Leland-Toft endogenous default

Merton, Black-Cox, and CreditGrades all treat the default boundary as exogenous: someone (analyst, regulator, indenture writer) tells the model when default occurs. Leland (1994) and Leland-Toft (1996) flip this on its head — equity holders choose the default boundary V_B^* to maximise their own claim, weighing the option value of continuing to pay coupons against the loss of the firm’s tax shield.

Setup

  • Assets follow GBM under the risk-neutral measure with dividend yield \delta: dV/V = (r - \delta)\,dt + \sigma_A\,dW.

  • Debt pays a continuous coupon C.

  • Tax rate \tau (coupon payments shielded from tax).

  • Bankruptcy cost \alpha (debt holders recover (1 - \alpha) V_B).

Optimal default boundary

Let x be the positive root of the characteristic quadratic \tfrac{1}{2}\sigma_A^2 y^2 + (r - \delta - \sigma_A^2/2)y - r = 0:

\[x = \frac{\sqrt{(r - \delta - \sigma_A^2/2)^2 + 2r\sigma_A^2} - (r - \delta - \sigma_A^2/2)}{\sigma_A^2}.\]

The smooth-pasting condition yields

\[V_B^* = \frac{(1 - \tau)\,C\,x}{r\,(1 + x)}.\]

Default probability

For V > V_B:

\[\mathrm{PD} = \left(\frac{V_B}{V}\right)^x \in (0, 1].\]

(Below the boundary, \mathrm{PD} = 1 — the firm is already in default.)

Equity and debt values

\[E(V) = V - \frac{(1 - \tau)C}{r}\,\bigl(1 - \mathrm{PD}\bigr) - (1 - \alpha)\,V_B\,\mathrm{PD},\]
\[D(V) = \frac{C}{r}\,\bigl(1 - \mathrm{PD}\bigr) + (1 - \alpha)\,V_B\,\mathrm{PD}.\]

The Modigliani-Miller decomposition E + D = V + \text{tax shield} - \text{bankruptcy cost} is preserved.

Example

from merton import Firm
from merton.extensions import LelandToftModel

firm = Firm(equity=30, debt_short=30, debt_long=40, equity_vol=0.50,
            rf=0.04, horizon=1.0)
result = LelandToftModel(
    coupon=5.0, tax_rate=0.25, bankruptcy_cost=0.30,
).fit(firm)
print(result.summary())
print("V_B*:", result.default_point)
print("PD :", result.pd)

When to prefer Leland-Toft

  • Capital-structure analytics. If you’re varying the coupon or tax treatment, the endogenous boundary is exactly what you want.

  • LBO and recap modeling. The optimal-stopping formulation captures how new debt issuance changes the default decision.

  • Bond pricing with embedded covenants. Pairing Leland-Toft with the Vasicek single-factor portfolio engine gives a self-consistent framework for both pricing and capital.

For “what’s this firm’s one-year PD?” Merton or KMV remain the simpler choices.