Credit risk of a leveraged firm in a controlled optimal stopping framework

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Content CREDIT RISK OF A LEVERAGED FIRM IN A CONTROLLED OPTIMAL STOPPING FRAMEWORK by Yuegang Zhou FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY August 2008 Copyright 2008 Yuegang Zhou (APPLIED MATHEMATICS) A Dissertation Presented to the Dedication Dedicated to my wife Xiaoyan, and my daughter Grace. They are the source of my inspirations. ii Acknowledgements My deepest gratitude goes to Professor Jianfeng Zhang for his inspiration, guidance and patience. He has encouraged me to explore questions that I nd interesting, to think broadly and to enjoy the process of learning. His intellent, kindness, and patience have been a source of inspiration. I own him more than words can express. I would like to thank Professor Jin Ma and Professor Yongheng Deng for serving my committee. I have beneted tremendously from their comments, and their input has vastly improved this dissertation. iii Table of Contents Dedication ii Acknowledgements iii List of Figures vi Abstract vii Chapter 1: Literature Review 1 1.1 The Evolvement of Modeling Individual Default Risk . . . . . . . . . . . . 2 1.1.1 Structural Models for Single Firm . . . . . . . . . . . . . . . . . . 4 1.1.2 Reduced-form Models: Modeling Credit Spreads, Default Probabil- ity Intensities and So on . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 The Correlated and Dependent Defaults . . . . . . . . . . . . . . . . . . . 21 1.2.1 Zhous (2001) Model: A Natural Start of Structural Models . . . . 21 1.2.2 Jumps in Volatilities due to Other Firms Default. . . . . . . . . . 26 1.2.3 Counterparty Risk Based on Individual Default Intensity . . . . . 28 Chapter 2: Credit Risk of a Leveraged Firm With Multiple Investment Projects 35 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.1 Asset Value Process . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Firms Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 The One-Project Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.1 Existence and Uniqueness of the Optimal Stopping . . . . . . . . . 50 2.3.2 Results of One-Project Cases . . . . . . . . . . . . . . . . . . . . . 63 2.4 Optimal Strategy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4.1 Properties of the Cost Function . . . . . . . . . . . . . . . . . . . . 67 2.4.2 Optimal Investment Strategy and Bankruptcy Boundary . . . . . . 76 2.5 Application of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.5.1 The Investment Decision of Two-Project Models . . . . . . . . . . 99 2.5.2 The E¤ects of Optional Project . . . . . . . . . . . . . . . . . . . . 104 iv 2.5.3 Debt Renegotiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.6 Conclusion and Future Research . . . . . . . . . . . . . . . . . . . . . . . 112 Bibliography 114 v List of Figures 1.1 The graph depicts the change of with respect to 12 and 21 , given 11 and 22 . 11 = 0:2, 22 = 0:3,0:5< 12 ; 21 < 0:5. . . . . . . . . . . . 25 2.1 Examples of investing in project one or two. r = 0:06;b 1 = 0:001;b 2 = 0:05: Solid line: 1 = 0:15; 2 = 0:22; choosing project two; dashed line: 1 = 0:18; 2 = 0:21; choosing project one: . . . . . . . . . . . . . . . . . 103 2.2 2 changes with 2 and b 2 . r = 0:06;b 1 = 0:001; 1 = 0:15. Thus 1 = 1:8983: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3 The relation between b 2 and 2 given project one. . . . . . . . . . . . . . 105 2.4 Without new project, the rm value, equity value and debt value change withassetvalueX duringthermsoperation. r = 0:06;b 1 = 0:001; 1 = 0:15: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5 The debt value as a function of asset value X for di¤erent 2 . r = 0:06; b 1 = 0:001; 1 = 0:15;b 2 = 0:005: . . . . . . . . . . . . . . . . . . . . . . 107 2.6 The left graph represents the equity value di¤erence in two project; the right graph represents the di¤erence between new debt value and the old debtvaluefordi¤erent 2 and asset valueX given currentstater = 0:06; b 1 = 0:001; 1 = 0:15;b 2 = 0:005. . . . . . . . . . . . . . . . . . . . . . . 108 2.7 The left graph is the equity value when the rm invest in project two, and the right graph plots the debt value for di¤erent asset value and 2 , given r = 0:06; b 1 = 0:001; 1 = 0:15; and b 2 = 0:005. . . . . . . . . . . . 110 2.8 The graph shows the value (F 2 (X)D 1 (X))V 2 (X), where V 2 (X) is the amount of equity value without compensation to debt-holders, and (F 2 (X)D 1 (X) is the actual amount of equity value, given r = 0:06; b 1 = 0:001; 1 = 0:15; and b 2 = 0:005. . . . . . . . . . . . . . . . . . . . 111 vi Abstract This is an investigation of how a rm allocates capital into multiple investment opportu- nities. Thisframeworkanalyzestherm-ownersbehaviorasoneofthefactorsinuencing the credit quality of the rm. Based on the equity value maximization, criterion is con- structedforthermsinvestmentstrategydecisions. Thecombinationofinvestmentport- folio and credit risk is an optimal stopping problem of a controlled di¤usion process. This modelindicatesthatthermsobjectiveisequivalenttominimizethediscountedexpected cost payout due to the debt issuance. There exists a constant bankruptcy-triggering asset level for the optimal stopping control problem. Although multiple projects exist, the rm will choose only a specic project, which has the lowest bankruptcy-triggering boundary. An investigation of the changes in security values when a rm switches its investment project will show that it is preferable for both the debt-holders and the rm-owners to take on a new investment opportunity if the asset value is low. For the case in which the investment switch will damage the debt-holdersbenets, this model provides a mecha- nism of debt-renegotiation to achieve the switch, which increases in both equity and debt values. vii Chapter 1: Literature Review Credit risk is a risk that debtors won.t ful.ll their obligations due to their abilities and/or willingness. If we think the obligor is not able to ful.ll his/her obligations, the credit risk model is usually classi.ed into exogenous type in which the obligor don.t have enough money to pay the interest and/or the principle. If we think the default is a result of the obligors decision making, the model is called endogenous type. The di¤erence between the two types is located at the default barrier. For exogenous model, the default barrier is usually a given scalar or a stochastic process. For endogenous models, however, the defaultbarrierisnotgivenbutafactorofthemodels. Amongtheresearchpapers,Merton (1974) applied the technology of option pricing developed by Black and Scholes (1973) intotheanalysisofdebtevaluation. Asan originalcreditriskmodel, Mertonsframework considers the hitting time of process at terminal time, which is extended to the rst- passage time model by Black and Cox (1976). All these models assume the barriers as exogenously given. By taking into account the optimal capital structure, Leland (1994) details the analysis of Merton (1969) and develops a framework in which the default barrier is endogenous. Leland and Toft (1996) further extends the model to allow nite debt maturity. 1 Fromtheangleofeconomics,wecanclassifythecreditriskmodelsintotwotypes. One is the so-called structural models; and the other is the reduced-form models. Structural models start with the rms asset value or cash storage to evaluate the debt and yield spreads. The Merton type models are representatives of the structural models including Geske(1977), HoandSinger(1982), Kim, RamaswamyandSundaresan(1992), Longsta¤ and Schwartz (1995), Zhou (2001a), Vasicek (1997) and others. The reduced-form models dealwithissuesrelatedtocreditrisksbymodelingthedefaultprobabilitydirectlywithout considering the causes of the default. The instensity-based models assume the absolute continuity of the default probability, which is related to exogenously speci.ed hazard rate process. TheyconsistofArtznerandDelbaen(1992),MadanandUnal(1994),Jarrowand Turnbull (1995), Jarrow, Lando and Turnbull (1997), Du¢ e and Singleton (1997), Lando (1998), Schönbucher (2000) and others. Bélanger, Shreve and Wong (2004) generalizes the credit risk model to include the case in which the default probability is not absolutely continuous. 1.1 The Evolvement of Modeling Individual Default Risk In a seminal paper, Black and Scholes (1973)[4] derived a valuation formula for options. Merton (1974)[36] originally applied the theory of Black and Scholes (1973) to pricing debt value. Mertons (1974) work opened a new way to investigate credit risk. Many researches have been inspired by Mertons (1974) paper. The term "structural models" has been used to name the models developed by Merton and the followers. These models start from the rms characteristics (e.g., rm value, equity value) to study the causes of credit risk. Structural models are intuitive in economics, and interpret credit risk in 2 a way according with peoples logic. Although Mertons model (or its modications) has been applied in practice, its limits are discussed widely. Most of the limits come from the di¢ culty that the fundamental of a rm is not observable. Some researchers make use of option pricing theory to circumvent the unobservable rm value, for example, one regards rms value as a function of equity value, and equity value (stock market value) can be obtained. However to apply structural model more successfully, there are many jobs needed to do. To avoid the di¢ culties faced by structural models, some other researchers aim to nd a new way, which is completely di¤erent. There the so-called "reduced-form models" are developed. The researchers of reduced-form models believe that they can nd the evolution (stochastic process) of some specic economic variables interested (e.g., default probability intensity, credit spread), by estimating coe¢ cients of their models. But im- plementation of these models critically depends on how well they describe the reality. Furthermore, even if the models are very consistent with the data obtainable, we still dont know how well they predict the future. Another very important di¢ culty faced by reduced-form model is located at the fact that some economic phenomena can not be depicted by some mathematical model or models, especially when we needed to take into consideration many economic variables at the same time. It is meaningless to say which type of models is better. What we should do is to im- provethemorinitiatenewmethodstomakepeopledealwithcreditriskmoresuccessfully in practice. So in this section (actually in the dissertation), we will review some of both structural and reduced-form models, which have been inspiring me to do my research. More details about these models, please refer to the corresponding papers and references therein, and books related to this area. 3 1.1.1 Structural Models for Single Firm In this subsection, we are going to study Mortons (1974) and Lelands (1994) papers, because we get a lot of inspirations from them. Here we are not going to just repeat the works like what many other researches have reviewed, but we want to go through them via our understanding of the real world. Merton s Creative Work Morton (1973) applied Black-Scholesoption pricing formula to analyze the pricing of corporate debt, which initiated the modern theory and its application in ance, especially in credit risk. Here we are in a position to study the seminal work. Under some ideal assumptions which guarantee an e¢ cient and non-restriction econ- omy 1 , it is assumed that the dynamics for the value of a rm V satises dV = (V C)dt+VdW; where is the instantaneous expected return rate, a constant; C is the coupon rate paid out continuously; 2 is the usual volatility of the asset return; W is a standard Brownian motion. The rm issues a zero-conpon debt with face value L and maturity T. We want to know what is price of the bond. Before evaluating the bond, we consider an arbitrary 1 Basically the conditions of the ideal economy include A.1-A.4 plus A.6 in Mertons paper [36]. 4 security X, which completely depends on the operation of the rm. 2 Thus we have the relation X t =F (V;t) for some functional F. X has a dynamic process represented by dX = ( x XC x )dt+ x XdW x ; (1.1) where x and 2 x are the expected instantaneous return rate and the volatility of return rate of the security X respectively; C x is a constant coupon payment to this security; W x is a standard Brownian motion need to determined. By Itos formula, we can derive X satisfying a stochastic process as follows, dX = 1 2 2 V 2 F VV +(V C)F V +F t dt+VF V dW; (1.2) where subscripts represent partial derivatives, and r; a constant, is independent risk-free interest rate. By comparing terms between (1.1) and (1.2), we can nd x X = x F = 1 2 2 V 2 F VV +(V C)F V +F t +C x ; x X = x F =VF V ; dW x = dW: Therstequationaboveindicatesthattheinstantaneousreturnofthesecurityisthesum of inow from the rm and the coupon; and the second and third equations implies that the uncertainty in the return of security is equivalent to that in the return of the rm. 2 The implication of complete dependence is that any uncertainty a¤ecting the rm valueV, has impact on the value of the security; and only the uncertainty a¤ecting the rm value, has impact on the security. We can think that the probability space of the rm value determines the security value. 5 Then we form a portfolio which has zero current cost. The portfolio consists of p 1 dollar invested in the rm, p 2 dollars in the security, and(p 1 +p 2 ) dollars in the money market. Let dz be the instantaneous return to the investment combination of p 1 and p 2 , then dz = p 1 dV +Cdt V +p 2 dX +C x dt X = (p 1 +p 2 x )dt+(p 1 +p 2 x )dW: If we want this combination is riskless and no arbitrage, the following condition must be satised p 1 +p 2 x = (p 1 +p 2 )r p 1 +p 2 x = 0: The conditions require that the instantaneous return rate of the combination (p 1 ;p 2 ) is r and no uncertainty. They are equivalent to r = x r x : (1.3) 6 This is a Sharpe-Ratio type equation, which implies that the extra return rates per unit volatility of the rm and the security are equal under no risk and no arbitrage condition. 3 Thus we get the following partial di¤erential equation of the value of security F, 1 2 2 V 2 F VV +(rV C)F V rF +F t +C y = 0; (1.4) or equivalently 1 2 2 V 2 F VV +(rV C)F V +F t +C y F =r: (1.5) ThereasonthatIrearrangeequation(1.4)tobeequation(1.5), although(1.4)isthePDF used to nd F in practice, is that (1.5) exhibits the economic implication, which is that the expected instantaneous return rate of the security is equal to r. Now we consider that the bond indenture only require the rm to pay L at maturity, or the rm will by taken over by the creditors. Let D be the debt value, it must satisfy 1 2 2 V 2 D VV +rVD V rDD t = 0: Note that in the PDE above, t is the time to maturity (or remaining maturity), but in (1.4) t is the position at time horizon. Lets derive the formula of value for equity E(V;t) =V t D(V;t); which satises 1 2 2 V 2 E VV +rVE V rEE t = 0; 3 The Sharpe-Ratio type equation is a result of what both the rm and the security are perfectly correlated. 7 subject to E(0;t) = 0; E(V;t) V; E(V;0) = maxf0;V Lg: All three restrictions are obvious for the one-debt and one-equity model. It is well-know that the problem above is the same as a European call option. Thus from Black-Scholes equation, we have E(V;t) =V(d 1 )Le rt (d 2 ); where () denotes the cumulative standard normal probability function, and d 1 = log(V=L)+ r + 1 2 2 t = p t; d 2 = d 1 p t: Then we get the evaluation formula for debt D, D(V;t) =Le rt h 2 l; 2 t + 1 l h 1 l; 2 t ; where l = Le rt =V; h 1 l; 2 t = 1 2 2 tlog(l) = p t; h 2 l; 2 t = 1 2 2 t+log(l) = p t: 8 It is convenient to derive the expression term structure of yields as R(t)r = 1 t log h 2 l; 2 t + 1 l h 1 l; 2 t ; where R(t) = log(D(V;t))log(L) t : Once we have explicit formula of the debt price, we can easily analyze how the debt value and yield spreads change with respect to the parameters such as V;L;t; 2 ; and r. In addition to the creative application of option theory to debt evaluation, the simplicity in pricing formula is another attractive advantage, which facilitates its implementation in practice with some improvements and modications. Leland s (1994) Extension In Merton (1974), Black and Cox (1976), and others, they didnt consider bankruptcy cost, tax, and they just took the default-triggering asset level to be exogenous. Leland (1994)[30] extended their models and constructed a model to evaluate corporate debts by taking into consideration capital structure. The main contribution of the article is that it got closed-form formulae which indicate many economic implications. It shows that the default threshold is endogenous not exogenous. To get the closed-form results, Leland (1994) had made fallowing assumptions. A rms activities have value V satisfying dV=V =(V;t)dt+dW; 9 which is the "assets value" of the rm, not the rm value, which is assumed by many other researchers. Any cash outows should be nanced by issuing new equity (not debt). This assump- tion says that any coupon and interest payment is not from the rm but from the wallet of the owner/manager. This assumption raises a critical question. It ignores the cost of the newly-issued equity to pay the coupon. But the owner/manager is also a investor, so he or she needs return for every investment. However as we have remarked, they want a explicit formula to explore economic implication. The interest rate which summarizes the market risk and preference is a constant r, under assumption that the market risk is independent of the credit risk. A claim F (V;t) on asset value V (attention: it is V, not the rm value) gains coupon payment C continuously only if the rm is solvent. The claims value satises the PDE 1 2 V 2 F VV (V;t)+rVF V (V;t)rF (V;t)+F t (V;t)+C = 0; 1 (1.6) Where subscripts denote partial derivatives. Under some boundary conditions which are usually payments at the maturity or in bankruptcy, we can simulate the value of the claim. But if F is time independent, there exist closed-form solution to (1.6) F (V) =A 0 +A 1 V +A 2 V X ; (1.7) 1 Similarly to the last subsection, we can rearrange it, and get 1 2 V 2 FVV (V;t)rVFV (V;t)+Ft(V;t)+C F (V;t) =r; i.e., the return of the claim is the same as the interest rate by no-arbitrage argument. 10 where X = 2r 2 : We can nd the expressions of the coe¢ cient by boundary conditions. ThematurityofthedebtisinnitewithconstantcouponC unlessthermbankrupts. Let V B be the lower bound of the asset value at which the rm declares bankruptcy. If bankruptcy occurs, the bankruptcy cost is V B for some constant , and debt-holders take over the rm. The boundary conditions of the debt are D(V) = (1)V B ; if V =V B ; D(V) ! C=r; if V !1: Thus by (1.7), we nd the coe¢ cients, A 0 = C=r; A 1 = 0; A 2 = (1)V B C r V X B : So the formula to debt price is D(V) = C r + (1)V B C r V V B X : (1.8) 11 So far we suppose that the bankruptcy-triggering asset level V B given. Boundary condi- tions imply that when the asset value V reaches V B ; the owner/manager has no incentive to continue this rm. 2 WecanviewthebankruptcycostsBC(V)asaclaimofV:Itsatisfy(1.6)andboundary conditions BC(V) = V B ; if V =V B ; BC(V) ! 0; if V !1: Thus the cost formula of bankruptcy is BC(V) =V B (V=V B ) X : LetTB(V)denotethetaxbenetsofissuingdebts. Sincethedebtcouponisconstant C, giventhetaxrate;thetax-shelteringvalueisC ifthermissolvent:Thetaxbenet is 0 once the rm announces bankruptcy. It satises the boundary conditions TB(V) = 0; if V =V B ; TB(V) = C=r; if V !1: Thus the formula to tax benet is TB(V) = C r C r V V B X : 2 This seems unrealistic, since the rm may be still solvent and it is possible that the rm will be protable in the future. 12 Let F (V) denote the total value of the rm. Then it is F (V) = V +TB(V)BC(V) = V + C r " 1 V V B X # V B V V B X : (1.9) F (V) is a strictly concave function of V. The value of equity E(V) is E(V) = F (V)D(V) = V (1r) C r + (1r) C r V B V V B X : (1.10) E(V) is convex in V for (1r) C r >V B . We adopt the Smooth-pasting condition, i.e., dE=dVj V=V B = 0:Thus the bankruptcy triggering level of asset value is given by di¤erentiating (1.10). V B = [(1)C=r][X=(1+X)] = (1)C= r + 1 2 2 : (1.11) This is the endogenous default boundary implied in above decision policy. Plugging (1.11) in formulae (1.8), (1.9), and (1.10), we have the values of the risky assets under endogenous bankruptcy rule, D(V) = C r " 1 C V X k # (1.12) F (V) = V +(C=r) h 1(C=V) X h i E(V) = V (1)(C=r) h 1(C=V) X m i 13 where m = [(1)X=r(1+X)] X =(1+X) h = [1+X +(1)X=]m k = [1+X(1)(1)X]m: Di¤erentiate (1.12) with respect to C, and set the derivative 0, we get C max (V) =V [(1+X)k] 1=X Substituting it into equation (1:12); we get the optimal debt value D max (V) =V h Xk 1=X (1+X) (1+1=X) i =r: Di¤erentiate (??) with respect to C, and set the derivative 0, we have C (V) =V [(1+X)h] 1=X : Note that h>k, implying C (V)<C max (V): Since in the model the information is symmetric, the investors have knowledge of the operation of the rm and the behavior of the owner/manager. The investors may not agree to the default boundary V B . Even though they agree to V B , they may not accept the coupon rate C ; since they expect higher coupon rate C max : 14 1.1.2 Reduced-form Models: Modeling Credit Spreads, Default Proba- bility Intensities and So on We are going to study several important papers on this eld in this subsection. They have been cited by many researchers. As we have remarked before, the structural models have been faced with many di¢ culties. Since the fundamental is not observable usually, it limits the application of structural models. Some empirical researchesinvestigations show that structural models tend to underestimate the default probability. Jarrow and Turnbull (1995)[24] took a benchmark 3 as given, and model the credit- risk spread of an interested security. Once the two processes can be estimated from data, security price can be calculated by using the martingale measure technology. To understandJarrowandTurnbulls(1995)model,letusconsideradefault-freezero-coupon bond with maturity T, face value 1 dollar, its price p 0 (t;T) at time t, and a defaultable zero-coupon bond with maturity T, face value 1 dollar, its price v 1 (t;T) at time t, for t T. It is easily understood that p 0 (t;t) = 1. 4 But what about the defaultable bond v 1 (t;t)? we denote e 1 (t) =v 1 (t;t); 3 The benchmark is charactered with the term structure of interest rate of a same security without default. 4 The price of buying a bond matured instantly with face value 1 dollar is obviously 1 dollar, if the bond is default-free. 15 and then does v 1 (t;t) = 1? If anyone can guarantee that the bond will not default instantly, then it is possible that v 1 (t;t) = 1. It is, however, a defaultable bond, thus there is no this guarantee. So we expect that e 1 (t)< 1. We dene p 1 (t;T) 4 =v 1 (t;T)=e 1 (t): It is easy to verify that p 1 (T;T) = 1, which implies that defaultable bonds have the same evaluation form as the default-free bonds, since defaultable bondsprices are scaled by themselves. We assume that the defaultable bonds and the default-free bonds are independent. Now dene f 0 (t;T) 4 =@logp 0 (t;T)=@T: Then the default-free interest rate is r 0 (t) 4 =f 0 (t;t); and the default-free saving account is B(t) = exp Z t 0 r 0 (s)ds : Similarly we can dene for defaultable bonds f 1 (t;T) 4 =@logp 1 (t;T)=@T; r 1 (t) 4 =f 1 (t;t); 16 and B 1 (t) = exp Z t 0 r 1 (s)ds : Now with these denitions we model the process of the two forward rates by df 0 (t;T) = 0 (t;T)dt+(t;T)dW 1 (t); and df 1 (t;T) = 8 > > > > > < > > > > > : [ 1 (t;T) 1 (t;T) 1 ]dt+(t;T)dW 1 (t) if t< 1 ; [ 1 (t;T) 1 (t;T) 1 ]dt+(t;T)dW 1 (t)+ 1 (t;T) if t = 1 ; 1 (t;T)dt+(t;T)dW 1 (t) if t> 1 ; where the coe¢ cients satisfy certain appropriate conditions; 5 W 1 is a standard Brownian motion; and 1 is a stopping time of default time of defaultable bonds, which is exponen- tially distributed with parameter 1 . Once the rm defaults, the debt-holders can only get a non-negative fraction 1 ( 1) of face value, i.e., e 1 (t) = 8 > < > : 1; if t< 1 ; 1 ; if t 1 : Thus we derive the stochastic processes for p 0 (t;T), v 1 (t;T), and B 1 (t;T) as follows, dp 0 (t;T)=p 0 (t;T) = [r 0 + 0 (t;T)dt+a(t;T)dW 1 (t)]; 5 They may be some smoothness and boundedness conditions to ensure that there are solution to the SDEs. 17 where 0 = Z T t 0 (t;u)du+ 1 2 a(t;T) 2 ; a(t;T) = Z T t (t;u)du: And dv 1 (t;T)=v 1 (t;T) = 8 > > > > > > > > > < > > > > > > > > > : [r 1 (t;T)+ 1 (t;T) 1 (t;T) 1 ]dt+a(t;T)dW 1 (t) if t< 1 ; [r 1 (t;T)+ 1 (t;T) 1 (t;T) 1 ]dt +a(t;T)dW 1 (t)+ 1 e 1 (t;T) 1 if t = 1 ; r 1 (t;T)dt+ 1 (t;T)+a(t;T)dW 1 (t) if t> 1 ; where 1 = Z T t 1 (t;u)du+ 1 2 a(t;T) 2 ; 1 (t;T) = Z T t 1 (t;u)du: Finally d[B 1 (t)e 1 (t)]=B 1 (t)e 1 (t) = 8 > > > > > < > > > > > : r 1 (t)dt, if t< 1 ; r 1 (t)dt+( 1 1), if t = 1 ; r 1 (t)dt, if t> 1 : 18 Essentially, Jarrow and Turnbull (1995) tried to nd the credit spreads in forward interest rate between default-free and defaultable bond with exactly same properties oth- erwise. This work is connected to the future researches which model the credit spread in instantaneous interest rates and default probability intensities. Since it is not easy to nd or construct the "default-free" bonds with the same properties with defaultable bonds, theapplicationofthemodelislimited. Inadditiontheindependenceassumptionbetween the two type of bonds is not realistic usually. Therearemanyworkshavedonewiththereduced-formmodels. Thefollowingcontent of this subsection based on Du¢ e et al. (1996)[11], Du¢ e and Singleton (1997)[9] [10], Lando (1998)[29], and others. If we can denote the uncertainty related to a rm by a ltration probability space ( ;G;F t ;P), whereF t G, for all t 0; is a ltration, andP is risk-neutral probability measure. As usual, let the default time to be , and then dene H t =I ftg : We denote H t = (H s :ut), which is the information about the rms status up to time t. Now we can explicit dene G t = H t _F t . The di¤erence between reduced-form models is mainly located in how the models assume the relationships among ;G t ;H t ;and F t . is not necessarily aF stopping time, but it denitely aG stopping time. IfthereexistsapositiveintensityprocessunderP,itisaF progressivelymeasurable such that M t 4 =H t Z t^ 0 s ds =H t Z t^ 0 h s ds;8t2 [0;T ]; 19 follows a G-martingale under P. In the expression above h t 4 = I ftg t , and T is the time horizon we are interested in a model. Now consider a debt with principle X and maturity T T , which is F t -measurable. If the rm defaults on the debt, the creditor can get Z, aF-predictable process, as a recovery process of X. Let r, a interest rate, be aF-adapted process, andB t = exp R t 0 r s ds is the saving account. Thus the price of the defaultable debt at time t is S t =B t E Z T t B 1 u dD u jG t ; (1.13) where D t =XI ft=T;Tg + Z t 0 Z s dH s : We can see that (1.13) can be rewritten as S t =B t E B 1 Z I ft<Tg +B 1 T XI fT<g jG t ; which is the expected discounted cash inows of the risk debt. Based on our assumption about the default probability intensity is t , and recovery process Z, we have the following proposition. Proposition 1 The process of debt price admits the expression as follows, S t =E Z T t (Z u h u r u S u )du+XI fT<g jG t : Proof. See Bielecki and Rutkowski [3] for a proof. 20 Lando (1998) built up a model to allow a dependence between the default-free term structure and rms default. Du¢ e and Singleton (1999) assumed a recovery scheme: recovery (loss L) fraction in market value, and derived a simple evaluation formula for risky debt value, where there exists adjusted discount factor R t = r t +h t L t . h t denote the hazard rate for default at time t. 1.2 The Correlated and Dependent Defaults It is observed that a decline in credit quality will result in an increase in the default correlation. Macroeconomic shocks which make the credit qualities deteriorated typically lead to a rise in the default correlation. 1.2.1 Zhou s (2001) Model: A Natural Start of Structural Models Zhou (2001) extended Mertons (1974) single rm model to the case with two rms. In the model the asset values of rms follow a two-dimensional geometric Brownian motion, d(lnV) =dt+dW; where lnV = [lnV 1 ;lnV 2 ] T6 , = [ 1 ; 2 ] T , W is a two-dimensional standard Brownian motion, and is a constant matrix such that T = 2 6 4 2 1 1 2 1 2 2 2 3 7 5: 6 Here and following superscript T means tanspose of a matrix or vector. 21 As in Mertons (1974), once a rms asset value declines to hit a given lower threshold level (contractual obligations), which is assumed to be C i =e i t K i ;i = 1;2, the rm has to declare default and is subject to liquidation. Consider rst the case in which i = i : 7 Denote i = min t;e i t V i (t)K i , which is the rst time when rm i falls into insolvency. Given a time horizon t, we are interested in the individual default probability and joint default probability. Then P (D i (t) = 1) = P ( i t), where D i (t) = 1 implies that rm i is in default state by time t. And P (D 1 (t) = 1 or D 2 (t) = 1) = P ( t), where = minf 1 ; 2 g: By Harrison (1990), we have for i = 1;2; P (D i (t) = 1) = 2 Z i p t ; where Z i 4 = ln(V i;0 =K i ) i is the normalized distance of rm i to its default state, since ln(V i;0 =K i ) i = ln(V i;0 )ln(K) i ; and () denotes the cumulative normal probability distribution. For the joint default probability, the following proposition is the result. 7 The implication of this assumptions is that the expected discounted return of the asset is the same as that of the obligation. Then the distance to default always follows a normal distribution with constant mean. 22 Proposition 2 Assuming that i = i , we have P (D 1 (t) = 1 or D 2 (t) = 1) = 1 2r 0 p 2t e r 2 0 4t X n=1;3;::: 1 n sin n 0 I1 2 ( n +1) r 2 0 4t +I1 2 ( n 1) r 2 0 4t ; where I v (z) is the modied Bessel function I with order v and = 8 > > < > > : tan 1 p 1 2 if < 0 +tan 1 p 1 2 otherwise ; 0 = 8 > > < > > : tan 1 Z 2 p 1 2 Z 1 Z 2 if < 0 +tan 1 Z 2 p 1 2 Z 1 Z 2 otherwise ; r 0 = Z 2 =sin( 0 ): Proof. See Zhou (2001) for a brief proof. Zhou (2001) also gave a more general formula of the joint default probability for the caseinwhich i 6= i . Theresultsofthejointdefaultprobabilityisevenmorecomplicated than that in the last proposition. The complication makes the implementation very computationally intensive, to say nothing of cases with more than two rms. For two rms case, the impact of i 6= i is relatively small. So Zhou (2001) focus more on the case in which i = i . As for the implications and applications of the model, see Zhou (2001). Below we will investigate the model in more details. 23 Now we denote as = 2 6 4 11 12 21 22 3 7 5; then 2 1 = 2 11 + 2 12 ; 2 2 = 2 21 + 2 22 ; = 21 11 + 22 12 p 2 11 + 2 12 p 2 21 + 2 22 : WecanthinkofuncertaintygeneratedbyW 1 (t)andW 2 (t)astwoindependentprobability spaces 1 and 2 . Then the correlation between the two rms is determined via their connectionstothetwoprobabilityspaces. Fromtheexpressionofaccordingtovolatility coe¢ cientsofthetwoassetvalues, theextendsofthermsdependenceonthetwospaces determine how the two rmsasset values are correlated. For example, if 12 and 21 are negative, and 11 and 22 are positive, then the correlation of the two rmsasset values is negative, < 0. In addition, if ii = ; i = 1;2, a constant, then = 1. Figure 1.1 shows how the correlation changes along with 12 and 21 , given 11 and 22 . In gure 1.1, we can see that can be any number between1 and 1 when domain of 12 and 21 is in interval [0:5;0:5]. From the analysis above, we can see the asset correlation is determined by the four coe¢ cients (two assetsvolatilities with respect to two probability spaces). But for any specic , there may be many di¤erent coe¢ cients. This provides us more details to look at the evolution of assets and the default correlation. 24 Figure 1.1: The graph depicts the change of with respect to 12 and 21 , given 11 and 22 . 11 = 0:2, 22 = 0:3,0:5< 12 ; 21 < 0:5. Zhou (2001) had concluded that the sign of asset correlation is same as that of default correlation, which is consistent with our intuition. In addition he also found that the default correlation between high quality rms is lower than that between low quality rms. This phenomenon is observed in practice. Furthermore, Zhou (2001) showed the default correlation is dynamic, which reects the reality. Although Zhou (2001) improved Mertons (1974) model, he just investigated the de- fault correlation due to the asset correlation. We can observe that there are many other factors orrelationship causingthe rmsdefault interactively. For example, if rm 2owns debts of rm 1, and if rm 1 bankrupts, then the asset value of rm 2 may jump down instantly. This example also shows that the continuity assumption of the rmsasset values may not follow the observation in practice. 25 1.2.2 Jumps in Volatilities due to Other Firm s Default Zhous(2001)modelessentiallyprovidesastructure,wherethereturnsofassetsaredriven by some common uncertainties. The coe¢ cients (volatilities ) measure the extends of impacts of uncertainties on the asset values. That is, there are not direct dependence between any two rms. It is because that if we remove any asset from a portfolio or any rm announces bankruptcy, the evolution pattern of the remaining asset values dont change. 8 However it is observed very often, once a rm falls into nancial distress, it may cause the parameters of asset return of other rm or rms change dramatically. Haworth et al (2006) develops a model, in which a rms bankruptcy may make a jump in volatilities of other asset return. They assume there aren rms, each of whose rm value follows a geometric Brownian motion dV i (t) = (r f q i )V i (t)dt+ i V i (t)dW i (t); i = 1;:::;n; (1.14) where r f denotes the risk-free interest rate, 9 q i dividend rate, i volatility, and all are constants. W i (t) are standard Brownian motions and satisfy cov(W i (t);W j (t)) = ij t; i;j = 1;:::;n; 8 Of course, the joint default probability of two alive rms may change. Although Zhou (2001) doesnt draw this conclusion, it may be a reasonable conjecture. 9 Here the model is constructed in a risk-neutral world. So the discounted asset return is a Martingale. 26 where ij are constant (dependent on i;j). As a rst-passage model, they assume that every rm has an exponential default barrier b i (t) =K i e i (Tt) ; for some constants K i ; i ;i = 1;:::;n. T represents the length of time period under consideration. In their model, they consider the default contagion as follows. If rm i defaults, the volatility of rm j, j6=i has a jump, that is j (V;t) = 8 > > > > > < > > > > > : j ; if V i (t)>b i (t); V j (t)>b j (t) j F ij ; if V i (t)b i (t); V j (t)>b j (t) 0; if V j (t)b j (t) ; for some constant F 1: 10 V = (V 1 ;:::;V n ) T . If taking one more rm k 6= i;j into consideration, we have the following relation j ! j F ij ! j F ij F kj : Following the same pattern, we can incorporate more rm defaults. 10 In Haworth et al (2006b), they assume F 1, but it is natural to extend the assumption to be F >1, since the default of one rm may make some other rm less risky, in which F <1. 27 To nd the default probability of a certain number of rms in a portfolio, letA denote the event of default, andL denote the innitesimal generator of (1.14), then we can nd the probability of the event A by solving LU = @U @t + n X i=1 i V i @U @V i + 1 2 n X i;j=1 a ij V ij @ 2 U @V i @V j = 0 U (V;T) = I A (V (T)); (1.15) where i = r f q i ; a ij = ij i j : Then by Feynman-Kac formula, the function U (V;t) is U (V;t) = EfI A (V (T))jV (t) =vg = P(V (T)2AjV (t) =v): The authors use nite-di¤erence method and a multigrid solver to solve (1.15). How- ever there is no explicit formula for even two rm case. Their numerical approach is not practicable for portfolio with three or more assets because of the computers speed and physical memory constraints. 1.2.3 Counterparty Risk Based on Individual Default Intensity Jarrow and Yu (2001)[25] extended Landos (1998)[29] intensity-based model of Cox processes to include counterparty risk, so that they can explain the possible cause of 28 the clustering defaults. It is well-known that there exist many connections among rms. The counterparty relations can be nancial relation (e.g. debtor-creditor), production relation (e.g. supply-demand), and administration relation (e.g. parent-subsidiary). The modeldevelopedbyJarrowandYu(2001)allowseachrmtofacecounterpartyriskraised by these relations. The counterparty structure helps to illustrate the clustering default, which seems not directly due to the macroeconomic recession. It is rational to see that, in an e¢ cient market, the prices of bonds issued by rms should reect these relations. So the term structure of credit spread should include the shift for the anticipation of counterparty risk. WhenextendingLandosmodel(1998)totakeintoaccountthecounterpartyrisk,Jar- rowandYuhavetodealwiththerstdi¢ culty,whichisabout"loops"inthecounterparty structure. To avoid this complexity, they make a "primary-secondary framework". This framework divides the rms I in consideration into two groups, S 1 and S 2 11 . S 1 is the subset containing only primary rms, and S 2 contains only secondary rms. The default intensities of primary rms depends only on F X t , which is the ltration generated by a processX uptotimet,andthedefaultintensitiesofsecondaryrmsdependsonbothF X t and the status (bankruptcy or not) of the rms in S 1 . Note that X 2R d is d dimensional macroeconomic state variable. Thus in the model, there are I rms. For rms in the primary set S 1 . We dene i = inf t : Z t 0 i s dsE i ; 1iS 1 ; 11 Without causing confusion, S1 and S2 denote the sets, and also the numbers of elements in the two sets. 29 where E i s are independent unit exponential random variables for 1 i S 1 , which are also independent of X t ; i t is the default intensity of rm i, 1iS 1 ; which is adapted toF X t . As showed in Lando (1998), P i >tjF X T = exp Z t 0 i s ds ; t2 [0;T ]; where T denotes the time horizon interested in the model. We let N i = I f i tg , i = 1;:::;I, the point process indicating the status of rm i, 1iS 1 , andF i t = N i s ;0st , the ltration generated by the default process of rm i. After this construction, we dene j = inf t : Z t 0 j s dsE j ; S 1 +1jI; where E j s are independent unit exponential random variables for S 1 +1 j I, which are also independent of both X t and i ;1iS 1 ; j t is the default intensity of rm j, S 1 +1jI, which is adapted toF X t _F 1 t _:::_F S 1 t 12 . So we get similarly P j >tjF X T _F 1 T _:::_F S 1 T = exp Z t 0 j s ds ; t2 [0;T ]: where we assume j t =a j 0;t + S 1 X k=1 a j k;t I ft k g ; S 1 +1jI; 12 Notation F_G represents the smallest eld cataining both F and G. 30 for some constants a j 0;t and a j k;t , k = 1;:::;S 1 . If there is not the primary-secondary structure, it is too complex to nd a easy way to derive the joint distribution of the rst jump times. Now we are ready to calculate prices of zero-coupon bonds. We rst assume that the term structure of default-free bond is independent of the portfolio we are interested. For single counterparty (two-rm model), let rm A be primary, and rm B secondary. For simplicity, A t =a> 0;for some constant a, and B t =b 1 +I ft A g b 2 ; where b 1 > 0 and b 2 are constant. b 2 can be negative, but it must guarantee B t > 0 ( B t = 0 is trivial). Suppose that the two rm issue zero-coupon bonds with maturity T T , and the recovery rate i , i = A;B is exogenous constant. We quote Jarrow and Yus (2001) a proposition. Proposition 3 At time t, the prices of zero-coupon bonds issued by A and B with ma- turity T are v A (t;T) p(t;T) = A + 1 A I f A >tg e a(Tt) ; and v B (t;T) p(t;T) = 8 > < > : B + 1 B I f B >tg b 2 e (a+b 1 )(Tt) ae (b 1 +b 2 )(Tt) b 2 a ; if b 2 6=a B + 1 B I f B >tg (a(T t)+1)e (a+b 1 )(Tt) ; if b 2 =a ; 31 if rm A has not defaulted by time t, and v B (t;T) p(t;T) = B + 1 B I f B >tg e (b 1 +b 2 )(Tt) ; if rm A has defaulted by time t. Proof. See Appendix B in Jarrow and Yu (2001) [25]. Using formula y(t;T) = 1 Tt ln v(t;T) p(t;T) ; we can compute the yield spread of a default- able bond v as a function of maturity T. We can see that when b 2 > 0, rm Bs yield spread increases as the default probability grows up; and when b 2 < 0, they exhibits the opposite properties. For the case in where there exist more than two counterparts, it is not easy to nd the formulae of the secondary bond prices. We have to pay attention to the interaction between primary rms. For example, rm A and B are primary rms, and rm C is a secondary rm. Let A t =a, B t =b, then the default intensity of C could be C t =c 0 +I ft A g c 1 +I ft B g c 2 +I ft A ;t B g c 3 ; for some appropriate constants c 0 ;c 1 ;c 2 ; and c 3 . The last one in right of the equation aboveistheinteractiontermbetweenrmsAandB. Evenforthesimpletwo-counterpart model, it is tedious to derive the price of bond issued by the secondary rm. This is a disadvantage of the model in implementation. In Jarrow and Yu (2001), for the three-rm two-counterpart model, they assume c 0 > 0, c 1 > 0, and c 2 < 0. Then the default probability of rm c increases if rm A defaults and decreases if rm B defaults. They give a scenario in which the rm C holds 32 a long-position of A-bonds and short position of B-bonds. But this rises some questions. Does short position denitely imply c 2 negative? It seems not exactly, since if the rm B defaults, the way that the debtors of rm B deal with the bond the rm B holds is critical. In addition, how to determine the values of c 2 and c 3 is still a undetermined problem, which needs further research. For the case in which the default is correlated with the default-free term structure, Jarrow and Yu (2001) assume the default probability intensity of primary rm A is A t = A 0 + A 1 r t ; for some constants A 0 and A 1 ,where r t is the default-free interest rate satisfying dr(t) =a(r(t)r(t))dt+ r dW (t); where W (t) is a Wiener process under P, and r(t) is a deterministic function. The probability intensity of secondary rm B is B t = B 0 + B 1 r t +cI ft A g ; for some constants B 0 , B 1 and c. Assuming a zero recovery rate and no default before t, we can nd the price of rm Bs bond with maturity T v B (t;T) = E t exp Z T t r s + B s ds = E t exp B 0 (T t) 1+ B 1 R t;T E t e c(T A )I f A Tg jF t _F r T ; 33 where R t;T = R T t r u du. Further calculation, we have v B (t;T) = E t e ( B 0 +c)(Tt)(1+ B 1 )R t;T 1+c Z T t e ( A 0 c)(st) A 1 Rt;s ds = e ( B 0 +c)(Tt)(1+ B 1 ) t;T +(1+ B 1 ) 2 2 t;T =2 1+c Z T t e ( A 0 c)(st) A 1 t;s +( A 1 ) 2 2 t;s =2+ A 1 (1+ B 1 )(t;s;T) ds ; where, let b(u;T) = ( r =a) 1e a(Tu) ; t;T = 1e a(Tt) a r(t)+ Z T t Z u t e a(su) ar(t)ds du; 2 t;T = var t Z T t r(u)du = Z T t b(u;T) 2 du; (t;s;T) = 2 r Z s t b(u;T)b(u;s)du: Note that t;T and 2 t;T are the mean and variance of R T t r(u)du respectively. The model developed by Jarrow and Yu (2001) doesnt take into consideration the correlation decay. It is rational to think of the phenomenon the instant jump in default probability intensities will decay as time passes by. As the manager of a secondary rm willadjusttheoperationandrelationshipwithotherrmswhenheorshefacesthedefault e¤ectofsomeprimaryrms,thedefaultprobabilityintensitymayrecovertopreviouslevel or some other level. 34 Chapter 2: Credit Risk of a Leveraged Firm With Multiple Investment Projects This research examines the optimal investment strategy of a rm with debts. When the rm possibly chooses to default its debts, and aims to maximize its equity value, we shows that the problem is a controlled optimal stopping problem. The rms objective is equivalent to minimize the discounted expected cost payout due to debt issuance. The rm seems to prefer to projects with low return rate and high risk. In the setup, there exists a constant endogenous bankruptcy-triggering asset level. We investigate the two- project models. For linear return-volatility model, the rm should invest all its capital in the risky project. If both projects are risky, at the bankruptcy state, the rm either invests all capital in the project with lower return (lower volatility) or all in that with higher return (higher volatility). The rm very likely chooses only the higher risk project alonglifetime. Iftheoptimalchoiceatdefaultisthelowerreturnproject,thermpossibly increases the proportion in project with higher return as the asset value increases away from bankruptcy-triggering level. And then if the asset value is too high, the rm will reduce the proportion in higher return project. 35 2.1 Introduction Some structural models including Black and Scholes (1973), Merton (1974), Black and Cox (1976), Longsta¤ and Schwartz (1995), and others, assume that a constant default boundary exists. A rm will fall into insolvency when its asset value hits the boundary fromabove. Ifso,thecreditorswilltakeovertherm. Thesemodelsexcludethebehaviors of the rms owners and managers. However, the rms owners and managers play an important role in operations and therefore bankruptcy probability. They may keep the rm alive by injecting capital if the owners and managers believe it protable to do so. Leland (1994), Leland and Toft (1996), and Anderson and Sundaresan (1996) take into accountthermsnancialdecisions;theirmodelsidentifythebankruptcy-triggeringasset level as endogenous rather than exogenously given. Leland (1994) assumes that the asset value of the rm follows a di¤usion process, and keeps the debt structure homogenous, so that the explicit formulae for debt value, rm value, and equity value, is obtained. In addition his model has a constant endogenous bankruptcy-triggering asset level. Leland and Toft (1996) extends Lelands (1994) model by allowing debts maturity to be nite, and also the closed-form formulae evaluating interesting securities have been derived. Most of the models above just take the asset value process as given, which follows a certain stochastic process. In a risk-neutral world, the drift term of the process is a risk-free instantaneous interest rate. Furthermore they assume that the volatility is a given constant. Under this construction, they make static analyses of the debt value with respect to the asset risk. In addition, these models with constant coe¢ cients can not cap- ture owner-managers behaviors in response to debt issuance and investment decisions. 36 the model in this research extends Leland (1994) to include multiple investment oppor- tunities and overcome the disadvantages of the models mentioned above. It is reasonable to think that a rm may have multiple projects to invest during the lifetime. Merton (1969) investigates the optimal portfolio and consumption strategy. Since then many improvements have been made. However, our model is di¤erent because our objective is not to maximize the lifetime consumption, but to maximize the expected equity value. Our model is also di¤erent from mean-variance analysis of risk-return relationship for portfolio investment. The main contribution of the research is application of the optimal constrolled problem to credit risk modelling. It is assumed in the model that a rm has N investment opportunities (projects), and each projects value satises a geometric Brownian motion. The rm can invest the capital, which consists of its own wealth and those borrowed from creditors, into the N projects. The rmaim is to reach the maximization of the equity value for every instant time,bydecidingonhowmuchcapitalitcaninvestineachproject. Itisapparentthatthe asset value process is a controlled di¤usion process. Krylov (1980) has investigated con- trolled di¤usion processes in a broad way. Based on Krylovs contribution, Mihailovskaya (1980) studies the optimal stopping problem for a controlled di¤usion process. It is not obvious what the fundamental is when a stopping time is reached. However, if the dif- fusion process, the strategy and the objective satisfy certain conditions, according to Mihailovskaya (1980), for each strategy, a unique optimal stopping time exists, and the objective function reaches the maximum when the rm chooses the best strategy. The asset value process of the rm in this model agrees with the conditions required in Mi- hailovskaya(1980). BecauseallavailableprojectsinourmodelfollowgeometricBrownian processes, it can be proved that a constant bankruptcy-triggering asset value level exists. 37 In general, it is hard to nd the optimal control and the value of the process at optimal stopping, but in the research we can nd the best strategy, which turns out to choose the project with the lowest bankruptcy-triggering boundary. Karatzas and Wang (2000) considers the utility maximization problems of mixed op- timal stopping and control, but they require the running utility (our running cost) and terminal utility (our terminal cost) to be strictly concave. Then they apply the duality approach and solve a family of related pure optimal stopping problems. In our model, running cost is a constant, and terminal cost is just asset value at stopping time. If the value of the asset value process at stopping time is a constant, the problem can be degenerated to become an optimal stopping problem with a constant lower boundary. If the optimal strategy is nice enough, we may determine the asset value at banktuptcy. In both cases the problem becomes simpler and closed-form solution may be obtained. Fortunately it turns out that the problem in this research has closed-form formula for debt evalution. We will see that a constant strategy is optimal, and the asset value is constant at the optimal stopping time. Toapplyourmodelanditsresultsinpractice,wefurtherconsiderthatanewpreferable project occurs sometime during the operation of the rm. The rm-owner must decide optimal investment strategy when it faces a di¤erent set of investment opportunities. We will see that the decision-making based on the model is the same as choosing a project from two options. Lets name the new optimal investment as the new project. From the perspective of the rm, it is always better to switch investment to the new project whenever the rm is in good or bad quality. However, for the debt-holders it depends on current level of the asset value. Keeping the debt structure, the debt-holders accept the new project if the current asset value is high, and they do not accept it without 38 compensation if the asset value is high. Thus when the asset value is high, the two counterparties have to renegotiate and reach an agreement in order to take the new project. This research provides a simple compensation mechanism that will make both the debt-holders and the rm-owner better o¤. The author has the following assumptions in this research. First, there is no cost caused by adjusting investment strategy, so that the owner-managers can revise their strategycontinuouslyifnecessary. Second,theprojectcanbedividedassmallaspossible, 1 so that every strategy in mathematics is admissible in reality. Third, there is no xed costs required to enter a project. The remainder of the article is organized as follows. In section 2 a multiple-project modelisdeveloped. Itcanbeseenthatthermsassetvaluefollowsacontrolleddi¤usion process. Afterconstructingahomogenousdebtstructure,thermvalue,equityvalueand debt value are dened. Then the objective function is formed, which is a cost function due to the debt issuance. In section 3, we analyze the optimal strategy of investment, and study the properties of the cost function. In this section we prove the existence of a constant bankruptcy-triggering asset value level. Then we nd the optimal strategy and derive formula of the endogenous boundary of asset value at bankruptcy. In section 4, we applyourmodeltoanalyzetheprocedureofdecidingtheinvestmentstrategy. Bystarting with the two-project models we can easily extend it to multiple-project cases. We also illustratehowtheequityvalueanddebtvaluechangewhenanewprojectisavailable, and learn how the security values change for di¤erent level in asset value and new projects volatility. In the last subsection a renegotiation mechanism is proposed. Finally, section 5 concludes the research with contributions and the possible future works. 1 This is similar to a property of nancial project, but not a real one. 39 2.2 Model Setup Lelands (1994) one project model is extended to multiple projects, in which a rm is providedwithseveralinvestmentopportunities. Inordertotakeadvantageoftaxbenets, the rm tends to issue debts to nance its investment. Thus the buyers of the debts bear credit risk due to this rms possible default. Debt price should have response to the possible debt depreciation. 2.2.1 Asset Value Process We consider that there areN risky projects and the evolution of the instantaneous return rate satises the following geometric Brownian process, dX i =X i = i dt+ i dW i t ;i = 1;:::;N (2.1) where W i is a one-dimensional standard Brownian motion; i is the expected instanta- neous rate of return, and 2 i denote the instantaneous variance of return on the project i. We assume that N Brownian motion are independent mutually, i.e., cov W i t ;W j t = 0, for all i6=j. We can denote and as = ( 1 ; 2 ;:::; N ) T ; = 2 6 6 6 6 6 6 6 6 6 4 2 1 0 ::: 0 0 2 1 ::: 0 ::: ::: ::: ::: 0 ::: ::: 2 N 3 7 7 7 7 7 7 7 7 7 5 : 40 is a diagonal matrix with constant diagonal elements. We assume the instantaneous default-free interest rate is a constant r. Without loss of generality, we have the ordering in elements of drifts such that r 1 < 2 <:::< N ; but it is not required that 1 < 2 < ::: < N . More generally we may think that 2R N R M ; W 2R M ; for some M. But we will not consider the general case in the research. Lets denote the ltered probability space by ;F;fF t g t0 ;P . F t is right contin- uous, and fF t g t0 is a ltration generated by the N-dimensional Brownian motion. P is a probability measure. F 0 consists of all theP-null sets in F. The probability space satisfying the usual conditions 2 . X 0 denotes the initial total asset value, which the owner-manager needs to collect from themselves and outside debt-buyers. For reasonable explanation to X 0 , we can imagine that there exists a rm which faces N potential investment opportunities, and its available asset capital is valued at X 0 dollars. Now an entrepreneur wants to buy the rm. Before buying the rm, the new owner wants to take advantage of tax benet of coupon payment, so he/she issues a certain amount of debts. During running the rm, the owner may declare bankruptcy at some future time . If the rm defaults, a fraction of the asset value X will be paid out as bankruptcy costs such as liquidation costs, legal costs and so on. The debt-holders get (1)X t and the rm owner gets nothing fordefault. Sothermshould notissue arbitrary manydebts due tothe bankruptcyloss. 2 See Yong and Zhou (1999, p17) for description of usual conditions. 41 Suppose that the rm collects D 0 dollars by issuing console 3 bonds with coupon rate c. Thus the owner-manager injects in (X 0 D 0 ) from his/her own wealth. X 0 is possibly distributed into the N projects at time 0. We will examine the proportions of D 0 and (X 0 D 0 ) for the rm to reach maximum rm value at t = 0 given X 0 . During its operation, we assume that the investment is stationary Markov. Denition 1 Astrategyu(t;x)issaidtobestationaryMarkovcontrolifu(t) =u(X(t)), t 0. Let u n t ;n = 1;::N; be the proportion of investment in project n at time t, which obviously areF t -adapted processes. Dene A = ( u :u2R n ; u> 0; N X n=1 u n = 1 ) [0;1] N ; which is the set of the admissible investment strategies for the rm; where u > 0 means u n 0 for all n = 1;:::;N; and at least one u i > 0 for some i. In general, we require that a control u(t;!) 2 A is progressively measurable with respect to fF t g. But under reasonable simplication, we just consider the control is stationary Markov in our model. The strategy u t is Markov, and depends only on X t (to get more understanding to this Markov properties of our control, refer to Krylov (1980)). For any instant time t, the rm can generate cash ow X t , where 0 1, a constant for simplication. The rm uses the cash to pay the debt coupon c to creditors, and dividend (X t (1)c) to equity holders if X t (1)c. In our model, the dividends paid to equity holders are consumed immediately, so that the dividends will not be reinvested into the rm again. If X t < (1)c, we assume that the owner 3 A console bond is a bond that has no maturity and pays a xed coupon. 42 issues equity to ll cash decit , so that we keep a constant. In Black and Cox (1976), Leland (1994), Leland and Toft (1996), and Uhrig-Homburg (2005), the coupon payment is assumed to be made by issuing new equity as in our model. Specially Uhrig-Homburg assumes that equity issuance entails costs if X t < (1)c. We dont take into account the transaction costs, although it isnt hard to extend our model to include the friction. Thermcanalwaysraiseenoughmoneybyissuingequityfreelytopaycouponifitwilldo so. Thus in this model, if the cash ow is less than the coupon, i.e., X t < (1)c, the rm issues an amount of (1)cX t new equity to pay coupon without costs. Notice that once new equity holders join the rm, they participate in rms decision making. We insert one more assumption E Z 1 0 e rs X i ds<1;for all i = 1;::;N; (2.2) where dX n X n = ( i )dt + n dW i , and is a constant dened above. This restriction implies that r+ n > 0 for all n = 1;:::;N, which excludes the explosion of the asset value. From the construction above, we know the rm invests u n t X t into project n, so the total asset value X t of the rm at time t satises dX t = N X n=1 u n t dX n t X t dt 43 By plugging in (2.1), and rearranging it, we get dX t = X t ( u t )dt+X t u t d f W t = X t b u t dt+X t u t d f W t (2.3) X(0) = X 0 : Where u t = N X n=1 u n t n ; b u t = N X n=1 u n t ( n ) u t = v u u t N X n=1 (u n t n ) 2 ; and f W t is a one-dimensional Brownian motion 4 . In following content, we still use W instead of f W. Equation (2.3) is the budget constraint equation of the investment under uncertainty. Letsrecallamoregeneralstateprocess,whichisaMarkovcontrolleddi¤usionprocess satisfying the stochastic di¤erential equation dX t =b(t;X t ;u t )dt+(t;X t ;u t )dW t ; (2.4) 4 This f W essentially is a combination of N independent Brownian motions W i , for i = 1;:::;N. Thus to say that fFtg t>0 is generated by W i , for i=1;:::;N, is equivalent to say that fFtg t>0 is generated by f W. 44 where u t belongs to a set A of admissible controls, which is a compact set; X 2R; b(t;X t ;u t ) and (t;X t ;u t ) are real scalars. For SDE (2.4) to have a unique solution, certain conditions must be satised. These conditions are constraints on the ltration (completion condition) and on coe¢ cients (Lipschitz condition) (see appendix A; for de- tails, refer to Krylov (1980), or Mihailovskaya (1980)). We can check our model satises those conditions. In our model, rst, completion condition is assumed; and second, since b(t;x;u) = x P N n=1 u n n and (t;x;u) = x q P N n=1 (u n t n ) 2 , so the Lipschitz con- dition is also satised. Condition 1 (completion condition) The algebras F t generated by W s , st, are complete. Condition 2 (Lipschitz condition) and b are continuous in (u;x); continuous in x uniformly with respect to u for each t, and Borel in (u;t;x). Furthermore, there are non-negative constant m and K; for all u2A, x;y2R, and t 0 j(t;x;u)(t;y;u)j+jb(t;x;u)b(t;y;u)j Kjxyj; j(t;x;u)+b(t;x;u)j Kj1+jxjj: Condition 3 The running cost function f (t;x) is continuous in x for each t, and boundary cost function g(t;x) is continuous in (t;x). In addition for some non-negative constants m and K, for all u2A; x2R, and t 0 jf (t;x)j+jg(t;x)jKj1+jxjj m : Herejj is absolute value of a scalar. 45 Checking our model for coe¢ cients: Since our assumption on F 0 and F t , the rst condition is satised automatically. For the second condition, we have b(t;x;u) = x P N n=1 u n n and (t;x;u) = x q P N n=1 (u n t n ) 2 , so for any x, y 0, (notice that we actually dont consider x< 0) j(t;x;u)(t;y;u)j+jb(t;x;u)b(t;y;u)j 0 @ N X n=1 u n n ! + v u u t N X n=1 (u n t n ) 2 1 A jxyj; and j(t;x;u)+b(t;x;u)j N X n=1 u n n ! + v u u t N X n=1 (u n t n ) 2 x < N X n=1 u n n ! + v u u t N X n=1 (u n t n ) 2 j1+xj: Since f (t;x) =c(1); and g(t;x) =x, jf (t;x)j+jg(t;x)j = c(1)+x c(1)j1+xj This nishes the checking. For st, equation (2.3) has a unique solution, X t =X s e R t s (b u l 1 2 u2 l )dl+ u l dW l ;ts: 46 Recall that the debt contract in our model requires the rm to pay constant coupon rate c, and maturity is innite. The coupon rate is o¤ered by the rm (but priced by the market)whenthermissetup. Thusthecouponrateisnotacontrolparameterast> 0, but the criterion to determine optimal coupon rate is to maximize the rm value when the rm is initiated att = 0. It is di¤erent from the criterion during the operation. Along its running, the rm optimally chooses its investment strategy u, bankruptcy-triggering asset level and default time . For the moment, we denote the bankruptcy-triggering asset value level by X . Let be set of all stopping times 0. Under the uniqueness of the solution to X t , we are able to make following denitions. As in Leland (1994), we assume the rm gets tax benet of coupon payment as long as it doesnt bankrupt. Let the tax rate be a constant . We have assumed that the rm will lose a proportion of the asset value X due to bankruptcy cost once it announces bankruptcy. Dene TB to be the tax benet, then TB t =E tx Z t e r(st) cds ; (2.5) which is the expected discounted value of tax rebate. E tx represents the expectation conditional on the asset value is x at time t. And the bankruptcy cost BC is BC t =E tx h e r(t) X i ; (2.6) 47 which is the expected discounted rm value lost due to its default. At default time , the asset value is not X , but (1)X : Now we are in a position to dene the rm value F t as F t =X t +TB t BC t (2.7) It is obvious that rm value F t can be greater than asset value X t , if the tax benet is morethanbankruptcycost. Thisistheideabehindthetrade-o¤theoremaboutcorporate nance. From the expression of (2.7), supposing the current time is t, provided that no bankruptcy happens by time t, if the undetermined bankruptcy asset value level X is lower, then the stopping time will be farther, and then the current rm value F t is higher. The debt value at t is dened by D t =E tx Z t e r(st) cds+(1)e r(t) X : (2.8) The relationship between debt value and bankruptcy-triggering asset level X is not ob- vious. When X < c r(1) ; the larger is , the higher D t is. If X > c r(1) , the smaller is ; the higher D t is. After we have made clear the costs and rm value, we are able to dene the rms aim, which is the objective function of the optimal stopping problem of controlled di¤usion process. 48 2.2.2 Firm s Objective It is intuitive to say that the rm aims to get as much return as possible during its lifetime by issuing debts and optimally investing in the projects. We dene equity value V (t;X t ;u;) =F t D t , and plug in the formulae obtained above, thus we have V (t;X t ;u;) =X t +TB t BC t D t = X t E t;x Z t e r(st) c(1)ds+e r(t) X ; (2.9) given X t = x. In the expression of equity value (2.9), the rst term is the current asset value, the second one represents the expected discounted cumulative cost due to debt obligation. R t e r(st) c(1)ds denotes the running cost of coupon payment minus tax rebate, and e r(t) X denotes the instant loss due to bankruptcy if it happens. In expression (2.9) we can see that is not involved in equity evaluation. At any instant time t, given X t , the owner wants to maximize his/her total equity value by choosing a stopping time and the investment portfolio u s , t<s<. From equation (2.9), in order to maximize V, the rm should try its best to lower the expected discounted cumulative cost, which is the second term of right hand side, so it is equivalent to solve the following controlled optimal stopping problem J (t;x) = min 2 us2A E t;x Z t e r(st) c(1)ds+e r(t) X ; (2.10) where is set of stopping time as dened above. When we say u s 2A in the argument, we mean u s 2 A, for all t s . Any other same notation follows the law made here in this research. The left-hand side of equation (2.10) is the conditional expectation 49 provided that X t =x. From this expression, we can see that the policy decision criterion of the rm is to minimize the expected cost due to debt issuance. As we mentioned above, at t = 0, the rm owners and creditors agree with rms value maximum by determining coupon rate c. This can be represented by maximizing F 0 . Since F 0 =X 0 +TB 0 BC 0 , given X(0) =X 0 , c = argmaxfX 0 +TB 0 BC 0 g; (2.11) whichwillbeafunctionofX 0 . (2.11)isequivalenttondcwhichmaximizes(TB 0 BC 0 ). We will see how we nd c later. 2.3 The One-Project Cases 2.3.1 Existence and Uniqueness of the Optimal Stopping For an one-project case, the rm-owner doesnt have choice of the investment portfolio, but she/he can decide when is the best time to stopping running the rm. Now the main problem (2.10) becomes J (t;x) = inf 2 E (t;x) Z t e r(st) c(1)ds+e r(t) X = c(1) r sup 2 E (t;x) e r(t) c(1) r X ; (2.12) 50 and X t follows a geometric brownian motion with coe¢ cients and . The problem is equivalent to g(t;x) = sup 2 E (t;x) e r(t) c(1) r X = sup 2 E (t;x) [g(;X )]: (2.13) Dene a two-dimensional Itô di¤usion Y t =Y (s;x) t inR 2 by ^ X t = 2 6 4 s+t X x t 3 7 5; t 0: Then the constraint follows the process d ^ X t = 2 6 4 1 b t X t 3 7 5dt+ 2 6 4 0 t X t dW t 3 7 5 = ^ bdt+ ^ dW t where ^ b = 2 6 4 1 b t X t 3 7 52R 2 ; ^ = 2 6 4 0 t X t 3 7 52R 2 : By this construction, we know ^ X s is an Itô di¤usion starting at ^ x = (t;x). Denote the new expectationE ^ x =E (t;x) , so in terms of ^ X s the problem (2.13) can be written as g (t;x) = sup 2 E (t;x) h g ^ X i =E (t;x) h g ^ X i ; 51 whichisatime-homogeneouscase. Sowestartourinvestigationwiththetime-homogenous case. We aim to investigate whether there exists a optimal stopping time, furthermore, if there does, whether it is unique. To do this, lets make the following denition. Denition 2 A stopping time = (x;!) is called optimal stopping time for fX t g, if it satises E x [g(X )] = sup 2 E x [g(X )]; for all x2R n : Here g(X ) is dened as 0 at the points ! 2 where (!) = 1, and the expectation w.r.t. Q x of the process X t for t 0 starting at X 0 =x2R n . We also need the following denition. Denition 3 We call a measurable function f :R n ! [0;1] supermeanvalued w.r.t. X t if f (x)E x [f (X )] for all stopping times and all x2R n . Moreover iff is also lower semicontinuous, thenf is called l.s.c. superharmonic w.r.t. X t . For any sequence f k g of stopping times such that k ! 0 a.s., by Fatou lemma, if f :R n ! [0;1] is lower semicontinuous, then f (x)E x [lim k!1 f (X k )] lim k!1 E x [f (X k )]: 52 From the denition above, if f is also l.s.c. superharmonic, then f (x) = lim k!1 E x [f (X k )]; for all x: Remark 1 Furthermore, if f 2 C 2 R 2 , by Dynkins formula, we can see that f is superharmonic w.r.t. X t if and only if Af 0 where A is the characteristic operator of X t . This is very useful for our problem. Lemma 4 (a)Ifff j g j2J isafamilyofsupermeanvaluedfunctions, thenf (x), inf j2J ff j (x)g is supermeanvalued if it is measurable. (b) If f 1 ;f 2 ;::: are superharmonic (supermeanvalued) functions and f k " f pointwise, then f is superharmonic (supermeanvalued). (c) If f is supermeanvalued and H is a Borel set, then ~ f (x),E x [f (X H )] is super- meanvalued, where H is the rst exit time of X t from H. Denition 4 If aR n valued measurable function h is superharmonic (or supermeanval- ued), and f h, then f is a superharmonic (or supermeanvalued) majorant of h w.r.t. X t . Dene function h(x) = inf f f (x); x2R n ; where inf is taken over all supermeanvalued majorants f of h, then h is the least super- meanvalued majorant of h. 53 In a similar way, we can dene least superharmonic majorant ^ h of h, if ^ h is a super- harmonic majorant of h and ^ h f, where f is any other superharmonic majorant of h. From the lemma (a) above, h is supermeanvalued if it is measurable. Moreover if h is l.s.c., then ^ h exists and ^ h = h. In our problem, we are interested in the situation g 0. If f is a supermeanvalued majorant of g, and is a stopping time, then f (x)E x [f (X )]E x [g(X )]; so f (x) sup E x [g(X )] =g (x): It implies that if ^ g exists, ^ g(x)g (x); for all x2R n . Our objective is to prove ^ g = g . Before that we need to nd ^ g. Now we have the denition. Denition 5 A function f :R n ! [0;1] is called excessive, if it is l.s.c., and satises f (x)E x [f (X s )]; for all s 0;x2R n : It is obvious that a superharmonic function must be excessive. The next theorem will establish the converse relation. 54 Theorem 5 Let f :R n ! [0;1]. f is excessive w.r.t. X t if and only if f is superhar- monic w.r.t. X t . Based on the theorem, to nd g is equivalent to nd ^ g. Now we are in a position to take advantage of a iterative procedure for the least superharmonic majorant ^ g of g: Theorem 6 Let g =g 0 be a nonnegative, l.s.c. function onR n , and dene inductively g m (x) = sup t2Sm E x [g m1 (X t )]; where S m =fk=2 m : 0k 4 m g; m = 1;2;:::. Then g m " ^ g and ^ g is the least superhar- monic majorant of g. Moreover, ^ g = g. Proof. Sincefg m g is a increasing sequence, we can dene ~ g(x) = lim m!1 g m (x). It is obvious ~ g(x)g m (x)E x [g m1 (X t )]; for all m and all t2S m : So ~ g(x) lim m!1 E x [g m1 (X t )] =E x [~ g(X t )] for all t2S,[ 1 m=1 S n . In addition, ~ g is l.s.c., sincefg m g is a sequence of l.s.c. functions. Now for any t2R, we can nd a sequenceft k g2S such that t k !t. By Fatou and lower semicontinuity, ~ g(x) lim k!1 E x [~ g(X t k )]E x [lim k!1 ~ g(X t k )]E x [~ g(X t )]: 55 This proves that ~ g is an excessive function, and so is superharmonic. Hence ~ g is a super- harmonic majorant of g. On the other hand, let f be any supermeanvalued majorant of g, we can see f (x)g m (x) for all m. So f (x) ~ g(x): This proves that ~ g is the least supermeanvalued majorant g of g, and ~ g = ^ g. With the denitions, lemmas and theorems, we are able to reach a main result which is about the existence of optimal stopping. Theorem 7 Let g be the optimal objective function and ^ g the least superharmonic ma- jorant of a continuous objective function g 0. (a) Then g (x) = ^ g(x): (2.14) (b) Dene R =fx :g(x)<g (x)g; which is called the continuation region of g. For M = 1;2;:::, denote g M = g ^ M, R M =fx :g M (x)< c g M (x)g. So R M R\g 1 ([0;M)); R =[ M R M . If R M <1 a.s. for all M then g (x) = lim M!1 E x h g X R M i : (2.15) (c) Particularly if R <1 a.s. and the family n g X R M o M is uniformly integrable w.r.t. Q x , then g (x) =E x [g(X R )] 56 and = R is an optimal stopping time. Proof. Assume that g is bounded and for "> 0 let R " =fx :g(x)< ^ g(x)"g: Denote " the rst exit time from R " . Dene ~ g " (x) =E x [^ g(X " )]: By lemma (c) above, ~ g is supermeanvalued. Claim that g(x) ~ g(x)+"for all x: (2.16) To prove this claim, suppose , sup x fg(x) ~ g " (x)g>": (2.17) Then for all > 0 we can nd x 0 such that g(x 0 ) ~ g " (x 0 ): (2.18) However, since ~ g " + is a supermanvalued majorant of g, we have ^ g(x 0 ) ~ g " (x 0 )+: 57 Thus we get ^ g(x 0 )g(x 0 )+: Case 1: " > 0 a.s. Q x 0 . Then g(x 0 )+ ^ g(x 0 )E x 0 [^ g(X t^" )]E x (g(X t )+")1 ft<"g for all t> 0. By Fatou lemma and lower semicontinuity of g g(x 0 )+ lim t!0 E x (g(X t )+")1 ft<"g E x lim t!0 (g(X t )+")1 ft<"g g(x 0 )+": There is a contradiction if <". Case 2: " = 0 a.s. Q x 0 . Then ~ g " (x 0 ) = ^ g(x 0 ), so g(x 0 ) ~ g " (x 0 ). This contradicts (2.18) for <. Therefore (2.17) leads to a contradition. This proves (2.16) and ~ g " +" is a superme- anvalued majorant of g. Thus ^ g ~ g " +" =E[^ g(X " )]+"E[(g +")(X " )]+"g +2", (2.19) and since " is arbitrary we get ^ g =g : 58 Now if g is not bounded, let g M = min(M;g); M = 1;2;::: and let c g M be the least superharmonic majorant of g M . Then g g M = c g M "h as M !1, where h ^ g since h is a superharmonic majorant of g. Thus h = ^ g =g and this proves (2.14). To prove (b) and (c) we again rst assume that g is bounded. Since " " R as "# 0 and R <1 a.s., we have E x [g(X " )]!E x [g(X R )] as "# 0, and by (2.14) and (2.19) g (x) =E x [g(X R )] if g is bounded. If g is not bounded, we can dene h = lim M!1 c g M . 59 Since h is superharmonic and c g M ^ g for all M, we have h ^ g. On the other hand g M c g M h for all M and therefore gh. ^ g is the lease superharmonic majorant of g we get h = ^ g: Thus g (x) = lim M!1 c g M (x) = lim M!1 E x h g M X R M i lim M!1 E x h g X R M i g (x): So we obtain (2.15). Since c g M N, from the denition of R M , if x2R M , g M (x)< c g M , thus g M (x)<N. Therefore g(x) = g M (x) < c g M (x) ^ g(x) and g M+1 (x) = g M (x) < c g M (x)\ g M+1 (x). So R M R\fx :g(x)<Mg and R M R M+1 for all M, M = 1;2;:::. Therefore R = lim M!1 R M . Finally by (2.15) an uniformly integrability we have ^ g(x) = lim M!1 c g M (x) = lim M!1 E x h g M X R M i = E x lim M!1 g M X R M =E x [g(X R )]: This completes our proof of the theorem. Since ^ g = g is lower semicontinuous and g is continuous, we nd the sets R, R " and R M are open. We will use the following corollary in the research. 60 Corollary 8 Let H be a Borel set and dene ~ g H (x),E x [g(X H )]: If ~ g H (x) is a supermeanvalued majorant of g, then g (x) = ~ g H (x); and = H is optimal. We have obtained the theorem about the existence of the optimal value and optimal stopping time. Next we need to investigate the uniqueness for the optimal stopping. It is included in the following theorem. Theorem 9 Let us denote R =fx :g(x)<g (x)g: Suppose that there is an optimal stopping time = (x;!) for all x. Then R for all x2R, (2.20) and g (x) =E x [g(X R )] for all x2R n . (2.21) Thus R is an optimal stopping time. 61 Proof. Take x2R. is anF t stopping time and assume Q x [ < R ]. Since g(X )< g (X R ) if < D and gg always, we have E x [g(X )] = Z < R g(X )dQ x + Z R g(X )dQ x < Z < D g (X )dQ x + Z R g 8 (X )dQ x = E x [g (X )]g (x) since g is superharmonic. This proves (2.20). In addition for x2R, since ^ g is superharmonic we have by (2.20) g (x) = E x [g(X )]E x [^ g(X )]E x [^ g(X R )] = E x [g(X R )]g (x); which proves (2.21) for x2 R. Now for x2 @R to be an irregular boundary point of R. Then D > 0 a.s. Q x . Let f k g be a suquence of stopping times such that 0 < k < R and k ! 0 a.s. Q x , as k!1: Then X k 2R, so by strong Markov property E x [g(X R )] =E x E X k [g(X R )] =E x [g (X k )], for all k. Thus by lower semicontinuity and the Fatou lemma g (x)E x [lim k!1 g(X )] lim k!1 E x [g (X k )] =E x [g(X R )]: If x2@R is a regular boundary point of R or if x = 2 R, we have R = 0 a.s. Q x and hence g (x) =E x [g(X D )]. This nishes the proof of the theorem. 62 Note that if g2C 2 (R m ); dene U =fx;Ag(x)> 0g; whereA is the characteristic operator of X. Then U R: 2.3.2 Results of One-Project Cases As we have seen in the subsection above J (t;x) = c(1) r sup 2 E (t;x) [g(;X )]; (2.22) where g(s;x) =e rs c(1) r x : (2.23) Thus to nd (2.22) is equivalent to nd a stopping time that maximizes g(;X ). The characteristic operator ^ A of the process ^ X s = (t+s;X s ) is given by ^ Af (t;x) = @f @t +bx @f @x + 1 2 2 x 2 @ 2 f @x 2 ; f 2C 2 R 2 . Hence ^ Ag(t;x) =re rt c(1) r x bxe rt =e rt (x(br)+c(1)): So U, n (t;x); ^ Ag(t;x)> 0 o = 8 > < > : R + R if rb n (t;x);x> c(1) rb o if r >b: 63 To exlude the explosion in cash in ow, we have an assumption r > b. From the form of U, if r b, we have U = R =RR + , then does not exist. If b > r then g = 1, which implies that the rm will never fall into insolvency state because of the high prot of the project. If r =b, g (t;x) =c(1)xe rs . It remains to examin the case b<r. First we establish that the region R is invariant w.r.t. time t, i.e., R+(t 0 ;0) =R, for all t 0 R+(t 0 ;0) = f(t+t 0 ;x)j(t;x)2Rg = f(s;x)j(st 0 ;x)2Rg = f(s;x)jg(st 0 ;x)<g (st 0 ;x)g = (s;x)je rt 0 g(s;x)<e rt 0 g (s;x) = f(s;x)jg(s;x)<g (s;x)g = R 64 It is left to show g (st 0 ;x) = e rt 0 g (s;x): In order to include the cases with multiple project, we put the strategy u in the following equations. We will not redo this in the next section. g (st 0 ;x) = sup u >st 0 u2A E st 0 e r( u +st 0 ) c(1) r X x u = e rt 0 sup u >st 0 u2A E st 0 E s e r( u +s) c(1) r X x u = e rt 0 sup u >st 0 u2A E st 0 8 < : sup u >s u2A E s e r( u +s) c(1) r X x u 9 = ; = e rt 0 g (s;x) Therefore the connected component of R that contains U must have the form R(x 0 ) =f(t;x);x>x 0 g for some x 0 c(1) rb . So f = ~ g is the solution of the boundary value problem @f @s +bx @f @x + 1 2 2 x 2 @ 2 f @x 2 = 0 for x>x 0 (2.24) f (s;x 0 ) = e rs (c(1)x 0 ): If we try a solution of (2.24) of the form f (s;x) =e rs (x) 65 we have r+bx 0 + 1 2 2 x 2 00 = 0 for x>x 0 (2.25) (x 0 ) = c(1) r x 0 The general solution of has the form (x) =C 1 x 1 +C 2 x 2 where C 1 , C 2 are constants being determined soon, and i = 1 2 2 4 1 2 2 b s b 1 2 2 2 +2r 2 3 5 (i = 1;2), 1 < 0< 2 . Since (x) is bounded as x ! 1, we have C 2 = 0. The boundary condition (x 0 ) = c(1)x 0 gives C 1 =x 1 0 (c(1)x 0 ). Thus we get the solution f of (2.25) is ~ g(s;x) =f (s;x) =e rs c(1) r x 0 x x 0 1 : To nd x 0 , we x (s;x) by rst order condition of maximizing ~ g(s;x), thus x 0 =x max = c(1) r 1 1 1 : Let us denote = 1 , = 1 2 2 4 b 1 2 2 + s b 1 2 2 2 +2r 2 3 5 (2.26) 66 thus > 0. We can rewrite x 0 = c(1) r 1+ : Note that here g (s;x) is the value we look at the point of time t = 0. So in order to transfer it to the value at "current" time, we will remove e sr , which is the discount part. Wecanalsousetherstpassagetimetosolvethisproblem(seethepertinentpartbelow). 2.4 Optimal Strategy Analysis The optimal strategy of the rm is to choose an investment portfolio and stopping time in order to minimize the expected discounted cumulative cost in the future life. But it is not clear what the asset value is at which the manager-owner is not willing to run the rm any more. In this section, we shall make a detailed analysis in this aspect. 2.4.1 Properties of the Cost Function For any strategy u and stopping time 2 , we dene J u; (t;x) =E u t;x Z t e r(st) f (s;x s )ds+e r(t) g(;X ) : where (t;x)2 [0;1)R, indicating "current" state;E u t;x denotes the conditional expec- tation for strategy u, given X t = x; and g(;X ) is the boundary cost function, in our model g(;X ) =X . We shall express the optimal value (2.10) as J (t;x) = inf u2A inf 2 J u; (t;x): (2.27) 67 This expression says that for every admissible strategy u in A, we can nd an optimal stopping time just as one-project case does, thus the corresponding minimum cost is determined; afterthat, theglobalminimumcanbeobtainedbylocatingthebeststrategy. To nd J (t;x), we need to examine all possible strategies and its stopping time. It in general is complicated, and there isnt explicit expression. Due to the homogeneity of our model,J (t;x) =J (x) 5 foranytimet. FromnowonwejustuseJ (x)withoutconsidering time. The properties of cost function in our model consist of the following propositions. Proposition 10 In our model J (x) is increasing and concave in x. And if the cost function J (x) obtained by (2.27) is di¤erentiable with respect to x, denoting @J(x) @x by J x , then 0J x 1. Furthermore lim x!1 J x = 0. Proof. First, lets prove that J (t;x) is increasing in x. For any x 1 < x 2 , from (2.10), J (t;x i ) = min 2 us2A E t;x i R 0 e rs c(1)ds+e r X t+ , for i = 1;2. For any xed stopping time and xed control strategy u; we have E t;x 1 Z 0 e rs c(1)ds+e r X x 1 t+ <E t;x 2 Z 0 e rs c(1)ds+e r X x 2 t+ since X s =X t e R s t (b l (u) 1 2 2 l (u))dl+ l (u)dW l ;st: E t;x 1 Z 0 e rs c(1)ds+e r X x 1 t+ = E t;x 1 Z 0 e rs c(1)ds+e r x 1 e R t (b l (u) 1 2 2 l (u))dt+ l (u)dW l < E t;x 2 Z 0 e rs c(1)ds+e r X x 2 t+ = E t;x 2 Z 0 e rs c(1)ds+e r x 2 e R t (b l (u) 1 2 2 l (u))dt+ l (u)dW l 5 The time invariance can be proved in our setup. 68 Thus J (t;x 1 )<J (t;x 2 ); sincebothsidesaredoubleminimizingoverthesameadmissiblestrategysetandthesame set of the stopping times. This proves J (t;x) is increasing in x. Second,WewillprovethatJ (t;x)isconcaveinx. Foranyinitialvaluesx 1 ;x 2 atwhich the rm is in solvent state, and for any 2 [0;1], there is a value x =x 1 +(1)x 2 , where the rm is still in solvency. For any "> 0, there are optimal policy u " and " , such that J (t;x )>J u " ; " (t;x )"; where J u " ; " (t;x) = E t;x Z " 0 e rs c(1)ds+e r " X (u " ) t+ = E t;x Z " 0 e rs c(1)ds + e r " (x 1 +(1)x 2 )e R t (b l (u " ) 1 2 2 l (u " ))dt+ l (u " )dW l o = E t;x Z " 0 e rs c(1)ds+x 1 e R t (b l (u " ) 1 2 2 l (u " ))dt+ l (u " )dW l + (1) Z " 0 e rs c(1)ds+x 2 e R t (b l (u " ) 1 2 2 l (u " ))dt+ l (u " )dW l = J u " ; " (t;x 1 )+(1)J u " ; " (t;x 2 ); 69 which is the cost function give strategy u and stopping time . Thus J (t;x ) > J u " ; " (t;x 1 )+(1)J u " ; " (t;x 2 )" > J (t;x 1 )+(1)J (t;x 2 )"; and let "! 0, we get J (t;x )>J (t;x 1 )+(1)J (t;x 2 ): This proves that J (t;x) is concave in x. Third, To prove J x < 1; it is su¢ cient to prove J(s;x 2 )J(s;x 1 ) x 2 x 1 < 1. For any small "> 0, there exists u " and , such that J u " ; (t;x 1 )<J (t;x 1 )+" where J u " ; (s;x 1 ) is the cost value given strategy u " and stopping time . we have J (t;x 2 )J (t;x 1 ) J u " ; (t;x 2 )J u " ; (t;x 1 )+" = (x 2 x 1 )E t n e R t (bs(u " )r 1 2 2 s (u " ))ds+s(u " )dWs o +" < x 2 x 1 +" Let "! 0, J (s;x 2 )J (s;x 1 )<x 2 x 1 , i.e., J(s;x 2 )J(s;x 1 ) x 2 x 1 < 1. So if J x exists, J x < 1. 70 Finally, we prove lim x!1 J x (x) = 0. Since J (x) is concave and increasing in x, it is su¢ cient to prove J (x) is bounded as x! +1. For any s 0, J (x) E Z s 0 c(1)e r d +e rs X s = E Z s 0 c(1)e r d +xe R s 0 (b r 1 2 2 )ds+ dW = Z s 0 c(1)e r d +E n xe R s 0 (b r 1 2 2 )ds+ dW o = c(1) 1e rs r +xe R s 0 (b r)d ! c(1) r ; ass! +1 The limit exists because b s r < 0 for all s 0. In the proof we obtain that J (x)j x!+1 = c(1) r . It means that if the asset value is very high, the default probability of the rm is very small. So the cost is the di¤erence between the value of default-free console bond ( c r ) and the tax benet ( c r ) due to the bond. This result also tells us that the cost wont explode, so that it is always preferable toissuebondtonanceitsinvestments. TheconcavityofcostfunctionJ (x)isequivalent to convexity of the equity value function (see (2.9)) in x. For special case in which there is only one investment project with geometric Brownian motion, the explicit formula for equity value can be found. The convexity is veried (see Leland (1994)). So this propositionillustratesthatasrmscurrentassetvalueincreases,theexpecteddiscounted cost will increase (J x > 0). But the amount of increasing is always less than that of increasing in asset value (J x < 1). Thus the rm has no reason to default its bonds when the asset value is very high. This is consistent with our observation in reality. Given debt structure, when the rms asset value rises, the rm is located in higher credit ranking, 71 because the equity-debt ratio is higher. From perspective of the debt-holders, they have fewer worries when the rm has higher asset value. Proposition 11 In our framework, if X 0 > 0, we can always form a console bond struc- ture with coupon c > 0 such that the equity value is greater than zero. But given X 0 > 0 (so c > 0), as the asset value x tends to zero, under assumption of smoothness, the rst derivative of J (x) converges to 1, i.e., lim x!0 J x (x) = 1: Proof. First to prove whenever X 0 > 0, we can form a console bond with coupon c> 0, so that the equity value is strictly positive. Given X 0 > 0, by equity value (2.9), we want V (0;X 0 ;u;) =X 0 E Z 0 e rs c(1)ds+e r X > 0: If = 0, we have V = 0, so we assume > 0. Since X t =X 0 e R t 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs , we denote M t =e R t 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs . So we need X 0 E Z 0 e rs c(1)ds+e r X 0 M = c(1) 1Ee r r +X 0 1E e r M > 0 It su¢ ces to nd a c> 0 such that for some c< rX 0 (1Efe r M g) (1)(1Ee r ) 72 It is not hard to check that for deterministic time t > 0, the right hand side of above inequality is strictly positive (remember < 1, and e rt M t is a supermartingale since b t (u)<r for all u2A). So such a c always exists. Next to prove lim x!0 J x (x) = 1, given X 0 ;c> 0, it su¢ ces to prove J x (0)> 1" for 8"> 0. That is, for8"> 0 there exists , such that8x satisfying 0<x<, we have J (x) (1")x: It is equivalent to showE R 0 e rs c(1)ds+e r X (1")x, for all . E Z 0 e rs c(1)ds+e r X E Z 0 e rs c(1)ds1 fg +e r X 1 f<g = I 1 +I 2 ; where1 fAg isanindicatorfunction,I 1 =E R 0 e rs c(1)ds1 fg ,andI 2 =E e r X 1 f<g . I 1 P( ) Z 0 e rs c(1)ds And I 2 = xP( <)+xE e r M 1 1 f<g xP( <)xE e r M 1 1 f<g xP( <)xE sup t< e rt M t 1 1 f<g = xP( <)xO() 73 where M t is dened as in (1), and O() implies that O()! 0, as ! 0. We can choose =("), such that O() " 2 . IfP( <) 1 " 2 , then it is obvious J (x)I 1 +I 2 x(1"): IfP( <) 1 " 2 , thenP( ) " 2 . Thus want " 2 Z 0 e rs c(1)ds 1 " 2 x; we just choose (") = 1 1"=2 " 2 Z (") 0 e rs c(1)ds; so that8x<("), J (x)I 1 x(1"): Let "! 0, we have J x (x)! 1. Given X 0 > 0, the rm doesnt default immediately as long as it doesnt nance its investment completely by debt issuance. If the rm uses only debt without equity, X 0 = D 0 , by formula (2.9) and non-negative equity value, the optimal stopping time is = 0, i.e., the rm should bankrupt right away. This excludes the possibility to buy a pure debt rm 6 . But a rm with positive market value is always preferrable by some investors. In the proof of the proposition, we dont use any information about asset value level at default, since it is not clear what X is as far as we mentioned. The purpose to investigate the properties is to conrm a fact that a rm will announce bankruptcy for sure when its asset value is very low. 6 A pure debt rm is a rm whose net capital is negative. 74 Inthenextsubsectionwewillinvestigatetheoptimalstrategyandendogenousbankruptcy- triggering asset leve. It will be indirectly proved that the result of J x (X B ) = 1 and the "smooth-pasting" condition mentioned in Leland (1994) and others under requirement of non-negative rm value. It is one of the most important results in the research. It turns out that there is a constant endogenous bankruptcy-triggering asset value level. Proposition 12 The cost function J (x) satises the following quasi-variational inequal- ity, inf inf u2A fc(1)+L u s J (x)rJ (x)g;xJ (x) = 0 (2.28) where L u s J (x) =J x (x)b(x;u)+ 1 2 2 (x;u)J xx (x); Proof. Here we just give a brief proof. Given current state (s;x); J (x) inf u2A E s;x n e rh c(1)h+J (x)++J x (x)e rh (x s+h x) + 1 2 J xx (x)e rh (x s+h x) 2 rJ (x)e rh h : Thus rearranging and dividing by h, and let h! 0; we have inf u2A c(1)+J x (x)b(x;u)+ 1 2 (x;u)J xx (x)rJ (x) 0: WhenJ (x)<x,theequalityholds. Andiftheinequalityaboveisstrict,thenJ (t;x) =x. Put this two situations together we obtain inf inf u2A fc(1)+L u s J (x)rJ (x)g;xJ (x) = 0 75 In our model 2 (x;u) = P N n=1 (u n t n ) 2 x 2 , and b(x;u) = P N n=1 u n t ( n )x: The two parameters do not depend on time t directly. The intuition behind (2.28) is as follows: If it is not optimal to stop right now, i.e., J (x) < x, then the rst term inf u2A fc(1)+L u s J (x)rJ (x)g = 0, which is a usual HJB equation. It says that there exists a strategy which makesc(1)+L u s J (x)rJ (x) to be zero. If it is optimal to stop now, the current expected cost J (x) is equal to x, and there is no strategy to make c(1)+L u s J (x)rJ (x) to be zero any more. From results in this subsection, we have known that the optimal cost function J (x) is increasing and continuous in x, so the solvency set G,fx;J (x)<xg is an interval (x B ;1) where x B , supfx;J (x) =xg. 2.4.2 Optimal Investment Strategy and Bankruptcy Boundary Intheone-projectcasewehaveshownthatthereexistsauniqueoptimalstoppingtimeand also the bankruptcy-triggering asset level can be found. Acturally this is an extension of rst-passage-time models. The only di¤erence is that we dont know the barrier of triggering default, and we just determine it only after we have got the explicit formula for the corporate debts. Before we investigate the optimal investment strategy, lets recall the rst-passage-time model of geometric Brownian process. This is useful to prove our main result in this research later on. 76 Let a strategy 2 U be a constant vector, so the corresponding b t () = b, and t () = for some constants b, . We have X s =X t e (b 1 2 2 )(st)+(WsWt) : Lets denote the asset value at default by x B corresponding to strategy , we need the distribution function of rst-passage time of geometric di¤usion process. A little modi- cation of Leland (1994), we can explicitly nd the formulae of tax benets, bankruptcy costs, rm value and bond price for the model with constant coe¢ cients. Assume the "currentstate is X t . We dene H s = I fs>g ; which is a hazard process indicating the rms status, and let f (s;X t ;x B ) denote the density of rst passage time s of asset value X s from X t to x B , and then the cumulative probability is F (s;X t ;x B ): Also we dene A(s;t) = Z s t e r(lt) f (l;X t ;x B )dl: We can get the explicit expression A(s;t) = X t x B a+z (q 1 (s;t))+ X t x B az (q 2 (s;t)); where () denotes the standard normal distribution function; a = b 2 =2 2 ; z = h a 2 2 +2r 2 i 1=2 2 ; 77 we denote =a+z, since here a, z are dependent on strategy ; and q 1 (s;t) = d B z 2 (st) p st ; q 2 (s;t) = d B +z 2 (st) p st ; d B = ln X t x B : Thus we have E t h e r(t) i = E t Z 1 t e r(st) dH s (2.29) = Z 1 t e r(st) dF (s;X t ;x B ) = A(1;t)A(t;t) = X t x B : Thus the tax benets are TB t = E tx " c 1e r(t) r # = c r 1 X t x B ! : The bankruptcy costs are BC t = X B E tx h e r(t) i = X B X t x B : 78 Note that here we can assume that x B is a given undetermined constant, so we can move X out of expectation sign in denition (2.6). So the rm value is F (X t ) =X t + c r 1 X t x B ! x B X t x B : And the bond price is D(X t ) = c r " 1 X t x B # +(1)x B X t x B : Thus nally the equity value is V (X t ) = F (X t )D(X t ) = X t (1) c r + X t x B h (1) c r x B i : (2.30) Since at bankruptcy state we have proved that J x (x) = 1, that is, @V(Xt) @Xt Xt=x B = 0, thus we get x B , x B = +1 c(1) r ; (2.31) where the superscript means that x B , and depend on specic constant strategy . Indeed we have got this result above by solving a optimal stopping problem. Here the di¤erence is that here we dont investigate that the existence and uniqueness of the stopping time and its corresponding asset value. Notice that in Leland (1994), = 2r 2 , thus x B = (1)c= r + 1 2 2 . It is because that in that model, there only has one investment project, and the fundamental process is measured in risk-neutral probability. But we keep everything in physical world in this research. From equation (2.31), we can 79 see that x B does not depend on the asset value X t , but only on the parameters b, , r, c and . Actually x v B depend only on the investment portfolio at the time of bankruptcy, even if the strategy is not constant. This implies the strategy before bankruptcy has no impact on the bankruptcy-triggering asset level as long as it keeps the rm in solvency. Proposition 13 For any xed strategy t (X) 2 U, we have the solvency set G = (x v B ;1), and x B = +1 c(1) r where = (b( 2 =2)) 2 + h (a 2 ) 2 +2r 2 i 1=2 2 , b = b( ) and = ( ), and is the optimal stopping time of a one-project model. Proof. The asset value process satises that dX t =b( t )X t dt+(v t )X t dW t : We aim to solve the problem g(s;x) = sup 2 E (s;x) e r(s) c(1) r X : Wehaveshowedinlastsectionthatthereexistsauniquestoppingtime v forthestrategy . Now we need to nd x B . For any "> 0, there exist a stopping time " < such that g " (s;x)>g(s;x)", 80 where g " (s;x),E (s;x) e r( " s) c(1) r X " : Then we keep a constant strategy t = for t2 [ " ;]. Denote g( " ;X "), sup 2 E ( " ;X ") e r( " ) c(1) r X ; which is a one-project model. We have solve the problem in the geometric Brownian process with constant coe¢ cients. And we know on [ " ;] that x B = +1 c(1) r . Since g(s;x) > g 0 (s;x)+E (s;x) n e r( " s) g( " ;X ") o > g(s;x)"+E (s;x) n e r( " s) g( " ;X ") o ; we getE (s;x) e r( " s) g( " ;X ") <". As "! 0, g 0 (s;x)!g(s;x) and " ! . Based on this proposition, the possible insolvency set can be restricted to the region D =fx u B ju2Ag: Since strategy setA = n u :u2R n ; u> 0; P N n=1 u n = 1 o is a compact set, for any strat- egy at the stopping time, the instant strategy must belong to A. So we know in our modelthesetD ofpossiblebankruptcy-triggeringassetlevelx B isdeterminedcompletely by strategy set A. x u B is dened by equation (2.31) by replacing with u. Since A is 81 compact, and x u B is a nice function of u, the setD is also compact. We can always nd a u 2A such that it minimizes x u B . We denote x = inffx u B ;u2Ag x = supfx u B ;u2Ag: Corollary 14 For any asset value level X t , if X t x, then the rm has to declare bankruptcy; and if X t > x, the rm is in solvency state for any strategy u t 2A, t 0: To search for the bankruptcy-triggering asset level, we only need to focus on the pair (t;X t ) for X t x. Here the question is: does the rm have to announce bankrupty once X t x? If it doesnt, then does the rm have to announce bankruptcy if X t x u(Xt) B , where x u(Xt) B is obtained by inserting the strategy u(X t ) into formula (2.31)? Answer to the later question looks positive. Basically it is what the proposition above says. Lets go forward a little more. What will happen if the strategy u(X t ) (at the time X t =x u(Xt) B ) becomes a new strategy u 0 (X t ) which makes x u 0 (Xt) B <X t ? A natural guess may be that the rm will not bankruptcy if we change the strategy. As we have known, J () is a monotonic continuous function of x, so this corollary is easy to get. In virtue of this corollary, our problem becomes a pure stochastic control one. For stochastic control, we have the following useful theorem. For 2 U; and 2C 2 0 (RR), dene (L )(x) = @ @t (x)+b @ @x (x)+ 1 2 @ 2 @x 2 : Theorem 15 (Hamilton-Jacobi-Bellman (HJB) equation II) 82 Let be a function in C 2 (G)\C G such that, for all 2U, c(1)+(L )(x) 0; x2G, with boundary values lim t! G (X t ) =X G =x B , a.s. Q x , andsuchthat (X ); stopping time, G isuniformlyQ x -integrableforallMarkov controls and all x2G. Then (x)J (x) for all Markov controls and x2G. Moreover, if for each y2G we have found (x) such that c(1)+ L (x) (x) = 0 and n X ; stopping time, G o is uniformly Q x -integrable for all x2G, then = (x) is a Markov control such that (x) =J u 0 (x) and hence if is admissible then must be an optimal control and (x) =J (x) is the optimal cost function. 83 For the stochastic control problem with Markov control, if we can nd a strategy and afunctionwhichsatisfytheconditionsinthelasttheorem,itmusttheoptimalinvestment strategy and optimal cost function. Theorem 16 Let M (x) = inffJ (x); =(x)g Markov control, and a (x) = inffJ (x); =(t;!)g F t adapted control. Suppose there exists an optimal Markov control = (X) for the Markov problem (i.e. M (x) = J (x) for all x2 G) such that all the boundary points of G are regular w.r.t. X t and that M is a bounded function in C 2 (G)\C G satisfying E x j M (X )j+ Z 0 jL u M (Y t )jdt <1 for all bounded stopping times G , all adapted controls u and all x2G. Then M (x) = a (x) for all x2G. The theorem guarantees that the optimal Markov control is the optimal one of all adapted controls. Without help of the knowledge of bankruptcy-triggering asset level x B , it is hard to ndoptimalstrategy. Foramoregeneraloptimalstoppingproblemofcontrolleddi¤usion process, quasi-variational inequalities are needed to investigate the property of optimal 84 strategy. But due to the simplicity of our model, the cost function is very special (2.10). We may nd the most important property of the optimal strategy. Proposition 17 The optimal strategy of the multiple-project model is a constant, i.e., t =u , for t 0. Furthermore, the asset level at bankruptcy x B =x. Proof. Since x B 2D, <1 a.s. Q x for any x2G. For8"> 0,9 " <N " <1 and " such that J ( " ; " ) (x)<J (x)+": Let ft i g I i=0 be an increasing stopping time, such that 0 = t 0 < t 1 < ::: < t I = N " . Construct strategy ~ " t = I1 X i=0 u i 1 [t i ;t i+1 ) , u i isF i measurable, u i 2A: (2.32) ~ " t = u 0 for tt I ; such that J (~ " ; " ) <J (x)+2": Let u 2A be the investment portfolio such that x =x u B = +1 c(1) r . And let ~ " be stopping time such that J (~ " ;~ " ) (x) = inf 2 J (~ " ; " ) (x): 85 ~ " is the optimal stopping time for strategy ~ " . We have J (~ " ;~ " ) (x) = E x Z ~ " 0 c(1)e rs 1 f~ " t I1 g ds+e r~ " X ~ " 1 f~ " t I1 g +E x Z ~ " 0 c(1)e rs 1 f~ " <t I1 g ds+e r~ " X ~ " 1 f~ " <t I1 g = A+B; where A;B represent the rst and the second expectation respectively. The idea to investigate the strategy u i is as following. We rst examine the strategy u I on time interval [t I ;1); then we search u i backward until we nd the best u 1 . Notice that the strategy obtained in this way is the best strategy under the construction (2.32). We start our search with u I . For ~ " >t I , we just consider J u (t I ;X t I ) =E Xt I Z ~ " t I c(1)e rs ds+e r~ " X ~ " : Tondthebeststrategyofu I ,sinceitisaconsantstrategyu 0 2A,wecantakeadvantage of the formula we have obtained. Given arbitrary strategy u2A;lets dene J u (X I ) = inf 2 E Xt I Z t I c(1)e rs ds+e r~ " X u Thus J u I (t I ;X t I ) J u I (X I ) = c(1) r X u I t I x 0 B 0 c(1) r x 0 B = c(1) r X u I t I 0 0 0 +1 c(1) r 0 c(1) r 1 0 +1 ; 86 where 0 = 1 2 (u 0 ) b(u 0 ) 1 2 2 (u 0 )+ q b(u 0 ) 1 2 2 (u 0 ) 2 +2r 2 (u 0 ) : Moreover J (t I ;X t I )J u I (X I ) inf u 0 2A J u 0 (X t I ) = inf 0 2R ( c(1) r c(1) r c(1) rX t I 0 0 0 +1 0 1 0 +1 ) : Since 0 0 +1 c(1) r < X t I , the rst-order derivative of the expression following the second equal sign w.r.t. 0 is positive, the minimum is obtained where 0 reaches its minimum in its domain. When 0 gets its mimimum, the strategy is u . Thus the strategy ~ " t =u for tt I . Now we want to nd the best constant strategy on time interval [t I1 ;t I ) given u on [t I ;1). It requires to minimize the second part of A. Conditional on ~ " t I1 ; we denote g(x) = c(1) r x h where h = c(1) r c(1) r +1 1 +1 : and dene J u I1 t I1 ;X t I1 = E Xt I1 [ Z ~ " t I1 c(1)e r(st I1 ) ds+e r(~ " t I1 ) g(X ~ " )1 f~ " >t I g +e r(~ " t I1 ) X ~ " 1 ft I >~ " >t I1 g i 87 Since x>g(x) for x>x, we have J u I1 t I1 ;X t I1 inf us2A t I1 s<t I E Xt I1 " Z ~ " t I1 c(1)e r(st I1 ) ds+e r(~ " t I1 ) g(X ~ " ) # inf us2A t I1 s<t I 2 E Xt I1 " Z t I1 c(1)e r(st I1 ) ds+e r(t I1 ) g(X ~ " ) # : Weneedtosearchwhatistheoptimalconstantstrategyu I1 onthetimeinterval[t I1 ;t I ) given ~ " t = u for t t I . It is a control problem with constant strategies on two distint intervalandonlythestrategyontherstintervalissubjecttobechoosen, butthesecond one is given. So we rst study the two-period problem ~ J (x) = inf us2A;0s<t E x Z 0 c(1)e rs ds+e r g(X ) It aims to nd the best strategy u on the rst time interval [0;t) given the strategy u on [t;1). Lets investigate a new setup in which the time invtervals are not determined but separated by a stopping time , that is, the strategy is u on [;1), and undetermined u on [0;). We denote H(x) = inf 2 ut2A;t< E x Z 0 c(1)e rs ds+e r g(X u ) (2.33) 88 thus H(x) = E x c(1) r 1e r +e r c(1) r (X u ) h = E x c(1) r he r (X u ) And dene g(s;x) =he rs x : The characteristic operator ~ A is ~ Af (s;x) = @f ds +bx @f @x + 1 2 2 x 2 @ 2 f @x 2 So ~ Ag(s;x) = rhe rs x bx he rs x 1 + 1 2 2 x 2 ( +1)he rs x 2 = he rs x r + b 1 2 2 1 2 2 2 We nd that ~ Ag(s;x) 0; since r + b 1 2 2 1 2 2 2 0 always holds in our framework. The condition @g ds +bx @g @x + 1 2 2 x 2@ 2 g @x 2 < 0, for all x> 0, implies that the optimal stopping time = 0. = 0 for all x>x B =x for problem (2.33). 89 Next it is easy to verify that for any u2A. ~ J (x)H(x) where H(x) is dened by (2.33). It is because that the optimal stopping is the optimal time in and t I 2 . Thus H(x)< ~ J (x)+"<J (x)+"; So as I ! 1;" ! 0, we have H(x) < J (x), but the other direction is obvious, so we haveH(x) =J (x). Thisprovesthatunderthetwo-periodsetup, thestrategyontherst time interval is u = u . Combined with the given strategy on the second time interval, we get that the strategy is always u for the two-period framework. So the strategy is u for t>t I1 . By backward induction, we can prove by the same mothod of two-period setup that u t =u for t> 0. Thus this completes the proof. In fact, we just compare all possible constant strategies on any time interval. When the asset value process satises a controlled geometric Brownian motion, if the regions of expected return and uncertainty are compact set, there must be a strategy which has minimum dened by (2.26). If we consider the current state is (t;x), we dene stopping time for some y >x B = minfs>tjX s =yg: 90 is the rst time the asset value hits the upper boundary of the unhealthy state from top. Thus the cost function is J (x) = min u E Z t c(1)e r(st) ds+e r( t) J (y) = min u c(1) r E h e r( t) i c(1) r J (y) : The last equality holds because J (y) is a constant. Our model is time-homogenous, J (y) doesntdependonstoppingtime . Noticethatthesecondtermofthelastlineispositive, since c(1) r > J (x). Remember that c(1) r is the cost if the rm will never bankrupt. J (x)isboundedby c(1) r . Sotheruleofchoosingstrategyforthemanageristomaximize E e r( t) . By this method, we can show that the main result in the last proposition can be obtained. Corollary 18 In our model, n and n , n = 1;:::;N, satisfy conditions in sections above, and admissible strategy set is A. Then for all asset value X > x B , the optimal strategy for the rm is to choose u for t<. The asset value process (2.3) becomes a geometric Brownian motion with constant coe¢ cients. Proof. Assume the current time is t and asset value is x, and for some y such that x>y >x B , dene = minfs>tjX s =y;X t =xg. 91 Then for time s< , X s >y, by Bellman principle J (x) = min u E Z t c(1)e (st) ds+e r( t) J (y) = min u E Z t c(1)e r(st) ds+e r( t) J (y) = min u E c(1) r 1e r( t) +e r( t) J (y) = min u E c(1) r e r( t) c(1) r J (y) We need to maximizeE e r( t) . If the rm choose the strategy constant u , we know E h e r( t) i = x y where = (u ), and u is the constant strategy corresponding to the bankruptcy- triggering asset value level x B :Now consider an alternative strategy , which is di¤erent from u only on [t; 1 ], where 1 = minfs>tjX s =y 1 g for some x > y 1 > y. And on [s; 1 ], = u 1 . u 1 is another constant strategy, so if we denote 1 = u 1 then 1 > . We need to show u is better than for t< . 92 Under strategy ; at time t, E h e r( t) i = E h e r( 1 )r( 1 t) i = E h e r( 1 ) e r( 1 t) i = E h e r( 1 t) E 1 e r( 1 ) i = x y 1 Y 1 y 1 y < x y 1 Y y 1 y = x y : The inequality holds because 1 > and x y 1 > 1. So we proved strategy u is better than . In the similar way we can show that any strategy di¤erent from u is not optimal for the rm when X s > y. Since y is arbitrary number greater than x B , the rm should choose the investment strategy u for all X x B . In the proof of the proposition, we consider an alternative strategy which is dif- ferent from u only on a time interval with stopping time endpoint. Although it is not homogenous, we dont need homogenous strategy. It is because if we let one more time interval with the same asset value as that in the rst time interval, thenE e r( t) will be even lower. Thus the homogenous strategy constructed by this way will be less prefer- able. So any other strategy in A di¤erent from u will be discarded. The proposition veries that the rm will prefer to strategy which makes asset value go down with most probability. When the available investment opportunities form a compact set, we can nd the optimal strategy in the set. Because of the constant strategy, it is convenient to analyze the default probability. Even when the strategy set A is more complicated than 93 ours, the proposition can provide a good benchmark of analysis. In practice, the available strategies in consideration are limited in many situations. Since the setA wont change, then x B can be obtained once we know the parameters of all projects. By minimizing (u) (see (2.26)), we will nd that (u) always reaches minimum at a project, that is, the rm invests only in one project. However if the admissible set A changes with time, we should calculate (u) sequentially so that the strategy may change accordingly. Generally,itisunclearwhatX is,sinceitmaybeavariate. Wecanviewtheproblem fromanotheranglebyassumingthattheassetvaluerisesfromalowlevel. Thedenation of the optimal stopping time is = infft> 0jJ (X) =Xg given X 0 in a solvent state. It is equal to dene by X B = supfx> 0jJ x (x) = 1g. We have had the knowledge that the asset value at default is a constant. x B is determined by the coupon rate c and the parameters of all the available projects. WhentheinitialassetvalueX 0 > 0isgiven,weknowcouponratecisstrictlypositive. Nowwetakethemasgiven,wedeneanewdenitionofbankruptcy-triggeringassetlevel X , and corresponding stopping time. The bankruptcy time is = minft> 0jJ x (X t ) = 1g; And the bankruptcy-triggering asset value is x B = supfx> 0jJ x (x) = 1g: (2.34) 94 If fx> 0jJ x (x) = 1g is not an empty set, we can see x B is a constant. Other than the propositions above, here lets study the problem from other angle. We have the following proposition. Proposition 19 In our model, if X 0 > 0, there exists a positive coupon rate c, and a constant bankruptcy-triggering asset level x B dened by (2.34). Proof. SinceX t =X 0 e R t 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs , andV (x) =xJ (x)> 0, V x (x)> 0 for all x X 0 , it is natural to assume that x B < X 0 . Here we just consider the asset value x is low enough, and then we examine the path Y t = xe R t 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs . Dene default time as = infft> 0jY t =x B g; and dene another stopping time 0 as 0 = infft> 0jY t =X 0 g; whichisthersttimethattheassetvaluereturnstotheinitialpoint. Itisobvious > 0 . We now determine the range of coupon value. Since J (X 0 )<X 0 , inf ;u E Z 0 e rs c(1)ds+e r X 0 e R 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs <X 0 ) inf ;u E 1e r u c(1) r +e r X 0 M u <X 0 95 where M u t =e R t 0 (bs(u) 1 2 2 s (u))ds+s(u)dWs . If = +1, left hand side of inequality above is c(1) r , so we can take any c< rX 0 1 . If < +1, we can nd a constant c> 0, s.t. c< rX 0 (1inf ;u E[e r M u ]) 1inf ;u Ee r : It is necessary to verify that X 0(1inf;uE[e r M u ]) 1inf;uEe r > 0. Because 0 < < +1, 1 inf t;u E[e r M u ]> 0 and 1inf t;u Ee r > 0. So such a c exists. Now for x is small, we want a x such that J (x)x: Denote Z t =e rt M t . Then Z t is a supermartingale and Z 0 = 1. J (x) = min ;u E 1e r u c(1) r +xZ u : It su¢ ces to nd x> 0 such that E 1e r c(1) r +xZ u x;8u;: But < 0 . Since P (0< 0 < +1) = 1; P (0< < +1) = 1. Thus x< c(1)(1Ee r ) r(1EZ u ) : Since Ee r < 1, and EZ u < EZ 0 = 1, the right hand side of the inequality above is strictly positive for 8u;. Let k = inf ;u c(1)(1Ee r ) r(1EZ u ) . So for all x < k, we have 96 J (x) x. Since in our model J (x) x, so J (x) = x for all x < k. So we know the denition x B = supfxjJ (x) =xg is well-dened, which is a constant. In the proof of the proposition, we just search the default-triggering asset level from down to top. We assume that if the asset value is at low level, it is better to announce bankruptcy immediately. We examine the behavior of rm by assuming the rms asset value goes up, until it reaches a point at which the owner is willing to run the rm. Althoughthermafterdefaultactuallydoesntexist, theimaginationisstillvalid. Along this imagination, we can nd a highest point in asset value where the rm is on verge of default, and it is a constant by denition (2.34). The denition is equivalent to x B = supfxjJ (x) =xg, since J (x) =x for allxx B . The analysis avoids disadvantage of the usual thought which is from top to down. The usual way cant tell us much information about x B , because it is dependent on the optimal strategy and optimal stopping time. Although in this denition J (x) depends on strategy and stopping time (so does x B ), we have successfully circumvent the searching optimal ones. Moreover as mentioned above, fromthisdenition,x B isaconstant. Anintuitionaboutthisapproachisthatforx<x B , no strategy is good, so we dont need to consider the choice of strategies. In the proof we just need to know the rangeA of strategies and the non-explosion condition (see (2.2)). By understanding the problem in this way, it becomes comparatively simpler. Once the range of strategies is given, we can nd the constant x B . For our model, because the Markovian properties, we dont need to consider all path of the asset value, we just examine the neighborhood of x B , although it isnt determined yet. 97 Through this analysis we are able to nd x B by starting with a constant strategy. As we have discussed as above, if a rm is in solvent state and furthermore its asset value X = 2 D, then it is health enough. There is no much worrying about bankruptcy. We consider the cases in which X 2 D. For a rm whose asset value satises X 2 D and X > x B , since the rm has to declare bankrupt immediately if it chooses a improper strategy. It is because there are investment portfoliosu such thatx u B X. The manager, however, can choose another strategy u 0 such that x u 0 B < X. Thus the rm doesnt need to default its bond, since J (X) < X under strategy u 0 . This implies that the rm can still keep solvent, and no bankrupt happens. So when X 2 D, the rm can survive by adjusting its strategy as long as X >x B . Once the rms asset value drops to x B , there is no available strategy to make the rm survive. We can call the setD as the collection of "unhealthy states", since the rm is in danger if its asset value is inD. The rm needs to take care of its strategy intensively as the asset value process X t evolves. From the results and analysis above, the rms endogenous bankruptcy-triggering asset level is that minimizing x u B for u 2 A. It is equivalent to minimize (u) when other parameters are given (see (2.31)). By substituting b = P N n=1 u n ( n ) and 2 = P N n=1 (u n n ) 2 into , we have (u) = P N n=1 u n ( n ) P N n=1 (u n n ) 2 1 2 (2.35) + 2 4 P N n=1 u n ( n ) P N n=1 (u n n ) 2 1 2 ! 2 + 2r P N n=1 (u n n ) 2 3 5 1=2 ; whereu2A. Bypluggingallu2Aintoexpression(2.35),wecanndtherangeof ,and thusrangeofx u B . Forthebankruptcy-triggeringassetvaluelevelx B , itisnothardtond 98 (u ), where u minimizes (u). The upper boundary x of x u B can be attained similarly. x is a important point where the manager monitors very often. As we have discussed above, if asset value X > x, the rm is healthy. But once X x, the rms manager should pay enough attention to the investment portfolio. It is reasonable to think that managers behavior is di¤erent for X x and X > x. However as we have known, the strategy is surprisingly the same for X > x and X < x. The purpose of emphasizing the point is to remind one the cases in which the objective function may not be the same as that here in our model, the strategy is usually not a constant, then x should play more important role than that here. 2.5 Application of the Model In last section we have proved there exists a constant bankruptcy-triggering asset level x B . SoifN projectsareavailable, andweknowtheirparameters, weareabletocalculate x B by formulae (2.31) and (2.35). In this section, we will start with two-project models asexamplestoinvestigatehowthesecuritiesaredi¤erentfromthoseinoneprojectmodel of Leland (1994). 2.5.1 The Investment Decision of Two-Project Models In this section, we consider simple cases in which there are only two projects. Their asset value processes satisfy dX i =X i i dt+X i i dW i t ;i = 1;2: 99 This is a special case of (2.1) for N = 2. But here the assumptions about and are di¤erent. We only require r < 1 < 2 ; and 2 1 < 2 2 ; since we have only two projects, we cant form an investment portfolio dominating any of the two projects. We also require that Brownian motion W 1 is independent of W 2 . Any other assumptions are the same as those of the general setup. Now the investment strategy, at time s, is u s = (1u;u);and 0 u 1. u is the proportion of capital invested in riskier project 7 . Thus the state variable satises the stochastic di¤erential equation dX = X((1u)b 1 +ub 2 )dt+X q (1u) 2 2 1 +u 2 2 2 d f W t = X(b 1 +(b 2 b 1 )u)dt+X q u 2 2 1 + 2 2 2 2 1 u+ 2 1 d f W t ; where f W is another Brownian motion as before, and b i = i ; for i = 1;2. By plugging in b(u) = b 1 + (b 2 b 1 )u; and 2 (u) = u 2 2 1 + 2 2 2 2 1 u + 2 1 into equation (2.35), Y has the form, 7 When we say a project is riskier, we mean the volatility of the project is larger, without considering the instantanous return rate. 100 (u) = b 1 +(b 2 b 1 )u u 2 2 1 + 2 2 2 2 1 u+ 2 1 1 2 (2.36) + 2 4 b 1 +(b 2 b 1 )u u 2 2 1 + 2 2 2 2 1 u+ 2 1 1 2 ! 2 + 2r u 2 2 1 + 2 2 2 2 1 u+ 2 1 # 1=2 : On the interval [0;1] ofu, (u) is increasing rstly and then decreasing asu increases. So the possible investment strategy at bankruptcy is to either invest all capital into project one or invest all capital into project two, which depends on condition (0)< (1) or not. Forcomparison, weconsider (0)> (1), i.e., thermwillchoosetoinvestallcapital in project two whenever it is available. So if only project one exists, given the current state (t;X), the equity value is V 1 (X) = X c 1 (1) r + X x B 1 1 c 1 (1) r x B 1 (2.37) = X c 1 (1) r 1c 1 1 X 1 1 ; where 1 = 1 (1) r( 1 +1) 1 1 1 +1 ; 101 where 1 isthevalueforprojectone;c 1 isoptimalcouponrate;andx B 1 isthebankruptcy- triggering asset level for project one. The second line is obtained by plugging in x B 1 = 1 1 +1 c 1 (1) r . The rm value is F 1 (X) =X + c r 1 c X 0 1 1 where 1 = h 1+ 1 +(1) 1 i 1 We can nd c 1 which maximizes F 1 (X 0 ) given the initial asset value X 0 . c 1 =X 0 [(1+ 1 ) 1 ] 1= 1 : The debt value is D =F V, that is, D 1 (X) = c 1 r 1 c 1 X 1 1 where 1 = (1+ 1 (1)(1) 1 ) 1 : And if project two is included such that 2 < 1 , we can nd the rm value, equity value, the optimal coupon rate, and debt value respectively for two-project model in the same way. The Figure 1 gives examples of two possibilities. The solid line shows that 2 < 1 , in which case the rm chooses the project two. The dashed line indicates 1 < 2 , in which case the rm prefers to project one. Notice that we only adjust the parameters of 102 volatilities of the two projects. From solid line to dashed line, we change 1 from 0:15 to 0:18 and 2 from 0:22 to 0:21. Keeping the drift terms unchanged, the rm prefer to riskier project other than safer one. This is consistent with our objective function. Figure 2.1: Examples of investing in project one or two. r = 0:06;b 1 = 0:001;b 2 = 0:05: Solid line: 1 = 0:15; 2 = 0:22; choosing project two; dashed line: 1 = 0:18; 2 = 0:21; choosing project one: Supposeatbeginningthereisonlyoneprojecttoinvest. Thenduringtheoperationthe rm may nd a new investment project. The rm should decide to switch its investment or not. Notice that the decision is not based on only drift term or volatility term, but on the total e¤ect of them, which is indicated by .In Figure 2 2 decreases as b 2 decreases and 2 increases. Given r = 0:06;b 1 = 0:001; 1 = 0:15, thus 1 = 1:8983: Whenever 2 < 1 , whichistheleft-front cornerin the Figure 2, the rm should shifttonewproject if it occurs. If there are more than two projects, we can compare their Y in pairs. For example, if 2 < 1 , then we discard 1 and compare 2 with 3 . Until we nd the project with 103 Figure 2.2: 2 changes with 2 andb 2 . r = 0:06;b 1 = 0:001; 1 = 0:15. Thus 1 = 1:8983: minimum in all available projects. The rm will put all its capital in the project with minimum . But in many cases, the occurrence of new project is a random event, and its arrival and parameters are unpredictable. We need to estimate the probability of occurrence of a better project in a certain future time interval. This will be analyzed in the future work. 2.5.2 The E¤ects of Optional Project Now lets investigate how it a¤ects the rms optimal capital structure, debt value, equity value change, when we have one more investment opportunity. If the new project is not better than old one, the rm will keep the old project, so that nothing is changed. We just examine the cases in which the new project has smaller . 104 Assume r = 0:06, the parameters of old project are b 1 = 0:001; 1 = 0:15 as in the last subsection. In Figure 3 we can see the curve of b 2 for di¤erent 2 to make 1 = 2 . The region above the curve is the cases in which the rm still keeps the old project since 1 < 2 . If the new project is located in the region below the curve, the rm should choose the new project. We consider the cases of new project located in the region below the curve.Now consider the rm start with X 0 = 100 in project one. The optimal coupon Figure 2.3: The relation between b 2 and 2 given project one. is c 1 = 7:06, debt value D 0 = 95:81, rm value F 0 = 130:83, and equity value E 0 = 35:02. If there is no new project available, we can plot E 1 (X); D 1 (X) and F 1 (X) as in Figure 4. All of them are increasing function of asset value, but equity value is concave, and the other two are convex. Notice that in the case the bankruptcy-triggering asset level x B 1 = 46:25, which is obtained once we have found the optimal coupon rate c. 105 Figure 2.4: Without new project, the rm value, equity value and debt value change with asset value X during the rms operation. r = 0:06; b 1 = 0:001; 1 = 0:15: But due to the occurrence of new investment opportunity, the manager has chance to switch the investment. In our model, we dont take into account the renegotiation of debt when changing the investment project. It means that the coupon rate is still c 1 even after implementing the new project. And also when we determine the coupon rate c 1 , we dont make any forecast of any new project. We will take into consideration this situation including arrival time and parameters later. Figure 3 tells us when the rm should switch its investment. For simplicity in illustration, we assume b 2 = 0:005 and allow 2 to be alterable. So we have x B = ( 2 ) ( 2 )+1 c 1 (1) r ; whichisafunctionofvolatilityofnewproject. Noticethatthecouponrateintheformula of x B is c 1 but not the implied optimal coupon rate c 2 of the new project. If there is renegotiation between the debt-holders and the rm-owner, the proper coupon rate may 106 be somenumberbetweenc 1 andc 2 . Want to accept new project, it needs 2 > 0:16, since then ( 2 )< 1 . The debt value will be like Figure 5. Figure 2.5: The debt value as a function of asset value X for di¤erent 2 . r = 0:06; b 1 = 0:001; 1 = 0:15;b 2 = 0:005: Figure 5 also illustrates that at low level of asset value X, the debt value will increase as the volatility 2 , but when asset value is high, the debt value will go down as 2 rises. The explanation to this phenomenon is fallowing. When asset value is so low that it is close to bankruptcy-triggering level of old project, higher volatility 2 of new project implies the current asset level is farther away to the new bankruptcy-triggering level. So the debt is more valuable, because it becomes safer. But if the current asset value is high, there isnt too much worry about the bankruptcy, if the rm chooses new project with high volatility, the risk will be higher, so that the debt will be less preferable. This is basically related to the short-term and long-term e¤ect of the project risk. 107 We are interested in the di¤erence in bond value between the new project and the old one. The margin is D 2 (X)D 1 (X) = c 1 r c 1 X Y 2 2 c 1 X 1 1 : As show in the left graph in Figure 6, we can see that V 2 (X) is greater than V 1 (X) Figure 2.6: The left graph represents the equity value di¤erence in two project; the right graphrepresentsthedi¤erencebetweennewdebtvalueandtheolddebtvaluefordi¤erent 2 and asset value X given current state r = 0:06; b 1 = 0:001; 1 = 0:15;b 2 = 0:005. for all 2 > 0:16. So the rm owner always prefers to this project switch. The right graph in Figure 6 shows how debt value di¤erence moves as asset value and volatility 2 change. From the right graph we can see when 2 = 0:16, the debt value di¤erence will not change with asset value X, that is, the debt value in old project is the same as that in new project. The reason is that 1 = 2 when 2 = 0:16, so that the two project have the same e¤ect on security evaluation. The right graph in Figure 6 also reveals opposite outcomes when the current asset value is at high and low level. The explanation to this 108 observation is similar to that of debt value of new project. When asset value is on the verge of nancial distress, the debt value will increase as volatility 2 rises. Because given the drift term b 2 = 0:005, from Figure 3 higher volatility 2 implies 2 is less than and further away from 1 . So the rm succeeds in getting away from the verge of bankruptcy, and thus as a consequence the credit quality of the rms debt can be improved instantly. This results in the debt value increasing for the rm with low asset value. From this we can also see that it is always benecial to both debt-holder and rm-owner to switch investment project when the rm is in nancial distress. It is easy for them to make a new agreement on debt renegotiation. But when the asset value is high, the conict between the debt-holder and the rm-owner occurs. A higher 2 implies that the new project is riskier and the rm can shift the risk to debt-holder by taking the new project. In Figure 6 it shows that the equity value will increase and debt value will decrease when X gets higher. To make the switch in project succeed the rm-owner has to transfer part of his/her benets to the debt-holders to compensate the higher risk. 2.5.3 Debt Renegotiation In the last subsection, we have known that there is conict between the debt-holders and rm-owner if the rm changes its investment project, especially when the asset value is high. So in order to invest in another project, the two counterparties will make a new contract about the debt. Our model provides a framework to reach an agreement. Let (t;X) be the rms state when a better new project occurs. Figure 6 shows that both rm-owner and debt-holders will accept the project switch when X is low. But if X is high, the debt value willdecrease if the rm changes its investment project. We assume 109 that the debt-holderscompensation guarantees that the new debt value is the same as that without project switch, i.e., D 2 (X)+d =D 1 (X): where D 2 (X) = c 1 r 1 c 1 X 2 2 , and d represents the switch compensation to the debt-holders. We plot V 2 and D 1 in Figure 7 given r = 0:06; b 1 = 0:001; 1 = 0:15; and b 2 = 0:005. From Figure 6 we know that D 2 D 1 is positive if X < 80. The negative part of D 2 D 1 is located in the region X > 80 and 2 > 0:2, which is what we are interested in. From the Figure 7 we see that by switching the investment to project two, both equity value and debt-value will be strictly positive and increasing in X. And also the rm-owner will not shift the risk to debt-holders. So the debt is protected.Figure 8 Figure 2.7: The left graph is the equity value when the rm invest in project two, and the right graph plots the debt value for di¤erent asset value and 2 , given r = 0:06; b 1 = 0:001; 1 = 0:15; and b 2 = 0:005. 110 is another angle to look at the compensation from the rm-owner to the debt-holders. In Figure 8 all value is negative since we plot (F 2 (X)D 1 (X))V 2 (X) = F 2 (X)V 2 (X)D 1 (X) = F 2 (X)V 2 (X)D 1 (X) = D 2 (X)D 1 (X); which is (d). Notice that the debt-holders require compensation only whenD 2 <D 1 , so dispositive. Wecanseethatthecompensationishigherwhen 2 getshigher. Thisimplies Figure 2.8: The graph shows the value (F 2 (X)D 1 (X))V 2 (X), where V 2 (X) is the amount of equity value without compensation to debt-holders, and (F 2 (X)D 1 (X) is the actual amount of equity value, given r = 0:06; b 1 = 0:001; 1 = 0:15; and b 2 = 0:005. that when the new project is riskier, the rm-owner should pay more compensation to debt-holder. We can also see in Figure 8 that the compensation is more when asset value is higher. The di¤erence is more obvious especially when 2 is large (right part of Figure 8). It shows that in order to switch their investment the rm-owner needs to pay more to 111 get the debt-holderspermission. Under this agreement mechanism, the debt-holders will not prevent the rm from taking riskier project since their debts are respected as when the rm is started. 2.6 Conclusion and Future Research We develop a model that extends the model in Leland (1994) by allowing multiple in- vestment opportunities. The rm in Leland (1994) just needs to optimally decide when to stop running the rm. Since allowing for multiple investment projects, our model has the characteristics of both optimal stopping and optimal control problem. Under the assumption of the returns of the projects following geometric Brownian motion, and the principleofequitymaximization,wendthat,asinLeland(1994),aconstantendogenous bankruptcy-triggering asset level exists. Once the asset value of the rm hits the bankruptcy-triggering asset level, the rm will announce default. The optimal strategy turns out to be a single project with the lowest default boundary. this model is useful to decide investment opportunity switch in practice. If a new project is available and is better than current project, the rm may want to take it. When the asset value is high, the rm must compensate debt-holders for investment switch. But both the debt-holders and the rm-owner will better o¤to take the new project when the asset value is low. We are interested in extending our model to consider possibility of the occurence of a newprojectinthefuture. Themostimportantissuesincludeestimatingtheprobabilityof arrivalofapreferablenewprojectinatimeinterval, andthedistributionofparametersof the stochastic return process. Since the debt-holders may take into account the possible 112 new project when they write a contract, it should be more reasonable that the debt evaluation reects the possibility of the future project. Another extension of our model is to investigate the optimal strategy when the ma- turity of debt is nite. If the debt structure is still homogeneous the formulae of security values may be explicit. However, it often involves debt issuance or renegotiation along witheachstrategyadjustment,sothatthehomogeneousdebtstructureisnotareasonable assumption. If so, more technical methods are required to reach interested results. Moreover frictions exist universally. Therefore, the rm can only adjust its strategy discretely. There should have break-even points in asset value at which the rm modies itsinvestmentportfolio. Otherwisethermwillkeeptheoldstrategy, becausethebenet of strategy adjustment is cancelled out by the friction cost. As we mentioned in context it often causes cost to a rm for equity issuance, so that the rm may not collect enough cash to cover the debt coupon. Some authors take the cash storage as the fundamental. When the cash storage of a rmcannotcoverthecouponandthematureprinciple, thermdeclaresdefault. Models with cash storage should take into consideration the decision of how much cash a rm should get instantaneously not only to keep the rm alive but also to get return as high as possible. 113 Bibliography [1] Artzner, P. and Delbaen, F. (1992), Credit Risk and Prepayment Option, ASTIN Bulletin 22: 81-96. [2] Bélanger, A., Shreve, S. and Wong, D. (2004), A General Framework for Pricing Credit Risk, Mathematical Finance, Vol. 14, No. 3, 317-350. [3] Tomasz Bielecki and Marek Rutkowski, Credit Risk Modelling: Intensity Based Ap- proach. [4] Black, F. and M. Scholes. 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-659. [5] Black, Fischer and Cox, John C., Valuing Corporate Securities: Some E¤ects of Bond Indenture Provisions, The Journal of Finance, Vol. 31, No. 2, (May, 1976), pp. 351-367. [6] Borkar,Vivek S., Controlled di¤usion processes, Probability Surveys, Vol. 2 (2005) 213244. [7] Boyle, Phelim and Tian, Weidong, Portfolio Management with Constraints, Mathe- matical Finance, Vol. 17, No. 3 (July 2007), 319?43. [8] DeMarzo, Peter M., and Sannikov, Yully, Optimal Security Design and Dynamic Capital Structure in a Continuous-Time Agency Model, The Journal of Finance, VOL. LXI, NO. 6 DECEMBER 2006. [9] Darrell Du¢ e and Kenneth J. Singleton, An Econometric Model of the Term Struc- ture of Interest-Rate Swap Yields, The Journal of Finance, Volume 52, Issue 4 (Sep., 1997), 1287-1321. [10] Darrell Du¢ e and Kenneth J. Singleton, Modeling Term Structures of Defaultable Bonds, The Review of Financial Studies, Volume 12, Issue 4 (1999), 687-720. [11] Darrell Du¢ e, Mark Schroder and Costis Skiadas, Recursive Valuation of Default- able Securities and the Timing of Resolution of Uncertainty, The Annals of Applied Probability 1996, Vol. 6, No. 4, 1075-1090. 114 [12] Eom, YoungHo; Helwege, JeanandHuang, Jing-Zhi, Structural Models of Corporate Bond Pricing: An Empirical Analysis,TheReviewofFinancialStudiesVol.17,No.2, pp. 499?44, 2004. [13] Friedman, Avner. Optimal Stopping Problems in Stochastic Control, SIAM Review, Vol. 21, No. 1. (Jan., 1979), pp. 71-80. [14] Geske, R. (1977). The Valuation of Corporate Liabilities as Compond Options, The Journal of Financial and Quantitative Analysis, pp. 541-552. [15] Goldstein, Robert; Nengjiu Ju, and Hayne Leland; An EBIT-Based Model of Dy- namic Capital Structure; Journal of Business, 2001, vol. 74, no. 4. [16] Harrison, J. M., 1990, Brownian Motion and Stochastic Flow Systems, Krieger Pub- lishing Company, Fla. [17] Helen Haworth, Christoph Reisinger and William Shaw, Modelling Bonds & Credit Default Swaps using a Structural Model with Contagion, 2006. [18] Helen Haworth and Christoph Reisinger, Modeling Basket Credit Default Swaps with Default Contagion, 2006. [19] Hilberink, Bianca and Rogers, L.C.G., Optimal capital structure and endogenous default, Finance Stochast. 6, 237-63 (2002). [20] Ho, T. and Singer, R. (1982), Bond Indenture Provisions and the Risk of Corporate Debt, Journal of Financial Economics (10): 375-406. [21] Jarrow, R., Lando, D. and Turnbull, S. (1997), A Markov Model for the Term Struc- ture of Credit Risk Spreads, The Review of Financial Studies 10(2): 481-523. [22] Jarrow, R. and Turnbull, S. (1995), Pricing Options on Financial Securities Subject to Default Risk, Journal of Finance 50: 53-86. [23] Jarrow, R. and Turnbull, S. (1992). Credit Risk: Drawing the Analogy.Risk Mag- azine 5(9). [24] Robert A. Jarrow and Stuart M. Turnbull, Pricing Derivatives on Financial Securi- ties Subject to Credit Risk, The Journal of Finance, Vol. 50, No. 1. (Mar., 1995), pp. 53-85. [25] Robert A. Jarrow and Fan Yu, Counterparty Risk and the Pricing of Defaultable Securities, THE JOURNAL OF FINANCE VOL. LVI, NO. 5 OCT. 2001. [26] Karatzas, Ioannis and Wang, Hui. 2000, Utility Maximization with Discretionary Stopping, SIAM J. CONTROL OPTIM., Vol. 39, No. 1, pp. 306329. 115 [27] Kim, J., Ramaswamy, K., and Sundaresan, S. (1992), The Valuation of Corporate Fixed Income Securities. [28] Krylov, N.V., Controlled Di¤usion Processes, Springer-Verlag, Berlin-New York, 1980. [29] Lando, David On Cox Processes and Credit Risky Securities, Review of Derivatives Research, 2, 99120 (1998). [30] Leland, Hayne E, Corporate Debt Value, Bond Covenants, and Optimal Capital Structure, The Journal of Finance, Vol.49, No.4. (Sep.,1994), pp.1213-1252. [31] Leland, Hayne E and Klaus Bjerre Toft, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, The Journal of Finance, Vol. 51, No. 3, 1996. (Jul., 1996), pp. 987-1019. [32] Longsta¤, F and Schwartz, E. (1995), A Simple Approach to Valuing Risky Fixed and Floating Rate Debt, The Journal of Finance 50(3): 789-819. [33] Madan, D. and Unal, H. (1994), Pricing the Risks of Default, Review of Derivatives Research 2(2/3): 121-160. [34] Merton,RobertC.,Lifetime Portfolio Selection under Uncertainty: The Continuous- Time Case, The Review of Economics and Statistics, Vol. 51, No. 3. (Aug., 1969), pp. 247-257. [35] Merton, Robert C., Theory of Rational Option Pricing, The Bell Journal of Eco- nomics and Management Science, Vol. 4, No. 1. (Spring, 1973), pp. 141-183. [36] Merton,RobertC., On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, The Journal of Finance, Vol. 29, No. 2, 1974. [37] Mihailovskaya, On the Optimal Stopping of a Controlled Di¤usion Process, Theory of Probability and its Applications, Volume XXV, Number 2, 1980. [38] Markowitz, Harry (1952), Portfolio Selection, The Journal of Finance, Vol. 1, NO. 1, pp 77-91. [39] Øksendal, Bernt, Stochastic Di¤erential Equations: An Introduction with Applica- tions, six edition, Springer, 2003. [40] Roy Radner and Larry Shepp, Risk vs. Prot Potential: A Model for Corporate Strategy, Journal of Economic Dynamics and Control 20 (1996). [41] Schönbucher, P. (2000), Credit Risk Modelling and Credit Derivatives, PhD Thesis. [42] Titman, Sheridan and Tsyplakov, Sergey, A Dynamic Model of Optimal Capital Structure, Review of Finance, 2007. 116 [43] Uhrig-Homburg, Marliese, Cash-ow shortage as an endogenous bankruptcy reason, Journal of Banking & Finance 29 (2005) 15091534. [44] Young, Jiongmin and Zhou, Xun Yu, Stochastic Controls: Hamiltonian Systems and HJB Equations (Stochastic Modelling and Applied Probability), Springer, 1999. [45] Yu,F.(2007), Correlated Defaults in Intensity-Based Models,MathematicalFinance, Vol. 17, No. 2 (April 2007), 155-73. [46] Zhou, C. (2001), The Term Structure of Credit Spreads with Jump Risk, Journal of Banking and Finance 25, 2015-2040. [47] Chunsheng Zhou, An Analysis of Default Correlations and Multiple Defaults, The Review ofFinancial Studies, Vol. 14, No.2. (Summer, 2001), pp. 555-576. 117

Abstract (if available)

Abstract This is an investigation of how a firm allocates capital into multiple investment opportunities. This framework analyzes the firm-owners' behavior as one of the factors influencing the credit quality of the firm. Based on the equity value maximization, criterion is constructed for the firm's investment strategy decisions. The combination of investment portfolio and credit risk is an optimal stopping problem of a controlled diffusion process. This model indicates that the firm's objective is equivalent to minimize the discounted expected cost payout due to the debt issuance. There exists a constant bankruptcy-triggering asset level for the optimal stopping control problem. Although multiple projects exist, the firm will choose only a specific project, which has the lowest bankruptcy-triggering boundary. An investigation of the changes in security values when a firm switches its investment project will show that it is preferable for both the debt-holders and the firm-owners to take on a new investment opportunity if the asset value is low. For the case in which the investment switch will damage the debt-holders' benefits, this model provides a mechanism of debt-renegotiation to achieve the switch, which increases in both equity and debt values.

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University of Southern California Dissertations and Theses

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Optimal and exact control of evolution equations

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Some topics on continuous time principal-agent problem

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Creator Zhou, Yuegang (author)

Core Title Credit risk of a leveraged firm in a controlled optimal stopping framework

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 06/09/2008

Defense Date 04/25/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag credit risk,endogenous bankruptcy,OAI-PMH Harvest,optimal stopping time,optimal strategy

Language English

Advisor Zhang, Jianfeng (committee chair), Deng, Yongheng (committee member), Ma, Jin (committee member)

Creator Email zyuegang@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1257

Unique identifier UC1205624

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Document Type Dissertation

Rights Zhou, Yuegang

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu