A Note on “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps”

• Author: JIAQI ZHANG,TONGSHU MA,RAVI JAGANNATHAN
• Date: 01 October 2019
• DOI: http://doi.org/10.1111/jofi.12824
• Published date: 01 October 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 5 •OCTOBER 2019

A Note on “Risk Reduction in Large Portfolios:

Why Imposing the Wrong Constraints Helps’’

RAVI JAGANNATHAN, TONGSHU MA, and JIAQI ZHANG∗

ABSTRACT

This note corrects an error in the proof of Proposition 2 of “Risk Reduction in Large

Portfolios: Why Imposing the Wrong Constraint Helps” that appeared in the Journal

of Finance, August 2003.

PROPOSITION 1(PAGE 1655) OF JAGANNATHAN AND MA(2003) shows that the

global minimum variance portfolio of a given sample covariance matrix when

there are “no short sale” constraints and upper bounds on portfolio weights

is the unconstrained global minimum variance portfolio when the covariance

matrix is adjusted by shrinking its elements that are large in magnitude in a

particular manner.

Proposition 2(page 1656), says that: (1) the adjusted covariance matrix in

Proposition 1is the maximum likelihood estimate of the covariance matrix

subject to the constraint that its implied global minimum variance portfolio

satisfies the portfolio weight constraints, and (2) the global minimum variance

portfolio of the constrained maximum likelihood estimate of the covariance ma-

trix is also the global minimum variance of the sample covariance matrix when

there are “no short sale” constraints and upper bounds on portfolio weights.

Unfortunately, there is an error in the proof of Proposition 2, and in this note

we correct the error. Equation (8) in Jagannathan and Ma (2003) should have

been equation (10) in this note. The two equations are the same except for the

fourth term on the right side of equation (10) in this note. The fourth term was

omitted in Jagannathan and Ma (2003) by mistake, and that mistake carried

over to the proofs of the propositions. Because of this error, the proofs in the

paper are correct only when there are lower bounds on portfolio weights (i.e., “no

short sale” constraints only) but not when there are upper bounds on portfolio

weights as well. We now have the fourth term incorporated, and the proofs of

the propositions have been corrected. The propositions as originally stated in

Jagannathan and Ma (2003) remain valid. In the following, we present a self-

contained discussion of the effects of portfolio weight constraints that corrects

the omission in Section I of the original paper.

∗Ravi Jagannathan is at Kellogg School of Management, Northwestern University and NBER,

ISB, SAIF. Jiaqi Zhang is at Kellogg School, Northwestern University. Tongshu Ma is with Bing-

hamton University.

DOI: 10.1111/jofi.12824

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