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

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

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  1. Binnenlandse Francqui Leerstoel VUB 2004-20055. 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 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 WACC VUB 05 Options and optimal capital structure

  5. 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

  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 VUB 05 Options and optimal capital structure

  7. 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

  8. 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

  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: 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

  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 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% VUB 05 Options and optimal capital structure

  12. 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

  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 Bankruptcy Maturity Today 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 VUB 05 Options and optimal capital structure

  15. 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

  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% 46,626 53,374 40% 48,506 51,494 +1,880 -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 Default point Time 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 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

  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. 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 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

  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. 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

  32. 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

  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) 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

  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

  36. VUB 05 Options and optimal capital structure

  37. VUB 05 Options and optimal capital structure

  38. VUB 05 Options and optimal capital structure

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  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. 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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