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Risky Debt: Understanding Credit Risk and Bond Valuation

Explore credit risk, bond valuation, rating agencies, and Merton's model in finance. Learn about spreads, default probabilities, and debt determinants.

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Risky Debt: Understanding Credit Risk and Bond Valuation

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

  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 Probability 48% (CreditGrades calculation) 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) EMF 2006 Risky debt

  4. 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 EMF 2006 Risky debt

  5. Spread over Treasury for Industrial Bonds 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 Source: Bodies, Kane, Marcus 2005 Table 14.3 EMF 2006 Risky debt

  7. Moody’s:Average cumulative default rates 1920-1999 % 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 EMF 2006 Risky debt

  9. 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 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 ↔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 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 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 EMF 2006 Risky debt

  12. EMF 2006 Risky debt

  13. EMF 2006 Risky debt

  14. 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) EMF 2006 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 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 = 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 EMF 2006 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% EMF 2006 Risky debt

  19. EMF 2006 Risky debt

  20. 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% EMF 2006 Risky debt

  21. 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 EMF 2006 Risky debt

  22. 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 = 1.327, d = 0.754rf = 2.47% per period =(1.05)1/2-1p = 0.473 EMF 2006 Risky debt

  23. Multiperiod valuation: details 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 EMF 2006 Risky debt

  25. 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 EMF 2006 Risky debt

  26. 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)) EMF 2006 Risky debt

  27. Black-Scholes using Excel 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 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 EMF 2006 Risky debt

  29. 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 × × + 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 EMF 2006 Risky debt

  31. 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 EMF 2006 Risky debt

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

  33. 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 EMF 2006 Risky debt

  34. 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 EMF 2006 Risky debt

  35. Determinant of the spreads Volatility Quasi debt PV(F)/V Maturity EMF 2006 Risky debt

  36. Maturity and spread Proba of no default - Delta of put option EMF 2006 Risky debt

  37. 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 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 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%) Volatility Equity Debt 30% 46,626 53,374 40% 48,506 51,494 +1,880 -1,880 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 = 100,000 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 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 = 100,000 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 EMF 2006 Risky debt

  42. 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 – No tax shield • Agency costs • The tradeoff model: Leland 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) EMF 2006 Risky debt

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

  46. VU Barrier VB Default point Time EMF 2006 Risky debt

  47. 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 EMF 2006 Risky debt

  48. 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) EMF 2006 Risky debt

  49. Year of bankruptcy – Frequency distribution EMF 2006 Risky debt

  50. Understanding pB Exact value Simulation N =number of simulations Yn = Year of bankruptcy in simulation n EMF 2006 Risky debt

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