Empirical analyses of the term structure of interest rates

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Empirical analyses of the term structure of interest rates
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Thesis (Ph. D.)--University of Florida,1986.
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Includes bibliographical references (leaves 92-97).
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by Robert Edwin Brooks.
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EMPIRICAL ANALYSES OF THE TERM STRUCTURE OF INTEREST RATES


BY


ROBERT EDWIN BROOKS

























A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1986












ACKNOWLEDGMENTS


I would like to thank the members of my committee for

their support and encouragement: Haim Levy (ChairmanIr

Miles Livingston, and Stephen Cosslett.

For personal reasons, special thanks are also due to Ann

Harris and to my mother, Dr. Cal Brookst for her labor of

editing and proof reading.












TABLE OF CONTENTS



PAGE

ACKNOWLFDGMFNTS .* .* * * i

ABSTRACT * . * v


CHAPTER

I. INTRODUCTION TO THE STUDY . .* 1

Problems Addressed . * * 2
Previous Research * * 5

II. AN EMPIRICAL ANALYSIS OF TERM PREMIUMS . 11

Previous Research * * * 16
Mean-Variance and 4ean-Variance-Skewness *. 18
Data . . o 18
Risk Measures . . * * 18
M-V Decision Rules . a a 21
Conventional Performance Indices . 23
Efficient Frontiers o o * o 24
Applying Stochastic Dominance . o 27
Stochastic dominance Rules . 27
Empirical Results a a a 28
Stochastic Dominance with a Riskless Asset 31
An Illustration . . . 31
Empirical Results * a * 34
Sensitivity to Risk-Free Rate . *. 35
Longer Holding Periods . . . 37
Summary .o * *. * 39

III. AN FMPIRICAL ANALYSIS OF THE COUPON EFFECT 1ON
TERM PREMIUMS . . . 41

Previous Research 0 0 a a a a a* a 0 42
Theoretical Research o * * 42
Empirical Research o a a * o 44
Empirical Results * * 45
Data *. . . * 45
Mean-Variance Rule a * 46
Applying SO and S)R Rules . * 48
Summary o a . o * * 50


iii







IV. AN EMPIRICAL ANALYSIS OF GOVERNMENT DEALER
SERVICES . . o o . 51

Previous Research 0 a I a a .* 53
Stocks 0 a * * * 53
Bonds e .* *. . 56
The Dealer's Inventory Problem * a 57
Theoretical Considerations . e 58
Empirical Results . . . 61
Data * *0 *. *. * 61
An Example . . .o 62
Regression Results . . . a 63
Summary a a o o o o* 68

V. AN EMPIRICAL ANALYSIS OF TRANSACTION COSTS AND
INFORMATION * * * 69

Previous Research * * a * 71
The Model * *. 72
Empirical Results a * * a 74
Data * * *. 74
Repression Analysis a * 75
Summary .* a * * 80

VI. AN EMPIRICAL ANALYSIS OF TRANSACTION COSTS ON
TERM PREMIUMS 0. a. *. * 81

Previous Research .* * * *a 83
Mean-Variance and Mean-Variance-Skewness *. 84
Data .* * * * * 84
M-V Decision Rules . a 84
Average a * * * 34
Investor * * * 85
Dealer *. * a e 7
Applying Stochastic Dominance . 38
Summary * a* * * * 99

VII. CONCLUSIONS . a a a * 90

REFERENCES .* a. a a* 92

BIOGRAPHICAL SKETCH o. . a a * 98












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



EMPIRICAL ANALYSES OF THE TERM STRUCTURE OF INTEREST RATES


By


Robert Edwin Brooks


May 1986


Chairman: Haim Levy
Major Department: Finance, Insurance, and Real Estate



The purpose of this study is to enhance the current

understanding of the term structure of interest rates. The

term structure of interest rates is defined as the

relationship between various bond yields that differ only by

maturity.

Five issues are analyzed in this study. The first two

problems concern term premiums. The latter three problems

examine bid-asked spreads

The first problem is to examine term premiums in Treasury

bill holding period returns. For most periods examined no

second order stochastic dominance is found between monthly

returns for the first five month maturities. When a







riskless asset is assumed, however, the first two maturities

are dominated. Therefore, term premiums are not

economically meaningful if there is no riskless borrowing

and lending. The term premiums are economically meaningful

when a riskless asset is available.

The second problem is to compare term premiums in the

Treasury bill holding period returns to term premiums in

Treasury note holding period returns. Stochastic dominance

is employed to evaluate whether the difference in returns of

these two securities is economically meaningful. Because

bills tend to dominate notes, it is concluded that the

impact of coupon payments gives bills an economically

meaningful increase in before tax returns.

The third problem is to examine reported bid-asked

spreads in the Treasury bill market. For most periods

examined, maturity had significant explanatory power in

accounting for the variation in the bid-asked spread at a

point in time.

The fourth problem is to examine the impact of bid-asked

spreads in Treasury bills on assessments of future spot

interest rates and future term premiums. The intent is to

determine whether incorporating transaction costs alters

conclusions regarding the information in forward interest

rates about future spot interest rates and time varying

premiums. Because no significant change was found by

incorporating the bid-asked spread, it was concluded that

incorporating transaction costs does not alter the results.







An analysis of the final problem showed that term

premiums are reduced dramatically by incorporating

transaction costs.


vi i












CHAPTER I
INTRODUCTION TO3 THE STUDY


The term structure of interest rates is the relationship

between various bond yields that differ only by maturity.

Major term structure theories can be categorized as either

an expectations hypothesis or a preferred habitat

hypothesis. Expectation hypotheses are based on either the

notion that anticipations of future interest rates influence

the term structure or that a bonds holding period returns

are the same. Preferred habitat hypotheses are based on

maturity preferences; that is, market participants require

a premium to be "lured" from a specific maturity bond into a

different maturity bond.

In empirical research on term structure behavior,

attempts have been made to establish whether forward rates

are unbiased estimates of expected spot rates in the future.

(See, for example, Fama (1976a) and Malkiel (19661.1 In

other related research attempts have been made to determine

whether expected holding period returns for various maturity

bonds are the same. (Seet for example, Fama (1984bl)

McCallum (1975), and Roll (1970, 19711.1 While much of this

research has been useful, our understanding of term

structure behavior continues to be somewhat nebulous.






Through the research reported in the present study it is

hoped that the term structure's behavior will be better

understood.

The beneficiaries of a better understanding of term

structure behavior would include monetary policy makers in

their attempts to influence interest rates, financial

planners in their need to forecast future interest rates

and macroeconomists in the study of multiperiod consumption-

investment decision making. Also those concerned with the

term structure as used in bond pricing and the pricing of

most other financial claims would be helped.



Prable1sMAddressed

While perusing the empirical literature related to the

term structure, five problems for further examination were

identified. The first two problems focus on term premiums;

the latter three deal with transaction costs. First,

several techniques are applied to determine whether term

premiums which are the differences between long-term bills

holding period return and the current spot rate, are

ecomonically meaningful. Secondly, stochastic dominance is

used to assess the differences in historical returns of U.S.

Treasury bills and U.S. Treasury notes. Thirdly, an

examination is made of factors influencing transaction costs

as measured by the reported bid-asked spreads. Fourthly, an

analysis is performed of whether there is useful information






in bid-asked spreads for forecasting future term premiums

and future spot rates. Finally, term premiums are re-

examined in light of transaction costs.

The first problem focuses on research methodology. In

tests of the local expectations hypothesis, which states

that holding period returns should equal the current spot

rate, inference is usually based solely on comparison of

means. For example, Fama (198hb) found that the average

term premiums (holding period returns less the spot rate

are significantly positive using Hotelling's T square test

and using Bonferroni multiple comparison test. Fama (1984b,

p. 539), however* reports monotonically increasing standard

deviations with respect to maturity; that is, accompanying

the higher premiums is higher variation. A more convincing

test of this hypothesis would be to determine if investor

preferences can be established between return distributions.

Specifically, if the local expectations hypothesis is true,

then stochastic dominance between return distributions

should not be observed unless the investor class is not

misspecified. (The first derivative of utility is positive,

the second derivative is negative, and so forth.I

The second problem emphasizes security characteristics.

Applying stochastic dominance to return distributions of

U.S. Treasury bills and U.S. Treasury notes, an effort is

made to determine whether the ex-post holding period return

distributions are distinguishable for these two securities






when they have the same maturity. This analysis seeks to

establish whether these two securities are pure substitutes

despite the notes being coupon-bearing.

The third problem is to analyze market frictions. Market

frictions are impediments in the markets which hinder the

efficient trading of securities* Impediments would include

transaction costs, as well as taxes and institutional

restrictions. Theoretical models of the term structure tend

to ignore market frictions; nonetheless, market frictions

have been identified as influencing the term structure.

Malkiel identifies transaction costs as his first amendment

to his "perfect certainty analysis" of the term structure

(1966, p. 103). Malkiel asserts that the only transaction

costs paid by major investors in default-free bonds is the

dealer bid-asked spread (1966, p. 105). To determine what

factors influence the magnitude of the bid-asked spread is

the problem addressed in this study.

The fourth problem focuses on the ability to predict

using reported data about the term structure. Specifically,

the bid-asked spread is incorporated into forecasts of

future term premiums and future spot interest rates in an

effort to provide better predicting power. It is hoped that

by incorporating bid-asked spreads that a more refined

estimate will be used reducing measurement error. Fama's

(1984a) regression approach is used.







The final problem is to consider the impact of

transaction costs on term premiums. The techniques to be

employed in the resolution of the first problem are applied

here.





The analysis of whether term premiums are economically

meaningful is derived from one version of the expectations

hypothesis; namely, the assertion that the return over the

next holding period is the same for all maturities.

Culbertson (1957) was the first to examine whether holding

period returns were the same for different maturities. He

compared Treasury bills with long-tern Treasury bonds for

one week and three month holding periods. He concluded that

the expectations hypothesis was not an adequate explanation

of the term structure of interest rates.

Michaelsen (1965) demonstrated theoretically that, if you

assume risk aversion, then the anticipated holding period

returns for longer maturities should be higher. However* he

remained inconclusive as to the empirical evidence. Roll

11970, 1971) presented a portfolio approach to explaining

the term structure using the Capital Asset Pricing Model.

He found some evidence of a risk premium in returns,

implying an upward-sloping term structure on average.

McCallum (19751 extended this work using Canadian bonds.

McCallum found that both the standard deviation and beta






increase with maturity but the expected return only

increased up to three years and leveled off.

Cox, Ingersoll, and Ross (1981) provided rigorous

theoretical support for the version of the expectations

hypothesis asserting that expected holding period returns

for all maturities are equal. These results, however, hinge

on the ability to form a riskless portfolio similar to that

of the Arbitrage Pricing Model.

The work presented here builds on and extends the work of

Fama (1976a* 1994b). Fama used statistical techniques in an

effort to determine the significance of the excess of

longer-term bonds holding period returns over the present

spot rate or the term oremium. He finds statistically

significant term premiums.

The effort to determine whether the return distributions

of notes and bills are similar stems from the need to assess

the effect of coupon payments on holding period returns. In

1969, Pye examined the effect of tax-exempt coupons and

capital gains on bond yields and brought up several issues

related to this present study. It is well known that part

of the return on bonds selling below par is capital gain

which is taxable at a lower rate. Pye noted a difference in

yields between high and low coupon issues where low coupon

bonds had lower yields than high coupon bonds. He

attributed this to the advantage of capital gains of low-

coupon bonds.







McCulloch (1975awbl Robichek and Niebuhr (L9701,

Livingston (1979avb) as well as others have demonstrated

that taxes impact yield to maturity which in turn influences

holding period returns. Livingston demonstrates when

before-tax zero coupon rates are the same for all maturities

the yield curve for coupon-bearing bonds will rise with

maturity. However if the after-tax zero coupon rates are

constant, then the coupon-bearing yield curve for non-oar

bonds will take a wide variety of shapes. Based on this

theoretical work, there exists no a prior reason to expect

before-tax zero coupon Treasury bill returns to be higher or

lower than Treasury note returns.

Recently, Fama (1984b) examined returns on U.S. Treasury

bills and on U.S. Government bond portfolios. He concludes

that reliable inferences are limited to maturities up to one

year (bills) because of the high variability of longer-term

bond portfolio returns. Fama found that the highest average

return for bonds was always less than four years. Also the

highest average return on a bond portfolio never exceeded

the highest average return on bills. Once again, the

research herein presented builds on and extends the work of

Fama (1976a, 1984b).

The analysis of market frictions in an effort to

determine what factors influence the size of the bid-asked

spread is not new. Demsetz (1968) identified the bid-asked

spread for stocks to be the cost to investors for immediacy.




8

That is* investors would be willing to pay a dealer in order

to transact immediately rather than to bear the risk of a

price change by waiting for a seller or buyer to arrive.

In 1985, Stoll performed an economic analysis of the

stock exchange specialist system. He identified three major

cost categories, order processing costs, inventory holding

costs and adverse information costs. Inventory holding

costs refer to the cost associated with a dealer unbalancing

his own portfolio thus giving him additional risk. Adverse

information costs refer to the losses to the specialist from

trading with people with superior information. Traders with

superior information that justifies a different price than

the quoted bid-asked price can expect to make profits at the

specialist expense, implying larger bid-asked spreads.

Grant and Whaley (19781 showed that a bond's risk, as

measured by its duration, is an important determinant of the

bid-asked spread. Using the par value of bonds outstanding

as a proxy for transaction volume, they found a significant

relationship between volume and the bid-asked spread.

The previous research on prediction capabilities of the

term structure of interest rate is somewhat mixed.

According to the unbiased expectations hypothesis, the

implied forward interest rate should be an unbiased

predictor of the subsequently observed spot interest rate.

Several papers have tested this hypothesis by examining the

difference between the implied forward rate today and the







future observed spot rate. These prediction errors have

been found to be positive on average and increasing with

maturity leading investigators to the rejection of the

unbiased expectations hypothesis.

Another way of testing the unbiased expectations

hypothesis, referred to as error-learning, has been

presented by Meiselman (1962). The Meiselman approach

correlates prediction errors between the nearest forward

rate and the subsequently observed spot rate with revisions

in distant forward rates* The prediction errors have been

found to be highly correlated with revisions of distant

forward rates with these correlations decreasing as

distance into the future increases. The Meiselman results

have been regarded as strong support for the unbiased

expectations hypothesis. Thus, there have been several

tests of the unbiased expectations hypothesis with opposing

conclusions. Fama (1984a) uses the regression technique to

make assessments about the information contained in the term

structure. Specifically, he assesses the ability of the

difference between the forward rate and the current spot

rate to predict time varying term premiums and changes in

the spot rates. He found predictive power for the short-

term (two to three months) but not for the longer term.

In a study examining Treasury bill futures contracts

ability to guage interest rate expectations, Poole (1978)

notes that the relationship of transaction costs to





10
term premiums has never been carefully investigated. (p. 9)"

He provides some preliminary evidence that the shape of the

term structure may be due in part to transaction costs.












CHAPTER II
AN EMPIRICAL ANALYSTS OF TERM PREMIUMS


The purpose of this chapter is to examine term premiums

in U.S. Treasury bill holding period returns. Several

techniques are employed to determine whether the observed

term premiums are economically meaningful. For most periods

examined no second order stochastic dominance was found

between monthly returns for the first five month maturities.

When a riskless asset is assumed, however, the first two

maturities were dominated. Therefore, though there are

statistically significant term premiums they are not

economically meaningful if there is no riskless borrowing

and lending. The term premiums are economically meaningful*

however* in the sense that the longer maturities dominate

the shorter maturities when a riskless asset is available.

The results are similar for longer holding periods.

Term premiums not only exist but are statistically

significant at least for short term Treasury bills; this

fact has been well documented. (See, for example, Roll

11970, 19711, Fama (1976a b, 1984a, b) and Startz (1982)1.

More recently, Fama reported statistically significant term

premiums as measured by the excess of the holding period

return on a multi-month bill over the current spot rate (the







yield to maturity on bonds with one month to maturity).

(See Fama l1984bl, p. 535, especially Table 2.1

Accompanying these positive though non-monotonically

increasing premiums, however, are monotonically increasing

standard deviations# where monotonic is with respect to

maturity. (See Fama (1984b)t Table 4, po 539.) Thus, even

if term premiums exist, there economic meaning is

questionable.

In this chapter, the problem is to determine whether

these statistically significant average term premiums are

economically meaningful* Unlike Choi (19851 who sought to

explain the term premium using the Arbitrage Pricing Theoryt

this chapter is to determine whether the term premium is

sufficient to conclude that a certain class of investors

(for example, all risk averters) will prefer one maturity

over another one. Stochastic dominance as well as other

criteria are used to make these economic inferences.

To illustrate the issue, suppose that the investor's

holding period is one month. Since the investor can buy a

bond which matures in one month* the return is given by the

one-month spot rate. However, the investor can buy a bond

with n months left to maturity where n = 2, 3, .* 12

and sell it after one month when n-l months to maturity are

left (only bonds with maturities up to 12 months are

considered. In this case the investor's return is a random

variable since the yield curve may shift from one period to




13

another. The investor, therefore, faces the random variable

R(ltn) where the one indicates that the holding period is

one month and n denotes the maturity of the bond purchased.

The ex-post distribution of the returns R(len) can be

measured by observing the returns R(ln) for each month

during the period under consideration. F(lnl) denotes the

distribution of R(ln). There are 12 distributions to

compare and to chose from, where the number of observations

for each distribution, in principle, is equal to the number

of months covered in the study.

Previous empirical studies have tested mainly whether the

means of F(ltn) are statistically different from each other.

(See* for example, Fama 11984bl.) If E(RIlln)) > E(Rlltn-

tll (0 < t < 121 for all n >1 and for all relevant t, and if

this difference is statistically significant one can

conclude that significant liquidity premiums exist in the

bond market. Fama (1984b) found that the difference

EIRIl1n) RI)), (n = 2, 3, 121 is positive and

significantly different from zero. There is a term premium,

therefore, in comparison to the one month spot rate.

In this chapter, whether there is an economic difference

between the 12 distributions F(ltn In = 19 2, 12)

is tested rather than testing for statistical difference of

the means of the distributions. An economic difference is

said to exist if investors are better off by choosing one

investment strategy over the other. For example, if one





14

finds that for maturity n = 4, distribution F(1,4) dominates

(by a certain decision rule) all other investment

strategies, one may conclude that irrespective of the means

of these distributions, this distribution is the most

desirable since it maximizes the investor's expected

utility. Efficient investment strategies are sought that

is, to recommend which maturities of bonds constitute the

efficient set of investment strategies.

Obviously, the content of the efficient set (or the

optimal choice) is a function of the assumptions one is

willing to make. The following alternative sets of

assumptions are made.

1. Returns are normally distributed, and investors hold

only a portfolio of bonds with a maturity of n months

(n = I, 2, o 12).

2. Returns are normally distributed and the Capital

Asset Pricing Model (CAPMI holds; namely a large portfolio

of risky assets is held with the specific bonds under

consideration.

3. Returns are not normal and investors consider mean,

variance and skewness in their decision making process.

4. No assumption is made regarding the return

distribution. In this case first, second, and third de.iree

stochastic dominance rules (FSD, SSD, and TSD) are applied

where u'>0 u'>0 and u"<0O and u'>0, u"0t

respectively, are assumed.






5. In addition to 4. as stated above# investors are

allowed to borrow and lend at the one-month spot rate

ignoring transaction costs. In this case stochastic

dominance rules with a riskless asset are applied which are

known in the literature as FSOR, SSDR, and TSOR, where R

stands for the existence of the riskless asset.

While there is no one framework which is the "true

framework" and each one has its pros and cons, in general

the more assumptions that are made, the smaller is the

number of investments included in the efficient set. Rather

than arguing which framework is superior to the other, the

efficient sets derived under alternate models are

investigated.

The format of this chapter is as follows. A brief review

of the literature is given in the next section. Then

monthly holding period returns are examined by using

performance measures and decision criteria which are based

on the mean-variance or the mean-variance-skewness approach.

Next these returns are examined using stochastic dominance

and finally stochastic dominance is used assuming a riskless

asset exists. The analysis is then extended to the

multiperiod case.









The analysis of the economic meaning of term premiums is

derived from one version of the expectations hypothesis.

The two major emphases of this hypothesis are 1t the return

over the next holding period is the same of all maturities

and 2) the return from holding a long-term bond is equal to

the return on a series of investments in short-term bonds.

The primary focus here is on 1).

Culbertson (1957) was the first to examine whether

holding period returns were the same for different

maturities. He compared Treasury bills with long-term

Treasury bonds for one week and three month holding periods.

He concluded that the expectations hypothesis was not an

adequate explanation of the term structure of interest

rates. Culbertson put forth what is now known as the market

segmentation hypothesis which asserts that investors have

strong maturity preferences and so bonds of varying

maturities are not substitutable. A more refined version of

this was put forth by Modigliani and Sutch (1966) which is

called the preferred habitat hypothesis. In this version

monotonically increasing premiums are not necessary.

Hicks (1946), based on the notion of risk preferences of

investors presented the "liquidity preference hypothesis."

This hypothesis asserts that forward rates are

systematically higher than the expected spot rates and

increasing in magnitude as the time to maturity is larger.





17

Michaelsen (1965) demonstrated theoretically that if you

assume risk aversion, then the anticipated holding period

returns for longer maturities should be higher. However, he

remained inconclusive as to the empirical evidence.

Roll (1970, 19711 presented a portfolio approach to

explaining the term structure using the Capital Asset

Pricing Model. Roll concludes* "the data did indicate that

portfolio risk components of Treasury bills as measured

by (beta)l increased with term-to-maturity. This

implies an upward-sloping term structure on average" (1971,

p. 65). McCallum (19751 extended this work using Canadian

bonds. He considered a three month holding period and

looked at bonds with maturities from three months to 240

months. He calculated both the standard deviation and beta

which both rely on the normality assumption to be a valid

risk measure* It will be shown later that bond holding

period returns are positively skewed which violates the

normality assumption. McCallum found that both risk

measures increase with maturity but the expected return only

increased up to three years and leveled off.

Cox, Ingersoll, and Ross (19811 provided rigorous

theoretical support for the version of the expectations

hypothesis asserting that expected holding period returns

for all maturities are equal. These results, however, hinge

on the ability to form a riskless portfolio similar to that

of the Arbitrage Pricing Model.




18

This present chapter builds upon and extends the work of

Fama l(976a, 1984b). Fama used statistical techniques in an

effort to determine the significance of the excess of

longer-term bonds holding period returns over the present

spot rate or the term premium. He finds statistically

significant term premiums.



Mryniaciance and Me aQ:ridnce:Skeaas


DIta


The data are taken from the

Tape, specifically the FAMAFILE

monthly holding period returns

May 1985. Periods in which

deleted entirely. Included

observation periods with each

holding period returns for the

structure.


1985 CRSP Government Bond

which contains Treasury bill

from September 1964 through

there is missing data are

in the sample are 239

period including monthly

first 12 months on the term


Fama (1984b) presented evidence that both the mean and

the standard deviation of holding period returns increase

with maturity. This seems reasonable since risk and return

go up simultaneously. The real question in testing if the

term premium is economically meaningful is how to adjust for

risk; that is, what is the relevant risk index. There are a

few alternatives each of which has different underlying

assumptions:







a) Treynor's 11965) performance index


I(T) = (EIRInl) r) / Binl


where r is the risk-free rate and B(n) is the beta from the

Capital Asset Pricing Model (CAPM). This performance

measure assumes risk aversion and the other assumptions of

the CAPM; particularly, riskless borrowing and lending and

normal distributions*

bl Jensen's (1968) excess return index aln), derived

from the regression


(R(n,t) rItll = a(ln + B(n)(Rl(mt)-r(t)) + etnti


where Rlmtl) is the "market" portfolio holding period return

observable at t and e(lnt) is the residual error. This

performance index makes the same assumptions of a) above*

however, it does not yield necessarily the same ranking.

c) Sharpe's (1966) index


I(S) = (EIRInlI rI / sinl


assumes normality of returns and that the risk index is

s(n), the standard deviation of the return on R(n), rather

than B(n)l that isv the investor holds only one risky asset

in his portfolio clean be a portfolio of bonds with n months

to maturity). See Levy and Sarnat (19d4) for details.

dl Arditti's (1971) index is similar to Sharpe's index,

but in order to have dominance Arditti requires also that





20

the skewness of the superior investment will be larger (by

Sharpers index) than the inferior one. Obviously, Arditti

implicitly assumes non-normal distributions since he

considers skewness and not only the mean and variance.

These four performance frameworks share one commonality:

they allow borrowing and lending at a risk-free interest

rate.

Table 2.1 presents the mean, standard deviation and

skewness for each of the 12 investment strategies. Note

that indeed a term premium exists so the mean return tends

to increase with maturity, though this increase is not

monotonict reaching its peak at n=9. A liquidity premium is

observed in comparison to one-month bonds since EIRIn)) >

E(RIll) for n = 2, 3, 12.



TA3LF 2.1

Univariate Statistics


Months to
Maturity
(when purchased)
1
2
3
4
5
6
7
8
9
10
11
12


Monthly Returns 9-64

Standard
Mean Deviation
.00559 .0024
.00590 .0026
.00617 .0030
.00622 .0033
.e0633 .0037
.00637 .0041
.10633 .0044
.00652 .0049
.00657* .0056
.00628 .0062
.00636 .0065
.00 48 .0070*


through 5-85


Skewness
1.155
1.351
1.546
1.944
2.125
2.203*
1.923
1.862
2.026
1.872
1.855
1.796


+,Peak







The standard deviation, unlike the mean, increases

monotonically with maturity, reaching its peak at n = l2,

where s = .0070. The third column of Table 2.1 reveals that

the skewness is always positive. While it also has a

tendence to increase with maturity, it fluctuates, reaching

its peak at n = 6. When using the M-V rules or the CAPM

normal distributions of returns are assumed. The positive

skewness appearing in Table 2.1 indicates that some other

rules which do not rely on the normality assumption are

required. If the investment is normal and the positive

skewness is due to sampling errors some positive skewness

and some negative skewness would be expected' (six positive

and six negative). Nonetheless, before turning to the

distribution-free decision rules, M-V decision rules and

conventional performance indices which do rely on the

normality assumption are presented.





Assuming no riskless asset one can apply the well-known

M-V rules to the figures of Table 2.1 to establish the

efficient and the inefficient sets. The M-V rule in this

context is as follows: A return distribution Fin) dominates

or is preferred to another return distribution Flm), where n

is not equal to m In = 1 2V 0 12 and




* Under the normality assumption, the probability to obtain
12 positive skewness is extremely low 10.000244, one half
raised to the twelveth power).




22

m = 1, 2, .* 12), by the M-V rule if and only if the

expected return of Fin) is greater than or equal to the

expected return of F)ml agd the variance of F(n) is less

than or equal to the variance of F(m) with at least one

strong inequality.

Rased on this criteria, months 1 59 8, and 9 are in the

efficient set (undominated). Months 6, 7, 10, lt, and 12

are in the inefficient set (dominated). Hence, if investors

are risk averse and returns are assumed normal, all

investors will prefer to invest in one of the portfolios in

the efficient set (1 5# 8, 9) rather than one in the

inefficient set (6, 7, 10, 11, 12). Notice that the

dominated securities (month 6 for example) are dominated by

shorter maturity securities. Of particular interest is the

large positive skewness exhibited in Table 2.1.

Arditti (1971) suggested considering both Sharpe's index

and skewness simultaneously when comparing securities (see

Table 2.2 for Sharpe's index). His rule is that one return

dominates another if it has both a higher Sharpe's index and

higher skewness. Based on this criterion and looking at

Tables 2.1 and 2.2, it is found that months 5 and 6 comprise

the efficient set.







CGaQentinoal_Perfor!mance_IndLcs

In Table 2.2 the values for the three performance indices

are presented. The proxy for the market portfolio employed

in calculating 3 (beta) is the value-weighted market

portfolio, including dividend which consisted of all NYSE

and AMFX stocks. This index is taken from the 1985 CRSP

Stock Tape. (The equally weighted index yields similar

results.) A stock-bond index was not used because of the

difficulty in justifying whatever index chosen. Thus for

the purposes here the value weighted index is sufficient.



TABLE 2.2

Performance Measures


Monthly Returns 9-64 through 12-14
Months to
Maturity Indices
(when purchased) Jensen* Treynor Sharpe
1 .0 .0 .0
2 .297 -.198 .115
3 .556 1.234"* .192
4 .594 .269 .186
5 .720 .099 .202*S
6 .705 .068 .181
7 .643 .047 .157
8 .830 .048 .179
9 .857** .039 .165
10 .559 .021 .102
11 .618 .021 .105
12 .822 .023 .119

*Jensen's indices are multiplied by 10)0
" Peak



From Table 2.2 it is concluded that the preferred

maturity depends on the index selected (or the assumptions





24

one is willing to make)* Using Sharpe's index investing in

a five month Treasury bill is the preferred investment.

Using Treynor's index investing in a three month bill is

preferred and using Jensen's index the nine month bill is

preferred.



EffilienltEranliaE

Another method to evaluate the quality of various return

distributions would be to calculate efficient frontiers and

examine which securities are contained in the portfolio at

various points on the frontier. If all twelve securities

are included in the portfolio then although some are

dominated by M-V they are useful in the portfolio context.

This approach assumes that investors consider both expected

return and variance in investment decision making and

usually results in investing in several securities. Returns

are assumed to be normal and also all the other assumptions

of the CAPM. A numerical method known as quadratic

programming is used which attempts to minimize the portfolio

risk (as measured by variance or standard deviation) at each

given return level, but additional nonneqativity constraints

are imposed which require that no maturity can be held in

negative proportions. The procedure can be expressed as

follows: Minimize portfolio variance subject to a given

portfolio's mean, all proportions must add to onet and all

proportions must be greater than or equal to zero. See Levy

and Sarnat (19841, chapter nine for more details.







Table 2.3 shows these frontiers with and without the

stock index. Under these assumptions the efficient sets

contain 1t 3, 12, and the index and It 5, and 12 without the

index. The efficient frontier is not much improved over

individual investments because of the extreme positive

correlation between these returns. The inclusion of

maturity 12 is unique to this approach. Maturity 12,

therefore, may be valuable in a portfolio context.



TABLE 2.3

Efficient Frontiers


A.

Standard
Deviation
.0349
.0134
.0062
.0045
.0036
.0032
.0030
.0022



Standard
Deviation
.0062
.0051
.0035
.0035
.0028
.0026


With Index
Percent Invested
Securities
Index 1 3
93.1
36.5
10.1 14.7
6.8 52.5
3.2 89.6
2.5 8.8 88.7
2.3 23.5 74.2
1.8 98.2

Without Index
Percent Inves
Securities
1 5


14.5
37.7
61.0
87.8


12
6.9
63.5
75.1
40.7
7.3
2.5


ted

12


30.6 69.4
70.2 29.8
85.5
62.3
39.0
12.2


Table 2.4 summarizes the results obtained thus far. Out

of the 12 investment strategies (the feasible set), the M-V


Mean
0.0075
0.0070
0.0067
0.0065
0.0063
0.0062
0.0061
0.0056




Mean
0.0066
0.0065
0.0063
0.0061
0.0059
0.0057




26

efficient set includes portfolios I 5, 8, and 9. Sharper

Jensen and Treynor selections are taken from the M-V

efficient set while Arditti's efficient set consists of one

investment taken from the M-V efficient set and one taken

from the inefficient set. This is not surprising since

Arditti considers skewness as well as variance. Also the

efficient frontier uses only securities t1 3, and 12 when a

market index is assumed and securities t1 5, and 12 when a

market index is not assumed.



TABLE 2.4

The Investment Selection by Various Rules


Months to Maturity

Feasible Set 1 2 3 4 5 6 7 8 9 10 11 12
M-V Efficient Set + * *
Jensen's Index
Treynor's Index +
Sh3rpe's Index
Arditti's Index 4 4
Efficient Frontiers
With Index 4 *
Without Index A



Therefore, up to this point, it is clear that the

conclusions drawn about the economic value of term premiums

depend heavily on the equilibrium model assumed or the

decision criteria employed. For this reason* the stochastic

dominance technique is advocated because it requires minimal

assumptions and is flexible enough to be able to add

assumptions to strengthen the test.








Stochastic_ mlajnce_3uies

Stochastic dominance criteria were developed in 1969-70

when four papers were published by Hadar and Russell (1969)1

Hanoch and Levy (1969), Rothschild and Stiglitz (19701 and

Whitmore (1970). The benefits of stochastic dominance (SO)

rules are that they allow inferences to be made with only

partial information about investor preferences and no

assumptions regarding the nature of the distribution. SO

rules are based on three classes of utility functions U(il

where i = lv 2, 3, where u belongs to U(l1 if u* > 0; u

belongs to U(2 if u" > 0 and u" < 0; and u belongs to U(3)

if u' > 0, u" < 0 and u'" > 0. The decision rules are

called first, second and third degree stochastic dominance

(FSOD SSD, and TSO, respectively) based on the utility class

Utilt i = 1, 2, 3, respectively. The decision rules are as

follows: Let F and G be the cumulative distribution of two

distinct options (say F(n) and Fll), where F(n) is the

holding period return distribution generated by purchasing

an n maturity bill (n = 2, 3, 12) and selling it one

month later and Fi1) is the distribution of one month

returns). Then F dominates G (FDG) by FSD, SSO and TSDO if

and only if


FSD: F(xl < G(xl for all x (2.1)


SSO: SIG(t) F(tlldt > 0 for all x (2.21


SS(G(t) F(tl)dtdv > 0 for all x (2.3)


TSD:








where S represents the integral from minus infinity to x (a

dummy argument)l The inside integral of TSD runs from minus

infinity to v (another dummy argument). (Also, in all rules

there is a strict inequality for at least one value x.)

If investors in these securities invest in only one

maturity and belong to UM(i! i = 1, 2, or 3 and term

premiums are not economically meaningfulI then no FSD (for i

= 1), SSD (for i = 2) or TSD (for i = 31 is anticipated. If

economically meaningful (under the above conditions) term

premiums exist, then dominance is anticipated. Namely, if

dominance existsT not only the mean increases with maturity,

but also the whole distribution changes such that the

investor's expected utility increases. If monotonically

increasing term premiums are expected (based on the

liquidity preference hypothesis)l and they are economically

meaningful, then longer maturity bills will dominate the

shorter maturity bills* and in particular the spot rate as

has been found by Fama (1934b).



Elmairical.Results

Results are presented in Table 2.5 along with the results

for selected subperiods. Subperiods are selected so that

comparison can be made with Fama's (1984b) Table 2 analyzing

the statistical significance of term premiums. Table 2.5

presents the efficient sets under the three various




29

assumptions about utility U(i), i = 1, 2, 3. The numbers

which appear in the table are the maturities which are

undominated. For example, for the overall time period (8/64

- 5/85) and for i = 2 (SSDI, the efficient set contains

maturities I 5, 8, and 9. This indicates that these

return distributions are not dominated. Therefore,

maturities 6, 7, and 10 12 are dominated by at least one

maturity taken from the efficient set*



T43LE 2.5

Efficient Sets for Treasury Bill Returns


Uli), (i = to1, 2, 31
Time Number of 1 2 3
Period Observations FSO SSD TSO
8/64-5/85 239 1-12 1-5,8,9 1-5,8,9
8/64-12/82 210 1-12 1-5,8,9 1-5,8,9
8/64-12/72 101 1-12 1-3,5-9,12 1-3,5-9*12
1/73-12/i2 109 1-12 1-598,9 1-5,8.9
1/78-12/82 54 1-12 1-5 1-5
6/30-5/95 58 2-12 2-5*8,9 2-5,8,9



In all cases (except maturity one for subperiod 6/80 -

5/85), maturities are dominated by a shorter maturity. This

is exactly the 2Da ijte result expected if the liquidity

preference hypothesis were true and the premiums were

economically meaningful. The significant lower return of

distribution Fill found by Fama (1984b) is not economically

meaningful since longer maturity distributions have other

desired properties; hence it is not relegated to the

inefficient set.





30

Notice that the efficient set of SSO for the whole period

is identical to the mean-variance efficient set presented

above. This result is startling in light of the fact that

the distributions are positively skewed. This is also true

for each subperiod. These results are derived without

making the invalid assumption of normality. Finally, the

assumption that u'* > 0 (U(3)) is not binding for these

periods.

Kroll and Levy (19RO0 evaluated the possible effects of

sampling errors on SO and MV rules. They used simulation

and covered a wide range of correlations and return

distributions (normal, lognormal* and uniform). Their main

result was that FSO is highly effected by sampling error and

not SSD, TSDO and MV. In many cases they found that

sampling error probabilities actually increased with an

increase in the sample size for FSD. The sampling error

probabilities decreased rapidly, however, for an increase in

sample size for SSD, TSOD and MV. The last two time periods

have higher probabilities of sampling error than the others

for SSOD TSO, and MV. The consistency of these results

across time periods indicates that sampling error is not a

problem.

Bawa (1783) examined the theoretical impact of sampling

error using a Bayesian framework. He shows) that

in a non-parametric context with sample data being the only

information available, empirical distribution functions are





31

the appropriate distribution functions to be used in making

optimal choice among unknown distributions. (po 57)"

These findings indicate that, based on the assumptions

that investors are risk averse and select bonds based on

their historical distributions, economically meaningful term

premiums do not appear to exist. In the next section, it is

assumed that investors are allowed to mix the risky bonds

with the riskless asset.





Levy and Kroll (19763 applied the quantile approach to

develop stochastic dominance criteria with a riskless asset

(SDRI. With the additional assumption that investors can

borrow and lend at a risk free interest rate, these criteria

(SOR) afford dominance in distributions that would not be

established by the SD criteria.



AD-IilustratioQ

It is possible, therefore, that distribution F does not

dominate distribution G by FSO but that such a dominance

does exist by FSDR. With a riskless asset, a smaller

efficient set may be obtained. To illustrate this point

suppose that one faces two distributions F and G given as

follows:







F G

Return Probability Return Probability

1.5 1/4 0 1/2

2.5 3/4 3 1/2

The return on the risk free asset is assumed to be 1.5.

Obviously, by FS3 neither F nor G dominates the other since

the two distributions cross (see Figure 2.1). However,

adding the possibility to borrow or lend money reveals that

dominance prevails. With this set of parameters, F clearly

dominates G by FSOR (first degree stochastic dominance with

a riskless asset).



Cumulative
Probability I
1.0 -1 I-- -- 1----I-----
I I I I
3/4 -I G F -1 G-1 1- F'
I I I I I
1/2 -I------------------------I----I I
I I I
1/4 -I I--------I ----------I
F -1- F'
0.0 -I ----1----1----I----1----1----1----1---1-
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Figure 2.1: Illustration of FSOR without FSO



To see this, create a new distribution F' where under

this strategy $1.00 is borrowed and $2.00 (the borrowed

money plus the initial wealth) is invested in F. The return

on F* is







F'

Return Probability

2 x 1.5 1.5 = 1.5 1/4

2 x 2.5 1.5 = 3.5 3/4

note that F' dominates G by FSOD (see Figure 2.11. The

claims that this is an unfair comparison since F is mixed

with the riskless asset and G is not, is not true since for

any arbitrary mix of G with the riskless asset, one can find

another mix of F with the riskless asset which dominates it.

For example, suppose that $1.00 is borrowed and a new

distribution G" is created:



Return Probability

2 x 0 1.5 = -1.5 1/2

3 x 2 1.5 = 4.5 1/2

Clearly F' does not dominate G* by FSO since the two

distributions cross each other. It is possible to create a

new distribution F" which dominates G*. For exaMDle borrow

$2.03 and invest in Ft resulting in

F"

Return Probability

3 x 1.5 2 x 1.5 = 1.5 1/4

3 x 2.5 2 x 1.5 = 4.5 3/4

and hence F" dominates G'. To find one mix of F and the

riskless asset which dominates G is to guarantee that any

mix of G and the riskless asset will be dominated by some





34

mix of F and the riskless asset. To be more specific, Levy

and Kroll (1976) proved specifically that if there is one x

such that xF + (1 x)r dominates G, then for any mix yG +

(1 ylr there is a mix wF + 11 wir which dominates it.

Having this illustration, it is clear that the SDR efficient

sets are subsets of the SD efficient sets (in the weak

sense).





The same set of data and the SOR algorithm developed by

Levy and Kroll (1979) are employed. Obviously, in

comparison to the last section* the riskless asset is added.

It was assumed to be the average spot rate for the period

examined. The results are in Table 2.6



TABLE 2.6

Efficient Sets with Riskless Asset


Uli), (i=1,2,3)
Time Risk-free
Period N* Rate FSOR SSDR TSDQ
8/64-5/85 239 .0056 1-12 3-5,8,9 3-5,8,9
8/64-12/82 210 .0054 1-12 3-5,8,9 3-5,8,9
8/64-12/72 101 .0039 1-9,11,12 3,5-9,12 3,5-9,12
1/73-12/82 109 .0068 1-12 3-5,8,9 3-5,8,9
1/78-12/82 54 .0088 2-12 3-5 3-5
6/90-5/85 58 .0085 2-12 3-5,8,9 3-5,8,9

*N stands for the number of observations



FSDO rule is very ineffective. For almost all periods

studied the efficient set is relatively large, and in three

periods out of six it includes all twelve maturities.






A second conclusion which is similar to the previous

results is that TSOR yields the same efficient set as the

SSDR. There is no need to assume that u*" > 0O since this

assumption does not yield any reduction in the efficient

set. The main result, however, is that unlike SSD, SSOR

reveals that the term 2gjeium is 2oanmicallM g1 Dmtangful.

Fama (1984b) showed that a significant term premium exists

when one compares EIR(n)) to E(R(1)) (for most values*

n = Z2 3, 12). Using M-V rule or SSD rules (risk

aversion) shows that F(t) is included in the efficient set.

Though E(RIMl < EIRlnll) In = 2, 12) more

information on the distributions under consideration is

utilized and M-V and SSO show that the term premium is not

economically meaningful. Usinq SSOR, however, supports the

previous temporal conclusion of Fama, namely that the term

premium is also economically meaningful since the

distributions Fll) and FM2) are not included in the

efficient set. (See Table 2.5.) Any investors who are risk

averse will be better off selecting investments from the

efficient set which does not include F(I) and F(2).



SQ ityitv tQRisk=FreRate

One item to consider is the selection of the average spot

rate to be the risk-free rate. A rate is needed which would

be representative of the market at the time the ex-post data

is collected. To assess the problem of possibly selecting







an improper risk-free rate an analysis is performed to

assess how sensitive the conclusions are to the choice of

the risk-free rate. Table 2.7 presents the efficient sets

for the entire period under various risk-free rate

assumptions. The range of the spot rate for this period was

0.002494 for 0.2494%) to 0.013593 (or 1.3593%) on a monthly

basis. As shown in Table 2.7, however, the results change

within these extreme bounds.



TABLE 2.7

Efficient Sets with Different Risk-Free Rates


Risk-Free
Rate
0.00225
0.00300
0.00400
0.00420
0.00440
0.00460
0.00480
0.00500
0.00520
0.00540
0.00580
0.00600
0.00620
0.00640
0.00660


Monthly Returns 8-64 through 5-85
U(il, (i = t1 2, 31
1 2 3
FSDR SSDR TSDR
1-2 1-2 1-2
1-6,8,9 2-5,8,9 2-5,8,9
1-6,8,9 1-5,8,9 1-5,8,9
1-6,8,9,12 1-5,8,9 1-5,8,9
1-6,8,9,12 2-5,8,9 2-5,8,9
1-6,9,9,12 2-5,8,9 2-5,8,9
1-9,12 3-5,8,9 3-5,8,9
1-9,11,12 3-5,8,9 3-5,8,9
1-12 3-5,8,9 3-5,8,9
1-12 3-5,8,9 3-5,8,9
1-12 5,8,9 5,8,9
1-12 5,8,9 5,8,9
1-12 5,8,9 5,8,9
2-12 9 9
2-12 -- --


From Table 2.7 it is clear that the same general

conclusions would be drawn if the risk-free rate was between

0.00440 to 0.00640 (or 5.4. to 8.0% on an annual basis).

Thus, the results are fairly insensitive to the choice of

the risk-free rate.








One may argue that investors do not have a holding period

of one month. Are the conclusions put forth above the same

for longer holding periods? To answer this question the

same analysis was performed as the proceeding two sections.

However, a refined data set was used due to compounding of

measurement error. The data used in the proceeding sections

was created by Fama (198tb); he selected a twelve month bill

and used the same bill in the calculation of returns in

future periods. For example, eight months after the bill

was issued it would be used in the calculation of a four

month holding period return. Starting on August 289 1973,

however, fifty-two week bills were issued every four weeks,

so precise monthly returns are not available after this

date. Because bills are issued weekly for a twenty-six week

maturity, a bill maturing near month-end is more likely to

be available for maturities with less than six months.

From Table 2.8 it is clear that the conclusions remain

exactly the same for holding periods of less than nine

months. When a riskless asset is not assumed there is no

dominance by longer maturity bills. Thus, term premiums do

not appear to be economically meaningful. When a riskless

asset is assumed* however, there is dominance by longer

maturity bills. Thus, term premiums do appear to be

economically meaningful.







TABLE 2.8

Efficient Sets for Longer Maturities


Purchase
Maturity
(Months)
2
3
4
5
6
7
8
9
10

*Number


Time Period 8/64

N* FSD SSO TSD
246 2-11 2-6,8,9 2-6,8.9
244 3-11 3-9 3-9
242 4-11 4-9 4-9
240 5-11 5-9 5-9
238 6-11 6-10 6-10
236 7-11 7-11 8-11
234 8-11 8-11 8-11
232 9-11 9-11 9-11
230 10911 10,11 11

of observations


The market is efficient in the sense that for SSD there

exist at least one holding period such that each maturity is

in the efficient set. Maturities I 5, 8, and 9 are in the

efficient set for monthly returns. Maturities 6 and 7 are

in the efficient set for all longer holding periods. For a

two month holding period, month 7 is included because

maturity 8 is purchased and held for two months. Maturities

10 and 11 are in the efficient set for holding periods of 7

months or longer. The market is inefficient in the sense

that for SSDP maturities I and 2 are not in the efficient

set (except for very long holding periods of 9 and 10

months).


- 5/85

FS OR
2-11
3-11
4-11
5-11
6-11
7-11
8-11
9-11
10,11


SSOR
6,8,9
7,8,9
8,9
9
9,10
11
11
9,11
1011l


TSOR
6,8,9
7,8,9
8,9
9
9,10
11
II
II
11









Comparing the return of long-term Treasury bills to the

return of short-term bills, a statistically significant term

premium has previously been found. In particular, Fama

(1984b) found that a term premium is observed in comparison

to the one month returns that is, the spot rate.

In this chapter the question addressed is whether this

significant term premium is also economically meaningful or

is there a term premium when variance skewness and other

relevant parameters are taken into account. The results

indicated that the economic value of observed term premiums

in the U.S. Treasury bill monthly holding period returns

depends on the set of assumptions one is willing to make.

If one believes that investors make decisions based on

the relative characteristics of each maturity bill (that is,

the return's distribution) then one would conclude that

though the term premiums are statistically significant they

are not economically meaningful. On the contrary, there is

a tendency for short maturities to dominate long maturities.

Thus, from an economic point of view* when the whole

distribution is examined and not only the means a negative

economic premium prevails.

If one believes that investors make decisions based on

the portfolio distribution of each maturity bill and the

riskless asset (that is, investors either lever their

positions or invest in both the bill and the risk-free





40

asset), then one would conclude that term premiums are both

statistically significant and economically meaningful. In

this case the premium exists not only with respect to the

one month maturity (as Fama documented) but the results are

even stronger since the two-month bills are also an

inefficient strategy. The results are also the same for

longer holding periods.











CHAPTER III
AN EMPIRICAL ANALYSIS OF THE COUPON EFFECT ON TERM PREMIUMS


Whereas the focus of Chapter II was to examine whether

the U.S. Treasury bill term premiums are economically

meaningful, in Chapter III the focus will be the comparison

of term premiums in U.S. Treasury bill holding period

returns and term premiums in U.S. Treasury note holding

period returns. Stochastic dominance is employed to

evaluate whether the difference in returns of these two

securities, for similar maturities is economically

meaningful. Bills dominate notes for maturities three,

five, six and seven and no dominance is found for other

maturities using second order stochastic dominance. When a

riskless asset is assumed only maturities onet two and four

bills do not dominate notes. Therefore, it is concluded

that the impact of coupon payments on notes gives bills an

economically meaningful increase in before tax return.

The purpose of this chapter is to determine whether the

ex-post return distributions for U.S. Treasury bills and

notes differ in an economically meaningful way. That is, by

evaluating the difference between return distributions in

the bill and note markets for similar maturities, the impact

of coupon payments is examined. The intent of this study is







to determine whether reliable inferences from the term

structure are limited to maturities of one year or less,

because bills are not issued for longer maturities.

The null hypothesis is as follows: If bills and notes

for the same maturity are pure substitutes, then no

economically meaningful difference between bill's and note's

returns will be found. If a difference is found between

bills and notes then using longer-term securities which are

coupon-bearing may produce unreliable inferences. If a

difference is not found then the conclusion is less clear.

The coupon effect increases in magnitude for longer

maturities. The lack of distinction between bills and notes

may be due to the short-term maturities used. Bills,

however, are only issued for one year maturities or less.

Fortunately, a difference is found between bills and notes.







Although the primary focus of Pye's (1969) study was

examining the effect of tax-exenmot coupons and capital gains

on bond yields, several issues are related to this present

study. It is well known that part of the return on bonds

selling below par is capital gain which is taxable at a

lower rate. Pye observes, "Low-coupon treasuries have

recently sold as much as a full percentage point below

otherwise comparable issues" (1969, p. 5623. Thus, the







advantage of capital gains of low-coupon bonds produces

lower yields. Pye also points out that "(blonds selling

above par are treated asymmetrically by the tax authorities.

On taxablesf each year the amortized premium is deducted

from taxable income, and also from the cost price (for

determining gain or loss). The effect of these

provisions is to make bonds selling above par good

substitutes for bonds selling at par" (1969, p. 562).

McCulloch (1975ab), Robichek and Niebuhr (1970),

Livingston (1979avb) as well as others have demonstrated

that taxes impact yield to maturity. Thus, this tax effect

also influences holding period returns. Livingston examined

"(tihe relationship between the yield curves for zero coupon

bonds and coupon-bearing bonds (1979a, p. 189) in a

world with differential taxation of coupons and capital

gains. Livingston demonstrates when before-tax zero coupon

rates are the same for all maturities the yield curve for

coupon-bearing bonds will rise with maturity. However, if

the after-tax zero coupon rates are constant, then the

coupon-bearing yield curve for non-par bonds will take a

wide variety of shapes. Livingston also demonstrates that

"(iln the more general case where actual coupons vary with

maturity and tax rates differ by maturity, no

inferences at all (about the shape of the zero coupon yield

curved should be made" (1979a, p. 189). Thus, based on this

research, there exists no a prior reason to expect zero







coupon Treasury bill yields to be higher or lower than

Treasury note yields. In a related study, Livingston

demonstrates that the tax treatment for premium bonds

" creates) the possibility that the "coupon effect*

can have a different sign for discount and premium bonds of

the same maturity" (1979b p. 526). Thus the relationship

between yields of notes and bills is further confounded.



EmaLElicalASssarch

Recently, Fama (1984b) examined returns on U.S. Treasury

bills and on U.S. government bond portfolios. He concludes

that reliable inferences are limited to maturities up to one

year (bills) because of the high variability of longer-term

bond portfolio returns.

By aggregating holding period returns for various

maturity ranges, Fama found that the highest average return

was always a maturity range of less than four years. He

also notes the following:

During periods where the bond file overlaps with
the bill file, the highest average return on a
bond portfolio never exceeds the highest average
return on a bill. We cannot conclude, however,
that longer-term bonds have lower expected returns
than short-term instruments. Like McCulloch
(1975), but with the advantage of an exhaustive
data base, we find that the high variability of
longer-term bond returns preempts precise
conclusions about their expected returns. The
bond data are consistent with maturity structures
of expected returns that are flat, upward sloping
or downward sloping beyond a year. (p. 5301

Though the evidence lacks statistical
precision, it is interesting that there is no
five-year subperiod of the 1953-82 sample period







during which average bond returns increase
systematically with maturity. The shortest-
maturity bond returns increase systematically with
maturity. The shortest-maturity portfolio (< 6
months) produces the largest average return in two
of the five-year subperiods. Average returns
never peak in maturity intervals beyond four
years. At least on an ex post basis, the thirty-
year period 1953-82 was not propitious for long-
term bonds. (p. 5381

There is one weakness to Fama*s approach. He qqggregated

bond returns by maturity groups. For the purpose here, it

is desired to compare specific maturities. The aggregation

may be the source of the higher variation noted by Fama and

not the bonds themselves.



aMicical aesulIts



The data used came from the 1985 CRSP Monthly Government

3ond Tape. This tape contains price data for all Treasury

securities at the end of each month since 1939. This

constrains the holding period to be one month or some

multiple of months. Twelve month bills were issued from

August 1963 to August 1973 at the end of each month. Fifty-

two week bills have been issued every fourth week starting

August 289 1973. 3y adjusting for this four week procedure,

price data are available for every maturity up to 12 months

since August 1964. However, monthly issues for two-year

Treasury notes began in the mid-1970s and data are available

beginning January 1977. Month 12 is deleted entirely due to

the high number of missing observations. The data set







selected began January 1977 and ended May 1985t because this

is the only period in which both bill and note returns are

available.

Percentage price changes are calculated using continuous

compounding. Because some securities do not mature

precisely at month-end and months vary in length, an

adjustment is made. The return is converted to a daily rate

and then multiply by 30.4. This provides a consistent

estimate of a monthly rate. (See Famat 1984a and 1984b.)





Table 3.1 gives the preliminary statistics. The M-V rule

in this context is as follows: A return distribution of

bills, say B(n), dominates the return distribution of notes,

say N(n), where n = tl 2, l11 by the M-V rule if,

and only if, the expected return of Sin) is greater than or

equal to the expected return of Nin) and the variance of

B(n) is less than or equal to the variance of N(nl with at

least one strong inequality. Obviously, the same approach

is used in examining whether notes dominate bills.

Applying the M-V rules to the figures in Table 3.1

several conclusions can be drawn. First, making a

comparison by maturity, notes do not dominate bills. Bills

dominate notes for maturities 3, and 5 8. Therfore, bills

appear to be preferable to notes when considering a one

month holding period and maturities 3, 5, 6t 7, and 8 given







TABLE 3.1

Preliminary Statistics


1/77 through 5/85
BILLS 4N1TES
Mat. Mean* Var.* Skewness Mean* Var.o Skewness
1 .7514 .2301 .607 .8125 .3101 .816
2 .8226 .2761 .825 .8375 .3240 1.149
3 .8451 .3221 1.293 .8093 .3653 .702
4 .8348 .3745 1.652 .8461 .3943 1.438
5 .8692 .4454 1.807 .8347 .4669 1.646
6 .8605 .5085 1.811 .8029 .5316 1.149
7 .8552 .5660 1.545 .8374 .5817 1.402
8 .8603 .6284 1.338 .8599 .6425 1.576
9 .8704 .7257 1.370 .8608 .6900 1.432
10 .8719 .8127 1.356 .8382 .7523 1.208
11 .8493 .8529 1.279 .8372 .7984 1.235

*Multiplied by 100



the assumptions of M-V criteria are not violated. That is,

if investors are risk averse and returns are assumed normal*

all investors will prefer to invest in bills rather than

notes for purchase maturities of 3, 5, 6, 79 or 8 months.

Looking only at bills the efficient set contains

maturities 1 3, 5, 9, and 10 so the inefficient set

contains maturities 4, 6, 7, 8, and 11. For notes these

sets are very similar; the efficient set contains maturities

1 2, 4, 8, and 9 leaving maturities 3, 5, 6, 7T 10, and 11

in the inefficient set. Thus taken separately there does

not appear to be strong evidence in support of a

economically meaningful term premium. Once again, the

existence of positive skewness may invalidate the h-V

results. Stochastic dominance criteria are applied in an







effort to incorporate this skewness into the decision-

making.



AnalvniDaso a)_SQR_Sules

The benefits of SD and SOR rules are that they allow

inferences to be made with only partial information about

investor preferences and require no assumption regarding the

nature of the return distributions to be compared. See

Chapter II for a description of these rules as well as a

discussion about them.

Stochastic dominance rules are applied in an effort to

establish a preference between one month holding period

returns for bills and notes. That is, for each maturity (1

through 11) SO and SOR criteria are applied to the two

distributions--bill and note. The results are given in

Table 3.2 for both SO and SDR criteria. FSD has no

discriminating power. A preference is established for FSOR

for only maturity 3. SSD and TSD provide the same results.

However, in the eighth maturity bills do not dominate notes

as the M-V criteria indicated. Looking at SSDR, bills

dominate notes for longer maturities. Using TSiRW

discrimination is made in every period where in only

maturities 1, 2, and 4 do notes dominate bills.

Finally all maturities for both notes and bills taken

together are considered. It is clear in Table 3.3 that the

majority of securities in the efficient set are bills for







TABLE 3.2

Efficient Sets by


Maturity


Maturity
1
2
3
4
5
6
7
8
9
10
11


SSO and TSD.

efficient set.


FSD
BN
9 ,N
BN
BN
B,N
39N
9,N
BN
B,N
BN


1/77 through
SSO TSO
BIN BN
BN B.N
B B
BiN RN
B B
B B
B B
B,N BN
3BN r1,N
B,N 3,N
B,N B,N


5/85
FSDR
B,N
B,N
B
B,N
3,N
8N

B,N
3,N
B,N
B,N

B,N
B,N


SSDR
B,N
B,N
B
B,N
B
B
B
B
B
B
B


TSDR
N
N
B
N
B
B
B
B
B
B
B


Using SSDR and TSaR only bills remain in the





TABLE 3.3

Efficient Set of bills and notes


1/77 through 5/85
Maturity at Purchase
Order bills notes
FSD 1-11 1-23
SSO 1-3,5 4
TSD 1-3,5 4
FSDR 2-10 1-9,13-23
SSDR 3,5 --
TSR 3,5 --


Again, the sample size is large enough (101 observations)

that by the simulation done by Kroll and Levy (1983)

sampling error probabilities are small. However, exact

probabilities are not possible because they assume specific

distributions which do not resemble the distributions here.









Based on previous research a clear preference for bills

or notes was not anticipated. The impact of coupons and

differential tax treatment between bills and notes clouds

the relationship of their before tax return distributions.

Examining monthly holding period returns of bills and

notes for the period January 1977 through July 1935

indicated a preference for bills* Both mean-variance and

stochastic dominance criteria made this indication.

Therefore, notes cannot be considered a pure substitute for

bills. Caution should be used when making inferences for

longer holding periods.











CHAPTER IV
AN EMPIRICAL ANALYSIS OF GOVERNMENT DEALER SERVICES


In this chapter reported bid-asked spreads in the U.S.

Treasury bill market are examined. Maturity is shown to be

a measure of the instantaneous price risk borne by dealers

in U.S. Treasury bills. Cross-sectional regressions are

employed to determine what factors explain dealer bid-asked

spreads. For the periods examined, maturity explained a

large proportion of the variation in the bid-asked spread at

a point in time.

In a perfectly competitive market, security dealer bid-

asked spreads should be forced down to the inventory costs

of dealers. If all dealers have the same cost functions,

there will be no monopoly profits available.

In a world of complete certainty, the dealer bid-asked

spreads should reflect three costs: I1) fixed costs (12)

non-financing variable costs per transaction, and (3) the

cost of financing the dealer's inventory versus the returns

earned from holding the inventory. In a world with risk and

uncertainty, the risk of inventory price depreciation will

be a fourth cost reflected in the bid-asked spreads.

Finally, if some traders have access to superior

information* then the bid-asked spread should incorporate







the expected loss from trading with these people*' Prior

studies of security dealer bid-asked spreads have focused on

these five costs, as well as the extent of competition among

dealers. Many studies have examined the question of the

efficiency of the New York Stock Exchange specialist system

or other stock trading systems.

This chapter examines the impact of risk ithe fourth cost

mentioned above) upon the bid-asked spreads in the U.S.

Treasury bill market. It is shown theoretically that the

dealer's price risk for holding a particular Treasury bill

in inventory must be largely a function of the maturity.

Empirically, differences in bid-asked spreads among Treasury

bill's are shown to be explained by this risk measure. This

implies that the Treasury bill market is a highly liquid

market, where dealer spreads represent what Demsetz (1958)

called the price of "immediacy."

Treasury bills have several desirable characteristics for

studying bid-asked spreads. First, Treasury bills do not

have default risk. In contrast, dealers in common stock are

faced with the risk that the firm may default on some of its

obligations with a resulting unexpected, but precipitous

drop in the common stock price. Secondly, due to the

immediate availability of information in this market,

traders would not have access to superior information. In

addition, there is no corporate insider problem. Thirdly,


I See Glosten and Milgrom (1985).





53

Treasury bills do not pay coupons, making it much easier to

measure their risk than the risk of coupon-bearing bonds,

which have been the focus of previous studies of fixed

income bid-asked spreads.2 Fourthly, there are a large

number of dealers in the Treasury bill market strongly

suggesting that competition will drive monopoly profits

towards zero. This implies that at a particular point in

time bid-asked spreads for different bills should be

explained by risk differences. Fifthly, the par value

outstanding of each Treasury bill issue is quite large and

trades are range frequent# implying that bid-asked

quotations will be relatively accurate. Sixthly, an

extensive and reliable data base is available.







Demsetz (19681 identified the bid-asked spread for stocks

to be the cost to investors for immediacy. That ist

investors would be willing to pay a dealer in order to

consummate a transaction immediately rather than to bear the

risk of a price change by waiting for a seller or buyer to

arrive. Demsetz's primary focus was the influence of the

scale of trading las measured by the number of transactions

per day) and of the number of markets on which a security is

listed upon the bid-asked spread for New York Stock Exchange




2 See Grant and Whaley (19781 and Tanner and Kochin (19711.




54

stocks. Demsetz found a significant negative relationship

between the number of transactions per day and the (absolute

and relative$ bid-asked spread; he found a negative but non-

significant relationship between the number of markets and

the spread.

Tinic and West (1972) were the first to examine the

impact of risk on the pricing of dealer services. They

found that risk as measured by the relative of stock prices

over a year3 was not significantly related to the spread.

Obviously this is not a good measure of risk so no

conclusions are made.

Benston and Hagerman (1974) examined whether dealer

markets are natural monopolies. In a study of 314 Over The

Counter stocks, they presented some evidence in support of

the existence of economies of scale. They showed that

"trading scale (measured by the number of shareholders) is

negatively related to spreads (a doubling in the number of

shareholders is associated with a 16*.5. decrease in spread)"

(p. 363). They observe that this does not indicate that

dealers are natural monopolist. They reported that

"competition (measured by the number of dealers) is

associated with lower per share spreads (a doubling of the

number of dealers is associated with a 26*~9 decrease in

spreads)" (p. 363). This implies that dealers are in a

decreasing cost industry with economies external to the



' They measured range by (high price low price)/mean
price.





55

Individual dealer. Benston and Hagerman also found that the

spread is significantly related to unsystematic risk, but

not to systematic risk of the CAPM.

Stoll (l1978a, bl theoretically and empirically addressed

the question of the appropriate structure of the securities

market by considering bid-asked spreads for NASDAQ stocks.

He also attempted to identify what determines the

aDpropriate number of dealers willing to make a market in

NASDAQ stocks.

More recently, Stoll (1985) performed an economic

analysis of the stock exchange specialist system. He

identified three major cost categories: order processing

costs, inventory holding costs, and adverse information

costs. Order processing costs include fixed costs such as

space, computers and equipment, most labor costs, and the

specialist time. Also fixed costs for any transaction which

may vary with the number of transactions such as per trade

computer terminal charges, clearing fees, and variable labor

charges. The last order processing costs are those which

vary with transaction size such as greater clearing charges

and special attention required of the specialist. Inventory

holding costs refer to the cost associated with a dealer

unbalancing his own portfolio (or inventory) thus giving the

dealer additional risk. Adverse information costs refer to

the losses to the specialist from trading with people with

superior information. Traders with superior information can





56

expect to make profits at the specialist's expense. Thus*

specialist will charge larger fees, implying larger bid-

asked spreads.





Tanner and Kochin (1971) were the first to examine the

spread determinants for government bonds. They examined

empirically the determinants of spreads for Canadian

government coupon-bearing bonIs with maturities of two

months to 28 years. Tanner and Kochin found that the spread

had a positive relationship with maturity* a negative

relationship with coupon, and a negative relationship with

the quantity of bonds outstanding. They also found that

spread was not related to yield to maturity.

Grant and Whaley (19781 re-specified Tanner and Kochin's

(1971) work in an attempt to theoretically justify the

included variables. They show that a bond's risk, as

measured by its duration* (0), is an important determinant

of the bid-asked spread MS). Using the par value of bonds

outstanding (VI as a proxy for transaction volume, the

following model was adopted:


S = a + b(t)V + b(21D





* 4acaulay's duration is equal to the derivative of bond
price divided by price times (1 + yield). This represents
the instantaneous percentage change in bond price as
interest rates change, assuming a flat term structure of
interest rates.





57

They found bill to be significantly negative and b(21) to be

significantly positive.



Ihe_DZaler's_tI vent orE.oleu

In a world of complete certainty, the dealer has an

inventory problem. He must minimize his total cost of

operating as a dealer. The dealer has fixed costs, which

would include the following: rent on offices and equipment,

communications expenses, such as telephone and subscription

to an electronic quotation system, and salaries. The dealer

also has non-financing variable costs, which depend upon the

number of transactions; these would include costs of

transferring the ownership of securities, incremental

telephone costs, and incremental salaries. The third cost

of a dealer is the cost of financing his inventory. Part of

the inventory will be financed by equity. Since most

dealers are highly levered, a large part of the financing

cost will be debt cost in the form of short-term loans and

repurchase agreements. It is expected that the financing

costs are very heavily influenced by the level of money

market rates, such as the repurchase rate and the short-term

bank loan rate. As interest rates rise, dealer spreads

should rise to reflect the higher cost of financing the

dealer's inventory. (See Stigum 119831.3

In a world with risk or uncertainty, the dealer should

quote bigger bid-asked spreads for securities that have

greater risk of price depreciation.







TbeLetical Consideratiogs

The discussion to follow will examine the risk of

Treasury bills in terms of yield to maturity.5 The following

notation will be used:

P = the current price of a Treasury bill

R(jl = the yield to maturity on a Treasury bill with a

maturity of j periods. j will be less than one for Treasury

bills.

Yield to maturity and price are related as follows:


P = I / (I + Rijl)*j


where **j indicates that i1 + R(j)l


is raised to the jth


power.


To see the risk of a particular Treasury bill, take the

derivative of price with respect to the shortest maturity

interest rate, which is denoted by R(ll.


dP / JR(l) = (-j / (I + R(j)**(j+l)l (dR(j) / dR(l))


Then divide by price to arrive at the percentage

instantaneous price change.


dP / dR(ll / P = Ij / (1 + RIjlll (dRijl / dR(l))



s Treasury bills are usually quoted in terms of discount
rates or bond equivalent yields* This practice appears to
be a historical accident. Since the market for Treasury
bills developed before electronic calculators, discount
rates and bond equivalent yields were much easier to
calculate than yield to maturity, which requires finding
the jth root of the reciprocal of price. The practice has
remained even though yield to maturity calculations are
now quite convenient.










Thus, the instantaneous price sensitivity to interest

rates depends on the bill's maturity, its yield to maturity*

and the sensitivity of the yield on a j period bill to

changes in the shortest maturity yield, R(I).

The term (I + R(jI) will be a number close to one for

every j. This term will not vary much for different j.

Thust it is expected to have a relatively small impact

compared to the maturity, j. This is examined in detail

below.

The last term, dRlji/dR(l), represents the sensitivity of

a particular bill's yield to maturity to changes in the

yield to maturity on the shortest term bill. This

sensitivity can be estimated by the following regression.


R(j) = a(j) + bIj)R(l) + e(jl


where bijl is an estimate of dR(ji/dRil) and e(j) is the

residual error. Table 4.1 shows estimates of this

regression for two 60 month periods (1) August 1964 through

July 1969 and (2) July 1980 through June 1985. These two

particular periods were chosen because in the 1960s interest

rate changes were small while in the 1980s interest rates

were volatile. Sixty months was chosen because it allows

enough observations for a good estimate. The bijl slooe

estimates are close to one for both periods but especially

for the less volatile interest rates in the 1960s. This




60

suggests that the last term in the risk expression will have

a relatively small impact, even though for the more volatile

period b(jl is significantly different from one for some

maturities. Consequently the instantaneous risk of a

Treasury bill should be largely a function of its maturity.

This is the hypothesis tested in the next section.

Price risk also depends upon the amount of time that

inventory is held by the dealer. The average time between

trades should be a function of the amount outstanding of a

particular Treasury bill. This aspect of risk is also

examined in the next section.







TABLE 4.1

Estimation of dR(jI/dRIl)


R(jI = a + b(jiR(lt

8/64 7/69 7/80 6/85


a b(ji


2 .49

3 .36

4 .34

5 .97

6 .55

7 .63

8 .64

9 .49

t10 .74

11 .78

12 .61


*Parenthetical


0.9296
(.0371)
0.9918
(.038)
1.0086
(.041)
0.9733
(.049)
0.9991
(.054)
0.9917
(.057)
0.9935
(.0621
1.0390
(.0661
0.9739
(.061)
0.9690
(.063)
1.0242
(.0681

amounts


R-Square a


0.92

0.92

0.91

0.87

0.36

0.84

0.82

0.31

0.82

0.81

0.80


b(ji R-Square


0.96

0.94

0.92

0.91

0.89

0.88

0.87

0.86

0.85

0.85

0.69


.92 0.9706
(.0261*
1.34 0.9571
(.0321
1.63 0.9398
(.035)
2.14 0.9079**
(.038)
2.33 0.8971*
(.0411
2.70 0.86590*
(.0431
2.82 0.8606**
(.0431
3.08 0.8424<+
(.045)
3.24 0.8321+*
1.047)
3.44 0.8139*1
(.0461)
2.95 0.8469*S
(.0741


are standard errors of the estimate.


A*Significantly different from one at the 5% level.


EmaiCi3LRaleults



The data are taken from the Cross-sectional File on the

1985 CRSP Government Bond Ti3e. The study is limited

exclusively to Treasury bills. For each observation point

all Treasury bills recorded on the CRSP tape are employed.

Because bills are discount securities, their prices are

inversely related to maturity. Therefore, the relative


Maturity
(Months)







spread is used as the dependent variable. The relative

spread is defined as follows:


S = (Ptal P(bll / P(ave)


where P(al) P(b) and P(avel are the asked price, bid price

and the average of the bid and asked price, respectively.



AnD-Examnla

Table 4.2 presents univariate statistics for the 32

Treasury bills observed on June 28, 1985. The three

variables considered are the maturity, the par value issued,

and the market yield to maturity. Spreads rise with respect

to maturity. The par value outstanding is larger for

shorter maturities because the government issues 13 and 26

week bills every week and 52 week bills every four weeks.

Thus, the shorter maturity issues add to the supply of the

longer maturity bills already outstanding. Spreads tend to

increase with a higher yield to maturity when yield curves

are rising, but not when yield curves are declining. This

point is considered later.



TABLE 4.2

Univariate Statistics for June 28, 1985


Standard Correlation
Mean Deviation Range Vol. a3t. Yield
Spread (1) 0.019 0.012 .004-.048 -.53 .99 .034
Volume Imil.) 119914 4,783 6,500-22,250 -.58 -.51
Maturity (days) 128 90 7-349 .91
Yield M1) 7.3 .285 6.6-7.8






B8ressiQOnEsuiits

Table 4.3 presents the results of the regressions


S = a + b(V)V + e


S = a + b(MIM + e


and


S = a + bIMIMtl + e


where V is the par value outstanding, M is the bill's

maturity, Ml is the bill's maturity divided by (I + R(ji),

and e is the residual error. Observations are made annually

starting the end of August* 1964. Notice that for all three

regressions the slope has the anticipated sign except for

insignificant coefficients for volume in Auqust 1980 and

1984 and M and Ml in August 1930. The spread is negatively

related to par value outstanding, although for some periods

this is not a significant relationship (for example* August

1978.1 The R-square for maturity is more than double the R-

square for par value outstanding for eight observation

points. Thus, taken individually, maturity (or risk)

appears to offer the best explanation for the variations in

spreads at a point in time. Incorporating the divisor of

the above risk measure (1 + R(jll did not significantly

change the results.

Table 4.4 presents summary statistics for the regression


S = a + b(V)V + b(MIM + e







TABLE 4.3

Single Variable Regressions


Par Value
Date Outstanding
(Slope)
August -1.08E-07
1964 (.0001)*1
August -8.87E-08
1965 (.00011
August -4.26E-07
1966 (.0002)
August -2.56E-07
1967 (.0001)
August -2.97E-07
1968 (.0001)
August -3.51E-07
1969 (.0001)
August -2.56E-07
1970 (.0001)
August -1.36E-07
1971 (.00011
August -1.76E-07
1972 (.00011
August -1.72E-07
1973 (.0007)
August -2.17E-08
1974 (.0009)
August -1.16E-08
1975 (.0001)
August -9.19E-08
1976 (.0001)
August -4.93E-08
1977 (.0001)
August -2.15E-08
1978 (.11821
August -3.94E-08
1979 (.0011)
August 1.03E-08
1990 (.3993)
Augsut -8.10F-08
1981 (.0004)
August -1.94E-08
1982 (.0216)
August -2.02E-08
1983 (.0015)
August 6.37E-09
1984 (.6319)


Ro*2 Maturity R**2
(Slope)
.52 7.15E-07 .54
(.0001)*
.34 8.93E-07 .64
(.00011
.32 3.65E-06 .47
(.00011
.45 1.35E-06 .20
(.0056)
.41 2.52E-06 .36
(.00011
.34 3.74E-07 .48
(.0001)
.58 2.41F-06 .60
(.0001)
.42 2.00E-06 .52
(.00011
.62 2.12E-06 .56
(.0001)
.28 3.84E-06 .64
(.00011
.25 4.07E-06 .47
(.00011
.47 2.29E-06 .65
(.90001)
.33 2.15E-06 .87
(.0001)
.41 9.73F-07 .88
(.0001)
.06 5.95E-07 .27
(.0007)
.25 6.55P-;7 .28
(.00051
.02 -8.54E-08 .00
(.7583)
.35 3.52E-06 .73
(.0001)
.16 1.53E-06 1.00
(.0001)
.27 1.53E-06 1.00
(.0001)
.01 1.39E-06 .79
(.000011


Maturity
over
Yield R*2Z

7.43E-07 .54
(.0001 1
9.32E-07 .64
(.00011
3.88E-06 .47
(.00011
1.43E-06 .21
(.0054)
2.67E-06 .36
1.0001)
4.05E-06 .48
(.00011
2.58E-06 .61
(.0001)
2.11E-06 .52
(.0001)
2.24E-06 .56
(.0001)
4.19E-06 .64
(.0001)
4.50E-06 .47
(.0001)
2.47E-06 .65
(.0001)
2.27E-06 .87
(.0001)
1.04E-06 .83
(.00011
6.45E-07 .27
(.0007)
7.26E-07 .29
(.00051
-9.43E-08 .00
(.75961
4.15F-06 .73
(.0001)
1.71E-06 1.0
(.0001t
1.70E-06 1.0
( .0001)
1.56E-06 .78
(.0001)


*The probability of getting the observed t value given


Obs.

37

37

38

36

37


0

0


that the parameter


is really equal to zero.







At the 1% level, par value outstanding is not significantly

different from zero in most cases. With four exceptions,

maturity is significantly positively related to the spread.

Also the R-square does not improve dramatically for this

multivariate regression over the single variable regression

using maturity.

For twelve observation points, there exists significant

positive first order autocorrelation. Other functional

forms were examined with no improvement in the

autocorrelation. For example, including an interaction term

(M/V) and also taking the logs of the independent variables.

However, the strong significance of the maturity variable

indicates it's strong explanatory power.

Tanner and Kochin 119711 argue that the spread

would be expected to vary directly with yield to

maturity because higher yields at any point in time on

assets of given maturity probably reflect the intuitive

feeling of investors about greater risk. High

spreads make higher yields on the purchase price necessary

to induce the purchase of the bond by any individual having

a nonzero prior estimate of his own probability of selling

the bond before maturity" (p. 377). They found a positive

but non-siginificant relationship.

Grant and Whaley (1978) assert that the derivative of

duration with respect to yield to maturity is negative.

Thus, they claim that as yield increases the price risk






TA3LE 4.4

Multivariate Regression Results


Par Value
Date Outstanding
(Slope)
August -6.22E-08
1964 (.0044)1
August -1.70E-08
1965 (.4110)
August -1.60E-07
1966 (.1800)
August -2.81E-07
1967 (.0004)
August -2.00E-07
1968 (.01181
August -1.43E-07
1969 (.1341)
August -1.50E-07
1970 (.0007)
August -5.76E-09
1971 (.10491
August -1.17E-07
1972 (.0004)
August -I.18E-08
1973 (.7843)
August -5.12E-08
1974 (.4401)
August -5.89F-03
1975 (.0014)
Auaust 3.21F-09
1976 (.79171
August 5.41E-10
1977 (.92991
August 1.13E-08
1978 (.4728)
August -2.22E-08
1979 (.1044)
August .2102-09
1980 (.4241)
August -8.68F-09
1981 (.6191)
August 8.42E-10
198? (.0052)
August 6.20E-10
1983 (.0113)
August 2.49E-10
1984 (.9682)

*The probability of getting


Maturity R
(Slope)
4.48F-07
(.0019)*
8.12E-07
(.00011
2.93E-06
(.0012)
2.81E-07
(.5544)
1.38E-06
(.04371)
2.91E-06
(.0014)
1.51E-06
1(.0003)
1.46E-06
(.0030)
1.07E-06
(.00811
3.73E-06
(.0001)
3.63E-07
(.0003)
1.75E-06
(.0001)
2.18E-06
(.0001)
9.71E-07
(.00011
6.92E-07
(.0022)
4.43E-07
(.04161
7.09E-08
(.8350)
3.35E-06
(.0001)
1.54E-06
(.00011
1.54E-06
(.00011
1.38E-07
(.0001)

the observed


-Square

.64

.65

.50

.46

.47

.51

.72

.55

.70

.64


Durbin-
Watson

1.1**

2.2

1.5

1.o 2

2.0

2.0

1.7


1.7

1.3'S*


2.0


.74

.88



.28

.34

.02


.73

1.00

1.00


.79


t value


.55"+

.450+

.47

.8909


1.5

*57**


2.0


given that


the parameter is really equal to zero.
'*oFirst-order autocorrelation is significant at the 5% level.




67

decreases. They do not explicitly examine the relationship

between spread and yield to maturity.

It is now shown that yield does not provide a meaningful

explanation of the spread at a point in time. If the level

of rates for different maturities is positively related to

spreads, this relationship should hold for all yield curve

shapes. In fact, it does not as shown by the following

examples. Three points in time are examined; a declining

yield curve (January 30, 1981), a relatively flat yield

curve (June 30, 1981), and a rising yield curve (June 30t

19 ?). Table 4.5 presents the correlations for these three

periods. In all three periods maturity is significantly

correlated with the spread. However, for a flat yield

curved yield to maturity is not significantly correlated

with spread. Also, note that for a declining yield curved

yield to maturity is negatively correlated with spread. For

a rising yield curved yield to maturity is positively

correlated with spread. Contrary to the Tanner and Kochin

(1971) hypothesis, it is found that the relationship between

spread and yield to maturity depends on the shape of the

yield curve.6









6 Also notice in Table 4.5 that the size of the average
spread is much higher for a declining yield curve than for
other shapes. Thus, a declining yield curve may indicate
more uncertainty in the market leading to higher spreads.






TABLE 4.5

Correlation Matrices for Various Yield Curves


January 30, 1981 June 30, 1981
(declining) (Flat)
Vol. Mat. Yield Vol. Mat. Yield
S -.69 .94 -.76 -.66 .99 .16
V -.68 .70 -.68 -.45
D -.79 .16
Observations 34 32
Yield (%)
Mean 15.5 15.7
Std. Dev. .495 .176
Spread
Mean .00039 .00019
Std. Dev. .00026 .00013


June 30, 1982
(Rising)
Vol. Mat. Yield
-.60 .99 .77
-.64 -.69
.81
32

13.8
1.05

.00016
.00011


Summaf z

In this chapter the pricing of government dealer services

in the U.S. Treasury bill market was examined for the period

August 1964 through June 1985. It was found that maturity

is a primary determinant of the variation in reported bid-

asked spreads at a point in time. It was also demonstrated

that, for a point in time, the market yield to maturity is

not a significant determinant of the spread.












CHAPTER V
AN EMPIRICAL ANALYSIS OF TRANSACTION COSTS AND INFORMATION


This chapter examines the impact of bid-asked spreads in

U.S. Treasury bills on assessments of future spot interest

rates and future term premiums. Several different

regressions are employed to determine the information in the

observable term structure. Specifically, these regressions

examine whether incorporating transaction costs alters

conclusions regarding the information of forward interest

rates. For most periods examined, incorporating the

reported bid-asked spread does not change the results.

Most empirical examinations of the term structure of

interest rates have used the average of the bid-asked spread

as a price from which to calculate returns. ISee, for

example, Roll (1970, 1971), Malkiel (1966), Kessel (1965),

Fama (1976at b, 1934a, b), and Startz (1992).) This may

stem from the common assumption of term structure

theoretical models that there are no market frictions.

However, market frictions have been identified as

influencing the term structure.

Malkiel (1966, p. 103) identifies transaction costs as

his first amendment to his "perfect certainty analysis" of

the term structure. He claims that the only transaction




70

costs paid by major investors in default-free bonds is the

dealer bid-asked spread (Malkiel, 1966, p. 1051. The bid-

asked spread is the difference between the price that a

dealer is willing to buy versus sell a particular security.

A Joint Economic Committee study in 1960 identified 17

dealers in U.S. Treasury securities. (See Meltzer (1950),

p. 2.) By 1974 the number of dealers had grown to 24. (See

Garbade and Silber (1976)t p. 722.) The existence of

transaction costs implies two separate participants in these

securities. Because dealers take positions in securities

and do not pay transaction cost, their holding period return

will be higher than an investor who must buy at the asked

price and sell at the bid price.

Two different assumptions can be made concerning the

dealers true holding period return. Either it can be

assumed that the dealers can buy at the bid price and sell

at the asked price or they can buy and sell at the average

of the bid and asked prices. These two assumptions are

based upon who is actually buying and selling securities and

result in different rates of return.

It has been suqqested that many major participants in the

Government securities market pay only a proportion of the

bid-asked spread. (See Malkiel (1966), p. 115.1 Garbade

and Silber based on interviews with dealers, concluded that

reported bid-asked prices are real prices at which

transactions take place for modest-size public orders,






although not for large institutional trades" (1976, p.

730).

The hypothesis tested below is as follows. If either the

dealers or the investors drive expectations in the bill

market* then incorporating the bid-asked spread in return

calculations will improve the prediction power from the term

structure.





According to the unbiased expectations hypothesis, the

implied forward interest rate should be an unbiased

predictor of the subsequently observed spot interest rate.

Several papers have tested this hypothesis by examining the

difference between the implied forward rate today and the

future observed spot rate. These prediction errors have

been found to be positive on average and increasing with

maturity. (See, for example, Fama 11976al and Startz

(1982).) These results have lead investigators to reject

the unbiased expectations hypothesis. For example Startz

(1982) concludes, "A planner interested in future short

rates would be well advised not to take today's implied

forward rate as an estimator" (p. 237).

Another way of testing the unbiased expectations

hypothesis, referred to as error-learning* has been

presented by Meiselman (19621). The Meiselman approach

correlates prediction errors between the nearest forward





72

rate and the subsequently observed spot rate with revisions

in distant forward rates. The prediction errors have been

found to be highly correlated with revisions of distant

forward rates, with these correlations decreasing as

distance into the future increases. The Meiselman results

have been regarded as strong support for the unbiased

expectations hypothesis.

Thus, there have been several tests of the unbiased

expectations hypothesis, with opposing conclusions. Brooks

and Livingston (1986) demonstrate that although error-

learning is found, the evidence is inconsistent with the

unbiased expectations hypothesis. Therefore, they conclude

that expectations affect the term structure, but not in the

way suggested by the unbiased expectations hypothesis.

This present chapter extends the work of Fama (198l4).

As will be described below* Fama uses the regression

technique to make assessments about the information

contained in the term structure.



Ibhemdel

Recently, Fama (1934bl assessed the information in the

term structure based on the following two regressions.


PH(t+lm) = a + b(F(tm)-R(tl)) + e(t+1,a) 15.11


R(t+m-,19l-R(t+l,ll) = c+dlF(tml-R(ttl) +f(t+m-lll (5.2)





73

where PHit+lsm)=Hit+lvm)-Ritl) is the holding period return

premium, H(t+ltm is the holding period return observable at

t+1 for a m maturity bill, R(ttl) is the monthly spot rate,

and Fttim) is the forward rate observable at t for the mth

month, a and c are intercepts. b and d are slopes and e and

f are the residuals. Under the unbiased expectations

hypothesis, b=O and d=l. That is because the unbiased

expectations hypothesis implies


Fltom) R(tlt = EIR(t+m-lil Rlttll) 15.31


As Fama (1984b) argues* if b does not equal 0, then the

forward rate contains information about premiums observed at

t+l. If d does not equal 1t then the forward rate contains

information about subsequent spot rates. By examining

different participants in the market (dealer and investor)

possibly better predictability will be found.

In addition to the above reqressions, Fama (1984b)

examined the "fine-tuned" rej'ssions


H(t+lnm)-H(t+lm-1) = a+b(F(ttm)-F(tem-l)+e(t+ll) (5.4)


R(t+l) Rtt+m-1) = c + d(F(tm)-F(tem-l1 + f(te-m-1 15.5)


Equation 5.5 assesses the ability of forward rates to

predict successively more distant one-month future

spot rates" (Fama, 1984b, p. 519). Equation 5.4 fine tunes

the available information about the *. variations in

the term structure of expected bill returns" IFamat 1984bt

p. 519).





74

These regressions are applied to an improved data set as

well as extended to incorporate transaction costs. Two

perspectives are taken, the dealer's and the investor's.

For the dealer it is assumed that either the dealer receives

the additional return of the bid-asked spread or the dealer

does not receive this extra profit. For the investor it is

assumed that he/she must pay the bid-asked spread thus

receiving a lower return than average. Forward rates are

calculated based on the average of the bid-asked spread.



EMaitical_Sesults



The data available are on the CRSP Monthly Government

Bond Tape. This tape contains price data for all Treasury

securities at the end of each month since 1939. This

constrains the holding period to be one month or some

multiple of months. Twelve month bills were issued from

August 1963 to August 1973 at the end of each month. Fifty-

two week bills have been issued every fourth week starting

August 28, 1973. Iy adjusting for this four week procedure,

price data are available for every maturity up to 12 months

since August 1964.

Percentage price changes are calculated using continuous

compounding. Because some securities do not mature

precisely at month-end and because months vary in length* an

adjustment is made. The return is converted to a daily rate







and then multiplied by 30.4. This provides a consistent

estimate of a monthly rate. (See Fama, l934a and 1984b.)





Table 5.1 summarizes estimated regressions of the premium

on the forward-spot differential. These results are similar

to Fama's (1984b, p. 5171 except that the slope is higher

for maturities two through six. Fama did not examine

maturities greater than six months and the time periods are

not exactly the same. The higher slopes are probably due to

the examination of different time periods and not to the

different data set. (These regressions were run on

subperiods yielding almost identical results as Fama

(1984b).

The more important result is the lack of a significant

change in the R-squared when bid-asked spreads were

incorporated. Although the R-squared from the investor's

perspective is slightly larger for early maturities this

phenomena did not appear for all the subperiods. R-squares

from the dealer's perspective tended to be smaller although

this too did not hold for the subperiods.

Table 5.2 summarizes the estimated regressions of the

change in the spot rate on the forward-spot differential.

Again these results are similar to Fama (1984b, p. 5171.

The lack of a significant change in the R-squared when bid-

asked spreads were incorporated implies that neither dealers







TABLE 5.1

Premiums on Forward-Spot Differential


Monthly Returns 9-64 through 5-85
P(mvt+l) = a + b(F(mt)-R(t+l1) + e
Average Dealer Investor
Maturity b R-Square b R-Square b R-Square
2 .62 .24 .57 .20 .68 .32
(.07* 1.07)* 1.06)*
3 .66 .11 .64 .09 .69 .13
.121) (.13) (.11)
4 .92 .08 .88 .07 .93 .10
(1.19) (.20) (.18)
5 .97 .10 .92 .09 .99 .13
(.181 (.19) (.16)
6 1.09 .09 1.00 .08 1.11 .12
(.221) (.21) (.19)
7 .77 .04 .81 .05 .78 .07
(.24) (.24) (.18)
8 .61 .02 .54 .02 .61 .02
(.301 (.23) (.29)
9 1.84 .13 1.87 .14 1.53 .11
(.311 1.30) (.27)
10 1.59 .09 1.82 .12 1.23 .07
(.32) (.31) (.29)
11 .55 .01 1.02 .03 .23 .00
(.36) (.361 (.281
12 .85 .02 .91 .02 .78 .02
(.43) (.42) (.41)

*Parenthetical amounts are standard errors.


nor investors drive expectations in the market. Like Fama

(1994a), it is found that the slope is not equal to zero.

This is not supportive of the unbiased expectations

hypothesis. That is, the forward rate contains information

about term premiums observable at t+l.

Tables 5.3 and 5.4 exhibit the results of the "fine

tuned" regressions. Here again the results are similar to

Fama (l984b). Also the R-squares do not significantly







TABLE 5.Z

Future Spot Change on Forward-Spot Differential


Monthly Returns 9-64 through 5-85


R(t+m)-Rttil) =
Average


Maturity


b R-Square


.12


.02

.02

.02

.00

.01

.02

.00

.02

.08

.04


amounts


.41
(.071)*


3 .20
(.10)
4 .31
(.131
5 .22
(.10)
6 .09
(.11)
7 .15
(.10)
8 .26
(.121
9 .10
(.11)
10 .20
(.10)
11 .51
(.111
12 .33
(.12)

"Parenthetical


a + b(F(mt)-R(t+I)l + e


Dealer


b R-Square
.36 .10
(.07)*
.16 .01
(.10)
.23 .01
(.131
.17 .01
(.11
.03 .00
(.10)
.04 .00
(.101
.07 .00
(.09)
.03 .00
(.11
.01 .00
(.101
.17 .01
(.12)
.20 .02
l(.11)

are standard errors.


improve when the bid-asked spread is incorporated. In fact,

the poor R-squares indicate that past maturity two there is

no prediction power in future spot rates.

Table 5.3 is not supportive of the unbiased expectations

hypothesis. There appears to be information today about

changes in time varying premiums in the future. This should

not be the case if the unbiased expectations hypothesis were

true.


Investor
b R-Square
.42 .13
(.071*
.22 .02
(.091
.34 .03
(.111
.23 .03
1.091
.12 .01
(.09)
*15 .02
(.08)
.36 .04
(.11)
.19 .02
(.10)
.31 .05
(.091
.51 .13
(.08)
.39 .06
(.11l







TABLE 5.3

Fine Tuned Premium Regression


Monthly Returns 9-64 through 5-35


Hlt+lm) Hlt4-lm-l
Average
Maturity b R-Squar
2 .62 .24
(.07)*
3 .53 .14
(1.08)
4 -.37 .06
(.10)
5 .60 .26
(.071)
6 .40 .13
(.06)
7 .50 .24
(.06)
8 .37 .17
(.05)
9 .60 .26
(.06)
10 .61 .25
(.07)
11 .49 .36
(.04)
12 .23 .02
(.12)

*Parenthetical amounts


= a + b(Fltrm)-Fltm-1l) + e


Dealer
e b R-Square
.57 .20
(.071*
.50 .12
(.09)
.33 .04
(.10)
.52 .19
1.071
.29 .07
(.071
.39 .13
(.06)
.10 .01
(.06)
.40 .13
(.06)
.68 .26
(.07)
.62 .39
(.051
.16 .01
(.15)

are standard errors.


Investor
b R-Square
.67 .32
(.061*
.57 .18
(.081
.42 .07
(.10)
.70 .33
(.061
.53 .23
(.061
.67 .46
(.051
.64 .40
(.051
.73 .37
(.06)
.59 .29
(.06)
.30 .13
(.051
.28 .04
(.10)


Table 5.4 is not supportive of the unbiased


expectations hypothesis.


The expected value of the slop-


was one if the unbiased expectations hypothesis is true.

With the exception of month two, the slope is not

significantly different from zero. For longer Deriods into

the future, the forward-spot differential appears to be

biased in its prediction of future changes in spot rates.







TABLE 5.4

Fine Tuned Spot Reqressions


Monthly Returns 9-64 through 5-85
R(t+m) RIt+m-1) = a + b(F(tm)i-F(tm-1)l + e


Maturity


b R-Square


.41
(.071)*
-.10

(.09)

.09
1.071
.05
(.06)
-.10
(.05)
.07
(.051
.10
(.05)
.09
(.05)
.12
(.04)
.08
1.05)


.12

.01

.01

.01

.00

.02

.01

.02

.02

.03

.01


b R-Square


.36
(.071*
-.12
(.09)
-.07
(.09)
.09
(.071
.03
(.06)
-.100
(.05)
.08
(.04)
.07
(.041
.05
(.051
.09
(.04)
.06
(.051


.10

.01

.02

.01

.00

.01

.01

.01

.01

.02

.01


b R-Square


.42
(.07)
-.08
(.09)
-.14
1.10)
.07
1.071
.06
(.06)
-.08
(.04)
.07
(.051
.11
(.06)
.1I1
(.05)
.12
(.041
.07
(1.04)


.13

.00

.01

.00

.00

.01

.01

.01

.02

.04

.01


*Parenthetical


amounts are standard errors.


Fama (1984a) showed that the autocorrelation in returns

is not a problem in these regressions. The construction of

equations 5.1, 5.2, 5.4, and 5.5 are first differences.

That is, they are the differences in returns as opposed to

using the actual returns. He reported significant first

order autocorrelation when actual returns are used. He also

reported that there is not significant first order

autocorrelation using differences in returns. Therefore,

the problem of autocorrelation was not addressed.






Summany

In this chapter the impact of bid-asked spreads in U.S.

Treasury bills on assessments of future spot interest rates

and future term premiums was examined* Regression analysis

was used to make this assessment. For most periods

examined, incorporating the reported bid-asked spread does

not change the results.

Based on this analysis expectations of the dealers do not

dominate the expectations of the investor nor do the

expectations of the investor dominate the expectations of

the dealers. Therefore transaction costs do not

significantly impact inferences made by examining the

information in forward rates about future spot interest

rates and time varying premiums.












CHAPTER VI
AN EMPIRICAL ANALYSIS OF TRANSACTION COSTS 04 TERM PREMIUMS


The impact of transaction costs on term premiums cannot

be ignored. Consequently, this chapter examines stochastic

dominance results incorporating bid-asked spreads. The

evidence suggests that term premiums are economically

meaningful from a dealers perspective but not from an

investors perspective.

Most empirical examinations of the term structure of

interest rates have used the average of the bid-asked spread

as a price from which to calculate returns. See, for

example, Roll (1970, 1971), Malkiel (19661, Kessel (19651,

Fama (1976a, b, 19a4a, bi, and Startz (1982). Fama (1976a,

1984b) and others have documented statistically significant

term premiums. Having examined the economic meaning of

these term premiums in Chapter II, the purpose here is to

analyze the differences in the conclusions drawn when bid-

asked spreads are considered. That difference can be

analyzed by considering two views of the bid-asked spread.

The dealer receives the benefit of the spread* whereas the

investor bears the cost. Three perspectives will be

analyzed: the dealer, the investor, and the average which

is used for comparitive purposes.




82

To illustrate this analysis, suppose that both dealers

and investors have a one month holding period. Both the

dealer and the investor face a choice between purchasing a

bill which matures in one month or purchasing a bill with n

months left to maturity where n = 2, 3, *. 12 and sell

it after one month where n-I months to maturity are left.

Because of transaction costs the return distributions for

the investor and the dealer are different. Therefore, it is

possible for dealers to prefer month 5 to month It but for

investors such a preference cannot be established.

In this chapter* whether there is an economic difference

between the 12 distributions is tested rather than using

statistical techniques. An economic difference exists if

investors for dealers) are better off by choosing one

investment strategy over another. Efficient investment

strategies are sought, that is, to recommend which

maturities of bills constitute the efficient set of

investment strategies. Clearly, the content of the

efficient set is a function of the assumptions one is

willing to make. Mean-Variance# Arditti's index, anj

stochastic dominance are applied in an effort to establish

the efficient set.









In a study examining Treasury bill futures contracts

ability to guaqe interest rate expectations, Poole (1973)

notes that *. the relationship of transaction costs to

term premiums has never been carefully

investigated" (p. 92). Poole (1978) claims the following:

The sharp drop in Roll's (19701 estimated marginal
term premium--the average difference between the
one-week implicit forward rate m weeks in the
future and the one--week spot rate realized in m
weeks--between the 13--and 14--week maturities
appears to be suspiciously related to the sharp
increase in the mean spread between the same two
maturities, (p. 93)

It is interesting to note that Roll (1970)
found the hypothesis of market efficiency well-
supported except for maturities of 4 to 8 weeks.
For these maturities yields seem to be too low, on
the average. We may conjecture, however, that the
apparent anomaly would disappear with a fuller
accounting of transactions costs. (p. 95)

The impact of transaction cost may dramatically change

the results of all previous studies which use the bid-asked

average rather than incorporate this transaction cost. A

partial list of these studies include Roll (1970, 1971),

Malkiel (1966), Kessel (1965), Fama (1976a, b, 1984a, b),


and Startz (1982).










The data used came from the 1985 CRSP Monthly Government

Bond Tape. This tape contains price data for all Treasury

securities at the end of each month since 1939. This

constrains the holding period to be one month or some

multiple of months.

Twelve month bills were issued from August 1963 to August

1973 at the end of each month. Fifty-two week bills have

been issued every fourth week starting August 28, 1973. By

adjusting for this four week procedure, price data are

available for every maturity up to 12 months since August

1964.

This data set is slightly different than that used in

Chapter II because the FAMAFILE was constructed using the

same bill from month 12 until it matured. The bills used

here are those closest to month-end at month 6 as well as

month 12 which yields more precise estimates of monthly

holding period returns, reducing the amount of sampling

error.






Aymardi. As described in Chapter II, the M-V rules are

applied to establish the efficient and inefficient sets

using the average bid-asked spread. (See Table 6.1 for the

means and variances.) Based on these criteria, months







I 3, 5, 6, 8, 9, 12 are in the efficient set and thus

months 4, 7T 109 11 are in the inefficient set. These

results are similar to those presented in Table 2.1. As

described above, the only difference between these two

tables is in the way monthly returns are estimated.



TABLE 6.1

Average Return Distribution


Monthly Returns 9/64 through 6/85
Standard Sharpe's
Maturity Mean Deviation Skewness Index
1 .5576* .2331* 1.156 0
2 .60?9 .2706 1.343 .1674
3 .6248 .2989 1.655 .2248
4 .6232 .3280 1.996 .2000
5 .6457 .3758 2.219 .2344*
6 .6467 .4176 2.258** .2134
7 .6452 .4539 2.022 .1930
8 .6529 .4937 1.857 .1930
9 .6591** .5616 1.868 .1807
10 .6322 .6297 1.844 .1185
11 .6357 .6521 1.756 .1198
12 .6755 .7489*C 1.987 .1574

-Percentage return on a monthly basis
**Peak



As in Table 2.2, by Sharpe's index, maturity five is the

preferred investment. Also Arditti's measure implies an

efficient set of maturities five and six.


IESytor.- By assuming the bills are purchased at the

asked price and sold at the bid price, Table 6.2 presents

the return distribution from the investors viewpoint.

Notice that the average term oremium is much smaller. For







example, for month 9 the term premium for the average

returns is .1015 (.6591 .5576), whereas for the investor

the premium is .0517 (.5951 .5434). In this case

transaction costs account for approximately 50% of the term

premium observed in average returns!



TABLE 6.2

Investors Return Distribution


Monthly Returns 9/64 through 6/85
Standard Sharpe's
Maturity Mean Deviation Skewness Index
1 .5434* .2343* 1.189 0
2 .5730 .2726 1.323 .1036
3 .5953 .3022 1.660 .1717**
4 .5879 .3333 1.970 .1335
5 .5975** .3826 2.155 .1414
6 .5936 .4261 2.209** .1178
7 .5830 .4645 1.964 .0853
3 .5877 .4995 1.830 .0887
9 .5951 .5642 1.838 .0916
10 .5570 .6263 1.736 .0217
11 .5515 .6526 1.722 .0124
12 .6003 .7433 3 1.941 .0766

*Percentage return on a monthly basis
*+P-ak


Because of this marked decrease in mean returns Sharpe's

index peaks at month three rather than month five as before.

Arditti's measure yields an efficient set of months 3, 59

and 6. By the M-V rules, months I 39 59 12 are in the

efficient set. Notice that by incorporating transaction

costs the efficient set is reduced by eliminating months 69

8# and 9.





87

Dealer. By assuming the bills are purchased at the bid

price and sold at the asked price, Table 6.3 presents the

return distribution from the dealers viewpoint. Here the

average term premium is much larger. The term premium for

month 9, in this case, is .1512 1.7230 .57183, which is

almost three times the term premium from the investors

perspective!



TABLE 6.3

Dealer Return Distribution


Monthly Return 9/64 through 6/85
Standard Sharpe's
Maturity Mean Deviation Skewness Index
1 *57140 .2325* 1.119 0
2 .6328 .2703 1.353 .2257
3 .6542 .2969 1.632 .2775
4 .6586 .3242 1.990 .2677
5 .6939 .3718 2.233 .3294+*
6 .6997 .4127 2.243+* .3099
7 .7073 .4508 2.009 .3006
8 .7181 *4943 1.785 .2q60
9 .7230 .5616 1.871 .2692
10 .7074 .6371 1.963 .2128
11 .7199 .6563 1.781 .2257
12 .7507* .*7570** 2.035 .2363

*Percentage return on a monthly basis
**Peak


Once again, Sharpe's index peaks at five months and

Arditti's measure implies an efficient set of months five

and six. The M-V efficient set is much larger consisting of

months I 9, and 12. Thus, the dealer market appears much

more efficient in the sense that only months 10 and 11 are




88

dominated. In the next section, stochastic dominance rules

are employed because these distibutions are clearly non-

normal.





The benefits of stochastic dominance rules are that they

allow only partial information about investor preferences

and no assumptions regarding the nature of the distribution.

This technique is developed in detail in Chapter 1[.

The results are in Table 6.4 where the riskless asset

assumed is the average spot rate from each perspective. As

indicated in the table, the FSO and FSDR criterion are very

ineffective. For the dealer, however* month one is

dominated by month two indicating that term premiums are

economically meaningful for a dealer. Assuming u"O > 0 is

not restrictive because SSD and TSDO, and SSDR and TSDR

contain the same securities in the efficient set.

Incorporating a riskless asset in all three cases imolies

that term premiums are economically meaningful. As with the

M-V criteria, the efficient set is smaller when transaction

costs are subtracted from holding period returns, and larger

when added. Because of the rising transaction costs with

respect to maturity, longer maturities are dominated for the

investor, whereas for the dealer longer maturities move

into the the efficient set.






TABLE 6.4

Stochastic Dominance Results


Monthly Returns 9/64 through 6/85
FSO SSD and TSD FSDR SSDR and TSOR
Investor 1-11 1-3,5 1-11 3,5
Average 1-11 1-305,6,8,9 1-11 3,596,8,9
Dealer 2-11 2-9 2-11 5-9




Summary

In this chapter, the influence of transaction costs on

term premiums in U.S. Treasury bill holding period returns

was examined. The existence of first-order stochastic

dominance from a dealers perspective indicates that dealer

term premiums are economically meaningful as well as

statistically significant.

Term premiums are reduced dramatically by incorporating

transaction costs, a fact which may vitiate most previous

research in this area. In some cases, transaction costs

account for 50% of the observed term premium. Assuming the

existence of a riskless asset, however, still supports the

existence of economically meaningful term premiums.













CHAPTER VII
CONCLUSIONS


The purpose of this study was to enhance the current

understanding of the term structure of interest rates.

Although our understanding of term structure behavior

continues to be somewhat nebulous, five issues are now

clear. First determining whether term premiums of U.S.

Treasury bills is economically meaningful depends on the

assumptions made. Secondly, U.S. Treasury note's and U.S.

Treasury bill's holding period returns are different, with

bills dominating notes for many maturities. Thirdly, a

primary determinant of reported bid-asked spreads at a point

in time is the bill's maturity. Fourthlyt incorporating the

bid-asked spread Joes not improve the prediction power

imbedded in the forward rates about future time varying

premiums or future spot rates. Finally, transaction costs

have a dramatic impact on term premiums.

Comparing the return of long-term bills to the return of

short-term bills, a statistically significant term premium

has previously been found by Fama (1984b). Whether this

significant term premium is also economically meaningful was

examined. If investors make decisions based on the relative

characteristics of each maturity bill then term premiums






are not economically meaningful. If investors make

decisions based on the portfolio distribution of each

maturity bill and the riskless asset, then term premiums are

economically meaningful.

The second problem was to establish a preference between

return distributions of bills and notes. The impact of

coupons and differential tax treatment between bills and

notes clouds the relationship of their before tax return

distributions. For the period examined, bills tend to

dominate notes; therefore, notes cannot be considered a pure

substitute for bills.

The third problem of examining the determinants of the

bid-asked spread resulted in finding maturity as a

significant determinant. A risk measure was developed which

consisted primarily of the bill's maturity.

The fourth problem was resolved in that no significant

change in R-squared was found in the regression analysis.

It was concluded that there is no additional information in

the bid-asked spread about time varying term premiums and

forecasts of future spot rates.

Finally, transaction costs were shown to account for a

significant proportion of the term premium. This result

cast doubt as to the validity of prior research which

ignores transaction costs.











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