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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 MeanVariance and 4eanVarianceSkewness *. 18 Data . . o 18 Risk Measures . . * * 18 MV 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 RiskFree 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 MeanVariance 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 MeanVariance and MeanVarianceSkewness *. 84 Data .* * * * * 84 MV 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 bidasked 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 bidasked spreads in the Treasury bill market. For most periods examined, maturity had significant explanatory power in accounting for the variation in the bidasked spread at a point in time. The fourth problem is to examine the impact of bidasked 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 bidasked 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 longterm 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 bidasked spreads. Fourthly, an analysis is performed of whether there is useful information in bidasked 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 expost 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 couponbearing. 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 defaultfree bonds is the dealer bidasked spread (1966, p. 105). To determine what factors influence the magnitude of the bidasked 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 bidasked 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 bidasked 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 longtern 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 upwardsloping 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 longerterm 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 taxexempt 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 beforetax zero coupon rates are the same for all maturities the yield curve for couponbearing bonds will rise with maturity. However if the aftertax zero coupon rates are constant, then the couponbearing yield curve for nonoar bonds will take a wide variety of shapes. Based on this theoretical work, there exists no a prior reason to expect beforetax 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 longerterm 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 bidasked spread is not new. Demsetz (1968) identified the bidasked 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 bidasked price can expect to make profits at the specialist expense, implying larger bidasked spreads. Grant and Whaley (19781 showed that a bond's risk, as measured by its duration, is an important determinant of the bidasked spread. Using the par value of bonds outstanding as a proxy for transaction volume, they found a significant relationship between volume and the bidasked 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 errorlearning, 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 multimonth 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 nonmonotonically 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 onemonth 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 nl 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 expost 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" respectively, are assumed. 5. In addition to 4. as stated above# investors are allowed to borrow and lend at the onemonth 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 meanvariance or the meanvarianceskewness 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 longterm bond is equal to the return on a series of investments in shortterm 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 longterm 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 termtomaturity. This implies an upwardsloping 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 longerterm 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 riskfree 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 nonnormal 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 riskfree 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 onemonth 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 964 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 585 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 MV 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 distributionfree decision rules, MV decision rules and conventional performance indices which do rely on the normality assumption are presented. Assuming no riskless asset one can apply the wellknown MV rules to the figures of Table 2.1 to establish the efficient and the inefficient sets. The MV 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 MV 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 valueweighted 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 stockbond 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 964 through 1214 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 MV 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 MV 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 MV efficient set while Arditti's efficient set consists of one investment taken from the MV 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 MV 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 196970 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 8/645/85 239 112 15,8,9 15,8,9 8/6412/82 210 112 15,8,9 15,8,9 8/6412/72 101 112 13,59,12 13,59*12 1/7312/i2 109 112 1598,9 15,8.9 1/7812/82 54 112 15 15 6/305/95 58 212 25*8,9 25,8,9 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 meanvariance 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 nonparametric 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. ADIilustratioQ 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  1I I I I I 3/4 I G F 1 G1 1 F' I I I I I 1/2 III I I I I 1/4 I II I F 1 F' 0.0 I 11I11111 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 Riskfree Period N* Rate FSOR SSDR TSDQ 8/645/85 239 .0056 112 35,8,9 35,8,9 8/6412/82 210 .0054 112 35,8,9 35,8,9 8/6412/72 101 .0039 19,11,12 3,59,12 3,59,12 1/7312/82 109 .0068 112 35,8,9 35,8,9 1/7812/82 54 .0088 212 35 35 6/905/85 58 .0085 212 35,8,9 35,8,9 *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 MV 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 MV 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 riskfree rate. A rate is needed which would be representative of the market at the time the expost data is collected. To assess the problem of possibly selecting an improper riskfree rate an analysis is performed to assess how sensitive the conclusions are to the choice of the riskfree rate. Table 2.7 presents the efficient sets for the entire period under various riskfree 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 RiskFree Rates RiskFree 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 864 through 585 U(il, (i = t1 2, 31 1 2 3 FSDR SSDR TSDR 12 12 12 16,8,9 25,8,9 25,8,9 16,8,9 15,8,9 15,8,9 16,8,9,12 15,8,9 15,8,9 16,8,9,12 25,8,9 25,8,9 16,9,9,12 25,8,9 25,8,9 19,12 35,8,9 35,8,9 19,11,12 35,8,9 35,8,9 112 35,8,9 35,8,9 112 35,8,9 35,8,9 112 5,8,9 5,8,9 112 5,8,9 5,8,9 112 5,8,9 5,8,9 212 9 9 212   From Table 2.7 it is clear that the same general conclusions would be drawn if the riskfree 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 riskfree 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, fiftytwo week bills were issued every four weeks, so precise monthly returns are not available after this date. Because bills are issued weekly for a twentysix week maturity, a bill maturing near monthend 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 246 211 26,8,9 26,8.9 244 311 39 39 242 411 49 49 240 511 59 59 238 611 610 610 236 711 711 811 234 811 811 811 232 911 911 911 230 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 211 311 411 511 611 711 811 911 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 longterm Treasury bills to the return of shortterm 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 riskfree 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 twomonth 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 expost 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 longerterm securities which are couponbearing 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 shortterm 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 taxexenmot 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, "Lowcoupon 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 lowcoupon 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 couponbearing bonds (1979a, p. 189) in a world with differential taxation of coupons and capital gains. Livingston demonstrates when beforetax zero coupon rates are the same for all maturities the yield curve for couponbearing bonds will rise with maturity. However, if the aftertax zero coupon rates are constant, then the couponbearing yield curve for nonpar 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 longerterm 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 longerterm bonds have lower expected returns than shortterm instruments. Like McCulloch (1975), but with the advantage of an exhaustive data base, we find that the high variability of longerterm 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 fiveyear subperiod of the 195382 sample period during which average bond returns increase systematically with maturity. The shortest maturity bond returns increase systematically with maturity. The shortestmaturity portfolio (< 6 months) produces the largest average return in two of the fiveyear subperiods. Average returns never peak in maturity intervals beyond four years. At least on an ex post basis, the thirty year period 195382 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 twoyear Treasury notes began in the mid1970s 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 monthend 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 MV 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 MV 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 MV 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 MV 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 hV 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 distributionsbill 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 MV 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 111 123 SSO 13,5 4 TSD 13,5 4 FSDR 210 19,1323 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 meanvariance 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 bidasked 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. Crosssectional regressions are employed to determine what factors explain dealer bidasked spreads. For the periods examined, maturity explained a large proportion of the variation in the bidasked 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 bidasked spreads should reflect three costs: I1) fixed costs (12) nonfinancing 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 bidasked spreads. Finally, if some traders have access to superior information* then the bidasked spread should incorporate the expected loss from trading with these people*' Prior studies of security dealer bidasked 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 bidasked 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 bidasked 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 bidasked 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 couponbearing bonds, which have been the focus of previous studies of fixed income bidasked 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 bidasked 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 bidasked quotations will be relatively accurate. Sixthly, an extensive and reliable data base is available. Demsetz (19681 identified the bidasked 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 bidasked 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$ bidasked 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 bidasked 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 couponbearing 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 respecified 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 bidasked 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 nonfinancing 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 shortterm 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 shortterm 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 bidasked 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 RSquare 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 RSquare 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 Crosssectional 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. AnDExamnla 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 6,50022,250 .58 .51 Maturity (days) 128 90 7349 .91 Yield M1) 7.3 .285 6.67.8 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 Rsquare 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 1.08E07 1964 (.0001)*1 August 8.87E08 1965 (.00011 August 4.26E07 1966 (.0002) August 2.56E07 1967 (.0001) August 2.97E07 1968 (.0001) August 3.51E07 1969 (.0001) August 2.56E07 1970 (.0001) August 1.36E07 1971 (.00011 August 1.76E07 1972 (.00011 August 1.72E07 1973 (.0007) August 2.17E08 1974 (.0009) August 1.16E08 1975 (.0001) August 9.19E08 1976 (.0001) August 4.93E08 1977 (.0001) August 2.15E08 1978 (.11821 August 3.94E08 1979 (.0011) August 1.03E08 1990 (.3993) Augsut 8.10F08 1981 (.0004) August 1.94E08 1982 (.0216) August 2.02E08 1983 (.0015) August 6.37E09 1984 (.6319) Ro*2 Maturity R**2 (Slope) .52 7.15E07 .54 (.0001)* .34 8.93E07 .64 (.00011 .32 3.65E06 .47 (.00011 .45 1.35E06 .20 (.0056) .41 2.52E06 .36 (.00011 .34 3.74E07 .48 (.0001) .58 2.41F06 .60 (.0001) .42 2.00E06 .52 (.00011 .62 2.12E06 .56 (.0001) .28 3.84E06 .64 (.00011 .25 4.07E06 .47 (.00011 .47 2.29E06 .65 (.90001) .33 2.15E06 .87 (.0001) .41 9.73F07 .88 (.0001) .06 5.95E07 .27 (.0007) .25 6.55P;7 .28 (.00051 .02 8.54E08 .00 (.7583) .35 3.52E06 .73 (.0001) .16 1.53E06 1.00 (.0001) .27 1.53E06 1.00 (.0001) .01 1.39E06 .79 (.000011 Maturity over Yield R*2Z 7.43E07 .54 (.0001 1 9.32E07 .64 (.00011 3.88E06 .47 (.00011 1.43E06 .21 (.0054) 2.67E06 .36 1.0001) 4.05E06 .48 (.00011 2.58E06 .61 (.0001) 2.11E06 .52 (.0001) 2.24E06 .56 (.0001) 4.19E06 .64 (.0001) 4.50E06 .47 (.0001) 2.47E06 .65 (.0001) 2.27E06 .87 (.0001) 1.04E06 .83 (.00011 6.45E07 .27 (.0007) 7.26E07 .29 (.00051 9.43E08 .00 (.75961 4.15F06 .73 (.0001) 1.71E06 1.0 (.0001t 1.70E06 1.0 ( .0001) 1.56E06 .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 Rsquare 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 nonsiginificant 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 6.22E08 1964 (.0044)1 August 1.70E08 1965 (.4110) August 1.60E07 1966 (.1800) August 2.81E07 1967 (.0004) August 2.00E07 1968 (.01181 August 1.43E07 1969 (.1341) August 1.50E07 1970 (.0007) August 5.76E09 1971 (.10491 August 1.17E07 1972 (.0004) August I.18E08 1973 (.7843) August 5.12E08 1974 (.4401) August 5.89F03 1975 (.0014) Auaust 3.21F09 1976 (.79171 August 5.41E10 1977 (.92991 August 1.13E08 1978 (.4728) August 2.22E08 1979 (.1044) August .210209 1980 (.4241) August 8.68F09 1981 (.6191) August 8.42E10 198? (.0052) August 6.20E10 1983 (.0113) August 2.49E10 1984 (.9682) *The probability of getting Maturity R (Slope) 4.48F07 (.0019)* 8.12E07 (.00011 2.93E06 (.0012) 2.81E07 (.5544) 1.38E06 (.04371) 2.91E06 (.0014) 1.51E06 1(.0003) 1.46E06 (.0030) 1.07E06 (.00811 3.73E06 (.0001) 3.63E07 (.0003) 1.75E06 (.0001) 2.18E06 (.0001) 9.71E07 (.00011 6.92E07 (.0022) 4.43E07 (.04161 7.09E08 (.8350) 3.35E06 (.0001) 1.54E06 (.00011 1.54E06 (.00011 1.38E07 (.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. '*oFirstorder 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 bidasked 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 bidasked spread does not change the results. Most empirical examinations of the term structure of interest rates have used the average of the bidasked 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 defaultfree bonds is the dealer bidasked 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 bidasked spread. (See Malkiel (1966), p. 115.1 Garbade and Silber based on interviews with dealers, concluded that reported bidasked prices are real prices at which transactions take place for modestsize 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 bidasked 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 errorlearning* 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,19lR(t+l,ll) = c+dlF(tmlR(ttl) +f(t+mlll (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+mlil 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 "finetuned" rej'ssions H(t+lnm)H(t+lm1) = a+b(F(ttm)F(teml)+e(t+ll) (5.4) R(t+l) Rtt+m1) = c + d(F(tm)F(teml1 + f(tem1 15.5) Equation 5.5 assesses the ability of forward rates to predict successively more distant onemonth 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 bidasked spread or the dealer does not receive this extra profit. For the investor it is assumed that he/she must pay the bidasked spread thus receiving a lower return than average. Forward rates are calculated based on the average of the bidasked 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 monthend 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 forwardspot 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 Rsquared when bidasked spreads were incorporated. Although the Rsquared from the investor's perspective is slightly larger for early maturities this phenomena did not appear for all the subperiods. Rsquares 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 forwardspot differential. Again these results are similar to Fama (1984b, p. 5171. The lack of a significant change in the Rsquared when bid asked spreads were incorporated implies that neither dealers TABLE 5.1 Premiums on ForwardSpot Differential Monthly Returns 964 through 585 P(mvt+l) = a + b(F(mt)R(t+l1) + e Average Dealer Investor Maturity b RSquare b RSquare b RSquare 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 Rsquares do not significantly TABLE 5.Z Future Spot Change on ForwardSpot Differential Monthly Returns 964 through 585 R(t+m)Rttil) = Average Maturity b RSquare .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 RSquare .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 bidasked spread is incorporated. In fact, the poor Rsquares 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 RSquare .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 964 through 535 Hlt+lm) Hlt4lml Average Maturity b RSquar 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)Fltm1l) + e Dealer e b RSquare .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 RSquare .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 forwardspot differential appears to be biased in its prediction of future changes in spot rates. TABLE 5.4 Fine Tuned Spot Reqressions Monthly Returns 964 through 585 R(t+m) RIt+m1) = a + b(F(tm)iF(tm1)l + e Maturity b RSquare .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 RSquare .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 RSquare .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 bidasked 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 bidasked 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 bidasked 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 bidasked 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 bidasked 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 nI 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. MeanVariance# 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 premiumthe average difference between the oneweek implicit forward rate m weeks in the future and the oneweek spot rate realized in m weeksbetween the 13and 14week 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 bidasked 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. Fiftytwo 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 monthend 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 MV rules are applied to establish the efficient and inefficient sets using the average bidasked 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 *+Pak 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 MV 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 MV 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 MV 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 111 13,5 111 3,5 Average 111 1305,6,8,9 111 3,596,8,9 Dealer 211 29 211 59 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 firstorder 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 bidasked spreads at a point in time is the bill's maturity. Fourthlyt incorporating the bidasked 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 longterm bills to the return of shortterm 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. 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