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Binnenlandse Francqui Leerstoel VUB 2004-2005 5. Options and Optimal Capital Structure

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### Binnenlandse Francqui Leerstoel VUB 2004-20055. Options and Optimal Capital Structure

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

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

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

VUB 05 Options and optimal capital structure

Weighted average cost of capital

V (=VU ) = E + D

Value of equity

rEquity

Value of all-equity firm

rAsset

rDebt

Value of debt

WACC

VUB 05 Options and optimal capital structure

Cost of equity

- 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

11%

WACC

rA

5%

rDebt

D/E

0.25

VUB 05 Options and optimal capital structure

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

VUB 05 Options and optimal capital structure

The Beta-CAPM diagram

Beta

L

βEquity

U

βAsset

r

rAsset

rDebt=rf

rEquity

0

D/E

rEquity

D/E

rDebt

WACC

VUB 05 Options and optimal capital structure

Limited liability: equity viewed as a call option on the company.Merton (1974): Review

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

Bankruptcy

VMarket value of comany

FFace value of debt

VMarket value of comany

FFace value of debt

VUB 05 Options and optimal capital structure

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: reviewUp and down factors:

V = 149,182E = 79,182D = 70,000

Risk neutral probability :

V = 100,000E = 34,854D = 65,146

V = 67,032E = 0D = 67,032

1-period valuation formula

Cost of borrowing:y = 7.45%

∆t = 1

VUB 05 Options and optimal capital structure

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

VUB 05 Options and optimal capital structure

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%

VUB 05 Options and optimal capital structure

Cost (beta) of debt

- Remember : D = PV(FaceValue) – Put
- Put = Deltaput× V + B (!! Deltaputis 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%

VUB 05 Options and optimal capital structure

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

Bankruptcy

Maturity

Today

Time

VUB 05 Options and optimal capital structure

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

VUB 05 Options and optimal capital structure

Cost of equity calculation

V = VU + TCD = E + D

Value of equity

rE

rA

Value of all-equity firm

rD

Value of debt

Value of tax shield = TCD

rD

VUB 05 Options and optimal capital structure

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

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% 46,626 53,374

40% 48,506 51,494

+1,880 -1,880

VUB 05 Options and optimal capital structure

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

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

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

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

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

Example

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) VUΔt + (.3464) VUΔW

1 path simulated for 100 years with Δt = 1/12

1,000 simulations

Result: Probability of bankruptcy = 0.677 (within the next 100 years)

Year of bankruptcy is a random variable

Expected year of bankruptcy = 25.89 (see next slide)

VUB 05 Options and optimal capital structure

Year of bankruptcy – Frequency distribution

VUB 05 Options and optimal capital structure

Understanding pB

Exact value

Simulation

N =number of simulations

Yn = Year of bankruptcy in simulation n

VUB 05 Options and optimal capital structure

Value of tax benefit

Tax shield if no default

PV of $1 if no default

Example:

VUB 05 Options and optimal capital structure

Present value of bankruptcy cost

PV of $1 if default

Recovery if default

Example:

BC(VU) = 0.50 ×25×0.25 = 3.13

VUB 05 Options and optimal capital structure

Value of debt

Risk-free debt

PV of $1 if default

Loss given default

VUB 05 Options and optimal capital structure

Endogeneous bankruptcy level

- If bankrupcy takes place when market value of equity equals 0:

VUB 05 Options and optimal capital structure

Notation

VU value of unlevered company

VBlevel of bankruptcy

C perpetual coupon

r riskless interest rate (const.)

σ volatility (unlevered)

α bankruptcy cost (fraction)

TCcorporate tax rate

Present value of $1 contingent on bankruptcy

Value of levered company:

Unlevered: VU

Tax benefit: + (TCC/r)(1-pB)

Bankrupcy costs: - αVB pB

Value of debt

Endogeneous level of bankruptcy

Leland 1994 - SummaryVUB 05 Options and optimal capital structure

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-Xwith X = 2r/σ²

- Constants A0, A1and A2determined by boundary conditions:
- At V = VB : D = (1 – α) VB
- At V→∞ : D→ C/r

VUB 05 Options and optimal capital structure

Black Scholes’ PDE and the binomial model

- We have:
- BS PDE : f’t + rS f’S + ½² f”SS = r f
- Binomial model: p fu + (1-p) fd = ert
- Use Taylor approximation:
- fu = f + (u-1) Sf’S + ½ (u–1)² S² f”SS + f’tt
- fd = f + (d-1) Sf’S + ½ (d–1)² S² f”SS + f’tt
- u = 1 + √t + ½ ²t
- d = 1 – √t + ½ ²t
- ert = 1 + rt
- Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes

VUB 05 Options and optimal capital structure

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

Example

VUB 05 Options and optimal capital structure

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. 53-70
- 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. 1213-

VUB 05 Options and optimal capital structure

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