Exogenous-Dynamic

Lastly, we consider a group of structural models in which the default boundary is as- sumed to vary stochastically through time, but unlike the endogenous models, time- variation in the default boundary is governed by an exogenously known stochastic pro- cess and not by shareholders seeking to maximise their wealth at each point in time.

The first group of related models that fits this description belong to models that share the assumption that the default boundary is a traded financial instrument. Nielsen et al. (1993) and Sa´a-Requejo & Santa-Clara (1999) let the default boundary be the exogenously determined market value of the firm’s liabilities. The default boundary value is specified as a geometric Brownian motion under risk neutrality with correlated innovations with the firm asset return and risk-free rate. The state variable, for pricing purposes, is the log of the ratio of firm value to the default boundary,x(t), which is shown to be an arithmetic Brownian motion with constant driftµx (Sa´a-Requejo & Santa-Clara 1999, Equation 5)

dx(t) =µxdt+σxdWx,t , (2.20)

whereσxis the volatility of the change in the solvency ratio,x(t), anddWx,t is a Brownian motion correlated with the risk-free rate innovations. Default is triggered on the first passage ofx(t) to zero. The constants in equation (2.20) summarise a larger number of parameters that we do not reproduce here.11 _{Since we only need to estimate the joint} process in equation (2.20), the model has been criticised for its unwarranted complexity by Uhrig-Homburg (2002, p. 54). Not surprisingly, the model shares some similarity with a constant rate version of the LS model. The key difference between the two models is that the drift rate does not vary with the risk-free rate, which it does in the LS model.12 A further difficulty with the model is that the market value of the firm’s liabilities must approach the value of the firm in default; at some stage the two market assetsV andK become one. This does not appear to have been treated in the model. Recently, Hsu, Saa- Requejo & Santa-Clara (2003) have redefined the meaning ofV to be the continuation value of the firm in a non-default state andK to be the value of the firm in bankruptcy. However, it still appears problematic to treatVandKas separate financial assets when at default they are the same asset, and it is conceptually problematic to correlate the return on the same underlying asset in two mutually exclusive states of the world; V and K cannot trade simultaneously. In contrast, K is usually treated as a threshold value ofV in the exogenous boundary literature, or alternatively, within the endogenous boundary literature the value of the firm’s liability is valued coherently with firm assets and equity. In neither case does the default boundary suffer from definitional problems.

11_{Refer Sa´a-Requejo & Santa-Clara (1999, Equations (6)-(8)).}

12_{If we were to fit the two models implicitly from market prices, we would find it impossible to dis-}

tinguish between Sa´a-Requejo & Santa-Clara (1999), with its correlated stochastic boundary and constant driftµx, and a constant short rate LS model with its constant boundary and drift(r−δv).

The other type of exogenous-dynamic boundary model assumes that the default boundary varies in a stochastic manner that mirrors the stylised facts of observed capital structure dynamics. Thus, the models permit expected management driven capital struc- ture changes, but the expected debt-ratio behaviour must be exogenously known before debt can be valued.

As discussed in Section 2.4.1.2, there is good theoretical reason to suggest that man- agement (acting on behalf of shareholders) dynamically adjust their firm’s debt-ratio over time. Rather than attempting to fully explain debt-ratio dynamics as a consequence of shareholder wealth maximisation, the exogenous-boundary models take an assumed debt-ratio behaviour as given. The theoretical complexity is reduced, but the disadvan- tage is that we must a priori form an opinion as to the most appropriate underlying stochastic process for the firm’s capital structure process.

Further, the process for the latent log-solvency ratio (the state variable,x(t)) must be parameterised. To date, the exogenous-dynamic empirical literature has parameterised expected capital structure dynamics from observed changes in book debt-ratios. How- ever, since we have shown that the firm’s book value of debt to be potentially unreliable as a measure of the default boundary, using capital structure parameters sourced from debt-ratio movements may also be flawed. Alternatively, we propose to parameterise the capital structure process implicitly from the credit spreads and avoid the potential measurement error introduced by use of proxy variables.

Taur´en (1999) propose a structural model in which the state variable is the firm’s ratio of book liabilities to its market value of assets that follows a mean reverting stochastic process. Thus, the default boundary, represented by the firm’s liabilities, is assumed to be stochastic and governed by a known process that represents, in a reduced manner, the dynamic behaviour of the firm’s management. Like the endogenous-dynamic literature, permitting mean-reversion in firm-leverage captures the additional risk to bondholders of management’s option to re-leverage the firm in the future. The effect of assuming mean- reversion in firm leverage results in higher forward default rates, and a flatter credit spread term structure relative to the Merton and LS models. Longer-term credit spreads are an increasing function of the firm’s target leverage and decreasing with respect to the speed of reversion. In addition, short-term spreads are more sensitive to the current level of leverage. It is argued by Taur´en that capital structure mean-reversion results in more realistic credit spread term structures.

Independently, Collin-Dufresne & Goldstein (2001) (hereafter CDG) suggested a similar model, which has become the most widely known and empirically studied target- leverage structural model, empirically fitted by EHH and HH. CDG differ from Taur´en in their choice of state variable and boundary dynamics. The default boundary is assumed to change dynamically over time. As in Taur´en the firm adjusts its level of debt, mean reverting to a long-run target level. Secondly, the firm will issue debt opportunistically

to time the debt market, issuing more (less) debt when the current risk-free rate is below (above) the long-run risk-free rate level and therefore expected to increase (decrease) in future. The default boundary is assumed to evolve as

dlnK(t) =κv lnV(t)₋lnK(t)₋ν₋φ(r(t)₋θ)

dt

=κv x(t)₋ν₋φ(r(t)₋θ)

dt. (2.21)

Thus, the default boundary is assumed to be a function of the current level of the log- solvency ratio, x(t), a long-run target level of the log-solvency ratio, ν, the speed of mean-reversion to the target in the absence of debt market timing,κv_≥0, the sensitivity of the firm’s debt issuance policy to the expected change in the risk-free risk-free rate, φ, and the trend in risk-free rate expectations given by the difference in the current short rate,r(t), and its expected long-run level,θ, as per Vasicek’s (1977) interest rate model. The LS model is nested within CDG. For estimation purposes we restate the models to be dependent on their log-solvency ratiosx(t)defined as the log of the ratio of the firm’s market value to its default boundary (refer Appendix A for derivation). The dynamic process for the log-solvency ratio in the LS model is given by

dx(t) = (r(t)₋δ₋σ_v2/2)dt+σvdW_vQ_,t, (2.22)

and for the CDG model it is dx(t) =κv r(t)₋δ₋σ2 v/2 κv +ν+φ(r(t)−θ) −x(t) dt+σvdWvQ,t, (2.23) wheredWv,t is correlated with the short rate under the Vasicek (1977) model.

From equation (2.22), it is evident that the LS model has no mean-reversion; share- holders do not invoke the option to re-leverage nor attempt to reduce bankruptcy costs. In the other models, there is an assumed continuous adjustment towards a long run solvency ratio target, that is symmetrical above and below the target. Implicit in this specification is that capital structure adjustments are continuous and that shareholders actively adjust debt-ratios downwards in response to rising bankruptcy costs. We can also see that the log-solvency ratio drift is time-varying and positively related to the risk-free rate. An increase in the short rate, ceteris paribus, decreases default risk and credit spreads. From equation (2.23), the CDG model’s log-solvency drift rate is shown also to be positively influenced by the level of target log-solvency, ν, and to the size of the risk-free term- structure slope; the latter is intended to capture management debt-timing behaviour.

Mueller (2000) shows that the drift rate in equation (2.21) can be expanded to in- clude multivariate factors influencing the direction of capital structure decisions. The resultant model has the disadvantage of a significant increase in numeric processing needed to solve the expected first passage crossing time. More recently, Demchuk &

Gibson (2006) adapt the CDG model without a substantial increase in numeric complex- ity. They assume that management adjusts debt levels continuously toward a target as per CDG, but with the target a stochastic function of by recent past equity returns.