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 2002This Draft: November 2003

Abstract

Existing models of the term structure of interest rate swap yields assume a unique regime for the datagenerating process and ascribe variations in swap-Treasury yield spread to default risk or to liquiditypremium. However, the interest rate swap market has been marked by economic events and institutionalchanges that might have significant effects on the data generating process, and thus on the relationshipbetween the swap spread and its determining factors. We investigate the stability of the relationshipbetween the swap spread and its determining factors with the structural change econometric techniquesof Bai and Perron (1998). The structural change tests produce endogenous break dates and associatedconfidence intervals. We trace the break dates to events related to liquidity, default, and institutionalchanges in the swap market. We find that default risk is an important source of variation of the swapspread at the beginning of the sample period, but is relatively less important at the end. Liquidity is, bycontrast, more important towards the end of the sample period. Since these results call into question theassumption of one regime, we propose and estimate a joint Treasury and swap term structure model thataccommodates regime switching. Evidence from the maximum likelihood estimation provides consider-able support for the regime switching model. Consequently, the implied swap spreads may differ greatlyacross regimes. This finding suggests that the failure to account for regime shifts may result in significantmispricing of corporate debt, mortgage-backed securities, as well as derivatives that increasingly use theswap spread as a benchmark for pricing and hedging.

JEL classification: G12; G13; G14

Keywords: Interest rate swaps; Liquidity; Default risk; Structural changes; Regime switch; Term struc-ture model

∗This research has benefitted greatly from the advice and direction of Geert Bekaert and Suresh Sundaresan and helpfulcomments 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, YangruWu, and participants at the 2003 FMA doctoral consortium. Needless to say that I am responsible for any remaining errors.

†PhD Candidate, Columbia Business School, Doctoral Program, 3022 Broadway, 311 URIS Hall, New York, NY 10027.E-mail: kma25@columbia.edu. Phone: (212) 853-9073

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1 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 150 basis points, sometimes moving violently. The obvious question is: why

do they fluctuate so much? To get a sense of these movements, Figure 1 displays both the

swap spreads and their weekly changes for the 2, 5, 7, and 10-year maturities, from April

1987 to December 2002. A casual examination of these graphs of the interest rate swap

spreads reveals at least three distinct patterns across all maturities. From April 1987 to

December 1989, the swap spreads are high and very volatile. There is a noticeable decrease

in magnitude and variability of the swap spreads from early 1990 to mid-1998. The behavior

of the swap spreads from late-1998 to the end of 2002 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 (1993), Cossin

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and Pirotte (1997), Duffie and Singleton (1997), and Mozumdar (1999), of the importance

of credit risk in pricing interest rate swap contracts. Duffie and Singleton (1997) 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 (1997) framework,

Liu, Longstaff and Mandell (2002) decompose the spread into liquidity and credit risk com-

ponents and find that both components vary significantly over time.

Second, Duffie and Huang (1996), Hentschel and Smith (1997), Minton (1997), and Grin-

blatt (2001) find weak or no evidence of counterparty credit risk pricing in swap spreads.

Collin-Dufresne and Solnik (2001) and He (2001) 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 (1991) find that swap spreads are not as cyclical as A-rated corporate spreads, while

Chen and Selender (1994) 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 (1998). 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

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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 (2000) 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 1980s that increased default risk in the banking sector; Treasury actions such as the

change in the long bond auction cycle in 1993, and the buyback program in spring 2000;

the aggressive cutting of the target rate by the Fed in the early 1990s; the Russian default

and LTCM crisis in 1998; the Y2K liquidity problem in 1999. 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 (2003)).

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 sacri-

ficing 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 struc-

ture 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.1 Specifically, we use the econometric

techniques developed by Bai, Lumsdaine, and Stock (1998) and Bai and Perron (1998) to

1More details on the structural break literature can be found in Nyblom (1989), Andrews (1993), Andrews and Ploberger(1994)

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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

(1998) test of structural changes. The Bai and Perron (1998) 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 (1998) 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 80 trillions dollars at the end of 2002.2 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 1990s,

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. (1986), Turnbull (1987), Arak et al. (1988), Kuprianov (1994), and Aragon

(2002) 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

2Bank for International Settlements, 2003, Regular OTC Derivatives Market Statistics.

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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 (1994), Nielsen and Ronn (1996), Grinblatt (2001), He (2001) and Cooper and

Scholtes (2002) 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)

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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 de-

fault 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

(1997). 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 move-

ments 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 deter-

minants of the IR swap spread through time by applying the structural break methodologies

of Bai and Perron (1998) and Bai, Lumsdaine and Stock (1998) 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.

2.1 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 (1993) shows that the de-

fault 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

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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 1999 survey, the International Swaps and Derivatives

Association (ISDA) finds a widespread use of collateral in swap transactions. Litzenberger

(1992) 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 (2003) point out that unlike a collateralized loan where the lender is automat-

ically 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 10

years generally contain credit triggers. A typical credit trigger specifies that if either coun-

terparty’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 (1993) argue that dealers’ credit reputation has an effect on

swap rates. Cooper and Mello (1991), Bollier and Sorensen (1994), Cossin and Pirotte (1997),

Mozumdar (1999) also find evidence of credit risk pricing in the IR swap market. However,

Duffie and Huang (1996) find that 100 basis points difference in debt rates correspond to

1 basis point difference in swap rates. Similarly, Hentschel and Smith (1997) 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 0.00025 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 1980s or the 1998 financial crisis, one could

expect economic agents to weigh more the default risk factor in pricing an IR swap contract.

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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 1980s and early 1990s, 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 (1991) and in Chen

and Selender (1994) that show significant differences between the time series properties of

corporate credit spreads and IR swap spreads, Grinblatt (2001) 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 (1996) and Jordan and Jordan (1997)

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

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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 (2002) 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 (1997) 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 (2002) 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.

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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

10 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, 1987 because of data limitations on

the IR swap rates, to December 27, 2002 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 1 we plot the IR swap spreads for all maturities. The

right Panel of Figure 1 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 10-year maturity. For the structural change tests, we focus

the analysis on the 10-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 1

and 2 of Figure 4 show the graphs of the 10-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 10-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

(1997) and Krishnamurthy (2002) and use the spread between the 10-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 1998 to the present, the off/on-the-run spread has significantly increased

and become more volatile. This increase in the demand for liquidity in 1998 corresponds

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to the “flight-to-quality” following the financial crisis in the second half of 1998 when in-

vestors 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 ex-

plained by the “flight-to-liquidity” during the Y2K liquidity crisis in 1999 and the decision of

the government to repurchase some Treasuries in 2000 (see Longstaff (2002)). 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 10-year off/on-the-run and the 10-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 10-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 (2001) 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 100) 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 (2001) 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 10-year

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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 (2001) 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 (2002) argues that the low levels and low variability of the IR swap

spreads in the early 1990s were due to the recovery of the banking sector from the S&L

crisis in the 1980s whereas the high levels and fluctuations of the IR swap spread in the

late 1990s were due to the perceived scarcity in the supply of U.S. Treasuries. Gupta and

Subrahmanyam (2000) 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

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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 (1960) 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 (1989), Andrews (1993), Andrews and Ploeberger (1994), Andrews,

Lee and Ploeberger (1996), Bai, Lumsdaine, and Stock (1998), and Bai and Perron (1998).

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 (1998) 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

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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 FT (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 FT (l + 1|l) test with the null of l breaks versus the alternative of l + 1

breaks starting with l = 1. One concludes for a rejection in favor of a model with (l + 1)

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

(1998) applied to a linear model of the swap spread and its determinants enumerated above.3

4.1 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 10-year off/on-the-run spread, the general corpo-

rate 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 + 1 regimes), where

SS denotes the 10-year IR swap spread, Treasury is the 6-month constant maturity Trea-

sury, LIBORdefault is the default risk in the Eurodollar market, Off/On is the 10-year

off/on-the-run spread, and VIX is the CBOE’s VIX index.

SS (t) = βj1+βj

2Treasury (t)+βj3LIBORdefault (t)+βj

4Off/On (t)+βj5V IX (t)+u (t) (1)

where t = Tj−1 + 1, ..., Tj, j = 1, ..., m + 1, with the convention that T0 = 0 and Tm+1 = T .

We proceed to the estimation of a full structural break model where all the coefficients in

3Summary of the multiple structural changes econometric method proposed by Bai and Perron (1998) is in Appendix A

15

the above equation are allowed to change. Equivalently, we test the null hypothesis of no

structural break, H0 : β1 = β2 = · · · = βm+1 where βj =(βj

1, βj2, β

j3, β

j4, β

j5

)′. Again, we

note that in this paper, we restrict our analysis to the 10-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 1980s and early 1990s.

• 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 R2?

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

16

the fundamentals. Moreover, the results in Gupta and Subrahmanyam (2000), who

show that there was systematic mispricing of IR swap rates in the late 1980s and the

early 1990s, 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 1998 crisis?

At the height of the 1998 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 0.15

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 (1998) structural change econometric procedure to

equation (1) while accounting for potential serial correlation and heteroscedasticity. Table 1

summarizes the main results.4

4The Gauss program used to do the structural break estimation was obtained from Pierre Perron, athttp://econ.bu.edu/perron/code.html

17

The values of the sup F − test statistic,5 which test the null of no break versus the

alternative of 1 to the maximum number of m breaks, are all significant at the 1 percent

level. Similarly, the values of the UDmax and the WDmax statistics which test for the null

of no break versus the alternative of an unknown number of breaks, are also significant at

the 1 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 (1). As for the number of

breaks in the model, the sequential procedure developed by Bai and Perron (1998) selects 3

breaks.

With m = 3 breaks, we proceed to estimate the break dates and their confidence inter-

vals. Under global minimization, the first break date is August 25,1989, with a 95 percent

confidence interval of August 11, 1989 to October 13, 1989. The second break date is May

8, 1992 with a 95 percent confidence interval of April 24, 1992 to May 22, 1992. August 14,

1998 is the estimated date for the third break with confidence interval of June 5, 1998 to

August 21, 1998. 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 10-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 sub-

periods 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

5See Appendix A for formal definitions of the statistics supF − test, UDmax and the WDmax that test whether we havestructural changes.

18

run spread(except for the volatility), and the VIX index. The characteristics of this sub-

period 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 10-year interest rate swap spread and the 6-month constant maturity Treasury

rate varies between a low of −0.64 in the first sub-period, to a high of 0.90 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) =[β1

1D1 (t) + β21D2 (t) + β3

1D3 (t) + β41D4 (t)

]+

[β1

2D1 (t) + β22D2 (t) + β3

2D3 (t) + β42D4 (t)

]Treasury (t) +

[β1

3D1 (t) + β23D2 (t) + β3

3D3 (t) + β43D4 (t)

]LIBORdefault (t) +

[β1

4D1 (t) + β24D2 (t) + β3

4D3 (t) + β44D4 (t)

]Off/On (t) +

[β1

5D1 (t) + β25D2 (t) + β3

5D3 (t) + β45D4 (t)

]V IX (t) +

u (t)

where Dj (t) are dummy variables that take the value of 1 if t is in sub-period j and 0

otherwise. The sub-periods are: April 3, 1987 to August 25, 1989, September 1, 1989 to

May 5, 1992, May 15, 1992 to August 14, 1998, and August 21, 1998 to December 27, 2002.

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 10-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 R2 of the regression

is approximately 60 percent.

The second Panel of Table 4 reports the results of the regression equation with dummy

19

variables. The adjusted R2 is 90 percent. The constant term varies widely across the sub-

periods: it has a very significant value of 122 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 1, 1989 to May 8,

1992. 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 9.8125 percent on May 5,

1989 to a low of 3 percent on September 4, 1992. 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 (2002) 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, 1989, 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 1989 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 15,

1992 to August 14, 1998 but insignificant in the last sub-period.

20

The coefficient of the liquidity factor, the 10-year off/on-the-run spread, is positive and in

general highly significant throughout, except for the last period where the t-statistic is 1.72.

This relatively low t-statistic may be due to multicollinearity problem since Panel E of Table

3 shows that the correlation between 10-year IR swap spread and the 10-year off/on-the-run

spread is 0.73. 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 1 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 R2. Doing a period by period

regression,6 we note indeed that the adjusted R2 of equation (1) 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

R2 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 (2000).

The third break matches exactly the height of the 1998 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 1998,

only the liquidity factor is important. This is consistent with the results of Liu, Longstaff

6By definition the results are the same as the dummy variable approach in Table 4 and are therefore not reported

21

and Mandell (2002) and He (2001) 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, 1992 is hard to pin to economic events. From Figure 1, the small notional size

underlines the low depth of the IR swap market prior to 1992. This microstructure charac-

teristic 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 R2 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 R2 of default risk shoots

up to about 30 percent from 11 percent in the first period, whereas the marginal R2 of the

liquidity factor increases to about 20 percent from 10 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.

7The marginal adjusted R2 are computed as follow: for each sub-period, we run a regression of IR swap spread on a constantand either the liquidity or the default factors and computed the adjusted R2. The marginal adjusted R2 of the default factors(of the liquidity factor) is the adjusted R2 of the multivariate regression of IR swap spread on a constant, on the liquidity, andon the default factors minus the adjusted R2 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 (1998). The Bai,

Lumsdaine, and Stock (1998) methodology allows us to specify a reduced form vector auto-

regression (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 13, 1989, 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 (1998) test to equation (1) 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 (1997, 1999), Collin-Dufresne and

Solnik (2001), Liu, Longstaff and Mandell (2002), Duffie, Pedersen and Singleton (2003),

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 (1997, 1999) show that

cash flows in the swap market can be discounted at the adjusted short rate Rt 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 (1997) show that for

an IR swap contract initiated at time t to exchange at every six-month t+0.5k, k = 1, 2, ...2τ ,

the preset six-month LIBOR Lt+0.5(k−1) against the fixed payment rate Sτt for τ years can

be priced as follows:

0 =2τ∑

k=1

EQt

[exp

(−

∫ t+0.5k

t

Rsds

) (Lt+0.5(k−1) − Sτ

t

)](2)

where Q denotes an equivalent martingale measure. The six-month LIBOR Lt+0.5(k−1) is

defined as:

Lt+0.5(k−1) = 2

(1− P (t + 0.5 (k − 1) , t + 0.5k)

P (t + 0.5 (k − 1) , t + 0.5k)

)(3)

and

P (t, t + 0.5k) = EQt

[exp

(−

∫ t+0.5k

t

Rsds

)](4)

24

is the price at time t of a 0.5k maturity risky discount bond. Manipulation of the above

equations yields the following expression for the fixed payment rate Sτt :

Sτt = 2

(1− P (t, t + τ)∑2τk=1 P (t, t + 0.5k)

)(5)

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 (1997) 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

5.1 The Model

Our model formulation is based on the framework of Ang and Bekaert (2003) who study

the term structure of risk-free rates. The model has three unobserved state variables: the

one-period short rate rt, the central tendency θt toward which the short rate adjusts,9 and

the spread process δt which is an adjustment for time-varying default risk and liquidity in

the IR swap market.

Let Xt ≡ (θt rt δt)′, the vector of state variables, follow the discrete time Gaussian regime

switching process under the data-generating or real-world measure:

Xt+1 = µ (st+1) + ΦXt + Σ (st+1) εt+1 (6)

where the regime variable st can be either liquidity (st = 1) or default (st = 2), and where

it follows a Markov chain with transition probability Π = pij, pij = Pr (st = j|st−1 = i),

8Indeed, it assumes for instance, symmetric counterparty credit risk, a homogeneous LIBOR-swap market credit qualitywhich is against the findings in Sun Sundaresan, and Wang (1993) and in Collin-Dufresne and Solnik (2001). See Duffie andSingleton (1997) for more details on the assumptions.

9See Jegadeesh and Pennacchi (1996) and Balduzzi, Das, and Foresi (1998)

25

εt =(εθr′t εδ

t

)′=

(εθt εr

t εδt

)′ ∼ N (0, I) and

µ (st) =

κθθ

0

κδδ (st)

, Φ =

1− κθ 0 0

κr 1− κr 0

κθδ κrδ 1− κδ

, Σ (st) =

σθ 0 0

0 σr 0

0 0 σδ (st)

(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 Rt is defined as Rt ≡ rt+δt.

We assume that the state variables rt and θt follow a Gaussian process and the spread factor

δt 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 (1988), Naik and Lee (1997),

Garcia and Perron (1996), Gray (1996), Landen (2000), Ang and Bekaert (2002a, 2002b,

2003), Bansal and Zhou (2002), Evans (2003), Dai, Singleton, and Yang (2003). Since most

of these papers focus on extended sample periods that encompass multiple monetary policy

regime changes (oil crisis in the early 1970s and the monetary experiment in the early 1980s

among others), and given the short time period for the analysis, 1987 to 2002, 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

(1979) show that the assumption of an arbitrage-free environment guarantees the existence

of a risk-neutral measure Q such that the price pt at time t of a claim to the cash flows ct+1

at time t+1 satisfies pt = EQt [e−rtct+1] where the expectation is taken under the risk neutral

measure Q. For a random variable dt+1 at time t + 1, we have:

EQt

[e−rtdt+1

]= Et

[ξt+1

ξt

e−rtdt+1

]= Et

[M r

t+1dt+1

](8)

where ξt+1 is the Radon-Nikodym derivative that converts the risk-neutral measure to the real

world or data-generating measure. By assuming the existence of ξt+1 ≡ ξt exp(−1

2λ′tλt − λtεt+1

)

or equivalently the existence of M rt+1 = ξt+1

ξte−rt , one can price any traded asset in the econ-

26

omy, particularly bonds. λt is the price of risk associated with the source of uncertainty εt+1

of the state variables because it determines the covariance between ξt+1 or M rt+1 and the

state variables. Therefore, we define the risk-free pricing kernel M r as:

mrt+1 = log

(M r

t+1

)= −rt − 1

2λθr′

t λθrt − λθr′

t εθrt+1 (9)

and similarly (see Duffie and Singleton (1997, 1999)), the spread-adjusted pricing kernel MR

takes the form:

mRt+1 = log

(MR

t+1

)= −rt − δt − 1

2λt (st+1)

′ λt (st+1)− λt (st+1)′ εt+1 (10)

where the price of risk is:

λt (st+1) =(λθr′

t , λδ0 (st+1)

)′=

(λθ

t , λrt , λδ

0 (st+1))′

=(λθ

0 + λθ1θt, λr

0 + λr1rt, λδ

0 (st+1))′

.

The prices of risk associated with the state variables rt and θt are time-varying but not regime

dependent and the price of risk associated with the spread state variable δt 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 (2003) 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 δt tries to encompass the

implications of our structural break results, while not sacrificing the analytical tractabil-

ity 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 long-

term mean of the state variable δt regime-dependent is consistent with our structural break

findings. It is also consistent with the evidence in Liu, Longstaff, and Mandell (2002) 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 ex-

pect 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 δt such that the coefficient of δt switches regimes, results

in the loss of a closed-form solution for the bond prices.10

5.2 Bond Prices

5.2.1 Risk-Free Bond Prices

Let bnt be the time t price of a risk-free discount bond that pays 1 at time t+n. The prices of

bonds are computed recursively using equation (8), bn+1t = Et

[M r

t+1bnt+1

], starting with the

price of a one-period bond. More explicitly, the price b1t 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 b0t = 1.

Therefore, we have:

b1t = Et

[M r

t+1b0t+1

]= Et

[M r

t+11]

= Et

[M r

t+1

]= exp (−rt) (11)

Given the formulation of the state variables θt and rt and of their respective prices of risk

above, our model falls within the affine class of term structure models (see Duffie and Kan

(1996)), and thus yields bond prices that are exponential affine functions of the state vari-

ables. Therefore, the risk-free bond prices are given by:

bnt = exp (Ar

n + Brnθt + Cr

nrt) (12)

where the loadings Arn, Br

n, and Crn follow the difference equations:11

Arn+1 = Ar

n + (κθθ − σθλθ0)B

rn − σrλ

r0C

rn +

1

2(σθB

rn)2 +

1

2(σrC

rn)2

Brn+1 = (1− κθ − σθλ

θ1)B

rn + κrC

rn

Crn+1 = −1 + (1− κr − σrλ

r1)C

rn (13)

with initial values deduced from equation (11).

10One alternative (see Ang and Bekaert (2003)), is to specify another independent state variable whose parameters do notswitch regime and whose role is to control the time-varying aspect of the prices of risk. In that case all the coefficients in theprices of risk can switch regimes.

11See Appendix B.1 for explicit derivation of the formula for the loadings Arn, Br

n, and Crn

28

5.2.2 Risky Bond Prices

Similarly, the time t price of a risky discount bond price, P nt (i), with promised payoff 1 at

time t + n conditional on regime st = i, is given by:

P nt (i) = exp

(AR

n (i) + BRn θt + CR

n rt + DRn δt

)(14)

where the loadings ARn (i), BR

n , CRn , and DR

n follow the difference equations:12

ARn+1(i) = (κθθ − σθλ

θ0)B

Rn − σrλ

r0C

Rn +

1

2

(σθB

Rn

)2+

1

2

(σrC

Rn

)2

+ log∑

j

pij expARn (j) + (κδδ(j)− σδ(j)λ

δ0(j))D

Rn +

1

2

(σδ(j)D

Rn

)2

BRn+1 = (1− κθ − σθλ

θ1)B

Rn + κrC

Rn + κθδD

Rn

CRn+1 = −1 + (1− κr − σrλ

r1)C

Rn + κrδD

Rn

DRn+1 = −1 + (1− κδ)D

Rn (15)

5.3 Econometrics and Estimation Results

We estimate the parameters of the model with standard maximum likelihood methodology

as in Chen and Scott (1993), Ang and Bekaert (2003), Dai, Singleton, and Yang (2003).13

we use both Treasury and swap yields to estimate the model.

5.3.1 Par Rates to Zero Rates

For this estimation, the data consists of Datastream weekly (Friday) cross-sectional obser-

vations of constant maturity Treasury, LIBOR and fixed-for-floating swap middle rates from

April 3, 1987, to December 27, 2002, 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 nt = 2

(1− bn

t∑n/26k=1 b26k

t

)(16)

12See Appendix B.2 for explicit derivation of the loadings ARn , BR

n , CRn , and DR

n13See also Duffie and Singleton (1997), Dai and Singleton (2000), Liu, Longstaff, and Mandell (2002)

29

for the constant maturity Treasuries, and the expression:

CMSnt = 2

(1− P n

t∑n/26k=1 P 26k

t

)(17)

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 10-year maturities, the six-month LIBOR,

and the 2-, 5-, 7-, and 10-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 (1993) 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. (1997) 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 10-year maturities, the six-month LIBOR, and the 2-,

5-, 7-, and 10-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 (1993) maximum likelihood estimation methodology requires the as-

sumption 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 vari-

ables 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

30

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 (12) and (14), the risk-free yield for maturity nk is given by:

− 1

nk

log (bnkt ) ≡ yrnk

t = −Arnk

nk

− Brnk

nk

θt −Cr

nk

nk

rt (18)

and similarly, the risky yield conditional on regime i is:

− 1

nk

log (P nkt (i)) ≡ yRnk

t (i) = −ARnk

(i)

nk

− BRnk

nk

θt −CR

nk

nk

rt −DR

nk

nk

δt (19)

By stacking the yields observed without error at time t into the N−vector R1t, we get

the following expression for R1t:

R1t = a1 (st) + b1Xt (20)

where a1 is the N−vector of the −Arnk

nkand −AR

nk(i)

nkterms, b1 is the N×N matrix of the −Br

nk

nk,

−Crnk

nk, −BR

nk

nk, −CR

nk

nk, and −DR

nk

nkterms corresponding to the yields in R1t. From equation (19),

we can easily extract the state variables of our model:

Xt = b−11 (R1t − a1 (st)) (21)

Substituting the expression of Xt in equation (6) into equation (21), and after rearranging,

we get:

R1t = c1 (s∗t ) + Ψ1R1t−1 + Ω (s∗t ) εt (22)

where c1 (s∗t ) = a1 (st) + b1µ (st) − b1Φb−11 a1 (st−1), Ψ1 = b1Φb−1

1 , Ω (s∗t ) = b1Σ (st), and s∗t

is defined as the state variable that counts all combinations of st and st−1 and a transition

31

probability matrix Π∗ =p∗ij

, p∗ij = Pr

(s∗t = j|s∗t−1 = i

). Similarly, we get the following

expression by stacking the remaining yields observed with error at time t into the M−vector

R2t :

R2t = Ru2t + ut (23)

where model implied or unobserved rates Ru2t = a2 (st) + b2Xt. We assume that the mea-

surement error ut is IID normal and uncorrelated across the yields measured with error,

ut ∼ N (0, V ) with V is a M × M diagonal matrix. Substituting equation (21) into

equation (23) yields the following equation for the dynamics of R2t:

R2t = c2 (s∗t ) + Ψ2R1t + ut (24)

where c2 (s∗t ) = a2 (st)− b2b−11 a1 (st), Ψ2 = b2b

−11 . Let Θ be the vector containing all the pa-

rameters of the model, and It =(R′

1t, R′1t−1, R

′1t−2, ..., R

′2t, R

′2t−1, R

′2t−2, ...

)be the econometri-

cian’s information set or a vector containing all observations through date t. From Hamilton

(1994), Ang and Bekaert (2003), Dai, Singleton, and Yang (2003), the log-likelihood function

is then:

L (IT , Θ) =T∑

t=2

log

∑

s∗t

f (R1t, R2t|s∗t , It−1; Θ) Pr (s∗t |It−1; Θ)

(25)

=T∑

t=2

log

∑

s∗t

f (R2t|R1t, s∗t , It−1; Θ) f (R1t|s∗t , It−1; Θ) Pr (s∗t |It−1; Θ)

where

f (R1t|s∗t , It−1; Θ) = (2π)−n2∣∣Ω (s∗t ) Ω (s∗t )

′∣∣− 12

× exp

−12(R1t − c1 (s∗t )−Ψ1R1t−1)

′

× (Ω (s∗t ) Ω (s∗t )

′)−1

× (R1t − c1 (s∗t )−Ψ1R1t−1)

32

is the probability density of R1t conditional on s∗t ,

f (R2t|R1t, s∗t , It−1; Θ) = (2π)−

m2 |V |− 1

2

× exp

−1

2(R2t − c2 (s∗t )−Ψ2R1t)

′ V −1 (R2t − c2 (s∗t )−Ψ2R1t)

is the probability density function of the measurement errors ut conditional on s∗t ,

Pr (s∗t = i|It−1; Θ) =∑

j

Pr(s∗t = i|s∗t−1 = j, It−1; Θ

)Pr

(s∗t−1 = j|It−1; Θ

)

and

Pr(s∗t−1 = j|It−1; Θ

)=

f(R1t−1, R2t−1|s∗t−1 = j, It−2; Θ

)Pr

(s∗t−1 = j|It−2; Θ

)∑

m f(R1t−1, R2t−1|s∗t−1 = m, It−2; Θ

)Pr

(s∗t−1 = m|It−2; Θ

)

5.3.3 Parameter Estimates

For identification, we follow Dai and Singleton (2000) 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 esti-

mation of a Gaussian term structure model, not all the price of risk parameters are easily

identified. Therefore, to facilitate econometric identification, we set λθ0 = 0 and λr

0 = 0. The

resulting model is identified. Following Hamilton (1994), s∗t is defined as:

s∗t st st−1

1 1 1

2 1 2

3 2 1

4 2 2

(26)

33

with associated transition probability matrix Π∗ =p∗ij

=

p11 0 p12 0

p11 0 p12 0

0 p21 0 p22

0 p21 0 p22

.

The log-likelihood function in equation (25) with the additional restrictions above is es-

timated using the standard Hamilton (1994) and Gray (1996) 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 param-

eters 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 p11 = Pr (st = 1|st−1 = 1) and p22 = Pr (st = 2|st−1 = 2) are 0.9763

and 0.9950 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 signifi-

cantly across regimes: δ is insignificant in regime 1, but high and statistically significant in

regime 2. Similarly, the constant price of risk of the spread factor varies significantly across

regimes. λδ0 is negative in the first regime, λδ

0 (st = 1) = −0.1687, but positive in the second

regime, λδ0 (st = 2) = 0.0760. 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 pa-

rameters across regimes at a significance level less than 1 percent. Also, Figure 6 shows the

graph of the constant loading term ARn (i) and we observe that AR

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 1, Pr (st = 1|IT ). The smoothed probabilities are used to classify observations into

regimes. The last two Panels of Figure 7 show the earlier graph of the marginal R2 of the

liquidity and default factors and the graph of the 10-year interest rate swap spread with

the dates of the structural changes estimated with the Bai and Perron (1998) methodology.

It is interesting to note that the graphs in the three Panels are consistent with each other.

Roughly, regime 1 covers the following periods: 04/03/1987 to 05/13/1988, 09/15/1989

to 06/29/1990, 11/05/1993 to 01/27/1995, and 08/07/1998 to 06/14/2002. The dates of

change of regimes coincide for the most part with the structural change dates of 08/25/1989,

05/08/1992, 08/14/1998, estimated in the first part of the paper. From the marginal R2

graph, we can call regime 1, the Liquidity regime. This observation is buttressed by the

fact that we are in regime 1 whenever there is a liquidity event. The announcement of the

Treasury in the fall of 1993 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 1998, the Y2K liquidity crisis in 1999, and the announcement by the Treasury

of its buyback program in spring 2000 are all situated in regime 1. 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 1998 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 (2002)), 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:

Xt+1 = µ(sL

t , sDt

)+ ΦXt + Σ

(sL

t , sDt

)εt+1

where Xt ≡ (θt rt δLt δDt)′ is the vector of the state variables, with δLt the liquidity adjustment

of the spread and δDt is the default adjustment. sLt ∈ 1, 2, ..., KL is the liquidity regime

variable with transition probability ΠL and sDt ∈ 1, 2, ..., KD is the default regime variable

with transition probability ΠD. Given that default and liquidity are very related concepts,

a switch in the liquidity regime can affect default risk and vice versa. Following Hamilton

(1994), and Ang and Bekaert (2003), we could incorporate the joint effects of sLt and sD

t by

defining an aggregate regime variable st to account for all combinations of sLt and sD

t . With

this new notation, the new model is equivalent to (6) with the regime variable st taking

values in 1, 2, ..., KL ∗KD, thus, yielding a KL ∗ KD × KL ∗ KD 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.

Prob (st = jk|st−1 = mn) ≡ Prob(sL

t = j, sDt = k|sL

t−1 = m, sDt−1 = n

)

= Prob(sL

t = j|sDt = k, sL

t−1 = m, sDt−1 = n

)

×Prob(sD

t = k|sLt−1 = m, sD

t−1 = 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 (1994)); 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 1990s

and in 2000, 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 150 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 peri-

ods. 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 regime-

switching 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 time-

varying 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 (1998): Estimating and Testing Linear Modelswith Multiple Structural Changes

A.1 Estimation of Break Dates

Using the notation of Bai and Perron (1998), the multiple linear regression model with m breaks (m + 1regimes) can be written as:

yt = x′tβ + z′tδj + ut t = Tj−1 + 1, ..., Tj (27)

for j = 1, ...,m + 1 and with the convention that T0 = 0 and Tm+1 = T . In equation (9), yt is the observeddependent variable at time t, xt and zt are the observed column vectors of explanatory variables at time tof dimension p and q respectively. The variance of ut needs not be constant. Using T observations on yt, xt

and zt, the purposes of the econometric method presented below are (i) to determine whether there is breakin the relationship between the variables yt and zt, (ii) to estimate the break points T1, T2, ..., Tm, and (iii)to construct the confidence intervals of the break points. In matrix notation, equation (9) can be rewrittenas:

Y = Xβ + Zδ + U (28)

where Y is a T × 1 vector (y1, y2, ..., yT )′, U is a T × 1 vector of the disturbances, X is a T × p matrix(x1, x2, ..., xT )′, Z = diag (Z1, Z2, ..., Zm+1) is a T × q (m + 1) matrix with Zi a ∆iT × q matrix of theobservations on zt in the ith regime,Zi =

[(z(Ti−1+1)1, z(Ti−1+2)1, ..., z(Ti)1

)′, ...,

(z(Ti−1+1)q, z(Ti−1+2)q, ..., z(Ti)q

)′], and ∆Ti = Ti − Ti−1. It isassumed in this section that the number of breaks m is known. In the next few sections, we will review themethods 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 (T1, ..., Tm)denoted by Tj, the corresponding least-squared estimates β (Tj) and δ (Tj) of β and δ respectively,are obtained by minimizing the sum of squared residuals

ST (T1, ..., Tm; β, δ) =(Y −Xβ − Zδ

)′ (Y −Xβ − Zδ

)

=m+1∑

i=1

Ti∑

t=Ti−1

(yt − x′tβ − z′tδi)2 (29)

Substituting β (Tj) and δ (Tj) into (11) and denoting the resulting sum of squared residuals as ST (T1, ..., Tm),

the estimated break points are such that

Tj

=

(T1, ..., Tm

)≡ arg min(T1,...,Tm)∈Πm

ST (T1, ..., Tm), where

Πm = (T1, ..., Tm) : Ti − Ti−1 ≥ h ≥ q. The regression parameter estimates are then β(

Tj

)and

δ(

Tj

)respectively.

A.2 Confidence Intervals of Break Dates

For i = 1, ..., m, and ∆T 0i = T 0

i −T 0i−1 where the superscript 0 refers to the true value of the parameter, define

∆i = δ0i+1 − δ0

i , Qi = p lim(∆T 0

i

)−1 ∑T 0i

t=T 0i−1+1

E (ztz′t), Ωi =

∑T 0i

r=T 0i−1+1

∑T 0i

t=T 0i−1+1

E (zrz′turut). Under

various assumptions, Bai and Perron (1998) obtain the following result about the limiting distribution14 ofthe break dates:

(∆′iQi∆i)

2

(∆′iΩi∆i)

(Ti − T 0

i

)=⇒ arg max

sV (i) (s) , (i = 1, ...,m) (30)

14The cumulative distribution of arg maxs V (i) (s) is in Bai (1997)

39

where V (i) (s) = W(i)1 (−s) − |s|/2, if s ≤ 0, V (i) (s) =

√ξi (φi,2/φi,1)W

(i)2 (−s) − ξi|s|/2, if s > 0, and

ξi = ∆′iQi+1∆i/∆′

iQi∆i, φ2i,1 = ∆′

iΩi∆i/∆′iQi∆i, φ2

i,2 = ∆′iΩi+1∆i/∆′

iQi+1∆i, W(i)1 (s) and W

(i)2 (s) are

independent Wiener processes defined on [0,∞), starting at 0 when s = 0. These processes are also inde-pendent across i.

A.3 Test of No Break versus a Fixed Number of Breaks

A supF − test is used to test the null hypothesis of no structural break m = 0 versus m = k breaks. Moreformally, it tests the hypothesis H0 : δ1 = δ2 = ... δk+1 of no break or equivalently H0 : Rδ = 0 where R

is a matrix with 1 on the main NW-SE diagonal, −1 at the ith row, and (q + i)th column and 0 elsewhere.Let (T1, ..., Tk) be a k-partition and define the break fraction λi = Ti/T or equivalently Ti = bTλic whereb·c denotes the greatest least integer. The F -statistic is defined as:

sup(λ1,...,λk)∈Λε

F ∗T (λ1, ..., λk; q) =

1T

(T − (k + 1) q − p

kq

)((Rδ

)′ (RV

(δ)

R′)−1 (

Rδ))

(31)

where V(δ)

is an estimate of the variance covariance matrix of δ that is robust to serial correlation andheteroscedasticity, Λε = (λ1, ..., λk) : |λi+1 − λi| ≥ ε, λ1 ≥ ε, λk ≤ 1− ε with ε an arbitrary small positivenumber. Intuitively, each break date is restricted to be asymptotically distinct and bounded from theboundaries of the sample. The limiting distribution of the test depends on the nature of the regressors andthe 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 Maximumtest considers the null of no structural break versus the alternative of an unknown number of breaks givensome upper bound M on the possible number of breaks. The Double Maximum test is specified as follow:

DmaxFT (M, q, a1, ..., aM ) = max1≤m≤M

am

sup

(λ1,...,λm)∈Λε

F ∗T (λ1, ..., λk; q)

(32)

where a1, ..., aM is a set of fixed weights. The distribution of DmaxFT follows from the distribution ofsup FT . The weights a1, ..., aM may reflect the imposition of some priors on the likelihood of variousnumber of breaks. An obvious candidate is to set all the weights equal to unity; this gives:

UDmaxFT (M, q) = max1≤m≤M

sup

(λ1,...,λm)∈Λε

F ∗T (λ1, ..., λk; q)

(33)

For the set of weights a1, ..., aM such that a1 = 1 and am = c (q, α, 1) /c (q, α, m) for m > 1, with c (q, α, m)the asymptotic critical value of the afore-mentioned test sup(λ1,...,λk)∈Λε

F ∗T (λ1, ..., λk; q) for a significancelevel α, the DmaxFT test becomes:

WDmaxFT (M, q) = max1≤m≤M

c (q, α, 1)c (q, α, m)

sup

(λ1,...,λm)∈Λε

F ∗T (λ1, ..., λk; q)

(34)

A.5 Number of Breaks

Both the sup F − test and the Double Maximum tests tell us whether there is a structural break in the timeseries model considered. To determine the actual number of breaks, one can use an information criterionsuch as the Bayesian Information Criterion defined as:

BIC (m) = ln(σ2

)+ p∗ ln (T ) /T (35)

40

where σ2 = T−1ST

(T1, ..., Tm

)and p∗ = (m + 1) q + m + p.

The sequential method is the third approach of determining the number of breaks. Given that the supF−testand the Double Maximum tests show the presence of structural break in the time series model, the methodto determine the number of breaks consists of sequentially applying the supFT (l + 1|l) test with the null of lbreaks versus the alternative of l+1 breaks starting with l = 1. At each step, the sequential method amountsto the application of (l + 1) tests of the null hypothesis of no structural change versus the alternative of asingle change; in other words the test is applied to each segment containing the observations Ti−1 + 1 toTi (i = 1, ..., l + 1) for the presence of an additional break, where T0 = 0, Tl+1 = T , and T1, ..., Tl denotethe estimated break dates for the model with l breaks. One concludes for a rejection in favor of a modelwith (l + 1) breaks if the overall minimal value of the sum of squared residuals (over all segments wherean additional break is included) is sufficiently smaller than the sum of squared residuals from the l breaksmodel. The break date thus selected is the one associated with this overall minimum. The test statistics isdefined by:

supFT (l + 1|l) =

ST

(T1, ..., Tm

)− min

1≤i≤linf

τ∈Λi,η

ST

(T1, ..., Ti−1, τ, T1, ..., Tl

)/σ2 (36)

For simplicity, the errors are assumed to be spherical and σ2 is a consistent estimate of the variance of the dis-turbances under the null hypothesis. ST

(T1, ..., Ti−1, τ, T1, ..., Tl

)is understood as ST

(τ, T1, ..., Tl

)for i = 1,

and as ST

(T1, ..., Ti−1, τ

)for i = l + 1. Finally, Λi,η =

τ : Ti−1 +

(Ti − Ti−1

)η ≤ Ti −

(Ti − Ti−1

)η

.As for the distribution of the test statistic, the authors showed that limT→∞ P [FT (l + 1|l) ≤ x] = Gq,η (x)with Gq,η (x) the distribution function of supη≤µ≤1−η ‖ Wq (µ)− µWq (1) ‖2 / (µ (1− µ)).This process is repeated, increasing l sequentially until the sup FT (l + 1|l) fails to reject the null hypothesisof no additional structural changes. The number of breaks is therefore the last l considered. The authorssuggest that the sequential method may be the best test to estimating the number of breaks.

B Bond Prices

B.1 Risk-Free Bond Prices

Given the formulation of the state variables θt and rt 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 (1996)), andthus yields bond prices that are exponential affine functions of the state variables. In this section, we showexplicitly how to derive the difference equations in the article.

Suppose that the price bnt at time t of an n-period risk-free bond price is an exponential affine function of

the state variables θt and rt and is given by

bnt = exp (Ar

n + Brnθt + Cr

nrt) (37)

From equation (3) the price b1t at time t of a one-period risk-free zero-coupon bond satisfies:

b1t = Et

[Mr

t+1b0t+1

]

= Et

[Mr

t+11]

= Et

[Mr

t+1

]

= Et

[exp(−rt − 1

2λθr′

t λθrt − λθr′

t εθrt+1)

]

= exp(−rt − 12λθr′

t λθrt )Et

[exp(−λθr

t εθrt+1)

]

= exp(−rt)= exp(Ar

1 + Br1θt + Cr

1rt) (38)

41

Simple matching of coefficients gives initial values Ar1 = 0, Br

1 = 0, and Cr1 = −1 for starting the recursive

equations. We easily verify that the price bn+1t at time t of an n + 1-period risk-free bond price also has an

exponential affine functional form:

bn+1t = Et

[Mr

t+1bnt+1

]

= Et

[exp

(−rt − 1

2λθr′

t λθrt − λθr′

t εθrt+1

)exp (Ar

n + Brnθt+1 + Cr

nrt+1)]

= exp(−rt − 1

2λθr′

t λθrt + Ar

n

)× Et

[exp

(−λθrt εθr

t+1 + Ψθrn Xθr

t+1

)]

= exp(−rt − 1

2λθr′

t λθrt + Ar

n + Ψθrn µθr + Ψθr

n ΦθrXθrt

)

× Et

[exp

(Ψθr

n Σθr − λθr′t

)εθrt+1

]

= exp −rt − 1

2λθr′t λθr

t + Arn + Ψθr

n µθr + Ψθrn ΦθrXθr

t

+ 12 (Ψθr

n Σθr − λθr′t )(Ψθr

n Σθr − λθr′t )′

= exp

Arn +

(κθθ − σθλ

θ0

)Br

n + 12 (σθB

rn)2 + 1

2 (σrCrn)2 − σrλ

r0C

rn

+((

1− κθ − σθλθ1

)Br

n + κrCrn

)θt

+(−1 + (1− κr − σrλr1) Cr

n) rt

= exp(Ar

n+1 + Brn+1θt + Cr

n+1rt

)(39)

where in the third and fourth equalities, we have Ψθrn =

[Br

n Crn

]and Xθr

t+1 =[

θt+1

rt+1

], and µθr, Φθr,

Σθrare partitions of parameter matrices in equations (6) and (6) corresponding to the state variables θt andrt. Again, matching coefficients yields the recursive equations:

Arn+1 = Ar

n +(κθθ − σθλ

θ0

)Br

n − σrλr0C

rn +

12

(σθBrn)2 +

12

(σrCrn)2

Brn+1 =

(1− κθ − σθλ

θ1

)Br

n + κrCrn

Crn+1 = −1 + (1− κr − σrλ

r1)Cr

n (40)

B.2 Risky Bond Prices: Regime-Switch

Suppose that the time t price Pnt (i) of a risky discount bond with promised payoff of 1 at time t + n,

conditional on regime st = i, is exponential affine functions of the state variables θt, rt, and δt, and is givenby:

Pnt (i) = exp

(AR

n (i) + BRn θt + CR

n rt + DRn δt

)(41)

From equation (8) the price P 1t (i) at time t of a one-period risky zero-coupon bond conditional on regime

st = i, satisfies:

P 1t (i) =

∑

j

pijEt

[MR

t+1P0t+1(j)|st+1 = j

]

=∑

j

pijEt

[Mr

t+1|st+1 = j]

=∑

j

pijEt

[exp(−rt − δt − 1

2λt (j)′ λt (j)− λt (j)′ εt+1)

]

=∑

j

pij exp(−rt − δt − 1

2λt (j)′ λt (j)

)Et

[exp

(−λt (j)′ εt+1

)]

= exp (−rt − δt)= exp

(AR

1 + BR1 θt + CR

1 rt + DR1 δt

)(42)

42

Simple matching of coefficients gives initial values AR1 (i) = 0, BR

1 = 0, CR1 = −1, and DR

1 = −1 for startingthe recursive equations. We easily verify that the time t price Pn+1

t (i) of a risky discount bond with promisedpayoff of 1 at time t + n + 1, conditional on regime st = i also has an exponential affine functional form:

Pn+1t (i) =

∑

j

pijEt

[MR

t+1Pnt+1(j)|st+1 = j

]

=∑

j

pijEt

[exp

(−rt − δt − 1

2λt (j)′ λt (j)− λt (j)′ εt+1

)]

× exp(AR

n (j) + BRn θt+1 + CR

n rt+1 + DRn δt+1

)

=∑

j

pij exp(−rt − δt − 1

2λt (j)′ λt (j) + AR

n (j))

×Et

[exp

(−λt (j)′ εt+1 + ΨnXt+1

)]

=∑

j

pij exp(−rt − δt − 1

2λt (j)′ λt (j) + AR

n (j) + Ψnµ(j) + ΨnΦXt

)

×Et [exp (ΨnΣ(j)− λt(j)′) εt+1]

=∑

j

pij exp −rt − δt − 1

2λt (j)′ λt (j) + ARn (j) + Ψnµ(j) + ΨnΦXt

+ 12 (ΨnΣ(j)− λt(j)′) (ΨnΣ(j)− λt(j)′)

′

= exp

κθθBRn + 1

2

(σθB

Rn

)2 + 12

(σrC

Rn

)2 − σθλθ0B

Rn − σrλ

r0C

Rn

+ log∑

j pij exp

ARn (j) + κδδ(j)DR

n + 12

(σδ(j)DR

n

)2 − σδ(j)λδ0(j)D

Rn

+((

1− κθ − σθλθ1

)BR

n + κrCRn + κθδD

Rn

)θt

+(−1 + (1− κr − σrλ

r1)CR

n + κrδDRn

)rt

+(−1 + (1− κδ)DR

n

)δt

= exp(AR

n+1(i) + BRn+1θt + CR

n+1rt + DRn+1δt

)(43)

where in the third equality, we have Ψn =[

BRn CR

n DRn

]. We deduce that the loadings AR

n (i), BRn , CR

n ,and DR

n follow the difference equations:

ARn+1(i) =

(κθθ − σθλ

θ0

)BR

n − σrλr0C

Rn +

12

(σθB

Rn

)2+

12

(σrC

Rn

)2

+ log

∑

j

pij exp

ARn (j) +

(κδδ(j)− σδ(j)λδ

0(j))DR

n +12

(σδ(j)DR

n

)2

BRn+1 =

(1− κθ − σθλ

θ1

)BR

n + κrCRn + κθδD

Rn

CRn+1 = −1 + (1− κr − σrλ

r1)CR

n + κrδDRn

DRn+1 = −1 + (1− κδ)DR

n (44)

43

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47

Figure 1: Interest Rate Swap Spreads Weekly time series of interest rate swap spread in basis points for maturity 2, 5, 7, and 10 years, for the period of April 1987 to December 2002. Interest rate swap spread is defined as the swap rate minus the corresponding constant maturity Treasury rate.

2-year Interest Rate Swap Spread

0

50

100

150

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

2-year Interest Rate Swap Spread Change

-40

-20

0

20

40

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

5-year Interest Rate Swap Spread

0

50

100

150

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

5-year Interest Rate Swap Spread Change

-40-2002040

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

7-year Interest Rate Swap Spread

0

50

100

150

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

7-year Interest Rate Swap Spread Change

-40

-20

0

20

40

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

10-year Interest Rate Swap Spread

0

50

100

150

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

10-year Interest Rate Swap Spread Change

-40

-20

0

20

40

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

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.

0

20

40

60

80

Dec-85

Dec-86

Dec-87

Dec-88

Dec-89

Dec-90

Dec-91

Dec-92

Dec-93

Dec-94

Dec-95

Dec-96

Dec-97

Dec-98

Dec-99

Dec-00

Dec-01

Dec-02

Treasury LIBOR

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

Figure 4: Drivers of Interest Rate Swap Spread. Interest rate swap spread is defined as the 10-year swap rate minus the 10-year constant maturity Treasury rate. Off/on-the-run spread, the measure of the liquidity factor, refers to the spread between the yield 10-year Treasury issued in the previous auctions and yield on the most recently auctioned 10-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 10-year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options.

0

50

100

150

Apr-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Swap

Spr

ead

0

4

8

12

Fed

eral

Fun

ds T

arge

t R

ate

10-year Swap Spread Federal Funds Target Rate

0

50

100

150

Apr-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Swap

Spr

ead

0

4

8

12

6-m

onth

Tre

asur

y

10-year Swap Spread 6-month Treasury

0

50

100

150

Apr-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Swap

Spr

ead

-20

-10

0

10

20

30

10-y

ear O

ff/O

n Sp

read

10-year Swap Spread 10-year Off/On Spread

0

50

100

150

Apr-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Swap

Spr

ead

-40

0

40

80

120

160

LIB

OR

Def

ault

Ris

k

10-year Swap Spread LIBOR Default

0

50

100

150

Apr-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Swap

Spr

ead

0

20

40

60

80

100

VIX

10-year Swap Spread CBOE's VIX

Figure 5: Explanatory Power of Liquidity and Default Marginal adjusted R2 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 10-year swap rate minus the 10-year constant maturity Treasury rate. The liquidity factor is proxied by the spread between the yield on 10-year Treasury issued in the previous auctions and yield on the most recently auctioned 10-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 10-year off/on-the-run 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 R2 of a specific sub-period is assigned to all dates of that sub-period.

0

10

20

30

40

50

Apr-87

Apr-89

Apr-91

Apr-93

Apr-95

Apr-97

Apr-99

Apr-01

Mar

gina

l Adj

uste

d R

-squ

are

Liquidity Default

Figure 6: Loadings of the Constant Term: ( )iARn The price of a risky bond price conditional on ist = is given by:

( ) ( )( )tRnt

Rnt

Rn

Rn

nt DrCBiAiP δθ +++= exp

-0.16

-0.12

-0.08

-0.04

0

0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25

Maturity (years)

st = 1 st = 2

Figure 7: Smoothed Probabilities [ ]Tt Isob |1Pr = The smoothed probabilities are used to classify observations into regimes. [ ]Tt Isob |1Pr = is the probability of being in regime 1 in the Gaussian regime-switching term structure model of interest rate swap yields in section 5.

Smoothed Probability for Regime 1

0

0.2

0.4

0.6

0.8

1

Apr

-87

Apr

-88

Apr

-89

Apr

-90

Apr

-91

Apr

-92

Apr

-93

Apr

-94

Apr

-95

Apr

-96

Apr

-97

Apr

-98

Apr

-99

Apr

-00

Apr

-01

Apr

-02

0

10

20

30

40

50

Apr

-87

Apr

-88

Apr

-89

Apr

-90

Apr

-91

Apr

-92

Apr

-93

Apr

-94

Apr

-95

Apr

-96

Apr

-97

Apr

-98

Apr

-99

Apr

-00

Apr

-01

Apr

-02

Liquidity Marginal Adjusted R-square Default Marginal Adjusted R-square

10-year Swap Spread with Structural Change Dates

0

50

100

150

Apr

-87

Apr

-88

Apr

-89

Apr

-90

Apr

-91

Apr

-92

Apr

-93

Apr

-94

Apr

-95

Apr

-96

Apr

-97

Apr

-98

Apr

-99

Apr

-00

Apr

-01

Apr

-02

Table 1: Bai and Perron (1998) Structural Change Test. Test for multiple structural breaks in the relationship between 10-year interest rate swap spread and its driving factors. This test allows for a maximum of 5 breaks with 15% trimming.

( ) ( ) ( ) ( ) ( ) ( )tutVIXtOnOfftltLIBORdefautTreasurytSS jjjjj +++++= 54321 / βββββ jj TTt ,,11 K+= − SS denotes the 10-year IRS spread, Treasury is the 6-month constant maturity Treasury rate, ltLIBORdefau is the default risk in the Eurodollar market, OnOff / is the 10-year off/on-the-run spread, and VIX is the CBOE's VIX index

SupF(1) SupF(2) SupF(3) SupF(4) SupF(5) UDmax WDmax 54.46*** 91.72*** 256.13*** 316.38*** 274.50*** 316.38*** 506.49***

SupF(2|1) SupF(3|2) SupF(4|3) SupF(5|4) 70.35*** 170.22*** 9.94 13.06

Number of Breaks Selected: 3 The Three Breaks from the Global Optimization Break Date 95% Confidence Interval Break 1 8/25/89 8/11/89 10/13/89 Break 2 5/8/92 4/24/92 5/22/92 Break 3 8/14/98 6/5/98 8/21/98 *** indicates significance at the 1% 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+1|l) tests the null of l breaks versus the alternative of l+1 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 10-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 (1998) multiple structural changes test. Interest rate swap spread is defined as the 10-year swap rate minus the 10-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 10-year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options.

Level First Difference Sample Period # Obs Mean Std

Auto- correlation Mean Std

Auto- correlation

Whole Sample: 04/03/1987 - 12/27/2002 822 10-year Interest Rate Swap Spread 65.45 26.51 0.97 -0.05 6.84 -0.37 6-month Constant Maturity Treasury 5.31 1.80 1.00 -0.01 0.13 -0.02 LIBOR Default Risk 38.18 27.10 0.88 -0.06 13.51 -0.35 10-year Off/On-the-run Spread 4.33 4.67 0.82 -0.01 2.80 -0.40 CBOE's VIX Index 21.51 8.10 0.89 0.01 3.85 -0.26 Sub-period 1: 04/03/1987 - 08/25/1989 126 10-year Interest Rate Swap Spread 90.21 13.94 0.69 -0.05 11.01 -0.45 6-month Constant Maturity Treasury 7.43 1.07 0.98 0.02 0.23 -0.05 LIBOR Default Risk 87.23 24.28 0.61 -0.19 21.52 -0.31 10-year Off/On-the-run Spread 4.29 2.08 0.65 0.03 1.75 -0.30 CBOE's VIX Index 25.00 11.05 0.78 -0.04 7.32 -0.23 Sub-period 2: 09/01/1989 - 05/08/1992 141 10-year Interest Rate Swap Spread 74.15 16.54 0.94 -0.26 5.92 -0.40 6-month Constant Maturity Treasury 6.58 1.47 1.00 -0.03 0.11 -0.01 LIBOR Default Risk 35.29 18.78 0.77 -0.27 12.79 -0.36 10-year Off/On-the-run Spread 5.00 2.25 0.90 -0.02 1.01 -0.05 CBOE's VIX Index 20.16 4.53 0.80 -0.02 2.84 -0.35 Sub-period 3: 05/15/1992 - 08/14/1998 327 10-year Interest Rate Swap Spread 39.71 6.65 0.73 0.04 4.92 -0.44 6-month Constant Maturity Treasury 4.78 1.03 1.00 0.00 0.09 -0.09 LIBOR Default Risk 26.84 10.52 0.62 0.11 9.21 -0.47 10-year Off/On-the-run Spread 2.58 4.10 0.93 0.01 1.59 -0.07 CBOE's VIX Index 16.41 4.92 0.92 0.05 2.03 -0.33 Sub-period 4: 08/21/1998 - 12/27/2002 228 10-year Interest Rate Swap Spread 83.52 23.13 0.96 -0.06 6.80 -0.19 6-month Constant Maturity Treasury 4.12 1.69 1.00 -0.02 0.12 0.04 LIBOR Default Risk 29.42 19.06 0.75 -0.12 13.56 -0.33 10-year Off/On-the-run Spread 6.47 6.31 0.72 -0.04 4.72 -0.47 CBOE's VIX Index 27.71 6.23 0.83 0.01 3.64 -0.26

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 (1998) test for multiple structural changes. Interest rate swap spread is defined as the 10-year swap rate minus the 10-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 10-year off/on-the-run spread. VIX is a weighted average of implied volatilities of near-the-money OEX put and call options. Level First Difference

10-year Swap

Spread 6-month Treasury

Spread LIBOR Default

Risk 10-year Off/On-the-run Spread

10-year Swap Spread

6-month Treasury Spread

LIBOR Default Risk

10-year Off/On-the-run Spread

Panel A: Whole Sample Period: 04/03/1987 - 12/27/2002 6-month Treasury 0.46 -0.07 LIBOR Default Risk 0.45 0.53 0.23 -0.41 10-year Off/On-the-run Spread 0.54 0.31 0.00 0.03 0.08 -0.47 VIX 0.48 -0.05 0.25 0.23 0.00 -0.28 -0.02 0.02 Panel B: Sub-period 1: 04/03/1987 - 08/25/1989 6-month Treasury -0.64 -0.13 LIBOR Default Risk 0.31 -0.10 0.34 -0.37 10-year Off/On-the-run Spread 0.46 -0.29 0.10 0.22 -0.09 -0.19 VIX 0.40 -0.49 0.16 0.35 -0.16 -0.47 -0.08 0.29 Panel C: Sub-period 2: 09/01/1989 - 05/08/1992 6-month Treasury 0.90 -0.07 LIBOR Default Risk 0.38 0.46 0.06 -0.30 10-year Off/On-the-run Spread 0.34 0.09 -0.20 0.07 -0.07 -0.22 VIX 0.32 0.42 0.45 -0.29 0.02 -0.08 0.01 -0.02 Panel D: Sub-period 3: 05/15/1992 - 08/14/1998 6-month Treasury 0.44 -0.05 LIBOR Default Risk 0.20 0.18 0.23 -0.51 10-year Off/On-the-run Spread 0.61 0.40 -0.14 0.13 0.05 -0.28 VIX 0.46 0.33 0.42 0.50 0.08 0.02 -0.01 0.03 Panel E: Sub-period 4: 08/21/1998 - 12/27/2002 6-month Treasury 0.85 -0.01 LIBOR Default Risk 0.26 0.32 0.18 -0.50 10-year Off/On-the-run Spread 0.67 0.73 0.03 -0.06 0.20 -0.81 VIX -0.24 -0.30 -0.02 -0.29 0.21 -0.18 0.08 -0.08

Table 4: Regression Analysis of the Drivers of Interest Rate Swap Spread For each sub-period estimated by the Bai and Perron (1998) structural change test, a dummy variable Dj is defined that takes a value of 1 or 0 depending on whether the current observation is in the relevant sub-period. The sub-periods are: 04/03/1987 to 08/25/1989, 09/01/1989 to 05/08/1992, 05/15/1992 to 08/14/1998, and 08/21/1998 to 12/27/2002. Interest rate swap spread is defined as the 10-year swap rate minus the 10-year Treasury rate. Off/on-the-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 10-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 1% level and ** indicates significance at the 5% level.

Variables Coefficient T-stat F-stat Panel A: Regression 1

Constant 4.5982 0.87 6-month Treasury 3.3501 4.67 LIBOR Default Risk 0.2396 5.22 Off/On-the-run Spread 2.2439 8.98 VIX 1.1257 5.91 Adjusted R2 = 0.60 Panel B: Regression 2 D1 122.4965 10.84 48.82*** D2 -2.1844 -0.56 D3 27.6377 13.31 D4 35.5798 4.39 (6-month Treasury) x D1 -6.9843 -6.18 110.13*** (6-month Treasury) x D2 9.6435 19.88 (6-month Treasury) x D3 1.0484 2.78 (6-month Treasury) x D4 10.4258 8.32 (LIBOR Default Risk) x D1 0.1312 4.47 2.38** (LIBOR Default Risk) x D2 0.0288 0.62 (LIBOR Default Risk) x D3 0.1468 3.17 (LIBOR Default Risk) x D4 0.0135 0.25 (Off/On-the-run Spread) x D1 1.8951 4.29 7.19*** (Off/On-the-run Spread) x D2 2.0556 7.99 (Off/On-the-run Spread) x D3 0.9079 6.46 (Off/On-the-run Spread) x D4 0.4403 1.72 (VIX) x D1 0.0026 0.02 0.09 (VIX) x D2 0.0853 0.53 (VIX) x D3 0.0472 0.43 (VIX) x D4 0.0668 0.31 Adjusted R2 = 0.90 ( ) ( ) ( ) ( ) ( ) ( )tutVIXtOnOfftltLIBORdefautTreasurytSS +++++= 5/4321 βββββ (Regression 1)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )tutVIXtDtDtDtD

tOnOfftDtDtDtDtltLIBORdefautDtDtDtD

tTreasurytDtDtDtDtDtDtDtDtSS

+++++

++++++++

+++++++=

4453

352

251

15

/4443

342

241

144

433

332

231

13

4423

322

221

124

413

312

211

11

ββββ

ββββββββ

ββββββββ

(Regression 2)

Table 5: Maximum Likelihood Estimates ( )′= tttt rX δθ ,, follows ( ) ( ) 1111 ++++ Σ+Φ+= ttttt sXsX εµ

The regime variable 2,1∈ts , the transition matrix ( )isjsobppppp

ttij ===

−

−=Π + |Pr,

11

12222

1111 ,

( )( )

( )( )

=Σ

−−

−=Φ

=

+

+

+

+

1

1

1

1

000000

,1

01001

,0

t

rt

r

rr

t

t

ss

ss

δ

θ

δδθδ

θ

δ

θ

σσ

σ

κκκκκ

κ

δκ

θκµ

The risk-free pricing kernel rtM 1+ and the spread-adjusted pricing kernel R

tM 1+ are defined respectively as:

( ) rt

rt

rt

rtt

rt

rt rMm θθθθ ελλλ 12

111 log +++

′−

′−−== and ( ) ( ) ( ) ( ) 11112

111 log ++++++

′−′−−−== tttttttttRt

Rt sssrMm ελλλδ

where the price of risk ( ) ( ) ( )( ) ( )( )′++=′

=′

′

= ++++ 10101010101 ,,,,, ttrr

ttrttt

rttt srsss δθθδθδθ λλλθλλλλλλλλ .

( )INt ,0~ε and ( )′=′

′

= δθδθ εεεεεε trttt

rtt ,,,

Parameter Estimate Std Error

θκ × 52 0.1961 0.04356

θ × 52 0.0560 0.02625

θσ × 52 0.0052 0.00145 θλ0 - - θλ1 46.8902 8.84248

rκ × 52 0.1824 0.05066

rσ × 52 0.0016 0.00002 r0λ - - r1λ -86.8035 32.34501

δκ × 52 0.0889 0.00373

θδκ × 52 -0.0149 0.00414

δκ r × 52 0.0163 0.00065 ( )1=tsδ × 52 0.0001 0.00052 ( )2=tsδ × 52 0.0323 0.00868 ( )1=tsδσ × 52 0.0013 0.00004 ( )2=tsδσ × 52 0.0013 0.00003 ( )10 =ts

δλ -0.1687 0.01311 ( )20 =ts

δλ 0.0760 0.01522

11p 0.9763 0.00203

22p 0.9950 0.00173 100 × Std Deviation of Yield Measurement Errors 6-month Treasury × 52 0.0036 0.000003 5-year Treasury × 52 0.0009 0.000000 7-year Treasury × 52 0.0010 0.000000 10-year Treasury × 52 0.0015 0.000001 2-year swap × 52 0.0020 0.000002 5-year swap × 52 0.0013 0.000001 7-year swap × 52 0.0007 0.000001