MINIMUM-VARIANCE

FUTURES HEDGING

UNDER ALTERNATIVE

RETURN SPECIFICATIONS

ERIC TERRY

It is widely believed that the conventional futures hedge ratio, isvariance-minimizing when it is computed using percentage returns or logreturns. It is shown that the conventional hedge ratio is variance-minimizingwhen computed from returns measured in dollar terms but not fromreturns measured in percentage or log terms. Formulas for the minimum-variance hedge ratio under percentage and log returns are derived. Thedifference between the conventional hedge ratio computed from percent-age and log returns and the minimum-variance hedge ratio is found to berelatively small when directly hedging, especially when using near-maturityfutures. However, the minimum-variance hedge ratio can vary significantlyfrom the conventional hedge ratio computed from percentage or logreturns when used in cross-hedging situations. Simulation analysis showsthat the incorrect application of the conventional hedge ratio in cross-hedging situations can substantially reduce hedging performance. © 2005Wiley Periodicals, Inc. Jrl Fut Mark 25:537–552, 2005

ssf�s2f ,

Some of the research for this paper was completed while the author was a Visiting Scholar at theCenter for Financial Engineering, National University of Singapore. A research grant from the Schoolof Business at Central Connecticut State University and the helpful comments by an anonymousreferee are acknowledged.For correspondence, Department of Finance, Vance Academic Center #427, 1615 Stanley Street,Central Connecticut State University, New Britain, Connecticut 06050; e-mail: terrye@ccsu.edu

Received January 2004; Accepted September 2004

� Eric Terry is an Associate Professor of Finance in the School of Business at CentralConnecticut State University in New Britain, Connecticut.

The Journal of Futures Markets, Vol. 25, No. 6, 537–552 (2005) © 2005 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.20153

538 Terry

INTRODUCTION

It is widely known that the variance-minimizing futures hedge is given bythe ratio of the conditional covariance of the futures and spot returnsto the conditional variance of the futures return. This standard result canbe found in virtually every leading derivatives or risk management textbook(Hull, 2002; Kolb, 2003). There is, however, much confusion over theconditions under which this result holds. This result has been asserted—either explicitly or implicitly—when returns are measured in dollarterms (Anderson & Danthine, 1981; Ederington, 1979; Johnson, 1960),percentage terms (Ceccetti, Cumby, & Figlewski, 1988; Lien, Tse, &Tsui, 2002), and log terms (Baillie & Myers, 1991; Brooks, Henry, &Persand, 2002; Gagnon & Lypny, 1995; Geppert, 1995; Harris & Shen,2003; Kroner & Sultan, 1993; Park & Switzer, 1995; Poomimars, Cadle, &Theobald, 2003).

This problem has its roots in the earliest papers on futures hedging.In their pioneering work on futures hedging, neither Ederington (1979)nor Johnson (1960) explicitly state the type of returns being used intheir derivation of the minimum-variance hedge ratio—although a care-ful reading of both articles shows that they were using dollar returns.The next generation of hedging papers repeat the derivations ofEderington (1979) and Johnson (1960) while assuming that they arevalid for percentage or continuously compounded returns (Baillie &Myers, 1991; Ceccetti, Cumby, & Figlewski, 1988; Geppert, 1995;Gagnon & Lypny, 1995; Kroner & Sultan, 1993; Park and Switzer,1995). More recent articles merely cite one or more of these earlierpapers and then proceed to apply the conventional hedge ratio toeither percentage or continously compounded returns (Brooks,Henry, & Persand, 2002; Harris & Shen, 2003; Lien, Tse, & Tsui, 2002;Poomimars, Cadle, & Theobald, 2003).

In this article, we examine the minimum-variance hedge ratio(MVHR) under alternative return specifications. Formulas for theMVHR are derived for cases in which returns are measured in dollarterms, percentage terms, and log terms. These formulas are shown not tobe sensitive to alternative definitions of variance minimization. It isfound that the conventional hedge ratio—given by the ratio of the condi-tional covariance of the futures and spot returns to the conditional vari-ance of the futures return—is variance-minimizing when computed fromreturns measured in dollar terms but not from returns measured in per-centage or log terms. The conventional hedge ratio computed from

Minimum-Variance Futures Hedging 539

percentage returns, however, does give the dollar amount of futures inthe minimum-variance hedge per dollar of spot position.

The difference between the conventional hedge ratio computed frompercentage and log returns and the MVHR is found to be relatively smallwhen directly hedging, especially when using near-maturity futures.Because all of the articles cited above have examined this specific case intheir empirical work, it is not surprising that the conventional hedge hasappeared to perform well when calculated from percentage or continouslycompounded returns. However, the MVHR can vary significantly from theconventional hedge ratio computed from percentage or log returns whenused in cross-hedging situations. Simulation analysis shows that theincorrect application of the conventional hedge ratio can substantiallyreduce hedging performance in cross-hedging situations.

HEDGING TO MINIMIZE PAYOFF VARIANCE

Consider a firm that is long one unit of a particular asset at time t � 1. Adecision is made to hedge this position until time t using a specifiedfutures contract. Let ht�1 represent the short position taken in the futuresmarket at time t � 1 under the adopted hedging strategy. Ignoring dailyresettlement, the net payoff to the hedging firm will be

Pt � St � ht�1[Ft � Ft�1] (1)

where St and Ft are the respective spot and futures prices at time t. It isassumed that the firm’s objective is to minimize the variance of this netpayoff. This objective is equivalent to mean-variance utility maximizationwhen the expectations hypothesis holds, i.e., the futures price equals theexpected future spot price. Under the additional condition that spot andfutures price changes follow a symmetric distribution, this objective isalso equivalent to minimizing value-at-risk.

Using Dollar Returns

Suppose the hedger prefers to compute the MVHR using dollar returns.Let and represent the spot and futuresreturns in dollar terms for period t. Using these definitions, the netpayoff to the hedging firm can be expressed as

Pt � St�1 � Rdst � ht�1R

dst

Rdft � Ft � Ft�1Rd

st � St � St�1

540 Terry

The conditional variance at time t � 1 of this payoff is

Taking the first-order necessary condition for a minimum and solving,we find that the minimum-variance hedge ratio for dollar returns isgiven by

(2)

Thus, when spot and futures returns are expressed in dollar terms,the MVHR is given by the ratio of the conditional covariance of thefutures and spot returns to the conditional variance of the futures return.In other words, the conventional hedge ratio is variance-minimizing whenreturns are measured in dollar terms. This merely restates the standardhedging result first shown by Johnson (1960) and Ederington (1979).

Using Percentage Returns

Now suppose the hedger prefers to compute the MVHR using percent-age returns. Let and repre-sent the spot and futures returns in percentage terms for period t. Itshould be noted that, because no investment is needed for the futuresposition beyond the required margin, technically represents the per-centage change in the futures price rather than the percentage return oneach futures contact. However, we follow the usual convention of refer-ring to this term as the percentage futures return for ease of exposition.This convention will also be followed when log returns are considered.

If we substitute for the end-of-period spot and future prices usingthe two return definitions above, net payoff (1) can be rewritten as

At time t � 1, the conditional variance of this payoff is

By taking the first-order necessary condition and solving, the MVHR forpercentage returns is found to be

(3)h*t�1 �St�1

Ft�1 �

Covt�1(Rpst, R

pft)

Vart�1(Rpft)

� h2t�1F

2t�1Vart�1(Rp

ft)

Vart�1(Pt) � S2t�1Vart�1(Rp

st) � 2ht�1St�1Ft�1Covt�1(Rpst, R

pft)

Pt � St�1(1 � Rpst) � ht�1Ft�1R

pft

Rpft

Rpft � (Ft � Ft�1)�Ft�1Rp

st � (St � St�1)�St�1

h*t�1 �Covt�1(Rd

st, Rdft)

Vart�1(Rdft)

Vart�1(Pt) � Vart�1(Rdst) � 2ht�1Covt�1(Rd

st, Rdft) � h2

t�1Vart�1(Rdft)

Minimum-Variance Futures Hedging 541

This differs from the conventional hedge ratio due to the inclusion of aterm reflecting the initial spot price relative to the initial futures price.Failure to account for this term will lead to computed hedge ratios thattend to underestimate the MVHR when the spot price is above thefutures price and overestimate it when the spot price is below the futuresprice. The greater is the difference between the spot and futures prices,the greater will be the error produced by applying the conventionalhedge ratio to percentage returns. Only when the spot and futures pricesare identical will the conventional hedge ratio be variance minimizing forpercentage returns.

Before proceeding to the final case, it should be noted thatEquation (3) implies that

(4)

where represents the dollar amount of futures in the optimal hedgeper dollar of spot position held. This demonstrates that the conventionalhedge ratio does have a valid interpretation when computed from per-centage returns. It describes the dollar amount of futures in theminimum-variance hedge per dollar of spot position rather than the num-ber of futures contracts per unit of asset being hedged. Under thisdefinition of the optimal hedge, the firm is completely hedged when

rather than one. Neither Ceccetti, Cumby, and Figlewski(1988) nor Lien, Tse, and Tsui (2002) use this representation of theminimum-variance hedge when applying the conventional hedge ratio topercentage returns.

Using Continuously Compounded Returns

Finally, suppose the hedger prefers to compute the MVHR using contin-uously compounded returns. Let and represent the spot and futures returns in log terms for period t. Substitu-ting these definitions into Equation (1), the hedger’s net payoff can beexpressed as

Unlike the previous two cases, an assumption about the conditional dis-tribution of the spot and futures returns is required in order to derive theMVHR. Following standard practice, we assume that they follow a

Pt � St�1 exp5Rcst6 � ht�1Ft�1[exp5Rc

ft6 � 1]

Rcft � ln(Ft�Ft�1)Rc

st � ln(St�St�1)

H*t�1 � Ft�1�St�1

H*t�1

H*t�1 �Ft�1

St�1 � h*t�1 �

Covt�1(Rpst, R

pft)

Vart�1(Rpft)

542 Terry

bivariate normal distribution. Assuming bivariate normality, the condi-tional variance at t � 1 of the above payoff is

Taking the first-order necessary condition and solving, the MVHR forcontinuously compounded returns is given by

(5)

This formula significantly differs from the conventional hedge ratio.The optimal hedge ratio for this case consists of four multiplicativeterms. The first term is the ratio of the initial spot price to the initialfutures price. The second and third terms capture the relative amountof drift and volatility that is expected in spot and futures prices overthe hedging period. The final term, similar to the conventional hedgeratio, measures the conditional covariance of the futures and spotreturns relative to the conditional variance of the futures return. Notethat hedging formula (5) involves the expected spot and futuresreturns as well as the conditional variance of the spot return. In con-trast, the conventional hedge ratio does not depend on any of thesethree values.

As noted earlier, the MVHR for continuously compounded returnsdepends upon the assumed distribution of the conditional spot andfutures returns. The presence of either skewness or kurtosis in the con-ditional log returns would cause the minimum-variance hedge to differslightly from that given in Equation (5). However, we are not aware ofany return distribution under which the MVHR would correspond to theconventional hedge ratio.

Discussion

Before examining the robustness of these results, several commentsshould be made. First, MVHR Equations (2), (3), and (5) should not beviewed as separate formulas but instead as the same underlying formula

h*t�1 �St�1

Ft�1 �

exp5Et�1 (Rcst)6

exp5Et�1(Rcft)6 �

exp5Vart�1(Rcst)�26

exp5Vart�1(Rcft)�26 �

exp5Covt�1(Rcst, R

cft)6 � 1

exp5Vart�1(Rcft)6 � 1

[exp5Vart�1(Rcft)6�1]� h2

t�1F2t�1 exp52Et�1(Rc

ft)6 exp5Vart�1(Rcft)6

� exp5Vart�1(Rcft)�26 [exp5Covt�1(Rc

st, Rcft)6 � 1]

� 2ht�1St�1Ft�1 exp5Et�1(Rcst)6 exp5Et�1(Rc

ft)6 exp5Vart�1(Rcst)�26

[exp5Vart�1(Rcst)6 � 1]Vart�1(Pt) � S2

t�1 exp 52Et�1(Rcst)6 exp 5Vart�1(Rc

st)6

Minimum-Variance Futures Hedging 543

expressed in three different forms. If we take hedge ratio (2) and makethe change of variables and we find that

which is hedge ratio (3). This demonstrates that hedge ratios (2) and(3) are equivalent. Similarly, the identities and

can be used to show that hedge ratios (2) and (5)are equivalent. Thus, MVHR formulas (3) and (5) are merely transformedversions of conventional hedge ratio (2).

Second, these results should not be interpreted as implying that it isbetter to compute the MVHR using dollar returns rather than percentagereturns or log returns. The choice of which type of returns to use when esti-mating hedge ratios depends on the perceived statistical properties of thesereturns. Most practitioners prefer to work with percentage returns or logreturns rather than dollar returns because expected returns and returnvolatility are generally considered to be proportional to price levels.An addi-tional reason for using continuously compounded returns is that the cost-of-carry formula is linear in log prices: ln(Ft) � ln(St) � c(T � t), where c isthe net cost-of-carry and T is the futures maturity date. Our results merelystate that the correct formulas to use when computing the MVHR frompercentage or log returns are Equations (3) and (5), respectively, ratherthan the conventional hedge ratio. The conventional hedge ratio corre-sponds to the MVHR only when it is computed using dollar returns.

Finally, it should be noted that our results are not dependent on anyspecific form for the joint spot and futures return process. No assump-tions were made about this process except that it has well-defined condi-tional first and second moments. The conditional means of the returnprocess could include autoregressive terms (Howard & D’Antonio, 1991),cointegration (Kroner & Sultan, 1993) or fractional cointegration (Lien &Tse, 1999) terms, time to maturity effects (Low, Muthuswamy, Sakar, &Terry, 2002), terms reflecting external information (McNew & Fackler,1994), or even dependences on other futures prices (Neuberger, 1999).Meanwhile, the conditional second moments could include GARCH

ln(1 � Rdft�St�1)Rc

ft �

Rcst � ln(1 � Rd

st�St�1)

�St�1

Ft�1 �

Covt�1(Rpst, R

pft)

Vart�1(Rpft)

�Covt�1(St�1R

pst, Ft�1 R

pft)

Vart�1 (F2t�1 R

pft)

h*t�1 �

Covt�1(Rdst, R

dft)

Vart�1(Rdft)

Rpft � Rd

ft�Ft�1,Rpst � Rd

st�St�1

544 Terry

(Baillie & Myers, 1991) or stochastic volatility (Lien & Wilson, 2001)effects, terms reflecting external information (McNew & Fackler, 1994),or time to maturity effects (Low, Muthuswamy, Sakar, & Terry, 2002).Thus, our results are applicable to all minimum-variance futures hedgingmodels.

HEDGING TO MINIMIZE RETURN VARIANCE

In the previous section, the MVHR when the hedging firm seeks to mini-mize the variance of its net payoff was analyzed. To examine the robustnessof these results, the MVHR when the hedger instead seeks to minimizereturn variance is next examined.

Minimizing Dollar Return Variance

Assume first that the objective of the hedging firm is to minimize thevariance of its dollar return between times t � 1 and t. Ignoring dailyresettlement, the dollar return to the hedging firm is given by

Substituting and into this equation, thehedger’s dollar return can be rewritten as

Comparing this with Equation (1), it is found that

i.e., the firm’s dollar return is equal to the firm’s net payoff less the initialspot price. Because the initial spot price is deterministic at time t � 1,this relationship implies that

Consequently, the objective of minimizing the conditional variance of thefirm’s dollar return is equivalent to minimizing the conditional variance ofthe firm’s net payoff. It immediately follows from our previous results thatthe MVHR is given by Equation (2) when it is computed using dollarreturns, Equation (3) when it is computed using percentage returns, andEquation (5) when it is computed using continuously compounded returns.Thus, under this alternative hedging objective the conventional hedge ratiocontinues to be variance-minimizing when returns are measured in dollarterms but not when they are measured in percentage or log terms.

Vart�1(Rdt ) � Vart�1(Pt)

Rdt � Pt � St�1

Rdt � [St � St�1] � ht�1[Ft � Ft�1]

Rdft � Ft � Ft�1Rd

st � St � St�1

Rdt � Rd

st � ht�1Rdst

Minimum-Variance Futures Hedging 545

Minimizing Percentage Return Variance

Assume now that the objective of the hedging firm is to minimize thevariance of its percentage return between times t � 1 and t. The per-centage return to the firm, Rt, is given by

(6)

where Pt, the net payoff from the hedger’s spot and futures position, isgiven by Equation (1). Because the initial spot price is deterministic attime t � 1, this equation implies that the return variance is related to thevariance of the firm’s net payoff by

As in the previous case, minimizing the conditional variance of the firm’spercentage return is equivalent to minimizing the conditional variance ofthe firm’s net payoff. Therefore, the MVHR under this alternative objec-tive continues to be given by Equation (2) when computed using dollarreturns, Equation (3) when computed using percentage returns, andEquation (5) when computed using continuously compounded returns.Thus, the conventional hedge ratio is once again variance-minimizingwhen returns are measured in dollar terms but not when they are meas-ured in percentage or log terms.

Note that the firm’s percentage return (6) can be rewritten as

where and represent thepercentage spot and futures returns, and Ht�1 � ht�1Ft�1�St�1 represents the dollar amount of futures in the hedge per dollar of spot position held.Taking the first-order necessary condition for a minimum and solving,the minimum-variance hedge ratio is given by

This is merely reiterates our previously shown result that the conventionalhedge ratio describes the dollar amount of futures in the minimum-variance hedge per dollar of spot position rather than the number offutures contracts per unit of asset being hedged when returns are meas-ured in percentage terms.

H*t�1 �Covt�1(Rp

st, Rpft)

Vart�1(Rpft)

Rpft � (Ft � Ft�1)�Ft�1Rp

st � (St � St�1)�St�1

Rpt � Rp

st � Ht�1Rpft

Vart�1(Rpt ) �

Vart�1(Pt)

S2t�1

�Pt

St�1� 1

Rpt �

[St � St�1] � ht�1[Ft � Ft�1]

St�1

546 Terry

Minimizing Log Return Variance

Finally, assume that the firm’s objective is to minimize the variance of itscontinuously compounded return over the hedging period. Ignoring dailyresettlement, the hedger’s log return will be

Because continuously compounded returns are not cross-sectionallyseparable, i.e., ln(1 � RA � RB) � ln(1 � RA) � ln(1 � RB), this returncannot divided into separate spot and futures returns. As a result, aclosed-form solution for the minimum-variance hedge ratio does notexist. However, the minimum-variance hedge ratio can be approximatedby applying a Taylor series expansion of the ln(�) function. The first-orderTaylor series approximation ln(1 � R) � R implies that the hedger’s logreturn can be approximated by its simple return:

Using the results from the previous subsection, it follows that the MVHRis approximately equal to conventional hedge ratio (2) when computedfrom dollar returns, Equation (3) when computed from percentagereturns, and Equation (5) when computed from continuously com-pounded returns.

Better approximations of the MVHR can be obtained by includinghigher-order terms in the Taylor series expansion. However, the resultingformulas for the MVHR continue to differ from the conventional hedgeratio when returns are measured in either percentage or log terms.Furthermore, the complexity of the resulting formulas increase geomet-rically as additional terms are included in the approximations. For thesetwo reasons, such higher-order approximations are not presented in thisarticle.

EMPIRICAL IMPLICATIONS

The previous sections have shown that the MVHR differs from the con-ventional hedge ratio when returns are measured in either percentage orlog terms. Furthermore, this result is not sensitive to alternative defini-tions of variance minimization. In this section, the economic signifi-cance of these results are examined.

Convergence of the futures basis implies that Ft � St for directhedging using near-term contacts. Applying this approximation to

Rct �

[St � St�1] � ht�1[Ft � Ft�1]

St�1

Rct � ln(St � ht�1[Ft � Ft�1]) � ln(St�1)

Minimum-Variance Futures Hedging 547

Equation (3), it is found that

Thus, the conventional hedge ratio computed using percentage returnswill closely approximate the MVHR when the firm is directly hedgingusing near-term futures contacts. Similar arguments show that the con-ventional hedge ratio computed using log returns also will closelyapproximate the MVHR for the case of direct hedging using near-termfutures contacts.

For direct hedging using longer-term futures contacts, spot andfutures prices can deviate from each other somewhat, but only withinthe bounds set by physical (or financial) arbitrage. Consequently, theconventional hedge ratio computed using either percentage or logreturns will differ from the MVHR by only a relatively small amount.Consider the case of directly hedging a portfolio that tracks the Nikkeiindex using the Nikkei futures contracts traded on the SingaporeGeneral Exchange (formerly SIMEX). Figure 1 shows the ratio of spot tofutures price as a function of the futures time to maturity over the period1987 through 1999 inclusive, excluding the week of and the weekfollowing the October 1987 stock market crash. The observed range forthe ratio of spot to futures price increases as the futures maturity

h*t�1 �

Covt�1(Rpst, R

pft)

Vart�1(Rpft)

FIGURE 1Ratio of spot to futures price for the Nikkei Index by futures maturity.

548 Terry

lengthens, but only within narrow bounds. For maturities of between20 and 25 weeks, this ratio never exceeds 1.02 or falls below 0.945.From Equation (3), this implies that the conventional hedge ratiocomputed using percentage returns would have overestimated theMVHR by (1 � .945)�.945 � 5.8% at most and underestimated it by(1.02 � 1)�1.02 � 2.0% over this period using futures maturities ofbetween 20 and 25 weeks.

For cross hedging, however, the minimum-variance hedge ratio foreither percentage or log returns can differ significantly from the conven-tional hedge ratio. Consider the case of hedging a spot position in Dubaicrude oil using the nearest light sweet oil futures contract traded onNYMEX. Using mid-week (Wednesday) closing spot prices and futuressettlement prices over the period 1992 to 2002 inclusive, the averageratio of spot to futures price was 0.861, with a maximum value of 0.966and a minimum value of 0.688. From Equation (3), this implies that theconventional hedge ratio computed using percentage returns would haveoverestimated the MVHR by an average of (1 � .861)�.861 � 16.2% overthis 11-year period, with the actual amount of this overestimation varyingfrom 3.5% to 45.4%.

To determine whether such deviations from the MVHR significantlyimpact hedging performance, a series of simulations were conducted.For direct hedging, the constant-correlation GARCH model of Kronerand Sultan (1993) was used:

where the conditional variance–covariance matrix for the residual errorsis given by

and the conditional variance terms follow the GARCH(1,1) processes

For cross hedging, the cointegration terms were removed from the abovemodel. The simulations were conducted using the parameters estimatedby Kroner and Sultan for the British Pound, Canadian Dollar, JapaneseYen, and Swiss Franc. (The estimated GARCH parameters for the

h2ft � cs � asP2

ft�1 � bsh2ft�1

h2st � cs � asP2

st�1 � bsh2st�1

at� a h2

st rhsthft

rhsthft h2ftb

Rft � mf � uf [ln(St�1) � ln(Ft�1)] � Pft

Rst � ms � us[ln(St�1) � ln(Ft�1)] � Pst

Minimum-Variance Futures Hedging 549

German Mark were mean-fleeting, which led to negative return volatili-ties arising frequently in the scenarios, and so this currency was omittedfrom the simulations.) No significant differences were noted in thesimulation results across the four currencies, which indicates that thesimulation results are not very sensitive to the assumed return parame-ters. Consequently, simulation results were averaged across the fourcurrencies.

The simulation results, given in Table I, were based on 1,000 repli-cations of a 250-week time series for each of the four currencies. Fordirect hedging, the results were not sensitive to the initial ratio of spotto futures price and so the results for S0�F0 � 1 are presented. Theconventional hedge ratio deviated from the MVHR by an average of0.60% when computed using percentage returns and by an averageof 0.52% when computed using log returns. Hedging performance, meas-ured by the conditional variance of the hedger’s return, fell by somewherebetween 0.08% and 0.11% when the conventional hedge ratio was used

TABLE I

Simulated Performance of the Conventional Hedge RatioWhen Computed From Percentage and Log Returns

Cross hedging

Direct S0�F0 S0�F0 S0�F0 S0�F0 S0�F0 S0�F0

hedging � 0.8 � 0.9 � 1.0 � 1.1 � 1.2 � 1.3

Hedge ratio computed from percentage returns

Average percentage deviation 0.07 24.17 5.42 �0.76 �9.70 �17.29 �23.65from MVHR

Mean percentage standard 0.60 24.59 11.21 3.62 10.23 17.54 23.81deviation from MVHR

Loss in hedging efficiencya

Dollar returns 0.11 59.24 25.21 4.38 22.26 43.22 57.36Percentage returns 0.11 58.97 24.12 3.78 21.82 43.25 57.31Log returns 0.10 58.98 24.17 3.81 21.98 43.43 57.51

Hedge ratio computed from log returns

Average percentage deviation 0.07 24.31 5.54 �0.66 �9.60 �17.20 �23.56from MVHR

Mean percentage standard 0.52 24.72 11.35 3.60 10.14 17.46 23.73deviation from MVHR

Loss in hedging efficiencya

Dollar returns 0.08 59.54 25.76 4.43 22.02 42.98 57.18Percentage returns 0.08 59.23 24.59 3.80 21.61 43.03 57.16Log returns 0.09 58.82 24.12 3.64 20.79 42.11 56.35

aGiven by the percentage reduction in the conditional variance of the hedger’s return under the MVHR versus theconventional hedge ratio.

550 Terry

instead of the MVHR, depending on the type of returns used to measurehedging performance and the type of returns used to compute the con-ventional hedge ratio. Thus, the economic loss from using the conven-tional hedge ratio computed from either percentage or log returns insteadof using the actual MVHR appears to be quite small.

The simulation results for cross hedging were highly dependent onthe initial ratio of spot to futures price. When this price ratio was set toone, the reduction in hedging efficiency from using the conventionalhedge ratio computed from percentage or log returns instead of usingthe MVHR was found to be between 3.6% and 4.4%, depending on thetype of returns used to measure hedging performance. As the initial spotto futures price deviated from one, the performance of the conventionalhedge ratio deteriorated rapidly. When the initial ratio of spot to futuresprice was 1.1, the conventional hedge ratio computed from percentageor log returns performed over 20% worse than the MVHR. The conven-tional hedge ratio computed using percentage or log returns performedover 42% worse than the MVHR when the initial spot to futures priceratio was 1.2 and over 56% worse when the initial spot to futures priceratio was 1.3. Similar results were found as the initial spot to futuresprice ratio fell below one. These results indicate that the conventionalhedge performs substantially worse for cross hedging than for directhedging when it is computed from percentage or continuously com-pounded returns. In addition, even relatively small deviations betweenthe spot and futures price can lead to a substantial reduction in hedgingefficiency when the conventional hedge ratio is incorrectly applied topercentage and log returns.

CONCLUSIONS

In this article, we examined the minimum-variance hedge ratio when itis computed from alternative return specifications. Formulas for theminimum-variance hedge were derived for cases in which returns aremeasured in dollar terms, percentage terms, and log terms. These formulaswere shown not to be sensitive to alternative definitions of variance mini-mization. It was found that the conventional hedge ratio—given by theratio of the conditional covariance of the futures and spot returns tothe conditional variance of the futures return—is variance-minimizingwhen returns are measured in dollar terms but not when they are meas-ured in percentage or log terms.

The difference between the conventional hedge ratio computedfrom percentage or log returns and the variance-minimizing hedge ratio

Minimum-Variance Futures Hedging 551

was shown to be relatively small when directly hedging using near-maturity futures. Because all of the articles cited herein have examinedthis specific case in their empirical work, it is not surprising that theconventional hedge has appeared to perform well when calculated usingpercentage or log returns. In contrast, it was shown that the minimum-variance hedge ratio can vary significantly from the conventional hedgeratio computed from percentage or log returns in cross hedging situa-tions. Simulation analysis indicates that the incorrect application of theconventional hedge ratio can substantially reduce hedging performancein such cross-hedging situations.

BIBLIOGRAPHY

Anderson, R. W., & Danthine, J. P. (1981). Cross hedging. Journal of PoliticalEconomy, 89, 1182–1196.

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