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HEDGING WITHI IMBE ESP RATE FUTURES: AN ERROR CORRECTION MODEL ASIM GMOSH
ASIM GHOSN is an assistant pro- fessor of finance at Rider College in
Lawrenceville, New Jersey.
olders of fixed-income securities are exposed to interest rate risk, which is a change in the price of a security as a result of a change in the interest H rate. Interest rate risk affects both issuers and
holders of debt securities. Changes in interest rates influence both the return on the interest rate-sensitive assets and the cost of acquiring the funds.
Since 1979, the Federal Reserve’s decision to emphasize control of the money supply rather than of interest rates has led to higher volatility in interest rates, especially short-term interest rates. To control or reduce short-term interest rate risk, short-term interest rate futures can be used as a hedge to ensure future borrowing cost or future investment returns.
Following Working’s [1953, 19621 seminal work on hedging, the theory of hedging was extended by Johnson [1960] and Stein [1961] within a portfolio framework. Ederington [1979] applies the theory of hedging using GNMAs, T-bills, and wheat and corn futures, and observes considerable hedging effective- ness. Kolb and Chiang [1981] apply a duration approach to hedge interest rate risk, using T-bill and T-bond futures. Gay, Kolb, and Chiang [1983] investi- gate the cross hedge between corporate bonds and T- bond futures, and find that T-bond futures are quite effective in reducing the risk of price changes.
Hilliard [1984] and Hilliard and Jordan [1989] apply the duration approach for hedging interest rate risk, which they find to be less accurate than the price change hedge ratio method. Koppenhaver [1984] finds that use of T-bill futures reduces the interest rate risk of banks quite significantly. Howard and D’Antonio [ 19861 and McCabe and Solberg [1989] investigate the hedging effectiveness of T-bill futures on spot T-bills
72 tEDGING WITH INTEREST RATE FUTURES: AN UWOR CORRECTION MODEL JUNE 1993
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with varying maturities. Overdahl and Starleaf [1986] study the relative hedging effectiveness of T-bill futures or CD futures to hedge CDs. Gosnell and Heuson [1990] apply the T-bill futures to hedge the interest rate risk of adjustable-rate mortgage lenders. Breeden [1991] examines dynamic hedging of mortgages with Treasury bond futures.
Because of the increased volatility of interest rates, the importance of hedging with interest rate futures cannot be overemphasized. There are two basic approaches to hedging with interest rate futures: the regression method and the duration method. The regression model attempts to find the hedge ratio by minimizing the variance of the hedged position. This model has some limitations, such as that dfferences in coupon rates or maturity dates between spot and futures instruments, if not properly accounted for, may give rise to inefficient hedge ratios.
Duration provides an alternative measure of the hedge ratio and is equal to SD,/FDF, where S is the price of the spot asset being hedged, D, is its duration, F is the price of the interest rate futures contract, and DF its duration. It too has its limitations, such as that moderate or large changes in interest rates and non- parallel shifts in the yield curve give a less accurate measure of the hedge ratio than the price change hedge ratio method.
According to the traditional theory of hedging, developed by Johnson-Stein-Ederington, the objective is to minimize the risk of the portfolio. This frame- work of a risk-minimizing hedge ratio will be used in this study. Following Ederington, the hedge ratio is estimated by regressing the cash price change on the futures price change. The optimal hedge ratio is given by the estimated regression coefficient, and the hedg- ing effectiveness is measured by R2. Many researchers, including Hill and Schneeweis [1982]; Figlewski [1984, 19851; Witt, Schroeder, and Hayenga [1987]; Myers and Thompson [1989]; Castelino [1990a, 1990b, 19921; and Viswanath and Chattejee [1992], have applied this framework in various contexts.
In most cases, the hedges will be cross hedges. A cross hedge is defined as a hedge where the asset underlying the futures contract is different from the cash market instrument. Because of this, the correla- tion between the price movements of the spot instru- ment being hedged and the futures contract is impor- tant. An unhedged trader wdl face price risk - the risk of adverse price movement of the spot price. A
hedger faces basis risk, where the term basis is defined as the difference between the cash price and the futures price.
Because of convergence of the cash price with the futures price on the maturity date of the futures contract, a portion of the basis risk is predictable to some extent. The hedger's objective will be to mini- mize the basis risk. To do so, the hedger must decide how many futures contracts need to be traded to reduce this risk as much as possible. The optimal hedge ratio is the most desirable combination of cash and htures positions that will minimize the risk of the portfolio.
Typically, the hedge ratio is estimated by the ordinary least squares (OLS) method, and the hedging effectiveness is measured by R2. Three different theo- retical specifications of the spot and futures price are used to estimate the optimal hedge ratio. The first one is the price level model, where the cash price is regressed against the futures price. The second specifi- cation is the price change model, where the change in cash price is regressed against the change in futures price. For the third specification, the percentage change in the cash price is regressed against the per- centage change in the futures price.
It is important to derive the optimal hedge ratio using sound theoretical reasoning. In the futures litera- ture, there is considerable disagreement as to which is the appropriate hedge ratio. Price-level regression is supported by Witt et al. [1987], price change regres- sion by Hill and Schneeweis [1982] and Wilson [1983] (and many others), and the percentage change regres- sion is supported by Brown [ 19851.
This leaves the prospective hedger in a state of confusion, and the outcome of the hedge will be less than satisfactory, because these methods are misspeci- fied. The objective of this study is to improve hedge ratios by correcting the statistical problems and apply- ing the recently developed theory of cointegration as an example to hedging thirty-day federal funds (FF) and one-month London Interbank Offer Rate (LIBOR) with the respective futures contracts.
Economic theory is based on equilibrium rela- tionships between and among variables. Statistical anal- yses are applied to economic and/or financial time series to test such relationships.
Classical statistical inference will be valid if the variables are stationary. Most financial time series are, however, not stationary, and for testing any of the
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regression models mentioned earlier, using nom-sta- tionary variables will result in rrtisspecification. To achieve stationarity, financial econclmists difyerence the variables and use these differenced variables in statisti- cal analysis. Valid statistical inference will be achieved, but at the expense of losing valuable long-run infor- mation.
The concept of cointegratjon, introduced by Granger [1981] and developed further by Engle and Granger [1987], incorporates the presence of non-sta- tionarity, long-term linkages, and the short-run prop- erties in the modeling process.
A financial time series is said to be integrated of order one, i.e., I(1), if it becomes stationary after first differencing. If two series are integrated of order one, they may have linear combinations that are stationary without differencing. If the linear combination of two such series is stationary, the series are said to be cointe- grated according to Engle and Granger.
These researchers show for a cointegrated sys- tem how to evaluate the existence of equilibrium rela- tionships, as implied by financial theory, within a dynamic specification framework. For price level regression, ignoring the non-stationarity of the vari- ables can lead to spurious regression according to Granger and Newbold [1974]. Price change regression specification disregards the error correction term and thereby fails to incorporate the impact of last period's
equilibrium error. Percentage change regression speci- fication ignores the short-run deviations by not including the lagged variables.
In what follows, we show how non-stationari- ty, long-run relationships, and short-run deviations are integrated within a dynamic specification frame- work.
I. DATA AND METHODOLOGJT
The data for this study consist of daily spot and nearby futures prices of the thirty-day FF and one- month LIBOR. The data from the Knight-Ridder News Service cover the period from December 3, 1990, through December 4, 1992. Futures contracts are rolled over to the next nearby contract, which we use because it is highly liquid and the most active.
There are some observations missing from the spot price series and/or the futures price series for both FF and LIBOR. The data are checked carefdly to ensure that a pair of spot and futures prices are
available for each day. After filtering the data, 493 and 467 observations remain available fcr analysis of FF and LIBOR, respectively.
Without guidance from theory on how many observations should be used for estimation and out-of- sample forecasting, an initial portion of the sample is used for estimation, with the remaining sample segre- ,gated for out-of-sample forecasting. 300 and 375 observations are employed for estimation, and 193 and '92 observations are used for out-of-sample forecasting, for FF and LIBOR, respectively. Cash and futures prices are denoted by C, and F,, respectively, at time t.
A variable yt is integrated of order one (i.e., Y(1)) if it requires differencing once to make it station- ary. Let us consider two time series xt and yt, which are both I(1). Usually, any linear combination of x, and y, will be I(1). But if there exists a linear combination Z, - yt - 01 - Pxt, which is I(O), then xt and yt are cointegrated, according to Engle and Granger, with the cointegrating parameter p. Cointegration'links the long-run relationship between integrated financial variables to a statistical model of those variables.
In order to test whether the series are cointe- grated, it is imperative to check that each series is I(1). Testing for unit roots is conducted by performing the augmented Dickey-Fuller (ADF) [1981] regression, which may be written as:
-
where p is large enough to ensure that the residual series E, is white noise.
For sufficiently large values of p, the ADF test loses its power. An alternative test proposed by Phillips and Perron (PP) [1988], which allows weak depen- dence and heterogeneity in disturbances, is performed using the following regression:
Yt = bo + blYt-1 + Ut (2)
where ut is white noise. Once it is found that each series contains a sin-
gle unit root, i.e., I(1), it is interesting to check whether they are cointegrated. If the series share a long-run relationship, then the series involved are cointegrated. Empirical existence of cointegration is tested by constructing test statistics fmm the residuals of the following cointegrating regression:
74 HEDGING WITH INTEREST RATE N N R E S : AN ERROR CORRECTION MODEL JUNE 1993
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C, = a + bF, + e,
If the two series are cointegrated, then 6, will be I(0). The ADF test is performed on the estimated residuals 6 , from Equation (3):
(3) EXHIBIT 1 Augmented Dickey-Fuller (ADF) and Phillips Perron (PP) Tests for Unit Roots in Autore- gression of the Logarithms of Daily Closing Prices of Thirty-Day Federal Funds and One-Month ]Lon- don Interbank Offer Rate
cri t ical (4) FF LIBOR Value
A D F PP ADF PP (10%)
where q is large enough to make v, white noise. The estimated residuals are also subject to a PP test: Spot -1.14 -1.75 -2.37 -2.37 -2.57
Futures -0.54 -1.23 -1.22 -0.89 -2.57
a, = a + pa,, + y,
where v, is white noise.
DifFerences:
Futures -7.40 -36.95 -4.81 -26.79 -2.57 Spot -9.60 -15.97 -5.60 -19.46 -2.57 (5)
Once it is established that the series are cointe- grated. their dvnamic structures can be exdoited for
Critical values taken from MacKinnon [1991]. " further investigation. Engle and Granger show that cointegration implies and is implied by the existence of an error correction representation of the series involved. An error correction model (ECM) abstracts the short- and long-run information in modeling the data. The relevant ECM to be estimated to generate the optimal hedge ratio is given by
specification of the dynamics is estimated by OLS in the second stage. The appropriate values of n and m are chosen by the Akaike information criterion (AIC) [1974].
Equations (6) and (7) provide the optimal hedge ratios P and b, respectively. Performance of these models is evaluated by means of a likelihood
i=l j=l ratio test by constructing the test statistic -21nh, where h is the ratio of the likelihood functions. The test statistic has a X distribution, with degrees of freedom equal to the number of restrictions. Effec- tiveness of the hedge ratios is judged by examining their out-of-sample forecasts.
L,
(6) m
A C ~ = ae,-, + ~ A F , + i y i ~ l - i + E ~ , A c , - ~ + U t
2 where n and m are large enough to make ut white noise.
In order to establish the superiority of this hedge ratio, we compare it with the one estimated from the traditional price change regression specifica- tion: II. EMPIRICAL EVIDENCE
AC, = a + bAF, + 6, (7)
where 5, is white noise. Equation (7) ignores the error correction term,
and thereby fails to take into account the last period's equilibrium error. Furthermore, it fails to capture the short-run dynamics by excluding the relevant lagged variables. The hedge ratio estimated from such a small- er information set is likely to be less informative.
Engle and Granger propose a two-step estima- tion procedure for the estimation of the parameters of Equation (6). First, C, is regressed on F, and the resid- uals are collected from Equation (3) by using ordinary least squares (OLS). The ECM with the appropriate
All the spot and nearby futures prices are tested to ensure they are I(1). The results of the ADF and PP test are shown in Exhibit 1. The level series demon- strate that they have a unit root in their autoregressive representations. This evidence indicates that the series are non-stationary.
Next the difference series are checked for the presence of a unit root. The ADF and PP tests clearly reject the null hypothesis of the presence of a unit root. This implies that the difference series are I(0). Therefore, the spot and futures prices are I(1) for FF and LIBOR.
Once it is established that each series is I(1), it is imperative to test whether there is a linear combina-
JUNE 1993 THE JOURNAL OF FIXED INCOME 75
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EXHIBIT 2 Phillips Perron (PP) Tests for Coiintegradsn
Augmented Dickey-Fuller (ADIF) and
Regressand Spot Regressor: Futures Critical FF LIBOR Value
ADF PP ADF PP (10% ~~ ~
- -3.05 FF -3.52 -12.06 - LIBOR - - -2.36 -6.39 -3.05
Critical values taken &om MacKinnon [1991]. -
tion of spot and corresponding fu.tures prices that is I(0) for FF and LIBOR. If there is a long-run relation- ship, they will be cointegrated. Results of the tests of cointegration are presented in Exhibit 2.
Both the ADF and PP tests reject the null hypothesis of no cointegration at the 10% level of sig- nificance for both FF and LIBOR, except the ADF
test for LIBOR. This observation reinforces the notion that cointegration unites the long-run relationship between the relevant variables.
Cointegration implies that the series have an error correction representation; conversely, an ECM implies that the series are cointegrated (Engle and Granger). The ECM (6) provides a better representa- tion of the stochastic dynamic relationship between the series, by expanding the information set. For example, the last period's equilibrium error is incorpo- rated through the error correction term. Short-run deviations in one period are adjusted through lagged variables in the next period. Traditional model (7) ignores the error correction term and the lagged vari- ables, thereby providing an inefficient hedge ratio.
Exhibit 3 presents the estimates of the parame- ters of Equations (6) and (7) for both FF and LIBOR. 'To evaluate whether Equation (6) is better than Equa- tion (7), the likelihood ratio test is performed. The likelihood ratio statistics for FF and LIBOR are 85.04
EXHIBIT 3 E Estimates of the Parameters from the Error Correction Moden (6) and Those f r ~ m the Price Change Hedge Ratio Model (7)
FF Regressand A c t
LIBOR Regressor Equation (6) Equation (7) Equation (6) Equation (7) Constant x 10-2 0.1727 -0.1649 -0.293 -0.2657
(0.49)a (-0.42) (-1.12) (-2.29) -0.4659 - -0.1393 - ut-1
(-8.48) (-5.72) AFt 0.7271 0.7275 0.9938 0.9939
(1 10.24) (96.7 5) (429.67) (403.4 1) - - 0.1785 - AFt-1
(3.65) - - - AFt-2 0.0835
(-2.1 11 -0.1831 (3.65)
- - - Act-2 -0.1257 - - - (-2.37)
Log-Lkelihood 421.98 379.46 918.01 892.73 Adiusted R2 0.9763 0.9690 0.9980 0.9977 L R ~ 85.04* 50.56**
=Numbers in parentheses are t-ratios. *he likelihood ratio test statistic:
LR = 2 [Log-Likelihood(6) - Log-Likelihood (7)] * * 3 *x;.o.os = 7.81 x 3 , O . O S = 7.81
76 HEDGING WITH INTEREST RATE FUTURES AN ERROR CORRECTION MODEL JUNE 1993
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and 50.56, respectively, both statistically significant at the 5% significance level.
These findings clearly show that Equation (6) is superior to Equation (7) for both interest rate instru- ments. Therefore, Equation (6) provides a better hedge ratio than the one obtained from Equation (7).
The intercepts from Equations (6) and (7) are found to be statistically insignificant. This appears to imply the absence of linear trend in the data genera- tion process. The error correction terms are negative and statistically significant for FF and LIBOR. These coefficients measure the speed with which the system moves toward equilibrium.
Lagged variables of AF, and AC, are statistically significant for both the interest rate-sensitive instru- ments. These findings show that the deviations in one period are adjusted in the next period.
The estimated coefficient 6 from Equation (6) brings together short- and long-run information in modeling the data. The traditional Equation (7) fails to incorporate these important sources of information.
The superiority of the optimal hedge ratio derived from Equdtion (6) over that obtained from Equation (7) is evidenced by the likelihood ratio test. Despite dfferential basis risk, the hedge ratio estimated by the ECM (6) will be more efficient in controlling and/or reducing the risk of the spot price change. The hedge ratios estimated by Equation (6) are slightly smaller than those estimated by Equation (7). This implies that the traditional model tends to overestimate the number of futures contracts needed to hedge the spot portfolio.
To the extent this overestimation of the hedge ratio is significant, the portfolio manager can incur sig- nificant loss by using the traditional hedge ratio. This provides evidence that the hedge ratio 6 estimated by ECM (6) significantly improves the performance of the hedging activity.
The empirical evidence establishes the superior- ity of Equation (6) over Equation (7) within the sam- ple period investigated in this study. The effectiveness of the hedge ratio, however, should be judged by examining the out-of-sample forecasts from both equations.
Forecasting statistics from both models are pre- sented in Exhibit 4. The root mean squared error (RMSE) of Equation (6), which is a measure of fore- casting performance, is lower than that of Equation (7) for both FF and LIBOR. ECM (6) reduces the RMSE
by 14% and 8% for FF and LIBOR, respectively. This evidence reinforces the superiority of Equation (6) over Equation (7).
To summarize, the evidence presented in this paper demonstrates that the hedge ratio estimated from the ECM (6) is superior to that derived from Equation (7). It is more effective in reducing and/or controlling the risk of the spot price changes. Out-of-sample fore- casts reinforce the superiority of Equation (6).
111. CONCLUSION
This article investigates the traditional price change hedge ratio estimation method in the case of interest rate futures contracts for FF and LIBOR. It is shown that this method is misspecified, because it ignores the error correction term and short-run dynamics.
This study differs in significant ways from past studies. Each series is tested for the presence of a unit root in its autoregressive representation. It is found that each series is integrated of order one. Each pair of series for each interest rate is tested for the presence of a long-run equilibrium relationship. The paired series are then cointegrated for each interest rate.
Cointegration implies and is implied by the existence of an error correction representation of the series. ECM integrates the short- and long-run infor- mation in modeling the data, which proves to be a superior modeling technique compared to the tradi- tional method.
EXHIBIT 4 W Summary Statistics
Statistic Model (6) Model (7) Model (6) Model (7) Mean 0.0069 -0.0016 0.0055 -0.0027 SD 0.0004 0.0378 0.0143 0.0015 RMSE 0.0681 0.0795 , 0.0391 0.0421
RMSE Ratios Ratio WF), A(LIBOR), Model (6)/Model (7) 0.86 0.92
Note: 193 one-step ahead forecasts for thirty-day federal hnds (FF) out-of-a-total of 493 observations and 92 one-step ahead forecasts for one-month London Interbank Offer Rate (LIBOR) from a total of 467 observations December 3, 1990, through December 4, 1992.
JUNE 1993 THE JOURNAL OF FIXED INCOME 77
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T h e evidence presented in this investigation shows that the optimal hedge ratio derived from 3CM (6) is superior to that obtained fi-om the traditional price change regression specification. ECM reduces the RMSE of the spot position for each interest rate by a considerable margin. T h e ECM hedge ratio
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