Manager skill and portfolio size with respect to a benchmark

AuthorAndrei Bolshakov,Ludwig B. Chincarini
Date01 January 2020
DOIhttp://doi.org/10.1111/eufm.12210
Published date01 January 2020
DOI: 10.1111/eufm.12210
ORIGINAL ARTICLE
Manager skill and portfolio size with respect to a
benchmark
Andrei Bolshakov1Ludwig B. Chincarini2,3
1Wedge Capital Management, 301 South
College Street, Suite 3800, Charlotte,
NC 28202
Email: ABolshakov@wedgecapital.com
2University of San Francisco School of
Management, 101 Howard Street, Suite
500, San Francisco, CA 94105
Email: chincarinil@hotmail.com;
lbchincarini@usfca.edu.
3United States Commodity Funds, 1999
Harrison Street #1530, Oakland, CA
94612
Abstract
Investment managers often manage a portfolio with respect
to a benchmark. Typically, they use a mean-variance opti-
mization framework to maximize the information ratio of
their portfolio. We develop an unconventional approach to
this question. Given a set of assumptions, we ask what
optimal percentage of the benchmark stocks the portfolio
manager should select. This optimal portfolio depends on
Fisher’s and Wallenius’s noncentral hypergeometric distri-
butions. We find that the optimal selectivity of a benchmark
universe varies from 50% to 80%. These results are provoca-
tive, given that many enhanced index portfolio managers se-
lect a low percentage of the benchmark universe.
KEYWORDS
enhanced indexing, information ratio, portfolio management, active man-
agement
JEL CLASSIFICATIONS
G0, G13
1INTRODUCTION
The investment world is broadly divided into active management and passive management. In recent
years, there has been a push toward passive management. One can think of portfolio management as
being very active or completely passive. For example, a portfolio manager who purchases every stock
in the benchmark with the same weights as the benchmark is not providing any security picking ability.
On the other hand, a portfolio manager who selects very few stocks with a variety of weighting schemes
is creating a portfolio that is very different than the benchmark, perhaps with the ultimate objective of
outperforming the benchmark.
Traditionally, we measure a portfolio’s distance from pure passive management by the tracking
error of the portfolio; that is, by the standard deviation of the difference between the portfolio’s and the
We would like to thank Bruce Levin and Richard Gonzalez for useful discussions. We also thank an anonymous referee for
valuable suggestions.
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2CHINCARINI
benchmark’s expected or historical returns.1In this paper, we examine the question of what percentage
of the benchmark portfolio is optimal to own for an enhanced index portfolio manager, given his skill.
We depart from the typical portfolio optimization formulation and consider a simple world in which
some stocks will do well and others will not and ask how much of the benchmark universe should
be selected. We find that the skilled portfolio manager should select between 50% and 80% of the
benchmark universe regardless of his skill level.
Our analysis of the optimal enhanced index selectivity is based on a series of assumptions that in
subsequent work might be relaxed. We also bring some interesting new tools to the enhanced asset
management framework, including the noncentral hypergeometric distributions of Fisher (FNCH) and
Wallenius (WNCH). To our knowledge, this is the first time that these statistical tools have been applied
to portfolio selection.
The paper is organized as follows. Section 2 discusses our framework for enhanced index portfolio
selection. Section 3 discusses the optimal selectivity of the enhanced index portfolio. Section 4 provides
a more general discussion about the assumptions in the model and suggests ideas for future work in
this area. Section 5 concludes.
2A FRAMEWORK FOR ENHANCED INDEXING
The fundamental law of active management (FLAM; Grinold, 1989) establishes the following relation-
ship between a manager’s skill (i.e. his information coefficient, đŒđ¶), the number of bets he is making
(i.e. his breadth, đ”đ‘…) and the resulting information ratio (đŒđ‘…) of a certain strategy:
đŒđ‘… â‰ˆđŒđ¶î€‚đ”đ‘…. (1)
There has been disagreement as to the meaning of ‘‘breadth.’’ Grinold (1989) did not precisely state
what it represented, except that it would be the number of independent bets of the portfolio manager.
Different interpretations have been given to the breadth, including the number of relevant factors (Chin-
carini & Kim, 2007) and the number of stocks in one’s universe (Clarke, Silva, & Thorley, 2002).2
One can imagine breadth as having two components: a cross-sectional dimension (the number of po-
tential bets made at a given time) and a time-series dimension (how many times bets are evaluated
during each year or period). Hallerbach (2014) considers breadth along a time dimension in terms of
the number of equal size periods in which a manager makes bets per year. He uses the variable 𝑁
for his time intervals per year (not to be confused with the number of stocks necessarily). Thus, his
expression for the fundamental law is đŒđ‘… â‰ˆđŒđ¶î€‚đ‘, where 𝑁is the number of betting intervals per
year.3
The information coefficient is also at times misunderstood and at times requires unrealistic esti-
mations of the correlation between the manager’s forecasts and subsequent returns.4Some academic
studies have been devoted to establishing a different metric for a manager’s skill. For example, Consta-
ble and Armitage (2006) offered the interpretation of skill as a binomial variable where a manager has
1There are problems with this type of analysis that have been documented (Roll, 1992; Jorion, 2003; Bertrand, 2010).
2Many papers have extended the discussion of the original FLAM. For example, Qian and Hua (2004) and Ye (2008) study the
effect of variation in the information coefficient. Buckle (2004) covers correlated forecasts, Zhou (2008a) focuses on estimation
risk, and Lam and Li (2004) and Zhou (2008b) analyze unconditional optimality in the context of the FLAM.
3With certain assumptions, it can be shown that 𝑁represents the number of securities in one’s universe (see Clarke et al., 2002,
for a derivation).
4Chincarini and Kim (2007) propose one way to interpret the parameters. The most commonly used formulation assumes that
đŒđ¶ is the same for all securities, despite being highly unrealistic.
HEATON 1151
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