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CREDIT RISK: Lecture III Lina El-Jahel 2006 1 Firm Value Models: Default prior to maturity 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 dVt = rV dt + V 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 continuous 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 dVt = (rV - b)dt + V 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 rF dt = bdt + Et [dF ] dF rF = b + Et [ ] dt Total return on the bond is equal to the risk free return Where, dF = FV dV + 1/2FV V (dV )2 dF = [rV FV + 1/2 2 V 2 FV V ]dt + V FV dz Et [dF ] = [rV FV + 1/2 2 V 2 FV V ]dt This implies that, 1/2 2 V 2 FV V + rV FV - rF + b = 0 (3) There is another important assumption made here: Debt is perpetual. 2.2 Solution The general solution is: b + C1 V 1 + C2 V 2 (4) r Where 1 and 2 are the negative and positive roots, respectively of the quadratic equation. F (V ) = ( - 1) 2 /2 + r = r (5) To determine C1 and C2 we need to impose appropriate boundary conditions. When V the possibility of default become highly unlikely so the b debt becomes risk free and F (V ) = r . This implies that C2 = 0. Otherwise 2 C2 V would explode. The second boundary condition comes from the fact that at the reorganisation point V the debt value is F (V ) = V - for some bankruptcy cost > 0. Hence we have b (6) V - = + C1 V 1 r 2 so V - - b/r V 1 The total solution can therefore be written as C1 = F (V ) = b V 1- r V 1 (7) + (V - ) V V 1 (8) V ( V )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. Equityholders Claims: V 1 E(V ) = V - L(V ) - V b b - V - = V - r r V V 1 The equiholders will choose V in order to maximise the value of their claim. Using this argument to derive the optimal V call it V E(V, V ) =0 V For V = V V =- 1 b (1 - 1 ) r (10) (9) 3 Stochastic inetrest rates Default occurs at a time when the firm's value falls below a constant default barrier (Longstaff and Schwartz). This also allows to capture the violation of bond covenants. Recovery in these models is also set a fraction of the principal of a safe bond with the same maturity as the defaultable bond which also allows to capture the violation of the absolute priority rule. 3 3.1 Longstaff and Schwartz The model by Longstaff-Schwartz(1995) develop a new approach to valuing risky debt by extending Black-Cox (1976) model in two ways. First this model incorporates both default risk and interest rate risk. Second, this approach explicitly allows for deviations from strict absolute priority. This model allows the term structure of credit spreads to be either monotone increasing or humped shaped. 3.1.1 Assupmtions and Valuation Framework The basic assumptions of this valuation framework parallel those of BlackScholes (1973), Merton (1974) and Black and Cox (1976). 1. Let V designate the total value of the assets of the firm. The dynamics of V are given by dV = V dt + V dW1 (11) where is a constant and dW1 is a standard Borwnian motion. 2. Let r denote the short-term riskless interest rate. The dynamics of r are given by dr = ( - r)dt + dW2 (12) where , and are constants and W2 is also a standard Brownian motion. The instantaneous correlation between dW1 and dW2 is dt. This assumption about the dynamics of r is drawn from the term structure model of Vasicek (1977). 3. The value of the firm is independent of the capital structure of the firm. This is standard assumption that the Modigliani-Miller Theorem holds. 4. Following Black-Cox (1976) there is a threshold value K for the firm at which financial distress occurs. As long as V is greater than K, the firm continues to be able to meet its contractual obligations. If V reaches K, however, the firm immediately enters financial distress, defaults on all its obligations and some form of corporate restructuring takes place. An important implication of this assumption is that default occurs for all debt contracts simultaneously. This is realistic since when a firm defaults on a debt issue, it typically defaults on other issues because of cross-default provisions, acceleration of principal provisions, 4 or injunctions against making coupon payments on other debt issues. Since financial distress is triggered when V = K, a reorganisation or bankruptcy is simply a mechanism by which total assets of K are allocated to the various classes of corporate claimants. The traditional approach to valuing corporate debt 1 assumes that strict absolute priority holds. However a growing amount of evidence shows that absolute priority rules are frequently violated in corporate restructurings. In addition, recent research suggests that the actual payoff to bondholders in a reorganization depends on a host of exogenous variables such as firm size, the bargaining power of the bondholders, the existence of an equity committee, and the strength of ties between managers and shareholders. In their paper Longstaff and Schwartz rather than trying to model the complex bargaining process among corporate claimants during a restructuring they take the allocation of the firm's assets as exogenously given. 5. If a reorganization occurs during the life of a security, the security holder receives 1 - times the face value of the security at maturity. An equivalent way of specifying the payoffs in the event of a default would be to assume that the security holder receives N riskless zerocoupon bonds at the time of the default, where N equals 1- times the face amount of the debt, and where the maturity date of the riskless bonds is the same as for the original debt. The equivalent specification is consistent with typical reorganizations in which security holders receive new securities rather than cash in exchange for their original claims. The factor represents the percentage write-down on a security if there is reorganization of the firm during the life of the security. For limited liability securities, 1. In general , will differ across the various bond issues and classes of securities in the firm's capital structure. When = 0 there is no write-down and the security holder is unimpaired. When = 1 the security holder receives nothing in a restructuring. If < 0 a security holder actually benefits from a restructuring. In practice the value of for a particular class of securities could be estimated form actuarial information. The only constraint on the value of is the adding-up constraint that the total settlement on all classes of claims cannot exceed K. Note that even when firms have many issues of debt outstanding, the bonds are usually grouped into a small handful of categories for purposes of reorganization. Thus, only two 1 see Merton (1974) and Black-Cox (1976) 5 or three different values of are usually necessary in valuing a firm's debt. 6. We assume perfect, frictionless markets in which securities trade in continuous time. This assumption allows us to invoke standard results to derive the fundamental partial differential equation defining the price H(V, r, T ) of any derivative security with payoff at time T contingent on the values of V and r. This partial differential equation is 1 2 2 2H 2H 1 2 2H H H H V +V + +rV +(-r) + -rH = 0 2 2 2 V V r 2 r V r t (13) where represents the sum of the parameter and a constant representing the market price of interest rate risk. The value of the derivative security is obtained by solving equation (13) subject to the appropriate boundary conditions. The value of a riskless zero-coupon bond plays an important role in the derivation of valuation expressions for corporate securities. In this framework the value of a riskless zero-coupon bond is given by Vasicek model we refer to it by D(r, T ). 3.1.2 Valuing Fixed-Rate Debt Let H(V,r,T) denote the price of a risky discount bond with maturity date T . The payoff on this contingent claim is 1 if default does not occur during the life of the bond, and 1 - if it does. This payoff function can be expressed as 1 - I (14) where I is an indicator function that takes value one if V reaches K during the life of the bond, and zero otherwise. In addition let X denote the ratio V /K. The value of the risky discount bond is the solution to the equation (13) subject to (14). It takes the following form: H(X, r, T ) = D(r, T ) - D(r, T )Q(X, r, T ) where n (15) Q(X, r, T, n) = i=1 qi i-1 (16) (17) qj N (bij ), j=1 q1 = N (a1 ) qi = N (ai ) - 6 i = 2, 3, ...n (18) ai = bij = - log(X) - M (iT /n, T ) S(iT /n) M (jT /n, T ) - M (iT /n, T ) S(iT /n) - S(jT /n) (19) (20) For the expressions for M (t, T ) and S(t) you are referred back to LongstaffSchwartz (1995) paper. 2 The closed-from expression for the risky zero-coupon bonds involves nothing more complex than the standard normal distribution function. Note that the qi terms in equation (16) are define recursively, which makes it straightforward to program this valuation expression and to calculate risky discount bond prices. Although Q(X, r, T ) is defined as the limit of Q(X, r, T, n) the convergence is rapid; numerical simulations show that setting n = 200 results in values of Q(X, r, T ) and Q(X, r, T, n) that are virtually indistinguishable. The above expression for the value of the risky zero-coupon bond depends on V and K only through their ratio X. Thus, X provides a summary measure of default risk of the firm and can be viewed as a proxy variable for the credit rating of the firm. An important implication of this is that risky debt can be valued without having to separately specify the values of V and K. From Equation (15) the price of a risky zero-coupon bond is an explicit function of X, r and T , and depends on the parameters , , , , and . This closed form expression has an intuitive structure. The first term in equation (15) represents the value the bond would have if it were riskless. The second term represents a discount for the default risk of the bond. The discount for default risk consists of two components. The first component, D(r, T ) is the present value of the write-down on the bond in the event of a default. The second component, Q(X, r, T ) is the probability under the risk neutral measure that a default occurs. It is important to recognize that the probability of a default Q(X, r, T ) under the risk neutral measure may differ from the actual probability of a default. This is because the upward drift of the actual process for V in equation (11) is V , while the upward drift of the risk neutral process depends on the value of r and is independent of . Now given the explicit solution for risky fixed-rate debt, Longstaff and Schwartz solve for the credit spread. This is defined as the difference between the yields of a risky and riskless bond with identical maturity dates and coupon rates. The term structure of credit spreads derived by this model can be monotone increasing as well as humped shaped. The term Q(X, r, T ) is the limit of Q(X, r, T, n) as n . N(.) denotes the cumulative standard normal distribution function. 2 7 4 Strategic Modelling: Anderson and Sundaresan (1996) This lecture will introduce strategic modelling of corporate debt in discrete time by studying the Anderson and Sundaresan (1996) model (Review of Financial Studies). Starting Points 1. Bankruptcies are costly both because of direct costs and because of disruption of firm's activities. 2. Bankruptcy procedures give a considerable scope for opportunistic behavior. 3. Deviations from the APR rule are common 4. Despite incentives to do so in practice it often proves difficult to renegotiate claims so that formal bankruptcies and liquidation often result. 5. Valuation models are dynamic but fail to endogenise contract provisions. 6. This article fills the gap: It could be used for valuation and it endogenises the default boundary 7. Equilibrium often will result in renegotiations with deviations from APR in favor of equity. 5 Assumptions 1. One owner manager who needs to raise debt to finance a project 2. When undertaken the project will give rise to a stream of rent indefinitely into the future 3. we have a homogeneous group of creditors 4. The terms of the debt contract entitle the bondholder to a payment of CSt in each period t = 0, 1, 2, . . . , T . Where T is the maturity of the debt contract. 5. Debt service is met out of the cash flows and any asset sales or issue of new securities require the explicit agreement of the creditors 8 6. The approach of Anderson and Sundaresan (1996) is based on a binomial model. In state j, the value of the firm's assets at time t either j j+1 increase to Vt+1 or falls to Vt+1 . j uVtj = Vt+1 1 u p Vtj d= 1-p j+1 dVtj = Vt+1 Underlying project value 7. All contracting parties have full information about the state of nature 8. The firm produces a flow of earnings ftj . The cash flows are proportional to the value of the project. ft = Vt where, is the payout ratio. 9. If the manager is risk neutral: Vtj j+1 j pVt+1 + (1 - p)Vt+1 = + ftj r puVtj + (1 - p)Vtj = + Vtj r r(1 - ) - d (21) u-d For a high growth company is small and therefore p is high. For a mature company is high and therefore p is small. p= 10. Once underway, control of the project can be transferred only at a cost 11. Direct cost (cost of the verification of the collateral) 12. Time and effort to find another management team that will be able to produce efficiently a high enough level of cash flows. 13. the cost of bankruptcy is denoted by K 9 14. All cash flows from the project are paid out in terms of dividends, debt service or to cover the bankruptcy cots. We therefore have the preservation property V = E + B + L. E denotes the equity value, B the debt value and K and L is the expected value of future bankruptcy costs. 6 The Game If there are bankruptcy costs, equity holders may be able to extract concessions, blackmailing bondholders into accepting a lower coupon that the originally contracted amount when the firm is close to bankruptcy. Bondholders may be willing to accept lower coupon since if they trigger bankruptcy, they will only get the value of the firm's assets, less any bankruptcy costs and this may be less than the continuation value of the debt, i.e. the value of future discounted coupons. At each period the firm generates a cash flow ft . Given ft the firm chooses a level of debt service St , St [0, ft ]. ft is an upper bound as no asset sales and no further issue of assets is allowed. If St CSt Creditors accept If St < CSt Creditors reject Firm is liquidated, creditors get Vt - K Game continues to next period Game continues to next period Nature Vtj / Vtj+1 ? ? CSt ~ CSt Liquidate Continue Continue Extensive Form Game 10 The model can be studied through the extensive form of the renegotiation game between debt- and equity-holders. Contrary to Merton's model bankruptcy is not only determined by nature, but also depends on the creditors and owner's actions. Owner - Chooses St Creditors - Accepts or rejects Therefore violating the debt covenant may not lead to automatic default as creditor may not initiate legal action To work out equilibrium involves stepping back from the terminal maturity of the bond T . V20 V10 V0 V11 V2 T = 2 is debt2 maturity Example At T , VT is given and the owner selects ST as a service flow. If ST CST the game ends. Otherwise , ST < CST . The creditor must decide whether to accept or liquidate. If he accepts, the payoffs to debt and equity are ST and VT - ST . If the service flow is rejected the payoffs are max (VT - K, 0), where K is the bankruptcy cost. It is assumed here that liquidation costs are deducted from the remaining firm value and hence the firm's liquidation value is non-negative. V21 7 Payoff Accept ST VT - ST Reject max (VT - K, 0) 0 Debt-holders Equity-holders 11 7.1 Equilibrium at T Equilibrium is derived when all groups take their decisions to maximise their payoffs. Equilibrium is formed by the decision rules of the creditors and the equityholders (owner) that constitute the best response in light of the payoffs If VT - K CST To maximise their payoffs it is in the interest of equityholders to pay CST if the firm value is high in particular if VT - K CST because the remaining value for the equityholders will be VT - CST which is higher than the value they would get if bankruptcy costs are paid after the liquidation procedures is initiated VT - CST - K. This is also true as the creditors knowing that the equityholders can afford to pay them their contractual claim they will reject any offer ST < CST If VT - K < CST (low VT ) the owner can under-perform on the debt service and the creditor may accept, Why? ST < CST the creditor has the right to initiate legal procedures and he will get from the liquidation of the company max (VT - K, 0). However if the owner offered the creditors ST = max (VT - K, 0) then the creditors will be indifferent between accepting and rejecting and will have no incentive to liquidate. 7.1.1 In Summary If the value of the firm is relatively high so that the liquidation value exceeds the contracted debt service, the owner is best honoring the contract. For relatively low values of the firm, the owner is best off by making the minimum debt service which just leaves the creditor indifferent between accepting and liquidating the firm. The creditors will reject any offer that will give them less than the liquidation value max (VT - K, 0) CST if VT - K > CST The owners best response is to set: ST = max (VT - K, 0) otherwise 7.2 Time T Payoffs B(VT ) = min (CST , max (VT - K, 0)) E(VT ) = VT - B(VT ) The payoffs to bond and equity holders may be summarised by 12 7.3 Equilibrium for t < T Same reasoning apply, BUT we have to include the continuation values when calculating the payoffs. The continuation value depends on the future realisations of V and these are uncertain. We therefore, need to calculate the expected continuation values using the probabilities on the binomial tree. Stepping back in time, suppose at time t,V = Vtj and the owner selects a debt service St . If St CSt the game continues. If St < CSt then the bondholder can liquidate and get max (Vt - K, 0). If he accepts, he gets: j j+1 pt B(Vt+1 ) + (1 - pt )B(Vt+1 ) St + r (22) where pt is the appropriate risk neutral probability. At time t - V Owner offers - S t t If - S CS t t Continue - < CS St t Accept Continue U Reject Liquidate Time t Creditor Choices High value of Vt The owner selects the service flow optimally so that for high Vt he pays CSt . Otherwise (if he pays less than he promised), the creditor will opt to liquidate the firm and get what their have been promised in their contracts and the owner will get the residual value minus the bankruptcy costs. This is suboptimal as the owner can keep all the residual value (without the deduction of the bankruptcy costs) by paying the creditor what he is owed in the first place. Low value of Vt 13 Continue Creditor accepts St + pB(uVtj )+(1-p)B(dVtj ) r Liquidate R Creditor Rejects max (Vt - K, 0) Time t Creditor Payoffs For low Vt the owner can under-perform and offer St < CSt because he knows how the creditor is going to react and he also knows that the decision of the creditor will depend on the payoff he gets. The creditor will only accept an offer St if pB(uVtj ) + (1 - p)B(dVtj ) max (Vt - K, 0) (23) r The owner also wants to pay the creditor as little as possible to maximise his own payoff. Therefore the owner will chooses St so that the bondholder is indifferent between declaring bankruptcy or not, i.e., St + pB(uVtj ) + (1 - p)B(dVtj ) St = max (Vt - K, 0) - r (24) 7.4 Service flow at time t The service flow can be summarised for all values of Vt as: S(Vtj ) = min CSt , max 0, max (Vt - K, 0) - j j+1 pt B(Vt+1 ) + (1 - pt )B(Vt+1 ) r (25) 14 7.5 Equilibrium Payoff j j+1 pt B(Vt+1 ) + (1 - pt )B(Vt+1 ) = + r j j+1 pt E(Vt+1 ) + (1 - pt )E(Vt+1 ) j j j E(Vt ) = ft - S(Vt ) + r Debt and equity values are then: B(Vtj ) S(Vtj ) Where recall that ftj is the firm's earnings flow in state j at time t 7.6 Conclusions 1-Equilibrium is such that rejecting never occurs unless ft is not enough to meet the required optimal payment to keep the creditors in the game. Only in that case liquidation occurs and the payoffs are: B(Vtj ) = max 0, min (Vtj - K, CSt + Pt ) E(Vtj ) = Vt - K - B(Vtj ) Pt is the principal outstanding at time t 2-In this model we have a strategic debt service and equilibrium results in deviation of the APR rule in favor of equity-holders. 8 End note All structural models suffer from generating low short credit spreads. This could be remedied by introducing jumps or incomplete information. We will try to tackle this after we introduce jump processes in future weeks (time permitting) 15

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