complex hedges: how well do they work?
TRANSCRIPT
Complex Hedges: How Well Do They Work?
Dwight Grant Mark Eaker
his paper examines empirically whether complex hedges reduce risk more than T naive or simple hedges?’ Anderson and Danthine (1980,1981) derived two of the complex hedges we study. They extended a simple hedging model by introducing hedges in multiple assets and hedges of portfolios of assets. Baesel and Grant (1982) de- veloped the third complex strategy by including trading at multiple dates.
Ederington (1979) and others show that the simple hedge is the expected value of a regression coefficient. The dependent variable is the spot position. The independent variable is the futures price. Anderson and Danthine identified a similar result when there are multiple futures. The variance-minimizing futures positions are the expected values of ordinary least-squares (OLS) regression coefficients. The dependent variable is the spot position. The independent variables are the multiple futures prices. Anderson and Danthine also examined portfolios composed of linear combinations of spot posi- tions. They showed that the risk-minimizing futures positions are linear combinations of the futures positions for the individual spot positions.
Baesel and Grant examined trading in a different contract at each of two dates. Their results extend to trading in multiple contracts at many dates. The variance-minimizing hedges are the expected values of a vector of regression coefficients. The dependent variable is always a “properly defined” spot position. The independent variables are the futures prices. The “properly defined” spot position at each trading date is the asset be- ing hedged plus the stochastic returns from futures positions at later dates.*
This paper examines the empirical value of these complex hedges which are based on more realistic assumptions. For example, the number of assets traded in futures markets is small relative to those traded in spot markets. Therefore, many potential hedgers will
‘A naive hedge is a one-to-one futures position in the same asset. A simple hedge is a risk-minimizing fu-
%e authors will provide on request a unified derivation of a11 of the complex hedges tested in this paper.
Grant and Eaker received financial support from the Chicago Board of Trade Foundation and the Business Foundation, University of North Carolina. Comments by referees for this Journal and by Chris Bany, Steve Manaster, Wally Thuman, and participants in workshops at Southern Methodist University, North Carolina State University, Concordia University, York University, and the University of New Mexico improved this pa- per. Ellen Berry, Jim Comer, and Randy York provided valuable computing help.
tures position in the same asset.
Dwight Grant is a Presidential Professor of Finance, University of New Mexico.
Mark Eaker is a Professor of Business Administration at the University of Virginia.
The Journal of Futures Markets, Vol. 9, No. 1, 15-27 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0270-7314/89/010015-13$04.00
not be able to trade in identical assets. They must consider cross-hedging. Multivariate cross-hedging follows directly. A combination of cross-hedges may eliminate more risk than a single cross-hedge. Likewise, because many potential hedgers hold portfolios of assets, portfolio hedging is potentially valuable. Finally, hedgers trade at many dates.
There have been few empirical tests of the risk-minimizing effectiveness of these complex hedges. Miller’s work on cross-hedging is relevant to our tests of multivariate cross-hedging. He found (Miller (1982)) that a forecasting model with both live hog fu- tures and corn futures produced a lower mean square error than one using live hog futures alone. In a later study (Miller (1985)) he investigated the cross-hedging relation- ship between millbrans and corn, oats, wheat, and soybean meal futures. This study confirmed the industry’s perception that the closest relationship is between millfeeds and corn futures. His study also indicated modest support for the idea of multiple cross- hedges, because corn and soybean meal futures together produced the lowest mean square error. Witt et al. (1987) estimated the risk-minimizing hedge ratios when corn fu- tures hedge sorghum and barley. Those authors were primarily interested in estimation techniques and did not test the effectiveness of their estimates in a subsequent period. Therefore, this paper demonstrates only a cross-hedging potential. It will be shown that the potential risk reduction of cross-hedges varies over time. Further, that potential is not always realized because the optimal size of the cross-hedge also varies over time.
This paper extends the evidence on the effectiveness of complex hedges by addressing 4 questions: (1) Do cross-hedges contribute to risk reduction when there is a futures con- tract in the spot position?; (2) Do cross-hedges contribute to risk reduction when there is not a futures contract in the spot position?; (3) Do portfolio hedges differ from the sum of hedges of the components of the portfolio?; and (4) Do multiperiod hedges reduce risk more than naive or simple hedges?
The next section of the paper presents empirical answers to each of these questions. Hedging effectiveness is measured by the reduction in variance. Both in-sample and out- of-sample calculations are made. In-sample measures are hypothetical because the hedge and its effectiveness are measured with the same data. The in-sample results measure the maximum possible variance reduction with a constant hedge. They serve as benchmarks for evaluating the out-of-sample results. The out-of-sample results are realistic simula- tions of hedging effectiveness, because the hedge is estimated with one set of observa- tions its performance measured with a subsequent set.
Data and Methodology
The multiperiod tests dictate the choice of data for this study. The effectiveness of trad- ing in two contracts and rebalancing the futures positions as one of those contracts expires is e ~ a m i n e d . ~ The goal is to provide maximum opportunity for the complex hedges to exhibit superior performance. Therefore, the holding periods are divided equally between the time both contracts trade and the time only one trades. Only corn (C), oats (O), and wheat ( W ) fit this structure with matching holding periods. Each commodity has futures contracts calling for delivery in March, May, July, September, and December. This permits five two-month holding periods with maturity dates be- tween the delivery dates of two contracts. The five holding periods are February 15 to
30nly two contracts are examined to keep the rnultiperiod tests manageable. Tests of the other complex strategies use only the two-month holding period. They examine trading only in a contract that spans the hold- ing period.
16 / GRANT AND EAKER
April 15 to June 15, June 15 to August 15, August 15 to October 15, and November 15 to January 15.4
The Statistical Annual reports of the Chicago Board of Trade provide the closing spot and futures price^.^ The spot prices are for No. 2 yellow corn, No. 2 heavy oats, and No. 2 soft red wheat. These are Chicago spot prices. The results for other markets will differ because the basis is different in each market. The effect of local prices on hedging is an important question that deserves study. The use of Chicago prices, however, should not influence the comparison of the performance of naive, simple, and complex hedges.
OLS regressions are used to estimate the hedges. The distributions of prices are not stable over time. Therefore, unexpected price changes are used rather than raw prices. These are the differences between the closing prices and their conditional expectations at the beginning of the period. It is assumed that the futures price is an unbiased predictor of itself. Therefore the independent variables are the changes in the futures prices.
The adjustment for spot prices is less obvious. There are 3 alternatives. One is to sub- tract a spot price forecast developed from an econometric model. Another is to subtract the start-of-period spot price, the first-differences method. The third is to subtract the start-of-period futures price. The first alternative was rejected because it might introduce a large random prediction error. The other two alternatives both omit the effect of the drift in spot prices. Extensive tests, not reported here, indicate that neither dominates as a forecast and they produce similar hedging results. We report the results generated us- ing the futures prices.6
The period of analysis is 16 years from 1968 to 1983 inclusive. The data are divided into 2 equal length intervals for most of the tests of hedging effectiveness. The first 8 years (40 holding periods) constitute the estimation period, and the second 8 years the test period. This procedure is satisfactory except for testing cross-hedges, which appear to vary over time. Therefore, the risk-minimizing cross-hedges are recalculated every period.
The remaining parts of this section of the paper compare the risk-reducing effective- ness of the alternative hedges. The effectiveness, e, is analogous to a coefficient of de- termination.' It is 1.0 minus the variance of the hedged portfolio divided by the variance of the unhedged portfolio. Its size will depend on 3 factors: the accuracy of hedge esti- mates, the intertemporal stability of the hedges, and the potential risk-reduction from hedging.
Hedging with Multiple Futures Contracts and Cross-Hedging
Table I reports the effectiveness of simple hedging. Cases I , 5, and 9, establish bench- marks. The other cases report the effectiveness of adding cross-hedges to a simple
Vhese dates make each holding period as similar as possible. There are unavoidable seasonal differences in the contracts. In addition, the first three holding periods terminate in the month prior to the next contract deliv- ery date, but the last two contracts mature two months prior. In this analysis the hedger has the following op- tions in the multiperiod case. On February 15 the hedger may trade in both March and May delivery contracts. On March 15 the hedger must close the March delivery contract and may adjust his position in May delivery futures.
'Prices from the Wall Street Journal were also used. To illustrate the data consider the first holding period of each year. It requires the March and May futures prices on February 15 and March 15, and the May futures price and the spot price on April 15.
qhere is considerable work that supports the use of futures as forecasts. For example, see Just and Rausser (1981). The empirical results for the first-differences approach are available on request.
'When using the risk-minimizing hedge out-of-sample it is no longer necessarily true that total variance can be partitioned into explained and unexplained components. Thus e is not identical to R 2 .
COMPLEX HEDGES: HOW WELL DO THEY WORK? / I 7
Tabl
e 1
HED
GIN
G E
FFE
CT
IVE
NE
SS W
ITH
MU
LT
IPL
E F
UT
UR
ES
CO
NT
RA
CT
S
Hed
ges
1968
-75
Hed
ges
1976
-83
k
cc \
Cas
e Sp
ot
R~
/DW
@
c 0
W
e+
R
2/D
W@
C
0
W
n 4
1.75
( .
044)
1.
76
( .054
)
n
1 C
0.
93
0.97
* 0.
90
0.90
0.
98*
2 C
0.
93
0.97
* -0
.01
0.90
0.
90
1 .ou*
-0
.03
3 C
0.
94
1.11
" -0
.11"
0.
88
0.90
0.
99"
-0.0
1
4 C
0.
94
1.04
" 0.
29
-0.1
5 0.
86
0.90
1 .o
o*
-0.0
3 -0
.00
5 0
0.77
0.
89"
0.65
0.
65
0.88
" 1.
17
(.078
) 1.
54
(. 1
03)
6 0
0.78
-0
.04
0.97
* 0.
65
0.65
-0
.02
0.90
*
7 0
0.78
1.
04"
-0.0
5 0.
66
0.67
0.
95*
-0.0
7
8 0
0.78
-0
.02
1.06
" -0
.05
0.59
0.
67
0.03
0.
93"
-0.0
8
9 W
0.
97
1.01
* 0.
85
0.85
0.
97"
10
W
0.97
-0
.18*
1.
10*
0.84
0.
85
-0.0
6 0.
99"
11
W
0.97
-0
.55*
1.
14*
0.84
0.
86
-0.2
6 1.
04*
12
W
0.97
-0
.08
-0.4
5"
1.15
* 0.
84
0.86
0.
07
-0.3
1 1.
02*
73
P z 9
Z tr
P
P
1.75
(.0
86)
(.187
) I .
75
(.077
) (.
114)
2.09
( .
070)
( .0
44)
1.76
(.0
71)
(.051
)
1.87
(.0
82)
(.198
) ( .
050)
1.
75
(. 08
6)
(.118
) (.0
53)
G
1.09
(.0
70)
(. 15
2)
1.52
(.0
99)
(. 14
6)
1.03
(.
147)
( .0
42)
1.41
(.
123)
(.
060)
1 .oo
(.074
) (.1
77)
(.045
) 1.
42
(.109
) (.
149)
(.0
67)
1.10
(.
03 1)
0.
97
(.063
)
1.33
(.0
80)
(.050
) 0.
96
(. 1
15)
(.083
)
1.41
(.1
84)
(.053
) 0.
99
(. 15
2)
( .07
4)
1.43
(.0
92)
( .22
0)
(.056
) 1.
03
(.134
) (.
183)
(.0
82)
*The
ast
eris
k id
entif
ies
coef
ficie
nts
whi
ch a
re si
gnifi
cant
ly d
iffe
rent
kom
0 a
t a 5
% c
onfi
denc
e le
vel.
The
stan
dard
err
ors o
f th
e co
effi
cien
ts a
re in
the
pare
nthe
ses.
/"
The D
urbi
n-W
atso
n st
atis
tic a
ppea
rs u
nder
the RZ
valu
e.
"Thi
s is
the
effe
ctiv
enes
s of
the
1968
-75
estim
ates
of h
edge
rat
ios
whe
n im
plem
ente
d du
ring
the
perio
d 19
76-8
3.
hedge. This table reports estimates of the risk-minimizing hedges and the proportion of risk they eliminate (R') for 1968-75. The proportion of risk these hedge levels eliminate in the following period, 1976-83, are reported as e . For comparison the risk-minimizing hedges are reported for the later period and the proportion of risk (R') they eliminate. This allows one to compare e to the RZ in the test period.
In the period 1968-75 the risk-minimizing hedges are 0.97, 0.89, and 1.01 for corn, oats, and wheat. All 3 are significantly different from 0.0 but none is significantly from 1.0 at a 5% confidence level.8 These hedges eliminate 93%, 77%, and 97% of the vari- ance of the corn, oats, and wheat spot positions. The effectiveness of these hedges in the second period 1976-83 is 90%, 65%, and 84%. These values are equal to the in-sample R2's in the second period. Two factors contribute to that. The estimated hedges for 1976-83 are 0.98, 0.88, and 0.97. These differ little from those for 1968-75.9 In addi- tion, the degree of risk reduction is not very sensitive to the hedge in the range of 1.0. This latter point explains why the naive or one-to-one hedges are also as effective at re- ducing risk. ''
The other 9 cases examined in Table I include cross-hedges. These involve a single spot position hedged by a futures position in the same commodity plus 1 or 2 cross- hedges. For example, case 4 involves a spot position in corn hedged by a futures posi- tion in corn and at the same time, cross-hedged by futures positions in oats and wheat. The potential marginal contribution of cross-hedging to out-of-sample risk reduction is quite small for these data: 0.10 for corn, 0.35 for oats, and 0.16 for wheat. Nevertheless cross-hedges do not realize any of this potential. None of the cross-hedges reduces risk in-sample by more than 1%. This is true even though 5 of 12 cross-hedges in 1968-75 are significantly different from 0.0. The out-of-sample measures of effectiveness are even less favorable to cross-hedging. The inclusion of cross-hedges increases the e values slightly in 6 cases. It has no effect in 2 cases. It decreases risk slightly in only 1 case.
Table I1 illustrates the results of pure cross-hedging. These results are hypothetical. They are calculated as if there were no futures in the spot commodity." There are 9 cases of pure cross-hedging of a single spot position. For example, the first case involves a spot position in corn cross-hedged by a futures position in oats. Six of these involve 1 cross-hedge. Three of these involve 2 cross-hedges. Only 1 hedge is significantly differ- ent from 0.0 in each of the 9 cases. The potential effectiveness (average R 2 ) of the pure cross-hedges in 1968-75 is 59%. This is less than the 89% achieved when we include hedges in the same commodity. Still, it is a promising level of risk reduction.
The risk reduction achieved by applying these hedges in the next period, an average of lo%, is less encouraging. There are 2 reasons for the low level of risk reduction. First, the potential for cross-hedging decreased materially in the period 1976-83. The
%e spot position is assumed to be a single unit long position. Consequently, most of the reported hedges are short positions. To simplify presentation of the results, the hedges are all multiplied by - 1 .O. Therefore, positive coefficients indicate short positions and negative coefficients indicate long positions. The intercept terms are not reported in this paper. Most of them are small and insignificantly different from zero.
"hese changes were tested for statistical significance by estimating regressions using all of the data and in- cluding both a dummy variable intercept term and a dummy variable slope shifter for the second half of the data. The standard errors for these changes in the hedges are 0.071, 0.031, and 0.067 respectively. Full reports on these regressions are available from the authors.
'%is insensitivity is consistent with Merington's findings. "These are not the ideal tests. One should consider, for example, how effectively corn hedges grain sor-
ghum. The authors think that these three crops are sufficiently similar to provide illustrative evidence on the likely effectiveness of cross-hedging.
COMPLEX HEDGES: HOW WELL DO THEY WORK? / I 9
average R 2 for this period is 32% as compared to 59% in the earlier sample. Only a por- tion of this decline occurs when direct hedges are available, 81% versus 89%. The sec- ond reason is the instability of the cross-hedges. For 5 of the 9 cases the risk-minimizing cross-hedges estimated for 1968-75 differ considerably from those for 1975-83.
To examine the instability of the cross-hedges the data are pooled for both periods. The hedges are reestimated using dummy variables for both the intercept and slope in the second period. The coefficients attached to the slope dummy measure the changes in the hedges between periods. When added to the coefficients for the first period, they produce the hedges for the second period. The changes between periods and the hedges in the second period appear in the last 6 columns on the right-hand-side of Table 11. All of the hedges that are significantly different from 0.0 in the first period are also signifi- cant in the second. One additional coefficient is also significant. The average of the ab- solute values of the changes in the coefficients is 0.531. This is much larger than the corresponding value, 0.115, found in Table I. It also has a much greater effect on the out-of-sample effectiveness of the hedges.
The most extreme and adverse instability occurs when oats futures hedge corn and wheat spot positions. In all 4 cases (1, 3, 8, and 9) the changes in the estimated hedges between periods are significantly different from 0.0. These are the only significant changes measured. These are also the 4 cases in which the e values are smallest relative to the R 2 values in the second period. Indeed, hedging wheat with oats increases rather than decreases risk.
The results in Table I1 are based on estimating the hedges from observations 1 to 40 and applying the estimated hedges, without adjustment, for observations 41 to 80. This procedure ignores useful data. For example, at the start of the 42nd period, the estimates of the hedge should include the 41st observation. This can be remedied by estimating a unique hedge for each period. The most recent set of 40 observations (a rolling regres- sion) can be used. Alternatively, one can increase the data to include all observations from the first to the last before implementation. Both approaches produce a series of hedges.
The first technique is preferable if the intertemporal instability reflects fundamental changes in the relationship between the 2 commodities. Using the most recent data will lead to the removal of data reflecting old patterns of price movements. The second method is better if the instability is due to random external shocks. In that case the larger number of observations will diminish the impact of outliers. In Table 111 the effec- tiveness measures are included for both of these methods for all of the univariate cross- hedges. For comparison the corresponding values of e from Table I1 are included.
The series of hedges are not more effective in 4 cases (2-5). These are the cases for which the variation of the hedges is not statistically significant. Where that variation is significant, cases 1 and 6, the series of hedges perform better. This is particularly true when oats hedge wheat, the case with the largest intertemporal change in hedges. In- cluding the most recent data in estimating the hedges improved the risk reduction mea- sure from -0.62 to -0.27 or -0.28. Even with the improved estimation procedure however, cross-hedging wheats with oats still increases risk.
The instability of the oats hedges raise doubts about the value of cross-hedges, Therefore, that instability is examined in more detail. When these hedges are calculated using the rolling regression there is a general downward trend over time. A dispropor- tionate share of the change occurs between the 2nd and 3rd and the 28th and the 34th hedges. The cause of these changes in the estimates becomes clear when selected data are examined.
20 / GRANT AND EAKER
z 0 E U
0
Tabl
e I1
E
FFE
CT
IVE
NE
SS O
F H
YPO
TH
ET
ICA
L C
RO
SS-H
ED
GE
S
Hed
ges
Hed
ges
1976
-83
and
S P l9
68-7
5 C
hang
es (A
) fro
m 1
968-
75
0 t
R2
C
0
W
e+
R'
AC
C
A0
0
AW
W
C
0.68
C
0.52
C
0.68
0
0.53
0
0.49
0
0.56
W
0.58
W
0.63
W
0.65
0.34
* (.0
52)
0.22
" (.0
86)
1.24
* (.
173)
0.50
(.
308)
1.79
* ( .2
00)
1.65
" (.3
83)
2.81
* (.3
53)
1.88
* (.6
67)
0.17
0.45
* 0.
36
( .07
0)
0.05
0.
25
(.110
) 0.
30
0.20
* 0.
13
(. 03
4)
0.09
0.
27
(.054
) 0.
24
-0.6
2
-0.1
8
0.43
0.36
0.51
0.30
0.13
0.30
0.33
0.19
0.33
-0.7
8*
(.275
)
0.93
* (.4
31)
0.07
0.
40*
(.107
) (.0
99)
0.18
0.
40"
(. 15
5)
(. 13
0)
-0.4
2 0.
82*
(.268
) (.
193)
- 1.
89"
(.472
) 0.
26
0.76
* -1
.75*
(.4
13)
(.275
) ( .7
67)
1.01
" (.1
89)
-0.0
0 (.
124)
0.
72*
0.21
(.2
12)
(.151
)
-0.0
2
-0.0
9 (.1
05)
(. 10
5)
0.92
* (.
306)
0.
13
( .40
6)
0.45
* (.
097)
0.
25*
(. 10
3)
0.19
* ( .
080)
0.
01
(.094
)
*The
ast
eris
k id
entif
ies c
oeff
icie
nts t
hat a
re s
igni
fican
tly di
ffer
ent f
rom
0 a
t a 5
% co
nfid
ence
leve
l. Th
e st
anda
rd er
rors
of
the
coef
ficie
nts a
re in
the
pare
nthe
ses.
"T
his
is th
e ef
fect
iven
ess o
f th
e 19
68-7
5 es
timat
es of
the
hed
ges
whe
n im
plem
ente
d du
ring
the
perio
d 19
76-8
3.
Table 111
ALTERNATIVE HEDGE ESTIMATION PROCEDURES THE EFFECTIVENESS OF CROSS-HEDGES FOR
Case spot Futures RZ 1976-83 el e2 e3
1 C 0 0.43 0.17 0.23 0.28 2 C W 0.36 0.36 0.35 0.35 3 0 C 0..30 0.30 0.27 0.29 4 0 W 0.13 0.13 0.12 0.12 5 W C 0.33 0.24 0.28 0.24 6 W 0 0.19 -0.62 -0.27 -0.28
“The effectiveness for 1976-83 of a single hedge estimated from 4 observations, 1968-75. ‘?The effectiveness for 1976-83 of hedges, each of which is estimated from the last 40 observations prior to
‘?The effectiveness for 1976-83 of hedges, each of which is estimated using all data from 1968 to the period its implementation.
prior to its implementation.
Table IV reports selected price changes. Consider observations 2 and 42. When the 3rd hedge is estimated using the rolling regression technique, observation 42 is added and observation 2 is dropped. For observation 42 the ratio of the corn spot price to the oats futures price is 0.52. For wheat the corresponding ratio is -0.49. These values are much smaller than the estimated hedges. Furthermore, the price changes are large and thus relatively influential in an OLS regression. The decline in the regression estimates reflects these influences. The corn hedge declines from 1.79 to 1.57. The wheat hedge declines from 2.83 to 2.26.
The other major change in the estimates of the hedges begins with the 29th estimate and continues through the 34th. Dropping the extreme observations 28-33 causes this. The period of time covered by those observations is June 15, 1973 to October 15, 1974. Information shocks associated with the 1972 trade agreement between the U.S. and the U .S .S.R. created unusual volatility during this period. Furthermore, this agreement focused on corn and wheat appears to have disrupted the historic price relationships between oats and corn and wheat.
The price changes between June 15, 1973 and October 15, 1974 may be statistical and economic anomalies from the vantage point of 1976. If so, one can justify ignoring these data when estimating hedges for the 2nd period. With this reduced data base the rolling regression estimates produce no substantial changes for the cases involving corn and wheat futures as cross-hedges. The effectiveness measures which correspond to cases 2, 3, 4, and 5 in Table 111 are, respectively, 0.34, 0.25, 0.10, and 0.25. When oats hedge corn and wheat the effects of dropping the data are more important. The cross-hedge for corn begins at 1.45 as opposed to 1.79. After the 2nd period it drops to 1.02 as opposed to 1.57 and varies around 1 .O. This series of hedges eliminates 37% of the risk as op- posed to the 23% reported in Table 111. When oats futures hedge a spot position in wheat the hedge begins at 2.31 as opposed to 2.81. It drops to 1.04 after the 2nd observation and varies between 1.25 and 0.94. This series of hedges increases risk by 4% as op- posed to increasing risk by 27% as reported in Table 111.’’ Thus, even when a degree of judgement is incorporated, cross-hedging wheat with oats is ineffective.
”Observation 42 is obviously influential in these calculations. There is no apparent explanation for it and no reason to disregard it in any calculation. Even when we omit it the oats cross-hedge of wheat eliminates only 14% of the risk.
22 / GRANT AND EAKER
e m F: U
0
Tabl
e IV
SELE
CTE
D D
ATA
ILL
UST
RA
TIN
G T
HE
REL
ATI
ON
SHIP
BET
WEE
N U
NE
XPE
CT
ED
C
HA
NG
ES IN
OA
TS F
UTU
RES
PR
ICES
AN
D C
OR
N A
ND
WH
EAT
SPO
T P
RIC
ES
Oat
s C
orn
Rol
ling
Hed
ge
Whe
at
Rol
ling
Hed
ge
Obs
erva
tion
Futu
res'
spot
' R
atio
' C
S /O
F
Spot
' R
atio
' W
S /O
F
2 3 28
29
30
31
32
33
34
42
-1.7
5 -8
.25
45.7
5 -2
6.75
27
.50
-52.
75
14.2
5 32
.75
18.7
5 47
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Portfolio Hedging
The portfolio characteristics of corn, oats, and wheat are important to a discussion of managing portfolio risk with futures. Table V documents these characteristics. It reports the correlations between the pairs of commodities. For the 1968-75 they fall between 0.67 and 0.74. For 1976-83 they range 0.28 to 0.57. Table V also reports the variances of 7 portfolios for 1976-83. The portfolios are the 3 individual commodities and equally weighted portfoiios of them.I3 In addition the proportion of risk reduced through diver- sification is included. This is the complement of the ratio of the variance of a portfolio divided by the variance if all spot positions were perfectly correlated. Diversification reduces the risk of the two commodity portfolios by approximately 25%. It reduces the risk of the 3 commodity portfolio by 40%. This reduces the risk that futures might eliminate.
The risk reduction of naive, simple, and complex portfolio hedges are measured. A naive hedge is a one-to-one matching with futures of each commodity in the portfolio. A simple hedge of a portfolio is a weighted average of the simple hedges of the individual spot positions. The weights are the proportions that each spot position represents in the portfolio. Complex hedges are the coefficients of a multivariate regression. The depen- dent variable is the unexpected price change in the portfolio of spot positions. The inde- pendent variables are the changes in the futures prices.
Table VI reports the hedging results for the equally weighted portfolios. In every case except 1 the naive, simple, and complex hedges are similar. The measures of effective-
Table V CORRELATION MATRIX FOR COMMODITY PRICES (1968-75)
Corn (C) Oats (0) Wheat ( W )
C 1 .oo 0 W
0.74 0.68 1 .oo 0.67
1 .oo (1976-83)
~~ ~
C 0 W C 1 .oo 0.51 0.57 0 1 .oo 0.28 W 1 .oo
Portfolio Effects (1976-83)
Variance ( G ~ ) 716 355
1333 398 792 540 50 1
Diversification' 0.00 0.00 0.00 0.24 0.21 0.29 0.41
'The benchmark level of variance is that which would have existed had the prices of the commodities been perfectly correlated. Column 3 is (1 - column 2/benchmark).
"The data dictate these portfolios. There is no reason to believe that the relative performance of naive. simple, and complex hedges will differ fix more realistic combinations of commodities.
24 / GRANT AND EAKER
Table VI HEDGE RATIOS AND THEIR EFFECTIVENESS FOR
ALTERNATIVE APPROACHES TO PORTFOLIO HEDGING
Hedges (1968-75) Hedges (1976-83)
Case Spot RZ C 0 W e+ R2 C o w Naive co 0.50 0.50 0.87 0.50 0.50 Simple CO 0.48 0.45 0.88 0.49 0.44 Complex CO 0.91 0.46 0.48 0.88 0.88 0.49 0.44 Naive cw 0.50 0.50 0.89 0.50 0.50 Simple CW 0.48 0.50 0.90 0.49 0.47 Complex CW 0.97 0.47 0.49 0.90 0.90 0.48 0.48 Naive ow 0.50 0.50 0.79 0.50 0.50 Simple OW 0.45 0.50 0.80 0.44 0.47 Complex OW 0.96 0.24 0.55 0.80 0.81 0.37 0.47 Naive cow 0.33 0.33 0.33 0.87 0.33 0.33 0.33 Simple COW 0.32 0.20 0.33 0.88 0.33 0.29 0.32 Complex COW 0.96 0.32 0.30 0.32 0.88 0.88 0.37 0.21 0.30
“‘This is the effectiveness of the 1968-75 hedge ratios when implemented during the period 1976-83.
ness do not vary by more than 1%. Only the portfolio of oats and wheat yields substan- tially different hedges. Even in this case, however, the effectiveness measures of the three alternatives do not vary by more than 1%.
Multiperiod Hedging
Multiperiod hedging by introducing futures contracts that trade only for the first month of the two-month holding periods are examined. This requires hedge rebalancing. To il- lustrate this, consider the holding period from February 15 to April 15. On February 15 the hedger can trade in both the March and May futures contracts. The hedger must close the March futures on March 15. On that date the hedger can adjust the position in the May futures.
The effectiveness of multiperiod hedging for spot positions in corn, oats, and wheat is measured. The risk-reducing effects of three alternative strategies are reported. The first is the simple approach. It requires a single position in the distant contract with no inter- mediate trading. It is the same as that reported in Table I and serves as a benchmark. The second is a single futures rolling-the-hedge strategy. At the beginning of the period the hedger takes a position in the near maturity futures contract only. At the end of one month the hedger closes it. He then opens a new position in the distant futures contract for the second month. The hedger rolls the hedge forward from one contract to the next. The third alternative is the general approach developed by Baesel and Grant. Initially, the hedger takes positions in both the near and distant contract. After one month he closes the former and rebalances the latter.
Each of these three strategies is implemented with naive hedges. Prior data is also used to estimate the hedges. Where there is rebalancing, the hedges are estimated in a backward recursive manner. The spot position, the dependent variable, includes the sto- chastic returns to hedges in the second half of the holding period. Table VII reports the results of these tests.
COMPLEX HEDGES: HOW WELL DO THEY WORK? /25
Table VII HEDGE RATIOS AND THEIR EFFECTIVENESS FOR
ALTERNATIVE APPROACHES TO MULTIPERIOD HEDGING ~- Hedges (1968-75) Hedges (1976-83)
Spot R 2 b0“ bod bld e+ RZ b0“ bod bld
C C C C 0.93 C 0.96 C 0.96 0 0 0 0 0.77 0 0.81 0 0.81 W W W W 0.97 W 0.95 W 0.97
0.00 1 .oo 0.50 0.00 1.14 0.88 0.00 1.00 0.50 0.00 0.83 0.80 0.00 1 .oo 0.50 0.00 0.98 0.00
1 .oo 0.00 0.50 0.97 0.00 0.30 1 .oo 0.00 0.50 0.89 0.00 0.04 1 .oo 0.00 0.50 1.01 0.00 1.06
1 .00 1 .oo 1 .oo 0.97 0.74 0.74 1 .oo 1 .oo 1 .oo 0.89 0.96 0.96 1 .oo 1 .oo 1 .oo 1.01 0.96 0.96
0.90 0.90 0.91 0.90 0.85 0.85 0.64 0.70 0.68 0.65 0.70 0.70 0.85 0.86 0.86 0.85 0.85 0.85
0.90 0.91 0.91
0.65 0.70 0.70
0.85 0.85 0.85
0.00 1.00 1.00 1 .oo 0.00 1.00 0.50 0.50 1.00 0.00 0.98 0.98 0.85 0.00 1.00 0.78 0.08 1.00 0.00 1.00 1.00 1 .oo 0.00 1.00 0.50 0.50 1.00 0.00 0.88 0.88 0.98 0.00 0.93 1.03 -0.05 0.93 0.00 1.00 1.00 1 .oo 0.00 1.00 0.50 0.50 1.00 0.00 0.94 0.94 0.96 0.00 0.86 1.52 0.46 0.86
b, is the size of the position in near maturity futures contract take initially. b, is the size of the position in the distant maturity futures contract take initially. b,d is the size of the position in the distant maturity futures contract take when the near maturity contract
e’, this is the effectiveness of the 1968-75 hedge ratios when implemented in the period 1976-83. expires.
For corn the naive models perform as well as, or better than the estimated models. Among the naive models a single position in the distant futures contract is as effective as trading in two contracts. Among the estimated models, a single position in the distant contract is more effective out-of-sample. For oats there is little difference between the effectiveness of the naive model and the estimated models. Trading in two contracts re- duces risk slightly more than trading in only one contact. For wheat there are no impor- tant differences in effectiveness whether one trades in just one or both contracts. Likewise it does not matter whether the hedges are selected naively or estimated from prior data.
Summary and Conclusions
How well do complex hedges work? The results of this study suggest “No better than naive or simple hedges.” Naive matching of spot and futures contracts in the same com- modity reduces risk as effectively as simple regression estimates. Both of these methods perform as well or better than multivariate hedges that include futures in different com- modities. The results indicate that the potential risk reduction from pure cross-hedging is on average low, 45% for these data. Effectiveness varies between commodities, 51% to 13%, and over time, 63% to 19%. Cross-hedges achieve only a portion of this potential
26 / GRANT AND EAKER
because they exhibit intertemporal instability. Pure cross-hedging reduces risk by 36% at most and sometimes it increases risk.
This cross-hedging conclusion is at odds with Miller’s results, which indicated that cross-hedges can reduce risk out-of-sample. His examples, however, derive from institu- tional knowledge; while our cross-hedges are ad hoc. This explanation is consistent with recent evidence from a different market. Eaker and Grant (1987) found that German mark futures contracts can reduce the risk of currencies that are not traded in futures markets. Most importantly, they found that cross-hedging effectiveness is related to trade flows. Taken together these results suggest that the risk reducing potential of cross- hedging depends on underlying economic relationships. Potential cross-hedges should be validated on a case-by-case basis, with particular attention to the issue of economic and statistical stability.
The risk of portfolios of spot positions in corn, oats, and wheat is considerably re- duced by diversification. Naive hedges eliminate 90% of the remaining risk. Multivariate estimates of hedges do not perform any better. Finally, multiperiod hedging is not supe- rior to single-period hedging. Furthermore, naive multiperiod hedges are as effective as the complex estimates.
The conclusion that naive and simple hedges are as effective as complex hedges has important implications. Consider the example of an elevator operator who intends to sell part of his position in March and part in May. Should he hedge in both March and May futures or in just one contract at a time? How should he estimate the risk-minimizing hedge? The results reported above indicate there is little to choose, in terms of risk re- duction. One-to-one matching is as effective as estimated hedges. Therefore, one should make such decisions on the basis of other considerations such as: economies in transac- tions costs, contract marketability, and the ability to most closely match desired posi- tions with an integer number of contracts.
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COMPLEX HEDGES: HOW WELL DO THEY WORK? 127