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Binnenlandse Francqui Leerstoel VUB 2004-2005 5. 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

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binnenlandse francqui leerstoel vub 2004 2005 5 options and optimal capital structure

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

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

outline of presentation
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
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
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
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 r equity increases with leverage
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
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

merton 1974 review
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
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
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
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
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
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
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
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
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
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
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
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
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
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

slide22

VU

Barrier VB

Default point

Time

VUB 05 Options and optimal capital structure

exogeneous level of bankruptcy
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
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
Year of bankruptcy – Frequency distribution

VUB 05 Options and optimal capital structure

understanding p b
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
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
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
Value of debt

Risk-free debt

PV of $1 if default

Loss given default

VUB 05 Options and optimal capital structure

endogeneous bankruptcy level
Endogeneous bankruptcy level
  • If bankrupcy takes place when market value of equity equals 0:

VUB 05 Options and optimal capital structure

leland 1994 summary
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 - Summary

VUB 05 Options and optimal capital structure

inside the model
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
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) Sf’S + ½ (u–1)² S² f”SS + f’tt
      • fd = f + (d-1) Sf’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

VUB 05 Options and optimal capital structure

unprotected and protected debt
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

example1
Example

VUB 05 Options and optimal capital structure

references
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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