Bruce A. Benet* Department of Finance
ERROR-CORRECTION MODEL-BASED HEDGE RATIO ADJUSTMENT AND FUTURES OUT-OF-SAMPLE HEDGING EFFECTIVENESS
by
Bruce A. Benet* Department of Finance
College of Business Administration
Central MichiganUniversity
and
Visiting Fellow, the Fred Arditti
Risk Management Center, DePaul University
third revision: March 30, 2017
ABSTRACT: This study investigates the ex ante performance of futures direct and cross hedges, where hedge ratios are adjusted to incorporate estimation period forecast error residuals. We examine the empirical question as to whether spot and futures price cointegration (adapting models of Davidson, Hendry, Serba and Yeo [7] and Engle and Granger [11]) merits hedging model adjustment-correction. ECM-based hedge ratio estimates offer potential improvements in econometric efficiency and hedge ratio stability, which may contribute to improved out-of-sample performance when compared to OLS
orwidely-acceptedunitaryhedgeratiostrategies.Usingdailydatefroma66-month(2006-2011)sample period, tested direct hedges experience lower out-of-sample performance variability; while ECM adjustedcrosshedgeeffectivenessincreases7%-14%whencomparedtoOLSapproaches.Gains
in ex ante hedging performance are even greater (18% to 46%) when compared to alternate strategies.
*Associate Professor of Finance. Please send comments to 302Sloan Hall, Mt. Pleasant, Michigan 48859 (). The author gratefully acknowledges various & suggestions from Fred Arditti, Adam Gehr, Tom Howard, and comments from anonymous JFQA referees; as well as expert data and computer help from Irina Krop and JohnRichard. This is a preliminary draft, and is not for quotation without the author's written permission. Comments are welcome at (616) 724-0066 or (312) 362-5153.
ERROR-CORRECTION MODEL-BASED HEDGE RATIO ADJUSTMENT; AND FUTURES OUT-OF-SAMPLE HEDGING EFFECTIVENESS
l. INTRODUCTION. Numerous studies in the hedging literature (e.g., [3] [8) [16]) compare
futures effectivenessexpostversusexante.Thesestudiesgenerallyreportpoorerorlessconsistentperformance for the more realistic out-of-sample hedges. Reduced ex ante hedging performance
may be attributable in part to intertemporal instability of hedge ratios.[1]Since a real-world hedger
(e.g., a firm repeatedly "anticipation hedging" against increases in the price of raw materials used
in production) cannot determine an optimal hedge ratio ex ante, it may simply use a "one-for-one"
hedge ratio strategy, or (more appropriately) forecast next period's ratio based upon historical
spot-futurescorrelations. Yet either such approach in the presence of changing correlations
causes the firm to under or over-hedge, resulting in unanticipated and unwanted gainsorlosses.
Period-to-period hedging performancewouldfluctuate, often dramatically as result of missing
ex post optimal hedge ratio targets.
Whyshouldratiosbaseduponstatistically-significantspot-futurescorrelationsfroman
earlierhedge period result in poorer risk-reduction some days orweekslater? Many researchers
have studiedthedynamic nature of futures prices and hedges in various frameworks. GARCH [5],
ARIMA [14] [25], binomial diffusion [28], conditional probability-based [33] and random walk [21]
time series processes have investigated to model time-dependent spot-futures price and covariance
relationships, with possible impactonhedgeratios. Yet these highly sophisticated statistical
andmathematical models havenotbeenthoroughly tested in an ex ante framework, wherestable
spot-futures correlations play a crucialrole across multiple hedging periods.
We investigatewhethercointegration[2] between realized spot prices and futures prices may
contribute to priceseriesnon-stationarity,andtherebyundermine intertemporalcorrelation stability
(andthereby bias estimated hedgeratios). Our study focuses on the error-correctionmodel (ECM)
as an extension to themoretraditional Ederington [9] methodology. We first empirically investi-
gate the spot-futures cointegration question in ahedgingcontext.Time series models related to
work by Davidson. Hendry, Srba and Yeo[7]andEngle and Granger [11], as well as test
statistics suggested by Phillips and Peron ([30] [32]) help identify cointegrated time series
and estimate the size of forecast error residuals.[3] These residuals are then used to construct
ECM-based adjusted hedge ratios. In-sample and out-of-sample effectiveness is compared between
adjusted, OLS, and one-for-one (unitary, sometimes described as “naïve”) hedge ratio benchmark
strategies. We focus upon the performance of ex ante hedges, since such outcomes are truly more
representative of hedge success actually available to real-world futures market participants. We
examine the effectiveness of sample foreign currency. interest rate and stock index futures direct
and crosshedges during a 5 ½ year sample period using daily spot (cash, index) and futures prices.
This study asserts two contributions to extant futures hedging literature. First, our explicit
examination of spot-futures cointegration focuseshedging success ontorealized future spot prices.
We argue this is an equally appropriate hedging model perspective to evaluate hedging performance.[4]
Secondly, we believe our empirical findings support the use
of “residual” ECM-based hedge ratio
adjustment techniques as means to increase stability of out-of-sample hedge ratio estimates,
and thereby improve ex ante hedging effectiveness.[5]
- METHODOLOGY. The ECM-based hedge ratio adjustment procedure used in this study is a
three-stage regression approach. First, price series of varying "realization lengths" are tested (using
methodsof DHSY [7], Engle and Granger [l1] and Phillips and Perron ([30] [32]) for spot-futures
price seriesintegration and cointegration; and to examine relative size of any forecast error
residuals. Integrationand cointegration parameterestimatesprovide information concerning
spot and futuresprice stationarity, as well as whether any non-stationarity may be corrected
via statistical adjustment techniques. Secondly, forecast error residuals are incorporated into
a hedge model similar to the Ederington[9] OLS approach. The resulting ECM-based hedge
model is applied in (immediately prior) estimationsub-periods to obtain adjusted hedge
ratio ex ante forecasts.[6] For comparison, OLS ‘unadjusted’ hedgeratios are also
estimated in this second stage. Third and finally, hedge ratio forecasts are next implemented
into subsequent (immediately subsequent) hedge test sub-periods. ECM-based hedge
ratio performance is compared (using multiple hedging effectiveness (HE) measures) to
results obtained using OLS techniques and traditional one-for-one hedges, which serve as HE
out-of-sample performance benchmarks. The balance of our paper section two describes the
adapted ECM-hedge ratio adjustment methodology and tests ingreater detail.
We begin by proposing the following unrestricted autoregressive[7](AR) model specifying relationships between spot and futures prices (commodity, stock index, FX and/or interest rates):
(1)
where: St+k; St+k-1 represent current period and last hedge period's (respectively) spot prices
realized at the hedging horizoninterval (k)
ft,k; ft-1,kare current period and ( beginning of) last hedge period'sfuturesprices for contracts with k days remaining in thehedge
vt+k represents a residual or disturbanceterm
α; βare parameter coefficients
Note that the above model is slightly different from other traditional hedge models, which
focus upon contemporaneous spot-futures prices relationships as a determinant of hedging effectiveness.[8]
Our model focus, however, is upon realized future spot price St+k as a principle HE determinant. Any deviation between ft,k (contracted at the beginning of the hedge period) and St+k (realized at hedge horizon) are reflected in basis changes due to (partial or complete) convergence. Basis changes, also
known as"basis risk" to hedgers, generate dollar gains or losses which determine practical success
or failure of a real-world hedge.10 Thusourforward-looking hedgingmodel emphasizesperformance athedgehorizondatek1 (e.g., when a firm's business-related required spot transaction must occur).
This more general model is not subject to the specific zero coefficient constraints on a.1 and fl:
such as in the Ederington model. However, Moser [24]proposes that the coefficient restrictions
α1+β1+ β2= +1.0, β2= -l+Ф, and α1= l-Ф be imposed onto model equation (1). Hisreasoning
for such parameter constraints and restrictions is likely presence of cointegration between current
futures prices ft,k and ‘realized’ spot prices at end of the hedging horizon St+k. Moser’s proposals
are consistent with those by Benet [3] in that “static” and “dynamic” hedge nature (and HE) of an
ex ante hedge are appropriately best preserved through such constraints on the cointegrating parameter.
Assuming cointegration, improved price series stationarity and model efficiency wouldbe
achieved by differencing ft-1,k and St+k-1 terms to estimate a prior hedge period forecast error
residual term. Rearranging equation (1) according to the above restrictions, we obtain an simplified
error-correction model (ECM) framework in testableform:
(2)
(NOTE:β1*=1.0(theoreticallyduetotheconstraints): St+k-1= St in a "rollover"hedge)
The prior hedge period forecast error residual (3rd RHS term) is an error-correction tothis
period's model. The ECM time series is assumed stationary. We next may compare equation (2)
above to the "levels" form of the Ederington [9] hedge model withouterror-correction:
(3)
Should spot-futures price comtegration be shown to exist, our alternative hedge model specification distills down to omitted variables and econometric efficiency questions. If β2(Ф-1) is
Statisticallysignificantduetotheeconomicimportanceofforecasterrorresidualstothemodel,β1*fromequation(2) above may be interpreted (and used) as an adjusted hedge ratio. We argue that this adjusted hedge ratio wouldbeamoreefficientelasticityestimatethan OLSrisk-minimizingratios
(as estimated from equation (3) abovebecause: 1)the model is more appropriately specified to include realized future spot price and additional information of the error-correction term[9]: and 2) the cointegration correction may result smaller standard errors for the adjusted hedge ratio, relative to traditional OLS estimates. Our second stage is in-sample estimation of ECM-adjusted hedge ratios.
A related empirical question is whether model specification changes and estimation efficiency
gains will favorably affect hedge ratio stability, and thus improve hedging effectiveness ex ante.
The third stage of our regression methodology is an empirical test and comparison of the
ECM-based hedge model equation (2) versus the OLS model equation (3) and traditional
one-for-one “naïve” hedges. For our out of-sample effectiveness tests, relative performance is
measured, where hedge ratio forecasts generated frompricecorrelationsinprioralternating
andnon-overlapping"estimation"sub-periodsareimplemented into (immediately) subsequent
"hedge test" sub-periods. Unlike GARCH or VAR labors, this three-step hedge ratio
adjustment procedure is relatively readily implemented by real world practitioners; yet is also
offered as alternative to in-sample beta and one-for-one ratio calculations commonly employed
in today's derivatives markets for hedging purposes. Note that our model and tests are consistent
with "conditional" hedge ratio models such as those proposed by McKnew and Fackler [23]
and Myers [26]. Suchapproachesassumethathedgersconsiderallavailableinformationat
thetimeofhedgeconstruction. The ECM-adjusted strategy proposed here incorporates
additional (potentially valuable) information of prior hedge period forecast errorsinto the
calculation ofnext period's ratio. Additionally, our focus upon ex ante strategies, which
use only price and covariance information actually available at the beginning of the hedge period
(as opposed to "unconditional" in-sample estimation found in much of the futures hedge literature)
qualifies our ratios as "conditional" in a very real sense, albeit in a somewhat competing
manner to those in the papers mentioned immediately above.[10]
Since there is no consensus at present as to the most theoretically-correct/appropriate
futures hedging effectiveness measure, we calculate two alternative performance measures
in this study. Our first hedging effectiveness measure 1s the traditional Ederington [9]
risk-minimization measure obtained from OLS regression estimation of equation (3)above:
HE1=cov(St.ft)2== r2 Var(St)Var(ft)
(4)
This effectiveness measure is generally accepted and is widely used in numerous hedging studies. It subject to criticism, however, since it is unconditional and its risk-minimization objective implicitly considers only one possibleoutcome from a variety of hedgers' risk-returnpreferences.Asecondmore general risk-return performance measure proposed by Howard-D'Antonio [16):
where:
HE2= |l- 2λρ+λ2
√1-ρ2
λ= _(rf/σf)
(r, - 1)/σs
(5)
and:
rf, rs = the expected one-period percentage change returns for the futures and spot.
σf,, σs =the standard deviationofone-periodpercentage returns for the futures and spot,respectively.
ρ = the correlation between futures and spot changes.
The HE2 performance measure Illustrates risk-versus-return tradeoffs of the chosen
hedge position in a manner similar to the Sharpe index common to the investments literature;
whileincorporating the spot-hedge correlation considered to be crucial to the success of futures
hedges.[11] Model prices are tested in logarithmic form, which (appropriately) permits interpretation
of estimated coefficients as elasticity-type hedge ratios. As mentioned above, out-of-sample
effectiveness results are emphasized inthis study.We believe such ex ante strategies provide
results more ngndicativeofsuccessavailable to actual futuresmarkethedgers. Ex post in-sample
results are also estimatedandpresentedfor econometric comparisonpurposes. Unlike forward
contracts, the limited number of available futures contract delivery dates requires us to focus our
hedge model upon a (admittedly arbitrary) hedging horizon date at time t+kfor "realization", rather
than delivery. Therefore, we examine one-week, two-week and four-week horizons.
3. DATA. We test ECM-based adjusted, OLS and one-for-one hedge ratio strategies focusing on
stock index, foreign currency, and interest rate futures contracts. For an available data sample
period of April. 2008 to September,2011, end-of day (last transaction, not ‘settlement’)
prices for "nearest" S&P 500 futures contracts are extracted froma "tick-to-tick" data tape
provided from the Chicago Mercantile Exchange (CME )[12]; while allmatched corresponding
‘underlying’ S&P 500 indicies are obtained from the Wall Street Journal. For greater realism
and more general applicability of our methodology, a stock index futures cross hedge is tested
in this study: Wilshire 5000 daily spot changes (as would approximate value fluctuations of a
widely-diversified index fund) are covered via S&P 500 futures price change hedges. We
deliberately construct this spot-futures mis-match to investigate cross hedge performance
and impact of what practitioners typically describe as “tracking error” upon our ECM-
based hedge ratio adjustments- and consequential out-of-sample performance- as
described in Section 2 above. Daily Wilshire 5000 index values are also collected from WSJ.
Foreign currency hedges are represented in this study by Japanese yen (spot and) futures.
Nearest contract prices weregatheredfrom the IMM Yearbook. We use daily JY settlement prices
from January, 2009 to December, 2011 in order to (roughly) correspond with our available stock
index futures data.Japanesespot currency rates and Thai baht spot exchanges rates were scanned
from the WSJ to test foreign exchange direct and cross hedges, respectively. Our study also examines interest rate futures hedges. U.S. Treasury Bond futures data was provided to us by the Chicago
Board of Trade (CBOT) for a December, 2006 to May, 2011 sample period; while 20-year, 8%[13]
(or closest available) spot T-Bond prices quotes came from a T-Bill/T-Bond data set obtained from
the Federal Reserve Bank of New York. Gaps in T-Bond spot and futures data were filled from the
Wall Street Journal.The interest rate cross hedge examined in this study is a U.S. T-Bond futures
hedge of the Merrill-Lynch U.S. Corporate Bond Index (l0+ year maturities). We believe this cross
hedge is fairly representative of approaches used to reduce interest rate risk exposure by certain real-world bondmarketparticipants. Cross hedges are examined in this study fortworeasons.
First, cross hedging (in the strictest technicaldefinition) is actually quite common, due to a diversity
of investor spot positions and the relatively small number of traded futures contracts. Secondly, we believe a prioi that some econometric benefit of ECM-based hedge ratio adjustment manifests in
form of improved intertemporal hedge ratio stability. Previousstudies[13][22]suggestthatdirect
hedgesarelesssusceptibletovariableratiosout-of-sample; while the instability problems seem more
severe for cross hedges. Thus we would anticipate (hope for?) more substantial ex antehedging
effectiveness improvements be demonstrated for the crosshedge strategies tested in this research.
Estimation and hedge test sub-periods are structured to be non-overlapping, alternating periods
across the entire sample. One-week, two-week and four-week estimation and hedge test periods
are examined (see illustration ONE). This econometric approach corresponds to (and readily permits
overall performanceevaluationof)rollingone,twoandfour-weekrollinghedgesinanout-of-
sampleframework. Partitioning the dataset in this manner parallels a common ex ante strategy
of using last period's estimated hedge ratio as a "best guess" as to the subsequent hedge period's
optimal hedge ratio. Although there is no consensus in the literature as to a standard out-of-sample
empirical test; we believe our approach is more realistic, and introduces less bias than previous
studies which partition price data into arbitrarily (and often very large/long) historical time segments.
Shorter sampling intervals are also more representativeof real-world hedging practices, such
as typical hedge rebalancing intervals and contract liquidity and maturity constraints. For the
numerical example at the end of the paper (see Table SIX and related discussion), an alternate
"calendar time" methodology is used, and dollar-based hedge performance (net hedge gains/losses)
are averaged across 26 successive two-week hedge test periods. We vary sub period window
intervals, not only for robustness checks, but to investigate the empirical relationship between
out-of-sample effectiveness and estimation period length. The stock index futures tape data is
limited to a 66-month time period which includes 1393 end-of-day price observations. Foreign
currency and interest rate futures spot and futures time series are 1517 observations in length.
4. EMPIRICAL RESULTS.4.1. Cointegration and Forecast Error Tests.Following Phillips
and Peron [30] [32], tests for time series integration and cointegration are performed, with results
reportedin Table ONE. The mechanics and properties of the Z-test statistic used are fully described
in the paper appendix. We first present integration test results for all of the ninetime series included
in our study in panel A. Using critical values for an adjusted t-statistic Z(t) suggested by Fuller [12 ],[14]
we cannot reject the null hypothesis of a unit root I(l ) for any spot or futures prices included
in this study. Presence of unit root is indicative of a non-stationary time series. which we contend
would impact hedge “beta” HR estimation, intertemporal instability of ratios. and resulting ex ante
hedging effectiveness. Results from panel A strongly suggest both spot and futures priceinstability.