DESIGNING A PROPER HEDGE: THEORY VERSUS PRACTICE

• DOI: http://doi.org/10.1111/jfir.12091
• Author: Chao Jiang,Paul D. Koch,Ira G. Kawaller
• Published date: 01 June 2016
• Date: 01 June 2016

DESIGNING A PROPER HEDGE: THEORY VERSUS PRACTICE

Chao Jiang

University of South Carolina

Ira G. Kawaller

Kawaller & Co.

Paul D. Koch

University of Kansas

Abstract

Determining the hedge ratio based on the slope coefficient of a regression on price

changes suffers from several critical shortcomings. First, it is difficult to assemble a

properly constructed data set. Second, results vary depending on the length of the change

interval. Third, the resulting ex post effective prices realized under this approach are

wholly uncertain, ex ante. We show that when the hedge ratio is determined with

reference to a regression on the respective price levels, rather than price changes, the

resulting hedge ratio solution is superior in that none of these shortcomings apply.

JEL Classification: G18, G32, G38

I. Introduction

The use of derivative instruments for hedging purposes is straightforward. For any

undesired exposure, find a closely related derivative instrument (e.g., a futures contract

that relies on the same or a similar underlying price) and then enter into a derivative

position as an overlay to the exposure being hedged. Once an acceptable derivative

instrument is identified, the next critical concern relates to sizing the hedge—that is, in

the parlance of the marketplace, determining the proper hedge ratio.

One well-established approach for setting the optimal hedge ratio that has

attracted much attention in the academic literature calls for minimizing risk, defined as

the variance of changes in the value of the hedged portfolio, inclusive of the hedged item

and the hedging derivative. Henceforth, we refer to this orientation as the “traditional

academic approach”to sizing a hedge. To minimize variance, this traditional approach

prescribes regressing changes in the cash price of the hedged item on changes in the

forward price of the related derivative, using either a simple regression or an error

correction model (ECM).

1

In both specifications, the length of the interval used to

We thank Scott Hein (editor) and an anonymous associate editor and referee for their helpful comments.

1

The cash price refers to the invoice amount paid or received for a physical purchase or sale for immediate

delivery. The ECM appends the simple regression on price changes to include a lagged error correction term that

accounts for the link between cash and futures price changes in a cointegrating relation. Juhl, Kawaller, and Koch

(2012) prove that if the two prices are cointegrated, these two specifications involving price changes should

produce the same estimate of the hedge ratio when the interval for measuring price changes is lengthened.

The Journal of Financial Research Vol. XXXIX, No. 2 Pages 123–144 Summer 2016

123

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measure price changes should reflect the length of the intended hedge horizon. Under

either specification, the resulting slope coefficient is the optimal hedge ratio that

minimizes the regression sum of squared errors (SSE; which conforms to this definition

of risk) and thus maximizes the R

2

(which often serves as a measure of the effectiveness

of this hedge).

2

We challenge the universal relevance and practicality of this traditional

academic approach for determining the optimal hedge ratio that minimizes variance. We

offer an alternative approach that strives to satisfy a different objective. In practice, rather

than seeking to minimize variance, hedgers often use futures, forwards, and swaps to lock

in prices for forthcoming transactions, subject to uncertainty about the basis (i.e., the

difference between cash prices of hedged items and the futures/forward prices implicit in

hedging derivatives). This objective, however, cannot generally be expected to be

realized with hedge ratios determined by a regression that relies on price changes.

Instead, we establish that the price-fixing objective requires estimating a simple linear

relation between the price levels associated with the exposure being hedged and the asset

underlying the derivative used as a hedge. The slope coefficient from this regression on

price levels then serves as the foundation for determining the proper hedge ratio to lock in

a price.

We illustrate this alternative approach and result by offering a case study

involving a cross hedge, in which the hedged item is not identical to the asset underlying

the hedging derivative. Our case study relates to the use of heating oil futures to hedge the

risk associated with a forthcoming purchase or sale of fuel oil. This case demonstrates

how a linear regression between the cash price level of the hedged item and the

underlying price level of the hedging derivative can be used to obtain the hedge ratio that

locks in a price for a forthcoming transaction, ex ante.

II. The Traditional Academic Approach for Determining the

Optimal Hedge Ratio

The variables are defined as follows:

Y

t

¼dependent variable (price of exposure) at time t;

X

t

¼independent variable (price of hedging derivative) at time t;

2

There are several alternative well-known hedging models to determine the proper hedge ratio, which also

attempt to minimize some form of risk. For example, alternative models include the price-sensitivity model, the

na€

ıve-hedge model, and the utility-based hedging model (see Johnson 2009, p. 321). There are also several forms of

the minimum-variance hedging model in the literature. Still, we note that numerous academic articles and

textbooks on risk management develop and build on what we call the traditional academic minimum-variance

approach, which relies on a simple regression on price changes to determine the proper hedge ratio. For example,

see Benet (1992), Charnes, Berkman, and Koch (2003), Chance and Brooks (2013, p. 373), Chou, Fan Denis, and

Lee (1996), Ederington (1979), Geppert (1995), Hill and Schneeweis (1981, 1982), Howard and D’Antonio(1991),

Hull (2014, pp. 59–61), Jarrow and Chatterjea (2013, pp. 320–324), Kawaller and Koch (2000, 2013), Lien (2000),

Lien and Luo (1993), and Stulz (2003, p. 168).

124 The Journal of Financial Research