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A Dynamic Optimal Capital Structure Model with Costly Adjustment Mechanisms. Hsien-Hsing, Liao Yi-Hsuan, Tung Tsung-kang Chen Department of Finance National Taiwan University 2008.12.12. Abstract.
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Hsien-Hsing, LiaoYi-Hsuan, TungTsung-kang ChenDepartment of FinanceNational Taiwan University 2008.12.12
where the growth rate, ,
and the volatility of cash flow, , are constants.
where μ = μP - θσ is the risk-neutral drift of the cash flow rate under risk neutral measure.
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Consider an otherwise identical firm whose management decides to choose a static leverage level that will maximize the wealth of current equity holders.
We assume a firm will issue a perpetual bond, promising a constant coupon payment C to debt holders as long as the firm remains solvent.
We define the firm chooses the default threshold as VB. If the firm value reaches VB , then an amount αVBwill be lost due to bankruptcy costs.
In general, any claim of firm’s derivative security must satisfy the partial differential equation (PDE) as follows:
where P is the cash flow.
Because of the issuance of perpetual debt, all claims will be time-independent. Thus the PDE reduces to an ordinary differential equation:
The solution is
and A0, A1 and A2 are constants, determined by boundary conditions.
Note that x>0, while y<0, so A1equals zero for all claims of interest.
Before proceeding, it is convenient to define pB (V) as the present value of a claim that pays $1 contingent on firm value reaching VB .
According to the boundary conditions
We define the value of this claim as Vsolv (V) as long as firm value remained above VB.
Taking into account the boundary conditions:
We find that
As long as the firm remains solvent, the coupon payment is C before the recapitalization. Thus, the value of the claim to interest payments during continued operations, Vint is
The boundary conditions are
We obtained (16)
The present value of the default claim Vdef (V) can be written as
with the boundary condition:
Note that the sum of the present value of the claim to funds during solvency (eq. ) and the present value of the claim to funds in bankruptcy (eq. ) is equal to the present value of firm, V.
Hence, value is neither created nor destroyed by changes in the capital structure; rather, it is only redistributed among the claimants. This invariance result is consistent with the “pie” model of Modigliani and Miller (1958), except that in this framework the claims of government and bankruptcy costs are also part of that pie.
Separating the value of continuing operations and default claim between equity, debt, government, and bankruptcy costs as follows:
A. Default Level
When the firm has the financial distress, which means the cash is less the promised coupon payment to the debtholders.
We can obtained the bankruptcy level VB by invoking the smoothing-pasting condition
In poor situation, the firm goes into bankruptcy, comparing with the debt holders get the residual firm value and the equity holders receive nothing.
Solving, we find
Observed that the firm value at which bankruptcy occurs
is proportional to the coupon, C ;
is independent of the current firm value, V ;
decreases as the risk-free interest rate, r, rises.
B. Optimal Coupon level
The objective of management is to maximize shareholder wealth. That is, current equity holders receive fair value for the debt claim sold.
We find that the optimal coupon level chosen when current firm value is V0, is