Macro Factors in the Term Structure of Credit Spreads

Transcription

1 Macro Factors in the Term Structure of Credit Spreads Jeffery D. Amato Bank for International Settlements Maurizio Luisi ABN AMRO Bank This version: November 2004 Preliminary. Comments welcome. Abstract We estimate arbitrage-free term structure models of US Treasury yields and corporate bond spreads. A novel feature of our analysis is the inclusion of macroeconomic variables, in addition to latent factors, as drivers of term structure dynamics. Our results point to a statistically significant role for real activity and inflation in determining both Treasury yields and spreads, although most of the time series variation across maturities is driven by the latent factors. We also relate the corporate factor in the model to various measures of financial conditions, and find that a newly constructed composite indicator of financial conditions explains about 70% of the variation in the corporate factor. JEL Classification Numbers: C13, C32, E44, E52, G12, G13, G14 Keywords: default risk, risk premia, interest rate rules, no-arbitrage The views are those of the authors and do not necessarily represent those of the BIS or ABN AMRO Bank. Corresponding author: Jeffery D. Amato, Bank for International Settlements, Monetary and Economic Department, Centralbhanplatz, 2, CH-4002 Basel, Switzerland. address: jeffery.amato@bis.org. ABN-AMRO Bank and University of Lugano (Ph.D candidate). Corresponding address: ABN AMRO Bank, 250 Bishopsgate, London EC2M 4AA, UK. address: maurizio.luisi@uk.abnamro.com.

2 1 Introduction In recent years there has been a significant surge of interest in the modelling and pricing of credit risk. This can be attributed to several factors: the greater role played by financial markets, as opposed to intermediaries, in the contracting of credit; rapid growth in credit derivatives markets; and regulatory developments, notably the recently signed Basel II accord governing bank capital requirements, surpervision and disclosure. It is surprising, then, that relatively little work has been done on the empirical relationship between credit spreads and the macroeconomy. Casual observation points to a close link between credit spreads and the business cycle. For example, the correlation between five-year BBB-rated industrial spreads and real activity at a monthly frequency is But apart from a few studies, the relationship between spreads and macroeconomic variables has largely been left unexamined. Some recent papers, such as Collin-Dufresne, Goldstein and Martin (2001) and Morris, Neal and Rolph (2001), have focused on explaining changes in spreads in a regression framework. By contrast, in this paper we provide new empirical evidence on the role of macroeconomic factors in an arbitrage-free model of the term structure of credit spreads. 1 While the literature on spreads and macroeconomic variables is relatively sparse, more work has been done examining the link between default risk and the macroeconomy. For example, default probabilities depend upon macroeconomic variables in two well-known risk management models, McKinsey s CreditPortfolioView (see Wilson (1997a, 1997b)) and Algorithmic s Mark to Future (see Dembo, Aziz, Rosen and Zerbs (2000)). 2 Cantor and Mann (2003) document the procyclicality of credit quality changes using a long history of Moody s data. Altman, Brady, Resti and Sironi (2003) show that there is a relationship between the correlation of default rates and loss in the event of default and the business cycle. From a theoretical perspective, a systematic relationship between, on the one hand, financial conditions and spreads, and on the 1 This stands in stark contrast to models of the term structure of default-free debt (e.g. US Treasury securities), in which a vast literature has developed over the past few decades. It is impossible to cite all of the relevant contributions. Recent work by Dai and Singleton (2001, 2003) and Piazzesi (2003) provide a good entry point into the literature. 2 For a more complete review of how macroeconomic factors are incorporated into credit risk models, see Allen and Saunders (2002). 1

3 other hand, output and inflation, can be explained by a general equilibrium model with a financial accelerator in investment and nominal price rigidities (Bernanke, Gertler and Gilchrist (1999)). In sum, both the empirical and theoretical literature suggest that real economic activity and inflation have a role to play in determining corporate yields and credit spreads. We specify and estimate multi-factor affine term structure models of Treasury yields and corporate spreads using monthly data for the United States over the period Affine models have been the workhorse in the empirical literature on no-arbitrage term structure models of default-free debt (see Piazzesi (2003) for a recent survey). For corporate spreads we estimate doubly-stochastic intensity-based models. In the light of the poor performance of firm-value (i.e. Merton-type) models in explaining spreads (see, e.g., Huang and Huang (2003) and Eom, Helwege and Huang (2004)), intensitybased models have become increasingly popular for pricing defaultable debt (Jarrow and Turnbull (1995), Lando (1998), Duffie and Singleton (1999)). 3 Following recent work on Treasury yield curves (Ang and Piazzesi (2003), Ang, Piazzesi and Wei (2003) and Dewachter and Lyrio (2003)), we introduce observable macroeconomic variables, along with latent factors, into the state vector of our affine model. In particular, spreads are assumed to be driven by all the factors that affect Treasury yields plus an additional latent corporate factor. Unlike in previous empirical studies on multi-factor intensity-based models of corporate spreads (e.g. Duffee (1999) and Driessen (2003)), we do not model the default process at the firm level. Instead, we focus on spread dynamics at the sector level and across credit rating categories. One advantage of focussing on sector-level spreads is that the noise of idiosyncratic firm-level shocks is eliminated, allowing for more efficient estimation of the role of macroeconomic variables in the term structure of spreads. 4 In addition, we are able to better document key differences across sectors and ratings. Our main results are based on spreads of BBB-rated industrial firms, which is one of the sector-rating classes with the largest number of outstanding issues in the market, but we also report how our conclusions change when we examine banks and speculative grade industrial firms. 3 See Duffie and Singleton (2003) for a broad overview of this class of models. 4 One disadvantage of our approach is that we are unable to assess the relative importance of firm shocks versus aggregate shocks for the pricing of bonds issued by individual obligors. 2

4 In a recent related paper, Yang (2003) also investigates the role of macroeconomic variables in a term structure model of defaultable debt. However, there are several important differences in our respective approaches, including the variables analysed, data sources, sample size and model specification. For instance, we use data from 1991 to 2004, whereas Yang s sample runs from 1995 to Since our focus is on the role of macro variables in affecting yields, the fact that our data spans more than one complete business cycle is an important advantage in our study. Second, there is significant empirical evidence to support the conclusion that the Federal Reserve systematically responds to movements in inflation and real activity in setting its target for the federal funds rate (Amato and Laubach (1999), Clarida, Gali and Gertler (2000)). As a consequence, we allow the instantaneous risk-free rate to depend upon both of these variables, whereas Yang omits these variables from his specification. It turns out that Treasury yields, especially at the short end of the maturity spectrum, are strongly affected by the macro factors. Third, we allow for a third latent factor to affect corporate spreads, and we examine the relationship between financial conditions and the factors driving corporate spreads. We find that the corporate factor can be interpreted as the level of credit spreads, along the lines of Litterman and Schienkman (1992). A composite indicator of financial conditions explains almost 70% of the variation in this factor. Moreover, we find that the slope of the term structure of spreads is closely related to the slope of the Treasury term structure, which may help explain the source of the residual common factor identified in the changes in spreads in Collin-Dufresne, Goldstein and Martin (2001). Finally, our results also shed new light on the role of macroeconomic variables in the term structure of Treasury yields. For instance, our data sample covers a period when inflation has been relatively low and stable, and therefore our results provide a test of the stability of estimates obtained by Ang and Piazzesi (2003), whose model was estimated using data over a period that includes the high inflation of the 1970s and early 1980s. We find that shocks to real activity have had a much stronger initial effect on the entire yield curve compared to inflation shocks. This is consistent with the widespread presumption that monetary policy has been much more credible during the past decade or so compared to the prior two decades, and hence that inflation has relatively little impact on yield curve dynamics. Ang and Piazzesi (2003) report contrasting results on the relative importance of inflation and real activity, depending 3

5 upon which model they examine. The plan of the paper is as follows. In the next section we layout the model. In section 3 we describe the data, including the construction of the macroeconomic factors. Section 4 provides results for our basline estimations based on BBB-rated industrial corporate bonds, while section 5 examines the sensitivity of the results with respect to rating and sector. Some concluding remarks and suggestions for future research are provided in the last section. Further information on the data and a description of our estimation methodology is provided in appendices. 2 Corporate Bond Pricing with Macro Factors Under the assumption that bond prices are arbitrage-free, the price at time t of a zero-coupon Treasury bond with τ periodslefttomaturityisgivenby: µ Z t+τ P t (τ) =E Q t exp r s ds (1) where r t is the risk-free rate and E Q t ( ) E Q ( I t ) is the expectation under the riskneutral measure Q conditional on the information set at time t. The pricing of defautable securities depends upon, among other things, the treatment of recovery in the event of default. We follow Duffie and Singleton (1999) and assume that recovery is determined as a fixed fraction of the market value of the bond just prior to default. This assumption, known as Recovery of Market Value (RMV), 5 implies that zero-coupon defaultable bond prices are given by µ Z t+τ V t (τ) =E Q t exp rs + h Q s L Q ds (2) s=t s=t where h Q t is the risk-neutral intensity and L Q istherisk-neutralrateoflossgivendefault. As noted by Duffie and Singleton (1999), it is very difficult to distinguish between h Q t and L Q empirically, so we simply assume that L Q is constant. Some consequences of this assumption are raised below in the discussion of our empirical results. Yields on Treasury and corporate bonds are given by yt T (τ) = ln P t (τ) /τ and yt C (τ) = 5 Two other common recovery assumptions in the literature are Recovery of Face Value and Recovery of Treasury. See Madan et al. (2001) and Duffie and Singleton (2003) for empirical analysis and a discussion of the relative attributes of these alternatives. 4

6 ln V t (τ) /τ, which implies that the spread at maturity τ is equal to S t (τ) yt C (τ) yt T (τ). To solve for P t (τ) and V t (τ) in (1) and (2), we need to specify the stochastic processes followed by r t and h Q t. In this paper we consider the class of multi-factor affine Gaussian term structure models. This means that both r t and h Q t are assumed to be affine functions of the state X t. Our approach is to specify the dynamics of X t under the physical measure P. The model is completed by specifying the market prices of risk to be a function of X t as well. In the context of the literature on credit spreads, one innovation of this paper is allowing the state to consist of two types of factors: unobservable (or latent) factors and observable macroeconomic variables. Specifically, we define: X t =[X 1,t,X 2,t,X 3,t,X y,t,x π,t ] 0 where the first three terms in X t,denotedbyx i,t (i=1,2,3), are latent factors and the last two terms, X y,t and X π,t, are observable factors corresponding to real activity and inflation, respectively. 6 X y,t and X π,t are the common factors of a set of observable real activity and inflation series, respectively (see section 3.3). We assume that X t follows an affine homoskedastic diffusion process under the physical measure P given by: dx t = KX t dt + ΣdW t (3) where W t =[W 1,t,W 2,t,W 3,t,W y,t,w π,t ] is a vector of independent Brownian motions, Σ is an identity matrix and k k 21 k K = 0 0 k (4) k 44 k k 54 k 55 There are several features of (3) worth highlighting. First, we have imposed the longrun means of all of the factors to be zero. This is done without any loss in generality, as the means cannot be separately identified from the constants in the equations for r t and h Q t given below. Similarly, we have normalised the variances of the factors to one, 6 In ongoing work we are extending the current model to include a fourth latent factor and observable financial variables in X t. 5

7 as these are not separately identified from the factor loadings in r t and h Q t. Second, the innovations to the factors are mutually independent. Third, we assume a block-diagonal structure in K, which restricts dynamic interactions between the latent and observable factors. This helps to reduce the dimensionality of the parameter space. 7 Finally, we have not allowed for heteroskedasticity in the factors. Recent evidence presented in Piazzesi (2003) suggests that heteroskedasticity has been a less important feature of Treasury yields data since the start of the 1990s. We use the 1-month Treasury yield as a proxy for the short rate r t, and assume that it is determined according to: r t = δ 0 + δ 1 X 1,t + δ 2 X 2,t + δ y X y,t + δ π X π,t δ 0 + δ > X t (5) This specification is similar to that used in recent studies on the role of macroeconomic factors in the term structure (Wu (2000), Ang and Piazzesi (2003), Rudebusch and Wu (2003) and Hordahl, Tristani and Vestin (2003)), and it encompasses standard latent factor models with one or two factors (e.g. Vasicek (1977)). Equation (5) implies that the one-month Treasury yield which can be considered a close proxy for the monthly average of the federal funds rate will have the form of a Taylor-type rule. Since the Federal Reserve is generally regarded as responding to forecasts of inflation (rather than current inflation), one advantage of using the composite indicator X π,t in (5) is it appears to provide more accurate forecasts of future consumer price inflation than predictions based on the current value of consumer price inflation itself (see appendix A). For any given sector-rating class, the risk-neutral intensity h Q t is specified to be h Q t = γ 0 + γ 1 X 1,t + γ 2 X 2,t + γ 3 X 3,t + γ y X y,t + γ π X π,t (6) The only difference between (5) and (6) is that we allow h Q t, but not r t, to depend on the third latent factor X 3,t. Since the factors are assumed to be independent, we can consider X 3,t to be a corporate factor. The relationship between X 3,t and observable financial variables is analysed below. Since we are modelling spreads at the broad sector level, X 3,t could reflect both market-wide and sector-specific shockstodefault risk and default risk premia. If γ y = γ π =0, then (6) is similar to the specification of 7 Our results appear to be robust to eliminating some of these restrictions in estimation results not reported. 6

8 the risk-neutral intensity employed by Duffee (1999). One key difference is that Duffee estimated an intensity equation for each firm in his sample, and so his third latent factor possibly incorporates firm-specific characteristics as well. 8 Finally, notice that correlation between r t and h Q t is captured by their common dependence on all of the factors except X 3,t. Equations (3), (5) and (6) imply that, in principle, both the risk-free rate and the default intensity could become negative at times. Of course, it is desirable to have processes for interest rates and credit spreads that are always positive. In the results reported below, it turns out that both r t and h Q t remain positive throughout the sample. 9 The final ingredient of the model is to specify the market prices of risk Λ t. We assume that Λ t is also an affine function of the factors: where λ 0 = λ 0,1 λ 0,2 λ 0,3 0 0, and λ 1 = Λ t = λ 0 + λ 1 X t (7) λ 1,(11) λ 1,(12) λ 1,(21) λ 1,(22) λ 1,(33) λ 1,(44) λ 1,(45) λ 1,(54) λ 1,(55) We impose zero-restrictions in λ 0 and λ 1 to reduce the dimensionality of the parameter space. Following Ang and Piazzesi (2003), we impose the restriction that the observable factors are not priced in steady state (λ 0,4 = λ 0,5 =0). We also assume a block-diagonal structure in λ 1 similar to that in K. The block-diagonal structure between X 3,t and the other factors justifies the consistency of our three-step estimation procedure (to be described below). Given the specification of the market price of risk in (7), our model belongs to the class of essentially affine models introduced by Duffee (2002) (see also Dai and Singleton (2002) and Duarte (2003)). In the general essentially affine EA m (N) model, the market prices of risk are given by: Λ t = Σ t λ 0 + Σ t λ 1 X t 8 Duffee also assumed that the latent factors follow CIR, instead of Gaussian, processes. 9 One main advantage of the Gaussian specification is tractability. For instance, convergence is achieved more quickly during estimation. In addition, the Gaussian model implies that maximum likelihood estimators are optimal when using the Kalman filtering based on mean squared error criteria. 7.

9 where Σ t and Σ t are diagonal matrices with elements defined by Σ t(ii) = p α i + β i X t, and Σ t(ii) = ( αi + β i X t 1/2, if inf (αi + β i X t) > 0 0, otherwise. In our EA 0 (N) model, α i =1(i =1,...,N) and β ij =0(i, j =1,..., N). Duffee (2002) and Dai and Singleton (2002) show that, despite having fixed volatility, the EA 0 (N) model fits Treasury yield curve dynamics nearly as well as more general specifications with m > 1, and better than the earlier generation of completely affine models analyzed by Duffie and Kan (1996) and Dai and Singleton (2000). 10 From Duffie and Kan (1996), the expectations in (1) and (2) can be solved to give the following expressions: P t (τ) =exp ³A T (τ)+b T (τ) > X t (8) and V t (τ) =exp ³A C (τ)+b C (τ) > X t (9) where A (τ) and B (τ) are obtained as solutions to a set of ordinary differential equations (see Appendix B). Yields on zero-coupon Treasury and corporate bonds are therefore given by yt T (τ) = ln P t (τ) = 1 ³A T (τ)+b T (τ) > X t τ τ and yt C (τ) = ln V t (τ) = 1 ³A C (τ)+b C (τ) > X t τ τ which implies that the spread at maturity τ is S t (τ) = 1 τ ³ A C (τ) A T (τ) + B C (τ) B T (τ) > Xt (10) (11) (12) 10 To our knowledge, no one has yet investigated the relative attributes of these two classes of models in fitting corporate yields or spreads. A proper investigation of this issue is beyond the scope of this paper. 8

10 3 Data 3.1 Treasury Yields We use data on zero-coupon constant maturity US Treasury yields to estimate the benchmark default-free curve in our model and to construct the spreads data. Data at various maturities is taken from interpolated yield curves available in the BIS DBS database, which have been constructed based on closing market bid yields on actively traded Treasury securities obtained by the Federal Reserve Bank of New York. The sources of all data series used in this paper are summarised in Table 1. In estimation, we use maturities of one, three, 12, 36, 60 and 120 month(s) (denoted 1M, 3M, 12M, 36M, 60M and 120M, respectively). Our sample period starts in April 1991 and ends in April 2004, giving 157 observations in total. A monthly time series of yields is assembled by taking month-end observations. Table 2 reports summary statistics on US Treasury yields, and the top panel of Figure 1 shows plots of these yields at 1M, 60M and 120M maturities. The unconditional means point to an upward-sloping yield curve on average from 3.76% at 1M to 5.88% at 120M. The term structure of the unconditional volatilities is hump-shaped, increasing from 1M to 12M, and then decreasing and flattening out at long maturities. Yield levels are highly persistent, with first-order serial correlations exceeding 0.95 for manymaturities.thereissomeevidencethatyieldsareplatykurticandhavenegative skewness, but the departures from normality are not strong. Pairwise correlations in Treasury yields at all maturities are high, with contemporaneous correlations for adjacent maturities often in excess of Table 3 shows the percentage of variation in yields explained by the six ordered principal components, on a marginal and cumulative basis. Most of the variability is accounted for by the first two components (over 99%). This suggests that a small number of common factors determine movements across the whole yield curve, consistent with many previous studies (see Litterman and Scheinkman (1991)). In a yields-only version of our model, we assume that two latent factors are sufficient. In our baseline model with real activity and inflation factors, we also include two latent factors, to make it a total of four factors affecting Treasury yields. By contrast, Ang and Piazzesi (2003) include three latent factors and two macro factors in their model of the Treasury curve. 9

11 3.2 Corporate Bond Yields Corporate spreads are constructed using data on corporate bond yields extracted from Bloomberg s Fair Market Value yield curves (see Table 1). These curves are constructed on a daily basis for various sectors and rating classes from a sample of Bloomberg Generic bond prices at market closing. Bonds with embedded options are adjusted to create option-adjusted yields. We utilise data on the curves for BBB- and B-rated industrial firms and A-rated banking firms. 11 As with Treasury yields, we create monthly time series using month-end observations. The BBB industrials sample runs from April 1991 to April 2004, while the B industrials sample starts in May 1992 and the sample on banking firms starts in September In estimation, we utilise maturities of 12M, 36M, 60M, 84M and 120M. 12 Credit spreads are calculated as the differences between the corporate yields and Treasury yields. Table 2 reports summary statistics on BBB-rated industrial yields and spreads, and Figure 1 plots the time series of these variables. As with Treasuries, the unconditional means of BBB corporate yields are increasing in maturity. The term structure of unconditional volatility is downward sloping. Similarly, the average level of spreads increases with maturity, from 5.47% at 12M to 7.12% at 120M, but unconditional volatilities are slightly higher at longer maturities. Spreads are positively skewed at long maturities. There is little evidence of excess kurtosis at all maturities. Thus, whilethereissomeevidenceofnon-normality in the distribution of spreads, our Gaussian model appears to be a reasonable approximation. Corporate spreads are also highly correlated across maturities. Table 3 shows that the first principal component accounts for almost 91% of the variation in the five BBB spreads included in our study, and over 99% of the variation is captured by the first three components together. In our model we allow spreads to be driven by five factors, but only one of these factors, X 3,t, is determined freely by the term structure of spreads. One reason we concentrate our analysis on the industrial sector is that many firms of this type issue corporate bonds. Figure 2 shows a breakdown by industry of the 11 Credit ratings are based on the Bloomberg composite rating, which is a blend of ratings of the major agencies. 12 Corporate bonds with less than one year to maturity are typically illiquid. As a consequence, the constructed yield curves at very short maturities are likely subject to greater estimation error. For this reason, we avoid using yields at the short end of the maturity spectrum in estimation. 10

12 number of bonds used to create the BBB-rated industrial curve on 24 August The industries with the greatest representation are transportation, food, forest products and oil & gas. 3.3 Macroeconomic Factors To construct the macro factors, we follow Ang and Piazzesi (2003); details are given in Appendix A. Each of X y,t and X π,t,plusafinancial activity factor which is used later in our analysis, are computed as a common factor (i.e. first principal component) from sets of observable time series. The main purpose of utilising common factors in our model, instead of the observable variables themselves, is to reduce the dimensionality of the state space. Table 1 summarises the data series used. The series correspondingtorealactivityandinflation are the same as in Ang and Piazzesi (2003), but our respective sample periods differ. For real activity these are the index of Help Wanted Advertising in Newspapers (HELP), unemployment rate (UE), the growth rate of employment (EMPLOY) and the growth rate of industrial production (IP). The inflation measures are the growth rates of the Consumer Price Index (CPI), Producer Price Index of finished goods (PPI) and a broad-based Commodity Prices Index (PCOM). All growth rates are measured as the 12-month difference in logs of the index. The financial activity factor is based on variables that represent leverage, interest coverage, cash flow and assets volatility. These quantities play a key role in firm-value models of credit risk and/or the ratings methodologies of the major ratings agencies (e.g. see Standard & Poor s (2003)). Leverage is measured as DEBT/PRO, where DEBT is Credit Market Debt and PRO is Profit After Tax; interest coverage is set equal to INT/GDP, where INT is Net Interest Payments and GDP is real Gross Domestic Product; a proxy for the ability of firms to generate cash flow is PRO/SALES, where SALES is Final Sales of Domestic Product; and the volatility of assets is proxied by call implied volatility on the S&P500 (IMPVOL) obtained from Bloomberg. 13 Data on DEBT, PRO, INT, and GDP refer to non-financial corporate business and are in real terms. 13 Bloomberg s data on call implied volatility begins in IMPVOL is extended back to 1991 using the VIX index. Values of the VIX index and implied volatility are almost identical in the month following the start of the latter, so there is no apparent break in the longer series. 11

13 Plots of the macro factors are shown in Figure 3. They are normalised to have mean zero and a standard deviation of one. At a broad level, fluctuations in each of the factors appear to be mainly driven by business cycle movements, particularly real activity as expected. For instance, the real activity factor increases for several years on the heels of the recession, and later falls significantly at the onset of the recession in The financial variable reflects the de-leveraging undertaken by firms at the start of the recovery in the early 1990s, and the subsequent rebuilding-up of leverage in the latter stages of the 90s boom, only to fall sharply again with the winding down of the recent recession. Table 4 reports correlations of the macro factors with Treasury yields and corporate spreads. Thesecorrelationsgivesomeideaonwhatrolewecanexpectthemacrofactors to play in our no-arbitrage term structure model. Both real activity and inflation are positively correlated with Treasury yields at all maturities, and in the case of real activity, they are particularly high at the short end of the curve (about 0.7). Yet, while real activity is negatively correlated with spreads, the correlation between inflation and spreads is almost nil. The financial factor is highly positively correlated with spreads. Interestingly, the correlations between the financial factor and Treasury yields are negative and significant. Finally, Table 5 reports regression results of the 1-month Treasury yield and Federal Funds Rate on the macro factors. The estimated coefficients are positive and significant on both real activity and inflation, in line with previous findings (Amato and Laubach (1999)). Moreover, they are almost identical in magnitude across the two equations, as are the R 2 statistics. This suggests that the equation for the 1-month Treasury yield in our no-arbitrage model mimics the Federal Reserve s reaction function, at least as measured with respect to our two macro factors. 4 Baseline Results: BBB-rated Industrial Spreads 4.1 Estimation Procedure The recursive structure of our model allows us to perform estimation sequentially in three steps. Since the macro factors are assumed to be exogenous with respect to yields and spreads, we estimate their dynamics in the first step using a vector autoregres- 12

14 sion (VAR). In addition, we can obtain consistent estimates of the coefficients in the equation for the short rate on real activity (δ y )andinflation (δ π )byols.theseare simply the estimates for the 1-month Treasury yield equation reported in Table 5, and reproduced in Table 6 for convenience. In the second step we estimate the Treasury portion of the model, and, conditional on these estimates, in step three we estimate the corporate spread portion of the model. This recursive procedure combines elements of similar procedures used by, on the one hand, Ang and Piazzesi (2003) for estimating a model with macro factors and Treasury yields, and, on the other hand, Duffee (1999) and Driessen (2003) for recursive estimation of a model of spreads. In both steps two and three, the cross-sectional and time-series properties of the observed yields and spreads are used in maximum likelihood estimation. The likelihood function, and filtered estimates of the latent factors, are constructed using the Kalman filter (see, e.g., Duan and Simonato (1995) and Lund (1997)). Appendix B gives further details on the estimation procedure. The VAR is specified to be of order one, which is consistent with the Gaussian affine model in (3) and (4). Including only one lag of each variable in the VAR not only reduces the number of parameters to be estimated in the VAR, but also keeps the dimension of the state vector small for estimation in steps two and three. OLS estimates of the VAR coefficients, expressed in continuous time as the parameters k 44,k 45,k 54 and k 55,are reported in Table 6. Real activity is very persistent with an autoregressive root equal to 0.98; inflation is also highly persistent, with a root equal to The coefficients on lagged cross terms are not statistically significant, although output has the expected positive effect on inflation (the coefficient on lagged output is 0.03 in discrete time). 4.2 Estimates of Treasury Yield Parameters Table 6 reports estimates of the parameters pertaining to the Treasury part of our model. To help assess the importance of the macro factors, we also estimate a yieldsonly version of the model, i.e. where the loadings on X y,t and X π,t in (5) and (7) are set to zero. The estimates of k 11 and k 22 imply that the two latent factors in the macro model are both highly persistent, but they are less persistent than in the yields-only model. This suggests that the macro factors account for some of the persistence in 13

15 the state variables needed to explain yields. 14 Estimates of the parameters governing the market prices of risk are significant in most cases. In particular, there is strong evidence to suggest that risk premia are time varying with respect to both real activity and inflation, as the estimates of λ 1,(45),λ 1,(54) and λ 1,(55) are all statistically significant. The in-sample fit of the models is very good, regardless of whether macro factors are included. Nonetheless, the macro model generally produces lower root mean squared errors (RMSEs) in fitting yields across maturities and a likelihood ratio test clearly rejects the yields-only model in favour of the macro model, with a p-value of zero to beyond four decimal places. Notably, the RMSE in fitting the 1-month yield (the policy reaction function) is 35% lower in the macro model. Following Litterman and Scheinkman (1991), we would like to relate the latent factors to the shape of the yield curve. We define the level of the curve to be the average of yt T (1M), yt T (36M) and yt T (120M), and the slope of the curve to be yt T (120M) yt T (1M). Figure 4 plots the the level and slope variables along with filtered estimates of the latent factors. It appears that what we call the second latent factor is closely related to the level of the term structure, and the firstlatentfactoris related to the slope of the term structure. 15 In fact, these relationships are stronger for the factors from the yields-only model than from the macro model. One implication is that the macro factors must have a strong impact on both the level and slope of the term structure. Figure 5 displays the factor loadings in Treasury yields, B T (τ) /τ, acrossthe maturity spectrum. This figure provides further evidence that the first two latent factors mainly affect the slope and level of the yield curve, respectively. For instance, notice that the loadings on Latent 1 decline monotonically with maturity, while the loadings on Latent 2 are flatter. Theimpactofinflation is strongest at short maturities and gradually declines to zero for maturities greater than 30 months. By contrast, the initial impact on yields of a shock to real activity is much higher than from inflation at all maturities, with the greatest effect at about a one-year maturity. Interestingly, theseresultsabouttherelativeeffects of the two macro factors are the opposite of what 14 Ang and Piazzesi (2003) report a similar finding. 15 In addition, the second latent factor is closely related to long-term inflation expectations. Using data on the average forecast of inflation ten years ahead from the Survey of Professional Forecasters, we find that the contemporaneous correlation with our estimated factor is

16 Ang and Piazzesi (2003) found in their VAR(1) model, but are closer to the results of their VAR(12) model. Finally, of all the factors, real activity has the largest impact at the short-end of the yield curve, whereas the level factor has the strongest effect at the long end. While the relative magnitudes of the factor loadings give the instantaneous effects of the factors on yields at various maturities, they do not indicate what proportion of the variability in yields is driven by innovations in each factor. The top panel in Table 8 reports variance decompositions of Treasury yields at 12M, 60M and 120M maturities and for forecast horizons of 12, 60 and 120 months. At a 12-month horizon, most of the variation in the 12M yield is due to real activity (68%), and then the level factor, latent 2 (31%). At higher maturities, the level factor becomes more important, whereas the impact of real activity drops off significantly and is replaced by the slope factor, latent 1. For example, 42% of the variation in the 120M Treasury yield can be attributed to the latent 1 factor. The patterns are broadly the same for longer forecast horizons of 60 or 120 months as well. It is noteworthy that the inflation factor accounts for virtually none of the variation in this set of Treasury yields. There are some significant differences in these results on variance decompositions compared to those obtained by Ang and Piazzesi (2003). In particular, these authors found that inflation accounts for a much larger proportion of the variation in 12M and 60M yields at horizons equal to or greater than 12 months (e.g., around 60-70% for 12M yields), whereas real activity accounts foronlyabout10% orlessofvariation in these yields. 16 The different sample periods of the two studies may be the main explanation, as Ang and Piazzesi data sample includes the 1970s and 1980s, which was a period of high and volatile inflation. By contrast, during our sample period, inflation was relatively low and much more stable. Thus, the inflation regime matters when drawing conclusions about the factors driving Treasury yields. 4.3 Estimates of Corporate Spread Parameters Table 7 reports estimates of the parameters in the corporate part of the model. Again, we also compute estimates of a yields-only version, i.e. where the macro factors do 16 Ang and Piazzesi (2003) report results for 1M, 12M and 60M Treasury yields only, at horizons of 1, 12, 60 and months (see Table 9 in their paper). 15

17 not affect the risk-neutral intensity in (6), as well as the risk-free rate and the market prices of risk. Both models fit spreads well: the average RMSEs are, in basis points, 9.1 (macro) and 10.0 (yields-only). All estimates of the intensity parameters are statistically significant, including the coefficients on the macro factors. The third latent factor, X 3,t, is very persistent in both models. We find that the market price of risk attached to X 3,t is negative and varies over time. Thus, all of the factors, including the macro factors, contribute to credit risk premia. 17 The right panel of Figure 5 reports the factor loadings B C (τ) B T (τ) /τ in spreads. The loadings on the corporate factor X 3,t are positive and fairly flat across maturities, and, except at very short maturities, this factor has the greatest impact on spreads. Consistent with this finding, below we interpret X 3,t to be the level of the term structure of spreads (in analogous fashion to level and slope in the Treasury curve). The second latent factor also seems to affect the level of spreads, but its impact is much smaller in magnitude than X 3,t. In fact, increases in X 2,t have a negative impact on spreads. We have interpreted X 2,t as the level of the Treasury curve. Thus, an upward shift in the Treasury curve is accompanied by a downward shift in spreads. This is consistent with Duffee s (1999) finding that the factors driving the risk-free rate are negatively related to the default intensity, and, hence, to spreads. Interestingly, changes in X 1,t appear to alter the slope of the term structure of spreads. Recall that, in the previous section, we showed that there is a close link between X 1,t and the slope of the Treasury yield curve. Thus, changes in the shape of both curves Treasury yields and corporate spreads appear to be driven by the same underlying source. Regarding the two macro factors, real activity has a hump-shaped effect on the spreads curve. The greatest impact is at about 24-months maturity, where a one percent increase in real activity leads to a 14 basis points decline in BBB-rated spreads. Similar to real activity, an increase in inflation leadstoadropinspreads, but the impact of inflation is strongest at the short end and declines monotonically (in absolute value) towards zero. Our interpretation of X 3,t and X 1,t as level and slope is confirmed in Figure 6, which displays the level and slope of the term structure of spreads, defined as (S t (12M) + S t (60M) +S t (120M))/3 and S t (120M) S t (12M), respectively. Also plotted with 17 Recall that the market prices of risk associated with the other factors are fixed at the estimated values obtained above in stage two of estimation. 16

18 the level term is X 3,t, andwiththeslopetermisx 1,t. Asanticipated,itisclearfrom the figure that X 3,t and the level of spreads are closely related; the contemporaneous correlation is The relationship between the slope and X 1,t is less tight, but still quite visible, and their correlation is Further evidence on the link between the level and slope of spreads and the factors is provided in Table 9. This table reports regressions of level (top panel) and slope (bottom panel) on each of the factors individually and all together. Not surprising, X 3,t has the highest explanatory power of all of the factors (R 2 is 0.88). 18 Real activity is the only other factor that appears to have a significant relationship with the level of spreads. Nonetheless, the regression on all of the factors has an R 2 of 0.99, whichisareflection of the very low RMSEs of the model in fitting spreads. The results for the slope series also confirm our earlier conjectures. The first latent factor has by far the highest explanatory power of slope movements. Still, the regression with all of the factors can explain about 68% of the variation in the slope, which is 13% more than whatx 1,t can explain alone. The fitted values from the two regressions with all the factors are plotted in Figure 6 along with the level and slope series. It is evident from the top panel that the sample paths of the fitted value and X 3,t are indeed very similar. As shown in the lower panel, there are greater discrepancies between X 1,t and the linear combination of factors from the regression. The fitted series based on all of the factors is much better at capturing the peaks and troughs in the slope series. The lower panel of Table 8 reports variance decompositions of spreads at 12M, 60M and 120M maturities. Consistent with the other results reported so far, innovations in the corporate factor (latent 3) are responsible for most of the variation in spreads at all of the maturities and horizons examined. The percentage of variation explained by this factor ranges from 61% for 12M spreads at a 12 month horizon to 95% for 60M and 120M spreads at 120M horizons. The next most important factor at the 12M maturity is the slope factor (latent 1), which is responsible for 26% of the variation at a 12 month horizon, although innovations to this factor are of trivial importance at longer maturities. Real activity accounts for close to 10% of the variation in 12M and 60M spreads at all horizons. Finally, as with Treasury yields, inflation accounts for very little variation in spreads at any maturity or horizon. 18 Since all variables have been scaled to have a standard deviation of one, the R 2 s in the univariate regressions are simply the correlation coefficients. 17

19 Figure 7 plots the estimated sample paths of the physical intensities from the macro and yields-only models. To extract an estimate of the physical intensity h t from the estimated risk-neutral intensity h Q t,wehavetomakeanassumptionaboutthesizeof the market price of jump-at-default risk J ξ,t.specifically, the risk-neutral and physical intensities are related according to: 19 h Q t = h t E t [1 + J ξ,t ] (13) where E t [ ] is an expectation taken with respect to the physical measure of the process for X t. If the conditions on conditional diversification as stated in Jarrow et al. (2003) are met, then J ξ,t =0,inwhichcaseh Q t = h t. These conditions do not literally hold in practice, and recent evidence suggests that E t [J ξ,n (t)] is fairly large on average and fluctuates over time (see Driessen (2003), Berndt et al. (2004) and Amato and Remolona (2004)). For instance, using data on credit default swaps, Berndt et al. find that E t [J ξ,t ] is approximately equal to one, which is the value we have imposed in constructing Figure 7. The intensities from the two models generally tend to follow each other, although it is clear that the inclusion of macro variables introduces some noticeable differences. In particular, the intensity in the macro model rises sooner during the second half of the 1990s, and flattens out and falls earlier during and after the recent recession, reflecting the positive loading on the real activity factor in h Q t. In intensity models, the K-period ahead conditional default probability is given by: µ Z t+k PD(t, t + K) =1 E t exp h s ds (14) One test of our model is to assess how well it can predict actual default rates in real time. 20 Figure 8 plots values of PD(t, t + K) generated from our models against the actual cumulative default rates in Moody s cohorts of BBB-rated industrial firms (K = 12 months (top panel), 36 months (middle) and 60 months (bottom)). A new cohort in a given sector and rating class is formed at the beginning of each calendar year, and its rating and default history is subsequently tracked through time. Both models generally overpredict actual default rates, although default probabilities from the macro model 19 Further discussion and a derivation of this relationship are provided in Piazzesi (2003). 20 This is not literally a real-time test, as the model has been estimated on the full sample of data. However, conditioning information only up to date t is used to form the conditional default probability PD(t, t + K). 18 s=t

20 are uniformly closer to actual default rates at all three horizons. However, the levels of the model-based probabilities depend, in part, on the assumed value for E t [J ξ,t ]. If,instead,weweretoimposealargervalueforE t [J ξ,t ], then, given our estimated process for h Q t, the average implied value for h t would be lower, and thus the level of PD(t, t + K) would match default rates more accurately. For instance, the value of E t [J ξ,t ] that equates the average value of PD(t, t +12) from the macro model to the average one-year default rate from Moody s is 4.63, which is even slightly larger than the estimate of 4.1 for the average market price of jump risk on BBB-rated US corporate issuers reported in Amato and Remolona (2004). 4.4 What is the Corporate Factor? The third latent factor in our model, which affects the risk-neutral intensity but not the risk-free rate by assumption, could be considered to be a corporate factor. We would like to understand what was driving the behavior of this variable during our sample period. A more structured approach to modelling credit risk for instance, firm-value models or rating agency methodologies suggests that financial variables such as those discussed in section 2 would affect the credit quality of obligors. In this sub-section we investigate the relationship between these variables and the estimated time series of the third latent factor, denoted ˆX 3,t. To begin, however, we would like to be able to interpret the factors that affect spreads, particularly X 3,t, in terms of the level and slope of the term structure of spreads. Table 9 reports regressions of level (Panel A) and slope (Panel B) on the five factors. The top panel clearly shows that the level of the term structure of spreads is related to the third latent factor, with 88% of the variation in the level explained by this factor. Real activity is the only other variable to explain a non-negligible proportion of the movements in the level of spreads. By contrast, Panel B shows that X 1,t has the highest explanatory power of the factors in explaining the slope of the term structure of spreads. Thus, the same factor appears to be the main driving force behind changes in the slope of both the Treasury yield curve and the spread curve. Note, however, that all of the factors together (column one) improves the R 2 from 0.55 to 0.68, and every factor is statistically significant in this regression. Now consider the relationship between the third latent factor and observable finan- 19

21 cial variables. Table 10 reports regression results of ˆX 3,t on leverage, interest coverage, profit-sales ratio, equity volatility and the financial activity factor. The top panel reports regressions with only the contemporaneous values of the explanatory variables; in the bottom panel, the first six lags of the variables are also included as regressors. Focussing on the top panel, all of the variables enter significantly and with the expected sign in the regressions on individual variables (columns 2 to 6). The financial factor has the highest explanatory power, with an R 2 of 0.68, yet leverage, interest coverage and the profit-sales ratio also explain about half of the variation in ˆX 3,t. It is perhaps not surprising that the financial factor explains a large proportion of the variation in ˆX 3,t given the high positive correlations between the financial factor and spreads reported in Table 4. When all four observable variables are included in the regression (first column), the R 2 only climbs to Thus,thefinancial activity variable we constructed contains almost all of the information available in the observable financial variables for explaining ˆX 3,t. It was noted above that ˆX 3,t is a very persistent series. Appendix A shows that the financial variables also exhibit rich dynamics. As a consequence, there may be a complex dynamic relationship between these variables, which is not adequately captured in regressions with contemporaneous variables only. The bottom panel of Table 10 shows that, in certain cases, it is beneficial to include lagged variables of the regressors as well. Notably, the explanatory power increases substantially in the regressions containing interest coverage and volatility. Figure 9 plots the unconditional correlations between each of the financial variables and spreads (top panel) or ˆX 3,t (bottom panel) at leads (positive axis) and lags of the financial variables. The figure shows that, in fact, the highest correlations are between the latent factor and the financial variables 12 months into the future. By contrast, the correlation with five-year spreads is generally highest at a lag (anywhere from one to 24 months). Furthermore, the correlations at leads and lags tend to be highest with the financial activity factor. These results are perhaps not surprising. For instance, the risk-neutral intensity is driven by the current value of X 3,t, whereas it is expected future values of the intensity (roughly put) that determine longer-maturity spreads. Overall, the results in Table 10 and Figure 9 indicate that the corporate factor is strongly related to variables that measure financial conditions and which are commonly believed to affect credit risk. Interest coverage seems to be most important, while 20