Swap variance hedging and efficiency: The role of high moments

• Published date: 01 September 2023
• Author: K. Victor Chow,Bingxin Li,Zhan Wang
• Date: 01 September 2023
• DOI: http://doi.org/10.1111/jfir.12328

Received: 28 February 2022

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Accepted: 6 April 2023

DOI: 10.1111/jfir.12328

ORIGINAL ARTICLE

Swap variance hedging and efficiency:

The role of high moments

K. Victor Chow

1

|Bingxin Li

2

|Zhan Wang

3

1

Department of Finance, West Virginia

University, Morgantown, West Virginia, USA

2

Department of Finance and Center for

Innovation in Gas Research and Utilization

(CIGRU), West Virginia University,

Morgantown, West Virginia, USA

3

Department of Finance, Shanghai Business

School, Shanghai, China

Correspondence

Zhan Wang, Department of Finance, Shanghai

Business School, Shanghai, China.

Email: zhanwang@sbs.edu.cn

Abstract

In this article, we propose a new theoretical approach for

developing hedging strategies based on swap variance

(SwV). SwV is a generalized risk measure equivalent to

a polynomial combination of all moments of a return

distribution. Using the S&P 500 index and West Texas

Intermediate (WTI) crude oil spot and futures price data, as

well as simulations by varying the distribution of asset

returns, we investigate the dynamic differences between

hedge ratios and portfolio performances based on SwV

(with high moments) and variance (without high moments).

We find that, on average, the minimizing‐SwV hedging

suggests more short futures contracts than minimizing‐

variance hedging; however, when market conditions

deteriorate, the minimizing‐SwV hedging suggests fewer

short positions in futures. The superior posthedge per-

formances of the mean‐SwV hedged portfolios over the

mean‐variance hedged portfolios in highly volatile or

extremely calm markets confirm the efficiency of the

mean‐SwV hedging strategy.

JEL CLASSIFICATION

G01, G11, G12, G13

J Financ Res. 2023;46:681–709. wileyonlinelibrary.com/journal/JFIR

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© 2023 The Southern Finance Association and the Southwestern Finance Association.

1|INTRODUCTION

A fundamental question in financial risk management is how to measure risk accurately. From the most commonly

used risk measure, variance, to a certain moment of a return distribution (skewness or kurtosis), to measurements

that focus on the downside of a return distribution, such as value‐at‐risk (VaR) and extended‐Gini, researchers and

practitioners have applied a variety of measures to account for the risks in the financial markets. These measures,

however, are either based on a quadratic utility function (jointly symmetric return distribution) that ignores

asymmetric higher moments or only on partial aspects of the return distribution and are not always consistent with

the framework of the utility function.

High moments of a return distribution have proved to be critical in asset pricing and financial risk management

(see, e.g., Bakshi et al., 2003; Harvey & Siddique, 2000; Peiro, 1999; Sears & Wei, 1985). Motivated by the literature

on the variance swap replication strategy (e.g., Neuberger, 1994), Jiang and Oomen (2008) propose a jump‐sensitive

risk measure, called swap variance (SwV). SwV combines variance and all higher orders of return moments and is a

powerful statistic in detecting distributional jumps. Chow, Sopranzetti, et al. (2020) build a theoretical framework of

portfolio theory and formally redefine SwV, which is equivalent to a polynomial combination of all return moments,

as a generalized and more complete measure of risk and uncertainty. This new risk measure, SwV, can be calculated

as twice the expected difference between arithmetic and logarithmic returns and therefore is ready to be used for

financial risk management and portfolio optimization.

However, discussions regarding whether SwV is a better risk measure and its possible applications are scarce in

theoretical and empirical studies. In this article, we investigate the properties of SwV as a risk measure and apply it

to hedging frameworks. By investigating the dynamics of hedge ratios and their subsequent performance, we

compare the SwV framework with traditional hedging strategies using variance to measure risks and shed light on

investors’dynamic hedging choices and hedging efficiency.

Hedging is one of the essential functions of futures contracts, in which investors form portfolios using futures

contracts and their underlying assets to reduce the risk of future price fluctuations. Previous studies on hedging

focus on the mechanism of hedge ratios and hedging efficiency to minimize investors’risks. Several objective

functions have been studied in the derivation of the optimal hedge ratio. The principal method is from Johnson

(1960), who builds an optimal hedging model that minimizes the variance of the hedged portfolio and demonstrates

that hedging through trading futures contracts reduces risks of adverse price movements. Ederington (1979) and

Myers and Thompson (1989) develop similar minimizing variance hedging methods using simple regression

approaches.

Although hedging aims to minimize associated risk, one cannot overlook return maximization. The minimizing‐

variance hedging strategy is generally inconsistent with mean‐variance efficiency unless the individuals are infinitely

risk averse (the risk aversion coefficient is so significant that expected returns are negligible) or the futures price

follows a pure martingale process. Literature that tries to improve the overall portfolio performance incorporates

both expected return and variance of the portfolio and maximizes expected utility functions (Cecchetti et al., 1988;

Howard & D'Antonio, 1984; Hsin et al., 1994). By considering both returns and risks of the portfolio, these methods

provide a better evaluation of portfolio performance. However, the risk measures used in these methods face the

same weakness as the minimizing‐variance framework, which is based on a quadratic utility function or jointly

symmetric return distribution.

Another stream of literature considers other risk measures to eliminate these strict assumptions regarding

utility functions and return distributions. For example, Cheung et al. (1990), Kolb and Okunev (1992,1993), Lien

and Luo (1993), Shalit (1995), and Lien and Shaffer (1999) apply methods of minimizing the mean‐extended‐Gini

coefficient, which is consistent with the concept of stochastic dominance and in the framework of expected utility

maximization. Crum et al. (1981), De Jong et al. (1997), Lien and Tse (1998,2000), and Chen et al. (2003) compute

hedge ratios based on generalized semi‐variance or lower partial moments minimization. They demonstrate that

these hedge ratios have another attractive feature consistent with the risk perceived by managers who care more

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