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RISKY DEBT VALUATION RISKY DEBT AND THE WACC USING EXCEL A. Farber This version: January 2006 This workbook contains several worksheets to illustrate the choice of a capital structure Riskless debt This worksheet illustrates the problem when debt is riskless. Can be used to illustrate MM 58 and MM 63 State prices In this worksheet, we identify options implicit in risky debt in a simple setting. Valuation is achieved using state prices. Binomial 1 period This worksheet uses the binomial model to value a bond with 1 year to maturity Binomial 4 periods In this worksheet, we generalize the binomial option model to several periods Merton Here use the Merton model (a binomial model with a large number to time steps) Leland The Merton model does not take into account interest tax shields and costs of bankruptcy. The Leland model is the first in a recent family to incorporate these dimensions. a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Introduction 3/1/2011 RISKY DEBT VALUATION Capital structure - Introduction - Certainty - Perpetuities Debt 5,000 (risk free) EBIT 1,500 EBIT 1,500 (perpetuity) Interest 200 ==> Debt = 5,000 Tax rate 34% Taxes 442 rD 4% (=risk free rate) Net Income 858 ==> Equity = 6,600 Beta Asset 1.00 V= 11,600 Mkt risk premium 6% rA 10.00% Value Exp.Ret. Beta Value Exp.Ret. Beta VU 9,900.00 85.3% 10.00% 1.00 Equity 6,600.00 56.9% 13.00% 1.50 V(TaxShield) 1,700.00 14.7% 4.00% 0.00 Debt 5,000.00 43.1% 4.00% 0.00 V 11,600.00 100.0% 9.12% 0.85 11,600.00 100.0% 9.12% 0.85 EBIAT =EBIT(1-Tc) 990 WACC 8.53% V 11600 Comments: This worksheet illustrates the standard approach found in textbooks. The assumptions underlying the calculations are: (1) EBIT and debt are constant perpetuities (2) debt is riskless. Therefore, the cost of debt is equal to the risk-free rate. a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Riskless debt 3/1/2011 RISKY DEBT VALUATION The worksheet can be used to illustate both Modigliani Miller 1958 (no taxes) and Modigliani Miller 1963 (corporate taxes) To illustrate MM58, set the tax rate equal to 0. Key points to notice: (1) the value of the company does not change (2) the beta of equity and the cost of equity increase (3) the WACC does not change To illustrate MM63, set the tax rate to a positive rate. Key points to notice: (1) the value of the company increases by the value of the tax shield. (2) the value of the tax shield is equal to TaxRate * Debt (3) the beta of equity and the cost of equity increase (4) the WACC decreases (5) the value of the company can be calculated as: V = VU + VTS V = Equity + Debt V = EBIAT/WACC a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Riskless debt 3/1/2011 RISKY DEBT VALUATION Risky debt - Introduction using state prices 1 period - No taxes Now Prosperity Depression Valuing options: Strike = 70.0 Probabilities 0.535 0.465 Now Prosperity Depression Gov Bond 100.0 105.0 105.0 Call 34.9 79.2 0.0 V unlevered 100.0 149.2 67.0 Put 1.5 0.0 3.0 State prices 0.44 0.51 Risk-free rate 5.0% Put call parity Mkt risk premium 6.00% Stock 100.0 149.2 67.0 VU Put 1.5 0.0 3.0 Debt face value 70 101.5 149.2 70.0 Now Prosperity Depression Call 34.9 79.2 0.0 Equity Debt 65.1 70.0 67.0 RFBond 66.7 70.0 70.0 Risk-free debt Equity 34.9 79.2 0.0 101.5 149.2 70.0 100.0 Beta Exp.Ret. Beta Exp.Ret. VU 100.0 100% 1.00 11.00% Equity 34.9 34.9% 2.76 21.58% Debt 65.1 65.1% 0.06 5.34% V 100.0 100% 1.00 11.00% 100.0 100.0% 1.00 11.00% YTM of debt 7.5% Spread (bp) 248 a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls State prices 3/1/2011 RISKY DEBT VALUATION Risky debt - 1 period binomial model VU 100 dt 1 Volatility 40.00% u 1.492 Beta asset 1.00 d 0.670 Risk-free rate 5% RNProba p 0.462 Mkt risk premium 6% Now Up Down Debt Face value 70 V 100 149 67 Maturity 1 (years) Equity 34.9 79 0 Debt 65.1 70 67 vu 0.44 1 0 vd 0.51 0 1 Value Beta Exp.Ret. Beta Exp.Ret. VU 100.00 100.0% 1.00 11.00% Equity 34.85 34.9% 2.77 21.59% Debt 65.15 65.1% 0.06 5.33% V 100.00 100.0% 1.00 11.00% 100.00 100.0% 11.00% Yield to maturity 7.45% Face value 70.0 Spread (bp) 245 Default probability 53.78% (risk neutral) Loss|Default 3.0 Risk-free debt 66.67 Expected loss|Default 1.6 Value of put option 1.52 Exp Recovery Rate 51.50% Value of risky debt 65.15 =(FaceValue - Default probability * Loss|D a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 1 period 3/1/2011 RISKY DEBT VALUATION Comments: This worksheet uses a 1-period binomial option pricing model to value risky debt. The time interval (dt) is 1 year. The value of the company can either go up or down. Up State Down State VU u*VU d*VU u and d are gross returns (=1 + return) in both states. u and d are related to the volatility of the company: u=exp(Volatility) d=1/u Valuation is achieved using risk-neutral valuation (see Ross Westerfield Jaffe 7th ed. p. 633) The risk-neutral probability of an up (p) is: p = (1+rf-d)/(u-d) Note that this is equivalent to: p*u+(1-p)*d=1+rf Using p, the expected (gross) return is equal to the (gross) risk-free rate. The 1-period risk-neutral valuation formula is: f = [p*fu + (1-p)*fd]/(1+rf) f is the current value of any security, fu and fd are the values in the two future states. [p*fu + (1-p)*fd] is the expected value in a risk neutral world. f is obtained by discounting this risk-neutral expected value at the risk-free interest rate. Future values for the equity and the debt are calculated in cells G9:H10. The equity is a call option on the company: Su = Max(0,Vu-F) and Sd = Max(0,Vd-F) where Su and Sd are the value of equity in the two states. For the debt: Bu=Min(Vu,F) and Bd = Min(Vd,F) a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 1 period 3/1/2011 RISKY DEBT VALUATION where Bu and Bd are the values of debt in the two states. Current values using risk neutral valuation appear in cells (F9:F10) RISK NEUTRAL VALUATION AND STATE PRICE COMPARED In this setting, the state price valuation formula is: f = vu*fu + vd*fd where vu and vd are state prices. Comparing the 1-period risk-neutral valuation formula with the state price valuation formula leads to: vu = p/(1+rf) vd = (1-p)/(1+rf) (see cells F11:F12 - state prices are the prices of digital options) BETA AND EXPECTED RETURN CALCULATIONS The beta of the equity (J15) is calculated using the following formula: BetaEquity = BetaAsset * DeltaEquity * (V/Equity) with DeltaEquity = (Su-Sd)/(Vu-Vd) (This is the standard formula for the delta of an option in the binomial model) Similarly, the beta of the debt (J16) is: BetaDebt = BetaAsset * DeltaDebt * (V/Debt) with DeltaDebt = (Bu-Bd)/(Vu-Vd) Conclusions (same as in the State prices worksheet) (1) MM 1958 holds - V = VU a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 1 period 3/1/2011 RISKY DEBT VALUATION (2) WACC is constant (3) Risky debt = Risk-free debt - Put option The put option is the price that shareholders pay to have the right to default. a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 1 period 3/1/2011 RISKY DEBT VALUATION - Default probability * Loss|Default)/(1+rf) a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 1 period 3/1/2011 RISKY DEBT VALUATION Risky debt - Binomial 4 periods VU 100 dt 0.50 Here I illustrate the valuation of a 2-year bond Volatility 40.00% u 1.327 using a binomial model with 2 steps per year. Beta asset 1.00 d 0.754 The calculation are longer but the underlying logic is the sam Risk-free rate 5% rF/period 2.47% Mkt risk premium 6% RNProba p 0.473 Debt Face value 70 Maturity (year) 2 Valuation: see below Value Beta Exp.Ret. Beta Exp.Ret. VU 100.00 100.0% 1.00 11.00% Equity 42.47 42.5% 2.02 17.11% Debt 57.53 57.5% 0.25 6.49% V 100.00 100.0% 1.00 11.00% 100.00 100.0% 11.00% Yield to maturity 10.31% Face value 70.0 Spread (bp) 531 Default probability 35.43% Loss if no recovery 70.0 Risk-free debt 63.49 Expected recovery 63.4 Value of put option 5.96 Expecte Loss|Default 18.6 Exp Recovery Rate 90.61% Value of risky debt 46.7 a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Binomial 4 periods 3/1/2011 RISKY DEBT VALUATION Risky debt Merton Model Input: Value of firm 100 Delta - N(d1) 0.8299 Dividend yield 0.00% RN Proba NoDefault N(d2) 0.3779 Volatility 40.00% Beta asset 1 Mkt risk premium 5.00% Face value of debt 70 Maturity (year) 10 Interest rate 5.00% Value Beta Exp.Ret. Beta Exp.Ret. VU 100 100.0% 1.00 10.00% Equity 66.9 66.9% 1.24 11.20% Debt 33.1 33.1% 0.51 7.57% V 100 100.0% 1.00 10.00% 100 100.0% 1.00 10.00% Yield to maturity 7.50% Face value 70 Spread (bp) 250 Default probability 67.10% Loss if no recovery 70 Riskless debt 48.52 Expected recovery 37.65 Put 15.47 Exp.Loss|Default 32.35 33.05 Exp Recovery Rate 53.78% Value of risky debt 31.29 a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Merton 3/1/2011 RISKY DEBT VALUATION Risky debt - Leland (1994) Data Value of unlevered firm 100 Level of bankruptcy VB 13.68 Volatility 30.00% PV of $1 if bankruptcy pB 0.110 Beta asset 1.00 pB' 0.001 Coupon 2.00 Risk free rate 5.00% Mkt risk premium 5.00% Bankruptcy cost 50% Corporate tax rate 35% Value Beta Exp.Ret Value Beta Exp.Ret VU 100.0 89.5% 1.00 10.00% Equity 75.4 67.5% 1.31 11.54% VTS 12.5 11.2% 0.14 5.68% Debt 36.4 32.5% 0.11 5.56% - VBC 0.8 -0.7% -1.11 -0.56% V 111.7 100.0% 0.92 9.59% V 111.7 100.0% 0.92 9.59% Leverage ratio D/VL 0.33 Yield on debt 5.50% Yield spread (bp) 50 Riskless debt 40.0 Put option 3.6 a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Leland 3/1/2011 RISKY DEBT VALUATION To EMF 2006 participants: this model illustrates the trade-off theory of the capital structure. This theory is explained in RWJ 7th ed. Chap 16 The model is complex but you can use this spreadsheet to get a feeling of how the data change the results. Comments: The Leland model is an extension of the Merton model that incorporate tax shields and costs of bankruptcy. Debt is a perpetuity paying a constant coupon. Bankruptcy takes place if the value of the assets drops below some level VB (calculated in cell I4) Bankruptcy costs are expressed as a fraction of the value if bankruptcy takes place. The value of the levered firm is equal to: the value of the unlevered firm + the present value of the tax shield (VTS) - the present value of bankruptcy costs a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls Leland 3/1/2011