Incorporating time‐varying jump intensities in the mean‐variance portfolio decisions

• Published date: 01 March 2020
• Author: Weidong Xu,Chunyang Zhou,Chongfeng Wu
• Date: 01 March 2020
• DOI: http://doi.org/10.1002/fut.22075

J Futures Markets. 2020;40:460–478.wileyonlinelibrary.com/journal/fut460

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© 2019 Wiley Periodicals, Inc.

Received: 14 March 2019

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Accepted: 6 November 2019

DOI: 10.1002/fut.22075

RESEARCH ARTICLE

Incorporating time‐varying jump intensities in the

mean‐variance portfolio decisions

Chunyang Zhou

1

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Chongfeng Wu

1

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Weidong Xu

2

1

Department of Finance, Antai College of

Economics and Management, Shanghai

Jiao Tong University, Shanghai, China

2

Department of Finance and Acccounting,

School of Management, Zhejiang

University, Zhejiang, China

Correspondence

Weidong Xu, School of Management,

Zhejiang University, Zhejiang 310058,

China.

Email: weidxu@zju.edu.cn

Abstract

This paper examines the role of time‐varying jump intensities in forming mean‐

variance portfolios. We find that compared with the no‐jump or constant‐jump

models, the model which incorporates time‐varying jump intensities better fits

the dynamics of the assets returns, and yields mean‐variance portfolios with

higher Sharpe ratios. Our research suggests that using a better econometric

model that captures non‐normal features in the data has benefits for portfolio

allocation even for a mean‐variance investor.

KEYWORDS

mean‐variance, optimal portfolio, time‐varying jump intensities

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INTRODUCTION

It is well documented that risky asset prices exhibit large jumps due to unexpected major events.

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Previous studies

showed that jump risk is an important factor that can affect the investors’asset allocation decisions. For instance, Liu,

Longstaff, and Pan (2003) showed that investors are unwilling to take leveraged positions in the presence of jump risks.

Das and Uppal (2004) documented that the market‐wide systematic jumps reduce the gains from portfolio

diversification. Aït‐Sahalia, Cacho‐Diaz, and Hurd (2009) provided a closed‐form solution to the consumption‐portfolio

selection problem considering both the Brownian risk and jump risk. Li and Zhou (2018) solved the optimal dynamic

portfolio problem under the double exponential jump diffusion distribution and examined how asymmetric jumps

affect the optimal dynamic asset allocation. Zhou, Wu, and Wang (2019) investigated dynamic portfolio problem with

time‐varying jump risks.

In this paper, we allow the jump intensity to be time‐varying when modeling financial returns. We contribute to the

literature by investigating whether incorporating time‐varying jump intensities in the return dynamics is useful to

improve portfolio performance under mean‐variance framework. To model the time‐varying jump intensities in the

risky asset returns, we follow Chan and Maheu (2002) and Maheu and McCurdy (2004) to specify the jump intensity to

follow an autoregressive process. That is, the jump intensity is a linear function of the lagged jump intensity, which can

capture the jump clustering effect, and an innovation term, which is the difference between the ex post filter of jump

counts and the lagged jump intensity. A number of studies relied on the autoregressive jump intensity (ARJI) process to

model the jump dynamics, and found that incorporating time‐varying jump intensities are important for volatilities

forecast (Chan & Maheu, 2002; Maheu & McCurdy 2004), risk measurement (Chang, Su, & Chiu, 2011; Nyberg &

Wilhelmsson, 2009; Su, 2014; Su & Hung, 2011), market equity risk premium (Maheu, McCurdy, & Zhao, 2013), and

dynamic portfolio allocation (Zhou et al., 2019).

Following Fleming, Kirby, and Ostdiek (2001) and Chou and Liu (2010), we conduct our analysis using futures

contracts to minimize the trading costs and avoid the short‐selling constraints. Two U.S. dollar‐denominated futures on

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See, for example, Eraker (2004), Barndorff‐Nielsen and Shephard (2006), S. Lee and Mykland (2008), and Aït‐Sahalia and Cacho‐Diaz (2015), among others.

the S&P 500 index and NIKKEI 225 index traded in Chicago mercantile exchange (CME) are considered in our

analysis.

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We capture the time‐varying jump intensities in two risky asset returns using the bivariate Baba‐Engle‐Kraft‐

Kroner with autoregressive jump intensity (BAJI) model. Like Chan (2004) and M.‐C. Lee and Cheng (2007), we allow

different assets prices to jump simultaneously or independently, where the simultaneous jumps capture the time‐

varying systematic jump risks while the independent jumps capture the time‐varying idiosyncratic jump risks. The

estimation results show that the jump intensities are time‐varying and persistent. The individual jump intensities of

S&P 500 index future are high during the early 2000s recession in the United States, and the individual jump intensities

of NIKKEI 225 index future are large during the 1990s recession in Japan. The individual jump intensities of the two

futures are relatively low during the 2008 subprime crisis in the United States, but their common jump intensities are

large, indicating that the U.S. subprime crisis has major influence on both the S&P 500 index futures and the NIKKEI

225 index futures.

We solve the portfolio optimization problem based on the mean‐variance framework of Markowitz (1952). That is,

the investors aim to minimize the portfolio variance for a given level of target expected return. To examine whether it is

important to consider time‐varying jump intensities in mean‐variance portfolio decisions, we describe the dynamics of

asset returns based on the BAJI model. We consider three alternative models, including the static model in which the

covariance matrix is calculated using the in‐sample covariance matrix estimate, the Baba‐Engle‐Kraft‐Kroner (BEKK)

model (Engle & Kroner, 1995) and Baba‐Engle‐Kraft‐Kroner with constant jump intensity (BCJI) model.

The in‐sample and out‐of‐sample results show that the BAJI model which incorporates time‐varying jump intensities

performs better, and produces higher Sharpe ratios than the other three models. The Sharpe ratios of BAJI model

remain highest after considering the trading cost. Based on the bootstrapping method, we find that the superiority of

BAJI model over the other three models is robust to the parameter uncertainties of expected returns, and increases

when the uncertainty level decreases.

Our paper is related to the previous literature regarding volatility timing. For instance, Fleming et al. (2001),

Fleminga, Kirby, and Ostdiek (2003), Chou and Liu (2010), and Nolte and Xu (2015) used different methods to model

the time‐varying volatilities. Under the mean‐variance framework, they showed that considering time‐varying

volatilities helps to improve the mean‐variance portfolio performance. Kirby and Ostdiek (2012) and Moreira and Muir

(2017) showed that strategies that adjust the portfolio weights according to time‐varying volatilities can yield better out‐

of‐sample performances.

Similar to Fleming et al. (2001) and Chou and Liu (2010), our paper is also based on the mean‐variance framework.

Different from the previous studies, in this paper we incorporate the time‐varying jump intensities when estimating the

covariance matrix. The covariance matrix of risky asset returns has two components: the time‐varying covariance

matrix of normal shocks and the time‐varying covariance matrix of discrete jump shocks. The empirical results show

that the model considering the time‐varying jump intensities better fits the dynamics of futures returns and improves

the Sharpe ratios of mean‐variance portfolios. Our research suggests that using a better econometric model that

captures non‐normal features in the data has benefits for portfolio allocation even for a mean‐variance investor.

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The paper is organized as follows. Section 2 introduces the methodologies of portfolio formation under the mean‐

variance framework, performance evaluations, and the BAJI model. Section 3 provides the empirical performance

comparisons between the BAJI model and three alternative models, including the static model, the BEKK model, and

BCJI model. Finally, Section 4 concludes the paper.

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OPTIMAL PORTFOLIO ALLOCATION IN A MEAN‐VARIANCE

FRAMEWORK

2.1

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The mean‐variance portfolio problem

Let

R

tdenote an n×

1

column vector of risky asset returns from

t

−

1

to

t

. The expected return and conditional

covariance matrix of

R

tare given by μER=(

)

tand ER μRμ

Σ

=Σ=[(−)( −)′|]

ttt t t t|−1−1

d, respectively, where

t−

1

d

2

The NIKKEI 225 index is designed to reflect the overall Japanese stock market and is calculated based on the common stock prices of 225 largest Japanese companies traded in the Tokyo Stock

Exchange. The futures on NIKKEI 225 index began trading in CME on September 25, 1990. Compared with equities trades, futures trades have very low commissions and have no short‐selling

constraints.

3

We thank the anonymous referee for pointing out the main idea of our paper.

ZHOU ET AL.

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