Structural Models for Corporate Bond Pricing: The Impact of the Interest Rate Dynamics

Transcription

1 Structural Models for Corporate Bond Pricing: The Impact of the Interest Rate Dynamics Steven Simon 1 Central Risk Management, Fortis 1 Central Risk Management, Fortis. Kreupelenstraat 10, 1000 Brussels, Belgium. steven.simon@fortis.com, phone: +32(0) , fax:+32(0)

2 Abstract We examine the impact of the interest rate dynamics on the performance of two different structural models for corporate bonds. While the physical dynamics of the spot rate are kept fixed, we allow for different risk-neutral dynamics. We find that the default spreads generated by the structural models are quite sensitive to the specification of the risk-neutral dynamics of the spot rate, as implied spreads on investment grade bonds can vary by as much as a factor four. Key words: Credit spreads, structural models for corporate bonds, interest rate dynamics. JEL classification: G12, G13.

3 1 Introduction We examine the impact of the interest rate dynamics on the performance of structural models for corporate bonds. While the physical dynamics of the spot rate are kept fixed, we allow for different risk-neutral dynamics. Our results indicate that the default spreads generated by structural models are quite sensitive to the specification of the risk-neutral dynamics of the spot rate. We find that implied spreads on investment grade bonds can vary by as much as a factor four. Several authors have found that credit spreads predicted by structural models are far below empirically observed spreads, see for instance Huang and Huang (2003). A notable exception is Eom et al. (2004), who find that some structural models generate too high yield spreads. Interestingly, different authors have found that structural models do a good job at explaining observed default frequencies across different rating classes and maturities, see for instance Leland (2002) and Churm and Panigirtoglou (2005). That is, whereas structural models typically generate too low yield spreads, they have no problem generating realistic values for implied default frequencies. This observation does not need to spell the failure of structural models. Rather, it could be seen as the modeling equivalent of the credit spread puzzle : the fact that empirically observed yield spreads are many times higher than can be expected on the basis of observed default frequencies, see Elton et al. (2001) and Collin-Dufresne et al. (2001). Part of the explanation seems to be that yield spreads do not only compensate investors for exposure to default risk, but hat they are also driven by non-default factors (such as a tax effect, and a liquidity premium), see for instance Driessen (2003). Thus, a structural model that does a good job at generating realistic default frequencies but does not account for the nondefault factors driving yield spreads, will imply values for yield spreads well below empirically observed levels. Especially the liquidity effect could account for a sizable fraction of observed credit spreads, see Houweling et al. (2004) and Longstaff et al. (2005). As structural models do not capture such non-default factors, they can not explain the full observed credit spread when accurately calibrated. However, the existence of these non-default factors does not fully explain the results from structural models. Calibrating a reduced-form model with a non-default factor to both credit-default swap data and corporate bond prices, Longstaff et al. (2005) find that default 1

4 risk accounts for at least 50% of the credit spread across all credit ratings, and as much as 83% of the observed spreads for BB-rated bonds. In contrast, Huang and Huang (2003) find that for a wide range of structural models the implied spreads on investment-grade bonds are well below 50% of the spreads observed in the market. Therefore, even after accounting for non-default factors, the yield spreads implied by structural models are still too low. One interpretation is that the physical dynamics of structural models, which determine the implied default frequencies, are well specified, but that something goes wrong with the risk-neutral dynamics, which determine the implied yield spreads. This view is supported by the observation that various structural models, each with their own physical dynamics and default mechanism 1, all show the same contrast between the realistic levels for the implied default frequencies and the low levels for the implied yield spreads. Despite the different specifications of the physical dynamics or default mechanisms, the different models perform equally well in replicating default frequencies. However, when the transformation to the risk-neutral dynamics is made, they all perform more or less equally badly. This seems to suggest that in order to improve the performance of structural models, we need to look at their risk-neutral dynamics. As we keep the assumption, common in the literature, that the total assets of the firm are traded, one element that has a strong impact on the risk-neutral dynamics of the structural model is the specification of the risk-neutral dynamics of the spot rate. Therefore, changing the specification of the risk-neutral dynamics of the spot rate could be one way to alter the risk neutral dynamics of the asset-value process in structural models. Moreover, any change to the term structure dynamics affects the different structural model in a similar way, in line with the observation that the different structural models all suffer from the same failure to generate high enough implied yield spreads. We keep the physical dynamics of the term structure model fixed at those of Vasicek (1977), while we consider three different specifications for the price-of-risk of the spot rate: 1) a constant price-of-risk (as in the original model of Vasicek (1977)), 2) the price-of-risk is a function of the spot-rate itself, and 3) the price-of-risk is driven by a second factor (which does not affect the physical dynamics of the spot-rate). Each of the three specifications of the term structure model is combined with two structural models with an exogenous default 1 e.g. endogenous versus exogenous default boundaries, the possibility of a mean-reverting leverage ratio,whether or not equityholders have the possibility of strategic default,... 2

5 boundary: the model of Longstaff and Schwartz (1995), with a constant default boundary, and the model of Collin-Dufresne and Goldstein (2001), with a mean-reverting leverage ratio. In the spirit of Leland (2002), the two structural models are calibrated to replicate observed default frequencies for 10 year bonds of various credit ratings. As the physical dynamics of the spot rate are being kept constant, any difference in implied yield spreads across the three specifications of the interest rate dynamics is completely due to the difference in risk-neutral dynamics of the spot rate. That is, our approach allows us to single out the effect of the different specifications of the price-of-risk of the spot rate in a structural model. Our results indicate that the default spreads generated by the two structural models are quite sensitive to the specification of the risk-neutral dynamics of the spot rate. Changing from a constant price-of-risk to one that is a function of the spot rate increases the implied yield spreads significantly. For instance, for the model of Longstaff and Schwartz (1995) the implied spread for an A-rated bond with a 10 year maturity increases from 19.1 basis points (bps) to 27.3 bps. The biggest effect is obtained when the price-of-risk for the spot rate is driven by an extra factor. In this case, the implied yield spread for the A-rated, 10-year bond is as high as 89.5 bps. This paper is organized as follows. Section 2 introduces the two structural models that are used in this paper. In Section 3, we analyze the role of interest rate dynamics in structural models and we introduce three related term structure models. The calibration of the term structure models is done in Section 4. Section 5 gives the details of the calibration of the structural models, and Section 6 discusses the results for the implied credit spreads. Finally, Section 7 contains the conclusions. 2 Two Structural Models Here we introduce the two structural models used in this paper; as all structural models for corporate bonds they are extensions of the original model of Merton (1974). 2.1 The Longstaff and Schwartz Model In the original Merton-model the total debt of a firm is given by a single bond, and default can only occur at the maturity date of this bond. It is also assumed that there is no interest rate risk. Longstaff and Schwartz (1995) extend the Merton-model by removing these two restrictions. They allow for early default, triggered by an exogenous default boundary for 3

6 the asset-value process, and they incorporate interest rate risk in the model. In the version of the used in this paper, the physical dynamics of the asset-value process V t of a firm are given by: dv t = (r t + π δ) V t dt + σ V V t dw V (t), (1) with π the risk premium for the assets of the firm, and δ the total pay-out rate. The total assets of the firm are assumed to be traded. The model does not allow for taxes, therefore there is no gain from leverage. For the moment we leave the dynamics of the spot rate r t unspecified. The total nominal value of the outstanding debt is given by F, and the default boundary K is equal to: K = βf, (2) with β < 1, i.e. default occurs when the value of the assets drops to a fraction β of the face value of total debt. In the event of default, the recovery rate ω is given by: ω = (1 α)β, (3) where α is the fraction of the value of the assets lost in bankruptcy. After default, creditors who hold a zero coupon bond with maturity T, receive a default-free bond with a face value equal to a fraction ω of the original bond and with the same maturity date as the original security. Therefore, a defaultable discount bond with maturity date T can be seen as a security that has a pay-off H(T ) at time T given by: H(T ) = 1 {τ>t } + ω1 {τ T } = 1 (1 ω)1 {τ T }, with τ the time of default of the firm. As a result, the price of a corporate zero-coupon bond with maturity date T is equal to the following expected value: P (T, t) = E Q [ e R T t r(u)du ( 1 (1 ω)1 {τ T } ) Ft ]. (4) The superscript Q indicates that the expectation is taken under the risk-neutral measure, with the spot rate as numéraire. In order to evaluate this expected value, we need the dynamics of the asset-value process V t under the risk-neutral measure Q. 4 Because of the

7 assumption that the total assets of the firm are traded, the risk-neutral dynamics are given by: dv t = (r t δ) V t dt + σ V V t d W V (t), (5) 2.2 Collin-Dufresne and Goldstein Model One problem with the model of Longstaff and Schwartz (1995) is the assumption of a constant default boundary, which creates the effect that the longer a firm has survived, the less likely it is that it will default in the next period. Collin-Dufresne and Goldstein (2001) solve this by developing a model with a mean reverting leverage ratio. The physical dynamics of the asset value of the firm are still given by equation (1). However, here it is more convenient to work with the log-firm value defined as: y t log(v t ). The physical dynamics of y t are: ( ) dy t = r t + π δ σ2 V dt + σ V dw V (t). (6) 2 Again, π is the risk premium for the assets of the firm, and δ is the total pay-out rate. Also in this model taxes are not considered, and the assets of the firm are assumed be traded. Default occurs when the value of the assets drops to the value of debt. In such an event, equity loses its value, and equityholders stop servicing debt. The central feature of this model is the the assumption that the log-debt level k t is driven by the value of the assets: dk t = κ [y t ν φ (r t r) k t ] dt, (7) with φ and ν positive constants. From the above equation we see that the log-debt process k t is mean-reverting, with the target level given by: y t ν φ(r t r). (8) The idea behind this mean-reversion is that firms have a target leverage ratio, such that they issue debt when the firm value increases, and vice versa. The log-leverage ratio l t itself (l t k t y t ) has the following physical dynamics: ) dl t = λ (l P (r t ) l t dt σ V dw V (t), (9) where l P (r t ) is given by: 5

8 l P (r) δ π + σ2 S /2 ( ) 1 ν + φθ r λ λ + φ. (10) Default is triggered when the process y t reaches the lower boundary k t, i.e. when the process l t increases to zero. Again, in the event of default, creditors who hold a zero coupon bond with maturity T receive a default-free bond with a face value equal to a fraction ω of the original bond and with the same maturity date as the original security. Here, the recovery rate ω is exogenously specified. Similar to the Longstaff and Schwartz model, the price at time t of a corporate zerocoupon bond with maturity date T is given by the following expectation under the risk-neutral measure Q: P (T, t) = E Q [ e R T t r(u)du ( 1 (1 ω)1 {τ T } ) Ft ]. (11) Where τ is now the hitting time for the log-leverage process l t for the upper boundary zero. Combining the definition of l t with the risk-neutral dynamics of V t gives us the risk-neutral dynamics of l t : Where l Q (r t ) is given by: ) dl t = λ (l Q (r t ) l t dt σ 1 d W 1 (t). (12) l Q (r) δ + σ2 /2 λ 3 Three Different Term Structure Models ( ) 1 ν + φθ r λ + φ. (13) In both structural models default occurs after a drop in the asset-value process. In the model of Longstaff and Schwartz (1995) the drop in the asset value itself triggers default, whereas in the model of Collin-Dufresne and Goldstein (2001) default occurs when a drop in the asset value causes the leverage ratio to reach the upper boundary level. For both of the structural models the physical dynamics of the asset-value process are given by: dv t = (r t + π δ) V t dt + σ V V t dw V (t), (14) and the risk-neutral dynamics are: dv t = (r t δ) V t dt + σ V V t d W V (t). (15) 6

9 At a first glance, the drift in the risk-neutral dynamics appears to be lower than in the physical dynamics, as the risk-premium π has been dropped. However, the dynamics of the spot rate are different in the two equations. In order to compare the drift terms, we need to know both the physical and the risk-neutral interest rate dynamics. We discuss this in more detail below. 3.1 The Base Case: The Model of Vasicek (1977) In the original term structure model of Vasicek (1977) the physical dynamics are given by: dr t = k P r (r P r t )dt + σ r dw t. (16) The assumption that the price-of-risk for the spot rate λ r is a constant leads to the following specification for the risk-neutral dynamics: dr t = ( k P r (r P r t ) + λ 0 rσ r ) dt + σr d W t (17) = k P r (r Q r t )dt + σ r d W t, (18) with: Prices for discount bonds are given in Appendix A. has: r Q = r P + λ0 rσ r kr P. (19) Observe that for the price P (t, T ) at time t of a zero-coupon bond with maturity T one P (t, T ) = E Q t [ ( T )] exp r(u)du < Et P t [ ( T )] exp r(u)du, (20) t where Q and P indicate that the expectations are taken under the risk-neutral and physical measure, respectively. The strict inequality follows from the fact that interest-rate risk is priced. The only way this inequality can be obtained in the Vasicek model is by assuming a strictly positive value for λ 0 r. This effectively increases the (conditional) mean of T 0 r(u)du under the risk-neutral measure Q relative to the one under the physical measure. However, as r(t) drives the drift of V (t) under the risk-neutral measure, this approach has an unwanted side-effect in the context of a structural model for corporate bonds. The higher the mean-reversion level of r(t) under the risk-neutral measure, the lower the risk-neutral default probability will be, and the smaller the implied credit spread. Therefore, a way to obtain the above inequality, other than by increasing the risk-neutral mean-reversion level of the spot rate, would be desirable. 7

10 We now introduce two alternative specifications for the price-of-risk of the spot rate. The resulting two term structure models belong to the class of the essentially affine term structure models, as defined by Duffee (2002). 3.2 An Essentially Affine One-Factor Model A first alternative for a constant price-of-risk λ r for the spot rate, is to specify the price-of-risk as linear function of r t itself: λ r (r t ) = λ 0 r + λ 1 rr t. (21) This way, the risk-neutral dynamics of r t are changed, while the physical dynamics are still given by equation (16). Specifying the price-of-risk as a function of the spot rate allows one to account for the fact that interest rate risk is priced, i.e. that the inequality equation (20) is strict, by changing both the mean-reversion level and the mean-reversion speed of r(t). The risk neutral dynamics of r t are now: dr t = ( kr P (r P r t ) + (λ 0 r + λ 1 ) rr t )σ r dt + σr d W t (22) = k Q r ( r Q r t ) dt + σr d W t, (23) where k Q r and r Q are given by: k Q r = k P r σ r λ 1 r (24) r Q = kp r r P + λ 0 rσ r k Q r (25) The prices for discount bonds are given in Appendix B. 3.3 An Essentially Affine Two-factor As in the previous specification, the price-of-risk for the spot rate is stochastic, but now it is driven by a second factor f t, rather than by r t itself: λ r (f t ) = λ 0 r + λ 1 rf t. (26) The risk-neutral dynamics of r t are equal to: dr t = ( k P r (r P r t ) + (λ 0 r + λ 1 rf t )σ r ) dt + σr d W t (27) = k P r (a 0 + a 1 f t r t )dt + σ r d W t, (28) 8

11 with: a 0 = r P + λ0 rσ r kr P, (29) a 1 = λ1 rσ r kr P. (30) The factor f t also follows an Ornstein-Uhlenbeck process, with physical dynamics equal to: df t = k P f (f P f t )dt + σ f dw f (t), (31) with dw r dw f = ρ. Just as for the spot rate, the price-of-risk for the process f t is a function of f t : λ f (f t ) = λ 0 f + λ1 f f t. (32) Combining equations (31) and (32) leads to the following specification for the risk-neutral dynamics for f t : df t = k Q f (f Q f t )dt + σ f d W f (t). (33) Where k Q f and f Q are given by: k Q f = k P f σ f λ 1 f, (34) f Q = kp f f P + λ 0 f σ f k Q f. (35) As in the previous model, with this specification of the price-of-risk for the spot rate one has the possibility to increase the conditional variance of the risk-neutral dynamics of r(t) in order to account for the fact that interest rate risk is priced. Whereas in the one-factor model this was obtained by changing the mean-reversion speed, we now allow for a stochastic mean-reversion level. The risk neutral dynamics of this model are equivalent to those of the stochastic-mean model of Balduzi et al. (2000). The prices for discount bonds of this two-factor model are given in Appendix C. 4 Calibration of the Three Term Structure Models Here we turn to the calibration of the three term structure models. As the physical dynamics are the same for all three models, they only need to be calibrated once. 9

12 4.1 The Data We calibrate the three term structure models to monthly US Treasury yield data for the period September 1976 until December 1997 i.e, 256 months in total. As a proxy for the spot rate process we use the 6-month interest rate. To estimate the different components of the price-of-risk vector, including the dynamics of the factor f t in the two-factor model, we use yields on discount bonds with maturities of 1,2,5,7 and 10 years. To obtain those yields, we used the following procedure. For each month we started from the par yields for the same five maturities, which are available from the website of the Federal Reserve. Using cubic-spline interpolation we obtained the par-yield curve at 6-month intervals. From this last curve we obtained the zero-coupon curve using a bootstrapping procedure. 4.2 Calibration of the Physical Dynamics of r t Calibration of the physical dynamics of the spot rate process to the observed 6-month interest rate is done by means of the Method of Moments. We use the following two expressions for the conditional expected value and conditional variance of r t : Et P [r t+1 ] = r P + e kp r t (r t r P ), (36) ( ) ) σ 2 2 r (1 e 2kP r t var P t [r t+1 ] = 2k P r.. (37) Using the exact expressions for Et P [r t+1 ] and vart P [r t+1 ] instead of discretizising equation (16) has the advantage that no approximation or discretization error is introduced into the estimation. Next, define: ɛ 1,t = r t+1 E P t [r t+1 ] (38) ɛ 2,t = (r t+1 ) 2 var P t [r t+1 ] + ( E P t [r t+1 ] ) 2. (39) Similar to Chan et. al (1992), the moment-equations are given by: E [ɛ 1,t Z t 1 ] = 0 E [ɛ 2,t Z t 1 ] = 0. 10

13 As instrumental variables we use a constant, the spot rate and the yield on a 7-year discount bond. This results in six moment conditions in three parameters, kr P, r P and σ r. The spectral density matrix S was estimated using the estimator of Newey-West (1987) with 12 lags. The results of the estimation are given in Table 1. Insert Table 1 here First, we observe that the diffusion coefficient σ r and the mean-reversion level r P are both significant at the 2.5% level but that the mean-reversion speed kr P is only significant at 5%. Secondly, the p-value for the χ 2 -test on over-identifying restrictions is equal to 0.78, indicating that the model is able to match the imposed moment conditions. 4.3 Calibration of the Price-of-risk Vector for the One-factor Models For both one-factor models the coefficients of the price-of-risk are estimated by calibrating the risk-neutral dynamics to prices of discount bonds with a 5-year maturity, given the value of the spot rate. The calibration is done by minimizing the sum of squared errors between the observed and the implied bond prices. For the original version of the Vasicek model one obtains the following value for λ 0 r: λ 0 r = , (40) which yields: r Q = (41) Notice the significant difference between the above value for r Q and the value of for r P. 11

14 For the essentially-affine one-factor model the estimates for the two parameters of the price-of-risk are equal to: λ 0 r = λ 1 r = This, in turn, leads to: k Q r = (42) r Q = (43) We see that, that as expected, the above risk-neutral value of the mean-reversion level is significantly lower in case of the original Vasicek model. 4.4 The Two-factor Version In order to complete the calibration of the two-factor model, we need to estimate the dynamics of the factor f t and the four price-of-risk parameters: λ 0 r, λ 1 r, λ 0 f and λ1 f The Physical dynamics of f t As the unobserved factor f t affects bond prices, it can be estimated from observed prices or yields. For this particular model, the factor f t can be obtained from observed bond prices, without first having to estimate any of parameters of its dynamics. All that is needed is an estimate for kr P, as demonstrated below. Let us select two bonds with fixed maturity dates T 1 and T 2 of which the prices are given by: P 1 (t, T 1 ) = exp (A(T 1 t) r t B(T 1 t) f t C(T 1 t)) and: P 2 (t, T 2 ) = exp (A(T 2 t) r t B(T 2 t) f t C(T 2 t)). A bit of algebra shows that the following equality holds: B(T 1 t) log(p (T 2 t)) B(T 2 t) log(p (T 1 t)) (44) = [B(T 1 t)a(t 2 t) B(T 2 t)a(t 1 t)] f t [B(T 2 t)c(t 1 t) B(T 1 t)c(t 2 t)]. 12

15 As in Balduzzi et al. (2000), the above equality allows us to construct a proxy for f t. Rewriting equation (44), one obtains the following expression for the factor f t : f t = b 0 + b 1 [(T 1 t)b(t 2 t)y(t 1 t) (T 2 t)b(t 1 t)y(t 2 t)], (45) where y(t 1 t) and y(t 2 t) are the yields to maturity of the two bonds, and b 0 and b 1 are two unknown constants. Because the function B(τ) is completely determined by the mean-reversion speed kr P of the spot rate, obtaining a linear transformation of the process f t once the physical dynamics of r t have been estimated, is a straightforward exercise. Of course, one still needs estimates for the constants b 0 and b 1 in order to obtain an estimate for the process f t itself. However, the feature of this model that both the price-of-risk λ f (f t ) for the process r t and the price-of-risk λ f (f t ) for the process f t are linear in f t, implies that we only need to determine the process f t up to a linear transformation. The impact of the unknown coefficients b 0 and b 1 on the risk-neutral dynamics of the spot rate r t is offset by changes in the four coefficients λ 0 r, λ 1 r, λ 0 f and λ1 f. Using the approach described above, we can obtain an estimate (up to a linear transformation) for the second factor from the yields on any two government bonds. To check for robustness, two different specifications for f t are calibrated. One estimate is based on the yields on the 1-year and 7-year discount bond, while the other uses the 2-year and 10-year discount bonds. From equation (45) we get a series for f t, which we use to estimate the remaining four parameters kf P, f P, σ f and ρ. Again, we use the analytical expressions for the moments involved: E0 P [f t ] = f P + e kp f t (f(0) f P ) (1 e 2kP t) f var0 P [f t ] = covar0 P [f t, r t ] = σ 2 f 2k P f σ f σ r ρ (1 e (kp f +kp )t) r kf P +. kp r Using a constant, the factor f t itself and the yield on a 7-year discount bond as instruments, leads to nine moment conditions in four parameters, kf P, f P, σ f and ρ. Again, the spectral density matrix S was estimated using the estimator of Newey-West (1987) with 12 lags. The results of the estimation are given in Table 2. 13

16 Insert Table 2 here We see that in contrast to the diffusion coefficient, the estimates for the two parameters of the drift term and the correlation ρ are not statistically significant for either specification of f t. The χ 2 -test on over-identifying restrictions leads to a p-value of 0.26 and 0.32 respectively, i.e., the model is able to match the moment conditions for both estimates for f t The Price-of-risk Vector for the Two-factor Model Having estimated the parameters for the physical dynamics of the factor f t, we need to obtain estimates for the values of the parameters of the price-of-risk for r t and f t : λ 0 r, λ 1 r, λ 0 f and λ 1 f. Similar to Balduzzi et al. (2000), we minimize the root mean squared price prediction error (RMSE) for discount bonds with maturities of 1, 2, 5, 7 and 10 years. For every month in our sample we compare the actual price for each of the five bonds with the price predicted by the model. We have five observations for each month for 256 months, resulting in 1280 observations. Table 3 gives the resulting values for the four coefficients which minimize the RMSE for either of the two specifications of the factor f t. Insert Table 3 here From Table 2 and Table 3 one sees that the calibration of the two-factor term structure model appears to be relatively insensitive to the choice for T 1 and T 2, the two maturities used to estimate the factor f t. Both the parameter values for the physical dynamics of f t, given by Table 2, and the estimated values for the price-of-risk parameters, in Table 3, seem to be fairly stable from one specification to the other. 14

17 Under the risk-neutral measure Q the mean-reversion level m t of the spot rate r t is itself stochastic, see equation (27). In turn, the risk-neutral mean-reversion level m Q of the process m t is given by: m Q = a 0 + a 1 f Q, (46) with a 0 and a 1 given by equations (29) and (30). For the two choices for the maturities T 1 and T 2 we obtain the following values for m Q : m Q T 1 =1,T 2 =7 = , (47) m Q T 1 =2,T 2 =10 = (48) When we compare these two values for m Q with the values for r Q obtained for the two onefactor models, we see that the two above values are quite low. Therefore, we expect the two structural models to generate the highest implied credit spreads when either of the two versions of the two-factor model is used for the dynamics of term structure. 5 Calibration of the Structural Models The default probabilities in this section are generated using Monte Carlo simulation based on a single set of sample paths (25000 paths antithetic paths), with 100 time steps per year. 5.1 The Model of Longstaff and Schwartz(1995) The calibration of the structural model follows the approach of Leland (2002). We calibrate the model to the observed 10-year default rate for each of the following three rating categories separately: A, BBB and B. The parameters for the physical interest rate dynamics are those obtained in Section 4.2, with r(t 0 ) being equal to Assuming that the asset value at t = 0 is equal to 100, we still need values for the following parameters of the structural model for each of the three rating categories: F : the nominal amount of debt δ: the pay-out rate of the firm π: the risk-premium for the asset-value process 15

18 β: the fraction the debt level that determines the default boundary α: the fraction of the assets lost in bankruptcy σ V : the asset volatility. ρ r,v : the correlation between W r and W V The parameters F, β and α are given the same values as in Leland (2002). The instantaneous correlation between the asset-value process and the spot rate is set equal to 0.2 for all three rating classes, a value close to what is used by Huang and Huang (2003). The value for the volatility σ V is obtained by calibrating the model to the observed default rate. The results are given in Table 4. Insert Table 4 here The implied asset volatilities in Table 4 are close to those of Leland (2002), who finds that the choice for the volatilities of 0.223, 0.23 and 0.32 generates a good fit for the default frequencies for the ratings A, BBB and B respectively. The implied asset volatilities are also close to those of Schaefer and Strebulaev (2004), who obtain estimates for asset volatilities of 0.22 for A-rated bonds, 0.22 for BBB-rated bonds and 0.31 for B-rated bonds, for the period December December Figure 1 plots the implied default frequencies for A and BBB-rated bonds for the parameter values in Table 4. Insert Figure 1 here 16

19 The implied default probabilities for the B-rated bonds are given in Figure 2. Insert Figure 2 here From Figure 1 and Figure 2, we see that the implied default frequencies converge to zero as the maturity decreases. However, observed default frequencies do not show this feature. For the B-rated bonds the difference with observed default frequencies is non-trivial. The implied one-year default frequency is equal to 2.5%, whereas the observed default rate is 6.5%. This effect is the result of the fact that the structural model does not allow for jumps in the asset-value process, which forces default frequencies to zero for decreasing maturities. 5.2 The Model of Collin-Dufresne and Goldstein (2001) Calibrating the structural model with the mean-reverting leverage ratio is done following the same approach as above. Here we need values for the following parameters: l 0 : the initial log-leverage ratio of the firm κ: the mean-reversion speed of the log-leverage process φ: the parameter determining the sensitivity of the target leverage-ratio to changes in the spot rate ν: the base level of the log-leverage ratio δ: the pay-out rate of the firm π: the risk-premium for the asset-value process ω: the fraction of the asset value that is lost in bankruptcy σ V : the asset volatility. 17

20 ρ r,v : the correlation between W r and W V The values for the pay-out ratio δ, the risk-premium π and the correlation ρ r,v are kept at those for the Longstaff and Schwartz (1995) model. The value for ω, the fraction of the asset value that is lost in bankruptcy, is set equal to , the value implied by the choices for α and β in the model of Longstaff and Schwartz (1995). As in Huang and Huang (2003), the value for the mean-reversion speed of the log-leverage process κ is 0.2 for all three rating categories. The value of φ is equal to 2.0, and the value of the parameter ν is set such that the implied mean-reversion level l for the log-leverage ratio is equal to 0.38, again as in Huang and Huang (2003). Table 5 gives the results. Insert Table 5 here The top panel in Table 5 gives the parameter values for the mean-reversion of the leverage ratio, where the value of the parameter ν is set such that the target mean-reversion level l is met. The lower panel of Table 5 gives the parameter values for the dynamics of the asset-value process, together with the asset volatility implied by the given default probability. Here, the parameter values for l 0, κ, φ and ν from the top panel are used as inputs. Comparing Table 4 and Table 5, we see that the implied values for asset volatilities do not differ much between the two structural models, the values being only slightly lower for the model of Collin-Dufresne and Goldstein (2001). The implied default rates for A and BBB-rated bonds are shown in Figure 3, and Figure 4 gives the implied default probabilities for the B-rated bonds. Insert Figure 3 here 18

21 Insert Figure 4 here As anticipated, we see that the implied default probabilities again converge to zero for decreasing maturities. Observe that whereas the cumulative default probabilities implied by the model of Longstaff and Schwartz (1995) are clearly concave over the entire range of maturities, the probabilities implied by the Collin-Dufresne and Goldstein (2001) model appear to be linear in the maturity of the bond for the upper range of the maturities. The intuition behind this difference can be found in the different default mechanisms of the two structural models. Because of the constant default boundary in the model of Longstaff and Schwartz (1995), the default intensity for high maturities is low. If a firm has already survived for a considerable number of years, it is likely to have moved away from the default boundary over that period. Hence, the conditional probability of default in the next period is low. In contrast, in the model of Collin-Dufresne and Goldstein (2001), when a firm has survived over a long period, its leverage will have been (partially) adjusted for the evolution of its asset value over that period. As the default boundary is determined by the leverage, the conditional probability of default in the next period will still be fairly high. 6 The Results: The Implied Yield Spreads The yield spreads in this section are generated using Monte Carlo simulations, based on the same set of sample paths as in the previous section, and again with 100 time steps per year. 6.1 The Results of the Model of Longstaff and Schwartz (1995) Table 6 gives the yield spreads implied by the model of Longstaff and Schwartz (1995) for bonds with a 10-year maturity under different assumptions about the interest rate dynamics. The parameters values are those of Table 4 in Section

22 Insert Table 6 here From Table 6, we see that the yield spreads implied under the original Vasicek model are tiny, particularly so for the two investment grade bonds. Allowing for the essentially affine version of the Vasicek model leads to somewhat higher implied yield spreads, but the effect is fairly weak. Looking at the two last columns, we see a very different picture. Whereas under the two one-factor interest rate models the implied yield spreads on the investment grade bonds are all well below 50%, they all are above 50% under the two versions of the two-factor model. Both Longstaff et al. (2005) and Churm and Panigirtoglou (2005) find that the default risk accounts for at least 50% of observed yield spreads across all rating classes. Longstaff et al. (2005) also find that as much as 83% of the observed yield spread on BBrated bonds is due to exposure to default risk. Therefore, also the high implied yield spreads for the B-rated bonds under the two versions of the 2-factor model, respectively 93.3% and 87.3% of observed yield spreads, seem realistic. Insert Figure 5 here Insert Figure 6 here 20

23 Insert Figure 7 here Figures 5, 6 and 7 show the yield spreads implied under the different interest rate dynamics for a wide range of maturities. As with the implied default probabilities, the implied yield spreads converge to zero for decreasing maturities, leading to an underestimation of yield spreads for small maturities in all three figures. Observe that for the all three rating classes the general shape of the credit spread curve is not affected by the choice for the interest rate dynamics. For the two investment grade bonds the spread curve initially increases, to flatten of at the high maturities. For the B-rated bond, the four spread curves also show a very similar pattern, all reaching a peak around four years and decreasing to much lower values at high maturities. These figures suggest that reducing the drift of the asset-value process under the riskneutral dynamics greatly improves the ability of the structural model of Longstaff and Schwartz (1995) to generate more realistic, i.e. higher, credit spreads. The strength of this result lays in the fact that the difference in the drift of the asset-value process across the four cases, and therefore the difference in implied yield spreads, is completely determined by the specification of the risk-neutral interest rate dynamics. The exposure to default risk is exactly the same across the four cases, as the structural model is only calibrated once, using as physical dynamics for the spot rate those of the original term structure model of Vasicek (1977). 6.2 The Results of the Model of Collin-Dufresne and Goldstein (2001) Table 7 gives the yield spreads implied by the model of Collin-Dufresne and Goldstein (2001) for bonds with a 10-year maturity under different assumptions about the interest rate dynamics. The parameters values are those of Table 5 in Section 5.2. Insert Table 7 here 21

24 We see that with the two versions of the one-factor term structure model implied yield spreads are again too low, lower even than those of the Longstaff and Schwartz (1995) model. Therefore, allowing for a mean-reverting leverage ratio itself does not seem to improve the performance of the structural model, a result also obtained by Huang and Huang (2003). In contrast, the credit spreads generated under the the two specifications of the two-factor model are higher than those of the Longstaff and Schwartz (1995) model. With overestimation being clearly the case for the specification of the two-factor model in the one but last column. Figures 8, 9 and 10 gives the implied yield spreads for A, BBB and B-rated bonds, respectively. Figures 8 and 9 confirm the results of Table 7. The two one-factor term structure models imply yield spreads that are too low, whereas the two versions of the two-factor model give rise to overshooting yield spreads for higher maturities on investment grade bonds, e.g. reaching 2% for the A-rated bond for a maturity of 20 years for one of the two specifications. For the B-rated bonds, Figure 10 shows that the model of Collin-Dufresne and Goldstein (2001) generates too high credit spreads for short-term maturities under all four specifications for the interest rate dynamics. Spreads range between 7% and 8% at 3 years, with the highest values being implied by the two specifications of the two-factor model. Insert Figure 5 here Insert Figure 6 here 22

25 Insert Figure 7 here We see that with the model of Collin-Dufresne and Goldstein (2001) the difference in implied yield spreads between the two one-factor interest rate models and the two versions of the two-factor model is even bigger than for the model of Longstaff and Schwartz (1995), now leading to overshooting in the case of the two investment grade bonds. 7 Conclusions This paper examines the impact of the choice of risk-neutral interest rate dynamics in structural models for corporate bonds. As argued in Section 3, the risk-neutral dynamics of the spot rate directly affect the drift term of the risk-neutral dynamics of the asset-value process, possibly driving the asset-value away from the default boundary. The results indicate that this problem might be solved by a careful choice of the interest rate dynamics. For the model of Longstaff and Schwartz (1995), the results show that implied yield spreads vary significantly across the four different choices for the risk-neutral interest rate dynamics. Particularly the two-factor term structure model gives rise to credit spreads much more in line with observed yield spreads. Therefore, in a structural model with a constant default boundary the right choice for the risk-neutral interest rate dynamics seems to solve the problem of too low implied credit spreads. The results for the model of Collin-Dufresne and Goldstein (2001) reinforce these findings. Compared with the yield spreads generated under the original model of Vasicek (1977) for the interest rate dynamics, both versions of the two-factor model lead to significantly higher implied yield spreads. For investment-grade bonds the two-factor term structure model even leads to overshooting of yield spreads. All in all, the results in this paper provide evidence that a careful specification of the risk-neutral interest rate dynamics can significantly improve the yield spreads implied by structural models for corporate bonds. 23

26 Appendix A. Prices of Zero-Coupon Bonds in the Model of Vasicek (1977) The price at time t of a discount bond with maturity date T is: P (t, T ) = exp {A(T t) B(T t)r t }, (49) with: B(τ) = 1 e kp r τ kr P, (50) ( (k ) (B(τ) 1) P 2 r r Q ( σ P ) ) 2 ( ) r /2 σ P 2 A(τ) = (kr P ) 2 r τb(τ) 2 4 (kr P. (51) ) Appendix B. Prices of Zero-Coupon Bonds in the Essentially Affine One- Factor Model The price at time t of a discount bond with maturity date T is given by: P (t, T ) = exp {A(T t) B(T t)r t }, (52) where now: B(τ) = 1 e kq r τ, (53) A(τ) = k Q r (B(τ) 1) ( ( ) 2 kr Q r Q ( ) σr P ) 2 /2 ( ) 2 kr Q ( σ P r ) 2 τb(τ) 2 ( ). (54) 4 kr Q Appendix C. Prices of Zero-Coupon Bonds in the Two-Factor Model Here we derive the expression for the price of a discount bond in the two-factor model. Observe that the risk-neutral dynamics of the spot rate are the same as in the stochastic-mean model of Balduzi et al. (2000). Combining their analytic results with some straightforward but tedious algebra leads to the following expression for the value P (t, T ) at time t of a zero-coupon bond with maturity date T : P (t, T ) = exp (A(T t) r t B(T t) f t C(T t)). (55) 24

27 The functions A(T ), B(T ) and C(T ) are given by: A(τ) γr 3 σf 2 = 4βr 3 βf 3 (β rβ f + 1) 2 + e2 βrβ f γr τ σ 2 f γ3 r γ r (β r 2 α f β f σf 2γ2 r + β r β f σ f ρσ r γr 2 βr 3 βf 3 (β rβ f + 1) 4βr 3 βf 3 (β rβ f + 1) 2 ) ( βrβ f β 2 +e γr τ r α f β f σf 2γ2 r + β r β f σ f ρσ r γr 2 βr 3 βf 3 (β rβ f + 1) ) γ r + γ3 r σ f ( σ f + β r β f ρσ r + ρσ r ) β r β f (β r β f 1) (β r β f + 1) 2 ) γ r ( β r 2 α f β f + σf 2γ2 r 2β r β f σ f ρσ r γr 2 σ f ρσ r γr 2 + βr 2 βf 2σ2 rγr 2 + β r β f σrγ 2 r 2 β r β f (β r β f + 1) ) γ +e 2τ r 3 γr γr 3 + +e τ γr ( σ 2 f 2β rβ f σ f ρσ r + 2σ f ρσ r + β 2 r β 2 f σ2 r + 2β r β f σ 2 r + σ 2 r 4σ f (β r β f + 1) 2 ( σf 2 2β rβ f σ f ρσ r 2σ f ρσ r + βr 2 βf 2σ2 r + 2β r β f σr 2 + σr 2 4 (β r β f + 1) 2 γ r ( β 2 r α f β f + σ 2 f γ2 r 2β r β f σ f ρσ r γ 2 r σ f ρσ r γ 2 r + β r β f σ 2 rγ 2 r β r β f (β r β f + 1) + 2β2 r α f β f + σf 2γ2 r 2β r β f σ f ρσγr 2 + βr 2 βf 2σ2 rγr 2 2βr 2 βf 2 τ e τ γr 3 σ f ( σ f + β r β f ρσ r + ρσ r ) γr β r β f (β r β f 1) (β r β f + 1) 2 ( ( ) ) α r β r β f e 1 γr τ 1 + e βr β f γr τ 1, (β r β f + 1) β r β f ) ) with: and: B(τ) = 1 e γrτ γ r, C(τ) = γ ( r e β f τ 1 ) β f (e γrτ 1), β f (β f γ r ) 25

28 where: α r = kr P a 0, β r = kr P a 1, γ r = kr P, α f = k Q f f Q, β f = k Q f. 26

29 References P. BALDUZZI, S.R. DAS, S. FORESI and R.K. SUNDARAM (2000) Stochastic Mean Models of the Term-Structure of Interest Rates, Chapter 5 in Advanced Tools for the Fixed-Income Professional, B. Tuckman and N. Jegadeesh, Eds, John Wiley and Sons, New York, K. C. CHAN, G. KAROLYI, F. LONGSTAFF and A. SANDERS (1992) An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of Finance 47, R. CHURM and N. PANIGIRTZOGLOU (2005) Decomposing Credit Spreads, Working Paper Nr. 253, Bank of England. P. COLLIN-DUFRESNE and R. GOLDSTEIN (2001) Do Credit Spreads Reflect Stationary Leverage Ratios?, Journal of Finance, 56, P. COLLIN-DUFRESNE, R. GOLDSTEIN and J. Martin (2001) The Determinants of Credit Spread Changes, Journal of Finance, 56, J. DRIESSEN (2003) Is Default Event Risk Priced in Corporate Bonds?, University of Amsterdam Working Paper. G.R. DUFFEE (2002) Term Premia and Interest Rate Forecasts in Affine Models, Journal of Finance, 58, E. ELTON, M. GRUBER D. AGRAWAL and C. MANN (2001) Explaining the Rate Spread on Corporate Bonds, Journal of Finance 56, Y.H. EOM, J. HELWEGE and J. HUANG (2004) Structural Models of Corporate Bond Pricing: An Empirical Analysis, Review of Financial Studies 17, J. HUANG and M. HUANG (2003) How Much of the Corporate-Treasury Yield Spread is Due to Credit Risk?, Working paper Stanford University. H. LELAND (2002) Predictions of Expected Default Frequencies in Structural Models for Debt, Working paper Haas Business School, University of California, Berkley. F. LONGSTAFF (2004) The Flight to Liquidity Premium in US Treasury Bond Prices, Journal of Business 77,

30 F. LONGSTAFF, E. NEIS and S. MITHAL (2005) Corporate Yield Spreads: Default Risk or Liquidity?, Journal of Finance 60, F. LONGSTAFF and E. SCHWARTZ (1995) Valuing Risky Debt: A New Approach, Journal of Finance 50, R.C. MERTON (1974), On the Pricing of Corporate Debt: the Risk Structure of Interest Rates,Journal of Finance 29, W. NEWEY and K. WEST (1987) A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica 55, I. STREBULAEV and S. SCHAEFER (2204) Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios, EFA 2004 Maastricht Meetings Paper No O. VASICEK (1977), An Equilibrium Characterization of the Term Structure, Journal of Financial Economics 5,

31 Figure 1: The PD s implied by the model of Longstaff and Schwartz (1995) for A and BBB rated bonds Figure 2: The PD s implied by the model of Longstaff and Schwartz (1995) for B rated bonds 29

32 Figure 3: The PD s implied by the model of Collin-Dufresne and Goldstein (2001) for A and BBB rated bonds Figure 4: The PD s implied by the model of Collin-Dufresne and Goldstein (2001) for B rated bonds 30

33 Figure 5: The yield spreads implied by the model of Longstaff and Schwartz (1995) for A rated bonds In the above plot, the solid line gives the yield spreads implied under the original Vasicek term structure model, the dashed line plots the spreads generated under the one-factor model in which the price of risk for the spot rate is a function of the spot rate itself. Finally, the dot-dashed line and the dotted line give the the yield spreads implied by the two-factor term structure model: the dot-dashed line with the maturities for the two bonds used to estimate the factor f t equal to 1 and 7 years, and the dotted line when these two maturities are 2 and 10 years. 31

34 Figure 6: The yield spreads implied by the model of Longstaff and Schwartz (1995) for BBB rated bonds In the above plot, the solid line gives the yield spreads implied under the original Vasicek term structure model, the dashed line plots the spreads generated under the one-factor model in which the price of risk for the spot rate is a function of the spot rate itself. Finally, the dot-dashed line and the dotted line give the the yield spreads implied by the two-factor term structure model: the dot-dashed line with the maturities for the two bonds used to estimate the factor f t equal to 1 and 7 years, and the dotted line when these two maturities are 2 and 10 years. 32

35 Figure 7: The yield spreads implied by the model of Longstaff and Schwartz (1995) for B rated bonds In the above plot, the solid line gives the yield spreads implied under the original Vasicek term structure model, the dashed line plots the spreads generated under the one-factor model in which the price of risk for the spot rate is a function of the spot rate itself. Finally, the dot-dashed line and the dotted line give the the yield spreads implied by the two-factor term structure model: the dot-dashed line with the maturities for the two bonds used to estimate the factor f t equal to 1 and 7 years, and the dotted line when these two maturities are 2 and 10 years. 33

36 Figure 8: The yield spreads implied by the model of Collin-Dufresne and Goldstein (2001) for A rated bonds In the above plot, the solid line gives the yield spreads implied under the original Vasicek term structure model, the dashed line plots the spreads generated under the one-factor model in which the price of risk for the spot rate is a function of the spot rate itself. Finally, the dot-dashed line and the dotted line give the the yield spreads implied by the two-factor term structure model: the dot-dashed line with the maturities for the two bonds used to estimate the factor f t equal to 1 and 7 years, and the dotted line when these two maturities are 2 and 10 years. 34

37 Figure 9: The yield spreads implied by the model of Collin-Dufresne and Goldstein (2001) for Baa rated bonds In the above plot, the solid line gives the yield spreads implied under the original Vasicek term structure model, the dashed line plots the spreads generated under the one-factor model in which the price of risk for the spot rate is a function of the spot rate itself. Finally, the dot-dashed line and the dotted line give the the yield spreads implied by the two-factor term structure model: the dot-dashed line with the maturities for the two bonds used to estimate the factor f t equal to 1 and 7 years, and the dotted line when these two maturities are 2 and 10 years. 35