1. Binnenlandse Francqui Leerstoel VUB 2004-2005 5
Binnenlandse Francqui Leerstoel VUB Options and Optimal Capital Structure
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Outline of presentation:
1. Modigliani Miller 1958: review
2. Merton Model: review
3. Interest tax shield
4. Bankruptcy costs and agency costs
5. The tradeoff model: Leland
VUB 05 Options and optimal capital structure

3. Modigliani Miller (1958)
Assume perfect capital markets: not taxes, no transaction costs
Proposition I:
The market value of any firm is independent of its capital structure:
V = E+D = VU
Proposition II:
The weighted average cost of capital is independent of its capital structure
WACC = rAsset
rAsset is the cost of capital of an all equity firm
Modigliani and Miller theorem were not the first to state that the Modigliani-Miller proposition of the irrelevancy of capital structure. They had been preceded by John Burr William. In his book The Theory of Investment Value (1938), he wrote:
“If the investment value of an enterprise as a whole is by definition the present worth of all its future distributions to security holders, whether on interest or dividend account, then this value in no wise depends on what the company’s capitalization is. Clearly, if a single individual or a single institutional investor owned all of the bonds, stocks and warrants issued by the corporation, it would not matter to this investor what the company’s capitalization was (except for details concerning the income tax). Any earnings collected as interest could not be collected as dividends. To such an individual it would be perfectly obvious that total interest- and dividend-paying power was in no wise dependent on the kind of securities issued to the company’s owner. Furthermore no change in the investment value of the enterprise as a whole would result from a change in its capitalization. Bonds could be retired with stock issues, or two classes of junior securities could be combined into one, without changing the investment value of the company as a whole. Such constancy of investment value is analogous to the indestructibility of matter or energy: it leads us to speak of the Law of the Conservation of Investment Value, just as physicists speak of the Law of the Conservation of Matter, or the Law of the Conservation of Energy.” (pp )
Reference
Rubinstein, M., Great Moments in Financial Economics: II Modigliani-Miller Theorem, Journal of Investment Management (Second Quarter 2003) (available at
VUB 05 Options and optimal capital structure

4. Weighted average cost of capital
V (=VU ) = E + D
Value of equity
rEquity
Value of all-equity firm
rAsset
rDebt
Value of debt
We now want the understand the implication of MM Proposition I on the weighted average cost of capital of a company.
Consider the balance sheet (using market values) of a company. It can be viewed either from the asset side or from the liability side.
When viewed from the liability side, the weighted average cost of capital is the expected return on a portfolio of both equity and debt.
Consider someone owning a portfolio of all firm’s securities (debt and equity) with XEquity = E/V and XDebt = D/V
Expected return on portfolio = rEquity * XEquity + rDebt * XDebt
This is equal to the WACC (see definition):
rPortoflio = WACC
But she/he would, in fact, own a fraction of the company. The expected return would be equal to the expected return of the unlevered (all equity) firm
rPortoflio = rAsset
The weighted average cost of capital is thus equal to the cost of capital of an all equity firm
WACC = rAsset
WACC
VUB 05 Options and optimal capital structure

5. Cost of equity D/E The equality WACC = rAsset can be written as:
Expected return on equity is an increasing function of leverage:
rEquity
12.5%
Additional cost due to leverage
This is another presentation of the equality: WACC = rAsset
This presentation shows that the cost of equity increases with leverage.
If a company replace equity with debt, two things happens:
The company saves money because the cost of debt is lower than the cost of equity
On the other hand, the remaining equity becomes more costly.
In the MM 58 framework, the two effects offset each other.
11% rA
WACC 5%
rDebt
D/E
0.25
VUB 05 Options and optimal capital structure

6. Why does rEquity increases with leverage?
Because leverage increases the risk of equity.
To see this, back to the portfolio with both debt and equity.
Beta of portfolio: Portfolio = Equity * XEquity + Debt * XDebt
But also: Portfolio = Asset
So: or
The reason why the cost of equity increases with leverage is because the risk is higher.
Remember that the beta of a portfolio is equal to the weighted average of the betas of the individual securities in the portfolio.
VUB 05 Options and optimal capital structure

7. The Beta-CAPM diagram     Beta L βEquity U βAsset r rEquity rAsset
rDebt=rf
D/E
The IS-LM diagram used by economist is the source of inspiration for this figure. For many years, I have tried to create a figure which would look as impressive as the IS-LM diagram. Here is the result.
The figure in the first quadrant shows the relationship between the debt-equity ratio and the beta equity. The figure is based on the assumption that the debt is riskless. In that case, the relationship is linear.
The security market line in illustrated in the second quadrant.
 The relationship between the debt-equity ratio and the cost of equity, the cost of debt and the WACC is in the third quadrant
 This is simply a -45% line.
This diagram can be used to show that investors can choose the level of leverage that they wish by rebalancing their portfolios.
If the company has no debt (point U), an investor can create leverage by borrowing at the risk-free interest to reach L, the risk and expected returns of a levered company.
If, on the hand, the company is levered (point L), an investor can allocate his money between the stock of the levered company and the risk free asset to undo the leverage and reach point U.  
rEquity
rDebt
D/E
WACC
VUB 05 Options and optimal capital structure

8. Merton (1974): Review
Limited liability: equity viewed as a call option on the company.
D Market value of debt
Risk-free debt - Put
E Market value of equity
Call option on the assets of the company
Loss given default F
Why is debt risky? The key reason is limited liability: a company can default on its obligation to pay the interests or to repay the principal. From the stockholders’ perspective, going bankrupt means that they lose the money that they have invested in the company. But their loss is limited. They are not required to pay additional amounts.
Black and Scholes, in their seminal paper, discuss the implications of their option pricing model for the bond valuation. This insight was later expanded by Robert Merton in 1974 (hence the name “Merton Model” for risky debt valuation models based on an option pricing model).
We present here the problem in a simplified setting:
1. The value of a the firm (V) is given and is independent of the capital structure (the MM 58 proposition is satisfied)
2. The firm issues one single debt: a zero-coupon giving the holder the right to a fixed sum of money at maturity.
3. There are no cost of bankruptcy.
Two main conclusions appear is this setting.
1. The market value of equity is equal to the value of a call option
2. The market value of the debt is equal to the value of the riskless debt minus a put option.
Bankruptcy
F Face value of debt
V Market value of comany
F Face value of debt
V Market value of comany
VUB 05 Options and optimal capital structure

9. Merton Model: example using binomial option pricing
Data:
Market Value of Unlevered Firm: 100,000
Risk-free rate per period: 5%
Volatility: 40%
Company issues 1-year zero-coupon
Face value = 70,000
Proceeds used to pay dividend or to buy back shares
Binomial option pricing: review Up and down factors:
V = 149,182 E = 79,182 D = 70,000
Risk neutral probability :
V = 100,000 E = 34,854 D = 65,146
We first use the binomial model to illustrate the Merton model. The calculation is a straightforward application of binomial option pricing. All securities are valued by discounting the risk-neutral expected value with the risk-free interest rate.
The analysis proceeds as follow:
1. Calculate the up and down factors
2. Calculate the risk neutral probabilities p and 1 – p
3. Calculate the value of the securities at maturity
4.Calculate the present value using the 1-period valuation formula.
Note that, in this example, the value of the risk-free debt would be:
PV(F) = 70,000 e-5% = 66,586
The difference between the value of the risk-free debt (66,586) and the value of the (risky) debt (65,146) is the value of the put option.
V = 67,032 E = 0 D = 67,032
1-period valuation formula
Cost of borrowing: y = 7.45%
∆t = 1
VUB 05 Options and optimal capital structure

10. Weighted Average Cost of Capital in Merton Model
(1) Start from WACC for unlevered company
As V does not change, WACC is unchanged
Assume that the CAPM holds
WACC = rA = rf + (rM - rf)βA
Suppose: βA = 1 rM – rf = 6%
WACC = 5%+6%× 1 = 11%
(2) Use WACC formula for levered company to find rE
We now wish to understand the effect of leverage on the cost of equity and the cost of debt when debt is risky.
Our starting point in the Modigliani Miller (1958) propositions. The value of the firm is unaffected by the capital structure. Hence, the weighted average cost of capital of the levered firm is equal to the cost of capital of the unlevered firm. Similarly, the weighted average beta of equity and debt is equal to the asset beta.
Combining the option pricing model with the CAPM, we can work out either the values of rE and βE. The cost of debt and the beta of the debt can than be calculated using the WACC formula.
VUB 05 Options and optimal capital structure

11. Cost (beta) of equity Remember : C = Deltacall × S - B
A call can is as portfolio of the underlying asset combined with borrowing B.
In Merton’s Model: E = DeltaEquity × V – B
The fraction invested in the underlying asset is X = (DeltaEquity × V) / E
The beta of this portfolio is X βasset
In example:
βA = 1
DeltaE = 0.96
V/E = 2.87
βE= 2.77
rE = 5% + 6% × 2.77
= 21.59%
To calculate the beta of equity, we proceed as follow.
1. The equity of the levered company is assimilated to a call option on the firm’value.
2. A call option is a portfolio composed of on a long position on delta shares combined with borrowing. The fraction of this portfolio invested in the shares is (delta V)/E.
3. The beta of a portfolio is equal to the weighted average of the beta of the underlying securities. In this setting, the betas of the underlying securities are the asset beta and the beta of borrowing in the replicating portfolio which is equal to zero.
4. Therefore βE = βA × Delta × (V/E)
This expression generalizes the formula that had been established when the debt is risk-free: βE = βA × (V/E)
Remember that 1≥Delta≥0. This implies that, for a given level of leverage, the equity is less risky with risky debt than when the debt is riskless.
Remember also that delta varies whenever V varies. As a consequence, the beta of equity will not be constant.
VUB 05 Options and optimal capital structure

12. Cost (beta) of debt Remember : D = PV(FaceValue) – Put
Put = Deltaput × V + B (!! Deltaput is negative: Deltaput=Deltacall – 1)
So : D = PV(FaceValue) - Deltaput × V - B
Fraction invested in underlying asset is X = - Deltaput × V/D
βD = - βA Deltaput V/D
In example:
βA = 1
DeltaD = 0.04
V/D = 1.54
βD= 0.06
rD = 5% + 6% × 0.06
= 5.33%
We now turn to the debt.
The beta of the debt is equal to minus the beta of the put option (remember that the beta of the put option is negative, therefore, the beta of the debt is positive).
An interesting point to note here is the difference between the cost of borrowing (y) calculated previously and the cost of debt (rD).
Remember that the cost of borrowing is defined as the yield to maturity on the debt whereas the cost of debt is equal to the expected return on the debt.
In our example, the cost of borrowing is 7.45% whereas the cost of debt is 5.33%.
To understand the difference, compare the formulas for y and rD.
where Π is the true probability of an up movement. This true probability can be calculated by solving the following equation:
rA = Π (u – 1) + (1 – Π) (d – 1)
In the example: Π = 0.535
The cost of borrowing calculation is based on the face value of the debt whereas the cost of debt calculation is based on the (true) expected future value.
VUB 05 Options and optimal capital structure

13. Toward Black Scholes formulas
Value
Increase the number to time steps for a fixed maturity
The probability distribution of the firm value at maturity is lognormal
What happens if we increase the number of steps of a given maturity?
The size of the binomial tree increases but the underlying logic remains unchanged. In a binomial tree with n steps (the length of each step is Δt=T/n), the firm can take n+1 different values at maturity depending on the number of ups (k) and downs (n – k):
Vk = ukdn – k k = 0, 1, …,n
The risk-neutral probability of Vk is:
Let fk be the value at maturity of a derivative if VT = Vk.
Risk-neutral pricing implies that the value at time 0 is:
f = e-rT (p0 f0 + p1 f1 + … + pk fk + … + pn+1 fn+1) [1]
As n→∞, the probability distribution of VT tends converges to a lognormal distribution.
ln(VT) ~ N[ln(V)+(r-0.5σ²)T, σ√T]
The Black-Scholes formulas is the continuous equivalent of [1] when the probability distribution of the underlying value at maturity is lognormal.
Bankruptcy
Today
Maturity
Time
VUB 05 Options and optimal capital structure

14. Corporate Tax Shield
Interest payments are tax deductible => tax shield
Tax shield = Interest payment × Corporate Tax Rate
= (rD × D) × TC
rD : cost of new debt
D : market value of debt
Value of levered firm
= Value if all-equity-financed + PV(Tax Shield)
PV(Tax Shield) - Assume permanent borrowing
V=VU + TCD
Interest payments are tax deductible. As a consequence, the taxes paid by a levered company are lower than if it were unlevered. The tax shield is the tax saving due to leverage. To see this, consider two companies with identical EBITs: company U has no debt whereas company L is levered. The corporate taxes paid by these two companies are:
TaxesU = EBIT × TC
TaxesL = (EBIT – Interest) × TC = EBIT× TC – Interest × TC
= TaxesU – Tax shield
As a consequence, the total cash flow payable to both stockholders (dividend) and debtholders (interest) is higher with leverage:
For U: DivU = EBIT(1-TC)
For L: DivL + Interest = (EBIT – Interest) ×(1-TC) + Interest = EBIT(1-TC) + Interest × TC
This is the reason why the value of the levered firm should be higher than the value of the unlevered.
The additional value due to leverage is the present value of the tax shield. Its calculation is tricky as: 1) the level of debt can change over time 2) an assumption is required on the risk of the tax shield to determine the discount factor to use.
In this lecture, we look at the simplest case analyzed by Modigliani and Miller in 1963: 1) Expected EBIT is a constant perpetuity 2) The level of debt is constant 3) The tax shield has the same risk as the debt.
VUB 05 Options and optimal capital structure

15. Cost of equity calculation
V = VU + TCD = E + D
Value of equity rA rE
Value of all-equity firm rD
Value of debt rD
Value of tax shield = TCD
Up to now, we have fixed the level of debt. We then calculated:
The value of the levered firm
The value of equity
The cost of equity
The WACC
We now want to start from the WACC to calculate the value of the levered firm.
VUB 05 Options and optimal capital structure

16. Still a puzzle…. If VTS >0, why not 100% debt?
Two counterbalancing forces:
cost of financial distress
As debt increases, probability of financial problem increases
The extreme case is bankruptcy.
Financial distress might be costly
agency costs
Conflicts of interest between shareholders and debtholders (more on this later in the Merton model)
The trade-off theory suggests that these forces leads to a debt ratio that maximizes firm value (more on this in the Leland model)
VUB 05 Options and optimal capital structure

17. Risk shifting
The value of a call option is an increasing function of the value of the underlying asset
By increasing the risk, the stockholders might fool the existing bondholders by increasing the value of their stocks at the expense of the value of the bonds
Example (V = 100,000 – F = 60,000 – T = 2 years – r = 5%)
Volatility Equity Debt
30% 46, ,374
40% 48, ,494
+1, ,880
VUB 05 Options and optimal capital structure

18. Underinvestment
Levered company might decide not to undertake projects with positive NPV if financed with equity.
Example: F = 100,000, T = 5 years, r = 5%, σ = 30%
V = 100,000 E = 35,958 D = 64,042
Investment project: Investment 8,000 & NPV = 2,000
∆V = I + NPV
V = 110,000 E = 43,780 D = 66,220
∆ V = 10,000 ∆E = 7,822 ∆D = 2,178
Shareholders loose if project all-equity financed:
Invest 8,000
∆E 7,822
Loss = 178
VUB 05 Options and optimal capital structure

19. Milking the property
Suppose now that the shareholders decide to pay themselves a special dividend.
Example: F = 100,000, T = 5 years, r = 5%, σ = 30%
V = 100,000 E = 35,958 D = 64,042
Dividend = 10,000
∆V = - Dividend
V = 90,000 E = 28,600 D = 61,400
∆ V = -10,000 ∆E = -7,357 ∆D =- 2,642
Shareholders gain:
Dividend 10,000
∆E -7,357
VUB 05 Options and optimal capital structure

20. Trade-off theory Market value PV(Costs of financial distress)
PV(Tax Shield)
Value of all-equity firm
Debt ratio
VUB 05 Options and optimal capital structure

21. Leland 1994
Model giving the optimal debt level when taking into account:
limited liability
interest tax shield
cost of bankruptcy
Main assumptions:
the value of the unlevered firm (VU) is known;
this value changes randomly through time according to a diffusion process with constant volatility  dVU= µVU dt + VU dW;
the riskless interest rate r is constant;
bankruptcy takes place if the asset value reaches a threshold VB;
debt promises a perpetual coupon C;
if bankruptcy occurs, a fraction α of value is lost to bankruptcy costs.
VUB 05 Options and optimal capital structure

22. VU Barrier VB Time Default point
VUB 05 Options and optimal capital structure

23. Exogeneous level of bankruptcy
Market value of levered company V = VU + VTS(VU) - BC(VU)
VU: market value of unlevered company
VTS(VU): present value of tax benefits
BC(VU): present value of bankruptcy costs
Closed form solution:
Define pB : present value of $1 contingent on future bankruptcy
VUB 05 Options and optimal capital structure

24. Example Simulation: ΔVU = (.06) VU Δt + (.3464) VU ΔW
Value of unlevered firm VU = 100
Volatility σ = 34.64%
Coupon C = 5
Tax rate TC = 40%
Bankruptcy level VB = 25
Risk-free rate r = 6%
Simulation: ΔVU = (.06) VU Δt + (.3464) VU ΔW
1 path simulated for 100 years with Δt = 1/12
1,000 simulations
Result: Probability of bankruptcy = (within the next 100 years)
Year of bankruptcy is a random variable
Expected year of bankruptcy = (see next slide)
VUB 05 Options and optimal capital structure

25. Year of bankruptcy – Frequency distribution
VUB 05 Options and optimal capital structure

26. Understanding pB Exact value Simulation N =number of simulations
Yn = Year of bankruptcy in simulation n
VUB 05 Options and optimal capital structure

27. Tax shield if no default PV of $1 if no default
Value of tax benefit
Tax shield if no default
PV of $1 if no default
Example:
VUB 05 Options and optimal capital structure

28. Present value of bankruptcy cost
Recovery if default
PV of $1 if default
Example:
BC(VU) = 0.50 ×25×0.25 = 3.13
VUB 05 Options and optimal capital structure

29. Value of debt Risk-free debt PV of $1 if default Loss given default
VUB 05 Options and optimal capital structure

30. Endogeneous bankruptcy level
If bankrupcy takes place when market value of equity equals 0:
VUB 05 Options and optimal capital structure

31. Leland 1994 - Summary 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
VUB 05 Options and optimal capital structure

32. F = A0 + A1V + A2 V-X with X = 2r/σ²
Inside the model
Value of claim on the firm: F(VU,t)
Black-Scholes-Merton: solution of partial differential equation
When non time dependence ( ), ordinary differential equation with general solution:
F = A0 + A1V + A2 V-X with X = 2r/σ²
Constants A0, A1 and A2 determined by boundary conditions:
At V = VB : D = (1 – α) VB
At V→∞ : D→ C/r
VUB 05 Options and optimal capital structure

33. Black Scholes’ PDE and the binomial model
We have:
BS PDE : f’t + rS f’S + ½ ² f”SS = r f
Binomial model: p fu + (1-p) fd = ert
Use Taylor approximation:
fu = f + (u-1) S f’S + ½ (u–1)² S² f”SS + f’t t
fd = f + (d-1) S f’S + ½ (d–1)² S² f”SS + f’t t
u = 1 + √t + ½ ²t
d = 1 – √t + ½ ²t
ert = 1 + rt
Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes
This result is explained in Rubinstein Derivatives: A PowerPlus Picture Book, Vol 1, Part B.
The proof involves a lot of tedious algebra.
Substituting the Taylor approximations in the binomial option pricing model:
f + [pu+(1-p)d]Sf’S – Sf’S + ½[p(u-1)² + (1-p)(d-1)²]S²f”SS + ft = f(1+rt)
But:
p u + (1-p) d = 1 + rt
and:
p (u-1)² + (1-p) (d-1)² = ² t
so that:
f + rt S f’S + ½ ² t S² f”SS + f’t t = f + r f t
Simplify and rearrange to obtain the PDE.
VUB 05 Options and optimal capital structure

34. Unprotected and protected debt
Unprotected debt:
Constant coupon
Bankruptcy if V = VB
Endogeneous bankruptcy level: when equity falls to zero
Protected debt:
Bankruptcy if V = principal value of debt D0
Interpretation: continuously renewed line of credit (short-term financing)
VUB 05 Options and optimal capital structure

35. Example
VUB 05 Options and optimal capital structure

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40. VUB 05 Options and optimal capital structure

41. References
Altman, E., Resti, A. and Sironi, A., Analyzing and Explaining Default Recovery Rates, A Report Submitted to ISDA, December 2001
Bohn, J.R., A Survey of Contingent-Claims Approaches to Risky Debt Valuation, Journal of Risk Finance (Spring 2000) pp
Merton, R. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates Journal of Finance, 29 (May 1974)
Merton, R. Continuous-Time Finance Basil Blackwell 1990
Leland, H. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure Journal of Finance 44, 4 (September 1994) pp
VUB 05 Options and optimal capital structure