Collin-Dufresne and Goldstein

In the CDG model, the dynamic term structure of credit spreads is assumed to be a func- tion of a bivariate stochastic differential process for the mean-reverting capital structure of the firm and the risk-free risk-free rate

dx(t) =κv r(t)₋δ₋σ2 v/2 κv +ν+φ(r(t)−θ) −x(t) dt+σvdW_vQ_,t dr(t) =κr(θ₋r(t))dt+σrdW_rQ_,t.

As per the implementation of the LS2 model, we follow the two-step estimation method of Duffee (1999) and fit the risk-free rate process separately from the log-solvency pro- cess, thereby ensuring that the same risk-free rate model parameters are applied equally to each firm. The risk-free rate is then assumed to be exogenously known and the log- solvency process implied from the observed credit spreads using the optimal estimates of the risk-free rate, fitted from a prior filtration. Consequently, the state-space framework

simplifies to be univariate and the transition equation is the generic transition equation (equation (3.12)) with the elements:

α(t) =x(t), c¯(t) =κvx¯(t)∆t,

¯

T =1₋κv∆t, R(t) =σv√∆t, (3.47)

where ¯x(t)is the risk-neutral target log-solvency ratio

¯ x(t) = r(t)₋δ₋σ2 v/2 κv +ν+φ(r(t)−θ) . (3.48)

The first-passage crossing time of the log-solvency state variable must be numerically solved due to the CDG model having two stochastic processes. The numeric grid method of CDG is used, details of which are described in Appendix B.1. The time involved in searching for an implicit solution for the CDG model is substantial with processing times, for a single firm solution, varying from 5 hours for the no-liquidity case, to 24 hours for the time-varying liquidity case.6 An important numerical simplification made is that we value only the principal cash flows and not the coupons of each bond. The simplification proved to be reasonable because of the panel nature of the data, there are sufficient bonds to span the term structure for each firm. Further, we seek to minimise the prediction error of the credit spread, valuing both the risky and risk-free values of the bond using the same set of promised cash flows. Thus, the simplification is present in the value of both sides of the credit spread calculation.7

To aid estimation, further simplifying assumptions can reasonably be made. Firstly, the asset payout rate is assumed to be zero following the finding of EHH, that the effect of the payout ratio on debt pricing in the CDG model is exactly cancelled by the inclusion of a target debt-ratio. The amount of payout in interest, dividends, capital raisings and share repurchases are subsumed into management’s choice of the speed of adjustment toward a target debt-ratio and level of the target.

Secondly, the asset-interest rate correlation is assumed to be zero. EHH report a correlation of -2 percent and find that CDG model’s credit spread prediction errors vary little with respect to correlation. Table 3.14 shows the correlation between daily changes in the 3-month CMT rate and daily changes in the observed log-solvency ratio to be 0.38 percent.

Finally we consider the problem of identifying the solvency-ratio dynamic param-

6_{In comparison, the LS1 model times range from 4 minutes to 38 minutes. All computations were}

performed in OX software using a desktop PC running Windows XP on a Pentium 4, 2.53 GHz processor.

7_{Testing showed no material difference in the estimates of the LS1 model when shifting from valuing all}

cash flows to just the principal cash flows. We can draw further comfort that no significant errors have been introduced by referring to the results shown in Table 4.7. The standardised step-ahead prediction errors are of comparable magnitude between the analytically calculated LS1 model with no cash flow simplification, and the numerically solved LS2 model with only the principal amounts valued.

eters, ν, φ, and κv. The extant empirical estimation method is exemplified by EHH and Suo & Wang (2006). It involves estimating the parameters of the CDG asset pro- cess from a regression of observed firm-specific leverage ratios and interest rates. In the remainder of this section we explore this regression method as a potential means of identifying initial parameter estimates.

The regression equation for the CDG model is obtained by first expressing the log- leverage process under the physical measure with a constant firm asset growth ofµ

dx(t) =κv µ −δ−σv2/2 κv +ν+φ(r(t)−θ) −x(t) dt+σvdW_vP_,t. (3.49) Letαx=µ₋δ₋σ_v2/2+κv(ν₋φθ)then, dx(t) = (αx+κvφr(t)₋κvx(t))dt+σvdW_vP_,t. (3.50)

Equation (3.50) is then discretised by the Euler approximation

x(t)₋x(t−1) = (αx+κvφr(t−1)₋κvx(t−1))∆t+σv√∆tη(t), (3.51)

whereη(t)_∼N(0,1). For estimation purposes, equation (3.51) is then expressed as a linear equation on the risk-free rate and the lagged observed log solvency ratio, S(t), with a normal i.i.d. error termε(t)

S(t)₋S(t₋1)

∆t =a+bS(t−1) +cr(t−1) +ε(t) (3.52) wherea_≡αx, b_{≡ −}κv, and c_≡κvφ. Parameter estimates are then obtained from the slope coefficients: ˆκv=₋b, and ˆφ=₋c/b. To find an estimate ofνfrom the regression method it is also necessary to know the firm’s expected asset return. Let

ˆ ν=φ θˆ +a−µ+δv+σ 2 v/2 ˆ κv . (3.53)

Then, µ can be further expressed as the sum of the risk-free rate and a market risk premium on the firm’s assets.

EHH regress 10 years of monthly data and find an average mean-reversion rate of 0.1. The expected asset return,µ, is obtained from the monthly 10 year historical firm value return and on average is 24 percent. They recognise that this is an ex-post measure not necessarily the market’s required return and find that their model prediction errors are sensitive to the choice ofµ. EHH suggest that the high absolute spread prediction errors they find with the CDG model may be a result of their estimation method. Alter- natively, they suggest fitting an implied level ofν from credit spreads. Unfortunately, no regression statistics or sample estimate ofν is reported to confirm the significance

of their result. Suo & Wang (2006) find a mean reversion rate close to zero, and conse- quently, their implementation of the CDG model is little different to the LS model.

As an alternative to regression, HH simply assume the long-run risk-neutral log- solvency level, ¯x, to be 0.38, which gives an estimate forν of 0.55 based on other as- sumed parameters including an asset risk premium.8 CDG similarly assumeνto be be- tween 0.5 and 0.6 which equates to a target debt-ratio of similar magnitudes. The target debt level chosen by CDG for illustrative purposes appears very conservative and may result in an over-estimation of long-term credit spreads. Opler & Titman (1994) report similar debt levels for firms in the top 20 percent of population gearing levels. For other firms in normal industry conditions the debt-ratio is 0.193 giving a log-solvency level of 1.65. More recently, using the same data sources and time period as our study, Davy- denko & Strebulaev (2004) report a higher mean debt-ratio of 0.322, which is equivalent to mean log-solvency ratio of 1.13.

The result of applying equation (3.52) to our sample of firms is shown in Table 3.13. Data is observed quarterly, over the sample period, to match the release of COMPUSTAT balance sheet debt figures. The rate of mean-reversion is found, on average, to be 0.79 per quarter across all firms, with estimates ranging from -0.54 to 3.56, with a median of 0.70. In most cases the estimate of mean reversion is not significantly different from zero. The average mean-reversion rate is much higher than the capital structure empirical literature has found for the debt-ratio dynamics in recent years. For example, Roberts (2002, Table 3) reports 0.16 using a similar definition of leverage. In comparison, CDG and HH assume the average firm mean-reversion rate to be 0.18.

Table 3.13 also shows that the observed sensitivity of the firm’s log-solvency to the slope of the yield curve is negative as often as it is positive. This observation contradicts the debt market timing hypothesis of CDG, which assumes that changes in the firm’s sol- vency are positively related to the slope of the risk-free yield curve slope. However, in no case is the estimated parameter, ˆφ, significant at the 5 percent level. The across-firm mean value of ˆφ is -12.67 with a very high standard deviation of 60.8 and a median of 0.64. EHH do not report their regression estimate. HH adopted an implicit assumption that φ is zero by omitting the parameter as evident in their specification of the CDG model (Huang & Huang 2003, Appendix A, equations (23) and (24)). For illustrative purposes CDG assume φ to be 2.8. We initialiseφ at the firm-wide median regression estimate of 0.64 and restrict its value to be positive consistent with the theoretical re- striction in the CDG model. However, given these results, it is not expected thatφ will be significant in the filtered model estimates.

Faced with limited results from the regression model we have little choice but to consider alternatives to the standard empirical estimation methods for the other asset pa-

8_{Estimate is based on HH assumptions of: r=8 percent,δ}

v=6 percent,κv=0.18,σv=26.5 percent,

Table 3.12: This table shows the average target log-solvency levels assumed in the CDG model imple- mentations. Data is sourced from Roberts (2002) and is matched to firms by second level SIC. The target log-solvency level is denotedνand the initial starting value of log-solvency is denotedx(0)and is the first observed log-solvency ratio in the sample period. Observed log-solvency is the sum of total book debt (COMPUSTAT items 45 and 51) and market equity capital (from CRSP), divided by total book debt.

Sector No. of Firms ν x(0)

Industrial 21 1.42 1.12

Finance 6 0.46 0.48

Utility 4 0.19 0.90

Total 31 1.36 0.94

rameters. Two alternatives were considered. The first uses the property that the LS model is nested in the CDG model. The mean filtered estimate ofx(t)from the LS2 model may be used as a reliable estimate of the average level of latent log-solvency, which we know will match the CDG model, as mean-reversion rate approaches zero. The advantage of this method is that the target log-solvency level is treated as a latent firm-specific vari- able. The disadvantage is that it is a measure of the sample firm-specific mean only, and not necessarily a target that is pursued by management. Only over a longer sample period, and across more firms, can we abstract sufficiently from idiosyncratic factors. The second method, which we adopt, uses an estimate ofνunder the mild assumption that debt timing is not material; an assumption well supported by our earlier regression of the observed changes in log-solvency as shown in Table 3.13. From equation (3.54) it can be seen thatνhas a directly observable physical interpretation. Ignoring debt-timing behaviour for simplicity, CDG posits that management changes the level of debt so that the log-solvency ratio mean-reverts toν where ν is a target level of log-solvency. To see this consider the dynamics of the default boundary in the absence of debt timing behaviour. The natural log of the default boundary then follows the process

dlnK(t) =κv(x(t)₋ν)dt+σdW_tP. (3.54) To use the sample specific mean level ofx(t) from the LS model would, for example, underestimate the ex-ante target level of solvency of a firm that ex-post suffered a dete- rioration in value. Rather, an industry-level observed log-solvency level is a better proxy for management’s desired target. The industry mean is preferred over the firm-specific sample averages since it is less influenced by idiosyncratic shocks from target, and when averaged over many firms and time, is more likely to represent a long-run equilibrium level. The use of industry is a natural conditioning variable since Opler & Titman (1994) find that target debt-ratio levels vary systematically with industry-specific business con- ditions. Each firm’s physical target level is set equal to the industry-specific 1980 to 1998 average of the observed log-solvency ratio as reported by Roberts (2002, Table

1). His study of debt-ratio dynamics is useful for our purposes because it draws on a similar sample of firms over the same time period. Firms are mapped to industries by their second level SIC codes. The rate of mean-reversion is then included in the hyper- parameter set using the Roberts’s (2002) industry-wide average to initialise, and fixingν to the industry average observed log-solvency ratio. Attempts at searching implicitly for both parameters was found to be infeasible due to multicollinearity. Our sample average value ofνis 1.36 implying a mean target debt-ratio of 0.27. In Table 3.12 the variation by industry sector is shown along with the initial values of log-solvency. For industrial firms, our initial parameterisation suggests a decrease in leverage over time, little change for finance companies, and an increase over time for utilities.

Conditional on the treatment of the non-credit component of the credit spread, the three hyperparameter sets are as follows after suppressing dependence on the firm for notational clarity: No liquidity: ψh_{(1) ={ln}σ2 v,lnσm2,lnκv,φ}, Constant liquidity: ψh_{(2) ={ln}σ2 v,lnσm2,lnκv,φ,lndi}, Time-varying liquidity: ψh_{(3) ={lnσ}2 v,lnσm2,lnκv,φ,lnβR,lndi}. (3.55)

Further parameters are:δv=0,ρv,r=0.