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Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt

Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt. Professor André Farber Solvay Business School Université Libre de Bruxelles. Today in the Financial Times. GM bond fall knocks wider markets GM’s debt downloaded to BBB- (just above junk status)

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Binnenlandse Francqui Leerstoel VUB 2004-2005 Options and risky debt

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  1. Binnenlandse Francqui Leerstoel VUB 2004-2005Options and risky debt Professor André Farber Solvay Business School Université Libre de Bruxelles

  2. Today 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 Probability 48% (CreditGrade calculation) VUB 04 Options and risky debt

  3. Fixed income markets Corporate bond market Companies Investors Assets Equity Debt Banks Loans Equity Deposits Credit derivatives Lannoo, K.and Levin, M. Toward a European Single Market for Financial Services, Presentation, CEPR 2004 VUB 04 Options and risky debt

  4. 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) VUB 04 Options and risky debt

  5. Rating Agencies • Moody’s (www.moodys.com) • Standard and Poors (www.standardandpoors.com) • Fitch/IBCA (www.fitchibca.com) • Letter grades to reflect safety of bond issue Very High Quality High Quality Speculative Very Poor Investment-grades Speculative-grades VUB 04 Options and risky debt

  6. Spread over Treasury for Industrial Bonds VUB 04 Options and risky debt

  7. 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, 1997-1999 Source: Bodies, Kane, Marcus 2002 Table 14.3 VUB 04 Options and risky debt

  8. Standard&Poor’s European Rating Distribution VUB 04 Options and risky debt

  9. Default Rate Calculation • Incorrect method: • Number defaults/Total number of bonds • Ignores growth/reduction of bond market over time • Ignores aging effect: takes time to get into trouble • Correct method: cohort style analysis • Pick up a cohort • Follow it through time VUB 04 Options and risky debt

  10. Moody’s:Average cumulative default rates 1920-1999 % VUB 04 Options and risky debt

  11. 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 VUB 04 Options and risky debt

  12. Limited liability: equity viewed as a call option on the company. Merton (1974) D Market value of debt E Market value of equity Loss given default F Bankruptcy VMarket value of comany FFace value of debt VMarket value of comany FFace value of debt VUB 04 Options and risky debt

  13. 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 ↔Stock The company is the underlying asset • E↔Call Equity is a call option on the company • F↔Strike The strike price is the face value of the debt • → D = PV(Strike) – Put • D = Risk-free debt - Put VUB 04 Options and risky debt

  14. 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 ∆t = 1 VUB 04 Options and 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 compouding): • In our example: y = 7.45% Spread = 7.45% - 5% = 2.45% (245 basis points) VUB 04 Options and risky debt

  16. Decomposing the value of the risky debt In our simplified model: F: loss given default if no recovery Vd : recovery if default F – Vd: loss given default (1 – p) : risk-neutral probability of default VUB 04 Options and 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 = 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 04 Options and risky debt

  18. Cost (beta) of equity • Remember : C = Deltacall× 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 = (Deltacall× S) / C • The beta of this portfolio is X βasset • When analyzing a levered company: • call option = equity • underlying asset = value of company • X = V/E = (1+D/E) In example: βA = 1 DeltaE = 0.96 V/E = 2.87 βE= 2.77 rE = 5% + 6%× 2.77 = 21.59% VUB 04 Options and risky debt

  19. 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% × 0.09 = 5.33% VUB 04 Options and risky debt

  20. Multiperiod binomial valuation Risk neutral proba u4V p4 • 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 u3V u²V 4p3(1 – p) u3dV uV u2dV 6p²(1 – p)² V udV u2d²V ud²V dV 4p(1 – p)3 ud3V Δt d²V d3V (1 – p)4 d4V VUB 04 Options and risky debt

  21. Multiperiod binomial valuation: example Firm issues a 2-year zero-couponFace value = 70,000V = 100,000Int.Rate = 5% (annually compounded)Volatility = 40%Beta Asset = 1 4-step binomial tree Δt = 0.50u = 1.332, d = 0.751rf = 2.47% per period =(1.05)1/2-1p = 0.471 VUB 04 Options and risky debt

  22. Multiperiod valuation: details VUB 04 Options and risky debt

  23. 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 VUB 04 Options and risky debt

  24. 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 04 Options and risky debt

  25. Black-Scholes: Review • European call option: C = S N(d1) – PV(X) N(d2) • Put-Call Parity: P = C – S + PV(X) • European put option: P = - S [N(d1)-1] + PV(X)[1-N(d2)] • P = - S N(-d1) +PV(X) N(-d2) Risk-neutral probability of exercising the option = Proba(ST>X) Delta of call option Risk-neutral probability of exercising the option = Proba(ST<X) Delta of put option (Remember: 1-N(x) = N(-x)) VUB 04 Options and risky debt

  26. Black-Scholes using Excel VUB 04 Options and risky debt

  27. 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 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 Using Black-Scholes formula Price of underling asset 100,000 Exercise price 70,000 Volatility s 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 VUB 04 Options and risky debt

  28. Valuing the risky debt • Market value of risky debt = Risk-free debt – Put Option D = e-rTF – {– V[1 – N(d1)] + e-rTF [1 – N(d2)]} • Rearrange: D = e-rTF N(d2) + V [1 – N(d1)] Discounted expected recovery given default Probability of default Value of risk-free debt Probability of no default × × + VUB 04 Options and risky debt

  29. Example (continued) D = V – E = 100,000 – 41,772 = 58,228 D = e-rT F – Put = 63,492 – 5,264 = 58,228 VUB 04 Options and risky debt

  30. Expected amount of recovery • We want to prove: E[VT|VT < F] = V erT[1 – N(d1)]/[1 – N(d2)] • Recovery if default = VT • Expected recovery given default = E[VT|VT < F] (mean of truncated lognormal distribution) • The value of the put option: • P = -V N(-d1) + e-rT F N(-d2) • can be written as • P = e-rT N(-d2)[- V erT N(-d1)/N(-d2) + F] • But, given default: VT = F – Put • So: E[VT|VT < F]=F - [- V erT N(-d1)/N(-d2) + F] = V erT N(-d1)/N(-d2) Put F Recovery Discount factor Expected value of put given Probability of default F Default VT VUB 04 Options and risky debt

  31. Another presentation Probability of default Loss if no recovery Discount factor Face Value Expected Amount of recovery given default Expected loss given default VUB 04 Options and risky debt

  32. Example using Black-Scholes DataMarket value unlevered company € 100,000Debt = 2-year zero coupon Face value € 60,000 Risk-free interest rate 5%Volatility unlevered company 30% Using Black-Scholes formula Value of risk-free debt € 60,000 x 0.9070 = 54,422 Probability of defaultN(-d2) = 1-N(d2) = 0.1109 Expected recovery given defaultV erT N(-d1)/N(-d2) = (100,000 / 0.9070) (0.05/0.11)= 49,585 Expected recovery rate | default= 49,585 / 60,000 = 82.64% Using Black-Scholes formula Market value unlevered company € 100,000Market value of equity € 46,626Market value of debt € 53,374 Discount factor 0.9070N(d1) 0.9501N(d2) 0.8891 VUB 04 Options and risky debt

  33. 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) Calculating borrowing cost VUB 04 Options and risky debt

  34. Determinant of the spreads Volatility Quasi debt PV(F)/V Maturity VUB 04 Options and risky debt

  35. Maturity and spread Proba of no default - Delta of put option VUB 04 Options and risky debt

  36. Inside the relationship between spread and maturity Spread (σ = 40%) d = 0.6 d = 1.4 T = 1 2.46% 39.01% T = 10 4.16% 8.22% Probability of bankruptcy d = 0.6 d = 1.4 T = 1 0.14 0.85 T = 10 0.59 0.82 Delta of put option d = 0.6 d = 1.4 T = 1 -0.07 -0.74 T = 10 -0.15 -0.37 VUB 04 Options and risky debt

  37. Agency costs • Stockholders and bondholders have conflicting interests • Stockholders might pursue self-interest at the expense of creditors • Risk shifting • Underinvestment • Milking the property VUB 04 Options and risky debt

  38. 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 04 Options and risky debt

  39. 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 04 Options and risky debt

  40. 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 04 Options and risky debt

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