- Email: [email protected]
The dynamics of Australian dollar bonds with different credit qualities
International Review of Financial Analysis 9 (2000) 389 ± 404
The dynamics of Australian dollar bonds with different credit qualities Jonathan Battena,*, Warren Hoganb,1, Seppo PynnoÈnenc,2 a
Deakin University, Melbourne Campus, Melbourne, Australia b University of Technology, Sydney, Australia c University of Vaasa, Vaasa, Finland
Abstract We investigate the long-term equilibrium relationship between Australian dollar bonds of different credit quality. Contrary to the expectations hypothesis, we find the yields of Eurobonds are not cointegrated with the equivalent maturity Government bond. Nevertheless, the results suggest that the yields of the different risk classes of Eurobonds are cointegrated with one another, with the higherrated bond yields tending to lead the lower-rated yields. The paper also demonstrates that the cointegration relationship can be utilised in modelling the dynamics of the spread changes between Eurobonds and Government bonds. D 2000 Elsevier Science Inc. All rights reserved. JEL classification: E43; G15 Keywords: Credit spreads; Eurobonds; Cointegration; Equilibrium correction
1. Introduction There are a number of studies that investigate the long-term relationship between bonds of different maturity and credit rating and across different markets. For example, Pederosa and Roll (1998) find that US credit spreads are cointegrated and leptokurtic, while Morris et al. (1999) also find that US securities are cointegrated but note the time-dependent nature of the * Corresponding author. Fax: +61-3-9251-7243. E-mail addresses: [email protected] (J. Batten), [email protected] (W. Hogan), [email protected] (S. PynnoÈnen) 1 Fax: +61-2-9281-0364. 2 Fax: +358-6-324-8557. 1057-5219/00/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 0 ) 0 0 0 3 6 - 3
390
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
price adjustment process. Other studies investigate the cointegration relationships in offshore markets such as the Hiraki et al. (1996) study on yen offshore rates. These studies are usually undertaken in the context of the expectations hypothesis (Hall et al., 1992) where, assuming stationary risk premiums, all bond yields, whether Eurobond or Government bonds, should be cointegrated. The objective of this study is to investigate these empirical relationships in the context of the Australian dollar denominated Eurobond and Australian Government bond markets. Specifically, we investigate the cointegration properties, from January 1986 to May 1998, between AAA-, AA- and A-rated3 Australian dollar Eurobonds with maturities of 2, 5, 7, and 10 years. The Australian dollar Eurobond market is small, accounting for only US$36.9 billion of the estimated outstanding Eurobonds of US$3657.3 billion in June 1998 (Bank of International Settlements, 1998). However, it is the eleventh most frequent currency of issue.4 Understanding the long-term equilibrium relationship between different credit classes and maturities of Australian bonds is important for four reasons. First, to price risky bonds, general market practice is to apply a yield spread to a risk-free bond of a specified maturity. This practice assumes an equilibrium relationship between different credit classes of bonds, with pricing based on a linear, but not necessarily constant, relationship between the risk free and risky interest rate for the specified maturity. In addition, derivatives based upon the prices of Australian Government securities serve as the benchmark for hedging the interest rate risk of portfolios of bonds in other credit classes. Second, both Australia and the US specify an overnight rate as the target rate for monetary policy purposes with intervention to ensure that the actual market rate does not deviate too far from the policy rate (Cohen, 1999). While there is a need to establish the interest rate equilibrium relationship to appreciate the volatility transmission mechanism between bonds of short- and long-term maturity, there is also a need to establish the relationship between bonds of different credit class. Specifically, cointegration analysis enables the construction of impulse response functions that describe the interest rate adjustment process between pairs, or groups, of bonds with the same maturity but different credit class. Third, the theoretical models of the pricing of corporate debt, such as Merton (1974), Longstaff and Schwartz (1995), and Jarrow et al. (1997), predict an inverse relationship between a change in the risk free rate and the credit spread on risky debt when markets are in equilibrium. As noted by Morris et al. (1999), these models do not specify the dynamics of the price adjustment process, or the time period over which a new equilibrium is established. 3
Bond credit rating agencies categorise bond issuers into nine major classes according to perceived credit quality. These ratings classes include investment grade issuers: AAA, AA, A, and BBB; and non-investment grades: BB, B, CCC, CC, and C. Bonds with ratings below C are bonds in default or of bankrupts. The two major agencies use slightly different notation to refer to equivalent credit risk categories. Standard and Poor use uppercase capitals (e.g., AAA), while Moody's Investor Services use an uppercase first character and have any remaining characters lower case (e.g., Aaa). This paper uses the Standard and Poor's notation. 4 The USD is the most frequent currency of issue with US$1673.4 billion, then the yen with US$407.1 billion, the Deutsche mark with US$369.4 billion, the pound with US$308.3 billion, the French franc US$191.3 billion, the Swiss franc US$141.5 billion, the Italian lire US117.9 billion, the Dutch guilder US$105.3 billion, the ECU US$99.3 billion, and finally the Luxembourg franc with US$37.8 billion in outstandings (Bank for International Settlements, 1998, Table 13B).
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
391
These models also predict that high-grade issuers face upward sloping credit yield curves (that is the spread over the risk free asset rises with maturity), whereas speculative firms face the reverse situation (downward sloping, or hump shaped but downward sloping curves). This suggests that the impact of changes in monetary policy will be experienced differently by different risk classes of borrowers. These different experiences of yield curve adjustment present important risk management implications for banks holding portfolios of risky assets. For example, simple value-at-risk models assume parallel yield curve shifts across all credit classes. Finally, for credit spread traders and sellers of credit spread options, it is important to determine the interday time-series properties of the data to ensure pricing decisions are made correctly. Risk emerges from the unpredictable component (innovation) of the changes in interday credit spreads. This is the risk component that traders need to manage. The unpredictable component is found after filtering all systematic interday dependencies. Cointegration analysis allows the identification of the unpredictable component in a correctly specified form and provides a model for the mean structure of the yield spreads, moments of distributions of the innovations and evidence of normality. The paper is structured as follows. Section 2 presents a theoretical framework implied by the term structure of interest rates for modelling the properties of credit spreads is developed. The data and the time-series properties of the Australian dollar Eurobond yield data are then described in section 3. This section contains descriptive statistics and correlation relationships. Since the series are all highly correlated we then investigate the linear relationship between different classes of bonds predicted by the expectations hypothesis using regression techniques. However, these simple equations fail to capture the complexity arising from cointegrated systems. Then in section 4, the cointegration analysis using the Johansen (1988) procedure is determined and a detailed example employing an error correction model for the 2-year AAA and Australian Government bond is developed. Section 5 allows for concluding remarks.
2. Theory of the term-structure of interest rates of risky bonds Denote the continuously compounded yield to maturity of a k-period pure discount bond as R*(k,t), (k = 1, 2,. . .) and let F(k,t) be the forward rate of a contract at time t to buy a oneperiod discount bond that matures at t + k. Then by Fisher±Hicks, k 1X F j; t; f or k 1; 2; 3; . . . 1 R k; t k j1 Consequently, F(1,t) = R*(1,t), but, generally, the forward rates F( j,t) differ from the future yields R*(1,t + jÿ1), and can be rather understood as rational expectations of the future one-period yields given the current information. Thus, we can write: F j; t Et R 1; t j ÿ 1 a j; t
2
where Et is the conditional expectation given information available at time t and
392
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
a( j,t), with a(1,t) = 0, is the premium accounting for possible risk about the future (Hall et al., 1992). Substituting Eq. (2) in Eq. (1) provides a general relationship between maturities of pure discount bonds as: k k 1X 1X Et R 1; t j ÿ 1 A k; t with A k; t a j; t: 3 R k; t k j1 k j1 Traditional theories of the term structure focus on the properties of the premiums A(k,t). The pure expectation hypothesis asserts that A(k,t) are zero and the expectation hypothesis that the premiums are constants. In the case of corporate bonds, there is a risk premium over an otherwise similar government bond. For a corporate bond, there is a risk that the coupon or principal payment may not be met, implying a distinction between the promised and expected return on the bond. Because of this uncertainty investors demand a risk premium covering the promised and expected return. Let D(k,t) denote the risk premium for a corporate bond over an otherwise similar government bond, then from Eq. (3), we get the following model for the yield of a corporate bond: R k; t R k; t D k; t:
4
Thus, using Eq. (4), we can write: R k; t ÿ R 1; t R k; t ÿ R 1; t D k; t ÿ D 1; t k 1X Et R 1; t j ÿ 1 ÿ Et R 1; t A k; t D k; t k j1
ÿ D 1; t j k ÿ1 X 1X k
Et DR 1; t i A k; t D k; t ÿ D 1; t
5
j1 i1
where DR*(1,t + i) = R*(1,t + i)ÿR*(1,t + iÿ1). Given that R*(1,t + i) is I(1) (integrated of order one), the differences DR*(1,t + i) are therefore stationary implying that the (double) summation term in the last line of Eq. (5) is stationary. Therefore, the important result predicted from the theory is that the yield spread R(k,t)ÿR(1,t) is stationary since the maturity premiums A(k,t) and default risk premiums D(k,t)ÿD(1,t) are stationary.
3. Australian dollar Eurobond data To investigate the dynamics of Australian dollar Eurobond and Government bond markets, we investigate the yields and the spreads, representing credit risk premiums, between different risk classes of aggregated Australian dollar Eurobonds and Australian Government bonds of equivalent maturity. For simplicity, this investigation excludes bonds with embedded options such as callable, puttable or convertible bonds. Daily data of three
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
393
different rating classes of Australian dollar Eurobond (non-government) bonds (AAA, AA, A), and with four different maturities (2, 5, 7, and 10 years) from 2 January 1995 to 31 August 1998 (954 observations) were investigated. This period was characterised by a substantial fall in bond yields from near 11% to 6% for AAA-rated bonds. The data5 was collected as a weighted-average daily yield for each credit rating across three or four industries. These included Industrial, Telecommunications, Banking and Finance, Utilities, and Cable and Broadcasting. There are limitations to this approach since both Longstaff and Schwartz (1995) and Duffee (1996) observe crosssectional variation in spreads across industry. However, both studies rely on industry aggregates with a mix of maturities, so some of the variation observed may have been introduced through differences in the average maturity of bonds issued by the different industry sectors. On the other hand, not averaging the yields introduces concerns over the liquidity of the bonds of some industry sectors given that the majority of bonds were issued by the banking and finance industries. The yields were collected from the credit-spread pages provided by Reuters' Information Services and were based on secondary market bond prices at 4 PM London time. The yields for the benchmark maturities for the different rated bonds were calculated from bond yields from various maturities and interpolated into a benchmark set of yields (t = 2, 5, 7, 10 years) using cubic-splinning techniques. Cubic splinning produces a zero curve, which is both smooth in its first derivation and continuous in the second-order derivation. The values of the second derivations can be solved using a set of tri-diagonal equations that enable the calculation of any interpolated value of t. In this instance, interpolated values of t = 2, 5, 7 and 10 were determined.6 The maturity structure of the different rated Australian dollar Eurobonds on the final day of the sample (31 August 1998) was obtained from the Reuters' Fixed Income Database, which comprises outstanding fixed-rate (zero-coupon and floating rate notes excluded) Eurobonds. We estimate this database captured about 90% of the US$36.9 billion in outstanding Australian dollar Eurobonds reported to the Bank for International Settlements at the 30 June 1998. We suggest that a large part of the remaining bonds were private issues and therefore not traded in secondary markets. Analysis of this data suggests, first, the Australian Government bond market, with outstandings of about A$67 billion, is larger (by about 30%) than the Australian dollar Eurobond market (with outstandings of about A$50.4 billion). Second, AAA-rated bonds have the largest outstandings (about 76%) and single A grade issues the least (4%). There were no fixed-rate issues for issuers with ratings below A (e.g., BBB and non-investment grade issuers), though there were a few floating rate issues for issuers with ratings below A. The distribution of Australian Government bonds was generally evenly spread across maturities from 0 to 10 years, though there were only 5.5% of issues with maturities 5
We thank Westpac Banking, Sydney Australia for providing the bond yield data. Cubic-splinning involves the following approach: Consider a zero-coupon rate Zi on a zero-coupon curve Z(ti) such that i = 1, 2, 3, . . ., n, then for the set t2(titi + 1), Z(t) = a + bt + gt2 + dt3 so that Z(ti) = Zi and Z(ti + 1) = Zi + 1, with Z(ti) = Ziand Z(ti + 1) = Zi + 1. Z(t) can now be expressed as follows: Z(t) = aZi + bZ(ti + 1) + cZi + dZ(ti + 1) where a = (ti + 1ÿt)/h, b = (tÿti)/h, c = 1/6(a3ÿa)h2, d = 1/6(b3ÿb)h2, and h = ti + 1ÿti. 6
394
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
greater than 10 years. The even spread of Australian Government bond issues is a conscious decision on the part of the authorities to maintain liquidity across the benchmark yield curve. By comparison, the greatest amount of outstandings in the Australian dollar Eurobond market is for maturities from 0 to 5 years (73% of total issues), though these issues largely carry AAA ratings (54% of total issues). This suggests there is very little liquidity in Australian dollar Eurobonds with longer maturities than 5 years and lesser credit ratings than AAA. 4. Empirical results The pricing of credit may be perceived to be a function of a number of factors. These include the market microstructure features within specific markets (e.g., regulatory differences including withholding taxes on non-resident income, which raise the cost of certain types of funds) and between markets (e.g., higher yields in domestic markets to embody sovereign risk), expectations by market participants, portfolio factors that provide diversification benefits to the investor, and, finally, credit risk perceived in terms of traditional statistical measures including yield volatility, correlation with other risky assets, default probabilities, and recovery rates. These last measures require sophisticated models and quality historical data to maintain their integrity. All these factors may be embodied in the spread between classes of different types of assets. For example, the Longstaff and Schwartz (1995) model for valuing the spread of corporate bonds considers a two-factor model that captures interest rate and asset default factors. When bonds are marketable, then the prices, yields, and default histories may be observed in secondary markets. Otherwise, this type of information will remain the proprietary information of the lending institution. For spread traders and sellers of credit spread options, it is important to determine the distributional and time-series qualities of the data to ensure pricing decisions are made correctly (see Batten & Hogan, 1999 for an analysis of credit derivatives in the Australian context). 4.1. Descriptive statistics Analysis of the bond yields initially involved determining the descriptive statistics and correlation relationships between the four different classes and maturities of bonds. These credit spreads represent the premium for credit risk between each risk group and maturity. Batten et al. (2000) provide a more detailed account of the time-series properties of the various Australian bond series. The data displayed a definite positively sloped yield curve (term structure) over the sample period, with the shorter-term maturities having a lower yield to the longer-term maturities at any one point in time. For example, AAA bonds had a 2-year yield mean of 7.0714% while the 10-year AAA yield mean was 8.0458% during the sample period. There was also evidence of a credit risk premium with the yields on the higher quality bonds for a specific maturity being lower than the yields for the equivalent maturity lower-rated bonds. This relationship between bond yields and default risk is referred to as the default-risk structure of
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
395
Table 1 Moments of the daily credit spreads of Australian dollar Eurobonds, 2 January 1995 to 31 August 1998 (expressed in basis points) Credit spread N = 954 in basis points
Mean
Standard deviation
Skewness
Excess kurtosis
AAA ± AA2 AAA ± AA5 AAA ± AA7 AAA ± AA10 AA ± A2 AA ± A5 AA ± A7 AA ± A10
7.622 6.904 9.484 11.022 13.908 15.865 12.724 11.923
5.177 3.652 5.756 8.427 5.285 7.128 8.433 11.157
1.5374 0.7922 0.7815 1.2703 ÿ0.0430 0.6270 1.3703 ÿ1.3983
2.4980 0.6103 0.1300 1.9012 ÿ0.2000 0.6373 4.0044 14.600
This table records the four moments of the daily credit spreads (AAA ± AA and AA ± A) between the different aggregated bond ratings with 2-, 5-, 7-, and 10-year maturities. The mean and standard deviation of the AAA ± AA bond spreads generally rose with increasing maturity (a positive credit term structure), while the mean spreads on the lower quality bonds generally fell with increasing maturity. The series were not significantly skewed though the AAA ± AA2, AAA ± AA10, AA ± A7, and AA ± A10 display significant excess kurtosis (leptokurtic). Note the AAA ± AA series suggests an upward sloping credit yield curve (i.e., 2-year spreads < 10-year spreads), while the AA ± A is a humped, downward sloping credit yield curve (i.e., 2-year spreads > 10-year spreads).
interest rates. For example, AAA 5-year bonds had a lower yield mean of 7.6125% compared with the A-rated 5-year bonds of 7.8402%.7 Table 1 provides descriptive statistics of the credit spreads between the different rating classes and maturities of bonds. These credit spreads represent the premium for credit risk between each group and maturity. For example, the variable AAA±AA2 with the mean result of 7.622 basis points (0.07622%) represents the mean margin between the AAA and AA 2year bonds over the sample period. Generally, the mean credit spreads over the sample period provide evidence in support of the theoretical behaviour suggested by Merton (1974). That is, the better quality credit AAA±AA spreads increased as maturity increased (an upward sloping credit yield curve), while the AA±A spreads decreased with an increase in maturity (a downward sloping curve). Both curves were humped around the 5-year maturity, but in different directions. The AAA±AA spread declined from 7.622 basis points (2 years) to 6.904 basis points (5 years), while the lower quality AA±A spread rose from 13.908 basis points (2 year) to 15.865 basis points (5 years). This may have been due to the greater liquidity in the AAA and AA 5-year bond market. The credit spreads for bonds with maturities greater than 5 years were better behaved. The mean credit spreads between the AAA to AA 5-year and the AAA to AA 7-year bonds increased from 6.904 to 9.484 basis points as maturity increased, while the AA to A spread declined from15.865 basis points for 5 year to 12.724 basis points for 7-year bonds. However, as the test results reported in Table 2
7
The behaviour of US credit risk premiums is discussed in Neal (1996).
396
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
Table 2 Augmented Dickey ± Fuller (ADF) and Phillip ± Perron tests for unit roots of Australian dollar Eurobonds and Government bonds, 2 January 1995 to 31 August 1998 (expressed in basis points) Series
ADF
PP
ADF trend
PP trend
First differences
ADF
PP
G2 G5 G7 G10 A2 A5 A7 A10 AA2 AA5 AA7 AA10 AAA2 AAA5 AAA7 AAA10
ÿ3.01 ÿ2.40 ÿ2.44 ÿ2.51 ÿ2.42 ÿ2.02 ÿ1.71 ÿ1.13 ÿ2.41 ÿ1.99 ÿ1.69 ÿ1.36 ÿ2.46 ÿ2.01 ÿ1.75 ÿ1.18
ÿ3.00 ÿ2.17 ÿ2.33 ÿ2.38 ÿ2.10 ÿ1.88 ÿ1.60 ÿ1.01 ÿ2.10 ÿ1.80 ÿ1.58 ÿ1.28 ÿ2.15 ÿ1.81 ÿ1.55 ÿ1.15
ÿ3.01 ÿ2.54 ÿ2.58 ÿ2.53 ÿ2.41 ÿ2.44 ÿ2.75 ÿ3.35 ÿ2.22 ÿ2.31 ÿ2.44 ÿ2.61 ÿ2.27 ÿ2.34 ÿ2.42 ÿ2.88
ÿ2.96 ÿ2.30 ÿ2.44 ÿ2.35 ÿ2.02 ÿ2.04 ÿ2.54 ÿ3.08 ÿ1.90 ÿ2.00 ÿ2.22 ÿ2.41 ÿ2.04 ÿ2.11 ÿ2.22 ÿ2.86
DG2 DG5 DG7 DG10 DA2 DA5 DA7 DA10 DAA2 DAA5 DAA7 DAA10 DAAA2 DAAA5 DAAA7 DAAA10
ÿ15.0 ÿ14.8 ÿ15.1 ÿ15.1 ÿ13.1 ÿ14.2 ÿ13.5 ÿ14.6 ÿ13.3 ÿ13.8 ÿ13.6 ÿ14.2 ÿ13.0 ÿ13.6 ÿ13.6 ÿ13.7
ÿ30.4 ÿ29.9 ÿ28.3 ÿ30.5 ÿ27.9 ÿ29.4 ÿ30.8 ÿ30.6 ÿ27.0 ÿ28.1 ÿ28.9 ÿ30.4 ÿ27.4 ÿ28.3 ÿ28.5 ÿ29.5
MacKinnon critical values for rejection of hypothesis of a unit root 1% 5% 10%
ÿ3.44 ÿ2.87 ÿ2.57
ÿ3.97 ÿ3.42 ÿ3.13
This table provides ADF and Phillips ± Perron unit root test of the Government bonds and three different ratings of Eurobonds with four maturities (2, 5, 7, and 10 years). Four lags for ADF and six lags for PP as suggested by Newey ± West. Tests for the levels include intercept (second and third columns) and intercept and trend (fourth and fifth columns). First differences include the intercept.
indicate, the yields are integrated with order one (i.e., they are I(1) processes) implying that they may wander as time goes by so that averages may not satisfactorily characterise the relationships between the yields unless the series are cointegrated. 4.2. Cointegration analysis between Australian dollar Eurobonds and Government bonds Given that the risk premiums, D(k,t), defined in section 2, can be assumed stationary, theory predicts that the series should be cointegrated with a cointegration vector (1,ÿ1). Generally, the cointegration analysis can be undertaken using a vector error correction (VEC) model of the form: 6 DYt m PYtÿ1 G1 DYtÿ1 : : : Gpÿ1 DYtÿp1 et ; where Yt in our case is a vector with two components, P = ab0 with prime denoting the transpose, and where: b0 b1 ; b2
7
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
397
is the cointegration vector in a two-vector system [Eq. (7)] and A = (a1,a2)0 contains the short-term adjusting parameters that drive the system back to the long-term equilibrium, D stands for the difference, Gi (i = 1, . . ., pÿ1) contains the short-term autoregressive parameters, and Et is the error (innovation) vector, assumed to be normally distributed. The intercept vector M imposes a non-zero drift in yields, which is usually not a reasonable model for interest rate behaviour. Consequently, a common practice is to restrict M to the cointegration space by augmenting Ytÿ1 to tÿ1 = (1,Y0tÿ1)0 Ä and the cointegration vector as b0 = (b0,b1,b2). Table 3 reports the unrestricted cointegration tests between the Australian dollar Eurobonds and the equivalent maturity Government bonds accompanied by tests of stationarity of the spreads. Non-stationarity of the spreads suggests the spreads follow I(1) processes, or equivalently, that the stochastic trends which play the dominating role in the behaviour of the yields are independent of one another. The five lags for parameter p in Eq. (6) proved to be sufficient to ensure that the residuals were not serially correlated. Surprisingly, the results
Table 3 Cointegration tests of Australian dollar Eurobonds with equivalent maturity Australian Government bonds, 2 January 1995 to 31 August 1998 Unit root tests of the spreads Likelihood ratio AAA2 ± G2 AAA5 ± G5 AAA7 ± G7 AAA10 ± G10 AA2 ± G2 AA5 ± G5 AA7 ± G7 AA10 ± G10 A2 ± G2 A5 ± G5 A7 ± G7 A10 ± G10
18.3 13.7 11.3 8.7 17.0 13.5 10.5 9.1 16.7 11.9 9.6 8.1
a
ADFb ÿ1.41 ÿ1.28 ÿ1.20 ÿ0.75 ÿ1.40 ÿ1.26 ÿ1.15 ÿ0.85 ÿ1.44 ÿ1.28 ÿ1.19 ÿ0.72
PPc ÿ1.22 ÿ1.09 ÿ0.93 ÿ0.47 ÿ1.18 ÿ1.07 ÿ0.92 ÿ0.57 ÿ1.22 ÿ1.09 ÿ0.98 ÿ0.35
Critical values for the likelihood ratiod
Critical values of ADF and PP tests
10% 5% 1%
10% 5% 1%
17.9 20.0 24.6
ÿ2.57 ÿ2.86 ÿ3.44
The table reports the results of Johansen's cointegration tests and stationarity of the credit spreads of AAA-, AA- and A-rated Eurobonds over equivalent maturity Government bonds. The stationarity of the credit spread would imply that the mean spread would serve as the common long-term equilibrium between the yields of the two bonds. a The null hypothesis is that the series are not cointegrated. b ADF test with four difference lags and intercept. c PP test with six lags and intercept (number of lags were defined as suggested by the Newey ± West procedure). d The intercept term is restricted into the cointegration space.
398
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
strongly indicate that the Eurobonds are not cointegrated with the Government bonds. This also means that the risk spreads, D(k,t) defined in Eq. (4), should be I(1) series implying that the Government bonds and the Eurobonds do not have any ties with one another and the two-bond markets are effectively independent in terms of pricing. This would be expected to be implausible in the long run, and this result may be due to the specific sample period with bond yields, especially the Eurobond yields, having fallen over the whole sample period. Analysis of a longer sample period may, however, shed further insights on this issue. Hence, these results challenging as they are should be treated with some caution. 4.3. Cointegration of Australian dollar Eurobonds with different risk classes We further investigate the long-term equilibrium relationships of the Australian dollar bonds of different credit class by investigating the relationships between the yields of the Eurobonds independently of the Government bonds. In principle, it is possible to use a multivariate (three-variate) test to include all the three ratings, however, we present only the bivariate test results since the three-variate residuals exhibited strong volatility clustering. Table 4 reports the cointegration tests in pairs of the same maturity Eurobonds. According to the unrestricted likelihood-tests, all the series should be cointegrated. The theory in section Table 4 Cointegration and restricted cointegration tests between Australian dollar Eurobonds with different ratings, 2 January 1995 to 31 August 1998 H0: (b1,b2) = (1,ÿ1) Likelihood ratio A2 ± AAA2 A5 ± AAA5 A7 ± AAA7 A10 ± AAA10 AA2 ± AAA2 AA5 ± AAA5 AA7 ± AAA7 AA10 ± AAA10 A2 ± AA2 A5 ± AA5 A7 ± AA7 A10 ± AA10
a
42.3 30.4 33.1 35.8 44.2 63.5 37.2 39.3 75.9 35.5 44.5 31.5
c2 (df = 1)
p Value
c2 (df = 1)b
p Value
17.0 0.03 0.4 1.8 9.7 6.7 11.2 5.2 20.9 0.2 2.4 0.2
0.000 0.855 0.530 0.186 0.002 0.010 0.001 0.022 0.000 0.686 0.124 0.690
18.8 0.2 0.9 0.6 8.8 5.2 10.7 4.4 17.2 0.2 2.4 0.003
0.000 0.643 0.341 0.440 0.003 0.022 0.001 0.036 0.000 0.665 0.124 0.954
Critical values for the likelihood ratio 10% 17.9 5% 20.0 1% 24.6 This table records the unrestricted cointegration tests in column 2 followed by testing whether the yields of the two bonds are independent. The variables are the daily yields with 2-, 5-, 7-, and 10-year maturities. a The null hypothesis is that the series are not cointegrated. b Test with constant correlation multivariate GARCH(1,1) errors as defined by Eq. (8)
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
399
2 predicts that the spread should establish the cointegration relationship, or, equivalently, that the spread is stationary, and, hence, independent of the yield levels. Columns 3 through 6 of Table 4 report the likelihood ratio tests for this hypothesis. The hypothesis is supported between AAA and A, as well as AA and A bonds in all but the 2-year maturity, whereas between AAA and AA bonds the hypothesis is rejected in each case. The numerical values of the coefficients do not depart much from unity, that is, they are all similar to those reported for the A±AAA2 bond in Table 6 below. The above test results do not change materially whether the models are estimated under a constant (conditional) residual variance, or under a bivariate GARCH(1,1) specification of the form: h11;t w10 a11 e21;tÿ1 a12 e22;tÿ1 b1 h11;tÿ1 h22;t w20 a21 e21;tÿ1 a22 e22;tÿ1 b2 h22;tÿ1 pp h21;t r h11;t h22;t
8
which captures the volatility clustering. In Eq. (8), hij,t are the conditional variances and Table 5 Weak exogeneity tests between different ratings of Australian dollar Eurobonds, 2 January 1995 to 31 August 1998 AAA 2
A2 ± AAA2 A5 ± AAA5 A7 ± AAA7 A10 ± AAA10
A
c (df = 1)
p Value
c2 (df = 1)
p Value
1.80 0.30 0.13 0.52
0.180 0.584 0.715 0.472
14.37 7.55 9.78 9.11
0.000 0.006 0.002 0.003
AAA 2
AA2 ± AAA2 AA5 ± AAA5 AA7 ± AAA7 AA10 ± AAA10
AA
c (df = 1)
p Value
0.21 0.16 0.97 3.04
0.644 0.693 0.326 0.082
AA 2
A2 ± AA2 A5 ± AA5 A7 ± AA7 A10 ± AA10
c2 (df = 1) 4.92 4.30 0.82 1.54
p Value 0.027 0.038 0.365 0.215
A
c (df = 1)
p Value
c2 (df = 1)
p Value
0.05 0.03 0.46 0.20
0.823 0.866 0.500 0.653
13.51 5.71 15.84 10.41
0.000 0.017 0.000 0.001
Large p values in the table indicate that the corresponding series is weakly exogenous with respect to the longrun equilibrium relation between the bond yields. In such a case, the other bond corrects its behaviour according to the changes in the yield of the weakly exogenous bond. That is to say the latter one is driving the movements of the bond yields.
400
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
Table 6 Equilibrium Correction Model for the Australian dollar A ± AAA Eurobond yields of different maturities, 2 January 1995 to 31 August 1998 A ± AAA2 Dx = DAAA2 xtÿ1 s.e. p Value Intercept s.e. p Value citÿ1 s.e. p Value Dxtÿ1 s.e. p Value Dytÿ1 s.e. p Value w s.e. p Value e2x,tÿ1 s.e. p Value e2y,tÿ1 s.e. p Value htÿ1 s.e. p Value r s.e. p Value
A ± AAA5 Dy = DA2
1.041 0.004 0.000 ÿ0.005 0.002 0.006
ÿ0.018 0.006 0.001 ÿ0.170 0.020 0.000 0.030 0.229 0.063 0.066 0.632 0.001 0.123 ÿ0.081 0.060 0.062 0.040 0.186 0.00037 0.00043 0.00008 0.00008 0.000 0.000 0.042 0.027 0.013 0.013 0.001 0.031 0.022 0.038 0.011 0.014 0.057 0.008 0.844 0.829 0.024 0.027 0.000 0.000 0.866 0.006 0.000
Diagnostic statistics Log-lik 743.4 Skew z ÿ0.140 p Value 0.079 Kurt z 3.77 p Value 0.000 Q(5) z 9.7 p Value 0.085 Q(10) z 11.0 p Value 0.354 Q(5) z2 2.4 p Value 0.790
ÿ0.246 0.002 4.12 0.000 6.4 0.267 15.7 0.109 4.1 0.536
Dx = DAAA5
A ± AAA7 Dy = DA5
1 ± ± ÿ0.005 0.002 0.031
0.017 0.004 0.000 ÿ0.093 0.014 0.000 0.055 0.233 0.058 0.066 0.341 0.000 0.059 ÿ0.110 0.055 0.060 0.282 0.066 0.00032 0.00031 0.00005 0.00005 0.000 0.000 0.061 0.046 0.014 0.008 0.000 0.000 0.016 0.016 0.011 0.007 0.143 0.013 0.852 0.875 0.021 0.016 0.000 0.000 0.838 0.007 0.000
656.5 0.005 0.953 4.73 0.000 10.2 0.071 12.5 0.253 2.5 0.777
0.169 0.034 4.12 0.000 4.2 0.527 11.7 0.303 4.2 0.520
Dx = DAAA7
A ± AAA10 Dy = DA7
1 ± ± ÿ0.006 0.002 0.004
0.012 0.003 0.000 ÿ0.088 0.010 0.000 0.138 0.233 0.051 0.046 0.007 0.000 ÿ0.021 ÿ0.156 0.047 0.056 0.659 0.005 0.00028 0.00022 0.00005 0.00003 0.000 0.000 0.067 0.019 0.010 0.006 0.000 0.001 0.012 0.063 0.007 0.009 0.082 0.000 0.877 0.887 0.009 0.008 0.000 0.000 0.754 0.009 0.000 565.5 0.144 0.069 3.71 0.000 8.8 0.116 13.9 0.176 2.3 0.803
0.308 0.000 5.61 0.000 6.3 0.277 12.5 0.253 9.8 0.081
Dx = DAAA10
Dy = DA10
1 ± ± ÿ0.008 0.002 0.001
0.010 0.003 0.000 ÿ0.077 0.007 0.000 ÿ0.034 0.006 0.033 0.012 0.306 0.624 0.082 0.029 0.038 0.035 0.033 0.407 0.00073 0.00017 0.00012 0.00002 0.000 0.000 0.083 ÿ0.007 0.010 0.001 0.000 0.000 0.016 0.080 0.009 0.007 0.069 0.000 0.754 0.893 0.034 0.009 0.000 0.000 0.697 0.010 0.000 533.3 ÿ0.151 0.058 5.39 0.000 9.7 0.085 17.7 0.061 4.2 0.515
ÿ0.092 0.249 5.25 0.000 7.8 0.167 19.6 0.034 1.4 0.927
(continued on next page)
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
401
Table 6 (continued ) A ± AAA2
Q(10) z2 p Value Corr(hx,hy)
A ± AAA5
A ± AAA7
A ± AAA10
Dx = DAAA2
Dy = DA2
Dx = DAAA5
Dy = DA5
Dx = DAAA7
Dy = DA7
Dx = DAAA10
Dy = DA10
5.1 0.885 0.9949
11.2 0.343
11.8 0.300 0.9958
30.5 0.001
7.0 0.725 0.9142
16.0 0.100
7.1 0.713 0.506
2.4 0.992
Rows 3 ± 6 report the estimates of the cointegration regression yt = b0 + b1xt + ut for the 2-year bond maturity with x denoting AAA2 and y A2. Relaxing the intercept term b0 freely estimated as parameters g1 and g2 in the VEC model (4) the cointegration relation becomes cit = ytÿb1xt with the parameter replaced by the estimate. For the longer maturity bond, the cointegration relationship is according to the results reported in Table 4 just the spread, i.e., cit = ytÿxt. In the estimation, also, the weak exogeneity results reported in Table 5 are utilised by restricting the coefficient of ci in the equation of Dx equal to zero. Conditional heteroskedasticity in the cointegration regression is of the form given by Eq. (10).
covariance of the residuals and r is the contemporaneous correlation of the residual series (volatility processes are considered more carefully below). Once we have found that the Australian dollar Eurobonds with different ratings are cointegrated an interesting question is then whether one of the bonds is driving the other. This is the case if one of the yields is not dependent on departures from the longrun equilibrium determined by the cointegration relationship (the series is then said to be weakly exogenous). In terms of model (6), this means that in the alpha-matrix, the corresponding row is equal to zero. Table 5 provides the test results. The results are clear-cut between AAA and A, as well as AA and A bonds, where the higher quality bond is in each case weakly exogenous. The relationship between AAA and AA is not that evident. At the short end the AAA bond seems to drive AA yields, but in the long end, the results are mixed. The overall result, however, is that at least in the short end, the AAA bonds seem to drive the yields of lower quality bonds. These results can be utilised in modelling the dynamics of spreads between Eurobonds and Government bonds. However, before undertaking this exercise, we must more closely consider the bivariate equilibrium model with conditional heteroskedastic yield innovations. 4.4. Equilibrium correction analysis of Australian dollar Eurobond yields Let x and y denote yields of Eurobonds in different credit classes. Then the cointegration of x and y implies an equilibrium correction (or error correction) model for the first differences of the variables as dxt g1 a1 citÿ1 lags dxt ; dyt eyt dyt g2 a2 citÿ1 lags dxt ; dyt ext
9
where ext and eyt are unpredictable innovation terms modelled by the conditional heteroskedasticity given by Eq. (8). As was noted previously, it is usually not feasible to include intercept terms (here g1 and g2) into the model for interest rates due to drifts in the
402
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
series. Nevertheless, in this particular case, the yields seem to exhibit trends in the sample period, so we include the drift terms as dummy parameters to capture these trends. Table 6 reports the estimation results for the A±AAA bonds, where we have estimated appropriate restricted models to utilise the weak exogeneity results presented in Table 5. The model with a one lag in differences (and bivariate GARCH(1,1) residuals) adequately captured serial dependencies in the yields. The negative equilibrium correction coefficients, reported on the ci-line, indicate the reverting behaviour once the A bond yield falls out of the equilibrium with respect to the AAA bond. In addition, the changes of A bond yields respond positively to the previous day changes of the AAA bond (denoted in the Table 6 by Dxtÿ1). Thus, the higher quality bonds seem to lead to the lower quality bonds, and there is a return spillover from the high quality bonds to the lower quality bonds such that a change in the higher quality bond yield predicts a change in the next days lower quality bond yield. Table 6 also shows the bivariate GARCH(1,1) estimation results. It is interesting to note the asymmetry in the volatility shocks. That is, shocks to the yields of the AAA bond (denoted by ex,t) are statistically more significant in the A-rated bond volatility equation, whereas A bond shocks (ey,t) do not affect the AAA bond volatility. This indicates that in addition to the yield dependency the AAA bonds are partially leading the volatility behaviour of the A-rated bonds. Finally, it is worth noting from Table 6 that the yield changes are highly correlated with the correlation ranging from 0.7 to 0.9. In addition, the last line of Table 6 indicates that the conditional volatility series are also very highly correlated. 4.5. An example: the dynamics of daily Australian dollar Eurobond credit spreads As was seen earlier, credit spreads between Australian dollar Eurobonds and Government bonds behave as I(1) series. In modelling the dynamics of the spreads, and also for trading of Eurobonds with different credit ratings, one can utilise the knowledge that the Eurobonds in different credit classes are cointegrated. Supposing further that the first differences of the government bond (denoted by z) can be modelled by an ARIMA model (see Batten et al., 2000), then because the spread st = ytÿzt is I(1), where y denotes the Eurobond, the first differences Dst = stÿstÿ1 are I(0). But we can write Dst = DytÿDzt. Using Eq. (9) and the ARMA structure of Dzt, we can obtain dst acitÿ1 lags dxt ; dyt ; dzt et ;
10
where et is an innovation process (which can follow a conditional heteroskedastic process). This model should provide a better description of the time-series behaviour of the spread than a simple ARIMA specification. As an example, consider the A2±G2 spread, which is not cointegrated. From Table 6, we can model the spread A2±AAA2 spread as ci = A2ÿ1.041AAA2. For comparison purposes, an AR-model was also estimated. In the estimation, we enhanced Eq. (10) with the intercept and retained only those explanatory series that were statistically significant. The results are reported in Table 7. Only the estimates of the spread and Government bond difference are statistically significant. None of the AR(2) model lags are significant, indicating that the series is in fact a pure innovation process with respect to its own history. Both models
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
403
Table 7 Estimates of the 2-year credit spread of the Australian dollar Eurobond with respect to the equivalent maturity Government bond DA2S
AR(2) model for DA2S
Coefficient
s.e.
t Value
p Value
ÿ1.291 ÿ0.147 0.143
0.407 0.044 0.044
ÿ3.172 ÿ3.318 3.279
0.002 0.001 0.001
GARCH estimates Constant 7.023 e2tÿ1 0.111 htÿ1 0.791
2.045 0.023 0.045
3.434 4.900 17.657
0.001 0.000 0.000
Diagnostic statistics Estimate
p Value
Estimate
p Value
0.000 0.000 0.086 0.178 0.457 0.658
0.001 ÿ0.004 ÿ3342.6 4.260 4.290 0.183 4.985 7.2 12.4 4.1 7.1
0.000 0.000 0.208 0.256 0.532 0.716
Intercept citÿ1 Dg2tÿ1
2
R Adj R2 Log-lik AIC SBC Skew z Kurt z Q(5) z Q(10) z Q(5) z2 Q(10) z2
0.017 0.012 ÿ3338.6 4.247 4.277 0.190 4.851 9.6 13.9 4.7 7.7
Intercept DA2Stÿ1 DA2Stÿ2
Coefficient
s.e.
t Value
p Value
ÿ0.286 ÿ0.023 ÿ0.058
0.245 0.038 0.034
ÿ1.166 ÿ0.598 ÿ1.720
0.244 0.550 0.086
7.443 0.107 0.790
2.155 0.022 0.045
3.453 4.955 17.514
0.001 0.000 0.000
The cointegration relation is taken according to the test results of Table 4 as the estimated cointegration regression given in Table 6 such that A2 = 1.041AAA2, so that ci = A2ÿ1.041AAA2. In the final estimation, the units of all variables are in basis points.
indicate that there is strong conditional heteroskedasticity in the residuals, and the Q-statistic on squared standardised residuals suggests that a GARCH(1,1) model fully captures the volatility clustering. 5. Conclusion Central to the management or trading of credit risk in its various forms by financial market participants, including recent innovations utilising credit derivatives is an appreciation of the distributional qualities and the long-term equilibrium relationships of the spreads between classes of risky bonds, and the processes generating the spreads. This contribution focused on the dynamics of Australian dollar Eurobond and Government yields and highlighted the cointegration relationships between the yields. The yield and yield spreads between government bonds and three classes of Australian dollar Eurobonds rated A, AA, and AAA, with four maturities (2, 5, 7, and 10 years) was examined. The empirical results strongly support
404
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
the proposition that yields in different risk classes of Australian dollar Eurobonds are cointegrated. However, the Eurobond yields and the Australian Government bond yields are not cointegrated. Analysis suggests that the higher-rated Eurobond yields tend to lead the lower-rated Eurobond yields. In addition to the mean dynamics, a bivariate GARCH model indicates that there is volatility transmission from higher-rated bonds to lower-rated bonds. The paper also demonstrates that the cointegration relationships can be utilised in modelling the dynamics of the spread changes between Eurobonds and Government bonds. Acknowledgments The authors are grateful for the valuable comments from Palle Anderson on an earlier draft of this paper. References Bank for International Settlements. (1998). The BIS Statistics on International Banking and Financial Market Activity, Monetary and Economic Department. Basel: Bank for International Settlements, November. Batten, J., Ellis, C., & Hogan, W. (2000). The time-series properties of credit spreads: evidence from Australian dollar Eurobonds. In: T. Fetherston (Ed.), Advances in Pacific Basin Financial Markets (Vol. 6, 261 ± 293). Stamford, CT: JAI Press. Batten, J., & Hogan, W. (1999). Credit derivatives: an appraisal for Australian financial institutions. Economic Paper 18(2), 19 ± 42. Cohen, B. (1999). Monetary policy procedures and volatility transmission along the yield curve. BIS Working Papers, Basle, Switzerland: Bank for International Settlements. Duffee, G. (1996). Treasury yields and corporate bond yields: an empirical analysis. Working paper, Washington D.C.: Federal Reserve Board. Hall, A., Anderson, H., & Granger, C. W. (1992). A cointegration analysis of treasury bill yields. The Review of Economics Statistics 74, 116 ± 126. Hiraki, T., Shiraishi, N., & Takezawa, N. (1996). Cointegration, common factors, and the term structure of Yen offshore interest rates. Journal of Fixed Income 6(3), 69 ± 75. Jarrow, R. A., Lando, D., & Turnbull, S. (1997). A Markov model for the term structure of credit risk spreads. The Review of Financial Studies 10, 481 ± 522. Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231 ± 254. Longstaff, F., & Schwartz, E. (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 29, 449 ± 470. Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449 ± 470. Morris, C., Neal, R., Rolph, D. (1999). Credit spreads and interest rates. Working Paper, Indiana University, Indianapolis, March. Neal, R. S. (1996). Credit derivatives: new financial instruments for controlling credit risk. Federal Reserve Bank of Kansas City Economic Review 81(2), 15 ± 27. Pederosa, M., & Roll, R. (1998). Systematic risk in corporate bond credit spreads. Journal of Fixed Income 8, 7 ± 26.
The dynamics of Australian dollar bonds with different credit qualities Jonathan Battena,*, Warren Hoganb,1, Seppo PynnoÈnenc,2 a
Deakin University, Melbourne Campus, Melbourne, Australia b University of Technology, Sydney, Australia c University of Vaasa, Vaasa, Finland
Abstract We investigate the long-term equilibrium relationship between Australian dollar bonds of different credit quality. Contrary to the expectations hypothesis, we find the yields of Eurobonds are not cointegrated with the equivalent maturity Government bond. Nevertheless, the results suggest that the yields of the different risk classes of Eurobonds are cointegrated with one another, with the higherrated bond yields tending to lead the lower-rated yields. The paper also demonstrates that the cointegration relationship can be utilised in modelling the dynamics of the spread changes between Eurobonds and Government bonds. D 2000 Elsevier Science Inc. All rights reserved. JEL classification: E43; G15 Keywords: Credit spreads; Eurobonds; Cointegration; Equilibrium correction
1. Introduction There are a number of studies that investigate the long-term relationship between bonds of different maturity and credit rating and across different markets. For example, Pederosa and Roll (1998) find that US credit spreads are cointegrated and leptokurtic, while Morris et al. (1999) also find that US securities are cointegrated but note the time-dependent nature of the * Corresponding author. Fax: +61-3-9251-7243. E-mail addresses: [email protected] (J. Batten), [email protected] (W. Hogan), [email protected] (S. PynnoÈnen) 1 Fax: +61-2-9281-0364. 2 Fax: +358-6-324-8557. 1057-5219/00/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 0 ) 0 0 0 3 6 - 3
390
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
price adjustment process. Other studies investigate the cointegration relationships in offshore markets such as the Hiraki et al. (1996) study on yen offshore rates. These studies are usually undertaken in the context of the expectations hypothesis (Hall et al., 1992) where, assuming stationary risk premiums, all bond yields, whether Eurobond or Government bonds, should be cointegrated. The objective of this study is to investigate these empirical relationships in the context of the Australian dollar denominated Eurobond and Australian Government bond markets. Specifically, we investigate the cointegration properties, from January 1986 to May 1998, between AAA-, AA- and A-rated3 Australian dollar Eurobonds with maturities of 2, 5, 7, and 10 years. The Australian dollar Eurobond market is small, accounting for only US$36.9 billion of the estimated outstanding Eurobonds of US$3657.3 billion in June 1998 (Bank of International Settlements, 1998). However, it is the eleventh most frequent currency of issue.4 Understanding the long-term equilibrium relationship between different credit classes and maturities of Australian bonds is important for four reasons. First, to price risky bonds, general market practice is to apply a yield spread to a risk-free bond of a specified maturity. This practice assumes an equilibrium relationship between different credit classes of bonds, with pricing based on a linear, but not necessarily constant, relationship between the risk free and risky interest rate for the specified maturity. In addition, derivatives based upon the prices of Australian Government securities serve as the benchmark for hedging the interest rate risk of portfolios of bonds in other credit classes. Second, both Australia and the US specify an overnight rate as the target rate for monetary policy purposes with intervention to ensure that the actual market rate does not deviate too far from the policy rate (Cohen, 1999). While there is a need to establish the interest rate equilibrium relationship to appreciate the volatility transmission mechanism between bonds of short- and long-term maturity, there is also a need to establish the relationship between bonds of different credit class. Specifically, cointegration analysis enables the construction of impulse response functions that describe the interest rate adjustment process between pairs, or groups, of bonds with the same maturity but different credit class. Third, the theoretical models of the pricing of corporate debt, such as Merton (1974), Longstaff and Schwartz (1995), and Jarrow et al. (1997), predict an inverse relationship between a change in the risk free rate and the credit spread on risky debt when markets are in equilibrium. As noted by Morris et al. (1999), these models do not specify the dynamics of the price adjustment process, or the time period over which a new equilibrium is established. 3
Bond credit rating agencies categorise bond issuers into nine major classes according to perceived credit quality. These ratings classes include investment grade issuers: AAA, AA, A, and BBB; and non-investment grades: BB, B, CCC, CC, and C. Bonds with ratings below C are bonds in default or of bankrupts. The two major agencies use slightly different notation to refer to equivalent credit risk categories. Standard and Poor use uppercase capitals (e.g., AAA), while Moody's Investor Services use an uppercase first character and have any remaining characters lower case (e.g., Aaa). This paper uses the Standard and Poor's notation. 4 The USD is the most frequent currency of issue with US$1673.4 billion, then the yen with US$407.1 billion, the Deutsche mark with US$369.4 billion, the pound with US$308.3 billion, the French franc US$191.3 billion, the Swiss franc US$141.5 billion, the Italian lire US117.9 billion, the Dutch guilder US$105.3 billion, the ECU US$99.3 billion, and finally the Luxembourg franc with US$37.8 billion in outstandings (Bank for International Settlements, 1998, Table 13B).
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
391
These models also predict that high-grade issuers face upward sloping credit yield curves (that is the spread over the risk free asset rises with maturity), whereas speculative firms face the reverse situation (downward sloping, or hump shaped but downward sloping curves). This suggests that the impact of changes in monetary policy will be experienced differently by different risk classes of borrowers. These different experiences of yield curve adjustment present important risk management implications for banks holding portfolios of risky assets. For example, simple value-at-risk models assume parallel yield curve shifts across all credit classes. Finally, for credit spread traders and sellers of credit spread options, it is important to determine the interday time-series properties of the data to ensure pricing decisions are made correctly. Risk emerges from the unpredictable component (innovation) of the changes in interday credit spreads. This is the risk component that traders need to manage. The unpredictable component is found after filtering all systematic interday dependencies. Cointegration analysis allows the identification of the unpredictable component in a correctly specified form and provides a model for the mean structure of the yield spreads, moments of distributions of the innovations and evidence of normality. The paper is structured as follows. Section 2 presents a theoretical framework implied by the term structure of interest rates for modelling the properties of credit spreads is developed. The data and the time-series properties of the Australian dollar Eurobond yield data are then described in section 3. This section contains descriptive statistics and correlation relationships. Since the series are all highly correlated we then investigate the linear relationship between different classes of bonds predicted by the expectations hypothesis using regression techniques. However, these simple equations fail to capture the complexity arising from cointegrated systems. Then in section 4, the cointegration analysis using the Johansen (1988) procedure is determined and a detailed example employing an error correction model for the 2-year AAA and Australian Government bond is developed. Section 5 allows for concluding remarks.
2. Theory of the term-structure of interest rates of risky bonds Denote the continuously compounded yield to maturity of a k-period pure discount bond as R*(k,t), (k = 1, 2,. . .) and let F(k,t) be the forward rate of a contract at time t to buy a oneperiod discount bond that matures at t + k. Then by Fisher±Hicks, k 1X F j; t; f or k 1; 2; 3; . . . 1 R k; t k j1 Consequently, F(1,t) = R*(1,t), but, generally, the forward rates F( j,t) differ from the future yields R*(1,t + jÿ1), and can be rather understood as rational expectations of the future one-period yields given the current information. Thus, we can write: F j; t Et R 1; t j ÿ 1 a j; t
2
where Et is the conditional expectation given information available at time t and
392
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
a( j,t), with a(1,t) = 0, is the premium accounting for possible risk about the future (Hall et al., 1992). Substituting Eq. (2) in Eq. (1) provides a general relationship between maturities of pure discount bonds as: k k 1X 1X Et R 1; t j ÿ 1 A k; t with A k; t a j; t: 3 R k; t k j1 k j1 Traditional theories of the term structure focus on the properties of the premiums A(k,t). The pure expectation hypothesis asserts that A(k,t) are zero and the expectation hypothesis that the premiums are constants. In the case of corporate bonds, there is a risk premium over an otherwise similar government bond. For a corporate bond, there is a risk that the coupon or principal payment may not be met, implying a distinction between the promised and expected return on the bond. Because of this uncertainty investors demand a risk premium covering the promised and expected return. Let D(k,t) denote the risk premium for a corporate bond over an otherwise similar government bond, then from Eq. (3), we get the following model for the yield of a corporate bond: R k; t R k; t D k; t:
4
Thus, using Eq. (4), we can write: R k; t ÿ R 1; t R k; t ÿ R 1; t D k; t ÿ D 1; t k 1X Et R 1; t j ÿ 1 ÿ Et R 1; t A k; t D k; t k j1
ÿ D 1; t j k ÿ1 X 1X k
Et DR 1; t i A k; t D k; t ÿ D 1; t
5
j1 i1
where DR*(1,t + i) = R*(1,t + i)ÿR*(1,t + iÿ1). Given that R*(1,t + i) is I(1) (integrated of order one), the differences DR*(1,t + i) are therefore stationary implying that the (double) summation term in the last line of Eq. (5) is stationary. Therefore, the important result predicted from the theory is that the yield spread R(k,t)ÿR(1,t) is stationary since the maturity premiums A(k,t) and default risk premiums D(k,t)ÿD(1,t) are stationary.
3. Australian dollar Eurobond data To investigate the dynamics of Australian dollar Eurobond and Government bond markets, we investigate the yields and the spreads, representing credit risk premiums, between different risk classes of aggregated Australian dollar Eurobonds and Australian Government bonds of equivalent maturity. For simplicity, this investigation excludes bonds with embedded options such as callable, puttable or convertible bonds. Daily data of three
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
393
different rating classes of Australian dollar Eurobond (non-government) bonds (AAA, AA, A), and with four different maturities (2, 5, 7, and 10 years) from 2 January 1995 to 31 August 1998 (954 observations) were investigated. This period was characterised by a substantial fall in bond yields from near 11% to 6% for AAA-rated bonds. The data5 was collected as a weighted-average daily yield for each credit rating across three or four industries. These included Industrial, Telecommunications, Banking and Finance, Utilities, and Cable and Broadcasting. There are limitations to this approach since both Longstaff and Schwartz (1995) and Duffee (1996) observe crosssectional variation in spreads across industry. However, both studies rely on industry aggregates with a mix of maturities, so some of the variation observed may have been introduced through differences in the average maturity of bonds issued by the different industry sectors. On the other hand, not averaging the yields introduces concerns over the liquidity of the bonds of some industry sectors given that the majority of bonds were issued by the banking and finance industries. The yields were collected from the credit-spread pages provided by Reuters' Information Services and were based on secondary market bond prices at 4 PM London time. The yields for the benchmark maturities for the different rated bonds were calculated from bond yields from various maturities and interpolated into a benchmark set of yields (t = 2, 5, 7, 10 years) using cubic-splinning techniques. Cubic splinning produces a zero curve, which is both smooth in its first derivation and continuous in the second-order derivation. The values of the second derivations can be solved using a set of tri-diagonal equations that enable the calculation of any interpolated value of t. In this instance, interpolated values of t = 2, 5, 7 and 10 were determined.6 The maturity structure of the different rated Australian dollar Eurobonds on the final day of the sample (31 August 1998) was obtained from the Reuters' Fixed Income Database, which comprises outstanding fixed-rate (zero-coupon and floating rate notes excluded) Eurobonds. We estimate this database captured about 90% of the US$36.9 billion in outstanding Australian dollar Eurobonds reported to the Bank for International Settlements at the 30 June 1998. We suggest that a large part of the remaining bonds were private issues and therefore not traded in secondary markets. Analysis of this data suggests, first, the Australian Government bond market, with outstandings of about A$67 billion, is larger (by about 30%) than the Australian dollar Eurobond market (with outstandings of about A$50.4 billion). Second, AAA-rated bonds have the largest outstandings (about 76%) and single A grade issues the least (4%). There were no fixed-rate issues for issuers with ratings below A (e.g., BBB and non-investment grade issuers), though there were a few floating rate issues for issuers with ratings below A. The distribution of Australian Government bonds was generally evenly spread across maturities from 0 to 10 years, though there were only 5.5% of issues with maturities 5
We thank Westpac Banking, Sydney Australia for providing the bond yield data. Cubic-splinning involves the following approach: Consider a zero-coupon rate Zi on a zero-coupon curve Z(ti) such that i = 1, 2, 3, . . ., n, then for the set t2(titi + 1), Z(t) = a + bt + gt2 + dt3 so that Z(ti) = Zi and Z(ti + 1) = Zi + 1, with Z(ti) = Ziand Z(ti + 1) = Zi + 1. Z(t) can now be expressed as follows: Z(t) = aZi + bZ(ti + 1) + cZi + dZ(ti + 1) where a = (ti + 1ÿt)/h, b = (tÿti)/h, c = 1/6(a3ÿa)h2, d = 1/6(b3ÿb)h2, and h = ti + 1ÿti. 6
394
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
greater than 10 years. The even spread of Australian Government bond issues is a conscious decision on the part of the authorities to maintain liquidity across the benchmark yield curve. By comparison, the greatest amount of outstandings in the Australian dollar Eurobond market is for maturities from 0 to 5 years (73% of total issues), though these issues largely carry AAA ratings (54% of total issues). This suggests there is very little liquidity in Australian dollar Eurobonds with longer maturities than 5 years and lesser credit ratings than AAA. 4. Empirical results The pricing of credit may be perceived to be a function of a number of factors. These include the market microstructure features within specific markets (e.g., regulatory differences including withholding taxes on non-resident income, which raise the cost of certain types of funds) and between markets (e.g., higher yields in domestic markets to embody sovereign risk), expectations by market participants, portfolio factors that provide diversification benefits to the investor, and, finally, credit risk perceived in terms of traditional statistical measures including yield volatility, correlation with other risky assets, default probabilities, and recovery rates. These last measures require sophisticated models and quality historical data to maintain their integrity. All these factors may be embodied in the spread between classes of different types of assets. For example, the Longstaff and Schwartz (1995) model for valuing the spread of corporate bonds considers a two-factor model that captures interest rate and asset default factors. When bonds are marketable, then the prices, yields, and default histories may be observed in secondary markets. Otherwise, this type of information will remain the proprietary information of the lending institution. For spread traders and sellers of credit spread options, it is important to determine the distributional and time-series qualities of the data to ensure pricing decisions are made correctly (see Batten & Hogan, 1999 for an analysis of credit derivatives in the Australian context). 4.1. Descriptive statistics Analysis of the bond yields initially involved determining the descriptive statistics and correlation relationships between the four different classes and maturities of bonds. These credit spreads represent the premium for credit risk between each risk group and maturity. Batten et al. (2000) provide a more detailed account of the time-series properties of the various Australian bond series. The data displayed a definite positively sloped yield curve (term structure) over the sample period, with the shorter-term maturities having a lower yield to the longer-term maturities at any one point in time. For example, AAA bonds had a 2-year yield mean of 7.0714% while the 10-year AAA yield mean was 8.0458% during the sample period. There was also evidence of a credit risk premium with the yields on the higher quality bonds for a specific maturity being lower than the yields for the equivalent maturity lower-rated bonds. This relationship between bond yields and default risk is referred to as the default-risk structure of
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
395
Table 1 Moments of the daily credit spreads of Australian dollar Eurobonds, 2 January 1995 to 31 August 1998 (expressed in basis points) Credit spread N = 954 in basis points
Mean
Standard deviation
Skewness
Excess kurtosis
AAA ± AA2 AAA ± AA5 AAA ± AA7 AAA ± AA10 AA ± A2 AA ± A5 AA ± A7 AA ± A10
7.622 6.904 9.484 11.022 13.908 15.865 12.724 11.923
5.177 3.652 5.756 8.427 5.285 7.128 8.433 11.157
1.5374 0.7922 0.7815 1.2703 ÿ0.0430 0.6270 1.3703 ÿ1.3983
2.4980 0.6103 0.1300 1.9012 ÿ0.2000 0.6373 4.0044 14.600
This table records the four moments of the daily credit spreads (AAA ± AA and AA ± A) between the different aggregated bond ratings with 2-, 5-, 7-, and 10-year maturities. The mean and standard deviation of the AAA ± AA bond spreads generally rose with increasing maturity (a positive credit term structure), while the mean spreads on the lower quality bonds generally fell with increasing maturity. The series were not significantly skewed though the AAA ± AA2, AAA ± AA10, AA ± A7, and AA ± A10 display significant excess kurtosis (leptokurtic). Note the AAA ± AA series suggests an upward sloping credit yield curve (i.e., 2-year spreads < 10-year spreads), while the AA ± A is a humped, downward sloping credit yield curve (i.e., 2-year spreads > 10-year spreads).
interest rates. For example, AAA 5-year bonds had a lower yield mean of 7.6125% compared with the A-rated 5-year bonds of 7.8402%.7 Table 1 provides descriptive statistics of the credit spreads between the different rating classes and maturities of bonds. These credit spreads represent the premium for credit risk between each group and maturity. For example, the variable AAA±AA2 with the mean result of 7.622 basis points (0.07622%) represents the mean margin between the AAA and AA 2year bonds over the sample period. Generally, the mean credit spreads over the sample period provide evidence in support of the theoretical behaviour suggested by Merton (1974). That is, the better quality credit AAA±AA spreads increased as maturity increased (an upward sloping credit yield curve), while the AA±A spreads decreased with an increase in maturity (a downward sloping curve). Both curves were humped around the 5-year maturity, but in different directions. The AAA±AA spread declined from 7.622 basis points (2 years) to 6.904 basis points (5 years), while the lower quality AA±A spread rose from 13.908 basis points (2 year) to 15.865 basis points (5 years). This may have been due to the greater liquidity in the AAA and AA 5-year bond market. The credit spreads for bonds with maturities greater than 5 years were better behaved. The mean credit spreads between the AAA to AA 5-year and the AAA to AA 7-year bonds increased from 6.904 to 9.484 basis points as maturity increased, while the AA to A spread declined from15.865 basis points for 5 year to 12.724 basis points for 7-year bonds. However, as the test results reported in Table 2
7
The behaviour of US credit risk premiums is discussed in Neal (1996).
396
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
Table 2 Augmented Dickey ± Fuller (ADF) and Phillip ± Perron tests for unit roots of Australian dollar Eurobonds and Government bonds, 2 January 1995 to 31 August 1998 (expressed in basis points) Series
ADF
PP
ADF trend
PP trend
First differences
ADF
PP
G2 G5 G7 G10 A2 A5 A7 A10 AA2 AA5 AA7 AA10 AAA2 AAA5 AAA7 AAA10
ÿ3.01 ÿ2.40 ÿ2.44 ÿ2.51 ÿ2.42 ÿ2.02 ÿ1.71 ÿ1.13 ÿ2.41 ÿ1.99 ÿ1.69 ÿ1.36 ÿ2.46 ÿ2.01 ÿ1.75 ÿ1.18
ÿ3.00 ÿ2.17 ÿ2.33 ÿ2.38 ÿ2.10 ÿ1.88 ÿ1.60 ÿ1.01 ÿ2.10 ÿ1.80 ÿ1.58 ÿ1.28 ÿ2.15 ÿ1.81 ÿ1.55 ÿ1.15
ÿ3.01 ÿ2.54 ÿ2.58 ÿ2.53 ÿ2.41 ÿ2.44 ÿ2.75 ÿ3.35 ÿ2.22 ÿ2.31 ÿ2.44 ÿ2.61 ÿ2.27 ÿ2.34 ÿ2.42 ÿ2.88
ÿ2.96 ÿ2.30 ÿ2.44 ÿ2.35 ÿ2.02 ÿ2.04 ÿ2.54 ÿ3.08 ÿ1.90 ÿ2.00 ÿ2.22 ÿ2.41 ÿ2.04 ÿ2.11 ÿ2.22 ÿ2.86
DG2 DG5 DG7 DG10 DA2 DA5 DA7 DA10 DAA2 DAA5 DAA7 DAA10 DAAA2 DAAA5 DAAA7 DAAA10
ÿ15.0 ÿ14.8 ÿ15.1 ÿ15.1 ÿ13.1 ÿ14.2 ÿ13.5 ÿ14.6 ÿ13.3 ÿ13.8 ÿ13.6 ÿ14.2 ÿ13.0 ÿ13.6 ÿ13.6 ÿ13.7
ÿ30.4 ÿ29.9 ÿ28.3 ÿ30.5 ÿ27.9 ÿ29.4 ÿ30.8 ÿ30.6 ÿ27.0 ÿ28.1 ÿ28.9 ÿ30.4 ÿ27.4 ÿ28.3 ÿ28.5 ÿ29.5
MacKinnon critical values for rejection of hypothesis of a unit root 1% 5% 10%
ÿ3.44 ÿ2.87 ÿ2.57
ÿ3.97 ÿ3.42 ÿ3.13
This table provides ADF and Phillips ± Perron unit root test of the Government bonds and three different ratings of Eurobonds with four maturities (2, 5, 7, and 10 years). Four lags for ADF and six lags for PP as suggested by Newey ± West. Tests for the levels include intercept (second and third columns) and intercept and trend (fourth and fifth columns). First differences include the intercept.
indicate, the yields are integrated with order one (i.e., they are I(1) processes) implying that they may wander as time goes by so that averages may not satisfactorily characterise the relationships between the yields unless the series are cointegrated. 4.2. Cointegration analysis between Australian dollar Eurobonds and Government bonds Given that the risk premiums, D(k,t), defined in section 2, can be assumed stationary, theory predicts that the series should be cointegrated with a cointegration vector (1,ÿ1). Generally, the cointegration analysis can be undertaken using a vector error correction (VEC) model of the form: 6 DYt m PYtÿ1 G1 DYtÿ1 : : : Gpÿ1 DYtÿp1 et ; where Yt in our case is a vector with two components, P = ab0 with prime denoting the transpose, and where: b0 b1 ; b2
7
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
397
is the cointegration vector in a two-vector system [Eq. (7)] and A = (a1,a2)0 contains the short-term adjusting parameters that drive the system back to the long-term equilibrium, D stands for the difference, Gi (i = 1, . . ., pÿ1) contains the short-term autoregressive parameters, and Et is the error (innovation) vector, assumed to be normally distributed. The intercept vector M imposes a non-zero drift in yields, which is usually not a reasonable model for interest rate behaviour. Consequently, a common practice is to restrict M to the cointegration space by augmenting Ytÿ1 to tÿ1 = (1,Y0tÿ1)0 Ä and the cointegration vector as b0 = (b0,b1,b2). Table 3 reports the unrestricted cointegration tests between the Australian dollar Eurobonds and the equivalent maturity Government bonds accompanied by tests of stationarity of the spreads. Non-stationarity of the spreads suggests the spreads follow I(1) processes, or equivalently, that the stochastic trends which play the dominating role in the behaviour of the yields are independent of one another. The five lags for parameter p in Eq. (6) proved to be sufficient to ensure that the residuals were not serially correlated. Surprisingly, the results
Table 3 Cointegration tests of Australian dollar Eurobonds with equivalent maturity Australian Government bonds, 2 January 1995 to 31 August 1998 Unit root tests of the spreads Likelihood ratio AAA2 ± G2 AAA5 ± G5 AAA7 ± G7 AAA10 ± G10 AA2 ± G2 AA5 ± G5 AA7 ± G7 AA10 ± G10 A2 ± G2 A5 ± G5 A7 ± G7 A10 ± G10
18.3 13.7 11.3 8.7 17.0 13.5 10.5 9.1 16.7 11.9 9.6 8.1
a
ADFb ÿ1.41 ÿ1.28 ÿ1.20 ÿ0.75 ÿ1.40 ÿ1.26 ÿ1.15 ÿ0.85 ÿ1.44 ÿ1.28 ÿ1.19 ÿ0.72
PPc ÿ1.22 ÿ1.09 ÿ0.93 ÿ0.47 ÿ1.18 ÿ1.07 ÿ0.92 ÿ0.57 ÿ1.22 ÿ1.09 ÿ0.98 ÿ0.35
Critical values for the likelihood ratiod
Critical values of ADF and PP tests
10% 5% 1%
10% 5% 1%
17.9 20.0 24.6
ÿ2.57 ÿ2.86 ÿ3.44
The table reports the results of Johansen's cointegration tests and stationarity of the credit spreads of AAA-, AA- and A-rated Eurobonds over equivalent maturity Government bonds. The stationarity of the credit spread would imply that the mean spread would serve as the common long-term equilibrium between the yields of the two bonds. a The null hypothesis is that the series are not cointegrated. b ADF test with four difference lags and intercept. c PP test with six lags and intercept (number of lags were defined as suggested by the Newey ± West procedure). d The intercept term is restricted into the cointegration space.
398
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
strongly indicate that the Eurobonds are not cointegrated with the Government bonds. This also means that the risk spreads, D(k,t) defined in Eq. (4), should be I(1) series implying that the Government bonds and the Eurobonds do not have any ties with one another and the two-bond markets are effectively independent in terms of pricing. This would be expected to be implausible in the long run, and this result may be due to the specific sample period with bond yields, especially the Eurobond yields, having fallen over the whole sample period. Analysis of a longer sample period may, however, shed further insights on this issue. Hence, these results challenging as they are should be treated with some caution. 4.3. Cointegration of Australian dollar Eurobonds with different risk classes We further investigate the long-term equilibrium relationships of the Australian dollar bonds of different credit class by investigating the relationships between the yields of the Eurobonds independently of the Government bonds. In principle, it is possible to use a multivariate (three-variate) test to include all the three ratings, however, we present only the bivariate test results since the three-variate residuals exhibited strong volatility clustering. Table 4 reports the cointegration tests in pairs of the same maturity Eurobonds. According to the unrestricted likelihood-tests, all the series should be cointegrated. The theory in section Table 4 Cointegration and restricted cointegration tests between Australian dollar Eurobonds with different ratings, 2 January 1995 to 31 August 1998 H0: (b1,b2) = (1,ÿ1) Likelihood ratio A2 ± AAA2 A5 ± AAA5 A7 ± AAA7 A10 ± AAA10 AA2 ± AAA2 AA5 ± AAA5 AA7 ± AAA7 AA10 ± AAA10 A2 ± AA2 A5 ± AA5 A7 ± AA7 A10 ± AA10
a
42.3 30.4 33.1 35.8 44.2 63.5 37.2 39.3 75.9 35.5 44.5 31.5
c2 (df = 1)
p Value
c2 (df = 1)b
p Value
17.0 0.03 0.4 1.8 9.7 6.7 11.2 5.2 20.9 0.2 2.4 0.2
0.000 0.855 0.530 0.186 0.002 0.010 0.001 0.022 0.000 0.686 0.124 0.690
18.8 0.2 0.9 0.6 8.8 5.2 10.7 4.4 17.2 0.2 2.4 0.003
0.000 0.643 0.341 0.440 0.003 0.022 0.001 0.036 0.000 0.665 0.124 0.954
Critical values for the likelihood ratio 10% 17.9 5% 20.0 1% 24.6 This table records the unrestricted cointegration tests in column 2 followed by testing whether the yields of the two bonds are independent. The variables are the daily yields with 2-, 5-, 7-, and 10-year maturities. a The null hypothesis is that the series are not cointegrated. b Test with constant correlation multivariate GARCH(1,1) errors as defined by Eq. (8)
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
399
2 predicts that the spread should establish the cointegration relationship, or, equivalently, that the spread is stationary, and, hence, independent of the yield levels. Columns 3 through 6 of Table 4 report the likelihood ratio tests for this hypothesis. The hypothesis is supported between AAA and A, as well as AA and A bonds in all but the 2-year maturity, whereas between AAA and AA bonds the hypothesis is rejected in each case. The numerical values of the coefficients do not depart much from unity, that is, they are all similar to those reported for the A±AAA2 bond in Table 6 below. The above test results do not change materially whether the models are estimated under a constant (conditional) residual variance, or under a bivariate GARCH(1,1) specification of the form: h11;t w10 a11 e21;tÿ1 a12 e22;tÿ1 b1 h11;tÿ1 h22;t w20 a21 e21;tÿ1 a22 e22;tÿ1 b2 h22;tÿ1 pp h21;t r h11;t h22;t
8
which captures the volatility clustering. In Eq. (8), hij,t are the conditional variances and Table 5 Weak exogeneity tests between different ratings of Australian dollar Eurobonds, 2 January 1995 to 31 August 1998 AAA 2
A2 ± AAA2 A5 ± AAA5 A7 ± AAA7 A10 ± AAA10
A
c (df = 1)
p Value
c2 (df = 1)
p Value
1.80 0.30 0.13 0.52
0.180 0.584 0.715 0.472
14.37 7.55 9.78 9.11
0.000 0.006 0.002 0.003
AAA 2
AA2 ± AAA2 AA5 ± AAA5 AA7 ± AAA7 AA10 ± AAA10
AA
c (df = 1)
p Value
0.21 0.16 0.97 3.04
0.644 0.693 0.326 0.082
AA 2
A2 ± AA2 A5 ± AA5 A7 ± AA7 A10 ± AA10
c2 (df = 1) 4.92 4.30 0.82 1.54
p Value 0.027 0.038 0.365 0.215
A
c (df = 1)
p Value
c2 (df = 1)
p Value
0.05 0.03 0.46 0.20
0.823 0.866 0.500 0.653
13.51 5.71 15.84 10.41
0.000 0.017 0.000 0.001
Large p values in the table indicate that the corresponding series is weakly exogenous with respect to the longrun equilibrium relation between the bond yields. In such a case, the other bond corrects its behaviour according to the changes in the yield of the weakly exogenous bond. That is to say the latter one is driving the movements of the bond yields.
400
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
Table 6 Equilibrium Correction Model for the Australian dollar A ± AAA Eurobond yields of different maturities, 2 January 1995 to 31 August 1998 A ± AAA2 Dx = DAAA2 xtÿ1 s.e. p Value Intercept s.e. p Value citÿ1 s.e. p Value Dxtÿ1 s.e. p Value Dytÿ1 s.e. p Value w s.e. p Value e2x,tÿ1 s.e. p Value e2y,tÿ1 s.e. p Value htÿ1 s.e. p Value r s.e. p Value
A ± AAA5 Dy = DA2
1.041 0.004 0.000 ÿ0.005 0.002 0.006
ÿ0.018 0.006 0.001 ÿ0.170 0.020 0.000 0.030 0.229 0.063 0.066 0.632 0.001 0.123 ÿ0.081 0.060 0.062 0.040 0.186 0.00037 0.00043 0.00008 0.00008 0.000 0.000 0.042 0.027 0.013 0.013 0.001 0.031 0.022 0.038 0.011 0.014 0.057 0.008 0.844 0.829 0.024 0.027 0.000 0.000 0.866 0.006 0.000
Diagnostic statistics Log-lik 743.4 Skew z ÿ0.140 p Value 0.079 Kurt z 3.77 p Value 0.000 Q(5) z 9.7 p Value 0.085 Q(10) z 11.0 p Value 0.354 Q(5) z2 2.4 p Value 0.790
ÿ0.246 0.002 4.12 0.000 6.4 0.267 15.7 0.109 4.1 0.536
Dx = DAAA5
A ± AAA7 Dy = DA5
1 ± ± ÿ0.005 0.002 0.031
0.017 0.004 0.000 ÿ0.093 0.014 0.000 0.055 0.233 0.058 0.066 0.341 0.000 0.059 ÿ0.110 0.055 0.060 0.282 0.066 0.00032 0.00031 0.00005 0.00005 0.000 0.000 0.061 0.046 0.014 0.008 0.000 0.000 0.016 0.016 0.011 0.007 0.143 0.013 0.852 0.875 0.021 0.016 0.000 0.000 0.838 0.007 0.000
656.5 0.005 0.953 4.73 0.000 10.2 0.071 12.5 0.253 2.5 0.777
0.169 0.034 4.12 0.000 4.2 0.527 11.7 0.303 4.2 0.520
Dx = DAAA7
A ± AAA10 Dy = DA7
1 ± ± ÿ0.006 0.002 0.004
0.012 0.003 0.000 ÿ0.088 0.010 0.000 0.138 0.233 0.051 0.046 0.007 0.000 ÿ0.021 ÿ0.156 0.047 0.056 0.659 0.005 0.00028 0.00022 0.00005 0.00003 0.000 0.000 0.067 0.019 0.010 0.006 0.000 0.001 0.012 0.063 0.007 0.009 0.082 0.000 0.877 0.887 0.009 0.008 0.000 0.000 0.754 0.009 0.000 565.5 0.144 0.069 3.71 0.000 8.8 0.116 13.9 0.176 2.3 0.803
0.308 0.000 5.61 0.000 6.3 0.277 12.5 0.253 9.8 0.081
Dx = DAAA10
Dy = DA10
1 ± ± ÿ0.008 0.002 0.001
0.010 0.003 0.000 ÿ0.077 0.007 0.000 ÿ0.034 0.006 0.033 0.012 0.306 0.624 0.082 0.029 0.038 0.035 0.033 0.407 0.00073 0.00017 0.00012 0.00002 0.000 0.000 0.083 ÿ0.007 0.010 0.001 0.000 0.000 0.016 0.080 0.009 0.007 0.069 0.000 0.754 0.893 0.034 0.009 0.000 0.000 0.697 0.010 0.000 533.3 ÿ0.151 0.058 5.39 0.000 9.7 0.085 17.7 0.061 4.2 0.515
ÿ0.092 0.249 5.25 0.000 7.8 0.167 19.6 0.034 1.4 0.927
(continued on next page)
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
401
Table 6 (continued ) A ± AAA2
Q(10) z2 p Value Corr(hx,hy)
A ± AAA5
A ± AAA7
A ± AAA10
Dx = DAAA2
Dy = DA2
Dx = DAAA5
Dy = DA5
Dx = DAAA7
Dy = DA7
Dx = DAAA10
Dy = DA10
5.1 0.885 0.9949
11.2 0.343
11.8 0.300 0.9958
30.5 0.001
7.0 0.725 0.9142
16.0 0.100
7.1 0.713 0.506
2.4 0.992
Rows 3 ± 6 report the estimates of the cointegration regression yt = b0 + b1xt + ut for the 2-year bond maturity with x denoting AAA2 and y A2. Relaxing the intercept term b0 freely estimated as parameters g1 and g2 in the VEC model (4) the cointegration relation becomes cit = ytÿb1xt with the parameter replaced by the estimate. For the longer maturity bond, the cointegration relationship is according to the results reported in Table 4 just the spread, i.e., cit = ytÿxt. In the estimation, also, the weak exogeneity results reported in Table 5 are utilised by restricting the coefficient of ci in the equation of Dx equal to zero. Conditional heteroskedasticity in the cointegration regression is of the form given by Eq. (10).
covariance of the residuals and r is the contemporaneous correlation of the residual series (volatility processes are considered more carefully below). Once we have found that the Australian dollar Eurobonds with different ratings are cointegrated an interesting question is then whether one of the bonds is driving the other. This is the case if one of the yields is not dependent on departures from the longrun equilibrium determined by the cointegration relationship (the series is then said to be weakly exogenous). In terms of model (6), this means that in the alpha-matrix, the corresponding row is equal to zero. Table 5 provides the test results. The results are clear-cut between AAA and A, as well as AA and A bonds, where the higher quality bond is in each case weakly exogenous. The relationship between AAA and AA is not that evident. At the short end the AAA bond seems to drive AA yields, but in the long end, the results are mixed. The overall result, however, is that at least in the short end, the AAA bonds seem to drive the yields of lower quality bonds. These results can be utilised in modelling the dynamics of spreads between Eurobonds and Government bonds. However, before undertaking this exercise, we must more closely consider the bivariate equilibrium model with conditional heteroskedastic yield innovations. 4.4. Equilibrium correction analysis of Australian dollar Eurobond yields Let x and y denote yields of Eurobonds in different credit classes. Then the cointegration of x and y implies an equilibrium correction (or error correction) model for the first differences of the variables as dxt g1 a1 citÿ1 lags dxt ; dyt eyt dyt g2 a2 citÿ1 lags dxt ; dyt ext
9
where ext and eyt are unpredictable innovation terms modelled by the conditional heteroskedasticity given by Eq. (8). As was noted previously, it is usually not feasible to include intercept terms (here g1 and g2) into the model for interest rates due to drifts in the
402
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
series. Nevertheless, in this particular case, the yields seem to exhibit trends in the sample period, so we include the drift terms as dummy parameters to capture these trends. Table 6 reports the estimation results for the A±AAA bonds, where we have estimated appropriate restricted models to utilise the weak exogeneity results presented in Table 5. The model with a one lag in differences (and bivariate GARCH(1,1) residuals) adequately captured serial dependencies in the yields. The negative equilibrium correction coefficients, reported on the ci-line, indicate the reverting behaviour once the A bond yield falls out of the equilibrium with respect to the AAA bond. In addition, the changes of A bond yields respond positively to the previous day changes of the AAA bond (denoted in the Table 6 by Dxtÿ1). Thus, the higher quality bonds seem to lead to the lower quality bonds, and there is a return spillover from the high quality bonds to the lower quality bonds such that a change in the higher quality bond yield predicts a change in the next days lower quality bond yield. Table 6 also shows the bivariate GARCH(1,1) estimation results. It is interesting to note the asymmetry in the volatility shocks. That is, shocks to the yields of the AAA bond (denoted by ex,t) are statistically more significant in the A-rated bond volatility equation, whereas A bond shocks (ey,t) do not affect the AAA bond volatility. This indicates that in addition to the yield dependency the AAA bonds are partially leading the volatility behaviour of the A-rated bonds. Finally, it is worth noting from Table 6 that the yield changes are highly correlated with the correlation ranging from 0.7 to 0.9. In addition, the last line of Table 6 indicates that the conditional volatility series are also very highly correlated. 4.5. An example: the dynamics of daily Australian dollar Eurobond credit spreads As was seen earlier, credit spreads between Australian dollar Eurobonds and Government bonds behave as I(1) series. In modelling the dynamics of the spreads, and also for trading of Eurobonds with different credit ratings, one can utilise the knowledge that the Eurobonds in different credit classes are cointegrated. Supposing further that the first differences of the government bond (denoted by z) can be modelled by an ARIMA model (see Batten et al., 2000), then because the spread st = ytÿzt is I(1), where y denotes the Eurobond, the first differences Dst = stÿstÿ1 are I(0). But we can write Dst = DytÿDzt. Using Eq. (9) and the ARMA structure of Dzt, we can obtain dst acitÿ1 lags dxt ; dyt ; dzt et ;
10
where et is an innovation process (which can follow a conditional heteroskedastic process). This model should provide a better description of the time-series behaviour of the spread than a simple ARIMA specification. As an example, consider the A2±G2 spread, which is not cointegrated. From Table 6, we can model the spread A2±AAA2 spread as ci = A2ÿ1.041AAA2. For comparison purposes, an AR-model was also estimated. In the estimation, we enhanced Eq. (10) with the intercept and retained only those explanatory series that were statistically significant. The results are reported in Table 7. Only the estimates of the spread and Government bond difference are statistically significant. None of the AR(2) model lags are significant, indicating that the series is in fact a pure innovation process with respect to its own history. Both models
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
403
Table 7 Estimates of the 2-year credit spread of the Australian dollar Eurobond with respect to the equivalent maturity Government bond DA2S
AR(2) model for DA2S
Coefficient
s.e.
t Value
p Value
ÿ1.291 ÿ0.147 0.143
0.407 0.044 0.044
ÿ3.172 ÿ3.318 3.279
0.002 0.001 0.001
GARCH estimates Constant 7.023 e2tÿ1 0.111 htÿ1 0.791
2.045 0.023 0.045
3.434 4.900 17.657
0.001 0.000 0.000
Diagnostic statistics Estimate
p Value
Estimate
p Value
0.000 0.000 0.086 0.178 0.457 0.658
0.001 ÿ0.004 ÿ3342.6 4.260 4.290 0.183 4.985 7.2 12.4 4.1 7.1
0.000 0.000 0.208 0.256 0.532 0.716
Intercept citÿ1 Dg2tÿ1
2
R Adj R2 Log-lik AIC SBC Skew z Kurt z Q(5) z Q(10) z Q(5) z2 Q(10) z2
0.017 0.012 ÿ3338.6 4.247 4.277 0.190 4.851 9.6 13.9 4.7 7.7
Intercept DA2Stÿ1 DA2Stÿ2
Coefficient
s.e.
t Value
p Value
ÿ0.286 ÿ0.023 ÿ0.058
0.245 0.038 0.034
ÿ1.166 ÿ0.598 ÿ1.720
0.244 0.550 0.086
7.443 0.107 0.790
2.155 0.022 0.045
3.453 4.955 17.514
0.001 0.000 0.000
The cointegration relation is taken according to the test results of Table 4 as the estimated cointegration regression given in Table 6 such that A2 = 1.041AAA2, so that ci = A2ÿ1.041AAA2. In the final estimation, the units of all variables are in basis points.
indicate that there is strong conditional heteroskedasticity in the residuals, and the Q-statistic on squared standardised residuals suggests that a GARCH(1,1) model fully captures the volatility clustering. 5. Conclusion Central to the management or trading of credit risk in its various forms by financial market participants, including recent innovations utilising credit derivatives is an appreciation of the distributional qualities and the long-term equilibrium relationships of the spreads between classes of risky bonds, and the processes generating the spreads. This contribution focused on the dynamics of Australian dollar Eurobond and Government yields and highlighted the cointegration relationships between the yields. The yield and yield spreads between government bonds and three classes of Australian dollar Eurobonds rated A, AA, and AAA, with four maturities (2, 5, 7, and 10 years) was examined. The empirical results strongly support
404
J. Batten et al. / International Review of Financial Analysis 9 (2000) 389±404
the proposition that yields in different risk classes of Australian dollar Eurobonds are cointegrated. However, the Eurobond yields and the Australian Government bond yields are not cointegrated. Analysis suggests that the higher-rated Eurobond yields tend to lead the lower-rated Eurobond yields. In addition to the mean dynamics, a bivariate GARCH model indicates that there is volatility transmission from higher-rated bonds to lower-rated bonds. The paper also demonstrates that the cointegration relationships can be utilised in modelling the dynamics of the spread changes between Eurobonds and Government bonds. Acknowledgments The authors are grateful for the valuable comments from Palle Anderson on an earlier draft of this paper. References Bank for International Settlements. (1998). The BIS Statistics on International Banking and Financial Market Activity, Monetary and Economic Department. Basel: Bank for International Settlements, November. Batten, J., Ellis, C., & Hogan, W. (2000). The time-series properties of credit spreads: evidence from Australian dollar Eurobonds. In: T. Fetherston (Ed.), Advances in Pacific Basin Financial Markets (Vol. 6, 261 ± 293). Stamford, CT: JAI Press. Batten, J., & Hogan, W. (1999). Credit derivatives: an appraisal for Australian financial institutions. Economic Paper 18(2), 19 ± 42. Cohen, B. (1999). Monetary policy procedures and volatility transmission along the yield curve. BIS Working Papers, Basle, Switzerland: Bank for International Settlements. Duffee, G. (1996). Treasury yields and corporate bond yields: an empirical analysis. Working paper, Washington D.C.: Federal Reserve Board. Hall, A., Anderson, H., & Granger, C. W. (1992). A cointegration analysis of treasury bill yields. The Review of Economics Statistics 74, 116 ± 126. Hiraki, T., Shiraishi, N., & Takezawa, N. (1996). Cointegration, common factors, and the term structure of Yen offshore interest rates. Journal of Fixed Income 6(3), 69 ± 75. Jarrow, R. A., Lando, D., & Turnbull, S. (1997). A Markov model for the term structure of credit risk spreads. The Review of Financial Studies 10, 481 ± 522. Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231 ± 254. Longstaff, F., & Schwartz, E. (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 29, 449 ± 470. Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449 ± 470. Morris, C., Neal, R., Rolph, D. (1999). Credit spreads and interest rates. Working Paper, Indiana University, Indianapolis, March. Neal, R. S. (1996). Credit derivatives: new financial instruments for controlling credit risk. Federal Reserve Bank of Kansas City Economic Review 81(2), 15 ± 27. Pederosa, M., & Roll, R. (1998). Systematic risk in corporate bond credit spreads. Journal of Fixed Income 8, 7 ± 26.