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Unformatted text preview: CREDIT RISK: Lecture III Lina ElJahel 2006 1 Firm Value Models: Default prior to ma turity The trouble with the Merton model is that it is hard to generate substantial default premia observed in actual markets. In this lecture we will look at three models 1. Black and Cox (1976) 2. Longstaff and Schwartz (1995) 3. Anderson and Sundaresan (1996) for strategic modelling 2 Black and Cox(1976) It is unrealistic to suppose that Default can only occur at maturity and that bonds pays no coupon dV t = rV dt + σV dz (1) Here we assume that the dynamics of the asset of the firm are risk adjusted. Default occurs when V hits a lower threshold V and the bond pays a contin uous coupon b . Also an additional important assumption is made. If V grows at r in a risk neutral economy then there is no net cash flows out of the firm. If bond holders are receiving b this implies that equity holders must be paying a negative dividend equal to b . The reason why this assumption is made is that if b were paid out of the firm then the dynamics for value would be dV t = ( rV b ) dt + σV dz (2) 1 This yields a differential equation with linear rather than proportional coefficients on the first derivative of the asset value. Solving such equations is rather harder than the case when the coefficient is proportional. 2.1 Valuation of Bondholders claim Total return on the bond is equal to the risk free return rFdt = bdt + E t [ dF ] rF = b + E t [ dF dt ] Where, dF = F V dV + 1 / 2 F V V ( dV ) 2 dF = [ rV F V + 1 / 2 σ 2 V 2 F V V ] dt + σV F V dz E t [ dF ] = [ rV F V + 1 / 2 σ 2 V 2 F V V ] dt This implies that, 1 / 2 σ 2 V 2 F V V + rV F V rF + b = 0 (3) There is another important assumption made here: Debt is perpetual. 2.2 Solution The general solution is: F ( V ) = b r + C 1 V λ 1 + C 2 V λ 2 (4) Where λ 1 and λ 2 are the negative and positive roots, respectively of the quadratic equation. λ ( λ 1) σ 2 / 2 + λr = r (5) To determine C 1 and C 2 we need to impose appropriate boundary condi tions. When V → ∞ the possibility of default become highly unlikely so the debt becomes risk free and F ( V ) = b r . This implies that C 2 = 0. Otherwise C 2 V λ 2 would explode. The second boundary condition comes from the fact that at the reorgan isation point V the debt value is F ( V ) = V δ for some bankruptcy cost δ > 0. Hence we have V δ = b r + C 1 V λ 1 (6) 2 so C 1 = V δ b/r V λ 1 (7) The total solution can therefore be written as F ( V ) = b r 1 V V λ 1 ! + ( V δ ) V V λ 1 (8) ( V V ) λ 1 could be interpreted as the expected default probability. 2.3 Equityholders choice of bankruptcy point So far we have taken V as given. Suppose instead that the bondholders can force bankruptcy when the equityholders cease to pat coupons. In this case the default point will be selected by the equityholders who will decide when to cease injecting money....
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 Spring '05
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 Bankruptcy, Debt

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