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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 company.

    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 company.

    • (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 company.

    • 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 company.

    • 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 company.

    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 company.

    • 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 company.

    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…. company.

    • 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 company.

    • 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 company.

    • 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 company.

    • 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 company.

    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 company.

    • 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


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

    V company.U

    Barrier VB

    Default point

    Time

    VUB 05 Options and optimal capital structure


    Exogeneous level of bankruptcy
    Exogeneous level of bankruptcy company.

    • 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 company.

    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 company.

    VUB 05 Options and optimal capital structure


    Understanding p b
    Understanding company.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 company.

    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 company.

    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 company.

    Risk-free debt

    PV of $1 if default

    Loss given default

    VUB 05 Options and optimal capital structure


    Endogeneous bankruptcy level
    Endogeneous bankruptcy level company.

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

    VUB 05 Options and optimal capital structure


    Leland 1994 summary

    Notation company.

    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 company.

    • 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 company.

    • 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 company.

    • 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 company.

    VUB 05 Options and optimal capital structure







    References
    References company.

    • 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