A Stability Approach to Mean‐Variance Optimization

• DOI: http://doi.org/10.1111/fire.12068
• Date: 01 August 2015
• Author: Apostolos Kourtis
• Published date: 01 August 2015

The Financial Review 50 (2015) 301–330

A Stability Approach to Mean-Variance

Optimization

Apostolos Kourtis

Norwich Business School, University of East Anglia

Abstract

I jointly treat two critical issues in the application of mean-variance portfolios, that is,

estimation risk and portfolio instability. I find that theory-based portfolio strategies,which are

known to outperform naive diversification (1/N) in the absence of transaction costs, heavily

underperform it under transaction costs. This is because they are highly unstable over time.

I propose a generic method to stabilize any given portfolio strategy while maintaining or

improving its efficiency. My empirical analysis confirms that the new method leads to stable

and efficient portfolios that offer equal or lower turnover than 1/N and larger Sharpe ratio,

even under high transaction costs.

Keywords: portfolio choice, stability, estimation risk, transaction costs

JEL Classifications: C13, C51, C61, G11

1. Introduction

Mean-variance analysis of Markowitz (1952) is an important portfolio choice

model in academia and investment practice. However, practical applications suffer

Corresponding author: Norwich Business School, University of East Anglia, Norwich Research Park,

Norwich, Norfolk NR4 7TJ, UK; Phone: +44 (0)1603 591387; E-mail: a.kourtis@uea.ac.uk.

I am grateful to the editor Robert Van Ness, an anonymous referee, Raphael Markellos and seminar

participants at the University of East Anglia, 3rd International Conference of FEBS and EFMA 2013

Annual Meetings for valuable comments. An earlier version of the paper was entitled “Stable and efficient

portfolios.”

C2015 The Eastern Finance Association 301

302 A. Kourtis/The Financial Review 50 (2015) 301–330

from two critical issues. First, the parameters that define mean-variance efficientport-

folios are unknown and need to be estimated in finite samples. Potential estimation

errors add risk to the portfolio selection process, coined as estimation risk in the liter-

ature, and negatively affect out-of-sample portfolio performance (e.g., see Michaud,

1989; Best and Grauer, 1991). Second, estimated portfolio weights tend to be very

unstable over time. This instability translates into high transaction costs and further

decreases portfolio returns. The recent work of Kirby and Ostdiek (2012) highlights

the relation between these two issues: portfolio instability tends to increase with esti-

mation risk. In this context, this paper treats estimation risk and portfolio instability

in a joint manner.

A vast literature has developed around proposing portfolio strategies that are

less sensitive to estimation risk (Brandt, 2010). This literature has been recently chal-

lenged by the influential work of DeMiguel, Garlappi and Uppal (2009) who find

that most sample-based strategies perform worse out-of-sample than the equally

weighted portfolio, also known as 1/N. In response to this finding, DeMiguel,

Garlappi, Nogales and Uppal (2009), Tu and Zhou (2011), Kirby and Ostdiek (2012)

and Kourtis, Dotsis and Markellos (2012) develop more efficient strategies that offer

significantly higher risk-adjusted returns than 1/N.

I find that 1/N still outperforms most existing sample-based strategies when

transaction costs are present. The reason is that most sample-based strategies are

very sensitive to small changes in the underlying sample. As new observations en-

ter the sample and portfolio is rebalanced, portfolio composition tends to change

dramatically. As a result, portfolio turnover and the associated transaction costs are

magnified. An empirical exercise similar to that of DeMiguel, Garlappi and Uppal

(2009) shows that the strategies in DeMiguel, Garlappi, Nogales and Uppal (2009),

Tu and Zhou (2011) and Kourtis, Dotsis and Markellos (2012) can even produce

negative average returns net of transaction costs. Such results question the practical

value of mean-variance optimization in the presence of transaction costs.

My findings motivate the main objective of this paper: to develop sample-based

strategies that are both efficient and stable. The traditional approach to promote sta-

bility is to explicitly incorporate proportional transaction costs in the mean-variance

framework.1However, the nonlinear form of proportional transaction costs does not

generally allow a closed-form solution to the portfolio problem. As a result, many

of the recent advances in the estimation risk literature cannot accommodate propor-

tional transaction costs, since they require an analytical representation of the optimal

weights (e.g., the methods in Kan and Zhou, 2007; Tu and Zhou, 2011; Kourtis,

Dotsis and Markellos, 2012). Also, a computational algorithm is required for the

derivation of the optimal portfolio, but such algorithms tend to be inefficient when

the number of assets is large.

1Woodside-Oriakhi, Lucas and Beasley (2013) review the recent literature that accounts for transaction

costs in the mean-variance framework. Alternatively, several studies investigate the problem of portfolio

choice under proportional transaction costs in a continuous-time setting (for a review,see Cvitani ´

c, 2001).