Estimating the Hedging Effectiveness of Treasury Bill Futures: An Alternative Approach

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acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

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 bills. These studies, as well as studies of the hedging effectiveness of livestock and grain futures (Heifner, 1973) and foreign currency futures (Dale, L981), 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 existinz studies to estimate the Johnson-Stein measure of hedging effectiveness 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 L3-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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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.

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-minimization. Suppose that at time t a trader has an existing spot market commitment,

x. that he plans to liquidate at some future date tts, prior to or coin-

j,t’ 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 (x; ) from t to tts.

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

the portfolio return. The return is given by

Reig ere) = Os tts Peet t+ tit _ £ | + (Ee45 t p> (1)

. . . . . ttr . : where P, + is the spot price of commodity j at time t; f is the price

Jo t at time t of the futures contract maturing at time t+T; and P, tts and > t+T : : : . tts are. the corresponding prices at time tts, O“ s ST. At time t the

trader views Re

(x. .,y,) as a random variable. His beliefs concerning +s * j,t°-t

the possible realizations of Riis; rey depends on his beliefs concerning ;

t+T t+T Pit and fets f. ). 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

Rigg be p> which can be expressed as a function of the subjective J;

~-4-

variances and covariances of the pees changes:

Var(R,, (x )) = F , Var@, )

- P tts j,t? Ye j,tts jot

2 t+T t+T + Ye Var(£e 45 - f. )

+ - 2x i ve Cov((P, tts Paw?

)). (2)

The risk-minimizing value of ye is determined by differentiating

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

Yee The solution is given byl! t+I .tt+T)) x Cov((P. t? (£ £ Ve *a,t WN itts iets “te Var(f, 4. - et ) (3)

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

hedge of the spot position Xt if and only if it satisfies equation (3). The effectiveness of the futures contract for hedging the spot

position x, , is defined as the proportional reduction, e(yr)s in the risk

that is achieved by forming the portfolio

of the spot position, x. |, j.t * e e(yy) = Var (Ro &, 429) - Var(R. 4, (x. ee): Var (Ris 49) (4)

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;

-~5~-

. 2 t+T t+T * - ety.) = SOY CO tts Pye)? Fes 7 fe)? t+T ttTr Var(Ps tts “Py Var (fi. - f. ) 5 (5) = P t+T t+T, °

, Ps tts Pa fg Fe )

IL. 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 (; tts “Py Pa > > and (ee -£*) 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

t+T t+T * CCP, -P. _),¢f Ye Xa ot j,tts it tts t (3")

otter ted V(t eas “fF )

and the measure of hedging effectiveness is the objective squared correlation

coefficient * 2 +, .ttT tt+T e(y,) =C°(@, 1. -P, 4) (fys -£, 6" tt+T t+T VPs tts “Pat V(Eeus “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

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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 (P -P, )Faty (fas f.

j,tts j,t ) +

c. j,tts.

+e : . 2 : : : The coefficient of determination (R') 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, Gi of publicity availab’e information 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 ge

That is the approach that is taken in this study. If a trader's

beliefs are identical to the objective distribution conditional on e. and

the spot and futures prices at time t, P.

t+T jt and f. » are elements of o> >

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

-9-

: . ous . t+T is a parameter in the conditional tat £ i Pp expectation o Potts given tts and O.- The conditional expectation is a linear function. = ' ECP, ag Fete) 72% + BEeyy +8" (6) where B= Cov(P, yoof re * | )/var(£,, 1 | )> _ ttt) -1 t+T 6 = (Var @, oP) CovPs tise Eee)? and tt+T = - - ‘ a ECs tts) BE (f 3 § EW). Thus, 8 is a parameter in the linear regression equation v4 Pitts a + Bey st 6 a + ®5 tts . (7) t+T : Given the assumption that f eo, equation (7) can be rewritten as t+T = Ty Pitts Tt B(fes = ™) + aT 6, 2 2¢ (8) “j.tts ’ ott where 9 = (£, %5,)> t+T

The assumption that cett ) is uncorrelated with the elements of P.

tte “ft

and, in particular, Poe implies that omission of any of the elements

of Pot from equation (8) does not affect the value of the coefficient on

t+T .ttT at 5 t+T o* (E444 ~f, ). Thus, letting - be any subset of S. such that f. € ne and a! *S letting ? = cert ore) list its elements, the slope coefficient, 8, on

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t+T . . . fits in the regression equation t+T t+T ‘ * = + ts) Putts TPE FOE FOG e. (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

ny

disturbances, are serially uncorrelated, the OLS estimator, ", has

* € j,tts’ an asymptotic normal distribution with mean 8. * Although the OLS estimator is consistent for any choice of eF such * t+T x, +e . . that $, co and f. € ne 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 *

Soe!

t+T t+T * A Var(P, L£ » f » D5.) AsyVar(B) = j,tts |! tts’ t 2t

t+T t

t+T Var(fi 4 - f )

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

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

(A3) there exists a variable, w -w_) is

such that (Ps tts t

t? uncorrelated with Sh

* then, if We is included in a. the inclusion of additional variables will not result in a more efficient estimator. This result reflects the fact that if

t+T

(fey

+ - f T) and (Py t wi) are uncorrelated with all elements of Be

+s

under the normality assumption the conditional distribution of P, tts and »tts

t+T

fits given a is identical to the marginal distribution of (P.

jytts “™t?

t+T

). An estimate of the parameter is best obtained b t y

t+T

f£ and ( tts f£ estimating the equation

_ t+T t+T ; we) e+ Blas f ) + Si.tts’

j,tts oe (10)

-7-

t+T of the future spot and future ic P. an : u P utures prices, j,tts d fits

t+T * ye = x, CovPs tts? tts

j,t er (3”) Var (fs | 3)

| 8.)

2 t+T * £ Y= Cov (Pa tts’ tts | 34)

t+T Var (Ps tts [8 Vary, 184)

e( (5”)

~ P t+T j,tts’? tts ls,

The princ:.pal contribution of this study is the development of techniques for estimating the ratio, 68, 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 vector, ,» which lists the information in 52 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

t+T . . . : * Potts, tts? 0.) is a multivariate normal, stationary time

(Al) ¢

series, and

t+T

t ), is uncorrelated

(A2) the change in the futures price, (fy -f with an

The normality assumption is frequently given as a justification

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

Pe ae ween

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 or if the futures market equilibrium price can be statec. in terms of the expectation of the spot price on the contract delivery date,

Pep? conditional on ee For example, if the futures price is an unbiased

(rational) expectation of the future spot price

etl - nep

t |,

tt+T t

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

tt+T_ t+T £ = E(t 45 | ¢

t

The martigale property implies that changes in the futures price are

uncorrelated with the elements of 32! Also, if the futures price

is a downward-biased forecast of the future spot price

t+r = >0 f= EC | 5p) + bps bp

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

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

t+T <

fe

t+T E(fi 4s |°.)-

The submartingale property also implies that changes in the futures price 4/

are uncorrelated with the elements of a7 .

When the joint distribution is multivariate normal the variance of

. : + . : the conditional distribution of (P, a) is not a function of the

j,tts’ fe

realization of e: In fact, the ratio, 8, that determines the hedge position

-1l1-

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

02 ter . Given the equality of the conditional distribution and

Ps otts’ tts 1%

«aad + ceeshutd t+T tt the bivariate distribution of (P, tts w,) and (fey, £. ), 0 t+T Py t+I .t+T t . Potts, tts |o, = acre tts wi.) (Ente ) ie ww Ds cet tT pitt, can be estimated by the sample correlation j,tts tts t

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

confidence interval for

te os 5/ Pip. “w Ds Come ett) is given by j,t Ctanh(z - 1.96/¥T - 2), tanh (z + 1.96/V, - 2)] (11) where

2 l-r 7 = the sample size.

zal log (Za)

Hedging 2ffectiveness is estimated by the square of the sample correlation coefficient which is, of course, the coefficient of determination (R2) 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 We which satisfies assumption A3, the best choice of a is not clear.

spaqe . . . t+T

One possibility is to include only the futures price, f. » and the

* spot price, P, be in oe An estimate of the parameter B is then obtained >

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by estimating the equation

* * * =o* + Bf. + 6 ettT 4 g*p +

et Pye * 85 tts: (12)

P. j,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 t+T t+T. t+T Cov(P. tis fess PPeeete 0 = Cov(Ps tis fits | &) and t+T t+ t+T = 3 Var (fi Peete ) Var(£. [é).

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

j,tts elements 6. from 5%; I t t t+T

= Var(Ps tis [Peete ) Var(Ps tts re

Together, these relationships imply that the square of the partial

+ correlation of P, and ie given Pi and fr T places a lower >

j,tts t

bound on the measure of hedging effectiveness:

2 2 = e( *) Op gttT P gttt <0 t+T Ye : j.tts’ “tts | j,t?-t j,tts’ “tts | as p t+I t+T can be consistently estimated by

Pa otts’ tts | jot’? t the sample partial correlation coefficient. Thus a consistent estimate of a lower bound on hedging effectiveness can be obtained by squarinz

the sample partial correlation coefficient. An asymptotic 95% confidence

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interval for this lower bound is given by:2/ Ctanh2(2 - 1.96/¥F2 4), tanh*(z + 1.96/¥F= 7%) I (13)

where, in this case, A

2 = ; log ( lt+er ) A l-r A r = 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-bil1) 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 (T-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 aire 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.

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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, ints, that is embodied in the term structure of rates of return on Treasury secu:cities satisfies assumption A3 . The implicit rate of return, re’, 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,

. JL Tes? and the spot rate on an (s + 13)-week T-bill, Te st13° wits = (stl3)r, 2413 7 SE g (14) t 13 tts tts

The implicit forward price, i, » is then obtained from TO

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 = i. Hedging effectiveness is measured by the Re 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 (st+13)-week T-bill. Only 13-week and 26-week T-bills are auctioned

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on a weekly basis. 52-week T-bills are auctioned at 4-week intervals. If 26-weelk: 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

i hedges of CP, E$, and CDs, the set oe consists of P,

jt and et, An 3

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 f T is used to estimate the 3

t+T and f iven P., j,tts tts § >

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 Icwer 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.

Point estimates and standard errors for the ratio, 8, that

determines the hedge position and point estimates and confidence intervals

~ 14 -

for the measure of hedging effectiveness (or lower bound thereof) ar2 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, ranginz from .453 to 1.696. As seen in Table 10 the estimates of 8 tend to »e 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 general 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 CP 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

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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 6, Hedging Effectiveness

Hedging Effectiveness

8 (or Lower Bound Thereof) By Insi:rument 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

- 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 well-correlated over sampling intervals less than one quarter .2/ 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 trarsactions in commercial paper, Eurodollar deposits, and certificates of deposit contradicts 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

- 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,

t+T *

P = f . By setting y.r7

ts tts the trader can form a riskless

Xie 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, Yee is often not equal in magnitude and opposite in sign to the spot position, i.e., y. # “Kak, From equation (3) above it can be seen that -x. t is the optimal

> position if and only if g=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

- 20 -

delivery of bills with a par value of $1 million, when f#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, = Fat might result in

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

in risk. If Ye = Fat the riskiness of the portfolio (ey ee “Xa og) is given by t+T 5 lvar tts | 2.) + var (fe 46 |.) - 2Cov(P, | tts? fi. mi yl. The proportional reduction in the risk of the spot position is ex, ) = 2Cov(P t+s’ aie a1 Se )- ~var (fry a ot Var (Ps tis | 3,) = Var(f 6.) (28 - 1) Var (fry | #28 - D (15) Var(P, tts | %,)

Using the same specifications of 2. as in Section 3, (15) can be consistently estimated by 2 gt S(-x, J = 46! i) Bic - 1)

j,t rr (16)

ci S 20, ou t)

2 : . ths . where s’( ) denotes the usual unbiased estimate of the conditional variance. Estimates of e(-x, D> for the spot positions considered in this >

study are reported in Table 11. These reveal that setting y= “%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

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- 21 -

A e(-x, Pe) 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 (£4. | §) are much smaller

than estimates of Var(P, The loss of effectiveness that results

,tts | 5,)-

from pursuing the suboptimal strategy is given by

ely.) - e(-x, ,) = Var(£ttT | 3.)

+s 2 (8-1). Var? s tts | & ) t Note that if var(£t*" | $.) = Var(P 6.) the loss of effectiveness is tts t j,tts | t

bounded by (8-1)°. In such a case, as long as 8 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.

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 Erior 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 C43 12 13 .918 052 39-Week Hedges , Weeks from 3 13 .978 C41 Delivery 6 13 1.010 045 Date (T-s) 9 13 991 C54 12

ll 944 . 60

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 6 214 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) 3 13 941 244 1

12 968 .270

Table 3

Estimates of 8

3-Month Eurodollar Deposits

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

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)

Sample Size (t)

ho WO OVW bo WO OV Nw OVW NV WO OW

mw dW

Coefficient Estimate

PRR Pee e al alo

PRR

278° -677 .194 .702

.254 .352 . 587 .718

.696 . 368 .543 .346

. 333 .953 394 .265

.300 395 . 266 .207

Standard Error

.255 ~245 .110 ~255

151 .214 .201 .253

.223 . 106 .134 .210

.162 .069 .085 .110

148 .094 0&7 .117

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)

123-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)

Sample Size (1)

how NL NOW OW No WwW OV ho W/O OV Oo

Nw anwW

Coefficient Estimate

Pee ed ee lela

Peer

241 . 783 .192 .810

.162 . 333 .261 . 868

. 300 424 254 .250

.374 442 .225 . 186

.182 .307 .195 .181

Standard

_Error_

.188 . 100 .101 .225

.118 . 167 .228 .202

. 162 .090 .090 .155

.116 .O71 .063 .092

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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 Forecasting 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

t+T

t+T 2 Var(f. - fe

) >o

is satisfied.

2/ It is assumed that $, contains a finite number of elements. 3/ See Geweke and Feige (1979).

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

See Anderson (1958), p. 77.

In tun =

/ 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).

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.