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Hedging strategy for crude oil trading and the factors influencing hedging effectiveness
ARTICLE IN PRESS Energy Policy 38 (2010) 2404–2408
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Hedging strategy for crude oil trading and the factors influencing hedging effectiveness Won-Cheol Yun a, Hyun Jae Kim b,n a b
Department of Economics and Finance, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea Division of Energy Industry, Korea Energy Economics Institute, 665-1 Naeson-dong, Euiwang-si, Kyunggi-do 437-713, South Korea
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 October 2009 Accepted 14 December 2009 Available online 18 January 2010
This study analyzes the hedging effectiveness of different hedge type and period by Korean oil traders. Both crude oil price and exchange rate risks are considered. Theoretical models are formulated to estimate the hedge ratios by separate and complex hedge types. The hedging period covers 1–12 months. This study also performs some statistical works to investigate the relationship between the hedging effectiveness and the crude oil price sensitivity to exchange rate. In addition, the relationship between the hedging effectiveness and the volatilities of crude oil price and exchange rate is analyzed. & 2009 Elsevier Ltd. All rights reserved.
JEL classification: C15 F31 G32 Keywords: Crude oil Complex hedge Hedging effectiveness
1. Introduction Recently, crude oil prices have been soaring up and climbed to their highest level ever, reaching over $110 per barrel in March 2008. Some analysts suggested that oil prices might soar to $200 per barrel. Since all of crude oil consumed in South Korea is imported from abroad, the recent crude oil price hikes have widespread and negative impact on the national economy. In addition, the exchange rate of Korean won to US dollar has been increasing. This depreciation of Korean won relative to US dollar would be another financial burden to the Korean oil traders importing crude oil from abroad. Under this circumstance, the Korean oil traders should find an appropriate measure to cope with exchange rate risk as well as crude oil price risk. One of them would be for them to enter derivatives markets and to make hedging positions. With respect to crude oil price risk, the exchanges such as the NYMEX in New York and the ICE in London have been offering hedging instruments recognized by world traders. Related to exchange rate risk, the Korea Exchange (KRX) has been trading a US dollar currency futures and option contracts.1 n
Corresponding author. Tel.: + 82 31 420 2228; fax: +82 31 420 2164. E-mail address: [email protected] (H. J. Kim). 1 Detailed specification for this contract is found in www.krx.co.kr. From May 26 of 2006, KRX started the trading of Japanese yen and euro currency futures contracts. Each contract covers 5 million yen and 50 thousand euro, respectively. We expect that these futures contracts would provide domestic and foreign market participants with a proper hedging tool for yen and euro currency risks as well as investment opportunities. 0301-4215/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enpol.2009.12.032
A considerable number of researches deal with the question of trade stability, especially, commodity price and exchange rate risks. The issue addressed in this book of literature is how nations and individual firms can reduce their exposure to these risks. Some suggest risk-conditioning policy strategies such as buffer stocks of major raw materials and foreign exchange pegging (Massell, 1969; Johnson and Summer, 1976). Others propose that futures market is a viable tool by which internationally trading countries can reduce their trade-related risks (Newbery and Stiglitz, 1981). Related to the risk management using futures markets, a growing literature deals with the optimal hedging strategies (Kamara, 1982). Some studies present analytical frameworks for offshore commodity hedging under floating exchange rates (Thompson and Bond, 1987; Liu et al., 2001). The other studies suggest various kinds of hedging strategies for domestic firms to deal with multiple risks such as commodity price and exchange rate (Fackler and McNew, 1993; Lapan and Moschini, 1994; Vukina et al., 1996; Li and Vukina, 1998; Nayak and Turvey, 2000; Liu et al., 2001; Zhang et al., 2007). Other studies explicitly consider the risk exposure of ocean freight rate with commodity price and exchange rate risks (Hauser and Neff, 1993; Haigh and Holt, 2000, 2002). The main objective of this study is to examine the hedging performance of multiple-risk hedging strategy. The strategy can be regarded as a complex hedge considering the intercorrelation between commodity price and exchange rate. In fact, crude oil price and exchange rate are likely to be positively or negatively correlated. As a result, a separate determination of the optimal
ARTICLE IN PRESS W.-C. Yun, H.J. Kim / Energy Policy 38 (2010) 2404–2408
hedge ratios for each of crude oil price and exchange rate could be less efficient compared with the simultaneous determination of hedge ratios accounting for the intercorrelation between these prices. In addition, this study tries to identify the factors affecting hedging effectiveness. For comparison, the hedging effectiveness is evaluated by different hedge type and period. The remainder of this paper is organized as follows. Section 2 lays out an analytical framework that highlights the derivation of separate and complex hedge ratios. Section 3 describes the data used, the empirical results for hedging effectiveness, and the statistical results for factors influencing the hedging effectiveness. Finally, Section 4 summarizes the main results.
2. Theoretical model This study analyzes the hedging effectiveness of a complex hedge type by a Korean oil trader. For comparison purpose, it also considers a separate hedge type compared to the no hedge one. For this purpose, this study sets up a theoretical model based on the well-known mean–variance framework. The optimal hedge ratios are basically the minimum variance ones that are designated to minimize the variances of revenue flows. This study simulates weekly revenue flows of Korean oil trader facing the risk exposures of import price and foreign exchange. Finally, it investigates the relationship between hedging effectiveness and crude oil price sensitivity to exchange rate change and the volatilities of crude oil price and foreign exchanges by hedge period. In a mean–variance framework, the decision maker is assumed to maximize the following objective function: ð1Þ maxEðpt Þ ¼ Eðpt 9Oti Þcðl=2Þvarðpt 9Oti Þ E( ) and var( ) denote the conditional expectation operator and the conditional variance operator, respectively; pt is the end-ofperiod return given the information available at t i (Ot i); c denotes the costs incurred in the transaction; l refers to the decision maker’s coefficient of risk aversion; t i and t stand for the time when a hedge is initiated and the time when the hedge is lifted, respectively. In this study, the hedge period is assumed to be from one month (i= 1) to 12 months (i =12).2
2405
derived as @ varðpt Þ=@Fti ¼ 2s2fe Fti 2sse;fe Qti ¼ 0
ð4Þ
HR1 ¼ Fti =Qti ¼ sse;fe =s2fe
ð5Þ
2.2. Separate hedge of foreign exchange risk only Assume that the Korean oil trader tries to hedge exchange rate risk only. Still the company’s measure of terminal return would be in terms of Korean won. Protecting it from the adverse changes in exchange rate, he chooses the currency futures quantity Xt i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðxti xt Þfti Xti c
ð6Þ
2 2 2 varðpt Þ ¼ s2se Qti þ s2x fti Xti 2sse;x Qti fti Xti
ð7Þ
where xt is the yen currency futures price at time t in terms of Korean won to US dollar. Assuming the unbiasedness of currency futures prices, minimizing Eq. (7) with respect to the decision variable Xt i given the spot amount Qt i, the minimum variance hedge ratio (HR2) is derived as 2 @varðpt Þ=@Xti ¼ 2s2x fti Xti 2sse;x Qti fti ¼ 0
ð8Þ
2 =Qti ¼ sse;x =s2x fti HR2 ¼ Xti
ð9Þ
2.3. Complex hedge of both price and foreign exchange risks Assume that the Korean oil trader tries to hedge both crude oil price and exchange rate risks. His terminal return would be measured in Korean won. Protecting it from the adverse changes in crude oil price and exchange rate, he chooses both crude oil futures quantity Ft i and the currency futures quantity Xt-i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðfti eti ft et ÞFti þ ðxti xt Þfti Xti c ð10Þ 2 2 2 2 þ s2fe Fti þ s2x fti Xti 2sse;fe Qti Fti varðpt Þ ¼ s2se Qti
2sse;x fti Qti Xti þ2sfe;x fti Fti Xti
ð11Þ
2.1. Separate hedge of crude oil price risk only Assume that the Korean oil trader measures its terminal return in Korean won and that the hedging decision is influenced by crude oil price only. In order to protect against an unfavorable spot crude oil price change, he would choose the crude oil futures quantity Ft i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þðfti eti ft et ÞFti c
ð2Þ
2 2 varðpt Þ ¼ s2se Qti þ s2fe Fti 2sse;fe Qti Fti
ð3Þ
Considering the interactions among the spot and futures prices of crude oil and exchange rate, the hedge ratios for the crude oil futures and the currency futures turn out to be more complex than HR1 in Eq. (5) and HR2 in Eq. (9). Assuming unbiasedness of both crude oil and currency futures prices, minimizing Eq. (11) with respect to the decision variables Ft-i and Xt-i given the spot amount Qt i, the minimum variance hedge ratios (HR3 and HR4) are derived as3 H31 fti H21 H32 fti H22 ð12Þ HR3 ¼ Fti =Qti ¼ A
where st and ft are the spot and futures price of crude oil at time t in US dollar, respectively; et denotes the exchange rate of Korean won to US dollar; s2x and sx,y are the variance of x variable and the covariance of variables x and y, respectively. Assuming the Martingale hypothesis or unbiasedness of crude oil futures prices, maximizing Eq. (1) with respect to the decision variable Ft-i given the spot amount Qt i is equivalent to minimizing Eq. (3). The minimum variance hedge ratio (HR1) is
where Hjk ¼ sjk =s2k , j ¼ fe ð1Þ; x ð2Þ; se ð3Þ, and k ¼ fe ð1Þ; x ð2Þ.
2 The hedge periods of 1–12 months correspond to the weeks of 4, 8, 13, 17, 21, 26, 30, 34, 39, 43, 47, and 52.
3 A detailed derivation of the complex hedge ratios is available in the Appendix.
and H11 H31 H12 H32 HR4 ¼ Xti =Qti ¼ A
ð13Þ
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3. Empirical results
Table 1 Descriptive statistics of sample data.
This study uses every Friday quotes for the spot and futures prices of crude oil and exchange rates of US dollar denominated in Korean won. The spot crude oil prices are Dubai (Fateh) prices (FOB), and the futures prices correspond to the nearby WTIFutures contracts traded on NYMEX.4 Both spot and futures prices are quoted in dollars per barrel. The data set is available from the database of Energy Information Administration of US Department of Energy (EIA/DOE) for crude oil prices. The KRX offers the spot and futures quotes of exchange rates. The sample covers the period January 2000–June 2007. This weekly data set has 389 observations, and is reduced to 337 observations by deleting the first 52 observations for differencing up to 52 weeks of the maximum hedge period. Table 1 shows the descriptive statistics of sample data. 3.1. Hedging effectiveness of separate and complex hedges In order to simulate the revenue flow each week, it is assumed that the Korean oil trader takes crude oil and currency futures positions based on the hedge ratios derived above. At the end of hedge period the hedging positions are offset, and the trading revenues without and with hedging are calculated. This process repeats each week until the end of sample period. The hedge periods considered range from 1 month to 12 months. The empirical analysis is performed for both in-sample data and outof-sample data. For the out-of-sample analysis, the whole data set is divided into the first sub-sample of January 2001–October 2005 (250 observations) and the second sub-sample of November 2005–June 2007 (87 observations). To estimate the hedge ratios by separate hedge type, this study adopts a simple OLS model. With the OLS estimation, the following equations are specified to estimate b^ 1 (HR1) for crude oil and b^ 2 (HR2) for foreign exchange, respectively: ðst sti Þ ¼ a^ 1 þ b^ 1 ðft fti Þ þ e1t
ð14Þ
ðet eti Þ ¼ a^ 2 þ b^ 2 ðxt xti Þ þ e2t
ð15Þ
For the estimation of complex hedge ratios (HR3 and HR4), the following equation is specified to estimate Hjk that appears in Eqs. (12) and (13): ^ jk Dk þ et Dj ¼ a^ þ H
ðj ¼ 1; 2; 3;
k ¼ 1; 2Þ
ð16Þ
where D1 = (ftet ft iet i), D2 = (xt xt i), and D3 =(stet st iet i). Following Johnson (1960) and Ederington (1979), the hedging effectiveness (HE) refers to the gain or loss in the variance of terminal revenue resulting from the price changes in an unhedged position relative to those in a hedged position defined as HE ¼
ðvaru ðpt Þvarh ðpt ÞÞ varu ðpt Þ
ð17Þ
where varu(pt) and varh(pt) are the variances for the unhedged and hedged position, respectively. Tables 2 and 3 present the ex-post and ex-ante simulation results by different hedge type and period. The estimated hedge ratios for crude oil turn out to be similar regardless of hedge type. 4 Since over 80% of imported crude oil by South Korea comes from Middle East countries, the Dubai crude oil prices are chosen as spot prices. However, until the launch of Dubai Mercantile Exchange (DME) as of June 2007 there was no exchange traded Middle East Sour Crude benchmark, which hindered risk management for Middle East Sour Crudes. In this context, the hedging scenario adopted in this paper is viewed as a cross-hedge. The Brent crude oil futures listed in the NYMEX and the ICE would be a possible candidate instead of using the WTI crude oil futures.
Variable
Mean
Std. dev.
Minimum
Maximum
Dubai-Spot WTI-Futures USD-Spot USD-Futures
36.36 41.07 1128.27 1129.63
15.10 15.88 121.96 123.54
16.55 18.00 920.30 919.50
71.35 77.03 1345.40 1367.00
Table 2 Ex-post results of variance changes by hedging period. Hedge period
1 2 3 4 5 6 7 8 9 10 11 12
Separate hedge
Complex hedge
HR-OIL
HR-USD
HE
HR-OIL
HR-USD
HE
0.5774 0.6793 0.7067 0.7227 0.7493 0.7447 0.7370 0.7346 0.7223 0.6999 0.6912 0.6993
0.9366 0.9688 0.9835 0.9826 0.9819 0.9814 0.9827 0.9827 0.9835 0.9826 0.9826 0.9822
51.34 56.31 56.68 54.97 51.82 51.27 50.83 50.02 48.27 40.48 37.39 40.52
0.5689 0.6829 0.7119 0.7305 0.7572 0.7504 0.7396 0.7304 0.7121 0.6836 0.6673 0.6719
0.2324 0.1024 0.1159 0.1638 0.2450 0.2529 0.2503 0.2353 0.2331 0.2494 0.2697 0.2815
55.51 66.48 69.23 70.86 72.53 73.95 76.11 75.91 76.93 74.64 73.97 75.23
Table 3 Ex-ante results of variance changes by hedging period. Hedge period
1 2 3 4 5 6 7 8 9 10 11 12
Separate hedge
Complex hedge
HR-OIL
HR-USD
HE
HR-OIL
HR-USD
HE
0.5486 0.6280 0.6509 0.6630 0.6902 0.6941 0.7006 0.7095 0.7157 0.6983 0.6911 0.7013
0.9496 0.9789 0.9877 0.9870 0.9874 0.9845 0.9856 0.9829 0.9818 0.9801 0.9782 0.9782
54.85 65.76 75.31 76.24 78.81 78.15 75.10 76.35 80.69 81.03 83.03 83.14
0.5280 0.6212 0.6461 0.6632 0.7013 0.7076 0.7137 0.7154 0.7141 0.6895 0.6705 0.6707
0.4233 0.1205 0.0727 0.0030 0.1467 0.2039 0.2276 0.2157 0.2250 0.2564 0.3362 0.4035
58.77 72.69 79.88 81.69 85.56 87.43 88.24 87.65 89.71 86.97 82.92 79.49
However, the hedge ratios for foreign exchange are shown to be quite different by hedge type and period. A noticeable finding is that the hedging effectiveness of complex hedge type turns out to be greater than that of separate hedge type. The exceptions are that the separate hedges are superior to the complex hedges for the hedge period of 11 and 12 months. The improvement of hedging effectiveness by complex hedge based on the ex-post analysis ranges about 4–34% by hedge period. The corresponding results of the ex-ante analysis are about 4–13% by hedge period. These findings imply that the ignorance of the interaction among variables based on the separate hedge might deteriorate the hedging effectiveness as a whole. Another interesting finding is the hedging effectiveness tends to gradually improve as the hedge period increases up to 7 months regardless of the ex-post and exante analyses. 3.2. Factors influencing hedging effectiveness In order to identify the factors influencing the hedging effectiveness derived above, the oil price sensitivity to foreign
ARTICLE IN PRESS W.-C. Yun, H.J. Kim / Energy Policy 38 (2010) 2404–2408
Table 4 Regression results of factors impacting hedging effectiveness. Hedge period 1 2 3 4 5 6 7 8 9 10 11 12
CONST
t-val
46.06 7.09 74.29 18.00 103.24 14.59 117.86 12.45 147.14 9.21 126.70 6.25 98.64 4.94 78.09 3.76 46.40 1.95 5.20 0.16 30.82 0.87 37.71 1.09
PRI
t-val
VOL1 t-val
VOL2
t-val
R2
12.76 5.96 7.27 9.64 9.64 24.39 24.48 21.57 11.30 8.39 5.32 6.50
3.94 2.06 1.17 1.39 0.97 1.67 1.85 1.61 0.85 0.57 0.36 0.40
0.64 0.23 0.46 1.17 2.17 2.70 2.81 3.75 5.53 7.69 7.91 7.15
1.21 2.60 7.74 12.94 21.89 20.16 15.52 15.96 17.93 18.84 15.68 11.29
1.68 3.18 3.08 4.06 5.00 3.83 3.30 3.57 4.05 3.31 2.77 2.26
0.14 0.18 0.39 0.57 0.68 0.62 0.62 0.68 0.77 0.79 0.80 0.77
3.67 1.51 0.98 2.04 2.70 2.54 2.92 4.13 7.10 10.24 11.39 9.94
Note: Critical values for 10%, 5%, and 1% are 1.656, 1.978, and 2.613, respectively.
exchange is analyzed first. For this purpose, the following simple OLS model is specified: _
_
Dpt ¼ a þ b Det þ et
ð18Þ
where pt and et are the crude oil price denominated in US dollar and the exchange rate of Korean won to US dollar, respectively. All the variables are log transformed, and then differenced by 1 week. The above regression model is estimated repeatedly by sequentially moving the sample data. That is, the 1st and 250th _
observations of the whole sample data are used to estimate b1 , _
the 2nd and 251st observations are used to estimate b2 , and so _
forth. This repeating process ends up with 138 estimates of b. The other factors impacting the hedging effectiveness include the volatilities of crude oil prices and exchange rates. The price volatility is defined as the standard deviation of weekly returns. This weekly volatility is converted to annual volatility by multiplying by the square root of 52. As with the above repeated regression, the series of volatilities of crude oil prices and exchange rates are calculated with the rolling-over sample data. The final version of regression model to analyze the impact of price sensitivity and volatilities on the hedging effectiveness is shown as follows: _
_
_
2407
and the hedge period of 1 month for VOL2t. According to this finding, the complex hedging can produce more favorable results when the crude oil market exhibits higher volatility. However, the higher volatility of exchange rates might deteriorate the hedging results.
4. Conclusions This study tries to examine the hedging effectiveness of different hedge type and period by the Korean oil trader facing with both crude oil price and exchange rate risks. For this purpose, the theoretical model is used to estimate the hedge ratios by different hedge type. In addition, some statistical works are performed to investigate the relationship between the hedging effectiveness and the crude oil price sensitivity to foreign exchange and the volatilities of crude oil prices and exchange rates. The empirical results of hedging effectiveness shed lights on some alternatives for the Korean oil traders to deal with the risk exposures related to crude oil price and exchange rate volatilities. The separate hedge dominates the speculation business without any hedging activity in terms of variance reduction. Compared with the separate hedge type, the complex hedge type turns out to be superior. In other words, considering the intercorrelation between commodity price and exchange rate movements would improve the hedging effectiveness. The hedging effectiveness tends to improve as the hedge period increases up to some points. Related to the factors influencing the hedging effectiveness, it is found that there exists an inverse relationship between hedging effectiveness and crude oil price sensitivity to exchange rate. The influences of volatilities of crude oil prices and exchange rates turn out to be opposite. That is, the hedging effectiveness tends to improve when crude oil prices becomes more volatile and/or exchange rate gets less fluctuated.
Acknowledgements This study was supported by the research fund of Hanyang University (HY-2007-1). The authors appreciate the valuable comments by an anonymous referee.
_
HEt ¼ a þ b1 PRIt þ b2 VOL1t þ b3 VOL2t þ et
ð19Þ
Here, HEt is the ex-post hedging effectiveness (in percent) by complex hedge type using the moving sample data. PRIt, VOL1t, and VOL2t are the crude oil price sensitivity to foreign exchange, the volatilities of crude oil prices, and exchange rates, respectively. Table 4 provides the regression results based on Eq. (19). Due to the characteristics of regression with overlapping observation, the t-values are adjusted by using heteroscedasticity and autocorrelation consistent covariance estimation (Newey and West, 1987). It is difficult to make any conclusion about the impact of crude oil price sensitivity to exchange rates on hedging effectiveness. Except for the hedge periods of 1, 2, and 12 months, the signs of estimates are shown to be negative, but are not statistically significant at 5% level. With the negative signs of PRIt, we argue that when the degree of price sensitivity is high, the hedging effectiveness would be relatively low. In contrast, as the crude oil price sensitivity to exchange rate turns to a low level, the hedging effectiveness tends to improve. The influences of volatilities of crude oil prices and exchange rates turn out to be positive and negative, respectively. The estimates are statistically significant at 5% level with some exceptions of the hedge periods of 2 and 3 months for VOL1t
Appendix A. Derivation of complex hedge ratios The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðfti eti ft et ÞFti þ ðxti xt Þfti Xti c ðA:1Þ 2 2 2 2 varðpt Þ ¼ s2se Qti þ s2fe Fti þ s2x fti Xti 2sse;fe Qti Fti
2sse;x fti Qti Xti þ 2sfe;x fti Fti Xti
ðA:2Þ
Assuming the unbiasedness of commodity and currency futures prices, minimizing Eq. (A.2) with respect to the decision variables Ft-i and Xt-i given the spot amount Qt-i gives @ varðpt Þ=@Fti ¼ 2sfe Fti 2sse;fe Qti þ 2sfe;x fti Xti ¼ 0
ðA:3Þ
2 @varðpt Þ=@Xti ¼ 2s2x fti Xti 2sse;x fti Qti þ 2sfe;x fti Fti ¼ 0
ðA:4Þ
Rearranging Eqs. (A.3) and (A.4) gives 2 fe Fti þ
s
sfe;x fti Xti ¼ sse;fe Qti
sfe;x Fti þ s2x fti Xti ¼ sse;x Qti
ðA:5Þ ðA:6Þ
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Dividing Eq. (A.5) and (A.6) by s2fe and s2x , respectively, gives Fti þðsfe;x =s
2 fe Þfti Xti
2 fe ÞQti
¼ ðsse;fe =s
ðsfe;x =s2x ÞFti þ fti Xti ¼ ðsse;x =s2x ÞQti
ðA:7Þ ðA:8Þ
Arranging into the simultaneous equation form gives H11 Fti þH21 fti Xti ¼ H31 Qti
ðA:9Þ
H12 Fti þH22 fti Xti ¼ H32 Qti
ðA:10Þ
where Hjk ¼ sjk =s2k , j ¼ fe ð1Þ; x ð2Þ; se ð3Þ, and k ¼ fe ð1Þ; x ð2Þ. In the matrix form of Ax= d this is " #" # " # Fti H31 Qti H11 fti H21 ¼ ðA:11Þ Xti H32 Qti H12 fti H22 Using Cramer’s rule, Xj ¼ Aj =A, and assuming Qt i = 1, we obtain H31 fti H21 H32 fti H22 ðA:12Þ Fti =Qti ¼ A and H11 H31 H12 H32 Xti =Qti ¼ A
ðA:13Þ
References Ederington, L., 1979. The hedging performance of the new futures markets. Journal of Finance 34, 157–170.
Fackler, P., McNew, K., 1993. Multiproduct hedging: theory, estimation, and an application. Review of Agricultural Economics 15, 521–535. Haigh, M., Holt, M., 2000. Hedging multiple price uncertainty in international grain trade. American Journal of Agricultural Economics 82, 881–896. Haigh, M., Holt, M., 2002. Hedging foreign currency, freight, and commodity futures portfolios: a note. Journal of Futures Markets 22, 1205–1221. Hauser, R., Neff, D., 1993. Export/import risks at alternative stages of U.S. grain export trade. Journal of Futures Markets 13, 579–595. Johnson, L., 1960. The theory of hedging and speculation in commodity futures. Review of Economic Studies 27, 139–151. Johnson, D., Summer, D., 1976. An optimization approach to grain reserves for development countries. In: Analysis of Grain Reserves: A Proceeding, US Economic Research Service, ERS-634, Washington, DC. Kamara, A., 1982. Issues in futures markets: a survey. Journal of Futures Markets 2, 261–294. Lapan, H., Moschini, G., 1994. Futures hedging under price, basis, and production risk. American Journal of Agricultural Economics 76, 465–477. Li, D., Vukina, T., 1998. The effectiveness of dual hedging with price and yield futures. Journal of Futures Markets 18, 541–561. Liu, K., Geaun, J., Lei, L.-F., 2001. Optimal hedging decisions for Taiwanese corn traders on the way of liberalization. Agricultural Economics 25, 303–309. Massell, B., 1969. Price stabilization and welfare. Quarterly Journal of Economics 83, 285–297. Nayak, G., Turvey, C., 2000. Simultaneous hedging of price, crop yield and currency risk. Canadian Journal of Agricultural Economics 48, 123–140. Newbery, D., Stiglitz, I., 1981. In: The Theory of Commodity Price Stabilization: A Study in the Economics of Risk. Clarendon, Oxford. Newey, W., West, K., 1987. A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Thompson, S., Bond, G., 1987. Offshore commodity hedging under floating exchange rates. American Journal of Agricultural Economics 69, 46–55. Vukina, T., Li, D., Holthausen, D., 1996. Hedging with crop yield futures: a mean variance analysis. American Journal of Agricultural Economics 78, 1015–1025. Zhang, Q., Reed, M., Maynard, L., 2007. Hedging decisions of importing firms for US commodity with multiple risks: the case of soybeans. In: Proceeding of American Agricultural Economics Association Annual Meeting, Portland, Oregon.
Contents lists available at ScienceDirect
Energy Policy journal homepage: www.elsevier.com/locate/enpol
Hedging strategy for crude oil trading and the factors influencing hedging effectiveness Won-Cheol Yun a, Hyun Jae Kim b,n a b
Department of Economics and Finance, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea Division of Energy Industry, Korea Energy Economics Institute, 665-1 Naeson-dong, Euiwang-si, Kyunggi-do 437-713, South Korea
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 October 2009 Accepted 14 December 2009 Available online 18 January 2010
This study analyzes the hedging effectiveness of different hedge type and period by Korean oil traders. Both crude oil price and exchange rate risks are considered. Theoretical models are formulated to estimate the hedge ratios by separate and complex hedge types. The hedging period covers 1–12 months. This study also performs some statistical works to investigate the relationship between the hedging effectiveness and the crude oil price sensitivity to exchange rate. In addition, the relationship between the hedging effectiveness and the volatilities of crude oil price and exchange rate is analyzed. & 2009 Elsevier Ltd. All rights reserved.
JEL classification: C15 F31 G32 Keywords: Crude oil Complex hedge Hedging effectiveness
1. Introduction Recently, crude oil prices have been soaring up and climbed to their highest level ever, reaching over $110 per barrel in March 2008. Some analysts suggested that oil prices might soar to $200 per barrel. Since all of crude oil consumed in South Korea is imported from abroad, the recent crude oil price hikes have widespread and negative impact on the national economy. In addition, the exchange rate of Korean won to US dollar has been increasing. This depreciation of Korean won relative to US dollar would be another financial burden to the Korean oil traders importing crude oil from abroad. Under this circumstance, the Korean oil traders should find an appropriate measure to cope with exchange rate risk as well as crude oil price risk. One of them would be for them to enter derivatives markets and to make hedging positions. With respect to crude oil price risk, the exchanges such as the NYMEX in New York and the ICE in London have been offering hedging instruments recognized by world traders. Related to exchange rate risk, the Korea Exchange (KRX) has been trading a US dollar currency futures and option contracts.1 n
Corresponding author. Tel.: + 82 31 420 2228; fax: +82 31 420 2164. E-mail address: [email protected] (H. J. Kim). 1 Detailed specification for this contract is found in www.krx.co.kr. From May 26 of 2006, KRX started the trading of Japanese yen and euro currency futures contracts. Each contract covers 5 million yen and 50 thousand euro, respectively. We expect that these futures contracts would provide domestic and foreign market participants with a proper hedging tool for yen and euro currency risks as well as investment opportunities. 0301-4215/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enpol.2009.12.032
A considerable number of researches deal with the question of trade stability, especially, commodity price and exchange rate risks. The issue addressed in this book of literature is how nations and individual firms can reduce their exposure to these risks. Some suggest risk-conditioning policy strategies such as buffer stocks of major raw materials and foreign exchange pegging (Massell, 1969; Johnson and Summer, 1976). Others propose that futures market is a viable tool by which internationally trading countries can reduce their trade-related risks (Newbery and Stiglitz, 1981). Related to the risk management using futures markets, a growing literature deals with the optimal hedging strategies (Kamara, 1982). Some studies present analytical frameworks for offshore commodity hedging under floating exchange rates (Thompson and Bond, 1987; Liu et al., 2001). The other studies suggest various kinds of hedging strategies for domestic firms to deal with multiple risks such as commodity price and exchange rate (Fackler and McNew, 1993; Lapan and Moschini, 1994; Vukina et al., 1996; Li and Vukina, 1998; Nayak and Turvey, 2000; Liu et al., 2001; Zhang et al., 2007). Other studies explicitly consider the risk exposure of ocean freight rate with commodity price and exchange rate risks (Hauser and Neff, 1993; Haigh and Holt, 2000, 2002). The main objective of this study is to examine the hedging performance of multiple-risk hedging strategy. The strategy can be regarded as a complex hedge considering the intercorrelation between commodity price and exchange rate. In fact, crude oil price and exchange rate are likely to be positively or negatively correlated. As a result, a separate determination of the optimal
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hedge ratios for each of crude oil price and exchange rate could be less efficient compared with the simultaneous determination of hedge ratios accounting for the intercorrelation between these prices. In addition, this study tries to identify the factors affecting hedging effectiveness. For comparison, the hedging effectiveness is evaluated by different hedge type and period. The remainder of this paper is organized as follows. Section 2 lays out an analytical framework that highlights the derivation of separate and complex hedge ratios. Section 3 describes the data used, the empirical results for hedging effectiveness, and the statistical results for factors influencing the hedging effectiveness. Finally, Section 4 summarizes the main results.
2. Theoretical model This study analyzes the hedging effectiveness of a complex hedge type by a Korean oil trader. For comparison purpose, it also considers a separate hedge type compared to the no hedge one. For this purpose, this study sets up a theoretical model based on the well-known mean–variance framework. The optimal hedge ratios are basically the minimum variance ones that are designated to minimize the variances of revenue flows. This study simulates weekly revenue flows of Korean oil trader facing the risk exposures of import price and foreign exchange. Finally, it investigates the relationship between hedging effectiveness and crude oil price sensitivity to exchange rate change and the volatilities of crude oil price and foreign exchanges by hedge period. In a mean–variance framework, the decision maker is assumed to maximize the following objective function: ð1Þ maxEðpt Þ ¼ Eðpt 9Oti Þcðl=2Þvarðpt 9Oti Þ E( ) and var( ) denote the conditional expectation operator and the conditional variance operator, respectively; pt is the end-ofperiod return given the information available at t i (Ot i); c denotes the costs incurred in the transaction; l refers to the decision maker’s coefficient of risk aversion; t i and t stand for the time when a hedge is initiated and the time when the hedge is lifted, respectively. In this study, the hedge period is assumed to be from one month (i= 1) to 12 months (i =12).2
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derived as @ varðpt Þ=@Fti ¼ 2s2fe Fti 2sse;fe Qti ¼ 0
ð4Þ
HR1 ¼ Fti =Qti ¼ sse;fe =s2fe
ð5Þ
2.2. Separate hedge of foreign exchange risk only Assume that the Korean oil trader tries to hedge exchange rate risk only. Still the company’s measure of terminal return would be in terms of Korean won. Protecting it from the adverse changes in exchange rate, he chooses the currency futures quantity Xt i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðxti xt Þfti Xti c
ð6Þ
2 2 2 varðpt Þ ¼ s2se Qti þ s2x fti Xti 2sse;x Qti fti Xti
ð7Þ
where xt is the yen currency futures price at time t in terms of Korean won to US dollar. Assuming the unbiasedness of currency futures prices, minimizing Eq. (7) with respect to the decision variable Xt i given the spot amount Qt i, the minimum variance hedge ratio (HR2) is derived as 2 @varðpt Þ=@Xti ¼ 2s2x fti Xti 2sse;x Qti fti ¼ 0
ð8Þ
2 =Qti ¼ sse;x =s2x fti HR2 ¼ Xti
ð9Þ
2.3. Complex hedge of both price and foreign exchange risks Assume that the Korean oil trader tries to hedge both crude oil price and exchange rate risks. His terminal return would be measured in Korean won. Protecting it from the adverse changes in crude oil price and exchange rate, he chooses both crude oil futures quantity Ft i and the currency futures quantity Xt-i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðfti eti ft et ÞFti þ ðxti xt Þfti Xti c ð10Þ 2 2 2 2 þ s2fe Fti þ s2x fti Xti 2sse;fe Qti Fti varðpt Þ ¼ s2se Qti
2sse;x fti Qti Xti þ2sfe;x fti Fti Xti
ð11Þ
2.1. Separate hedge of crude oil price risk only Assume that the Korean oil trader measures its terminal return in Korean won and that the hedging decision is influenced by crude oil price only. In order to protect against an unfavorable spot crude oil price change, he would choose the crude oil futures quantity Ft i given the spot quantity Qt i. The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þðfti eti ft et ÞFti c
ð2Þ
2 2 varðpt Þ ¼ s2se Qti þ s2fe Fti 2sse;fe Qti Fti
ð3Þ
Considering the interactions among the spot and futures prices of crude oil and exchange rate, the hedge ratios for the crude oil futures and the currency futures turn out to be more complex than HR1 in Eq. (5) and HR2 in Eq. (9). Assuming unbiasedness of both crude oil and currency futures prices, minimizing Eq. (11) with respect to the decision variables Ft-i and Xt-i given the spot amount Qt i, the minimum variance hedge ratios (HR3 and HR4) are derived as3 H31 fti H21 H32 fti H22 ð12Þ HR3 ¼ Fti =Qti ¼ A
where st and ft are the spot and futures price of crude oil at time t in US dollar, respectively; et denotes the exchange rate of Korean won to US dollar; s2x and sx,y are the variance of x variable and the covariance of variables x and y, respectively. Assuming the Martingale hypothesis or unbiasedness of crude oil futures prices, maximizing Eq. (1) with respect to the decision variable Ft-i given the spot amount Qt i is equivalent to minimizing Eq. (3). The minimum variance hedge ratio (HR1) is
where Hjk ¼ sjk =s2k , j ¼ fe ð1Þ; x ð2Þ; se ð3Þ, and k ¼ fe ð1Þ; x ð2Þ.
2 The hedge periods of 1–12 months correspond to the weeks of 4, 8, 13, 17, 21, 26, 30, 34, 39, 43, 47, and 52.
3 A detailed derivation of the complex hedge ratios is available in the Appendix.
and H11 H31 H12 H32 HR4 ¼ Xti =Qti ¼ A
ð13Þ
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3. Empirical results
Table 1 Descriptive statistics of sample data.
This study uses every Friday quotes for the spot and futures prices of crude oil and exchange rates of US dollar denominated in Korean won. The spot crude oil prices are Dubai (Fateh) prices (FOB), and the futures prices correspond to the nearby WTIFutures contracts traded on NYMEX.4 Both spot and futures prices are quoted in dollars per barrel. The data set is available from the database of Energy Information Administration of US Department of Energy (EIA/DOE) for crude oil prices. The KRX offers the spot and futures quotes of exchange rates. The sample covers the period January 2000–June 2007. This weekly data set has 389 observations, and is reduced to 337 observations by deleting the first 52 observations for differencing up to 52 weeks of the maximum hedge period. Table 1 shows the descriptive statistics of sample data. 3.1. Hedging effectiveness of separate and complex hedges In order to simulate the revenue flow each week, it is assumed that the Korean oil trader takes crude oil and currency futures positions based on the hedge ratios derived above. At the end of hedge period the hedging positions are offset, and the trading revenues without and with hedging are calculated. This process repeats each week until the end of sample period. The hedge periods considered range from 1 month to 12 months. The empirical analysis is performed for both in-sample data and outof-sample data. For the out-of-sample analysis, the whole data set is divided into the first sub-sample of January 2001–October 2005 (250 observations) and the second sub-sample of November 2005–June 2007 (87 observations). To estimate the hedge ratios by separate hedge type, this study adopts a simple OLS model. With the OLS estimation, the following equations are specified to estimate b^ 1 (HR1) for crude oil and b^ 2 (HR2) for foreign exchange, respectively: ðst sti Þ ¼ a^ 1 þ b^ 1 ðft fti Þ þ e1t
ð14Þ
ðet eti Þ ¼ a^ 2 þ b^ 2 ðxt xti Þ þ e2t
ð15Þ
For the estimation of complex hedge ratios (HR3 and HR4), the following equation is specified to estimate Hjk that appears in Eqs. (12) and (13): ^ jk Dk þ et Dj ¼ a^ þ H
ðj ¼ 1; 2; 3;
k ¼ 1; 2Þ
ð16Þ
where D1 = (ftet ft iet i), D2 = (xt xt i), and D3 =(stet st iet i). Following Johnson (1960) and Ederington (1979), the hedging effectiveness (HE) refers to the gain or loss in the variance of terminal revenue resulting from the price changes in an unhedged position relative to those in a hedged position defined as HE ¼
ðvaru ðpt Þvarh ðpt ÞÞ varu ðpt Þ
ð17Þ
where varu(pt) and varh(pt) are the variances for the unhedged and hedged position, respectively. Tables 2 and 3 present the ex-post and ex-ante simulation results by different hedge type and period. The estimated hedge ratios for crude oil turn out to be similar regardless of hedge type. 4 Since over 80% of imported crude oil by South Korea comes from Middle East countries, the Dubai crude oil prices are chosen as spot prices. However, until the launch of Dubai Mercantile Exchange (DME) as of June 2007 there was no exchange traded Middle East Sour Crude benchmark, which hindered risk management for Middle East Sour Crudes. In this context, the hedging scenario adopted in this paper is viewed as a cross-hedge. The Brent crude oil futures listed in the NYMEX and the ICE would be a possible candidate instead of using the WTI crude oil futures.
Variable
Mean
Std. dev.
Minimum
Maximum
Dubai-Spot WTI-Futures USD-Spot USD-Futures
36.36 41.07 1128.27 1129.63
15.10 15.88 121.96 123.54
16.55 18.00 920.30 919.50
71.35 77.03 1345.40 1367.00
Table 2 Ex-post results of variance changes by hedging period. Hedge period
1 2 3 4 5 6 7 8 9 10 11 12
Separate hedge
Complex hedge
HR-OIL
HR-USD
HE
HR-OIL
HR-USD
HE
0.5774 0.6793 0.7067 0.7227 0.7493 0.7447 0.7370 0.7346 0.7223 0.6999 0.6912 0.6993
0.9366 0.9688 0.9835 0.9826 0.9819 0.9814 0.9827 0.9827 0.9835 0.9826 0.9826 0.9822
51.34 56.31 56.68 54.97 51.82 51.27 50.83 50.02 48.27 40.48 37.39 40.52
0.5689 0.6829 0.7119 0.7305 0.7572 0.7504 0.7396 0.7304 0.7121 0.6836 0.6673 0.6719
0.2324 0.1024 0.1159 0.1638 0.2450 0.2529 0.2503 0.2353 0.2331 0.2494 0.2697 0.2815
55.51 66.48 69.23 70.86 72.53 73.95 76.11 75.91 76.93 74.64 73.97 75.23
Table 3 Ex-ante results of variance changes by hedging period. Hedge period
1 2 3 4 5 6 7 8 9 10 11 12
Separate hedge
Complex hedge
HR-OIL
HR-USD
HE
HR-OIL
HR-USD
HE
0.5486 0.6280 0.6509 0.6630 0.6902 0.6941 0.7006 0.7095 0.7157 0.6983 0.6911 0.7013
0.9496 0.9789 0.9877 0.9870 0.9874 0.9845 0.9856 0.9829 0.9818 0.9801 0.9782 0.9782
54.85 65.76 75.31 76.24 78.81 78.15 75.10 76.35 80.69 81.03 83.03 83.14
0.5280 0.6212 0.6461 0.6632 0.7013 0.7076 0.7137 0.7154 0.7141 0.6895 0.6705 0.6707
0.4233 0.1205 0.0727 0.0030 0.1467 0.2039 0.2276 0.2157 0.2250 0.2564 0.3362 0.4035
58.77 72.69 79.88 81.69 85.56 87.43 88.24 87.65 89.71 86.97 82.92 79.49
However, the hedge ratios for foreign exchange are shown to be quite different by hedge type and period. A noticeable finding is that the hedging effectiveness of complex hedge type turns out to be greater than that of separate hedge type. The exceptions are that the separate hedges are superior to the complex hedges for the hedge period of 11 and 12 months. The improvement of hedging effectiveness by complex hedge based on the ex-post analysis ranges about 4–34% by hedge period. The corresponding results of the ex-ante analysis are about 4–13% by hedge period. These findings imply that the ignorance of the interaction among variables based on the separate hedge might deteriorate the hedging effectiveness as a whole. Another interesting finding is the hedging effectiveness tends to gradually improve as the hedge period increases up to 7 months regardless of the ex-post and exante analyses. 3.2. Factors influencing hedging effectiveness In order to identify the factors influencing the hedging effectiveness derived above, the oil price sensitivity to foreign
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Table 4 Regression results of factors impacting hedging effectiveness. Hedge period 1 2 3 4 5 6 7 8 9 10 11 12
CONST
t-val
46.06 7.09 74.29 18.00 103.24 14.59 117.86 12.45 147.14 9.21 126.70 6.25 98.64 4.94 78.09 3.76 46.40 1.95 5.20 0.16 30.82 0.87 37.71 1.09
PRI
t-val
VOL1 t-val
VOL2
t-val
R2
12.76 5.96 7.27 9.64 9.64 24.39 24.48 21.57 11.30 8.39 5.32 6.50
3.94 2.06 1.17 1.39 0.97 1.67 1.85 1.61 0.85 0.57 0.36 0.40
0.64 0.23 0.46 1.17 2.17 2.70 2.81 3.75 5.53 7.69 7.91 7.15
1.21 2.60 7.74 12.94 21.89 20.16 15.52 15.96 17.93 18.84 15.68 11.29
1.68 3.18 3.08 4.06 5.00 3.83 3.30 3.57 4.05 3.31 2.77 2.26
0.14 0.18 0.39 0.57 0.68 0.62 0.62 0.68 0.77 0.79 0.80 0.77
3.67 1.51 0.98 2.04 2.70 2.54 2.92 4.13 7.10 10.24 11.39 9.94
Note: Critical values for 10%, 5%, and 1% are 1.656, 1.978, and 2.613, respectively.
exchange is analyzed first. For this purpose, the following simple OLS model is specified: _
_
Dpt ¼ a þ b Det þ et
ð18Þ
where pt and et are the crude oil price denominated in US dollar and the exchange rate of Korean won to US dollar, respectively. All the variables are log transformed, and then differenced by 1 week. The above regression model is estimated repeatedly by sequentially moving the sample data. That is, the 1st and 250th _
observations of the whole sample data are used to estimate b1 , _
the 2nd and 251st observations are used to estimate b2 , and so _
forth. This repeating process ends up with 138 estimates of b. The other factors impacting the hedging effectiveness include the volatilities of crude oil prices and exchange rates. The price volatility is defined as the standard deviation of weekly returns. This weekly volatility is converted to annual volatility by multiplying by the square root of 52. As with the above repeated regression, the series of volatilities of crude oil prices and exchange rates are calculated with the rolling-over sample data. The final version of regression model to analyze the impact of price sensitivity and volatilities on the hedging effectiveness is shown as follows: _
_
_
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and the hedge period of 1 month for VOL2t. According to this finding, the complex hedging can produce more favorable results when the crude oil market exhibits higher volatility. However, the higher volatility of exchange rates might deteriorate the hedging results.
4. Conclusions This study tries to examine the hedging effectiveness of different hedge type and period by the Korean oil trader facing with both crude oil price and exchange rate risks. For this purpose, the theoretical model is used to estimate the hedge ratios by different hedge type. In addition, some statistical works are performed to investigate the relationship between the hedging effectiveness and the crude oil price sensitivity to foreign exchange and the volatilities of crude oil prices and exchange rates. The empirical results of hedging effectiveness shed lights on some alternatives for the Korean oil traders to deal with the risk exposures related to crude oil price and exchange rate volatilities. The separate hedge dominates the speculation business without any hedging activity in terms of variance reduction. Compared with the separate hedge type, the complex hedge type turns out to be superior. In other words, considering the intercorrelation between commodity price and exchange rate movements would improve the hedging effectiveness. The hedging effectiveness tends to improve as the hedge period increases up to some points. Related to the factors influencing the hedging effectiveness, it is found that there exists an inverse relationship between hedging effectiveness and crude oil price sensitivity to exchange rate. The influences of volatilities of crude oil prices and exchange rates turn out to be opposite. That is, the hedging effectiveness tends to improve when crude oil prices becomes more volatile and/or exchange rate gets less fluctuated.
Acknowledgements This study was supported by the research fund of Hanyang University (HY-2007-1). The authors appreciate the valuable comments by an anonymous referee.
_
HEt ¼ a þ b1 PRIt þ b2 VOL1t þ b3 VOL2t þ et
ð19Þ
Here, HEt is the ex-post hedging effectiveness (in percent) by complex hedge type using the moving sample data. PRIt, VOL1t, and VOL2t are the crude oil price sensitivity to foreign exchange, the volatilities of crude oil prices, and exchange rates, respectively. Table 4 provides the regression results based on Eq. (19). Due to the characteristics of regression with overlapping observation, the t-values are adjusted by using heteroscedasticity and autocorrelation consistent covariance estimation (Newey and West, 1987). It is difficult to make any conclusion about the impact of crude oil price sensitivity to exchange rates on hedging effectiveness. Except for the hedge periods of 1, 2, and 12 months, the signs of estimates are shown to be negative, but are not statistically significant at 5% level. With the negative signs of PRIt, we argue that when the degree of price sensitivity is high, the hedging effectiveness would be relatively low. In contrast, as the crude oil price sensitivity to exchange rate turns to a low level, the hedging effectiveness tends to improve. The influences of volatilities of crude oil prices and exchange rates turn out to be positive and negative, respectively. The estimates are statistically significant at 5% level with some exceptions of the hedge periods of 2 and 3 months for VOL1t
Appendix A. Derivation of complex hedge ratios The end-of-period return and its variance are given by
pt ¼ ðst et sti eti ÞQti þ ðfti eti ft et ÞFti þ ðxti xt Þfti Xti c ðA:1Þ 2 2 2 2 varðpt Þ ¼ s2se Qti þ s2fe Fti þ s2x fti Xti 2sse;fe Qti Fti
2sse;x fti Qti Xti þ 2sfe;x fti Fti Xti
ðA:2Þ
Assuming the unbiasedness of commodity and currency futures prices, minimizing Eq. (A.2) with respect to the decision variables Ft-i and Xt-i given the spot amount Qt-i gives @ varðpt Þ=@Fti ¼ 2sfe Fti 2sse;fe Qti þ 2sfe;x fti Xti ¼ 0
ðA:3Þ
2 @varðpt Þ=@Xti ¼ 2s2x fti Xti 2sse;x fti Qti þ 2sfe;x fti Fti ¼ 0
ðA:4Þ
Rearranging Eqs. (A.3) and (A.4) gives 2 fe Fti þ
s
sfe;x fti Xti ¼ sse;fe Qti
sfe;x Fti þ s2x fti Xti ¼ sse;x Qti
ðA:5Þ ðA:6Þ
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Dividing Eq. (A.5) and (A.6) by s2fe and s2x , respectively, gives Fti þðsfe;x =s
2 fe Þfti Xti
2 fe ÞQti
¼ ðsse;fe =s
ðsfe;x =s2x ÞFti þ fti Xti ¼ ðsse;x =s2x ÞQti
ðA:7Þ ðA:8Þ
Arranging into the simultaneous equation form gives H11 Fti þH21 fti Xti ¼ H31 Qti
ðA:9Þ
H12 Fti þH22 fti Xti ¼ H32 Qti
ðA:10Þ
where Hjk ¼ sjk =s2k , j ¼ fe ð1Þ; x ð2Þ; se ð3Þ, and k ¼ fe ð1Þ; x ð2Þ. In the matrix form of Ax= d this is " #" # " # Fti H31 Qti H11 fti H21 ¼ ðA:11Þ Xti H32 Qti H12 fti H22 Using Cramer’s rule, Xj ¼ Aj =A, and assuming Qt i = 1, we obtain H31 fti H21 H32 fti H22 ðA:12Þ Fti =Qti ¼ A and H11 H31 H12 H32 Xti =Qti ¼ A
ðA:13Þ
References Ederington, L., 1979. The hedging performance of the new futures markets. Journal of Finance 34, 157–170.
Fackler, P., McNew, K., 1993. Multiproduct hedging: theory, estimation, and an application. Review of Agricultural Economics 15, 521–535. Haigh, M., Holt, M., 2000. Hedging multiple price uncertainty in international grain trade. American Journal of Agricultural Economics 82, 881–896. Haigh, M., Holt, M., 2002. Hedging foreign currency, freight, and commodity futures portfolios: a note. Journal of Futures Markets 22, 1205–1221. Hauser, R., Neff, D., 1993. Export/import risks at alternative stages of U.S. grain export trade. Journal of Futures Markets 13, 579–595. Johnson, L., 1960. The theory of hedging and speculation in commodity futures. Review of Economic Studies 27, 139–151. Johnson, D., Summer, D., 1976. An optimization approach to grain reserves for development countries. In: Analysis of Grain Reserves: A Proceeding, US Economic Research Service, ERS-634, Washington, DC. Kamara, A., 1982. Issues in futures markets: a survey. Journal of Futures Markets 2, 261–294. Lapan, H., Moschini, G., 1994. Futures hedging under price, basis, and production risk. American Journal of Agricultural Economics 76, 465–477. Li, D., Vukina, T., 1998. The effectiveness of dual hedging with price and yield futures. Journal of Futures Markets 18, 541–561. Liu, K., Geaun, J., Lei, L.-F., 2001. Optimal hedging decisions for Taiwanese corn traders on the way of liberalization. Agricultural Economics 25, 303–309. Massell, B., 1969. Price stabilization and welfare. Quarterly Journal of Economics 83, 285–297. Nayak, G., Turvey, C., 2000. Simultaneous hedging of price, crop yield and currency risk. Canadian Journal of Agricultural Economics 48, 123–140. Newbery, D., Stiglitz, I., 1981. In: The Theory of Commodity Price Stabilization: A Study in the Economics of Risk. Clarendon, Oxford. Newey, W., West, K., 1987. A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Thompson, S., Bond, G., 1987. Offshore commodity hedging under floating exchange rates. American Journal of Agricultural Economics 69, 46–55. Vukina, T., Li, D., Holthausen, D., 1996. Hedging with crop yield futures: a mean variance analysis. American Journal of Agricultural Economics 78, 1015–1025. Zhang, Q., Reed, M., Maynard, L., 2007. Hedging decisions of importing firms for US commodity with multiple risks: the case of soybeans. In: Proceeding of American Agricultural Economics Association Annual Meeting, Portland, Oregon.