Advanced Finance 2007-2008 Options and Optimal Capital Structure

1. Advanced Finance2007-2008Options and Optimal Capital Structure Professor André Farber Solvay Business School Université Libre de Bruxelles

2. Where are we? • 1. Modigliani Miller 1958 • V = E + D = VU • WACC = rA • 2. Debt and taxes: PV(Interest tax shield) • V = E + D = VU +VTS • WACC < rA • 3. Risky debt : Merton model • Agency costs • The tradeoff model: Leland Advanced Finance 2008 Optimal capital structure

3. 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) Advanced Finance 2008 Optimal capital structure

4. Trade-off theory Market value PV(Costs of financial distress) PV(Tax Shield) Value of all-equity firm Debt ratio Advanced Finance 2008 Optimal capital structure

5. 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. Advanced Finance 2008 Optimal capital structure

6. VU Barrier VB Default point Time Advanced Finance 2008 Optimal capital structure

7. 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 Advanced Finance 2008 Optimal capital structure

8. 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) Advanced Finance 2008 Optimal capital structure

9. Year of bankruptcy – Frequency distribution Advanced Finance 2008 Optimal capital structure

10. Understanding pB Exact value Simulation N =number of simulations Yn = Year of bankruptcy in simulation n Advanced Finance 2008 Optimal capital structure

11. Value of tax benefit Tax shield if no default PV of $1 if no default Example: Advanced Finance 2008 Optimal capital structure

12. Present value of bankruptcy cost PV of $1 if default Recovery if default Example: BC(VU) = 0.50 ×25×0.25 = 3.13 Advanced Finance 2008 Optimal capital structure

13. Value of debt Risk-free debt PV of $1 if default Loss given default Advanced Finance 2008 Optimal capital structure

14. Endogeneous bankruptcy level • If bankrupcy takes place when market value of equity equals 0: Advanced Finance 2008 Optimal capital structure

15. 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 Advanced Finance 2008 Optimal capital structure

16. 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 Advanced Finance 2008 Optimal capital structure

17. 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 Advanced Finance 2008 Optimal capital structure

18. 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) Advanced Finance 2008 Optimal capital structure

19. Example Advanced Finance 2008 Optimal capital structure

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25. Developed by S. Myers (1984) Starts with asymmetric information: Managers know more than outside investors Use equity if stock overvalued Use debt if stock undervalued Issuing equity is a signal of overvaluation =>stock price drops Main implication: stock issues costly Order of preference for financing: 1.Internal funds 2. Debt 3. Stock issue The Pecking Order Theory Consider the following story: The announcement of a stock issue drives down the stock price because investors believe managers are more likely to issue when shares are overpriced. Therefore firms prefer internal finance since funds can be raised without sending adverse signals. If external finance is required, firms issue debt first and equity as a last resort. The most profitable firms borrow less not because they have lower target debt ratios but because they don't need external finance. Advanced Finance 2008 Optimal capital structure

26. Implications of the pecking order theory • Firms do not have target debt ratios • Debt absorbs difference between retained earnings and investments • Debt increases when investments > retained earnings • Debt decreases when investments < retained earnings Advanced Finance 2008 Optimal capital structure

27. The message from CFO’s: debt Advanced Finance 2008 Optimal capital structure

28. Trade-off theory Corporate interest deduction moderately important Cash flow volatility important 44% have strict or somewhat strict target/range But: Expected distressed costs not important Personal taxes not important Pecking order theory Firm value flexibility Issue debt when internal funds are insufficient Equity issuance affected by equity undervaluation But: Equity issuance decision unaffected by ability to obtain funds from debt,… Debt issuance unaffected by equity valuation Survey evidence and capital structure theories Advanced Finance 2008 Optimal capital structure

29. Event studies Source: Smith, C. Raising Capital: Theory and Evidence Against tradeoff story Advanced Finance 2008 Optimal capital structure

30. Problems with empirical studies • Require data basis + computing capacities • Accounting convention obscure relevant variables • Problem for isolating capital structure decisions from other decisions • Which econometric techniques to use? • What are the testable hypothesis? • How to measure the relevant variables? • Contradictory results • Harris & Ravis (1990) “The second major trend in financial structure has been the secular increase in leverage.” (p.331) • Barclay, Smith, Watts (1995) “When viewed over the entire 30-year period, however, both market leverage ratios and dividend yields appear to be remarkably stable.” (p. 5) Advanced Finance 2008 Optimal capital structure

31. Rajan Zingales 1995 • International data – 1987-1991 • Large listed companies • Difference in accounting rules: pensions, leases • Do leverage ratios vary across countries? • Are determinants of leverage identical across countries? Advanced Finance 2008 Optimal capital structure

32. Table II - Balance Sheets for Non-Financial Firms - 1991 Advanced Finance 2008 Optimal capital structure

33. Table III Leverage in different countries Median debt to total capital in 1991 Adjusted debt = Net Debt = Debt – Cash Book: using book equity, Market: using market value of equity Advanced Finance 2008 Optimal capital structure

34. Determinants of leverage • Tangibility of assets: Fixed Assets/Total Assets Debt • Collateral => lower agency cost of debt • More value in liquidation • Market to book Debt • Growth opportunities - underinvestment • Costs of financial distress • Size Debt • Lower probability of bankruptcy • Less asymmetry of information • Profitability • Myers Majluf: profitable companies prefer internal funds Advanced Finance 2008 Optimal capital structure

35. Table IX Factors Correlated with Debt to Market Capital Advanced Finance 2008 Optimal capital structure

36. 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- Advanced Finance 2008 Optimal capital structure