What Drives Interest Rate Swap Spreads?

Transcription

1 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 kma25@columbia.edu. Phone: (22)

2 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

3 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

4 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

5 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 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

6 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

7 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

8 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

9 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

10 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.

11 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

12 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

13 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

14 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

15 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 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

16 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

17 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 7

18 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

19 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

20 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 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

21 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

22 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