A Portfolio Optimization Approach with a Large Number of Assets: Applications to the US and Korean Stock Markets

• Author: Miyoung Lee,Jihun Kim,Sekyung Oh
• Date: 01 October 2018
• DOI: http://doi.org/10.1111/ajfs.12233
• Published date: 01 October 2018

A Portfolio Optimization Approach with a

Large Number of Assets: Applications to

the US and Korean Stock Markets*

Miyoung Lee

Department of Business Administration, Konkuk University, Republic of Korea

Jihun Kim

KB Research, KB Financial Group, Republic of Korea

Sekyung Oh**

Department of Business Administration, Konkuk University, Republic of Korea

Received 17 August 2017; Accepted 26 April 2018

Abstract

We propose a new approach for determining optimal portfolio weights in a mean-variance

framework in cases when there are more assets than observations. We empirically show that

our approach attains a higher out-of-sample Sharpe ratio, M

2

measure, and CEQ return than

other methods. Our approach is also free from problems related to the estimation of the

covariance matrix because it does not need and solves the so-called “corner-solution prob-

lem” of the Markowitz model by minimizing the 2-norm of the portfolio weight vector.

Keywords Mean-variance portfolio optimization; Singular covariance matrix; Least squares;

Out-of-sample performance

JEL Classification: C13, G11, G15

1. Introduction

In his seminal paper, Markowitz (1952) proposed a mean-variance portfolio opti-

mization approach that minimizes the variance of a portfolio while achieving the

expected return. To implement the Markowitz portfolio, the means and covarian ces

of asset returns need to be estimated. Traditionally, sample means and covariances

*This paper was supported by a research grant at Konkuk University in 2014. We thank the

editor, Professor Kwangwoo Park, the associate editor and two anonymous referees for their

valuable comments and suggestions.

**Corresponding author: Department of Business Administration, School of Business, Kon-

kuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul 05029, Korea. Tel: +82-2-450-3641,

Fax: +82-2-3436-6610, email: skoh@konkuk.ac.kr.

Asia-Pacific Journal of Financial Studies (2018) 47, 634–659 doi:10.1111/ajfs.12233

634 ©2018 Korean Securities Association

have been used, but it is well known that the portfolios based on the sample

estimates perform poorly out-of-sample due to estimation errors (Frost and Savar-

ino, 1986, 1988; Litterman, 2003; Cochrane, 2014). Since the estimation of means is

more difficult and has a larger impact on portfolios’ out-of-sample performance

than the estimation of covariances (Merton, 1980; Jorion, 1986, 1991), stud ies in

empirical financial economics that apply the Markowitz portfolio selection mod el

have focused on the minimum-variance portfolio rather than the mean-variance

portfolio (Jagannathan and Ma, 2003; Ledoit and Wolf, 2003; DeMiguel et al.,

2009). Several studies have used forward-looking information to overcome the

problems related to relying on historical data when estimating the means and

covariances of a portfolio (Kempf et al., 2015; Park and Rhee, 2017).

Martellini and Ziemann (2010) extend the literature, which has focused on the

covariance matrix, by introducing improved estimators for the coskewness and

cokurtosis parameters. Basak and Chabakauri (2010) consider the dynamic mean-

variance portfolio problem and derive its time-consistent solution by extending

studies that have determined only myopic or precommitment solutions. Garleanu

and Pedersen (2013) also provide a dynamic portfolio selection model with pre-

dictable returns and transaction costs.

Even if the minimum-variance framework is used to avoid the estimation prob-

lem of means, many problems remain. When the number of observations Texceeds

the number of assets N, the number of estimates needed to fill the covariance

matrix grows exponentially as the number of assets increases and the errors in the

estimation of covariance matrix parameters lead to unstable and extreme portfolio

weights over time. Even worse, the sample covariance matrix becomes singular

either when Tis less than Nor when asset returns are linearly dependent, in which

case the inverse of the sample covariance matrix cannot be obtained. In real-world

applications, this case occurs frequently because the number of assets need ed to

construct stock portfolios can be greater than the number of observations. For

example, the number of stocks listed on the New York Stock Exchange (NYSE)

totaled more than 2700 in 2017, while the number of observations totaled just 1200

at a monthly frequency, even if the sample period were 100 year s.

1

Many attempts have been made to find an invertible estimator of the covariance

matrix when Nis larger than T. The pseudoinverse estimators of the covariance

matrix are used by Sengupta (1983) and Pappas et al. (2010), and the shrinkage

estimators of the covariance matrix are suggested by Ledoit and Wolf (2003). Ledoit

and Wolf (2003) propose estimating the covariance matrix by an optimally

weighted average of two existing estimators: the sample covariance matrix and sin-

gle-index covariance matrix.

Although both approaches find an invertible estimator of the covariance matrix

of the Markowitz model, they have a conceptual problem, as shown in Figure 1.

1

We can solve this problem by using higher-frequency data, an approach widely used among

practitioners.

Portfolio Optimization with numerous Assets

©2018 Korean Securities Association 635