1. Executive Master in Finance Risky debt Professor André Farber Solvay Business School Université Libre de Bruxelles

2. June 30, 2015 EMF 2006 Risky debt |2 Recently in the Financial Times GM bond fall knocks wider markets GM’s debt downloaded to BBB- (just above junk status) Stock price: $29(MarketCap $16.4b) Debt-per-share: $320(Total debt $300b) Cumulative Default Probability48% (CreditGrades calculation)

3. June 30, 2015 EMF 2006 Risky debt |3 Credit risk Credit risk exist derives from the possibility for a borrower to default on its obligations to pay interest or to repay the principal amount. Two determinants of credit risk: Probability of default Loss given default / Recovery rate Consequence: Cost of borrowing > Risk-free rate Spread = Cost of borrowing – Risk-free rate (usually expressed in basis points) Function of a rating –Internal (for loans) –External: rating agencies (for bonds)

4. June 30, 2015 EMF 2006 Risky debt |4 Rating Agencies Moody’s (www.moodys.com)www.moodys.com Standard and Poors (www.standardandpoors.com)www.standardandpoors.com Fitch/IBCA (www.fitchibca.com)www.fitchibca.com Letter grades to reflect safety of bond issue S&PAAAAAABBBBBBCCCD Moody’sAaaAaABaaBaBCaaC Very High Quality High Quality Speculative Very Poor Investment-gradesSpeculative-grades

5. June 30, 2015 EMF 2006 Risky debt |5 Spread over Treasury for Industrial Bonds

6. June 30, 2015 EMF 2006 Risky debt |6 Determinants of Bonds Safety Key financial ratio used: –Coverage ratio: EBIT/(Interest + lease & sinking fund payments) –Leverage ratio –Liquidity ratios –Profitability ratios –Cash flow-to-debt ratio Rating Classes and Median Financial Ratios, 1998-2000 Rating Category Coverage Ratio Cash Flow to Debt % Return on Capital % LT Debt to Capital % AAA21.484.234.913.3 AA10.125.221.728.2 A6.115.019.433.9 BBB3.78.513.642.5 BB2.12.611.657.2 B0.8(3.2)6.669.7 Source: Bodies, Kane, Marcus 2005 Table 14.3

7. June 30, 2015 EMF 2006 Risky debt |7 Moody’s:Average cumulative default rates 1920-1999 % 12345101520 Aaa0.00 0.020.090.201.091.892.38 Aa0.080.250.410.610.973.105.616.75 A0.080.270.600.971.373.616.137.47 Baa0.300.941.732.623.517.9211.4613.95 Inv. Grade0.160.490.931.431.974.857.599.24 Ba1.433.455.577.8010.0419.0525.9530.82 B4.489.1613.7317.5620.8931.9039.1743.70 Spec. Grade3.356.769.9812.8915.5725.3132.6137.74 All Corp.1.332.764.145.446.6511.4915.3517.79

8. June 30, 2015 EMF 2006 Risky debt |8 Modeling credit risk 2 approaches: Structural models ( Black Scholes, Merton, Black & Cox, Leland..) –Utilize option theory –Diffusion process for the evolution of the firm value –Better at explaining than forecasting Reduced form models ( Jarrow, Lando & Turnbull, Duffie Singleton ) –Assume Poisson process for probability default –Use observe credit spreads to calibrate the parameters –Better for forecasting than explaining

9. June 30, 2015 EMF 2006 Risky debt |9 Merton (1974) Limited liability: equity viewed as a call option on the company. E Market value of equity F Face value of debt V Market value of comany Bankruptcy D Market value of debt F Face value of debt V Market value of comany F Loss given default

10. June 30, 2015 EMF 2006 Risky debt |10 Using put-call parity Market value of firm: V = E + D Put-call parity (European options) Stock = Call + PV(Strike) – Put In our setting: V ↔StockThe company is the underlying asset E↔CallEquity is a call option on the company F↔StrikeThe strike price is the face value of the debt → D = PV(Strike) – Put D = Risk-free debt - Put

11. June 30, 2015 EMF 2006 Risky debt |11 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 V = 100,000 E = 34,854 D = 65,146 V = 67,032 E = 0 D = 67,032 V = 149,182 E = 79,182 D = 70,000 ∆t = 1 Binomial option pricing: review Up and down factors: Risk neutral probability : 1-period valuation formula

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15. June 30, 2015 EMF 2006 Risky debt |15 Calculating the cost of borrowing Spread = Borrowing rate – Risk-free rate Borrowing rate = Yield to maturity on risky debt For a zero coupon (using annual compounding): In our example: y = 7.45% Spread = 7.45% - 5% = 2.45% (245 basis points)

16. June 30, 2015 EMF 2006 Risky debt |16 Decomposing the value of the risky debt In our simplified model: F: loss given default if no recovery V d : recovery if default F – V d : loss given default (1 – p) : risk-neutral probability of default

17. June 30, 2015 EMF 2006 Risky debt |17 Weighted Average Cost of Capital (1) Start from WACC for unlevered company –As V does not change, WACC is unchanged –Assume that the CAPM holds WACC = r A = r f + (r M - r f )β A –Suppose: β A = 1 r M – r f = 6% WACC = 5%+6%× 1 = 11% (2) Use WACC formula for levered company to find rE

18. June 30, 2015 EMF 2006 Risky debt |18 Cost (beta) of equity Remember : C = Delta call × S - B –A call can is as portfolio of the underlying asset combined with borrowing B. The fraction invested in the underlying asset is X = (Delta call × S) / C The beta of this portfolio is X β asset When analyzing a levered company: –call option = equity –underlying asset = value of company –X = V/E = (1+D/E) In example: β A = 1 Delta E = 0.96 V/E = 2.87 β E = 2.77 r E = 5% + 6% × 2.77 = 21.59%

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20. June 30, 2015 EMF 2006 Risky debt |20 Cost (beta) of debt Remember : D = PV(FaceValue) – Put Put = Delta put × V + B (!! Delta put is negative: Delta put =Delta call – 1) So : D = PV(FaceValue) - Delta put × V - B Fraction invested in underlying asset is X = - Delta put × V/D β D = - β A Delta put V/D In example: β A = 1 Delta D = 0.04 V/D = 1.54 β D = 0.06 r D = 5% + 6% × 0.09 = 5.33%

21. June 30, 2015 EMF 2006 Risky debt |21 Multiperiod binomial valuation V uV u²V u3Vu3V u4Vu4V dV d²V udV u 2 dV u 3 dV u 2 d²V ud 3 V d4Vd4V ud²V d3Vd3V p4p4 4p 3 (1 – p) 6p ² (1 – p)² 4p (1 – p) 3 (1 – p) 4 ΔtΔt Risk neutral proba For European option, (1) At maturity, calculate - firm values; - equity and debt values - risk neutral probabilities (2) Calculate the expected values in a neutral world (3) Discount at the risk free rate

22. June 30, 2015 EMF 2006 Risky debt |22 Multiperiod binomial valuation: example Firm issues a 2-year zero-coupon Face value = 70,000 V = 100,000 Int.Rate = 5% (annually compounded) Volatility = 40% Beta Asset = 1 4-step binomial tree Δt = 0.50 u = 1.327, d = 0.754 r f = 2.47% per period =(1.05) 1/2 -1 p = 0.473

23. June 30, 2015 EMF 2006 Risky debt |23 Multiperiod valuation: details

24. June 30, 2015 EMF 2006 Risky debt |24 Multiperiod binomial valuation: additional details From the previous calculation, we can decompose D into: Risk-free debt Risk-neutral probability of default Expected loss given default Expected value at maturity: Risk-free debt = 70,000 Default probability = 0.354 Expected loss given default = 18,552 Risky debt = 70,000 – 0.354 × 18,552 = 63,427 Present value: D = 63,427 / (1.05)² = 57,530

25. June 30, 2015 EMF 2006 Risky debt |25 Toward Black Scholes formulas Increase the number to time steps for a fixed maturity The probability distribution of the firm value at maturity is lognormal Time Value Today Bankruptcy Maturity

26. June 30, 2015 EMF 2006 Risky debt |26 Black-Scholes: Review European call option: C = S N(d 1 ) – PV(X) N(d 2 ) Put-Call Parity: P = C – S + PV(X) European put option: P = + S [N(d 1 )-1] + PV(X)[1-N(d 2 )] P = - S N(-d 1 ) +PV(X) N(-d 2 ) Delta of call option Risk-neutral probability of exercising the option = Proba(S T >X) Delta of put option Risk-neutral probability of exercising the option = Proba(S T <X) (Remember: 1-N(x) = N(-x))

27. June 30, 2015 EMF 2006 Risky debt |27 Black-Scholes using Excel

28. June 30, 2015 EMF 2006 Risky debt |28 Merton Model: example Data Market value unlevered firm €100,000 Risk-free interest rate (an.comp): 5% Beta asset 1 Market risk premium 6% Volatility unlevered 40% Company issues 2-year zero-coupon Face value = €70,000 Proceed used to buy back shares Using Black-Scholes formula Price of underling asset 100,000 Exercise price 70,000 Volatility  0.40 Years to maturity 2 Interest rate 5% Value of call option 41,772 Value of put option (using put-call parity) C+PV(ExPrice)-Sprice 5,264 Details of calculation: PV(ExPrice) = 70,000/(1.05)²= 63,492 log[Price/PV(ExPrice)] = log(100,000/63,492) = 0.4543  √t = 0.40 √ 2 = 0.5657 d1 = log[Price/PV(ExPrice)]/  √ + 0.5  √ t = 1.086 d2 = d1 -  √ t = 1.086 - 0.5657 = 0.520 N(d1) = 0.861 N(d2) = 0.699 C = N(d1) Price - N(d2) PV(ExPrice) = 0.861 × 100,000 - 0.699 × 63,492 = 41,772

29. June 30, 2015 EMF 2006 Risky debt |29 Valuing the risky debt Market value of risky debt = Risk-free debt – Put Option D = e -rT F – {– V[1 – N(d 1 )] + e -rT F [1 – N(d 2 )]} Rearrange: D = e -rT F N(d 2 ) + V [1 – N(d 1 )] Value of risk-free debt Probability of no default Probability of default × × Discounted expected recovery given default +

30. June 30, 2015 EMF 2006 Risky debt |30 Example (continued) D = V – E = 100,000 – 41,772 = 58,228 D = e -rT F – Put = 63,492 – 5,264 = 58,228

31. June 30, 2015 EMF 2006 Risky debt |31 Expected amount of recovery We want to prove: E[V T |V T < F] = V e rT [1 – N(d 1 )]/[1 – N(d 2 )] Recovery if default = V T Expected recovery given default = E[V T |V T < F] (mean of truncated lognormal distribution) The value of the put option: P = -V N(-d 1 ) + e -rT F N(-d 2 ) can be written as P = e -rT N(-d 2 )[- V e rT N(-d 1 )/N(-d 2 ) + F] But, given default: V T = F – Put So: E[V T |V T < F]=F - [- V e rT N(-d 1 )/N(-d 2 ) + F] = V e rT N(-d 1 )/N(-d 2 ) Discount factor Probability of default Expected value of put given F F Default Put Recovery VTVT

32. June 30, 2015 EMF 2006 Risky debt |32 Another presentation Discount factor Face Value Probability of default Expected loss given default Loss if no recovery Expected Amount of recovery given default

33. June 30, 2015 EMF 2006 Risky debt |33 Example using Black-Scholes Data Market value unlevered company € 100,000 Debt = 2-year zero coupon Face value € 60,000 Risk-free interest rate5% Volatility unlevered company30% Using Black-Scholes formula Market value unlevered company € 100,000 Market value of equity € 46,626 Market value of debt € 53,374 Discount factor0.9070 N(d 1 )0.9501 N(d 2 )0.8891 Using Black-Scholes formula Value of risk-free debt € 60,000 x 0.9070 = 54,422 Probability of default N(-d 2 ) = 1-N(d 2 ) = 0.1109 Expected recovery given default V e rT N(-d 1 )/N(-d 2 ) = (100,000 / 0.9070) (0.05/0.11) = 49,585 Expected recovery rate | default = 49,585 / 60,000 = 82.64%

34. June 30, 2015 EMF 2006 Risky debt |34 Calculating borrowing cost Initial situation Balance sheet (market value) Assets 100,000 Equity 100,000 Note: in this model, market value of company doesn’t change (Modigliani Miller 1958) Final situation after: issue of zero-coupon & shares buy back Balance sheet (market value) Assets 100,000 Equity 41,772 Debt 58,228 Yield to maturity on debt y: D = FaceValue/(1+y)² 58,228 = 60,000/(1+y)² y = 9.64% Spread = 364 basis points (bp)

35. June 30, 2015 EMF 2006 Risky debt |35 Determinant of the spreads Quasi debt PV(F)/V Volatility Maturity

36. June 30, 2015 EMF 2006 Risky debt |36 Maturity and spread Proba of no default - Delta of put option

37. June 30, 2015 EMF 2006 Risky debt |37 Inside the relationship between spread and maturity Probability of bankruptcy d = 0.6d = 1.4 T = 10.140.85 T = 100.59 0.82 Delta of put option d = 0.6d = 1.4 T = 1-0.07-0.74 T = 10-0.15 -0.37 Spread (σ = 40%) d = 0.6d = 1.4 T = 12.46%39.01% T = 104.16% 8.22%

38. June 30, 2015 EMF 2006 Risky debt |38 Agency costs Stockholders and bondholders have conflicting interests Stockholders might pursue self-interest at the expense of creditors –Risk shifting –Underinvestment –Milking the property

39. June 30, 2015 EMF 2006 Risky debt |39 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%) VolatilityEquityDebt 30%46,62653,374 40%48,50651,494 +1,880-1,880

40. June 30, 2015 EMF 2006 Risky debt |40 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 = 100,000E = 35,958D = 64,042 Investment project: Investment 8,000 & NPV = 2,000 ∆V = I + NPV V = 110,000E = 43,780D = 66,220 ∆ V = 10,000 ∆E = 7,822 ∆D = 2,178 Shareholders loose if project all-equity financed: Invest8,000 ∆E 7,822 Loss = 178

41. June 30, 2015 EMF 2006 Risky debt |41 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 = 100,000E = 35,958D = 64,042 Dividend = 10,000 ∆V = - Dividend V = 90,000E = 28,600D = 61,400 ∆ V = -10,000 ∆E = -7,357 ∆D =- 2,642 Shareholders gain: Dividend10,000 ∆E -7,357

42. June 30, 2015 EMF 2006 Risky debt |42 Where are we? 1. Modigliani Miller 1958 V = E + D = V U WACC = r A 2. Debt and taxes: PV(Interest tax shield) V = E + D = V U +VTS WACC < r A 3. Risky debt : Merton model – No tax shield Agency costs The tradeoff model: Leland

43. June 30, 2015 EMF 2006 Risky debt |43 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)

44. June 30, 2015 EMF 2006 Risky debt |44 Trade-off theory Market value Debt ratio Value of all-equity firm PV(Tax Shield) PV(Costs of financial distress)

45. June 30, 2015 EMF 2006 Risky debt |45 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 (V U ) is known; –this value changes randomly through time according to a diffusion process with constant volatility  dV U = µV U dt +  V U dW; –the riskless interest rate r is constant; –bankruptcy takes place if the asset value reaches a threshold V B ; –debt promises a perpetual coupon C; –if bankruptcy occurs, a fraction α of value is lost to bankruptcy costs.

46. June 30, 2015 EMF 2006 Risky debt |46 VUVU Default point Time Barrier V B

47. June 30, 2015 EMF 2006 Risky debt |47 Exogeneous level of bankruptcy Market value of levered company V = V U + VTS(V U ) - BC(V U ) –V U : market value of unlevered company –VTS(V U ): present value of tax benefits –BC(V U ): present value of bankruptcy costs Closed form solution: Define p B : present value of $1 contingent on future bankruptcy

48. June 30, 2015 EMF 2006 Risky debt |48 Example Value of unlevered firm V U = 100 Volatility σ = 34.64% Coupon C = 5 Tax rate T C = 40% Bankruptcy level V B = 25 Risk-free rater = 6% Simulation: ΔV U = (.06) V U Δt + (.3464) V U ΔW 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)

49. June 30, 2015 EMF 2006 Risky debt |49 Year of bankruptcy – Frequency distribution

50. June 30, 2015 EMF 2006 Risky debt |50 Understanding p B Exact value Simulation N =number of simulations Y n = Year of bankruptcy in simulation n

51. June 30, 2015 EMF 2006 Risky debt |51 Value of tax benefit Tax shield if no default PV of $1 if no default Example:

52. June 30, 2015 EMF 2006 Risky debt |52 Present value of bankruptcy cost Recovery if default PV of $1 if default Example: BC(V U ) = 0.50 ×25×0.25 = 3.13

53. June 30, 2015 EMF 2006 Risky debt |53 Value of debt Risk- free debt Loss given default PV of $1 if default

54. June 30, 2015 EMF 2006 Risky debt |54 Endogeneous bankruptcy level If bankrupcy takes place when market value of equity equals 0:

55. June 30, 2015 EMF 2006 Risky debt |55 Leland 1994 - Summary Notation V U value of unlevered company V B level of bankruptcy Cperpetual coupon rriskless interest rate (const.) σvolatility (unlevered) αbankruptcy cost (fraction) T C corporate tax rate Present value of $1 contingent on bankruptcy Value of levered company: Unlevered: V U Tax benefit: + (T C C/r)(1-p B ) Bankrupcy costs: - α V B p B Value of debt Endogeneous level of bankruptcy

56. June 30, 2015 EMF 2006 Risky debt |56 Inside the model Value of claim on the firm: F(V U,t) Black-Scholes-Merton: solution of partial differential equation When non time dependence ( ), ordinary differential equation with general solution: F = A 0 + A 1 V + A 2 V -X with X = 2r/σ² Constants A 0, A 1 and A 2 determined by boundary conditions: At V = V B : D = (1 – α) V B At V→∞ : D→ C/r

57. June 30, 2015 EMF 2006 Risky debt |57 Unprotected and protected debt Unprotected debt: Constant coupon Bankruptcy if V = V B Endogeneous bankruptcy level: when equity falls to zero Protected debt: Bankruptcy if V = principal value of debt D 0 Interpretation: continuously renewed line of credit (short-term financing)

58. June 30, 2015 EMF 2006 Risky debt |58 The Pecking Order Theory 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 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.

59. June 30, 2015 EMF 2006 Risky debt |59 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

60. June 30, 2015 EMF 2006 Risky debt |60 The message from CFO’s: debt

61. June 30, 2015 EMF 2006 Risky debt |61 Survey evidence and capital structure theories 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

62. June 30, 2015 EMF 2006 Risky debt |62 Event studies Security Issued Security Retired Two-Day Announcement Period Retun Leverage Increased Stock RepurchaseDebtCommon21.9% Exchange offerDebtCommon14.0% Exchange offerPreferredCommon8.3% Leverage reduced Exchange offerCommonDebt-9.9% Security SalesCommonDebt-4.2% Conversion-forcing callCommonConvertible-0.4% Conversion-forcing callCommonPreferred-2.1% Source: Smith, C. Raising Capital: Theory and Evidence Against tradeoff story

63. June 30, 2015 EMF 2006 Risky debt |63 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)

64. June 30, 2015 EMF 2006 Risky debt |64 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?

65. June 30, 2015 EMF 2006 Risky debt |65 Table II - Balance Sheets for Non-Financial Firms - 1991

66. June 30, 2015 EMF 2006 Risky debt |66 Table III Leverage in different countries BookBook adjusted MarketMarket adjusted EBITDA/ Interest United States37%33%28%23%4.05x Japan53%37%29%17%4.66x Germany38%18%23%15%6.81x France48%34%41%28%4.35x Italy47%39%46%36%3.24x United Kingdom28%16%19%11%6.44x Canada39%37%35%32%3.05x Median debt to total capital in 1991 Adjusted debt = Net Debt = Debt – Cash Book: using book equity, Market: using market value of equity

67. June 30, 2015 EMF 2006 Risky debt |67 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

68. June 30, 2015 EMF 2006 Risky debt |68 Table IX Factors Correlated with Debt to Market Capital

69. June 30, 2015 EMF 2006 Risky debt |69 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-