Exogenous-Static

We use the term exogenous-static to group models in which the firm’s default boundary is determined independently of shareholder preferences and is a constant function of time.

The simplest version of this model is to assume that the firm is funded by a single bond and its default boundary is determined by the bond’s indenture. For example, the Merton model assumes that the default boundary is equal to the face value of debt payable only at maturity. Black & Cox (1976) permit early default where they let the default barrier be equal to a minimum solvency level, as contained in the bond indenture, which is assumed to grow over time at the risk-free rate. A more common method to modelling the triggering of early default was first suggested by LS. Their model assumes the default barrier is constant through time but are silent on the level of the barrier. Importantly, they show that the ratio of firm value over the default barrier is a sufficient state variable to value bonds with coupons and across multiple bonds issued from the same firm. The spanning of multiple securities by a predefined underlying state process, is the defining feature that makes the exogenous-boundary model form useful in finance, and is the property we exploit further in Section 3.2 to to derive estimates of the state process. To illustrate this property we first consider the LS model in detail.

LS define default as the first passage of firm asset valueV(t)across a constant default boundary, K. Unexpected shocks inV are correlated byρr_,V with a stochastic risk-free rater(t)that follows a mean-reverting stochastic process as per Vasicek (1977), thereby characterising LS as a two-factor model of the log-solvency ratiox(t) =lnV(t)/K

dx(t) = (r(t)₋δ₋σ 2 v 2 )dt+σvdW Q V,t (2.16) dr(t) =κr(θ₋r(t))dt+σrdW_rQ_,t, (2.17) where, W_VQ_,t and WrQ,t are Weiner processes, κr is the speed of mean-reversion for the instantaneous short rate,θis the long-run level ofr(t), andσris the short rate volatility. Defining the first passage stopping time byτ=inf_{t_≥0 :x(t) =0_}then the proba- bility of default betweent=0 andT is

Q(0,T;x(0),r(0),Θ) =Pr(τ_≤T_|τ_≥t=0). (2.18) With no explicit modelling of the firm’s cash flows and debt covenants, there is no spe- cific bankruptcy cause ascribed toK; it is simply the value of the firm at which default is triggered. This simplification enables LS to value complex debt structures. Att=τ all debt is assumed to default under cross-collateralisation rules. Different priority lev- els between debtors is accommodated by varying the writedown rateω. Valuation of a zero-coupon bond then proceeds as the risk neutral expected payoffs in default and non-

default states. The assumption that payment to bondholders occurs only at the original bond maturity facilitates the valuation of coupon paying bonds by valuing each con- tracted payment as a zero-coupon bond and summing together as a ‘portfolio of zeros’.

The division of assets in the event of default, is also exogenously specified. A pro- portion(1₋ω)ofFis assumed to be paid to bondholders at the original maturity of the debt.7 The writedown rate,ω, representsK/F, being the expected outcome from strate- gic default and negotiation, or bankruptcy and liquidation including expected breaches of APR. However, as noted by Briys & de Varenne (1997), there is nothing to limit the payment to bondholders to be no greater than the value of the firm nor to ensure that the value of the firm is sufficient to cover the payment of the bond at maturity. They suggest a more structured barrier equal to the present value of the firm’s single-zero coupon li- ability adjusted for expected APR breaches. Unfortunately, the adjustment by Briys & de Varenne (1997) prohibits valuation of complex debt structures since the boundary is made a function of the face value of debt.8

A further weakness of the LS model is evident from examination of the latent log- solvency process in equation (2.17). Ifr(t)is on average greater than(δ₋σ2

v/2), then the firm is assumed to deleverage ad infinitum. Such behaviour is not expected in the presence of tax shield benefits (nor observed empirically).

The LS model has been extended to consider alternative asset processes. Zhou (1997) extends LS to a jump diffusion model. The motivation to consider downward jumps in firm value comes from the empirical observation that structural models under- state short term credit spreads. The potential for the firm value at default, to jump below the face value of debt, gives an endogenous variation in recovery rates. The first-passage crossing time is solved by Monte Carlo. HH describe an alternative jump-diffusion model with the assumption of a constant risk-free rate. They adopt the same exogenous default boundary and recovery assumptions of LS but let the asset value evolve with a double-exponential distribution such that a semi-analytic solution for the crossing time is known.9 The calibration of the jump component is difficult considering that the firm asset process is unobserved. HH and Delianedis & Geske (2001) demonstrate that by ad- justing jump parameters the predicted credit spreads on short tenor bonds can be made close to those observed, but the resultant jump parameters are found to be unrealistic. Further, jump parameters have most effect at short tenors and without mean-reversion in the leverage level, the previous criticism of the LS model remains. A simple compari- son of Merton against a jump-diffusion equivalent derived from Merton (1976) by Hull, Nelken & White (2004) showed that in all cases the non-jump Merton model provided

7_{Termed a ‘Treasury at Default’ assumption. Alternative writedown specifications are ‘Recovery of Par}

at Default’ and ‘Recovery of Market Value’. The differences are explained further in Guha (2003).

8_{Formally,K=αexp(−rT)Fwhere 0<α<1 is a scalar to accommodate expected breaches of APR.} 9_{The probability of default requires numeric methods to solve for a Laplace inversion. Refer to Huang}

significantly better predictions of default probabilities and credit spreads.

Drawing from agency theory, Barone-Adesi & Colwell (1999) propose a model for valuing zero-coupon bonds where the volatility of the firm’s assets increases as the firm’s value approaches the default boundary. This is consistent with Jensen & Meckling’s (1976) theory of asset-substitution that proposes that it is in the interests of shareholders to take greater business risks, thus increasing the volatility of the firm, the closer to the firm is to default. Using a constant barrier and risk-free rate, a closed-form value for a zero-coupon bond is obtained under the assumption that the return on the firm follows a constant elasticity of variance (CEV) process as first described by Cox (1975). Under the CEV model the firm’s asset value has a local volatility that is a deterministic function of solvency.

Barone-Adesi & Colwell (1999) propose the firm follows the CEV process

dX(t) = (r₋δ)X(t)dt+σv¯ X(t)ρdW(v,t)Q, (2.19)

whereX(t) =V(t)₋K is the firm’s net worth, as measured by its equity value, and ¯σv is a constant scale factor for the instantaneous volatility. The local asset return volatility is given by ¯σvX(t)(ρ−1)_{and is therefore time-varying withX(t). Default occurs at the} first passage ofX(t)to zero;τ =inf_{t_≥0 :X(t) =0_}. For the case where(ρ₋1)<0, volatility increases with default risk and declines with solvency, which is the agency theory predicted relationship. Because the firm’s level of solvency is time-varying, it follows that the firm’s asset return volatility is also time-varying. However, the func- tional relationship is constant through time, fixed by the elasticity parameter ρ, and so the firm’s volatility is assumed to have a rigid volatility skew.

Usefully, the probability of the first passage time is known analytically for a 100 percent drop in value to zero. This result has been used by Campi & Sbuelz (2005) and Albanese & Chen (2005) to value equity default swap contracts, and by Campi, Polbennikov & Sbuelz (2005) to value bonds and credit derivatives.10

A weakness of the extant CEV models is that default occurs only when the firm is market-value insolvent. This precludes the possibility of strategic default or default with the firm having positive net worth. A more general approach is achieved in our estimation method by letting K be the earliest unobservable default threshold whether triggered by insolvency or strategic default. This implies that volatility increases as the point of default is reached, but permits the value of equity to be strictly positive.

The aforementioned exogenous-boundary models assume the writedown rate,ω, is exogenously determined and unrelated to the firm’s asset value. Intuitively, it would

10_{Equity default swaps are a recent financial innovation used as an alternative to credit default swaps.}

Normally these instruments pay 50 percent of notional value if the firm’s equity price drops by 30 percent. Campi & Sbuelz (2005) value the special case of a benchmark equity default swap that pays nothing in the event of a 100 percent value drop. The instruments are described in more detail in Medova & Smith (2004).

seem reasonable that the firm’s post-default recovery prospects would be tied to the stochastic process that led to default. However, the LS model and subsequent exogenous- boundary literature, define the default barrier as an absorbing state for the firm value process; the expected value of bond recovery ceases to be informed from the ongoing dynamics of the firm post default, even though bankruptcy proceedings may take several years to complete and the firm may not be liquidated. The distinction between imme- diate liquidation, and continuation under Chapter 11 with court imposed renegotiation, was first suggested by Francois & Morellec (2004) in the context of extending the Leland (1994) model. Under U.S. Bankruptcy Code firms can either liquidate assets immedi- ately under Chapter 7, or renegotiate with their creditors under Chapter 11. The latter is the predominate option chosen. Upon entering Chapter 11, the court grants the firm a period of observation, protected from the actions of bondholders to liquidate assets, during which time the firm renegotiates its debt. Consequently, liquidation does not arise at the moment of first passage under Chapter 11 bankruptcy. At the end of this period, the court decides whether the firm continues as a going concern or not (Francois & Morellec 2004, page 390).

Following Francois & Morellec (2004), Moraux (2002) separates default timing from liquidation. The former remains as specified under LS so that default remains exogenous and is triggered by the first passage of firm value to a constant boundary. Un- like LS, once the default boundary is hit the firm remains trading and the state process continues for the duration of the period the firm remains in administration under Chap- ter 11. Liquidation is then a separate uncertain event that occurs if the cumulative time spent below the reorganisation barrier exceeds a given fixed period of time. Francois & Morellec (2004) assume liquidation occurs when the unbroken period of time spent in default exceeds a fixed period. Moraux (2002) defines liquidation by the total cumula- tive time the firm value is below the reorganisation boundary. He shows that the effect of delayed liquidation is bounded between the results of two well known models that have analytic solutions. With infinite delay, there is no early liquidation of the firm, and the model approaches Merton; with coincident default and liquidation, the model ap- proaches Black & Cox (1976). A further refinement is suggested by Galai et al. (2005) to trigger liquidation after the weighted cumulative time in default exceeds a maximum time, where the weight is the distance of the firm value from the reorganisation bound- ary. The model therefore weights the severity of the financial distress. However, debt can only be valued numerically. While promising in the suggestion that the writedown rate should be endogenously related to the stochastic state process, no empirical tests of these models appears to have been attempted. The models are limited to simple capi- tal structures and involve time-intensive numeric solutions that discourage econometric estimation of parameters.