- 19 -

less likely to be effective than own hedges. For example, Arak and McCurdy
(1979) claim that

When the cash asset is different from the security

specified in the futures contract, the transaction

... provides much less protection than an exact

hedge (p. 39).
It is true that if the spot position involves securities deliverable
against the futures contract, a nearly perfect hedge is assured. For

example, if the planned liquidation date of a 13-week T-bill position

is a contract delivery date (s = T), then, ignoring transactions costs,

p= eit

tts tts * the trader can form a riskless

By setting x = ie
portfolio. But only rarely is a spot position in T-bil!s deliverable
against the contract. T-bill futures contracts are available for only
four delivery dates per year. There is no a priori reason to believe
that hedges of spot positions in T-bills that are not deliverable are
any more effective than crosshedges . 20/

Finally, as noted in the previous empirical studies the T-bill
futures market, the optimal hedge position, we is often not equal in
magnitude and opposite in sign to the spot position, i.e., vy # “ye.
From equation (3) above it can be seen that -x, is the optimal

j,t
position if and only if p=1. But for 24 of 80 hedges considered in this
study the estimated value of 8 is significantly different than 1.

This is potentially important for two reasons. First, in much
of the descriptive literature on hedging in trade journals and exchange
publications a hedge position is defined as a futures positions that is
equal in magnitude and opposite in sign to the spot position. —!/ Second,

the money markets are wholesale markets where spot transactions are for

multiples of a million dollars. Since the T-bill, futures market calls for


FOOTNOTES

%*/ Economist, Division of International Finance, Board of Goverrors of
the Federal Reserve System. The views expressed in this paper are solely
those of the author; they should not be interpreted as those of the Board
or other members of its staff. The paper is from the author's doctoral
dissertation ("The Usefulness of Treasury Bill Futures For Forecast ing
and Hedging," University of Wisconsin-Madison, 1981). The author is
greatly indebted to his thesis advisor, John Geweke, for his advice and
support throughout the preparation of the study. Helpful advice was
also provided by Donald Hester and Dale Henderson.

1/ The sufficient second-order condition for a minimum

HT _ gttT

2 Var (fis t

) >o
is satisfied.

2/ It is assumed that a contains a finite number of elements.

Jeo

/ See Geweke and Feige (1979).

4/ See Roll (1970), pp. 82 - 83.

5/ See Anderson (1958), p. 77.

6/ See Anderson (1958), pp. 79 and 85.

7/ | For a derivation of this result see Parkinson (1981), Appendix A, or
Roll (1970), p. 19.

8/ Ederington used a weekly average of daily prices for 13-week T-bills
for the spot price P, _. Frankle argued that (1) the maturity of an
inventory of T-bills”~’~ changes over time so that if the maturity is 13
weeks at the liquidation date it must be (s + 13) weeks at the time is
initiated and (8) weekly averages of prices should not be used. In the
approach taken in this paper, the choice of a specification for P, t is
based on the information it provides concerning P, tts? not on J?
the basis of maturity. Frankle's contention thatJ**"Suse of weekly
averages should be avoided is correct.

9/ This contention, although part of financial economic folklore, is
usually not well-documented. One source where it is well-documented is
Chicago Mercantile Exchange (1980), p. 14.

10/ In fact, they should be considered crosshedges.

11/ For example, see Chicago Mercantile Exchange (1976).


-13-

interval for this lower bound is given py:2/
[eanh@(2 = 1.96/y¥F 24), tanh@(2 + 1.96/19 =%)] (13)
where, in this case, A
z= 3 log ( Ltr )
1-f

q
il}

the sample partial correlation coefficient.

IV. The Treasury Bill Futures Contract
A. Specification

The procedures outlined in the previous section are used to
examine the effectiveness of the 13-week U.S. Treasury bill (T-bill)
futures contract traded on the IMM for hedging spot positions in 13-week
T-bills, 3-month commercial paper (CP), 3-month Eurodollar deposits (E$), and
3-month negotiable certificates of deposit (CDs). For each of these
securities hedges with a number of different durations, s, and intervals
(I-s), between the planned liquidation date and the nearest contract
delive:ry date are investigated.

The durations chosen for investigation are 2, 4, 13, 26, and
39 weeks. The selection of 2-week and 4-week durations facilitates the
compar:ison of the results of this study with the previous studies of the
T-bill futures market. The choice of particular horizons for the longer
hedges is arbitrary. For each duration selected, hedges with planned
liquidation dates 3, 6, 9, and 12 weeks from the nearest contract delivery
date are considered. Since the T-bill futures market features quarterly
delivery dates up to two years forward, for hedges of duration of 39 weeks
or less the nearest date is always within 13 weeks of the planned date of

liquidation of the spot position.


- 21 -

A
ex, =) lie outside the confidence interval for the measure of the
>

*
effectiveness of the hedged position, e(y,). These results reflect that

the fact that, in general, estimates of var (£7 | 5) are much smaller

than estimates of Var, The loss of effectiveness that results

stts | 2e)+
from pursuing the suboptimal strategy is given by

ew) - ex, ro) = var(£t#T | 5)

ts
VarPs tts | 3)

2
(B-1)” .
Note that if var (ett? | 3.) < Var(P 3.) the loss of effectiveness is
tts | Pe q tts | %e
bounded by (p-1)". In such a case, as long as § is in the interval [-1.3,1.3],
the constraint imposed by the uniform $1 million contract size will result

is a loss of hedging effectiveness of less than 10 percent.


-4-

variances and covariances of the price changes:

») = x; t Var(P.

Var Ry 4, jatts 7 Pyye

q,t? Ye )

t+T.

t?

2 t+T
+ YE Var(Ee 4, -f

+ 2x, Cov((P, >
Bet « 45 )

tts ~ Poot

(ett. gt¥T)), (2)

tts t

The risk-minimizing value of Ye is determined by differentiating

(2) with respect to Ye setting the result equal to zero, and solving for

Yer The solution is given byl!

t+T t+T ))
* - -
Ye = “%y t Cov(Ps tig Pet ets fe
? t+T t+T
Var(£.4,- f, ) (3)

Thus, in the Johnson-Stein framework a trader's futures position is a

hedge of the spot position x, t if and only if it satisfies equation (3).

>
The effectiveness of the futures contract for hedging the spot
position x; t is defined as the proportional reduction, eye), in the risk
>
of the spot position, x

¢

ft? that is achieved by forming the portfolio
>

eID!
*.
= Var; 420) ~ Var poy):

Var(Ri 4,5 ,0)) (4)

tts

ey)

Substitution for the subjective variances in (4) using (2) and (3) reveals
that the effectiveness of the hedge can be measured by the square of the

subjective correlation between the spot and futures price changes:


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- 20 -

delivery of bills with a par value of $1 million, when ##1 the optimal
hedge position will, in general, be impossible to achieve. The position
suggested in the descriptive literature may be the most attractive chat
is feasible.

A priori it is possible that setting y= might result in

“x
jst

a much smaller reduction in risk than the optimal strategy or even an increase

in risk. If y= “ye the riskiness of the portfolio 5 ee 8404) is
given by
2 t+T
6
xf Vary tas | re) + Var(Ee ye | 3.)
t+T
~ 20VP 5 reerfeas | Fe))
The proportional reduction in the risk of the spot position is
t+T t+T
eCx, = 2CovePs tisefees | P)-VarGiys |)
>
Var@®, tis |e)
= t+ .
= Var (fp, |.) 28 - D (15)
var? tts |)

Using the same specifications of a. as in Section 3, (15) can be

consistently estimated by

2) ,ttT 4

=. = 8 (£4. $8.) (28 - 1)

eC *5 ot) re t - > (16)
Os tae! Fe

2 . ape .
where s’( ) denotes the usual unbiased estimate of the conditional variance.
Estimates of ex, 2) for the spot positions considered in this
>
study are reported in Table 11. These reveal that setting y= “*, t
>
in most cases does not result in a much riskier portfolio than does

the optimal strategy. In only 8 of 80 cases does


2-, 4-, 13-, 26-, and 39-Week Hedges

Table 3

Estimates of 8

3-Month Eurodollar Deposits

January 1976 - December 1979

2-Week Hedges
Weeks from

Delivery
Date (T-s)

4-Week Hedges
Weeks from

Delivery
Date (T-s)

13-Week Hedges
Weeks from

Delivery
Date (T-s)

26-Week Hedges

Weeks from
Delivery
Date (T-s)

39-Week Hedges

Weeks from
Delivery
Date (T-s)

Coefficient Standard

Sample Size (t) Estimate Error
3 16 1.278 .255
6 16 -677 -245
9 16 1.194 -110
12 15 -702 .255
3 16 1.254 -151
6 15 1.352 .214
9 15 1.587 -201
12 15 .718 .253
3 15 1.696 .223
6 15 1.568 .106
9 15 1.543 -134
12 14 1.346 -210
3 13 1.533 -162
6 14 1.553 . 069
9 14 1.394 -085
12 13 1.265 -110
3 13 1.300 .148
6 13 1.395 094
9 13 1.266 . 087
12 12 1.207 .117



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-17-

comparison of hedges of different spot positions. The standard errors
are large relative to the differences between the point estimates
involved in the comparison; the confidence intervals for the parameters
overlap considerably. In addition, it must be remembered that in most
cases the estimate of hedging effectiveness is actually an estimate of a
lower bound of the true measure. To the extent that the amount by which
the true value exceeds the lower bound varies across different spot

positions, the comparisons can be very misleading.

Table 10

Averages of Point Estimates of 8,
Hedging Effectiveness

Hedging Effectiveness

8 (or Lower Bound Thereof)
By Instrument
U.S. Treasury Bills +907 - 8867
Commercial Paper -926 .7317
Eurodollars 1.291 .8173
Certificates of Deposit 1.209 .8730
By Interval Between
Delivery Date, Planned
Liquidation Date
3 Weeks 1.141 -8061
6 Weeks 1.172 -8649
9 Weeks 1.110 +8954
12 Weeks -909 7244
By Length of Hedge
2 Weeks +940 -7193
4 Weeks 1.014 7589
13 Weeks 1.167 8574
26 Weeks 1.176 -9116

39 Weeks 1.119 - 8889


-7-

. ce t+,
of the future spot and futures prices, Pitts and fea?
t+T
*
yr nx,  OVPs tts fas | 5.)

. if t+ ul
ee 3!
var (£545 | 6.) (3°)

*-¥5 568
2 t+T
* £
e(y,) = Cov tts? tts | Fe)
var (P [a Dvar(Ety: | 34)
jatts let tts |Fe 5")
_ 2
p t+T

Pitts fets |

The princ:.pal contribution of this study is the development of techniques

for estimating the ratio, 8, that determines the hedge position (3”) and

the squared correlation coefficient that measures hedging effectiveness (5").
In estimating those two magnitudes two basic problems must be

confronted. First, the conditional distribution of the future spot and

futures prices depends, in general, on the realization of the random

2/

vector, 5 which lists the information in a . Second, as a practical
matter, all of the elements of a cannot be listed. In order to make

the estimation problems tractable it is assumed that

+ * . .
(Al) @, tts at, %,) is a multivariate normal, stationary time
»tts,

series, and

t+T

t ), is uncorrelated

(A2) the change in the futures price, cect -£
ith ?
with t

The normality assumption is frequently given as a justification

for mean-variance portfolio analysis. If a trader's subjective joint


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-6-

objective covariance of spot and futures price changes to the object:ive
variance of the futures price change is the slope coefficient in the
population linear regression of the spot change on the futures price
change. A consistent estimate of that ratio can be obtained from ordinary
least squares (OLS) estimation of the regression equation

t+T.

t

t+r
deaty (fy -f ) +e

Pytts “Pi,e jytts.

The coefficient of determination (Rr?) from the OLS estimation of this
equation provides a consistent estimate of the objective squared correlation

coefficient (5').

II. Estimation: An Alternative Approach

At any time t there exists a set, Bee of publicity availabe infor-
mation which a trader can use to forecast spot and futures price changes.
In representing a trader's beliefs concerning spot and futures price
changes by the objective unconditional distribution, the existing empirical
studies ignore the availability of such information. As a result, they
fail to distinguish reductions in the variability of the return on a
portfolio from reductions in the riskiness of a portfolio. The riskiness
of a portfolio should be measured by the variance of the portfolio
return, conditional on Se

That is the approach that is taken in this study. If a trader's
beliefs are identical to the objective distribution conditional on a and

ttT
the spot and futures prices at time t, PB and fF » are elements of as
>

t
then the hedge position (3) and the measure of hedging effectiveness (5)

can be stated in terms of conditional variances and covariances of the levels


2
* Cc P
ecyg) = SOY OO tte “Pit Ee
t
Var? s tts “Py Var(feys - £, )
(5)
=? t+T_, CHT)"

, 5 tts mPa Cag t

II. Estimation: The Existing Studies

The trader's subjective variances and covariances, which determine
the hedge position (3) and the effectiveness of the hedge (5), are, of course,
unobservable. In the existing empirical studies of hedging effectiveness

based on the Johnson-Stein measure it is implicitly assumed that a trader's

subjective joint probability density for the price changes (P, -P, .)
j,tts “j,t
and eet -£e't) is identical to the true, objective joint density function.

Thus, using a "V" to denote an objective variance and a "C" to denote an

objective covariance, the hedge position is

t4T tet
* - -
y, = -x, COPS tts Pt ets fe ») '
t i,t + 7 GB")
vcettt ttt)
tts t

and the measure of hedging effectiveness is the objective squared correlation

coefficient
*. 2 t+T t+T.
ey) = OCP, eye Py gd eas “Ae? 65
2 2 E 5
t+T t+T.
VO tts Pye) Mts “fe +

Given the additional assumption that the price changes are
realizations of a covariance stationary time series, estimates of the
hedge position and hedging effectiveness can be obtained from a time

series of past realizations of the price changes. The ratio of the


- 18 -
V. Discussion

The first published study of the hedging effectiveness of the
T-bill futures contract (Ederington, 1979) concluded that it was a
relatively poor instrument for hedging spot positions in T-bills for
durations of 2 or 4 weeks. A subsequent study (Frankle, 1980) disputed
this conclusion and suggested that it resulted from misspecification of
spot T-bill prices .2/ This study, based on an alternative methodology,
confirms the effectiveness of this contract for hedging spot positions
of T-bills for durations of 2 and 4 weeks. It also suggests that hedges
of spot positions in T-bills and other money market instruments for
durations of up to 39 weeks are quite effective.

Indeed, the effectiveness of hedging is found, in general, to
increase with the length of the hedge. That result is really not surprising.
In general, changes in prices of different financial instruments are not

8/

well-correlated over sampling intervals less than one quarter.— One
possible explanation for this phenomenon is that investor's portfolio
choices are subject to increasing marginal adjustment costs. Given such
costs, in the short run investors will find it unprofitable to make the
portfolio adjustments necessary to arbitrage away discrepancies between
various asset prices. However, over the long ran the adjustments will
occur and movements in prices of similar financial instruments will be
highly correlated.

On the other hand the effectiveness of hedges of spot transactions
in commercial paper, Eurodollar deposits, and certificates of deposit contra-
dicts widely-held views. In the existing futures market literature hedges

of transactions in commodities other than those for which the futures market

9
exists are termed crosshedges .—! It is often argued that crosshedges are


-10-

t+T

tots in the regression equation
t+T t+T ‘ *
= + 3)
Putts “FT SE FOE Fo Ge, (9)

has the same value as in equation (8). Equation (9) can be estimated by

ordinary least squares (OLS). Given the additional assumption that the

nN

disturbances, +5? are serially uncorrelated, the OLS estimator, *, has

*
e
j,t

an asymptotic normal distribution with mean 8.
*
Although the OLS estimator is consistent for any choice of a such
* t+T * a . .
that a Cc a and f, € Ley its efficiency, as measured by its asymptotic
variance, is not independent of the specification, More specifically,

under assumptions Al and A2 the asymptotic variance is the following function of
*

: *
2t a Var(P, a a)

! >
Asyvar(8) = j,tts | ‘tts’ “t
)

5

t+T t+T.
Var(fi 45 - f,

*
In general, the more elements that are included in Pots the smaller

A
is the asymptotic variance of g. However, if it is assumed that

-w,) is

(A3) there exists a variable, we such that C5 tts t

uncorrelated with a >

*
then, if wy is included in a the inclusion of additional variables will not

result in a more efficient estimator. This result reflects the fact that if

are - et) and 5 tts ~w) are uncorrelated with all elements of an
under the normality assumption the conditional distribution of Potts and
ptt given 4, is identical to the marginal distribution of (P, -w.)
tts t j,tts “et
and ant - a). An estimate of the parameter is best obtained by
estimating the equation
@5 ets Me) 7+ acer - “") + ey eae: (40)


Appendix

Table 1
Estimates of 8
13-Week U.S. Treasury Bills
2-, 4-, 13-, 26-, and 39-Week Hedges
January 1976 - December 1979

Coefficient Standard

Sample Size (t) Estimate Error
2-Week Hedges
Weeks from 3 16 1.107 .093
Delivery 6 16 1.047 -099
Date (T-s) 9 15 1.071 .079
12 15 -527 . 106
4-Week Hedges
Weeks from 3 16 -941 -107
Delivery 6 15 -900 -142
Date (T-s) 9 15 .780 042
12 15 .896 -145
13-Week Hedges
Weeks from 3 15 : +931 -103
Delivery 6 15 .857 -140
Date (T-s) 9 15 . 766 -075
12 14 571 - 106
26-Week Hedges
Weeks from 3 13 - 946 -040
Delivery 6 14 -979 042
Date (T-s) 9 14 971 .043
12 13 -918 .C52
39-Week Hedges :
Weeks from 3 13 -978 C41
Delivery 6 13 1.010 -045
Date (T-s) 9 13 991 C54
12 11 944 0160



International Finance Discussion Papers
Number 196

January 1982

ESTIMATING THE HEDGING EFFECTIVENESS OF TREASURY BILL FUTURES:
AN ALTERNATIVE APPROACH

by

Patrick M. Parkinson

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that he has had access to unpublished material)
should be cleared with the author or authors.


-ll-

Estimation of (10) also provides an estimate of hedging effectiveness.
*
Equation (5”) states that hedging effectiveness, e(y,)» is measured by

0? ter . Given the equality of the conditional distribution and

Pitts’ tts lee
a t+T.

the bivariate distribution of Cs tts -w, fa) and (fey, - ee )»;
o 0 t+T_ HT.
rag
Paotts,ftts | Pe 7 °@;, tts Me)? Ferg fe”
p t+T ettt can be estimated by the sample correlation
yet? Feis fe?

coefficient r (the maximum likelihood estimator). As asymptotic 95%

confidence interval for

ss ot 5/
p t+T gttt is given by =
Oi W> Ces fe
Leanh(z - 1.96/¥ 7 - 2), tanh (2 + 1.96/V,, - 2)] (11)
where

z= log Li)
2 1

q = the sample size.

Hedging 2ffectiveness is estimated by the square of the sample correlation

coefficient which is, of course, the coefficient of determination (R?) from

estimation of equation (10). A 95 percent confidence interval for hedging

effectiveness is obtained by squaring the end points of the interval (11).
In cases in which it is not assumed that there exists a variable

ve which satisfies assumption A3, the best choice of aad is not clear,

One possibility is to include only the futures price, eft, and the

*
spot price, az te in a An estimate of the parameter 8 is then obtained
>


a

-8-

density functions for spot and futures price ‘changes is a bivariate normal

density, his preferences concerning alternative portfolios can be expressed

in terms of preferences for combinations of subjective means and variances.
The change in the futures price will be uncorrelated with the

elements of a if the futures market equilibrium price can be state¢. in

terms of the expectation of the spot price on the contract delivery date,

» conditional on a For example, if the futures price is an urbiased

Pat

(rational) expectation of the future spot price

t+
)

£ = E(P

t \2

t+T t

then the futures price will follow a martigale, i.e.,

t:
t

t+T

+T_
= Efi.

f en

The martigale property implies that changes in the futures price are
uncorrelated with the elements of af, Also, if the futures price
is a downward-biased forecast of the future spot price

ter
Ee = ER 5p) + Lye Dy 7

and the downward bias decreases as the delivery date approaches, then

the futures price will follow a submartigale, i.e.,

t4T < t+T
f E(fsis |%)-

t

The submartingale property also implies that changes in the futures price

4/

are uncorrelated with the elements of a .

When the joint distribution is multivariate normal the variance of

the conditional distribution of @,; en) is not a function of the

»tts?

realization of o.- In fact, the ratio, 8, that determines the hedge position


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A 4 ; t+T
is a parameter in the ditional tat i
pi con onal expectation of Pitts given fois and Oe.

The conditional expectation is a linear function.

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ECs ts Frese) = % + Bey, 75" Oy (6)
where
- t+ t+
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= t+2,\-1 t+
6 = (Var @ | ae) CovPy nse Fees)? and
= t+T
a= ECs tts) - BE (Ee45) - SEW).

Thus, 8 is a parameter in the linear regression equation

_ tH
Pritts + Bf tO Fes ts. q@)

Given the assumption that eet eh, equation (7) can be

rewritten as

= t+ ttt t+T
P, Fa + BCE = EO) +658 +6

,
j tts Poe (8)

+85 tts °
an t+T '
where 9 = (£, 7, %5,)-

t4+E ot

The assumption that (Eras -£t'T) is uncorrelated with the elements of Pr»

and, in particular, Poe implies that omission of any of the elements
of Poe from equation (8) does not affect the value of the coefficient on

* +. *
cet'T -et#7)_ thus, letting *F be any subset of 8, such that f°" «31, and

tts “t

t+T

ink! cu tnd
letting ve = ica > oe) list its elements, the slope coefficient, 8, on


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3-Month Certificates of Deposit

2-, 4-, 13-, 26-, and 39-Week Hedges

Table 4

Estimates of 8

January 1976 - December 1979

2-Week Hedges
Weeks from
Delivery
Date (T-s)

4-Week Hedges
Weeks from
Delivery
Date (T-s)

13-Week Hedges
Weeks from
Delivery
Date (T-s)

26-Week Hedges
Weeks from
Delivery
Date (T-s)

39-Week Hedges
Weeks from
Delivery
Date (T-s)

Coefficient Standard
Sample Size (1) Estimate Error
3 16 1.241 . 188
6 16 - 783 .100
9 16 1.192 -101
12 15 -810 .225
3 16 1.162 .118
6 15 1.333 . 167
9 15 1.261 .228
12 15 - 868 .202
3 15 1.500 . 162
6 15 1.424 -090
9 15 1.254 -090
12 14 1.250 .155
3 15 1.374 .116
6 14 1.442 .071
9 14 1.225 .063
12 13 1.186 -092
3 13 1.182 -105
6 13 1.307 - 086
9 13 1.195 -057
12 12 1.181 -096



I. The Johnson-Stein Measure

The Johnson-Stein analysis uses the concepts of mean-variance
portfolio theory to formulate a precise definition of hedging and obtain
a measure of hedging effectiveness. Hedging is identified with risk-minimiza-
tion. Suppose that at time t a trader has an existing spot market commitment,
ioe? that he plans to liquidate at some future date tts, prior to or coin-
cident with the futures contract delivery date. The trader is said to be
hedging this spot commitment if his purchases or sales of futures contracts,
Y,> are chosen to minimize the price risk associated with holding the
portfolio of spot and futures positions (ye, from t to tts.

Price risk is defined in terms of the trader's beliefs concerning

the portfolio return. The return is given by

Res 0%) = Py tts Py eit
t+
t+T £
+ -
(fein 7 tO pe re)
* . . + t+I .
where P, ¢ is the spot price of commodity j at time t; fo is the price
i>
at time t of the futures contract maturing at time t+T; and Py tts and
>
t+T + , ‘
feis are the corresponding prices at time tts, 0“ s ST. At time t the

trader views Ress (x ) as a random variable. His beliefs concerning

4 te

the possible realizations of R.

+3 2¥4) depends on his beliefs concerning

x,

‘ j.t
t+T t+T

~ Poe and Ets f°).

t The latter set of

the price changes Cs tts

beliefs is represented by a subjective joint probability density function.

The riskiness of the portfolio is measured by the subjective variance of

Ras e ¥)> which can be expressed as a function of the subjective


-u4-

Two different specifications of the subset, a of the set of
publicly available information are employed. For hedges of 13-week T-bills
for durations of 2, 4, and 13 weeks it is assumed that the implicit forward
price for delivery of 13-week T-bills s weeks in the future, ifs, that
is embodied in the term structure of rates of return on Treasury securities
satisfies assumption A3 . The implicit rate of return, res, on a forward
contract for delivery of a 13-week T-bill s weeks in the future can be

approximated by a linear function of the spot rate on an s-week T=bill,

. 2/
Tes? and the spot rate on an (s + 13)-week T-bill, Te gt13!
rtts = (st13)r, st13 7 S"tis (14)
t 13
tts tts

The implicit forward price, i, , is then obtained from re
Parkinson (1981) tested and failed to reject the hypothesis
that the implicit forward price is an unbiased expectation of the spot
13-week T-bill price on the contract delivery date. As noted above, A3
is implied by the unbiased expectations hypothesis. Thus, for hedges
of T-bills for durations of 2, 4, and 13 weeks the hedge position is
determined by estimating equation (10) with we = ages, Hedging effective-
ness is measured by the Rr? from this equation. A 95 percent confidence
interval for the measure of hedging effectiveness is computed on the basis
of (11).
For liquidation dates more than 13 weeks in the future implicit
forward prices for 13-week T-bills often cannot be computed. As equation
14 indicates, the computation of an implicit forward rate for the delivery

of a 13-week T-bill s weeks in the future requires rates on an s-week T-bill

and an (s+13)-week T-bill. Only 13-week and 26-week T-bills are auctioned


for the measure of hedging effectiveness (or lower bound thereof) are
reported in Tables 1 through 9 in the appendix. Averages of the poiat
estimates for various categories of spot positions are presented in
Table 10 below.

An examination of the point estimates of 8 reported in the
appendix reveals that they exhibit a great deal of variation, ranging
from .453 to 1.696. As seen in Table 10 the estimates of 8 tend to 3e
larger for hedges of spot positions in E$ and CDs than for T-bills and CP.
There does not appear to be a strong systematic relationship between the
estimates of § and the duration of the hedge or the proximity of the
contract delivery date to the planned liquidation date.

The estimates of hedging effectiveness indicate that in gezeral
the T-bill futures contract is a highly effective instrument for reducing
risk; for 58 of the 80 spot positions considered hedging eliminates at
least 75 percent of the initial risk. Hedges of spot positions in C? are
on average somewhat less effective than those in the other instruments
considered. Both the length of the hedge and the proximity of the planned
liquidation date to a contract delivery date are important determinaats
of hedging effectiveness, Hedges of 2 and 4 weeks are generally less
effective than hedges of longer durations. As the interval between the
planned liquidation date and the contract delivery date increases from 9
to 12 weeks, hedging effectiveness drops off sharply.

Examination of the standard errors of the estimates of B aad
confidence intervals for the measure of hedging effectiveness suggests

that these results be viewed cautiously. This is particularly true for


-2-

Eurodollars for durations of 2 to 39 weeks. The results suggest that the
Treasury bill futures contract is a highly effective instrument for
hedging prices risks associated with spot transactions in all of these

money market instruments.


REFERENCES

Arak, M. and C. J. McCurdy, 1980, "Interest Rate Futures," Federal Reserve
Bank of New York Quarterly Review, Winter, 33-46.

Anderson, T. W., 1958, An Introduction to Multivariate Statistical Analysis
(New York: John Wiley and Sons, Inc.)

Chicago Mercantile Exchange, 1980, Justification for the Chicago Mercantile
Exchange Proposed Futures Contract in Eurodollars (Chicago, IL).

Chicago Mercantile Exchange, 1976, Treasury Bill Futures: Opportunities in
Interest Rates (Chicago, IL).

Dale, C., 1981, "The Hedging Effectiveness of Currency Futures Markets,"
Journal of Futures Markets 1, 77-88,

Ederington, L. H., 1979, "The Hedging Performance of the New Futures Markets,"
Journal of Finance 34, 157-170.

Frankle, C. T., 1980, "The Hedging Performance of the New Futures Markets:
Comment," Journal of Finance, 35, 1273-1279.

Geweke, J. and E, Feige, 1979, "Some Joint Tests of the Efficiency of Markets
for Forward Foreign Exchange," Review of Economics and Statistics 61,
334-341.

Heifner, R., 1973, "Hedging Potential in Grain Storage and Livestock Feeding,"
Agricultural Economics Report No. 238, Economic Research Service,
U.S. Department of Agriculture.

Johnson, L. L., 1960, "The Theory of Hedging and Speculation in Commodity
Futures," Review of Economic Studies 27, 139-151.

Parkinson, P., 1981, "The Usefulness of Treasury Bill Futures For Forecasting
and Hedging," unpublished Ph.D. dissertation, University of
Wisconsin-Madison.

Roll, R., 1970, The Behavior of Interest Rates (New York: Basic Books).

Stein, J. L., 1961, "The Simultaneous Determination of Spot and Futures
Prices," American Economic Review 51, 1012-1025.



Table 2
Estimates of 8
3-Month Commercial Paper
2-, 4-, 13-, 26-, and 39-Week Hedges
January 1976 - December 1979

Coefficient Standard

Sample Size (t) Estimate Error

2-Week Hedges
Weeks from 3 16 -975 -340
Delivery 6 16 -957 -121
Date (T-s) 9 16 1.023 -050
12 15 -453 .153

4-Week Hedges
Weeks from 3 15 -672 281
Delivery 6 15 1.239 +158
Date (T-s) 9 15 -620 -109
12 15 -647 -181

13-Week Hedges
Weeks from 3 15 -996 .180
Delivery 6 15 1.201 .100
Date. (T-s) 9 15 «918 .111
: 12 14 «847 .165

26-Week Hedges
Weeks from 3 13 . 888 . 166
Delivery 6 14 1.258 .146
Date (T-s) 9 14 1.008 .158
12 13 -882 .201

39-Week Hedges
Weeks from 3 13 -865 .166
Delivery 6 13 1.166 -168
Date(T-s) 9 13 -941 2244
12 12 -968 .270



ESTIMATING THE HEDGING EFFECTIVENESS OF TREASURY BILL FUTURES:
AN ALTERNATIVE APPROACH

by Patrick M. Parkinson*

Recent studies by Ederington (1979) and Frankle (1980) examined
the effectiveness of Treasury bill futures contracts as instruments for
hedging price risks associated with spot market transactions in Treasury
pills. These studies, as well as studies of the hedging effectiveness of
livestock and grain futures (Heifner, 1973) and foreign currency futures
(Dale, 1981), employed a common set of procedures to estimate a measure of
hedging effectiveness developed by Johnson (1960) and Stein (1961).

The present study argues that the procedures employed by the
existing studies to estimate the Johnson-Stein measure of hedging effective-
ness fail to distinguish reductions in price risk from reductions in price
variability. The principal contribution of this study is to develop
alternative procedures for estimating the Johnson-Stein measure that
capture this distinction.

The new procedures are used to examine the hedging effectiveness
of the 13-week Treasury bill futures contract that is traded on the
Internazional Monetary Market (IMM) of the Chicago Mercantile Exchange.

An additional contribution of the study is that it considers hedges of

a much larger set of spot transactions than that considered in the existing
studies of Treasury bill futures. Whereas the existing studies restricted
attention to hedges of spot transactions in Treasury bills for durations

of 2 and 4 weeks, this study examines hedges of spot transactions in

Treasury bills, commercial paper, negotiable certificates of deposit, and


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-12-

by estimating the equation

* *.
=o* + gett 4 g*ettT 4 g%p *
tts

ve aye 7 85 t45° (12)

Py tts
For this specification it is not clear how hedging effectiveness

can be consistently estimated. Nonetheless, a lower bound on hedgirg

effectiveness can be consistently estimated. Assumptions Al and A2 imply

that
tT ter, = ter
Cov(Ps tisefets | Py ref: Cove tts? ftts | ee)
and
tT t4T t+r
= 3
Var (fi 46 | Poets ) Var (fe 45 | 51).

However, the conditional variance of P. is affected by the omission of

j,tts

*
ts 6 f: ar:
elements _ frome.

ttt, >
Var(Py tas IPs pete 2 Var (Py tts | o.)-

Together, these relationships imply that the square of the partial

. tt+I t+T
correlation of Pitts and flag given Pat and ft places a lower

bound on the measure of hedging effectiveness:

2 2 = e( *)

o> gti) p — gttT <0 tor ¥,)-
j.tts?fets | jt t j,tts?ftts | %e

p t t+T can be consistently estimated by

tT
P,
Py tts’ tts | gett
the sample partial correlation coefficient. Thus a consistent estimate

of a lower bound on hedging effectiveness can be obtained by squaring

the sample partial correlation coefficient. An asymptotic 95% confidence


-15-

on a weekly basis. 52-week T-bills are auctioned at 4-week intervals. If
26-week T-bills and 52-week T-bills matured on the same day of the week
outstanding issues could be combined to form implicit forward contracts for
13-week T-bills for delivery on certain dates up to 26 weeks in the future.
However, until November 1979 52-week T-bills matured on a Tuesday, whereas
13-week and 26-week T-bills matured on a Thursday. For the other money
market instruments considered in this study, implicit forward rates are
difficult to compute for any forecast horizon. Spot rates are difficult
to obtain for maturities other than 1, 3, or 6 months.

Thus, for 26-week and 39-week hedges of spot T-bills and all
hedges of CP, ES, and CDs, the set x consists of Pit and et An
estimate of the optimal, risk-minimizing futures position is obtained by
estimating equation (12). The square of the sample partial correlation

coefficient of P, and he given P

a
and £¢*7 is used to estimate the
j,tts t

j,t
square of the population partial correlation coefficient, which is a lower
bound of the measure of hedging effectiveness. A confidence interval for
this lcwer bound is given by (13).

All estimates are based on a quarterly sampling interval. The
sample period is from January 1976 to December 1979. The futures and spot

T-bill prices are daily closing prices on Thursday of week t, obtained

from the International Monetary Market Yearbook and the Wall Street Journal.

The spot prices of the other money market instruments are weekly averages

of daily prices, obtained from the Federal Reserve Bulletin.

EB. Results
Point estimates and standard errors for the ratio, B, that

determines the hedge position and point estimates and confidence intervals