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Unformatted text preview: CREDIT RISK: Lecture II Lina El-Jahel 20056 1 Firm Value Models In this part of the course we will introduce Firm Valuation Models. We will begin with an extremely simple model of the stochastic behavior of the market value of assets, equity and debt. In that model equity and debt may be thought of as derivatives on the total market value of the firm and then priced accordingly (Merton (1974)). Later we will introduce some market imperfections and discuss the impact of default prior to maturity (Longstaff and Schwartz(1997)) and (Briys and Varenne (1997)) and finally increase the degree of control that may be exercised by holders of equity and debt (Anderson and Sandaresan (1996)). Of course as we introduce more elements of reality the theory will become more complex and less like a derivative valuation model. 2 Merton (1974): Assumptions 1. No transaction costs, taxes or problem with indivisibility of assets. 2. Each investors can buy/sell as much of an asset as he wants. Borrowing and lending happen at the same rate. 3. Short sales of all assets with full use of the proceeds is allowed. 4. Trading takes place continuously in time. 5. MM obtains. 6. Term structure is flat therefore the price of a riskless discount bond with unit principal can be written as P t = exp- r ( T- t ). 1 7. The firm future cash flows have a total market value at time t given by V t , where V is a geometric Brownian motion, satisfying dV t = V dt + V dz (1) for constant and > 0. and dz is the increment of the Wiener process. One refers to V t as the assets of the firm. 8. The firm has two classes of claims: A single homogenous class of debt A residual claim (equity) 9. The indenture of the bond contract contains the following provisions and restrictions: The firm promises to pay a total of $ B to the bond- holders at date T In the event of the payment being not met the bond- holders immediately take over the firm the firm cannot issue any new senior or equivalent rank claims nor can it pay dividends or do share repurchases prior to maturity 2.1 Pricing risky debt The valuation will use the contingent claims approach. In effect will assume that either there is a security whose market value is V t or we can find and asset or portfolio of assets that spans V t . In the event that the total value V T of the firm at maturity is less than the contractual payment B due on the debt, the firm defaults, giving its future cash flows worth V T to debtholders. Let F represent the value of debt F ( V,t ). Using Itos Lemma we can write: dF = F V dV + 1 / 2 F V V ( dV ) 2 + F t dt = (1 / 2 2 F V V + V F V + F t ) dt + V F v dz = y Fdt + y Fdz Where y F = 1 / 2 2 V 2 F V V + V F V + F t y F = V F V Now lets form a portfolio =Firm +debt +Riskless Bond....
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- Spring '05