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Hedging foreign exchange exposure in the Japanese Yen
HEDGING FOREIGN EXCHANGE EXPOSURE IN THE JAPANESE YEN A.F. Herbst D.D. Kare S.C. Caples
During the 198Os, the exchange rate between the U.S. dollar and the Japanese yen (Y) had been subject to extreme volatility. The early years of the decade were marked by a strong dollar and the exchange rate hovered around 250 yen to the dollar. However, a slow erosion in the strength of the dollar occurred due to mounting trade deficits, and by the end of the decade, the exchange rate was about 125 yen per dollar. These dramatic fluctuations in the dollar-yen exchange rate have increased the foreign exchange risk borne by financial institutions, exporters and importers engaged in bilateral trade. The need for hedging the foreign exchange exposure against an adverse exchange rate movement is therefore obvious. Fortunately, the Japanese yen contracts in the foreign currency futures market and the forward foreign currency market provide the means for hedging foreign exchange exposure in the Japanese yen. One of the first attempts to examine the financial futures market for hedging effectiveness was a study by Ederington [7], who built upon the foundation developed by Johnson [14] and Stein 1201. Ederington investigated the Government National Mortgage Association (UNMAN and T-Bill futures contracts within the context of Markowitz Portfolio Theory (MET). Ederington showed that the optimal hedge ratio in most cases is significantly different from the traditional one-to-one ratio of futures to spot position holdings. He suggested that even pure risk minimizers should hedge only a portion of their entire spot portfolios. Franckle [8] pointed out some errors in Ederington’s study. He showed that a crucial assumption for using the model is a predetermined hedge period. Also, he extended Ederington’s work by attempting to estimate the effects of changing maturities on the valance-minimiz~g hedge ratio in the T-Bill market, and found that by matching futures prices with the correct T-Bill prices the market should be shown to be much more efficient than Ederington’s results. Franckle A.F. Herb& Department of Economics & Finance, The University of Texas at El Paso, El Paso, TX 79968, and King Fahd University of Petroleum and Minerals, Department of Finance & Economics, Dhahran, Saudi Arabia, 31261; D.D. Kare Department of Accounting and Finance, University of North Florida, Jacksonville, FL 32216; SC. Caples Department of Finance and Economics, McNeese State University, Lake Charles, LA 70609. Global Finance Journal, ISSN: 1044-0283
2(3/4), 243-253
Copyright D 1991 by JAI Press, Inc. All rights of reproduction in any form reserved.
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showed that the use of daily prices in a study of the futures market was superior to the use of weekly or monthly prices, such as Friday settlement. The use of daily prices yields a higher estimate of the optimal hedge ratio and causes a larger reduction in the variance of the portfolio. Hill and Schneeweis [ll] explored the hedging potential of the foreign currency futures market by estimating optimal hedge ratios and hedging effectiveness measures for five foreign currencies and twenty futures contracts. They concluded that the British pound, the German mark, and the Swiss franc futures contracts have consistently high hedging effectiveness, and that the hedging effectiveness increased when the duration of the hedge was increased. In contrast, they showed that short-term contracts in the Canadian dollar and contracts of all maturities in the Japanese yen were inferior hedging instruments, but were nevertheless superior to an unhedged position. Hill and Schneeweis also examined the hedging effectiveness and the optimal hedge ratios for forward contracts in the British pound, the German mark, and the Japanese yen. They found that the mark and pound futures and forward contracts were indistinguishable in terms of hedging effectiveness for both 90and 180-day positions. The hedge ratio for the yen futures contracts were found to be substantially lower compared with the hedge ratios for forward contracts, and forward contracts were found to be more effective in hedging the yen exposure. Hill and Schneeweis attributed this to the newness and thinness in the futures market. Swanson and Caples [21] estimated the optimal hedge ratios and hedging effectiveness of forward foreign currency contracts in two currencies: the British pound and the German mark. They criticized previous studies on foreign currency hedging for ignoring the existence of serially correlated error terms. Swanson and Caples also used daily data on spot and forward contracts as opposed to weekly data used by Hill and Schneeweis. Herbst, Kare, and Caples [lo] compared the hedging effectiveness for five currency futures contracts for OLS hedge ratios versus Box-Jenkins [2] transfer function coefficients. They found that the Box-Jenkins transfer function coefficients yielded lower variance of hedge outcomes than did the OLS coefficients.
I. PURPOSE
& METHODOLOGY
The purpose of this study is to derive improved estimates of optimal hedge ratios and hedging effectiveness in the Japanese yen market using statistical techniques which were specifically developed to analyze time series data. For hedge ratios will also be estimated using the purpose of comparison, Ederington’s technique (i.e., OLS regression). Early studies made no attempt to determine the extent of autocorrelation in the spot and futures prices used. The data on spot and futures prices also violated some other assumptions of OLS regression, leading to statistically inconsistent estimates of optimal hedge ratios. The sample sizes tended to be small in those studies, thus making generalization of their results tenuous. This study
Hedging Foreign Exchange Exposure in the Japanese Yen
245
uses time series analysis techniques developed by Box and Jenkins [2] to arrive at unbiased estimates of optimal hedge ratios and hedging effectiveness.
OLS Regression In his work with financial futures, Ederington [7] first showed that OLS regression can be used to estimate the optimal hedge ratio between spot and futures contracts of U.S. Treasury bills. The most effective hedge is obtained when the portfolio variance (i.e., the exchange-rate risk) is reduced to a minimum. Ederington showed that the minimum variance (minimum risk) portfolio is obtained when the relative investment in futures contracts, b”, is b” = u /a2 $f
f
(1)
where osf = covariance between spot rates and futures rates 02 = variance of futures rates. f This is the slope of the line that regresses rates, i.e.,
futures contract rates against
SP, = a + b* FU, + e,
spot
(2)
where SF’, is the spot rate for the underlying currency, FL& is the futures contract rate for the currency, e, is the random error term, and b* is the optimal hedge ratio. Ederington further showed that the effectiveness of this hedge can be measured by the fraction of the variance of the unhedged portfolio that is eliminated by hedging. The effectiveness of the hedge turns out to be the coefficient of determination (X2 in the estimation of equation (2). A similar equation can be used to determine the parameters of hedging using forward contracts. Ederington’s technique of estimating optimal hedge ratios using OLS regression will yield statistically unbiased and consistent estimators only when the data used for estimation satisfy the following stringent assumptions of OLS regression: (1) (2) (3) (4)
Normality: e, (the error term) is normally distributed. Zero mean: E(e,) = 0. Homoscedasticity: the error terms have a constant variance. Nonautocorrelation: E(eiej) = 0 (i is not equal to j).
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Nonstochastic independent variable: the independent variable is a nonstochastic variable with values fixed in repeated samples.
However, as Franckle pointed out, time series data on spot and futures rates of foreign currencies show significant serial correlation and therefore, Ederington’s technique yields statistically inconsistent results. One way to account for the presence of serial correlation in data is to use an autoregressive model which yields consistent estimators in the presence of autoregressive error terms. An autoregressive model was tried, but the error terms in this estimation were still found to be correlated across time (the results are not presented here to preserve the journal space). Consequently, this technique was abandoned in favor of Box-Jenkins auto-regressive integrated moving average (ARIMA) model.
Box-Jenkins
Model
The second methodology used in this study is a technique developed by Box and Jenkins [2]. This methodology can be used to obtain superior results when time series possesses high autocorrelation. The Box-Jenkins method is a threestep approach to model building. The first stage is model identification or specification in which the appropriate structural form is selected to fit the time series data. The second stage is estimation of parameters from the data. Finally, the model selected is tested (diagnostic checking) to see if it actually fits the data properly; if the data being tested do not fit the model, the process is repeated. The Box-Jenkins process, an autoregressive integrated moving average model (ARIMA), is a process considered to be driven by a series of random shocks (or white noise). Each observation of a white noise process is assumed to be drawn randomly and independently from a normal distribution with a zero-mean and constant variance. A white noise shock enters an ARIMA (p, d, 4) model, passes through a series of filters or “black boxes, ” and leaves as a time series observation. These “black boxes” introduce correlation structure or memory into the time series. The process involved is to reconstruct the proper ARIMA (p, d, 4) model to describe the time series. The parameter p describes the number of error terms included in the estimation of the equation to eliminate the serial correlation. The parameter d is the level of differencing employed on the time series data. The parameter 4 is the number of moving average terms which are accounted for in the estimation. The ARIMA model is a series of filters that determine the properties of the output time series. The choice of filters is based on the knowledge of the process generating the time series data. An ARIMA (0, 1, 1) model will be used to describes a series in which differenced data behaves as if it were generated by a moving average process. An ARIMA (1, 1, 0) filter, on the other hand, describes a series that, when differenced, behaves as if it were generated by an autoregressive process. For more complex time series data, the proper values of the param-
Hedging Foreign Exchange Exposure in the Japanese Yen
247
eters p, d and 9 will be decided through a trial and error procedure. Those values which best fit the data to the estimated function are normally used. The mathematical model used in this study to estimate optimal hedge ratios is (1 - B)SP, = Y(B) FU, + e(g)/~(g)e~ where I? is the backshift
operator,
(3)
such that B(X,) = X,_,
Y(B) is the transfer function, B(B) is the autoregressive operator, (P(B) is the moving average operator. Both B(B) and a,(B) are polynomials in B. The nature of these polynomials is determined by the lag structure and the moving average structure in the data used. The appropriate models used in the case of futures contracts and forward contracts are presented in Table 2. The above model can be also used to estimate optimal hedge ratios and hedging effectiveness for forward contracts.
DATA Data on the Japanese yen spot, forward and futures rates were collected for the period beginning with January 1, 1988, and ending with July 31, 1989. Data on spot rates and 90-day forward contract rates were collected from the l&l2 Street Joournal. The data on perpetual contracts were purchased from Commodity Systems, Inc., of Boca Raton, Florida. The choice between using daily data and weekly/monthly data warrants some discussion. One drawback of using weekly data is that very few observations can be used in a study. But, more importantly, Fran&e 181, in his critique of the Ederington study, showed that a greater amount of risk reduction (i.e., variance reduction) can be achieved using daily data compared with the risk reduction achieved using weekly data. This is because there is a greater covariance between daily spot and futures prices than the covariance in weekly (or monthly) data. Based on the above reasons, it was decided to use daily data on spot, forward and futures rates. One problem encountered in any analysis involving futures contracts is that the individual contracts expire .I This introduces what may be termed as the “maturity bias” in the estimates of the optimal hedge ratio. For example, if a four week hedge was placed using a 90-day futures contract, by the time the hedge is lifted the contract only has 62 days left till maturity. The relations~p between spot rates and futures rates (i.e., the basis) will keep changing in a hedge of this type. Both Ederington [7] and Fran&e [S] recognized this problem and suggested that the optimal hedge ratios be modified to account for the change in basis. To address this problem, the concept of perpetual contract was developed
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by Pelletier 1171. A perpetual contract rate is calculated for a specific day in the future by taking a weighted average of the rates of the two futures contracts with expiry dates closest to the specified day. This artificial device helps in constructing a hedge that has no maturity bias and the basis remains constant. Further, because the perpetual contract has no maturity date, a large number of daily observations could be used in this study. This study uses 372 observations, compared with 14 observations used by Hill and Schneeweis (hereafter HS). In studies using weekly or monthly futures market data, the rate on a particular day must be selected arbitrarily as the basis for comparison. That choice may bias the results of data analysis signific~tly. The use of daily data and perpetual contracts can eliminate that problem. Another problem facing a researcher using daily data on futures markets is the problem of missing values for holidays. Box-Jenkins ARIMA technique can estimate the missing values. This feature is extremely important in statistical analysis of time series data, because missing values would tend to mask any periodicities (as well as holiday effects) that might be present in the data.
III. RESULTS Table 1 presents the resuhs of OLS regression performed on futures and forward price data. Estimates of optimal hedge ratios using OLS regression appear to be close to one in all cases. However, estimates for both futures and forward contracts are significantly different from 1.0 at the 99 percent confidence level. The R2 statistic, which measures hedging effectiveness, also appears to be quite high, close to one in all cases. The interpretation of the R2 statistic warrants some caution, because of the presence of autocorrelation in the data.
RESULTS Equation: Equation: Estimates a
b* X2 D-W stat.
Table 1 OF OLS REGRESSION
SP, = d + b*FU, + e, (Futures Contract) SP, = d + b”FOR, + e, (Forward Contract) Futures Contracts 0.1431 (0.98) 1.0542 (64.7) 0.9543 1.1320
Forward Contracts 0.1154 (1.432) 1.0080 (202.6) 0.9911 1.0980
(T’-statistics in parentheses. T-statistics for b”, the optimal hedge ratio, are statistically significant at the 99% confidence level.)
Hedging Foreign Exchange Exposure in the Japanese Yen
RESULTS Futures contracts: Forward contract: Hedging
Table 2 OF ARIMA
249
MODELS
(1 - B) SP, = b” (1 - B) FLI, + (1 - .983B)e, (1 - B) SP, = b” (1 - B) FOR, + (1 - .957B)e,
Device
Opt. Hedge Ratio (b”)
T-Statistic
.9856 .9744
33.26” 128.11*
1. FUTURES CONTRACT 2. FORWARD CONTRACT
(” Indicates that the t-statistic is significant at the 99.99 percent confidence
level.)
A major problem with the use of OLS regression is highlighted by the DurbinWatson statistic, which is significant in both cases. This indicates the presence of autoregressive disturbances in data used in this study. Once again, the presence of autoregressive disturbances violates a basic assumption of OLS regression analysis, causing overestimation of the X2 statistic and yielding a biased estimate of the optimal hedge ratio (i.e., the slope coefficient). It also causes an underestimation of the variance of error terms. The presence of autoregressive disturbances in the data for both forward and futures contracts justifies the search for a statistical technique that can adjust for it. Autoregressive models proved to be no improvement, because error terms showed “infinite memory.” Table 2 shows the results of estimates using ARlMA models applied to futures and forward contract price data. Estimates of optimal hedge ratios using this technique are lower than those in Table 1. The R* statistics are not presented in Table 2, because in the case of the ARIMA technique this statistic is not very meaningful (Box and Jenkins [2J). This is because the parameter estimation in the ARIMA technique is carried out through a trial and error procedure until the best possible fit is achieved. Therefore, the X2 statistic in the ARIMA analysis is always very close to 1. The estimated optimal hedge ratios are close to 1 .O but they were found to be statistically different from 1 .O at the 99 percent confidence level. Table 3 presents a comparative analysis of optimal hedge ratios estimated
COMPARISON Hedging
Device
1. FUTURES CONTRACT 2. FORWARD CONTRACT”
Table 3 OF OPTIMAL
HEDGE
RATIO
OLS Est.
ARIMA Est.
% Reduction
1.0542 1.0080
.9856 .9744
6.51 3.33
(” Indicates that the estimates from two techniques from each other at the 99 percent confidence level.)
are significantly
different
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Table 4 COMPARISON OF HEDGING EFFECTIVENESS (VARIANCE OF RESIDUALS x lo61 Hedging
Device
OLS Est.
ARIMA Est.
% Reduction
1. FUTURES CONTRACT 2. FORWARD CONTRACT
57.45 36.87
21.58 14.63
62.44 60.32
using the two alternative techniques. Optimal hedge ratios estimated using ARIMA models are lower in both cases, the reduction is 6.51 percent for the futures contract and 3.33 percent for the forward contract. Table 4 lists estimates of the variance of residuals (error terms) in both OLS regression and ARIMA technique. Again, ARIMA estimates are substantially lower compared to OLS estimates. The use of R2 to measure hedging effectiveness as suggested by Ederington creates a couple of problems. In OLS regression, the R* statistic is overestimated because of the presence of autoregressive disturbances. The R2 statistic in ARIMA models, on the other hand, may lead to misleading interpretation, because of the reasons cited above. It is therefore proposed that a better comparison of hedging effectiveness could be made by comparing the reduction in the variance of the hedged position (to estimate the minimum risk portfolio). Reduction in the variance of the hedged position is inversely related to the variance of residuals; i.e., the larger the reduction of variance in the hedged position, the smaller the variance of the residuals. This suggests that the variance of residuals should be used as the measure ofhedging effectiveness. In the case of both forward and futures contracts, ARIMA models yield estimates of residual variances that are substantially lower compared to those in OLS regression. The reduction on average is 61 percent. Table 5 presents a comparison between results from this study and the results from the Hill-Schneeweis (HS) study. It is worth noting that the optimal hedge
THIS
Table 5 COMPARATIVE RESULTS STUDY V/S HILGSCHNEEWEIS
1. THIS STUDY (OLS) 2. THIS STUDY (ARIMA) 3. H-S STUDY (90 day)
STUDY
Futures
Forward
1.0542* .9856* .505
1.0080* .9744* 1.152
(*Indicates that the estimates are significantly from 1.0 at the 99 percent confidence level.)
different
Hedging Foreign Exchange Exposure in the Japanese Yen
251
ratios obtained in the HS study were not significantly different from 1.0 for both the forward and futures contracts. Optimal hedge ratios from this study are lower in both cases, and significantly different from 1.0 at the 99 percent confidence level. A meaningful comparison of the reduction in variance cannot be made because data on residuals of the HS study are unavailable. For the reasons cited previously, a comparison of the Ra statistic is not presented. Various explanations can be offered for the difference in the two sets of results. This study uses 372 observations, while the HS study was based on 14 observations. This study uses daily data as opposed to weekly data used in the HS study. Finally, the periods over which data were collected are also different.
IV. ECONOMIC
IMPLICATIONS
The finding that the use of ARIMA models to account for autoregressive disturbances in data causes a reduction in estimated optimal hedge ratios for the Japanese yen has important economic implications for a hedger. The extent of the reduction is between 3.3 to 6.5 percent. This means that hedgers in the Japanese yen really need a smaller investment in the futures contracts or forward contracts of the same currency. Considering the amounts typically involved in foreign currency transactions, this reduction in optimal hedge ratio would result in significant benefits to the hedger. Second, the comparison of variances of residuals, which should be considered the proper measure of hedging effectiveness, shows that the optimal hedge ratios estimated using ARIMA models succeed in creating portfolios which have substantially lower variance of residuals compared to portfolios created using OLS optimal hedge ratios. This implies that a hedger using hedge ratios based on ARIMA estimates would be exposed to an exchange rate risk which is substantially lower compared to a hedger using OLS estimates. Also, since fewer futures contracts would be needed, margin and commission costs would be reduced. ARIMA models appear to yield estimates of optimal hedge ratios which not only reduce the costs of hedging a spot position, but also reduce the foreign exchange exposure by nearly 60 percent.
V. CONCLUSION The results of this investigation of spot, forward and futures price data on the Japanese yen show that the optimal hedge ratios calcuIated using the methodology common to prior studies (i.e., OLS regression) were overestimated. The reason for this overestimation may be the presence of autoregressive disturbances in the exchange rate data and the use of weekly (or monthly) data. The presence of autoregressive disturbances violates a basic assumption underlying OLS regression leading to inconsistent estimates, and as Franckle [S] pointed out, the use of weekly or monthly data also leads to the overestimation of optimal hedge ratios.
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This investigation departs from prior studies in many respects: (1) It uses daily data as opposed to the use of weekly or monthly data in past studies; (2) It employs the concept of constant maturity perpetual contracts, allowing the use of a large number of observations to model the behavior of price data. In contrast, previous researchers based their results on the analysis of a small number of observations; (3) The statistical technique used in this study generates missing values for holidays and weekends, while past studies ignored missing values; (4) Finally, the most important departure from previous studies is that this study uses the ARIMA methodology developed by Box and Jenkins to model the relationship between spot and futures prices. This technique successfully solves the vexing problem of autoregressive disturbances in the data which renders OLS regression technique of dubious value. Results yielded by ARIMA methodology show significant improvement over results from OLS regression. Estimates of optimal hedge ratios were lower. The reduction in optimal hedge ratios was 3.3 percent for futures contracts and 6.51 percent for forward contracts. Considering the large dollar amounts generally involved in foreign currency transactions, this reduction in optimal hedge ratios will yield a substantial reduction in the margin deposit required of hedgers and a corresponding reduction in transaction costs. The minimum risk portfolio created using ARIMA estimates is subject to a considerably lower risk compared to portfolios created using OLS regression. The risk reduction achieved ranges between 60 and 62 percent. NOTES 1. A particular problem exists when 90-day futures contracts are compared with 90-day forward contracts because futures contracts are written only for four delivery months a year, but forward contracts are written for delivery on any day 90 days in the future. In some studies that compare the two contracts (most notably Hill and Schneeweis [ll]) only thirteen to eighteen observations were available for each currency, even though the study involved several years. BIBLIOGRAPHY [l] B&on, John F.O., “Rational Expectations and the Exchange Rate,” in Jacob A. Frenkel and Harry G. Johnson (eds.), The Economics of Exchurzge Rates, Addison-Wesley: Reading, MA, 1978. [2] Box, George E.P., and G.M. Jenkins, Time Series Analysis: Forecasting and ControI, Revised Edition, Holden-Day: San Francisco, 1976. [3] Black, F., “The Pricing of Commodity Contracts,” JournuZ Financial Economics, January/March 1976, pp. 167-179. [4] Cornell, Bradford, “Spot Rates, Forward Rates and Market Efficiency,” Joottrml of Financial Economics, August 1977, pp. 55-65. “Forward and Futures Prices: Evidence [5] Cornell B., and M. Reinganum,
of
Hedging Foreign Exchange Exposure in the Japanese Yen
[6] [7] [8] [9] [lo]
[ll] [12] [13] [14] [15] [16]
[17] 1181 [19]
[20] [21]
[22]
[23]
[24]
253
from the Foreign Exchange Markets,” Journal @Finance, December 1981, pp. 1035-1045. Cox, J., J. Ingersoll, and S. Ross, “The Relation Between Forward Prices and Futures Prices,” Journal of FinanciuI Economics, December 1981, pp. 321-346. Ederington, L., “The Hedging Performance of the New Futures Markets,” Journal of Finance, March 1979, pp. 157-170. FranckIe, CT., “The Hedging Performance of the New Futures Market: Comment,” Jotlrnal of Finance, December 1980, pp. 1273-1279. French, K.R., “A Comparison of Futures and Forward Prices,” IournuZ of FinunciuZ Economics, December 1983, pp. 311-342. Herbst, A.F., D.D. Kare, and S.C. Caples, “Hedging Effectiveness and Minimum Risk Hedge Ratios in the Presence of Autocorrelation: Foreign Currency Futures,” JournuZ of Futures Markets, June 1989, pp. 185-197. Hill, J., and F. Schneeweis, “Forecasting Effectiveness of Foreign Currency Futures,” Business Economics, May 1981, pp. 42-46. Hill, J., and F. Schneeweis, “The Hedging Effectiveness of Foreign Currency Futures,” Journal of Financial Research, Spring 1982, pp. 95-104. Jarrow, Robert, and George Oldfield, “Forward Contracts and Futures Contracts,” Journal of Financial Economics, December 1981, pp. 373-382. Johnson, Leland L., “The Theory of Hedging and Speculation in Commodity Futures,” Review of Economic Studies, June 1960, pp. 139-151. Kolb, R., and R. Chang, “Improving Hedging Performance Using Interest Rate Futures,” Financial Management, Autumn 1981, pp. 27-29. Panton, D.P., and O.M. Joy, “Empirical Evidence on International Monetary Market Currency Futures,” Journal of International Business Studies, Fall 1978, pp. 59-68. Pelletier, Robert, “Contracts That Don’t Expire Aid Technical Analysis,” Commodities, March 1983, pp. 71-75. Rendleman, R., and C. Carabini, “The Efficiency of the Treasury Bill Futures Markets,” Journal of Finance, September 1979, pp. 895-914. Richard, Scott, and M. Sundaresan, “A Continuous Time Equilibrium Model of Forward Prices and Futures Prices in a Multigoal Economy,” JournaZ of Financial Economics, December 1981, pp. 265-286. Determination of Spot and Futures Stein, Jerome, “The Simultaneous Prices,” American Economic Review, December 1961, pp. 1012-1025. Swanson, Peggy E., and S. Caples, “Hedging Foreign Exchange Risk Using Forward Foreign Exchange Markets: An Extension,” Journal of InternationuZ Business Studies, Spring 1987, pp. 75-82. Vignola, A., and C. Dale, “The Efficiency of the Treasury Bill Futures Market: An Analysis of Alternate Specifications,” JournaZ of FinanciuZ Research, Fall 1980, pp. 169-188. Wardrep, B., and J. Buck, “The Efficiency of Hedging with Financial Futures: A Historical Perspective,” The Journal of Futures Markets, Fall 1982, pp. 243-254. Working, H., “Futures Trading and Hedging,” American Economic Review, June 1953, pp. 314-343.
During the 198Os, the exchange rate between the U.S. dollar and the Japanese yen (Y) had been subject to extreme volatility. The early years of the decade were marked by a strong dollar and the exchange rate hovered around 250 yen to the dollar. However, a slow erosion in the strength of the dollar occurred due to mounting trade deficits, and by the end of the decade, the exchange rate was about 125 yen per dollar. These dramatic fluctuations in the dollar-yen exchange rate have increased the foreign exchange risk borne by financial institutions, exporters and importers engaged in bilateral trade. The need for hedging the foreign exchange exposure against an adverse exchange rate movement is therefore obvious. Fortunately, the Japanese yen contracts in the foreign currency futures market and the forward foreign currency market provide the means for hedging foreign exchange exposure in the Japanese yen. One of the first attempts to examine the financial futures market for hedging effectiveness was a study by Ederington [7], who built upon the foundation developed by Johnson [14] and Stein 1201. Ederington investigated the Government National Mortgage Association (UNMAN and T-Bill futures contracts within the context of Markowitz Portfolio Theory (MET). Ederington showed that the optimal hedge ratio in most cases is significantly different from the traditional one-to-one ratio of futures to spot position holdings. He suggested that even pure risk minimizers should hedge only a portion of their entire spot portfolios. Franckle [8] pointed out some errors in Ederington’s study. He showed that a crucial assumption for using the model is a predetermined hedge period. Also, he extended Ederington’s work by attempting to estimate the effects of changing maturities on the valance-minimiz~g hedge ratio in the T-Bill market, and found that by matching futures prices with the correct T-Bill prices the market should be shown to be much more efficient than Ederington’s results. Franckle A.F. Herb& Department of Economics & Finance, The University of Texas at El Paso, El Paso, TX 79968, and King Fahd University of Petroleum and Minerals, Department of Finance & Economics, Dhahran, Saudi Arabia, 31261; D.D. Kare Department of Accounting and Finance, University of North Florida, Jacksonville, FL 32216; SC. Caples Department of Finance and Economics, McNeese State University, Lake Charles, LA 70609. Global Finance Journal, ISSN: 1044-0283
2(3/4), 243-253
Copyright D 1991 by JAI Press, Inc. All rights of reproduction in any form reserved.
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GLOBAL FINANCE JOURNAL
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showed that the use of daily prices in a study of the futures market was superior to the use of weekly or monthly prices, such as Friday settlement. The use of daily prices yields a higher estimate of the optimal hedge ratio and causes a larger reduction in the variance of the portfolio. Hill and Schneeweis [ll] explored the hedging potential of the foreign currency futures market by estimating optimal hedge ratios and hedging effectiveness measures for five foreign currencies and twenty futures contracts. They concluded that the British pound, the German mark, and the Swiss franc futures contracts have consistently high hedging effectiveness, and that the hedging effectiveness increased when the duration of the hedge was increased. In contrast, they showed that short-term contracts in the Canadian dollar and contracts of all maturities in the Japanese yen were inferior hedging instruments, but were nevertheless superior to an unhedged position. Hill and Schneeweis also examined the hedging effectiveness and the optimal hedge ratios for forward contracts in the British pound, the German mark, and the Japanese yen. They found that the mark and pound futures and forward contracts were indistinguishable in terms of hedging effectiveness for both 90and 180-day positions. The hedge ratio for the yen futures contracts were found to be substantially lower compared with the hedge ratios for forward contracts, and forward contracts were found to be more effective in hedging the yen exposure. Hill and Schneeweis attributed this to the newness and thinness in the futures market. Swanson and Caples [21] estimated the optimal hedge ratios and hedging effectiveness of forward foreign currency contracts in two currencies: the British pound and the German mark. They criticized previous studies on foreign currency hedging for ignoring the existence of serially correlated error terms. Swanson and Caples also used daily data on spot and forward contracts as opposed to weekly data used by Hill and Schneeweis. Herbst, Kare, and Caples [lo] compared the hedging effectiveness for five currency futures contracts for OLS hedge ratios versus Box-Jenkins [2] transfer function coefficients. They found that the Box-Jenkins transfer function coefficients yielded lower variance of hedge outcomes than did the OLS coefficients.
I. PURPOSE
& METHODOLOGY
The purpose of this study is to derive improved estimates of optimal hedge ratios and hedging effectiveness in the Japanese yen market using statistical techniques which were specifically developed to analyze time series data. For hedge ratios will also be estimated using the purpose of comparison, Ederington’s technique (i.e., OLS regression). Early studies made no attempt to determine the extent of autocorrelation in the spot and futures prices used. The data on spot and futures prices also violated some other assumptions of OLS regression, leading to statistically inconsistent estimates of optimal hedge ratios. The sample sizes tended to be small in those studies, thus making generalization of their results tenuous. This study
Hedging Foreign Exchange Exposure in the Japanese Yen
245
uses time series analysis techniques developed by Box and Jenkins [2] to arrive at unbiased estimates of optimal hedge ratios and hedging effectiveness.
OLS Regression In his work with financial futures, Ederington [7] first showed that OLS regression can be used to estimate the optimal hedge ratio between spot and futures contracts of U.S. Treasury bills. The most effective hedge is obtained when the portfolio variance (i.e., the exchange-rate risk) is reduced to a minimum. Ederington showed that the minimum variance (minimum risk) portfolio is obtained when the relative investment in futures contracts, b”, is b” = u /a2 $f
f
(1)
where osf = covariance between spot rates and futures rates 02 = variance of futures rates. f This is the slope of the line that regresses rates, i.e.,
futures contract rates against
SP, = a + b* FU, + e,
spot
(2)
where SF’, is the spot rate for the underlying currency, FL& is the futures contract rate for the currency, e, is the random error term, and b* is the optimal hedge ratio. Ederington further showed that the effectiveness of this hedge can be measured by the fraction of the variance of the unhedged portfolio that is eliminated by hedging. The effectiveness of the hedge turns out to be the coefficient of determination (X2 in the estimation of equation (2). A similar equation can be used to determine the parameters of hedging using forward contracts. Ederington’s technique of estimating optimal hedge ratios using OLS regression will yield statistically unbiased and consistent estimators only when the data used for estimation satisfy the following stringent assumptions of OLS regression: (1) (2) (3) (4)
Normality: e, (the error term) is normally distributed. Zero mean: E(e,) = 0. Homoscedasticity: the error terms have a constant variance. Nonautocorrelation: E(eiej) = 0 (i is not equal to j).
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Nonstochastic independent variable: the independent variable is a nonstochastic variable with values fixed in repeated samples.
However, as Franckle pointed out, time series data on spot and futures rates of foreign currencies show significant serial correlation and therefore, Ederington’s technique yields statistically inconsistent results. One way to account for the presence of serial correlation in data is to use an autoregressive model which yields consistent estimators in the presence of autoregressive error terms. An autoregressive model was tried, but the error terms in this estimation were still found to be correlated across time (the results are not presented here to preserve the journal space). Consequently, this technique was abandoned in favor of Box-Jenkins auto-regressive integrated moving average (ARIMA) model.
Box-Jenkins
Model
The second methodology used in this study is a technique developed by Box and Jenkins [2]. This methodology can be used to obtain superior results when time series possesses high autocorrelation. The Box-Jenkins method is a threestep approach to model building. The first stage is model identification or specification in which the appropriate structural form is selected to fit the time series data. The second stage is estimation of parameters from the data. Finally, the model selected is tested (diagnostic checking) to see if it actually fits the data properly; if the data being tested do not fit the model, the process is repeated. The Box-Jenkins process, an autoregressive integrated moving average model (ARIMA), is a process considered to be driven by a series of random shocks (or white noise). Each observation of a white noise process is assumed to be drawn randomly and independently from a normal distribution with a zero-mean and constant variance. A white noise shock enters an ARIMA (p, d, 4) model, passes through a series of filters or “black boxes, ” and leaves as a time series observation. These “black boxes” introduce correlation structure or memory into the time series. The process involved is to reconstruct the proper ARIMA (p, d, 4) model to describe the time series. The parameter p describes the number of error terms included in the estimation of the equation to eliminate the serial correlation. The parameter d is the level of differencing employed on the time series data. The parameter 4 is the number of moving average terms which are accounted for in the estimation. The ARIMA model is a series of filters that determine the properties of the output time series. The choice of filters is based on the knowledge of the process generating the time series data. An ARIMA (0, 1, 1) model will be used to describes a series in which differenced data behaves as if it were generated by a moving average process. An ARIMA (1, 1, 0) filter, on the other hand, describes a series that, when differenced, behaves as if it were generated by an autoregressive process. For more complex time series data, the proper values of the param-
Hedging Foreign Exchange Exposure in the Japanese Yen
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eters p, d and 9 will be decided through a trial and error procedure. Those values which best fit the data to the estimated function are normally used. The mathematical model used in this study to estimate optimal hedge ratios is (1 - B)SP, = Y(B) FU, + e(g)/~(g)e~ where I? is the backshift
operator,
(3)
such that B(X,) = X,_,
Y(B) is the transfer function, B(B) is the autoregressive operator, (P(B) is the moving average operator. Both B(B) and a,(B) are polynomials in B. The nature of these polynomials is determined by the lag structure and the moving average structure in the data used. The appropriate models used in the case of futures contracts and forward contracts are presented in Table 2. The above model can be also used to estimate optimal hedge ratios and hedging effectiveness for forward contracts.
DATA Data on the Japanese yen spot, forward and futures rates were collected for the period beginning with January 1, 1988, and ending with July 31, 1989. Data on spot rates and 90-day forward contract rates were collected from the l&l2 Street Joournal. The data on perpetual contracts were purchased from Commodity Systems, Inc., of Boca Raton, Florida. The choice between using daily data and weekly/monthly data warrants some discussion. One drawback of using weekly data is that very few observations can be used in a study. But, more importantly, Fran&e 181, in his critique of the Ederington study, showed that a greater amount of risk reduction (i.e., variance reduction) can be achieved using daily data compared with the risk reduction achieved using weekly data. This is because there is a greater covariance between daily spot and futures prices than the covariance in weekly (or monthly) data. Based on the above reasons, it was decided to use daily data on spot, forward and futures rates. One problem encountered in any analysis involving futures contracts is that the individual contracts expire .I This introduces what may be termed as the “maturity bias” in the estimates of the optimal hedge ratio. For example, if a four week hedge was placed using a 90-day futures contract, by the time the hedge is lifted the contract only has 62 days left till maturity. The relations~p between spot rates and futures rates (i.e., the basis) will keep changing in a hedge of this type. Both Ederington [7] and Fran&e [S] recognized this problem and suggested that the optimal hedge ratios be modified to account for the change in basis. To address this problem, the concept of perpetual contract was developed
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by Pelletier 1171. A perpetual contract rate is calculated for a specific day in the future by taking a weighted average of the rates of the two futures contracts with expiry dates closest to the specified day. This artificial device helps in constructing a hedge that has no maturity bias and the basis remains constant. Further, because the perpetual contract has no maturity date, a large number of daily observations could be used in this study. This study uses 372 observations, compared with 14 observations used by Hill and Schneeweis (hereafter HS). In studies using weekly or monthly futures market data, the rate on a particular day must be selected arbitrarily as the basis for comparison. That choice may bias the results of data analysis signific~tly. The use of daily data and perpetual contracts can eliminate that problem. Another problem facing a researcher using daily data on futures markets is the problem of missing values for holidays. Box-Jenkins ARIMA technique can estimate the missing values. This feature is extremely important in statistical analysis of time series data, because missing values would tend to mask any periodicities (as well as holiday effects) that might be present in the data.
III. RESULTS Table 1 presents the resuhs of OLS regression performed on futures and forward price data. Estimates of optimal hedge ratios using OLS regression appear to be close to one in all cases. However, estimates for both futures and forward contracts are significantly different from 1.0 at the 99 percent confidence level. The R2 statistic, which measures hedging effectiveness, also appears to be quite high, close to one in all cases. The interpretation of the R2 statistic warrants some caution, because of the presence of autocorrelation in the data.
RESULTS Equation: Equation: Estimates a
b* X2 D-W stat.
Table 1 OF OLS REGRESSION
SP, = d + b*FU, + e, (Futures Contract) SP, = d + b”FOR, + e, (Forward Contract) Futures Contracts 0.1431 (0.98) 1.0542 (64.7) 0.9543 1.1320
Forward Contracts 0.1154 (1.432) 1.0080 (202.6) 0.9911 1.0980
(T’-statistics in parentheses. T-statistics for b”, the optimal hedge ratio, are statistically significant at the 99% confidence level.)
Hedging Foreign Exchange Exposure in the Japanese Yen
RESULTS Futures contracts: Forward contract: Hedging
Table 2 OF ARIMA
249
MODELS
(1 - B) SP, = b” (1 - B) FLI, + (1 - .983B)e, (1 - B) SP, = b” (1 - B) FOR, + (1 - .957B)e,
Device
Opt. Hedge Ratio (b”)
T-Statistic
.9856 .9744
33.26” 128.11*
1. FUTURES CONTRACT 2. FORWARD CONTRACT
(” Indicates that the t-statistic is significant at the 99.99 percent confidence
level.)
A major problem with the use of OLS regression is highlighted by the DurbinWatson statistic, which is significant in both cases. This indicates the presence of autoregressive disturbances in data used in this study. Once again, the presence of autoregressive disturbances violates a basic assumption of OLS regression analysis, causing overestimation of the X2 statistic and yielding a biased estimate of the optimal hedge ratio (i.e., the slope coefficient). It also causes an underestimation of the variance of error terms. The presence of autoregressive disturbances in the data for both forward and futures contracts justifies the search for a statistical technique that can adjust for it. Autoregressive models proved to be no improvement, because error terms showed “infinite memory.” Table 2 shows the results of estimates using ARlMA models applied to futures and forward contract price data. Estimates of optimal hedge ratios using this technique are lower than those in Table 1. The R* statistics are not presented in Table 2, because in the case of the ARIMA technique this statistic is not very meaningful (Box and Jenkins [2J). This is because the parameter estimation in the ARIMA technique is carried out through a trial and error procedure until the best possible fit is achieved. Therefore, the X2 statistic in the ARIMA analysis is always very close to 1. The estimated optimal hedge ratios are close to 1 .O but they were found to be statistically different from 1 .O at the 99 percent confidence level. Table 3 presents a comparative analysis of optimal hedge ratios estimated
COMPARISON Hedging
Device
1. FUTURES CONTRACT 2. FORWARD CONTRACT”
Table 3 OF OPTIMAL
HEDGE
RATIO
OLS Est.
ARIMA Est.
% Reduction
1.0542 1.0080
.9856 .9744
6.51 3.33
(” Indicates that the estimates from two techniques from each other at the 99 percent confidence level.)
are significantly
different
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Table 4 COMPARISON OF HEDGING EFFECTIVENESS (VARIANCE OF RESIDUALS x lo61 Hedging
Device
OLS Est.
ARIMA Est.
% Reduction
1. FUTURES CONTRACT 2. FORWARD CONTRACT
57.45 36.87
21.58 14.63
62.44 60.32
using the two alternative techniques. Optimal hedge ratios estimated using ARIMA models are lower in both cases, the reduction is 6.51 percent for the futures contract and 3.33 percent for the forward contract. Table 4 lists estimates of the variance of residuals (error terms) in both OLS regression and ARIMA technique. Again, ARIMA estimates are substantially lower compared to OLS estimates. The use of R2 to measure hedging effectiveness as suggested by Ederington creates a couple of problems. In OLS regression, the R* statistic is overestimated because of the presence of autoregressive disturbances. The R2 statistic in ARIMA models, on the other hand, may lead to misleading interpretation, because of the reasons cited above. It is therefore proposed that a better comparison of hedging effectiveness could be made by comparing the reduction in the variance of the hedged position (to estimate the minimum risk portfolio). Reduction in the variance of the hedged position is inversely related to the variance of residuals; i.e., the larger the reduction of variance in the hedged position, the smaller the variance of the residuals. This suggests that the variance of residuals should be used as the measure ofhedging effectiveness. In the case of both forward and futures contracts, ARIMA models yield estimates of residual variances that are substantially lower compared to those in OLS regression. The reduction on average is 61 percent. Table 5 presents a comparison between results from this study and the results from the Hill-Schneeweis (HS) study. It is worth noting that the optimal hedge
THIS
Table 5 COMPARATIVE RESULTS STUDY V/S HILGSCHNEEWEIS
1. THIS STUDY (OLS) 2. THIS STUDY (ARIMA) 3. H-S STUDY (90 day)
STUDY
Futures
Forward
1.0542* .9856* .505
1.0080* .9744* 1.152
(*Indicates that the estimates are significantly from 1.0 at the 99 percent confidence level.)
different
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ratios obtained in the HS study were not significantly different from 1.0 for both the forward and futures contracts. Optimal hedge ratios from this study are lower in both cases, and significantly different from 1.0 at the 99 percent confidence level. A meaningful comparison of the reduction in variance cannot be made because data on residuals of the HS study are unavailable. For the reasons cited previously, a comparison of the Ra statistic is not presented. Various explanations can be offered for the difference in the two sets of results. This study uses 372 observations, while the HS study was based on 14 observations. This study uses daily data as opposed to weekly data used in the HS study. Finally, the periods over which data were collected are also different.
IV. ECONOMIC
IMPLICATIONS
The finding that the use of ARIMA models to account for autoregressive disturbances in data causes a reduction in estimated optimal hedge ratios for the Japanese yen has important economic implications for a hedger. The extent of the reduction is between 3.3 to 6.5 percent. This means that hedgers in the Japanese yen really need a smaller investment in the futures contracts or forward contracts of the same currency. Considering the amounts typically involved in foreign currency transactions, this reduction in optimal hedge ratio would result in significant benefits to the hedger. Second, the comparison of variances of residuals, which should be considered the proper measure of hedging effectiveness, shows that the optimal hedge ratios estimated using ARIMA models succeed in creating portfolios which have substantially lower variance of residuals compared to portfolios created using OLS optimal hedge ratios. This implies that a hedger using hedge ratios based on ARIMA estimates would be exposed to an exchange rate risk which is substantially lower compared to a hedger using OLS estimates. Also, since fewer futures contracts would be needed, margin and commission costs would be reduced. ARIMA models appear to yield estimates of optimal hedge ratios which not only reduce the costs of hedging a spot position, but also reduce the foreign exchange exposure by nearly 60 percent.
V. CONCLUSION The results of this investigation of spot, forward and futures price data on the Japanese yen show that the optimal hedge ratios calcuIated using the methodology common to prior studies (i.e., OLS regression) were overestimated. The reason for this overestimation may be the presence of autoregressive disturbances in the exchange rate data and the use of weekly (or monthly) data. The presence of autoregressive disturbances violates a basic assumption underlying OLS regression leading to inconsistent estimates, and as Franckle [S] pointed out, the use of weekly or monthly data also leads to the overestimation of optimal hedge ratios.
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This investigation departs from prior studies in many respects: (1) It uses daily data as opposed to the use of weekly or monthly data in past studies; (2) It employs the concept of constant maturity perpetual contracts, allowing the use of a large number of observations to model the behavior of price data. In contrast, previous researchers based their results on the analysis of a small number of observations; (3) The statistical technique used in this study generates missing values for holidays and weekends, while past studies ignored missing values; (4) Finally, the most important departure from previous studies is that this study uses the ARIMA methodology developed by Box and Jenkins to model the relationship between spot and futures prices. This technique successfully solves the vexing problem of autoregressive disturbances in the data which renders OLS regression technique of dubious value. Results yielded by ARIMA methodology show significant improvement over results from OLS regression. Estimates of optimal hedge ratios were lower. The reduction in optimal hedge ratios was 3.3 percent for futures contracts and 6.51 percent for forward contracts. Considering the large dollar amounts generally involved in foreign currency transactions, this reduction in optimal hedge ratios will yield a substantial reduction in the margin deposit required of hedgers and a corresponding reduction in transaction costs. The minimum risk portfolio created using ARIMA estimates is subject to a considerably lower risk compared to portfolios created using OLS regression. The risk reduction achieved ranges between 60 and 62 percent. NOTES 1. A particular problem exists when 90-day futures contracts are compared with 90-day forward contracts because futures contracts are written only for four delivery months a year, but forward contracts are written for delivery on any day 90 days in the future. In some studies that compare the two contracts (most notably Hill and Schneeweis [ll]) only thirteen to eighteen observations were available for each currency, even though the study involved several years. BIBLIOGRAPHY [l] B&on, John F.O., “Rational Expectations and the Exchange Rate,” in Jacob A. Frenkel and Harry G. Johnson (eds.), The Economics of Exchurzge Rates, Addison-Wesley: Reading, MA, 1978. [2] Box, George E.P., and G.M. Jenkins, Time Series Analysis: Forecasting and ControI, Revised Edition, Holden-Day: San Francisco, 1976. [3] Black, F., “The Pricing of Commodity Contracts,” JournuZ Financial Economics, January/March 1976, pp. 167-179. [4] Cornell, Bradford, “Spot Rates, Forward Rates and Market Efficiency,” Joottrml of Financial Economics, August 1977, pp. 55-65. “Forward and Futures Prices: Evidence [5] Cornell B., and M. Reinganum,
of
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[6] [7] [8] [9] [lo]
[ll] [12] [13] [14] [15] [16]
[17] 1181 [19]
[20] [21]
[22]
[23]
[24]
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from the Foreign Exchange Markets,” Journal @Finance, December 1981, pp. 1035-1045. Cox, J., J. Ingersoll, and S. Ross, “The Relation Between Forward Prices and Futures Prices,” Journal of FinanciuI Economics, December 1981, pp. 321-346. Ederington, L., “The Hedging Performance of the New Futures Markets,” Journal of Finance, March 1979, pp. 157-170. FranckIe, CT., “The Hedging Performance of the New Futures Market: Comment,” Jotlrnal of Finance, December 1980, pp. 1273-1279. French, K.R., “A Comparison of Futures and Forward Prices,” IournuZ of FinunciuZ Economics, December 1983, pp. 311-342. Herbst, A.F., D.D. Kare, and S.C. Caples, “Hedging Effectiveness and Minimum Risk Hedge Ratios in the Presence of Autocorrelation: Foreign Currency Futures,” JournuZ of Futures Markets, June 1989, pp. 185-197. Hill, J., and F. Schneeweis, “Forecasting Effectiveness of Foreign Currency Futures,” Business Economics, May 1981, pp. 42-46. Hill, J., and F. Schneeweis, “The Hedging Effectiveness of Foreign Currency Futures,” Journal of Financial Research, Spring 1982, pp. 95-104. Jarrow, Robert, and George Oldfield, “Forward Contracts and Futures Contracts,” Journal of Financial Economics, December 1981, pp. 373-382. Johnson, Leland L., “The Theory of Hedging and Speculation in Commodity Futures,” Review of Economic Studies, June 1960, pp. 139-151. Kolb, R., and R. Chang, “Improving Hedging Performance Using Interest Rate Futures,” Financial Management, Autumn 1981, pp. 27-29. Panton, D.P., and O.M. Joy, “Empirical Evidence on International Monetary Market Currency Futures,” Journal of International Business Studies, Fall 1978, pp. 59-68. Pelletier, Robert, “Contracts That Don’t Expire Aid Technical Analysis,” Commodities, March 1983, pp. 71-75. Rendleman, R., and C. Carabini, “The Efficiency of the Treasury Bill Futures Markets,” Journal of Finance, September 1979, pp. 895-914. Richard, Scott, and M. Sundaresan, “A Continuous Time Equilibrium Model of Forward Prices and Futures Prices in a Multigoal Economy,” JournaZ of Financial Economics, December 1981, pp. 265-286. Determination of Spot and Futures Stein, Jerome, “The Simultaneous Prices,” American Economic Review, December 1961, pp. 1012-1025. Swanson, Peggy E., and S. Caples, “Hedging Foreign Exchange Risk Using Forward Foreign Exchange Markets: An Extension,” Journal of InternationuZ Business Studies, Spring 1987, pp. 75-82. Vignola, A., and C. Dale, “The Efficiency of the Treasury Bill Futures Market: An Analysis of Alternate Specifications,” JournaZ of FinanciuZ Research, Fall 1980, pp. 169-188. Wardrep, B., and J. Buck, “The Efficiency of Hedging with Financial Futures: A Historical Perspective,” The Journal of Futures Markets, Fall 1982, pp. 243-254. Working, H., “Futures Trading and Hedging,” American Economic Review, June 1953, pp. 314-343.