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Structural Models I The Merton Model Stephen M Schaefer London Business School Credit Risk Elective Summer 2012 The Black-Scholes-Merton Option Pricing Model “… options are specialized and relatively unimportant financial securities …”. Robert Merton – Nobel prize winner for work on option pricing – in 1974 seminal paper on option pricing • Great hope for the new theory was the valuation of corporate liabilities, in particular: equity corporate debt Structural Models 1 2 Equity is a call option on the firm • Suppose a firm has borrowed €5 million (zero coupon) for 5 years (say) and that at the maturity of the loan there are two possible scenarios: Scenario I: the assets of the firm are worth €9 million: lenders get €5 million (paid in full) equity holders get residual: €9 - €5 = €4 million Scenario II: the assets are worth, say, €3 million firm defaults, lenders take over assets and get €3 million equity holders receive zero • Payments to equity holders are those of a call option written on the assets of the firm with a strike price of €5 million, the face value of the debt Structural Models 1 3 Payoffs to Debt and Equity at Maturity • Firm has single 5-year zero-coupon bond outstanding with face value B= € 5 (million) Equity is a call option on the assets of the firm Payoff on risky debt looks like this 8 8 7 6 Bond Payoff at Maturiy Equity Payoff at Maturiy 7 5 no default 4 ` 3 2 1 default 1 2 3 5 no default 4 ` 3 default 2 1 0 0 6 4 5 6 7 8 9 value of assets of firm at maturity (€ million) Structural Models 1 10 11 0 0 1 2 3 4 5 6 7 8 9 10 value of assets of firm at maturity (€ million) 4 11 Prior to maturity … Equity as a Call Option • Face value B= € 5 (million); riskless PV of debt = € 3.5 (million) 10 Equity Value (€ million) 8 value of equity when assets are risky ` 6 4 2 value of equity if assets are riskless max(V - PV(B), 0) 0 0 1 2 3 4 5 6 7 8 9 10 value of assets of firm (€ million) Structural Models 1 5 Prior to maturity … Value of Debt and Equity 10 9 7 € million • value of the debt is value of firm’s assets less the value of the equity (a call) asset value 8 6 5 value of debt 4 3 2 value of equity 1 0 0 1 2 3 4 5 6 7 8 9 10 value of assets of firm (€ million) Structural Models 1 6 • The sensitivity of a credit risky bond to the value of the collateralising assets (the “firm value”) is a useful way of thinking about credit exposure. Structural Models 1 7 What is the price discount on credit risky debt? put-call parity Modigliani-Miller underlying = riskless - put + call option asset option bond value of = firm assets bond value + equity value • Since equity is a call option value of value of put riskless = riskydebt option on assets bond • Merton model uses Black-Scholes to value the (default) put. Structural Models 1 8 Limited Liability and the “Default Put” • Limited liability of equity means that no matter how bad things get, equity holders can walk away from firm’s debt in exchange for payoff of zero • Limited liability equivalent to equity holders: issuing riskless debt BUT lenders giving equity holders a put on the firm’s assets with a strike price equal to the face amount of the debt (“default put”) Structural Models 1 9 Relation between Bond Spread and Value of Equity • Bond value increases and the spread declines with the value of firm assets (left hand panel below) • Equity value increases with the value of firm assets (right hand panel below) • Combining these two, the spread also declines with the equity value (next slide) Equity Value Bond Value and Bond Spread 4.0 6 25% 3.5 4 2.5 15% Bond Value (LHS) Bond Spread (RHS) 2.0 10% 1.5 € million € million 5 20% 3.0 3 2 1.0 5% 1 0.5 - 0% 0 1 2 3 4 5 6 7 8 value of assets of firm (€ million) Structural Models 1 9 10 0 0 1 2 3 4 5 6 7 8 9 value of assets of firm (€ million) 10 10 Bond Spread vs. Equity Value 25% Bond Spread (%) 20% 15% 10% 5% 0% 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 equity value (€ million) Structural Models 1 11 Understanding Credit • The idea that corporate debt and equity can be viewed as derivatives written on the assets of the firm is of fundamental importance • This idea is the basis of the structural approach – one of the two most useful ways of thinking about and analysing credit risky instruments Structural Models 1 12 The Merton Model Structural Models 1 13 Valuation Theory: Merton Model • Merton model: value credit risky bond as value of equivalent riskless bond minus Black-Scholes value of put on assets • Assumptions as those of Black-Scholes model lognormal distribution for value of assets of firm no uncertainty in interest rates • Merton model: basis for all structural models has been generalised and extended Structural Models 1 14 The Merton Model: Assumptions • Parameters constant interest rate: constant volatility of firm value • Structure of debt zero coupon bond is only liability – dealing with realistically complex capital structure is problematic • Nature of bankruptcy costless bankruptcy: “undramatic” – simply allocation of property rights between equity and debt holders no loss of value in default (e.g., nothing for the lawyers) strict priority of claims preserved: defines recovery rate (1-L) bankruptcy triggered only at maturity when value of assets falls below face value of debt: defines default event Structural Models 1 15 Bond Prices in the Merton Model Structural Models 1 16 Black-Scholes Formula for Call Option Value • The Black-Scholes formula for value of a call option on a stock with current price S and exercise price X, is: C = SN ( d ) − PV ( X ) N ( d ) 1 2 1 2 ln( S / X ) + ( r + σ )T 2 S d = 1 σS T d 2 = d −σ S 1 and T Structural Models 1 17 Bond Prices in the Merton Model • The Black-Scholes Value for a call on the firm assets (V) with exercise price B, i.e., the value of equity, E, is: 1 2 ln(V / B ) + ( r + 2σ V )T and d = d − σ V E = VN ( d ) − PV ( B ) N ( d ); d = 1 2 1 2 1 σV T T • Since the bond value, D, is the firm value minus the equity value: D =V −E = V − VN ( d1 ) − PV ( B ) N ( d 2 ) = V (1 − N ( d )) + PV ( B ) N ( d ) 1 2 = VN ( − d ) + PV ( B ) N ( d ) 1 2 Structural Models 1 since 1 − N ( x ) = N ( − x ) 18 Credit Spreads in the Merton Model • The promised yield on the bond in the Merton model is y = r + s where s is the "credit spread" and y is defined by: D≡e and − yT B=e − ( r + s )T B=e − sT PV ( B ) D = VN ( − d ) + PV ( B ) N ( d ) 1 2 • If we now define the “quasi leverage ratio”, L, as PV(B)/V, i.e., the debt-to-firm value ratio, but valuing the debt using the riskless rate , then we can express the spread simply as a function of L and σ√Τ s=− 1 1 ln(1/ L) 1 + σ T , and d 2 = d1 − σ T ln N ( −d1 ) + N (d 2 ) , where d1 = 2 T L σ T Structural Models 1 19 What are Reasonable Values of Leverage and Asset Volatility? All AAA Mean Std.Dev. 0.34 0.21 0.10 0.08 Mean Std.Dev. 0.32 0.13 0.25 0.06 Mean Std.Dev. 0.22 0.08 0.22 0.05 AA A BBB Quasi-Market Leverage 0.21 0.32 0.37 0.19 0.20 0.17 Equity Volatility 0.29 0.31 0.33 0.10 0.11 0.13 Estimated Asset Volatility 0.22 0.21 0.22 0.07 0.08 0.08 BB B 0.50 0.23 0.66 0.22 0.42 0.19 0.61 0.19 0.23 0.08 0.28 0.08 • Quasi-market leverage ratio Book Value of Debt (Compustat items 9 and 34) Book Value of Debt + Market Value of Equity • Estimated asset volatility 2 2 2 σ Ajt = (1 − L jt ) 2 σ Ejt + L2jtσ Djt + 2 L jt (1 − L jt )σ ED , jt Source: Schaefer & Strebulaev (2009) Structural Models 1 20 The Merton Model, contd. • For a given maturity, the credit spread in the Merton model depends on only two variables: asset volatility and the quasi-leverage ratio: L=PV(B)/V • When the likelihood of default is high (right-hand panel) the term structure of spreads is likely to be downward sloping; • When the risk of default is lower (LH panel) it will be upward sloping or hump shaped. 14.0% 3.0% L=0.65 L=1.1 12.0% 2.5% 10.0% Volatility = 20 spread (% p.a.) 1.5% Volatility = 25 8.0% Volatility = 30 6.0% 1.0% 4.0% Volatility = 20 Volatility = 25 0.5% 2.0% Volatility = 30 0.0% 0.0% 0 5 10 15 20 25 0 30 5 10 15 20 25 30 maturity (years) maturity (years) Structural Models 1 21 Merton model: spreads vs. leverage and volatility • In the Merton model the spread over riskless rate increases with volatility and leverage (L) 4.5% 4.0% 3.5% 3.0% spread (% p.a.) spread (% p.a.) 2.0% L= 30% maturity = 10 years L= 50% L= 70% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 10.00 15.00 20.00 25.00 30.00 35.00 volatility (% p.a.) Structural Models 1 22 Hedging Corporate Debt with Equity Structural Models 1 23 Hedging Corporate Debt with Equity • An important implication of the Black-Scholes framework is that it is possible to hedge (in principle perfectly) an option with the underlying stock. • In the same way, the same framework implies that it should be possible to hedge the credit risk on a corporate bond with a position in the equity. • This is fundamental to the model because the idea is that both the debt and equity values are driven by changes in the firm value. • This implies that it should be possible to hedge debt with equity (or vice versa) Structural Models 1 24 Hedging Corporate Debt with Equity • For a call option that has been sold (bought) the amount of the underlying stock that must be bought (sold) to hedge the position is given by the option delta ∆ = N(d1) • For a long position in a bond in the Merton model the amount of the underlying stock that must be sold per unit investment in the bond is given by: 1 E − 1 , where E is the value of equity, D is the value of debt ∆ D and ∆ is the delta of the firm's equity (call) - N (d1 ) - against the value of the firm's assets (V ) Structural Models 1 25 Estimating Hedge Ratios Via Regression • Running a regression of the rate of return on a bond (say, monthly) against the return on equity will give a good idea of how well (or badly) this hedge will work. Rbond = α + β REquity + ε • If the Merton model worked perfectly the R-squared in this regression would be very high (although not 100% because the theoretical value of β in the Merton model changes as the equity value changes) • In practice the R-squared is much less than 100% Structural Models 1 26 The Lucent Exercise (A) • The Lucent Exercise (A) will give you the opportunity to see how well or badly the idea of hedging debt with equity works in practice. • The objective is to get a sense of the relation between debt and equity returns and whether this relation is consistent with the predictions of the Merton model. Structural Models 1 27 Merton - Takeaways • Important first step in modelling default • Idea of credit risky debt as riskless debt minus a put AND using Black-Scholes to value put. basis for all structural models of credit risk • Predictions of credit spreads appear too low (more later) • Regression as a way of investigating the effectiveness of hedging corporate debt with equity • Occurrence of default only at maturity is major limitation BUT, inclusion of early default does not necessarily increase spreads. Structural Models 1 28