Decomposing the Impact of Portfolio Constraints Aug 2009

8/8/2019 Decomposing the Impact of Portfolio Constraints Aug 2009

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Decomposing the Impact of

Portfolio Constraints | August 2009

MSCI Barra Research 2009 MSCI Barra. All r ights reserved. 1 of 9Please refer to the disclaimer at the end of this document.

Jennifer Bender

Jyh-Huei Lee

Dan Stefek

Introduction

Portfolio managers may use an array of constraints when employing mean-variance optimizationfor portfolio construction. These include the long-only constraint, turnover constraints, sectorconstraints, size constraints, and beta constraints, to name a few. However, using constraintsmay potentially prevent a manager from getting the most out of his return forecasts in portfolioconstruction.

This paper analyzes the impact of constraints on portfolio return and risk, extending the insightsof previous research in this area, including Clark et al (2002), Grinold and Easton (1998), andScherer and Xu (2007). We show that constraints move a managers portfolio away from the

optimal unconstrained portfolio in two ways. First, they may rein in or increase the risk of theportfolio without impairing its information ratio. Second, they may force the portfolio to takeunwanted bets that incur risk but yield no return.

As a result, a constrained portfolio consists of positions that are aligned with the managersalphas and positions that are orthogonal to the alphas but are adopted to satisfy the constraints.We illustrate how to measure the risk and return arising from each of these sources and how todrill down to examine the contributions of individual constraints.

The Basic Framework

The basic framework follows Grinold and Easton (1998). In the presence of constraints, the

standard active optimization problem is:

2

Maximize h h h

st Ah b

(1)

where h represents the vector of active portfolio holdings, is a vector of the managers

forecast returns, is the asset covariance matrix and is the aversion to risk. The matrix A

and vector b represent the asset bounds as well as any other linear constraints1

11U

h

=

.

If there were no constraints, the solution to the optimization problem would be:

(2)

This portfolio achieves the maximum ex-ante information ratio,U

IR , for the given alphas. In the

presence of constraints, the optimal solution to the problem in (1) becomes:

1Note that our methods may be easily extended to include more general, convex constraints.


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Decomposing the Impact of Portfolio Constraints

| August 2009

MSCI Barra Research 2009 MSCI Barra. All r ights reserved. 2 of 9Please refer to the disclaimer at the end of this document. RV0609

( )11

Ch A

= (3)

where is the vector of dual or shadow prices of the constraints. Each shadow price, k ,represents the change in utility per unit increase in the constraint bound,

kb .

Substituting in Equation (2), we get:

AhhUC

=11

(4)

Equation (4) shows that the optimal constrained portfolioC

h is the difference between two

portfolios: the unconstrained portfolioU

h that reflects the managers information, and the

constraint portfolio

AhX =11 . This is depicted in Figure 1.

Figure 1: Decomposing the Constrained Portfolio

The constraint portfolioX

h can be rewritten as a weighted sum of the k individual constraint

portfolios:, X k X k

h h= . The k th constraint portfolio is:

1 '

,

1 X k k

h A

= (5)

In equation (5),,X k

h is the portfolio with the smallest risk per unit exposure to the constraint k .

Only binding constraints contribute to the constraint portfolio. If a constraint is not binding, theshadow price of the constraint is zero and the corresponding constraint portfolio makes no

contribution toX

h .

Ch

Uh

AhX

=11


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A Closer Look

So far we have decomposed the constrained portfolioC

h into two parts: the unconstrained

optimal portfolioU

h and the constraint portfolioX

h . We are interested in how constraints force a

manager to take bets that are not aligned with his alphas (i.e. with the unconstrained optimalportfolios). We can see this by resolving the constraint portfolio into a part that is aligned with the

optimal unconstrained portfolio,,X A

h , and a part that is orthogonal to it,, .X O

h 2

, ,C U X A X Oh h h h= +

, as shown in

Figure 2. We now have:

(6)

To get the constrained solution we first subtract,X A

h from the unconstrained optimal portfolio.

Since,X A

h is aligned withUh , the result is a portfolio with a different risk and return but with the

same information ratio (IR) as the unconstrained optimal portfolio. We then must add,X O

h .

Since,X O

h is orthogonal toUh , adding it increases the risk of the portfolio but does not

change its return. Thus,,X O

h represents the unwanted bets forced on the manager by the

constraints.

Figure 2: The Impact of Constraints

Like Clarke et al. (2002), who show that the optimal constrained portfolio can be written as the

sum of two uncorrelated portfolios, we see that the constrained portfolio,C

h,is the sum of the

two components:,

C I X Oh h h= + . The portfolio

Ih represents the managers information. The

portfolio,X O

h reflects the distortions or inefficiencies created by the use of constraints. The two

portfolios are calculated as follows: (see Appendix for details)

UUCIhh

,= (7A)

2By orthogonal, we mean uncorrelated:

,0

X O U h h =

C

hX

h

Ih

Uh

OXh

,

AXh

,


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, , X O C C U U h h h= (7B)

whereUC,

is the beta of the constrained portfolio to the unconstrained portfolioU

h .

To better understand how each constraint contributes to the uncompensated risk, we can write

,X Oh as a sum of k individual constraint portfolios,

, , X O k h s, which contribute risk without any

return:

, ,

, , ,

X O k

X O k U U X k k

kh

h h h =

(8)

where

1

,

, 2 1

X k U k

k U

U

h h A

= =

for each constraint k

This detailed decomposition allows us to isolate the contribution of each individual constraint tothe distortionary effect of the set of constraints.

Attributing Risk and Return

We can see how much ex-ante risk and return of an optimal portfolio comes from the managersinformation and how much comes from the constraints by using the basic Litterman (1996)decomposition. Active risk is decomposed as follows:

,0

information constraints

X cI c

c

c c

h hh h

= +

(9)

The overall constraint risk is the sum of the risks arising from each individual constraint k :

,0 ,0,

X c X k c

kc c

h h h h

= (10)

Turning to return, we see that all of the expected alpha comes from the managers information:

Ih . In this decomposition, there is no return associated with the constraints.


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An Illustration

We illustrate the decomposition framework with an example. We start with an active portfolio

optimized as of March 2008 using the Barra Short-Term US Equity Model (USE3S). The MSCIUS Prime Market 750 Index is used as the universe and benchmark. We use the followingconstraints:

All stock weights must sum to 1 ("Budget Constraint")

Each stock's weight must lie between 0 and 1 ("Long-Only Constraint")

The active portfolio should have no exposure to the Barra Earnings Variability factor.

The optimized constrained portfolioC

h has an active return of 1.62% and annualized risk of

3.69%. In contrast, the optimal unconstrained portfolioU

h , has an active return of 4.99% and

annualized risk of 7.45%.

The decomposition in Table 1 isolates the two components of the constrained portfolio:I

h ,which

is in line with the managers information, andOX

h , , which is not. Recall that the constrained

portfolioC

hcan be written as a sum of the two components:

OXIChhh

,+= . We see that more

of the active risk is coming from constraints than from the managers information!

Table 1: The Information Content of Constraints: How Much Risk Without Return3

Decomposition of theConstrained Portfolio

AnnualizedActive Return

Contributionto Active

Risk

Information Portfolio Ih 1.62% 1.59%

Constraint PortfolioOX

h,

0.00% 2.10%

- Long-only 0.00% 1.91%

- Earnings Variability 0.00% 0.22%

- Budget 0.00% -0.03%

The table further shows the contributions of individual constraints to risk and return. Notsurprisingly, the Long-Only constraint has the largest impact with respect to non-compensatedrisk. The Earnings Variability constraint has the next largest impact followed by the Budgetconstraint.

3Note that we combine the individual binding holdings constraints into the Long-Only Constraint category.


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Lets consider another example where the portfolio has the same level of active risk as ourprevious example, but we allow the optimizer to take up to a 0.3% short position in each asset.Table 2 gives the results.

Table 2: Relaxing the Long-Only Constraint Reduces Its Contribution to Active Risk

Decomposition of theConstrained Portfolio

AnnualizedActive Return

Contributionto Active

Risk

Information PortfolioI

h 2.30% 3.19%

Constraint PortfolioOX

h,

0.00% 0.50%

- Long-Only 0.00% 0.19%

- Earnings Variability 0.00% 0.28%

- Budget 0.00% 0.04%

Compared to the first example, this loosening of the Long-Only Constraint reduces its contributionto active risk significantly from 1.91% to 0.19% - below that of Earnings Variability. Most of theactive risk now comes from the Information Portfolio, reflecting the better alignment of the overallportfolios risk with the managers information.

Conclusion

Portfolio constraints may prevent an asset manager from getting the most out of return forecastsin portfolio construction. We show that the use of constraints in optimization has two effects: oneis to rein in (or increase) the risk of the portfolio without diminishing its IR, and the other is toforce the portfolio to take additional bets that incur risk but garner no return. As a result, the active

risk of the constrained portfolio comes from both positions that are aligned with the managersinformation and positions that are orthogonal to the information but are taken to satisfy theconstraints. We describe how to measure the risk and return coming from each of these sources.Although our focus has been on ex-ante risk and return, the analysis extends to ex-post return aswell.

References

Clark, Roger, Harindra de Silva, and Steven Thorley (2002), Portfolio Constraints and theFundamental Law of Active Management, Financial Analysts Journal, 2002.

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

Litterman, Robert, Hot Spots and Hedges, The Journal of Portfolio Management, pages 52-75,December. 1996.

Scherer, Bernd and Xiadong Xu (2007), The Impact of Constraints on Value-Added, TheJournal of Portfolio Management, 33(4):4554, 2007.


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Appendix

This appendix provides further details underlying the discussion in the section The Information

Content of Constraints. There we show how the constraints alter the risk of an optimized portfolioand force it to take bets that are not aligned with the managers alphas. The full decompositionwith equations is shown below in Figure A1.

Figure A1: The Decomposition of Constraint Effects

where,

c u

C U

u u

h h

h h

=

We can resolve,X O

h into individual constraints portfolios as follows:

1

, , ,

1 '

, , ,

1

1

X O C C U U U C U U

k U k k k U U X k k

k k

h h h h A h

A h h

= =

= =

(A1)

where

1

,

, 2 1

X k U k

k U

U

h h A

= =

for each constraint k

k


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| August 2009


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Decomposing the Impact of Portfolio Constraints Aug 2009

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