the impact of skewness in the hedging decision

THE IMPACT OF SKEWNESS

IN THE HEDGING DECISION

SCOTT GILBERTSAMUEL KYLE JONES*GAY HATFIELD MORRIS

The impact of skewness in the hedger’s objective function is tested using amodel of hedging derived from a third-order Taylor Series approximationof expected utility. To determine the effect of price skewness upon hedgingand speculation, analytical results are derived using an example of cottonstorage. Findings suggest that when forward risk premiums and priceskewness in the spot asset have opposite signs, speculation increases rela-tive to the mean-variance model. When the signs are identical, speculationwill decrease, contradicting findings of mean-variance models. © 2006Wiley Periodicals, Inc. Jrl Fut Mark 26:503–520, 2006

Professor Morris contributed to this paper while at the University of Mississippi as the AssistantDean of the School of Business and Associate Professor of Finance.The authors thank an anonymous reviewer for helpful comments.*Correspondence author, Department of Economics and Finance, Stephen F. Austin StateUniversity, Box 13009—SFA Station, Nacogdoches, Texas 75962; e-mail: [email protected]

Received November 2004; Accepted June 2005

� Scott Gilbert is an Associate Professor of Economics in the Department of Economics atthe Southern Illinois University Carbondale in Carbondale, Illinois.

� Samuel Kyle Jones is an Assistant Professor of Finance in the Department of Economicsand Finance at Stephen F. Austin State University in Nacogdoches, Texas.

� Gay Hatfield Morris is an Associate Professor of Finance at Al Akhawayn University inIfrane, Morocco.

The Journal of Futures Markets, Vol. 26, No. 5, 503–520 (2006)© 2006 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com).DOI: 10.1002/fut.20201

504 Gilbert, Jones, and Morris

Journal of Futures Markets DOI: 10.1002/fut

1The issue of non-normal distributions for asset prices has been investigated in the asset pricing lit-erature with the development of models incorporating skewness. Three-moment portfolio modelshave been developed by Kraus and Litzenberger (1976), Simkowitz and Beedles (1978), Conine andTamarkin (1981), Kane (1982), and Sears and Trennepohl (1983). Arditti (1967) has shown thatinvestors prefer positive skewness. Further, assets with positive skewness have low returns thatwould be suboptimal in a two-moment model. Beedles (1979) shows that portfolio skewness isinversely related to the size of a portfolio. Conine and Tamarkin (1981) demonstrate that, for vari-ous combinations of specific utility functions and levels of risk aversion, it can be shown that aninvestor might choose to hold a nondiversified portfolio for the sake of retaining positive skewness.Skewness has been shown to decrease as portfolio size increases (Beedles, 1979; Simkowitz &Beedles, 1978). Further, these studies show that it is rational behavior for investors to reduce port-folio size to preserve positive skewness. This makes skewness a matter of concern for nondiversifiedor poorly diversified investors.

INTRODUCTION

Minimum variance hedging considers the use of forward and futuresmarkets solely as a means for managing financial risk. However, the der-ivation of minimum variance hedge ratios implicitly assumes thathedgers are infinitely risk-averse. For less risk-averse firms, optimalhedge positions can be selected with a risk-return trade-off to maximizeexpected utility. Hedge ratios computed from utility-based models areaffected by the distribution of all random variables common to themodel. A standard assumption applied to utility-based hedging models isthat these random variables are normally distributed, allowing the prob-lem to be specified solely in terms of mean and variance. While this sim-plifies the model, mounting evidence suggests that many randomprocesses are non-normal and often asymmetric.

Until recently, the literature on hedging has virtually ignored theissue of skewness. On the other hand, portfolio theory has producedsome interesting theoretical and empirical results. Research in this areahas shown that positive skewness is desirable to risk-averse investorsholding long positions in securities. That is, investors holding an assetlong would be willing to trade expected return for positive skewness—everything else being equal. Additionally, skewness diminishes rapidlywith portfolio size (Simkowitz & Beedles, 1978). In related research,Tsiang (1972) shows that when the proportion of wealth at risk is rela-tively small standard mean-variance analysis is adequate, but that whenthe proportion of wealth put at risk becomes large it is necessary toconsider higher order moments such as skewness.1

An important implication from this literature is that skewnessshould become a relevant concern for poorly diversified investors. Manyfarm operations are nondiversified, with farmers holding the bulk oftheir wealth in their business. Several recent studies have provided both

Skewness in the Hedging Decision 505


direct and indirect evidence that farmers prefer skewness in spot com-modity prices. Using survey data, Patrick, Musser, and Eckman (1998)show that hedging activity is less than that predicted by mean-variancemodels. Instead, they show that farmers tend to treat forward markets asa means of obtaining a price more favorable than the expected spot priceat harvest. That is, rather than using forward contracts for hedging,farmers use them for speculative purposes. Goodwin and Schroeder(1994) provide a similar result. Kenyon (2001) provides reasoning forthis behavior by showing that farmers tend to generate positive skewedforecasts concerning expected spot prices at harvest, overestimating theactual harvest time spot price. Roe, Sporleder, and Belleville (2004)provide further anecdotal evidence of preference for price skewness.They analyze the preferences of farmers participating in cooperatives,showing a much greater preference for increases in price ceilings thanfor increases in price floors, which is suggestive of a preference forupside gain or positive skewness in spot prices. Direct evidence of priceskewness preference is provided by Groom, Koundouri, Nauges, andThomas (2002), finding that the variance and skewness of farm profitcan influence the behavior of farmers. They show that farmers are will-ing to pay, in the form of lower profits, a premium to lower risk and toincrease skewness. Because of this literature, it is of practical concern toask how skewness might affect the optimal decision making of suchfarmers.

The purpose of this study is to determine the theoretical impact thatunconditional skewness in spot prices has on hedge decisions. To allowfor asymmetric price risk the hedger’s objective function is developed asa Taylor’s series approximation of direct expected utility. To gauge theimpact of skewness upon hedging decisions, the model is applied to theexample of cotton storage and hedging. Numerical solutions show thatwhen forward prices provide unbiased estimates of expected spot prices,the distribution of random spot prices will not impact the hedging deci-sion. When forward prices are biased away from expected spot prices,the firm will forward contract an amount of the spot asset different fromthe quantity held in storage. This action gives rise to speculative behav-ior, defined herein as hedging any amount different from the quantityheld in storage. The results show that when the risk premium on the for-ward contract and skewness in spot prices have opposite signs, skewnessincreases the amount of speculation. When the forward risk premiumand skewness on the spot asset have the same sign, an increase in skew-ness in spot prices reduces speculation.



2Extensions of this work show the conditions under which the separation property does not hold.Grant (1985) adds production risk, while Antonovitz and Nelson (1988), Heaney and Poitras(1991), and Lapan, Moschini, and Hanson (1991) hedge with futures, thereby introducing basisrisk. These studies show that the separation of production and hedging decisions that results withspot price risk and forward contracting will not hold when production or basis risk is also present.3Empirical testing for skewness in price distributions is extensive. A small sample of this workincludes Bodie and Rosansky (1980), Helms and Martell (1985), and Junkus (1991), who provideevidence of skewness in the prices of agricultural futures. Yang and Brorsen (1992) show that spotprices of agricultural commodities also have asymmetric distributions.

RELATED LITERATURE

An extensive body of literature is focused on the theoretical determina-tion of the optimal joint production and hedging decisions for a risk-averse, utility maximizing investor in a single-period model. Baron(1970) and Sandmo (1971) show that the optimal level of output in thepresence of spot price risk will fall below the level that would be optimalif future spot prices were known with certainty. Danthine (1978),Holthausen (1979), and Feder, Just, and Schmitz (1980) show that theuse of forward contracting eliminates spot price risk, making the outputdecision independent of the price distribution and the firm’s degree ofrisk aversion.2 Output is increased until the marginal cost of output isequal to the known forward price. In contrast, the distribution of therandom spot prices and the firm’s level of risk aversion will affect thehedging decision. With spot price risk and forward contracting, the opti-mization problem is similar to that of portfolio selection models. Byselection of the optimal hedge ratio, the firm determines the position inthe risk-free asset as given by the quantity of the commodity sold forwardat the certain forward price, and the position in the risky asset as givenby the quantity of the commodity left unhedged.

When the forward price provides an unbiased estimate of theexpected spot price, the firm will fully hedge output because expectedprofits from speculation will be zero. When the forward price is a biasedestimate of the expected spot price, expected profits provide the incentiveto speculate in forward markets. Consequently, the firm will choose tohedge an amount of the commodity different from the amount produced.

Empirical and analytical applications of expected utility modelshave been largely limited to the mean-variance case by incorporating theassumption that spot prices are normally distributed. Not only does evi-dence exist to the contrary, but theory does not justify the strict use ofmean-variance analysis when used in practical applications.3 Accordingto Samuelson (1970) when asset prices follow a diffusion process suchthat portfolios are rebalanced continuously, mean-variance analysisprovides an adequate approximation of expected utility. As a practical



4According to Arrow (1970), these properties are (a) positive marginal utility, (b) risk aversion,(c) non-positive absolute risk aversion, and (d) non-negative relative risk aversion. Further, Arditti(1967) shows that non-increasing absolute risk aversion requires that be satisfied. Thisimplies an investor preference for positive skewness and aversion to negative skewness.5According to Tsiang (1972) utility functions of the hyperbolic, absolute risk aversion (HARA) classare acceptable. These include negative exponential utility, logarithmic utility, and power utility.

u‡ � 0

matter, continuous rebalancing is not possible. As the interval overwhich portfolio decisions are made is increased, mean-variance approxi-mations can lose accuracy, which may be improved upon by the additionof statistical moments of higher order than variance.

Quantifying the decision-making process in terms of mean, vari-ance, and skewness can be accomplished by several different approaches.Focusing on the distribution of asset returns, Beedles (1979) shows thatwhen this distribution is known, stochastic dominance can be employedto select among alternative decisions. If only the moments of the distri-bution are known, then a form of utility specified by the first threemoments of the distribution is required. According to Levy (1969),investor preferences are defined exactly in terms of the first threemoments only in the case of cubic utility. However, Tsiang (1972) showsthat the use of any polynomial utility function to represent investorbehavior is unacceptable, as polynomials cannot simultaneously displayall properties considered desirable for utility functions.4 These propertiescan only be satisfied with certain non-polynomial utility functions.5 As analternative, Tsiang (1972) proposes approximating utility in terms ofmean, variance, and skewness by using a Taylor’s series expansion of anonpolynomial utility function of the HARA class.

The method proposed by Tsiang (1972) has been used in portfolioselection models that have explicitly accounted for skewness. Kraus andLitzenberger (1976), Conine and Tamarkin (1981), Kane (1982), andSears and Trennepohl (1983) have used third-order Taylor’s seriesapproximations of HARA utility functions to show the impact of skew-ness in asset pricing as well as the antidiversification incentive thatpositive skewness imparts on portfolio decisions.

Extensions of these results to the hedging literature have been sparse.Numerical analysis of optimal hedging by Karp (1987) and Martinez andZering (1992) is couched in terms of mean-variance decision making.However, they propose that skewness will become important in the deci-sion making process as risk aversion becomes large, and that the impact ofskewness should increase as the level of investor risk aversion increases.They use this assumption to cap the upper limit of risk aversion in theirmodel to avoid introducing skewness into the analysis.



Poitras (1993) explicitly models the objective function for a risk-averse, utility maximizing hedger as the third-order Taylor’s seriesapproximation to expected utility. It is shown that when skewness isincluded in the model and there are multiple-choice variables, closed-form solutions are not possible for the optimal hedge decision. Poitras(1993) proposes the use of numerical simulations as well as empiricalanalysis of this model to assess the impact of skewness on hedging, butleaves such analysis open as an avenue for future research.

MODEL

In this section a partial equilibrium model of hedging price risk is pre-sented defined in terms of the first three central moments of the distri-bution of random spot prices. The model assumes that a risk-averse,competitive firm faces an uncertain future spot price. Using a static,two-period setting decisions are made at the beginning of the period soas to maximize the expected utility of end-of-period profit, . The firmmust choose some quantity, q, of a single output to store with randomend-of-period price, p1. The cost of storing this output is given by thecost function . Price risk can be hedged by selling for-ward an amount, h, of the stored commodity for end-of-period delivery atthe certain forward price, b.

The risk management problem is summarized by the followingobjective function, where the firm seeks to maximize expected utility ofend-of-period profit:

(1)

subject to the following constraint on end-of-period profit:

(2)

First-order conditions are derived by maximizing expected utility ofend-of-period profit with respect to the quantity stored, q, and the quan-tity sold forward, h:

(3)

(4)

Adding Equations (3) and (4) gives the following result:

(5)

For a risk-averse firm, marginal utility is positive, so it must holdthat As Holthausen (1979) shows, the optimal quantity ofb � c�(q*).

(b � c�)E[u�(p~1)] � 0

E[u�(p~1)(b � p~1)] � 0

E[u�(p~1)(p~1 � c� )] � 0

p~1 � p~1(q � h) � bh � c(q)

maxq,h

E[u(p~1)]

c(q)(c�, c– � 0)

p~1



spot asset to put into storage is set to the level at which marginal storagecost is equal to the forward contract price. With a deterministic costfunction and access to a forward market, production decisions are keyedto the known forward price and are made as if spot prices were knownwith certainty. The firm’s level of risk aversion as well as the distributionof spot prices has no influence on the optimal level of storage.

In contrast to the production decision, the optimal level of forwardcontracting can be a function of both risk aversion and the distributionof end-of-period spot price. Holthausen (1979) shows that if theexpected spot price is equal to the known forward price, then the firmwill fully hedge the spot asset such that In this case, the hedgedecision is independent of risk aversion and the spot price distribution.Only when the expected spot price deviates from the forward price willthe firm speculate by hedging a quantity different from the amountstored. That is, the distribution of random spot prices can only affect thehedging decision when forward prices are biased estimates of expectedspot prices.

As a result, the optimal quantity of storage to forward contract,must be solved for when there exists a bias between forward and expectedspot prices. However, because the quantity stored is a function of deter-ministic parameters only, the optimization problem can be simplified tothat of an optimization in terms of the single choice variable, :

(6)

While optimization of direct expected utility produces optimalhedge ratios, it assumes that the hedging firm knows the exact distribu-tion of spot prices. In practice, the true distribution may not be knownand only estimates of the first several moments of the distribution may beavailable. Further, use of direct expected utility does not make explicitthe effects of skewness from that of higher-order distributional momentsupon optimal hedge ratios. To gain some quantitative guidance as to theeffects of skewness on hedging decisions, it is necessary to formulateexplicitly the objective function in terms of skewness.

Following Tsiang (1972), utility of end-of-period profit is expandedaround the expected value of end-of-period profit by means of a third-order Taylor series expansion:

(7)�13!

u‡(p1)(p1i �p1)3

u(p~1) � u(p1) � u�(p1)(p1i � p1) �12!

u–(p1)(p1i � p1)2

maxh

E[u(p~1)]

h*

h*,

h*� q*.



where is the ith derivative of the utility of end-of-period profit.Taking expectations on both sides of Equation (7), expected utility is

approximated by the derivatives of the utility function and the centralmoments of the distribution of spot prices.

(8)

To derive analytical solutions it is necessary to specify the exact formof the nonpolynomial utility function for the expansion. Logarithmic util-ity and power utility have the preferable property of decreasing absoluterisk aversion, though both are problematic when the possibility of nega-tive profits exists. As an alternative, the model used here assumes that thefirm has preferences represented by a negative exponential utilityfunction:

(9)

with constant absolute risk aversion, k. Substituting Equations (2) and(9) into Equation (8) defines the objective function for an investor con-cerned with the mean, variance, and skewness of expected profit:

(10)

A closed-form solution to this third-order expected utility functionis given by the first derivative of Equation (10) with respect to the quan-tity sold forward, h. Upon simplifying, the following cubic equation isderived:

(11)

where X � (q � h). In general, there can be as many as three uniquevalues for X that can solve Equation (11), though one value will maxi-mize expected utility.

For the purpose of comparison, a mean-variance model of hedgingis used. Given assumptions that the firm’s preferences are represented bya negative exponential utility function and that the same firm believesrandom spot prices to be normally distributed, the following mean-variance objective function is derived:6

(12)maxh

E[u(p~1)] � mp �k2sp

2

X3 � 3 c 1k(b � p)

�s2

p

km3pdX2 �

6s2p

k2m3p(b � p)

X �6

k3m3p

� 0

maxh

E[u(p~1)] � �Exp(�kp1) e1 �k2(q � h)2s2

p

2�

k3(q � h)3m3p

6f

u(p~1) � �Exp(�kp~1)

E[u(p~1)] � u(p1) �12!

u–(p1)sp2 �

13!

u‡(p1)mp3

ui(p1)

6See Baron (1970).



7The economic usefulness of this study relies on investors exhibiting preference for skewness, andthat such preferences are reflected in asset prices. Empirical evidence of investor preference forskewness has been found in electricity markets (Bessembinder & Lemmon, 2002) and foreignexchange markets (Aggarwal, 1990). As such, Aggarwal’s (1990) model was applied to spot cottonprices to test for skewness preference. The coefficients in the model had signs consistent with thosepredicted by Aggarwal, with a skewness coefficient that was statistically significant at the 10% level.This finding suggests that spot cotton prices reflect skewness preference.8This figure assumes an average sized farm in the United States during 1987–1996 producing anaverage quantity of cotton. Data on farm size is obtained from the U.S. Department ofAgriculture/National Agricultural Statistics Service (USDA/NASS), Livestock and EconomicsBranch. Data on cotton production is obtained from the USDA/NASS, Crops Branch.9Spot cotton prices are end of month prices of strict low middling 11�16 cotton as reported by theUSDA’s Agricultural Marketing Service and supplied by the National Cotton Council of America.Spot prices are adjusted to reflect the cost of carrying the spot asset from the spot date to thefollowing May.

–

Substituting Equation (2) into Equation (12) gives the followingobjective function in terms of spot prices:

(13)

Differentiation of Equation (13) with respect to the choice variableh leads to the closed-form solution for optimal mean-variance hedging:

(14)

NUMERICAL EXAMPLE: COTTON STORAGEAND FORWARD CONTRACTING

The impact of skewness upon the hedging decision is measured by com-puting hedge values, h, from Equation (11) and comparing them tomean-variance hedge values computed from Equation (14). Comparisonof optimal three-moment and mean-variance hedges is done using anexample of cotton storage and forward contracting.7 In this example afirm has a fixed quantity, , of cotton at the end of harvest. This asset isheld in storage for the interval between production seasons, beginning inNovember and continuing until May. Price risk on the stored cotton canbe hedged by forward contracting at known forward price b with Maydelivery. Cotton not sold forward will receive the uncertain future spotprice in May. The firm holds an inventory of pounds ofcotton in storage at the end of harvest in November.8

Based on estimated November to May spot cotton prices from 1978to 2005, the mean and standard deviation are set to 0.63 $/lb and0.13 $/lb, respectively.9 Skewness in the price of spot cotton variesthroughout the year, with monthly estimates for normalized skewness

q* � 300,000

q*

hmv � q � c b � p1

ks2pd

maxh

E[u(p~1)] � p1(q � h) � bh � c[q] � ak2b (q � h)2sp

2



10In this example, skewness refers to normalized skewness (or the coefficient of skewness) �

11See Cromwell, Labys, and Terraza (1994, p. 19) for an explanation of these tests.12As pointed out by an anonymous referee, if skewness preference is valued by the marginal hedger,then it should be reflected in the relationship between forward and expected spot prices. A generalequilibrium framework could be used that would allow one to make the forward bias endogenous.This avenue for future research might offer some interesting results.

E[(p~ � p)3]�s3.

ranging from �0.54 to �0.68 $/lb.10 The monthly skewness estimatestend towards negative values from June to November, with maximum neg-ative spot price skewness occurring in August just prior to the beginningof the cotton harvest period. From November to June skewness is posi-tive, reaching its maximum value in March, which is approximately thebeginning of the planting season. For the November to May storage inter-val normalized skewness is estimated at �0.30 $/lb, which is significant atthe 1% level.11 The test for kurtosis for the November to May period is notsignificant at the 10% level. A Jarque-Bera test is performed as a generaltest of normality. The JB statistic is 20.372, which is significant at the 1%level. Thus, it seems reasonable that a hedging model applied to this datashould include skewness, while higher order moments (e.g., kurtosis) canbe safely omitted.

The forward price is exogenously specified as a function of theexpected spot price:

(15)

where b is the forward price bias.12 When the bias equals unity, the for-ward price provides an unbiased estimate of the expected future spotprice. For this case Holthausen (1979) shows that the optimal hedgewill always exactly equal the quantity of spot asset in storage. In thiscase, the distribution of spot prices will have no affect on hedge deci-sions. When the bias is less than unity, the forward price is less thanthe expected spot price and the firm will hedge less than the amount ofspot asset held in storage. If the forward price is sufficiently less thanthe expected spot price, then the firm will buy additional cotton in theforward market. This case corresponds to the Keynesian view of hedg-ing known as normal backwardation. Investors that hedge are willing togive up some anticipated profit on the spot asset to reduce their level ofprice exposure. As such, investors pay risk premiums to speculators asinducements to enter into the transaction. When the forward price isgreater than the expected spot price, a situation referred to as contango,Holthausen (1979) shows that the hedger will sell forward a quantity ofspot asset greater than the amount held in storage. This will result in a

b � E(p~1)b



13Financing cost is proxied by the 6-month T-Bill rate reported by the St. Louis Federal ReserveBank in the FRED II database. The monthly storage cost is taken from a prior study by Fama andFrench (1987) who estimated the total monthly storage and warehousing costs for cotton to be0.45%. The convenience yield is excluded since it is not directly observable.14Futures prices are end-of-month prices for the May contract on No. 2 Cotton futures trading onthe New York Cotton Exchange.15Hedge values were computed for values of the risk-aversion parameter outside of this range, butwere not reported. For values greater than 1.0, all hedges were approximately equal to the fullhedge. For values below 0.00001 the effect of skewness on the optimal hedge increased at a rapidlyincreasing rate, resulting in levels of speculation that could be considered unrealistic.

shortfall between the amount of the spot asset required for delivery inMay on the forward contract and the quantity held in storage. Theforward sale of cotton is justified by the investor’s expectation ofincreasing profits by purchasing the shortfall at the lower anticipatedspot price in May.

To estimate the bias in Equation (15) a specific functional form isrequired for the expected spot price. Forward and spot prices areassumed to follow a standard cost-of-carry relationship with deviationsresulting from the forward price bias:

Ft,T � Ste(r�u�y)(T�t)bt (16)

where Ft,T is the forward price at date t for delivery at T, St is the spotprice at date t, r is the monthly cost of financing the spot asset over thestorage period, u is the monthly storage cost, and y is the convenienceyield on the spot asset.13 Forward contract prices are proxied by theprices of the May futures contract.14 The forward price bias, bt, is esti-mated from Equation (16) for each month, with estimated values rang-ing from 0.947 to 1.074. Consequently, optimal two and three-momenthedge values are reported for a forward price bias of 5% and 7.5% aboveand below the expected spot price. The monthly estimates of the forwardprice bias reveal a pattern where the bias is at its minimum in June andreaches its maximum value in May. This is consistent with the seasonalpatterns that have been observed for other agricultural commodities(Fama & French, 1987).

In specifying the level of risk aversion, little guidance is offered bythe literature on the subject. Values for the level of risk aversion are setin the range 1.0 to 0.00001. This selection, while arbitrary, is made toallow for sufficient impact of skewness upon hedging decisions.15

Further, this range of values is consistent with those used in similarmodels, albeit under mean-variance analysis, by Rolfo (1980), Martinezand Zering (1992), and Karp (1987).



TABLE I

Optimal Hedge Positions

Risk Mean Normalized skewness in spot pricesaversion variance(k) hedge �1.00 �0.70 �0.35 0.00 0.35 0.70 1.00

Bias � 1.0751.0 300,003 300,005 300,004 300,003 300,003 300,003 300,003 300,0030.1 300,028 300,045 300,037 300,033 300,030 300,028 300,027 300,025

0.01 300,280 300,449 300,371 300,328 300,301 300,281 300,266 300,2540.001 302,796 304,490 303,711 303,283 303,010 302,811 302,655 302,545

0.0001 327,959 344,895 337,114 332,830 330,099 328,107 326,551 325,4480.00001 579,586 748,854 671,137 628,302 600,989 581,073 565,507 524,479

Bias � 1.051.0 300,002 300,002 300,002 300,002 300,002 300,002 300,002 300,0020.1 300,019 300,023 300,021 300,020 300,019 300,018 300,018 300,017

0.01 300,186 300,227 300,214 300,202 300,192 300,184 300,177 300,1720.001 301,864 302,273 302,140 302,019 301,922 301,842 301,773 301,721

0.0001 318,639 322,731 321,396 320,189 319,221 318,417 317,730 317,2140.00001 486,391 527,314 513,963 501,886 492,209 484,166 477,304 472,144

Bias � 1.001.0 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,0000.1 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,000

0.01 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,0000.001 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,000

0.0001 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,0000.00001 300,000 300,000 300,000 300,000 300,000 300,000 300,000 300,000

Bias � 0.951.0 299,998 299,998 299,998 299,998 299,998 299,998 299,998 299,9980.1 299,981 299,983 299,983 299,982 299,981 299,980 299,979 299,977

0.01 299,814 299,828 299,823 299,816 299,808 299,799 299,786 299,7730.001 297,136 298,279 298,227 298,158 298,078 297,981 297,760 297,727

0.0001 281,361 282,786 282,270 281,583 280,779 279,811 278,604 277,2690.00001 113,609 127,856 122,696 115,834 107,791 98,114 86,037 72,686

Bias � 0.9251.0 299,997 299,997 299,997 299,997 299,997 299,997 299,996 299,9950.1 299,972 299,975 299,973 299,972 299,970 299,967 299,963 299,955

0.01 299,720 299,746 299,734 299,719 299,699 299,672 299,629 299,5510.001 297,204 297,455 297,345 297,189 296,990 296,717 296,289 295,910

0.0001 272,041 274,552 273,449 271,893 269,901 267,170 262,886 255,1050.00001 20,414 45,521 34,493 18,927 (989) (28,302) (71,137) (148,954)

Note. Hedge positions are presented in total pounds of cotton hedged. Optimal storage is 300,000 pounds of cotton for allcases. Computations assume an expected spot price of $0.63 and standard deviation of $0.13 per pound. Mean variancehedge values are computed from Equation (14). All other hedge values are computed by solving Equation (11) for x, anddetermining the optimal hedge as h � q � x. Positive (negative) values for h indicate a short (long) forward position.Normalized skewness �E [(p~ � p)3]�s3.

NUMERICAL RESULTS

Numerical results are reported in Table I. Optimal hedge positions areshown for varying levels of risk aversion, forward price bias, and normal-ized skewness in spot prices. The optimal mean-variance hedge, computed



from Equation (14), is reported in column two. All other hedge com-putations are derived from Equation (11) by first solving for the optimalvalue of X, and then determining the optimal three-moment hedge bysubtracting this value out of the optimal cash position, q. Positive valuesfor the hedge position, h, indicate that the firm has sold the commodity for-ward, whereas negative values indicate that the firm has purchased units ofthe commodity in the forward market.

Anderson and Danthine (1981) show that utility-maximizing hedgescan be separated into a pure hedge term and a speculative term, the lat-ter of the two being inversely related to the firm’s level of risk aversion.Consistent with Anderson and Danthine (1981), our results show thatdecreasing the risk aversion coefficient increases the difference betweenthe quantity of spot asset held in storage and the size of the hedgeposition. By including skewness in the model, there is in a change in themagnitude of the optimal three-moment hedge relative to the mean-variance hedge.

For a risk aversion coefficient of 1.0, the results show all hedge posi-tions, for both models and across all distributions, converge to the fullhedge. At this level of risk aversion the firm is acting more like a mini-mum variance hedger, seeking to sell forward an amount of cotton equalto the quantity held in storage. As the degree of risk aversion decreases,the firm becomes increasingly risk tolerant and the distribution ofexpected spot prices begins to impact the hedge decision through thespeculative component of the hedge ratio. However, as long as forwardprices are unbiased estimates of expected spot prices, the firm will refrainfrom speculating.

The results show that a change in skewness in spot prices can eitherincrease or decrease speculation, depending on the direction of the for-ward bias and the degree of skewness. When forward prices are biasedbelow expected spot prices and the distribution of expected spot prices isnegatively skewed, the optimal hedge will increase as it converges to afull hedge such that In this case, the firm is underhedging suchthat a portion of the stored commodity will be sold at the uncertain Mayspot price. However, as spot prices become increasingly negativelyskewed, there is greater chance that the May spot transaction price willbe very low with respect to the current forward price. The firm will beless likely to leave the stored cotton exposed to such a low price and willchoose to hedge a greater portion of the commodity.

A similar scenario occurs when the forward price is greater than theexpected spot price and expected spot prices are positively skewed. In thiscase, the optimal hedge will converge from above toward a full hedge.

h* � q*.



Economically, the firm has “overhedged,” with the expectation of cover-ing the excess quantity hedged, (h � q), by purchasing cotton in the Mayspot market to fulfill delivery requirements on the short forward con-tracts. With positive skewness in the distribution of May spot prices, it ismore likely that the firm will have to buy cotton at a very large price andcould thereby lose money when delivering on the forward contract. Thus,the firm will be less likely to overhedge and will reduce the quantity ofcotton sold forward.

In both cases, speculation decreases as skewness increases. This isan important amendment to the results under mean-variance analysiswhich holds that speculation is, all else held equal, an increasing func-tion of the forward price bias. In our analysis, it is possible that hedgersare less likely to speculate despite the existence of any bias betweenforward and expected spot prices.

For the other two cases, a normal backwardation in the forwardprice combined with positive spot price skewness, and contango in theforward price combined with negative spot price skewness, the introduc-tion of spot price skewness has the effect of reinforcing the impact of theforward price bias. Increasing the amount of spot price skewnessincreases the level of speculation, producing optimal hedge positionsthat are increasingly divergent from the full hedge result.

To provide economic justification, when the forward price is below theexpected spot price the firm is willing to underhedge to gain a more favor-able price on the commodity in the May spot market. When the spot priceis positively skewed, the possibility of very large May prices increases, sothe firm underhedges to a greater degree, thereby increasing the level ofspeculation. On the other hand, when the forward price is above theexpected spot price the firm would be expected to overhedge and cover inthe spot market. When spot prices are negatively skewed, there is a greaterchance of very small May spot prices, so the gains to speculation increaseand the firm increases the degree of overhedging.

For agricultural commodities it is generally regarded that arbitragewill keep the forward price from rising above the expected spot price, butcannot keep the reverse from occurring. Consequently, normal backwarda-tion is the more likely scenario in commodity markets. This implies that themost relevant cases are those corresponding to the lower half of Table Iwhere the forward price bias has a value equal to and less than unity. Onthe other hand, it is possible for forward prices to exceed expected spotprices when convenience yields on stocks of the commodity held in storageare sufficiently high to exceed the total costs of storage of the commodity.In this case, a contango on the forward contract, represented by a bias



greater than unity in Table I, is possible. According to Fama and French(1987) the forward prices of agricultural commodities tend to exceedexpected spot prices during the storage season following harvest, whilethey tend to fall below expected spot prices prior to harvest. This suggeststhat the hedge values in Table I where the forward bias is above one aremore representative of hedging during storage intervals, whereas the lowerhalf of Table I best represents the forward bias during preharvest periodswhen expected spot prices tend to exceed forward prices.

Like most agricultural commodities, the cotton production cyclehas four distinct phases: planting, growing, harvest, and storage. Thoughthe starting dates of these phases vary geographically, they tend to beginon March, June, August, and November, respectively. Casual inspectionof our data reveals a pattern in the forward bias and spot price skewnessrelated to these phases. During the cotton growing phase the forwardbias is below one and spot price skewness is negative. During the subse-quent harvest phase of production, the bias moves above one and skew-ness remains negative. In both the storage and planting phases of theproduction cycle the forward bias is above one while spot prices displaypositive skewness. Consequently, this pattern suggests that the hedger isin the lower left quadrant of Table I during the growing phase, moving tothe upper left quadrant during the harvest phase, and finally to the upperright quadrant during storage and planting phases. Thus, with the excep-tion of the harvest phase of production, this implies that hedgers whoinclude spot price skewness in their model will speculate less thanrecommended by the traditional mean-variance hedge model.

CONCLUDING REMARKS

This study presents a theoretical model of hedging that accounts explic-itly for the effects of asymmetric price risk in the hedger’s decision mak-ing. The results of this research demonstrate that skewness can be animportant concern for a relatively undiversified firm such as a farm.16

These findings show that skewness can increase the magnitude of specu-lative trading in forward markets when there also exists a price bias onforward contracts. When there is no bias, both two and three-momentmodels prescribe an optimal hedge policy of selling forward the spotasset in an amount equal to the quantity held in storage.

16To the extent that systematic skewness may exist in aggregate indexes (e.g., the Dow JonesIndustrial Average), skewness might be a relevant concern for investors holding diversified portfo-lios. Kraus and Litzenberger (1976) give empirical evidence of the applicability of skewness in aportfolio context. Their findings show that a significant amount of asset pricing is attributable tosystematic skewness.



Contrary to past theoretical results under direct expected utility, andempirical tests using mean-variance models, some combinations ofskewness and forward price bias can cause the firm to reduce its levelof speculation. In particular, combinations of positive skewness andcontango, and negative skewness and normal backwardation cause theoptimal three-moment hedge to converge toward the full hedge.

It is also interesting that the effect of skewness occurs in a directionopposite of that predicted by Karp (1987). His analysis suggests thatskewness becomes increasingly important as risk aversion increases.However, the results from this study show that skewness becomes a rele-vant concern as risk aversion decreases. This makes intuitive sense giventhat hedgers, who are highly risk-averse, are less likely to take on specu-lative positions. On the other hand, recent research casts some doubt onthe proposition that the use of forward markets by farmers is done soentirely for risk management purposes. Instead, there is some evidencethat farmers enter into forward contracts out of speculative motivationand value payoffs that provide positive skewness (Goodwin & Schroeder,1994; Patrick, Musser, & Eckman, 1998; Kenyon, 2001; Groom et al.,2002; Roe, Sporleder, & Belleville, 2004).

A useful extension to this model would be to include random out-put. For farmers this would be especially important, as Day (1965), in anempirical study, found that distributions of production yield on severalfield crops are skewed. Other useful extensions would allow for optionsas tools to hedge asymmetric price and yield risk. It would also be inter-esting to determine if negative skewness in either crop yields or spotprices is a concern to farmers where there exists government crop insur-ance, crop loans and subsidies, and price supports.

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