3.11 A/8/J

h o . A R O 5 ~

FUTURES-FORWARD PRICE DIFFERENCES AND

EFFICIENCY IN THE TREASURY BILL

FUTURES MARKET

DISSERTATION

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Alan Wong, M.B.A.

Denton, Texas

May, 1986

\ V )

Wong, Alan, Futures-Forward Prioe Differences and

Efficiency in the Treasury Bill Futures Market. Doctor of

Philosophy (Finance), May, 1986, 8 tables, bibliography,

58 titles, 97 pp.

This study addressed two issues. First, it examined

the ability of two models, developed by Cox, Ingersoll and

Ross (CIR), to explain the differences between futures and

implicit forward prices in the thirteen-week T-bill market.

The models imply that if future interest rates are stochastic,

futures and forward prices differ; the structural difference

is due to the daily settlement process required in futures

trading. Second, the study determined the efficiency of the

thirteen-week T-bill futures market using volatility and

regression tests. Volatility tests use variance bounds to

examine whether futures prices are excessively volatile for

the market to be efficient. Regression tests investigate

whether futures prices are unbiased predictors of future

spot prices.

The study was limited to analysis of the first three

futures contracts, using weekly price data as reported m

the Wall Street Journal from March, 1976 to December, 1984.

Testing of the first CIR model involved determination of

whether changes in futures-forward price differences are

related to changes in local covariances between T-bill

p i i \ ) f

futures and bond prices. The same procedure applied in

testing the second model with respect to changes in futures-

forward price differences, local covariances between T-bill

spot and bond prices, and local variances of bond prices.

Volatility tests of market efficiency involved comparison

of mean variances on both sides of two inequality equations.

Regression tests involved determination of whether slope

coefficients are significantly different from zero.

The results provide no evidence that the CIR models

can adequately explain T-bill futures-forward price differ-

ences. Measurement errors could have contributed to the

inconsistent results. It is also possible that the models

are valid only with non—financial underlying assets. The

volatility and regression test results indicate that the

T-bill futures market is efficient. However, the regression

test results show that autocorrelation exists in the nine-

month futures contract data, probably due to a missing

variable related to information costs or default risk.

TABLE OF CONTENTS

Page

LIST OF TABLES v

Chapter

I. INTRODUCTION . . 1

Background and Significance of the Problem

Usefulness of the Study Theoretical Models Outline

II. THE TREASURY BILL FUTURES MARKET 10

Emergence of the Treasury Bill Futures Market

Characteristics of the Treasury Bill Futures Market

Advantages of Forward Trading in Futures Markets

III. REVIEW OF LITERATURE 20

Futures-Forward Price Difference and Transaction Costs

Futures-Forward Price Difference, Daily Settlement and Taxes

Variance Tests of Market Efficiency Summary

IV. METHODOLOGY 46

Hypotheses Procedures Data Summary

V. RESULTS 67

Tests of the Cox-Ingersoll-Ross Propositions

i n

Chapter Page

Intra-Sample Variations Among Futures-Forward Price Differences

Variance Tests of Market Efficiency Regression Tests of Market Efficiency Summary

VI. SUMMARY AND CONCLUSIONS 87

Implications for Further Research Conclusion

BIBLIOGRAPHY 92

IV

LIST OF TABLES

Table Page

I. Data Behavior During Sample Period 59

II. Autocorrelations of T-Bill Spot Price 62

III. Autocorrelations of Spot and Futures Percentage Price Changes 64

IV. Test Results of CIR Models 68

V. Intra-Sample Variations of Futures-Forward

Price Differences 72

VI. Variance Test Results of Equation [20] . . . . 75

VII. Variance Test Results of Equation [23] . . . . 77

VIII. Regression Test Results 80

CHAPTER I

INTRODUCTION

Background and Significance of the Problem

Commodity futures and forward markets have long served

a very important economic function—hedging away risk.

Hedging buyers and sellers can pass the risks they intend

to avoid on to the professional speculator. The fact that

commodity exchanges have been growing for more than a

century indicates the importance of the risk-averting func-

tion. Speculators relieve commodity producers and businesses

of their buying and selling risks so that they can concen-

trate on efficient production. In spite of the attention

futures and foward markets have received, many commodity

traders and economists do not recognize that these two

types of markets are fundamentally different. Although the

two markets look similar, they are not identical. This

study explains the underlying contractual difference between

forward and futures markets; profits or losses from a

futures contract are settled daily between participating

parties while those from a forward contract are settled at

maturity. This paper empirically investigates the differ

ence between T-bill futures and implied forward contracts

using two models—developed by Cox, Ingersoll and Ross (1),

hereafter CIR. CIR propose that stochastic interest rates

contribute to futures-forward price differences due to the

daily settlement requirement in futures trading.

This study also tests the efficiency of the thirteen-

week T-bill futures market. Since the International Monetary

Market of the Chicago Mercantile Exchange first introduced

the T-bill futures contract on January 6, 1976, a number of

researchers have tested the futures market's efficiency.

These efficiency tests dealt mostly with possible arbitrage

opportunities by comparing future yields with the corre-

sponding adjusted forward yields implied in the term struc-

ture of the spot T-bill rates. The implicit forward yields

were adjusted for market imperfections—mainly transaction

costs and taxes. Any significant and sustained differential

between the futures and the forward yields suggests profit-

able arbitrage opportunities and, therefore, market ineffi-

ciency. The assumption of different arbitrage conditions

and the choice of different sample periods are some of the

shortcomings of the past studies. More important, the fail-

ure to subject data of the same period to alternative

techniques impairs the validity of the conclusions. Each

technique has its limitations, and some may not effectively

detect certain complicated futures and forward price move-

ments—other tests might. Since one cannot compare the

power of different techniques, alternative techniques should

be used to determine whether the results are technique-

sensitive. This study examines the efficiency of the

thirteen-week T-bill futures market using regression and

variance tests. Regression tests assume that futures prices

are unbiased predictors of future spot prices. The variance

tests involve formulation of variance bounds to determine

whether T-bill futures prices are too volatile for the market

to be efficient.

This study, thus, addresses the following questions.

1. Are CIR able to explain the differences between

T-bill futures and implicit forward prices?

2. Is the thirteen-week T-bill futures market

efficient, based on the use of regression and

variance tests?

The study finds that the CIR models are unable to explain

the price differences and that the market is efficient.

Usefulness of the Study

Many practitioners apparently believe that futures and

forward contracts are synonymous and regard any discrepancy

between futures and forward prices as evidence of market

inefficiency—the presence of profitable arbitrage oppor-

tunities. If futures and forward markets differ fundamen-

tally, the supposed market inefficiency might lead

arbitrageurs to take up trading positions that would not be

beneficial. Moreover, a trader who wishes to hedge by

taking a long position in the futures market and a short

position in the forward market does not necessarily take a

riskless position. If interest rates are stochastic, such

a forward-futures hedge is risky. This study seeks to

empirically clarify an obscure, but basic, difference

between futures and implied forward contracts in the thirteen-

week T-bill futures market. Some have examined the theoreti-

cal implications of the difference, but empirical tests of

the theory have not yet been published.

The T-bill futures contract is relatively simple to

price due to the uniformity and liquidity of the contract's

underlying asset. Traders make bids and offers on bill

futures contracts by open outcry on the floor of the futures

exchange. Given the ease of pricing and contract and a

highly competitive system, one would expect the T-bill

futures market to be highly efficient. However, past studies

produce conflicting results. An inefficient bill futures

market might impair governmental borrowing capacity. More-

over, T-bill futures data provide valuable help to many. It

provides economic theorists with testable data on the forma-

tion and predictive accuracy of expectations. One weakness

of earlier empirical studies of term structure is the absence

of such observable data. Observable interest rate forecasts

also provide policymakers and forecasters a counter-check on

their econometric forecasts. If the pricing of the T-bill

futures contracts is inefficient—prices do not fully

incorporate available information—they would provide biased

forecasts of subsequently observed spot prices. The useful-

ness of bill futures price data to economic theorists,

policymakers and other interested parties depend on pricing

efficiency.

The efficiency of the T-bill futures market is, there-

fore, important. Since past authors have mostly investi-

gated the issue with arbitrage arguments, this study examines

it with alternative approaches—variance and regression

tests.

Theoretical Models

The regression and variance tests are based on the

efficient market model. This study uses the Cox-Ingersoll-

Ross models to derive tests that can determine whether the

futures-forward price differences are attributable to sto-

chastic interest rates. The CIR model is only briefly dis-

cussed here. It is necessary to leave a more detailed

exposition of CIR's work for the literature review chapter.

Market Efficiency Model

The efficient market model is a fair game model. A

fair game implies that, on average, the actual price of an

asset in the future is equal to its expected price, based on

information at the time. Symbolically,

E{xt+1 " E(xt+l/3:t)} = °' [1]

where E(.) = expectations operator,

xt +l = price at time t+1, and

It = information set at time t.

In other words, expectations are not biased and they result

in zero expected profits.

Efficiency in futures markets is often stated in terms

of a fair game. The relationship between any futures price

and the expectations of its realization is

Ft,n = V W ' n>t 121

where E (.) = expectations operator,

1^ = information set at time t,

Sn = spot price at time n, and

Ft n

= futures price at time t, quoted for delivery

at time n.

An equivalent expression of Eq. [2] is

S = F +U. [31 n t,n t,n 1 J

where U. = S - E (S ). t, n n t n

According to Fama (2), Eq. [2] is an efficient market

model if the forecast error is a fair game. Mathematically

speaking,

Vut,„' = °' W

The forecast error is orthogonal to any element of the

information set; therefore, it is serially uncorrelated.

Eq. [2] is the model for the tests in this study.

Cox-Ingersoil-Ross Models

Proposition 6 in CIR theoretically establishes the

futures-forward price differential in terms of the relation

between futures and bond prices. Assuming a continuous-

time, continuous state economy, the proposition states that

the futures-forward price difference is equal to s

F(t,s) - f (t, s) = PVfc {-£F(U) [covF(u),B(u)]du/B(t)}, [5]

where cov{F(u),B(u)} = local covariance of the percentage

changes in futures and bond prices,

f(t,s) = forward price at time t for a con-

tract that matures at time s,

PVt s = current value at time t of a payoff

amount obtained at time s,

F(t,s) = futures price at time t for a contract

that matures at time s, and

B(t) = price of a default-free discount bond

at time t paying $1 at time s.

Proposition 9 in CIR expresses the difference between

futures and forward prices in terms of spot and bond prices.

Assuming the underlying asset does not make any payout

between t and s in a continuous-time, continuous-state

economy, the futures-forward price difference can be

expressed as s s

F (t, s) - f (t, s) = p vt f S C ~ { e x p [ £ l o g R ( u ) d u ] £ [ S (u)/B(u) ]

{cov[S(u),B(u)] - var[B(u)]}d (u)}}, [6]

where R(u) = 1 + spot interest rate for all u, and

S(u) = spot price of the underlying asset.

Eqs. [5] and [6] suggest that futures and forward prices

differ structurally if future interest rates are stochastic.

They contain implications that are testable in the T-bill

futures market.

Outline

The organization of the remainder of this study is as

follows. Chapter II provides an overview of the T-bill

futures market. Chapter III reviews literature on T-bill

futures market efficiency based on arbitrage arguments and

explains the structural difference between futures and for-

ward prices. Because variance tests of market efficiency

are relatively new, the chapter also provides a review of a

few studies that use the technique. Chapter IV develops

statements of hypotheses. Subsequent sections provide the

methodology to test the hypotheses. Chapter V presents the

results. Chapter VI provides a summary and discusses the

importance of the findings.

CHAPTER BIBLIOGRAPHY

Cox, John, Jonathan Ingersoll and Stephen Ross, "The Relation Between Forward Prices and Future Prices," Journal of Financial Economics, IX (December, 1981) , 321-346.

Fama, Eugene, "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, XXV (May, 1970), 383-415.

CHAPTER II

THE TREASURY BILL FUTURES MARKET

Emergence of the Treasury Bill Futures Market

During the early 1970s, interest rates, which had been

fairly stable for years, became highly erratic. The sharp

interest rate swings created a need for price protection.

Some mechanism was necessary to hedge against interest rate

fluctuations. The success of commodity futures prompted

development of financial instrument futures. Such futures

were first applied to the Government National Mortgage

Association securities (GMNAs). A few months later

(January 6, 1976), the International Monetary Market (IMM)

started trading United States T-bill futures. The three-

month T-bill futures contract serves the function of trans-

ferring short-term interest rate risk. It won immediate

acceptance and its growth has been explosive. One reason

for this acceptance is that the yield movements of its

underlying instrument closely correlate with those of other

short-term financial instruments. Changes in the bill

yields reflect interest rate movements free of credit

constraints that affect some other money market instruments.

Consequently, T-bill futures provide hedges against yield

10

11

volatility in T-bills and all other short—term instruments

(e.g., Commercial Paper, CDs, and Bankers Acceptances).

Subsequent introduction of futures trading in other money

market instruments have met very limited success; the T-bill

futures contract seems to have taken care of much of the need

to transfer short-term interest rate risk.

Characteristics of the Treasury Bill Futures Market

This section discusses the specifics of the T-bill

futures contract and the safeguards provided by the organ-

ized exchange on which the futures contract is traded—the

International Monetary Market.

Contract Specifics

A T-bill futures contract is a standardized forward

contract, traded on an organized exchange, to sell or pur-

chase a fixed quantity of T-bills at a set price in the

future. Each T-bill futures contract calls for delivery of

T-bills maturing thirteen weeks from the delivery day with a

par value of one million dollars at bill maturity. The

delivery months are March, June, September and December.

The life of a futures contract can extend up to two years

into the future. At any one time, eight T-bill futures

contracts will be outstanding—eight contract delivery months

at quarterly intervals for eight quarters into the future.

At the delivery date, the seller can deliver any United

12

States T-bill with thirteen weeks to maturity. However,

only a small percentage of contracts traded end up in

delivery. Any trader with a long position who wishes to

cancel it, can do so by taking a short position of the same

amount. Most traders terminate their contracts before

expiration by taking an offsetting position.

Pricing of T-bill futures contracts is based on the IMM

quotation. Because the underlying instrument is quoted on a

yield basis, bids are higher than offers due to the inverse

relationship between yield and price. This quoting method

is contrary to normal market practice where bids are normally

lower than offers. To reduce confusion, the IMM established

a different method of quoting T-bill futures prices. The

IMM quotation is the difference between the T-bill yield

and 100. The IMM method conforms to the traditional quoting

method where bid prices are lower than offer prices. The

quotation, based on an annualized rate, is not the real

price of the thirteen-week spot T-bill. It serves only to

facilitate understanding by traders.

The exchange requires an initial margin deposit of

$1,500 per T-bill futures contract. The limit for daily

price changes is normally sixty basis points above or below

the final settlement price on the previous day. The IMM

also requires a minimum daily price change for T-bill

futures contract of one basis point which changes the value

13

of one futures contract by twenty-five dollars. This study

provides more detailed discussion on minimum margins and

change limits in the next subsection.

Safeguards

To insure that futures markets function smoothly and

free from abuses and disorder, the IMM established safe-

guards .

Clearing House.—The Clearing House is part of the

organized exchange and it serves as the opposite party in

all transactions. After the buyer and seller have initiated

a T-bill futures contract on the exchange floor, the Clear-

ing House acts as a go-between. As a result, no obligatory

relationship exists between the two parties. Each party is

subsequently free to act independently of the other. The

creditworthiness of the opposite party to the transaction is

of no concern because the Clearing House guarantees each

contract. In return, members of the exchange contribute to

a fund which backs the Clearing House.

Settlement of all transactions takes place daily at the

end of the trading day. Before trading can resume the next

business day, each account is "marked to market." The

daily settlement process involves adjustment of accounts to

reflect profits or losses from futures price changes.

Accounts are credited with profits and debited with losses;

14

the net balance of the Clearing House ledger is zero before

trading starts the next day.

Daily price limits.—To limit market participants'

exposure on any trading day, the exchange limits maximum

price changes from the settlement price of the previous day.

Price ranges are determined based on experience in the under-

lying cash market. If the price change in any trading day

is more than sixty basis points above or below the preceding

day's settlement price, the exchange restricts trading; no

T-bill futures trades would be cleared if the contract's

price changes by more than $1,500 (in either direction) from

the previous day's level. If trading restriction applies

for two successive days, widening of the price range is

necessary to accommodate the increased price volatility.

The normal range (sixty basis points) is reinstated if

trading is within the expanded limits. On the last trading

day, the price limit safeguard does not apply.

In addition to limiting participants' daily exposure,

the limits provide the market time to evaluate the causes

of unusual conditions. Manipulative practices may exert

artificial price pressure. The exchange has the authority

to take immediate corrective action in such situations.

Margin requirements.—Margin requirements play a key

role in the clearing system. They insure that market par-

ticipants can withstand adverse price movements. The

15

initial margin ($1,500) is not a downpayment; it is security

to cover losses from price changes that move against the

participants positions. Traders can post the initial margin

either in cash, selected securities, or bank letters of

credit. Of the initial margin, $1,200 is the maintenance

margin. As mentioned, margin accounts are debited (losses)

and credited (profits) when accounts are marked to market

daily. If losses reduce an account below the $1,200 mainte-

nance margin, the accountholder has to bring it back to the

$1,500 initial level (cash only) before trading starts the

next day. If the margin falls below $1,500 but not below

$1,200, the participant does not have to deposit the differ-

ence. Traders can withdraw profits and reinvest the money

elsewhere. The price for determining profits and losses in

margin accounts is the daily settlement price, which is the

average price in the last thirty seconds of trading.

Advantages of Forward Trading in Futures Markets

The T-bill futures market provides a way of hedging

risks. It also provides an opportunity for speculators to

try to make a profit. However, the cash market can also

provide the same functions through a series of transactions.

If so, why is there a need for a futures market? This sec-

tion examines some forward trading techniques and their

drawbacks (1). The discussion should make clear why T-bill

futures are useful.

16

Short selling is one way of forward trading. A short

sale is the sale of borrowed securities (in hope of a price

decline). To hedge security holdings in a short sale,

profits from the sale can offset the drop in value of

security holdings in the case of a price decline. The

ability to hedge security holdings is important to T-bill

dealers. Bill dealers can continue to make a market even

in risky periods by shorting bills. Any dealer constantly

absent from risky markets may create resentment and lose

customers. In the highly competitive bill market, dealers

depend on repeat business.

There are two ways to borrow securities in a short

sale. One is on a demand basis and the other is through a

reverse repurchase agreement for a designated time period.

Borrowing securities on demand subjects the borrower to the

risk of having to close out the position sooner than

expected. The use of a reverse repo—a repurchase agreement

initiated by the lender of funds—can lock in the desired

time period. In this case, the short seller is also subject

to the risks of the repo market which include the credit

risk of the repo seller and the risk of changes in the

T-bill collateral value. If the seller defaults, the buyer

may lose because the T-bill has fallen.

Cash market traders can effect forward trading through

long forward commitments. A long forward commitment is a

17

contract to purchase T-bill securities at a specified date

in the future at an agreed-upon price; it enables a trader

to profit from price increases without holding the securi-

ties. The long forward contract is not standardized and

involves credit risks with regard to the other side of the

transaction. Since such contracts are non-standard, it may

be difficult to locate another party with exactly the oppo-

site need.

Traders can also trade T-bills forward on a when-issued

basis. A when-issued commitment is a promise to purchase or

sell T-bills on delivery day during the period between the

auction of the bills and the settlement day. There is a

time lapse between the acceptance of competitive bids and

the settlement day when buyers make payments to the Federal

Reserve. On a when-issued trade, no cash changes hands

until the completion of the transaction. The major dis-

advantage of trading on a when-issued basis is that it tends

to be short term.

There are several substantial drawbacks to forward

trading in the cash market (e.g., creditworthiness of the

other party, short duration problem, and unequal access of

less active participants to forward trading transactions

or techniques). The T-bill futures market provides the

same function without the accompanying limitations. IMM

safeguards largely eliminate the creditworthiness problems.

18

The most distant T-bill futures contract is two years,

solving the short duration problem. Trading in the futures

market is by open outcry auction. It puts less active,

smaller participants on equal footing with major T-bill

dealers.

CHAPTER BIBLIOGRAPHY

* Inside T—Bill Futures, Chicago, Chicago Mercantile Exchange, 1983.

19

CHAPTER III

REVIEW OF LITERATURE

Empirical studies of the T-bill markets reveal a dif-

ference between futures and forward prices. Such studies

generally attribute the difference either to transaction

costs, taxes, the daily settlement process in futures trad-

ing, or a combination of these factors. The unaccounted—for

price difference implies market inefficiency (arbitrage

opportunities). Another approach to evaluate market effi-

ciency is testing for excessive variance in spot and futures

prices. This chapter briefly reviews literature on (1)

futures-forward price differences attributable to transac-

tion costs, (2) futures-forward price differences attribut-

able to the daily settlement process and different tax

treatments, and (3) variance tests of market efficiency.

Futures-Forward Price Difference and Transaction Costs

There have been several studies of the efficiency of

T-bill futures market. They examine the bill futures market

efficiency by comparing futures with the corresponding for-

ward rates implied in the term structure of T-bill interest

rates. Without an explicit T-bill forward market, the for-

ward rate is computed from the appropriate long and short

20

21

interest rates. The discrepancy between futures and forward

rates suggests arbitrage opportunities. Persistent arbi-

trage opportunities imply blatant inefficiency--that the

market is inefficient in the weak form as suggested by Kolb

(10). This section reviews market efficiency tests using

arbitrage arguments.

Poole (15) does one of the earliest T-bill futures

market studies. Poole hypothesizes that the inequality

between futures and forward rates is attributable to higher

transaction costs in the spot market. The implied forward

rate should be higher than the futures rate. Poole investi-

gates the issue by defining the upper and lower critical

points for profitable arbitrage, taking into account com-

mission costs and margin requirements. Profitable arbitrage

opportunities should not exist if futures rates were between

the critical points. Using daily data from January 6, 1976

through June 23, 1977, Poole finds very few profitable arbi-

trage opportunities. Poole's findings support his hypothe-

sis futures rates should be lower than the implied forward

rates. Poole studies only the nearest-to-maturity futures

contract, but he extrapolates his findings to other, more

distant maturity futures contracts. The implication of

Poole's findings is that the T-bill futures market is

efficient in the weak form.

Land and Rasche (11) also compare T-bill futures

yields with corresponding forward yields implied by the term

22

structure. Using Poole's formulas to calculate upper and

lower arbitrage points, they extend the analysis to futures

contracts with longer maturities. Futures maturities are as

long as twenty-four months. Lang and Rasche examine all

eight T-bill futures contracts. T-bills comparable in

length of maturity to the four longest futures contracts do

not exist. Lang and Rasche use yields on treasury coupon

securities to calculate forward rates. Sample data are

collected from March, 1976 through March, 1978. The authors

divide the overall period into three different sub-sample

periods. They find substantial differences between futures

and forward rates for contracts with longer maturities.

Futures rates are higher than forward rates and the rate

differential increases with time to maturity in their sample.

Their findings do not support Poole's hypothesis that futures

rates should be below forward rates.

Lang and Rasche conclude that the substantial differ-

ence between the two rates is not attributable to the ini-

tial low volume in the relatively new bill futures market.

The rate differential has not narrowed consistently over

time. The market has not become more efficient over time.

There appear to be frequent arbitrage opportunities with

longer maturity contracts. Such findings suggest that the

T-bill futures market is inefficient.

Puglisi (16) compares rates of return on two alterna-

tive strategies—futures and outstanding T-bills in the cash

23

market. The bills-only strategy involves purchase of a

short-term T-bill while the bills-futures strategy involves

purchase of a longer term bill while shorting a T-bill

futures contract. The latter strategy involves making

delivery on the futures contract (when it matures) with the

longer term bill. The hypothesis is that if the market is

efficient, the bills-only strategy and the bills-futures

strategy should earn the same return. Assuming a round-trip

commission of sixty dollars per futures contract, the author

finds a significant difference in returns between the two

strategies. Arbitrage profits can be earned—suggesting

market inefficiency. Puglisi's results indicate that market

inefficiency has a tendency to decrease over time, contrary

to the Lang-Rasche's findings.

Capozza and Cornell (2) investigate the efficiency of

the three futures contracts nearest to maturity. Comparing

futures rates with corresponding implied forward rates, test

results indicate significant deviations between rates in the

two markets. Deviations for the near contract are generally

small. The deviations tend to increase with length to

maturity. For the contracts with twenty-six and thirty-nine

weeks to maturity, the differential between futures and for-

ward rates is greater and there is no tendency for it to

decline over time. Moreover, the results show that futures

rates consistently exceed forward rates for those two con-

tracts .

24

Capozza and Cornell argue that arbitrage opportunities

do not necessarily imply that profitable arbitraging is pos-

sible. Arbitrage costs (costs of initiating and closing a

futures position, purchasing and selling spot T-bills, and

short selling in spot bills) could offset any gain. The

cost of short selling, the most significant of the three

costs, is a function of time. Consequently, the differen-

tial between forward and futures rates can increase with

maturity without attracting arbitrageurs. In economic

sense, the T-bill futures market may be efficient even if

arbitrage opportunities do exist.

Rendleman and Carabini (17), henceforth RC, determine

a range within which profitable arbitrage opportunities

would be absent. The bounds of the range are defined as

L = 100P®/P^ - .006, and [7]

U = 100P^/P® + .006, [8]

where L = lower bound,

U = upper bound,

P = spot price per $100 face value for T-bills ill

maturing at time m,

Pn = spot price per $100 face value for T-bills

maturing at time n, and

A and B = asked and bid prices, respectively.

RC analyze the price performance of the first three

futures contracts. A pure arbitrage opportunity refers to

25

a. trader making an arbitrage profit without taking an

initial position. With no pre-existing portfolio of spot

bills, arbitrageurs must short spot T-bills—incurring

heavy borrowing cost.

Taking into account transaction costs, the results

indicate that no profitable pure arbitrage opportunities are

available during the sample period. However, their results

also indicate that profitable quasi-arbitrage opportunities

are frequently available. A quasi-arbitrage opportunity

exists for an arbitrageur who has a pre-existing portfolio

of spot T-bills. RC conclude that the market might not be

efficient. The costs of training traders and policymakers

in financial institutions and of monitoring the futures

market, as well as the reluctance of financial institutions

to change the maturity structure of their portfolios, might

be enough to offset the quasi-arbitrage opportunities.

Futures-Forward Price Difference,

Daily Settlement and Taxes

There are several studies that offer other possible

explanations for the futures-forward differential. Morgan

(14) contends that the price difference results from the

daily marking-to-market mechanism required on futures

contracts. Futures prices could not equal forward prices,

even in an efficient market. The daily settlement process

requires the realization of a profit or loss with the

26

adjustment of the futures price change in the investor's

account at the end of each trading day. In case of a loss,

interest expense is incurred until the delivery date. With

a profit, withdrawing it creates reinvestment interest

income. Morgan's equation in defining the relation between

forward and futures prices is

P D - pd 4. v_1^A-nD-ts D-t for ~ fut ^_1 fut D-t' W

where P°Qr = forward price at D days prior to delivery,

Pf u t = futures price at D days prior to delivery,

= backward difference operator, and

D-t n ^ „

rD-t ~ D _ t day interest rate observed at D-t days

prior to delivery.

This equation shows that forward and futures prices

will not be equal as long as a probability exists that

futures price will change through time. The expected course

of interest rates over the time until the contract expires

is extremely important to futures pricing and it contributes

to the futures-forward differential. The differential can-

not be arbitraged away. Morgan provides no empirical sup-

port for his theory.

Cox, Ingersoll, and Ross (5) derive a model in which

forward and futures prices are unequal when interest rates

are stochastic. Forward and futures markets are fundamen-

tally different. Assuming perfect markets, no taxes and

transaction costs, and the existence of an implicit forward

27

market, CXR formulate a theory that highlights the cause of

the difference—marking-to-market mechanism. In a forward

contract, no payments are made until maturity; the payoff on

a forward contract equals the difference between the forward

price and the maturity spot price. The payoff on a futures

contract equals the difference between the initial futures

price and the maturity spot price. However, the payoff on a

futures contract is received in installments throughout the

period of the contract due to the daily settlement process.

With a forward contract, the total payoff is received at

maturity. The forward contract is comparable to investing

in a long-term security that matures at the same time as the

forward contract. However, the futures contract is compara-

ble to reinvesting continuously in one-day securities until

the futures contract matures. Forward and futures contracts

are dissimilar if interest rates are stochastic because the

futures price will then be a function of the unknown one—day

interest rates over the period of the futures contract. The

futures-forward price differential should not be interpreted

as evidence of market inefficiency, but rather as a result

of structural difference between futures and forward con-

tracts. This price differential cannot be arbitraged away.

Jarrow and Oldfield (9), henceforward JO, independently

reach the same conclusion as CIR—the forward price does not

equal the futures price unless default-free interest rates

28

are nonstochastic. Forward and futures contracts are

deceptively similar, but the difference arises due to the

daily marking-to-market mechanism in the futures market.

JO use a series of arbitrage positions to show that the

forward price is equal to the futures price only if interest

rates are deterministic.

JO also argue that the relationship between forward and

futures contract values depends upon the default-free rate.

JO define the relationship between forward and futures con-

tract values as follows:

f (p(s,t*) ;k(t) ft*) = F(P(s,t*) ?K(t) ,t*) .B(s,t*) , [10]

where f = forward contract value at time s,

F = futures contract value at time s,

t* = delivery date,

k(t) = exercise price of forward contract,

p(s,t*) = forward price at time s,

K(t) = exercise price of futures contract,

P(s,t*) = futures price at time s, and

B(s,t*) = spot price at time s of $1 received with

certainty at delivery date.

If the default-free rate is deterministic, the equation

above describes the relationship between forward and futures

values. As the equation shows, the forward contract value

is a proportion of the futures contract value at time s.

The discrepancy between f and F is due only to the different

29

cash flow patterns. Therefore, a futures contract is not

equal to a forward contract rewritten at the end of each day

even though the default-free rate is deterministic and the

forward and futures prices are equal. If the default-free

rate is stochastic, there is no relationship between f and

F.

Both CIR and JO show that futures and forward prices

are not equal if interest rates are stochastic. Neither

provides empirical evidence. French (7) empirically

examines two of the CIR propositions. According to CIR,

the futures-forward price difference should be positive if

the covariance between the daily percentage changes in the

futures price and the bond price is negative, and vice versa.

Further, the futures-forward price difference should be

negative if the bond price variance is smaller than its

covariance with the spot price, and vice versa. French uses

futures and forward prices for silver and copper to test

these two CIR propositions.

French encounters several serious complications in his

empirical work. Prices are collected from 1968 through

1980, which includes the years of 1979 and 1980 when the

economy experienced astronomical interest and inflation

rates. Another problem is matching futures and forward

prices—the futures prices reflect trading on the Commodity

Exchange and the Chicago Board of Trade in the United States

30

and the forward prices reflect trading on the London Metal

Exchange in Great Britain. Trading time differences might

introduce serious measurement errors; prices in Great

Britain are observed several hours earlier than those in the

United States. Other problems are changes in exchange rates,

transportation costs and international trade restrictions.

French finds some support for the first proposition;

support for the second proposition is more substantial.

Generally, the evidence is consistent with the CIR pre-

dictions, suggesting that the observed difference between

the forward and futures prices may not be evidence of market

inefficiency. This difference may be caused by structural

differences between futures and forward contracts. It is

unclear whether such conclusions apply to the T-bill futures

market.

Another explanation for the futures-forward differen-

tial is tax treatment. Several studies investigate whether

the difference between the T-bill futures price and the

implicit forward price might be attributable to taxes.

Cornell (3) employs a methodology that does not require

computation of implicit forward prices. The methodology is

similar to the one used to study the effect of taxes on

stock prices—examining returns on ex-dividend date when no

dividend is available to be taxed as ordinary income. In

November, 1978, the Internal Revenue Service ruled that all

31

futures contracts would be considered capital assets. The

holding period for long-term gains is six months for long

positions while all gains and losses on short positions are

considered short-term, regardless of the length of the

holding period. This asymmetric tax treatment should

encourage investors to take long positions with over six

months to maturity. If gains occur on the last trading day,

immediately liquidating futures contracts would lock in long-

term gains. When losses exist, taking delivery of the bills

at maturity and immediately reselling the bills in the cash

market establishes ordinary losses.

The tax ruling makes taking a long position in bill

futures with a maturity of more than six months attractive.

If taxes affect the futures-forward differential, futures

prices should drop when the maturity of bill futures con-

tracts changes from six months and one day to exactly six

months. Using a sample period of September, 1976 to March,

1980, the empirical study finds no tendency for futures

prices to drop when the critical maturity is reached.

Arak (1) also examines the effect of differential tax

treatment of bill futures and implicit forward contracts.

The sample period is January, 1976 to March, 1982. By

comparing futures and forward rates, Arak finds that the

November, 1979 IRS ruling has a significant impact on

futures prices. After the ruling, prices of more distant

32

maturity contracts rose significantly above forward, prices.

Even after the passage of the Economic Recovery Tax Act of

1981 (ERTA), the spread between the futures and forward

prices persisted. ERTA requires that for all positions

established after June 23, 1981, any net gain is taxed as if

60 percent is long term and 40 percent is short term, regard-

less of the holding period. ERTA also treats T-bills as

capital assets. As a result, taking delivery in order to

convert capital losses into ordinary losses is no longer

possible. However, there is insufficient data to draw firm

conclusions on the impact of ERTA.

Most empirical studies of the T-bill futures market

conclude that the difference between futures and implied

forward prices is attributable to either transaction costs,

marking-to-market procedures, or taxes. Cornell and Reinga-

num (4) examine all three factors by using data from the

foreign exchange market. Using a sample period of June,

1974 to June, 1979, the authors collect data on six major

currencies.

The empirical results indicate that the discrepancy

between forward and futures prices observed in the foreign

exchange market is not similar to the discrepancy found in

the T-bill market. The foreign exchange data show the mean

difference to be significantly different from zero from both

statistical and arbitrage points of view. Since the CIR

33

effect is negligible in both the foreign exchange and T-bill

markets, the only remaining explanation for the T-bill

results are differential tax treatment and transaction costs

unique to the T-bill futures market. These transaction

costs include the cost of short selling (due to the absence

°f a T-bill forward market), Cornell and Reinganum do not

attrioute the difference between the T—bill futures and

implicit forward prices to market inefficiency; they suggest

tax treatment and transaction costs.

Variance Tests of Market Efficiency

A traditional test of market efficiency has been to run

a regression to determine whether innovations are forecast^

able. If a market is efficient, only the arrival of unantici-

pated information results in price changes. Arrival of new

information is random so that price changes are not fore-

castable.

Recently, a few studies have used a different approach

to evaluate market efficiency. The studies use volatility

measures to determine whether prices show excessive varia-

tion to be accounted for in terms of the random arrival of

new information. The variance tests are a relatively new

efficient markets methodology. A chronological review of

some studies using this approach seems appropriate.

Shiller (18) was the first to use variance tests to

test bond market efficiency. The rational expectations

34

theory implies that a long-term interest rate is an average

of expected short-term rates. Therefore, long-term rate

series should be smoother than the short-term rate series.

The variance of the long-rate series should be limited by

the variance of the short rate series. It has been sug-

gested that long-term interest rates in rational expecta-

tions models are too volatile to be explained in terms of

the averaging inherent in the models. If long—term interest

rates are too volatile, it implies that the market is inef-

ficient. The conditional average (long rate), representing

rational expectations of future short-term rates, should

change only with arrival of significant new information.

However, if the long-term rate movements are "too big" rela-

tive to actual subsequent events, the movements cannot be

realistically attributed to new information. If this phe-

nomenon occurs too often, it implies a kind of forecastabil—

ity for long rates. In such case, public expectations are

not rational as expounded by expectations models.

Shiller uses the bond market, generally regarded as

efficient, to show that long-term rates are too volatile to

be consistent with rational expectations theory. The vola-

tility of long—term rates, as measured by the variance of

short-term holding yields on long-term bonds, should not

exceed bounds imposed by expectations models. Shiller finds

that the holding-yield variance generally exceeds its

35

theoretical oounds. This excess volatility implies that

either* (1) the expectations models of the term structure

are too restrictive, or (2) economic agents do not form

^^tional expectations and the bond market is inefficient.

Shiller (20) also examines whether stock prices move

more tnan justified by subsequent changes in dividends.

According to Shiller, the simple efficient markets model

that relates stock price to rationally—expected future divi-

dends is as follows:

CO

P t = k=o t < 1 / 1 + r > k + l E t < d t + k' }' 1111

where P = stock price at time t,

Dt+k = d i v i d e n d a t time t+k,

= mathematical expectations operator conditional

on information at time t, and

r = constant discount rate.

If stock prices equal the present value of expected

future dividends, any movement in stock prices is attribut-

able to random arrival of new information about future

dividends. Shiller finds that the volatility of stock price

is five to thirteen times too high to be attributed to new

information about future dividends. The upper bound, repre-

sented by the measure of the observed variability in divi-

dends, has been so dramatically violated by stock price

volatility that it would be unfair to attribute the failure

of the efficient market model to problems such as data

36

errors. Shiller argues that if the stock market is

efficient, the sample standard deviation of the movements

of dividends must understate the real uncertainty about

future dividends. The stock market might have rightfully

feared larger movements of future dividends than are actu-

ally realized.

Shiller (19) compares regression and variance tests

for evaluating market efficiency. He suggests several

advantages that volatility analysis has over regression

analysis. One of them is that a volatility test is more

robust and simple. Shiller further justifies the use of

volatility measures in assessing market efficiency because

they are insensitive to the problem of data misalignment.

In assessing the efficiency of the stock market, the price

movements during the sample period may reflect new data

about dividends that materialize after the sample period.

On the other hand, price movements reflecting information

that occurs during the sample period may take place before

the sample period. Regression tests are impossible when the

data are totally misaligned. Unaligned observations do not

affect variance tests. Shiller implies that variance tests

are more powerful than regression tests.

LeRoy and Porter (13) independently derive market

efficiency tests based on implied variance bounds by using

a present-value relation model. The model is as follows:

37

n Y = Z B3x®(j), [12]

j=0 ^

where Y = the stock price series,

0

xt (j)

= E^xt+j//lt^ wfiere It i

s the information on xt

and zt up to and including time t, = the earning series,

z - = a nY variable other than past earnings that are

used to predict earnings, and

B = discount factor and less than one.

Using the above model, the authors come out with three

theorems about the variance of the dependent variable process

relating to the variance of the independent variable process.

Theorem 1 states that the coefficient of dispersion of

Yfc is less than the standard deviation-mean ratio found in

• The theorem is theoretically sound because the present-

value model defines stock price as a weighted average of

expected earnings. With the average smoother and less vola-

tile than its various components, and the tendency of

expected earnings to move toward a mean in the increasingly

distant future, the coefficient of dispersion of stock price

should be relatively less.

Theorem 2 states that

CD of Y* > CD of Yfc >_ CD of Y , [13]

where CD = coefficient of dispersion, *

Yt = stock price series if future earning changes

are known with certainty.

38

Yfc = stock price series defined in Eq. [12], and

~ stock price series if future earning changes

are unknown completely.

By definition, equals (Y_ + U .) ; U_ is the forecast error

depending only on earning changes occurring after time t.

If the market is efficient, the forecast error cannot be

correlated with Yt—any correlation implies a biased fore-

cast of Yt> Assuming market efficiency, var(Y ) = var(Y ) + t t

var(Ut) if the covariance between Y and U is zero. It

follows that var(Yt) > var(Yt); the variance of Y* estab-

lishes the upper bound for the variance of Y . Likewise, ^ A

var(Yt) = var(Yt) + var (Ut). Assuming Yfc contains at least

the information on the past history of earnings, var(Y ) > A t —

var(Y^) ' the variance of Y_ establishes the lower bound on

the variance of Y .

Theorems 1 and 2 are similar to the restrictions that

Shiller obtains on the variance of long rates. Theorem 3

has already been stated in discussing Theorem 2. To restate

the first part of Theorem 3, the variance of Y equals the

sum of the variance of Y and U . L. L

The three theorems can be used as tests of market

efficiency because the present-value relation between stock

price and earnings implies market efficiency. Using a data

set based on the Standard and Poor's Composite Index of

stock prices and related earnings, the authors test all

39

three theorems. They find a clear rejection of market

efficiency during the sample period, 1955 to 1973. Stock

prices seem to be more volatile than is consistent with

market efficiency. This is contrary to the large body of

market efficiency literature using regression to determine

the presence or absence of autocorrelations in rates of

return. Since the autocorrelation tests and volatility tests

are derived from the same basic model, their different out-

comes regarding market efficiency puzzle the authors. One

explanation could be that the volatility or dispersion tests

have more power than the regression or autocorrelation tests.

Regression and variance tests of market efficiency do

not always produce conflicting results. Huang (8) studies

the efficiency of the foreign market. He rejects the null

hypothesis of market efficiency with both the regression and

variance tests. The model examined is the monetary approach

to exchange rate determination. Market efficiency tests are

necessarily joint tests of market efficiency and model

validity. Rejecting the null hypothesis does not necessarily

mean that the foreign exchange market is inefficient, but

that the model is misspecified.

Studies in asset-price volatility suggest that price

variance is too great to be consistent with conventional

security valuation models. LeRoy and LaCivita (12) ques-

tion these variance test conclusions. They argue that the

40

conventional models are based on risk-neutrality or near

risk-neutrality. Risk-aversion leads to greater asset price

volatility than risk neutrality. It is then possible that

if risk-aversion is taken into account, more volatility is

permitted which might change interpretation of the results

on asset-price dispersion.

LeRoy and LaCivita point out that efficient market

tests are joint tests of three assumptions: stationarity,

rational expectations, and present-value relation. A

restricted (two-moment) stationarity assumption requires

that

var(Pt) =var(P t + 1), [14]

where P = price or yield, and

var (.) = variance operator.

The rational expectations assumption requires the opti-

mal forecast of future spot prices to be made, based on

available information. The present-value relation assump-

tion is the element of the joint hypothesis that the authors

emphasize in their study. The assumption is equivalent to

the martingale assumption (that the future spot price is

equal to the current spot price) in the stock market. They

maintain that the assumption is valid only under risk-

neutrality . LeRoy and LaCivita conclude that there is a

positive connection between risk-aversion and price volatil-

ity. It still remains an open empirical question as to what

41

degree of risk-aversion is adequate to account for the

observed volatility in asset prices.

Flavin (6) also reassesses the empirical evidence

regarding excess volatility in the financial markets. The

seeming violation of market efficiency using variance mea-

sures may simply reflect small sample bias. Variance tests

tend to be biased toward rejecting the null hypothesis of

market efficiency. In variance analysis, variances are

expressed in deviations from the sample mean rather than the

population mean. Obtaining deviations from the sample mean

creates a downward bias in the sample variance, due to

asymptotic sample properties. Thus, implied upper variance

bounds should be higher if the population means were known a

priori.

Summary

Previous studies offer various explanations for the

price discrepancy in the T-bill spot and futures markets.

In general, pure arbitrage opportunities are rare. However,

observation of quasi-arbitrage opportunities are frequent—

possibly due to the absence of the short selling portion of

transaction costs. The futures market is inefficient in the

quasi-arbitrage sense.

A competing explanation for the futures-forward price

differences is the daily marking-to-market feature in the

futures market. Morgan argues that such a feature would

42

create futures-forward price differences even in an

efficient T-bill futures market. Cox-Ingersoll-Ross and

Jarrow-Oldfield both provide theoretical justification for

the futures-forward price differences: they could be attrib^

utable to stochastic interest rates. French obtains empiri-

cal support for two of CIR's propositions in a study of

copper and silver, but some serious data problems undermine

the effectiveness of the study.

A third explanation for the T-bill futures-forward price

differences is the differential tax treatment. Cornell finds

no evidence that taxes have contributed to the futures-

forward differential. Arak finds evidence to the contrary.

In an effort to determine which of the three competing

explanations for the bill futures-forward price differential

is most promising, Cornell and Reinganum examine all three.

They suggest that only the differential tax treatment and

transaction costs explanations seem reasonable. The effect

of the daily marking should be negligible.

The differences between futures and implicit forward

prices suggest arbitrage opportunities. Unsatisfactory

explanations for the differences infer market inefficiency.

The conflicting findings of previous studies leave the issue

of T-bill futures market efficiency unresolved,

Variance-bound tests suggest excess volatility in the

financial markets. Shiller points out that long-term rates

43

and stock prices are too volatile to be consistent with

their corresponding models. LeRoy and Porter conclude that

excess stock price volatility contradicts the market effi-

ciency hypothesis. Huang finds the foreign exchange market

inefficient. These findings suggest that either the models

at hand are not valid or the markets studied are inefficient.

However, both LeRoy and LaCivita, and Flavin caution that

apparent evidence of market inefficiency may be attributable

to the unrealistic assumption of risk-neutrality (which is

the basis for conventional security valuation models) and

small sample bias.

This chapter reviewed literature on theoretical

futures-forward price differences and efficiency tests based

on arbitrage arguments. The issue of T-bill futures market

efficiency needs further examination using different method-

ological approaches. The next chapter describes the alterna-

tive methodologies. The methodology chapter also includes

operational statements on the empirical testing of the T-

bill futures-forward difference.

CHAPTER BIBLIOGRAPHY

1. Arak, Marcelle, "The Effect of the Tax Treatment of Treasury Bill Futures on Their Rates," Journal of Futures Markets, III (Spring, 1983), 65-73.

2. Capozza, Dennis and Bradford Cornell, "Treasury Bill Pricing in the Spot and Futures Markets," Review of Economics and Statistics, LXI (November^ 1979), 513-520.

3. Cornell, Bradford, "A Note on Taxes and the Pricing of Treasury Bill Futures Contracts," Journal of Finance, XXXVI (June, 1981), 1169-1176.

4. Cornell, Bradford and Marc Reinganum, "Forward and Futures Prices: Evidence from the Foreign Exchange Markets," Journal of Finance, XXXVI (December, 1981), 1035-1045.

5. Cox, John, Jonathan Ingersoll and Stephen Ross, "The Relation Between Forward Prices and Futures Prices," Journal of Financial Economics, IX (December, 1981), 321-346.

6. Flavin, Marjorie, "Excess Volatility in the Financial Markets: A Reassessment of the Empirical Evidence," Journal of Political Economy, XCI (December, 1983), 929-956.

7. French, Kenneth, "A Comparison of Futures and Forward Prices," Journal of Financial Economics, XII (November, 1983), 311-342.

8. Huang, Roger, "The Monetary Approach to Exchange Rate in an Efficient Foreign Exchange Market: Tests Based on Volatility," Journal of Finance, XXXVI (March, 1981), 31-41.

9. Jarrow, Robert and George Oldfield, "Forward Contract and Futures Contracts," Journal of Financial Economics, IX (December, 1981), 373-382.

10. Kolb, Robert, Interest Rate Futures: A Comprehensive Introduction, Richmond, Virginia,-Robert F. Dame, Inc., 1982.

44

45

11. Lang, Richard and Robert Rasche, "A Comparison of Yields on Futures Contracts and Implied Forward Rates," Federal Reserve Bank of St. Louis Review, LX (December, 197B), 21-30.

12. LeRoy, Stephen and C. J. LaCivita, "Risk Aversion and the Dispersion of Asset Prices," Journal of Business, LIV (October, 1981), 535-547.

13. LeRoy, Stephen and Richard Porter, "The Present Value Relation: Tests Based on Implied Variance Bounds," Econometrica, XLIX (May, 1981), 555-574.

14. Morgan, George, "Forward and Futures Pricing on Treasury Bills," Journal of Banking and Finance, V (December, 1981), 483-496.

15. Poole, William, "Using T-Bill Futures to Gauge Interest-Rate Expectations," Federal Reserve Bank of San Francisco Economics Review, Spring, 1978, pp. 7-16.

16. Puglisi, Donald, "Is the Futures Market for Treasury Bills Efficient?," Journal of Portfolio Management, IV (Winter, 1978), 64-67.

17. Rendleman, Richard and Christopher Carabini, "The Efficiency of the Treasury Bill Futures Market," Journal of Finance, XXXIX (September, 1979), 895-914.

18. Shiller, Robert, "The Volatility of Long-Term Interest Rates and Expectations Models of the Term Structure," Journal of Political Economy, LXXXVII (December, 1979), 1190-1219.

19. , "The Use of Volatility Measures in Assessing Market Efficiency," Journal of Finance, XXXVI (May, 1981), 291-304.

20. , "Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?," American Economic Review, LXXI (June, 1981), 421-436.

CHAPTER IV

METHODOLOGY

This chapter begins by developing the hypotheses to be

tested. Subsequent discussion focuses on the procedures and

data used to test the hypotheses. The final section of the

chapter is a summary.

Hypotheses

Futures-Forward Price Difference and Stochastic Interest Rates

Eqs. [5] and [61 show the futures-forward price differ-

ences in terms of the relations between futures and bond

prices, and between spot and bond prices. The futures and

forward prices are equal only if interest rates are non-

stochastic. These models assume no taxes or transaction

costs.

Eq. [5] implies that if the local covariance (localized

to individual contract periods) between the percentage

changes in the futures price and the percentage changes in

the bond price is positive, the integral m the equation

positive. This makes the right side of the equality sign

negative; the futures price should be less than the forward

price. If the covariance is negative, the futures price

46

47

should be higher than the forward price. To test this

hypothesis, the average futures-forward price difference

is compared with the average covariance between the futures

and bond percentage price changes. The null and alternative

hypotheses for a significant mean price difference are

H q: diff(F, f) = 0

H;l: diff(F,f) > 0, and

H 2: diff(F,f) < 0,

where diff(F,f> is the average futures-forward price differ-

ence. The null and alternative hypotheses for a significant

mean local covariance are

HQ: cov(F,B) = 0,

: cov(F',B) > 0, and

H2: COV (F,B) < 0,

where cov(F,B) is the mean covariance between the futures

and bond percentage price changes.

With unknown population standard deviation and the

small sample size of local covariances, the appropriate

statistical test to use is the Student's t. The study

assumes normal distribution for the population values.

The significance level is at 5 percent. ihe definition

the sampling distribution for t is

t = (x - 0)/(s//n) with n-1 degrees of freedom,

where x = variable mean,

s = standard deviation, and

n = number of observations.

48

The decision rule is to reject the null hypothesis if t is

less than the left critical value or more than the right

critical value.

Eg. [6] implies that if the local variance of the bond

percentage price changes is larger than the local covariance

between the spot and bond percentage price changes, the

futures-forward price difference should be positive. If the

variance is smaller than the covariance, the futures-forward

price difference should be negative. Testing this hypothesis

requires comparing the average futures-forward price differ-

ence with the difference between the mean bond price variance

and the mean spot-bond covariance. The null and alternate

hypotheses for a significant mean bond price variance are

HQ : var(B) = 0,

H i var(B) > 0, and

H2*. var(B) < 0.

The hypotheses for a significant mean spot-bond covariance

are also stated likewise. The null hypothesis testing

procedure associated with Eq. [5] also applies for testing

the above null hypotheses.

The empirical result supports the model in Eq. [5] if

diff (F,f) is significantly different from zero and positive,

and cov(F,B) is significantly different from zero and nega-

tive. The same is true if these conditions are reversed.

There is also empirical support for the model m Eq. [6] if

diff (F,f) is significantly different from zero and positive,

49

and [cov(s,B)-var(B)l is significantly different from zero

and negative. The same is true if the signs are reversed.

Variance Tests of Futures Market Efficiency

This subsection begins with the derivation of variance

inequalities. The development of hypotheses follows.

Derivation.—The variance tests of market efficiency

assume weak stationarity in the variables analyzed. Weak

stationarity implies that there are no systematic changes m

the variance. A process is strictly stationary only if all

its moments are constant—same probability distribution fox-

all time. This study assumes only weak stationarity.

Eqs. [2] and [3] in Chapter I provide the basis for the

derivation of two inequality equations that are used as

variance tests of market efficiency. Given Eq. [3], an

efficient market implies that

c°v ( Ft,n

Ut,n> = °' 1 1 5 1

where F = futures price quoted at time t for delivery t f n

at time n, and

U = forecast error given the available information t,n

at time t,

since the forecast error is orthogonal to any element of the

information set. Computing the variance for each variable

in Eq. [3] gives

var(Sn) = var(Ft,n} + var(Ut,n)* [161

50

Since positive semi-definiteness requires var(U^n) > 0, it

follows that

var(S ) > var(F ).

n — n

The futures price, an estimate of the expected spot price on

the maturity date of the contract, should be less volatile

than the realized spot price. Theorem 3 in Leroy-Porter

proposes that var(Yfc) is equal or less than var(Yt), where

Y* represents a time series that contains more information

on the future expected spot price than Yt< Market partici-

pants possess more information at time t+1 than at time t on

the future spot price at time n. It can be expressed as (Sn ' Ft,n' = (Sn " Ft+l,n> + U' 1 1 8 1

where U = additional forecast error due to less information

at time t than at time t+1.

Following the manner in which Eq. [17] was obtained,

var(Sn - F t f n) > var(Sn - F t + 1, n) • [ 1 9 ]

The greater the time distance between the date when the

futures price is quoted and the contract maturity date, the

less information there is to estimate S n and the bigger the

forecast error (as measured by the variance); i.e.,

v a r ( sm ' Ft,m» i v a r ( S n ' Ft,n>' n < m 1201

Following Huang (1), Eq. [2] can be used to derive

another variance inequality. S t is known at time t. If

markets are efficient,

< Ft,n - St> = Et ( Sn ' St>" U U

51

Taking the variances of both sides, Eq. [21] can be general-

ized to obtain the following inequality.

v a r ( Sn'" St ) ~ v a r ( Ft,n ' St) -

var(Sn-St) - var(F

t+i,n ~ St) ' 0 < i < n '-221

With the variability of ( Ft +i ; n ~

st^ a c c o u n ti n <? f°r a

greater portion of the variability of the spot price at time

n,

v a r ( Ft +i,n " St' > v a r ( Ft,n " St»" 1231

Hypotheses for variance tests.—This study uses Eqs.

[20] and [23] to test market efficiency. The null and

alternative hypotheses associated with Eq. [20] are

K0 var(S - F ) > var(S - F ), and

Ha : v a r ( Sm " Ft,m) < v a r ( Sn " "

This study uses the Pitman statistic to test the null hypoth

esis. The test (3) is defined as

R = <si/s2 " X ) /y/ (S2/S2 + l)2 - ^ ( S ^ / S p

with n-2 degrees of freedom,

where S2 = sample estimate of variance,

S2 = sample estimate of smaller variance, and

r = correlation coefficient between two variables.

The statistic is used for a one-tailed test. The probaoility

of committing a Type One error is 5 percent. The null (mar-

ket efficiency) hypothesis is rejected if the computed R is

52

less than the critical R value. The null hypothesis is not

otherwise rejected.

The null and alternative hypotheses associated with Eq.

[23] are

Hq: var<Ft+i(I1 - Sfc) > var(Ft_n - Sfc) , and

Ha: var(Ft+i,n ' St> < var(Ft,n " St>"

The above null hypothesis is tested using the same testing

procedure as described earlier.

Regression Tests of Futures Market Efficiency

Prices in an efficient market reflect all available

information. Only the arrival of unpredicted information

can make traders adjust their forecasts. The futures price,

an estimate of the future spot price and conditional on the

information set at time t, should be unbiased if the market

is efficient. Eq. [2] in Chapter I describes the relation-

ship between the futures price and the expected spot price.

As Eq. [4] has shown, Eq. [2] is an efficient market model

if the forecast error is orthogonal to any element of the

information set. Putting it differently,

VSn " -WV " °' U 4 1

The same is true for futures prices that are quoted at

different dates of the same contract period; they are all

unbiased estimates of the same future spot price.

53

Empirical testing of futures market efficiency can take

the following regression form:

S = a + bF. _ + u . [25] n t, n t

where a = intercept term,

b = slope coefficient, and

u = S - E (S ) when E (S ) = F . t n t n t n

Assuming that ufc are dependent and identically distributed

with zero mean and a2, the null and alternative hypotheses

are

Hq: b = 1, and

H : b f 1. a

This study uses the t statistic to test the null hypothesis

at the 5 percent level of significance. The decision rule

is to reject the null hypothesis if the computed t value is

smaller (greater) than the lower (higher) critical value.

Failure to reject the null hypothesis implies that the

futures market is efficient.

Procedures

The scope of this study is limited to analyzing the

first three T-bill futures contracts. Going beyond the

third contract necessitates the use of Treasury coupon

securities that might introduce measurement problems. The

next subsection presents the procedure for testing the CIR

propositions regarding the difference between futures and

implicit forward contracts. The last chapter subsection

54

deals with the procedure for testing market efficiency in

the three T-bill futures contracts nearest to maturity.

Tests of the Cox-Ingersoll-Ross Models

•phe futures—forward price differences are computed from

price data during the first thirteen-week periods of the

three-, six- and nine-month contracts to avoid overlapping

data. This study defines the futures-forward percentage

price difference as

(Ft " ft ) / ft x 1 0 0' [26]

where = futures price at time t, and

f = implied forward price at time t.

Expressing the price difference in percentage gives the

relative magnitude of the difference. The average futures-

forward price difference is obtained for each of the three

contracts.

Estimating the mean covariance consists of two steps.

First, the local covariance of a given maturity-period con-

tract is estimated over each contract period in the sample.

Second, averaging local covariances across contract periods

estimates the mean covariance. The mean local covariance of

the daily percentage changes in the bond price is obtained

similarly. Subsequent comparisons of the estimated vari-

ables would then determine whether there is support for the

two CIR models.

55

This study further examines the ability of the CIR

theory to explain T-bill futures-forward price differences.

Although the ability of the theory to discriminate among

price differences is not one of the issues in this study,

it is useful to determine whether their theory explains

intra-sample variations. The procedures in this study are

similar to those used by French; he examined the models'

usefulness in describing intra-sample variations in the

copper and silver markets. After the estimation of local

covariances over each three—, six- and nine—month contract

period, the contracts are then assigned to two groups.

Using the first CIR model, the basis of assignment is whether

a contract has a negative or positive covariance estimate.

If changes in covariance and price differences are related,

there should be a positive futures-forward mean price differ-

ence for the group with negative covariance estimates. The

reverse should also be true. A second CIR proposition is

tested in similar fashion. The futures-forward price differ-

ences should be positive in the group where the bond price

variance is greater than the bond-spot price covariance.

The reverse should also be true.

Market Efficiency Tests

This study tests Eq, [20] for n = 13 and 26 weeks, m =

n+13 weeks and t = 1, 4, 7, 10 and 12 weeks. For a given n,

the variance of the absolute price differences between the

56

expected T-bill spot price and the futures price at time t

is computed for each T-bill futures contract period in the

sample. Based on Eg. [20], the mean variances on both sides

of the inequality sign are then compared.

This study tests Eg. [23] for n = 13, 26 and 39 weeks

and t = 0, 3, 6, and 9 weeks. The subscript of i in the

equation represents a period length of four weeks. The pro-

cess of finding the average variance of the absolute price

difference (futures price - bill spot price at time t) is

similar to that used for Eq. [20].

A second approach to test futures market efficiency

involves performing regression tests for n = 13, 26 and 39

weeks and t = 1, 4, 7, 10 and 12 weeks. The expected spot

price at the futures contract maturity date is regressed

against the futures price at time t. In addition to calcu-

lating significance statistics for the regression coeffi-

cients, R values (and F tests) and Durbin-Watson statistics

are reported.

Data

The Wall Street Journal is the source of weekly futures

price data. This study uses the Wednesday closing prices as

reported in the Journal: since the last day of trading is

two business days after the third weekly T—bill auction dur-

ing the contract month for any specific futures contract,

the spot bill auction usually falls on Monday and futures

57

trading for any contract usually terminates on Wednesday.

Moreover, the choice of Wednesday's prices avoids roost

holidays. To maintain compatibility with futures prices,

T-bill spot prices are also computed from Wednesday's clos-

ing bid yields as reported in the Wall Street Journal.

This study uses weekly bid prices because practitioners

assert that bid prices more accurately reflect market prices

The sample period begins March, 1976 and runs through

December, 1984. The overall period is divided into three

subperiods to facilitate more in-depth analysis of the

sample data. Table I shows the length of each subperiod.

T—bill futures trading started on January, 1976, but the

first two months of trading v/ere relatively light. Data

from the first two months of trading are, therefore, omit-

ted. This study also omits data in cases where no spot

T-bills matured at or near the maturity date of the T-bill

futures contract concerned.

Data used in testing the CIR models consist of the

weekly Tuesday-Wednesday price changes, except for those

used in finding the futures-forward price differences. The

bond percentage price change is obtained by measuring the

percentage price change in the spot T-bill that matures at

the same time as the relevant futures contract. The

measurement of percentage changes in the spot T-bill that

matures thirteen weeks after the relevant futures contract

58

maturity date provides the spot percentage price change data

for this study. Wednesday's T-bill bid quotations provide

the basis for calculating implicit forward prices. The

traditional formula to compute implicit forward prices

involves extensive calculations. Roll's formula (2) is used

in this study:

„ fQ1 = 100 - [nRT - (n - 91)R_]/91, [27] n-m 91 L b

where f~, = forward price on a 91-day T-bill to begin n-m 91 r

in n-m periods,

n = number of days to long bill maturity date,

m = number of days to short bill maturity date,

R = spot rate on long bill, and Li

Rc = spot rate on short bill.

Roll's formula may generate different, but inconsequential,

forward prices.

Before generating the results and analyzing them, it is

helpful to examine the general behavior of the sample price

data. Table I presents the comparison of the behavior of

the futures, forward, bond and spot prices. The sample

prices were lowest during the second subperiod (December,

1978 to December, 1981). They were highest during the first

subperiod. The variance statistics indicate that the prices

varied the most during the second subperiod. The smallest

variance value is 3.6 in the nine-month futures contract.

59

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60

The level and variability of these prices suggest that the

second subperiod was very turbulent. The subperiod includes

the volatile years of 1979 and 1980. Prices were relatively

stable during the first subperiod. The largest variance

value is only 1.86 in the six-month forward contract. There

might be an association between price volatility and the

price curve patterns. Prices were generally upward sloping

during the second subperiod, downward sloping during the

first subperiod, and mixed during the third subperiod. It

implies that short-term interest rates are generally higher

than long-term rates during periods of great uncertainty

and volatility. These periods of great volatility usually

occur at interest-rate peaks when the economy is entering

or just about to enter the recessionary phase. Investors

prefer to lengthen the maturity of their debt security

portfolios in order to reap potential capital gain. The

strategy generally results in short rates higher than long

rates at cyclical peaks.

Organized futures exchanges impose a limit on futures

price changes in a particular day. The limit could create a

constraint on futures price movements. However, price

variances of futures and forward contracts do not indicate

any evidence of such a constraint. In many instances, the

futures price variance was greater than the forward price

variance. The nine-month contract provides an example

during the third subperiod, where the futures and forward

61

price variances are 2.5 and 1.91, respectively. In compar-

ing the variances between bond and spot prices, the overall

sample data statistics in Table I show that the bond price

variance was greater for all three contracts. The bond price

variances are 9.7, 9.03 and 7.06, whereas the corresponding

spot price variances are 9.03, 7.06 and 6.47. On average,

price variances decreased as maturity increased.

Since the CIR models' tests require the stationarity

assumption, the T-bill price series is examined for station-

arity. Autocorrelations for the T-bill spot series are

generated for ten lags. The series is nonstationary if the

series has large autocorrelations that decay slowly as lag

increases. If they quickly approach insignificance or are

not significantly different from zero, the series is

stationary. Table II shows the time series is nonstationary

for all maturity periods. The autocorrelations decrease

slowly as lag increases. It is necessary to determine the

minimal degree of differencing so that the tests in this

study do not violate the stationarity assumption. With one

degree of differencing, the serial correlations are insig-

nificantly different from zero at all lags. The correlation

coefficients are mostly within two standard errors (figures

in parentheses equal one standard error). Thus, no further

differencing is needed.

The nonstationarity of the original T-bill spot price

series does not imply the existence of profit opportunities.

62

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63

Table III contains the autocorrelation coefficients of T-bill

spot and futures percentage price changes. The coefficients

of spot price changes are low and statistically insignificant

at the first lag. The largest coefficient is only -0.081.

The same pattern is also observed for future price changes

in all maturity contracts. The largest autocorrelation

coefficient of futures price changes is -0.084.

Summary

The CIR models show that the futures and forward prices

differ if future interest rates are stochastic. If these

models are correct, this study should yield the following

results:

(1) If cov(F,B) and [cov(S,B) - var(B)] are signifi-

cantly negative, the futures-forward price differ-

ence should be significantly positive; and

(2) If cov(F,B) and [cov(S,B) - var(B)] are signifi-

cantly positive, the futures-forward price differ-

ence should be significantly negative.

Thus, empirical testing of the CIR hypotheses involves

testing whether cov(F,B), [cov(S,B) - var(B)] and diff(F,f)

are significantly different from zero.

Two inequality equations—var(S - F. ) > var(S - F^ ) 1 2 n m t ,m - n t,n

and var(F . - S, ) > var (F - S )—are derived to test "C. • i f n "c t f in t

whether the T-bill futures market is efficient. The Pitman

64

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65

statistic provides the significant testing of the inequali-

ties. Regression tests of market efficiency involve testing

whether the slope coefficient is significantly different

from zero.

Testing the CIR models and the T-bill futures market

efficiency hypothesis requires the estimation of average

covariance and variance. The local covariances and

variances over each contract period in the sample are com-

puted and they are then averaged over all contract periods.

The mean of diff(F,f) is estimated from the weekly futures-

forward price difference data.

This study is confined to analyzing the first three

T-bill futures contracts. Weekly data have been collected

from the Wall Street Journal. The sample period begins on

March, 1976 and runs through December, 1984. The division

of the overall sample period into three subperiods permits

an examination of the sample data behavior. The stationarity

of the relevant price series is also examined.

CHAPTER BIBLIOGRAPHY

Huang, Roger, "Efficiency and the Volatility of Flexible Exchange Rates," unpublished manuscript, University of Pennsylvania, 1980.

Roll, Richard, Behavior of Interest Rates, New York, Basic Books, 1970.

Snedecor, George and William Cochran, Statistical Methods, Ames, Iowa, Iowa State University Press. 1980.

66

CHAPTER V

RESULTS

This chapter presents and interprets the results of

the statistical tests to determine whether they are con-

sistent with the hypothesized theory. The first section

gives the Cox-Ingersoll-Ross models' test results and

interpretation. The next section reports the findings and

discusses the CIR models' ability to discriminate among the

price differences. Subsequent sections cover the results

from the variance and regression tests of market efficiency.

Tests of the Cox-Ingersoll-Ross Propositions

Table IV summarizes the results of the tests of Eqs.

[5] and [6]. (All covariances and variances are magnified

by 100 times to reduce the large number of decimal places.)

The results for Eq. [5] indicate that the mean futures-

forward price differences in all contracts are significantly

different from zero. On average, futures prices in the

three- and six-month contracts are respectively 0.31 and

0.12 greater than the corresponding forward prices. How-

ever, forward prices exceed futures prices by 0.28 in the

nine-month contract. The null hypothesis of an insignificant

mean price difference is, therefore, rejected for all three

67

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69

contracts. The null hypothesis of an insignificant mean

local covariance between the futures and bond percentage

price changes is also rejected for all futures contracts.

The mean local covariances are 0.99, 1.64 and 0.77.

A comparison between the mean price difference and the

mean covariance for the three-month contract reveals no

support for the CIR proposition embodied in Eq. [5]. With a

positive covariance of 0.99, the mean price difference should

be negative. The summary statistics for the six-month con-

tract also indicate no support for the proposition. However,

the nine-month contract's results are consistent with the

CIR models' prediction. With a positive mean covariance of

0.77, the mean price difference is negative as hypothesized.

There is little evidence to suggest that the CIR model

explains why the futures and forward prices differ.

There is more support for the CIR proposition embodied

in Eq. [6]. Table IV again shows that all mean price differ-

ences, covariances and variances are significantly different

from zero. It also shows the mean price values are an

inverse function of length to contract maturity. The mean

values of 6.31, 1.2 and 4.6 in the three-month contract

decrease to -0.28, 0.36 and 1.11, respectively, in the nine-

month contract. The figures for the first two contracts

show support for the CIR proposition in Eq, [6], With the

mean variances greater than the mean covariances, the mean

70

price differences are positive. The results are consistent

with the model's prediction. However, the mean price differ-

ence of -0.28 in the nine-month contract is inconsistent

with what Eq. [6] would predict.

The evidence in Table XV is mixed. If the underlying

CIR proposition were valid, the empirical test results of

Eqs. [5] and [6] should have led to similar conclusions.

Both equations express the same proposition—but in differ-

ent ways. The evidence in this study is inconsistent and

contradictory.

There are several possible explanations for the incon-

sistent test results. One possible reason is the invalidity

of the CIR models. The futures-forward price differences

may not be attributed to the daily settlement process and

stochastic interest rates. The price differences might not

be related to market participants' expectations of the rele-

vant covariances and variances. Other factors may cause the

price discrepancy.

A second possibility is that the CIR models are valid

only for non-financial underlying assets. The first CIR

model indicates that the futures-bond covariance should be

negative in order to obtain a positive futures-forward price

difference. This is highly improbable for financial assets.

Since this study deals with T-bills and it specifies that

the bond price is the present spot price of a T-bill

71

expiring on the relevant futures contract's maturity date,

the futures and bond prices would always exhibit a positive

covariance. Prices of financial spot and futures assets

move inversely to interest rates. Accordingly, the futures-

forward price difference should be negative. The possibility

of a positive price difference is remote unless there is a

basic change in the economic relationship between financial

assets' prices and interest rates. The point is that the

CIR models may not adequately describe the price behavior of

financial assets and their corresponding futures contracts.

The inconsistency of the test results may also be

attributed to measurement errors. There is no explicit

forward market for T-bills. The implicit forward prices may

not accurately represent such prices. Factors that affect

explicit prices could be completely ignored in determining

implicit prices. Moreover, spot prices that are used in

computing implicit forward prices may introduce further

measurement errors. Because of the matching problem between

futures contracts and the relevant T-bills, it is necessary

to use the nearest spot prices as substitutes. The problem

is especially acute in determining the implicit nine-month

forward contract prices.

Intra-Sample Variations Among Futures-Forward Price

Differences

The ability of Eq. [5] to explain futures-forward price

differences is questionable. The results in Table V provide

72

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73

no evidence that the futures "bond, cova.iria.nce is useful in

discriminating among the futures-forward price differences.

With positive covariances, the average price difference in

the three-month contract (third column) should be negative.

However, the observed average price difference is 0.34 and

significantly positive. For the group with negative

covariances, the average price difference in the three—month

contract is 0.075 and insignificant (only four observations).

For the six-month contract, none of the observations has

negative covariance. For those assigned to the group with

positive covariances, the average price difference is 0.122

and insignificant. The only average price difference that

is significant and has the right sign is that of the nine-

month contract in the positive covariance group. The mean

price difference is -0.292.

Table V also shows that Eq. [6] is not useful in

explaining the intra-sample variations. Observations with

smaller spot-bond covariances than bond variances are

assigned to the second column. The rest are assigned to the

third column. If the C1R model proves useful, the average

futures-forward price difference in the second and third

columns should be positive and negative, respectively. The

average price difference in the three—month contract (second

column) is 0.31 and significant. However, the price differ-

ences of 0.12 and -0.26 in the six-month and nine-month

contracts are not different from zero. No result is

74

obtained for the group with larger spot-bond covariances

than bond variances because of inadequate data.

The results in Table V do not present a consistent

pattern across all contracts. The results reject the CIR

models. The models do not seem able to discriminate among

the price differences. The models' predictions could be

more accurate with better proxies of the market expectations

of the relevant variances and covariances. This study uses

ex—post variances and covariances as proxies for market

expectations. The ex-post estimates of expectations may

contain measurement errors. Moreover, the estimates are not

obtained from observations across the whole six-month and

nine-month contract periods. To avoid overlapping estima-

tion periods, this study uses data from only the first three

months of the six-month and nine-month contracts. Thus, this

study implicitly assumes that the variance and covariance

are constant across the relevant contract period. This

assumption may not hold. These measurement errors could

contribute to the inconsistent results reported in Table V.

Variance Tests of Market Efficiency

The variance tests indicate that the futures market is

efficient. Tables VI and VII present the test results for

Eqs. [20] and [23], respectively. Table VI shows that the

mean values of Var(Sm - F ^ ) are greater than those of

Var(S - F ) for all t's in the first set of variance n t,n

75

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76

comparisons, where ni — 26 weeks and. n — 13 weeks. The case

of t = 1 exemplifies the findings where Var(Sm - and

VarfS - F ) are 7.67 and 5.31, respectively. The Pitman n t ,n

statistics indicate that all variance differences are

significant; hence the inability to reject the null hypothe-

sis for all t's. The table also shows similar findings from

the other two sets of variance comparisons, except for two

cases. In the second set where t = 1, Var(Sm - F

t^ m) equals

6.83 and is less than Var(S„ - F ). In the third set n l. f ii

where t = 1, the inequality is insignificant—the computed

Pitman statistic is only 0.25.

In general, the variance value is directly related to

length to maturity. The variances are largest where m = 39

weeks and are computed as 6.83, 7.77, 8.28, 7.14 and 8.18.

In contrast, where m = 13 weeks, the corresponding variances

are 5.31, 3.32, 1.37, 0.67 and 0.25. The variables, for

which the variances are computed, are forecast errors. The

error terms become more variable as maturity lengthens.

The results also indicate that the correlations between

the variables in the first set are smaller than those in the

second set. The forecast errors of longer-maturity con-

tracts are probably more closely related because information

differences between them is less than the difference between

nearby contracts. Error terms are least correlated in the

third set between contracts in the third set where m = 39

weeks and n = 13 weeks.

77

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78

The test results for Eq. [23] in Table VII reinforce

the findings from Table VI—the futures market efficiency

hypothesis (null hypothesis) cannot be rejected. The values

of Var(F,,• n - S ) exceed those of Var(F - Sfc), except t+ifn

f

for t = 0 in the six- and nine-month contracts. The case

of t = 3 i n the three-month contract provides an example,

where Var(Ft+i>n - Sfc) and Var(F^ n - Sfc) equal 1.66 and

0.19, respectively. The table also shows that the correla-

tion coefficients are also a direct function of length to

maturity on an inter-contract basis. The correlation

coefficients of 0.33, 0.72, 0.59 and 0.75 in the nine-month

contract are greater than those in the three- or six-month

contracts. In Eq. [20], the correlation is computed between

forecast errors in different contracts. In Eq. [23],

however, the correlation is computed between price differ-

ences at two different time points (four weeks apart) m the

same contract. The lack of information for contracts farther

from maturity may be a possible cause for the greater cor-

relation. Market participants are less able to distinctly

distinguish between the prices at two time points for longer-

maturity contracts. The difference between prices becomes

less obvious to them due to less information. The market

sees the prices at different time points becoming more

related for longer~roa.turity contracts.

79

Regression Tests of Market Efficiency

Before examining the regression results, residuals are

tested to determine the regression model's appropriateness

for the data. An effective way to examine a set of residuals

is through graphic analysis. Residuals are plotted against

the relevant independent variables for all contracts and t's.

All the residual plots (fifteen of them) show no presence of

any systematic trends. There is no evidence of error variance

nonhomogeneity or nonlinearity. To determine whether the

data are substantially non-normal, this study generates

residual normal probability plots for all contracts and t's.

The evidence from the skewness and kurtosis coefficients is

mixed, but the plots show no indication of serious departures

from normality.

The residual analysis indicates no serious violation of

regression assumptions—homocedasticity, normality, and

linearity. No data transformations are necessary. The

residual independence assumption is examined during the

review of the regression results•

The regression results (Table VIII) indicate that the

null hypothesis (b=l) cannot be rejected for the first four

t's associated with the three-month futures contract. The

t statistics (in parentheses) of -1,64, -0.7, -0.29 and

-0.21 are less than the critical value. However, the slope

coefficient for t = 12 weeks is not randomly distributed

30

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82

about one. The intercept terms are not significantly

different from zero, except for t = 12 where the t statistic

is 2.42. The R-squares are highly significant and increase

as the futures contracts approach maturity. The R-square

for t = 1 is 0.49, but it increases to 0.98 for t = 12. As

length to maturity decreases, more information is available.

It leads to better estimates of future spot prices, hence

greater R-squares. Table VIII also indicates no serious

autocorrelation problem with the three-month contract. The

significance level for the DW statistics is 5 percent.

The results for the six-month futures contract indicate

that the null hypothesis cannot be rejected for t = 4, 7 and

10 weeks. The relevant t statistics are -1.38, -1.3 and

-1.35. The intercept terms for the above t's are distributed

randomly about zero. The R-square values are smaller, but

they follow a similar pattern observed in the three-month

contract. Some autocorrelation exists for the one and four

week regressions where the DW statistics are both 1.3. It

is minor, however, and should not seriously affect the

regression estimates or the related statistical tests.

It appears that the null hypothesis cannot be rejected

for t = 1, 4 and 10 weeks in the nine-month futures con-

tract, with the relevant t statistics of -1.83, -2 and -1.84.

The intercept coefficients for the above t s are not

significantly different from zero. However, Table VIII shows

that substantial serial correlation exists for all t s in the

83

nine-month contract. Consequently, the parameter estimates

are not efficient, although still unbiased. More seriously,

the standard errors are underestimated. Thus, it understates

the ability to reject the null hypothesis. To adjust for

this problem, re-estimation is done using Yule-Walker (1)

equations. The adjusted estimates indicate the slope coef-

ficients for all t's are significantly different from one.

The relevant t statistics are -2.42, -2.81, -3.85, -3.2 and

-3.75. The null hypothesis is rejected at the 5 percent

level.

There are several possible reasons why serial correla-

tion exists in the nine-month contract. There could be

systematic measurement errors in the relevant variables.

This is not a strong argument in this case because the

market is generally efficient for the three- and six-month

contracts. The market should be inefficient throughout,

unless the errors are only in the nine-month contract data.

Another reason could be the omission of a significant

variable related to information costs. More information

and a longer learning process are necessary to obtain longer-

term forecasts. The variable becomes relevant for longer-

maturity contracts—that may explain the existence of auto

correlation for all t's in the nine-month contract and for

t = 1 and 4 in the six-month contract.

A third possible, but unlikely, explanation might be

default risk. This might also be relevant for longer-maturity

84

futures contracts. The underlying asset is default-free,

but the futures contract is not. Although organized

futures exchanges could promise settlement, future economic

conditions could dictate otherwise. Defaults on futures

contracts have happened. Prices of longer-maturity futures

contracts may contain default risk premiums, leading to

serial correlation in the residuals.

The R—square values in the nine—month contract do not

increase monotonically as observed in the three- and six-

month contracts. It implies that more information is not

readily available to the market in making better estimates

as length to maturity decreases from thirty-nine to twenty-

seven weeks. The longer-term forecasts may require more

information than the passage of time allows during that

period (thirty-nine to twenty-seven weeks before maturity).

Summary

This chapter presents three sets of results and their

accompanying interpretation. There is not much support for

the two CIR models. Although there is a little more support

for the second model, it is not known why the test results

from the first CIR model contradict those from the second

model--both are derived from the same underlying hypothesis.

The models' inability to explain intra-sample variations

among futures-forward price differences further shows the

lack of support for the CIR models.

85

The second and third sets of results pertain to the

efficient market hypothesis. The variance tests indicate

that the T-bill futures market is highly efficient. The

regression test results generally support the efficient

market hypothesis.

CHAPTER BIBLIOGRAPHY

1. SAS Institute, Inc., SAS/ETS User's Guide, 1982 ed., Cary, North Carolina, SAS Institute, Inc., 1982

86

CHAPTER VI

SUMMARY AND CONCLUSION

This study examines two aspects of the T-bill futures

market. First, it tests whether two Cox-Ingersoil-Ross

models might explain price differences between futures and

implied forward contracts. CIR hypothesize that stochastic

interest rates contribute to the futures-forward price

differences. The effect of interest rate stochasticity on

the futures—forward differential is a result of the daily

settlement process.

This study also investigates the efficiency of the

futures market using variance and regression tests. The

variance tests investigate whether futures prices are too

erratic for the market to be efficient. The regression

tests examine whether futures prices are unbiased predictors

of future spot prices.

Expressing the futures-forward price difference in

terms of futures and bond prices, the first CIR model

implies that if the local covariance between the futures

percentage price changes and bond percentage price changes

is positive, the futures-forward price difference should be

negative. If the covariance is negative, the model predicts

87

88

a positive futures-forward price difference. The test

results provide little support for the model.

The second CIR model expresses the same hypothesis in

terms of spot and bond prices. If the local bond variance

of the percentage price changes is larger than the local

covariance between the spot and bond percentage price

changes, the model predicts a positive futures-forward price

difference. If the variance is smaller, it predicts a nega-

tive price difference. Test results do not support the

model. The results cast doubt on the validity of both

models. The models could be correct, with the findings

being due to measurement errors or futures market ineffi

ciency. Futures market efficiency is the second issue in

this study.

Efficiency in the T—bill futures market is stated in

terms of a fair game. On average, the futures price should

equal the expected spot price; i.e., the forecast error is

orthogonal to any element of the information set. The

futures price should vary only with the random arrival of

new information. If the futures price movements are con-

sistently excessive relative to actual subsequent events,

the market is inefficient. The inefficiency may be

attributable to either futures prices not reflecting all

available information, or the market rationally and

persistently expecting events that do not materialize.

89

Krasker (1) refers to the latter as the "Peso problem.' The

market is efficient, but it may be concerned with an adverse

effect that does not happen. Although the probability of

the event may be low, the market rightfully reflects it

period after period. Even if the event has not yet happened,

it may and should, therefore, be impounded in prices.

The theoretical basis for variance testing of market

efficiency is that the forecast variance should be lower

than the realized variance. The tests require that the

relevant time series be stationary. Using two inequality

equations, this study obtains results that strongly indicate

the T-bill futures market is efficient for all three con-

tracts. Market participants take into consideration all

available information in generating their forecasts (futures

prices).

The regression test equation is derived from the same

basic model that generates the variance test equations. The

T-bill futures price should be an unbiased predictor of the

future spot price if the market is efficient. The regres-

sion tests posit a slope coefficient of one for an efficient

market. The test results indicate the efficient market

hypothesis cannot be rejected for the three- and six-month

contracts. They also show that the market for the nine-

month contract is inconsistent with market efficiency. The

existence of autocorrelation might be due to either measure-

ment errors, omission of a time—related variable, or a

90

default-risk factor. The regression tests could be sensi-

tive to one (or more) of these factors while the variance

tests are not. That would explain the difference in results

for the nine-month contract.

The R-square values increase as the three- and six-

month contracts approach maturity. Although the R—squares

in the nine—month contract do not increase monotonically,

they are less than those in the three— and six—month con-

tracts. As time passes, better information is available

through accumulation. The market obtains better estimates

with better-quality information. A lack of good quality

information does not imply inefficiency as long as prices

impound all available information existing then.

Implications for Further Research

This study's results suggest several avenues for fur-

ther research. This study tests the CIR models in the

T—bill futures market. French has done work in the copper

and silver futures markets. To date, the evidence is mixed.

Testing other futures markets would provide further evidence

of the CIR models' validity.

This study suggests work in futures markets that have

corresponding explicit forward markets. It eliminates the

necessity of generating implicit forward prices that may

not be representative.

91

Regression tests in this study examine whether futures

prices are effective and unbiased predictors of future spot

prices. A possible avenue for further research is examining

whether implied forward prices provide unbiased estimates of

future spot prices. A related area for further exploration

is determining which prices (futures versus implied forward)

are better spot price estimates.

Another possible area for further research is deter-

mining why some regression and variance test results differ.

Although both types of tests are derived from the same

underlying model, one could be more powerful than the other.

This study uses data in the first thirteen weeks of

any contract period to avoid overlapping data. It may be

desirable to expand the sample period so that adequate non-

overlapping data can be obtained. Since futures trading in

T-bills started in 1976, it could be several years before

enough non-overlapping data can be obtained for an adequate

sample size.

Conclusion

The results of this study support two conclusions;

(1) CXR do not adequately explain futures price structure,

and (2) the T-bill futures market does appear to be effi-

cient even though the prices do not behave as CIR would

expect.

CHAPTER BIBLIOGRAPHY

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92

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Unpublished Materials

Huang, Roger, "Efficiency and the Volatility of Flexible Exchange Rates," unpublished manuscript, University of Pennsylvania, 1980.

Brochures

Inside T-Bill Futures, Chicago, Chicago Merchantile Exchange, 1983

The International Monetary Market, Chicago, Chicago Mercantile Exchange, 1982.

Treasury Bill Futures, Chicago, Chicago Mercantile Exchange, 1977.