Hedging effectiveness and minimum risk hedge ratios in the presence of autocorrelation: Foreign...

Hedging Effectiveness and Minimum Risk Hedge Ratios in the

Presence of Autocorrelation: Foreign Currency Futures

A.F. Herbst D . D . Kare S. C. Caples

ne of the first attempts to examine the financial futures market for hedging effec- 0 tiveness was a study by Ederington (1979), who built upon the foundation developed by Johnson (1960) and Stein (1961). Ederington investigated the Government National Mortgage Association (GNMA) and T-Bill futures contracts within the context of Markowitz Portfolio Theory (MPT). He concluded that hedging decisions in the futures market are no different from any other investment decision and that traditional hedging theory and Working’s (1960) theory are special cases of the broader portfolio theory.

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 not hedge their entire spot portfolio but only a portion of it because minimum risk (i.e., variance) is achieved with a ratio of less than one-to-one.

Franckle (1980) pointed out some errors in Ederington’s study, among them that a two-week hedge of T-Bill futures was far more efficient than Ederington had shown. He extended Ederington’s work by attempting to estimate the effects of changing maturities on the variance-minimizing hedge ratio in the T-Bill market, and found that by matching futures prices with the correct T-Bill prices the market could be shown to be much more effective than Ederington’s results indicated. He also showed that a crucial assumption for using the model is a predetermined hedge period. Franckle indicated several reasons why 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 most important among them is that 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.

A . F. Herbst is the Charles and Dorothy Carter Professor of Finance at the University of Texas at El Paso.

D . D . Kare is a n Assistant Professor of Finance at the University of North Florida.

S. C . Caples is a n Associate Professor of Finance at McNeese University.

The Journal of Futures Markets, Vol. 9, No. 3, 185-197 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0270-73 14/89/030185- 13$04.00

Hill and Schneeweis (1982) explored the hedging potential of the foreign currency futures market. They estimated optimal hedge ratios and hedging effectiveness measures for five foreign currencies and twenty futures contracts. They found 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. They also showed that short-term contracts in the Canadian Dollar, and contracts of all maturities in the Japanese Yen, were inferior hedging in- struments compared to those for the aforementioned currencies, but were nevertheless superior to an unhedged position.

Wardrep and Buck (1982) drew attention to the need to explore time series character- istics of futures market data. It is the aim of this study to take the first step in that direc- tion by investigating foreign currency futures markets using time series analysis.

I. PURPOSE & METHODOLOGY The purpose of this study is to derive improved estimates of optimal hedge ratios and hedging effectiveness in the foreign currency market using statistical techniques which were specifically developed to analyze time series data. For the purpose of comparison, hedge ratios will also be estimated using Ederington’s technique (i.e., OLS regression). The currencies investigated in this study are the British Pound, the German Mark, the Canadian Dollar, the Swiss Franc, and the Japenese Yen: the five which are actively traded in the futures market.

Early studies made no attempt to determine the extent of autocorrelation in the spot and futures prices used. The sample sizes also tended to be small in those studies, thus making generalization of their results tenuous. This study uses time series analysis techniques developed by Box and Jenkins (1976) to arrive at unbiased estimates of optimal hedge ratios and hedging effectiveness.

OLS Regression

In his pioneering work with financial futures, Ederington first showed that OLS regression can be used to estimate the optimal hedge ratio between spot and futures contracts of U.S. Treasury bills. His approach was based on application of Markowitz Portfolio Theory (MPT) to hedging. An investor who holds a spot position in a foreign currency is subject to exchange rate risk. He can eliminate a significant part of this risk by hedging his spot position using futures contracts. When hedging a spot position using futures contracts he forms a portfolio of two assets: the spot position and the futures contract. MPT allows one to calculate the variance of this two-asset portfolio based on the indi- vidual asset variances, fractions of investments in each asset and the correlation between the two assets. 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* = vsf/c$ ( 1 )

where

This can be recognized as the slope of the line that regresses futures rates against spot rates, i.e.,

(2)

= covariance between spot rates and futures rates a,Z = variance of futures rates.

SP, = a + b* * FU, + e,

186 I HERBST, KARE, AND CAPLES

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, and that the effectiveness of the hedge is equal to the coefficient of determination (R’) in the estimation of eq. (2).

But Ederington’s technique of estimating optimal hedge ratios using OLS regression will yield unbiased estimators only when the data used for estimation satisfy stringent assumptions of OLS regression: normality, constant variance (homoscedasticity), uncor- related error terms among others. But as Franckle pointed out, time series data on spot and futures rates of foreign currencies show significant serial correlation, and therefore, Ederington’s technique would yield biased results.

Various statistical techniques are available which account for the presence of serial correlation in data, and yield unbiased results. An autoregressive model was tried, but the error terms in this estimation were still found to be correlated across time. Therefore this technique was abandoned in favor of Box-Jenkins autoregressive integrated moving average (ARIMA) model.

Box-Jenkins Model

The second methodology used in this study is a technique developed by Box and Jenkins (1976). This methodology can be used to obtain superior results when time series pos- sess high autocorrelation. The Box-Jenkins method is a three-step 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 ran- domly and independently from a normal distribution with a zero-mean and constant variance. A white noise shock enters an ARIMA ( p , d, q) model, passes through a series of filters or “black boxes” and leaves as a time series observation. These “black boxes” in- troduce correlation structure or memory into the time series. The process involved is to reconstruct the proper ARIMA ( p , d, q) model to describe the time series. The ARIMA model is a series of filters that determine the properties of the output time series. An ARIMA (0 , 1, 1) model describes a series in which a random shock passes through a differencing filter (d = 1) and a moving average filter (q = l), but not an autoregressive filter ( p = 0). An ARIMA (1 , 0, 1) filter describes a series that passes through a moving average filter an autoregessive filter but was not differenced. The output of the two such systems will be quite different.

The mathematical model used in this study to estimate optimal hedge ratios for foreign currencies is:

where B is the backshift operator, such that B(X,) = Xt-, r ( B ) is the transfer function @(B) is the autoregressive operator @(B) is the moving average operator.

Both e ( B ) and @(B) are polynomials in B. The coefficients of B in these polynomials are determined by the lag and moving average structure of the error terms.

FOREIGN CURRENCY FUTURES / 187

The model coefficients are estimated using data for a specified period. Then, using a subsequent and independent set of data (i.e., holdout sample) the model results are judged vis-a-vis those of the OLS model.

11. DATA Data on the five foreign currency spot and futures contract rates were collected for the period beginning January 1, 1982 and ending August 3 1, 1984. Data on spot rates were collected from the Wall Street Journal. Data on futures contracts were purchased from Commodity Systems, Inc., Boca Raton, Florida.

One problem encountered in any analysis involving futures contracts is that the indi- vidual contracts expire.’ To address this problem, the concept of perpetual contract was developed by Pelletier (1983). A perpetual contract rate is calculated for a day that is a specific time into the future by taking a weighted average of the rates for futures contracts that expire before and after the focal data. The weights are based on the respective contract’s number of days from the presente2

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 significantly. However, the use of daily data and perpetual contracts3 can eliminate that problem.

Another problem facing a researcher using daily data on futures markets is the problem of missing values for holidays. The Box-Jenkins ARIMA technique can estimate the missing values. This feature is extremely important in statistical analysis of time series data in general, and not in futures markets alone, because missing values would tend to mask any periodicities (as well as holiday effects) that might be present in the data.

When Franckle (1980) used daily data to recalculate the hedge ratios obtained by Ed- erington (1979) with weekly data, he found significantly different hedge ratios. Franckle found that use of daily prices caused a substantial reduction (by almost one-third) in the variance of the hedged portfolio. He suggested that this is due to the increase in the covariance between daily spot prices and daily futures prices. Since daily prices are more volatile, there is more variance to reduce in the first place. However, the increase in the covariance is so great that there is an actual reduction in the variance of the hedge portfolio using daily prices compared to the variance of the hedged portfolio using weekly prices.

Among other things, both Ederington (1979) and Franckle (1980) point out that the value of the optimal hedge ratio, b*, should be adjusted because of the bias introduced by the “maturity” effect. Consider a four-week hedge. At the beginning of the hedge, a 90-day futures contract is used to place the hedge. As the hedge is lifted the futures con-

‘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, 1982) only thirteen to eighteen observations were available for each currency, even though the study involved several years.

’For example, using contracts that expire N and F days in the future to create a 91-day ahead synthetic contract, we would use the following relationship: P9, = PF - [(F - 91)/(F - N ) ] ( P F - PN), where the sub- scripted P s denote the corresponding prices to those points in time, and N < 91 < F.

’In principle, a hedger could create synthetic perpetual contracts by using a properly weighted number of actual futures contracts that straddle the target date of, for example, 90-days. The main effect on the basis change in such a synthetic perpetual contract would then result from changes in the term structure of interest rates. However, because the actual contracts employed could be no further distant than the more liquid futures contracts, with their reasonably high open interest, term structure effects should be less than the basis effects manifest with ordinary contracts.


tract has a maturity of only 62 days. There is an imperfect correlation between the 90- day futures price and 62-day futures price. Generally, as a futures contract matures, the futures price converges to the price of the underlying cash commodity or financial in- strument; i.e., the basis approaches zero. In addition to this basis bias there is another effect. As the futures contract approaches expiration the futures price change becomes more closely associated with that of the underlying commodity; with diminishing time to expiration, market response to expectations tends to affect the futures price like the cash price, and uncertainty tends to attenuate as contract expiration becomes imminent. The use of constant maturity contracts would tend to eliminate both effects.

111. RESULTS Table I contains descriptive statistics for the currency variables used in this study. Table I1 presents the results of OLS regression performed on futures data for the five foreign currencies under investigation. Because the raw data exhibited considerable heteroscedasticity, thereby violating a basic assumption of OLS regression, a natural log transformation was used to stabilize the variance of raw data.

Estimates of optimal hedge ratios using OLS regression appear to be close to one in all cases. However, the magnitude of the ?-statistics indicates that in four of five currencies the estimated hedge ratios are significantly different from 1.00. The R 2 statistic, which measures hedging effectiveness, also appears to be quite high, close to one in all cases. The hedge ratios presented here are higher compared to those estimated in previous studies. Several reasons may be cited for this difference. This study uses daily data

Table I DESCRIPTIVE STATISTICS OF THE CURRENCY VARIABLES

spot Future

Currency Mean Std. Dev. Mean Std. Dev.

BP 1.2358 .lo270 1.2364 .09670 DM .3349 .02189 .3311 .02072 CD .7515 .01636 .7530 .01552 SF .4025 .02880 .3970 .02684 JY .4099 .01589 .4056 .01475

All exchange rates in terms of U.S. $ per unit of foreign currency, except in the case of the Japanese Yen, where the rate is U.S. $ per 100 Yen.

Table I1 RESULTS OF OLS REGRESSION

~~

Currency Opt. Hedge Ratio R' D-W Stat. Std. Err. I-Stat.

1. British Pound .9893 .9979 0.9980 0.0184 0.58 2. Canadian Dollar .9622 .9658 1.3980 0.0074 5.13* 3. German Mark .9800 .9945 0.6600 0.0030 6.74* 4. Swiss Franc .9627 .9918 0.8450 0.0036 10.51* 5. Japanese Yen .9812 .9834 0.9720 0.0052 3.62*

The ?-statistic tests the null hypothesis that the optimum hedge ratio is not statistically different from 1 .00. An * indicates rejection at the 99 percent level.


and perpetual contracts which have no maturity dates; previous studies used monthly or weekly data and contracts with fixed expiration dates. This study uses hedges of 90 day duration; previous studies used one- to four-week hedges.

A major problem with the use of OLS regression is highlighted by the Durbin-Watson statistic, which is significant in all five cases. This indicates the presence of autoregressive disturbances in the data used in this study. Once again, the presence of autoregressive disturbances violates a basic assumption of OLS regression analysis, causing overestimation of the R 2 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 all five foreign currencies justifies the search for a statistical technique that can adjust for it. Autoregressive models were tried, but proved to be no improvement, because the error terms showed “infi- nite m e m ~ r y ” . ~

Table I11 shows the results of estimates using ARIMA models’ applied to futures data. Estimates of optimal hedge ratios using this technique are lower than those in Table 11. Based on the magnitude of the t-statistics the estimates of optimal hedge ratios for all currencies are significantly different from 1 .OO. The R 2 statistics presented in Table I11 also are quite high, but they should be interpreted with caution. Box and Jenkins (1976) suggested that in some forms of ARIMA models, the R 2 statistic may be misleading.

Table IV presents a comparative analysis of optimal hedge ratios estimated using the two alternative techniques. Optimal hedge ratios estimated using ARIMA models are

Table I11 RESULTS OF ARIMA MODELS

Currency Opt. Hedge Ratio RZ Std. Err. t-Stat.

1. British Pound .9235 ,9980 0.0138 5.56* 2. Canadian Dollar ,8928 .9781 0.0193 5.55* 3. German Mark .9450 .9933 0.0121 4.75* 4. Swiss Franc ,9180 .9955 0.0117 7.01* 5. Japanese Yen .9740 .9905 0.0120 2.17*

The t-statistic tests the null hypothesis that the optimum hedge ratio is not statistically different from 1 .OO. An * indicates rejection at the 99 percent level.

Table IV COMPARISON OF OPTIMAL HEDGE RATIOS

Currency OLS Est. Arima Est. % Reduction

1. British Pound ,9893 .9235 7.13 2. Canadian Dollar .9622 .8928 7.77 3. German Mark .9800 ,9450 3.57 4. Swiss Franc .9627 .9180 4.87 5. Japanese Yen .9812 .9740 0.90

&Stat.

4.05* 5.10* 4.25* 5.17* 0.78

The t-statistic tests the null hypothesis that the OLS estimate is not statistically different from the Box- Jenkins estimate. An * indicates rejection at the 99 percent level.

T e n lags were used with the SAS procedure AUTOREG. Since the error terms were not “dying out” it

’In the model identification phase 42 lags were used with SAS ETS procedure ARIMA. The authors’ expe- would not have been worthwhile to examine more lags than this.

rience with both procedures suggests that the BMDP 2T would have performed comparably.


lower in all five cases, the reduction ranging from approximately 1 percent for the Japa- nese Yen to 7.77 percent for the Canadian Dollar. In the cases of four out of five currencies the estimates of optimal hedge ratios obtained using the ARIMA technique are significantly lower compared to those from the OLS technique.

Table V 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 R to measure hedging effectiveness as suggested by Eder- ington creates a couple of problems. In OLS regression, the R 2 statistic is overestimated because of the presence of autoregressive disturbances. The R 2 statistic in ARIMA models, on the other hand, may lead to misleading interpretation. It is therefore proposed that a better comparison of hedging effectiveness could be made by comparing the reduction in the variances of the hedged positions. 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 of hedging effectiveness. In the case of all five currencies, ARIMA models yield estimates of residual variances that are substantially lower compared to those in OLS regression.

F-statistics testing the null hypothesis that the residual variances from the ARIMA technique and those from the OLS technique are not statistically different are significant at the 95 per cent confidence level. This indicates rejection of the null hypothesis. In other words, the optimal hedge ratios obtained from the ARIMA technique succeed in reducing significantly the foreign exchange exposure of traders.

A possible criticism about the use of the ARIMA technique is that ARIMA models are fitted to the data used in a study. How valid are the results and conclusions based upon those results when the model is applied to a new set of observations? To answer such a question, models estimated from both the OLS and ARIMA technique were used to forecast perpetual contract prices based on spot price data for the period May 1984 to April 1985. Forecast errors were then calculated by subtracting the forecast prices from the actual perpetual contract prices. The variances of these errors (i.e., residuals) are presented in Table VI. Once again the F statistics testing the null hypothesis about variances being equal are sufficiently large in magnitude to reject the null hypothesis.

This provides strong evidence to support the assertion made earlier in this paper: The optimal hedge ratios obtained using the ARIMA technique do a much better job of re-

Table V COMPARISON OF HEDGING EFFECTIVENESS USING THE DATA

SET FROM WHICH THE COEFFICIENTS WERE OBTAINED

(Variance of Residuals x lo6)

Currency OLS Est. ARIMA Est. % Reduction F -Stat.

1 . British Pound 41.82 24.01 42.6 1.742** 2. German Mark 2.46 1.93 21.5 1.275* 3. Canadian Dollar 5.24 2.28 56.5 2.298** 4. Swiss Franc 5.24 2.40 54.0 2.183** 5 . Japanese Yen 6.65 3.46 48.0 1.922**

The F-statistic tests the hypothesis that the residual variances from both the OLS and the Box-Jenkins methods are the same, with 600 degrees of freedom in numerator and denominator.

*Indicates the rejection of the null hypothesis at the 95% level. **Indicates the rejection of the null hypothesis at the 99% level.


Table VI RESIDUAL VARIANCES OBTAINED FROM THE HOLDOUT SAMPLE WHEN

THE PREVIOUSLY ESTIMATED OPTIMAL HEDGE RATIOS ARE USED

(Variance x lo6)

Currency OLS BJ F-Statistic

BP 68.81 DM 3.23 CD 3.82 SF 10.31 JY 3.20

50.41 1.25 2.88 8.16 2.17

1.36" 2.584** 1.33* 1.26* 1.48**

The F-statistic tests the hypothesis that the residual variances from both the OLS and the Box-Jenkins methods are the same, with 250 degrees of freedom in numerator and denominator.

*Indicates the rejection of the null hypothesis at the 95% level. **Indicates the rejection of the null hypothesis at the 99% level.

ducing the foreign exchange exposure of a cash position than those obtained from the OLS technique. Incidentally, the means of forecast errors using the ARIMA model were also substantially lower than the means of the OLS forecast errors. However, since no economic implications are attached to the mean errors, the results are not provided.

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 foreign currencies has important economic implications for a hedger. The extent of the reduction averages 5 percent. This means that on average hedgers in foreign currencies really need a 5 percent smaller investment in the futures contracts of the same currency. Considering the magnitude 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 ex- posed 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 com- mission costs would be reduced.

In summary, it appears that ARIMA models yield estimates of optimal hedge ratios which not only reduce the costs of hedging a spot position, but also reduce the risk exposure.

V. CONCLUSION The results of this study of spot and futures price data on five foreign currencies show that the estimates of optimal hedge ratios calculated using the methodology common to prior studies (i.e., OLS regression) were upward biased. The reason for this bias stems from the presence of autoregressive disturbances in the exchange rate data, which violates a basic assumption underlying OLS regression.

192 / HERBST, KARE, AND CAPLES

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 handful of observations. (3) The statistical technique used in this study generates missing values for holidays and weekends, while past studies ig- nored 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 the OLS regression technique of dubious value.

Results yielded by ARIMA methodology show significant improvement over results from OLS regression. Estimates of optimal hedge ratios are lower; the reduction in optimal hedge ratios ranging from approximately 1 percent for the Japanese Yen to 7.77 percent for the Canadian Dollar. 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 while providing better hedging results. The minimum risk portfolio created using ARIMA estimates is subject to a considerably lower risk compared to portfolios created using OLS regression.

Appendix BRITISH POUND

OLS: BPSP, = 0.016 + 0.98929 BPFUT, + e,

(0.02963) (0.0 1838)

with R 2 = 0.9979 and e2 = 41.82*10-6 (5 )

The Durbin-Watson Statistic is 0.998, and the estimated degree of first order autocorrelation is 0.499.

An autoregression procedure was used next to adjust for the presence of autocorrelation, yielding the following model:

AR: BPST, = 0.033 + 0.9785 BPFUT,

(0.0 1057) (0.00564)

with R 2 = 0.9739 and e2 = 25.73*10-6 (6)

Box-Jenkins methodology produced the following model for the British pound spot and futures data:

B-J: (1 - B) BPSP, = 0.9235 (1 - B) BPFUT, + (1 - 0.768B) e,

(0.01375) (0.02648)

with R”= 0.9980 and e2 = 24.6*1OW6 (7)

The B-J model has a pure moving average error term with a lag of one period. The hedge ratio is 0.9235, and the residuals are free from autocorrelation.


GERMAN MARK Because data on the D-mark indicate heteroscedasticity, a natural log transformation was taken to stabilize the variance before first differences were taken to make the data series stationary.

Optimal hedge ratios were first estimated using ordinary least squares regression in which the D-mark spot rate was regressed against the futures rates:

GMSP, = 0.00395 + 0.98000 GMFUT, + e, (0.00119) (0.00297)

with R 2 = 0.9945 and 6’ = 2.46*10-6 (8)

This model indicates an optimal hedge ratio of 0.9800 to 1, and the R 2 of 0.9945 implies a very high hedging efficiency. However, the Durbin-Watson Statistic is 1.398, indicating the presence of autocorrelation, and the estimated degree of first order autocorrelation is 0.300.

An autoregression procedure was used next to adjust for the presence of autocorrelation, yielding the following model:

GMSP, = 0.00865 + 0.96830GMFUT, (0.00317) (0.00785)

with R 2 = 0.9622 and (i2 = 1.96*10-6 (9)

This indicates a hedge ratio of 0.9683 and a measure of effectiveness of 0.9622. How- ever, the residuals remain correlated, suggesting that an ARIMA model is appropriate.

The Box-Jenkins methodology for the D-mark spot and futures data produced the following model:

( 1 - B ) GMSP, = 0.94495 (1 - B ) GMFUT, + (1 - 0.845B) e, (0.0 1205) (0.02207)

and b2 = 1.93*10-6 t 10)

As in the case of the British Pound, this is a simple moving average filter with a lag of one period. The hedge ratio is now 0.9450, less than previously indicated. The residuals appear to be free from autocorrelation, and the p-values are also acceptable. This model shows no significant autocorrelation.

CANADIAN DOLLAR The next currency examined was the Canadian Dollar. These data were also heteroscedastic, so the natural log transformation was taken to stabilize the variance of the white noise before first differences of the data were taken to make the series stationary.

Next the OLS optimal hedge ratio was calculated by regressing spot rates against futures rates, and the following model obtained:

CDSP, = 0.03184 + 0.96211 (CDFUT,) + e, (0.00595) (0.00737)

with R 2 = 0.9658 and 8’ = 5.24*10-6 (11)

This implies a hedge ratio of 0.9621 to 1 , and an effectiveness ratio (R2> of 0.9658. The Durbin-Watson Statistic for this model is 0.66, indicating the presence of autocorrelation.


The hedge ratio indicated by the autoregression procedure is 0.9036. The model is given by:

CDSP, = 0.07920 + (0.90355) CDFUT, (0.01506) (0.01836) (12)

For the Canadian Dollar futures and spot rates, the appropriate Box-Jenkins transfer function is estimated to be:

(1 - B)(1 - 0.0182B) GDSP, = 0.8928 CDFUT, (0.3074) (0.0193)

+ (1 - 0.6909B - 0.1185B2) e, (0.3054) (0.2387) (13)

This model has an error structure that consists of a differenced (MA2) filter. The indicated hedge ratio of 0.8928 is lower than previously indicated. The autocorrelations of the residuals appeared to be insignificant.

SWISS FRANC For the Swiss franc also, the data were heteroscedastic, so a log transformation was again taken to stabilize the variance before the series was differenced to make it a stationary process.

The hedge ratio was first calculated using OLS regression:

SFSP, = 0.01080 + 0.96270 SFFUT, + e,

The R 2 is 0.9918 and 6’ is 5.24*1OW6. This indicates a hedge ratio of 0.9627 and a very high effectiveness. However, the Durbin-Watson statistic is 0.8450 and the estimated first order autocorrelation 0.5750. Autoregression modelling gives the foliowing model:

SFSP, = 0.02780 + 0.9283 SFFUT,

(0.00 174) (0.00355) ( 14)

(0.00524) (0.0 106) (15)

The R 2 is 0.9270 and 6’ = 2.56*10-6, the indicated hedge ratio is 0.9283. The continued presence of autocorrelation, however, makes the estimates questionable.

The Box-Jenkins model given by the transfer function procedure is:

(1 - B)(1 - 0,15498’) SFSP, = 0.9180 SFFUT, (0.04948) (0.0117)

(1 - 0.8493B + 0.0312B5) e, (0.0306) (0.0262) (16)

This model is a simple MA filter of the differenced residuals. There is an observed one day moving average effect plus a weekly effect.

The appropriate hedge ratio is shown to be 0.9180, lower than previously indicated. The residuals are clean (and reduce to white noise), and the p-values also indicate the absence of autocorrelation.

JAPANESE YEN The last currency tested was the Japanese yen. Again, the data were heteroscedastic SO

the log transformation was taken before differencing the data to make it stationary.


Optimal hedge ratios estimated using OLS are:

JYSP, = 0.003478 + 0.98120 JYFUT, + e, (0.00218) (0.00519)

with R 2 = 0.9834 and e2 = 6.65*10-6 (16)

The optimal hedge ratio is indicated to be 0.9812 to 1, with the measure of effectiveness estimated at 0.9834. The Durbin-Watson statistic is 0.972, which indicates the presence of autocorrelation, which was estimated to be 0.5 140.

JYSP, = 0.01247 + (0.9597) JYFUT,

The model given by autoregression analysis is:

(0.00568) (0 .O 1325)

with R 2 = 0.8979 and e2 = 3.46*10-6 (17)

For the Japanese yen futures and spot rate, the estimated Box-Jenkins transfer function model is:

( I - B)(JYSP,) = 0.97400 (1 - B ) JYFUT, (0.0 1199) + (1 - 0.8926B + 0.0808B’0) e,

(0.0 187) (0.0413) (18)

The model structure is MA(1) with lags at one and ten periods. The indicated hedge ratio is 0.9740 which is the only case where the ARIMA model yields a hedge ratio higher than the other methodologies. However, though the hedge ratio is larger than for the autoregression model, it is nevertheless less than the OLS hedge ratio. Again, the autocorrelations of the residuals appear to be insignificant.

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