A Multifactor Model of Credit Spreads

A Multifactor Model of Credit Spreads Ramaprasad Bhar School of Banking and Finance University of New South Wales r.bhar@unsw.edu.au Nedim Handzic University of New South Wales & Tudor Investment Corporation nedim.handzic@tudor.com November 2006

Abstract We use a state-space model to represent the time-variation of credit spread indices by rating as a function of latent factors of the Vasicek form. By application of the Kalman Filter we simultaneously estimate the factor realizations, their process parameters, and the exposure of each observed credit spread series to each factor. A three-factor model is found to capture most of the time-variation in credit spreads across ratings, for any given maturity. Most significantly, the extracted factor series are closely correlated with the long bond rate, the implied volatility index (VIX), and the S&P500 level, suggesting that the three variables can explain the systematic variation in credit spreads across the quality spectrum. 2

I. Introduction The theoretical link between credit spreads and market variables is established by structural models of default. Models such as Merton (1974) and Longstaff and Schwartz (1995) are based on the economic definition of default as the event where a firm s value falls below the face value of its outstanding debt. The unobservable value of the firm is assumed to follow a risk-neutral diffusion, allowing the calculation of default probabilities and an endogenous recovery rate. Credit spreads are attributed entirely to the risk-neutral expected default loss, which is positively related to leverage and volatility in the firm value. An increase in the firm value through positive equity performance has the effect of reducing leverage and credit spreads. Given the risk-neutrality assumption, an increase in treasury yields raises the drift rate of the firm value process, leading to lower credit spreads. In practice, structural models tend to underestimate short-term credit spreads. The use of smooth processes to represent the firm value may exclude the possibility of default by high grade issuers in the short term, which is inconsistent with the observed role of surprise in credit markets. In contrast, reduced-form models are flexible enough to empirically fit the term structure of credit spreads, but do not provide an economic interpretation of default. Reduced form models such as Jarrow and Turnbull (1995) and Duffie and Singleton (1999), define exogenous stochastic processes for the arrival time of default and exogenous recovery rates. An additional class of models combines the advantages of both structural and reduced-form approaches by incorporating exogenous effects such as jump-diffusions (Zhou 1997) in the firm value process to allow for surprise default. 3

To the extent that credit spreads reflect expectations on future default and recovery, we would expect broad-based spread indices to vary with macroeconomic variables such as interest rates, stock market returns and market volatility. In general, low-grade bond spreads are observed to be closely related to equity market factors (Huang and Huang 2002) while highgrade bonds behave like treasuries. Campbell and Taksler (2003) show credit spreads to be positively related to the market average of firm-level volatility, and the increase in market and firm volatility documented by Campbell et al. (2001) is consistent with the steady rise in credit spreads throughout the 1990s. Duffee (1998) finds that yield spreads vary inversely with treasury yields, with the effect being strongest for callable bonds. As treasury yields fall, prices of callable bonds increase by a lower proportion than treasuries due to the higher value of embedded issuer calls, leading to wider credit spreads. Given that the proportion of callable bonds is greater among low grade issuers, we expect the sensitivity to interest rates to be higher for indices of low-grade bonds. The evidence on the relationship between observed spreads and their theoretical determinants is mixed. Using principal components analysis, Colin- Dufresne et al. (2001) find that changes in individual bond spreads are driven by a single common systematic factor, unaccounted for by theoretical variables. A stream of recent literature demonstrates that credit risk accounts for a minor portion of spreads, with most variation due to alternative risk factors or a risk premium similar to that in the equity markets. A significant portion of spread levels has attributed to the positive difference between tax rates on corporate and treasury bonds. Elton, Gruber, et al. (2001) find 4

that expected loss accounts for less than 25% of the observed corporate bond spreads, with the remainder due to state taxes and factors commonly associated with the equity premium. Similarly, Delianedis and Geske (2001) attribute credit spreads to taxes, jumps, liquidity and market risk. Liquidity risk itself has been found to be a positive function of the volatility of a firm assets and its leverage, the same variables that are seen as determinants of credit risk (Ericsson and Renault 2001). Based on existing evidence, we take the view that the time-variation in credit spreads is driven by two classes of factors that are non-stationary and mean-reverting, respectively. The first group of factors affects credit risk which changes with macroeconomic conditions and has low rates of meanreversion. The second group relates to liquidity premia that change with noisy short-term supply and demand shocks. Given that credit risk explains a lower proportion of high-grade spreads than low-grade spreads, it is intuitive that high-grade spreads should have stronger mean-reversion that reflects changes in liquidity due to supply/demand. In figure 1 sub-investment grade bond spreads appear to be non-stationary while investment grade spreads revert to a long-run mean. While the two bond classes behave in fundamentally different ways, their shared sources of variation are evident in the common shocks. This study assumes that the time-variation in credit spreads across ratings classes is driven by a common set of unobservable factors to which each spread is exposed with some unknown sensitivity. We aim to answer the following questions: 1) how many factors can explain the evolution of ratingsbased spread indices, 2) what are the exposures of individual indices to each 5

of the factors, and 3) what economic variables, if any, could form proxies for the factors. State-space representation has the advantage of simultaneously allowing for both time-series and cross-sectional data in establishing dependency of observed series on latent factors. Given a parametric process form for the latent factors, the Kalman Filter Maximum Likelihood method can be applied to simultaneously estimate 1) the parameters of each factor process, 2) the exposures of each observed series to the individual factors, and 3) the most likely realizations of the factor series. The Vasicek (1977) normal form is chosen for the factors since, depending on the size of its mean-reversion coefficient, it is suitable for representing both non-stationary (macroeconomic) as well as (microeconomic) determinants of credit spreads. A multi-factor Vasicek form is supported by the findings of Pedrosa and Roll (1998) that Gaussian mixtures can capture the fat-tailed distributions of credit spreads. An additional advantage of the Vasicek form is that the conditional normality of the states leads to an exact likelihood function that can be maximized to obtain the model parameters under the state-space approach. Previous applications of the state-space model have focused on the term structure of treasury rates. Babbs and Nowman (1999) find that a threefactor Vasicek model adequately captures variations in the shape of the treasury yield curve, with two factors providing most of the explanatory power. Chen and Scott (1993) and Geyer and Pichler (1999) reach similar conclusions based on a multifactor CIR (1985) model, and find the factors to be closely related to the short rate and the slope of the curve. This study aims to relate the factors driving credit spreads to variables from both equity and 6

interest rate markets. 7

II. Method A. The Multifactor Vasicek Model in State Space Form For a given term to maturity, each of the n observed credit spread indices by rating R t = {R 1t, R 2t,..., R nt } is expressed as a function of m independent latent factors or states X t = {X 1t, X 2t,..., X mt } of the Vasicek form. In continuous time n dr it = a ij dx jt i = 1, 2,..., n (1) k=1 dx jt = κ j (θ j X jt ) dt + σ j dw jt j = 1, 2,..., m (2) The application of the Kalman Filter algorithm to estimate the factor loadings a ij, the state process parameters ψ = {κ j, θ j, σ j } and the state vector ˆX, requires that the model is expressed in the state space form. State space representation consists of the measurement and the transition (or state) equations R t = Z(ψ) X t + ɛ t ɛ t N(0, H(ψ)) (3) X t = C(ψ) + Φ(ψ) X t 1 + η t η t N(0, Q(ψ)) (4) The measurement equation (3) maps the vector of observed credit spreads R t (n 1) to the state vector X t (m 1) via a measurement matrix Z(ψ)(n m). Unexpected changes in credit spreads and errors in the sampling of observed series are allowed for through jointly normal error terms ɛ j, with zero conditional mean and covariance matrix H(ψ). Since the computational 8

burden of estimating a full error covariance matrix H(n n) increases rapidly with additional observed series, most studies assume error independence. In state-space models of the treasury curve, (Chen and Scott (1993), Geyer and Pichler (1996), and Babbs and Nowman (1999)) a diagonal matrix with elements h 1, h 2,..., h n was used to capture the effects of differences in bidask spreads across n maturities. In this study we choose the same form to allow for different bid-ask spreads across n bond quality groups. The state equation (4) represents the discrete-time conditional distribution of the states, with terms following from the discrete form of the Vasicek model for interval size t X j,t+ t = θ j ( 1 e κ j t ) + e κ j t X j,t + η j,t (5) ( η j,t N 0, σ2 ( j )) 1 e 2κ j t 2κ j (6) Gaussian innovations of the states are represented by the noise vector η t, with covariance matrix Q. It is assumed that the sources of noise in the state and measurement equations are independent. B. Kalman Filter Estimation of the Model The optimal estimator of the latent state vector X t is its conditional mean E t 1 [X t ]. At each time step t, the filtered estimate ˆX t consists of a predictive component ˆXt t 1, based on information to time t 1, and an updating component incorporating new observations from R t. For t = 1, 2,..., T ˆX t t 1 = E t 1 [X t ] = C(ψ) + Φ ˆX t 1 (7) 9

The covariance Σ t t 1 of ˆXt t 1 is given by Σ t t 1 = E t 1 [ (Xt ˆX t t 1 )(X t ˆX t t 1 ) ] = ΦΣ t 1 Φ (8) Σ t = E t [ (Xt ˆX t )(X t ˆX t ) ] = ( Σ 1 t t 1 + Z H 1 Z ) 1 (9) The estimate ˆX t is calculated as the sum of ˆXt t 1 and an error-correcting innovation matrix, v t, weighted by the Kalman Gain matrix K t. v t = R t ( C(ψ) + Φ(ψ)X t 1 ) (10) ˆX t = ˆX t t 1 + K t v t (11) K t = Σ t t 1 Z [ ZΣ t t 1 Z + H ] 1 (12) The recursive equations are started by setting the initial state vector estimate X 0 to its unconditional mean. To ensure quick adjustment of the state vector estimates to early observations, we set the initial Kalman Gain close to unity by choosing an arbitrarily high initial covariance Σ 0. With additional observations the covariance terms and the Kalman gain drop to reflect the true uncertainty about the state vectors, and the sensitivity of the estimates to new observations becomes progressively lower. The initial burn-in phase of time steps that is required for the Kalman Gain to stabilize is typically excluded when drawing conclusions about the state vector realizations. The parameter set ψ is obtained by maximizing the log-likelihood function (less a constant) (13) that follows directly from the prediction error decomposition. 10

logl(r 1, R 2,..., R T ; ψ) = 1 T log F t 1 T v 2 t=1 2 tf t 1 v t (13) t=1 F 1 t = H 1 H 1 Z ( Σ t t 1 + Z H 1 Z ) 1 Z H 1 (14) F t = H Σt t 1 Σ 1 t t 1 + Z H 1 Z (15) 11

III. Results One, two, and three-factor models are estimated using month-end data for the period 30-Apr-96 to 31-Mar-03. Yield spreads across ratings classes are defined as differences between the J.P. Morgan 10-year Corporate Bond Index and the 10-year benchmark bond yields. Table I shows the estimates for the mean-reversion speed (κ), mean (θ), and volatility (σ) of each Vasicek factor. The log-likelihood, AIC, and BIC criteria are highest for the three-factor model, under which all parameters (with the exception of one mean) are highly significant. The marginal improvement in the log-likelihood from the addition of a third factor is far smaller than for a second factor, suggesting that a 3-factor model is sufficient in capturing the common sources of variation in credit spreads. For comparison, the log-likelihoods for the one, two, three, and four-factor models are 1246.0, 1840.6, 2040.6, and 2100.10, respectively. The parameter estimates for factor 4 in a four-factor model are largely insignificant, supporting the choice of the three-factor model which is presented. The extracted factor under the one-factor model (Figure 2) can be interpreted as an average of the 14 observed series. Allowing for a second factor (Figure 3) reveals two distinct smooth processes as the drivers of the crosssection of credit spreads, while in the three-factor model an additional more noisy process is identified (Figure 4). The factors under the three-factor model are compared to well-known economic time-series in figures 5 to 7. Under the three-factor model, the 1st factor resembles the (negative of) 10- year bond rate, the 2nd the VIX, and the 3rd the S & P 500 level. Based on the full sample of monthly observations, the correlation between Factor 12

1 and the long rate is -0.74, between Factor 2 and the VIX is 0.08, and 0.92 between Factor 3 and the S &P 500. Excluding a burn-in phase of the first 12 months under the Kalman Filter approach increases the correlation between the VIX and Factor 1 to 0.47. The estimated loadings of the observed series to each factor are shown in Table III and Figure 8. The loading of the first observed series (Aaa) to each of the factors is assumed equal to one, with the remaining loadings and the factors scaled accordingly by the estimation process. Are the sensitivities to the factors consistent with theory? The positive loadings on Factor 1 appear counter-intuitive since lower rates should improve issuer ability to repay debt. But the result is consistent with the market feature that call provisions are more common among lower-rated bonds. As interest rates fall, the value of both government and corporate bonds increases, but credit spreads rise as non-callable government bond prices increase by a greater proportion than callable corporates. As expected, this effect on spreads is strongest for the lowest-grade indices. The shape of the loadings on the 2nd factor suggests that equity volatility risk has a positive impact on credit spreads and that exposure to it increases with declining credit quality. The sharpest increase occurs at the crossing from investment to sub-investment grade bonds. To the extent that equity volatility is a proxy for a firm s asset value volatility, this result is consistent with the prediction of Merton (1974) that the probability of default and credit spreads both increase with volatility in the value of assets. The sensitivities to Factor 3, which is closely correlated to the S&P500, change sign from positive to negative as bonds move from investment to sub- 13

investment grade. The suggested positive relationship between equity market performance and high-grade credit spreads is at odds with the Merton (1974) model. According to the model, higher equity values increase the value of a firm s assets relative to its debt, lowering its probability of default. A possible explanation is that the positive equity performance throughout the 1990s is a proxy for rising leverage ratios during the same period. The negative effect on spreads of higher asset values may have been more than offset by a higher values in debt. The greater ability of high-quality issuers to raise leverage ratios may explain the difference in the signs of the loadings on factor 3 for investment and sub-investment grade issuers. Another possibility is that, for all but the worst credits, positive equity market performance may have contributed more to the substitution out of risky debt, in favor of equity, than to an increase in its value through improved creditworthiness. The results suggest that all credit spreads vary in response to three common systematic risks that are reflected in the macroeconomic risk variables VIX, S&P500 and the long bond rate. However, the ability of these factors to explain observed spreads can rapidly decline during exogenous events, as shown by the conditional density likelihoods in figure 9. During the LTCM crisis of 1998, the log-likelihoods fall to the same level for all 3 models. The lack of explanatory power added by a second or third factor confirm that the event came as a surprise independent of macroeconomic conditions that accompanied it. 14

IV. Conclusion This study concludes that most of the variation in credit spreads across ratings is explained by three independent factors. These vary broadly with the long bond rate, the VIX and the S&P500. The sensitivities of credit spreads to each of the factors are in agreement with the theoretical predictions of the Merton (1974) model, on an aggregate level. By making no prior assumption about the determinants of credit spreads, the state-space model provides an independent verification of existing theory. 15

Figure 1: 10-year Credit Spread Indices by Rating: Monthly Apr-96 to Mar-03 16

Figure 2: One Factor Model Figure 3: Two Factor Model Figure 4: Three Factor Model 17

Figure 5: Factor One and the 10-year Treasury Rate Figure 6: Factor Two and the Implied Volatility Index (VIX) Figure 7: Factor Three and the S&P500 Level 18

Table I: The Log-Likelihood, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) for the one, two, and three-factor models One Factor Two Factor Three Factor LogL 1,246.0 1,840.6 2,040.6 AIC 2,556.0 3,781.2 4,217.1 BIC 2,633.4 3,902.1 4,381.6 κ 1 0.391 0.411 0.171 (0.281) (0.161) (0.021) κ 2 0.351 0.421 (0.111) (0.041) κ 3 2.981 (0.231) θ 1 0.791 0.601 0.641 (0.211) (0.141) (0.171) θ 2 0.161 0.161 (0.141) (0.111) θ 3 0.251 (0.021) σ 1 0.181 0.141 0.081 (0.021) (0.011) (0.011) σ 2 0.141 0.131 (0.021) (0.011) σ 3 0.121 (0.011) 19

Table II: Estimated Standard Deviations of Measurement Errors for the one, two, and three-factor models h1,1 One Factor Two Factor Three Factor 0.0791 0.0691 0.0661 h2,2 (0.0071) (0.0061) (0.0031) 0.0971 0.0641 0.0641 h3,3 (0.0081) (0.0051) (0.0031) 0.0931 0.0521 0.0481 h4,4 (0.0081) (0.0051) (0.0021) 0.0821 0.0401 0.0251 h5,5 (0.0081) (0.0051) (0.0011) 0.0721 0.0471 0.0471 h6,6 (0.0071) (0.0041) (0.0021) 0.0451 0.0571 0.0541 h7,7 (0.0061) (0.0061) (0.0031) 0.0751 0.0901 0.0851 h8,8 (0.0081) (0.0081) (0.0041) 0.1211 0.1201 0.1071 h9,9 (0.0111) (0.0091) (0.0051) 0.4591 0.3561 0.0871 h10,10 (0.0371) (0.0331) (0.0041) 0.5241 0.1911 0.1741 h11,11 (0.0421) (0.0201) (0.0071) 0.6231 0.1501 0.1461 h12,12 (0.0481) (0.0181) (0.0061) 0.7071 0.1521 0.0751 h13,13 (0.0571) (0.0181) (0.0051) 0.8911 0.3251 0.2131 h14,14 (0.0731) (0.0311) (0.0091) 1.0931 0.3151 0.3031 (0.0901) (0.0361) (0.0141) 20

Table III: Factor Loadings for the Three-Factor Model Factor 1 Factor 2 Factor 3 a 1,i 1 1 1 - - - a 2,i 0.8831 1.1001 1.3211 (0.0441) (0.0421) (0.0531) a 3,i 0.8861 1.5261 1.5571 (0.0451) (0.0431) (0.0491) a 4,i 1.0491 1.7661 1.7131 (0.0421) (0.0361) (0.0431) a 5,i 1.4981 1.9301 1.7271 (0.0521) (0.0411) (0.0491) a 6,i 1.9101 2.1471 1.6681 (0.0471) (0.0421) (0.0551) a 7,i 2.2851 2.3811 1.4661 (0.0441) (0.0561) (0.0591) a 8,i 2.8721 2.7031 1.3971 (0.0791) (0.0611) (0.0811) a 9,i 6.7141 3.0691 0.0141 (0.1201) (0.0891) (0.0951) a 10,i 5.9651 5.7611-0.0491 (0.0811) (0.0791) (0.0871) a 11,i 6.5211 6.9821-0.2241 (0.0761) (0.0711) (0.0851) a 12,i 6.2731 8.5891-0.0851 (0.0611) (0.0771) (0.0741) a 13,i 6.7851 10.8661 0.1451 (0.0711) (0.0651) (0.0721) a 14,i 10.6411 11.7021-0.7631 (0.1091) (0.0791) (0.1311) 21

Figure 8: Factor Loadings for the One, Two, and Three-Factor Models 12 10 Factor Loading 8 6 4 2 0 2 1 1 2 Factor 1 2 3 AAA AAA1A2A3 BB1 BB2 BB3 B1B2B3 BBB3 BBB1 BBB2 Rating 22

Figure 9: Conditional Density Log-likelihoods for the One, Two, and Three-Factor Models 23

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