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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: [email protected]. Phone: (212) 853-9073
1
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
2
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
3
is the most important determinant of the swap spread, and a second regime in which the
liquidity in the Treasury market is the most important determinant of the swap spread.
The presence of these different regimes coincides with well-known economic events: evidence
in Gupta and Subrahmanyam (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)
4
estimate whether there are breaks in the time series properties of swap spreads and to date
the break points accordingly. The events cited above provide the motivations to do the
break tests. We first test the hypothesis of no break against the alternative of at least one
break. Second, we do a sequential test of one break versus the alternative of two breaks.
Given that we do not reject two breaks, we test the null of two breaks versus the alternative
of three breaks, etc. until we cannot reject the null. This constitutes the Bai and Perron
(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.
5
affect as they please the characteristics of their cash flows.
The paper is organized as follows. In the next section, we discuss the determinants of
the IR swap spread. Section 3 contains the data description. In section 4, we present the
results of the structural change tests. The regime-switching term structure model and its
estimation results are in section 5 and we conclude in the final section.
2 Determinants of Swap Spreads
To avoid a kitchen-sink type approach, we review below the arguments in Brown, Harlow,
and Smith (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)
6
The above relationship shows that the IR swap spread is approximately a function of the
LIBOR-Treasury rate (GC) spread. The LIBOR-Treasury rate spread encompasses the 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
7
one of the counterparties defaults, the other counterparty is automatically relieved from the
rest of its obligations. Also, throughout the history of the IR swap market, and much more
so recently, there have been credit enhancement innovations such as transaction with only an
approved list of clients, the use of collateral, and marking-to-market to explicitly deal with
the counterparty default risk. In a 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.
8
We investigate whether the sensitivity of IR swap spread to counterparty default risk is
time-varying. This issue is particularly important for agents who are deciding whether to
hedge their corporate debt portfolios with Treasuries or IR swaps.
Second, given that the IR swap spread is a function of the LIBOR-Treasury rate spread,
it also reflects the default risk in the banking sector. Since LIBOR is the rate on short
term loans to banks rated A to AA on average, the default risk in the LIBOR market could
be very small in normal times. In other words, the LIBOR does reflect the default risk
of highly rated banks and not the default risk of banks with serious credit risk problems
because banks with deteriorating credit risk are simply removed from the calculation of the
LIBOR. However, in turbulent times like the S&L crisis in the 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
9
convenience yield is lost to an investor wishing to receive fixed rate payments, who, in lieu
of purchasing a Treasury note, enters into an IR swap contract to receive fixed payments.
Liu, Longstaff, and Mandell (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.
10
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
11
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
12
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
13
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
14
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
References
[1] Andrews, W. K. Donald, 1993, Tests for Parameter Instability and Structural Change with UnknownChange Point, Econometrica, 61, 821-856.
[2] Andrews, W. K. Donald, and Werner Ploberger, 1994, Optimal Tests When a Nuisance Parameter isPresent Only Under the Alternative, Econometrica, 62, 1383-1414.
[3] Andrews, W. K. Donald, Inpyo Lee, and Werner Ploberger, 1996, Optimal Changepoint Tests for NormalLinear Regression, Journal of Econometrics, 70, 9-38.
[4] Ang, Andrew and Geert Bekaert, 2002a, Regime Switches in Interest Rates, Journal of Business andEconomic Statistics, 20, 163-182.
[5] Ang, Andrew and Geert Bekaert, 2002b, Short Rate Nonlinearities and Regime Switches, Journal ofEconomic Dynamics and Control, 26, 7-8, 1243-1274.
[6] Ang, Andrew and Geert Bekaert, 2003, The Term Structure of Real Rates and Expected Inflation,working paper, Columbia University.
[7] Ang, Andrew and Monika Piazzesi, 2003, A No-Arbitrage Vector Autoregression of Term StructureDynamics with Macroeconomic and Latent Variables, Journal of Monetary Economics, 50, 745-787.
[8] Aragon, O. George, 2002, Interest Rate Swaps: An Asset Allocation Perspective, working paper, BostonCollege
[9] Arak, Marcelle, Arturo Estrella, Laurie Goodman, and Andrew Silver, 1988, Interest Rate Swaps: AnAlternative Explanation, Financial Management, Summer, 12-18.
[10] Backus David, Silverio Foresi, and Chris Telmer, 1998, Discrete-Time Models of Bond Pricing, WorkingPaper, NBER.
[11] Bai, Jushan, 1997, Estimation of a Change Point in Multiple Regression Models, Review of Economicand Statistics, 79, 551-563.
[12] Bai, Jushan, Robin L. Lumsdaine, and James H. Stock, 1998, Testing for and Dating Breaks in Sta-tionary and Nonstationary Multivariate Time Series, Review of Economic Studies, 65:3, 395-432.
[13] Bai, Jushan, and Pierre Perron, 1998, Estimating and Testing Linear Models with Multiple StructuralChanges, Econometrica, 66, 47-78.
[14] Bai, Jushan, and Pierre Perron, 2001a, Computation and Analysis of Multiple Structural Change Mod-els, Working Paper, Boston College.
[15] Bai, Jushan, and Pierre Perron, 2001b, Multiple Structural Change Models: A Simulation Analysis,Working Paper, Boston College.
[16] Balduzzi Pierluigi, Sanjiv R. Das, and Silverio Foresi, 1998, The Central Tendency: A Second Factor inBond Yields, The Review of Economics and Statistics,
[17] Bank for International Settlements, 1986, Recent Innovations in International Banking, Basle, April.
[18] Bank for International Settlements, 2000, The Global OTC Derivatives Market Continues to Grow,Press Release, 36/2000E.
[19] Bansal, R., and H. Zhou, 2002, Term Structure of Interest Rates with Regime Shifts, Journal of Finance,57, 5, 1997-2043.
[20] Bansal, R., G. Tauchen, and H. Zhou, 2003, Regime-Shifts, Risk Premiums in the Term Structure, andthe Business Cycle, working paper, Duke University.
44
[21] Bekaert, Geert, Campbell R. Harvey, and Robin L. Lumsdaine, 2002a, Dating the Integration of WorldEquity Markets, Journal of Financial Economics, 65, 203-247.
[22] Bekaert, Geert, Campbell R. Harvey, and Robin L. Lumsdaine, 2002b, The Dynamics of EmergingMarket Equity Flows, Journal of International Money and Finance, 21, 295-350.
[23] Bessembinder, Hendrik, 1991, Forward Contracts and Firm Value: Investment Incentive and ContractingEffects, Journal of Financial and Quantitative Analysis, 26, 519-32.
[24] Bicksler, J., and A. H. Chen 1986, An Economic Analysis of Interest Rate Swaps, Journal of Finance,41, 645-55.
[25] Bodnar, Gordon M., Gregory S. Hayt, Richard C Marston, 1998, 1998 Wharton Survey of FinancialRisk Management by US Non-financial Firms, Financial Management, 27, 70-91.
[26] Brown, C. Keith, W. V. Harlow, Donald J. Smith, 1994, An Empirical Analysis of Interest Rate SwapSpreads, Journal of Fixed Income, March 1994, 61-78.
[27] Brown, C. Keith and Donald J. Smith, 1988, Recent Innovations in Interest Rate Risk Managementand the Reintermediation of Commercial Banking, Financial Management, Winter, 45-58.
[28] John Y. Campbell and Glen B. Taksler, 2003, Equity Volatility and Corporate Bond Yields, forthcoming,Journal of Finance.
[29] Campbell, Y. John, Andrew W. Lo, and A. Craig Mackinlay, 1997, The Econometrics of FinancialMarkets, Princeton, NJ, Princeton University Press.
[30] Chen, Andrew and Arthur K. Selender, 1994, Determination of Swap Spreads: An Empirical Analysis,Working paper, Southern Methodist University.
[31] Chen, Ren-Raw, and Louis Scott, 1993, Maximum Likelihood Estimation for a Multi-factor EquilibriumModel of the Term Structure of Interest Rates, Journal of Fixed Income, 3, 14-31.
[32] Chow, Gregory, 1960, Tests of Equality Between Sets of Coefficients in Two Linear Regressions, Econo-metrica, 28, 3, 591-605.
[33] Cochrane, John, 2001, Asset Pricing, Princeton, NJ, Princeton University Press.
[34] Collin-Dufresne, Pierre, Robert S. Goldstein, J. Spencer Martin, 2001, The Determinants of CreditSpread Changes, Journal of Finance, 56, 2177-2208.
[35] Collin-Dufresne, Pierre, Bruno Solnik, 2001, On the term Structure of Default Premia in the Swap andLIBOR markets, Journal of Finance, 56, 1095-1115.
[36] Cooper, Neil and Cedric Scholtes, 2001, Government Bond Market Valuations in an Era of DwindlingSupply, BIS Papers 5, October 2001.
[37] Cossin, Didier and Hugues Pirotte, 1997, Swap Credit Risk: An Empirical Investigation on TransactionData, Journal of Banking and Finance, 21, 1351-1373.
[38] Crosbie, Peter and Jeffrey Bohn, 2002, Modeling Default Risk, KMV, LLC.
[39] Dai, Qiang, and Kenneth. J. Singleton, 2000, Specification Analysis of Affine Term Structure Models,Journal of Finance, 55, 5, 1943-1978.
[40] Dai, Qiang, and Kenneth. J. Singleton, 2002, Expectation Puzzles, Time-Varying Risk Premia, andAffine Models of the Term Structure, Journal of Financial Economics, 63, 415-41.
[41] Dai, Qiang, and Kenneth. J. Singleton, 2003, Term Structure Dynamics in Theory and Reality, forth-coming, Review of Financial Studies.
45
[42] Dai, Qiang, and Kenneth. J. Singleton, and Wei Yang, 2003, Are Regime Shifts Priced in U.S. TreasuryMarkets? working paper, NYU.
[43] Duffie, Darrell and Ming Huang, 1996, Swap Rates and Credit Quality, Journal of Finance, 51, 921-949.
[44] Duffie, Darrell and Rui Kan, 1996, A Yield-Factor Model of Interest Rates, Mathematical Finance, 6,379-406.
[45] Duffie, Darrell and Kenneth J. Singleton, 1999, Modeling Term Structure Models of Defaultable Bonds,Review of Financial Studies, 12, 1999, 687-720.
[46] Duffie, Darrell and Kenneth J. Singleton, 1997, An Econometric Model of Term Structure of InterestRate Swap Yields, Journal of Finance, 52, 1287-1321.
[47] Evans, E. and Giola Parente Bales, 1991, What Drives Interest Rate Swap Spreads?, in Interest RateSwaps, Carl Beidleman, ed. 280-303.
[48] Evans, M. D. D., 2003, Real Risk, Inflation Risk, and the Term Structure, forthcoming EconomicJournal.
[49] Fenn, W. George, Mitch Post, and Steven A. Sharpe, 1996, Debt Maturity and the Use of Interest RateDerivatives by Nonfinancial Firms, Capital Markets Section, Federal Reserve Board, Washington, D.C.
[50] Filardo, Andrew, 1994, Business-Cycle Phases and Their Transitional Dynamics, Journal of Businessand Economic Statistics, 12, 299-308.
[51] Gray, Stephen F., Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Pro-cess, Journal of Financial Economics, 42, 27-62.
[52] Grinblatt, Mark, 2001, An analytical Solution for Interest Rate Swap Spreads, Review of InternationalFinance, forthcoming.
[53] Guedes, Jose and Tim Opler, 1996, The determinants of the Maturity of Corporate Debt Issues, Journalof Finance, 51, 1809-1833.
[54] Gupta, Anurag and Marti G. Subrahmanyam (2000), An Empirical Examination of the Convexity Biasin the Pricing of Interest Rate Swaps, Journal of Financial Economics, 55, 239-279.
[55] Hamilton, James, 1994, Time Series Analysis, Princeton, NJ, Princeton University Press.
[56] Harper, T. Joel and John R. Wingender, 2000, An Empirical Test of Agency Cost Reduction UsingInterest Rate Swaps, Journal of Banking and Finance, 24, 1419-1431.
[57] He, Hua, 2001, Modeling Term Structures of Swap Spreads, working paper, Yale School of management,Yale University.
[58] Hull, C. John., 2000, Options, Futures, & Other Derivatives. Upper Saddle River, NJ, Prentice Hall,4th edition.
[59] Hubner, Georges, 2001, The Analytical Pricing of Asymmetric Defaultable Swaps, Journal of Bankingand Finance, 25, 295-316.
[60] Jegadeesh, Narasimhan and George G. Pennachi, 1996, The Behavior of Interest Rates Implied by theTerm Structure of Eurodollar Futures, Journal of Money, Credit, and banking, 28, 426-451
[61] Johannes, Michael and Suresh Sundaresan, 2003, Pricing Collateralized Swaps, working paper, ColumbiaUniversity.
[62] Kocic, Aleksandar, Carmela Quintos, Francis Yared, 2000, Identifying the Benchmark Security in amultifactor Spread Environment, Fixed Income Derivatives Research, Lehman Brothers.
46
[63] Kuprianov, Anatoli , 1994, The Role of Interest Rate Swaps in Corporate Finance, Economic Quarterly,Federal Reserve Bank of Richmond, 80/3 Summer, 49-68.
[64] Landen, Camilla., 2000, Bond Pricing in a Hidden Markov Model of the Short Rate, Finance andStochastics, 4, 371-389.
[65] Liu, Jian, Shiying Wu, and James V. Zidek, 1997, On segmented Multivariate Regresions, StatisticaSinica, 7, 497-525.
[66] Liu, Jun, Francis A. Longstaff, Ravit E. Mandell, 2002, The Market Price of Credit Risk: An EmpiricalAnalysis of Interest Rate Swap Spreads, working paper, UCLA.
[67] Minton, A. Bernadette, 1997, An Empirical Examination of Basic Valuation Models for Plain VanillaU.S. Interest Rate Swaps, Journal of Financial Economics, 44, 251-277.
[68] Mozumdar Abon, 1999, Default Risk of Interest Rate Swaps: Theory & Evidence, working paper,Virginia Tech.
[69] Naik, Vasanttilak, and Moon Hoe Lee, 1994, The Yield Curve and Bond Option Prices with DiscreteShifts in Economic Regimes, working paper, University of British Columbia.
[70] Phillips, Aaron L., 1995, 1995 Derivatives Practices and Instruments Survey, Financial Management,Summer, 115-125.
[71] Nyblom, Jukka, 1989, Testing for the Constancy of parameters Over Time, Journal of the AmericanStatistical Association, 84, 223-230.
[72] Reinhart, Vincent and Brian Sack, 2002, The Changing Information Content of Market Interest Rates,Board of Governors of the Federal Reserve System.
[73] Samant, Ajay, 1996, An Empirical Study of Interest Rate Swap Usage by Nonfinancial CorporateBusiness, Journal of Financial Services Research, 10, 43-57.
[74] Smith, Clifford W., Charles W, Smithson, and Lee Macdonald Wakeman, 1986, The Evolving Marketfor Swaps, Midland Corporate Finance Journal, Winter, 20-32.
[75] Sun, Tong-sheng, Suresh Sundaresan, and Ching Wang, 1993, Interest Rate Swaps: An EmpiricalInvestigation, Journal of Financial Economics, 34, 77-99.
[76] Sundaresan, Suresh, 2001, Fixed Income Markets and Their Derivatives, Cincinnati, OH, South-WesternCollege Publications, 2nd edition.
[77] Timmermann, Allan, 2000, Moments of Markov Switching Models, Journal of Econometrics, 96, 1,75-111.
[78] Tuckman, Bruce, 2002, Fixed Income Securities: Tools for Today’s Market, Hoboken, NJ, John Wiley& Sons, Inc., 2nd edition.
[79] Turnbull, M. Stuart, 1987, Swaps: A Zero Sum Game?, Financial Management, Spring, 15-21.
[80] Wall, D. Larry, 1989, Interest Rate Swaps in an Agency Theoretic Model with Uncertain Interest Rates,Journal of Banking and Finance, 13, 261-270.
[81] Wall, D. Larry and John J. Pringle, 1989, Alternative Explanations of Interest Rate Swaps: A Theo-retical and Empirical Analysis, Financial Management, 18, 59-73.
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