tier

Spr

ead

Realized Active Risk

16% 8%

Page 28: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

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

Real

ized

IR

Realized Turnover

US2AxiomaMH CRM+AAF

Page 29: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

The Opportunity Cost of Misalignment:

Experiments with Client Data*

Copyright © 2012 Axioma * With permission from Madison Square Investors

Page 30: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 31: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Improvements From Alignment Are Also Significant for More Complex Alpha Models

Copyright © 2012 Axioma

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

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

Real

ized

Act

ive

Retu

rn

Realized Active Risk

US2AxiomaMH CRM CRM+AAF

Page 32: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

When are FAP solutions most

valuable?

Copyright © 2012 Axioma

Page 33: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

The Outperformance of CRM+AAF Is Very Strongly Correlated With the Latent Volatility of the Orthogonal Component of Implied Alpha

Copyright © 2012 Axioma

20%

30%

40%

50%

60%

70%

-4%

-2%

0%

2%

4%

6%

8%

10%

Feb-

01Ju

n-01

Oct

-01

Feb-

02Ju

n-02

Oct

-02

Feb-

03Ju

n-03

Oct

-03

Feb-

04Ju

n-04

Oct

-04

Feb-

05Ju

n-05

Oct

-05

Feb-

06Ju

n-06

Oct

-06

Feb-

07Ju

n-07

Oct

-07

Feb-

08Ju

n-08

Oct

-08

Feb-

09

Latent Voltility (Im

plied Alpha)

Perf

orm

ance

Diff

eren

tial

Performance Differential Latent Volatility

Page 34: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

The Performance Differential Using AAF in Different Market Regimes Has 60% Correlation With Latent Volatility of the Orthogonal Part of Implied Alpha

Copyright © 2012 Axioma

-4%

-2%

0%

2%

4%

6%

8%

10%

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

Perf

orm

ance

Diff

eren

tial

Latent Volatility (Implied Alpha)

Performance differential vs Latent Volatility (Implied Alpha)

Page 35: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Theoretical Foundations: Pushing Frontier Theorem (Saxena and Stubbs)

Theorem

Copyright © 2012 Axioma

• 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

Page 36: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

Lessons Learned From

Client Implementations

Copyright © 2012 Axioma

Page 37: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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)

Copyright © 2012 Axioma

Page 38: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 39: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 40: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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

Copyright © 2012 Axioma

Page 41: Factor Alignment for Equity Portfolio Managementcfe.columbia.edu/.../CeriaAlignmentPresentationColumbiaFEC2012.pdf · h h. Qh. h. T 2. max αT ... What are the sources of factor alignment

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.

Copyright © 2012 Axioma