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Advanced Finance 2006-2007 Risky debt (2)

This advanced finance course explores the Merton Model for valuing risky debt and examines the relationship between spreads and maturity. Professor André Farber from Solvay Business School, Université Libre de Bruxelles, guides students through Black-Scholes formulas and techniques using Excel. Gain insights into risk-neutral probabilities, expected recovery rates, and agency costs. Text language is English.

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Advanced Finance 2006-2007 Risky debt (2)

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  1. Advanced Finance2006-2007Risky debt (2) Professor André Farber Solvay Business School Université Libre de Bruxelles

  2. 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 Advanced Finance 2007 Risky debt - Merton

  3. 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)) Advanced Finance 2007 Risky debt - Merton

  4. Black-Scholes using Excel Advanced Finance 2007 Risky debt - Merton

  5. 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 Advanced Finance 2007 Risky debt - Merton

  6. 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 × × + Advanced Finance 2007 Risky debt - Merton

  7. Example (continued) D = V – E = 100,000 – 41,772 = 58,228 D = e-rT F – Put = 63,492 – 5,264 = 58,228 Advanced Finance 2007 Risky debt - Merton

  8. 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 Advanced Finance 2007 Risky debt - Merton

  9. Another presentation Probability of default Loss if no recovery Discount factor Face Value Expected Amount of recovery given default Expected loss given default Advanced Finance 2007 Risky debt - Merton

  10. 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 Advanced Finance 2007 Risky debt - Merton

  11. 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 Advanced Finance 2007 Risky debt - Merton

  12. Determinant of the spreads Volatility Quasi debt PV(F)/V Maturity Advanced Finance 2007 Risky debt - Merton

  13. Maturity and spread Proba of no default - Delta of put option Advanced Finance 2007 Risky debt - Merton

  14. 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 Advanced Finance 2007 Risky debt - Merton

  15. Agency costs • Stockholders and bondholders have conflicting interests • Stockholders might pursue self-interest at the expense of creditors • Risk shifting • Underinvestment • Milking the property Advanced Finance 2007 Risky debt - Merton

  16. 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 Advanced Finance 2007 Risky debt - Merton

  17. 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 Advanced Finance 2007 Risky debt - Merton

  18. 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 Advanced Finance 2007 Risky debt - Merton

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