The Increasing Price Efficiency of the US Treasury Market

Transcription

1 The Increasing Price Efficiency of the US Treasury Market Miles Livingston University of Florida Warrington College of Business Department of Finance, Insurance and Real Estate Gainesville, FL Phone: (352) Yanbin Wu Emory University Guizueta Business School Department of Finance Atlanta Georgia Lei Zhou Northern Illinois University College of Business Department of Finance DeKalb, IL Phone: (815) /16/2018

2 Abstract While largest and most liquid in the world, US Treasury market still has frictions which can lead to special pricing for some securities. Previous studies have documented specific market frictions and their impacts on Treasury pricings. We propose a linear programming (LP) based measure of aggregate special pricing and find an average special pricing of $0.11 per $100 par of T-notes and T-bonds from 1980 to 2016, or 130% of the average bid-ask spread. The special pricing is not transient and primarily driven by persistent factors. Further, we document a dramatic improvement in price efficiency since Average special pricing decreased from $0.25 in the early 1980s to less than $0.05 since 2010.

3 The Increasing Price Efficiency of the US Treasury Market In a frictionless market with no impediments to arbitrage, the law of one price should hold, i.e., securities or portfolios of securities with the same future cash flows should have the same prices. However, market frictions can result in discrepancies in the prices of securities or portfolios of securities with the same cash flows. In other words, some securities can be more expensive, or have special pricing, than other securities or portfolios with the same cash flows. While the US Treasury market is probably the largest and most liquid in the world, market frictions still can exist, resulting in special pricing of some Treasury securities. An example of special pricing in the Treasury market is the well-examined on-the-run effect. Many studies document lower yields for the on-the-run Treasuries relative to otherwise similar off-the-run Treasuries and attribute the on-the-run effect to liquidity differences and/or repo specialness (Warga, 1992, Duffie, 1996). Furthermore, impediments to arbitrage can prevent potential arbitrageurs from exploiting special pricing or price inefficiency (Hu, Pan, and Wang, 2013). 1 While prior studies examine particular aspects of market frictions and their impacts on the relative pricing of different Treasury securities, no study has quantified the aggregate special pricing in the Treasury market and its intertemporal variation. Over the last four decades, there have been several institutional developments and changes in tax law that likely have reduced market frictions and made arbitrage easier and less costly. For example, the Tax Reform Act of 1986 has reduced the preferential tax treatment of discount bonds. The introduction of the Treasury STRIPS program in the mid-1980s has increased arbitrageurs ability to take advantage of potential price inefficiency. These changes and institutional developments, therefore, are 1 The terms special pricing and price inefficiency are used interchangeably throughout the paper. They are defined as the difference in prices of portfolios of Treasury notes and bonds with the same future cash flows. 1

4 likely to have improved the price efficiency of the US Treasury market. This study quantifies the aggregate special pricing in the marketable long-term Treasury bond market and examines its trend over the last four decades. Most prior studies on Treasury market price efficiency rely on yield curve fitting models, making any documented special pricing theoretical in nature. 2 We propose a linear programming (LP) based measure of special pricing. The LP model assumes a perfect market without any obstacles to arbitrage. A tax-free arbitrageur tries to find an immediate profit without investing any funds or having to incur any net outflows in the future. The arbitrageur is allowed to choose combinations of T-bonds and T-notes and use the future cash inflows from one bond to make payments to others. 3 The theoretical arbitrage profit derived from the LP solution measures the aggregate exploitable price inefficiencies or special pricing that would be eliminated if the market were free of frictions and impediments to arbitrage. The LP approach has several advantages. First, it does not require any model to estimate the zero-coupon yield curve from the observed prices of coupon-bearing bonds, making its empirical results independent of specific models and free of potential model misspecifications. 4,5 2 An exception is several studies that take advantage of triplets of Treasury securities with the same maturity dates (Jordan and Jordan, 1991, Longstaff, 1992, Elton and Green, 1998). 3 In equity markets, risk-free arbitrage is straightforward. It involves buying a security that is underpriced in one market and simultaneously selling the exact same security in the market where it is overpriced. The transaction is closed out immediately for a profit and the actions of an arbitrageur tend to push the prices together, resulting in the Law of One Price. In bond markets, risk-free arbitrage is typically more complicated because bonds have contractual payments at many future dates. 4 Since yield to maturity is a complicated average of the zero-coupon interest rates, fitted yield curves have potential errors. First, coupon level on a specific bond can have a significant impact on the yield to maturity as shown by Robichek and Niebuhr (1970). Second, for longer maturities, the averaging effect becomes stronger, leading to flattening of yield curves and a potentially sizable disparity between the zero-coupon rate for a particular long maturity and the yield to maturity as shown by Livingston and Jain (1982). 5 Sack (2000) examines two commonly used yield curve fitting models: Nelson-Siegel and Fisher-Nychka-Zervos models. He finds that one assumption of the Nelson-Siegel model is consistently violated. In addition, the Fisher- Nychka-Zervos model performs poorly when there are large gaps in maturities of available Treasury securities. 2

5 Second, the LP model directly measures special pricing in dollar terms. Previous studies measure special pricing by the difference between actual bond yields and model-fitted yields, implicitly and incorrectly assuming that, for example, a 5-basis-point price inefficiency on a 2- year note and a 30-year bond have the identical special pricing in dollar terms. Third, isolating the effect of bid-ask spreads is straightforward in a LP model. In contrast, previous studies based on yield curve fitting models use either bid price, ask price, or mid-price. This advantage of the LP model is particularly important in examining price efficiency for a long sample period, during which the bid-ask spread has decreased significantly. Finally, compared with studies using several bonds with the same maturity date but different coupons, the LP model allows for combinations composed of all Treasury securities. Thus, the results derived from the LP model represent the upper bounds of the aggregate special pricing in the market. Using month-end bid and ask prices from the CRSP Daily Treasury database, we estimate aggregate Treasury market special pricing at the end of each month from 1980 to The average month-end LP-based special pricing for the sample period is about $16.51 and the average special pricing per Treasury security is about $0.11 per $100 par value As a comparison, the average bid-ask spread for the sample period is about $0.08. Thus, the average special pricing per Treasury is about 130% of the average bid-ask spread. Second, we document a dramatic increase in price efficiency from the 1980s to more recent years, suggesting significant reductions in market frictions and impediments to arbitrage in the marketable US Treasury market over the last four decades. The average month-end special pricing is more than $28 and special pricing per security is $0.25 for the 1980 to 1984 subsample period. In contrast, the average aggregate month-end special pricing and per security special pricing has decreased to about $10 and less than $0.05 respectively since Thus, the 3

6 US Treasury market has been moving increasingly closer to the perfect market as assumed by the LP model. This substantial improvement in the price efficiency of the US Treasury market coincides with several developments in the market that tend to reduce market frictions and make arbitrage easier and less costly. 6 Furthermore, our findings document a spike in special pricing during the financial crisis, consistent with findings in Hu, Pan, and Wang (2013) and Musto, Nini and Schwarz (2016). While these two studies offer different explanations for the spike in price inefficiencies, both suggest market frictions and/or limits to arbitrage increased during financial market disruption. This supports the use of LP-based theoretical arbitrage profits as a measure of the aggregate effects of market frictions and limits to arbitrage on price efficiency. Finally, the LP-based month-end special pricing is persistent for up to 10 trading days into the following month. In other words, overpriced (underpriced) securities tend to remain overpriced (underpriced). This finding indicates that the majority of special pricing is driven by inherent market frictions that are hard to be arbitraged away. The remainder of the paper is organized as follows. Section I reviews existing literature on the market frictions and their impacts on Treasury market price efficiency. Given the voluminous extant literature, this review is not intended to be exhaustive, but just to highlight the various market frictions documented by prior research. In addition, the section discusses changes in tax law and several market and institutional developments that are expected to improve price efficiency. Section II introduces the Linear Programming model and describes our 6 As we will describe in more details later, most of these institutional developments and changes occurred around the same time, making it impossible to disentangle their individual impacts on the Treasury market price efficiency. 4

7 sample. Section III presents and discusses the empirical results from and Section IV concludes the paper. I. Market Frictions and Evolution of the US Treasury Market A. Literature on Treasury Market Frictions 1. Asymmetric Tax Treatments for Discount and Premium Bonds Prior to the 1986 Tax Reform Act, discounts on market discount bonds were treated as capital gains at the maturity or disposal of the bonds. This created two tax advantages for discount T-notes and T-bonds. First, the market discount was taxed at the lower capital gain tax rate. Second, the tax on the discount was deferred until the disposal or maturity of the bond. Numerous studies have documented the impact of the asymmetrical tax treatment on US Treasury pricing and the tax clientele effects (Jordan, 1984, Litzenberger and Rolfo, 1984, Ronn, 1987, and Kamara, 1994). 2. On-the-run Effect A well-documented special pricing in the Treasury market is the on-the-run effect. Warga (1992) finds that returns of on-the-run Treasuries, or the most recently issued Treasury securities, average 55 basis points lower than otherwise similar off-the-run Treasuries, suggesting overpricing of the on-the-run issues. The extant literature offers three explanations for the on-the-run effect. First, on-the-run issues command a higher price because of higher liquidity. Sundaresan (2002) documents that on-the-run issues enjoy almost 10 times higher trading volume compared to the off-the-run issues. Flemming (2002) finds that the bid-ask spreads of the on-the-run Treasury bills are much smaller than those of the off-the-run bills. 5

8 Second, on-the-run issues are special in the repo market, i.e., the repo rates of the on-therun issues are significantly lower than the general collateral repo rates. Duffie (1996) reports an average repo specialness of 66 basis points for on-the-run issues. The lower repo rates of the onthe-run issues reduce the financing costs for dealers and investors to hold on-the-run-issues, resulting in higher prices of these Treasuries versus similar off-the-run issues. Third, Vayanos and Weill (2008) proposes a search-externality explanation for the onthe-run effect. In their theoretical model, the authors show that the higher liquidity and repo specialness of the on-the-run issues are driven by the concentration of shorting activities on these issues due to search externalities and the constraint on short-sellers to deliver the same assets they borrowed. Regardless of the differences in the three explanations, they all suggest that the on-therun effect is driven by market frictions, such as search costs and liquidity differences. 3. Liquidity Effect In addition to the liquidity difference between the on-the-run and off-the-run issues, there is also a significant variation in liquidity among the off-the-run issues. Amihud and Mendelson (1991) compare T-bills and one-year T-notes with six months left to maturity. They find that T- notes have higher bid-ask spreads, suggesting lower liquidity for T-notes. Further, T-notes have significantly higher yields than T-bills, indicating an illiquidity premium on T-notes. In the same vein, Musto, Nini and Schwarz (2016) find that bid-ask spreads on T-bonds are twice those on 10-year T-notes, and the yields on T-bonds are higher than T-notes with similar time left to final maturity. 4. Aging Effect 6

9 Closely related to the liquidity effect is the aging effect of US Treasury securities, documented by Fontaine and Garcia (2012) and Diaz and Escribano (2017). As Treasury securities age over time, a larger and larger fraction may be acquired by long-term buy-and-hold investors, leading to lower liquidity. Diaz and Escribano (2017) find higher yields on older T- notes, indicating an illiquidity premium. 5. Deliverability Effect Several studies have shown that Treasury securities eligible for delivery under futures contracts have higher prices than otherwise similar non-deliverable securities. Garbade (1985) and Simpson and Ireland (1985) document small but statistically significant price premiums on deliverable Treasury bills. Kuipers (2008) has similar findings on deliverable 30-year T-bonds. Furthermore, Sack (2000) finds that the cheapest-to-deliver Treasury securities have significantly lower yields than the implied yields based on coupon Treasury strips. B. Impediments to Arbitrage In a perfect world without arbitrage restrictions or costs, any special pricing should be arbitraged away. In actual practice, there are significant impediments or costs to risk-free arbitrage. A good example of the limiting effects of costs to arbitrage is the repo specialness of the on-the-run issues (Duffie, 1996, Krishnamurthy, 2002). To profit from the price difference between an expensive on-the-run issue and an otherwise similar but cheaper off-the-run issue, an investor would short the on-the-run issue and long the off-the-run issue. To establish the short position, the investor engages in a reverse repo on the on-the-run issue, earning the lower special repo rate on the on-the-run issue. In the meantime, the investor finances the long position by a repo agreement on the off-the-run issue, paying the higher general collateral repo rate. The cost 7

10 of this strategy is the difference between the (lower) special repo rate on the on-the-run issue and the (higher) general collateral repo rate. This difference prevents arbitrage from completely eliminating the price premium for the on-the-run issues. Other direct costs to arbitrage include transaction costs, bid-ask spreads, and haircuts in the repo market. Availability of equity capital for arbitrage positions and leverage constraints on arbitrageurs pose further limitations to arbitrage (Shleifer and Vishny, 1997, Gromb and Vayanos, 2010). Through a theoretical model, Liu and Longstaff (2004) demonstrate that collateral requirements on short positions can make some textbook arbitrage strategies infeasible or money-losing. C. Changes in Tax Law and Institutional Development since Expansion of the US Treasury Market Since the 1980s, the US Government has been running sizable deficits resulting in the US Treasury borrowing very large amounts of money. To meet these borrowing needs the US Treasury has increased both the issuing frequency and issue size. Figure 1 illustrates the increases in the size of the long-term marketable Treasury security market. Panel A shows the number of non-callable, non-flower, fixed coupon and fixed principal T-notes and T-bonds and the total amount of par value outstanding at the end of each month from 1980 to There has been a steady increase in the number of Treasury securities except for the years between the late 1990s to early 2000s, a period when the US Federal Government had fiscal surpluses. Similarly, the total size of the market, in terms of the dollar amount outstanding, has increased dramatically since Panel B reports the average amount outstanding per Treasury note or bond. The significant increases in both the total market size and the size of individual Treasury securities likely have led to substantial improvements in market liquidity over the last four decades. 8

11 Several studies have shown that higher liquidity helps to improve price efficiency in the equity, credit and option markets (Chordia, Roll, and Subrahmanyam, 2008, Nashikkar, Subrahmanyam, and Mahanti, 2011, Deville and Riva, 2007). Thus, the increased sizes of individual issues and of the total market have likely improved the pricing efficiency of the US Treasury market as well. In addition, the large increase in the size of Treasury issues combined with the increase in the number of securities with various maturities and different coupon levels has increased the opportunities for large institutional bond investors to switch from a relatively high priced security to a similar, but more attractively priced security. With a large number of securities available, bond investors (who typically hold portfolios of bonds) can examine individual bonds in their portfolio and use linear programming to find whether some bonds in their portfolio should be switched for other bonds with the same or higher future cash flows for a lower total cost (Durand, 1959, Hodges and Schaefer, 1977, and Ronn, 1987). The consequence of portfolio switching by large institutions is the downward (upward) price adjustment of bonds with relatively high (low) prices, similar to risk-free arbitrage. Thus, portfolio switches by institutions and individuals can help to improve price efficiency. 2. Decreases in the Average Bid-ask Spreads and Variations in Bid-ask Spreads Consistent with the increased sizes of both the total Treasury security market and individual Treasury securities, the bid-ask spreads of Treasury securities have decreased significantly over the last four decades. Figure 2 depicts the monthly average bid-ask spreads of T-notes and T-bonds at the end of each month. In the early 1980s, the average bid-ask spread was about $0.25. It decreased to about $0.10 by the early 1990s and has remained below $0.05 9

12 since The dramatic decrease in bid-ask spreads further suggests great improvement in the overall liquidity of the Treasury security market. Furthermore, lower bid-ask spreads reduce the transaction costs to potential arbitrageurs and likely contribute to market price efficiency. In addition to the significant decreases in the average bid-ask spreads, the cross sectional variation in bid-ask spreads between different Treasury securities has declined over the last four decades as well. 8 This suggests a significant decrease in the liquidity differences between different Treasury securities. Reduced cross sectional liquidity variation is expected to mitigate special pricing due to liquidity differences. 3. Tax Reform Act of 1986 The Tax Reform Act of 1986 significantly reduced the preferential tax treatment of discount bonds. For discount bonds issued after July 18, 1984, or issued on or before that date but purchased after April 30, 1993, the market discount is treated as interest income and taxed at the ordinary income rate. While investors still have the option to defer the income tax on market discounts until the disposal or maturity of the bond, the change in the asymmetric tax treatment of discount and premium bonds has significantly diminished the appeal of discount bonds and likely has reduced special pricing on discount bonds. 4. Increased Holdings by Potentially Tax-exempt Institutional Investors 7 The cliff-like drop in average bid-ask spreads in late 1996 is likely due to the change in the primary data provider of CRSP Treasury Database on Oct The main data source of CRSP Treasury Database from 1962 to 1996 was the Federal Reserve Bank of New York, which discontinued the data series on Oct Since then, CRSP Treasury Database relies on GovPX as its the primary data source. The steep drop in the average bid-ask spreads around Oct is likely due to differences in data collection methodologies of the two data sources (Jordan and Kuipers, 2005). 8 A plot of the monthly cross sectional standard deviations of bid-ask spread shows a significant decline from 1980s to recent years, almost mirroring the decrease in average bid-ask spread. For the sake of brevity, we do not report the plot. 10

13 Over the last four decades, there has been a significant increase of holdings of marketable Treasury securities by potentially tax-exempt institutions. Figure 3 depicts the percentages of marketable US Treasury securities owned by foreign institutions, state and local governments, and pension funds. 9 State and local governments as well as pension funds are exempt from federal income tax. The majority of foreign holdings of US Treasuries are by foreign official institutions which are exempt from US federal income tax. The 2015 Report on Foreign Portfolio Holdings of US Securities shows that about 70% (60%) of foreign holdings of longterm US Treasuries (Treasury bills) are by foreign official institutions from 2008 to 2015 (Tables 10 and 11 on page 15). Similar patterns are observed in the earlier years. 10 As shown in Figure 3, potentially tax-exempt investors held less than one third of privately owned US Treasury securities in the early 1980s. The percentage increased to about half by the mid-1990 and stayed close to 70% in more recent years. It is very plausible that the Tax Reform Act of 1986 and the increased participation of tax-exempt investors in the US Treasury market have reduced significantly or even completely eliminated the price distortion from tax effects. Indeed, Green and Odegaard (1997) find the tax effect on Treasury pricing disappears after 1986 and the implicit marginal tax rate for a representative investor in the US Treasury market became close to zero. 5. Introduction and Growth of Treasury STRIPS In 1985, the U.S. Treasury allowed stripping of Treasury securities by book entry in the STRIPS (separately trading of interest and principal of securities) program, i.e., trading of the 9 Holdings by the monetary authority are excluded. We obtain the Treasury holdings data from the US Treasury Bulletins. 10 Detailed data on foreign holdings by official institutions are not available prior to However, the 2003 Report indicates the percentages of holdings by foreign official institutions are relatively stable from early 1980s to the present at about 60% (Table 10 on page 18). 11

14 coupons and principals of some T-notes and T-bonds separately as zero-coupon Treasury strips. Reconstitution of strips was initiated in All 10-year T-notes and T-bonds issued after November 15, 1984 were eligible for the STRIPS program either upon their original issuance or after their first interest payment date. Between 1987 and 1997, many strippable 10-year T-notes and T-bonds were issued and the total amount of par value of Treasury securities held in stripped form increased steadily. All marketable fixed-rate T-notes and T-bonds issued on and after September 30, 1997 are strippable (Bulter, Livingston and Zhou, 2014). The Treasury STRIPS program increases the number of investment choices and makes portfolio improvements more feasible for institutional bond investors. With sufficient supplies of strips available, Treasury security dealers are able to rapidly complete arbitrage transactions between strips and coupon bearing Treasuries at minimal or no cost. If the market price of a bond is higher (lower) than the market price of a portfolio of strips with the same pretax cash flows, dealers are able to strip the bond and sell its cash flows as individual strips (reconstitute the strips and sell it as a bond). Indeed, Jordan, Jorgensen, and Kuipers (2000) finds potential arbitrage profits between coupon-bearing Treasury securities and their corresponding portfolios of strips are rare and generally not statistically significant, indicating a great integration of the two markets and high pricing efficiency. Furthermore, arbitrages by dealers between coupon bearing bonds and strips do not depend upon the tax rates. Although dealers are liable for income tax on arbitrage profits, there is no tax liability for future cash inflows since all positions are closed immediately. This further reduces any potential impact of different tax treatments of discount and premium bonds on price efficiency. 12

15 II. The Linear Programming Model and Sample. A. Linear Programming Model The Linear Programming model assumes a tax-free arbitrageur operating in a frictionless market with no limit or costs to arbitrage. Specifically, it assumes the following: 1. Transactions take place at the bid and ask prices. 2. No other trading costs (i.e., no brokerage commission and price impact). 3. Securities can be short sold and the proceeds be used without any restriction (i.e., no collateral or margin requirements for short positions.) 4. There are sufficient supplies of securities to be short sold. A short sale can be for the full value of the securities without any so-called haircut (or discount). 5. There are no defaults on any payment or delivery obligations. 6. The tax rate on the arbitrageur is zero. 7. Positive net cash flows at a future point in time can be rolled over into subsequent points in time at a zero rollover rate (or forward rate). 11 The above assumptions basically rule out any market frictions and impediments to arbitrage (e.g., brokerage commission, differential tax treatment, liquidity difference, etc.). However, the LP model explicitly incorporates bid-ask spreads. Thus, the special pricing from the LP model is net of the effect of bid-ask spreads. The LP model has the following objective function max = (, ) ( ) (1) 11 Since the horizons of these arbitrage solutions are very long, up to 30 years, we do not assume a reinvestment rate if the cumulative cash flow at a future point in time is positive. One reason is that there is no forward market for long horizons. Thus the arbitrageur cannot lock in a reinvestment rate with complete certainty. Second, predictions of the future reinvestment rate are subject to a great deal of uncertainty. 13

16 where ( ) is the bid (ask) price of bond j, ( ) is bond j s Short Weight (Buy Weight), or the unit of bond j to be shorted (bought), and H is the total number of Treasury securities in the feasible set. and for t>1, The maximization function in (1) is subject to the following constraints: = ( = + ( ) ( ) 0 (2) ) ( ) 0 (3) where Ct is the net cumulative cash flows at time t, is the coupon or principal payment of bond j at time t, and t =1,, T, and 0 1 (4) 0 1 (5) The objective function in (1) tries to find a combination of long and short positions of bonds to maximize the cash flow at time zero, with the constraint that the cumulative net cash flows at each subsequent period from this long-short portfolio are never negative. Since all T- notes and T-bonds in our sample make coupon payments and mature at either the end or the 15 th of the month, the payment period is defined as semi-monthly. Given the longest maturity of 30 years, the maximum t in inequality (3) is 719 (30*12*2-1). Inequalities (4) and (5) require that an arbitrageur can buy or short no more than 1 unit of each security to make sure I is not unbounded. Thus, the total number of constraints of the LP model is 719+ (2* the number of Treasuries). 14

17 The symbol I in the objective function (1) is the theoretical arbitrage profit that can be achieved by a tax-free arbitrageur. In a frictionless market, competition among arbitrageurs should reduce the theoretical arbitrage profit to zero. Thus, any theoretical arbitrage identified by the LP measures the aggregate special pricing, or market price inefficiency due to market frictions and limits or costs to arbitrage. LP approaches have been used in bond portfolio optimization (Hodges and Schaefer, 1977) and in studies of the tax-specific term structure (Schaefer, 1981, 1982, Ronn, 1987). B. Data Collection and Sample Descriptive Statistics The CRSP Daily Treasury data from 1980 to 2016 are used as the data source. The sample includes all non-callable, non-flower, fixed coupon and fixed principal US T-notes and T-bonds. Several types of Treasury securities are excluded: inflation protected securities (TIPS), when-issued securities, Treasury strips and T-bills. T-bills are heavily used as monetary policy instruments. 12 Treasury strips are not included for several reasons. First, there are no Treasury strips in the earliest part of the sample. Second, most prior studies on Treasury market price efficiency do not include Treasury strips. Third, prior studies have shown the uniqueness of principal strips as compared to the generic coupon strips (Daves and Ehrhardt, 1993, Jordan, Jorgensen, and Kuipers, 2000). The quoted bond prices are adjusted for accrued interest. Besides prices, the CRSP bond data file is the source of the total amount of each bond outstanding, the coupon rate, and the maturity. Table I reports the descriptive statistics of the Treasury securities used in the study. Over the whole sample, there are a total of 1,422 T-notes and T-bonds with an average coupon rate of 12 Studies based on yield-curve fitting models include T-bills to estimate short term yields. 15

18 5.472%. The average issue size is about $18.7 billion and average maturity is about 6 years. Panel B (C) reports the descriptive statistics for the ( ) subsample period. Treasuries included in the latter half of the sample period are noticeably larger in issue sizes and lower in coupon rates. In addition, there are more T-notes and T-bonds in the second half of the sample period. 13 These results are consistent with the general patterns observed in Figure 1. III. Empirical Results At the end of each month from 1980 through 2016, the Linear Programming model uses daily quoted bid and ask prices, adjusted for accrued interest, to find the maximum theoretical arbitrage profit, or I in the objective function (1), which will be referred to as Total Month-end Special Pricing in the remainder of the paper. The bid and ask prices are based on a $100 par value. As argued earlier, the Total Month-end Special Pricing measures the Treasury market special pricing due to market frictions and limits or costs to arbitrage. A. Descriptive Statistics Table II reports the descriptive statistics for the monthly LP results. The sample average Number of Treasuries (i.e., the number of T-notes and T-bonds each month in the feasible set of the LP model) is 176 with a minimum of 91 and maximum of 305. The mean (median) Total Month-end Special Pricing for the whole sample period is $16.51 ($14.53) with a standard deviation of $ Given the large number of T-notes and T-bonds, the Special Pricing per Treasury, defined as the Total Month-end Special Pricing divided by the month-end Number of Treasuries, provides a better sense of the magnitude of the special pricing. The mean (median) 13 The sum of the numbers of Treasuries in panels B and C is larger than the number of Treasuries in panel A due to some US Treasuries spanning both subsample periods. Treasuries spanning both subsample periods have maturity longer than one year, resulting in the average maturities of both subsamples higher than that of the full sample. 16

19 Special Pricing per Treasury is $0.106 ($0.077) per $100 par value with a standard deviation of $ As a comparison, the mean (median) bid-ask spread for the Treasury securities during the sample period is $0.078 ($0.063). Thus, the average Special Pricing per Treasury is about 130% of the average bid-ask spread. B. Intertemporal Variations in Treasury Market Price Efficiency As discussed in Section II, several reforms and institutional developments since 1980 might have reduced market frictions and lowered costs to arbitrage. Thus, Treasury market price efficiency is likely to have improved over the sample period. To examine the intertemporal variations in price efficiency, we divide the whole sample into eight subsamples of 60 months each, except the subsample period. Table III reports the subsample means of the two measures of special pricing. The average Total Month-end Special Pricing was $25.58 during the subsample period. This measure of special pricing was halved a decade later to $14.52 during the subsample period, and has remained at around $10 since The decline in Special Pricing per Treasury was even more dramatic. During the period, the Special Pricing per Treasury was $0.25. It dropped to $0.03 in the two most recent years. Two other interesting patterns emerge from Table III. First, the gains in price efficiency were very steep from 1980 to1994, but have become more gradual since then. This pattern is consistent with the fact that most market efficiency-enhancing reforms and institutional developments occurred during 1980 to 1994, including the Tax Reform Act of 1986 and the introduction of the Treasury STRIPS program. In addition, as Figure 2 shows, the average bidask spreads decreased dramatically from 1980 to

20 Second, the gains in price efficiency over time are not monotonic and there are two subperiods ( , and ) where the market pricing became less efficient than the previous sub-period. The period experienced the Asian and Russian financial crises as well as the collapse of the Long Term Capital Management. The time period overlapped significantly with the recent global financial crisis. An extensive literature suggests that financial crises are often coupled with disruptions in financial intermediation and increases in financial market frictions (Gertler and Kiyotaki, 2010). To further investigate if the price inefficiency is related to financial crises, we plot the two month-end measures of special pricing over the sample period in Figures 4 and 5. Despite significant month-to-month variation, there is a clear downward trend in these measures of Treasury special pricing, particularly from the 1980 to early 1990s, consistent with the pattern observed in Table III. There is indeed a significant spike in both measures of price inefficiency during late 2008 and early However, the spike is quite short-lived, and the level of inefficiency came back down quickly. Hu, Pan, and Wang (2013) and Musto, Nini and Schwartz (2016) document a similar jump in Treasury special pricing during the financial crisis. Hu, Pan and Wang (2013) presents evidence that severe lack of risk-arbitrage capital during the crisis period kept potential arbitrageurs from exploiting significant price inefficiency. On the other hand, Musto, Nini and Schwartz (2016) argues that heterogeneous investors preference for liquidity drives the spike in special pricing. Both explanations suggest increased market frictions and limitations on arbitrage during the financial crisis, supporting our basic premise that the LPbased approach captures the aggregate effects of market frictions and limits/costs to arbitrage on price efficiency. 18

21 For the time period, there is no similar spike in special pricing, but the level of price inefficiency seemed to be elevated at the peak of Asian and Russian financial crises in C. Persistence in Special Pricing The LP model is designed to detect any special pricing, whether due to temporary and transient factors or inherent and persistent market frictions. An alternative explanation for the large improvement in price efficiency over the last four decades is large decreases in temporary or transient factors of price inefficiency. For example, due to generally lower liquidity in the early years, prices of some Treasury securities might be temporarily deviate from their fundamental values due to short-term imbalances in buy and sell orders. This transientinefficiency hypothesis suggests that special pricing is random and can be arbitraged away quickly. High price inefficiency at the end of a given month might attract more arbitrage trading in the following trading days which would result in improvement in price efficiency. In addition, the special pricing identified by the LP model could also be a result of errors in reported prices, particularly for the earlier sample period. Erroneous reported prices of some Treasury securities could result in large LP special pricing and corrections of the reported price errors in the following trading days would reverse the special pricing. This explanation also suggests that the documented decline in price inefficiency over the last four decades could be a result of improved data quality and more accurately reported Treasury prices over time. On the other hand, if the special pricing is a manifestation of some inherent hard-toarbitrage-away market frictions, overpriced (underpriced) securities would tend to remain overpriced (underpriced) over time. In other words, the special pricing should be persistent after each month-end LP model. We call this the persistent-inefficiency hypothesis. 19

22 To distinguish these two hypotheses, we examine special pricing on a daily basis, based on the month-end LP Buy and Short Weights. 14 Since all US T-notes and T-bonds in our sample mature either at the end or the middle of the month, there is no change in the composition of T- notes and T-bonds from the end of the month when the LP is run and the following 10 trading days. To detect the potential persistence in relative special pricing, we multiple the month-end LP weights by the subsequent daily price of each respective Treasury security and sum up the products as in the following equation: = (,, ), ) (6) (, H is the total number of Treasury securities., (, ) is bond j s Short Weight (Buy Weight), from the LP model at the end of month k., (, ) is the bid (ask) price of bond j at the i th trading day after the end of month k. Thus, measures the special pricing at the i th trading day after the end of month k. The transient-inefficiency hypothesis implies that would quickly converge to zero or even negative as 1) special pricing is arbitraged away, 2) reported pricing errors are corrected, 3) overpriced securities at end of month k might become less overpriced or even underpriced at date k+i, and vice versa. On the other hand, the persistent-inefficiency hypothesis predicts that will remain positive and relatively constant over time, i.e., overpriced (underpriced) securities remain overpriced (underpriced). Panel A of Table IV gives the empirical results for the whole sample period. Trading Day 0 represents the end of each month and Trading Days 1 to 10 are the first to the 10 th trading 14 Note that we do not run the LP model on a daily basis. We could not infer whether the special pricing from daily runs of LP model is driven by the same set of mispriced securities overtime. In other words, we could not determine the persistence or transience in special pricing of individual securities overtime with daily LP model runs. 20

23 days of the following month. The statistics for Trading Day 0 are LP results, already reported earlier but reproduced in the Table for easy comparison. The statistics for Trading Days 1 to 10 are based on equation (6). The average Total Month-end Special Pricing decreases from $16.51 at the month end to $11.70 on the first trading day of the following month, suggesting that about 29% of the LP special pricing is transient. The special pricing decreases slowly over the next nine trading days to $8.98 by day 10. The standard deviations of daily special pricing are much larger than that of the Month-end Special Pricing. Even so, the 10 th percentile of the daily special pricing remains positive until the ninth trading day. To examine the robustness of the patterns observed above, Panels B and C report the empirical results for two subsample periods: the and the periods. 15 Similar patterns of persistent special pricing are observed for both subsample periods, even though their initial Month-end Special Pricing differs significantly. One noticeable difference, however, is that the component of transient special pricing in the early sample period is much smaller, accounting for less than 5% of the Total Month-end Special Pricing. On the other hand, for the subsample period, the transient special pricing accounts for more than 35% of the Total Month-end Special Pricing. This finding suggests that reduced market frictions and impediments to arbitrage in more recent years not only enhance the overall market price efficiency, but also make special pricing easier to be arbitraged away. 15 The two subsample periods have the highest and lowest Total Month-end Special Pricing respectively as reported in Table III. Other subsample periods have similar empirical results and, for brevity, are not reported. 21

24 Overall, the empirical results suggest that the majority of the special pricing identified by the LP model at the month end is persistent and hard to arbitrage away, consistent with the persistent-inefficiency hypothesis. D. Robustness Checks Numerous studies have documented the on-the-run effect for the relative pricing of US Treasuries. A natural question is to what extent the one-the-run effect contributes to the overall Treasury market special pricing. To answer this question, we remove all the on-the-run issues from the feasible set and re-run the LP model every month. The Total Month-end Special Pricing and Special Pricing per Treasury without the on-the run issues are reported in the third column of Table V. Column Two reproduces the results for the whole sample in Table II for ease of comparison. The Total Month-end Special Pricing (Special Pricing per Treasury) without the on-therun issues is $ ($0.096). Thus, about 90% of the special pricing is not due to the on-therun effect. That said, the on-the-run effect is salient. On average, there are about 8 on-the-run issues each month, or less than 5% of the total number of Treasuries, but they account for about 10% of the aggregate special pricing. There is a concern about possible stale prices in the CRSP bond data. As noted by prior research, some old T-notes and T-bonds do not trade often in the secondary market and the quoted prices from CRSP might not be actual tradeable prices. For example, Sarig and Warga (1989) find a significant percentage (2% to 10%) of CRSP Treasury price quotes at the end of month exactly match the quotes at the end of the previous month, from 1926 to The authors call these price runs and argue that they could be used as a measure of liquidity. 22

25 To address this concern, we check the end-of-month price quotes against the price quotes on the previous trading date. On average, about 8% of quotes represent daily price runs. 16 The LP model is re-run without Treasuries with daily price runs in the feasible set. The results are reported in the fourth column of Table V. The Total Month-end Special Pricing (Special Pricing per Treasury) decreases to $ ($0.102) after exclusion of Treasury securities with a daily price run. Thus, our results are not dramatically affected by potentially stale prices. Some previous studies based on yield-curve fitting model generally exclude 30-year T- bonds due to difficulty in estimating long term yields (see, for example, Hu, Pan and Wang, 2013). In addition, Musto, Nini and Swartz (2016) find large price differentials between T-notes and 30-year T-bonds with 10 or less years left to maturity. To check if our main results are robust to the exclusion of 30-year T-bonds, we re-run the LP model without T-bonds in the feasible set and report the results in the fifth column of Table V. The Total Month-end Special Pricing decreases to $14.607, but Special Pricing per Treasury remains largely unchanged. Finally, the LP model assumes that an arbitrageur can trade at the bid or ask price without moving the market. Large buy or sell orders can possibly move the prices, making some potential arbitrage impractical. To address this, we artificially increase the quoted bid-ask spread by 100%. The implicit assumption is that a buy (sell) order will push the price higher (lower) by 50% of the quoted spread. This approach takes into consideration the relation between liquidity and price impact since less liquid bonds tend to have both large bid-ask spreads and high price impacts. 16 We also checked for monthly price runs for our sample and find them much rarer, at less than 1%, than that documented by Sarig and Warga (1989), suggesting greater improvement in the liquidity of the US Treasury market. 23

26 The empirical results based on the inflated bid-ask spreads are reported in the last column of Table V. The Total Month-end Special Pricing (Special Pricing per Treasury) decreases to $ ($0.093). While it is not surprising that higher bid-ask spreads lower potential arbitrage profit since they represent costs to arbitrage, the impact is fairly limited. Thus, our main empirical results are robust to potential price impacts of large trades. IV. Conclusion Market frictions can lead to price inefficiency, i.e., the violation of the Law of One Price. Limitations on arbitrage prevent potential arbitrageurs from fully exploiting price inefficiencies and making markets completely efficient. A linear programming model is used to examine the price efficiency of the marketable long term Treasury security market from 1980 to T-notes and T-bonds have had an average special pricing of about $0.11 per $100 par value, more than 30% higher than the average bid-ask spreads. However, there has been a significant improvement in price efficiency over the last four decades. Average special pricing per Treasury security deceased from $0.25 from to less than $0.05 since This dramatic improvement in the price efficiency coincides with several institutional and market developments and tax changes that tend to reduce market frictions and lower costs to arbitrage. In recent years, the US Treasury market has become very efficient in pricing and is close to the perfect market condition as assumed by the Linear Programming model. 24

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