Futures Hedge Profit Measurement, Error-Correction Model vs. Regression Approach Hedge Ratios, and Data Error Effects.
* There are a number of recent surveys concerning firms' financial risk management practices (e.g., Berkman, Bradbury, and Magan, 1997; Bodnar, Hayt, and Marston, 1996 and 1998; Bodnar, Hayt, Marston, and Smithson, 1995; Gay and Nam, 1998; and Howton and Perfect, 1998). Typically, about half of the respondents use derivatives for hedging. They also generally report that futures are the most commonly used commodity hedging instruments and are among the most commonly used currency, interest rate, and equity hedging instruments.
A primary futures hedger concern is whether the hedge can be expected to reduce their risk. The baseline hedging effectiveness (defined as the percentage risk reduced by the hedge) forecast is typically assumed to be what it would have been over a recent period.
Along with the expected hedging effectiveness, the hedge ratio must be determined. The regression method is the most popular for determining hedge ratios. It is discussed in mainstream corporate finance texts (e.g., Brealey and Myers, 1996). This paper proposes that and explains why hedge profits and regression approach hedge ratios should be calculated using cost-of-carry-adjusted price changes. This Modified Regression Method for determining hedge ratios is denoted by MRM. The paper discusses the Error-Correction Model as it has been applied (denoted by ECM), discusses how it should be applied (denoted by MECM), and relates each to the MRM. It finds that estimated MRM and ECM hedge ratios are similar, though MRM hedge ratios tend to be slightly smaller and more variable. It also finds that the two approaches' hedging performances do not appear to be economically, nor statistically, significantly different. Data errors can account for the MRM hedge ratios being smaller and more variable.
The paper is organized as follows. Section I discusses the regression method. Section II reviews the ECM, presents the MECM, and shows their relation to the MRM. Section III presents the tests comparing the MRM and ECM hedge ratios and hedging performances. It also gives the test results and argues that data errors can account for the MRM hedge ratios being smaller and more variable than the ECM hedge ratios. The last section has the conclusions. Additional results are presented in Appendix A.
I. The Regression Approach
Ederington (1979), building on the foundation laid in part by Stein (1961), derived the regression approach hedge ratio using the inventory carrier's economic profit formulation. The hedger is assumed to minimize the variance of his or her profit ([pi]) conditional on the current period information. The definition of [pi] is important. It aids in specifying the conditional information set for estimating the hedge ratio and represents the foundation for measuring hedging performance. Profit from time t to time t+1 for a portfolio consisting of a long-spot position and a short position (i.e. hedge ratio) of N maturity-date-T futures contracts is given as:
[pi] = ([P.sub.t+1] - [P.sub.t] - [C.sub.St]) - N([F.sub.Tt+1] - [F.sub.Tt]) (1)
where [pi], ([P.sub.t+1] - [P.sub.t]),[C.sub.St] , and ([F.sub.Tt+1]- [F.sub.Tt] denote the portfolio's profit, the change in the spot price, the spot asset cost of carry, and the change in the maturity-date-T futures contract price, respectively, from time t to t+1.  From Equation (1), the MRM variance-minimizing hedge ratio estimate is the [beta] in the following OLS regression:
MRM hedge ratio regression:
([P.sub.t+1] - [P.sub.t] - [C.sub.St]) = [alpha] + [beta]([F.sub.Tt+1] - [F.sub.Tt]) + [[epsilon].sub.t+1] (2)
Traditional Regression Method, denoted TRM, hedge ratios are also calculated this way but assume [C.sub.St] is zero. They have been estimated by many authors (e.g., Chou, Denis and Lee, 1996; Ederington, 1979; Ghosh, 1993; Ghosh and Clayton, 1996; and Kroner and Sultan, 1993).
If the spot asset cost-of-carry were constant, MRM and TRM regression [beta]s would be the same. However, because the spot asset's cost-of-carry depends on the spot price level and the financing rate, where both are stochastic and part of the conditional information set, the spot asset cost-of-carry cannot be treated as a constant. Hence, the TRM regression [alpha] is not constant and its [beta] is not the minimum-variance unbiased estimator.
Leistikow (1992, 1994) finds that the MRM's hedging performance is statistically significantly better than the TRM's for debt, currencies, and equities using this paper's tests and data.
The Castelino (1990) TRM criticism does not apply to the regression method, as it should be implemented (MRM). The criticism was that a "to-maturity" hedge should have a hedge ratio of one due to the maturity date spot and futures price convergence, but would only be one if the hedge beginning date spot and futures prices were equal.
The cost-of-carry model cannot be used to control for changes in the hedge ratio caused by changes in the futures contract time-to-maturity or other changes in the conditioning information set (such as those discussed in Bell and Krasker, 1986; and Leistikow, 1993; and those empirically verified in Leistikow, 1989). In this paper, to control for the possibility that the hedge ratio is a function of the futures contract's time-to-maturity, the futures contract times-to-maturity for all the observations used in the [beta] calculation are the same. Moreover, their time-to-maturity is that for the futures contract in which the hedge is to be implemented. Leistikow (1992, 1994) finds that this method statistically significantly improves hedging performance.
II. The Error-Correction Model Approach
Recently, time series analysis and the ECM have been used to determine futures hedge ratios (Chou et al. 1996; Ghosh, 1993; Ghosh and Clayton, 1996; and Kroner and Sultan, 1993). These authors find that the ECM's hedging performance beats the TRM's.  However, the ECM's hedging performance should be compared to the MRM's and, in contrast to what has been done, the comparison should be based on cost-of-carry-adjusted hedge profits. Also these studies have only partially controlled for the hedge ratio being a possible function of the futures contract time to maturity, by considering only the nearest to expiration futures contract. The observations used to determine their hedge ratios do not have the same time to maturity, and their time-to-maturity is not that for the futures contract in which the hedge is to be implemented.
The ECM approach begins with a time-series model, such as Equation (3) (Kennedy, 1992):
[P.sub.t+1] = [alpha] + [[beta].sub.1][F.sub.Tt+1] + [[beta].sub.2][F.sub.Tt] + [[beta].sub.3][P.sub.t] + [[epsilon].sub.t+1] (3)
In practice, spot and futures price lag terms in addition to [[beta].sub.2][F.sub.Tt] and [[beta].sub.3][P.sub.t] may be included. On the other hand, no lag terms are needed if [P.sub.t+1] and [F.sub.Tt+1] are each stationary, or equivalently they do not have unit roots. If they are stationary, they are said to be integrated of order zero, denoted as I(0). To determine if [P.sub.t+1] and [F.sub.Tt+1] are I(0), a unit root test is performed on each. Generally, the unit-root null hypothesis is accepted for each. This was the case for the debt, equity, commodity, and currencies analyzed in this study and in Chou et al. (1996), Ghosh (1993), Ghosh and Clayton (1996), and Kroner and Sultan (1993). (Results are available upon request.) When the unit-root null hypothesis is accepted for the price levels, the first differences ([P.sub.t+1] - [P.sub.t]) and ([F.sub.Tt+1] - [F.sub.Tt]) are tested. Generally, the first differences are stationary. This was always the case for the debt, equity, commodity, and currencies analyzed in this study and in Chou et al (1996), Ghosh (1993), and Ghosh and Clayton (1996). (Results are available upon request.) When the first differences are stationary, [P.sub.t+1] and [F.sub.Tt+1] are said to be integrated of order one, denoted as I(1).
Imposing the standard long-run equilibrium ECM constraint that [[beta].sub.1] + [[beta].sub.2] + [[beta].sub.3] = 1, Equation (3) can be manipulated to produce ECM Equation (4) (Hendry, Pagan, and Sargan, 1984; and Kennedy, 1992): 
ECM hedge ratio regression:
([P.sub.t+1] - [P.sub.t]) = [alpha] + [[beta].sub.1]([F.sub.Tt+1] - [F.sub.Tt]) + (1 - [[beta].sub.3])([F.sub.Tt] - [P.sub.t]) + [[epsilon].sub.t+1] (4)
In ECM parlance, the ([F.sub.Tt] - [P.sub.t]) term is known as the error-correction term, denoted as ECT (e.g. Kroner and Sultan (1993)).  According to the ECM literature (Chou et al. (1996), Ghosh (1993), Ghosh and Clayton (1996), and Kroner and Sultan (1993)), the ECT is a measure of long-run disequilibrium spot and futures price responses.
However, in the more mainstream, longer standing, and more widely accepted cost-of-carry literature (Working, 1949, is one of the seminal papers), the ([F.sub.Tt] - [P.sub.t]) term is known as the basis. The cost-of-carry model equilibrium has the basis equal to the spot asset cost-of-carry from period t to T. This can be seen from the discussion in Brealey and Myers (1996).  From the cost-of-carry perspective, the (1 - [[beta].sub.3])([F.sub.Tt]-[P.sub.t]) ECM term is the spot cost-of-carry from t to t+1. Thus, from this perspective, Equations (2) and (4) are nearly equivalent. The ECM essentially estimates the proportionate-hedge period (i.e., the hedge period divided by the hedge beginning date futures contract time-to-maturity) as an independent variable and multiplies it by the basis to derive the cost-of-carry as an independent variable, whereas the MRM's dependent variable is explicitly cost-of-carry adjusted. On theoretical grounds, the latter is preferred. The ECM also does not suggest that out-of-sample hedge profits should be cost-of-carry adjusted and that hedging performance measures (such as hedging effectiveness) should be based on cost-of-carryadjusted hedge profits. Despite the similarities noted above, the cost of carry and ECM disequilibrium interpretations are incompatible. In the cost-of-carry model, there is disequilibrium when the basis differs from the cost of carry to the futures contract maturity, not when the basis is zero.
However, reinterpret the spot price as the spot price inclusive of the cost of carry to the futures contract maturity in Equation (4) and denote this version as the MECM.  Its left side is the cost-of-carry-adjusted spot price change, as called for in the MRM.  In addition, the MECM error correction term is ([F.sub.Tt] - [P.sub.t] - [[C.sup.T].sub.St]) as called for by the cost-of-carry model, where [[C.sup.T].sub.St] is the cost-of-carry to the futures contract maturity. The MECM error correction term could make a contribution over the MRM. However, the disequilibrium is not "long-run." There should not be a MECM error term contribution in the low frequency data applied to the liquid (within three months to expiry futures) markets considered in this study and the literature. Liquid markets and relatively low frequency data are studied in the literature because they are the ones likely to be of interest to hedgers. Disequilibrium would be eliminated within minutes in these liquid markets. It should not be a factor in the daily differences studied by Ghosh (1993) and Ghosh and Clayton (1996), the weekly differences considered in Kroner and Sultan (1993), the up to two-week differences studied in this paper, or the up to five-week differences considered in Chou, et al. (1996).
Before the ECM can be applied, the ECT (i.e., basis) must be tested for stationarity. When the ECT is stationary, the spot and futures prices are said to be cointegrated. Cointegrated series can be represented by ECMs (Engle and Granger, 1987). The ECT is generally found to be stationary for storable assets. (This was always the case for the debt, equity, commodity, and currencies analyzed in this study  and in Kroner and Sultan, 1993.)  This is sensible since 1) for futures to be actively traded, they have to be effective as hedging tools and 2) for hedges to be effective, spot and futures prices have to be cointegrated (i.e., significant shocks effect each similarly). Since [P.sub.t+1] and [F.sub.Tt+1] are I(1) and the ECT is stationary, Equation (4) is estimated and the ECM futures hedge ratio is given by [[beta].sub.1].
III. Methodology and Results
To examine the above issues, COMEX gold futures and spot price data were gathered.  Weekly Friday matched spot and futures closing/settlement price data were collected. When there was no Friday trading in one or both of the markets (perhaps due to a holiday), the preceding Thursday was used. If there was also no trading on Thursday in one or both of the markets, the following Monday was used. This strategy was employed so that each spot/futures price change pair encompassed about seven days and each member of a price change pair encompassed the same economic events. Days for which futures prices were nonmarket prices, due to futures price limits, were treated as if they were holidays (see the discussion above).
The futures contracts' times to maturity were approximately between two months and one week to ensure that the futures prices were the most liquid possible. Seven weekly spot/futures price change pairs were calculated for each futures contract and were denoted by the futures contract's time to maturity. The seventh weekly price change for each contract represents the price change for which the futures contract had approximately from 52 to 45 days to maturity, the sixth price change represents the price change from 45 to 38 days-to-maturity, and so on, down to the first price change which represents the price change for which the futures contract had from 10 to 3 days to maturity. There was no time overlap between price change pairs; in particular, there was no time overlap between low time-to-maturity price change pairs for one contract and high time-to-maturity price change pairs of the subsequent contract. Thus, there was no hedge profit dependence across periods induced by time overlap. In addition to one -week price changes, two-week price changes were generated from the above data and analyzed.
Six futures contract maturities per year were examined (February, April, June, August, October, and December). The data correspond to the April 1978 through February 1991 futures contracts.
The following spot-price change cost-of-carry adjustment was used. The cost of carry is the interest on the spot price compounded for the period of the hedge at the yield-to-maturity of the approximately two week-to-maturity T-bill (reported in the Wall Street Journal).  The two week-to-maturity T-bill is appropriate for the one- and two-week hedges being considered. The hedge-ratio regressions are across contracts (i.e., time to maturity is fixed.) Because it is unclear how many observations should be used to calculate the hedge ratio, the hedge ratios were calculated twice, once using 10 matched spot and futures price change pairs and a second time using 20 matched spot and futures price change pairs. When 10 observations were used to calculate the hedge ratio, the hedge ratio to be implemented for each contract-price change pair is calculated from the same time-to-maturity price change of the previous ten contracts. Therefore, when ten observations were used to calculate the hedge ratio for which there are 78 futures contracts, 68 (=78-10) hedges are implemented for each time to maturity beginning with contract 11. Similarly, when 20 observations were used to calculate the hedge ratio, 58 (=78-20) hedges are implemented for each time to maturity beginning with contract 21. Because there are seven different times-to-maturity for the one-week price changes, there are 7x68=476 hedges implemented when the hedge ratio is calculated using 10 observations and 7x58=406 hedges when the hedge ratio was calculated using 20 observations. Similarly, because there are only three different times to maturity for the two-week price changes, there are 3x68=204 and 3x58=174 hedges implemented when 10 and 20 observations are used to calculate the hedge ratio, respectively.
Table 1 compares MRM and ECM hedge ratios. Panel A presents their means, variances, and parametric tests for differences in their means and variances. Columns 3 and 5 present the MRM and ECM hedge ratio means, respectively. The ECM mean is slightly higher but the difference does not appear to be economically significant. This is particularly true because futures contract indivisibility prevents the implementation of the exact estimated hedge ratios. To determine if the means are statistically significantly different, a standard t-test is performed on the hedge ratio differences (MRM HR minus ECM HR) and presented in column seven. The t-values are generally not significant at the 0.05 level. Columns four and six present the MRM and ECM hedge ratio variances, respectively. The difference in variances does not appear to be economically significant. To determine if the hedge ratio variances are statistically significantly different, we use the Morgan (1939) parametric paired t-test for equality of variance. Its t-statistic is calculated as:
[t.sub.n-2] = ([[[sigma].sup.2].sub.a] - [[[sigma].sup.2].sub.b])/[[sigma].sub.a] [[sigma].sub.b] (1 - [[r.sup.2].sub.ab]) (n-2/2) (5)
where [[[sigma].sup.2].sub.a] is the ECM hedge ratio variance, [[[sigma].sup.2].sub.b] is the MRM hedge ratio variance, [r.sub.ab] is the correlation coefficient between the two hedge ratio series, and n equals the number of observations. We use the Morgan (1939) t-test rather than the better-known F-test because of the high positive correlation between the two hedge ratio series. In three of the four cases, the t-value presented in column eight is statistically significant at the 0.01 level. The MRM hedge ratios are more variable.
Nonparametric binomial tests comparing the ECM and MRM hedge ratios are presented in Panel B. Column three presents the number of MRM and ECM hedge ratio pairs. Column four shows the percentage of times that the MRM hedge ratio exceeds the ECM hedge ratio. The statistical significance from a binomial test of the percentages is presented in column five. The MRM hedge ratio is generally smaller (but not statistically significantly smaller) than the ECM hedge ratio. A second binomial test calculates the percentage of the time that the hedge ratio absolute value deviation from its median is larger for the ECM than the MRM. The percentage is presented in column six and its statistical significance is in column seven. The ECM hedge ratio's absolute value deviation from its median is smaller than the MRM's and, in most cases, it is statistically significantly smaller.
The hedging performances of the MRM and ECM hedge ratios are compared in Panel C. In each case, the hedge profit is calculated as the spot cost-of-carry adjusted price change minus the product of the futures price change and the hedge ratio. Hedging performance is measured in two ways--by the hedge profit absolute value deviation from its median and by the hedging effectiveness or its close relative, the variance of the hedge profit. The smaller the hedge profit absolute value deviation from its median, the higher the hedging effectiveness, or the lower the hedge profit variance, the better the hedging performance. The number of hedges is given in column three. The percentage of the time that the hedge profit absolute value deviation from its median is smaller when the MRM approach hedge ratio is used rather than the ECM hedge ratio is presented in column four. A percentage above 50 indicates that the MRM hedge ratio is superior in minimizing risk, while a percentage below 50 indicates it is inferior. The binomial test Z value for the number of times in which the hedge profit absolute value deviation from its median is lower when the hedge profit is calculated using the MRM hedge ratio rather than the ECM hedge ratio is given in column five. The (out-of-sample) hedging effectiveness of the MRM and the excess of the MRM hedging effectiveness over that of the ECM approach is presented in columns six and seven, respectively. The Morgan (1939) t-value for the difference in the variances of hedge profit calculated using the MRM and ECM hedge ratios is given in column eight. The hedging performances do not appear to be economically or statistically significantly different. 
Similar Panel A-C results are found for debt (T-Bills), foreign exchange (German Mark, British Pound, Japanese Yen, and Canadian Dollar), and equities (S&P 500) for one-week and two-week hedge ratios and hedging performances where both 10 and 20 observations were used to estimate the hedge ratios. Some of these results are presented in Appendix A. The others' results are available upon request from the authors.
In practice, data errors can occur because the cost-of-carry rates, spot prices, and futures prices are not recorded contemporaneously or because of recording mistakes. The effect of data errors on the MRM and ECM hedge ratios can be assessed by calculating their values based on data simulated with and without normally distributed random errors. The simulation results described here are robust and are available from the authors upon request. Two-week hedge beginning spot prices are simulated from the same lognormal distribution. The hedge ending spot price is the beginning spot price compounded at the simulated cost-of-carry rate for two weeks plus a normal variate. Beginning and ending futures prices are their respective spot prices compounded up at the simulated cost-of-carry rate to the futures maturity. Data errors were simulated at a 5% probability rate for beginning cost-of-carry rates and beginning and ending prices. Data errors cause the hedge ratios to be smaller as the correlation between the dependent variable (the spot price change and the cost-of-carry-adjusted spot price change for the ECM and MRM, respectively) and the futures price change series are smaller. Data errors also cause ECM intercept decreases and basis coefficient increases. The extent to which the basis coefficient exceeds the ratio of the hedge period to the hedge beginning date futures contract time-to-maturity reflects the extent of data errors.
Data errors cause the MRM hedge ratio to decrease by more than the drop in the ECM hedge ratio for three reasons. The ECM has negative beginning spot prices on each side of its regression equation. Thus, errors in the beginning spot price are somewhat offset in the ECM. Likewise, it has positive and negative beginning futures prices on the right side of its regression equation that are somewhat offsetting. Finally, there is no explicit ECM cost-of-carry variable through which cost-of-carry rate data errors can enter. The MRM hedge ratios are more variable as these data error rates, magnitudes, and sources vary. However, ending spot and futures price errors affect the MRM and ECM hedge ratio estimates in the same way because they do not have any offsets in the regression equations.
In contrast to the literature, hedge profits, hedging performance measures, and regression approach hedge ratio regressions should be based on cost-of-carry-adjusted price changes. Castelino's TRM criticism only arose because the spot price change was not cost-of-carry-adjusted. It does not apply to the regression method as it should be implemented (i.e., the MRM).
The ECM, as it has been applied, is flawed on theoretical grounds. It should be applied as the MECM where the spot price is inclusive of the cost of carry to the futures contract maturity. While, the MECM should be an improvement over the MRM when they are applied in illiquid markets using high-frequency data, this type of application is likely of little practical interest.
For the storable assets considered here, the estimated MRM and ECM hedge ratios are similar, though the MRM hedge ratios tend to be slightly smaller and more variable. The two approaches' hedging performances do not appear to be economically, nor statistically, significantly different. Data errors can account for the MRM hedge ratios being smaller and more variable. On theoretical and practical grounds, the MRM is preferred to the ECM unless there are significant data errors.
Robert Ferguson is President of Axiomatic Systems. Dean Leistikow is an Associate Professor of Finance at Fordham University.
The authors would like to thank Michael Desaulnier, Iftekhar Hasan, David Lee, Jim Lothian, Ira Seeman, the Editors, and two anonymous referees for their help in the preparation of this paper. All errors are the responsibility of the authors.
(1.) See Brennan (1958) for the classic cost of carry discussion. Brealey and Myers (1996) discuss it as well. The spot cost of carry includes a risk premium, if one exists. Also, any futures risk premium should be deducted from the futures price change. Given that hedge ratios are about 1, in this paper and generally, any spot and futures risk premia will offset each other for all practical purposes. Nothing is lost by excluding any spot risk premium from the spot cost of carry and excluding any futures risk premium. This is done in the following empirical analysis. The resulting profit can be characterized as the unanticipated profit in a risk-neutral valuation sense.
(2.) Actually, Kroner and Sultan compare the TRM to a method that incorporates both the ECM and time-varying risk. Thus, it is unclear whether their approach's superiority stems from the ECM's superiority over the TRM, the incorporation of time-varying risk, or both. However, the Kroner and Sultan improvement appears to stem from the ECM's superiority over the TRM because: 1) Leistikow (1992, 1994) shows that the MRM is superior to the TRM using the data and tests employed in this paper; 2) this paper finds that the ECM and MRM provide the same hedging results; and 3) McNew and Fackler (1994) find that hedge ratios do not have time-varying risk.
(3.) Technically, Equation (4) should be estimated without an intercept (as in Hendry et al., 1984). However, as in MRM Equation (2), we estimate ECM Equation (4) with an intercept in order to be consistent with the rest of the literature. Moreover, our intercept estimates are insignificant and the hedge ratio estimates are negligibly changed when Equation (4) is estimated without an intercept.
(4.) Their ECT futures price is multiplied by a scale factor but they find it is equal to one empirically.
(5.) This can also be seen in Leistikow (1993) by adding together his Equations (4) and (5).
(6.) Like the futures price, this cost-of-carry-adjusted (to futures maturity) spot price will be I(1).
(7.) However, it would contain an extra term, the unanticipated change in the cost of carry to the futures contract maturity. This term is likely to be very small given that the cost of carry itself is small relative to the spot price. The cost of carry is small because the carry period is less than three months and the annual cost-of-carry rate is small.
(8.) Results are available upon request. It should be noted that in our procedure, each basis of the basis-lagged basis pairs employed in the stationarity test has the same time to maturity to be consistent with the practice of estimating a hedge ratio for a given time to maturity.
(9.) Given that the basis is stationary, the MECM ECT is surely stationary.
(10.) The futures price data came from the Dow Jones News Retrieval. Up to 1985, the spot gold data came from COMEX Yearbooks--thereafter, they were retrieved using a Bloomberg terminal.
(11.) To the extent a convenience yield exists, it should be small and nearly constant for gold.
(12.) Nevertheless, technically the ECM performance is generally very slightly better. It should also be noted that the binomial and Morgan (1939) t-tests did find that the MRM is superior to the TRM for the same equity, debt, and currency data (Leistikow, 1992, 1994). Thus, the tests are powerful enough to distinguish any differences that do exist.
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Modified Regression Method (MRM) vs. Error-Correction Model (ECM) Hedge Ratios
In this table, HR denotes hedge ratio. The MRM-hedge-ratio estimation equation is: [P.sub.t+1]- [P.sub.1] - [C.sub.St] = [alpha] + [beta][[F.sub.Tt+1]-[F.sub.Tt]] + [[epsilon].sub.t+1]
The ECM-hedge-ratio estimation equation is: [P.sub.t+1] - [P.sub.1] = [alpha] + [[beta].sub.1][[F.sub.Tt+1] - [F.sub.Tt]] + (1-[[beta].sub.3])[[F.sub.Tt] - [P.sub.t]] + [[epsilon].sub.t+1]
Panel A. Means, Variances, and Parametric Tests for Differences in their Means and Variances Number of Obs. Used T-Value for Number of in the Hedge the Hedge Weeks in Ratio MRM HR ECM HR Ratio the Hedge Calculation Mean Variance Mean Variance Difference 1 10 0.84023 0.05258 0.84304 0.02865 -0.3229 1 20 0.82662 0.01682 0.83939 0.01487 -2.5106 [**] 2 10 0.88271 0.03276 0.89695 0.01352 -1.1612 2 20 0.89385 0.01429 0.90599 0.00735 -1.6990 [*] T-Value for the Difference in Number of the Hedge Weeks in Ratio the Hedge Variances 1 -8.2473 [***] 1 -1.6291 2 -6.9944 [***] 2 -5.6634 [***] Panel B. Nonparametric Test for Differences in Their Sizes and Dispersions Number of % When the Z-Value for % When HR Obs. Used MRM Hedge the Number Absolute Value in the Ratio of Times that Deviation from the Number of Hedge Number Exceeded the the MRM HR Median for the Weeks in Ratio of Hedge ECM Hedge Exceeds the ECM Exceeds that the Hedge Calculation Ratios Ratio ECM HR for the MRM 1 10 476 48.95 -0.4583 41.39 1 20 406 51.72 0.6948 47.04 2 10 204 48.53 -0.4201 27.45 2 20 174 45.40 -1.2130 22.99 Z-Value for Number of Times that HR Absolute Value Number of Deviation from the Weeks in Median for ECM the Hedge Exceeds that for MRM 1 -3.7585 [***] 1 -1.1911 2 -6.4413 [***] 2 -7.1261 [***] Panel C. Parametric and Nonparametric Tests for Differences in their Hedging Performance % When Hedge Profit Number of Absolute Value Obs. Used in Deviation from Median Number of the Hedge Number is Lower Using the Weeks in the Ratio of MRM HR Rather than Hedge Calculation Hedges the ECM HR Z Value 1 10 476 47.69 -1.0084 1 20 406 50.99 0.3970 2 10 204 47.06 -0.8402 2 20 174 43.68 -1.6678 [*] T-Value for the Difference in Number of Hedging the Variance Weeks in the Effectiveness of Hedge Hedge MRM MRM-ECM Profits 1 69.66 -13.40 -10.5800 [***] 1 74.58 -0.34 -0.6502 2 96.62 -0.07 -0.5672 2 89.28 0.90 0.9498
(***.)Significant at the 0.01 level.
(**.)Significant at the 0.05 level.
(*.)Significant at the 0.10 level.
MRM vs ECM Two-week, 20-Observation Hedge Ratios for Additional Assets
The MRM-hedge-ratio estimation equation is: [P.sub.t+1] - [P.sub.1] - [C.sub.St] = [alpha] + [beta][[F.sub.Tt+1] + [F.sub.Tt]] + [[epsilon].sub.t+1]
The ECM-hedge-ratio estimation equation is: [P.sub.t+1] - [P.sub.t] = [alpha] + [[beta].sub.1][[F.sub.Tt+1] - [F.sub.Tt]] + (1-[[beta].sub.3])[[F.sub.Tt] - [P.sub.t]] + [[epsilon].sub.t+1]
The March, June, September, and December futures contract maturities were examined. The currency contracts begin with the March 1978 contract and end with the September 1992 contract. The S&P 500 contracts begin with the September 1978 contract and end with the June 1991 contract. The futures maturities have between approximately three months plus two weeks and two weeks to maturity. The cost of carry is calculated as the basis times the proportion that the hedge period is relative to the futures contract maturity.
Panel A. Means, Variances, and Parametric Tests for Differences in their Means and Variances T-Value for the MRM HR ECM HR Hedge Ratio Asset Mean Variance Mean Variance Difference BPound 0.96756 0.00157 0.97716 0.00119 -8.4868 [***] GMark 0.97495 0.00185 0.98245 0.00150 -7.5537 [***] S&P500 0.95257 0.00517 0.96482 0.00467 -6.4188 [***] T-Value for the Difference in the Hedge Ratio Asset Variances BPound -4.7540 [***] GMark -4.5518 [***] S&P500 -1.9255 [*] Panel B. Nonparametric Tests for Differences in their Sizes and Dispersions Z-Value for the % When HR Absolute % When the Number of Times Value Deviation from Number of MRM HR that the MRM HR the Median for the ECM Hedge Exceeded the Exceeds the Exceeds that for the Asset Ratios ECM HR ECM HR MRM BPound 234 26.07 -7.3217 [***] 33.33 GMark 234 27.78 -6.7987 [***] 37.18 S&P 500 144 28.47 -5.1667 [***] 39.58 Z-Value for Number of Times HR Absolute Value Deviation from the Median for ECM Asset Exceeds that for MRM BPound -5.0990 [***] GMark -3.9223 [***] S&P 500 -2.5000 [**] Panel C. Parametric and Nonparametric Tests for Differences in their Hedging Performance % When Hedge Profit Absolute Deviation from Number Median is Lower Using of the MRM HR rather than Hedging Effectiveness Asset Hedges the ECM HR Z-Value MRM MRM-ECM BPound 234 47.86 -0.6537 97.80 0.01 GMark 234 45.30 -1.4382 97.87 -0.01 S&P500 144 47.92 -0.5000 96.73 0.05 T-Value for the Difference in the Variance of Hedge Asset Profits BPound 0.2707 GMark -0.1596 S&P500 0.5998 (***.)Significant at the 0.01 level. (**.)Significant at the 0.05 level. (*.)Significant at the 0.10 level.
There are two popular ways of estimating futures hedge ratios. One way is to regress spot price changes against futures price changes (denoted by TRM for Traditional Regression Method). Another is a more sophisticated statistical procedure termed the Error Correction Model (ECM). In effect, this adds the basis to the TRM as an additional explanatory variable. Unfortunately, these approaches to determining hedge ratios are mispecified. The source of mispecification is similar for each. Hedge profits, hedging performance measures, and regression approach hedge ratios should be based on cost-of-carry adjusted price changes (denoted MRM for Modified Regression Method), not on actual price changes as in the TRM. Similarly, ECM hedge ratios should be calculated incorporating the cost-of-carry.
The MRM and ECM approaches to determining hedge ratios are similar both theoretically and empirically. For the storable assets considered in this paper, the estimated MRM hedge ratios tend to be slightly smaller and more variable. Data errors can account for the MRM hedge ratios being smaller and more variable. The hedging performances of the two approaches do not appear to be economically, nor statistically, significantly different. However, the MRM is the preferred approach for the following reasons. The ECM requires the laborious additional steps of testing the price levels, price changes, and basis for unit roots. The MRM calculates the hedge ratio using price changes as given by the cost-of-carry model, whereas in effect the ECM uses the basis times the proportionate hedge period (i.e. the hedge period divided by the hedge beginning date futures contract time-to-maturity) to control for the anticipated spot price change. Unlike the ECM, the MRM makes it clear that out-of-sample hedge profits should be measured using cost-of-carry adjusted price changes.
The ECM should be applied as the Modified ECM, denoted MECM, where the spot price is inclusive of the cost-of-carry to the futures contract maturity. However, while the MECM should be an improvement over the MRM when they are applied in illiquid markets using high frequency (i.e. minute by minute or higher frequency) data, this type of application is likely of little practical interest for two reasons. First, as a rule, hedgers are interested only in liquid markets because the transaction costs are lower and the hedging effectiveness is greater. Second, hedgers are not interested in hedges of such a short duration (in contrast to the high frequency hedges referred to above, the most common hedge period studied in the literature is two weeks).
Finally, Castelino's TRM criticism only arose because the spot price change was not cost-of-carry adjusted. It does not apply to the MRM. The criticism was that a "to-maturity" hedge should have a hedge ratio of one due to the maturity date spot and futures price convergence, but it would only be one if the hedge beginning date spot and futures prices were equal.
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|Author:||Ferguson, Robert; Leistikow, Dean|
|Article Type:||Statistical Data Included|
|Date:||Dec 22, 1999|
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