Endogenous-Static

In an early extension of the Merton model, Black & Cox (1976) consider the valuation of debt when management choose the timing of default, acting to optimise the value of shareholder equity. In doing so the early default assumption of Merton is relaxed and the level of the default barrier is determined endogenously. It is assumed that the firm is financed by equity and a single consol (infinite maturity) bond paying a continuous coupon. For large firms with many debt issues, this choice of bond payment is a more re- alistic representation of the firm’s aggregate going-concern debt financing requirements than Merton’s single zero-coupon bond assumption. At each point in time, management choose whether to pay coupons or otherwise default and pass the assets of the firm to bondholders. Default will be avoided provided that the value of equity, after the coupon payment, is not less than the coupon payment. Thus, default occurs if the value of the firm falls to a point where new equity cannot be raised to service debt; in continuous time equivalent to the value of equity equal to zero. An important result achieved by Black & Cox (1976) is to identify the shareholder wealth maximising default boundary as

K= c

r+σv2

2

(2.14) wherec is the coupon rate, andr is the risk-free rate. Equation (2.14) shows that the default barrier is independent of firm value, and decreases as asset volatility and the risk-free rate increase.

Leland (1994) extends Black & Cox (1976) to include the effects of tax and bankruptcy costs on the default boundary. Taxation presents management with an opportunity to increase firm value by utilising tax savings on interest payments. An optimum value maximising level of debt exists, at which point the marginal benefit of the tax shield is equalled by the marginal cost of increased bankruptcy risk. Like Black & Cox (1976), Leland (1994) assume that management seek to maximise the value of the shareholder’s

claim, and not the value of the firm. They find that the default boundary of the firm is

K=(1−τ)c

r+σv2

2

, (2.15)

whereτ is the tax rate.

Equation (2.15) shows that the default barrier is positively related to the after-tax coupon rate, and is negatively related to the risk-free rate and firm asset volatility. The default boundary is unaffected by bankruptcy costs, which are borne by the bondholders in the event of default, and not by the shareholders, and therefore do not enter into con- sideration by management when setting the firm’s debt-ratio. Bankruptcy costs reduce the overall value of the firm, but leave the value of equity unchanged with the cost passed to bondholders in a reduced value of the bond.

LT extends Leland (1994) and Black & Cox (1976) with the more realistic assump- tion that the firm issues finite maturity debt. To find a tractable solution for the value of the firm’s debt, they assume debt is ‘rolled over’, i.e. refinanced, in perpetuity at a constant maturity, T maintaining a constant level of principal. The firm’s capital struc- ture is assumed to be time-homogeneous and management choose the optimal debt level only initially leaving the aggregate level of debt static thereafter. In addition to the ex- planatory variables in equation (2.15), the default boundary is found to be an increasing function of the debt-ratio, and bankruptcy costs, and a decreasing function of debt ma- turity (refer equation (3.35)). Default only occurs when new equity cannot be raised, which will generally occur when debt service costs equal the expected equity return.5

The LT model provides a plausible theoretical basis for Davydenko’s (2005) obser- vation that firms default with negative net equity and not immediately at a zero equity default boundary. The reason is related to the maturity of debt and expected equity return. With long-term debt, the default boundary will typically be less that the debt principal due to the potential for equity to appreciate before the debt is rolled over. The longer the maturity of the debt and the higher the expected equity return, the greater the opportunity for the firm to attract additional equity and avoid bankruptcy despite imme- diate negative net worth. However, as maturity approaches zero, new equity will only be attracted if the value of the firm, after bankruptcy costs, exceeds the par value of debt. Thus, the default boundary is predicted to approach K=P/(1₋α)asT →0, whereP is the debt principal andα_≥0 is the bankruptcy cost. Thus, default is predicted to occur when the firm has positive net worth if bankruptcy costs are non-zero.

5_{Debt service is defined as the intertemporal change in firm value due to leveraging; includes after tax}

cost of coupons and principal repayments at par, less funds from new debt issued at market value and cash available for payout to shareholders generated from operations.