HEDGING, LIQUIDITY,AND THE COMPETITIVE

FIRM UNDER PRICE

UNCERTAINTY

KIT PONG WONG

In this paper, the behavior of the competitive firm under price uncertaintywhen the firm has access to an intertemporally unbiased futures market isexamined. Futures contracts are marked-to-market and thus requireinterim cash settlement of gains and losses. The firm is subject to a liquid-ity constraint in that it is forced to prematurely close its futures positionon which the interim loss incurred exceeds a threshold level. It is shownthat the liquidity constrained firm optimally opts for an under-hedgeshould it be prudent. Furthermore, the prudent firm cuts down its optimallevel of output in response to the presence of the liquidity constraint. Assuch, the liquidity risk created by the interim funding requirement of afutures hedge adversely affects the hedging and production decisions ofthe competitive firm under price uncertainty. © 2004 Wiley Periodicals,Inc. Jrl Fut Mark 24:697–706, 2004

I would like to thank Axel Adam-Müller, Donald Lien, Robert Webb (the editor), and an anonymousreferee for their helpful comments and suggestions. Any remaining errors are of course mine.For correspondence, Kit Pong Wong, School of Economics and Finance, University of Hong Kong,Pokfulam Road, Hong Kong; e-mail: kpwong@econ.hku.hk

Received April 2003; Accepted August 2003

� Kit Pong Wong is an Associate Professor of Finance in the School of Economics andFinance at the University of Hong Kong.

The Journal of Futures Markets, Vol. 24, No. 7, 697–706 (2004) © 2004 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.10133

698 Wong

1Culp and Hanke (1994) report that “four major European banks called in their outstanding loans toMGRM when its problems became public in December 1993. Those loans, which the banks hadpreviously rolled-over each month, denied MGRM much needed cash to finance its variation mar-gin payments and exacerbated its liquidity problems.”2One might argue that the interim losses on the futures position are coupled with an increase in thevalue of the spot position. The firm at worst faces only a temporary liquidity problem and thusshould be able to persuade its creditors to extend financing for the resulting deficit. However, it ishard to believe that creditors could understand what the firm is doing in general, and distinguishbetween a hedged position and a speculative one in particular. Concerns about moral hazard prob-lems are thus likely to stop creditors from providing funds at a time when the firm needs them most,as evident from the Metallgesellschaft case.3Loosely speaking, prudence refers to the propensity to prepare and forearm oneself under uncer-tainty, vis-à-vis risk aversion which is how much one dislikes uncertainty and would turn away fromit if one could.

INTRODUCTION

In 1993, MG Refining and Marketing, Inc. (MGRM), the U.S. subsidiaryof the German industrial firm, Metallgesellschaft A. G., offered its cus-tomers fixed prices on oil and refined oil products up to 10 years into thefuture. To hedge against its exposure to oil prices, MGRM took on largepositions in energy derivatives, primarily oil futures. When oil pricesplummeted in 1993, the margin calls on MGRM’s futures positions wereso substantial that MGRM was unable to meet due to lack of sufficientliquidity.1 This resulted in a $2.4 billion rescue package coupled with apremature liquidation of the futures positions en masse so as to keepMetallgesellschaft A. G. from going bankrupt (Culp & Miller, 1995).

The case of Metallgesellschaft A. G. vividly demonstrates the impor-tance of taking liquidity risk into account when devising risk manage-ment strategies. Indeed, the Committee on Payments and SettlementSystems (1998) identifies liquidity risk as one of the risks that users ofderivatives and other financial contracts must consider. The purpose ofthis paper is thus to study the impact of liquidity risk on the behaviorof the competitive firm under price uncertainty (Sandmo, 1971). To thisend, the firm is restricted to use futures contracts for hedging purposes.Liquidity risk arises from the marking-to-market process of the futurescontracts and from the inability of the firm to raise external funds tofinance huge interim losses on its futures position.2

In the presence of the liquidity risk, the firm is shown to optimallyopt for an under-hedge and cut down its production should it be prudentin the sense of Kimball (1990; 1993).3 These results are similar to thefindings of Paroush and Wolf (1989), who incorporate basis risk due tothe imperfect correlation between spot and futures prices into the theoryof the competitive firm under price uncertainty. The theme in commonis as follows. The presence of additional source of risk, basis risk as inParoush and Wolf (1989) or liquidity risk as in this paper, as long as it is

Hedging, Liquidity, and the Competitive Firm 699

4Throughout the paper, random variables have a tilde while their realizations do not.(~)

not hedgeable via futures trading, would adversely affect the hedging andproduction decisions of the competitive firm under price uncertainty.

Liquidity risk is a concern to the firm only when losses are substan-tial. Basis risk, on the other hand, affects the firm equally in both up anddown states. Due to the asymmetric nature of liquidity risk, risk aversionalone is not powerful enough to yield intuitively appealing results. In thispaper, it is shown that prudence à la Kimball (1990; 1993) is a usefulconcept in analyzing decision making under multiple sources of uncer-tainty wherein some risk is of a truncated sort.

THE MODEL

To incorporate marking to market into Sandmo’s (1971) model of thecompetitive firm under price uncertainty, the static one-period set-up isextended into a dynamic one. Succinctly, there are two periods withthree dates (indexed by t � 0, 1, and 2). At t � 0, the firm produces asingle output, Q, according to a cost function, C(Q), compounded tot � 2, where and . The firm sells itsentire output, Q, at t � 2 at the then prevailing spot price, , which isnot known ex ante.4

To hedge against its exposure to the random spot price, the firmtrades contracts in a futures market, each of which calls for delivery ofone unit of its output at t � 2. Let F0 be the initial futures price and H bethe short position established by the firm at t � 0. The futures price att � 1, , is a random variable distributed over support accordingto a cumulative distribution function, G(F1), where .Conditioned on the realized futures price at t � 1, , the spot out-put price at t � 2, which is also the futures price at t � 2, can be speci-fied as , where is a random variable. To focus on the hedg-ing motive, vis-à-vis the speculative motive, of the firm, the futuresmarket is assumed to be intertemporally unbiased. That is, F0 is set equalto the unconditional expected value of and is set equal to a zero-mean random variable conditionally independent of , thereby render-ing F1 equal to the conditional expected value of . Let be thecumulative distribution function of over support , where

.All futures contracts are marked to market at the end of each period.

If F1 � F0 at t � 1, the firm suffers a loss in its short futures position ofthe amount, . Following Deep (2002) and Lien (2003), thefirm is liquidity constrained in the sense that it has to close its futuresposition at t � 1 whenever the loss at that time exceeds a predetermined

(F1 � F0)H

�F1 � P � 0 � P � �

[P, P]P~°(P)P

~2

F~

1

P~P~

2

P~P~

2 � F1 � P~

F~

1 � F1

0 � F1 � F2 � �

[F1, F2]F�1

P�2

C–(Q) � 0C(0) � 0, C(Q) � 0,

700 Wong

5If the firm is risk neutral, hedging with the intertemporally unbiased futures contracts adds novalue to the firm. The assumption of risk aversion can be justified by the prevalence of corporatetaxes, costs of financial distress, or capital market imperfections (Stulz, 1996).

threshold level, K, i.e., whenever . Conditioned on therealized futures price at t � 1, the firm’s random profit at t � 2is therefore given by

(1)

(2)

where subscripts C and D indicate that the firm’s futures position is helduntil t � 2 and prematurely liquidated at t � 1, respectively.

The firm is risk averse and possesses a von Neumann–Morgensternutility function, , defined over its profit at t � 2, .5 Risk aversionimplies that and . At t � 0, anticipating the liq-uidity constraint at t � 1, the firm has to choose a level of output, Q, and afutures position, H, so as to maximize the expected utility of its randomprofit at t � 2:

(3)

where is the expectation operator with respect to , and andare defined in Equations (1) and (2), respectively.Using Leibniz’s rule, the first-order conditions for program 3 are

given by

(4)

(5)� EP[U(~*D0) � U(

~*C0)]g(F0 � K�H*)(K�H*2) � 0

� �F1

F0�K�H*

EP[U(~*D)(F0 � F1)] dG(F1)

�F0�K�H*

F1

EP[U(~*C)(F0 � F1 � P~)] dG(F1)

� �F1

F0�K�H*

EP5U(~*D)[F1 � P~ � C(Q*)]6 dG(F1) � 0

�F0�K�H*

F1

EP5U(~*C)[F1 � P~ � C(Q*)]6 dG(F1)

~

D

~

C°(P)EP�

maxQ, H

EU � �F0�K�H

F1

EP[U(~

C)] dG(F1) � �F1

F0�K�H

EP[U(~

D)] dG(F1)

U–( ) � 0U( ) � 0U( )

if F0 � K�H � F1 � F1

~

D � (F1 � P~)Q � (F0 � F1)H � C(Q)

if F1 � F1 � F0 � K�H

~

C � (F1 � P~)Q � (F0 � F1 � P~)H � C(Q)

F�1 � F1,(F1 � F0)H � K

Hedging, Liquidity, and the Competitive Firm 701

6In Lien (2003), this term is mistakenly omitted. Omitting this term is inconsequential only whenthe utility function is quadratic, i.e., .U‡( ) K 0

where is the probability density function of , and an aster-

isk (*) indicates an optimal level. The second-order conditions forprogram 3 are satisfied given risk aversion and the strict convexity ofC(Q).

OPTIMAL HEDGING DECISION

Partially differentiating EU in program 3 with respect to H and evaluatingthe resulting derivative at H � Q � Q* yields

(6)

where Since and is equal to theexpected value of Equation (6) can be reduced to

(7)

Since , it follows from risk aversion and Jensen’s inequalitythat and thus the second term on theright-hand side of Equation (7) is unambiguously negative. The first termon the right-hand side of Equation (7) is however indeterminate withoutknowing the sign of .6

Kimball (1990; 1993) defines as prudence. If the firmis prudent, it follows from and Jensen’s inequality that

. Since the integrand of the first term onthe right-hand side of Equation (7) is taken over , thisterm is unambiguously negative should . Using Equation (5)U‡( ) � 0

[F0 � K�Q*, F1]EP[U(*C1 � P~ Q*)] � U(*C1)

EP(P~) � 0U‡( ) � 0

U‡( )

U(*C1) � EP[U(*C1 � P~ Q*)]EP(P~ ) � 0

� U(*C1)6 g(F0 � K�Q*)(K�Q*2)

� U(*C1)6(F0 � F1) dG(F1) � 5EP[U(*C1 � P~ Q*)]

0EU0H`H�Q�Q*

� �F1

F0�K�Q*

5EP[U(*C1 � P~ Q*)]

F~

1,F0EP(P~ ) � 0*C1 � F0Q* � C(Q*).

� 5EP[U(*C1 � P~ Q*)] � U(*C1)6 g(F0 � K�Q*)(K�Q*2)

� �F1

F0�K�Q* EP[U(*C1 � P~ Q*)(F0 � F1)] dG(F1)

0EU0H`H�Q�Q*

� �F0�K�Q*

F1

EP[U(*C1)(F0 � F1 � P~)] dG(F1)

F�1g(F1) � G(F1)~*C0 �

~*D0 � P~ H*,

~*D0 � (F0 � K�H* � P~)Q* � K � C(Q*),

702 Wong

7Tailing has the similar effect on inducing hedgers to use fewer futures contracts initially than theywould if forward contracts were used. A tail alters a futures position in a way that the adjusted futuresposition behaves exactly like a forward position. The tailing adjustment is greater the longer the posi-tion to be held and the higher the riskless rate of interest (Figlewski, Lanskroner, & Silber, 1991).

and the second-order conditions for program 3, it follows that H* � Q*if .

Proposition 1. If the futures market is intertemporally unbiasedand if the competitive firm facing the liquidity constraint is prudent,then the firm’s optimal futures position is an under-hedge, i.e.,H* � Q*.

It should be evident that an under-hedge remains optimal when thefirm’s utility function is quadratic, i.e., . To see the intuitionof Proposition 1, refer to Equations (1) and (2). If the liquidity constraintfaced by the firm is absent, the firm’s profit at t � 2 is delineated solely byEquation (1). In this case, a full-hedge (i.e., H � Q) is optimal since iteliminates all the price risk. In the presence of the liquidity constraint,however, a full-hedge is no longer optimal due to the residual risk, ,arising from the premature liquidation of the futures position at t � 1, asevident from Equation (2). If the firm has a quadratic utility function, itis well known that its optimal futures position is the one that minimizesthe variability of its random profit at t � 2. Since the residual risk dueto the premature liquidation of the futures position at t � 1 prevails onlywhen the firm has incentives to shrink this intervalby setting H below Q. As such, the firm finds an under-hedge optimalwhen its utility function is quadratic.

Given prudence, the firm is more sensitive to low realizations of itsrandom profit at t � 2 than to high ones (Kimball, 1990; 1993). Noticethat the low realizations of its random profit at t � 2 occur when the firmprematurely closes its futures position at t � 1 and the realized valuesof are negative. Inspection of Equations (1) and (2) reveals that thefirm can avoid these realizations by shorting less of the futures contracts.Thus, the under-hedging incentive as described under quadratic utilityfunctions is reinforced when the firm becomes prudent in the sense ofKimball (1990; 1993).7

OPTIMAL PRODUCTION DECISION

If the liquidity constraint faced by the firm is absent, it is easily shownthat the optimal level of output, Q**, solves C(Q**) � F0. To compareQ* with Q**, Equation (5) is added to Equation (4). Rearranging terms

P~

F1 � [F0 � K�H, F1],

P~ Q

U‡( ) K 0

U‡( ) � 0

Hedging, Liquidity, and the Competitive Firm 703

of the resulting equation yields

(8)

Risk aversion implies that for all Thus, the second term on the right-hand side of Equation (8) is unam-biguously positive. The sign of the first term on the right-hand side ofEquation (8) is the same as that of .

To sign , only the case in which the firmis prudent is considered. Propositions 1 and 2 imply that 0 � H* � Q*.Let be the cumulative distribution function of . Usingthe change-of-variable technique (Hogg & Craig, 1989), has sup-port and where

Thus, one can write

(9)

Likewise, let be the cumulative distribution function of . Usingthe change-of-variable technique, has support

andThus, one can write

(10)

Subtracting Equation (9) from Equation (10) yields

(11)

which follows from the fact that for all It is shown in the

Appendix that is a mean preserving spread of in the sense ofRothschild and Stiglitz (1970). Thus, according to Rothschild andStiglitz (1971), the right-hand side of Equation (11) is unambiguously

®(X)£(X)M � P(Q* � H*)] ´ [M � P(Q* � H*), M � PQ*].

X � [M � PQ*,d®(X) � 0

EP[U(~

*D0)] � EP[U(~

*C0)] � �M�PQ*

M�PQ*

U(X) d[£(X) � ®(X)]

EP[U(~

*C0)] � �M�P(Q*�H*)

M�P(Q*�H*)

U(X) d®(X)

(Q*� H*)].®(*C0) � £[*C0 � (*C0 � M)H*�M � P(Q* � H*)]P(Q* � H*),[M �

~*C0

~*C0®(*C0)

EP[U(~

*D0)] � �M�PQ*

M�PQ*

U(X) d£(X)

M � (F0 � K�H*)Q* � K � C(Q*).£(*D0) � °[(*D0 � M)�Q*],[M � PQ*, M � PQ*]

~

*D0

~

*D0£(*D0)

EP[U(~

*C0)] � EP[U(~

*D0)]EP[U(

~*C0)] � EP[U(

~*D0)]

F1 � [F1, F2].CovP[U(~

*D), P~] � 0

� �F1

F0�K�H*

CovP[U(~

*D), P~] dG(F1)

� EP[U(~

*C0) � U(~

*D0)] g(F0 � K�H*)(K�H*2)

� �F1

F0�K�H*

EP[U(~

*D)] dG(F1) f[F0 � C(Q*)] e �F0�K�H*

F1

EP[U(~

*C)] dG(F1)

704 Wong

negative given risk aversion. It then follows from Equation (8) thatC(Q*) � F0. The strict convexity of C(Q) immediately implies thatQ* � Q**.

Proposition 2. If the futures market is intertemporally unbiased and ifthe competitive firm is prudent, then imposing the liquidity constraintonto the firm reduces the firm’s incentive to produce, i.e., Q* � Q**.

To see the intuition of Proposition 2, refer to Equations (1) and (2).If the liquidity constraint faced by the firm is absent, the firm’s profit att � 2 is given by Equation (1) only. Inspection of Equation (1) revealsthat the firm could have completely eliminated its exposure to the ran-dom spot price had it chosen H � Q within its own discretion.Alternatively put, the degree of risk exposure to be assumed by the firmshould be totally unrelated to its production decision. The optimal levelof output is then chosen to maximize F0Q � C(Q), which yields Q**. Inthe presence of the liquidity constraint, setting H � Q cannot eliminateall the price risk due to the residual risk, , arising from the prematureliquidation of the futures position at t � 1, as evident from Equation (2).Such residual risk, however, can be reduced by lowering Q. Since theprudent firm is more sensitive to low realizations of its profit at t � 2than to high ones, it has incentives to avoid the low realizations of itsprofit at t � 2, which occur when the firm prematurely closes its futuresposition at t � 1 and the realized values of are negative. Inspection ofEquations (1) and (2) reveals that the firm can achieve this goal byreducing Q. Thus, the presence of the liquidity constraint entices theprudent firm into cutting down its production.

CONCLUSIONS

The focus of this paper has been the behavior of the competitive firmunder price uncertainty (Sandmo, 1971) when there is an intertemporallyunbiased futures market. Futures contracts are marked-to-market andthus require interim cash settlement of gains and losses. As in Deep(2002) and Lien (2003), there is a liquidity constraint that forces the firmto prematurely close its futures position on which the interim lossincurred exceeds a threshold level. Within this context, this paper hasshown that the liquidity constrained firm optimally opts for an under-hedge if it is prudent in the sense of Kimball (1990; 1993). Furthermore,the prudent firm cuts down its production as an optimal response to theimposition of the liquidity constraint. As such, the liquidity risk created bythe interim funding requirement of a futures hedge bestows perverse

P~

P~ Q

Hedging, Liquidity, and the Competitive Firm 705

µ

incentives on the firm regarding its hedging and production decisions.The firm hedges less not only in the absolute sense due to the reduction inoutput, but also in the relative sense in that the hedge ratio is strictly lessthan unity.

APPENDIX

Consider the function, for allThen, one can write

T(X) �

which follows from the fact that for allDifferentiating T(X)

with respect to X and using Leibniz’s rule yields

It follows from that if Hence, T(X) is strictly increasing for all

and strictly decreasing for all .Note that

� 0

� �M�P(Q*�H*)

M�P(Q*�H*)

Y d®(Y) � �M�P

Q*

M�P(Q*�H*)

dY

� M � PQ* � �M�PQ*

M�PQ*

Y d£(Y) � [M � P(Q* � H*)]

T(M � PQ*) � �M�PQ*

M�P Q*

£(Y) dY � �M�P(Q*�H*)

M�P (Q*�H*)

®(Y) dY � �M�PQ*

M�P (Q*�H*)

dY

X � (M, M � PQ*)X � (M � PQ*, M)X � (� ) M.®(X) � (� ) 0

£(X) �®(X) � £[X � (X � M)H*�(Q* � H*)]

T(X) � •£(X) if M � PQ* � X � M � P(Q* � H*),£(X) � ®(X) if M � P(Q* � H*) � X � M � P(Q* � H*),£(X) � 1 if M � P(Q* � H*) � X � M � PQ*

M � P(Q* � H*)]´ [M � P(Q* � H*), M � PQ*].X � [M � P Q*,d®(X) � 0

if M � P (Q* � H*) � X � M � P Q*

�XM�P Q* £(Y) dY � �M�P(Q*�H*)

M�P (Q*�H*) ®(Y) dY � �XM�P (Q*�H*) dY

if M � P (Q* � H*) � X � M � P(Q* � H*),

�XM�P Q* £(Y) dY � �X

M�P (Q*�H*) ®(Y) dY

if M � P Q* � X � M � P(Q* � H*),

�XM�P Q* £(Y) dY

[M � PQ*, M � PQ*].X �T(X) � �X

M�PQ*[£(Y) � ®(Y)] dY,

706 Wong

where the second equality follows from integration by parts and the factthat and

Since and T(X) is first increasing and then decreasing in X, it must be true thatT(X) � 0 for all . In other words, is amean preserving spread of in the sense of Rothschild and Stiglitz(1970).

BIBLIOGRAPHY

Committee on Payment and Settlement Systems. (1998). OTC derivatives:Settlement procedures and counterparty risk management. Basel,Switzerland: Bank for International Settlements.

Culp, C. L., & Hanke, S. H. (1994). Derivative dingbats. The InternationalEconomy, 8, 12.

Culp, C. L., & Miller, M. H. (1995). Metallgesellschaft and the economics ofsynthetic storage. Journal of Applied Corporate Finance, 7, 62–76.

Deep, A. (2002). Optimal dynamic hedging using futures under a borrowingconstraint. Unpublished manuscript, Bank for International Settlements,Basel, Switzerland.

Figlewski, S., Lanskroner, Y., & Silber, W. L. (1991). Tailing the hedge: Why andhow. The Journal of Futures Markets, 11, 201–212.

Hogg, R. V., & Craig, A. T. (1989). Introduction to mathematical statistics(4th ed.). New York: Macmillan.

Kimball, M. S. (1990). Precautionary saving in the small and in the large.Econometrica, 58, 53–73.

Kimball, M. S. (1993). Standard risk aversion. Econometrica, 61, 589–611.Lien, D. (2003). The effect of liquidity constraints on futures hedging. The

Journal of Futures Markets, 23, 603–613.Paroush, J., & Wolf, A. (1989). Production and hedging decisions in the pres-

ence of basis risk. The Journal of Futures Markets, 9, 547–563.Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk: I. A definition. Journal

of Economic Theory, 2, 225–243.Rothschild, M., & Stiglitz, J. E. (1971). Increasing risk II: Its economic conse-

quences. Journal of Economic Theory, 3, 66–84.Sandmo, A. (1971). On the theory of the competitive firm under price uncer-

tainty. American Economic Review, 61, 65–73.Stulz, R. M. (1996). Rethinking risk management. Journal of Applied Corporate

Finance, 9, 8–24.

®(X)£(X)M � PQ*)X � (M � P Q*,

T(M � PQ*) � 0T(M � P Q*) �®[M � P (Q* � H*)] � 1.£(M � PQ*) �£(M � PQ*) � ®[M � P(Q* � H*)] � 0