Corporate Financial Policy Calculating an optimal capital structure Professor André Farber Solvay Business School 2003-2004 04/06/2019 Cofipo 2003-2004 Merton Leland

References 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- 04/06/2019 Cofipo 2003-2004 Merton Leland

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 Rate, Journal of Finance, 29 (May 1974) Leland, H., Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, Journal of Finance, 44, 4 (September 1994) pp. 1213 - 04/06/2019 Cofipo 2003-2004 Merton Leland

V Market value of comany Merton (1974) Limited liability: equity viewed as a call option on the company. Mkt value of equity: E = V - D Put-call parity (European options) Call = Stock + Put - PV(Strike) Conclusion: D = PV(Strike) - Put E Market value of equity Market value of risky debt Market value of risk-free debt Bankruptcy Market value of put option (option to default) F Face value of debt V Market value of comany 04/06/2019 Cofipo 2003-2004 Merton Leland

V Market value of company Risky debt Market value of debt F Face value of debt V Market value of company 04/06/2019 Cofipo 2003-2004 Merton Leland

Merton Model: example using binomial option pricing Data: Market Value of Unlevered Firm: 100,000 Risk-free rate: 5% (cont.compounding) Volatility: 40.55% u = 1.50 d = 0.667 Company issues 1-year zero-coupon Face value = 70,000 Proceeds used to pay dividend Binomial option pricing - Review Up and down factors: Risk neutral probability: 1-period valuation formula: V = 150,000 E = 80,000 D = 70,000 V = 100,000 E = ? D = ? V = 66,667 E = 0 D = 66,667 ∆t = 1 04/06/2019 Cofipo 2003-2004 Merton Leland

Binomial valuation Risk neutral probability p =(1.051 - 0.67)/(1.50-0.67) = 0.46 Market value of equity (a call option) E =( 0.46 x 80,000 + 0.54 x 0)/1.051=35,048 Market value of debt: B = V - E = 100,000 - 35,048 = 64,952 Market value of riskless debt = 66,586 Value of put option = 1,634 Borrowing rate = 70,000/64,952 - 1 = 7.77% Spread = 2.77% 04/06/2019 Cofipo 2003-2004 Merton Leland

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)) 04/06/2019 Cofipo 2003-2004 Merton Leland

Merton Model: example using Black-Scholes Data Market value unlevered firm €100,000 Risk-free interest rate: 5% Beta asset 1 Market risk premium 6% Volatility unlevered 30% Company issues 2-year zero-coupon Face value = €60,000 Proceed used to buy back shares Details of calculation: PV(ExPrice) = 60,000/(1.05)²= 54,422 log[Price/PV(ExPrice)] = log(100,000/54,422) = 0.6084 √t = 0.30 √ 2 = 0.4243 d1 = log[Price/PV(ExPrice)]/ √ + 0.5 √ t = 0.6084/0.4243 + (0.5)(0.4243) = 1.6462 d2 = d1 - √ t = 1.6460 - 0.4243 = 1.2219 N(d1) = 0.95 N(d2) = 0.89 C = N(d1) Price - N(d2) PV(ExPrice) = 0.95 × 100,000 - 0.89 × 54,422 = 46,626 Using Black-Scholes formula Price of underling asset 100,000 Exercise price 60,000 Volatility ó 0.30 Years to maturity 2 Interest rate 5% Value of call option 46,626 Value of put option (using put-call parity) C+PV(ExPrice)-Sprice 1,047 04/06/2019 Cofipo 2003-2004 Merton Leland

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 buyback Balance sheet (market value) Assets 100,000 Equity 46,626 Debt 53,374 Decomposition of market value of debt: Risk-free debt 54,422 - Put option 1,047 Yield to maturity on debt y: D = FaceValue/(1+y)² 53,374 = 60,000/(1+y)² y = 6.03% Spread = 103 basis points (bp) 04/06/2019 Cofipo 2003-2004 Merton Leland

Cost of equity (1) Start from WACC for unlevered company As V does not change, WACC is unchanged For unlevered company: WACC = rA = rf + (rM - rf)βassets = 5%+6%× 1 = 11% (2) Use WACC formula for levered company to find rE 04/06/2019 Cofipo 2003-2004 Merton Leland

Beta of equity Remember : C = Deltacall × S - B A call can be viewed as portfolio of the underlying asset combined with some 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: βasset = 1 Delta = 0.95 D/E = 1.14 βequity= 2.03 rE = 5% + 6% x 2.03 = 17.20% 04/06/2019 Cofipo 2003-2004 Merton Leland

Beta of debt Remember : D = PV(FaceValue) - Put Put = Deltaput × V + B !! Deltaput is negative So : D = PV(FaceValue) - Deltaput × V - B Fraction invested in underlying asset is X = - Deltaput × V/D βput = - βasset Deltaput V/D In example: βasset = 1 Deltaput = -.05 V/D = 1.87 βD= 0.09 rD = 5% + 6% x 0.09 = 5.56% 04/06/2019 Cofipo 2003-2004 Merton Leland

Agency costs Stockholders and bondholders have conflicting interests Stockholders might pursue self-interest at the expense of creditors Risk shifting Underinvestment Milking the property 04/06/2019 Cofipo 2003-2004 Merton Leland

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 04/06/2019 Cofipo 2003-2004 Merton Leland

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 04/06/2019 Cofipo 2003-2004 Merton Leland

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 04/06/2019 Cofipo 2003-2004 Merton Leland

More on Merton Market value of risky debt = Risk-free debt – Put Option Exercise price = Face value of debt (F) Underlying asset = Value of company (V) D = PV(F) – [- V N(-d1) + PV(F) N(-d2)] = PV(F) – N(-d2) [PV(F) – V N(-d1)/N(-d2)] Default probability Expected discounted loss given default 04/06/2019 Cofipo 2003-2004 Merton Leland

Another presentation D = PV(F) – N(-d2) [PV(F) – V N(-d1)/N(-d2)] = e-rT F - N(-d2) [e-rT F – V N(-d1)/N(-d2)] = e-rT {F - N(-d2) [F – V erT N(-d1)/N(-d2)]} Discount factor Face Value Probability of default Loss if no recovery Expected Amount of recovery Expected loss given default 04/06/2019 Cofipo 2003-2004 Merton Leland

Expected amount of recovery Recovery if default = VT Expected recovery = E[VT|VT < F] (mean of truncated lognormal distribution) We want to prove: E[VT|VT < F] = V erT N(-d1)/N(-d2) To see why, notice that 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 Discount factor Probability of default Expected value of put given Recovery F Default VT 04/06/2019 Cofipo 2003-2004 Merton Leland

Example using Black-Scholes Data Market value unlevered company € 100,000 Debt = 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 default N(-d2) = 1-N(d2) = 0.1109 Expected recovery given default V 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 € 100,000 Market value of equity € 46,626 Market value of debt € 53,374 Discount factor 0.9070 N(d1) 0.9501 N(d2) 0.8891 04/06/2019 Cofipo 2003-2004 Merton Leland

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) is known; this value changes randomly through time according to a diffusion process with constant volatility  dV= µVdt + VdW; 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. 04/06/2019 Cofipo 2003-2004 Merton Leland

V Barrier VB Time Default point 04/06/2019 Cofipo 2003-2004 Merton Leland

Exogeneous level of bankruptcy Market value of levered company VL = V + PVTS(V) - BC(V) V: market value of unlevered company PVTS(V): present value of tax benefits BC(V): present value of bankruptcy costs Closed form solution: Define pB : present value of $1 contingent on future bankruptcy 04/06/2019 Cofipo 2003-2004 Merton Leland

Tax shield if no default Value of tax benefit Tax shield if no default 04/06/2019 Cofipo 2003-2004 Merton Leland

Present value of bankruptcy cost 04/06/2019 Cofipo 2003-2004 Merton Leland

Value of debt 04/06/2019 Cofipo 2003-2004 Merton Leland

Endogeneous bankruptcy level If bankrupcy takes place when market value of equity equals 0: 04/06/2019 Cofipo 2003-2004 Merton Leland

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