Nonparametric Kernel Method to Hedge Downside Risk

• DOI: http://doi.org/10.1111/irfi.12257
• Author: Jinbo Huang,Yong Li,Ashley Ding
• Published date: 01 December 2019
• Date: 01 December 2019

Nonparametric Kernel Method to

Hedge Downside Risk*

JINBO HUANG

†

,ASHLEY DING

‡

AND YONG LI

§

†

School of Finance, Collaborative Innovation Development Center of Pearl River

Delta Science &Technology Finance Industry, Guangdong University of Finance and

Economics, Guangzhou, China

‡

Centre for Corporate Sustainability and Environmental Finance, Macquarie

University, Sydney, Australia and

§

School of Banking and Finance, University of International Business and Economics,

Beijing, China

ABSTRACT

We propose a nonparametric kernel estimation method (KEM) that deter-

mines the optimal hedge ratio by minimizing the downside risk of a hedged

portfolio, measured by conditional value-at-risk (CVaR). We also demonstrate

that the KEM minimum-CVaR hedge model is a convex optimization. The

simulation results show that our KEM provides more accurate estimations

and the empirical results suggest that, compared to other conventional

methods, our KEM yields higher effectiveness in hedging the downside risk

in the weather-sensitive markets.

JEL Codes: G11; G23; C14; C15; C61

Accepted: 31 January 2019

I. INTRODUCTION

Risk minimizing hedging strategies aim to construct a spot-future portfolio with

minimal downside risk. In this paper, we use conditional value-at-risk (CVaR)

to measure the downside risk, which is a coherent risk measure and overcomes

the shortcoming of value-at-risk (VaR) (Acerbi and Tasche 2002). We adopt the

nonparametric KEM approach to propose new CVaR hedge models. Unlike the

* This paper has benefited from the comments and suggestions received from Tom Smith and Mar-

tina Linnenluecke. This paper was supported by the National Natural Science Foundation of China

(Nos. 71603058, 71721001, 71273066, 71573056); the Social Science Research Foundation of the

National Ministry of Education of China (No. 16YJC790033); the Natural Science Foundation of

Guangdong Province of China (No. 2016A030313656); the Philosophy and Social Science Program-

ming Foundation of Guangdong Province of China (No. GD15XYJ03); Ordinary University Innova-

tion Team Project of Guangdong Province of China (No. 2017WCXTD004); and Guangdong

University of Finance & Economics Big Data and Educational Statistics Application Laboratory

(No. 2017WSYS001); the Fundamental Research Funds for the Central Universities in UIBE

(No. 18YB11); Accounting and Finance Association of Australia and New Zealand Grant

(2016-2017).

© 2019 International Review of Finance Ltd. 2019

International Review of Finance, 19:4, 2019: pp. 929–944

DOI: 10.1111/irfi.12257

previous literature that presumes the distribution of asset returns

(e.g., Alexander and Baptista 2004; Harris and Shen 2006), such approach does

not require prior information on distributions and estimators are driven by

market data (Li and Racine 2007). Another problem with the conventional

methods such as the classic empirical distribution method (EDM) (Cao et al.

2010) is that there is no guarantee for their hedging models a convex optimiza-

tion considering the difficulty of obtaining the first- and second-order condi-

tions of hedging models. We contribute to demonstrate KEM CVaR-minimizing

hedge model is a convex optimization and there exists a global optimal hedge

ratio. Given that kernel functions are smooth in contrast to empirical distribu-

tion functions, the kernel estimator of CVaR is smooth and differentiable func-

tions of hedge ratio. Therefore, various optimization methods can be applied to

obtain KEM hedge ratios. In addition, compared with the semi-parametric

Cornish–Fisher method (CFM) (Harris and Shen 2006), KEM can perform large

sample properties (Cai and Wang 2008). Monte Carlo simulations show KEM

outperforms EDM and CFM, especially when distribution is asymmetric with

heavy tails. Results from the agricultural and energy commodities and futures

support the efficiency of our KEM approach in hedging the downside risk in

these markets.

II. METHODOLOGY

A. Settings

We assume that spot and future assets returns are r

1

and r

2

, respectively, where

the mean and standard deviation of r

1

are μ

1

and σ

1

; the mean and standard

deviation of r

2

are μ

2

and σ

2

. We aim to find an optimal hedge ratio h, such that

a hedge portfolio (a long spot and hshort futures) has minimum risk. Let this

hedge portfolio’s return r

p

be

rp=r1−hr2ð1Þ

Then the expected return of this hedge portfolio is μ

p

(h)=μ

1

−hμ

2

, and vari-

ance is σ2

phðÞ=σ2

1+h2σ2

2−2hσ12, where σ

12

is the covariance between spot and

futures returns.

Let probability density function of r

p

be f(x,h) that depends on the hedge

ratio h. We do not impose any assumption on the functional form of f(x,h).

Instead, we use sample data to estimate them. r1,t



T

t=1 and r2,t



T

t=1 are sample

data of spot and futures returns and rp,t



T

t=1 is the hedge portfolio return’s

sample data, where r

p,t

=r

1,t

−hr

2,t

. Then let ^σ2

phðÞ=^σ2

1+h2^σ2

2−2h^σ12 be the

sample variance of the hedge portfolio return, where ^σ2

1,^σ2

2are the sample vari-

ances of spot and future returns and ^σ12 is the sample covariance.

© 2019 International Review of Finance Ltd. 2019930

International Review of Finance