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4 Optimal portfolios from ordering
Robert Almgren and Neil Chriss
Modern portfolio theory produces an optimal portfolio from estimates of expected returns
and a covariance matrix. We present a method for portfolio optimization based on replacing
expected returns with sorting criteria that is, with information about the order of the
expected returns but not their values. We give a simple and economically rational definition
of optimal portfolios that extends Markowitz’ definition in a natural way; in particular,
our construction allows full use of covariance information. We give efficient numerical
algorithms for constructing optimal portfolios. This formulation is very general and is easily
extended to more general cases, where assets are divided into multiple sectors or there are
multiple sorting criteria available, and may be combined with transaction cost restrictions.
Using both real and simulated data, we demonstrate dramatic improvement over simpler
4.1 Introduction
This chapter presents a framework for portfolio selection when an investor possesses
information about the order of expected returns in the cross-section of stocks, but not
the values of the expected returns. Even in the simplest case of a complete ordering, there
has previously been no rational way to form an optimal portfolio that makes full use of
covariance information; for the first time we give a complete solution to this important
In general, ordering information may be any set of inequality beliefs about the expected
returns, such as the order of expected returns across all stocks, sorts within sectors or
other subdivisions, decile rankings, sorts with sign beliefs, multiple incompatible sorts,
or incomplete information. We give a simple and general method of producing portfolios
that are optimal with respect to this information.
Portfolio selection as introduced by Markowitz (1952) constructs portfolios by max-
imizing expected return subject to a set of constraints. His key contribution was the
observation that an optimizing investor should want to invest only in efficient portfolios
that deliver the maximum level of expected return for a given level of risk. In the absence
of expected returns it is not clear how to generalize this approach. But ordering
mation has become increasingly important to the financial literature and the investment
process: many researchers and practitioners have associated both firm characteristics and
recent price history to expected returns in a manner that naturally gives rise to ordering

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