A Linearization of the Portfolio Optimization Problem with General Risk Measures Under Multivariate Conditional Heteroskedastic Models

• DOI: http://doi.org/10.1111/ajfs.12218
• Author: Tze‐Yun Lin,Shih‐Feng Huang
• Published date: 01 June 2018
• Date: 01 June 2018

A Linearization of the Portfolio

Optimization Problem with General Risk

Measures Under Multivariate Conditional

Heteroskedastic Models*

Shih-Feng Huang**

Institute of Statistics, National University of Kaohsiung, Taiwan

Tze-Yun Lin

Institute of Statistics, National University of Kaohsiung, Taiwan

Received 2 October 2016; Accepted 25 December 2017

Abstract

This study presents an example of the linearization of a complex mean-risk investment

problem. The spectral risk measure is employed as a measure of risk and assets are assumed

to have autocorrelation and conditionally heteroskedastic volatilities. Simulation results indi-

cate that the proposed method saves a great deal of computational time. Empirical studies

show that this strategy, implemented with certain trading frequency constraints, outperforms

the equal-weighted portfolio, the classical mean-variance method, and the corresponding

market index in Taiwan, the US, and Japan when considering transaction costs and different

economic conditions.

Keywords Conditional heteroskedastic model; Expected shortfall; Linear programming; Port-

folio selection; Spectral risk measure; Transaction cost

JEL Classification: C61, D81, G11

1. Introduction

In accordance with Sharpe’s (1971a) work on linear approximation to the mean-

variance model, many studies have been devoted to the development of linear pro-

gramming (LP) solvable models, where the role of the standard deviation in the

Markowitz model (Markowitz, 1952, 1959) is replaced by other risk measures.

*The authors thank the anonymous reviewers for their insightful comments and suggestions.

This research was partially supported by the grant MOST 106-2118-M-390-003-MY2 from

the Ministry of Science and Technology, Taiwan.

**Corresponding author: Institute of Statistics, National University of Kaohsiung, 700,

Kaohsiung University Rd., Nanzih District, Kaohsiung 811, Taiwan. Tel: +886-7-591-9167,

Fax: +886-7-591-9344, email: huangsf@nuk.edu.tw.

Asia-Pacific Journal of Financial Studies (2018) 47, 449–469 doi:10.1111/ajfs.12218

©2018 Korean Securities Association 449

Throughout this paper, we call out the portfolio optimization with risk measures

other than standard deviation (or variance) by mean-risk models. For instance, the

mean-risk model considered by Sharpe (1971b) used mean absolute deviation

(MAD) in asset allocation. Konno and Yamazaki (1991) established the LP solvable

optimization for portfolios with MAD. Michalowski and Ogryczak (2001) further

extended the MAD portfolio optimization model to an m-MAD model for incorpo-

rating downside risk aversion. Rockafellar and Uryasev (2000, 2002) proposed a lin-

earization approach for the mean-risk problem when expected shortfall (ES; also

called conditional VaR, CVaR) is employed as the risk measure. Mansini et al.

(2003a,b) provided a systematic overview of the LP solvable models for portfolio

selection. Adam et al. (2008) extended Rock afellar and Uryasev’s (2002) lineariza-

tion technique to the mean-risk model with the spectral risk measure (SRM).

However, the aforementioned studies did not consider serial correlation within

each return process. Since ARMA-GARCH-type models are capable of depicting

many important features of financial data, such as autocorrelation, negative skew-

ness, kurtosis, conditional heteroskedasticity, and tail dependence, this study’s

objective is to linearize a mean-risk problem under an ARMA-GARCH framework.

Additionally, Hahn and Kwon (2015) found that the degree of investors’ risk aver-

sion has significant influence on the effects of ambiguity on price volatility. Because

the SRM is capable of reflecting the magnitude of the risk aversion of investors and

is a general family of coherent risk measures, which includes the ES as a special case

(Acerbi, 2002; Acerbi and Simonetti, 2002; Adam et al., 2008), this study employs

the SRM as the risk measure in our mean-risk problem, denoted by mean-SRM.

The main challenge in linearizing the mean-SRM problem under an ARMA-

GARCH model is twofold. First, it is not trivial to extend the linearization tech-

nique directly from the no serial correlation framework to an ARMA-GARCH

model. Second, the SRM is a nonlinear function of the holding weights of the

assets, which also increases the difficulty of establishing a linear approximation of

mean-risk models. Our simulation results indicate that the proposed linearization

method obtains more accurate and optimal holding weights than the simulation

based method proposed by Harris and Mazibas (2013). The computational costs of

the proposed method are remarkably less than those in the simulation-based

method, particularly when the number of underlying assets increases.

For practical implementation, a self-financing trading strategy combined with a

momentum-type criterion for preventing excessive rebalancing is proposed to

reduce the effect of transaction costs. It has been reported that the incorporation of

transaction costs into an optimal portfolio framework leads to a significant reduc-

tion of profit and a stabilization of the optimal portfolio (Georgiev et al., 2015). To

address this problem, we propose a momentum-type criterion to decide whether

the holding weights of the portfolio should be rebalanced when processing the self-

financing framework. Herein, we denote the proposed trading strategy by SFM. To

investigate the investment performance of the SFM, the daily prices of the con-

stituents of the Taiwan Mid-Cap 100 Index, S&P100 Index, and Nikkei 225 Index

S.-F. Huang and T.-Y. Lin

450 ©2018 Korean Securities Association