Jennifer Bender, PhD Vice President, Applied Research, Americas Optimization Analytics.

Jennifer Bender, PhDVice President, Applied Research, Americas

Optimization Analytics Optimization Analytics

© 2010. All rights reserved.2

Role of constraints

Analyzing the impact of constraints on risk and return

- Old and new

Ex-Ante and Ex-Post Analysis

Sensitivity

OutlineOutline


Asset managers use constraints for a variety of reasons, including:

- correct for overly large/small positions

- limit exposure to certain sources of risk which are either undesirable or for which they have no information

- achieve a certain risk profile

Why Do Managers Use Constraints?Why Do Managers Use Constraints?


And to:

- lower trading costs

- comply with institutional requirements, such as no-shorting

- reduce influence of errors in input estimates

Constraints may impair performance

Why Do Managers Use Constraints?Why Do Managers Use Constraints?


Unconstrained problem:

Optimal portfolio:

Portfolio OptimizationPortfolio Optimization

2

Maximize h h h

11 0

U Uh h

Risk

Return

Uh


Constrained problem:

For each constraint i, we get the a shadow price, , which is the rate at which the portfolio utility increases as we relax the constraint. (These apply only for small changes)

The optimal constrained portfolio satisfies:

Adding ConstraintsAdding Constraints

2

Maximize h h h

Ah b

i

0Ch A


Now,

The Optimal Constrained PortfolioThe Optimal Constrained Portfolio

11

X

C U

h

h h A

Ch

Uh

Xh


Adding ConstraintsAdding Constraints

Ch

Uh

Active frontier without constraint

Active frontier with constraint(s)

Risk

Alpha

11A


The constraint portfolio is the sum of individual constraint portfolios:

is the portfolio with the smallest risk per unit exposure to

constraint i

Contribution of Individual ConstraintsContribution of Individual Constraints

1 2

1 1 11 1 2 2

1 1 1...

X X Xk

X k k

h h h

h A A A

11iA


MSCI US Prime Market 750 Index is universe and benchmark (March 2008)

Alpha is based on Barra US Short-Term Model (USE3S) Earning Yield factor

Risk model = USE3S

Constraints

- Long-Only

- Size Factor Neutral

- Budget (Holdings must sum to 1)

Illustrating the Basic FrameworkIllustrating the Basic Framework


Illustrating the Basic FrameworkIllustrating the Basic Framework

VALERO ENERGY CORP 4.5 1.02 -0.20 0.83BERKSHIRE 3.9 1.19 -0.11 1.08NATIONWIDE FINL SVCS 2.0 0.06 -0.08 -0.02CITIGROUP INC 1.1 -0.03 -0.85 -0.88GENERAL ELECTRIC CO -0.1 -0.77 -0.83 -1.60MICROSOFT CORP -0.1 0.52 0.41 0.93INTEL CORP -0.3 0.54 -0.18 0.35R H DONNELLEY CORP -0.4 0.03 -0.03 0.00

Asset α (%)

h U

(%)

h X

(%)

h C

(%)

0.05 -0.25 0.00-0.07 -0.04 0.00-0.17 0.10 0.000.20 -1.11 0.060.71 -1.60 0.060.33 0.19 -0.110.39 -0.45 -0.12-0.03 0.03 -0.02

h LongOnly

(%)

h Size

(%)

h Budget

(%)


We can attribute return to the manager’s information and to the constraints:

- Unconstrained portfolio:

- Constraints:

Return AttributionReturn Attribution

Uh

1 2...

kX X X Xh h h h

Active Return

Constrained Portfolio 2.2%

- Unconstrained Portfolio 7.5%

- Long-Only Constraint -5.5%

- Size Constraint 0.1%

- Budget Constraint 0.1%


Risk Contributions:

Constraints:

Risk AttributionRisk Attribution

1 2 ... kC XC X C XC X

C C C C

h hh h h hh h

Unconstrained Constraints Portfolio

C U X C U C XC

C C C

h h h h h h h

Active Risk

Constrained Portfolio 4.0%

Contribution to Active Risk

- Unconstrained Portfolio 5.7%

- Long-Only Constraint -1.7%

- Size Constraint 0.0%

- Budget Constraint 0.0%


Constraints act in two ways:

- To alter the risk and return without changing the information ratio

- To add risk but no return

A Closer LookA Closer Look

,X Oh

,X Uh

Ch

Uh

Xh

8.9%U

4.0%C

,X Oh

,X Uh


The constrained portfolio = Information + Noise:

A New DecompositionA New Decomposition

Ch

Ih

,X Oh

Uh

C UIR TC IR

2

, 1 X O

C

TC Transfer Coefficient

The information ratio is:


New holdings decomposition:

A New DecompositionA New Decomposition

, C I X Oh h h

, , , ,1, 1,

Constraint k Constraint k

kX O X O k X k U U

k K k K

h h h h

, ,X O kh

Uh

kXh

,k U Uh

VALERO ENERGY CORP 0.83 0.30 0.53BERKSHIRE 1.08 0.34 0.74NATIONWIDE FINL SVCS -0.02 0.02 -0.04CITIGROUP INC -0.88 -0.01 -0.87GENERAL ELECTRIC CO -1.60 -0.22 -1.38MICROSOFT CORP 0.93 0.15 0.78INTEL CORP 0.35 0.15 0.20R H DONNELLEY CORP 0.00 0.01 -0.01

h X,O

(%)Asset

h C

(%)

h I

(%)

Individual constraints:


Constraints consume the risk budget

New Risk and Return DecompositionNew Risk and Return Decomposition


5.7%

-1.7%

0.0%

0.0%

Original

MethodActive Return Active Risk

Constrained Portfolio 2.2% 4.0%

Active Return Contribution to Active Risk

- Information Portfolio 2.2% 1.6%

- Long-Only Constraint 0.0% 2.6%

- Size Constraint 0.0% -0.1%

- Budget Constraint 0.0% -0.1%


We can attribute a manager’s ex-post performance to the information and the constraints

Each period, we compute:

We compute average realized returns to the information and constraint portfolios

To determine the risk contribution from each source, we compute its empirical risk contribution. For example,

Ex-Post AnalysisEx-Post Analysis

, , ,X O k C

kC

Cov r rCTR

, ,1,

C I X O kk K

r h r h r h


MSCI US Prime Market 750 Index is universe and benchmark

Alpha is based on USE3S Earning Yield

Risk model = USE3S

Manager keeps active risk at 3% forecast active risk every month

Constraints

- Long-Only

- Neutral to Earnings Variability

- Budget (Holdings must sum to 1)

Backtest period: January 2000 to December 2008

Ex-Post analysisEx-Post analysis


ResultsResults

Ex-Ante

Ex-Post

Active Return (%)

Active Risk (%)

Constrained Portfolio (IR=0.89) 2.67 3.00

Active Return (%)


(%) - Information Portfolio 2.67 1.35 - Long-Only Constraint 0.00 1.76 - Earnings Var. Constraint 0.00 -0.09 - Budget Constraint 0.00 -0.02

Active Return (%)

Active Risk (%)


Active Return (%)


(%) - Information Portfolio 2.79 1.61 - Long-Only Constraint 1.75 2.55 - Earnings Var. Constraint -0.66 -0.33 - Budget Constraint -1.36 -0.26


Removing ConstraintsRemoving Constraints

Remove Long-only Constraints

Remove Earnings Variability ConstraintActive Return

(%)Active Risk

(%)Constrained Portfolio (IR=0.80) 2.92 3.66

Active Return (%)


(%) - Information Portfolio 2.81 1.65 - Long-Only Constraint 1.11 2.25 - Earnings Var. Constraint 0.00 0.00 - Budget Constraint -1.01 -0.23

Was 2.52

Was 3.56

Was -0.33

Was -0.66

Active Return (%)

Active Risk (%)


Active Return (%)


(%) - Information Portfolio 4.02 3.20 - Long-Only Constraint 0.00 0.00 - Earnings Var. Constraint -0.07 0.10 - Budget Constraint 0.10 0.03

Was 2.55

Was 1.75

Was 1.61


We allow some shorting and reign in the industry bets

Reducing Industry BetsReducing Industry Bets

Active Return (%)

Active Risk (%)

Constrained Portfolio (IR=0.67) 2.15% 3.23%

Active Return (%)


(%) - Information Portfolio 1.79% 2.06% - 0.2% Shorts Allowed -0.31% 0.93% - Industry Constraint±1% 0.69% 0.19% - Budget Constraint -0.01% 0.04%


As we relax constraint “i” by a little, :

where

If the constraint portfolios have little covariance, mainly changes !

Sensitivity – A Look Under the HoodSensitivity – A Look Under the Hood

1 1 1 11 1 1 1 1

1 1 1 1 ... ...

k k k k kA A A A

ib

11

0

0iA A b

i


Constraints may force a manager to take unintended bets and incur risk that are unrelated to his information

Managers may want to know

- Which constraints are the most “costly”?

- What is the effect of constraint(s) on realized performance ?

We show how to analyze the impact of individual constraints on the ex-ante and ex-post risk and return of the portfolio

SummarySummary


Grinold, Richard and Kelly Easton (1998), “Attribution of Performance and Holdings,” in Worldwide Asset and Liability Modeling, eds. W.T. Ziemba, John M. Mulvey, Isaac Newton

Scherer, Bernd and Xiadong Xu (2007), “The Impact of Constraints on Value-Added,” The Journal of Portfolio Management, 2007

ReferencesReferences

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Jennifer Bender, PhD Vice President, Applied Research, Americas Optimization Analytics.

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