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ESCP-EAP Working Paper
No. 25
May 2007
Are Credit Spreads and Interest Rates
co-integrated? Empirical Analysis in the
USD Corporate Bond Market
Ulrich Pape
Matthias Schlecker
Authors: Editor:
Prof. Dr. Ulrich Pape ESCP-EAP
Dipl.-Kfm. Matthias Schlecker Europäische Wirtschaftshochschule Berlin
Chair of Finance Heubnerweg 6
ESCP-EAP 14059 Berlin
Europäische Wirtschaftshochschule Berlin Germany
Heubnerweg 6 T: ++49(0)30 / 32007 147
14059 Berlin F: ++49(0)30 / 32007 108
Germany workingpaper-berlin@escp-eap.de
T: ++49(0)30 / 32007 134 www.escp-eap.de
F: ++49(0)30 / 32007 110
ulrich.pape@escp-eap.de
ISSN: 1619-7658
Abstract: Structural models for pricing risky debt imply a negative relationship be-
tween interest rates and credit spreads. In contrast, credit default swap pricing mod-
els assume independence between credit risk and the term structure of interest rates.
So far, empirical studies have focused on first differences of time series and con-
firmed the negative relation implied by structural models based on Merton (1974).
Nevertheless, the co-integration analysis of Morris, Neal and Rolph (1998) finds a
positive long-run relation between the credit spread and the risk free interest rate
which is not supported by the theoretical literature.
This paper analyses the relationship between different forms of credit spreads and
interest rates using the co-integration approach developed by Engle and Granger
(1987). Our study is based on weekly USD corporate bond index data which is con-
trolled for rating ranging from 1994 to 2006. We confirm the negative relation be-
tween credit spreads and interest rates with high statistical significance. Furthermore,
we find that the S&P 500 stock index and the spread between the swap curve and
the Treasury yield curve can be considered as additional factors explaining the credit
spread.
Key Words: Interest Rate, Credit Spread, Co-Integration, Structural Model
J. E. L. Classification Codes: C32, E4, G15
Acknowledgments: Earlier versions of this paper were presented at the Campus for
Finance Research Conference 2007 in Vallendar/Koblenz and the 30th annual meet-
ing of the German Classification Society 2006 in Berlin. The authors are grateful for
the detailed and helpful comments from the discussants and conference participants.
Contents
1 Introduction ....................................................................................................... 1
2 Credit Spread Methodology.............................................................................. 5
2.1 Calculation of the Spreads ........................................................................... 5
2.2 Different Forms of the Credit Spread............................................................ 7
2.3 Information Captured in the Swap Spread ................................................... 8
3 Review of the Literature.................................................................................. 11
3.1 Evidence in Theoretical Models ................................................................. 11
3.2 Empirical Studies........................................................................................ 12
4 Empirical Analysis .......................................................................................... 15
4.1 Motivation................................................................................................... 15
4.2 Data and Methodology ............................................................................... 16
4.3 Empirical Results........................................................................................ 19
4.3.1 Swap Spread....................................................................................... 19
4.3.2 Credit Spread ...................................................................................... 22
4.3.3 Spread over Swap............................................................................... 24
4.4 Discussion.................................................................................................. 27
5 Summary and Conclusions ............................................................................ 29
References .............................................................................................................. 31
Appendix ................................................................................................................. 37
Are Credit Spreads and Interest Rates co-integrated? 1
1 Introduction
From 2004 to 2006 corporate bonds have been trading at very low levels at US-
American as well as European bond markets. Market participants have perceived
credit spread levels to be very low, too. In February 2006, a “single A” rated corpo-
rate bond with a maturity of 5 years was quoted with a credit spread (CS) of 75 basis
points (bps) which is comparable to credit spread levels in the early ‘90s (see figure
1). Merely from 1998 to 2004 a 5-year “single A” corporate bond was priced up to
170 bps higher than default risk free Treasury bonds because markets were very
volatile. Treasury yields and the 3-month Libor dropped from high levels (6%) in 1995
to 4.5% in 2006.
1,8 8
1,6 7
1,4
Credit Spread in 100 BP
6
1,2
Yield to Maturity
5
1
4
0,8
3
0,6
2
0,4
0,2 1
0 0
94
95
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Credit Spread, A, 5y Libor 10y Treasury Yield
Figure 1: Credit spread, Libor and 10y Treasury yield
Central banks in the US and in Europe raised key interest rates in 2006 which led to
a change in the Treasury term structure and directly affected the price of bonds. For
corporate bond investors and issuers it is additionally important to know if changes in
the risk free term structure influence the credit spread and in which direction. The
change in interest rates can directly affect and change the credit spread or it can indi-
rectly influence the market’s perception of default risk which has an impact on credit
Are Credit Spreads and Interest Rates co-integrated? 2
spreads, too. In general, rising interest rates result in lower bond prices. The simulta-
neous change of the credit spread can weaken (negative relationship) or strengthen
(positive relationship) this effect, as shown in figure 2.
Positive Relationship: Negative Relationship:
Interest Rate Interest Rate
Credit Spread: Credit Spread:
Bond Price: Bond Price:
Bond Price: Bond Price:
Bond Price: Bond Price:
Figure 2: Effect of rising interest rates on bond prices when credit spreads and interest
rates have a positive or negative relationship.
Financial theory has developed two frameworks for pricing risky debt, the structural
and the reduced form models. Structural models which are based on the contingent
claim approach by Black and Scholes (1973) and Merton (1974) use option pricing
theory to derive the default premium of a corporate bond. According to the structural
models, the credit spread is determined, among other factors, by the short-term risk
free rate and the value of the firm’s assets. Iben and Litterman (1991), Jarrow and
Turnbull (1995) and Jarrow, Lando and Turnbull (1997) developed the reduced form
models. Credit risk is determined by the occurrence of default (stochastic or determi-
nistic) and the recovered amount at default.1 Correlation between default risk and
interest rates has not been modelled until Duffie and Singleton (1999) introduce a
negative correlation following the empirical results of Duffee (1998).2 Credit default
swap pricing models which are often based on reduced form models assume inde-
pendence between interest rates and credit risk.3 Unfortunately, the models show
contradictory results. It depends on the theoretical framework applied (structural or
1
See Houweling/Vorst (2005), p. 1203.
2
See Duffie/Singleton (1999), p. 702.
3
See e.g. Duffie (1999) or Hull/White (2000).
Are Credit Spreads and Interest Rates co-integrated? 3
reduced form model) and the type of spread considered (credit spread or credit de-
fault swap) whether you can find a relationship.
An analysis of the relation between credit spreads and interest rates has to take into
account that both variables are related with further factors which might change at the
same time. Therefore the analysis cannot only focus on the dynamics of the interest
rate and the credit spread; it has to incorporate other variables as well.4 The struc-
tural models suggest the use of a proxy for the value of the firm’s assets.5 Our study
uses the S&P 500 stock market index as a proxy for the value of the firm’s assets.
The probability of default of a corporate bond does not remain constant over time
within one rating class and therefore the default risk premium in the credit spread
cannot be constant. The credit spread of a corporate bond is at least partly due to
credit risk. Other factors include illiquidity, call features, tax treatment and a risk pre-
mium.6 In our study, we use the swap spread which is the yield difference between
the swap curve and the government yield curve as an explanatory variable which
captures information related with the market’s perception of corporate bond risk. The
relationship between the analysed variables has to be stable over time in order to
parameterize risk management models. Therefore we focus on long-term dynamics.
Our study focuses on two main questions:
1. Are credit spreads and interest rates related as assumed by the structural
models?
2. How can the dynamics of related variables be incorporated into the analysis?
In other words, we are interested in modelling a long-term equilibrium between
the credit spread and other related variables, including the interest rates.
We assume that the first question can be answered by modelling the long-term equi-
librium for credit spreads which includes interest rates. This paper uses the co-
integration approach developed by Engle and Granger (1987) which allows to study
the impact of dynamics of the interest rate (Libor), the S&P 500 and the swap spread.
4
See Heinke (1998), pp. 262-266 and Herbst (2005), pp. 177-180 for a detailed overview on empiri-
cal studys which analyse credit spread determinants.
5
See also Norden/Weber (2004) pp. 4-9 for a discussion of literature on stock and bond market
relationships.
6
See Höfling/Kiesel/Löffler (2004), Huang/Huang (2003), Elton et al. (2001).
Are Credit Spreads and Interest Rates co-integrated? 4
The Engle/Granger approach is very robust and the coefficients allow a direct inter-
pretation in contrast to other co-integration methods.7
We examine weekly USD corporate bond index data from Bloomberg which covers
the period from December 1994 till February 2006. Our study is based on three dif-
ferent rating classes (AA, A and BBB) and index data with 3, 5 and 10 years to ma-
turity which we analyse separately. In total we use nine different corporate bond yield
indices which allow a detailed analysis.
Our empirical results show that interest rates and credit spreads are co-integrated
and that the relation is negative. We develop a long term equilibrium model which is
able to explain the level of credit spreads by the Libor, a stock market index and the
swap spread.
The remainder of this paper is structured as follows. The second part discusses dif-
ferent forms of credit spreads and explains the information captured in the swap
spread. In the third part, we examine the structural models which state a negative
relation between credit spreads and interest rates. That section is followed by a brief
overview of empirical studies focusing on credit spreads and interest rates. As we will
see, most of the studies examine the short-run relationship by using changes in in-
terest rates to explain changes in credit spreads. In section four we will analyse the
common stochastic trend, thus the long-run relation, of credit spreads, interest rates
and other variables using the co-integration approach by Engle and Granger (1987).
The analysis is carried out for three different forms of the credit spread, i.e. the swap
spread, the credit spread and the spread over swap and it is followed by a discussion
of the results.
7
Another widely used method was developed by Johansen (1988).
Are Credit Spreads and Interest Rates co-integrated? 5
2 Credit Spread Methodology
2.1 Calculation of the Spreads
The credit spread is the yield difference between a bond with default risk and a de-
fault risk free bond. From a theoretical point of view, the risk free rate should reflect
three components:8 (1) a rental rate which compensates the lender who cannot use
the funds for consumption, (2) a premium to compensate for inflation and (3) a pre-
mium for changes of the interest rates. It is obvious that the risk free rate includes the
risk of interest rate changes but no default risk.9 The best proxy for such a risk free
rate is the Treasury yield, which is almost unanimously agreed upon in academia.10
Government bonds from the US, Germany or the UK can be considered as default
risk free; although even AAA rated government bonds bear a minimum credit risk.11
In recent years, government bonds lost their predominant role as risk-free bench-
mark. Practitioners prefer the Libor12 as a proxy for the short-term risk-free rate13 and
focus on the swap rate as a proxy for longer maturities.14 A change in the relevant
benchmark occurred already in the money market where the 3-month US Treasury
bill lost its pre-eminent role as a basis for pricing. Since the late ‘80s the government
security was substituted by the corporate 3-month Eurodollar future that was intro-
duced in Chicago in 1982.15 There are several reasons explaining the shift from the
government yield curve to the swap curve as a proxy for the risk-free term structure.
Due to US budget surpluses, the supply of US Treasury securities decreased rapidly
in the late ‘90s, especially on the long end of the yield curve.16 Compared to the swap
8
See Brooks/Yong Yan (1999), p. 72.
9
SeeCollin-Dufresne/Solnik (2001), p. 1100.
10
Brooks/Yong Yan (1999), p. 72, see also Collin-Dufresne/Solnik (2001), p. 1100.
11
In legal terms sovereigns or states cannot file for bankruptcy and therefore default. The credit risk
emerges from the possibility that the state might not be willing to meet its obligations. See Isensee
(2004), p. 232.
12
The Libor is the interbank interest rate for Eurodollar deposits by one bank with another.
13
See Houweling/Vorst (2005), p. 1209.
14
The swap rate is the fixed rate in an interest rate swap contract. In an interest rate swap contract
two counterparties agree to exchange fixed and variable interest payments. The swap rate is ad-
justed that the initial market value of the swap is zero.
15
See Ron (2004), pp. 631-632, McCauley (2001), pp. 40-42.
16
See Golub/Tilman (2000), p. 44.
Are Credit Spreads and Interest Rates co-integrated? 6
market, the regulated government bond markets are regarded as less liquid.17 Swap
markets are typically very liquid with narrow bid-ask spreads. Market efficiency is
high and swaps with comparable features have similar prices. Banks who offer
swaps are not able to ask for high margins. The swap curve can be estimated omit-
ting the coupon effect which is due to taxation. Finally, the introduction of the Euro
was another catalyzing factor.18 In contrast to the government bond market, the swap
curve actually reflects a credit quality which is comparable between countries.19 From
an investor’s point of view, the swap market offers an unlimited number of synthetic
instruments, allowing investors to go long or short any desired amount and increases
the flexibility for the portfolio composition.
As mentioned above, the credit spread is calculated as yield difference between cor-
porate and government bonds or a risk-free proxy such as the swap curve. The sen-
sitivity of bond prices to changes in the term structure is dependant on the coupon.
Government bonds pay a lower coupon and have a higher duration. Therefore Gov-
ernment bonds are more sensitive to changes of the interest rate than corporate
bonds. Some researchers suggest using the yields of bonds with same duration in-
stead of same maturity to control for interest rate sensitivity.20 The limited number of
outstanding bonds aggravates the calculation of constant maturity credit spreads.
Another approach to derive credit spreads is the use of spot rates.21 This method
evades the coupon bias and reduces the difficulties caused by limited bond price
data. The use of spot rates requires the estimation of the term structure of govern-
ment and corporate bonds.22 The accuracy of the credit spread estimation depends
on the estimation method and the pricing error, especially in illiquid markets. For
humped or s-shaped term structures23 the simultaneous estimation of government
17
See Fehle (2003), p. 348.
18
See Houweling/Vorst (2005), p. 1209.
19
Credit quality between sovereigns is not equal. In the Euro-area there are significant differences
which lead to spreads between government bonds, see Düllmann/Windfuhr (2000).
20
See Buberl (2002), pp. 109, 118, Heinke (1998), p. 379, Fons (1994), p. 29, Iben/Litterman (1991),
p. 53.
21
See Iben/Litterman (1991).
22
See McCulloch (1971), McCulloch (1975), Schaefer (1981), Vasicek/Fong (1982), Nelson/Siegel
(1987), Langetieg/Smoot (1989). For advanced estimation see Steeley (1990), Steeley (1991),
Svensson (1994), Linton et al. (2001), Subramanian (2001).
23
See Geyer/Kossmeier/Pichler (2004).
Are Credit Spreads and Interest Rates co-integrated? 7
and corporate term structures provides more specific and better results.24 In our pa-
per, we use constant maturity zero-bond indices. This is comparable to the spot rate
approach.
2.2 Different Forms of the Credit Spread
Taking into account the importance of the swap curve for market practitioners, we
can derive different types of credit spreads as shown in figure 3.
Yield to Maturity
Corporate Bond
Yield Curve
Spread over Swap
Credit Swap Curve
Spread
Swap Spread
Treasury
Yield Curve
Libor
Libor Spread
Maturity
Figure 3: Credit spread methodology
We denote the excess return of corporate bonds over Treasuries as credit spread.
The swap spread is the yield difference between the swap curve and Treasury yields.
Finally, we define the market driven yield difference between the corporate bond
curve and the swap curve as spread over swap. Libor bonds carry an AA or better
default risk and represent a special case of 6 month corporate AA bonds.25 The dif-
ferential between the Libor rate and the money-market government rate is called Li-
bor spread. The Libor and the short term swap rate are very similar, but not neces-
sarily equal.
24
See Jankowitsch/Pichler (2004), Houweling/Hoek/Kleibergen (2001).
25
See Collin-Dufresne/Solnik (2001), p. 1103.
Are Credit Spreads and Interest Rates co-integrated? 8
2.3 Information Captured in the Swap Spread
Our study uses the swap spread as explanatory variable. First, we try to explain the
information reflected by the swap spread and briefly elaborate on the construction
and pricing of plain vanilla interest rate swaps. Secondly, we provide determinants of
the swap spread which are discussed and empirically analysed in the literature.
1,4
1,2
1
Spread in 100 BP
0,8
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5y SS 5y AA CS 09
Figure 4: USD 5-year swap spread (SS) and AA corporate bond credit spread (CS)
The counterparties of an interest rate swap exchange a fixed payment stream
against a floating payment stream. The notional amount is not exchanged. The float-
ing leg is based on a short-term interest rate, usually the interbank interest rate for
the relevant currency which is referred to as USD or GBP Libor or as Euribor (EUR).
In order to avoid arbitrage, the present value of the known fixed-rate stream of pay-
ments has to be equal to the present value of the expected floating-rate stream.26
26
See Brown/Harlow/Smith (1994), p. 63. The discount factors used for the pricing of the swap can
be derived from the Treasury yield curve. Duffie/Singleton (1997), p. 1291, propose that the dis-
count rate has to be adjusted for the default risk.
Are Credit Spreads and Interest Rates co-integrated? 9
The floating leg can therefore be seen as a portfolio of Libor-forward contracts.27 The
fixed leg of the swap is the market rate for the swap (swap rate) and reflects a credit
quality that is AA or better.28 Figure 4 shows the 5-year swap spread and the credit
spread for AA-rated corporate bonds. The credit spread is higher than the swap
spread which reflects different credit qualities as well as different determinants and
dynamics.
Using expectation theory it can be shown that the swap rate is higher if the floating
leg is indexed on the Libor and not on money-market Treasury bills. Thus the con-
struction of an interest rate swap enforces a swap spread which is not zero. A higher
difference between the Libor and money-market government bills (Libor spread) re-
sults in higher swap spreads.29 Therefore swap spreads are related with the default
risk represented in the Libor rate. Brown, Harlow and Smith (1994) and Eom,
Subrahmanyam and Uno (2000) show empirically the positive relation of the swap
spread and the Libor spread. However, both studies do not control for spurious re-
gression.
The plain vanilla interest rate swap is virtually free of counterparty default risk, as
only net interest payments are paid in a swap. The notional is used as a scaling fac-
tor. The potential loss is limited by the posting of collateral and regular mark-to-
market. The default probability is reduced because it is jointly determined by the
probability of counterparty default and the probability of the swap having a negative
market value for the defaulting counterparty. It is questionable if weaker counterpar-
ties have access to the swap market without strict collateral requirements. Thus, the
influence of counterparty default risk on the swap rate is very limited.30
Research has focussed on the determinants of swap spreads. Following factors are
usually used to explain the swap spread.
Term structure: The term structure of the default risk-free interest rate is linked
with the swap spread. Malhotra, Bhargava and Chaudhry (2005) show in a co-
27
For longer maturities, there is no Libor forward rate available. The forward rates can be computed
using Eurodollar future rates less a convexity adjustment. See Zelenko/Kobor/Shi (2005), p. 7 for
discussion of the literature.
28
See Collin-Dufresne/Solnik (2001), p. 1095.
29
See Brown/Harlow/Smith (1994), p. 64.
30
See Zelenko/Kobor/Shi (2005), p. 7, Fehle (2003), p. 349, Collin-Dufresne/Solnik (2001), p. 1098
and the mentioned studies.
Are Credit Spreads and Interest Rates co-integrated? 10
integration approach that the slope of the term structure is negatively corre-
lated with the swap spread. The Treasury yield, however, is not a significant
variable and therefore has no relationship with the swap spread.31 Lekkos and
Milas (2001) find that the level and the slope have a negative effect on swap
spreads.32 The negative relationship between swap spreads and the slope is
affirmed by Fehle (2003).33 Eom, Subrahmanyam and Uno (2000) show that
the level of the risk-free term structure is negatively correlated with the swap
spread.34
Repo rate: Swap dealers can hedge their swap exposure by buying a Treas-
ury bond and repoing it over time. A higher repo rate results in higher hedging
costs. Swap dealers will ask for higher swap rates to compensate the cost of
their hedges. However, Brown, Harlow and Smith (1994) are not able to prove
this relationship statistically.35
Corporate bond spread: The swap rate reflects an AA credit quality. It is not
surprising that AA credit spreads are related with swap rates, although swap
rate and AA credit spread are not equal (see figure 4). Some empirical studies
show a positive relationship between corporate bond credit spreads and the
swap spread.36 Other studies using corporate bond yields can show the same
positive effect between AA bond yields and the swap spread.37
Supply of government bonds, liquidity and taxes: It is sometimes argued that
government bond markets are more liquid than swap markets which is some-
how contradictory to the opinion of practitioners mentioned above who prefer
the swap market because of higher liquidity. Nevertheless, investors can eas-
ily repo government bonds and have a cheap access to funds which makes
government bonds kind of liquid. Treasury bonds are exempt from local and
31
See Malhotra/Bhargava/Chaudhry (2005), p. 702.
32
See Lekkos/Milas (2001), p. 765.
33
See Fehle (2003), p. 371.
34
See Eom/Subrahmanyam/Uno (2000) , p. 15.
35
See Brown/Harlow/Smith (1994), p. 71.
36
See Brown/Harlow/Smith (1994), p. 73; Klein (2004), p. 54.
37
See Minton (1997), p. 267; Eom/Helwege/Huang (2004), p. 19.
Are Credit Spreads and Interest Rates co-integrated? 11
state taxes. Investors do not ask for tax compensation.38 Institutional investors
can be restricted to certain (risk-free) securities and therefore attribute a
higher demand for Treasury bonds. Examples for supply and demand effects
with widening spreads are the Russia and LTCM crisis (fall 1998) and the year
2000 concerns. Here we can see widening spreads because an excess de-
mand for government bonds (flight into quality) dropped the risk-free yield.
Monetary expansion since 2001 led to widening spreads.
The swap spreads seem not to be independent from credit spreads, the term struc-
ture or the short-term Libor spread. Following the study of Brown, In and Fang (2002)
we will analyse the explanatory power of swap spreads on credit spreads and
spreads over swap in a co-integration analysis.
3 Review of the Literature
3.1 Evidence in Theoretical Models
The contingent claim approach by Merton (1974) is based on the Black and Scholes
(1973) option pricing model. Equity is modelled as a long call on the firm’s assets;
debt can be seen as a risk free bond combined with a short put on the firm’s assets.
As changes in the risk free rate lead to changes in the value of the put, credit
spreads are sensitive to interest rate changes. Rising interest rates result in decreas-
ing values of the put which reduce the default premium. Merton’s model states a
negative relation between the risk free rate and credit spreads.
The diffusion process of the asset’s value is modelled with a drift term which is per-
fectly and positively correlated with the risk free rate in the risk-neutral world. Positive
changes in the risk free rate will result in higher future values of the firm’s assets. The
firm value will drift away at a faster rate from the default boundary.39 Although this
effect reduces credit risk it is offset by higher discount rates, and the present value of
the firm’s assets remains constant. However, if an increase in interest rates triggers a
38
See Zelenko/Kobor/Shi (2005), p. 7; Fehle (2003), pp. 349-350; Brooks/Yong Yan (1999), p. 82.
39
See Longstaff/Schwartz (1995), p. 798.
Are Credit Spreads and Interest Rates co-integrated? 12
drop in firm value, credit spreads rise.40 Only in this case credit spreads and interest
rates are positively related.
The underlying of the put is the value of the firm’s assets. The strike price is the face
value of the debt. Therefore it is straight forward that an increase in the value of the
underlying, i.e. the firm’s value, reduces the put’s price which results in a lower credit
spread. The higher the value of the assets, the lower is the default risk and the credit
spread.
Structural models derived from Merton’s model propose a negative relation between
credit spreads and interest rates, too. Cox, Ingersoll and Ross (1985) introduced a
stochastic firm value into the model which was extended by Kim, Ramaswamy and
Sundaresan (1993) on the default of the coupon. Credit spreads of noncallable and
callable corporate bonds tend to be negatively related with interest rates.41 Longstaff
and Schwartz (1995) developed an approach to value risky debt which implies the
same negative relation between credit spreads and the level of the short-term risk
free interest rate.42 Finally, the valuation model of Leland and Toft (1996) confirms
that credit spreads fall when the default-free interest rate rises.43
In general, the analysis of the relation between interest rates and credit spreads has
to distinguish between callable and noncallable bonds. Corporations have so far pre-
ferred to issue callable bonds. The value of the embedded call option is positively
correlated with the interest rate. This is the reason why the negative relation between
interest rates and credit spreads of callable bonds is stronger than in the case of non-
callable bonds.44
3.2 Empirical Studies
The negative relationship between interest rates and credit spreads which is stated
by the structural models can only hold if the pricing models for risky debt mentioned
above show high accuracy in empirical tests. Eom, Helwege and Huang (2004) test
40
See Morris/Neal/Rolph (1998), p. 9.
41
See Kim/Ramaswamy/Sundaresan (1993), pp. 125, 128.
42
See Longstaff/Schwartz (1995), pp. 798, 815.
43
See Leland/Toft (1996), p. 1003.
44
See Duffee (1998), p. 2226.
Are Credit Spreads and Interest Rates co-integrated? 13
five structural models45 and show that they all lack accuracy. The Merton model pre-
dicts too low credit spreads while the other structural models generate credit spreads
that are too high on average. The credit spreads estimated by these models “are of-
ten either ludicrously small or incredibly large while the average spread prediction
error is not particularly informative.”46 The authors also investigate the impact of in-
terest rates on the prediction error and show that the Longstaff and Schwartz (1995)
model is the most sensitive one concerning the interest rate. The role of the interest
rate is not clear in any of the tested models.47
So far, there has been little empirical research on the correlation of interest rates and
credit spreads. One of the first studies on this issue was published by Longstaff and
Schwartz (1995) who regress monthly changes in credit spreads against changes in
the 30-year Treasury bond yield. The coefficients are significant and clearly nega-
tive.48 Nevertheless, there are inconsistencies in the methodology applied. The au-
thors use the monthly average yields of Moody’s Bond Record for the corporate bond
yield from 1977 to 1992. The credit spread is computed by taking the average of the
10-year and 30-year Treasury yields that matches the maturity of the corporate yield
average for the month. This approach neglects the skewness of the term structure.
The results seem to be significant and show a high R-squared. However, the validity
may be somewhat limited due to the used data.
The study of Duffee (1998) confirms a negative relationship between changes in the
interest rate level, measured by the 3-month Treasury bill, and changes of credit
spreads on different maturities and ratings. The results for callable bonds are
stronger than for noncallable bonds.49 Duffee includes stock returns in his analysis
which are negatively correlated with the credit spread. Although the coefficients of his
regression are significant, the adjusted R-squared is quite low which indicates that
important variables are omitted and coefficients may be biased.
45
Merton (1974), Geske (1977), Longstaff/Schwartz (1995), Leland/Toft (1996) and Collin-
Dufresne/Goldstein (2001).
46
Eom/Helwege/Huang (2004), p. 502.
47
See Eom/Helwege/Huang (2004), p. 532.
48
See Longstaff/Schwartz (1995), p. 810.
49
See Duffee (1998), pp. 2232, 2239.
Are Credit Spreads and Interest Rates co-integrated? 14
Collin-Dufresne, Goldstein and Martin (2001) conducted a study on monthly corpo-
rate bond data without embedded options ranging from 1988 to 1997. Credit spreads
are calculated with an interpolated and therefore inaccurate Treasury yield curve.
The interpolation does not take the curvature of the term structure into account. Nev-
ertheless, the results are in line with the theoretical models and other empirical stud-
ies. The authors investigate changes and show a negative relationship between
credit spreads and the 10-year Treasury yield with mediocre statistical quality and a
very low R-squared. The study shows a negative relationship between stock returns
and credit spreads.
Papageorgiou and Skinner (2006) use spot rates to calculate the credit spreads. The
empirical analysis explains credit spread changes with changes in the level and the
slope of the risk free term structure. Although the authors adjust for first-order auto-
correlation, the results are similar to Duffee’s (1998) findings. Level and slope are
negatively correlated. The coefficients are significant, but the R-squared is low point-
ing out poor overall statistical quality.
The authors of the studies mentioned above based their analyses on changes, i.e.
the first difference of a time series ∆y t = y t − y t −1 . This procedure avoids the statistical
problem of spurious correlation but the information of the level of the time series is
not included in the analysis. Changes in the time series can be seen as short-run dy-
namics which cannot be used as a basis for the inference of long-run equilibriums.
For a long-run analysis, the information which is captured in the level of a time series
has to be taken into account. Therefore, the above mentioned studies give evidence
on short-run, but not on long-run relationships between interest rates and credit
spreads. Finally, results of the analysis of changes will be biased if the variables are
co-integrated.50
The paper of Kao (2000) extends its research on several variables and separately
analyses the relationship of credit spread changes with interest rate parameters, the
swap market and the equity markets. However, the results of Kao lack statistical ac-
curacy but the directions of the relationships in his paper are confirmed by our com-
prehensive modelling.
50
See Morris/Neal/Rolph (1998), p. 11.
Are Credit Spreads and Interest Rates co-integrated? 15
A long-run analysis should concentrate on modelling the long-run equilibrium be-
tween the variables and focus on the transition path. Morris, Neal and Rolph (1998)
conduct a co-integration analysis. They use monthly averages of daily rates for 10-
year constant maturity Treasury bonds and Moody’s seasoned bond indices51 from
1960 to 1997. Using the co-integration approach of Johansen and Juselius (1990),
the study cannot reject the null hypothesis that there is no co-integration vector. The
data confirms a positive relationship between interest rates and credit spreads, al-
though the use of monthly averages can be questioned. This positive relationship
which is confirmed by the statistically more robust co-integration method of Engle
and Granger (1987) is not in line with the theoretical models mentioned above.
A more precise co-integration analysis based on month-end data was conducted by
Joutz, Mansi and Maxwell (2001). Credit spreads of corporate bonds without embed-
ded options are analysed for co-integration with a 15-year Treasury index, the slope
of the yield curve and the risk factors of Fama and French (1993). The study points
out that the relationship between interest rates and credit spreads is rather com-
plex.52 The analysis shows a positive relationship for the level of the term structure of
interest rates.
4 Empirical Analysis
4.1 Motivation
Regarding the various empirical studies, the relationship between the level of the risk
free interest rate and credit spreads remains unclear. Structural bond pricing models
suggest a negative relationship which is partly confirmed by empirical analysis focus-
ing on changes of interest rates and credit spreads. Co-integration analysis by
Morris, Neal and Rolph (1998) and Joutz, Mansi and Maxwell (2001) asserts a posi-
tive relation.
51
This index contains non-financial corporate bond data with seasonal business activity. See Xiao
(2001), p. 19.
52
See Joutz/Mansi/Maxwell (2001), p. 4.
Are Credit Spreads and Interest Rates co-integrated? 16
The term structure of interest rates is not the sole determinant of the credit spread. It
is intuitive that the macroeconomic environment drives the business cycle which has
an impact on equity returns as well as on the interest rate level. Furthermore, the
business cycle can explain changes of the creditors’ probability of default.53 The
swap spread shows pro-cyclical behaviour.54 As mentioned above, the firm value is
one determinant for the credit spread in the structural models. Equity prices are often
used as a proxy. Figure 5 illustrates some important determinants of the credit
spread.
Economic outlook and
business cycle
Yield Curve
Stock Market
(Level, Slope)
Credit Spread
Figure 5: Major determinants of the credit spread
In our empirical analysis we use the co-integration approach developed by Engle and
Granger (1987). The co-integration approach is able to discover and to test for long-
term equilibriums between several variables. We focus on the co-integration relation-
ship of credit spreads for different maturities and rating classes with interest rates, a
stock market index and the swap spread. We do not model the business cycle be-
cause we use weekly data and important indicators (e.g. the GDP) are only available
on a quarterly basis. However, as the swap spread is pro-cyclical this information is
captured through the swap spread.
4.2 Data and Methodology
We use weekly Bloomberg data (581 data points) ranging from December 1994 to
February 2006. Credit spreads are derived as the difference between the constant
53
See Wagatha (2005).
54
See Lang/Litzenberger/Luchuan (1998), p. 1528.
Are Credit Spreads and Interest Rates co-integrated? 17
maturity discount yield indices for USD corporate zero bonds and the Treasury spot
curve with the same maturity. For every rating class (AA, A and BBB) we use an ag-
gregated index. The indices have a constant maturity of 3, 5 and 10 years. As a
proxy for the level of the risk free rate we use the 3-month USD Libor. The equity in-
dex is the S&P 500 index which we divide by 1000. The descriptive statistics are
shown in table 4 (p. 37) in the appendix.
2
Credit Spread in 100 BP
1.5
1
0.5
0
94
95
96
97
98
99
00
01
02
03
04
05
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29
27
26
25
24
22
21
20
19
17
16
CS, A, 3y CS, A, 5y CS, A, 10y
Figure 6: USD credit spreads
Figure 6 shows the development of the USD A credit spreads for different maturities.
In August and September 1998 credit spreads widened abruptly. The reason is the
devaluation of the Russian Ruble and the default of Russian government bonds with
a volume of 13.5 Billion USD. Afterwards a flight into quality could be observed and
yields of US Treasuries declined. After the crisis in Russia, the hedge fund LTCM
faced a huge drop in value of its assets which forced the Federal Reserve Bank to
support the fund. The LTCM crisis was the beginning of a rather volatile period on the
capital markets including the defaults of Enron and Worldcom. Since mid-2003 credit
spreads became tighter returning to the levels of the years 1995-1997.
Are Credit Spreads and Interest Rates co-integrated? 18
To avoid spurious regression55 the time series have to be stationary. Two random
walks which are non-stationary by nature can explain a third one.56 A time series is
said to be stationary if expectation value, variance and covariance of a time series
are time independent. One test for stationarity is the Dickey Fuller Test57 which is a
unit root test. Stationary time series do not have a unit root. Economic time series
often follow an autoregressive process of higher order. In this case the Augmented
Dickey Fuller Test is appropriate. Using an Augmented Dickey Fuller we test the null-
hypothesis that the time series has a unit root and is therefore non-stationary. As
shown in table 4 (appendix), the null-hypotheses cannot be rejected. The time series
are non-stationary,58 except the Libor spread and the AA spreads over swap.
Non-stationary time series could be first-differenced in order to get stationary time
series which can be included into a regression analysis. The shortcoming of this ap-
proach is the loss of information captured in the level of the time series. Co-
integration analysis requires non-stationary time series as input data and therefore is
able to use this information.
Time series are co-integrated if they follow a long-run equilibrium relationship. That
means that the variables cannot drift independently far away from each other. Follow-
ing Engle and Granger (1987) non-stationary variables are co-integrated if a linear
combination of the time series has a lower order of integration.59 The single co-
integration vector can be easily estimated by the ordinary least square method.60 The
residual of the regression is stationary if the variables are co-integrated. Therefore
we perform on the residual an Augmented Dickey Fuller Test. The test statistic is a
measure to asses the quality of the model. A small value implies a stronger co-
integrating relationship. However, the test statistics cannot be compared with the
typical critical values of the unit root tests. The critical values of the residual-based
55
One of the first publications in this field was on wheat price index and cumulated rainfall in Green-
wich by Yule (1926).
56
See Granger/Newbold (1974), p. 115-117.
57
See Fuller (1976), Dickey/Fuller (1979).
58
It seems strange that interest rates or credit spreads could explode which is implied by a unit root
process. The statistical power of the Dickey Fuller Test is low against near unit root time series.
See Pedrosa/Roll (1998), p. 9.
59
See Engle/Granger (1987), p. 252-253.
60
See Kirchgässner/Wolters (2006), p. 188-189. For statistical properties of the co-integration re-
gression see Hassler (2004), pp.101-104.
Are Credit Spreads and Interest Rates co-integrated? 19
unit root tests are smaller. These critical values were first published by MacKinnon
(1991).61
4.3 Empirical Results
Our study analyses three types of credit spreads: the swap spread, the credit spread
and the spread over swap. Some of the time series are highly correlated. We exclude
the swap rate, Treasury yields and the slope of the term structure to avoid multicol-
linearity. The excluded time series do not have a high explanatory power in the re-
gression analysis. The information is partly captured by the swap spread.
4.3.1 Swap Spread
For the swap spread we analyse the following two models. The studies of Brown,
Harlow and Smith (1994) and Eom, Subrahmanyam and Uno (2000) suggest to use
the Libor spread (LS), the S&P 500 Index (EQX) and a constant (C) to explain the
swap spread (SS) for different maturities (M):
SS M = β1LS + β 2 EQX + C
The R-squared is very low and only the residual for the 3-year swap spread is signifi-
cant in the co-integration test (Augmented Dickey Fuller Test on the residual). There-
fore we cannot confirm that Libor spread and swap spread follow a common long-
term equilibrium. The Libor spread is stationary whereas the other time series are
non-stationary. In the case of stationary and non-stationary time series the co-
integrating relationship is weak. The value of the test statistic is close to zero.
With reference to the literature regarding the swap spread and its determinants, we
substitute the Libor spread (LS) with the AA credit spread (CS) with the same matur-
ity to improve the estimation:62
SS M = β1CS AA, M + β 2 EQX − C
The co-integration relationship is significant on a 1% basis. Including the Libor
spread only improves the 10-year swap spread model. The results of the analysis are
61
See MacKinnon (1991), p. 275.
62
See Eom/Helwege/Huang (2004), p. 19; Klein (2004), p. 54; Minton (1997), p. 267;
Brown/Harlow/Smith (1994), p. 73.
Are Credit Spreads and Interest Rates co-integrated? 20
shown in table 1. Figure 7 plots the empirical 5-year swap spread, the modelled long-
term relationship (co-integration) and the residual. We can clearly see that the long-
term dynamics are very well captured in the graph.
Table 1: Swap Spread Co-integration Model63
Model Variables Coefficient t value R² Adj. R² DW ADF Signif.
SS 03 U03AACS 0.570 31.517 86.85 86.810 0.684 -8.435 1%
UEQX 0.360 32.256
C -0.241 -20.912
SS 05 U05AACS 0.626 38.550 87.99 87.950 0.499 -6.716 1%
UEQX 0.345 24.415
C -0.270 -20.208
SS 10 U10AACS 0.502 27.502 81.44 81.340 0.345 -5.835 1%
UEQX 0.291 13.543
ULS 0.427 19.943
C -0.329 -16.789
1,1
0,9
0,7
Spread in 100 BP
0,5
0,3
0,1
-0,1
95
96
97
98
99
00
01
02
03
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05
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27
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24
22
21
20
19
17
16
-0,3
5y SS Co-integration Residual
Figure 7: Co-integration analysis for 5-year swap spread (SS)
63
The shown t-values and the R-squared are biased, because of the co-integration relation. See
Banerjee et al. (1986), p. 262.
Are Credit Spreads and Interest Rates co-integrated? 21
As expected, the AA credit spread and the swap spread are positively correlated (see
also figure 4). However, the positive relationship between the equity index and the
swap spread is rather surprising. The correlation of the 5-year swap spread and the
equity index is ρ = 0,75 and the plot (see figure 8) confirms the similar pattern. The
swap spreads has a pro-cyclical behaviour as already stated in the literature.64
The estimated model is smoother than the empirical time series. This suggests that
the co-integration is able to fit well the long-term relationship but lacks accuracy for
short term dynamics. Especially the Russia crisis is not captured in the model. One
interpretation can be that such a shock is priced in the swap spread while the credit
spread remains constant.
1.6 1800
Swap Spread in 100 BP
1.4 1600
1.2 1400
Equity Index
1200
1
1000
0.8
800
0.6
600
0.4 400
0.2 200
0 0
5
6
.1 97
8
.1 99
0
1
.1 02
3
/9 4
5
20 199
19 199
17 199
14 200
13 200
10 200
0
00
19
19
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/2
2.
2.
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2.
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12
22
18
15
12
U10SS UEQX
Figure 8: USD 10-year swap spread (SS) and S&P 500 equity index (UEQX)
The Libor is not included into the model because it does not improve the model qual-
ity which we measure with the test statistics of the Augmented Dickey Fuller Test on
the residuals. Therefore we cannot confirm a long-run relationship between swap
spread and the Libor. Probably, this indicates an independence of the swap spread
64
See Lang/Litzenberger/Luchuan (1998), p. 1528.
Are Credit Spreads and Interest Rates co-integrated? 22
from interest rates. We can interpret the swap spread as a standardised default risk
premium. In this case, the market’s opinion on the price of credit risk does not
change when interest rates change although there might be the economic environ-
ment and business cycle as a common factor.
4.3.2 Credit Spread
The credit spread (CS) is modelled for each rating class (R) and each maturity (M) as
a function of the swap spread (SSM) with same maturity, the 3-month Libor (LIBOR),
the S&P 500 index (EQX) and a constant (C):
CS R , M = β1SS M − β 2 LIBOR − β 3 EQX + C
Using 3 rating classes and 3 maturities we analyse 9 time series. The results of the
co-integration are shown in table 2. For each time series we prove statistical signifi-
cance by comparing the result of the Augmented Dickey Fuller Test (ADF) on the
residual with the critical value of MacKinnon (1991). The hypothesis that there is no
co-integration can be rejected in 8 of 9 cases at the 1% level. Only the 3-year BBB
credit spread lacks statistical significance. Thus, we can conclude that the variables
are co-integrated. The credit spread can be explained with the Libor, the stock index
and the swap spread.
The Engle and Granger (1987) approach allows a direct interpretation of the coeffi-
cients for the different variables. The credit spread is negatively related with the 3-
month Libor, but the sensitivity is weak. However, the model quality deteriorates
slightly if the Libor is omitted. The value of the Libor’s coefficient increases with de-
clining credit quality and with increasing maturity. This result is in line with the struc-
tural models. The stock market index is also negatively related which parallels with
literature, too. The swap spread is positively related. With lower credit quality and
higher maturities the coefficients of the swap spread rise. This is not surprising be-
cause credit spreads increase with higher maturities and lower credit quality. The
constant is supporting this behaviour.
Are Credit Spreads and Interest Rates co-integrated? 23
Table 2: Credit Spread Co-integration Model
Model Variables Coefficient t value R² Adj. R² DW ADF Signif.
CS 03 AA U03SS 1.110 32.715 73.36 73.21 0.576 -4.610 5%
UIBOR 0.012 6.737
UEQX -0.288 -12.987
C 0.304 17.951
CS 03 A U03SS 1.792 27.240 65.94 65.76 0.319 -5.210 1%
UIBOR -0.017 -4.848
UEQX -0.465 -10.795
C 0.540 16.460
CS 03 BBB U03SS 2.676 25.823 67.93 67.77 0.263 -3.777 -
UIBOR -0.079 -14.094
UEQX -0.671 -9.891
C 1.012 19.571
CS 05 AA U05SS 1.217 41.830 82.08 81.98 0.500 -4.948 1%
UIBOR -0.019 -8.736
UEQX -0.320 -13.242
C 0.467 22.878
CS 05 A U05SS 1.761 44.833 84.31 84.29 0.489 -6.740 1%
UIBOR -0.044 -14.783
UEQX -0.449 -13.769
C 0.665 24.142
CS 05 BBB U05SS 2.433 42.110 84.29 84.29 0.410 -6.085 1%
UIBOR -0.110 -25.123
UEQX -0.564 -11.776
C 1.122 27.685
CS 10 AA U10SS 1.368 34.572 77.94 77.83 0.320 -5.414 1%
UIBOR -0.089 -24.477
UEQX -0.337 -9.419
C 0.805 25.575
CS 10 A U10SS 1.698 38.925 81.67 81.58 0.321 -5.452 1%
UIBOR -0.101 -25.034
UEQX -0.403 -10.212
C 0.985 28.384
CS 10 BBB U10SS 2.131 40.384 84.97 84.89 0.320 -5.446 1%
UIBOR -0.167 -34.200
UEQX -0.400 -8.366
C 1.472 35.036
The estimated co-integration relationship is able to fit the empirical credit spread time
series very well. Figure 9 shows the estimated (Co-integration) and empirical (5y A
CS) time series for the single A 5-year credit spread. Sometimes, the estimated
spread drifts away from the empirical credit spread, but the co-integration system
returns rather quick to the long-run equilibrium. It is interesting to note that from 2004
Are Credit Spreads and Interest Rates co-integrated? 24
to 2006 the empirical credit spread was too low compared to the estimated credit
spread levels which is in line with market opinion when bond market participants ex-
pected credit spreads to rise again after a period with too tight credit spread levels.
The difference between the empirical and modelled credit spread of 10 to 15 bps
might be explained by supply and demand effects. Institutional investors searching
for return in fixed income bonds might be willing to bear credit risk for quite a low pre-
mium.
2
1,5
Credit Spread in 100 BP
1
0,5
0
94
95
96
97
98
99
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01
02
03
04
05
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29
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26
25
24
22
21
20
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17
16
5y A CS Co-integration Residual
Figure 9: Co-integration analysis for 5-year A credit spread (CS)
The model captures the major credit events very well. This means that the swap
spread contains the necessary information for example the widening of spreads dur-
ing the Russia crisis.
4.3.3 Spread over Swap
In recent years the government yield curve lost its function as the risk-free bench-
mark. Market participants prefer the swap curve as the benchmark for pricing securi-
ties and for deriving the spread over swap. The information reflected in the credit
spread and the spread over swap should be related because both of them are ex-
cess returns above (quasi) risk free benchmarks. Both spreads can be seen as risk
premiums for default risk. A correlation analysis (see table 5, appendix) indicates that
Are Credit Spreads and Interest Rates co-integrated? 25
credit spreads and spreads over swap of lower credit qualities (A and BBB) are
highly correlated (up to ρ = 0,94 ). BBB spreads are highly correlated across maturi-
ties while single A credit spreads tend to have higher correlations with spreads of the
same maturity. With increasing spreads the difference between credit spreads and
spreads over swap is of minor importance.
Table 3: Spread over Swap Co-Integration Model
Model Variables Coefficient t value R² Adj. R² DW ADF Signif.
U03 AA OS UIBOR 0.012 6.737 39.18 38.86 0.576 -4.613 5%
U03SS 0.110 3.237
UEQX -0.288 -12.987
C 0.304 17.951
U03 A OS UIBOR -0.017 -4.848 22.89 22.49 0.319 -5.215 1%
U03SS 0.792 12.039
UEQX -0.465 -10.795
C 0.540 16.460
U03 BBB OS UIBOR -0.079 -14.094 46.22 45.94 0.263 -3.779 -
U03SS 1.676 16.172
UEQX -0.671 -9.891
C 1.012 19.571
U05 AA OS UIBOR -0.019 -8.736 28.87 28.50 0.510 -4.952 1%
U05SS 0.217 7.469
UEQX -0.320 -13.242
C 0.467 22.878
U05 A OS UIBOR -0.044 -14.783 45.45 45.17 0.488 -6.746 1%
U05SS 0.761 19.377
UEQX -0.449 -13.769
C 0.665 24.142
U05 BBB OS UIBOR -0.110 -25.123 66.46 66.29 0.408 -6.090 1%
U05SS 1.433 24.803
UEQX -0.564 -11.776
C 1.122 27.685
U10 AA OS UIBOR -0.084 -19.979 44.68 44.39 0.221 -4.585 5%
U10SS 0.152 3.326
UEQX -0.136 -3.284
C 0.723 19.867
U10 A OS UIBOR -0.096 -22.035 47.66 47.39 0.244 -4.850 1%
U10SS 0.482 10.245
UEQX -0.202 -4.737
C 0.902 24.110
U10 BBB OS UIBOR -0.161 -31.155 67.64 67.47 0.251 -4.907 1%
U10SS 0.916 16.304
UEQX -0.198 -3.899
C 1.389 31.076
Are Credit Spreads and Interest Rates co-integrated? 26
If the spread over swap and the credit spread have similar dynamics, the same fac-
tors should determine the both spreads. Thus, we analyse the spread over swap by
using the variables of the credit spread models in section 4.3.2:
SoS R , M = β1SS M − β 2 LIBOR − β3 EQX + C
Table 3 reports the results. The coefficients are negative for the Libor and positive for
the swap spread. The spread over swap model is in line with the results of the credit
spread model. Especially the negative relationship between spread over swap and
Libor is confirmed. The stock market index has a negative coefficient. For 8 of 9 time
series the hypothesis of co-integration cannot be rejected. Compared to the credit
spread results, the test statistics of the Augmented Dickey Fuller Test (see columns
ADF) are smaller. This indicates that the model quality is better. However, the R-
squared is mediocre.
1,5
1,3
1,1
0,9
Spread in 100 BP
0,7
0,5
0,3
0,1
-0,1
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16
-0,5
10y A SoS Co-integration Residual
Figure 10: Spread over swap (SoS) analysis, 10-year maturity, A rating
The plot of the single A 10-year spread over swap model (figure 10) shows that the
co-integration analysis is able to capture the main level of the spread over swap, but
the empirical spread is quite volatile and the deviation from the long-term equilibrium
can be rather big (up to 70 basis points). As the co-integration analysis is not able to
model these peaks, other factors have to be taken into account. The three peaks can
Are Credit Spreads and Interest Rates co-integrated? 27
probably be explained by credit driven events which mainly affect the spread over
swap. At the end of 2000 some telecom companies increased their debt often by is-
suing corporate bonds leading to excess supply; in fall 2001 Enron defaulted as did
Worldcom in 2003. These events are not reflected in the swap spread.
When analysing the correlation of the spread over swap with the level of the term
structure, we can see that the negative coefficients increase as credit quality de-
creases and as maturity increases. This means that the spread over swap is sensi-
tive to changes in the Libor.
4.4 Discussion
Through our analysis we developed a long-term equilibriums for three different credit
spreads:
1. Swap spread: SS M = β1CS AA, M + β 2 EQX − C
2. Credit spread: CS R , M = β1SS M − β 2 LIBOR − β3 EQX + C
3. Spread over swap: SoS R , M = β1SS M − β 2 LIBOR − β3 EQX + C
Long-term equilibrium: The pattern of the co-integration relationship is similar for all
three credit spreads. We show that the credit spread can be explained by a stock
market index and a risk premium proxy. While the swap spread is independent from
the interest rates, the credit spread and the spread over swap are negatively related
to the Libor which is a proxy for the level of the interest rate term structure. We focus
on the long-term dynamics which proof to be statistically significant. The long-term
equilibrium of credit spreads can be modelled with a rather parsimonious set of vari-
ables. The plots of the models show that the empirical spreads return regularly to the
long-term equilibrium.
Stronger sensitivity: With poorer credit quality and/or longer maturity the sensitivity
of the variables increases. The result is not surprising as it confirms again the accu-
racy of the models. Especially bonds with a longer time to maturity, i.e. with a high
duration, are more sensitive to interest rate changes. A higher coupon which is typi-
cal for lower rated corporate bonds also increases the duration.
In line with structural models: The variables which are included into the models
can also be found in the structural models. Especially the negative correlation of the
Are Credit Spreads and Interest Rates co-integrated? 28
credit spreads with the interest rate and the stock market index which is a proxy for
the company’s firm value is in line with the theoretical framework. As we use the
same stock index for all corporate bonds, we need a further measure to differentiate
for different classes of default risk. This is done by analysing different rating class
indices.
Term structure: Following other empirical work on credit spreads our work lacks
variables for the term structure of the interest rates, especially the slope. First, these
variables are not significant when included in the analysis. Second, the swap spread
reflects some of the information captured in the term structure.65
Swap spread as standardised risk premium: The credit spread as well as the
spread over swap model use the swap spread as an explanatory variable. The
weight of the coefficient increases with poorer credit quality and longer maturity. We
can therefore interpret the swap spread as a standardised risk premium. It reflects
the highly liquid and standardised swap market with a rather constant default risk (AA
risk) which is priced in the fixed leg of swap contracts. Changes in the market’s per-
ception of risk are reflected in the swap spread. When investors ask for a higher risk
premium, the swap spread is likely to widen. The credit spread and the spread over
swap should follow this movement. This might be one explanation why the swap
spread is an important variable in our models.
The non-explained peaks in the spread over swap model in figure 10 are another
interesting issue. The related credit events (e.g. Worldcom, Enron) are only reflected
in the spread over swap but not in the swap spread which is an explanatory variable.
One hypothesis could be that individual credit events lead to higher corporate bond
spreads while the standardised risk premium, the swap spread, remains rather un-
changed. Another explanation could be a kind of overshooting of the markets which
might be a topic for further research.
65
See discussion in section 2.3.
Are Credit Spreads and Interest Rates co-integrated? 29
5 Summary and Conclusions
This paper analyses the relationship between credit spreads and interest rates by
developing long-term equilibrium models based on the co-integration approach of
Engle and Granger (1987). We include into the analysis a stock market index and the
swap spread as a standardised risk premium and show that these variables follow a
common stochastic trend. One of the main findings of this paper is that credit
spreads and interest rates are negatively correlated which contradicts the assump-
tions of credit default swap pricing models.
So far, empirical studies have focused on first differences of time series and confirm
the negative relationship implied by structural models. In contrast to these findings,
the co-integration analysis of Morris, Neal and Rolph (1998) estimates a positive
long-run relation. Most of the credit spread papers focus on two variables, neglecting
common dynamics with other determinants.
The credit spread is often defined as the yield difference between the risk-free
Treasury yield and the corporate bond yield. Capital markets, however, recently pre-
fer to use the swap curve instead of Treasury yields as a benchmark for the risk-free
rate. Thus, different types of credit spreads can be derived and analysed. We de-
velop a spread framework which contains the swap spread, the credit spread and the
spread over swap.
In our analysis, we firstly focus on the swap spread and show a positive correlation
with AA credit spreads and the stock market index. The swap spread seems to be
independent from the risk-free term structure which we proxy with the Libor. Second,
we use the credit spread as the dependant variable. Our study confirms the negative
relationship between the credit spread and the Libor. We include the S&P 500 stock
index (negative correlation) and the swap spread (positive correlation) as risk factors
and the results are statistically significant. This is in line with theory, especially the
structural models, and other empirical findings. Our results contradict the credit de-
fault swap pricing models which assume independence between credit spreads and
interest rates. This contradiction could be subject of further research.
Finally, we analyse the spread over swap. Using the factors of our credit spread
analysis we can confirm a long-term equilibrium between spread over swap, Libor,
the stock market index and the swap spread. As expected, the spread over swap is
Are Credit Spreads and Interest Rates co-integrated? 30
negatively correlated with the Libor. But, the spread over swap is very volatile. Thus,
the short-term dynamics are very dominant in the spread over swap analysis. An er-
ror correction model could be applied in further research to improve model quality.
It is important to note that only the credit spread and the spread over swap are sensi-
tive to the level of the term structure while the swap spread is independent. Interest
rate risk emerging from changes in the risk-free term structure is partly offset by
credit spread changes which are usually negatively related. This holds as long as the
market’s opinion on AA credit risk which is captured in the swap spread remains con-
stant. Investors can use this information for their portfolio’s risk management which
has to consider interest rate risk, credit default risk and the correlation of interest rate
and spread changes.
Are Credit Spreads and Interest Rates co-integrated? 31
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Appendix
Table 4: Descriptive Statistics and Augmented Dickey Fuller Test
Series Mean Median Max Min Std Dev Obs ADF Signif.
U03MM 3.85 4.63 6.38 0.83 1.77 581 -1.09 n.s.
U02GOV 4.45 4.89 7.70 1.08 1.66 581 -1.95 n.s.
U03GOV 4.65 4.96 7.79 1.32 1.53 581 -2.06 n.s.
U05GOV 4.93 4.96 7.87 2.03 1.28 581 -2.29 n.s.
U10GOV 5.30 5.29 7.86 3.11 0.99 581 -2.53 n.s.
UIBOR 4.26 5.25 6.85 1.02 1.91 581 -1.14 n.s.
U02SWP 4.85 5.31 8.14 1.28 1.69 581 -1.95 n.s.
U03SWP 5.10 5.42 8.20 1.64 1.52 581 -2.06 n.s.
U05SWP 5.49 5.78 8.21 2.37 1.32 546 -1.96 n.s.
U10SWP 5.86 6.01 8.25 3.46 1.07 581 -2.27 n.s.
U03AA 5.21 5.62 8.36 1.80 1.55 581 -2.02 n.s.
U05AA 5.61 5.84 8.46 2.49 1.29 581 -2.21 n.s.
U10AA 6.17 6.23 8.55 4.05 0.96 581 -2.49 n.s.
U03A 5.44 5.83 8.44 2.17 1.50 581 -2.02 n.s.
U05A 5.85 6.09 8.59 2.85 1.27 581 -2.18 n.s.
U10A 6.43 6.52 8.78 4.36 0.97 581 -2.39 n.s.
U03BBB 5.83 6.23 8.69 2.64 1.41 581 -2.06 n.s.
U05BBB 6.25 6.48 8.88 3.42 1.20 581 -2.25 n.s.
U10BBB 6.89 7.00 9.04 4.89 0.92 581 -2.40 n.s.
UEQX 1039.62 1100.65 1527.46 459.27 263.79 581 -2.04 n.s.
UEQXDEL 0.11 0.14 0.51 -0.33 0.18 581 -2.16 n.s.
USTEIGOV 0.85 0.50 2.73 -0.51 0.86 581 -0.87 n.s.
ULS 0.40 0.36 1.57 0.09 0.22 581 -5.16 1%
U03SS 0.45 0.44 0.87 -0.03 0.17 581 -1.93 n.s.
U05SS 0.52 0.45 1.06 -0.02 0.22 581 -1.67 n.s.
U10SS 0.58 0.48 1.34 0.12 0.25 581 -1.76 n.s.
U03AACS 0.56 0.49 0.97 0.30 0.16 581 -2.58 n.s.
U05AACS 0.68 0.56 1.25 0.36 0.23 581 -1.82 n.s.
U10AACS 0.87 0.78 1.64 0.43 0.30 581 -1.87 n.s.
U03ACS 0.79 0.64 1.55 0.43 0.28 581 -1.55 n.s.
U05ACS 0.92 0.76 1.70 0.51 0.33 581 -1.31 n.s.
U10ACS 1.13 0.99 2.06 0.64 0.37 581 -1.43 n.s.
U03BBBCS 1.18 0.99 2.12 0.56 0.45 581 -1.11 n.s.
U05BBBCS 1.32 1.16 2.20 0.65 0.49 581 -1.14 n.s.
U10BBBCS 1.59 1.55 2.65 0.86 0.49 581 -1.28 n.s.
Are Credit Spreads and Interest Rates co-integrated? 38
Series Mean Median Max Min Std Dev Obs ADF Signif.
U03AAOS 0.11 0.10 0.45 -0.18 0.11 581 -2.87 10%
U05AAOS 0.16 0.15 0.51 -0.17 0.12 581 -3.24 1%
U10AAOS 0.31 0.29 1.09 -0.13 0.22 581 -3.18 1%
U03AOS 0.34 0.32 0.95 0.04 0.19 581 -2.40 n.s.
U05AOS 0.40 0.36 0.98 0.07 0.18 581 -2.29 n.s.
U10AOS 0.57 0.50 1.22 0.19 0.24 581 -2.82 10%
U03BBBOS 0.73 0.60 1.57 0.23 0.35 581 -1.58 n.s.
U05BBBOS 0.81 0.71 1.62 0.28 0.33 581 -1.68 n.s.
U10BBBOS 1.03 1.03 1.82 0.42 0.36 581 -2.02 n.s.
Descriptive statistics for the time series employed with MM = money market, GOV =
government bond, U = USD denominated, UIBOR = USD Libor, SWP = swap rate,
AA, A and BBB = yield corporate bond index for rating specified, EQX = equity index,
EQXDEL = 1-year performance of equity index, STEIGOV = slope of term structure,
LS = libor spread, SS = swap spread, CS = credit spread, OS = spread over swap.
Table 5: Credit Spread and Spread over Swap Correlation
U03BBBOS
U05BBBOS
U10BBBOS
U03AAOS
U05AAOS
U10AAOS
U03AOS
U05AOS
U10AOS
U03AACS 0.238 0.634 0.678 0.198 0.652 0.699 0.150 0.421 0.513
U03ACS 0.161 0.797 0.865 0.306 0.816 0.879 0.305 0.601 0.710
U03BBBCS 0.094 0.771 0.940 0.343 0.834 0.958 0.454 0.707 0.830
U05AACS 0.092 0.640 0.719 0.330 0.722 0.753 0.237 0.522 0.619
U05ACS 0.078 0.704 0.797 0.264 0.788 0.836 0.261 0.573 0.690
U05BBBCS 0.019 0.706 0.875 0.256 0.792 0.923 0.370 0.650 0.796
U10AACS -0.008 0.583 0.742 0.266 0.693 0.790 0.488 0.683 0.751
U10ACS 0.000 0.630 0.763 0.244 0.731 0.811 0.371 0.663 0.742
U10BBBCS -0.086 0.614 0.807 0.210 0.731 0.868 0.420 0.694 0.835
Working Paper Series
ESCP-EAP Europäische Wirtschaftshochschule Berlin
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Nr. 18 Schmid, Stefan/Kretschmer, Katharina (2006): Performance Evaluation of
Foreign Subsidiaries – A Contingency Framework.
Nr. 19 Festing, Marion/Lassalle, Julius (2006): Determinanten des psychologischen
Vertrags – Eine empirische Untersuchung am Beispiel von Alumni der ESCP-
EAP Europäische Wirtschaftshochschule Berlin.
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schen Forschung? – Objektivität, Reliabilität und Validität in der Diskussion.
Nr. 21 Schmid, Stefan/Daniel, Andrea (2006): Measuring Board Internationalization
– Towards a More Holistic Approach.
Nr. 22 Festing, Marion/Eidems, Judith/ Royer, Susanne/Kullak, Frank (2006): When
in Rome Pay as the Romans Pay? – Considerations about Transnational
Compensation Strategies and the Case of the German MNE.
Nr. 23 Schmid, Stefan/Daub, Matthias (2007): Embeddedness in International Busi-
ness Research – The Concept and Its Operationalization.
Nr. 24 Wrona, Thomas/Klingenfeld, Daniel (2007): Current Approaches in Entrepre-
neurship Research: Overview and Relevance for Management Research.
Nr. 25 Pape, Ulrich/Schlecker, Matthias (2007): Are Credit Spreads and Interest
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