The Leland Model

In the Leland (1994) model, financial distress is triggered when shareholders no longer find that running a company is profitable, given the revenue produced by the assets, to continue servicing debt. Bankruptcy is determined endogenously rather than by the imposition of a positive net worth condition or by a cash flow constraint.

Denote any claimF( tV, )on the firm that continuously pays a nonnegative coupon, C , per instant of time when the firm is solvent. Leland (1994) provides the solution of the perpetual debt. LetV_Bdenote the constant level of asset value at which bankruptcy is declared. If bankruptcy occurs, a fraction0≤α≤1of asset value will be lost to bankruptcy cost. The closed-form solution of risky debt is

X

Next, Leland (1994) derives the total value of the firm,v(V), which reflects three terms: the firm’s asset value, plus the value of the tax deduction of coupon payments, less the value of bankruptcy costs.

The value of equity is the total value of the firm less the value of debt as follows:

X By maximize the value of equity at any level ofV , the equilibrium bankruptcy-triggering asset value V_B is determined endogenously by the smooth-pasting condition ( ; , ) =0

)

Note thatV_B in (A.10) is independent of time and it confirms the assumption of the constant bankruptcy-triggering asset levelV_B.

Default Probability of Risky Debt

The cumulative probability of the firm going bankrupt over the period( Tt, ]is



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i See Section 2 for the summary of these empirical studies.

ii For the comprehensive analysis of these models, see Crouhy, Galai, and Mark (2000) and Saunders and Allen (2002).

iii See Lando (2004) for the review of reduced-form models.

iv There exists an extensive literature on default or bankruptcy prediction, readers who are interested in this subject can refer to the papers by Shumway (2001) and Duffie, Saita, and Wang (2007). In this paper, we have no intension to incorporate those previously identified variables, such as firm’s trailing stock return, trailing S&P 500 returns, and U.S. interest rates, into our analysis but rather focus on various default boundary assumptions in the structural models.

v Campbell, J. Y., J. Hilscher, and J. Szilagyi (2004) also show similar results that failure risk cannot be adequately summarized by a measure of distance to default by the KMV-Merton model.

vi In the context of structural credit risk modeling, the equity value is an option of asset valueV_t. Therefore, for example, under Merton’s model,S_t =V_tN(d_t)−Ke^−rTN(d_t−σ_v T). Also by Itô’s Lemma,

S v V V S S _t^{t t}_tσ

σ =_∂^∂ . SinceS_t =g(V_t;σ_v)is a one to one function ofV , the inverse exists. One can then first estimate _t the equity volatilityσ using historical data, and the two unknown variables asset value_S V_tand asset volatilityσ _v left can be solved by the above two-equation system.

vii Duan and Simonato (2002) further develop a MLE method for the two unobserved variables, namely, the firm valueV and the instantaneous interest rate_t r . In this case,_t θ also contains parameters of interest process and its correlation with firm value process. Thus, one needs to modify the log-likelihood function in Step 2 to incorporate this change.

viii See Bruche (2005) for the issues of some other estimation methods not presented here.

ix The KMV method is a simple two-step iterative algorithm which begins with an arbitrary value of the asset volatility and repeats the two steps until the convergent criterion is reached. The default barrier of the KMV method is assumed as the sum of short-term liabilities plus one-half long-term liabilities. See Crosbie and Bohn (2003) and Vasslou and Xing (2004) for details.

x A similar approach is adopted by Chen, Hu, and Pan (2006) using distant to default (DD) instead of default probability. However, this relationship cannot be applied in the barrier option framework since the default probability is not merely a transformation of distant of default. Therefore, we use the default probability directly in our study. The same argument is also addressed by Leland (2004).

xi There are several reasons choosing this “default point”: First, KMV has observed from a large sample of several hundreds companies that firms default when the asset value reaches a level somewhere between the value of total liabilities and the value of short-term debt. Therefore, as argued in Crouhy, et al. (2000), the probability of the asset value falling below the total face value may not be an accurate measure of the actual default probability. Secondly, as pointed out by Vassalou and Xing (2004), it is important to include long-term debt in the calculation because firms need to service the long-term debt, and these interest payments are part of the short-term liabilities. Furthermore, the size of the long-term debt may affect the ability of a firm to roll over its

xii We lost some samples due to convergent issue in the MLE maximization process of the Brockman and Turtle, the Black and Cox, and the Leland models. We lost 9, 10, and 6 firms of the in-sample, six-month out-of-sample, and one-year out-of-sample tests, respectively.

xiii The physical default probability here is under the assumption of constant asset risk premium.

xiv See Kim, Ramaswamy, and Sundaresan (1993) and Anderson and Sundaresan (1996) for the models assuming cash-flow or liquidity covenant.

xv Note that for the Brockman and Turtle and the Black and Cox models, a firm can default either before maturity or at maturity. Therefore, we also need to use incorporate the default probability at maturity for each firm.

xvi See Lin (2006) for the correction of the typographical error in this formula. The corrected formula is presented in (A.5).

xvii To incorporate payout, one needs to modify x as ₂

2 2 2

2/2) ( /2) 2

(

σ

σ σ

σ r g r

g

r− − + − − +

in (A.7)

andλ as

2

V2

V g σ

µ − − in (A.11).