What Drives Interest Rate Swap Spreads?

What Drives Interest Rate Swap Spreads? An Empirical Analysis of Structural Changes and Implications for Modeling the Dynamics of the Swap Term Structure Kodjo M. Apedjinou Job Market Paper First Draft: December 22 This Draft: November 23 Abstract Existing models of the term structure of interest rate swap yields assume a unique regime for the data generating process and ascribe variations in swap-treasury yield spread to default risk or to liquidity premium. However, the interest rate swap market has been marked by economic events and institutional changes that might have significant effects on the data generating process, and thus on the relationship between the swap spread and its determining factors. We investigate the stability of the relationship between the swap spread and its determining factors with the structural change econometric techniques of Bai and Perron (998). The structural change tests produce endogenous break dates and associated confidence intervals. We trace the break dates to events related to liquidity, default, and institutional changes in the swap market. We find that default risk is an important source of variation of the swap spread at the beginning of the sample period, but is relatively less important at the end. Liquidity is, by contrast, more important towards the end of the sample period. Since these results call into question the assumption of one regime, we propose and estimate a joint Treasury and swap term structure model that accommodates regime switching. Evidence from the maximum likelihood estimation provides considerable support for the regime switching model. Consequently, the implied swap spreads may differ greatly across regimes. This finding suggests that the failure to account for regime shifts may result in significant mispricing of corporate debt, mortgage-backed securities, as well as derivatives that increasingly use the swap spread as a benchmark for pricing and hedging. JEL classification: G2; G3; G4 Keywords: Interest rate swaps; Liquidity; Default risk; Structural changes; Regime switch; Term structure model This research has benefitted greatly from the advice and direction of Geert Bekaert and Suresh Sundaresan and helpful comments from Andrew Ang, Ruslan Bikbov, Jean Boivin, Anna Bordon, Mike Chernov, Loran Chollete, Andrew Dubinsky, Mira Farka, Stephen Figlewski, Li Gu, Raghu Iyengar, Michael Johannes, Stephan Siegel, Maria Vassalou, Vikrant Vig, Yangru Wu, and participants at the 23 FMA doctoral consortium. Needless to say that I am responsible for any remaining errors. PhD Candidate, Columbia Business School, Doctoral Program, 322 Broadway, 3 URIS Hall, New York, NY 27. E-mail: kma25@columbia.edu. Phone: (22) 853-973

Introduction The plain vanilla interest rate swaps are agreements to periodically exchange fixed for floating payments based on a fixed notional amount or principal. The floating payment is usually indexed to the LIBOR (London Interbank Offer Rate). The fixed payment is based on the swap rate which is defined as the yield of a recently issued Treasury of the same maturity as the swap contract, plus the so-called swap spread. Arguably, the central empirical issue surrounding swaps is what determines interest rate (IR hereafter) swap spreads. These spreads have varied from a low of roughly 25 basis points to more than 5 basis points, sometimes moving violently. The obvious question is: why do they fluctuate so much? To get a sense of these movements, Figure displays both the swap spreads and their weekly changes for the 2, 5, 7, and -year maturities, from April 987 to December 22. A casual examination of these graphs of the interest rate swap spreads reveals at least three distinct patterns across all maturities. From April 987 to December 989, the swap spreads are high and very volatile. There is a noticeable decrease in magnitude and variability of the swap spreads from early 99 to mid-998. The behavior of the swap spreads from late-998 to the end of 22 mirrors that of the early part of the sample period. The wild time series pattern of the swap spread can mean that its explanatory factors display a similar behavior while their coefficients remain unchanged. However, it can also mean that in addition to any time series pattern changes of the factors, their coefficients also change over time. In this paper, we examine the stability of the relationship between the swap spread and its drivers through structural change tests. The factors often considered by existing models in explaining the variations of the spread are the counterparty default risk, the default risk in the LIBOR market, and the liquidity premium in the Treasury market. While there is general agreement on the relevance of these swap spread factors, there is disagreement on their relative importance. First, we have empirical evidence provided by Sun, Sundaresan, and Wang (993), Cossin 2

and Pirotte (997), Duffie and Singleton (997), and Mozumdar (999), of the importance of credit risk in pricing interest rate swap contracts. Duffie and Singleton (997) use this empirical finding to develop a term structure model of swap yields, where the cash flows in a swap contract are discounted at the liquidity- and default-adjusted short rate. In their framework, swap rates become par bond rates of an issuer who remains at a LIBOR credit quality throughout the life of the swap. Using the Duffie and Singleton (997) framework, Liu, Longstaff and Mandell (22) decompose the spread into liquidity and credit risk components and find that both components vary significantly over time. Second, Duffie and Huang (996), Hentschel and Smith (997), Minton (997), and Grinblatt (2) find weak or no evidence of counterparty credit risk pricing in swap spreads. Collin-Dufresne and Solnik (2) and He (2) argue that the many credit enhancement devices, used by swap market participants to mitigate credit risk, have essentially rendered the swap contract risk-free. These authors propose a model of the term structure of swap yields, where they discount the cash flows in the swap contract by the risk free short rate. As for the default risk in the Eurodollar market (LIBOR default risk), researchers have shown that swap spreads behave very differently from corporate bond spreads: Evans and Bales (99) find that swap spreads are not as cyclical as A-rated corporate spreads, while Chen and Selender (994) find weak explanatory power of AA-AAA corporate bond spreads for the swap spreads. Based on these conflicting findings, and the observed time series properties of the swap spread, we investigate whether there have been changes in the IR swap spread data generating process. In other words, we investigate the stability of the relationship between the swap spread and its determining factors using the structural change methodology of Bai and Perron (998). We find that the relative importance of these factors changes over time. The first part of the paper then tries to reconcile the different findings in the literature. We attempt to disentangle the liquidity and default components in the swap spread and their relative importance through time. We show that the liquidity and default factors do play different roles in different periods. Specifically, we identify a regime where default risk 3

is the most important determinant of the swap spread, and a second regime in which the liquidity in the Treasury market is the most important determinant of the swap spread. The presence of these different regimes coincides with well-known economic events: evidence in Gupta and Subrahmanyam (2) of mispricing of the swap contract in the early part of the sample period; change in the swap market microstructure; the S&L crisis in the late 98s that increased default risk in the banking sector; Treasury actions such as the change in the long bond auction cycle in 993, and the buyback program in spring 2; the aggressive cutting of the target rate by the Fed in the early 99s; the Russian default and LTCM crisis in 998; the Y2K liquidity problem in 999. Also, institutional changes, such as credit enhancement innovations in the swap market, affect not only the relative importance of counterparty default risk but also the characteristics of the swap contract itself (see Johannes and Sundaresan (23)). Given these findings, the second part of the paper follows naturally: To the best of our knowledge, this is the first paper to formally investigate a regime-switching term structure model of the swap yields that is consistent with these empirical findings. We draw on affine term structure, regime-switching, and reduced form models of risky bond price literatures, to formally propose a three-factor swap term structure model with regime shifts. The model is formulated to incorporate the implications of the structural change tests, while not sacrificing the analytical tractability usually afforded by traditional affine term structure models. Specifically, we posit the existence of both a default and liquidity regime in our term structure model. The results of this model are consistent with the early empirical findings. we were able to match the smoothed regime probabilities to the sub-periods found through the structural change tests. Besides the literature on the determinants of the IR swap spread and the term structure of swap yields, this paper is also related to the econometric studies that examines issues of structural changes in a linear regression model. Specifically, we use the econometric techniques developed by Bai, Lumsdaine, and Stock (998) and Bai and Perron (998) to More details on the structural break literature can be found in Nyblom (989), Andrews (993), Andrews and Ploberger (994) 4

estimate whether there are breaks in the time series properties of swap spreads and to date the break points accordingly. The events cited above provide the motivations to do the break tests. We first test the hypothesis of no break against the alternative of at least one break. Second, we do a sequential test of one break versus the alternative of two breaks. Given that we do not reject two breaks, we test the null of two breaks versus the alternative of three breaks, etc. until we cannot reject the null. This constitutes the Bai and Perron (998) test of structural changes. The Bai and Perron (998) test allows us to divide the full sample period by the break points and examine the behavior of the IR swap spread in each sub-period. By examining the behavior of IR swap spreads before and after a break, one can investigate the changes a break induces in the stochastic process governing the variables in the model. If structural breaks were not taken into account, any inference about the time series properties of IR swap spreads using the full sample would be invalid. For robustness, we apply the Bai, Lumsdaine, and Stock (998) test to a reduced-form model of the determinants of swap spreads. This is needed in order to distinguish between breaks in the joint properties of the determinants of the IR swap spread and breaks in the relationship between the IR swap spread and its determinants. This is an interesting project because, first, in terms of notional amount outstanding, interest rate swaps are the largest derivative contracts in the world with a global notional size of roughly 8 trillions dollars at the end of 22. 2 Second, swap contracts have become important financial instruments for managing interest rate risk. Previously, Treasuries were the main vehicle for hedging; however, when the government retired debt in the late 99s, hedgers increasingly turned to the swap market. Third, failure to account for regime shifts may result in significant mispricing of swap yield sensitive securities, such as corporate debt, mortgage-backed securities, as well as other fixed income securities and derivatives. Lastly, Smith et al. (986), Turnbull (987), Arak et al. (988), Kuprianov (994), and Aragon (22) argue that the introduction of interest rate swap contracts brought an additional non-redundant financing choice to the market which allows both borrowers and lenders to 2 Bank for International Settlements, 23, Regular OTC Derivatives Market Statistics. 5

affect as they please the characteristics of their cash flows. The paper is organized as follows. In the next section, we discuss the determinants of the IR swap spread. Section 3 contains the data description. In section 4, we present the results of the structural change tests. The regime-switching term structure model and its estimation results are in section 5 and we conclude in the final section. 2 Determinants of Swap Spreads To avoid a kitchen-sink type approach, we review below the arguments in Brown, Harlow, and Smith (994), Nielsen and Ronn (996), Grinblatt (2), He (2) and Cooper and Scholtes (22) among others, that link the swap spread to its fundamental drivers. Indeed, consider the following zero value portfolio: Short sell P dollars of government bonds with maturity of T years, trading at par and yielding the fixed coupon rate of C paid semiannually. Invest the proceeds in six-month general collateral (GC) repo and roll over at each six-month interval over the life of the government bond above. Enter into an IR swap contract to receive fixed swap rate S and pay six-month LIBOR at every six-month interval on the notional amount of P dollars for T years. For simplicity, we assume that the counterparties involved have the same degree of credit worthiness and will maintain this level of credit worthiness throughout T years; this implies that there is no compensation for credit risk such as posting of collateral. Finally, we assume that there are no transaction and information costs to entering the IR swap and repo markets. See Figure 3 for a diagram of the above transactions. Every six months, the above portfolio yields the cash flows ((S C) (LIBOR GC)) P. The existence of no arbitrage implies: Present Value (S C) = Present Value (LIBOR GC) 6

The above relationship shows that the IR swap spread is approximately a function of the LIBOR-Treasury rate (GC) spread. The LIBOR-Treasury rate spread encompasses the default risk in the banking sector and the liquidity of the Treasury market. The IR swap spread is also a function of the discount rate used in the present value calculation which reflects both counterparty default risk and some adjustment for liquidity as in Duffie and Singleton (997). In summary, the IR swap spread depends on a short rate, on default risk factors, and on a liquidity factor. Different authors have found different explanatory power of the above factors for movements in the IR swap spread. In this paper, we are not proposing a new model of the IR swap spread; instead, we are investigating the relative importance of these established determinants of the IR swap spread through time by applying the structural break methodologies of Bai and Perron (998) and Bai, Lumsdaine and Stock (998) to a model of the IR swap spread and its explanatory factors. In other words, we will be testing the hypothesis that the importance of the different factors is period dependent or time-varying. Below, we review the determinants of the IR swap spread. 2. Default Risk From the above analysis, swap spreads could be impacted by two sources of default risk. First, IR swap contracts are traded over-the-counter and unlike futures or other select derivatives, are not explicitly backed by a clearing corporation or by an exchange. Therefore, IR swap contracts are subject to counterparty default risk. The question often posed by researchers is how much counterparty default risk is priced into the swap spread. Even though the cash flows in an IR swap contract are equivalent to the cash flows in a bond transactions, Sun, Sundaresan, and Wang (993) shows that the default premium required in the IR swap market must be much less than the default premium in the bond market because of some important differences between the IR swap market and the bond market. For instance, the principal in the IR swap market is just notional whereas in the bond market, the principal has to be exchanged. Moreover, in an IR swap contract, if 7

one of the counterparties defaults, the other counterparty is automatically relieved from the rest of its obligations. Also, throughout the history of the IR swap market, and much more so recently, there have been credit enhancement innovations such as transaction with only an approved list of clients, the use of collateral, and marking-to-market to explicitly deal with the counterparty default risk. In a 999 survey, the International Swaps and Derivatives Association (ISDA) finds a widespread use of collateral in swap transactions. Litzenberger (992) notes that weaker credit-rated counterparties are either simply rejected or required to collateralize the IR swap contracts, rather than be quoted higher spreads. Johannes and Sundaresan (23) point out that unlike a collateralized loan where the lender is automatically prevented from liquidating the collateral by the filing of a bankruptcy petition, the collateral supporting a swap may be liquidated and applied by the solvent counterparty to offset a positive settlement amount. Also, long-term swaps with maturities in excess of years generally contain credit triggers. A typical credit trigger specifies that if either counterparty s credit rating falls below investment grade (BBB), the other counterparty has the right to have the swap cash-settled. The evidence of the impact of counterparty default risk on the IR swap spread is mixed. Sun, Sundaresan, and Wang (993) argue that dealers credit reputation has an effect on swap rates. Cooper and Mello (99), Bollier and Sorensen (994), Cossin and Pirotte (997), Mozumdar (999) also find evidence of credit risk pricing in the IR swap market. However, Duffie and Huang (996) find that basis points difference in debt rates correspond to basis point difference in swap rates. Similarly, Hentschel and Smith (997) present a theoretical model of the counterparty default risk in swap and estimate conservatively the expected annual loss rate in the swap market to be.25 percent of the notional amount. We investigate whether counterparty credit risk might have been an important determinant of the swap spread in the early stage of the IR swap market and whether current industry practices have essentially removed this component from the IR swap spread. After a major default crisis such as the S&L crisis in the late 98s or the 998 financial crisis, one could expect economic agents to weigh more the default risk factor in pricing an IR swap contract. 8

We investigate whether the sensitivity of IR swap spread to counterparty default risk is time-varying. This issue is particularly important for agents who are deciding whether to hedge their corporate debt portfolios with Treasuries or IR swaps. Second, given that the IR swap spread is a function of the LIBOR-Treasury rate spread, it also reflects the default risk in the banking sector. Since LIBOR is the rate on short term loans to banks rated A to AA on average, the default risk in the LIBOR market could be very small in normal times. In other words, the LIBOR does reflect the default risk of highly rated banks and not the default risk of banks with serious credit risk problems because banks with deteriorating credit risk are simply removed from the calculation of the LIBOR. However, in turbulent times like the S&L crisis in the 98s and early 99s, the LIBOR does reflect high default premium since all banks are affected by a generalized credit problem. 2.2 Liquidity Premium Again, since the no-arbitrage argument above shows that the IR swap spread is a function of the LIBOR-Treasury rate spread, it follows that the IR swap spread reflects the relative liquidity of Treasuries. Using the empirical findings in Evans and Bales (99) and in Chen and Selender (994) that show significant differences between the time series properties of corporate credit spreads and IR swap spreads, Grinblatt (2) argues that swap spreads are not at all due to credit risk and that liquidity is a more plausible determinant of IR swap spreads than credit risk. The author models swap spreads as compensation for the convenience yield to Treasury notes associated with their relative liquidity and potential to go on special in the repo market. Indeed, Duffie (996) and Jordan and Jordan (997) document that holders of Treasury bonds that go on special can borrow at below market rates, known as special repo rates, using Treasuries as collateral. Treasury securities are one of the basic vehicles for hedging interest rate sensitive positions. Investors that own Treasuries and are sophisticated enough to participate in the repo market by lending out their Treasuries to hedgers typically receive loans at abnormally low interest rates. This 9

convenience yield is lost to an investor wishing to receive fixed rate payments, who, in lieu of purchasing a Treasury note, enters into an IR swap contract to receive fixed payments. Liu, Longstaff, and Mandell (22) show additional support for liquidity risk as a primary determinant of swap spread changes. Indeed, after decomposing the swap spread into a liquidity and a default risk component, the authors find that even though the default risk component is typically the largest component of swap spreads, the liquidity component, however, is much more volatile and can often exceed the size of the default risk component. Therefore, most of the variations in swap spreads are attributable to changes in the relative liquidity of swaps and Treasury bonds. Furthermore, they show that the historically high swap spreads recently observed in the financial markets are largely due to an increase in the liquidity of Treasury securities rather than to a decline in the credit worthiness of the financial sector. In a VAR model, Duffie and Singleton (997) find that a shock to a standard measure of liquidity has a positive and statistically significant long term effect on the swap spreads. 2.3 The Short Rate In addition to the factors of liquidity and default considered above, we also include in our regression model a measure of the risk-free short rate. The swap spread is a function of the short rate not just because the short rate is needed to discount the cash flows of a swap contract but also because the short rate plays a first order role in a corporation decision to hedge its interest rate risk. Tuckman (22) argues that recently, a lot of the sharp movements in IR swap spreads can be attributed to the activities of hedgers in the mortgage backed securities (MBS) market. Indeed, when interest rates fall, the duration of MBS falls; therefore, to increase duration, the hedgers usually enter into a swap contract to receive fixed swap rate, and thus negatively affecting the magnitude of the swap spread. The reverse is true when interest rates rise. This effect is important because the of size of the MBS market.

3 Data Description To analyze the IR swap spread, we obtained from Datastream weekly (ending on Friday) observations of IR swap rates and constant maturity Treasury rates of maturity 2, 5, 7, and years. Also from Datastream are the 6-month constant maturity Treasury rate and the 6-month LIBOR rate. We start the analysis from April 3, 987 because of data limitations on the IR swap rates, to December 27, 22 for a total of 822 observations. IR swap spreads are calculated as the difference between the IR swap rates and the constant maturity Treasury rates of the same maturity. In Figure we plot the IR swap spreads for all maturities. The right Panel of Figure shows the graphs of the weekly changes in the spreads. As noted in the introduction, the IR swap spreads are very volatile at the beginning of the sample period, stay fairly constant in the middle of the sample, and become more volatile recently. The average of the IR swap spreads goes from 43 basis points for the 2-year maturity to just over 65 basis points for the -year maturity. For the structural change tests, we focus the analysis on the -year maturity IR swap because it is one of the most liquid IR swap contracts. We use the 6-month constant maturity Treasury rate to proxy for the short rate. Panels and 2 of Figure 4 show the graphs of the -swap IR swap spread with the 6-month constant maturity and the Federal Funds target rate respectively. Except for the beginning of the sample period, the -year swap spread moves generally in the same direction as the constant maturity Treasury and the Federal Funds target rate. As a measure of the liquidity factor, we follow standard practice as in Duffie and Singleton (997) and Krishnamurthy (22) and use the spread between the -year off-the-run and the on-the-run Treasury bond yields. The on-the-run and the off-the-run Treasury rates are obtained from a bank. The on-the-run yields are the yields on the most recently auctioned Treasuries and the off-the-run yields are the yields on the Treasuries issued in the previous auctions. From late 998 to the present, the off/on-the-run spread has significantly increased and become more volatile. This increase in the demand for liquidity in 998 corresponds

to the flight-to-quality following the financial crisis in the second half of 998 when investors moved their capital to the safest possible assets such as the newly issued government Treasuries. The recent increase in volatility of this measure of liquidity could also be explained by the flight-to-liquidity during the Y2K liquidity crisis in 999 and the decision of the government to repurchase some Treasuries in 2 (see Longstaff (22)). The average off/on-the-run spread is about 4 basis points with a high of nearly 25 basis points. Panel 3 of Figure 4 shows the graph of the -year off/on-the-run and the -year IR swap spread. It is usually argued that the Treasury-Eurodollar spread or the TED spread has two components: the default risk in the banking sector and the relative liquidity associated with Treasury. Since we have in the -year off/on-the-run spread, a clean measure of the liquidity associated with Treasury, we can extract the other component of default risk from the TED spread. Therefore, we proxy the banking default risk with the residual obtained from the regression of the 6-month TED spread on our liquidity factor. The 6-month TED spread is the difference between the 6-month LIBOR and the 6-month Treasury rate. For systematic corporate default risk, we follow Collin-Dufresne, Goldstein, and Martin (2) and proxy it with the Chicago Board Options Exchange s VIX index. VIX is a weighted average of implied volatilities of near-the-money OEX (S&P ) put and call options and was obtained from Datastream. The natural measure of a firm s default risk is its probability of default during the life of the contract considered. An aggregate measure of default risk such as the average expected default frequency (EDF) by Moody s KMV should be considered. The probability of default is an increasing function of the volatility of the firm s assets. More intuitively, a corporate debt is a combination of a risk-free bond less a put option on the firm s assets with the strike price equal to the face value of the debt. Ceteris paribus, a firm with more volatile asset value is more likely to reach the default boundary condition. Therefore, default risk is an increasing function of volatility. This could also proxy for counterparty default risk. VIX has been used similarly in Collin-Dufresne, Goldstein, and Martin (2) in the context of explaining corporate bond spread. We tried other measures of aggregate default risk and the results do not differ qualitatively. Panels 4 and 5 of Figure 4 show the graph of the -year 2

swap spread with the banking default and the aggregate default factors respectively. 4 Structural Change Tests Most empirical models of the IR swap spread assume the model generating the spread to have constant parameters. However, there is anecdotal evidence that suggests structural changes in the data generating process. Indeed, it has been common knowledge (or at least many researchers suspect) that there have been breaks in the time series properties of IR swap spreads due to the multiple events enumerated earlier. For example, He (2) suspects that counterparty credit risk is much less important today in the IR swap market than it was at the beginning of the market because of credit enhancement innovations in the IR swap market. Tuckman (22) argues that the low levels and low variability of the IR swap spreads in the early 99s were due to the recovery of the banking sector from the S&L crisis in the 98s whereas the high levels and fluctuations of the IR swap spread in the late 99s were due to the perceived scarcity in the supply of U.S. Treasuries. Gupta and Subrahmanyam (2) show that there has been mispricing of IR swap contracts during the early years. Moreover, as mentioned earlier, different researchers find different factors affecting IR swap spreads. This paper is an attempt to reconcile these different findings and investigate their implications for the dynamics of the term structure of swap yields. Since structural changes could blur the results of any empirical analysis, in modeling a time series process with potential breaks in the parameter values of the model, one can deal with the temporal instability of parameters by choosing a fairly short period of time so that variations in the parameter values of the model are negligible. With that approach, one can then be fairly certain that a rejection of a tested model is not due to the breaks in the parameter values. However, this solution is not applicable to the IR swap spread because of the relatively short history of the swap market. Instead, in the empirical analysis, we conduct formal tests of structural changes for a number of reasons. First, the break tests can fail to reject the null hypothesis of no structural break and failure to reject the null hypothesis 3

suggests that the economic events and new market institutional features enumerated above have had little impact on the data generating process of the IR swap spread; in that case, the break tests solidify the IR swap spread as a strong benchmark with respect to which other assets can be priced. Second, we want to possibly motivate a regime switching term structure model of IR swap yields. A regime-switching model may be appropriate if the events that cause the structural changes are recurring, as is the case for some liquidity and default risk events. However, some of the changes engendered to the IR swap market may be irreversible. For example, it is hard to imagine a future state where there is no use of collateral in the IR swap transactions or where the IR swap market becomes a thin market. The Chow (96) F test is one of the earliest techniques that test for structural breaks in a linear regression model. The main drawback of the Chow F test is that the break date has to be known exactly. Its simplicity is particularly attractive in the case where the date of the event causing the break is widely accepted. However, it is hard to apply in the case where the break date is not known precisely. This is relevant to the case at hand where the dates of some of the events potentially causing the breaks in the time series properties of IR swap spreads are not easily identifiable. For instance, the Chow F test cannot help us answer the question of whether the increase in the use of collateral has had any effect on the IR swap spreads. Recently, considerable attention has been paid to the case where the break date is not known. See Nyblom (989), Andrews (993), Andrews and Ploeberger (994), Andrews, Lee and Ploeberger (996), Bai, Lumsdaine, and Stock (998), and Bai and Perron (998). Instead of assuming a priori the number of breaks and their respective dates, econometric techniques have been developed to endogenously estimate the break date(s). The econometric technique of Bai and Perron (998) is well suited for our purpose of investigating structural breaks in the relationship between the IR swap spread and its determining factors because it encompasses tests that determine whether a break occurs, the number of breaks given that there is a break, and inference about each break date and its confidence interval. More specifically, in a multiple linear regression model, if we know the exact number of 4

breaks but not their actual dates, the methodology can estimate the break dates through the least-squares principle. The idea is to pick the partition of the sample period that minimizes the sum of square residuals. The partition thus selected consists of the break dates. There are two types of test to determine whether there is a structural change. There is the sup F T (k) test that tests the null of no breaks versus the alternative of k breaks and the double maximum test that tests the null of no breaks versus the alternative of an unknown number of breaks. The method to determine the number of breaks consists of sequentially applying the sup F T (l + l) test with the null of l breaks versus the alternative of l + breaks starting with l =. One concludes for a rejection in favor of a model with (l + ) breaks if the overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of squared residuals from the l breaks model. Below are the results of the structural break test of Bai and Perron (998) applied to a linear model of the swap spread and its determinants enumerated above. 3 4. Empirical Specification From section 2, we assign variations in IR swap spreads to four main sources: the short rate proxied by the 6-month constant maturity Treasury rate, the default risk in the Eurodollar market, the liquidity factor proxied by the -year off/on-the-run spread, the general corporate default risk proxied by the CBOE s VIX index. In the empirical analysis, we assume the following simple multiple linear regression model with m breaks (m + regimes), where SS denotes the -year IR swap spread, T reasury is the 6-month constant maturity Treasury, LIBORdefault is the default risk in the Eurodollar market, Off/On is the -year off/on-the-run spread, and VIX is the CBOE s VIX index. SS (t) = β j +β j 2T reasury (t)+β j 3LIBORdefault (t)+β j 4Off/On (t)+β j 5V IX (t)+u (t) () where t = T j +,..., T j, j =,..., m +, with the convention that T = and T m+ = T. We proceed to the estimation of a full structural break model where all the coefficients in 3 Summary of the multiple structural changes econometric method proposed by Bai and Perron (998) is in Appendix A 5

the above equation are allowed to change. Equivalently, we test the null hypothesis of no structural break, H : β = β 2 = = β m+ where β j = (. β, j β2, j β3, j β4, j β5) j Again, we note that in this paper, we restrict our analysis to the -year IR swap spread because it is the most widely transacted contract among all IR swap contracts. Results using IR swap spreads of different maturities are similar, and thus are not reported. From the break tests, we are interested in answering the following questions about the determinants of IR swap spread in order to help resolve the conflicting findings enumerated earlier. Is the coefficient of the LIBOR default risk factor the most significant at the beginning of the sample period? An affirmative answer to this question will confirm the results that the default risk embedded in LIBOR was the most important determinant of the IR swap spread at the beginning of the sample period because of the S&L crisis in the 98s and early 99s. Is default risk much more important at the beginning of the sample period than at the end? This is related to the first point above but also takes into account the counterparty default risk factor. A positive answer to this question will help resolve two issues. First, it will be consistent with the argument that counterparty default risk does not impact IR swap spread anymore at any significant degree. Second, it will confirm the results that the default risk in the LIBOR market is no longer priced into the IR swap spread. The correlation of IR swap spread with credit risk factors has implication for deciding whether to hedge portfolios of corporate debt with either Treasuries or IR swap contracts. Does the regression on the early part of the sample period have the lowest adjusted R 2? From Figure 2, the small notional size at the beginning of the sample period hints at the low depth of the market for IR swap contracts in that time frame. This microstructure feature could potentially affect the time series properties of IR swap spread in the sense that, given that the depth of the market was low, IR swap spreads may not reflect 6

the fundamentals. Moreover, the results in Gupta and Subrahmanyam (2), who show that there was systematic mispricing of IR swap rates in the late 98s and the early 99s, could also contribute to the factors being poor explanatory sources for the variations in the spread. The authors argue that the swap rate mispricing was due to ignoring the convexity correction in the swap curve construction techniques. Are the liquidity and counterparty default risk factor coefficients most significant after the 998 crisis? At the height of the 998 financial crisis, there was an important flight-to-quality following the Russian default and the collapse of LTCM. Both the aggregate corporate default and liquidity premia must increase because people were worried about default risk, and thus sought riskless securities such as Treasuries. What is the relative importance of the liquidity and default risk factors in different sub-periods? With the widespread use of collateral and other credit enhancement devices to mitigate counterparty default risk, and with no banking crisis, we would expect default risk to have less explanatory power than liquidity in affecting variations in IR swap spreads. Is liquidity relatively more important than default towards the end of the sample period? 4.2 Break Dates and their Confidence Intervals The procedure for testing whether there is a structural change, determining the number of breaks, and estimating the break dates and their confidence intervals, consists of choosing the maximum number of breaks m and a corresponding trimming value k taken to be.5 for the case m = 5, with all possible break dates taking values between k T and T k T. We applied the above Bai and Perron (998) structural change econometric procedure to equation () while accounting for potential serial correlation and heteroscedasticity. Table summarizes the main results. 4 4 The Gauss program used to do the structural break estimation was obtained from Pierre Perron, at http://econ.bu.edu/perron/code.html 7

The values of the sup F test statistic, 5 which test the null of no break versus the alternative of to the maximum number of m breaks, are all significant at the percent level. Similarly, the values of the UDmax and the W Dmax statistics which test for the null of no break versus the alternative of an unknown number of breaks, are also significant at the percent level. Given the significance of the above three test statistics, we conclude that there exists a structural break in our IR swap spread model (). As for the number of breaks in the model, the sequential procedure developed by Bai and Perron (998) selects 3 breaks. With m = 3 breaks, we proceed to estimate the break dates and their confidence intervals. Under global minimization, the first break date is August 25,989, with a 95 percent confidence interval of August, 989 to October 3, 989. The second break date is May 8, 992 with a 95 percent confidence interval of April 24, 992 to May 22, 992. August 4, 998 is the estimated date for the third break with confidence interval of June 5, 998 to August 2, 998. All break dates are therefore precisely estimated with very tight 95 percent confidence. Note also that the confidence intervals can be asymmetric and this comes from the limiting distribution of the break dates (see Appendix A). 4.3 The Causes of the Breaks and their Implications In Table 2, we report summary statistics of the -yr IR swap spread and its determinants across the four different sub-periods estimated through the structural change test. There is a wide variation in both the means and volatilities of the variables. For instance, the IR swap spread mean and standard deviation in basis points in the last sub-period are more than double and quadruple the mean and standard deviation, respectively, of IR swap spread in the third sub-period. The volatilities of the IR swap spread in the first and second subperiods are equally high. One salient observation from Table 2 is that the third sub-period is the quietest sub-period; in general, it has the lowest mean and volatility for the IR swap spread, the 6-month Treasury (except the mean), the LIBOR default risk, the off/on-the 5 See Appendix A for formal definitions of the statistics sup F test, UDmax and the W Dmax that test whether we have structural changes. 8

run spread(except for the volatility), and the VIX index. The characteristics of this subperiod are in contrast to the characteristics of the other three sub-periods. Table 3 reports the correlation structure of all the variables in the model across the different sub-periods. We readily see time variation in the correlation coefficients. For example, the correlation between the -year interest rate swap spread and the 6-month constant maturity Treasury rate varies between a low of.64 in the first sub-period, to a high of.9 in the second sub-period. This time variation in the correlation structure confirms the earlier results of structural changes. Formally, we test the significance of the coefficients of the explanatory variables across the different sub-periods by estimating the following model: SS (t) = [ βd (t) + βd 2 2 (t) + βd 3 3 (t) + βd 4 4 (t) ] + [ β 2 D (t) + β2d 2 2 (t) + β2d 3 3 (t) + β2d 4 4 (t) ] T reasury (t) + [ β 3 D (t) + β 2 3D 2 (t) + β 3 3D 3 (t) + β 4 3D 4 (t) ] LIBORdefault (t) + [ β 4 D (t) + β 2 4D 2 (t) + β 3 4D 3 (t) + β 4 4D 4 (t) ] Off/On (t) + [ β 5 D (t) + β 2 5D 2 (t) + β 3 5D 3 (t) + β 4 5D 4 (t) ] V IX (t) + u (t) where D j (t) are dummy variables that take the value of if t is in sub-period j and otherwise. The sub-periods are: April 3, 987 to August 25, 989, September, 989 to May 5, 992, May 5, 992 to August 4, 998, and August 2, 998 to December 27, 22. Before proceeding to the estimation of the above model, we turn off the dummy variables and report in the first Panel of Table 4 the results of the simple regression of the -year swap spread on its determining factors using the whole sample. All coefficients are positive and, except for the constant term, are statistically significant. The adjusted R 2 of the regression is approximately 6 percent. The second Panel of Table 4 reports the results of the regression equation with dummy 9

variables. The adjusted R 2 is 9 percent. The constant term varies widely across the subperiods: it has a very significant value of 22 basis points in the first sub-period, becomes negative and insignificant in the second sub-period, and becomes positive and significant in the last two sub-periods. The coefficients of the short rate proxied by the 6-month constant maturity Treasury also vary over time. It is puzzlingly negative and statistically significant in the first sub-period, but becomes positive and significant thereafter. The coefficient of the 6-month Treasury is the most significant in the period from September, 989 to May 8, 992. This period corresponds to the period of aggressive cutting of the Federal Funds target rate. Indeed the Federal Funds target rate went from a high of 9.825 percent on May 5, 989 to a low of 3 percent on September 4, 992. The IR swap spread decreases significantly in the same time period. This significant positive relationship between the IR swap spread and a measure of the short rate is consistent with the argument of Tuckman (22) that a fall of interest rates increases the demand for swaps by hedgers in the MBS market to receive fixed, and thus leads to the tightening of the IR swap spread. The reverse is also true. The first two Panels of Figure 4 show that, except for the beginning of our sample period, there is a strong positive relationship between the IR swap spread and the risk free short rate. The first break date, August 25, 989, corresponds exactly to the date of the enactment of the Financial Institutions Reform Recovery and Enforcement Act (FIRREA). Indeed, the FIRREA was enacted in August 989 to address the S&L crisis and create the Resolution Trust Corporation to bail out insolvent S&Ls. The implication is that, before this break, the default risk in the LIBOR market or in the banking sector as a whole is an important determinant of variations in the IR swap spread. After this break date, the default risk in the LIBOR market should matter less. Effectively, we document that the coefficient of our measure of the default risk in the LIBOR market is the most significant prior to the enactment of FIRREA and insignificant immediately after. This addresses the first point above that the importance of the LIBOR default risk factor is conditional on a crisis in the banking sector. However, this measure is also significant in the third sub-period of May 5, 992 to August 4, 998 but insignificant in the last sub-period. 2

The coefficient of the liquidity factor, the -year off/on-the-run spread, is positive and in general highly significant throughout, except for the last period where the t-statistic is.72. This relatively low t-statistic may be due to multicollinearity problem since Panel E of Table 3 shows that the correlation between -year IR swap spread and the -year off/on-the-run spread is.73. Also, multicollinearity might have affected the coefficient of the VIX index across the sub-periods since the coefficient of the VIX index is highly significant in the full sample regression but insignificant in the dummy variable analysis. For each factor, we do a simple F test to illustrate that parameters of the IR swap model effectively change across the sub-periods. Individually, the F test rejects, at the percent significance level, the null that the constant terms, the coefficients of the risk-free short rate, and the coefficients of the liquidity factor are equal across the four sub-periods. Similarly, we reject, at the 5 percent significance level, the null that the coefficients of LIBOR default risk factor are constant across sub-periods. We were unable, however, to reject the null that the coefficients of the VIX index are the same across all sub-periods. This is not surprising since the coefficients of the VIX index are insignificant. This does not however mean that the VIX index is unimportant. Another evidence of structural changes in the relationship between IR swap spread and its drivers is the time-variation in adjusted R 2. Doing a period by period regression, 6 we note indeed that the adjusted R 2 of equation () varies between a high of 89 percent in the second sub-period to a low of 46 percent in the third sub-period. The adjusted R 2 of the first period is higher than that of the third sub-period. This result does not allow us to effectively address the third question above about the impact of mispricing on the IR swap spread in that sub-period as documented by Gupta and Subrahmanyam (2). The third break matches exactly the height of the 998 financial crisis that engendered an important flight-to-quality because of the LTCM and Russian default. Because of this ensued flight-to-quality, we try to determine whether both the liquidity and counterparty default risk factor coefficients became much more significant. Table 4 shows that, after 998, only the liquidity factor is important. This is consistent with the results of Liu, Longstaff 6 By definition the results are the same as the dummy variable approach in Table 4 and are therefore not reported 2

and Mandell (22) and He (2) that recently liquidity is a more important driver of IR swap spreads than default risk. As the market of IR swap expands, new practices such as the Master Swap Agreement which encompasses collateral agreement, marking-to-market, and rating trigger, sprouted out to facilitate transactions and mitigate default risk. With these new practices, one could reasonably expect shocks to default risk to matter relatively less in determining the behavior of the IR swap spread. Unlike the first and last breaks that coincide with well known financial events, the break on May 8, 992 is hard to pin to economic events. From Figure, the small notional size underlines the low depth of the IR swap market prior to 992. This microstructure characteristic could contribute to the structural change we observe in the relationship between the IR swap spread and its determinants. Figure 5 shows the marginal adjusted R 2 of both default and liquidity factors in explaining variations in IR swap spread. 7 We conclude from this graph that the relative importance of liquidity and default risk in affecting IR swap spread is regime-dependent. In the early part of the sample period, both the liquidity and default factors have the same relative importance in affecting the IR swap spreads with the default factor doing slightly better. In the second sub-period, the joint explanatory power of the default and liquidity factors increase very significantly but the gap between the explanatory power of liquidity and default factors widens in favor of the default factor. Indeed, the marginal R 2 of default risk shoots up to about 3 percent from percent in the first period, whereas the marginal R 2 of the liquidity factor increases to about 2 percent from percent in the first period. The third period saw the explanatory power of the default factors plummeting to about 8 percent and decreasing further to 5 percent in the last sub-period. The importance of default risk factor decreases significantly probably because of the increase in the use of credit enhancement innovations and the stability in the banking sector. As time passes, the default factors lose their explanatory power whereas the liquidity factor becomes much more important. 7 The marginal adjusted R 2 are computed as follow: for each sub-period, we run a regression of IR swap spread on a constant and either the liquidity or the default factors and computed the adjusted R 2. The marginal adjusted R 2 of the default factors (of the liquidity factor) is the adjusted R 2 of the multivariate regression of IR swap spread on a constant, on the liquidity, and on the default factors minus the adjusted R 2 of the regression of IR swap spread on just a constant and the liquidity factor (default factors). 22

We have therefore documented the existence of a regime where default risk is the most important determinant of the IR swap spread and a second regime in which the liquidity in the Treasury market is the most important determinant of the swap spread. This is strong motivation for a Markov regime-switching term structure model that we explore in the next section. 4.4 Robustness of the Structural Break Results The structural break methodology above assumes that the detected breaks are due to the changing relationship between the IR swap spread and its explanatory factors of the short rate, the liquidity, and default factors. One might instead argue that the breaks are detected as a result of breaks in the process generating these explanatory variables. To answer this concern, we test whether there is any significant structural change in the dynamics of the explanatory variables with the techniques of Bai, Lumsdaine, and Stock (998). The Bai, Lumsdaine, and Stock (998) methodology allows us to specify a reduced form vector autoregression (VAR) of the short rate, liquidity, and default factors and test whether there is a break in their dynamics. This test yields a break date on January 3, 989, with a very tight confidence region of two weeks. Given that this break date occurs way before all the break dates found in the relationship between swap spread and its drivers, it does not affect any of the conclusions. Overall, this analysis gives us confidence that the breaks we detect by applying the Bai and Perron (998) test to equation () result from the changing relationship between IR swap spreads and its determining factors rather than shifts in these explanatory variables alone. Also, in the linear model, we detect the presence of structural breaks even after including the lag of the IR swap spread and some nonlinearities in the explanatory variables. Thus, these results attest to the presence of structural changes in the relationship between IR swap spread and its determinants. 23

5 Regime-Switching Model of the Term Structure of Interest Rate Swap Yields our intent is to derive and estimate a parsimonious model of the term structure of IR swap yields that is consistent with the earlier empirical results of structural changes in the time series properties of IR swap spreads. We document the presence of a liquidity and default regimes. We follow the approach of Duffie and Singleton (997, 999), Collin-Dufresne and Solnik (2), Liu, Longstaff and Mandell (22), Duffie, Pedersen and Singleton (23), and jointly model the term structure of IR swap and Treasury yields using a three-factor affine framework. Under mild technical regularity conditions, Duffie and Singleton (997, 999) show that cash flows in the swap market can be discounted at the adjusted short rate R t which is interpreted as the default- and liquidity-adjusted short rate. This framework allows us to use existing techniques for term structure models for risk-free rates or Treasuries to develop a term structure model for IR swap yields. Indeed, Duffie and Singleton (997) show that for an IR swap contract initiated at time t to exchange at every six-month t+.5k, k =, 2,...2τ, the preset six-month LIBOR L t+.5(k ) against the fixed payment rate S τ t for τ years can be priced as follows: = 2τ E Q t k= [ ( t+.5k ) ) ] exp R s ds (Lt+.5(k ) St τ (2) t where Q denotes an equivalent martingale measure. The six-month LIBOR L t+.5(k ) is defined as: L t+.5(k ) = 2 ( ) P (t +.5 (k ), t +.5k) P (t +.5 (k ), t +.5k) (3) and P (t, t +.5k) = E Q t [ ( t+.5k )] exp R s ds t (4) 24

is the price at time t of a.5k maturity risky discount bond. Manipulation of the above equations yields the following expression for the fixed payment rate S τ t : ( ) St τ P (t, t + τ) = 2 2τ k= P (t, t +.5k) IR swap rates are thus par bond rates of an issuer who remains at LIBOR credit quality throughout the life of the contract. The Duffie and Singleton (997) framework thus offers a simple window through which we analyze the term structure of IR swap yields even though some of its underlying economic assumptions are a bit strong. 8 (5) 5. The Model Our model formulation is based on the framework of Ang and Bekaert (23) who study the term structure of risk-free rates. The model has three unobserved state variables: the one-period short rate r t, the central tendency θ t toward which the short rate adjusts, 9 and the spread process δ t which is an adjustment for time-varying default risk and liquidity in the IR swap market. Let X t (θ t r t δ t ), the vector of state variables, follow the discrete time Gaussian regime switching process under the data-generating or real-world measure: X t+ = µ (s t+ ) + ΦX t + Σ (s t+ ) ɛ t+ (6) where the regime variable s t can be either liquidity (s t = ) or default (s t = 2), and where it follows a Markov chain with transition probability Π = {p ij }, p ij = Pr (s t = j s t = i), 8 Indeed, it assumes for instance, symmetric counterparty credit risk, a homogeneous LIBOR-swap market credit quality which is against the findings in Sun Sundaresan, and Wang (993) and in Collin-Dufresne and Solnik (2). See Duffie and Singleton (997) for more details on the assumptions. 9 See Jegadeesh and Pennacchi (996) and Balduzzi, Das, and Foresi (998) 25

ɛ t = ( ɛ θr t ɛ δ t) = ( ɛ θ t ɛ r t ɛ δ t) N (, I) and µ (s t ) = κ θ θ κ δ δ (s t ), Φ = κ θ κ r κ r κ θδ κ rδ κ δ, Σ (s t) = σ θ σ r σ δ (s t ) (7) The first two state variables drive the risk-free or Treasury bond prices, and all three state variables drive the prices of risky bonds. The adjusted short rate R t is defined as R t r t +δ t. We assume that the state variables r t and θ t follow a Gaussian process and the spread factor δ t follows a regime-switching process. The model therefore implies that the risk-free bond price is not regime-dependent but the risky bond price is. There is a vast literature on the regime-switching of risk-free rates. See for example Hamilton (988), Naik and Lee (997), Garcia and Perron (996), Gray (996), Landen (2), Ang and Bekaert (22a, 22b, 23), Bansal and Zhou (22), Evans (23), Dai, Singleton, and Yang (23). Since most of these papers focus on extended sample periods that encompass multiple monetary policy regime changes (oil crisis in the early 97s and the monetary experiment in the early 98s among others), and given the short time period for the analysis, 987 to 22, we posit that there is no regime switching in the dynamics of the risk-free short rate. We complete the model with the specification of the pricing kernels. Harrison and Kreps (979) show that the assumption of an arbitrage-free environment guarantees the existence of a risk-neutral measure Q such that the price p t at time t of a claim to the cash flows c t+ at time t+ satisfies p t = E Q t [e r t c t+ ] where the expectation is taken under the risk neutral measure Q. For a random variable d t+ at time t +, we have: E Q t [ ] [ e r t ] ξt+ d t+ = Et e r t [ ] d t+ = E t M r ξ t+ d t+ t (8) where ξ t+ is the Radon-Nikodym derivative that converts the risk-neutral measure to the real world or data-generating measure. By assuming the existence of ξ t+ ξ t exp ( 2 λ tλ t λ t ɛ t+ ) or equivalently the existence of Mt+ r = ξ t+ ξ t e r t, one can price any traded asset in the econ- 26

omy, particularly bonds. λ t is the price of risk associated with the source of uncertainty ɛ t+ of the state variables because it determines the covariance between ξ t+ or M r t+ and the state variables. Therefore, we define the risk-free pricing kernel M r as: m r t+ = log ( M r t+ ) = rt 2 λθr t λ θr t λ θr t ɛ θr t+ (9) and similarly (see Duffie and Singleton (997, 999)), the spread-adjusted pricing kernel M R takes the form: m R t+ = log ( M R t+) = rt δ t 2 λ t (s t+ ) λ t (s t+ ) λ t (s t+ ) ɛ t+ () where the price of risk is: λ t (s t+ ) = ( λ θr t, λ δ (s t+ ) ) = ( λ θ t, λ r t, λ δ (s t+ ) ) = ( λ θ + λ θ θ t, λ r + λ r r t, λ δ (s t+ ) ). The prices of risk associated with the state variables r t and θ t are time-varying but not regime dependent and the price of risk associated with the spread state variable δ t is not time-varying but a function of the regime variable. The formulation of this model is different from the framework of Dai, Singleton, and Yang (23) because among other things, we assume that the market price of regime shift is zero. The chosen specification of the parameters of the state variable δ t tries to encompass the implications of our structural break results, while not sacrificing the analytical tractability usually afforded by traditional affine term structure models. The regime-independent specification of the autoregressive coefficient matrix (mean-reversion) Φ is needed to insure a closed-form solution of the bond prices. However, making both the volatility and longterm mean of the state variable δ t regime-dependent is consistent with our structural break findings. It is also consistent with the evidence in Liu, Longstaff, and Mandell (22) that find different magnitude and volatility of the liquidity and default premia; since the relative importance of liquidity and default risk on the IR swap spread is time-varying, one would expect different size and volatility of the IR swap spread depending on the relative importance of liquidity and default in that regime. In the above model, allowing a linear specification 27

of the price of risk associated with δ t such that the coefficient of δ t switches regimes, results in the loss of a closed-form solution for the bond prices. 5.2 Bond Prices 5.2. Risk-Free Bond Prices Let b n t be the time t price of a risk-free discount bond that pays at time t+n. The prices of bonds are computed recursively using equation (8), b n+ t = E t [ M r t+ b n t+], starting with the price of a one-period bond. More explicitly, the price b t at time t of a one-period risk-free zero-coupon bond is determined by noting that the price of a zero-period bond is b t =. Therefore, we have: b t = E t [ M r t+ b t+] = Et [ M r t+ ] = E t [ M r t+ ] = exp ( rt ) () Given the formulation of the state variables θ t and r t and of their respective prices of risk above, our model falls within the affine class of term structure models (see Duffie and Kan (996)), and thus yields bond prices that are exponential affine functions of the state variables. Therefore, the risk-free bond prices are given by: b n t = exp (A r n + B r nθ t + C r nr t ) (2) where the loadings A r n, B r n, and C r n follow the difference equations: A r n+ = A r n + (κ θ θ σ θ λ θ )B r n σ r λ r C r n + 2 (σ θb r n) 2 + 2 (σ rc r n) 2 B r n+ = ( κ θ σ θ λ θ )B r n + κ r C r n C r n+ = + ( κ r σ r λ r )C r n (3) with initial values deduced from equation (). One alternative (see Ang and Bekaert (23)), is to specify another independent state variable whose parameters do not switch regime and whose role is to control the time-varying aspect of the prices of risk. In that case all the coefficients in the prices of risk can switch regimes. See Appendix B. for explicit derivation of the formula for the loadings A r n, B r n, and C r n 28

5.2.2 Risky Bond Prices Similarly, the time t price of a risky discount bond price, P n t (i), with promised payoff at time t + n conditional on regime s t = i, is given by: P n t (i) = exp ( A R n (i) + B R n θ t + C R n r t + D R n δ t ) (4) where the loadings A R n (i), B R n, C R n, and D R n follow the difference equations: 2 A R n+(i) = (κ θ θ σ θ λ θ )B R n σ r λ r C R n + 2 + log{ j ( ) σθ Bn R 2 ( ) + σr Cn R 2 2 p ij exp{a R n (j) + (κ δ δ(j) σ δ (j)λ δ (j))d R n + 2 ( ) 2}} σδ (j)dn R B R n+ = ( κ θ σ θ λ θ )B R n + κ r C R n + κ θδ D R n C R n+ = + ( κ r σ r λ r )C R n + κ rδ D R n D R n+ = + ( κ δ )D R n (5) 5.3 Econometrics and Estimation Results We estimate the parameters of the model with standard maximum likelihood methodology as in Chen and Scott (993), Ang and Bekaert (23), Dai, Singleton, and Yang (23). 3 we use both Treasury and swap yields to estimate the model. 5.3. Par Rates to Zero Rates For this estimation, the data consists of Datastream weekly (Friday) cross-sectional observations of constant maturity Treasury, LIBOR and fixed-for-floating swap middle rates from April 3, 987, to December 27, 22, for a total of 822 observations. Both the constant maturity Treasury rates (CMT ) and swap rates (CMS) represent par rates and have the following expression: CMT n t = 2 ( b n t n/26 k= b26k t ) (6) 2 See Appendix B.2 for explicit derivation of the loadings A R n, B R n, C R n, and D R n 3 See also Duffie and Singleton (997), Dai and Singleton (2), Liu, Longstaff, and Mandell (22) 29

for the constant maturity Treasuries, and the expression: CMS n t = 2 ( ) Pt n n/26 k= P t 26k (7) for the swap rates, where 26 represents the numbers of weeks in six months. We use constant maturity Treasury for six-month, 2-, 5-, 7-, and -year maturities, the six-month LIBOR, and the 2-, 5-, 7-, and -year maturities for the swap rates. Given the regime-switching framework and given that the par rates are a very complicated function of the state variables, applying Chen and Scott (993) maximum likelihood methodology requires computationally intensive routines to extract the state variables from the par rates, and thus needlessly complicates the parameters estimation. We focus instead on zero rates. We extract from the par rate data set the corresponding zero yields through a bootstrap technique which complements the observable bonds by interpolating the par rates for each semi-annually separated maturity from six months through ten years. The technique uses the twenty par rates obtained from interpolation and sequentially extracts the zero-coupon rate that would give rise to the observable par rates. We experiment with both linear interpolating and piecewise cubic Hermite spline (see Anderson et al. (997) for more details) and the estimation results do not change significantly. We therefore convert the constant maturity Treasury for six-month, 2-, 5-, 7-, and -year maturities, the six-month LIBOR, and the 2-, 5-, 7-, and -year maturities for the swap rates, into the same corresponding maturity zero yields. The likelihood function for the zero yields is much more tractable than the likelihood function for the par rates. 5.3.2 Likelihood Function The Chen and Scott (993) maximum likelihood estimation methodology requires the assumption that we have the same number of yields measured or priced without error as the number of latent factors, N = 3. This allows us to solve for the three unobserved state variables in the model. The rest of the yields, M, are assumed to be measured with error. These additional yields provide additional cross-sectional pricing information or over-identifying 3

restrictions for the estimation of the parameters of the term structure model. Specifically, to construct the likelihood function, we assume that the two-year Treasury, the six-month LIBOR, and the ten-year swap rates are measured without error. The six-month LIBOR and the ten-year swap rates are the most liquid maturities and are therefore the most likely to be measured without error. The two-year Treasury rate is also assumed to be measured without error, in order to match exactly a yield on the risk-free curve. From the bond price equations (2) and (4), the risk-free yield for maturity n k is given by: log (b n k t ) yr n k t = Ar n k n k n k and similarly, the risky yield conditional on regime i is: n k log (P n k t (i)) yr n k t (i) = AR n k (i) n k Br n k n k θ t Cr n k n k r t (8) BR n k n k θ t CR n k n k r t DR n k n k δ t (9) By stacking the yields observed without error at time t into the N vector R t, we get the following expression for R t : R t = a (s t ) + b X t (2) where a is the N vector of the Ar n k n k Cr n k n k, BR n k n k and AR n k (i) n k terms, b is the N N matrix of the Br n k n k,, CR n k n k, and DR n k n k terms corresponding to the yields in R t. From equation (9), we can easily extract the state variables of our model: X t = b (R t a (s t )) (2) Substituting the expression of X t in equation (6) into equation (2), and after rearranging, we get: R t = c (s t ) + Ψ R t + Ω (s t ) ɛ t (22) where c (s t ) = a (s t ) + b µ (s t ) b Φb a (s t ), Ψ = b Φb, Ω (s t ) = b Σ (s t ), and s t is defined as the state variable that counts all combinations of s t and s t and a transition 3

probability matrix Π = { pij}, p ij = Pr ( s t = j s t = i ). Similarly, we get the following expression by stacking the remaining yields observed with error at time t into the M vector R 2t : R 2t = R2t u + u t (23) where model implied or unobserved rates R2t u = a 2 (s t ) + b 2 X t. We assume that the measurement error u t is IID normal and uncorrelated across the yields measured with error, u t N (, V ) with V is a M M diagonal matrix. Substituting equation (2) into equation (23) yields the following equation for the dynamics of R 2t : R 2t = c 2 (s t ) + Ψ 2 R t + u t (24) where c 2 (s t ) = a 2 (s t ) b 2 b a (s t ), Ψ 2 = b 2 b. Let Θ be the vector containing all the parameters of the model, and I t = ( R t, R t, R t 2,..., R 2t, R 2t, R 2t 2,... ) be the econometrician s information set or a vector containing all observations through date t. From Hamilton (994), Ang and Bekaert (23), Dai, Singleton, and Yang (23), the log-likelihood function is then: L (I T, Θ) = = T log f (R t, R 2t s t, I t ; Θ) Pr (s t I t ; Θ) (25) s t T log f (R 2t R t, s t, I t ; Θ) f (R t s t, I t ; Θ) Pr (s t I t ; Θ) t=2 t=2 s t where f (R t s t, I t ; Θ) = (2π) n 2 Ω (s t ) Ω (s t ) 2 (R 2 t c (s t ) Ψ R t ) exp ( Ω (s t ) Ω (s t ) ) (R t c (s t ) Ψ R t ) 32

is the probability density of R t conditional on s t, f (R 2t R t, s t, I t ; Θ) = (2π) m 2 V 2 { exp } 2 (R 2t c 2 (s t ) Ψ 2 R t ) V (R 2t c 2 (s t ) Ψ 2 R t ) is the probability density function of the measurement errors u t conditional on s t, Pr (s t = i I t ; Θ) = j Pr ( s t = i s t = j, I t ; Θ ) Pr ( s t = j I t ; Θ ) and Pr ( s t = j I t ; Θ ) = f ( R t, R 2t s t = j, I t 2 ; Θ ) Pr ( s t = j I t 2 ; Θ ) m f ( R t, R 2t s t = m, I t 2 ; Θ ) Pr ( s t = m I t 2 ; Θ ) 5.3.3 Parameter Estimates For identification, we follow Dai and Singleton (2) in the formulation (6) of the term structure model with latent variables by setting the conditional covariance matrix to be diagonal and setting the mean-reversion matrix Φ to be lower triangular in equation (7). Theoretically, the model is identified; however, since we use zero-coupon yields in our estimation of a Gaussian term structure model, not all the price of risk parameters are easily identified. Therefore, to facilitate econometric identification, we set λ θ = and λ r =. The resulting model is identified. Following Hamilton (994), s t is defined as: s t s t s t 2 2 (26) 3 2 4 2 2 33

p p 2 with associated transition probability matrix Π = { pij} p p 2 =. p 2 p 22 p 2 p 22 The log-likelihood function in equation (25) with the additional restrictions above is estimated using the standard Hamilton (994) and Gray (996) algorithm. The maximum likelihood estimates of the parameters are reported in Table 5. For robustness, we check the estimation results by starting the optimization routine at a wide array of initial values. Table 5 also reports the asymptotic standard errors of the parameter estimates. The parameters of the risk free rates are significant and agree generally with estimates in other papers. In summary, all the parameters estimates are reasonable and statistically significant. The transition probabilities p = Pr (s t = s t = ) and p 22 = Pr (s t = 2 s t = 2) are.9763 and.995 respectively and are highly significant. The magnitudes of these probabilities underline the high persistence of the regimes which is consistent with our earlier empirical findings of structural changes. One of the determinants of the long-term mean of the spread factor, δ, changes significantly across regimes: δ is insignificant in regime, but high and statistically significant in regime 2. Similarly, the constant price of risk of the spread factor varies significantly across regimes. λ δ is negative in the first regime, λ δ (s t = ) =.687, but positive in the second regime, λ δ (s t = 2) =.76. Therefore, the implied swap yields differ greatly across the regimes. However, the volatility of the spread factor is not much different across regimes. To show that the dynamics of the state variables driving the swap term structure switch regimes, we conduct a likelihood ratio test of equality of the regime-switching parameters across regimes. We easily reject the null hypothesis of equality of the regime-switching parameters across regimes at a significance level less than percent. Also, Figure 6 shows the graph of the constant loading term A R n (i) and we observe that A R n (i) is very different across regimes. Therefore, the term structure of swap yields is better explained through a model with regime shifts. 34

The top Panel of Figure 7 shows the graph of the smoothed probability of being in regime, Pr (s t = I T ). The smoothed probabilities are used to classify observations into regimes. The last two Panels of Figure 7 show the earlier graph of the marginal R 2 of the liquidity and default factors and the graph of the -year interest rate swap spread with the dates of the structural changes estimated with the Bai and Perron (998) methodology. It is interesting to note that the graphs in the three Panels are consistent with each other. Roughly, regime covers the following periods: 4/3/987 to 5/3/988, 9/5/989 to 6/29/99, /5/993 to /27/995, and 8/7/998 to 6/4/22. The dates of change of regimes coincide for the most part with the structural change dates of 8/25/989, 5/8/992, 8/4/998, estimated in the first part of the paper. From the marginal R 2 graph, we can call regime, the Liquidity regime. This observation is buttressed by the fact that we are in regime whenever there is a liquidity event. The announcement of the Treasury in the fall of 993 to increase the auction cycle of the long bond from a quarterly interval to a semiannual interval, the flight-to-quality following the Russian Default and the LTCM crisis in 998, the Y2K liquidity crisis in 999, and the announcement by the Treasury of its buyback program in spring 2 are all situated in regime. The process is in regime 2 for most of the middle period. We cannot however associate regime 2 with default risk. It is instead a possible combination of liquidity and default risk. 5.4 Possible Extensions It is often the case that aggregate default risk and liquidity events are simply different faces of the same coin. For example, the financial crisis of 998 encompasses both a default risk event and a liquidity event. However, the Y2K and the U.S. government decision to reduce the supply of Treasury securities were entirely liquidity events (Longstaff (22)), not preceded or followed up by a default event. The specification in (6) allows only two regimes, and thus restricts the aggregate default risk and liquidity processes to share the same regimes. Also, the spread factor is subject to only one source of shocks. An alternatively richer model of the term structure of swap yields should incorporate the possibility of different but correlated 35

liquidity and default risk regimes. Also, it should include two different factors for the spread. For example, we can have the following model: X t+ = µ ( s L t, s D t ) + ΦXt + Σ ( s L t, s D t ) ɛt+ where X t (θ t r t δ Lt δ Dt ) is the vector of the state variables, with δ Lt the liquidity adjustment of the spread and δ Dt is the default adjustment. s L t {, 2,..., K L } is the liquidity regime variable with transition probability Π L and s D t {, 2,..., K D } is the default regime variable with transition probability Π D. Given that default and liquidity are very related concepts, a switch in the liquidity regime can affect default risk and vice versa. Following Hamilton (994), and Ang and Bekaert (23), we could incorporate the joint effects of s L t and s D t by defining an aggregate regime variable s t to account for all combinations of s L t and s D t. With this new notation, the new model is equivalent to (6) with the regime variable s t taking values in {, 2,..., K L K D }, thus, yielding a K L K D K L K D transition matrix. An accurate estimation of the full model will be hard to achieve because of the large size of the parameter dimension it engenders. Restrictions on the realizations of both regimes are needed. P rob (s t = jk s t = mn) P rob ( s L t = j, s D t = k s L t = m, s D t = n ) = P rob ( s L t = j s D t = k, s L t = m, s D t = n ) P rob ( s D t = k s L t = m, s D t = n ) One would need a joint model of default and liquidity to be able to impose any restrictions on the above probabilities. Furthermore, the model would be richer but far more complicated if the regime transition probabilities are time-varying (Filardo (994)); both liquidity and default risk would help predict transitions in the regimes. This framework also offers an excellent opportunity to study the under-explored topic of the relationship between aggregate default and liquidity. Further details of this model and the study of the relationship between default risk and liquidity in the fixed income market are left for future work. 36

6 Conclusion In terms of outstanding notional, interest rate swap contracts are the largest derivative contracts in the world. After the government decision to retire some debt in the late 99s and in 2, market participants have increasingly turned to the swap market for hedging issues. Swap rates are also used as a benchmark for pricing other fixed income securities and derivatives. In spite of this importance of the swap market, we still lack a solid understanding of the time series dynamics of the swap spreads. These spreads have experienced wild fluctuations, varying from a low of about 25 basis points to more than 5 basis points. The central empirical issue surrounding swaps is what drives interest rate swap spread. While there is general agreement that interest rate swap spreads are driven by counterparty default risk, default risk in the Eurodollar market, and liquidity in the Treasury market, there is disagreement on their relative importance. This paper is an attempt to fill the gap in our understanding of the components of the swap spread and their relative importance. Given the history of the swap market marked by multiple microstructure, liquidity, and default risk events, we show that there are structural changes in the relationship between swap spread and its driving factors of default risk and liquidity. We show that the liquidity and default factors play different roles in different periods. Specifically, we identify a regime where default risk is the most important determinant of the swap spread and a liquidity regime in which the liquidity in the Treasury market is the most important determinant of the swap spread. The presence of these different regimes coincides with well-known economic events, institutional changes and innovations. Given these findings, the second part of the paper follows naturally: we propose the first regimeswitching term structure model of the swap yields that is consistent with these empirical findings. We posit the existence of both a default and liquidity regime in the term structure model. The results of this model are consistent with the early empirical analysis. The finding that the relative importance of the components of the swap spread is timevarying can be useful when using the swap spread as a benchmark for pricing other securities, or when using swap contracts for hedging. One immediate extension of the paper is to 37

explicitly show how the failure to account for regime shifts in the term structure of swap yields affects positions in swap yield sensitive securities such as corporate debt, asset-backed securities and other fixed income securities or derivatives. 38

Appendix A Bai and Perron (998): Estimating and Testing Linear Models with Multiple Structural Changes A. Estimation of Break Dates Using the notation of Bai and Perron (998), the multiple linear regression model with m breaks (m + regimes) can be written as: y t = x tβ + z tδ j + u t t = T j +,..., T j (27) for j =,..., m + and with the convention that T = and T m+ = T. In equation (9), y t is the observed dependent variable at time t, x t and z t are the observed column vectors of explanatory variables at time t of dimension p and q respectively. The variance of u t needs not be constant. Using T observations on y t, x t and z t, the purposes of the econometric method presented below are (i) to determine whether there is break in the relationship between the variables y t and z t, (ii) to estimate the break points T, T 2,..., T m, and (iii) to construct the confidence intervals of the break points. In matrix notation, equation (9) can be rewritten as: Y = Xβ + Zδ + U (28) where Y is a T vector (y, y 2,..., y T ), U is a T vector of the disturbances, X is a T p matrix (x, x 2,..., x T ), Z = diag (Z, Z 2,..., Z m+ ) is a T q (m + ) matrix with Z i a i T q matrix of the observations [ on z t in the i th regime, ) ( ] Z i = (z(ti +), z (Ti +2),..., z (Ti),..., z(ti +)q, z (Ti +2)q,..., z (Ti)q), and T i = T i T i. It is assumed in this section that the number of breaks m is known. In the next few sections, we will review the methods for testing for the presence of structural breaks and for estimating the number of structural breaks. The break dates are estimated through the least-squares principle. Specifically, for each m-partition (T,..., T m ) denoted by {T j }, the corresponding least-squared estimates ˆβ ({T j }) and ˆδ ({T j }) of β and δ respectively, are obtained by minimizing the sum of squared residuals S T (T,..., T m ; β, δ) = ( Y Xβ Zδ ) ( Y Xβ Zδ ) = m+ i= T i t=t i (y t x tβ z tδ i ) 2 (29) Substituting ˆβ ({T j }) and ˆδ ({T j }) into () { and } denoting ( the resulting sum of squared residuals as S T (T,..., T m ), the estimated break points are such that ˆTj = ˆT,..., ˆT ) m arg min (T,...,T m) Π m S T (T,..., T m ), where Π m = {(T,..., T m ) : T i T i h q}. The regression parameter estimates are then ˆβ ({ }) ˆTj and ({ }) ˆδ ˆTj respectively. A.2 Confidence Intervals of Break Dates For i =,..., m, and Ti = Ti T i where the superscript refers to the true value of the parameter, define i = δi+ δ i, Q i = p lim ( ) Ti T i t=ti + E (z tz t), Ω i = T i T r=ti i + t=ti + E (z rz tu r u t ). Under various assumptions, Bai and Perron (998) obtain the following result about the limiting distribution 4 of the break dates: ( i Q i i ) 2 ( ) ˆTi ( i Ω Ti = arg max V (i) (s), (i =,..., m) (3) i i ) s 4 The cumulative distribution of arg max s V (i) (s) is in Bai (997) 39

where V (i) (s) = W (i) ( s) s /2, if s, V (i) (s) = ξ i (φ i,2 /φ i, ) W (i) 2 ( s) ξ i s /2, if s >, and ξ i = i Q i+ i / i Q i i, φ 2 i, = i Ω i i / i Q i i, φ 2 i,2 = i Ω i+ i / i Q i+ i, W (i) (s) and W (i) 2 (s) are independent Wiener processes defined on [, ), starting at when s =. These processes are also independent across i. A.3 Test of No Break versus a Fixed Number of Breaks A sup F test is used to test the null hypothesis of no structural break m = versus m = k breaks. More formally, it tests the hypothesis H : δ = δ 2 =... δ k+ of no break or equivalently H : Rδ = where R is a matrix with on the main NW-SE diagonal, at the i th row, and (q + i) th column and elsewhere. Let (T,..., T k ) be a k-partition and define the break fraction λ i = T i /T or equivalently T i = T λ i where denotes the greatest least integer. The F -statistic is defined as: sup (λ,...,λ k ) Λ ɛ { F T (λ,..., λ k ; q) = T ( T (k + ) q p kq ) ( ( ( Rˆδ) R ˆV (ˆδ) R ) ( Rˆδ) )} (3) where ˆV (ˆδ) is an estimate of the variance covariance matrix of ˆδ that is robust to serial correlation and heteroscedasticity, Λ ɛ = {(λ,..., λ k ) : λ i+ λ i ɛ, λ ɛ, λ k ɛ} with ɛ an arbitrary small positive number. Intuitively, each break date is restricted to be asymptotically distinct and bounded from the boundaries of the sample. The limiting distribution of the test depends on the nature of the regressors and the presence or absence of serial correlation and heteroscedasticity in the residuals. A.4 A Double Maximum Test Unlike the sup F test above that requires the specification of the number of breaks, the Double Maximum test considers the null of no structural break versus the alternative of an unknown number of breaks given some upper bound M on the possible number of breaks. The Double Maximum test is specified as follow: { } DmaxF T (M, q, a,..., a M ) = max m M a m sup FT (λ,..., λ k ; q) (λ,...,λ m) Λ ɛ where {a,..., a M } is a set of fixed weights. The distribution of DmaxF T follows from the distribution of sup F T. The weights {a,..., a M } may reflect the imposition of some priors on the likelihood of various number of breaks. An obvious candidate is to set all the weights equal to unity; this gives: { } UDmaxF T (M, q) = max m M sup FT (λ,..., λ k ; q) (λ,...,λ m ) Λ ɛ For the set of weights {a,..., a M } such that a = and a m = c (q, α, ) /c (q, α, m) for m >, with c (q, α, m) the asymptotic critical value of the afore-mentioned test sup (λ,...,λ k ) Λ ɛ FT (λ,..., λ k ; q) for a significance level α, the DmaxF T test becomes: W DmaxF T (M, q) = A.5 Number of Breaks max m M { c (q, α, ) c (q, α, m) sup FT (λ,..., λ k ; q) (λ,...,λ m ) Λ ɛ Both the sup F test and the Double Maximum tests tell us whether there is a structural break in the time series model considered. To determine the actual number of breaks, one can use an information criterion such as the Bayesian Information Criterion defined as: } (32) (33) (34) BIC (m) = ln (ˆσ 2) + p ln (T ) /T (35) 4

( where ˆσ 2 = T S T ˆT,..., ˆT ) m and p = (m + ) q + m + p. The sequential method is the third approach of determining the number of breaks. Given that the sup F test and the Double Maximum tests show the presence of structural break in the time series model, the method to determine the number of breaks consists of sequentially applying the sup F T (l + l) test with the null of l breaks versus the alternative of l+ breaks starting with l =. At each step, the sequential method amounts to the application of (l + ) tests of the null hypothesis of no structural change versus the alternative of a single change; in other words the test is applied to each segment containing the observations ˆT i + to ˆT i (i =,..., l + ) for the presence of an additional break, where ˆT =, ˆTl+ = T, and ˆT,..., ˆT l denote the estimated break dates for the model with l breaks. One concludes for a rejection in favor of a model with (l + ) breaks if the overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of squared residuals from the l breaks model. The break date thus selected is the one associated with this overall minimum. The test statistics is defined by: ( sup F T (l + l) = {S T ˆT,..., ˆT ) ( m min inf S T ˆT,..., ˆT i, τ, ˆT,..., ˆT )} l / ˆσ 2 (36) τ Λ i,η i l For simplicity, the errors are assumed to ( be spherical and ˆσ 2 is a consistent estimate of the variance of the disturbances under the null hypothesis. S T ˆT,..., ˆT i, τ, ˆT,..., ˆT ) ( l is understood as S T τ, ˆT,..., ˆT ) l for i =, ( and as S T ˆT,..., ˆT ) { i, τ for i = l +. Finally, Λ i,η = τ : ˆT ( i + ˆTi ˆT ) i η ˆT ( i ˆTi ˆT ) } i η. As for the distribution of the test statistic, the authors showed that lim T P [F T (l + l) x] = G q,η (x) with G q,η (x) the distribution function of sup η µ η W q (µ) µw q () 2 / (µ ( µ)). This process is repeated, increasing l sequentially until the sup F T (l + l) fails to reject the null hypothesis of no additional structural changes. The number of breaks is therefore the last l considered. The authors suggest that the sequential method may be the best test to estimating the number of breaks. B Bond Prices B. Risk-Free Bond Prices Given the formulation of the state variables θ t and r t and of their respective prices of risk in equation (7) to (9), our model falls within the affine class of term structure models (see Duffie and Kan (996)), and thus yields bond prices that are exponential affine functions of the state variables. In this section, we show explicitly how to derive the difference equations in the article. Suppose that the price b n t at time t of an n-period risk-free bond price is an exponential affine function of the state variables θ t and r t and is given by b n t = exp (A r n + B r nθ t + C r nr t ) (37) From equation (3) the price b t at time t of a one-period risk-free zero-coupon bond satisfies: b [ t = E t M r t+ b ] t+ [ = E t M r t+ ] [ ] = E t M r t+ [ = E t exp( r t ] 2 λθr t λ θr t λ θr t ɛ θr t+) = exp( r t 2 λθr t = exp( r t ) λ θr t [ )E t exp( λ θr t ɛ θr t+) ] = exp(a r + B r θ t + C r r t ) (38) 4

Simple matching of coefficients gives initial values A r =, B r =, and C r = for starting the recursive equations. We easily verify that the price b n+ t at time t of an n + -period risk-free bond price also has an exponential affine functional form: b n+ [ t = E t M r t+ b n ] t+ ( = E t [exp r t ) ] 2 λθr t λ θr t λ θr t ɛ θr t+ exp (A r n + Bnθ r t+ + Cnr r t+ ) ( = exp r t ) 2 λθr t λ θr t + A r [ ( n E t exp λ θr t ɛ θr t+ + Ψ θr n Xt+ θr )] ( = exp r t ) 2 λθr t λ θr t + A r n + Ψ θr n µ θr + Ψ θr n Φ θr Xt θr [ {( E t exp Ψ θr n Σ θr λ θr ) }] t ɛ θr t+ { } rt = exp 2 λθr t λ θr t + A r n + Ψ θr n µ θr + Ψ θr n Φ θr Xt θr + 2 (Ψθr n Σ θr λ θr t )(Ψ θr n Σ θr λ θr t ) A r n + ( κ θ θ σ θ λ) θ B r n + 2 (σ θbn) r 2 + 2 (σ rcn) r 2 σ r λ r Cn r = exp + (( κ θ σ θ λ) θ B r n + κ r Cn) r θt + ( + ( κ r σ r λ r ) Cn) r r t = exp ( A r n+ + Bn+θ r t + Cn+r r ) t (39) where in the third and fourth equalities, we have Ψ θr n = [ B r n C r n [ ] ] and X θr θt+ t+ =, and µ r θr, Φ θr, t+ Σ θr are partitions of parameter matrices in equations (6) and (6) corresponding to the state variables θ t and r t. Again, matching coefficients yields the recursive equations: A r n+ = A r n + ( κ θ θ σ θ λ θ ) B r n σ r λ r Cn r + 2 (σ θbn) r 2 + 2 (σ rcn) r 2 Bn+ r = ( κ θ σ θ λ θ ) B r n + κ r Cn r Cn+ r = + ( κ r σ r λ r ) Cn r (4) B.2 Risky Bond Prices: Regime-Switch Suppose that the time t price Pt n (i) of a risky discount bond with promised payoff of at time t + n, conditional on regime s t = i, is exponential affine functions of the state variables θ t, r t, and δ t, and is given by: Pt n (i) = exp ( A R n (i) + Bn R θ t + Cn R r t + Dn R ) δ t (4) From equation (8) the price P t (i) at time t of a one-period risky zero-coupon bond conditional on regime s t = i, satisfies: P t (i) = j p ij E t [ M R t+ P t+(j) s t+ = j ] = j = j = j p ij E t [ M r t+ s t+ = j ] p ij E t [exp( r t δ t ] 2 λ t (j) λ t (j) λ t (j) ɛ t+ ) ( p ij exp r t δ t ) 2 λ t (j) [ ( λ t (j) E t exp λt (j) )] ɛ t+ = exp ( r t δ t ) = exp ( A R + B R θ t + C R r t + D R δ t ) (42) 42

Simple matching of coefficients gives initial values A R (i) =, B R =, C R =, and D R = for starting the recursive equations. We easily verify that the time t price Pt n+ (i) of a risky discount bond with promised payoff of at time t + n +, conditional on regime s t = i also has an exponential affine functional form: P n+ t (i) = j = j p ij E t [ M R t+ P n t+(j) s t+ = j ] p ij E t [exp ( r t δ t 2 )] λ t (j) λ t (j) λ t (j) ɛ t+ exp ( A R n (j) + Bn R θ t+ + Cn R r t+ + Dn R ) δ t+ = ( p ij exp r t δ t ) 2 λ t (j) λ t (j) + A R n (j) j [ ( E t exp λt (j) )] ɛ t+ + Ψ n X t+ = p ij exp ( r t δ t 2 ) λ t (j) λ t (j) + A Rn (j) + Ψ n µ(j) + Ψ n ΦX t j E t [exp {(Ψ n Σ(j) λ t (j) ) ɛ t+ }] = { rt δ p ij exp t 2 λ t (j) } λ t (j) + A R n (j) + Ψ n µ(j) + Ψ n ΦX t + j 2 (Ψ nσ(j) λ t (j) ) (Ψ n Σ(j) λ t (j) ) ( ) κ θ θbn R + 2 σθ B R 2 ( ) n + 2 σr C R 2 { n σθ λ θ Bn R σ r λ r Cn R { + log j p ( ) }} ij exp A R n (j) + κ δ δ(j)dn R + 2 σδ (j)d R 2 n σδ (j)λ δ (j)dn R = exp + (( ) κ θ σ θ λ) θ B R n + κ r Cn R + κ θδ Dn R θt + ( ) + ( κ r σ r λ r ) Cn R + κ rδ Dn R rt + ( ) + ( κ δ ) Dn R δt = exp ( A R n+(i) + Bn+θ R t + Cn+r R t + Dn+δ R ) t (43) where in the third equality, we have Ψ n = [ Bn R Cn R Dn R and Dn R follow the difference equations: ]. We deduce that the loadings A R n (i), B R n, C R n, A R n+(i) = ( κ θ θ σ θ λ θ ) B R n σ r λ r Cn R + ( σθ B R ) 2 n + 2 2 + log j ( σr Cn R ) 2 { p ij exp A R n (j) + ( κ δ δ(j) σ δ (j)λ δ (j) ) Dn R + } ( σδ (j)dn R ) 2 2 B R n+ = ( κ θ σ θ λ θ ) B R n + κ r C R n + κ θδ D R n Cn+ R = + ( κ r σ r λ r ) Cn R + κ rδ Dn R Dn+ R = + ( κ δ ) Dn R (44) 43

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Figure : Interest Rate Swap Spreads Weekly time series of interest rate swap spread in basis points for maturity 2, 5, 7, and years, for the period of April 987 to December 22. Interest rate swap spread is defined as the swap rate minus the corresponding constant maturity Treasury rate. 2-year Interest Rate Swap Spread 2-year Interest Rate Swap Spread Change 5 4 5 2-2 -4 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- 5-year Interest Rate Swap Spread 5-year Interest Rate Swap Spread Change 5 5 4 2-2 -4 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- 7-year Interest Rate Swap Spread 7-year Interest Rate Swap Spread Change 5 4 5 2-2 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- -4 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- -year Interest Rate Swap Spread -year Interest Rate Swap Spread Change 5 4 5 2-2 -4 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr-

Figure 2: Notional Size of the Global Interest Rate Swap Market The size data was obtained from http://www.swapsmonitor.com and from the Bank for International Settlements (BIS) publications of the Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity and the Regular OTC Derivatives Market Statistics. The size is in trillions of U.S. Dollars. 8 6 4 2 Dec-85 Dec-86 Dec-87 Dec-88 Dec-89 Dec-9 Dec-9 Dec-92 Dec-93 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec- Dec-2

Figure 3: Relationship Between LIBOR-General Collateral Repo Rate and IRS spread A diagram of the following zero value portfolio: Short sell $P of government bonds with maturity of T years, trading at par and yielding the fixed coupon rate of C paid semiannualy. Invest the proceeds in six-month GC repo and roll over at each six-month interval over the life of the government bond above. Enter into an IRS contract to receive fixed swap rate S and pay six-month LIBOR rate on the notional amount of $P for T years. No-arbitrage Present Value (Swap Spread) = Present Value (LIBOR GC Repo Rate). GC Repo Rate Treasury + Swap Spread Firm Treasury LIBOR

Figure 4: Drivers of Interest Rate Swap Spread. Interest rate swap spread is defined as the -year swap rate minus the -year constant maturity Treasury rate. Off/on-the-run spread, the measure of the liquidity factor, refers to the spread between the yield -year Treasury issued in the previous auctions and yield on the most recently auctioned -year Treasury. The LIBOR default is the sum of the residuals and constant term from the regression of the 6-month LIBOR-6-month constant maturity Treasury rate spread on the -year off/on-the-run spread. VIX is a weighted average of implied volatilities of nearthe-money OEX put and call options. -year Swap Spread Federal Funds Target Rate Swap Spread 5 5 Apr-87 Mar-89 Mar-9 Mar-93 Mar-95 Mar-97 Mar-99 Mar- 2 8 4 Federal Funds Target Rate -year Swap Spread 6-month Treasury 5 2 Swap Spread 5 8 4 6-month Treasury Apr-87 Mar-89 Mar-9 Mar-93 Mar-95 Mar-97 Mar-99 Mar- -year Swap Spread -year Off/On Spread Swap Spread 5 5 Apr-87 Mar-89 Mar-9 Mar-93 Mar-95 Mar-97 Mar-99 Mar- 3 2 - -2 -year Off/On Spread

-year Swap Spread LIBOR Default Swap Spread 5 5 Apr-87 Mar-89 Mar-9 Mar-93 Mar-95 Mar-97 Mar-99 Mar- 6 2 8 4-4 LIBOR Default Risk -year Swap Spread CBOE's VIX 5 8 Swap Spread 5 6 4 2 VIX Apr-87 Mar-89 Mar-9 Mar-93 Mar-95 Mar-97 Mar-99 Mar-

Figure 5: Explanatory Power of Liquidity and Default Marginal adjusted R 2 of both liquidity and default factors in affecting interest rate swap spread. The marginal adjusted R² of the default factor (of the liquidity factor) is the adjusted R² of the multivariate regression of IRS spread on a constant, on the liquidity, and on the default factors minus the adjusted R² of the regression of IRS spread on just the constant and the liquidity factor (default factor). Interest rate swap spread is defined as the -year swap rate minus the -year constant maturity Treasury rate. The liquidity factor is proxied by the spread between the yield on -year Treasury issued in the previous auctions and yield on the most recently auctioned -year Treasury. The default factor include both the default risk in the LIBOR market (the sum of the constant term and the residuals from the regression of the spread between the 6-month LIBOR and the 6-month constant maturity Treasury rate on the -year off/on-therun spread) and the VIX index which is a weighted average of implied volatilities of near-the-money OEX put and call options on the CBOE and serves as a measure of default risk. In the graph, the marginal R 2 of a specific sub-period is assigned to all dates of that sub-period. Liquidity Default 5 4 3 2 Apr-87 Apr-89 Apr-9 Apr-93 Apr-95 Apr-97 Apr-99 Apr- Marginal Adjusted R-square

Figure 6: Loadings of the Constant Term: A R n ( i) The price of a risky bond price conditional on n R R R R P i = exp A i + B θ + C r + D δ t () () ( ) n n t n t n t s t = i is given by: st = st = 2 -.4 -.8 -.2 -.6.25.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25 Maturity (years)

Figure 7: Smoothed Probabilities Pr ob [ s t = I T ] The smoothed probabilities are used to classify observations into regimes. ob [ ] Pr s t = I T is the probability of being in regime in the Gaussian regime-switching term structure model of interest rate swap yields in section 5. Smoothed Probability for Regime.8.6.4.2 Apr-87 Apr-88 Apr-89 Apr-9 Apr-9 Apr-92 Apr-93 Apr-94 Apr-95 Apr-96 Apr-97 Apr-98 Apr-99 Apr- Apr- Apr-2 Liquidity Marginal Adjusted R-square Default Marginal Adjusted R-square 5 4 3 2 Apr-87 Apr-88 Apr-89 Apr-9 Apr-9 Apr-92 Apr-93 Apr-94 Apr-95 Apr-96 Apr-97 Apr-98 Apr-99 Apr- Apr- Apr-2 -year Swap Spread with Structural Change Dates 5 5 Apr-87 Apr-88 Apr-89 Apr-9 Apr-9 Apr-92 Apr-93 Apr-94 Apr-95 Apr-96 Apr-97 Apr-98 Apr-99 Apr- Apr- Apr-2

Table : Bai and Perron (998) Structural Change Test. Test for multiple structural breaks in the relationship between -year interest rate swap spread and its driving factors. This test allows for a maximum of 5 breaks with 5% trimming. SS() t = β j + β 2 jtreasury() t + β 3 j LIBORdefault( t) + β 4 joff / On( t) + β 5 jvix ( t) + u( t) t = T j +, K, T j SS denotes the -year IRS spread, Treasury is the 6-month constant maturity Treasury rate, LIBORdefau lt is the default risk in the Eurodollar market, Off / On is the -year off/on-the-run spread, and VIX is the CBOE's VIX index SupF() SupF(2) SupF(3) SupF(4) SupF(5) UDmax WDmax 54.46*** 9.72*** 256.3*** 36.38*** 274.5*** 36.38*** 56.49*** SupF(2 ) SupF(3 2) SupF(4 3) SupF(5 4) 7.35*** 7.22*** 9.94 3.6 Number of Breaks Selected: 3 The Three Breaks from the Global Optimization Break Date 95% Confidence Interval Break 8/25/89 8//89 /3/89 Break 2 5/8/92 4/24/92 5/22/92 Break 3 8/4/98 6/5/98 8/2/98 *** indicates significance at the % level The SupF(k) is used to test the null hypothesis of no structural break versus a fixed number of k breaks The double maximum test statistics UDmax and WDmax considers the null of no break versus the alternative of an unknown number of breaks The SupF(l+ l) tests the null of l breaks versus the alternative of l+ breaks. All the tests and/or confidence intervals allow for the possibility of autocorrelation and heteroskedasticity in the disturbances

Table 2: Summary Statistics Summary statistics of the -year interest rate swap spread and of its driving factors for the whole sample period and for each sub-period estimated by the Bai and Perron (998) multiple structural changes test. Interest rate swap spread is defined as the -year swap rate minus the -year constant maturity Treasury rate. Off/on-the-run spread, the measure of liquidity in the fixed income market, refers to the spread between the yields on Treasuries issued in the previous auctions and yields on the most recently auctioned Treasuries. The LIBOR default risk is the sum of the constant term and the residuals from the regression of the spread between the 6-month LIBOR and the 6-month constant maturity Treasury rate on the -year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options. Sample Period # Obs Mean Std Whole Sample: 4/3/987-2/27/22 822 Level Autocorrelation Mean First Difference Std Autocorrelation -year Interest Rate Swap Spread 65.45 26.5.97 -.5 6.84 -.37 6-month Constant Maturity Treasury 5.3.8. -..3 -.2 LIBOR Default Risk 38.8 27..88 -.6 3.5 -.35 -year Off/On-the-run Spread 4.33 4.67.82 -. 2.8 -.4 CBOE's VIX Index 2.5 8..89. 3.85 -.26 Sub-period : 4/3/987-8/25/989 26 -year Interest Rate Swap Spread 9.2 3.94.69 -.5. -.45 6-month Constant Maturity Treasury 7.43.7.98.2.23 -.5 LIBOR Default Risk 87.23 24.28.6 -.9 2.52 -.3 -year Off/On-the-run Spread 4.29 2.8.65.3.75 -.3 CBOE's VIX Index 25..5.78 -.4 7.32 -.23 Sub-period 2: 9//989-5/8/992 4 -year Interest Rate Swap Spread 74.5 6.54.94 -.26 5.92 -.4 6-month Constant Maturity Treasury 6.58.47. -.3. -. LIBOR Default Risk 35.29 8.78.77 -.27 2.79 -.36 -year Off/On-the-run Spread 5. 2.25.9 -.2. -.5 CBOE's VIX Index 2.6 4.53.8 -.2 2.84 -.35 Sub-period 3: 5/5/992-8/4/998 327 -year Interest Rate Swap Spread 39.7 6.65.73.4 4.92 -.44 6-month Constant Maturity Treasury 4.78.3...9 -.9 LIBOR Default Risk 26.84.52.62. 9.2 -.47 -year Off/On-the-run Spread 2.58 4..93..59 -.7 CBOE's VIX Index 6.4 4.92.92.5 2.3 -.33 Sub-period 4: 8/2/998-2/27/22 228 -year Interest Rate Swap Spread 83.52 23.3.96 -.6 6.8 -.9 6-month Constant Maturity Treasury 4.2.69. -.2.2.4 LIBOR Default Risk 29.42 9.6.75 -.2 3.56 -.33 -year Off/On-the-run Spread 6.47 6.3.72 -.4 4.72 -.47 CBOE's VIX Index 27.7 6.23.83. 3.64 -.26

Table 3: Correlation Coefficients Correlation coefficients between interest rate swap spread and its driving factors for the whole sample and for each sub-period. The sub-periods were estimated by the Bai and Perron (998) test for multiple structural changes. Interest rate swap spread is defined as the -year swap rate minus the -year constant maturity Treasury rate. Off/on-the-run spread, the measure of liquidity in the fixed income market, refers to the spread between the yields on Treasuries issued in the previous auctions and yields on the most recently auctioned Treasuries. The LIBOR default risk is the sum of the constant term and the residuals from the regression of the spread between the 6-month London Interbank Offer Rate (LIBOR) and the 6-month constant maturity Treasury rate on the -year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options. -year Swap Spread 6-month Treasury Spread Level LIBOR Default Risk -year Off/Onthe-run Spread -year Swap Spread 6-month Treasury Spread First Difference LIBOR Default Risk -year Off/Onthe-run Spread Panel A: Whole Sample Period: 4/3/987-2/27/22 6-month Treasury.46 -.7 LIBOR Default Risk.45.53.23 -.4 -year Off/On-the-run Spread.54.3..3.8 -.47 VIX.48 -.5.25.23. -.28 -.2.2 Panel B: Sub-period : 4/3/987-8/25/989 6-month Treasury -.64 -.3 LIBOR Default Risk.3 -..34 -.37 -year Off/On-the-run Spread.46 -.29..22 -.9 -.9 VIX.4 -.49.6.35 -.6 -.47 -.8.29 Panel C: Sub-period 2: 9//989-5/8/992 6-month Treasury.9 -.7 LIBOR Default Risk.38.46.6 -.3 -year Off/On-the-run Spread.34.9 -.2.7 -.7 -.22 VIX.32.42.45 -.29.2 -.8. -.2 Panel D: Sub-period 3: 5/5/992-8/4/998 6-month Treasury.44 -.5 LIBOR Default Risk.2.8.23 -.5 -year Off/On-the-run Spread.6.4 -.4.3.5 -.28 VIX.46.33.42.5.8.2 -..3 Panel E: Sub-period 4: 8/2/998-2/27/22 6-month Treasury.85 -. LIBOR Default Risk.26.32.8 -.5 -year Off/On-the-run Spread.67.73.3 -.6.2 -.8 VIX -.24 -.3 -.2 -.29.2 -.8.8 -.8

Table 4: Regression Analysis of the Drivers of Interest Rate Swap Spread For each sub-period estimated by the Bai and Perron (998) structural change test, a dummy variable Dj is defined that takes a value of or depending on whether the current observation is in the relevant sub-period. The subperiods are: 4/3/987 to 8/25/989, 9//989 to 5/8/992, 5/5/992 to 8/4/998, and 8/2/998 to 2/27/22. Interest rate swap spread is defined as the -year swap rate minus the -year Treasury rate. Off/onthe-run spread, the measure of liquidity, is the spread between the yields on Treasuries issued in the previous auctions and yields on the most recently auctioned Treasuries. The LIBOR default risk is the sum of the constant term and the residuals from the regression of the spread between the 6-month LIBOR and the 6-month Treasury rate on the -year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options. Newey-West t-stat are reported. A significant F-stat rejects the null that the coefficients of a specific variable are equal across the different sub-periods. *** indicates significance at the % level and ** indicates significance at the 5% level. Variables Coefficient T-stat F-stat Panel A: Regression Constant 4.5982.87 6-month Treasury 3.35 4.67 LIBOR Default Risk.2396 5.22 Off/On-the-run Spread 2.2439 8.98 VIX.257 5.9 Adjusted R 2 =.6 Panel B: Regression 2 D 22.4965.84 48.82*** D 2-2.844 -.56 D 3 27.6377 3.3 D 4 35.5798 4.39 (6-month Treasury) x D -6.9843-6.8.3*** (6-month Treasury) x D 2 9.6435 9.88 (6-month Treasury) x D 3.484 2.78 (6-month Treasury) x D 4.4258 8.32 (LIBOR Default Risk) x D.32 4.47 2.38** (LIBOR Default Risk) x D 2.288.62 (LIBOR Default Risk) x D 3.468 3.7 (LIBOR Default Risk) x D 4.35.25 (Off/On-the-run Spread) x D.895 4.29 7.9*** (Off/On-the-run Spread) x D 2 2.556 7.99 (Off/On-the-run Spread) x D 3.979 6.46 (Off/On-the-run Spread) x D 4.443.72 (VIX) x D.26.2.9 (VIX) x D 2.853.53 (VIX) x D 3.472.43 (VIX) x D 4.668.3 Adjusted R 2 =.9 () t = + β2treasury() t + β3libordefault() t + β4off / On( t) + β5vix( t) u( t) 2 3 4 2 () t = β D () t + β D 2 () t + β D 3 () t + β D 4 () t + β 2 D t + β 2 D 2 t + 2 3 4 + β 3 D() t + β 3 D2 () t + β 3 D3 () t + β 3 D4 () t LIBORdefault t + β 4 D t 2 3 4 + β D () t + β D () t + β D () t + β D () t VIX t + u t SS β + (Regression ) SS 3 4 { } { ( ) ( ) β 2 D 3( t) + β 2 D 4 ( t) } Treasury() t 2 3 4 { } () { () + β 4 D2 () t + β 4 D3() t + β 4 D4 () t } Off / On() t { } () () 5 5 2 5 3 5 4 (Regression 2)

Table 5: Maximum Likelihood Estimates X = ( ) t θ t, rt, δ t follows X t+ = µ ( st+ ) + ΦX t + Σ( st+ ) ε t+ p p The regime variable s {, 2}, the transition matrix Π =, p = Pr ob( s + = j s i) t ij t t = p p, 22 22 κθθ κθ σ θ µ ( s ) Φ = Σ( ) = t+ =, κ r κ r, st+ σ r ( ) ( ) κ δδ st+ κθδ κ rδ κ δ σ δ st+ r R The risk-free pricing kernel M t + and the spread-adjusted pricing kernel M t + are defined respectively as: r r θr θr θr θr R R mt+ log ( M t+ ) = rt 2 λt λt λt ε t+ mt+ = log M t+ = rt δ t 2 λt st+ λt st+ λt st+ ε t+ where the price of risk ( ) ( ) = ( ( )) = ( + + ( )) θr δ θ r δ θ θ r r δ λt st+ = λt, λ st+ λt, λt, λ st+ λ λθ t, λ λ rt, λ st+. ε t ~ N(, I ) and = ( ) θr δ θ r δ ε t = ε t, ε t ε t, ε t, ε t = and ( ) ( ) ( ) ( ) Parameter Estimate Std Error κ 52.96.4356 θ θ 52.56.2625 σ θ 52.52.45 θ λ - - θ λ 46.892 8.84248 κ r 52.824.566 σ r 52.6.2 r λ - - r λ -86.835 32.345 κ δ 52.889.373 κ θδ 52 -.49.44 κ r δ 52.63.65 δ ( s t = ) 52..52 δ ( s t = 2) 52.323.868 σ δ ( s t = ) 52.3.4 σ δ ( s t = 2) 52.3.3 δ λ ( s t = ) -.687.3 δ ( s t ) λ = 2.76.522 p.9763.23 p 22.995.73 Std Deviation of Yield Measurement Errors 6-month Treasury 52.36.3 5-year Treasury 52.9. 7-year Treasury 52.. -year Treasury 52.5. 2-year swap 52.2.2 5-year swap 52.3. 7-year swap 52.7.