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Advanced Finance 2006-2007 Risky debt (2)

Advanced Finance 2006-2007 Risky debt (2). Professor André Farber Solvay Business School Université Libre de Bruxelles. 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.

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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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