1. Advanced Finance 2006-2007 More on structural model
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
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In the following slides I provide additional details on Leland’s structural models.
Four different issues are analyzed.
Is VTS a concave function of leverage (stated differently could VTS decrease when leverage increases)?
Where does the endogeneous level of bankruptcy come from?
What happens when the unlevered firm has a policy of paying a dividend to its stockholders?
What is the impact of the maturity of the debt?

2. 1. Could VTS decrease when debt increases?
We adress the first question: what happens to VTS when the level of debt goes up? Is there a maximum value for VTS when debt is risky?
We will analyze this question using the Leland JoF 1994 model.
To put the issue in perspective, remember that if debt if riskfree, VTS=TCC/r increases with leverage.
Does this result hold when debt is risky?

3. Why does VTS drops? Trade-off theory Market value
PV(Costs of financial distress)
PV(Tax Shield)
Value of all-equity firm
Here is the slide that caused the first question: could the VTS decrease at high level of debt?
As explained in class, I didn’t use a model to draw this graph. Therefore, the question is worth looking at more closely.
Debt ratio
Advanced Finance 2007 Optimal capital structure

4. Here I provide a graph similar to the previous one but based on the Leland model.
As can be observed, VTS is a concave function of the level of coupon. Beyond some level of the coupoun, VTS goes down.
We thus need to understand why this might be so.
Advanced Finance 2007 Optimal capital structure

5. In this slide, I represent VTS’, the marginal value of the tax shield (the derivative of VTS with respect to C).
As VTS is a concave function of C (see previous slide) , its derivative (VTS’) goes down as C increases (an opportunity to remind you what a concave function is).
The graph also shows that VTS’ becomes negative when C is above some level (C>7.8 in our setting).
Advanced Finance 2007 Optimal capital structure

6. Determinants of marginal VTS
We now try to understand why VTS is concave.
Start from the formula of VTS and take the first derivative with respect to C.
Remember that pB is a function of the level of bankruptcy VB which itself a function of C.
Two factors determine VTS’:
The coupon effect: (TC/r )(1-pB)>0: forr pB fixed, increasing the coupon leads to a higher VTS. However, as pB is positively related to C (see next slide), this effect is more important for low levels of the coupon (see blue curve in the graph)
The pB effect: increasing the coupon causes an increase of pB (dpB/dC>0) which reduces the value of the tax shield (see green curve in the graph)
VTS’ is zero (red curve in the graph) when these two effects are equal.
Advanced Finance 2007 Optimal capital structure

7. pB vs Coupon Here we show that pB is positively related to C.
For x given (=2r/σ² in Leland):
pB is a function of VB
and VB is a function of C[=(1-TC)C/(r+.5σ²) in Leland]
Therefore, pB is a increasing function of C.
In this graph, pB is convex in C because x>1 (2r>σ²). For x=1 (2r=σ²), the relation is linear. For x<1 (2r<σ²), the relation is concave.
Advanced Finance 2007 Optimal capital structure

8. pB vs Coupon
In this slide we have closer look at the relationship between pB and C.
As VB = (1-TC)C/(r+.5σ²), VB is a linear function of C (see graph on the right) and therefore dVB/dC is a positive constant.
The graph on the right illustrates the convexity of pB’ since pB is a power function of VB
Advanced Finance 2007 Optimal capital structure

9. 2. Understanding endogeneous bankruptcy
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
We now adress the second question: what is the underlying logic leading to an endogeneous level of bankruptcy?

10. Leland JoF 1994 – Constant perpetual coupon
Notation
VU value of unlevered company
VB level of bankruptcy
C perpetual coupon
r riskless interest rate (const.)
σ volatility (unlevered)
α bankruptcy cost (fraction)
TC corporate tax rate
Present value of $1 contingent on bankruptcy
Value of levered company:
Unlevered: VU
Tax benefit: (TCC/r)(1-pB)
Bankrupcy costs: - α VB pB
Value of debt
Endogeneous level of bankruptcy
Diffusion process of asset value:
This is a summary of Leland model with constant perpetual coupon.
An important point to stress is that in this first model, the company does not pay anything to its bondholders and stockholders. All the cash flows generated by the assets of the company remain in the company.
Stockholders provide the cash required to pay coupons.
Note: in this model, coupon payments are financed by equity.
Advanced Finance 2007 Optimal capital structure

11. Value of equity
In this slide, I derive a closed form formula for the value of the equity.
The derivation is straightforward: simply replace VTS, BC and D by their respective expressions and simplifiy.
To understand the formula, remember that with riskfree debt, the market value of E would be:
E =V – D = VU + VTS – D = VU + TCC/r – C/r = VU – (1-TC)C/r
Advanced Finance 2007 Optimal capital structure

12. Unlimited liability: VB = 0  pB = 0
Here we show that VB has to be positive in order to have limited liability.
If VB = 0, the value of equity might be negative: shareholders have to pay the coupon even when VU is very low.
Advanced Finance 2007 Optimal capital structure

13. Limited liability
Bankruptcy occurs when the firm cannot meet the required coupon payment by issuing additional equity: that is when equity value falls to zero.
Moreover:
As:
E is convex in VU for VB < (1-TC)/r
As shown in the previous slide, VB should be positive.
The calculation of the endogeneous level of bankruptcy is based on one condition:
E should be positive for all values of VU > VB
On the other hand, E is a convex function of VU for VB <(1-TC)C/r
The graph illustrates a situation which does not verify the general condition.
I have set the bankruptcy level equal to 12. But then we see that, as E is convex, it becomes negative for some values of VU>12. Therefore, the general principle of positive values of E is not satisfied for this level of bankruptcy.
Should not happen VB
Advanced Finance 2007 Optimal capital structure

14. Lowest VB with limited liability
Smooth-pasting condition:
Slope = 0
In order for E to be positive for VU>VB, the derivative of E with respect to VU should be equal to zero at VU = VB
This gives us the recipe to calculate VB (see next slide for calculation) VB
Advanced Finance 2007 Optimal capital structure

15. Some algebra Set VB = VU smooth-pasting condition is:
This slides goes through the details of the calculation of VB. A nice slide to look at for those of you who are nostalgic of their math classes.
Advanced Finance 2007 Optimal capital structure

16. 3. Cash payouts by the firm
The following slides generalize the model to the case of a company that pays a dividend to it shareholders if unlevered.

17. Cash payouts by the firm
If no cash payout by the firm:
If cash outflow = fraction δ of asset value
The only change is that the formula for x is more complicate than in the previous case.
The good new is that all other formula are the same.
Solutions for all security value (VTS,BC,D,E) remain the same.
Advanced Finance 2007 Optimal capital structure

18. Example
δ = 0%
δ = 3%
This illustrate the impact of a cash payout by the firm.
The firm value drops because bankruptcy is more likely with cash payouts (see in higher value of pB and the higher value of bankruptcy costs).
The spread rises from 16 to 42 bps
Advanced Finance 2007 Optimal capital structure

19. 4. Finite maturity: Leland exponential model
We now generalize the model by considering debt with finite maturity.

20. Debt with finite maturity
Reference: Leland unpublished WP Princeton lecture I See also Leland and Toft JoF 1996
Debt principal at time 0 = P , coupon = C
Debt retired at a proportional rate m
Remaining value at time t:
Cash flow to debtholders (if firm alive):
In this model, a constant fraction of the currently outstanding debt is retired annually and replaced with newly issued debt. The average maturity of the debt is equal to the inverse of the retirement rate.
Average maturity:
Retired debt replaced with new debt having the same principal and the same coupon => total P and C are constant
Advanced Finance 2007 Optimal capital structure

21. Closed form solution
Here are the formulas to show you that they exist.
Advanced Finance 2007 Optimal capital structure

22. Example This is a numerical illustration of the model.
Advanced Finance 2007 Optimal capital structure

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