Jennifer Bender, PhD Vice President, Applied Research, Americas Optimization Analytics.
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Transcript of 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
© 2010. All rights reserved.3
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?
© 2010. All rights reserved.4
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?
© 2010. All rights reserved.5
Unconstrained problem:
Optimal portfolio:
Portfolio OptimizationPortfolio Optimization
2
Maximize h h h
11 0
U Uh h
Risk
Return
Uh
© 2010. All rights reserved.6
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
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Now,
The Optimal Constrained PortfolioThe Optimal Constrained Portfolio
11
X
C U
h
h h A
Ch
Uh
Xh
© 2010. All rights reserved.8
Adding ConstraintsAdding Constraints
Ch
Uh
Active frontier without constraint
Active frontier with constraint(s)
Risk
Alpha
11A
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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
© 2010. All rights reserved.10
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
© 2010. All rights reserved.11
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
(%)
© 2010. All rights reserved.12
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%
© 2010. All rights reserved.13
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%
© 2010. All rights reserved.14
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
© 2010. All rights reserved.15
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:
© 2010. All rights reserved.16
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:
© 2010. All rights reserved.17
Constraints consume the risk budget
New Risk and Return DecompositionNew Risk and Return Decomposition
Contribution to Active Risk
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%
© 2010. All rights reserved.18
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
© 2010. All rights reserved.19
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
© 2010. All rights reserved.20
ResultsResults
Ex-Ante
Ex-Post
Active Return (%)
Active Risk (%)
Constrained Portfolio (IR=0.89) 2.67 3.00
Active Return (%)
Contribution to Active Risk
(%) - 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 (%)
Constrained Portfolio (IR=0.71) 2.52 3.56
Active Return (%)
Contribution to Active Risk
(%) - 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
© 2010. All rights reserved.21
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 (%)
Contribution to Active Risk
(%) - 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 (%)
Constrained Portfolio (IR=1.21) 4.04 3.34
Active Return (%)
Contribution to Active Risk
(%) - 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
© 2010. All rights reserved.22
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 (%)
Contribution to Active Risk
(%) - 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%
© 2010. All rights reserved.23
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
© 2010. All rights reserved.24
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
© 2010. All rights reserved.25
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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