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The use of absorbing boundaries in the analysis of bankruptcy Hildebrand, Paul 1998

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The use of Absorbing Boundaries in the Analysis of Bankruptcy by Paul Hildebrand B.Sc. University of British Columbia, 1977 L.L.B., University of British Columbia, 1980 M.A., University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1998 © Paul Hildebrand, 1998 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. The Department of Economics The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date : October 14, 1998 11 Abstract An explicit solution is given for the value of a risk neutral firm with stochastic revenue facing the possibility of bankruptcy. The analysis is conducted in continuous time. Uncertainty is modeled using an Ito process and bankruptcy is modeled as an absorbing boundary. The analysis yields an ordinary differential equation with a closed form solution. The value function is used to calculate the firm's demand for high interest rate loans, showing a positive demand at interest rates which appear intuitively to be excessive. A value function is also derived for a risk neutral lender advancing funds to the firm. The borrowing and lending value functions are then used to examine various aspects of lender-borrower transactions under different bargaining structures. In a competitive lending market, the model shows that credit rationing occurs inevitably. In a monopoly lending market, the lender sets interest rates and maximum loan levels which reduce the borrower to zero profit. When a second borrower is introduced, the lender must allocate limited funds between two borrowers. A lender is shown to squeeze the smaller "riskier" borrower out of the market when the lender's overall credit constraint is tight. Under each bargaining structure, the model is also used to examine changes in the respective "salvage" recoveries of the lender and borrower on bankruptcy. Accepted: I l l TABLE OF CONTENTS Abstr a c t i i i L i s t of Tables i v L i s t of Figures v I. INTRODUCTION 1 I I . THE VALUATION PROBLEM 8 1. Statement of the Problem 8 2. D e r i v a t i o n of the S o l u t i o n 12 3. The Marginal Value of C a p i t a l 24 4. L i m i t i n g Values 26 5. Related Functions 29 APPENDIX 4 0 I I I . THE DEMAND FOR HIGH INTEREST RATE LOANS 71 IV. PROJECT FINANCING - VALUE FUNCTIONS 7 9 1. Value Function of the Borrower / Pr o j e c t Owner 80 2. Value Function of the Lender 84 3. The f u n c t i o n M(B,R) 94 V. COMPETITION AMONG LENDERS 104 1. Statement of the Problem 105 2. The Lender's Constraint 107 3. The Borrower's Rate of Return 114 4. C r e d i t R a t i o n i n g 116 5. S e c u r i t y L i m i t a t i o n s 127 6. Bankruptcy L e g i s l a t i o n 139 VI. MONOPOLY IN THE LENDING MARKET - SINGLE BORROWER 147 1. A n a l y t i c a l I n t e r p r e t a t i o n - C r e d i t Rationing 149 2. C r e d i t R a t i o n i n g - A Numerical Example. 156 3. Comparison - Maximization of the Lender's Value Function 162 I V 4. Change i n S e c u r i t y L e g i s l a t i o n 167 5. Bankruptcy L e g i s l a t i o n 169 V I I . SINGLE LENDER, MULTIPLE BORROWERS 174 1. The Value Functions of the Borrower and Lender 176 2. The Existence of C r e d i t R a t i o n i n g 188 3. The Lender's Maximization Problem. 191 4. A change i n the lending c o n s t r a i n t 194 5. A change i n the salvage r a t i o . 196 V I I I . CONCLUSION 201 REFERENCES 2 04 ••V LIST OF TABLES Table 1 - Borrower's values: w=l 75 Table 2 - Borrower's values: w=3 76 Table 3 - Borrower's values: w=5 76 Table 4 - Borrower's values: w=10 77 Table 5 - Values of MR, MBR 103 Table 6 - Borrower's Returns 158 Table 7 - Lender's Returns 159 Table 8 - Lenders' Values 163 Table 9 - Maximizing Debt Levels f o r the Lender's Value Function 164 Table 10 - Borrower's values f o r w = 50 179 Table 11 - Borrower's values f o r w = 100 180 Table 12 - Lender's values f o r w = 50 181 Table 13 - Lender's values f o r w = 100 182 Table 14 - Debt l e v e l s f o r d i f f e r e n t Lender's c o n s t r a i n t s 195 Table 15 - Lender's values (and corresponding i n t e r e s t rates) f o r " l a r g e " lender (w=100) at d i f f e r e n t values of Sa 199 Table 16 - Lender's values (and corresponding i n t e r e s t rates) f o r "small" lender (w=50) at d i f f e r e n t values of Sa 200 v i LIST OF FIGURES Figure 1 - Value Function 20 Figure 2 - Graph of g(x) 31 Figure 3 - Graph of h(x) 33 Figure 4 - Graph of J(x) 35 Figure 5 - Graph of L(B,R)=0 113 Figure 6 - Borrowing w i t h C r e d i t R a t i o n i n g 120 Figure 7 - C r e d i t R a t i o n i n g Without Borrowing 121 Figure 8 - No Borrowing, No C r e d i t R a t i o n i n g 122 Figure 9 - C r e d i t R a t i o n i n g 152 Figure 10 - No C r e d i t R a t i o n i n g 154 Figure 11 - Maximizing Debt Levels 160 Figure 12 - Maximizing Debt Levels 166 Figure 13 - Increase i n Sa 169 Figure 14 - Borrower's values 183 Figure 15 - Lender's values: w=50 184 Figure 16 - Lender's values: w=100 184 X CHAPTER I - INTRODUCTION This t h e s i s uses the methods of continuous time finance to analyse the impact of p o t e n t i a l bankruptcy on corporate d e c i s i o n making, and on t r a n s a c t i o n s between borrowers and lenders. Bankruptcy i s analysed from the perspect i v e of a r i s k n e u t r a l f i r m , and a l s o from the per s p e c t i v e of a r i s k n e u t r a l lender advancing funds to the f i r m . D i f f e r e n t bargaining s t r u c t u r e s are imposed on the borrower and lender to determine r e s u l t i n g debt l e v e l s and i n t e r e s t r a t e s . I t o processes are used to model u n c e r t a i n t y i n the firm's revenue. Ordinary d i f f e r e n t i a l equations are de r i v e d to c a l c u l a t e the value of the f i r m and of debt s e c u r i t i e s which i t i s s u e s . The assumptions of the a n a l y s i s are d e l i b e r a t e l y kept as simple as p o s s i b l e . For example, a l l agents use the same exogenously imposed discount r a t e , which i s equal to the r a t e of re t u r n on p h y s i c a l c a p i t a l . These assumptions r e s u l t i n some l o s s of g e n e r a l i t y . However, the o f f s e t t i n g b e n e f i t i s that e x p l i c i t c l o sed form s o l u t i o n s are a v a i l a b l e f o r most of the v a l u a t i o n problems posed. The closed form s o l u t i o n s permit both q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s , and generate i n t u i t i o n about the d e c i s i o n making of borrowers and lenders. 2 The a n a l y s i s c a r r i e d out i n t h i s t h e s i s i s cl o s e i n s p i r i t to that of Leland (1994), who a l s o examines the value of a r i s k n e u t r a l f i r m f a c i n g p o t e n t i a l bankruptcy. Leland uses many s i m i l a r s i m p l i f y i n g assumptions, such as the exogenous discount rat e , and considers s i m i l a r consequential issues to some of those addressed here. A point of departure from Leland i s the s t o c h a s t i c process used to model the f i r m ' s wealth. Leland uses a s t o c h a s t i c process based on conventional geometric brownian motion, whose instantaneous standard d e v i a t i o n i s p r o p o r t i o n a l to wealth. This t h e s i s uses a s t o c h a s t i c process whose standard d e v i a t i o n i s independent of the wealth l e v e l . I t i s argued that t h i s s t o c h a s t i c process models more r e a l i s t i c a l l y the revenue of a f i n a n c i a l l y d i s t r e s s e d f i r m - there i s n e i t h e r t h e o r e t i c a l nor e m p i r i c a l support f o r the p o s i t i o n that the u n c e r t a i n t y i n f i r m revenue decreases as wealth decreases, or as the f i r m comes cl o s e to bankruptcy. As a r e s u l t of the d i f f e r e n t s t o c h a s t i c process used, the s o l u t i o n s derived i n t h i s t h e s i s d i f f e r s i g n i f i c a n t l y from Leland's. Chapter I I set s out the fi r m s ' v a l u a t i o n problem and derives i t s 3 clo s e d form s o l u t i o n . Chapter I I a l s o examines p r o p e r t i e s of the s o l u t i o n , which are used i n subsequent chapters, and compares the s o l u t i o n to those found i n other r e l a t e d papers, such as Leland (1994) and Milne and Robertson (1996) . Chapter I I I analyses a d i s t r e s s e d f i r m ' s demand f o r high i n t e r e s t r a t e loans. The f i r m can borrow a d d i t i o n a l c a p i t a l at an i n t e r e s t r a t e which exceeds the firm's own rat e of r e t u r n . Borrowing thus imposes a net i n t e r e s t cost on the f i r m . The o f f s e t t i n g b e n e f i t i s that borrowed c a p i t a l provides an a d d i t i o n a l cushion from the firm's bankruptcy th r e s h o l d . The clo s e d form s o l u t i o n derived i n Chapter I I i s used to c a l c u l a t e the f i r m ' s demand f o r loan funds at s p e c i f i c i n t e r e s t r a t e s . The r e s u l t s are set out i n t a b l e s which show a p o s i t i v e demand f o r loan funds at i n t e r e s t r a t e s which appear i n t u i t i v e l y to be excessive. Chapter IV r e f i n e s the v a l u a t i o n problem of Chapter I I , and a l s o i t s s o l u t i o n , to analyse a firm's optimal c a p i t a l s t r u c t u r e . The model now r e q u i r e s the f i r m to choose a combination of debt and eq u i t y to finance a p r o j e c t of f i x e d s i z e . Bankruptcy i s assumed to occur when the firm ' s wealth has dropped to the amount of i t s 4 debt, thus endogenizing the bankruptcy th r e s h o l d . The i n t e r e s t r a t e on debt i s now l e s s than the f i r m earns on c a p i t a l i t i n v e s t s . Under these assumptions, a higher debt-equity r a t i o has both advantages and disadvantages f o r the f i r m . On the one hand, more debt increases leverage. On the other hand, more debt increases the r i s k of bankruptcy. Chapter IV d e r i v e s the value f u n c t i o n which r e s u l t s from these assumptions. In a d d i t i o n , Chapter IV incorporates the p e r s p e c t i v e of the lender, who must value the discounted stream of i n t e r e s t payments which he expects to r e c e i v e from the borrower. The lender's v a l u a t i o n problem a l s o has a closed form s o l u t i o n , whose p r o p e r t i e s are examined. Chapter V uses the r e s u l t s of Chapter IV to analyse borrower -lender e q u i l i b r i u m i n a competitive lending market. Competition among lenders i s modelled by a zero p r o f i t c o n d i t i o n . Changes i n the loan amount and i n t e r e s t r a t e a f f e c t both the lender's r e t u r n and the r i s k of d e f a u l t , c r e a t i n g a schedule of loan amounts and i n t e r e s t r a t e s s a t i s f y i n g the zero p r o f i t c o n s t r a i n t . The s o l u t i o n to the maximization problem i s the optimal loan f o r the borrower, subject to the lender's c o n s t r a i n t . The a n a l y s i s of Chapter V leads to a s t r a i g h t f o r w a r d c o n d i t i o n 5 f o r c r e d i t r a t i o n i n g (defined as a c o n d i t i o n i n which the borrower i s unable to borrow a l l that he wishes at the p r e v a i l i n g i n t e r e s t r a t e ) , which i s shown to be i n e v i t a b l e i n a competitive len d i n g market. Chapter V a l s o attempts to analyse the e f f e c t s of changes i n the "salvage" value recovered by a lender on the borrower's bankruptcy. Changes i n the salvage recovery are used to model changes i n l e g i s l a t i o n governing secured lending t r a n s a c t i o n s (eg. by e n a b l i n g the borrower t o grant more e f f e c t i v e s e c u r i t y ) and i n bankruptcy l e g i s l a t i o n (eg. by re-a l l o c a t i n g assets remaining at the time of bankruptcy as between the borrower and l e n d e r ) . An increase i n the a b i l i t y of a borrower to grant s e c u r i t y i s shown to increase loan l e v e l s , but to have an ambiguous e f f e c t of i n t e r e s t r a t e s . No unambiguous r e s u l t s are a v a i l a b l e when bankruptcy l e g i s l a t i o n r e - a l l o c a t e s salvage recovery from the lender to the borrower, but both i n t e r e s t r a t e s and loan l e v e l s increase i f o n l y the borrower's salvage recovery i n c r e a s e s . Chapter VI examines the a c t i o n s of a monopolist lender. The lender sets both an i n t e r e s t r a t e and a maximum loan l e v e l f o r the borrower. The lender knows the borrower's value f u n c t i o n , and can thus p r e d i c t how much he w i l l borrow at any given 6 i n t e r e s t r a t e . The borrower can borrow any amount, up to the lender's s p e c i f i e d maximum, at the i n t e r e s t r a t e set by the lender. With the i n t r o d u c t i o n of a monopolist lender, a new issue a r i s e s : can e q u i l i b r i u m be sustained w i t h the borrower earning p o s i t i v e p r o f i t ? The a n a l y s i s of Chapter VI answers t h i s question i n the negative. Chapter VI a l s o considers the issue of c r e d i t r a t i o n i n g , but u n l i k e the competitive model, the monopoly model produces no unambiguous p r e d i c t i o n . F i n a l l y , Chapter VI a l s o analyses the e f f e c t s of changing salvage c o n d i t i o n s . In the monopoly lending market, the r e s u l t s t u r n out to be h e a v i l y dependent on whether c r e d i t r a t i o n i n g was present i n the i n i t i a l e q u i l i b r i u m . Chapter VII examines the conduct of a lender f a c i n g two competing borrowers. The lender i s given an exogenous c e i l i n g on h i s t o t a l loans, and must a l l o c a t e the a v a i l a b l e funds between the borrowers. The two borrowers have d i f f e r e n t s t a r t i n g wealth l e v e l s . The s t o c h a s t i c processes d e f i n i n g t h e i r revenue streams are modified so that the standard d e v i a t i o n that a p p l i e s to the smaller borrower i s p r o p o r t i o n a t e l y g r e a t e r than that of the l a r g e r borrower. Chapter VII thus simulates competition f o r loan funds between a sm a l l e r " r i s k i e r " borrower and a l a r g e r "less 7 r i s k y " borrower. Some of the r e s u l t s of Chapter VII are a n a l y t i c a l , but others (owing the complexity of the equations) are numerical only. A n a l y t i c a l l y , i t i s shown that the assumptions of Chapter VII l e a d to both c r e d i t r a t i o n i n g and zero p r o f i t s f o r borrowers. A numerical a n a l y s i s i s used to i n v e s t i g a t e the impact of changes i n the aggregate loan l i m i t imposed on the lender. That a n a l y s i s p r e d i c t s (as i s often commonly asserted) that the smaller borrower w i l l be squeezed out of the market f i r s t under t i g h t e n i n g c r e d i t c o n d i t i o n s . 8 CHAPTER I I - THE VALUATION PROBLEM This chapter derives the s o l u t i o n to the fundamental v a l u a t i o n problem considered i n t h i s t h e s i s . Section 1 s t a t e s the problem to be solved and s e c t i o n 2 derives the s o l u t i o n . S e c tion 3 examines the marginal value of wealth to a f i r m f a c i n g bankruptcy. Section 4 gives some b a s i c mathematical r e s u l t s which are used throughout the t h e s i s i n examining p r o p e r t i e s of the value f u n c t i o n s that are derived. 1. Statement of the Problem Consider a f i r m whose revenue i s s t o c h a s t i c , and whose wealth thus evolves s t o c h a s t i c a l l y over time. The f i r m ' s s t a r t i n g wealth w i l l be denoted as w. I t s wealth, as a f u n c t i o n of time, w i l l be denoted as x ( t ) and w i l l evolve according t o : dx(t) = (rx ( t ) + Y)dt + odz ; x(0)=w. (1) r represents a ra t e of r e t u r n to c a p i t a l , and Y represents a 9 f i x e d rate revenue o c c u r r i n g i n time dt. The term adz i s the source of u n c e r t a i n t y . dz i s a wiener process, and i s the conventional means of modelling u n c e r t a i n t y i n continuous time. dz represents a random v a r i a b l e with mean zero and w i t h standard d e v i a t i o n \/dt. The f i r m faces a r i s k of bankruptcy, which i s assumed to occur when the f i r m ' s wealth reaches the value x(t)=A. I t i s assumed that A<w, and a l s o that rA+Y>0 ( i . e . even at the p o i n t of bankruptcy, the f i r m s t i l l has p o s i t i v e expected cash f l o w ) . The l a t t e r assumption ensures that bankruptcy imposes some p o s i t i v e cost on the owners of the f i r m , beyond l o s i n g a s p e c u l a t i v e hope of p r o f i t a b l e o peration i n the f u t u r e . The value A can be given a number of economic i n t e r p r e t a t i o n s . I t could represent a wealth l e v e l at which secured c r e d i t o r s become e n t i t l e d to l i q u i d a t e the firm's assets. A l t e r n a t i v e l y , i t could represent a minimum wealth l e v e l imposed under bankruptcy l e g i s l a t i o n . For purposes of t h i s chapter, i t i s not necessary to adopt a s p e c i f i c i n t e r p r e t a t i o n of A. Nor i s i t h e l p f u l to consider an endogenous l e v e l f o r A, as the a n a l y s i s 10 does not yet include the p e r s p e c t i v e of a lender to the f i r m . In subsequent chapters, the bankruptcy thr e s h o l d w i l l be r e l a t e d to s p e c i f i c f i r m d e c i s i o n s , and w i l l be endogenized. The impact of bankruptcy can be analysed using theorems on absorbing boundaries of s t o c h a s t i c processes. The e f f e c t of an absorbing boundary i s that once x ( t ) reaches the boundary x=A, x remains at A t h e r e a f t e r . Mathematically, a process s u b j e c t i o n to absorption, x ( t ) , can be defined i n the f o l l o w i n g way. Let x*(t) represent a s t o c h a s t i c process without absorption, and define t ' as: t ' = i n f { t : x* (t) < A }, i f that q u a n t i t y i s f i n i t e = °° otherwise. Then: x(t ) = x*(t) f o r t < t ' ; = A f o r t > t ' . t ' represents the time at which the process x* (t) reaches the 11 boundary x*(t)=A f o r the f i r s t time. For a l l p o i n t s of time t<t', x ( t ) and x * ( t ) are equal. However, f o r t>t', x ( t ) and x*(t) diverge, w i t h x ( t ) remaining f i x e d at A. The v a l u a t i o n problem i s examined from the pe r s p e c t i v e of f i r m management, r a t h e r that from the perspec t i v e of an i n v e s t o r . Firm management acts i n a r i s k n e u t r a l way, v a l u i n g the f i r m as the discounted value of future revenue. The discount r a t e i s imposed exogenously as r . On bankruptcy the f i r m i s assumed to lose a l l i t s value. Accordingly, the fir m ' s value can be expressed as the l i m i t : E [lim t_. f (x(t) ) ] where f ( x ( t ) ) i s defined by: f (x(t) ) = e" r cx(t) f o r t < t ' = 0 f o r t > t ' with t ' defined as above. This value i s c l e a r l y a f u n c t i o n of s t a r t i n g wealth, w, and w i l l be represented by v(w). 12 2. D e r i v a t i o n of the S o l u t i o n An e x p l i c i t s o l u t i o n to the v a l u a t i o n problem e x i s t s . The s t a r t i n g p o i n t i s Theorem 3.22 of Gihman and Skorohod. I f x ( t ) s a t i s f i e s dx(t) = a ( x ( t ) ) d t + dz(t) i n the region 0 < x < 1, t > 0 , wi t h absorption at x=0 and x=l, and i f a(x) i s twice continuously d i f f e r e n t i a b l e i n (0,1), then: 1. The f u n c t i o n : where f (x) and <$> (x) are twice c o n t i n u o u s l y d i f f e r e n t i a b l e with f(0)=f(1)=0 i s a s o l u t i o n of: u[w,t) =E[f(x[t)) exp[ H ) ( x ( u ) ) d u ] ] o du [w, t) du (w, t) 1 d2u ( w, t) + <$> (w) u (w, t) dt - a(w) dw + 2 dw2 w i t h the boundary c o n d i t i o n s : u(w,t) approaches f(w) as t approaches 0 u(w,t) approaches 0 as w approaches 0 or 1. 2. The f u n c t i o n s p x (w,t)=Prob{x(t)=0} and 13 p 2 (w,t)=Prob{x(t)=1} are s o l u t i o n s t o : dp (w,t) dp [w, t) d2p (w, t) X = a ( w) dw + 2 d f o r i = 1, 2. The a p p l i c a b l e boundary c o n d i t i o n s are-. p_(w,t) approaches 1 as w approaches 0, and 0 as w approaches 1. p 2(w,t) approaches 0 as w approaches 0, and 1 as w approaches 1. p x(w,t), p 2(w,t) approach 0 as t approaches 0. This theorem must be extended i n two ways to provide a s o l u t i o n to the v a l u a t i o n problem. F i r s t , the theorem must be extended to apply to "one-sided" boundaries. This leads to Result 1: I f x ( t ) s a t i s f i e s x(0)=w, dx(t) = (rx(t)+Y)dt + dz(t) i n the regi o n 0 < x t > 0, wi t h absorption at x=0, then: Part 1. The f u n c t i o n 14 u ( w, t) = E[ (x ( t) ) exp [- r t ] ] i s a s o l u t i o n of: du{w,t) du{w,t) 1 dzu{w,t) = ( r w + Y) = + " ru(w,t) dt dw 2 d 2 w with the boundary c o n d i t i o n s : u(w,t) approaches w as t approaches 0; and u(w,t) approaches 0 as w approaches 0. Part 2. The f u n c t i o n Q(w,t)=Prob{x(t)=0} i s a s o l u t i o n of dQ(w, t) dQ(w, t) 1 d Q (w, t) {rw+Y) = + dt dw 2 d w 2 with boundary c o n d i t i o n s : Q(w,t) approaches 1 as w approaches 0 Q(w,t) approaches 0 as t approaches 0. The proof of Result 1 i s given i n the Appendix t o Chapter I I . (Note that most of the proof a p p l i e s to any E [e _ r t f (x) ] where f (: i s twice c o n t i n u o u s l y d i f f e r e n t i a b l e , f(x) has s u i t a b l e growth behaviour as x approaches i n f i n i t y , and f(0)=0. However, the boundary c o n d i t i o n s are examined only f o r the p a r t i c u l a r case f (x)=x) . 15 Second, i t i s necessary to examine the l i m i t i n g behaviour of the s o l u t i o n as t approaches i n f i n i t y . This leads to Result 2 (also proven i n the Appendix): I f x ( t ) s a t i s f i e s the same cond i t i o n s as i n Result 1, define v(w) as: v(w) = l i m E[e rtx (t) ] where x(0) -w t - o o v(w) must s a t i s f y : 1 d2v dv + (rw+Y) - rv = 0 2 dw2 d w subject to v ( 0 ) = 0 , and v(w) approaches w + Y/r as w approaches i n f i n i t y . These r e s u l t s are t r i v i a l l y extended to the s i t u a t i o n where absorption occurs at a non-zero value (A), and a l s o to the s i t u a t i o n where the c o e f f i c i e n t of d i s p e r s i o n = o * 1. I f x ( t ) i s a process f o l l o w i n g dx(t) = (rx(t)+Y)dt + odz w i t h absorption at x=A, then i t i s simply necessary to consider the process x l ( t ) = ( x ( t ) - A ) / o . x l ( t ) c l e a r l y s a t i s f i e s the c o n d i t i o n s of both Result 1 and Result 2. A l s o : 16 - r t x ( t ) - A - r t X ( t ) 1 im e E [ ] = l i m e E [ o o I t f o l l o w s that i f x ( t ) i s a process with absorption at x=A s a t i s f y i n g x(0)=w, dx(t) = (rx(t) + Y)dt + adz, and i f f(x) and v(w) are defined as above, v(w) s a t i s f i e s : subject to v(A)=0 and v(w) approaches w+Y/r as w approaches i n f i n i t y . Equation 2 has two general s o l u t i o n s . The f i r s t i s t r i v i a l . By i n s p e c t i o n , the f u n c t i o n f1(w) = rw + Y solves the equation. The second general s o l u t i o n can then be found by the process of order r e d u c t i o n , which permits a second order d i f f e r e n t i a l equation to be reduced to a f i r s t order equation i f one s o l u t i o n i s known.1 The second s o l u t i o n i s equal t o : The process of order reduction i s explained i n Boyce and d i Prima (1977). Consider a second order d i f f e r e n t i a l equation: a(t)x"(t) + b( t ) x ' ( t ) + c( t ) x ( t ) = 0 to which one solution, x l ( t ) , i s known. Assume that the second s o l u t i o n takes the form x2(t) = y ( t ) x l ( t ) . D i f f e r e n t i a t i n g twice, then s u b s t i t u t i n g into the o r i g i n a l equation and rearranging terms y i e l d s (with the argument t suppressed): 1 2 d2v dv + [rw+Y) - r v = 0 dw (2) 17 2 oo e X P ( " 2"> c ro f2(w) = (rw+Y) J ds rw + Y S The general s o l u t i o n thus takes the form v(w)= cl*f1(w)+c2*f2(w), where c l and c2 are constants of i n t e g r a t i o n determined by-boundary c o n d i t i o n s . With the boundary c o n d i t i o n s set out above, v(A ) = 0 and v(w) approaches w+Y/r f o r la r g e w, the s o l u t i o n becomes: ( Y+r e x p [ ] ro -ds v (w) = w + — -r (3) / e x p [ ] rO •ds y [ a x l " + b x l ' + c x l ] + y" [ a x l ] + y ' [ 2 b x l ' + b x l ] = 0 By t h e d e f i n i t i o n o f x l , t h e f i r s t t e r m i s z e r o . M a k i n g t h e s u b s t i t u t i o n u = y ' , t h e above e q u a t i o n t h u s r e d u c e s t o : [ a x l ] u ' + [ 2 b x l ' + b x l ] u = 0 S i n c e x l i s known, t h i s i s j u s t a f i r s t o r d e r e q u a t i o n i n u , w h i c h c a n be found b y o r d i n a r y i n t e g r a t i o n me thods . u c a n t h e n be u s e d t o s o l v e f o r y , and hence x 2 . 18 This expression can be s i m p l i f i e d using the f o l l o w i n g i n t e g r a t i o n by p a r t s : 7 e x p ( - s 2 ) e x p ( - x 2 ) "r , I ds - - 2 j e x p ( - s ) ds J 2 x J Equation (3) then becomes: ( Y+rw) 2 2 ( Y + rw) °°r s e x p [ - ] - J e x p [ - ] d s Y rO ro Y+ru rO v(w) = w + — ~ L s e x p [ ] / ra1 2 Y + rA s •ds (4) ( Y + rw) 2 ( Y + rw) °°r s e x p [ - ] - I e x p [ -]ds Y rO rO Y + r w rO - w + — -r (Y+rA)2 e x p [ ] r O 2 r s r f ~ / e x p [ ] d s ] ™ ^ 2 ^ The value f u n c t i o n can thus be reduced to expressions i n v o l v i n g the standard e r r o r f u n c t i o n . I n t u i t i v e i n t e r p r e t a t i o n of the f u n c t i o n v(w) i s a s s i s t e d by d e f i n i n g the q u a n t i t y M as f o l l o w s : 19-2 S exp [ -] ro ( Y+rw) ds 2 S M = 2 S (5) exp [ ] ro ( Y + rA) ds 2 S As w i l l be shown i n Chapter IV, s e c t i o n 3, M represents the discounted sum ( i . e . i n t e g r a l ) of the future p r o b a b i l i t i e s of bankruptcy. Using the q u a n t i t y M, the value f u n c t i o n (3) can then be re-w r i t t e n i n the f o l l o w i n g way: v(w) = w + Y/r - (A + Y/r)M = w - AM + (Y/r)(1-M) S t a r t i n g wealth i s w. However, with discounted p r o b a b i l i t y M, A of that w i l l be l o s t . In a d d i t i o n , the f i r m enjoys the value of revenue Y, whose discounted present value i s equal to Y/r. However, t h i s revenue stream may a l s o be l o s t , w ith p r o b a b i l i t y M, and i s thus "discounted" by the f a c t o r (1-M) . The f o l l o w i n g graph gives a p i c t o r i a l r e p r e s e n t a t i o n of the value f u n c t i o n . The value f u n c t i o n approaches w + Y/r a s y m p t o t i c a l l y as w approaches i n f i n i t y , reducing the l i k e l i h o o d of bankruptcy. Both the d i f f e r e n t i a l equation (2) and i t s s o l u t i o n (3) are s i m i l a r to other such equations and s o l u t i o n s found i n the finance l i t e r a t u r e . However, there are s i g n i f i c a n t d i f f e r e n c e s as w e l l . Merton (1992, chapter 11) d e r i v e s the f o l l o w i n g fundamental p a r t i a l d i f f e r e n t i a l equation f o r the value (F) of a contingent 21 c l a i m to an un d e r l y i n g asset with value V: 0 = 1/2 G 2V 2F W + rVF v - F t - rF. A more g e n e r a l i z e d equation, a l l o w i n g f o r payouts from the f i r m , i s presented i n Black and Cox (1976). Leland (1994) uses the i n f i n i t e time v e r s i o n of the equation s t a t e d by Black and Cox ( i n which the time v a r i a b l e becomes i r r e l e v a n t ) to value f i r m debt paying C per p e r i o d : 0 = 1/2 o 2V 2F w + rVF v + C - rF Leland goes on to i n v e s t i g a t e p r o p e r t i e s of the e x p l i c i t s o l u t i o n to h i s equation, which takes the form F (V) = A 0 + AjV + A 2V X (where X i s determined by parameters and boundary c o n d i t i o n s ) . D i r e c t a p p l i c a t i o n of the equations used to value contingent claims (as i n Leland) presents another means of d e r i v i n g equation (2). That method has not been used here, as the method used here proceeds d i r e c t l y from fundamental theorems on s t o c h a s t i c d i f f e r e n t i a l equations, and thus makes e x p l i c i t many of the 22 assumptions regarding convergence which would normally be i m p l i c i t i n the a p p l i c a t i o n of contingent claims a n a l y s i s . The d i f f e r e n t i a l equations of Merton, Black and Cox, and Leland are d i f f e r e n t from (2) i n t h e i r second order term. Equation (2) omits the f a c t o r V2 from that term. The reason i s that Merton, Black and Cox, and Leland base t h e i r analyses on an underlying s t o c h a s t i c process which e x h i b i t s geometric brownian motion: dx = rxdt + oxdz. Presence of the f a c t o r "x" i n the s t o c h a s t i c term (which i s not present i n (1), above) e x p l a i n s both the d i f f e r e n t forms of equation, and the d i f f e r e n t form of s o l u t i o n d e r i v e d by Leland. For the purposes of t h i s t h e s i s , the s t o c h a s t i c process which has been chosen appears more r e a l i s t i c than one based on geometric brownian motion. There i s no reason to b e l i e v e that a firm's revenue becomes any more c e r t a i n or r e l i a b l e as i t comes c l o s e r to bankruptcy. I f anything, the opposite i s t r u e . Milne and Robertson (1996) i n v e s t i g a t e the discounted value 23 (F(x), where x represents instantaneous cash balances) of the dividend payments from a f i r m which faces bankruptcy. The problem they solve i s thus c l o s e l y r e l a t e d to the problem under c o n s i d e r a t i o n here. In a d d i t i o n , Milne and Robertson consider the same s t o c h a s t i c process as set out i n (1). However, t h e i r model requires that the r e t u r n to c a p i t a l (r) be l e s s than the discount r a t e (p). M i l n e and Robertson thus a r r i v e at the f o l l o w i n g d i f f e r e n t i a l equation which i s h i g h l y s i m i l a r to (2) : 0 = 1/2 + (rx+u)F x - pF Milne and Robertson then go on to consider the circumstances under which an e x p l i c i t s o l u t i o n to t h e i r equation e x i s t s . They i d e n t i f y such a s o l u t i o n ( e s s e n t i a l l y a r a t i o of exponential values) i n the p a r t i c u l a r case r=0. However, the model of Milne and Robertson does not a l l o w f o r the case r=p, and thus does not y i e l d the closed form s o l u t i o n presented i n (3). The closed form s o l u t i o n derived i n t h i s t h e s i s can be considered a l i m i t i n g case, as the discount r a t e approaches the r a t e of r e t u r n on c a p i t a l from above, of the numerical r e s u l t s set out i n Milne and Robertson. 24 3. The Marginal V a l u e . o f - C a p i t a l A q u a n t i t y of i n t e r e s t i s v'(w), the marginal value of wealth. I n t u i t i v e l y , the marginal value of wealth i s the combination of three f a c t o r s , two p o s i t i v e and one negative: 1. The b a s i c revenue stream from the marginal u n i t of wealth -a p o s i t i v e f a c t o r . 2. The r i s k of l o s i n g the revenue stream from the marginal u n i t of wealth owing to bankruptcy - a negative f a c t o r . 3. Added p r o t e c t i o n from bankruptcy - a p o s i t i v e f a c t o r . The d e r i v a t i v e of the value f u n c t i o n i s : 2 s exp [ ] ( Y + rw) 2 exp [ -dv (6) = 1 + 2 2 S S exp [ .2 exp [ -2 25 For w>A, dv/dw must be p o s i t i v e , as the second term i n (6) i s l e s s than one. Equation (6) can a l s o be s i m p l i f i e d using i n t e g r a t i o n by parts t o : dv dw 1 + rO Y + rw s I exp [ ] ds ro I exp [ ] rO •ds 1 + rO Y + rw s / exp [ -] irw ro ds exp [ -(Y+rA) ro Y + rA rO oo Z I exp [ ] ds r i r A ro' (7) The d e r i v a t i v e of the value f u n c t i o n , l i k e the value f u n c t i o n i t s e l f , can thus be reduced to expressions i n v o l v i n g the standard e r r o r f u n c t i o n . Expression (7) a l s o d i s c l o s e s that v'(w) i s no l e s s than 1 - the "saving" f a c t o r i d e n t i f i e d as 3, above, outweighs the negative f a c t o r i d e n t i f i e d as 2. F i n a l l y , d i f f e r e n t i a t i o n of (7) with respect to w y i e l d s : 26 2 (Y+rw)2 - - e x p [ - — ] d v o rO dw2 (Y+rA)2 e x p [ ] r o 2 /• s f e x p [ ] ds Y+rA r n 2 J r n r o y + r A r o which i s unambiguously negative, confirming that v(w) i s concave i n w. The t h r e a t of bankruptcy induces l o c a l r i s k a version i n t o a value f u n c t i o n which would otherwise be r i s k n e u t r a l . Both v(w)and v'(w) have i n t e r e s t i n g l i m i t i n g values. Consider f i r s t the l i m i t i n g value of v(w) as the c o e f f i c i e n t of d i s p e r s i o n , o, becomes a r b i t r a r i l y l a r g e . Reparameterizing with z 2 = s 2 / r o 2 i n equation (4) leads t o : ( Y + rw) 2 2 (Y + rw) r 2 e x p [ ] - I e x p [ ~ z ]dz o 2 o^/r Yirw Y o/i v(w) = w + — - — r (Y+rA) e x p [ ] r O r o ^ r 2 r [ - e x p [ - z ]dz] Y+rA a FE J a^fr 27 For large o, the exponential terms i n the numerator and denominator both approach 1, while the i n t e g r a l s i n the numerator and denominator both approach zero. I t i s thus apparent that the l i m i t i n g value of v(w) becomes: w + Y/r - (A + Y/r) = w - A. With i n f i n i t e variance, the value f u n c t i o n depends only on the s t a r t i n g d i s t a n c e from the boundary. The r e s u l t i s i n t u i t i v e . A large variance e l i m i n a t e s any i n f l u e n c e from e i t h e r the constant d r i f t f a c t o r , Y, or the r e t u r n on c a p i t a l , r. I t f o l l o w s that as o approaches i n f i n i t y , v'(w) approaches 1, a r e s u l t which can a l s o be d e r i v e d d i r e c t l y from Equation (7). A second l i m i t i n g value of i n t e r e s t i s the marginal value of a d d i t i o n a l c a p i t a l as when the f i r m i s on the verge of bankruptcy. As w approaches A, v'(w) approaches: 28 (Y+rA) exp [ ] dv rO (A) d w (Y + rA)2 2 (Y + rA) °r s2 e x p [ - ] - - J exp [ - ] ds ro" ro" YtrA ro2 2 W 2 (Y+rA) r s J exp f - ] ds ro2 YtrA ro = 1 + (Y+rA) 2(Y+rA) r s e x p [ - ] - I exp [ -]ds ro ro y + r A r o Reparameterizing again w i t h z 2 = s 2 / r o 2 leads t o : 2(Y+rA) OJr j exp [ - z 2 ] dz —— (A) - i + y-d w Y+rA 2 2 (Y+rA) r 2 exp [ - ( ) ] - exp [ - z ]dz o 7^ o^F The marginal value of c a p i t a l as w approaches A can thus be w r i t t e n as a f u n c t i o n of the s i n g l e q u a n t i t y (Y+rA)/o/r. As t h i s q u a n t i t y approaches 0, i t i s apparent that v'(A) approaches 1. As (Y+rA)/o/r becomes a r b i t r a r i l y l a r g e , v'(A) becomes i n f i n i t e as w e l l . Further i n f o r m a t i o n concerning v'(A) i s given below i n s e c t i o n 5, Related Functions. 29 Related Functions There are s e v e r a l f u n c t i o n s r e l a t e d to v(w) which appear i n the proofs and d i s c u s s i o n s below. Some of the functions appear as components of the value f u n c t i o n , i n one or more of i t s various expressions. Others are used i n e v a l u a t i n g the signs of d e r i v a t i v e s of the value f u n c t i o n and other r e l a t e d f u n c t i o n s . In some cases, i t i s the l o g a r i t h m i c d e r i v a t i v e of an expression appearing a value f u n c t i o n which i s of i n t e r e s t . This s e c t i o n of t h i s t h e s i s i n v e s t i g a t e s the p r o p e r t i e s of these r e l a t e d f u n c t i o n s . At the end of t h i s s e c t i o n , some of the r e s u l t s are a p p l i e d i n a f u r t h e r i n v e s t i g a t i o n of v'(w), the marginal value of wealth. (I) The Function i ( x | = j e x p [ - s 2 ] ds x Consider f i r s t the f u n c t i o n i ( x ) = j e x p [ - s 2 ) d s and l e t g (x) x represent the negative of i t s l o g a r i t h m i c d e r i v a t i v e , - I ' ( x ) / I ( x ) . g(x) i s thus equal t o : 30 exp[-x ] g( x) = OO yexp [ -s 2] ds x Subject to constants, I(x) i s the standard normal d i s t r i b u t i o n f u n c t i o n , and appears d i r e c t l y i n expressions f o r v(w) and v'(w) (equations (3) and (5) of t h i s chapter). The p r o p e r t i e s of I(x) are w e l l known and need no e l a b o r a t i o n . g(x) i s used i n subsequent chapters i n e v a l u a t i n g the d e r i v a t i v e s of the value f u n c t i o n and r e l a t e d f u n c t i o n s . I t s p r o p e r t i e s are not obvious, and r e q u i r e a n a l y s i s . At x = 0, g(x) i s c l e a r l y equal to 2/V~n. For lar g e x, l ' H o p i t a l ' s r u l e i s necessary, as both the numerator and the denominator approach 0. Taking d e r i v a t i v e s y i e l d s the r a t i o : 2 xexp [ - x ' ] exp[~x ] From t h i s expression i s apparent that f o r l a r g e x, g(x) approaches 2x. The f o l l o w i n g graph shows the locus of g(x)= - I ' (x)/I (x) . 31 Figure 2 - Graph of g(x) (TT) The Function h(x) = g (x) - 2x = - I ' ( x ) / I ( x ) - 2x Consider next the f u n c t i o n h(x) = g(x) - 2x = - I ' ( x ) / I ( x ) - 2x. h(x) does not appear d i r e c t l y i n any of the expressions considered thus f a r . However, i t i s important i n analysing the f u n c t i o n J ( x ) , which i s def i n e d and analysed below. I t can be shown that h(x) i s monotonically decreasing i n x. 32 (Note that the r e s u l t s already given show that h(x) converges to 0 f o r large x ) . To save on no t a t i o n , l e t E=exp[-x 2], and l e t I represent the i n t e g r a l x= [exp[-s2]ds . The o b j e c t i v e i s thus to show that h(x)= E/I-2x i s a monotonically decreasing f u n c t i o n of x. The d e r i v a t i v e of h(x) i s h' (x) = E'/I - E I ' / I 2 - 2, but since E'=-2xE, and since I'=-E, h'(x) can be r e w r i t t e n as h' (x) = E 2 / I 2 - 2xE/l - 2 Thus, h(x) w i l l be monotonically decreasing i f : E 2 / I 2 2xE/l < 2, or E/I - 2x < 2I/E, or h(x) < 2I/E (9) I t has already been shown that E/I i s monotonically i n c r e a s i n g , from which i t f o l l o w s that the RHS of (9) i s monotonically decreasing. The f o l l o w i n g graph now i l l u s t r a t e s why h(x) must be 33 monotonically decreasing. Figure 3 - Graph of h(x) x I f the graph of the f u n c t i o n were to cross the graph of 2I/E (as i n the case of the dash l i n e ) , that would imply that h(x) > 2I/E, which would i n t u r n imply that h(x) i s i n c r e a s i n g . But with h(x) i n c r e a s i n g and 2I/E decreasing, the c o n d i t i o n h(x) > 2I/E would continue to h o l d - f o r a l l p o i n t s to the r i g h t of the poin t of i n t e r s e c t i o n (marked A i n the graph). h(x) would thus have to be i n c r e a s i n g f o r a l l p o i n t s to the r i g h t of the i n t e r s e c t i o n , 34 c o n t r a d i c t i n g the r e s u l t that h(x) approaches 0 f o r large x. From the foregoing d i s c u s s i o n , i t a l s o f o l l o w s that the f u n c t i o n h(x)/x + 2, or: r 2 n exp[ -x J xjexp[-s2]ds i s monotonically decreasing i n x as w e l l . J L I X I /6 X p l " S J . ds, which i s s c l e a r l y p o s i t i v e . J(x) appears d i r e c t l y i n expressions f o r the value f u n c t i o n and i t s d e r i v a t i v e s (equations (2) and (4)). I t i s of fundamental importance i n many of the r e s u l t s derived i n the f o l l o w i n g chapters. Using i n t e g r a t i o n by p a r t s , J(x) can be r e w r i t t e n as: J[x) = exp [ -x 2] - 2x jexp[-s2] ds This expression shows that J(0)=1, and i s a l s o e a s i l y 35 d i f f e r e n t i a t e d to o b t a i n : dJ r 2 = -2 exp[ - s ]ds = -2I (x) dx J X which i n tu r n implies that J" (x) = 2exp(-x 2) > 0. J(x) i s thus s t r i c t l y decreasing and convex. In s p e c t i o n of the above equations a l s o shows that J(x) must approach 0 f o r large J(x) i s thus g r a p h i c a l l y shown as f o l l o w s . Figure 4 - Graph of J(x) J(x) 1 x Now l e t p(x) = - J ' ( x ) / J ( x ) = 2I(x)/J(x) and l e t q(x) = l/p (x) J ( x ) / I ( x ) . q(x) can be r e - w r i t t e n i n the f o l l o w i n g way: 36 q(x) = -J[x) / j'(x) oo exp [ -x2] - 2x Jexp [-s 2] ds _ X 2^ e xP[ _s 2]ds X 1 exp[-x 2] = j l - — -2" /exp[-s ]ds X The term i n brackets i n the f i n a l expression was defined as h(x) i n the previous s e c t i o n , and was shown to be monotonically decreasing. I t f o l l o w s that p(x) =. l/q(x) = 2I(x)/J(x) i s monotonically i n c r e a s i n g , and thus that the (negative of) the lo g a r i t h m i c d e r i v a t i v e of J(x) i s a monotonically i n c r e a s i n g f u n c t i o n . I t a l s o f o l l o w s that the f u n c t i o n x l ( x ) / J ( x ) , as the product of two monotonic f u n c t i o n s , i s monotonically i n c r e a s i n g as w e l l . (TV) A p p l i c a t i o n to the a n a l y s i s of v'(w) An example of the a p p l i c a t i o n of some of these fu n c t i o n s comes determining the l i m i t i n g value of v'(A) as Y+rA becomes large, v'(A) measures, i n e f f e c t , the value of an a d d i t i o n a l u n i t of c a p i t a l to a f i r m on the verge of bankruptcy. A lar g e value of Y+rA i m p l i e s that the fi r m ' s a b i l i t y to generate p o s i t i v e cash 37 flow i s strong near bankruptcy. v'(A) thus measures what the f i r m would pay to preserve t h i s a b i l i t y to generate cash flow. Evaluated at w=A, equation (8) gives the marginal value of c a p i t a l as: (Y+rA) e x p [ ] dv ro ( A) = - = (10) d w [Y+rA) 2 (Y+rA) r s e x p [ - ] - J e x p [ -]ds ro ro Y+rA r ° As both the numerator and denominator vanish f o r large x, i t i s necessary to use l ' H o p i t a l ' s r u l e . To economize on notation, l e t E represent the numerator i n the above expression, and l e t I represent the i n t e g r a l i n the denominator. D i f f e r e n t i a t i o n of both the numerator and denominator with respect to (rA+Y) then y i e l d s the r a t i o : 2 ( rA+ Y) dv l i r a [ — (A) ] A - » dw rO -2 (rA + Y) ro E --2 (rA + Y) rO rO which i s equal to (rA+Y)E/I. However, as E ( x ) / l ( x ) approaches 38 2x/ro 2 f o r l a r g e x, t h i s expression i s i n turn equal to 2(rA+Y) 2 / r o 2 . Accordingly, the marginal value of c a p i t a l at the bankruptcy boundary i s p r o p o r t i o n a l to the square of the marginal cash flow at the boundary, as the l a t t e r value becomes l a r g e . (V) A p p l i c a t i o n to the a n a l y s i s of a change i n o Another question a r i s i n g from the value f u n c t i o n d e r i v e d above i s whether a f i r m on the verge of bankruptcy w i l l have an in c e n t i v e to increase the inherent r i s k l e v e l of i t s business. D i f f e r e n t i a t i o n of the value f u n c t i o n (3) with respect to o, the c o e f f i c i e n t of d i s p e r s i o n , y i e l d s : 2 2 oo e x p t - - ^ ] . e x p [ - -] v r ra r s r rO r s - 2 ( W + I ) j i d s j e x p [ - _ ] " J -2 ds J exp[-OV(W) r Y + rA S Y+rw f -S Y + rw S Y>rA rU = [ ] exp [ -] r a r r u 2 [ / ds] Y+rA s This expression reduces to an expression i n the form: -2( (rw+Y)I((rw+Y)/J(rw+Y) - (rA+Y)I((rA+Y)/J(rA+Y) ) 39 By the r e s u l t s of (III), above, xl(X)/J(x) i s monotonically decreasing. Since rw+Y>rA+Y, i t f o l l o w s that even near the bankruptcy th r e s h o l d , v i s a decreasing f u n c t i o n of a. The f i r m thus has no i n c e n t i v e to "gamble f o r r e s u r r e c t i o n " by switching to p r o j e c t w i t h higher o. 40 APPENDIX TO CHAPTER I I I. PROOF OF RESULT 1 A proof w i l l be given f o r Part 1. No proof i s given f o r Part 2 because the proof given i n Gihman and Skorohod (theorem 22.3) ap p l i e s d i r e c t l y to the case under c o n s i d e r a t i o n . Proof of p a r t 1. The proof f o l l o w s Gihman and Skorohod, theorem 22.3, and begins with a s i m i l a r r e s u l t a p p l i c a b l e to processes without absorption. The proof then defines a f u n c t i o n g ( x ( t ) ) which i s p o s i t i v e f o r x<0 and 0 f o r x>0. The f u n c t i o n g i s used to create an m u l t i p l y i n g f a c t o r of exp [-njg(x(u)]du. For l a r g e n, the m u l t i p l y i n g f a c t o r vanishes f o r a l l sample paths which cross the boundary. This feature of the m u l t i p l y i n g f a c t o r allows the r e s u l t f o r processes without absorption to be adapted to processes w i t h absorption. Notation: 1. x w - s* (t) i d e n t i f i e s the s t o c h a s t i c process f o r wealth at time 41 t>s, without absorption, given that x w s*(s)=w. Thus, dx w, s*(t) = r x w > s * ( t ) d t + Ydt + dz; x w s * (s) = w. 2 x W i S ( t ) i d e n t i f i e s the corresponding process w i t h absorption. dx w s (t) = r x w s ( t ) d t + Ydt + dz; x w s(s)=w, wi t h absorption at x = 0. 3. Q(w,t) denotes the p r o b a b i l i t y that x ( t ) has reached the boundary x(t)=0 at or before time t , given that x(0)=w. Assume t h a t : 1. g(x) has continuous, bounded d e r i v a t i v e s up t o and i n c l u d i n g second order; g(x)=0 f o r x>0; g(x)>0 f o r x<0; g(x) i s bounded. 2. f(x) i s nonnegative, has bounded continuous f i r s t and second d e r i v a t i v e s , and has s u i t a b l e growth r e s t r i c t i o n s f o r large x. A l s o , f(0)=0. 3. h(x) i s twice continuously d i f f e r e n t i a b l e w i t h h(x)=0 i f x l i e s o u t s i d e [0,b] f o r some a r b i t r a r y b>0. A l s o , h(0)=h' (0)=h"(0)=0 and h(b)=h' (b)=h"(b)=0. Define. u (w,s) =E[f{x *(t)) exp[-n jg(x *{u))du]]. From n w, s J w, s Gihman and Skorohod (#11, remark 1), each u n s a t i s f i e s : 42 du (w, s) du (w, s) d u (w, s) n n ng [w) u (w, s) (A) = (rw+Y) dw + 2 dw2 n subject to u n(w,t) = f(w). Define u Q(w,s) = limn_„un (w, s) , which l i m i t e x i s t s since u n(w,t) i s non-increasing as n approaches i n f i n i t y . From the d e f i n i t i o n of u 0, i t f o l l o w s that u Q(w,t) = f (w) . Using g(w)h(w)=0, i n t e g r a t i n g (A) wi t h respect to s from s1 to t , then m u l t i p l y i n g by h(w), then i n t e g r a t i n g w i t h respect to w from minus i n f i n i t y to i n f i n i t y , and then i n t e g r a t i n g by p a r t s with respect to w y i e l d s : u (w, t) h (w) dw + I u (w, s ) h (w) dw -n 1 a u (w, s) - 3 - ( (rw+Y) h(w) ) dwds 1 £ " j 2 I f f d + —j I u (w, s) h(w)dwds A l l i n t e g r a l s i n t h i s expression w i l l be f i n i t e , since h(w)=0 outside of some a r b i t r a r y i n t e r v a l [0,b]. L e t t i n g n go to i n f i n i t y : 43 j f {w) h {w) dw + ju^(w, s^) h (w) dw -r r d I d 2 u ( w, s) [ - ( (rw+Y) h(w) ) + — - h ( w) ] d w d s J J o dw 2 dw2 As h(w) i s 0 outs i d e of [0,b], i n t e g r a t i o n w i t h respect to w i n t h i s equation can be l i m i t e d to that i n t e r v a l . D i f f e r e n t i a t i n g w ith respect t o sx then y i e l d s : b b d 1 [u (w, s ) h (w) dw = [u ( w, s ) [ - - = - ( (rw+Y) h(w) ) + — h " ( w) ] dw J o I J o I dw 2 ds 1 o b ^ d 2 u du du fh(w) [ — —~-!L + (rw+Y) - T ^ + T ^ ] d w = 0 J 2 ^2 dw d s n O W i I n t e g r a t i o n of t h i s expression by p a r t s y i e l d s : This i m p l i e s that u 0(s,w) i s a g e n e r a l i z e d s o l u t i o n of d d I d ' - 3 - u (w,s) = (rw+Y) — u ( w , s ) + — -u (w,s) ( B ) ds o dw o 2 o This equation has thus been proven f o r a l l w i n an a r b i t r a r y i n t e r v a l (0,b). As b i s a r b i t r a r y , the equation must hold f o r 44 a l l W E (0 , °°) . I t can now be shown that u 0(w,t) = E [ f ( x ( t ) ] . Note that e x p [ - n f g ( x * (u))du] - 0 as n -« i f there i s any u i n [s,t] J w, s s such that x w s*(u)<0. Conversely, i f x w s* (u) >0 f o r a l l u i n [ s , t ] , then f g ( x * ( u ) ) d u = 0. Also, Prob {min s s u s tx* w s (u) =0 } = 0 . Thus, since J w, s ' s f (0)=0: t Prob ( l i m f(x * ( t ) ) e x p [ - n [ g (x *(u))du] *f(x ( t ) } r j - o o W, S J W f S W r S S < Prob ( m i n x * { u ) =o} = 0 s<u<t w,s Thus, w i t h p r o b a b i l i t y 1: t l i m f ( x * ( t ) ) e x p [ -n [g(x* (u))du] =f(x ( t ) ) n^°° w, s J w, s w, s so that l i m £ [ f ( x * ( t ) ) e x p [ -n f g ( x * { u ) ) d u ] ] = £ [ f ( x ( t ) ) ] n- ° ° w, s J w, s w, s But the LHS of t h i s expression i s u 0(w,t). This proves that equation (B) i s solved by E[f ( x W ; S ( t ) ] f o r w i n the i n t e r v a l (0,b). As the proof has been given f o r an a r b i t r a r y i n t e r v a l 45 (0,b), the equation must hold f o r the e n t i r e domain w>0. Two f u r t h e r steps are necessary to complete the f i r s t part of the r e s u l t . I t i s necessary to incorporate the discount f a c t o r , e~rt, and t o e l i m i n a t e the "backwards" nature of the equation. Since s t o c h a s t i c process f o r x does not depend e x p l i c i t l y on time : e-r(t-slu0(w,s) = E [ e " r ( t - s ) f (x w, s(t) ) ] = E [ e ~ r < t s ) f ( x w 0 (t - s) ) ] (c) D e f i n i n g u(w,t) = E [e~ r tf ( x w 0 (t) ) ] , equation (C) i m p l i e s t h a t : u(w,t-s) = e- r ( t" s )u 0 (w, t-s) . D i f f e r e n t i a t i n g both s i d e s with respect to s y i e l d s : du (w, t~s) du ( w, t- s) -r(t-s) - r ( t - s ) 0 = r e u (w, t~s) + e ds 0 Since du/dt - - du./ds, t h i s can be r e w r i t t e n as: 46 du ( w, t~w) a t r e u ( w, t-s) + o du -r(t-s) 0 w, t- s) du [w, t-s) ru(w,t~s) + e du -r(t-s) 0 W, t- S) S u b s t i t u t i n g expression (B) f o r du/ds leads t o : a , , . au [w, t~s) -, a 2 u (w, t-s) du(w,t-S) -rlt-3) . o 1 -<t-s> o - r u ( w , t - s ) - e (rw+50 5 - — e a t aw 2 a ^ 2 5 u ( w , t - s ) a 2 u ( w , t ~ s ) = r u ( w, t-s) ~ ( rw+Y) iw dw' Reparameterizing w i t h t in s t e a d of t-s leads to the p a r t i a l d i f f e r e n t i a l equation: a a i a2 — u(w,t) = ( rw+ Y) u ( w, t) + — -u{w,t) - r u ( w , t ) dt dw 2 d w * which i s the equation to be proved. The f i n a l step i n the f i r s t part of the proof i s establishment of 47 the boundary c o n d i t i o n s . I t i s now p o s s i b l e to r e s t r i c t the a n a l y s i s to the f u n c t i o n i n question, name f(x)=x. The f i r s t c o n d i t i o n , l i m t _ 0 u(w,t) = f (w) = w, i s s e l f - e v i d e n t . To e s t a b l i s h the second boundary c o n d i t i o n , namely that l i n v 0 u ( w , t ) = 0, consider the process Xw* (t) = x w*e~ r t (a process without a b s o r p t i o n ) . X w*(t) then s a t i s f i e s : dX w*(t) = Ye" r tdt + e- r cdz. Define the process z* (t) =w+ e~rsdz, so that Z* w(t) f o l l o w s dZ* w = e~ r tdz. Now define Z w(t) to be a process corresponding to Z * w ( t ) , but w i t h absorption at Z=0. Theorem 3, #12 of Gihman and Skorohod e s t a b l i s h e s that X w*(t) i s a b s o l u t e l y continuous w i t h respect to the process Z * w ( t ) , and that the d e n s i t y of the former wi t h respect t o the l a t t e r process i s : o o 48 I f A i s some Borel set i n (0,°°), then: Prob {0 < Xw* (u) f o r 0<u<t ; X w * ( t ) e A} = E [pLA] where L A = 1 i f 0<Z*w(u) f o r 0<u<t, Z* w(t)e A, and Lfl = 0 otherwise. The LHS of t h i s expression represents the p r o b a b i l i t y Prob{X w(t)6 A}, and the RHS can be expressed as E[K A(Z w(t)p] where KA i s the i n d i c a t o r f u n c t i o n f o r set A. Using these p r o b a b i l i t i e s , f o r any f u n c t i o n f such that f (0)=0: E[f {X it)] = E[ f iZ it) ) exp I \ YdZ* is) - — (Y2 ds] ] w w J w 2 J 0 0 or, i n the case f(x) = x: t ^ t E IX it) ] = E HZ it) ) exp [ I YdZ* is) - — I Y2 ds]] w w J w 2 J To prove the boundary c o n d i t i o n lim w_ 0u(w,t) =0, i t i s thus necessary t o show that the RHS of t h i s expression goes to 0 as w goes to 0. As the second term i n brackets on the RHS i s a constant f o r any given value of t , i t s u f f i c e s to analyze the expression: 49 E[X (t)] = E[[Z ( t ) ) e x p [ f YdZ* (s) ] ] w w J w 0 By Holder's i n e q u a l i t y : t t E [ (Z ( t ) ) e x p [ [ YdZ* (s)]] < (El(Z ( t ) ) q ] ) 1 / g ( £ [ e x p [ p /* 5 f d 2 * ( s ) ] ] ) 0 0 f o r l / q + 1/p = 1 and p, q p o s i t i v e . I t thus s u f f i c e s to prove that the f i r s t f a c t o r on the r i g h t hand side goes to 0 as w approaches 0, and that the second f a c t o r of the r i g h t hand side i s bounded. To analyze the second f a c t o r on the RHS, consider a s t o c h a s t i c process: dR = e"rtYRdz; R(0)= w. From Arnold, c o r o l l a r y 8.4.3, R(t) i s equal to: 50 t 2 t R(t) = wexp [ J— e ~ 2 r t ds + jVe~ r t dz] o 2 t 2 wexp [ j" — e ~ 2 r t ds + ^YdZ* (s)] o 2 From Arnold, Theorem 8.4.5, R(t) has a f i n i t e pth order moment, i n d i c a t i n g that the e x p e c t a t i o n : t {E [ exp [pf YdZ*Js) ] ] ) 1 / P must be f i n i t e . Turning next to the e x p e c t a t i o n (E [ (Zw (t) ) q] ) 1 / q , from the d e f i n i t i o n of Z w(t) : t) w + tAi dz where i represents the f i r s t e x i t time to -w f o r the process J V r s d z . Thus: 51 E\z dz(s) \q] < K [wq + E[\ [ e q J dz{s) \q] ] f o r a constant Kq. w q c l e a r l y approaches 0 as w approaches 0. To analyse the second term i n brackets, define Z* 0(t) as the process: with Z 0(t) as the corresponding process w i t h absorption at Z* 0(t)=-w. Thus, Prob{Z 0 (t) >-w} < Prob{inf 0 s u s c [Z* 0 (u) ] > -w} From Arnold, theorem 8.2.10, Z* 0(t) i s normally d i s t r i b u t e d w i t h mean 0. The variance solves: K' (t) = e~2rt , w i t h K(0) =0. Z* ( t) = e o J o This i m p l i e s that the variance at time t i s l / 2 r [ l - e"2rt] , so tha t : 52 = Probi m i n [ Z* ( u) > - w]) 0 < u <t 0 = Probi m a x [ Z * (u ) < w]} OSuSt 0 2 7 2 7 /• 2 r y 2 < v e x p - [ — — ] d y y 2 n [ l - e ^ 2 r t ] o [ l - e " 2 r t ] The l a t t e r expression c l e a r l y approaches 0 as w approaches 0, proving the second boundary c o n d i t i o n . I t fol l o w s that as w approaches 0, E [ Z w ( t ) ] q approaches 0, which completes the proof of the second boundary c o n d i t i o n . QED g2 C o r o l l a r y 1. The p a r t i a l d e r i v a t i v e - Q ( w , t ) ^ o . dw2 Proof: I t i s s e l f evident that - 5 - Q ( t ) £o . S i m i l a r l y , i t i s c l e a r that -^-Q{w, t)2o» since f o r s<t, the set {min u < g x(u)<0} i s subset of the set {min u < t x(u)<o} But Q(w,t) = 2 [-=— Q{w, t) - (rx + Y) - 3 - Q ( vr, t) ], which must a w 2 dt 0 w c l e a r l y be non-negative. QED C o r o l l a r y 2. For any given w>0, the p a r t i a l d e r i v a t i v e - 3 - Q ( W I T ) remains bounded f o r large t This f o l l o w s immediately from C o r o l l a r y 1, as — Q { W i t ) i s both i n c r e a s i n g and bounded from above by 0. QED. I I . PROOF OF RESULT 2 53 Notation: 1. x * ( t ) i d e n t i f i e s the s t o c h a s t i c process f o r wealth, without absorption, with s t a r t i n g value w: dx*(t) = r x * ( t ) d t + Ydt + dz; x*(0) = w. 2. x ( t ) i d e n t i f i e s the corresponding process w i t h absorption. Thus, dx(t) = r x ( t ) d t + Ydt + dz with absorption at x = 0 ; x (0) = w. 3. V(w,t) = u(w,t) from the proof of Result 1, so that V(w, t) = E [e" r tx (t) ] . 4 . v n (w) = V (w, n) . 5. hn(w) = V t(w,n). Thus, since V s a t i s f i e s : K V w w + (rw+Y)Vw - rV = Vt. I t f o l l o w s that v n(w) and hn(w) s a t i s f y : l/2v„"(w) + (rw+Y)vn' (w) - rv n(w) = h n (w) . 6. v(w) i s defined as v(w) = l i m t„„V(w,t), which l i m i t i s shown to e x i s t i n p r o p o s i t i o n 8. The proof uses Result 1 and proceeds as a s e r i e s of p r o p o s i t i o n s . The p r o p o s i t i o n s u l t i m a t e l y show uniform convergence of each of v n" (w) , v n' (w) , vn(w) , and hn(w) . Proof of the r e s u l t f o l l o w s 54 e a s i l y , once uniform convergence i s shown. P r o p o s i t i o n 1: V(w,t) i s bounded as t approaches i n f i n i t y . S p e c i f i c a l l y , V(w,t)< w + Y/r ( l - e - r t ) Proof: V(w,t) i s equal t o : V(w, t ) = / Y P o (w, t , dy) o where Po(w,t,dy) i s the t r a n s i t i o n p r o b a b i l i t y : Prob{e- r tx (t) <= dy / x(0) = w} To evaluate t h i s i n t e g r a l , i t i s necessary to consider the corresponding i n t e g r a l f o r the process without absorption, x * ( t ) . From Arnold, theorem 8.2.10, x* (t) i s normally d i s t r i b u t e d . The mean value of x * ( t ) s o l v e s the d i f f e r e n t i a l equation m' (t) = rm(t) + Y ; m(0) = w which i m p l i e s that m(t) = w e r t + Y/r [e r t - 1] . The variance of x*(t) solves the d i f f e r e n t i a l equation 55 K'(t) = 2rK(t) + 1, K(0) = 0 which i m p l i e s that K(t) = (l / 2 r ) [e 2 r t - 1] These r e s u l t s imply that x*e~ r t i s d i s t r i b u t e d as: N(w + Y / r [ l - e"rt] , (l / 2 r ) [1 - e"2rt] ) Let Po*(w,t,dy)= Prob {e" r tx* (t) c dy / x* (0) = w} , the t r a n s i t i o n p r o b a b i l i t y f o r x * ( t ) , the process without absorption. V(w,t) can be r e w r i t t e n as: V(w,t) = | y [Po* [w, t, dy) -Probix* ( t) c d y D m i n x*(s ) < o l ] V(wrt) = j yPo* (w, t, dy) ~ J yPo* (w, t, dy) ~ J yProb''x* (t) c d y >> m i n 0 < s < t x * (s) $0 For y<0, Prob{x* (t) cdy} = Prob {x*(t)cdy n min 0 < s < t x* (s) < 0}. Thus, V(w,t) can be w r i t t e n as: which can a l s o be w r i t t e n as: 56 This f i r s t i n t e g r a l i n t h i s expression i s j u s t the expected value of e~rtx* (t) , which i s equal to w+Y/r (l-e~ r t) . As w e l l , i t i s c l e a r that the second i n t e g r a l i s p o s i t i v e . To see t h i s , consider any sample path of x*(t) which i n t e r s e c t s the boundary at t=s, so that x*(s)=0. At the point when x*(s)=0, dx*(s) = r x * ( s ) + Ydt + odz = Ydt + odz. Accordingly, the process has p o s i t i v e d r i f t at t h i s p o i n t s=t, implying that f o r t>s, E [x*(t)/x*(s)=0] must be p o s i t i v e . In the r e s u l t , V(w,t)< w+Y/r (l-e- r c) . QED P r o p o s i t i o n 2: V(w,t) i s an i n c r e a s i n g f u n c t i o n of t . Proof: V(w,t+dt) - V(w,t) = E [e~ r ( t + d t )x (t+dt) - e- r tx(t)] Neglecting terms 0(dt 2) and higher, = (e" r t - e" r trdt) E[x(t+dt)] - e'rt E [ x ( t ) ] = e"rt; { E[x(t+dt)] - E [ x ( t ) ] - rdtE [x (t+dt) ] } I t i s c l e a r l y s u f f i c i e n t to show that the f u n c t i o n i n braces i s non-negative. E[x(t+dt)] - E ( x ( t ) ] - rdtE [x (t+dt) ] = / y P(w,t+dt,dy) - / y P(w,t,dy) - r d t / y P(w,t+dt,dy) ( A) 57 where P(w,t,dy) i s the t r a n s i t i o n p r o b a b i l i t y f o r the process x ( t ) : P(w,t,dy) = Prob { x ( t ) c dy / x(0)=w}. The expression / y P(w,t+dt,dy) can be replaced by: // z PI (w, t , dy, dt, dz) where PI(w, t , dy, dt, dz) i s the t r a n s i t i o n p r o b a b i l i t y : Prob{x (t+dt) <= dz n x ( t ) c dy / x(0)=w} The l a s t i n t e g r a l i s i n turn equal to: J7 z P(y,dt,dz) P(w,t,dy) = / E[x(t+dt) / x(t)=y] P(w,t,dy) S u b s t i t u t i n g t h i s l a s t expression i n t o (A) leads t o : E[x(t+dt)] - E ( x ( t ) ] - rdtE [x(t+dt) ] = / P(w,t,dy){E[x(t+dt)]/x(t)=y] - y - rdtE[x(t+dt)/x(t)=y]} I t thus s u f f i c e s to show t h a t : (1 - rdt) E[x(t+dt)/x(t)=y] - y > 0 The expression E[x(t+dt)/x(t)=y] i s equal to: / z P(y,dt,dz) 58 = / z P2*(y,dt,dz) where P2*(y,dt,dz) i s the t r a n s i t i o n p r o b a b i l i t y : Prob{x*(dt)c dz n min 0 s s s d t{x* (s) } >0 / x*(0)=y} = / z P*(y,dt,dz) - J z P3*(y,dt,dz) where P3*(y,dt,dz) i s the t r a n s i t i o n p r o b a b i l i t y : Prob{x*(dt)c dz n min 0< s s d t{x* (s) <0} / x*(0)=y} Consider f i r s t the i n t e g r a l J z P*(y,dt,dz). I t can be shown that t h i s i n t e g r a l i s equal to y + rydt + Ydt + o ( d t ) . Using the same method as i n the proof of P r o p o s i t i o n 1, the p r o b a b i l i t y d i s t r i b u t i o n of P*(y,dt,dz) i s normal: N( ye r d t + Y / r ( e r d t - l ) , l / 2 r (e 2 r d t-1) ) The i n t e g r a l i n question i s thus 2r~ 7 1 2r(z-yerdt-Y/rerdt + Y/r)2 1 z e x p [ - — ( ) ] dz { 2 l e 2 r d t _ 1 Usinq the s u b s t i t u t i o n FTr ^ J t the [ z - y e -Y/re + Y/r] ^ 2rdt i n t e g r a l becomes: 59 / 2 r d t _ J e [ u-i +ye +Y/re -Y/r]du J2^ rr- ' J 2 r W r i t i n g t i n place of dt, and using L e i b n i z ' Rule, t h i s expression can be d i f f e r e n t i a t e d w i t h respect to t to y i e l d : ' 2 0 2 r t 2 ru e r — true r t r t j e [ ———— +rye + Ye ] du J _ / 2 r t _ r=— V {-yect -Y/ceTt + Y/ r) V V As t approaches 0, t h i s expression approaches ry+Y, proving th; the i n t e g r a l under c o n s i d e r a t i o n i s y + rydt + Ydt + o ( d t ) . Now consider the i n t e g r a l : / z P 3 *(y,dt,dz) I t can be shown that t h i s i n t e g r a l i s o ( d t ) . To show t h i s , consider the p r o b a b i l i t y (again w r i t i n g t i n s t e a d of d t ) : Prob {min 0< s s t{x* (s) }<0} / x(0)=y} f o r any f i x e d y. This p r o b a b i l i t y i s simply the p r o b a b i l i t y that the process x* reaches the boundary of 0 during the i n t e r v a l (0, t ] . To show the order of t h i s p r o b a b i l i t y , i t i s s u f f i c i e n t to 60 examine a process without d r i f t x ^ ( t ) , where d x A ( t ) = dz. This process c l e a r l y has a greater p r o b a b i l i t y of reaching the 0 boundary than a process with p o s i t i v e d r i f t . An e x p l i c i t s o l u t i o n e x i s t s f o r the p r o b a b i l i t y of x ~ ( t ) reaching the boundary i n ( 0 , t ] , given that x A(0)=y. The s o l u t i o n , given i n Ross at p. 190, i s equal t o : o 2 2 r x — — / exp [ - - — ] dx J2fl ^ 2 The d e r i v a t i v e of t h i s i n t e g r a l with respect to t i s : 2 y -y , e x P [ - ^ r - ] The l i m i t of t h i s expression as t approaches 0 i s 0. This shows that the p r o b a b i l i t y of x A ( t ) reaching the 0 boundary i n (t,t+dt] i s o ( d t ) . The same must h o l d true f o r x*(t) and x ( t ) . Using these r e s u l t s , expression (C) now becomes: y + rydt + Ydt + o(dt) and the LHS of equation (B) can thus be w r i t t e n : 61 ( 1 - r d t ) ( y + rydt + Ydt + o(dt)) - y = Y + o(dt) which i s c l e a r l y p o s i t i v e f o r small dt. QED P r o p o s i t i o n 3. The p a r t i a l d e r i v a t i v e of V w i t h respect to t , Vt i s bounded. S p e c i f i c a l l y , Vt < Ye~rt. Proof: For a given w, dV = [E [e~ r ( t + d t )x (t+dt) ] - E [e _ r tx (t) ] ] dt = e"rt [ (1-rdt) E [x (t+dt) ] - E [ x ( t ) ] ] n e g l e c t i n g terms o(dt) = e" r t[ (1-rdt) / E[x(t+dt) / x(t)=y] P(w,t,dy)- / y P(w,t,dy)] = e"rt /P(w,t,dy) [(1-rdt) [/ zP*(w,dt,dz) - J" zP3* (w, dt, dz) ] -y] using equation (C) from P r o p o s i t i o n 3. Using the r e s u l t s of P r o p o s i t i o n 3, t h i s can a l s o be w r i t t e n as: = e-rt J" P(w,t,dy) [(1-rdt) [y + rydt + Ydt + o(dt) -o(dt) - y] = e'rt / P(w,t,dy) [Ydt + o(dt)] I t i s c l e a r , however, that \ P(w,t,dy) < 1, as the i n t e g r a l represents the p r o b a b i l i t y that x has not reached the boundary a 62 time t . I t f o l l o w s that V t(w,t) < Y e"rt. QED P r o p o s i t i o n 4 . The sequence of func t i o n s v n' (w) i s non-decreasing i n n, i . e . v n + 1' (w) > v n' (w) . (For ease of n o t a t i o n , t ' s h a l l be used to i n d i c a t e t+dt) Proof: Let x l ( t ) and x2(t) denote, r e s p e c t i v e l y , processes with s t a r t i n g values w and w+dw, i . e . x l ( 0 ) = w, x2(0) = w+dw. To compare v n + 1' (x) and v n' (x) , i t s u f f i c e s t o look at the d i f f e r e n c e : E [e _ r t'x2 (t' ) - e- r t ' x l ( t ' ) - {e" r tx2(t) - e " r t x l ( t ) } ] = e"rt E[x2(t') - x2(t) - r d t x 2 ( t ' ) + r d t x l ( t ' ) + x l ( t ) - x l ( t ' ) ] a f t e r n e g l e c t i n g terms of 0(dt 2)and higher order. E l i m i n a t i n g the f a c t o r e"rc: = (1-rdt) / z P(w+dw,t', dz) - / z P(w+dw,t,dz) 63 - [ ( l - r d t ) / z P(w,f,dz) - / z P(w,t,dz)] = / / ( l - r d t ) z P(y,dt,dz) P(w+dw,t,dy) - / y P(w+dw,t,dy) - // ( l - r d t ) z P(y,dt,dz) P(w,t,dy) + / y P(w,t,dy) = // ( l - r d t ) z P(w+dw,t,dy) [P*(y,dt,dz) - P3* (y, dt, dz) ] - // ( l - r d t ) z p(w,t,dy) [P*(y,dt,dz) - P3* (y, dt, dz) ] - J y P(w+dw,t,dy) + J y P(w,t,dy) = / P(w+dw,t,dy) [ / ( l - r d t ) z [P*(y,dt,dz) - P3*(y,dt,dz) -y] - /' P(w,t,dy) [ / ( l - r d t ) z [P*(y,dt,dz) - P3*(y,dt,dz) -y] From the proof of P r o p o s i t i o n 2: J z P*(y,dt,dz) = y + rydt + Ydt + o(dt) J z P3*(y,dt,dz) = o(dt) S u b s t i t u t i n g these r e s u l t s and e l i m i n a t i n g terms of o ( d t ) , the above expression becomes: = JP(w+dw,t,dy) [ ( l - r d t ) (y+rydt+Ydt+o(dt)-o(dt)-y) ] - |P(w,t,dy) [ ( l - r d t ) ( y+rydt+Ydt+o(dt)-o(dt)-y) ] 64 = Ydt [ / P(w+dw,t,dy) - / P(w,t,dy) ] The two i n t e g r a l s represent, r e s p e c t i v e l y , the p r o b a b i l i t i e s that x2 and x l have not reached the 0 boundary by time t . As x2 has a higher s t a r t i n g value, the d i f f e r e n c e between the i n t e g r a l s i s unambiguously p o s i t i v e . In the r e s u l t , the d i f f e r e n c e : E [e"rt'x2 ( t 1 ) - e- r t ' x l ( t ' ) - {e"rtx2 (t) - e " r t x l ( t ) ) ] i s unambiguously p o s i t i v e . QED P r o p o s i t i o n 5 . Each of the f u nctions hn(w) = vt(w,n) i s non-decreasing i n w. Proof: As i n the proof of p r o p o s i t i o n 4, l e t x l ( t ) and x2(t) denote, r e s p e c t i v e l y , processes with s t a r t i n g values w and w+dw, i . e . x l ( 0 ) = w, x2(0) = w+dw. Again, t+dt w i l l be denoted t ' . To compare hn(w+dw) and hn(w) , i t s u f f i c e s to look at the d i f f e r e n c e : E [e"rt'x2 (t' ) - e" r tx2(t) - (e- r t'xl (t • ) - e " r t x l ( t ) ) ] 65 = E [e~rt'x2 (t' ) - e- r t'xl(t') - (e"rcx2 (t) - e - r t x l ( t ) ) ] The l a t t e r expression was already shown to be non-negative i n the proof of p r o p o s i t i o n 4. QED P r o p o s i t i o n 6. The functions hn(w) converge uniformly to h(w)=0 on any closed, bounded i n t e r v a l [a,b] w i t h b>a>0. Proof: P r o p o s i t i o n 1 shows that f o r any w, V(w,t) i s bounded as t approaches i n f i n i t y . P r o p o s i t i o n 2 shows that f o r any w, V(w,t) i s i n c r e a s i n g i n t . I t fol l o w s that V t(x,t) approaches 0 as t approaches i n f i n i t y , showing pointwise convergence of hn(w) to 0. To show u n i f o r m i t y of convergence, p r o p o s i t i o n 6 shows that each hn(w) i s non-decreasing i n w, or h n(w)<h n(b) f o r w i n [a,b] . Since h n(b) converges to zero, f o r every e>0, there e x i s t s N such that n>N i m p l i e s hn(w) < h n(b) < e f o r a l l w i n [a,b]. QED P r o p o s i t i o n 7. The sequence vn(w) i s un i f o r m l y convergent on any closed i n t e r v a l [a,b] with b>a>0. 66 Proof: I t f o l l o w s from p r o p o s i t i o n 3 that n + / v Y . . . / - r s , x r - r n - r ( n + * r ) . x - r n (w) < I Ye ds - — [e -e ] < — e n J r r n The f i n a l value i s independent of w, and can be made a r b i t r a r i l y small by choosing n l a r g e . I t f o l l o w s that the sequence vn(w) s a t i s f i e s the Cauchy c r i t e r i o n f o r uniform convergence. QED P r o p o s i t i o n 8. v n' (w)>l. Proof: From the boundary c o n d i t i o n s a p p l i c a b l e to V(w,t), V(w,0)=w, from which i t f o l l o w s that 6V(w,0) /6w = 1. P r o p o s i t i o n s 4 and 5 show that d2V(w, t) /dwdt > 0, from which i t f o l l o w s that 8 v(w,t ) / 3 w ^ 1. P r o p o s i t i o n 9. vn"(w)<0 Proof: From the d e f i n i t i o n of v n(w), each v n(w) s a t i s f i e s : v (w) -n +k vn"(w) = 2[h n + r v n - (rw+Y) v'n(w) ] 67 From p r o p o s i t i o n 3, hn<Ye~rn. As w e l l , from P r o p o s i t i o n 1, V(w,n)< (w+Y/r (l-e~ r n) ) . Thus, using the r e s u l t from P r o p o s i t i o n 8 that v'n(w)>1: vn"(w) < 2 [Ye"rn + rw + Y - rwe"rn - Ye"rn - rw - Y] < 2[-rwe" r n] < 0 P r o p o s i t i o n 10. For each w, the sequences v n' (w) and v n" (w) are bounded as n approaches i n f i n i t y . Proof: Consider f i r s t the sequence v n' (w) and assume that there e x i s t s some w* such that v n' (w*) i s unbounded f o r la r g e n. For any M>0, there must then e x i s t some j f o r which V j' (w*)>M/w*. From P r o p o s i t i o n 9, V j "(w)< 0 f o r w i n (0,w*), from which i t f o l l o w s that v.,' (w) >M/w* f o r a l l w i n (0,w*) . This i n turn i m p l i e s that Vj(w*)>M, which would imply that vn(w) i s unbounded. This c o n t r a d i c t s the r e s u l t of P r o p o s i t i o n 1. This completes the proof that v n' (w) i s bounded. Boundedness of vn"(w) f o l l o w s e a s i l y . Each vn"(w) s a t i s f i e s : 68 vn"(w) = 2 [ h n + r v n ( w ) - (rw+Y) v „ ' (w) ] As each term on the RHS i s bounded f o r larg e n, the same must hold f o r v n" (w) . P r o p o s i t i o n 11. The sequence vn'(w) i s unifo r m l y convergent on any i n t e r v a l [a,b] wit h b>a>0. Proof: P r o p o s i t i o n 10 e s t a b l i s h e s that v n' (w) i s bounded as n increases. P r o p o s i t i o n 4 e s t a b l i s h e s that v n + 1' (w) > v n' (w) . These two r e s u l t s e s t a b l i s h pointwise convergence to a fu n c t i o n v*(w) . P r o p o s i t i o n 9 e s t a b l i s h e s that each vn'(w) i s monotonic, so the same must hold f o r v*(w). As a monotonic f u n c t i o n , v*(w) can have only jump d i s c o n t i n u i t i e s . In the neighbourhood of any sue d i s c o n t i n u i t y , the sequence v n" (w) would have t o diverge. This, however, c o n t r a d i c t s the second r e s u l t of P r o p o s i t i o n 10, that the sequence v n" (w) i s bounded. I t fo l l o w s that v* (w) i s continuous. Uniform convergence of v n' (w) now f o l l o w s from D i n i ' s Theorem, 69 which holds that a monotonic sequence of continuous functions converging to a continuous l i m i t f u n c t i o n must converge uniformly on a closed i n t e r v a l . PROOF OF RESULT 2. Result 2 can now be proved s h o r t l y from the above p r o p o s i t i o n s . Each v n(w) s a t i s f i e s : vn"(w) + (rw+Y)vn' (w) - r v n = h n(w). (F) By P r o p o s i t i o n 6, h n (w) converges uniformly to 0 on any i n t e r v a l [a,b] . v n(w) converges unif o r m l y by P r o p o s i t i o n 7 , t o l i m i t f u n c t i o n v(w). By p r o p o s i t i o n 11 v n' (x) converges unif o r m l y as w e l l . Thus, by Apostol, theorem 9.13, vn'(w) converges uniformly to v'(w). S i m i l a r l y , (rw+Y)v n' (w) converges unif o r m l y to (rw+Y)v' (w), as both f a c t o r s are bounded on any i n t e r v a l [a,b]. Each vn"(w) s a t i s f i e s : 70 vn"(w) = 2{ hn(w) + rv n(w) - (rw+Y) v n ' (w) } . Since each term on the RHS converges uniformly, so must vn" (w) . By Apostol, theorem 9.13, vn"(w) converges to v" (w) . I t i s thus p o s s i b l e to take l i m i t s on both side s of equation (F), which y i e l d s the equation to be proved: Vi v"(w) + (rw+Y)v' (w) - rv(w) =0. This equation has been proved f o r any i n t e r v a l [a,b], with b>a>0. Since both a and b are a r b i t r a r y , the equation must hold f o r w > 0 . F i n a l l y , the a p p l i c a b l e boundary c o n d i t i o n s f o l l o w e a s i l y . Since v n(0)=0 f o r a l l n, i t i s c l e a r that v(0) = 0. As w e l l , since each v n (w) approaches w + Y / r [ l - e ~ r n ] f o r l a r g e w, i t follows that v(w) approaches w + Y/r f o r large w. QED 71 CHAPTER I I I - THE DEMAND FOR HIGH INTEREST RATE LOANS The r e s u l t s of Chapter I I can be used to analyse the w i l l i n g n e s s of a n e a r l y bankrupt f i r m to borrow at high i n t e r e s t r a t e s . In t h i s chapter, the value f u n c t i o n derived i n Chapter I I i s adapted to account f o r such borrowing. The r e s u l t i n g value f u n c t i o n i s then used t o c a l c u l a t e numerical examples, which show a p o s i t i v e demand f o r loans at s t r i k i n g l y high i n t e r e s t r a t e s . The a n a l y s i s i s most apt f o r small to medium s i z e d firms, which often have few options to d i v e r s i f y r i s k or r a i s e new c a p i t a l when f a c i n g f i n a n c i a l d i s t r e s s . In p r a c t i c a l terms, such firms must o f t e n accept high i n t e r e s t r a t e s to r a i s e operating c a p i t a l r e q u i r e d t o stave o f f bankruptcy. (Note that while finance l i t e r a t u r e o f t e n r e f e r s to the era of the "junk bond", t h i s chapter does not r e a l l y i l l u s t r a t e that phenomenon. Junk bonds are seldom f l o a t e d as a means of avoiding imminent bankruptcy. This chapter more a c c u r a t e l y p o r t r a y s the demand f o r s e r v i c e s of loan sharks, or the high i n t e r e s t r a t e lenders operating i n secondary l e n d i n g markets.) As i n Chapter I I , assume the f i r m has steady revenue Y + odz, and 72 faces bankruptcy when i t s wealth drops to a s p e c i f i e d l e v e l A. In absence of borrowing, the fi r m ' s wealth evolves according t o : dx(t) = r x ( t ) d t + Ydt + odz; x(0)=w Now assume i n a d d i t i o n t h a t : 1. The f i r m i s able to borrow funds (B) at an i n t e r e s t r a t e R>r. 2 . The borrowed funds can be in v e s t e d at the same r a t e as the firm' s own c a p i t a l , producing a r e t u r n at rat e r . 3 . The a d d i t i o n a l borrowing does not change the bankruptcy t h r e s h o l d , which s t i l l occurs at x(t)=A. I f A represents a minimum wealth l e v e l imposed by secured c r e d i t o r s , the a d d i t i o n a l borrowing can be i n t e r p r e t e d as unsecured debt. The new lender earns h i s high i n t e r e s t r a t e , R, by accepting the r i s k of an unsecured loan. The new debt creates debt s e r v i c e charges equal to RBdt. By 73 assumption, t h i s i s greater than the a d d i t i o n a l revenue the f i r m can earn (rBdt) by i n v e s t i n g the borrowed funds. The o f f s e t t i n g advantage to the f i r m i s that the borrowed funds provide an e x t r a cushion against bankruptcy. As bankruptcy s t i l l occurs at the point x(t)=A, the firm's i n i t i a l margin from bankruptcy i s w+B-A, instead-of w-A. The borrower's wealth now evolves according to: dx(t) = r x ( t ) dt + Ydt - RBdt + odz with absorption at x=A. The discount rate i s again r. Define q=w+B as the t o t a l i n i t i a l assets of the c o r p o r a t i o n , i n c l u d i n g borrowed funds, so that x ( 0)=q. The borrower's value f u n c t i o n i s then: v(q) = v(w;B,R) = l i m t_„ E [e~ r t x ( t ) ] As a f u n c t i o n of q, v must now s a t i s f y 74 2 1 d v dv + (rq+Y-RB) - rv = 0 2 0 2 dg 2 d<7 subject to v(A)=0; v tending to q + Y/r - RB/r f o r large q. Expressed as a f u n c t i o n of q, the s o l u t i o n i s : Y RB v[q) = — ) (1 r r e x p [ ] r o ds rq+Y-RB e x p [ ] r O •ds This expression can be r e w r i t t e n as a f u n c t i o n of w, B, and R as f o l l o w s : oo e x p [ ] r o •ds Y RB rw + Y-{R-r)B v(w;RrB) = (w + B + —- ) (1 -r r i 0) / e x p [ ] r O -ds 75 The corporation's demand f o r loan funds can be c a l c u l a t e d using (1). For given s t a r t i n g wealth, w, and f o r a given borrowing r a t e , R, the c o r p o r a t i o n chooses the value of B which maximizes v. D i f f e r e n t i a t i o n of (1) y i e l d s expressions too d i f f i c u l t to manipulate i n t o a meaningful f i r s t order c o n d i t i o n . However, the f o l l o w i n g t a b l e s give numerical examples which i l l u s t r a t e the magnitude of the demand f o r high i n t e r e s t rate loans. The t a b l e s give values of v f o r d i f f e r i n g l e v e l s of w, B, and R. Other parameters are as f o l l o w s : A=0, Y=5, o=10, and r=.1. Q u a n t i t i e s marked with a s t e r i s k s represent maximizing values of B (from among those c a l c u l a t e d ) f o r the given borrowing i n t e r e s t r a t e , R. Table 1 - Borrower's values: w=l R = 11% 13% 15% 20% 30% 40% B = 1 13 12 . 9 12 . 9 12 . 7 12 .4 12 3 22 . 5 22 . 1 2 1 . 8 2 0 . 9 19 . 3 17 . 7 5 29 .4 28 .6 27 . 9 26 . 1 2 2 . 7 19 .6* 10 3 9 . 5 3 7 . 7 35 . 8 3 1 . 5 * 23 . 5* 16 . 6 20 46 .1 42 . 3 * 38 .4* 29 . 2 14 5 . 3 30 4 7 . 0 * 41 .4 35 . 6 22 :4 6 . 7 -76 Table 2 - Borrowers' values: w=3 R = 11% 13% 15% 20% 30% 40% B = 1 23 .4 23 . 7 23 .2 2 3 . 2 22 . 7 22 .1 3 31 3 0 . 5 30 .1 29 26 . 8 24 . 8 5 36 .2 35 .4 34 . 5 32 . 5 28 . 5* 24 . 8* 10 43 . 9 4 1 . 9 40 .1 3 5 . 5 * 26 . 8 19.3 20 48 .8 4 4 . 8 * 4 1 . 0 * 31 .6 15.6 6 30 49 .4* 43 .5 37 .7 24 . 2 7 .3 -Table 3 - Borrower's values: w=5 R = 11% 13% 15% 20% 30% 40% B = 1 32 . 6 32 . 5 32 . 3 32 31 .2 30 .5 3 38 .1 37 .6 37 . 1 35 .8 33 . 3 3 0 . 9 * 5 42 41 40 .2 37 . 9 3 3 . 6 * 29.4 10 47 .8 45 . 8 43 . 8* 3 9 . 1 * 30 21 . 9 20 51 . 3 4 7 . 3 * 43 . 4 33 . 8 17 .3 6 . 8 30 5 1 . 5* 4 5 . 6 39.8 26 8 -77 Table 4 - Borrower's values: w=10 R = 11% 13% 15% 20% 30% 40% B = 1 49 48 . 7 48.5 48 47 .0* 4 6 . 1 * 3 51. 5 50 . 9 50.3 48 . 8 45 . 8 42 . 9 5 53 . 3 52.3 51.3 48 . 8* 43 . 8 39 .1 10 55 . 9 53 . 9* 51.8* 46 . 8 37 . 1 28 . 1 20 57 . 1* 53 . 1 49.1 39.3 21.5 9 . 08 30 56 . 8 50.8 44 . 9 30.6 9 . 9 -These tab l e s demonstrate a strong demand f o r loan funds, even at what seem l i k e very high i n t e r e s t r a t e s . For example, a borrower with s t a r t i n g wealth equal to 5 would appear to have s u b s t a n t i a l s a f e t y from bankruptcy, without a d d i t i o n a l borrowing. His s t a r t i n g distance from the boundary t h r e s h o l d i s equal to a/2. In a d d i t i o n , he has expected p o s i t i v e cash flow equal to 5 + 5 ( . l ) =5.5. In s p i t e of t h i s , even at i n t e r e s t r a t e s of 30% and 40% he w i l l borrow, r e s p e c t i v e l y , amounts equal to 100% and 60% of h i s s t a r t i n g wealth. I t i s r e a d i l y apparent that the a b i l i t y to borrow w i l l be most valuable to a c o r p o r a t i o n close to bankruptcy. The marginal value of a d d i t i o n a l borrowing can be c a l c u l a t e d from (1), by f i x i n g w and R, and by c o n s i d e r i n g v as a f u n c t i o n of B. 78 D i f f e r e n t i a t i o n of (1) at the poin t w=A, B=0 y i e l d s : dv 2 (rA + Y) rO ] ds ro 1 + ( rA + Y) 2 ( rA + Y) e x p [ - ] -ro rO I e x P t ] ds rO This i s the same expression that was deriv e d i n Chapter I I , equation (8) as the marginal value of wealth at the bankruptcy boundary. As bankruptcy becomes imminent, avoiding bankruptcy becomes the paramount c o n s i d e r a t i o n . I t does not matter whether the wealth used to do that comes from borrowed funds or from the corporation's own endowed wealth. I t i s a l s o worth no t i n g that t h i s expression i s independent of the borrowing r a t e R. There i s no t h e o r e t i c a l maximum r a t e that the c o r p o r a t i o n w i l l pay. Facing imminent bankruptcy, i n f i n i t e s i m a l a d d i t i o n a l wealth w i l l be borrowed at any i n t e r e s t r a t e i f bankruptcy can be f o r e s t a l l e d . 79 CHAPTER IV - PROJECT FINANCING - VALUE FUNCTIONS This and f o l l o w i n g chapters w i l l examine s e v e r a l a d d i t i o n a l aspects of lender-borrower t r a n s a c t i o n s . These w i l l i n clude the' choice of an optimal debt-equity r a t i o , the e f f e c t s of a change i n the a b i l i t y of a borrower to grant s e c u r i t y , and the e f f e c t s of changes i n bankruptcy l e g i s l a t i o n . To complete the a n a l y s i s , both borrower and lender value f u n c t i o n s must be d e r i v e d . The value f u n c t i o n s of t h i s chapter are d e r i v e d f o r a p r o j e c t of f i x e d s i z e , w, which must be financed by a combination of debt, B, and equity, w-B. As the t o t a l s i z e of the p r o j e c t i s f i x e d , the value f u n c t i o n s can be used to i n v e s t i g a t e the f i r m ' s optimal c a p i t a l s t r u c t u r e , but not the optimal s c a l e f o r the p r o j e c t . S e c t i o n 1 adapts the borrower's value f u n c t i o n , c a l c u l a t e d i n Chapter I I , to a p r o j e c t financed p a r t l y by debt and p a r t l y by e q u i t y . S e c t i o n 2 d e r i v e s a value f u n c t i o n f o r the lender, based on the discounted value of h i s expected i n t e r e s t revenue. Under the maintained assumptions, t h i s problem as w e l l has a closed form s o l u t i o n . F i n a l l y , s e c t i o n 3 i n v e s t i g a t e s mathematical 80 p r o p e r t i e s of the value f u n c t i o n s which are used i n subsequent chapters. In a d d i t i o n to c a l c u l a t i n g value f u n c t i o n s , t h i s chapter also c a l c u l a t e s r a t e s of r e t u r n f o r both the borrower and lender. I t turns out that i n s e v e r a l of the subsequent problems ( p a r t i c u l a r l y those i n v o l v i n g a s i n g l e lender and a s i n g l e borrower), more i n t e r e s t i n g r e s u l t s are d e r i v e d i n maximizing the p a r t i e s ' r e s p e c t i v e r a t e s of r e t u r n , as opposed t o t h e i r value functions simpliciter. 1. Value Function of the Borrower / P r o j e c t Owner The s t r u c t u r e of the borrower's problem i s now s l i g h t l y d i f f e r e n t . The borrower i s seeking to finance a p r o j e c t of f i x e d s i z e w. w i s thus both the s i z e of the p r o j e c t and the borrower's t o t a l s t a r t i n g wealth. To finance the p r o j e c t , the borrower must chose a combination of debt, B, and equity, w-B. As before, the borrower's invested wealth earns a rate or r e t u r n r, and i n t e r e s t i s payable on debt at r a t e R. However, i t i s now assumed that R<r i . e . the i n t e r e s t 81 rate payable on debt i s l e s s than the p h y s i c a l rate of retu r n to c a p i t a l . (In the analyses of some problems, i t w i l l a l s o be assumed that the borrower inc u r s t r a n s a c t i o n costs i n arranging a loan. These w i l l be incorporated i n t o the value f u n c t i o n l a t e r on) . Bankruptcy no longer occurs at an a r b i t r a r y l e v e l A. Instead, bankruptcy occurs when the borrower's e q u i t y i s l o s t , or when wealth f a l l s to the l e v e l of the borrower's debt, B. In the r e s u l t , the bankruptcy t h r e s h o l d i s now endogenous. These assumptions confront the borrower with r e a l i s t i c t r a d e o f f s i n choosing a debt e q u i t y r a t i o . More debt increases the borrower's leverage, i n that the borrower i s able to invest borrowed funds at a r e t u r n (r) higher than h i s borrowing rate (R). However, incre a s e d debt a l s o increases the r i s k of bankruptcy, and hence the r i s k that s t a r t i n g e q u i t y w i l l be l o s t . As before, x ( t ) w i l l represent the e v o l u t i o n of the borrower's wealth over time, given that x(0) = w. w w i l l be the nominal argument of the value f u n c t i o n . In d e r i v i n g the value f u n c t i o n the debt l e v e l , B, and the i n t e r e s t r a t e on debt, R, w i l l be 82 taken as parametric. As a p r a c t i c a l matter, however, R and B w i l l be considered as the choice v a r i a b l e s . Once the value f u n c t i o n i s derived, the i n v e s t i g a t i o n s of i n t e r e s t w i l l be i n determining the e f f e c t of d i f f e r e n t values f o r the "parameters", B and R. The s t o c h a s t i c component of revenue i s s t i l l equal to odz. A l l agents are r i s k n e u t r a l , and value t h e i r assets as the.discounted expected value of future.revenue. The discount r a t e i s again set exogenously at r. From the assumptions set out above, x(t) evolves according to: dx(t) = r x ( t ) d t - RBdt + odz. Let v(w;B,R) represent the value f u n c t i o n of the borrower: v(w;B,R) = l i m t-.„ E[e" r t x ( t ) ] . v(w;B,R) must then s a t i s f y : 83 1 2 d v — O + {rw-RB) - - rv = 0 2 d w 2 dw The boundary con d i t i o n s are: v(w;B,R) approaches 0 as w approaches B v(w;B,R) approaches w-RB/r as w becomes a r b i t r a r i l y l a r g e . Using the same general s o l u t i o n s as were derived i n Chapter I I , v(w;B,R) i s equal to: RB v(w;B,R) = [ (w- ) (1 r / exp [ ] r O •ds ) ] (1) exp [ ] r O •ds To allo w f o r t r a n s a c t i o n c o s t s equal to cb, i t i s simply require to subtract t h i s q u a n t i t y from the value given above. To c a l c u l a t e the borrower's r a t e of re t u r n , the value f u n c t i o n i s di v i d e d by (w-B). The borrower's rate of r e t u r n , w i t h t r a n s a c t i o n costs included, i s thus: 84 exp [ ] ro -ds v(w;B,R) 1 RB rw-RB — = [ ] [ ( w - ) (1 -w-B w~B r ) ~ cb] (2) / exp [ ] r O • ds 2. Value Function of the Lender The value f u n c t i o n of the lender i s more invo l v e d , and requires the a p p l i c a t i o n of d i f f e r e n t mathematical r e s u l t s . (a) Assumptions The assumptions concerning the lender's revenue and cost s t r u c t u r e are as f o l l o w s . 1. The lender r e c e i v e s i n t e r e s t revenue equal to RB u n t i l bankruptcy intervenes. I n t e r e s t revenue ceases on bankruptcy. 2. The lender a l s o receives "salvage value" on bankruptcy. The salvage value i s equal to SaB. 85 3. The lender's cost of c a p i t a l i s at rate p<R<r. Bankruptcy-does not r e l i e v e the lender of t h i s cost. 4. The lender faces t r a n s a c t i o n costs equal to c l i n arranging a loan. (b) The B a s i c Value Function - The V a l u a t i o n of I n t e r e s t Revenue The value f u n c t i o n must be derived i n steps. The f i s t step i s to derive the b a s i c value of expected future i n t e r e s t revenue (l 1(w;B,R)), l e a v i n g salvage value and t r a n s a c t i o n costs f o r l a t e r . I n t e r e s t revenue at time t i s equal to RB i f bankruptcy has not occurred p r i o r to time t , but 0 i f bankruptcy has occurred p r i o r to time t . The expected discounted value of i n t e r e s t revenue at time t i s thus equal t o : e"rt RB (l-Q (w, t) ) where Q(w,t;B,R) i s the p r o b a b i l i t y that bankruptcy occurs at or 86 before time t , given s t a r t i n g wealth equal to w. The discounted value of a l l future i n t e r e s t revenue, l1{w;E,R), i s thus equal t o : 1 iw;B,R) = je~rtRB [1-Q(w, t; B, R) ) dt (3) (For the remainder of t h i s s e c t i o n , to economize on not a t i o n , B and R w i l l be suppressed as arguments of Q and 1 J . To evaluate t h i s expression, note t h a t : dl -dw J o dQ RB { 1 - - = — ) dt Ow RB r _ r t dQ - e RB^dt r J ow o and: = / e ' ' K B d - — ) d t dw o ow RB 7" -rt d2Q RB d t dw' Using these expressions: 87 1 , d 1i d l i RB r - r t 1 2 d2Q dQ — o 2 + (rw-B[R-r) ) = - e [ — O + [rw~B[R-r) ) -g-2 d w 2 dw r { 2 d w 2 dw However, by Result 1 of Chapter I I , the expression i n square brackets i s equal to dQ/dt. Using i n t e g r a t i o n by pa r t s y i e l d s - re dQ f e ' r t - ^ d t = - Q(w,0) •+ f re'rtQ(w, t) dt o By the boundary c o n d i t i o n s f o r Q, Q(w,0)=0. As w e l l , the f i n a l term i n t h i s expression i s equal to r l x (w) . Thus, 11 (w) must s a t i s f y : 1 rf2i d l x — O2 + {rw-B (R-r) ) - rl =0 2 dw2 d w This equation i s i d e n t i c a l to equation (2) of Chapter I I , and has the same general s o l u t i o n s . The boundary c o n d i t i o n s a p p l i c a b l e to l x are: lx(B)= 0 lx approaches RB/r as w becomes a r b i t r a r i l y l a r g e . Using the s o l u t i o n s d e r i v e d i n Chapter I I , 11 i s thus equal t o : 88 [rw-RB) j RB RB rw-RB 1 (w;B,R) = - [ i r r exp [ -] rO ds (4) exp [ -] [rB-RB) j rB-RB rO ds (c) Salvage Value To incorporate salvage value i n t o the lender's v a l u a t i o n problem, i t i s necessary to consider the p r o b a b i l i t y that bankruptcy occurs at a s p e c i f i c time, t . I f Q(w,t;B,R) represents the p r o b a b i l i t y that bankruptcy has occurred at or before time t , then dQ/dt must represent the p r o b a b i l i t y that bankruptcy occurs at time t . Let l2(w,-B,R) represent the expected discounted value of the salvage recovered by the lender. I f the lender recovers SaB when bankruptcy occurs, l 2(w;B,R) must be equal t o : 89 f - r s dQ(w,s;B,R) 1 (w;BrR) = le SaB = ds 2 J ds o _rs dQ{w, s;B, R) SaBle = ds oo o However, i t i s c l e a r than both Q(w,0;B,R)=0 and limt_.„ e _ r t[Q(w,t;B,R) ] =0. I n t e g r a t i o n of the above expression by-par t s thus y i e l d s : 1 (w;R,B) = le Q(w,s;B,R)di 2 r J o The i n t e g r a l i n t h i s expression i s the same i n t e g r a l encountered i n equation (3), i n determining the value of the stream of i n t e r e s t income. Following the same steps used to solve (3), the value of the salvage i s thus: 2 s exp [ ] ro r ro {rw-RB) I ds 1 (w;B,R) = S a B [ ] (5) 2 s exp [ -] ro f r  rB-RB) I ds rB-fiB S 90 Two more components are r e q u i r e d to complete the value f u n c t i o n . F i r s t , i t i s assumed that the lender has a cost of c a p i t a l equal to p , with p<R<r. Bankruptcy does not a f f e c t the lender's cost of c a p i t a l - he must continue to bear that cost even i f h i s own loan to the borrower goes bad. The discounted cost of the c a p i t a l used to fund a loan of s i z e B i s thus p B / r . Second, i t i s assumed that the lender i n c u r s t r a n s a c t i o n costs equal to c l i n n e g o t i a t i n g the loan. Again, t r a n s a c t i o n costs are independent of the s i z e of the loan. The lender's value f u n c t i o n , l(w;B,R) i s thus equal t o : l(w,;B,R) = l ^ w ^ R ) + l 2(w;B,R) - p B/r - c l . S u b s t i t u t i n g expressions given above f o r l x and 12, l(w;B,R) i s equal t o : 91 2 S exp [ -] ro r ro [rw-RB) I ds RB OB RB rw-RB s w;B,R) = - + (SaB- ) ( ; ) (6) r r r s 2 exp [ ] °r r O [rB-RB) J ds rB-RB s The lender's r a t e of r e t u r n , L(w;B,R) w i l l be equal t o : l[w;B,R) 1 RB pB RB ———— = [ - ] [ - + [SaB~ ) ( B B r r r [rw-RB) J exp [ -] ro -ds ) - cl] exp [ -] c ra [rB-RB) I ds Henceforth, w w i l l be suppressed as an argument of both 1 (the value function) and L (the r a t e of r e t u r n f u n c t i o n ) , which w i l l be w r i t t e n r e s p e c t i v e l y as 1(B,R) and L(B,R). Having derived the value f u n c t i o n s , i t i s worthwhile to make e x p l i c i t some of the r e s t r i c t i o n s that have been incorporated i n t o the un d e r l y i n g assumptions. F i r s t , the bankruptcy t h r e s h o l d has been endogenized, but only 92 p a r t l y so. The f i r m can a l t e r i t s bankruptcy t h r e s h o l d by s e l e c t i n g a d i f f e r e n t value f o r B. However, once B i s chosen the bankruptcy t h r e s h o l d i s f i x e d - n e i t h e r the borrower nor lender can change i t . Other assumptions, p e r m i t t i n g e i t h e r the lender or borrower to set the bankruptcy threshold, could be used. For example, Leland (1994) examines cases i n which the borrower has complete autonomy i n s e t t i n g the bankruptcy t h r e s h o l d , and other cases when lenders have c o n t r a c t u a l r i g h t s to fo r c e l i q u i d a t i o n . There i s no reason i n p r i n c i p a l why the value functions of t h i s chapter could not be adapted to f o l l o w Leland's a n a l y s i s , and that represents a p o s s i b l e extension of t h i s t h e s i s . Second, the form of contract between the borrower and lender i s imposed exogenously. There i s nothing i n the a n a l y s i s that makes a debt contract optimal - i t i s simply assumed as the form of c o n t r a c t . T h i r d , the a n a l y s i s does not permit e i t h e r r e n e g o t i a t i o n of the terms of the contract or dynamic adjustment of the fi r m ' s c a p i t a l s t r u c t u r e over time. Dynamic adjustment of a firm's c a p i t a l s t r u c t u r e was i n v e s t i g a t e d by Mauer and T r i a n t i s (1994), who note the mathematical complications introduced by a v a r i a b l e c a p i t a l 93 s t r u c t u r e , and who r e s o r t t o numerical a n a l y s i s i n the absence of a c l o s e d form s o l u t i o n . Debt r e n e g o t i a t i o n has been examined both i n the case of sovereign debt (eg. Krugman (1988) and Eaton and G e r s o v i t z (1981)) and i n the case of a f i r m (eg. M e l l a - B a r r a l and Perraudin (1997)). The l a t t e r paper begins w i t h an a n a l y s i s that f o l l o w s Leland (1994), and then considers what re n e g o t i a t i o n s might occur near bankruptcy, when bankruptcy costs give the lender an i n c e n t i v e to avoid l i q u i d a t i o n . A s i m i l a r a n a l y s i s using the value f u n c t i o n s of t h i s t h e s i s represents another p o s s i b l e extension of t h i s t h e s i s . I t i s l i k e l y that with bankruptcy costs of s u f f i c i e n t magnitude, r e n e g o t i a t i o n could increase the value f u n c t i o n s of both borrower and lender. As i n M e l l a - B a r r a l and Perraudin, i t i s a l s o l i k e l y that the d i s t r i b u t i o n of any gains from trade would depend on the bargaining s t r u c t u r e imposed on the p a r t i e s . While the assumptions of t h i s t h e s i s may seem r e s t r i c t i v e , many other analyses of bankruptcy i n v o l v e some form of l i m i t a t i o n on a firm's a c t i o n s . For example, Leland (1994) assumes that the f i r m r a i s e s new e q u i t y to pay debt s e r v i c e charges, but r a i s e s no a d d i t i o n a l e q u ity, even when bankruptcy i s near. Milne and Robertson (1996) assume that the f i r m can pay out dividends, but 94 that r a i s i n g new equi t y ( i . e . a negative dividend) i s impossible. Some l i m i t a t i o n on f i r m a c t i o n may be i n e v i t a b l e i n any a n a l y s i s of bankruptcy - a f i r m with u n r e s t r i c t e d access t o complete c a p i t a l markets would l i k e l y never go bankrupt. 3. The function M(B.R) Following the n o t a t i o n of Chapter I I (and t r e a t i n g w as parametric), define the f u n c t i o n M(B,R) as f o l l o w s : exp [ 2 s rc7 r r u [rw-RB) I ds rw-RB s M[B,R) = (7) s e x p [ - -] r o r r o rB-RB) J ds rB-RB S A s l i g h t l y d i f f e r e n t v e r s i o n of the f u n c t i o n M (under s l i g h t l y d i f f e r e n t model s p e c i f i c a t i o n s ) was p r e v i o u s l y given i n Chapter I I , i n a n a l y s i n g the value f u n c t i o n derived i n that chapter. The f u n c t i o n l 2(w;B,R), d e r i v e d i n s e c t i o n 2(c) as the salvage value r e c e i v e d by the lender on bankruptcy, a s s i s t s i n i n t e r p r e t i n g the f u n c t i o n M. Equation (5) defines 1 2 as: 95 (rw-RB) j" exp [ ] ro •ds 1 ( V ; B,R) = SaB [ 2 (rB-RB) j exp [ ] ro ds From (7), t h i s expression 1 2 can be r e w r i t t e n as 1 2 = SaBM. Since SaB i s the amount r e c e i v e d by the lender when bankruptcy occurs, M must represent the discounted sum ( i . e . i n t e g r a l ) of the p r o b a b i l i t i e s of bankruptcy o c c u r r i n g at a l l f u t u r e times. Again using (7), the value f u n c t i o n of the lender (6) can be w r i t t e n as: l(w;B,R) = RB/r - p B/r + B(Sa-R/r)M This expression can be f u r t h e r r e w r i t t e n as RB/r(1-M) - p B / + BsaM, an expression which provides f u r t h e r i n s i g h t i n t o the lender's value f u n c t i o n . The terms p B/r + BSaM have already been explained. However, the term RB/r(1-M) re q u i r e s f u r t h e r explanation. Absent bankruptcy 96 considerations, the "fundamental value" of the lenders' revenue stream would be RB/r. This fundamental value i s discounted by the value (1-M), w i t h M representing the discounted sum of the p r o b a b i l i t i e s of bankruptcy o c c u r r i n g at d i f f e r e n t p o i n t s i n the fu t u r e . At the p o i n t of bankruptcy, M=l and the "fundamental value" of the revenue stream disappears. A s i m i l a r p a t t e r n appears manner of the other functions which w i l l be considered l a t e r , and i n the d e c i s i o n making r u l e s f o r borrowers and lenders. The equations i n question r e f l e c t a "fundamental" value which becomes i r r e l e v a n t near bankruptcy, l e a v i n g as relevant only c o n s i d e r a t i o n s d i r e c t l y a f f e c t i n g the p r o b a b i l i t y of bankruptcy. The f u n c t i o n M appears i n s u f f i c i e n t l y many places that i t s p r o p e r t i e s merit i n v e s t i g a t i o n . F i r s t , i t should be p o i n t e d out that M permits of a simple "normalization". Making the s u b s t i t u t i o n z 2 = s 2/(ro 2) i n both integrands above, M can be w r i t t e n as: 97 — 2 rw-RB r exp[~z ] dz / ~ , ^ 2 rw-RB ° J r rw-RB Z J( ) M = ^ = ?J1 (8) . 2 , rB-RB v / rB-RB r exp[~z ] dz J( O j r rR — RR Z 2 o^r" where J(x) i s the f u n c t i o n defined i n Chapter I I , s e c t i o n 5: r exp [ -s 2] J(x) - xl ds s 2 To analyse the d e r i v a t i v e s of M, i t i s convenient to define the q u a n t i t i e s E l , F l , I I and J l as f o l l o w s : exp [ El = exp[ (rw-RB) rO 11 oo j exp [ ] ds Fl Jl rw-RB) r O rw-RB) (9) exp [ r w - R B r O (rw-RB) J -rw-RB rO •ds R e c a l l that by i n t e g r a t i o n by p a r t s , J l = E l - 2(rw-RB)11/(ro 2) . This expression i s equal to 1 i f (rw-RB) i s equal to zero. Also 98 (a) i f the f u n c t i o n expressed as J I i s d i f f e r e n t i a t e d with respect to the lower l i m i t of i t s i n t e g r a l , (rw-RB), the r e s u l t i s -211/ (ro 2) ; and (b) i f the f u n c t i o n expressed as I I i s d i f f e r e n t i a t e d with respect to the lower l i m i t of i t s i n t e g r a l (rw-RB), the r e s u l t i s - E l . E2, F2, 12 and J2 w i l l have d e f i n i t i o n s corresponding to E l , F l I I and J I , w i t h (rw-RB) replaced by (rB-RB). The r e l a t i o n s of the preceding paragraph apply e q u a l l y to E2, F2, 12, and J2. Using the d e f i n i t i o n s given above, i t i s easy to show that MB>0 MB = 2(RJ2I1 + (r-R)JH2) / (ro 2J2 2) > 0. Ev a l u a t i o n of MR i s only s l i g h t l y more complex: MR = 2B(J2I1 - J1I2) / (ro 2J2 2) = 2BJ1 ( I l / J l - I2/J2) / (ro 2J2) By the r e s u l t s of Chapter I I , s e c t i o n 5, I ( x ) / J ( x ) i s an 99 i n c r e a s i n g f u n c t i o n of i t s argument. As rw-RB > rB-RB, i t fo l l o w s that MR>0. The i n e q u a l i t y i s not s t r i c t , as J1=J2, 11=12 at the boundary w=B. Thus, MR=0 at t h i s boundary. E v a l u a t i o n of the second p a r t i a l d e r i v a t i v e s of M i s more d i f f i c u l t . D i f f e r e n t i a t i o n of MB w i t h respect to B y i e l d s : MBB= [J2 ( 2 R 2 J 2 E l - 2 ( r - R ) 2 J1E2) + ( 4 ( r - R ) 1 2 / ( r a 2 ) ) (2RJ2I1 + 2 ( r - R ) J 1 I 2 ) ] / ( J 2 3 r a 2 ) (10) I f i t i s assumed that r< 2R ( i . e . the lender's lending r a t e i s greater than one h a l f the inherent r e t u r n on c a p i t a l ) , the f i r s t two terms of the numerator of (10) can be analysed as f o l l o w s : 2R 2J2E1 - 2 ( r - R ) 2 J l E 2 > 2R 2(J2E1 - J1E2) = 2R 2J1J2 ( E l / J l - E2/J2) However, by the r e s u l t s of Chapter I I , s e c t i o n 5, E/J i s an in c r e a s i n g f u n c t i o n . I t f o l l o w s that MBB i s p o s i t i v e . 100 The second d e r i v a t i v e MRR i s ambiguous i n s i g n . Numerical e v a l u a t i o n confirms that MRR tends to be p o s i t i v e at low values of B, f a r away from bankruptcy, but negative f o r high values of B, near bankruptcy. The source of the ambiguity i n s i g n can be i d e n t i f i e d . A f t e r e l i m i n a t i n g the f a c t o r 2B/(ro 2) , d i f f e r e n t i a t i o n of MR with respect to R y i e l d s : MRR = [J2 2E1 - J1J2E2 - 4J2I1I2/(ro 2) + 4J1I2 2/ (ro 2) ] [B/J2 3] { n) Combining the second and f o u r t h terms y i e l d s : 4J1I2 2 / (ro 2) - J1J2E2 = J I (4I2 2/(ro 2) - J2E2) ( 1 2 ) To analyse t h i s expression i t i s necessary to consider the i n t e g r a l 12: Making the s u b s t i t u t i o n z 2 = s 2/(ro 2) , t h i s i n t e g r a l becomes: 101 7 2 (rB-RB) 12 = | exp[ - z ]dz = I( ) rB-RB °\fr where the f u n c t i o n I(x) i s as de f i n e d i n Chapter I I , s e c t i o n 5. The d i f f e r e n c e i n (12) thus becomes 4 I ( x ) 2 - E ( x ) J ( x ) , where x represents the q u a n t i t y (rb-RB) / (oV"r) . From the r e s u l t s of Chapter I I , s e c t i o n 5, t h i s d i f f e r e n c e i s p o s i t i v e . However, the remaining d i f f e r e n c e i n (11), J2 [ E1J2 - 4I1I2 / (ro 2) ] , cannot be signed unambiguously, and i s c l e a r l y negative when w=B. F i n a l l y , the mixed second p a r t i a l d e r i v a t i v e MBR, i s a l s o ambiguous i n s i g n . From above, MR = 2B(J2I1 - J1I 2 ) / J 2 2 (ro 2) . Inspection of t h i s expression shows that MR=0 at the two extreme values B=0 and B=w. Since MR assumes p o s i t i v e values f o r intermediate values of B, MBR must assume both p o s i t i v e and negative values. Again i t i s u s e f u l to i n v e s t i g a t e the reason f o r the ambiguity i 102 s i g n . D i f f e r e n t i a t i n g MB w i t h to R y i e l d s : MBR = ( l / J 2 4 ( r o 2 ) ) [J2 2{2J2I1-2J1I2 + 4RBI2I1/ (ro 2) + 2RBJ2E1 + 4 (r-R) BI1I2/(ro 2) + 2(r-R)BJlE2} - 8BRJ2 2I1I2/(ro 2) - 8 (r-R) BJ1J212 2/(ro 2) ] which, a f t e r grouping terms, y i e l d s : MBR = (1/J2 3 (ro 2) ) [2J2 (J2I1 - J1I2) + 2RBJ22E1 + 4 (r-R) BJ2I1I2/(ro 2) + 2 (r-R) BJ2J1E2 - 4BRJ2I1I2/(ro 2) - 8 (r-R) BJ1I2 2/ (ro 2) ] = ( l / J 2 3 ( r o 2 ) ) [ 2J2 (J2I1 - J1I2) + 2 ( r - R ) B J l (E2J2 - 2I 2 2 / ( r a 2 ) ) + 4 (r-R) BI2/(ro 2) (J2I1 - J1I2) + 2RBJ2(E1J2 - 21112/(ro 2)] The f i r s t and t h i r d terms are p o s i t i v e . The second term i s negative, and the f o u r t h term i s ambiguous. The f o l l o w i n g t a b l e g i ves sample values of MR and MBR, at d i f f e r e n t l e v e l s of B, f o r the f o l l o w i n g parameter values: w=100, r=.08, R=.06, o=30. 103 Table 5 - Values of MR, M, B = 0 B = 2 0 B = 4 0 B = 6 0 B = 8 0 B = 1 0 0 M 0 . 1 0 6 0 . 1 7 5 0 . 2 8 1 0 . 4 4 0 . 6 7 1 1 MR 0 0 . 4 1 4 0 . 9 8 5 1 . 5 2 1 . 5 2 0 M B R 0 . 0 1 6 0 . 0 2 5 0 . 0 3 1 0 . 0 1 9 - 0 . 0 2 7 - 0 . 1 3 9 104 CHAPTER V - COMPETITION AMONG LENDERS To make use of the value functions (and ra t e s of return) that have been derived, i t i s necessary to impose a bargaining s t r u c t u r e on the borrower and lender. In t h i s chapter, the lender w i l l be assumed to operate i n a competitive lending market. In t h i s context, the value f u n c t i o n s of Chapter IV l e a d to an explanation f o r c r e d i t r a t i o n i n g (as defined below). In a d d i t i o n , they can be used t o p r e d i c t changes i n loan l e v e l s and i n t e r e s t r a t e s r e s u l t i n g from changes i n a b i l i t y of a borrower to grant e f f e c t i v e s e c u r i t y , or i n the r u l e s governing the d i s p o s i t i o n of a bankrupt company. Competition among lenders w i l l be modelled by assuming that the lender's value f u n c t i o n must equal zero. For t h i s purpose, i t does not matter whether i t i s the lender's value f u n c t i o n or r a t e of r e t u r n which i s used - both are zero simultaneously. Mathematically, the f u n c t i o n e a s i e s t to manipulate turns out t o be the lender's r a t e of r e t u r n m u l t i p l i e d by the s c a l a r r. For the purposes of t h i s chapter, the f u n c t i o n L(B,R) w i l l represent t h i s q u a n t i t y . w i s suppressed as an argument of t h i s f u n c t i o n as the s i z e of the p r o j e c t i s taken as given. 105 The borrower a l s o seeks to maximize h i s rate of r e t u r n , and again i t i s the rate of r e t u r n m u l t i p l i e d by r which turns out to be the e a s i e s t to manipulate. For purposes of t h i s chapter, V(B,R) w i l l represent t h i s q u a n t i t y . No t r a n s a c t i o n costs are included i n the functions i n t h i s chapter, as they are an unnecessary complication. Section 1 w i l l s t a t e the borrower's maximization problem and the value f u n c t i o n s to be used. Sections 2 and 3 w i l l examine p r e l i m i n a r y r e s u l t s concerning the value functions and the i m p l i c a t i o n s of the lender's c o n s t r a i n t L=0. S e c t i o n 4 w i l l examine c r e d i t r a t i o n i n g . Sections 5 and 6 w i l l examine the i m p l i c a t i o n s of changes to the l e g a l r u l e s governing, r e s p e c t i v e l y , a lender's salvage recovery and bankruptcy l e g i s l a t i o n . Under the assumptions s t a t e d above, the borrower's maximization problem can be s t a t e d as: max B i RV(B,R) subject to L(B,R)=0 106 From Chapter IV, the rates of r e t u r n to be analysed are: [rw-RB) j" exp t ] ro • ds L[B,R) = R - p + [rSa - R) (1) (rB-RB) exp [ ] ro •ds / 2 S exp [ -2 1 rw-RB s V[B,R) = [rw-RB) (1 r O 2 d s [ w - B) s 2 exp [ - — i yA> / 2 S r o 2 — d s 1 [ (rw-RB) ~ [ rB-RB)M] w-B) Inspection of (1) i d e n t i f i e s a r e s t r i c t i o n which must be placed on the parameters immediately. I f bankruptcy occurs immediately ( i . e . the s t a r t i n g wealth l e v e l i s B), the lender's r a t e of re t u r n w i l l be: L = r S a - p 107 As i t makes no sense f o r the lender to p r o f i t from immediate bankruptcy, t h i s q u a n t i t y w i l l be assumed negative. As i n the previous chapter, the symbols E l , F l , J I and I I w i l l be defined as f o l l o w s : exp [ (rw-RB) El = e x p [ - ] ; Fl rO rw-RB) r O (rw-RB) II j" exp [ ds ; JI = (rw-RB) r O exp [ -] r o ds rw-RB S E2, F2, 12, and J2 w i l l have d e f i n i t i o n s corresponding to E l , F l , I I , and J I , w i t h (rw-RB) replaced by (rB-RB). F i n a l l y , as i n Chapter IV, M w i l l denote the r a t i o J1/J2. 2^ The Lender's C o n s t r a i n t S u b s t a n t i a l i n f o r m a t i o n about the problem can be gained by examining the set of p o i n t s s a t i s f y i n g L(B,R)=0. The f i r s t step i n t h i s examination i s an a n a l y s i s of the p a r t i a l d e r i v a t i v e s dh/dB and dL/dR. 108 (a) The P a r t i a l D e r i v a t i v e dL/dB I t can be shown that the p a r t i a l d e r i v a t i v e of L(B,R) with respect to B i s unambiguously negative. The i n t u i t i o n i s simple. A higher debt l e v e l b r i n g s the f i r m c l o s e r to bankruptcy, which reduces the expected r e t u r n . Mathematically, d i f f e r e n t i a t i o n of (1) w i t h respect to B y i e l d s : 3L Jl 2R rO II 2 (r-R) rO 12 = (rSa-R) dB J2 Jl J2 As t h i s f i r s t f a c t o r on the RHS i s negative and the others are p o s i t i v e , the p a r t i a l d e r i v a t i v e must be negative. (b) The P a r t i a l D e r i v a t i v e d h / d R The s i g n of the p a r t i a l d e r i v a t i v e of L(B,R) wi t h respect to R i s ambiguous. The i n t u i t i v e reason i s that an increase i n the i n t e r e s t rate has both negative and p o s i t i v e i n f l u e n c e s on the lender's rate of r e t u r n . The p o s i t i v e i n f l u e n c e of greater b a s i c 109 i n t e r e s t revenue i s o f f s e t by the negative influence of the increased r i s k of the borrower's bankruptcy. The ambiguity of sig n can be confirmed by numerical examples. However, i t p o s s i b l e to show that dh/dR must be p o s i t i v e at the in t e r c e p t p o i n t B=0. D i f f e r e n t i a t i n g (1) wi t h respect to R y i e l d s : 2BI1 2BI2 3L { B , R) Jl Jl ro2 rO2 = = ( 1 - ) + (rSa-R) [ - ] dR J2 J2 Jl J2 At B=0, t h i s i s equal to 1-J1. By the r e s u l t s of Chapter I I , s e c t i o n 5, J l i s l e s s than 1 f o r p o s i t i v e (rw-RB), showing that the p a r t i a l d e r i v a t i v e i s p o s i t i v e . L t _ a n c a p J i s _ With knowledge about the signs of dL/dB and dh/dR, i t i s po s s i b l e to i n v e s t i g a t e the poi n t s at which the L=0 locus i n t e r c e p t s the two boundaries B=0 and R=r. 110 Consider f i r s t the boundary B=0 and l e t R* (assuming i t e x i s t s ) represent a value of R such that L(0,R*)=0. S e t t i n g L(B,R) and B both equal to zero i n (1) y i e l d s : 2 s exp [ ] °f ro R* - p +{rSa-R*) (rw) / ds = 0 rw 3 L e t t i n g N denote (rw) times the i n t e g r a l i n f i n a l term of can be i s o l a t e d i n (3) as f o l l o w s : R* = (p-rSaN)/(1-N) I f R* e x i s t s , i t i s c l e a r that i t must be unique. This f o l l o w s from the f a c t t h a t along the boundary B=0, dL/dR i s p o s i t i v e . I t remains t o show that R* e x i s t s , but a simple argument shows that t h i s must be the case f o r any s e n s i b l e set of parameters. I t i s c l e a r t h a t i f L i s negative everywhere along the boundary B=0, i t w i l l be negative f o r a l l values of B (as dh/dB i s negative). Since no lending w i l l occur i f L i s everywhere negative, i t i s necessary to assume that L i s p o s i t i v e f o r at l e a s t some p o i n t s along the boundary B=0. However, i t i s a l s o (3) (3) , R* I l l c l e a r that L i s always negative f o r the extremal point of the B=0 locus (B=0,R=p). By c o n t i n u i t y and the intermediate value theorem, i t f o l l o w s that L must be equal to zero f o r at l e a s t one point on the locus B=0 - i . e . a s o l u t i o n f o r R* must e x i s t i f any lending can occur. I t f o l l o w s that i f any lending can take place, there i s a unique poin t at which the L=0 locus i n t e r c e p t s the boundary B=0. The other m a t e r i a l i n t e r c e p t f o r the L=0 locus i s the point at which i t meets the R=r locu s . Another simple argument shows that t h i s i n t e r c e p t must e x i s t , and that i t must be unique. Uniqueness f o l l o w s d i r e c t l y from the r e s u l t s t a t e d above, that dh/dB i s p o s i t i v e . Existence f o l l o w s d i r e c t l y from c o n t i n u i t y , the intermediate value theorem, and the f a c t s t h a t : (a) L(B=w,R=r) i s negative, from i n s p e c t i o n of (1) and the assumption that rSa - p < 0; and (b) L(B=0,R=r) i s p o s i t i v e , from the proceeding a n a l y s i s of the boundary B=0. 112 I f B* represents the value of B f o r which L(B*,r)=0, i t f o l l o w s from (1) th a t : 2 s exp [ ] ro r r o r - p + ( r S a - r ) ( r w - r B * ) I ds = 0 which can be r e w r i t t e n as: 2 s exp [ ] 7 r o 2 r-p (4) (rw-rB*) I ds = J 2 r-rSa rw-rB* & Equation (4) confirms c o n d i t i o n s which were imposed on the parameters of the problem i n s e c t i o n 1. The LHS of (4) i s equal to 1 i f w=B, and i s not gre a t e r than 1 i n any event (Chapter I I , s e c t i o n 5). I t must thus be true that 1-p/r < 1-Sa, which i n tu r n i m p l i e s the same c o n d i t i o n derived p r e v i o u s l y , rSa < p. 113 (ci) The locus T.(B,R)=0 From the foregoing r e s u l t s , the L(B,R)=0 locus meets the B=0 and R=r boundaries at unique p o i n t s . As w e l l , the slope of the L=0 locus, (-dL/dR) / (dL/dB) , i s ambiguous i n s i g n , but i s p o s i t i v e at B=0. Thus, the L(B,R)=0 locus must have the shape of one of the examples shown below. Figure 5 - Graph of L(B,R)=0 B r R I t i s now p o s s i b l e to see that while the slope of the L(B,R)=0 114 locus i s ambiguous, the borrower w i l l never s e l e c t at point on that locus where i t s slope i s negative. For any such point, there i s always another poi n t on the locus w i t h the same debt l e v e l , but w i t h a lower i n t e r e s t r a t e on debt. I t i s thus c l e a r that at the p o i n t which solves the borrower's constrained maximization problem, dL/dR w i l l be p o s i t i v e and dL/dB w i l l be negative. 3. The Borrower's Rate of Return The r e q u i r e d information about the p a r t i a l d e r i v a t i v e s of V(B,R) i s more e a s i l y obtained. (a) dV(B,R)/dR dV(B,R)/dR i s unambiguously negative. I n t u i t i v e l y , higher i n t e r e s t r a t e s impose higher debt s e r v i c e c o s t s on borrowers with no o f f s e t t i n g advantage. To prove t h i s r e s u l t , i t i s h e l p f u l to r e w r i t e the f u n c t i o n V (equation (2), above) i n the f o l l o w i n g way: 115 V(B,R) = [l/(w-B)] [rw-RB - (rB-RB)J1/J2] I t i s c l e a r that the f a c t o r 1/(w-B) can be neglected. D i f f e r e n t i a t i o n then y i e l d s : 2BI1 2BI2 dV JI JI rO2 r O 2 {W-B)-dR- = - B { 1 - ^ 2 ] ~ (rB-RB)-J-2[—nr ~ ~^J2-] The f i r s t term i s c l e a r l y negative, and the second i s negative as w e l l by the r e s u l t s of Chapter I I , s e c t i o n 5. Ih) d V ( B . R ) / d B Just l i k e dL(B,R)/dR, dV(B,R)/dB i s ambiguous i n s i g n . The i n t u i t i v e reason i s s i m i l a r . Increasing debt l o a d increases the borrower's leverage on e x i s t i n g e q u i ty, but a l s o increases the r i s k of bankruptcy. However, i t can be shown that dV(B,R)/dB must be negative as B approaches w. I t i s c l e a r that V(B,R) i s p o s i t i v e f o r a l l B<w. However, a p p l i c a t i o n of l ' H o p i t a l ' s r u l e to (2), above shows that V(B,R) 116 approaches 0 as B approaches w. I t f o l l o w s that dV(B,R)/dB must be negative as B approaches w. Using the r e s u l t s of s e c t i o n s 1, 2, and 3, the f i r s t aspect of lender borrower r e l a t i o n s to be examined i s c r e d i t r a t i o n i n g . L i t e r a t u r e on c r e d i t r a t i o n i n g has not been con s i s t e n t i n i t s terminology. T r a d i t i o n a l l y , the term "Type 1 c r e d i t r a t i o n i n g " has r e f e r r e d t o a c o n d i t i o n i n which a f i r m can not borrow as much as i t wishes at the going i n t e r e s t r a t e . "Type 2 c r e d i t r a t i o n i n g " has r e f e r r e d to a c o n d i t i o n where, among i d e n t i c a l borrowers, some can borrow and some can not (Blanchard and F i s h e r (1989), p. 479). More r e c e n t l y , F r e i x a s and Rochet (1997) have argued that circumstances i n which a borrower would l i k e to borrow more at the i n t e r e s t rate he i s paying should not p r o p e r l y be c a l l e d c r e d i t r a t i o n i n g at a l l . F r e i x a s and Rochet use the term "apparent c r e d i t r a t i o n i n g " to describe t h i s c o n d i t i o n . "Credit r a t i o n i n g " , as used i n t h i s t h e s i s , means a c o n d i t i o n i n which a borrower would l i k e to borrow more at the i n t e r e s t rate 117 he has negotiated with h i s lender. Whether t h i s c o n d i t i o n c o n s t i t u t e s "Type 1 c r e d i t r a t i o n i n g " or merely "apparent c r e d i t r a t i o n i n g " i s an issue which awaits a uniform d e f i n i t i o n of terms. Models of c r e d i t r a t i o n i n g o f t e n r e l y on some form of market imperfection, such as asymmetric information (eg. S t i g l i t z and Weiss, (1981)) . The analyses presented below w i l l concentrate on determining whether c r e d i t r a t i o n i n g can be explained w i t h l e s s severe assumptions. More p a r t i c u l a r l y , the analyses w i l l focus on whether r i s k of the borrower's bankruptcy can e x p l a i n c r e d i t r a t i o n i n g . Using the r e s u l t s of sections 2 and 3, the borrower's maximization problem can be set up as a lagrangean. The problem i s : Max V(B,R) subject to L(B,R)=0, R>R*, R<r where R* i s i n t e r e s t rate s o l v i n g L(0,R*)=0 (see s e c t i o n 2 ( c ) , above). The lagrangean i s : 118 V ( B , R ) - AL(B,R) - A2 ( R * - R) - A3(R - r) where A2 and A3 must be non-negative. Some s i m p l i f i c a t i o n i s p o s s i b l e immediately. I t i s c l e a r that the borrower w i l l never borrow at an i n t e r e s t r a t e equal to r, that A3 must be equal to zero. With t h i s s i m p l i f i c a t i o n , the f i r s t order c o n d i t i o n s f o r maximization are: dV/dR = AdL/dR - A2 dV/dB = Xdh/dB A2 ( R * - R ) = 0 Consider f i r s t the case of an i n t e r i o r maximum, where A2=0. From s e c t i o n 2(d) i t i s known that at the e q u i l i b r i u m p o i n t , dV/dR i s negative and dh/dR i s p o s i t i v e . These two c o n d i t i o n s imply that A i s negative. As w e l l , i t i s known that dh/dB i s negative. With A negative as w e l l , dV/dB must be p o s i t i v e . 119 This, however, means that c r e d i t must be r a t i o n e d . At the e q u i l i b r i u m debt l e v e l and i n t e r e s t r a t e , the borrower would l i k e to borrow more. At a n o n - i n t e r i o r maximum, no unambiguous r e s u l t can be stated. However, i t i s a l s o true that at a n o n - i n t e r i o r maximum, there i s no borrowing i n e q u i l i b r i u m . The r e s u l t i s thus that i f no borrowing i s observed, one can not t e l l whether i t i s because the borrower chooses not to borrow at a l l , or because the lender w i l l not lend. The f o l l o w i n g f i g u r e s i l l u s t r a t e three d i f f e r e n t r e s u l t s . In the f i r s t there i s p o s i t i v e borrowing i n e q u i l i b r i u m , and c r e d i t r a t i o n i n g as w e l l . The amount of the c r e d i t r a t i o n i n g i s measured by the distance between the e q u i l i b r i u m debt l e v e l (Be) and the borrower's maximizing debt at the e q u i l i b r i u m i n t e r e s t r a t e (Bm). Tangency occurs at Bm between an isoquant of the borrower and the v e r t i c a l l i n e through Re, the e q u i l i b r i u m i n t e r e s t r a t e . In the second f i g u r e , there i s no borrowing i n e q u i l i b r i u m , but there i s c r e d i t r a t i o n i n g . 120 In the t h i r d f i g u r e , there i s no borrowing i n e q u i l i b r i u m , but that i s not the r e s u l t of c r e d i t r a t i o n i n g . 121 Figure 7 - C r e d i t Rationing Without Borrowing [22 Figure 8 - No Borrowing, No C r e d i t Rationing B In the r e s u l t , there i s a r e l a t i v e l y s t r a i g h t f o r w a r d explanation f o r c r e d i t r a t i o n i n g , without r e s o r t i n g to asymmetric information. The i n t u i t i v e e x planation i s that the borrower has an i n c e n t i v e to r a i s e debt l e v e l s , as he keeps the b e n e f i t of a d d i t i o n a l leverage i f bankruptcy i s avoided. He i s thus w i l l i n g to accept a greater r i s k of bankruptcy. The lender r a t i o n s c r e d i t because he i s l e s s w i l l i n g to r i s k bankruptcy. The borrower's a d d i t i o n a l leverage from higher debt l e v e l s provides no corresponding advantage to the lender. 123 While the preceding a n a l y s i s has been done i n the context of s p e c i f i c value f u n c t i o n s , i t i s e q u a l l y a p p l i c a b l e to any ge n e r a l i z e d value f u n c t i o n s f o r a borrower (V*(B,R)) and lender (L*(B,R)). I f a zero p r o f i t c o n d i t i o n i s imposed on the lender, and i f an i n t e r i o r maximum i s assumed, the f o l l o w i n g problem r e s u l t s : maxB R V*(B,R) subject to L*(B,R) = 0 with the r e s u l t i n g f i r s t order c o n d i t i o n s : V*B(B,R) = AL*B(B,R) V*R(B,R) = AL*R(B,R) For obvious reasons, V*R and L* R w i l l g e n e r a l l y be negative and p o s i t i v e , r e s p e c t i v e l y , i m p l y i n g that A<0. This i n t u r n i m p l i e s that V*B(B,R) and L*B(B,R) w i l l be opposite i n s i g n . I f L* B (B, R) i s negative V*B(B,R) w i l l be p o s i t i v e , implying c r e d i t r a t i o n i n g . I t i s reasonable to expect L* B(B,R) to be negative when i n c r e a s i n g debt l e v e l s increase the threat of the borrower's bankruptcy, and thus the t h r e a t of loan d e f a u l t . 124 This a n a l y s i s adds i n s i g h t i n t o the r e s u l t s obtained by de Meza and Webb (1992). De Meza and Webb consider the f o l l o w i n g expected p r o f i t f u n ctions f o r the entrepreneur (e) and a bondholder (B). n e = p [ f (k,9 H) - (l+r)k] + (1-p) max [f (k,9L) - (1 + r) k , 0] n B = p ( l + r ) k + (l-p)maxtf (k,6 L) , ( l + r)k] - ( l + p)k Debt l e v e l i s represented by the v a r i a b l e k, and the entrepreneur's c a p i t a l s t r u c t u r e i s assumed to be 100% debt. p i s the cost of c a p i t a l , f i s a production f u n c t i o n , and 9 i s a random v a r i a b l e which takes a high value with p r o b a b i l i t y p and a low value w i t h p r o b a b i l i t y (1-p). De Meza and Webb impose a zero p r o f i t c o n d i t i o n on the lender, and go on to conclude that the existence of c r e d i t r a t i o n i n g depends on whether f(k,6) e x h i b i t s decreasing returns to s c a l e , i . e . f k<f/k. However, w i t h the p r o f i t f u n c t i o n s used by de Meza and Webb, t h e i r "decreasing returns" c o n d i t i o n i s equivalent to c o n d i t i o n that the d e r i v a t i v e of the lender's f u n c t i o n w i t h respect to k be negative. Assuming that f (k, 9L) < (1 + r) k (which i s the case of i n t e r e s t ) the d e r i v a t i v e of n B with respect to k i s equal t o : 125 n B , k = pd+r) + ( i - p ) f k ( k , e L ) - (i+p) . M u l t i p l y i n g t h i s expression by k and rearranging y i e l d s : k n B f k = [ p ( l + r ) k + ( l - p ) f ( k , 6 L ) - ( l + p)k] - ( l - p ) f ( k , 9 L ) + (1-p) f k ( k , e L ) k However, the term i n square brackets are equal t o n B which i s equal to zero by assumption. Thus: k n B k =(i-p) [ - f ( k , e L ) + f k ( k , e L ) k ] This expression shows that the decreasing r e t u r n s c o n d i t i o n derived by de Meza and Webb i s i d e n t i c a l to a c o n d i t i o n that the d e r i v a t i v e of the lender's p r o f i t f u n c t i o n w i t h respect to k be negative. The r e s u l t s of de Meza and Webb have been subject to c o n f l i c t i n g i n t e r p r e t a t i o n . A c c o r d i n g l y , the f o l l o w i n g should be noted i n comparing the r e s u l t s of t h i s chapter to those of de Meza and Webb. 126 1. Freixas and Rochet (1997) have argued that the model of de Meza and Webb does not i l l u s t r a t e c r e d i t r a t i o n i n g at a l l , but rather "apparent c r e d i t r a t i o n i n g . " F r e i x a s and Rochet argue that the term " c r e d i t r a t i o n i n g " should be reserved f o r c o n d i t i o n s i n which borrowers are able to f u l f i l the terms of t h e i r c o n t r a c t s , but are s t i l l denied c r e d i t . This i s not the case i n de Meza and Webb, where the r i s k of def a u l t d r i v e s at l e a s t some of the r e s u l t s . 2. The a n a l y s i s set out above omits an important feature of the model of de Meza and Webb. In the model of de Meza and Webb, the amount inv e s t e d i n the firm ' s production process i s optimal i n s p i t e of the c r e d i t r a t i o n i n g (or apparent c r e d i t r a t i o n i n g ) . In t h i s respect, de Meza and Webb argue, the presence of c r e d i t r a t i o n i n g i s not evidence of market f a i l u r e . No measure of e f f i c i e n c y i s a v a i l a b l e i n the a n a l y s i s presented here. While there are unresolved issues surrounding the d e f i n i t i o n of c r e d i t r a t i o n i n g , i t i s submitted that the a n a l y s i s presented here makes two p o i n t s . F i r s t , the lagrangean a n a l y s i s shows how simple i t i s to e x p l a i n c r e d i t r a t i o n i n g i n a competitive lending 127 market. With the signs of some d e r i v a t i v e s being obvious, the a n a l y s i s leads to one s t r a i g h t f o r w a r d c o n d i t i o n : c r e d i t r a t i o n i n g w i l l be present i f dL/dB i s negative. The assumptions leading t o t h i s c o n d i t i o n are so general that i t can be regarded as both necessary and s u f f i c i e n t . Second, i t i s submitted that the lagrangean a n a l y s i s demonstrates a more general c o n d i t i o n underlying the r e s u l t s of de Meza and Webb. While the lagrangean co n d i t i o n s include no e f f i c i e n c y c r i t e r i o n , they must be s a t i s f i e d at an i n t e r i o r maximum. They are, i t i s submitted, more general and l e s s model s p e c i f i c than the equivalent decreasing returns c o n d i t i o n s t a t e d i n de Meza and Webb. 5^ S e c u r i t y L i m i t a t i o n s The above a n a l y s i s can be extended to consider l i m i t a t i o n s on the a b i l i t y of a borrower to provide s e c u r i t y . A l l Canadian j u r i s d i c t i o n s (as w e l l as the Federal government) have w i t h i n the l a s t twenty years enacted l e g i s l a t i o n designed to streamline the g r a n t i n g of s e c u r i t y i n t e r e s t s by borrowers. Often, the l e g i s l a t i o n attempts to make as many forms of property as p o s s i b l e amenable to s e c u r i t y i n t e r e s t s . Examples include: 128 (a) Section 88 of the Bank Act of Canada, which i s designed to allow banks, with r e l a t i v e l y l i t t l e documentation, to take broadly based s e c u r i t y over the assets of an operating business ,-(b) The Personal Property S e c u r i t y Act of B r i t i s h Columbia, enacted i n the l a t e 1980's. This act (and s i m i l a r acts passed i n most other Canadian provinces) was intended to s i m p l i f y procedures f o r g r a n t i n g , r e c o r d i n g and enforcing s e c u r i t y i n t e r e s t s i n personal property. A change i n the range of assets over which a borrower can grant e f f e c t i v e s e c u r i t y can be modelled as a change i n the parameter Sa. An increase i n Sa represents an increase i n the amount recovered by the lender on l i q u i d a t i o n , i . e . an increase i n the scope of the borrower's assets over which s e c u r i t y has been obtained. The f o l l o w i n g a n a l y s i s assumes that the lender and borrower are at an i n t e r i o r maximum to the borrower's maximization problem, and L w i l l now be considered as f u n c t i o n s of (B,R;Sa). 129 At an i n t e r i o r maximum, the f i r s t order conditions are: VR = AL R V B = AL B (6) L(B,R;Sa) = 0 . Combining the f i r s t two c o n d i t i o n s y i e l d s V RL B = V BL R. T o t a l d i f f e r e n t i a t i o n of t h i s expression and the expression L=0 y i e l d s : LBdB + LRdR + L S adSa = 0 (7) dR(L BV R R + V RL B R - L RV R B - V BL R R) + dB (L BV R B + V RL B B - L RV B B - V BL B R) + dSa(V RL B s a + L BV R S a - V BL R S a - L RV B S a) = 0 The c o e f f i c i e n t s i n the f i r s t expression are c l e a r . L B has been shown negative, L R has been shown p o s i t i v e (at e q u i l i b r i u m ) and L s a = rM>0. The c o e f f i c i e n t s i n the second expression of (7) are more complex. Let X, Y, and Z represent, r e s p e c t i v e l y , the c o e f f i c i e n t s of dR, dB, and dSa i n the second expression. 130 Consider f i r s t Z, the c o e f f i c i e n t of dSa: Z = V RL B S a + L BV R S a - V BL R S a - L RV B S a As V does not depend i n any way on Sa, the second and f o u r t h terms are c l e a r l y zero. A l s o : L B S a = rM B >0 L R S a = rM R >0. Since VR <0 and VB> 0 ( i n e q u i l i b r i u m ) , Z must be negative. X and Y are more d i f f i c u l t to s i g n . Some of the components can be signed unambiguously. Others, however, are s u f f i c i e n t l y complicated that unambiguous expressions cannot be derived. In some cases, the best that can be done i s to show that "fundamental" values (which w i l l p r e v a i l f a r from bankruptcy) are of the c o r r e c t s i g n , and that the d e s i r e d s i g n i s a l s o achieved near the bankruptcy boundary. These conclusions must be r e l i e d upon t o conclude that the term i n question w i l l have the d e s i r e d s i g n i n general. Sometimes i t i s also, necessary to derive r e s t r i c t i o n s on parameters to ensure that the d e s i r e d signs are 131 maintained. X and Y can be r e w r i t t e n as: X = L B V R R + V R L B R - L R V R B - V B L R R = L B ( V R R - A L R J - L R ( V R B - A L R B ) (8) Y = L B V R B + V R L B B - L R V B B - V B L B R = - L R ( V B B - A L B B ) + L B (Vffi - A L R B ) The q u a n t i t i e s ( V R R - A L R R ) and ( V B B - A L B B ) must be non-positive (and i n general negative), as they are the f i r s t and second diagonal elements of the bordered hessian matrix f o r the maximization problem, which must be negative s e m i - d e f i n i t e at an i n t e r i o r maximum. Thus, si n c e L R >0 and L B <0, X and Y w i l l both be non-negative (and p o s i t i v e i n general) i f i t can be shown that the term ( V R B - A L R B ) i s negative. As A < 0 , i t thus s u f f i c e s to show that V R B , L R B <0. Consider f i r s t L R B . As L = R - p + (rSa-R)M, i t f o l l o w s that L R B = -MB + (rSa-R) MBR. From the previous chapter, MB>0, but MRB i s ambiguous i n si g n . The expression L R B = -MB + (rSa - R) MBR can be s i m p l i f i e d to some degree by invoking the zero p r o f i t c o n d i t i o n imposed on the 132 lender. Since L = R - p + (rSa-R)M = 0, the expression (rSa-R) can be replaced by (p-R)/M. Expanding the expression L R B = ~ M B + MBR(p-R)/M then y i e l d s : LBR= - (1/J2 2) [2RJ2I1/ (ro 2) + 2 (r-R) J1I2) / (ro 2) ] -(R-p)/M(J2 4) [ J2 2 { 2 J 2 I l / ( r o 2 ) - 2J1I2/(ro 2) + 4RBI2I1/(ro 2) 2 + 2RBJ2E1/(ro 2) + 4 (r-R) B l 112/(ro 2) 2 + 2 (r-R) BJ1E2/ (ro 2) } - 2J2I2/(ro 2) {2RJ2I1/(ro 2) + 2 (r-R) J1I2 ) / (ro 2) ] As shown i n Chapter IV, s e c t i o n 3, MBR achieves negative values only i n the v i c i n i t y of bankruptcy, and achieves i t s most negative values at the boundary w=B. Thus, t h i s boundary can be examined as a worst case scenario. D e f i n i n g J=J1=J2, 1=11=12, and E=E1=E2 (which c o n d i t i o n s a l l hold at the boundary w=B) the above expression s i m p l i f i e s t o : L B R = (1/J 2) [ - 2 r J l / ( r o 2 ) - (R-p) {2rBEJ/(ro 2) - 4 r B I 2 / ( r o 2 ) 2 } ] By the assumptions placed e a r l i e r on parameters, (R-p)<r/2. I t thus s u f f i c e s to examine the expression: - 2 r J l / ( r o 2 ) - r{rBEJ/(ro 2) - 2 r B I 2 / ( r o 2 ) 2 } 133 Making a s u b s t i t u t i o n of v a r i a b l e s i n the i n t e g r a t i o n s i n t h i s expression y i e l d s : ( r / ( o / r ) ) [J(x)I(x) - (rw/(o/r) )E(x) J(x) + (2rw/(o/r) ) I (x) 2] where x=(rw-Rw)/(o/r) and I ( x ) , J ( x ) , and E(x) are the functions defined i n Chapter I I , s e c t i o n 5. This expression contains only two "normalized" v a r i a b l e s , x and rB/(oV~r) and can be evaluated numerically. At the p o i n t w=B, r=R, x becomes equal to 0, and the above expression w i l l be negative i f rw/(oVr)<3.1. Numerical a n a l y s i s of the expression f o r other values of x and rw/(o/r) confirms that t h i s r e s t r i c t i o n on parameters i s s u f f i c i e n t to ensure that L B R remains negative along the e n t i r e boundary B=w. This r e s t r i c t i o n on parameters i s e a s i l y s a t i s i f e d by a l l numerical examples used i n t h i s paper, and i t i s d i f f i c u l t to generate meaningful r e s u l t s without such a r e s t r i c t i o n . As w e l l , i t a r i s e s from a "worst case" scenario, the p o i n t B=w. At the a c t u a l e q u i l i b r i u m p o i n t , where B<w, L B R i s even l e s s l i k e l y to be p o s i t i v e . I t can thus be taken s a f e l y that L B R i s negative. 134 Turning f i n a l l y to VBR, r e c a l l that V can be w r i t t e n as: V = [(rw-RB) - (rB-RB) M] / (w-B) . L e t t i n g D represent the q u a n t i t y (rw-RB) - (rB-RB)M, V can be w r i t t e n as: V = D/(w-B). V B R i s then equal t o : V B R = D R/(w-B) 2 + DBR/(w-B) DR = -B(l-M) - (rB-RB)M R which i s unambiguously negative. However, DRB = - l ( l - M ) - (r-R) MR + BMB - (rB-RB) MBR, which i s ambiguous i n s i g n . The "fundamental" value f o r DRB i s -1, which i s of the d e s i r e d s i g n . However, near bankruptcy t h i s term disappears and the expression i s dominated by other terms whose cumulative s i g n i s ambiguous. I t i s p o s s i b l e , however, to use l ' H o p i t a l ' s r u l e to show that even near bankruptcy, V B R w i l l be negative. 135 To apply l ' H o p i t a l ' s r u l e , i t i s f i r s t necessary to r e w r i t e the l i m i t of V B R as f o l l o w s : D D l i m V B-w BR l i m [ B-W~ w-B (w-B)2 D (9) D +-l i m [ BR (w-B) w-B I t i s c l e a r that the denominator of (9), w-B, approaches 0 as B approaches w. The numerator a l s o approaches 0, s i n c e : (i) DR = -R(l-M) - (rB-RB) MR approaches 0 at B=w; ( i i ) w-B = 0 at w=B; and ( i i i ) by l ' H o p i t a l ' s r u l e , DR/(w-B) approaches -DBR at w=B. Equation (9) can thus i t s e l f be evaluated by l ' H o p i t a l ' s r u l e , so that: 136 D f9 D R O R D + l i m -*r-[D +-BR (W~B) B - W 5B BR (W~B) l i m [—: ] = 3 B-w W~ B O l i m ( w-B) B-w (JB D D R RB l i m [D + - + ] (\Q\ B-w B R B [W-B)  W ~ B -1 D D R R B , - l i m [D ] - l i m [ - + - ] B-w BRB B-w ( W - B ) W~B - l i m [ D ] - l i m [ V ] B-w B R B B-w B R From (10) i t f o l l o w s that: l i m [V ] = - l i m [D ] - l i m [V ] B-w B R B-w BRB B-w B R or 1 l i m [ V ] = - ( —) l i m [D ] B-w B R 2 B-w B R B I t can thus be shown that V B R approaches a negative value at B=w i f i t can be shown that DBRB approaches a p o s i t i v e value. D i f f e r e n t i a t i o n of DBR with respect to B y i e l d s : D B R B = B M B B - 2 (r-R) M B R + 2MB - (rB-RB) M B R B 137 The a n a l y s i s of Chapter IV, s e c t i o n 3 shows that at the boundary B=w, MBB>0, MBR<0, MB>0, and MBRB<0. I t fo l l o w s that DBRB i s p o s i t i v e near the boundary B=w, and thus that VBR<0. With V B R having a negative fundamental value away from bankruptcy, and a l s o a negative l i m i t i n g value near bankruptcy, i t can be concluded that V B R w i l l be general i n negative. Returning t o the system (7): L BdB + L RdR + L S adSa = 0 XdR + YdB + ZdSa = 0 i t f o l l o w s from the foregoing a n a l y s i s that i n the second equation, both X and Y are p o s i t i v e , while Z i s negative. E l i m i n a t i n g dR from the system y i e l d s : dB [Y - XL B/L R] = dSa [-Z + XL S a/L R] Both c o e f f i c i e n t s are p o s i t i v e , w i t h the r e s u l t that B increases with Sa. 138 E l i m i n a t i n g dB from the system y i e l d s : dR [X - YL R/L B] = dSa [-Z + YL S a/L B] In t h i s expression, the c o e f f i c i e n t of dR i s p o s i t i v e , but the c o e f f i c i e n t of dSa i s ambiguous. In the r e s u l t : (i) the indebtedness l e v e l , B, w i l l increase as Sa increases; and ( i i ) the e f f e c t on the i n t e r e s t r a t e , R, i s ambiguous. I t has thus been shown that p e r m i t t i n g borrowers to grant e f f e c t i v e s e c u r i t y w i l l increase the l e v e l of borrowing, but w i l have an ambiguous e f f e c t on the i n t e r e s t r a t e . These r e s u l t s c o i n c i d e with i n t u i t i o n . With l e s s r i s k of l o s s r e s u l t i n g from bankruptcy, the lender i s more w i l l i n g to lend, which increases the loan amount. The i n t e r e s t r a t e must adjust to ensure that the c o n d i t i o n L=0 continues to be met, and t h i s may r e s u l t i n e i t h e r an increase or decrease. 139 5^ Bankruptcy L e g i s l a t i o n S i g n i f i c a n t d i f f e r e n c e s e x i s t between the bankruptcy l e g i s l a t i o n p r e v a i l i n g i n Canada and the United States. The most notable d i f f e r e n c e i s Chapter 11 of the United States Bankruptcy Code. Chapter 11 i s designed to permit the r e o r g a n i z a t i o n and s u r v i v a l of corporations which would otherwise face l i q u i d a t i o n . For corporations able to invoke i t s p r o t e c t i o n , Chapter 11 provides broad p r o t e c t i o n from c r e d i t o r s . The most obvious j u s t i f i c a t i o n f o r Chapter 11 i s that i t permits corporations to preserve t h e i r value as a going concern, which w i l l f a r exceed t h e i r break-up value. No s i m i l a r l e g i s l a t i o n e x i s t s i n Canada. The c l o s e s t analog i s the Companies C r e d i t o r s Arrangements Act. Under that l e g i s l a t i o n , q u a l i f y i n g c o r p o r a t i o n s can obtain a st a y of l e g a l proceedings while a r e o r g a n i z a t i o n plan i s prepared and presented to c r e d i t o r s . The CCAA i s not, however, amenable to p r o t r a c t e d operation of a f i n a n c i a l l y d i s t r e s s e d business under bankruptcy p r o t e c t i o n , as has occurred under Chapter 1 1 . Under the CCAA, a plan of r e o r g a n i z a t i o n must be presented and accepted w i t h i n a l i m i t e d p e r i o d , or t r a d i t i o n a l bankruptcy r e s u l t s . 140 Chapter 11 has been c r i t i c i z e d . Adler (1992) and Bradley and Rosenzweig (1992) argue that i t motivates corporate management to undertake p r o j e c t s with excessive r i s k . Bradley and Rosenzeig base t h e i r argument on the contention that corporate managers act i n the knowledge that i n s o l v e n c y w i l l not cost them t h e i r jobs, which i n t u r n removes the i n c e n t i v e f o r prudence. Adl e r adopts a l e s s c y n i c a l view of corporate managers, arguing that excessive r i s k t a k i n g i s a r e s u l t of the r e - a l l o c a t i o n i n favour of shareholders which r e s u l t s i n many Chapter 11 r e o r g a n i z a t i o n s . Warren (1992) has disputed the conclusions of Bradley and Rosenzweig, arguing that they have m i s i n t e r p r e t e d e m p i r i c a l evidence upon which they r e l y . More importantly, Warren argues that d e t r a c t o r s of Chapter 11 do not acknowledge the important r e d i s t r i b u t i v e e f f e c t s of Chapter 11, which are perceived to be part of i t s inherent j u s t i c e . The model developed i n t h i s t h e s i s can be used to examine the question of how bankruptcy p r o t e c t i o n w i l l a f f e c t the d e c i s i o n s of borrowers and lenders. Assume now that the model analysed above i s a l t e r e d to provide that upon bankruptcy, part of the remaining value of the f i r m i s 141 recovered by the lender, while part i s recovered by the borrower as w e l l . In the context of the previous d i s c u s s i o n , both the borrower and lender recover a "salvage" value. The e f f e c t of bankruptcy l e g i s l a t i o n i s t o apportion the salvage value between the borrower and lender. Assume that on bankruptcy, the lender w i l l r eceive SaB, while the borrower r e c e i v e s (l-Sa)B. The lender's r a t e of r e t u r n i s unchanged: L = R - p + (rSa-R)M However, the borrower's rate of r e t u r n ( m u l t i p l i e d by r, f o r convenience) i s now: V = [(rw-RB) - (rB-RB)M + (r-rSa)MB]/(w-B) As before, the f i r s t order c o n d i t i o n s are: VR = X L R ( 1 1 ) 142 L(B,R;Sa) = 0. As before, t o t a l d i f f e r e n t i a t i o n of the f i r s t order c o n d i t i o n s y i e l d s : L ndB + L„dR + L q adSa = 0 XdR + YdB + ZdSa = 0 ( 1 2 ) where: X = L BV R R + V RL B R - L RV R B - V BL R R Y = L BV R B + V RL B B - L RV B B - V BL B R ( 1 3 ) Z = V RL B s a + L BV R S a - V BL R S a - L RV B S a X and Y remain p o s i t i v e , as i n the previous a n a l y s i s . However, the s i g n of Z has now become ambiguous. P r e v i o u s l y , V was independent of Sa, with the r e s u l t that the second and f o u r t h terms of Z were zero. As the f i r s t and t h i r d terms were, r e s p e c t i v e l y , negative and p o s i t i v e , the s i g n of Z was unambiguously negative. 143 With Z now depending on Sa, the second term of Z, L BV R S a, i s now equal t o : L BV R S a = L B [-BMR/(w-B)] >0. The f o u r t h term, L RV B S a, i s equal to L R [-M/(w-B) - BMB/(w-B) - MB/ (w-B)2] < 0 With the second term of Z p o s i t i v e and the f o u r t h negative, the sign of Z i s ambiguous. E l i m i n a t i n g dR, and then dB, from system (12) y i e l d s : dB [Y - XL B/L R] = dSa[-Z + XL S a/L R] and dR[X - YL R/L B] = dSa[-Z + YL S a/L B] The c o e f f i c i e n t s of dB and dR are p o s i t i v e , as i n the a n a l y s i s of s e c t i o n 5 . However, with ambiguity i n the s i g n of Z, the c o e f f i c i e n t s of dSa cannot be signed. In the r e s u l t , a change i n Sa has an ambiguous e f f e c t on both loan l e v e l s and i n t e r e s t r a t e s . 144 The ambiguous e f f e c t on loan l e v e l s i s i n t u i t i v e . An increase i n Sa increases the lender's recovery on bankruptcy making him more w i l l i n g to lend. However, i t a l s o decreases the borrower's recovery on bankruptcy, making him l e s s w i l l i n g to borrow. The change i n loan l e v e l s w i l l depend on which e f f e c t i s greater. The ambiguous e f f e c t on i n t e r e s t r a t e s i s l e s s i n t u i t i v e , as i t would appear that the lender w i l l be w i l l i n g to accept lower i n t e r e s t r a t e s , and that the borrower w i l l only be able to pay lower i n t e r e s t r a t e s . However, the ambiguity i n dR can be explained. As e x p l a i n e d above, the change i n lending l e v e l s i s unpredictable. As a r e s u l t , the borrower and lender may be moved to a higher lending l e v e l , which corresponds to a higher i n t e r e s t r a t e on the locus L(B,R ) = 0 . The r e s u l t can be made unambiguous i f the change i n bankruptcy l e g i s l a t i o n a f f e c t s o n l y the borrower's recovery. As only d i f f e r e n t i a l s are r e l e v a n t , a change i n the borrower's recovery alone can be modelled by assuming that the lender recovers nothing on bankruptcy, so that h i s r a t e of r e t u r n i s now. L = R - p - RM 145 The borrower's rate of r e t u r n i s s t i l l : V = [(rw-RB) - (rB-RB)M + (r-rSa)MB]/(w-B) The f i r s t order c o n d i t i o n s have the same form as before. The t o t a l d i f f e r e n t i a l s are now: LBdB + LRdR = 0 XdR + YdB + ZdSa = 0 As L no longer depends on Sa, Z i s equal t o : Z = L BV R S a - L RV B S a which i s p o s i t i v e . S o l v i n g f o r dB and dR now y i e l d s : dB [Y - XL B/L R] = dSa[-Z] dR[X - YL R/L B] = dSa[-Z] The c o e f f i c i e n t s of dB and dR are both p o s i t i v e . The c o e f f i c i e n t 146 of dSa i s negative. Accordingly, both B and R r i s e as Sa f a l l s . A decrease i n Sa represents an increase i n the borrower's recovery on l i q u i d a t i o n . This makes him more w i l l i n g to borrow. His increased borrowing demand i s s p l i t between an increased q u a n t i t y of borrowing and a increased cost of borrowing ( i . e . a higher i n t e r e s t r a t e ) . As the lender's r a t e of r e t u r n i s now independent of Sa, the L=0 locus, on which any e q u i l i b r i u m must l i e , i s unchanged. The e q u i l i b r i u m p o i n t moves upward along the p o s i t i v e l y s l o p i n g L=0 locus, t o higher l e v e l s f o r both debt and i n t e r e s t r a t e s . 147 CHAPTER VI - MONOPOLY IN THE LENDING MARKET - SINGLE BORROWER This Chapter adopts the opposite extreme assumption about the lending market - that the borrower faces a s i n g l e monopolist lender. The lender occupies a p o s i t i o n a k i n to a Stackelberg leader. He quotes the borrower an i n t e r e s t rate and a maximum loan amount. The lender sets the i n t e r e s t r a t e and loan c e i l i n g w ith f u l l knowledge of the borrower's value f u n c t i o n . The borrower can borrow any amount, up to the lender's imposed maximum, at the p r e s c r i b e d i n t e r e s t r a t e . Forced borrowing i s not allowed, i n that the borrower need not borrow the maximum i f he p r e f e r s to borrow l e s s . Transaction costs of n e g o t i a t i n g a loan are now inc l u d e d f o r both the borrower and lender to r u l e out degenerate s o l u t i o n s . The monopoly assumption r a i s e s an issue not present i n analysing a competitive lending market: can an e q u i l i b r i u m e x i s t i n which the borrower earns p o s i t i v e p r o f i t s ? S e c tion 1 answers t h i s question i n the negative. S e c t i o n 1 a l s o deals w i t h c r e d i t r a t i o n i n g , which does not i n e v i t a b l y occur i n the case of a monopolist lender. 148 For most of the f o l l o w i n g a n a l y s i s , the lender and borrower w i l l seek to maximize t h e i r r e s p e c t i v e r a t e s of return, as the model contains only a s i n g l e borrower and s i n g l e lender. For purposes of comparison, some examples w i l l be considered i n which the lender maximizes the value of h i s loan, as opposed to h i s r a t e of r e t u r n . A n a l y t i c a l l y , the r e s u l t s of t h i s chapter are more complicated than those of Chapter IV. A c c o r d i n g l y , s e c t i o n 2 provides a numerical example of c r e d i t r a t i o n i n g , with the lender maximizing h i s r a t e of r e t u r n . S e c t i o n 3 provides a comparison to s e c t i o n 2, i n which the lender maximizes the value of h i s l o a n as opposed to h i s rate of r e t u r n . F i n a l l y , s e c t i o n s 4 and 5 consider the e f f e c t s of changes i n s e c u r i t y and bankruptcy l e g i s l a t i o n . I t turns out that e f f e c t s of changes i n the salvage recovery of the borrower or lender depend l a r g e l y on whether c r e d i t r a t i o n i n g was present i n the o r i g i n a l e q u i l i b r i u m . 149 1_. A n a l y t i c a l T n t e r p r e t a t i o n - C r e d i t R a t i o n i n g The f i r s t s t e p i s t o d e r i v e c e r t a i n r e s u l t s r e l a t i n g t o the Lender's r a t e of r e t u r n f u n c t i o n . W i t h t r a n s a c t i o n c o s t s of c l i n c l u d e d , t h i s f u n c t i o n i s : L = R/r - p / r + (Sa-R/r)M - c l / r B . I t s f i r s t and second d e r i v a t i v e s w i t h r e s p e c t t o B a r e : L B = (S a - R / r) MB + c l / r B 2 L B B = (Sa-R/r)M B B - 2 c l / r B 3 S i n c e M B B i s p o s i t i v e , L B B i s unambiguously n e g a t i v e . The b o r r o w e r ' s r a t e of r e t u r n f u n c t i o n i s e q u a l t o : V = [(w-RB/r) - (B-RB/r)M - cb]/(w-B) Any v a l u e s l e s s t h a n 1 f o r V r e p r e s e n t s a l o s i n g p r o p o s i t i o n , which the b o r r o w e r has no i n c e n t i v e t o u n d e r t a k e . 150 A c r i t i c a l f i g u r e i n the a n a l y s i s i s the maximum rat e at which the borrower can borrow, while s t i l l breaking even. Let R* represent t h i s value. For the f o l l o w i n g reasons, i t i s c l e a r that R*<r: 1. The borrower receives only a r e t u r n of r on h i s invested c a p i t a l , and thus could only break even on such a loan i f there were no r i s k of bankruptcy. The prospect of bankruptcy ensures that a loan at i n t e r e s t r a t e r must be a l o s i n g p r o p o s i t i o n . 2 . Transactions costs a l s o prevent the borrower from recovering a f u l l e f f e c t i v e rate of r on borrowed c a p i t a l which he i n v e s t s . For each i n t e r e s t r a t e R, l e t B* V(R) represent the borrower's optimal l e v e l of borrowing, subject to the p r o v i s o that B*V(R) i s only d e f i n e d f o r those values of R f o r which V ( R ) > 1 , i . e . the borrower does not lose money by t a k i n g the loan. By d e f i n i t i o n V(B* V(R*) ,R*)= 1 -Define B* L(R) s i m i l a r l y , as the lender's optimal loan l e v e l , f o r 151 a given i n t e r e s t rate R. Two cases must now be considered, namely the case B*L(R*) < B*V(R*) and the case B*L(R*) > B*V(R*) • Consider f i r s t the case B*L(R*) < B* V(R*), i . e . at the maximum rate the borrower can pay, h i s p r e f e r r e d debt l e v e l s t i l l exceed what the lender wishes to loan. This c o n d i t i o n i s s u f f i c i e n t to ensure that c r e d i t i s r a t i o n e d , and al s o that the borrower can not earn p o s i t i v e p r o f i t s i n e q u i l i b r i u m . The s i t u a t i o n i s most con v e n i e n t l y shown g r a p h i c a l l y . 152 Figure 9 - C r e d i t R a t i o n i n g B BL* (R) Bv* (R) Q P R* R E q u i l i b r i u m i n v o l v i n g p o s i t i v e borrowing must l i e somewhere on the locus V=l, between p o i n t s P and Q. This i s seen by the f o l l o w i n g reasoning: 1. No p o i n t to the l e f t of the V=l locus can be an e q u i l i b r i u m , as the lender always has an i n c e n t i v e to increase the i n t e r e s t r a t e . U n t i l he reaches h i s p o i n t of i n d i f f e r e n c e (V=l), the borrower w i l l accept the higher i n t e r e s t r a t e . 2. Any p o i n t l y i n g above the V=l locus i s l e s s advantageous f o r the lender than any p o i n t on the V=l locus. This i s because the V=l locus, i n the region i n question, l i e s above the 153 locus B* L(R), along which L B = 0 . As L B B < 0 , L B <0 throughout the region above the locus V=l. F i n a l l y , i t can a l s o be shown that an e q u i l i b r i u m along the locus from P to Q cannot be Q i t s e l f - i . e . e q u i l i b r i u m cannot be p r e c i s e l y at the maximum i n t e r e s t r a t e at which the borrower can su r v i v e . This f o l l o w s from the f a c t that slope of the V=l locus (-VR/VB) at t h i s p o i n t i s i n f i n i t e , since V B = 0 . A c c o r d i n g l y , a s l i g h t r eduction i n the len d i n g l e v e l along the V=l locu s , r e s u l t s i n the f o l l o w i n g change to L: dL = LBdB + LRdR = dB [L B - VB/VR L R] = LBdB >0 f o r dB<0 These r e s u l t s support the f o l l o w i n g conclusions, which apply to any e q u i l i b r i u m i n v o l v i n g p o s i t i v e borrowing when B*L(R*) < B*V(R*) = 1. C r e d i t w i l l be r a t i o n e d to the borrower. 2 . The borrower w i l l be reduced to zero p r o f i t s . However, the e q u i l i b r i u m i n t e r e s t r a t e w i l l be l e s s than the maximum i n t e r e s t r a t e at which the borrower can s u r v i v e . 154 3. The e q u i l i b r i u m can be described as the s o l u t i o n to the f o l l o w i n g maximization problem: max B R L (B, R) subject to V(B,R)=1. Consider now the case B*L (R*) > B* V(R*), i . e . t h a t the lender's p r e f e r r e d debt l e v e l l i e s above that of the borrower. G r a p h i c a l l y , the s i t u a t i o n i s now: Figure 10 - No C r e d i t R a t i o n i n g Lender's isoquant R* R The c e n t r a l question i s whether any e q u i l i b r i u m i n v o l v i n g p o s i t i v e borrowing w i l l n e c e s s a r i l y occur at p o i n t Q, the max 155 i n t e r e s t rate which the borrower can pay. No unambiguous a n a l y t i c a l answer can be given to t h i s question. To see t h i s consider a small decrease i n the i n t e r e s t r a t e , dR<0. As the lender's o p t i m i z i n g curve l i e s above the borrower's o p t i m i z i n g curve, the new debt l e v e l w i l l s t i l l be on the borrower's o p t i m i z i n g curve, but at a s l i g h t l y higher debt l e v e l . The borrower's o p t i m i z i n g curve i s c h a r a c t e r i z e d by VB=0, and thus has slope -VBR/VBB. The r e s u l t i n g change i n the lender's rate of r e t u r n w i l l thus be: dL = VBdB + VRdR = dR(V R - V BV B R/V B B) The components of the c o e f f i c i e n t of dR are s u f f i c i e n t l y complex that no unambiguously signed expression can be deri v e d . There i s , however, a f a i r l y c l e a r i n t u i t i v e argument showing that any e q u i l i b r i u m i n v o l v i n g p o s i t i v e borrowing must be at Q. The lender's r a t e of r e t u r n i s l i k e l y to be h i g h l y s e n s i t i v e to i n t e r e s t r a t e changes, while the slope of the locus VB=0 i s u n l i k e l y to be. In other words, the lender's isoquants are l i k e l y to be steeper than the VB=0 locus, as shown i n Figure 10 156 above. I f t h i s i s the case, a decrease i n the i n t e r e s t rate w i l l reduce the lender's rate of r e t u r n , so that the lender w i l l p r e f e r p o i n t Q i n the f i g u r e to any l e s s e r i n t e r e s t r a t e . E q u i l i b r i u m w i l l t h erefore be at point Q, at which the borrower i s s t i l l reduced to a c o n d i t i o n of zero p r o f i t , but does not experience c r e d i t r a t i o n i n g . 2. C r e d i t R a t i o n i n g - A Numerical Example. Owing to the complexity of the a n a l y t i c a l r e s u l t s , a numerical example i s i n s t r u c t i v e . The r a t e s of r e t u r n of the borrower and lender are: V = [(w-RB/r) - (B-RB/r)M - cb]/(w-B) L = R/r - p/r + (Sa-R/r)M - c l / B Consider the f o l l o w i n g set of parameter values. The magnitude of the p r o j e c t , w, equals 100. The r e t u r n to c a p i t a l invested i n the p r o j e c t , r, equals 8%. The lender's cost of c a p i t a l , p , i s 4%. The c o e f f i c i e n t of the lender's salvage recover, Sa, equals .5. F i n a l l y , the variance, o, i s 20 and the t r a n s a c t i o n s costs 157 of the borrower and lender, cb and c l , are each equal to 5 . The f o l l o w i n g t a b l e s set out the returns f o r the borrower and lender r e s p e c t i v e l y . For the borrower, any value l e s s than 1 represents a l o s i n g p r o j e c t , which w i l l not be undertaken at the i n t e r e s t r a t e and debt combinations given. For the lender, negative values have the same connotation. F i n a l l y , f i g u r e s w i t h an a s t e r i s k represent maximizing debt l e v e l s , f o r i n t e r e s t r a t e s given. 158 Table 6 - Borrower's Returns R=5% 5.5% 6% 6 . 5% 6 . 8% 6 . 9% 7% 7 . 5% B=30 1. 07 1. 05 1 . 02 1.00 0 . 98 0 . 98 0 . 97 0 . 95* 35 1. 07 1. 07 1 . 04 1.01 0 . 99 0 . 99 0 . 98 0 . 95 40 1 .13 1 . 09 1 . 05 1 . 02 1.00 1. 00 0 . 98 0 . 95 45 1 . 16 1 .15 1 . 07 1. 03 1. 00 1. 00 0 . 99 0 . 95 50 1. 19 1 . 14 1.09 1 . 04 1.01 1. 00 0 . 99* 0 . 94 55 1.23 1 . 17 1 . 11 1 . 05 1 . 01 1.00 0 .99 0 . 94 60 1.27 1 .20 1 . 12 1-. 06 1. 02* 1.00* 0 . 99 0 . 93 65 1.31 1.22 1 . 14 1 . 06 1 . 02 1. 00 0 . 99 0 . 92 70 1.35 1.25 1 . 15 1.06* 1 .01 0 . 99 0 . 98 0 . 90 75 1.39 1.27 1 . 16* 1 . 05 1 .00 0 . 98 0 . 96 0 .87 80 1.41 1.28* 1 . 15 1 . 03 0 . 97 0 . 94 0 . 92 0 . 83 85 1.42* 1.26 1 . 11 0 . 97 0 . 90 0 . 88 0 .85 0 . 75 90 1.34 1 . 16 0 . 99 0 . 84 0 . 75 0.73 0 .70 0 .59 95 0 . 95 0 . 73 0 . 54 0 .36 0 . 27 0 .25 0 .22 0 .10 159 Table 7 - Lenders' Returns R=5% 5 . 5% 6% 6.5% 6.8% 6 . 9% 7 . 0% 7 . 5% B = 30 - 0 . 0 5 5 0 . 000 0 . 053 0 .120 0 . 137 0 .148 0 .158 0 .209 35 -0 . 034 0 .019 0 . 071 0 .121 0 . 151 0 . 162 0 . 172 0 .222 40 - 0 . 0 1 9 0 . 032 0 . 082 0 . 130 0 .159 0 . 168 0 .177* 0 .222* 45 - 0 . 0 0 9 0 . 040 0 . 088 0 .133* 0 .160* 0 .168* 0 . 177 0 .219 50 - 0 . 0 0 2 0 . 045 0 .089* 0 .131 0 .156 0 . 164 0 . 172 0 .210 55 0 . 002 . 046* 0 . 088 0 .127 0 .149 0 .156 0 . 163 0 . 197 60 0 . 004 0". 045 0 . 083 0 .118 0 .138 0 . 144 0 . 150 0 .180 65 0 .004* 0 . 041 0 . 074 0 .106 0 . 124 0 . 129 0 . 134 0 .159 70 0 . 002 0 . 035 0 . 065 0 . 092 0 .106 0 . I l l 0 . 115 0 . 135 75 -0 . 001 0 . 027 0 . 052 0 . 075 0 . 086 0 . 090 0 . 093 0 .108 80 -0 . 007 0 . 016 0 . 037 0 . 055 0 . 063 0 . 066 0 .069 0 . 079 85 -0 . 015 0 . 004 0 . 019 0 . 032 0 . 038 0 . 040 0 . 042 0 . 048 90 -0 . 024 - 0 . 0 1 7 - 0 . 0 0 1 0 . 007 0 . O i l 0 . 012 -0 . 013 0 . 016 95 -0 . 036 -0 . 030 - 0 . 0 2 4 -0 . 021 -0 .019 -0 . 018 - 0 . 0 1 8 - 0 . 0 1 7 The f o l l o w i n g graph sets out the "maximization" debt l e v e l s f o r both borrower and lender, at d i f f e r e n t i n t e r e s t r a t e s . (Dashed l i n e s represent maximizing values which w i l l not be undertaken, as they represent l o s i n g p r o p o s i t i o n s . ) 160 Figure 11 - Maximizing Debt Levels B 90 Borrower 80 70 60 ^ ^ N ^ 4 0 Lender 30 20 10 0 5% 5.5% 6% 6.5% 7% 7.5% 8% R Two s p e c i f i c p o i n t s should be noted from the lender's "maximization" curve: 1. The lender's "maximization" curve i s downward s l o p i n g . As the i n t e r e s t r a t e i n c r e a s e s , so does the r i s k of the loan. The lender reduces debt l e v e l to o f f s e t the increased r i s k r e s u l t i n g from the higher i n t e r e s t r a t e . 161 2. The lender's "maximization" curve tends towards a p o s i t i v e debt value as the i n t e r e s t r a t e approaches 8%, the borrower's r e t u r n on invested c a p i t a l . In other words, the lender i s s t i l l w i l l i n g to lend a p o s i t i v e amount as the maximum allow a b l e i n t e r e s t r a t e i s approached (although there i s no guarantee that the borrower w i l l borrow at t h i s i n t e r e s t r a t e ) . From the lender's p e r s p e c t i v e , there i s s t i l l p r o f i t to be made from the loan, even though the borrower i s not earning a margin on borrowed c a p i t a l . The f i g u r e s given i n the t a b l e s demonstrate a simple example of c r e d i t r a t i o n i n g . Under the bargaining s t r u c t u r e set out above, the e q u i l i b r i u m i n t e r e s t r a t e w i l l be 6.9% (the maximum rate at which the borrower w i l l borrow). C r e d i t r a t i o n i n g occurs. With an i n t e r e s t r a t e of 6.9%, the borrower wishes to borrow 60 u n i t s of c a p i t a l . But the lender lends only 45. The i n t u i t i v e reason f o r the c r e d i t r a t i o n i n g i s simple. The borrower enjoys the b e n e f i t of lev e r a g i n g o f f of increased debt l e v e l s . He r e t a i n s a l l of the upside b e n e f i t should bankruptcy be avoided. 162 There i s no corresponding b e n e f i t to the lender. He cannot recover more than co n t r a c t i n t e r e s t - any upside beyond that i s f o r the account of the borrower only. Accordingly, he l i m i t s debt to give added s a f e t y to h i s re t u r n . Debt i s not reduced to i n f i n i t e s i m a l l e v e l s because of the f i x e d c o s t s involved - f i x e d costs g r e a t l y reduce the r e t u r n on very small loans. 3. Comparison - Maximization of the Lender's Value Function I t i s u s e f u l to compare the r e s u l t s of the e x e r c i s e when the lender seeks to maximize h i s value f u n c t i o n , as opposed to h i s rate of r e t u r n . The f o l l o w i n g t a b l e sets out the lender's value f u n c t i o n at d i f f e r e n t loan l e v e l s and i n t e r e s t r a t e s , using the same parameter values used i n s e c t i o n 2 . 163 Table 8 - Lenders' Values R=5% 5.5% 6% 6 . 5% 6 . 8% 6 . 9 7 . 0% 7 . 5% B = 30 - 1 . 6 5 0 . 00 1 .59 3 .60 4 .11 4 . 11 4 . 74 6 . 27 35 - 1 . 1 9 0 .665 2 .48 4 .23 5 .28 5 . 67 6 . 02 7 . 77 40 -0 . 76 1 .28 3 .28 5.20 6.36 6 . 72 7 .08 8 . 88 45 - 0 . 4 1 1 . 80 3 . 96 6 . 00 7 .20 7 . 56 7 .96 9 . 86 50 - 0 . 1 0 2 .25 4 . 4 5 6 .55 7 .80 8 .20 8.60 10 .5 55 0 . 11 2 . 53 4 . 84 6 . 98 8 .19 8 . 58 8 . 96 10 . 84* 60 0 . 24 2 . 70* 4 . 98* 7 . 08* 8 .28* 8 . 64* 9 . 0* 10 . 80 65 0 . 26* 2 . 66 4 .44 6 .36 7 .44 8.38 8 .71 10 . 33 70 0 . 14 2 .45 4 . 55 6 .44 7 .42 7 . 77 8 . 05 9.45 75 - 0 . 0 7 5 2 . 02 3 . 90 5.62 6 .45 6 . 75 6 . 97 8 . 10 80 -0 . 056 1.28 2 . 96 4 .40 5 . 04 5 .28 5 . 52 6 . 32 85 - 1 . 2 5 0.34 1.61 2 .72 3 .23 3 .40 3 . 57 4 . 08 90 -2 . 16 -1 . 53 - 0 . 9 0 - 0 . 6 3 0 . 99 1 . 08 1 . 17 1 . 44 95 -3 .42 -2 . 85 -2 . 28 - 1 . 9 9 - 1 . 8 0 - 1 . 7 1 -1 . 71 - 1 . 6 1 The lender's "maximization" curve i s now n e a r l y h o r i z o n t a l , approximately equal to the B=60 locus. There i s now no c r e d i t r a t i o n i n g , when the lender maximizes h i s value f u n c t i o n . The e q u i l i b r i u m i n t e r e s t r a t e i s again 6.9%, the maximum r a t e at which the borrower w i l l borrow. But now, the debt l e v e l i s equal to 60, the p r e f e r r e d l e v e l f o r both agents. 164 I t i s easy to demonstrate, however, that t h i s r e s u l t i s not general, and that c r e d i t r a t i o n i n g can appear when parameter values are changed. The simplest parameter to change i s Sa, the lender's recovery on bankruptcy. This parameter does not enter the borrower's value f u n c t i o n i n any way, and thus a f f e c t s the d e c i s i o n of the lender only. The f o l l o w i n g t a b l e sets out the debt l e v e l s that maximize the lender's value f u n c t i o n using the same parameter values as above, but wi t h v a r y i n g l e v e l s of Sa. Table 9 - Maximizing Debt Levels f o r the Lender's Value Function R = 5% 5 . 5% 6% 6 . 5% 7% 7 . 5% Sa = .5 * 60 60 60 60 55 0 .45 * 50 50 55 55 55 0 .4 * * 50 50 50 50 0.35 * * 45 45 45 50 0 . 3 * * * 45 45 45 0 . 25 * * * 40 40 45 0 . 2 * * * 35 40 40 0 . 15 * * " * 35 35 40 0 . 1 * * + * 35 35 The f o l l o w i n g graph compares the r e s u l t s of Tables 6 and 9, to 165 show the maximizing debt l e v e l s of the borrower and lender. The graph shows the lender's "maximization" curve f o r d i f f e r e n t l e v e l s of Sa, to show that the absence of c r e d i t r a t i o n i n g i s not a general r e s u l t . 166 Figure 1 2 - Maximizing Debt Levels 5% 5 .5% 6% 6.5% 7% 7 .5% 8% R 167 The graph makes i t c l e a r that c r e d i t r a t i o n i n g must r e s u l t f o r r e l a t i v e l y low l e v e l s of Sa. There i s no i n t e r e s t r a t e at which the lender w i l l lend the amount that the borrower would choose. As i n Chapter V, the model under c o n s i d e r a t i o n can be used to evaluate the consequences of a change i n lending l e g i s l a t i o n . Consider f i r s t the case i n which c r e d i t r a t i o n i n g i s present (and i n which the borrower i s a l s o operating at zero p r o f i t ) . From s e c t i o n 1, e q u i l i b r i u m i s the s o l u t i o n t o : maxB RL (B, R) subject to V(B,R) =1 with lagrangean: L - AV The f i r s t order c o n d i t i o n s are: L R = AV, B 168 L R = XVR V = l D i f f e r e n t i a t i o n of the f i r s t order c o n d i t i o n s y i e l d s XdR + YdB + ZdSa = 0 VRdR + VBdB = 0 where: X = - L R R V B + L B V R R + V R L B R - L R V B R = - V B [ L R R - A V R J + V „ L B R - L R V B R > 0 Y = V R L B B - L R V B B + LBVRB - VBLRB = V E [ L R R - X V R R ] + - V^RB > 0 Z = V R L B S A - V B L R S A < 0 E l i m i n a t i n g f i r s t dR, and then dB, from the above system y i e l d s dB [Y - XVB/VR] = -ZdSa dR[X - YVR/VB] = -ZdSa Both dB and dR can be signed unambiguously. As Sa increases, both B and R increase as w e l l . The f o l l o w i n g graph i l l u s t r a t e s the reason: B 169 Figure 13 - Increase i n Sa R * R When no c r e d i t r a t i o n i n g i s present, as i n Figure 10, above, i t i s c l e a r that an incremental change i n the lender's salvage recovery w i l l not a f f e c t the e q u i l i b r i u m p o i n t , which w i l l remain at Q. :r_upj Following the a n a l y s i s of Chapter V, s e c t i o n 6, consider f i r s t the s i t u a t i o n i n which bankruptcy l e g i s l a t i o n i s a l t e r e d to s h i f t recovery on bankruptcy from the lender to the borrower. 170 The borrower's and lender's rates of r e t u r n are now: V = [(w-RB/r) - (B-RB/r)M + MB(l/r-Sa) -cb] / (w-B) L = R/r - p/r + (Sa-R/r)M - c l / r B D i f f e r e n t i a t i o n of the f i r s t order c o n d i t i o n s now y i e l d s : XdR + YdB + ZdSa = 0 VRdR + VBdB + V S adSa = 0 where: X = - L R R V B + L B V R R + V R L B R - L R V B R = - V B [ L R R - A V R R ) + V R L B R - L R V B R > 0 Y = V R L B B - L R V B B + L B V ^ - VBLm = V R [ L R R - X V R R ] + L B V R B - V B L R B > 0 Z = V R L B S A + V R S A L B - V B L R S A - V B S A L R As i n Chapter V, s e c t i o n 6 , the s i g n of Z i s now ambiguous. I t fo l l o w s that n e i t h e r dB nor dR can be signed ambiguously when a change i s imposed on dSa. The s i t u a t i o n i s d i f f e r e n t i n the s i t u a t i o n where c r e d i t r a t i o n i n g does not e x i s t . E q u i l i b r i u m under these c o n d i t i o n s i s 171 at the maximum i n t e r e s t r a t e at which the borrower can s u r v i v e . This i s point Q i n Figure 10, and i s c h a r a c t e r i z e d by: V = 0 V B = 0 (1) Since the borrower i s at a maximization p o i n t with respect to debt l e v e l , i t must a l s o be the case that VBB<0. To t a l d i f f e r e n t i a t i o n of the system (1), and then e l i m i n a t i n g r e s p e c t i v e l y dR and dB, y i e l d s : V B B VS/VB] V B R VS/VR] In the f i r s t equation, the c o e f f i c i e n t s of dR and dSa are, r e s p e c t i v e l y , negative and p o s i t i v e . An increase i n Sa represents a decrease i n the borrower's recovery on bankruptcy, reducing h i s w i l l i n g n e s s t o borrow and the maximum i n t e r e s t r a t e he can pay. In the second equation, the c o e f f i c i e n t of dSa i s ambiguous, so that no e f f e c t on debt l e v e l s can be p r e d i c t e d . dR [V B R - V B B VR/VB] = -dSa[V B S a -dB[V B B - V B R VB/VR] = -dSa[V B S a -172 The f i n a l issue i s what happens when only the borrower's recovery on bankruptcy i s a l t e r e d , so that the r e s p e c t i v e r a t e of r e t u r n f u n c t i o n s are now: V = [(rw-RB). - (rB-RB) M + MB(l-rSa) - cb] / (w-B) L = R - p -RM - c l / B When c r e d i t r a t i o n i n g i s present, d i f f e r e n t i a t i o n of the f i r s t order c o n d i t i o n s y i e l d s : XdR + YdB + ZdSa = 0 VRdR + VBdB + V S adSa = 0 where: X = - L R R V B + L B V R R + V R L B R - L R V B R = - V B [ L R R - A V R J + V R L B R - L R V B R > 0 Y = V R L B B - L R V B B + LBVRB - V B L R B = V R [ L R R - A V R R ] + L^V^ - V B L R B > 0 2 = V R S A L B - V B S A L R > 0 E l i m i n a t i n g f i r s t dB and then dR y i e l d s : dR[X - YVR/VB] = -dSa[Z - YVSa/VB] 173 dB [Y - XVB/VR] = -dSa[Z - XVSa/VR] I f Sa increases, R decreases and the e f f e c t on B i s ambiguous. F i n a l l y , i t i s c l e a r that a n a l y s i s of the case without c r e d i t r a t i o n i n g i s the same as above ( i . e . a r e a l l o c a t i o n from the lender t o the borrower) as the r e s u l t of that a n a l y s i s depended only the c h a r a c t e r i s t i c s of the borrower's ra t e of r e t u r n f u n c t i o n . 174 CHAPTER VII - SINGLE LENDER, MULTIPLE BORROWERS This chapter considers the d e c i s i o n making of a s i n g l e lender who must a l l o c a t e loanable funds between two borrowers. The lender i s given an exogenous maximum on the t o t a l funds he can lend. He i s not forced to lend a l l of the a v a i l a b l e funds, i f holding some back i s c o n s i s t e n t w i t h p r o f i t maximization. The lender acts with knowledge of the borrowers' o b j e c t i v e f u n c t i o n s , and sets both an i n t e r e s t r a t e and a maximum loan l e v e l f o r each borrower. The borrowers act independently, and are unable to co l l u d e . To avoid degenerate s o l u t i o n s , the borrowers a l s o i n c u r t r a n s a c t i o n c o s t s . A p r i n c i p a l purpose behind the a n a l y s i s i s to examine the a l l o c a t i o n of loanable funds between borrowers of d i f f e r e n t s i z e s . I t i s commonly a s s e r t e d that small businesses are r i s k i e r than l a r g e r ones. In Canada, i t has a l s o been ass e r t e d that bank loans are not s u f f i c i e n t l y a v a i l a b l e to small business. The borrower's maximization problem, and the s t o c h a s t i c process which u n d e r l i e s i t , are modified t o i n v e s t i g a t e these i s s u e s . Each borrower i s endowed w i t h a given wealth l e v e l , wL. He can 175 borrow a d d i t i o n a l c a p i t a l , B I F at the i n t e r e s t rate s p e c i f i e d by the lender, RA. He earns a r e t u r n equal to r>Ri on both borrowed and e q u i t y c a p i t a l , with r being common to the borrowers. The borrower's t r a n s a c t i o n costs are equal to c, regardless of the amount borrowed. To accommodate d i f f e r e n t "sized" borrowers, a borrower's u n c e r t a i n t y now v a r i e s with the amount borrowed. The standard d e v i a t i o n b u i l t i n t o the s t o c h a s t i c process f o r the borrower's wealth i s equal to A+oB±, wit h A being common to the borrowers. S e t t i n g the standard d e v i a t i o n i n t h i s way ensures that a "small" borrower ( i . e . one wi t h lower s t a r t i n g wealth, w) has a p r o p o r t i o n a t e l y higher standard d e v i a t i o n ( i . e . i n r e l a t i o n to t o t a l wealth) f o r a given debt-equity r a t i o . Bankruptcy s t i l l occurs when a borrower has l o s t h i s e q u i t y c a p i t a l , i . e . when h i s t o t a l " wealth has f a l l e n to BL. The lender and borrowers a l l seek to maximize t h e i r value f u n c t i o n s , which are deri v e d i n s e c t i o n 1 . The assumptions of t h i s Chapter lead to a number of unambiguous r e s u l t s . S e c tion 2 shows that c r e d i t r a t i o n i n g f o r a l l borrowers occurs i n e v i t a b l y , and s e c t i o n 3 shows that a l l borrowers are 176 reduced to a c o n d i t i o n of zero p r o f i t . Section 4 examines the e f f e c t of changes i n the aggregate funds a v a i l a b l e t o the lender. Here, only numerical r e s u l t s are a v a i l a b l e , as a n a l y t i c a l r e s u l t s are too complex. The i m p l i c a t i o n of the numerical r e s u l t s i s c l e a r , however. When o v e r a l l c r e d i t i s s u f f i c i e n t l y t i g h t , i t i s the small borrower who i s squeezed out of the market. Section 5 i n v e s t i g a t e s the e f f e c t of changes i n the salvage recovery of the borrower and lender. Again, only numerical r e s u l t s are a v a i l a b l e . 1^ The Value Functions of the Borrower and Lender Consider a borrower with "equity" w. I f he borrows B, h i s s t a r t i n g wealth w i l l be w+B. Under the assumptions s t a t e d above, the s t o c h a s t i c process r e p r e s e n t i n g the borrower's wealth over time, x ( t ) , w i l l evolve according t o : dx(t) = r x ( t ) d t - RBdt + (A+oB)dz (1) 177 with absorption at x(t)=B. Define k = w+B as the s t a r t i n g value f o r x ( t ) , i . e . x ( 0 ) = k. As a f u n c t i o n of k, the borrower's value f u n c t i o n must solve: 1 2 d2v dv (2) — (A+OB) + (rk-RB) - rv = 0 v 7 2 dk2 d k with v defined as: v(k) = l i m £ [ e " r t ( x ( t ) -S) ] = l i m £ [ e " r t x ( t ) ] ; x ( 0) =x The boundary c o n d i t i o n s f o r v are: RB l i m v(k) =0 ; l i m v{k) =k-k-B r As B i s parametric and k = w+B, i t i s c l e a r l y p o s s i b l e to express t h i s value f u n c t i o n as v(w;B,R). Following the a n a l y s i s of Chapter I I , the s o l u t i o n to the d i f f e r e n t i a l equation i s given by: 2 S exp [ ] r{A+oB) (rw+rB-RB) I ds RB RB rw + rB-RB s v(w) = w + B - - ( B - ) " C (3) 2 s exp [ ] 7 r ( A + o S ) 2 ( rB-RB) J ds rB-RB 178 which can be r e w r i t t e n as: v(w) = w + (B-RB/r)(1-M) - c, where M i s now defined by: 2 s exp[ r(A+OB)2 2 d s r w + r B - R B S M exp [ 2 S r(A+OB)2 (rB-RB) J -rB-RB S (4) I f the lender recovers SaB on l i q u i d a t i o n , the lender's value f u n c t i o n can be expressed i n the same form as i n Chapter I I : 1(B,R) = RB/r - pB/r + (Sa-R/r)BM The f o l l o w i n g t a b l e s set out lender and borrower values f o r various l e v e l s of R and B. Two borrowers are considered, a "small" borrower w i t h w=50 and a "large" borrower w i t h w=100. In a d d i t i o n , the f o l l o w i n g parameters are used: p=.04, r=.08, A=30, o=.3, Sa=.4. 179 Table 10 - Borrower's values f o r w = 50 R=4 . 5 R=5 . 0 R=5.5 R=6.0 R=6 . 5 R=7 . 0 R=7.5 R=7.7 R=7 . 9 B=10 47 . 57 47 . 2 46 . 83 46.46 46 . 09 45.72 45.36 45.21 45.07 B=20 49 . 9 49 .19 48.47 47 . 77 47 . 07 46 . 37 45.68 45.41 45.13 B=30 52.01 50 . 98 49.95 47 . 77 47 . 94 46 . 94 45.96 45.58 45.19 B = 40 53 . 93 52 . 6 51.29 49 . 99 48 . 72 47.46 46 .22 45.73 45 . 24 B = 50 55 . 68 54 . 07 52 . 5 50.95 49 . 42 47 . 92 46.44 45.86 45.28 B=60 57 . 27 55.42 53 . 6 51.81 50 . 06 48 . 34 46.65 45 . 98 45.32 B=70 58 . 74 56.65 54.6 52.6 50.63 48 . 71 46 . 83 46.09 45.36 B=80 SO.09 57.78 55.52 53 . 32 51.16 49 . 05 47 46.19 45 .39 B=90 61. 34 58 . 82 56.37 53 . 97 51. 64 49 . 36 47 . 15 46.28 45 . 42 B = 100 62 .49 59.78 57.14 54 . 58 52 . 08 49 . 65 47 . 29 46 . 36 45.45 B=110 63 . 55 60 . 66 57 . 86 55.13 52 .47 49 . 9 47 . 29 46 .43 45 .47 B=120 64 . 53 61.48 58.51 55.63 52 . 84 50 .13 47 . 52 46 . 5 45.49 B=130 65.44 62 .23 59 .11 56 . 09 53 . 17 50.34 47 . 62 46.56 45 . 51 B=140 56 . 28 62 . 92 59 . 66 56 . 51 53 .46 50.53 47 . 71 46 . 61 45 . 53 B=150 67 . 06 63 . 55 60.16 56.89 53 . 73 50 . 7 47 . 79 46 . 65 45 . 54 B=160 67 . 77 64.13 60.62 57.23 53.98 50.85 47 .86 46 . 7 45.56 B=170 68 .43 64 . 67 61. 04 57 . 55 54 .19 50 . 98 47 . 92 46 . 73 45.57 B = 180 69 . 04 65 .15 61.42 57.83 54 .39 51.1 47 . 97 46.76 45.58 B=190 69 . 59 65.6 61.76 58 .08 54 . 56 51. 21 48 .02 46.79 45 . 59 B=200 70 .1 66 62 . 07 58 .31 54 .71 51. 3 48 . 06 46 . 81 45 . 59 B=210 70.56 66 . 37 62 . 35 58 .51 54 .85 51.37 48 . 09 46 . 83 45 . 6 B=220 70.98 66 . 7 62 . 6 58 . 68 54 . 96 51.44 48.12 46.84 45.6 B=230 71. 37 66 . 99 62 . 82 58 . 84 55 . 06 51. 5 48 . 14 46 . 86 45.61 B=240 71. 71 67.26 63 . 01 58 . 97 55 .15 51. 54 48 . 16 46 . 86 45 . 61 B=250 72 . 02 67 . 5 63 .01 59 . 09 55.22 51. 58 48 . 17 46 . 87 45.61 B=260 72 . 31 67 . 71 63 . 33 59.19 55.28 51. 61 48 .18 46 . 88 45.61 B=270 72 . 56 67 . 89 63.46 59 . 27 55.33 51.63 48 .18 46 . 88 45 . 61 180 Table 11 - Borrower's values f o r w = 100 R=4 . 5 R = 5 . 0 R=5 .5 R=6 .0 R=6 .5 R=7 . 0 R=7 .5 R=7 .7 R=7 . 9 B=10 9 8 . 8 9 8 . 2 5 97 . 7 97 .16 96 . 62 96 . 08 9 5 . 5 3 9 5 . 3 2 9 5 . 1 B=20 102 .3 1 0 1 . 3 1 0 0 . 2 99 .19 98 .13 97 . 08 96 . 04 9 5 . 6 2 95 . 2 B=30 1 0 5 . 7 104 .1 102 . 6 101 99 . 54 98 . 02 96 . 5 9 5 . 9 95 . 3 B=40 108 . 8 106 . 8 102 . 6 102 . 8 1 0 0 . 8 98 . 89 96 . 93 96 .16 95 . 38 B=50 1 1 1 . 8 109 . 3 106 . 9 104 . 4 102 9 9 . 6 9 97 . 33 96 . 39 9 5 . 4 6 B=S0 114 . 6 1 1 1 . 7 108 . 8 106 103 . 2 1 0 0 . 4 97 . 7 96 . 61 95 . 53 B=70 1 1 7 . 2 113 . 9 1 1 0 . 6 107 . 4 104 . 2 1 0 1 . 1 9 8 . 0 4 9 6 . 8 1 95 . 6 B = 80 1 1 9 . S 115 . 9 1 1 2 . 3 1 0 8 . 7 1 0 5 . 2 1 0 1 . 7 98 . 35 97 9 5 . 6 6 B=90 1 2 1 . 9 117 . 9 113 . 9 110 106 .1 1 0 2 . 3 98 . 64 97 .18 95 .72 B=100 124 .1 119 . 7 115 . 4 1 1 1 . 1 107 102 . 9 98 . 91 97 . 34 9 5 . 7 7 B=110 126 .1 1 2 1 . 4 116 . 7 112 . 2 107 . 7 103 .4 99 . 16 97 . 48 95 . 82 B=120 128 123 118 113 . 2 108 . 5 103 . 9 99 . 39 9 7 . 6 2 9 5 . 8 6 B=130 129 . 8 124 . 5 1 1 9 . 3 114 . 1 109 . 2 104 .3 99 . 6 97 . 74 95 . 91 B=140 1 3 1 . 5 125 . 9 120 . 4 115 109 . 8 1 0 4 . 7 99 . 79 97 . 86 95 . 94 B=150 133 .1 1 2 7 . 2 1 2 1 . 4 115 . 8 1 1 0 . 4 105 . 1 99 . 97 97 . 96 95 . 98 B=160 134 . 6 128 .4 122 . 4 116 . 6 110 . 9 105 . 4 100 .1 98 . 06 96 . 01 B=170 135 129 . 6 123 . 3 117 . 3 1 1 1 . 4 1 0 5 . 7 100 . 2 98 .14 96 . 04 B=180 137 . 3 1 3 0 . 6 124 . 2 117 . 9 1 1 1 . 9 106 100 .4 98 . 22 96 . 06 B=190 138 . 5 1 3 1 . 6 124 . 9 118 . 5 1 1 2 . 3 1 0 6 . 3 100 . 5 9 8 . 2 9 9 6 . 0 8 B=200 1 3 9 . 6 132 . 5 125 .7 119 1 1 2 . 6 106 . 5 1 0 0 . 6 9 8 . 3 5 96 .1 B=210 1 4 0 . 7 133 .4 126 .3 1 1 9 . 5 113 1 0 6 . 7 1 0 0 . 7 98 .4 96 .12 B=220 1 4 1 . 6 134 . 2 126 . 9 120 113 . 3 106 . 9 100 . 8 98 .45 96 .13 B=230 1 4 2 . 6 134 . 9 127 . 5 1 2 0 . 4 113 .6 107 100 . 8 98 .49 9 6 . 1 5 B = 240 143 .4 1 3 5 . 5 128 1 2 0 . 7 113 .8 1 0 7 . 2 100 . 9 9 8 . 5 2 9 6 . 1 6 B=250 144 .2 136 . 2 128 . 5 1 2 1 . 1 114 1 0 7 . 3 101 9 8 . 5 5 96 .17 B=260 144 . 9 136 .7 128 . 9 1 2 1 . 4 114 . 2 107 . 4 101 98 . 58 96 . 17 B=270 1 4 5 . 5 1 3 7 . 2 129 . 2 1 2 1 . 6 114 .4 107 . 5 101 98 . 59 9 6 . 1 8 181 Table 12 - Lender's values f o r w = 50 R = 4 . 5 R = 5 . 0 R = 5 . 5 R = 6 . 0 R = 6 . 5 R = 7 . 0 R = 7 . 5 R = 7 . 7 R = 7 . 9 B = 1 0 - 0 . 0 4 2 0 . 3 2 4 0 . 6 8 8 1 . 0 5 1 . 4 1 1 . 7 7 2 . 1 3 2 . 2 7 2 . 4 1 B = 2 0 0 . 1 7 7 0 . 5 1 4 1 . 2 1 . 8 8 2 . 5 5 3 . 2 2 3 . 8 9 4 . 1 5 4 . 4 1 B = 3 0 - 0 . 3 9 4 0 . 5 8 8 1 . 5 6 2 . 5 2 3 . 4 7 4 . 4 5 . 3 3 5 . 7 6 . 0 7 B = 4 0 - 0 . 6 8 2 0 . 5 6 1 1 . 7 9 2 . 9 7 4 . 1 8 5 . 3 6 6 . 5 1 6 . 9 7 7 . 4 2 B = 5 0 - 0 . 1 0 3 0 . 4 4 6 1 . 9 3 . 3 3 4 . 7 3 6 . 1 1 7 . 4 6 8 8 . 5 3 B = 6 0 - 1 . 4 4 0 . 2 5 1 . 9 1 3 . 5 4 5 . 1 3 6 . 6 9 8 . 2 1 8 . 8 2 9 . 4 1 B = 7 0 - 1 . 8 9 0 1 . 8 4 3 . 6 4 5 . 4 7 . 1 2 8 . 8 9 . 4 6 1 0 . 1 B = 8 0 - 2 . 3 9 - 0 . 3 3 1 . 6 9 3 . 6 5 5 . 5 6 7 . 4 2 9 . 2 3 9 . 9 4 1 0 . 6 B = 9 0 - 2 . 9 2 - 0 . 7 1 . 4 6 3 . 5 7 5 . 6 1 7 . 6 9 . 5 2 1 0 . 3 1 1 B = 1 0 0 - 3 . 5 - 1 . 1 3 1 . 1 7 3 . 4 1 5 . 5 7 7 . 6 7 9 . 6 9 1 0 . 5 1 1 . 2 B = 1 1 0 - 4 . 1 - 1 . 6 8 . 3 1 3 . 1 8 5 . 4 5 7 . 6 4 9 . 7 5 1 0 . 6 1 1 . 4 B = 1 2 0 - 4 . 7 4 - 2 . 1 1 0 . 4 3 3 2 . 8 8 5 . 2 5 7 . 5 2 9 . 7 1 0 . 5 1 1 . 4 B = 1 3 0 - 5 . 4 - 2 . 6 6 0 2 . 5 3 4 . 9 7 7 . 3 2 9 . 5 5 1 0 . 4 1 1 . 3 B = 1 4 0 - 6 . 0 9 - 3 . 2 5 0 2 . 1 1 4 . 6 3 7 . 03 9 . 3 2 1 0 . 2 1 1 . 1 B = 1 5 0 - 6 . 8 - 3 . 8 6 - 1 . 0 4 1 . 6 5 4 . 2 2 6 . 6 7 9 9 . 9 1 0 . 8 B = 1 6 0 - 7 . 5 4 - 4 . 5 1 - 0 . 1 6 1 . 1 3 3 . 7 6 6 . 2 4 8 . 6 9 . 5 1 0 . 4 B = 1 7 0 - 8 . 2 9 - 5 . 1 9 - 2 . 2 3 0 . 5 7 2 3 . 2 3 5 . 7 5 8 . 1 2 9 . 0 3 9 . 9 1 B = 1 8 0 - 9 . 0 7 - 5 . 9 - 2 . 8 9 0 2 . 6 6 5 . 2 7 / 5 8 8 . 4 9 9 . 3 6 B = 1 9 0 - 9 . 8 6 - 6 . 63 - 3 . 5 8 - 0 . 6 8 2 . 0 4 4 . 6 1 6 . 9 8 7 . 8 8 8 . 7 5 B = 2 0 0 - 1 0 . 6 7 - 7 . 3 9 - 4 . 2 9 - 1 . 3 6 1 . 3 8 3 . 9 4 6 . 3 2 7 . 2 1 8 . 0 8 B = 2 1 0 - 1 1 . 5 - 8 . 1 8 - 5 . 0 4 - 2 . 08 0 . 6 7 5 3 . 2 3 5 . 6 6 . 4 9 7 . 3 5 B = 2 2 0 - 1 2 . 3 4 - 8 . 9 9 - 5 . 8 1 - 2 . 83 - 0 . 1 2 . 4 9 4 . 8 4 5 . 7 2 6 . 5 6 B = 2 3 0 - 1 3 . 2 1 - 9 . 8 - 6 . 6 1 - 3 . 6 1 - 0 . 8 5 1 . 7 4 . 0 3 4 . 9 5 . 7 3 B = 2 4 0 - 1 4 . 0 7 - 1 0 . 6 - 7 . 2 4 - 4 . 4 3 - 1 . 6 6 0 . 8 7 8 3 . 1 8 4 . 0 3 4 . 8 5 B = 2 5 0 - 1 4 . 9 6 - 1 1 . 5 - 8 . 2 7 - 5 . 2 6 - 2 / 5 0 0 . 0 2 2 . 2 9 3 . 1 3 3 . 9 4 B = 2 6 0 - 1 5 . 8 5 - 1 2 . 4 - 9 . 1 2 - 6 . 1 2 - 3 . 3 7 - 0 . 8 7 1 . 3 8 2 . 2 3 B = 2 7 0 - 1 6 . 7 6 - 1 3 . 3 - 1 0 - 7 - 0 . 2 6 - 1 . 8 9 4 . 2 7 1 . 2 3 2 182 Table 13 - Lender's values f o r w = 100 R=4 . 5 R=5 . 0 R=5.5 R=6 . 0 R=6.5 R=7 . 0 R=7 . 5 R=7 . 7 R=7.9 B=10 0.411 0.95 1.49 2 . 03 2 . 57 3.1 3 . 64 3 . 9 4 . 07 B = 20 0.741 1.78 2 . 83 3 . 87 4 . 9 5.93 6 . 95 7.36 7.77 B=30 0.99 2 . 5 4 . 01 5.52 7 8.49 9.96 10 . 5 11.1 B=40 1.16 3 .12 5 . 06 6 . 98 8 . 9 10 . 7 12 . 7 13 .4 14 .1 B = 50 1.26 3 . 63 5 . 97 8.28 10.6 12 . 8 15.1 16 16.8 B=60 1. 29 4 . 04 6 . 74 9 .42 12 .1 14 . 7 17 . 2 18 . 2 19.3 B=70 1.26 4.36 7 . 41 10.42 13 .4 16 . 3 19.1 20.3 21.4 B=80 1.17 4 . 59 7 . 97 11.28 14.5 17 . 7 20 . 9 22 .1 23 .3 B=90 1. 02 4 . 76 8.42 12 . 01 15.5 19 22.4 23.7 25 B=100 0 . 823 4 . 84 8.78 12 . 63 16 .4 20 . 1 23.7 25.1 26 . 5 B=110 0.578 4 . 86 9 . 05 13 .14 17 .1 21 24 . 8 26.3 27 . 7 B=120 0 . 287 4 . 82 9 .25 13 . 56 17.7 21. 8 25 . 8 27 . 3 28 . 9 B=130 -0.05 4 . 72 9 . 35 13 . 87 18.3 22 . 5 26 . 6 28 . 2 29 . 8 B=140 -0.419 4 . 56 9 . 38 14 . 09 18 . 6 23 27.3 28 . 9 30.6 B=150 -0.831 4 . 34 9 . 37 14 .23 18 . 9 . 23 . 4 27 . 8 29 . 5 31. 1 B=160 -1.28 4 . 08 9.27 14 . 28 19.1 23 . 7 28 . 2 29 . 9 31.6 B=170 -1.76 3 . 77 9 .1 14 . 25 19.2 23 . 9 28.4 30.2 31.9 B=180 -2.23 3 . 4 8 .88 14 . 14 19.2 24 28 . 6 30.3 32 B=190 -2.83 3 8.6 13 . 96 19.1 24 28.6 30.4 32.1 B=200 -3.4 2 . 5 8 . 26 13.72 19.4 23.8 28 . 5 30.3 32 B=210 -4.1 2.06 7 . 87 13 .4 18 .7 23 . 6 28 . 3 10 31. 8 B = 220 -4 . 65 1. 53 7.43 13 . 03 18.3 23 . 3 18 29 . 8 31. 5 B=230 -5.31 0 . 961 6 . 94 12 . 6 17 . 9 22 . 9 27 . 6 29.4 31.1 B=240 -6 . 01 0.358 6.4 12 . 11 17 . 5 22 . 5 27.1 28 . 9 30 . 2 B=250 -6.72 -0.278 5.82 11.57 16 . 9 22 26.6 28.3 30 B=260 -7.45 -0.945 5.21 10 . 99 16 .4 21.4 26 27.7 29 . 3 B=270 -8.21 -1 . 64 4 . 56 10 . 36 15.7 20 . 7 25 . 2 27 28 . 6 Selected r e s u l t s ( i . e . f o r i n t e r e s t rates R=5%, 6%, 7%, and 7.5%) are presented i n the f o l l o w i n g graphs. 184 Figure 15 - Lender's values: w=50 50 100 150 200 250 300 B 185 While t a b l e s 10 to 13 are not extensive enough to show i t , c a l c u l a t i o n of the borrower's value f u n c t i o n f o r l a r g e r debt l e v e l s confirms that e v e n t u a l l y v B becomes negative. This i s the i n t u i t i v e r e s u l t , given the assumption of t h i s chapter that the magnitude of the s t o c h a s t i c component of revenue, (A+oB)dz, increases w i t h B. For l a r g e enough debt l e v e l s , i n c r e a s i n g v a r i a b i l i t y of revenue increases the r i s k of bankruptcy, to the point where that r i s k outweighs the b e n e f i t s of increased leverage. The f u n c t i o n M(B,R), defined above i n equation (4), i s more complex than i n previous chapters. However, some unambiguous r e s u l t s can be sta t e d . As i n Chapter I I I , s e c t i o n 3, define the functions E l , F l , I I , and J l as: (cw+rB-RB)2 e x p [ - — ] (rw+rB-RB) r{A+OB) El = e x p [ - ] ; Fl -r(A+OB)2 (rw+rB-RB)2 II = 1 e x p [ - - ] d s ; Jl = (rw+rB-RB) j r » * r B - « 8 r(A + OB)2 r» + rB-2 S e x p [ - -] r(A+oB) ds 2 s E2, F2, 12, and J2 have corresponding meanings w i t h rw+rB-RB replaced by rB-RB. 186 The expression f o r MR i s s i m i l a r to that obtained i n Chapter I I : MR = 2BJ1 ( I l / J l - I2/J2) / (r(A+oB) 2) (J2) which i s p o s i t i v e as I ( x ) / J ( x ) i s an i n c r e a s i n g f u n c t i o n , (see Chapter I I , s e c t i o n 5 ) . However, i t i s no longer p o s s i b l e to s i g n MB unambiguously f o r a l l parameter values. A f t e r considerable manipulation and s i m p l i f i c a t i o n , MB i s equal t o : MB = 2M [ ( r w o - A ( r - R ) ) I l / J l + A(r-R)I2/J2 ] / r(A+oB) 3 I t i s p l a i n that t h i s f u n c t i o n cannot be signed a p r i o r i , as I l / J l > I2/J2. I n t u i t i v e l y , an increase i n B now has two o f f s e t t i n g e f f e c t s on the p r o b a b i l i t y of f u t u r e bankruptcy. One the one hand, increased leverage increases expected p o s i t i v e cash flow, decreasing the r i s k of bankruptcy. On the other hand, an increase i n B increases the magnitude of the s t o c h a s t i c component of revenue, which increases the r i s k of bankruptcy. I f the " v a r i a b i l i t y " a s s o c i a t e d w i t h "equity" c a p i t a l i s the same 187 as that a s s o c i a t e d w i t h "debt" c a p i t a l , so that A=ow, then the expression f o r MB reduces t o : MB = 2M [ RAI1/J1 + (r-R)AI2/J2 ] / r(A+oB) 3 which i s unambiguously p o s i t i v e . I t f o l l o w s that MB w i l l only be negative i f the " v a r i a b i l i t y " a s s o c i a t e d with e q u i t y c a p i t a l i s much greater than that a s s o c i a t e d with debt c a p i t a l , i . e . A>>ow, •which i s an i n h e r e n t l y u n l i k e l y p r o p o s i t i o n . In a d d i t i o n , numerical a n a l y s i s shows that MB i s p o s i t i v e f o r a l l of the values c a l c u l a t e d i n t a b l e s 10 - 13. The p o s s i b i l i t y that MB i s negative can thus be s a f e l y ignored. Using these r e s u l t s both: v B = (1-R/r) (1-M) - MB (B-RB/r) and 1 B = R/r - p/r + (Sa-R/r) (M + BMB) are ambiguous i n value. The p a r t i a l d e r i v a t i v e of v wit h respect to R: 188 v R = (-B/r)(1-M) - MR(B-RB/r) i s unambiguously negative. F i n a l l y , the p a r t i a l d e r i v a t i v e of 1 w i t h respect t o R: 1 R = (B/r) (1-M) + (Sa-R/r) (BMR) cannot be signed unambiguously. However, i t s "fundamental" value, B/r, i s c l e a r l y p o s i t i v e , and a negative r e s u l t i s only p o s s i b l e i n the v i c i n i t y of bankruptcy, a p o s s i b i l i t y which can be ignored i n e q u i l i b r i u m s o l u t i o n s . 2 . The Existence of C r e d i t R a t i o n i n g Under the model set out above, i t can be shown r e l a t i v e l y e a s i l y that c r e d i t r a t i o n i n g must e x i s t . I t turns out that the existence of c r e d i t r a t i o n i n g f o l l o w s r e a d i l y from the value f u n c t i o n s of a s i n g l e borrower and s i n g l e lender, and i s not dependent on: (a) the presence of a second borrower, and the r e s u l t i n g 189 a l l o c a t i o n problem f o r the lender; or (b) the existence of an exogenous l i m i t on the lender's funds, or on such a c o n s t r a i n t being b i n d i n g . Consider f i r s t the borrower's value f u n c t i o n . (As i n previous chapters, the borrower's equity, w, w i l l be taken as parametric and suppressed. The choice v a r i a b l e s are now B and R). v(B,R) = w + (B-RB/r)(1-M) - c For any given r a t e of i n t e r e s t , the borrower's p r e f e r r e d debt l e v e l w i l l be c h a r a c t e r i z e d by the c o n d i t i o n v B = 0 . D i f f e r e n t i a t i n g v w i t h respect to B and s e t t i n g the r e s u l t to 0 y i e l d s : (1-R/r)(1-M) - (B-RB/r)M B = 0, which im p l i e s i n t u r n t h a t : M + BMR = 1. 190 Now consider the lender's value f u n c t i o n : 1 = RB/r - pB/r + (Sa-R/r)BM. D i f f e r e n t i a t i o n w i t h respect to B y i e l d s : 1 B = R/r - p/r + (Sa - R/r) (M + BMB) As shown above, at the p o i n t vB= 0, M+BMB = 1 , so that 1 B reduces to: 1 B = R/r - p/r + (Sa - R/r) = Sa - p/r. In the a n a l y s i s of previous chapters, the q u a n t i t y Sa-p/r was assumed to be negative, and f o r an obvious reason. I f the salvage value obtained by a lender on bankruptcy exceeds the cos of h i s loan, a lender can never lose money, even i f bankruptcy occurs immediately. So long as an immediate bankruptcy poses a r i s k of l o s s f o r the lender, the q u a n t i t y Sa-p/r must be negative, and thus 1 B must be negative at the poin t where vB=0. By i m p l i c a t i o n , the lender, i s always motivated to reduce c r e d i t from the borrower's d e s i r e d l e v e l - i . e . to r a t i o n c r e d i t . 191 Using the r e s u l t s of se c t i o n s 1 and 2, the lender's maximization problem can now be f o r m a l l y set up as a lagrangean. Assume that there are two borrowers w i t h wealth endowments wl and w2. The borrowers value functions v l ( B l , R l ) and v2(B2,R2). The corresponding value f u n c t i o n s f o r the lender are l l ( B l , R l ) and 12(B2,R2). Assume a l s o that B l * ( R l ) and B2*(R2) represent the re s p e c t i v e p r e f e r r e d borrowing l e v e l s f o r the f i r s t and second borrowers, f o r given values of R l and R2. As forced borrowing i s not allowed, the c o n d i t i o n s B1<B1* and B2<B2* must p r e v a i l . As w e l l , since no borrower would borrow i f the r e s u l t would be to reduce h i s net wealth, the lender's problem i s subject t o the c o n s t r a i n t s vl>wl, v2>w2. F i n a l l y , the lender's funds are constrained, so that Bl+B2<Bmax, where Bmax i s exogenous. The lender must now choose B l , R l , B2 and R2 to: maximize l l ( B l , R l ) + 12(B2,R2) subject t o : B l ( R l ) < B l * ( R l ) ; 192 B2(R2) < B2* (R2) ; v l ( B l , R l ) > w l ; (5) v2(B2,R1) > W2; B1+B2 < Bmax. The Lagrangean i s equal to 11 + 12 - a l ( w l - v l ) - a2(w2-v2) - y K B l - B l * ) - y2(B2-B2*) -A(Bl+B2-Bmax) with f i r s t order c o n d i t i o n s : l l B 1 + c<lvl B 1 - yl - X = 0 12 B 2 + ce2v2B2 - y2 - X = 0 11 R 1 + a l v l R 1 + Y1B1* R 1 = 0 12 R 2 + a2v2 R 2 + y2B2* R 2 = 0 (6) a l ( w l - v l ) = 0 a2(w2-v2) = 0 y l ( B l - B l * ) = 0 Y2(B2-B2*) = 0 X (Bl+B2-Bmax) = 0 As shown above, however, c r e d i t r a t i o n i n g must p r e v a i l , implying 193 that each of yl and y2 must be zero. The f i r s t order conditions can thus be reduced to the f o l l o w i n g : l l B 1 + a l v l B 1 - A = 0 12 B 2 + a2v2 B 2 - A = 0 l l R 1 + a l v l R 1 = 0 12 R 2 + a2v2 R 2 = 0 (7) a l ( w l - v l ) = 0 a2(w2-v2) = 0 A(B1+B2-Bmax) = 0 Since the d e r i v a t i v e of the lender's value f u n c t i o n s with respect to R l and R2 w i l l be non-zero i n e q u i l i b r i u m , i t fo l l o w s that each of a l and a2 must be non-zero. From t h i s i t follows that wl=vl and w2=v2. In other words, each of the borrowers i s d r i v e n to the zero p r o f i t c o n d i t i o n , and t h i s r e s u l t i s independent of whether the lender's o v e r a l l lending c o n s t r a i n t i s binding or not. In the r e s u l t , the lagrangean formulation of the lender's problem shows that the c o n d i t i o n s of c r e d i t r a t i o n i n g and "zero p r o f i t " f o r borrowers are automatic p r o p e r t i e s of the model. 194 4. A change i n the lending c o n s t r a i n t A n a t u r a l question t o ask i s what the r e s u l t of a change i n Bmax w i l l be. The expressions f o r the borrowers' and lender's value functions are too complex to produce a n a l y t i c a l r e s u l t s . The f o l l o w i n g numerical r e s u l t s i l l u s t r a t e both what trends can be expected, and why a n a l y t i c a l r e s u l t s are u n l i k e l y to be a v a i l a b l e . Using the data i n t a b l e s 10 to 13 (and i n t e r p o l a t i n g with a d d i t i o n a l c a l c u l a t i o n s where necessary), the f o l l o w i n g table sets out the loan amounts that w i l l be a l l o c a t e d to the "small" and "large" borrowers under c o n s i d e r a t i o n . The maximizing values are rounded t o the nearest m u l t i p l e of 10. 195 Table 14 - Debt l e v e l s f o r d i f f e r e n t Lender's c o n s t r a i n t s Bmax B l B2 Bmax B l B2 10 0 0 170 120 50 20 20 0 180 130 50 30 30 0 190 130 60 40 40 0 200 140 60 50 50 0 210 140 70 60 60 0 220 150 70 70 70 0 230 150 80 80 80 0 240 160 90 90 90 0 250 170 90 100 100 0 260 180 90 110 110 . 0 270 180 90 120 120 0 280 180 100 130 130 0 290 180 110 140 140 0 300 190 110 150 150 0 310 190 120 160 160 0 320 190 120 While the above t a b l e corresponds to a s p e c i f i c set of parameters, there i s no reason to expect that the trends i t shows would not g e n e r a l i z e . A loan to the smaller borrower i s r e l a t i v e l y more r i s k y , w i t h the r e s u l t that the lender w i l l a l l o c a t e a l l c r e d i t to the w e a l t h i e r borrower, when the amount of loanable funds i s sma l l . As the w e a l t h i e r borrower's debt l e v e l s grow, the marginal r e t u r n he pays on a d d i t i o n a l loan funds goes 196 down. The reason i s that higher debt l e v e l s b r i n g higher r i s k of bankruptcy, w i t h r e s u l t i n g l o s s of the lender's c a p i t a l . Accordingly, funds s t a r t to become a v a i l a b l e to the smaller borrower. There i s a t h r e s h o l d l e v e l at which the small borrower enters the market. Owing to h i s t r a n s a c t i o n costs, he must enter with a d i s c r e t e , r a t h e r than i n f i n i t e s i m a l , debt l e v e l . In the r e s u l t , the c r e d i t a v a i l a b l e to the large borrower takes a d i s c r e t e drop when the small borrower enters the market. F i n a l l y , f o r values of Bmax i n excess of the small borrower's "threshold" l e v e l , i t i s c l e a r that the b e n e f i t s of i n c r e a s i n g c r e d i t (or a l t e r n a t i v e l y the burden of t i g h t e r c r e d i t c o n s t r a i n t ) i s shared between the small and l a r g e borrower. 5 . A change i n the salvage r a t i o . In previous chapters, s e v e r a l changes i n the l e g a l r e l a t i o n s between borrowers and lenders were modelled by v a r y i n g Sa, the parameter governing the lender's recovery on insolvency. I t i s not proposed to d u p l i c a t e each a n a l y s i s here, as the maximization 197 problem i s too complex to y i e l d a n a l y t i c a l r e s u l t s . This can be seen from the system of f i r s t order c o n d i t i o n s (7). The f i r s t four equations from that system reduce to the f o l l o w i n g e q u i l i b r i u m c o n d i t i o n : 11 B - ( v l B / v l R ) l l R = 12 B - ( v 2 B / v 2 R ) l 2 R (8) This c o n d i t i o n s t a t e s , i n e f f e c t , that the t o t a l incremental value of an incremental loan must be the same f o r each borrower i n e q u i l i b r i u m . The f i r s t term i n each expression i s the d i r e c t increment i n value f o r an incremental loan, while the second term i s the i n d i r e c t increment i n value, r e s u l t i n g from the consequential change i n the i n t e r e s t r a t e . The change i n the i n t e r e s t r a t e ensures that the borrower remains i n a c o n d i t i o n of "zero p r o f i t " at the new loan l e v e l . Equation (8) i l l u s t r a t e s why i t i s not f e a s i b l e to derive a n a l y t i c r e s u l t s i n t h i s s e c t i o n - the expressions governing the r e s p e c t i v e changes i n the l e f t and r i g h t hand side s of (8), when the value of Sa i s changed, are h i g h l y complex. I t i s c l e a r that at or near a corner s o l u t i o n , a change i n Sa w i l l have a dramatic change i n e q u i l i b r i u m loan l e v e l s . I f 198 e q u i l i b r i u m l i e s at the th r e s h o l d p o i n t , where the small borrower has j u s t entered the market, a reduction i n Sa w i l l force him back out of the market, r e s u l t i n g i n a d i s c r e t e r e a l l o c a t i o n of c r e d i t to the l a r g e lender. Conversely, an increase i n Sa could allow the small borrower i n t o the market, r e s u l t i n g i n a d i s c r e t e r e a l l o c a t i o n of c r e d i t away from the lar g e borrower, i f the e x i s t i n g e q u i l i b r i u m only m a r g i n a l l y excludes the small borrower. Numerical r e s u l t s suggest, however, that at e q u i l i b r i u m s o l u t i o n s that are not at or near a corner, a change i n Sa i s l i k e l y to have l i t t l e e f f e c t on e q u i l i b r i u m loan l e v e l s . The f o l l o w i n g values have been computed to show how the e q u i l i b r i u m value f o r Bmax=180, as shown i n t a b l e 14, change when Sa = .3, .4, or .5. 199 Table 16 - Lender's values (and corresponding i n t e r e s t rates) f o r "large" lender (w=100) at d i f f e r e n t values of Sa Sa = .3 Sa = .4 Sa = .5 B=120; R=.0743 20 .3 25 .25 30 .21 B=121; R=.0743 20 .32 25 .38 30.36 B=122; R=.0744 20 .42 25 . 5 30 .59 B=123; R=.0744 20 .43 25 . 58 30 . 74 B=124; R=.0744 20 .45 25 .66 30 .88 B=125; R=.0745 20 . 54 25 .83 31 .11 B=126; R=.0745 20 .56 25 .91 31.26 B=127; R=.0745 20 . 57 25 . 98 31.4 B=128; R=.0745 20 . 58 26 .06 31 .55 B=129; R=.0746 20 .66 26 .22 31 .77 B=130; R=.0746 20 .67 26 .29 31 .91 200 Table 17 - Lender's values (and corresponding i n t e r e s t rates) f o r "small" lender (w=50) at d i f f e r e n t values of Sa Sa = .3 Sa = .4 Sa = . 5 B=50; R=.0631 0 . 1 4 . 2 6 . 84 B=51; R=.0634 1 4 . 33 7 . 03 B=52; R=.0636 1 . 66 4 . 24 7 . 19 B=53; R=.0638 1 .69 4 . 52 7 . 35 B=54; R=.0640 1 . 72 4 .62 7 . 51 B=55; R=.0642 1 . 75 4 .71 7 . 68 B=56; R=.0644 1 . 78 4 . 81 7 . 84 B=57; R=.0646 1.81 4 . 91 8 B=58; R=.0648 1 . 84 5 8 . 17 B=59; R=.0650 1 . 87 5 . 1 8 .33 B=60; R=.0652 1 . 9 5 .2 8 . 5 A n a l y s i s of these f i g u r e s shows that f o r Bmax=180, the value of Sa i s l a r g e l y i r r e l e v a n t t o the optimal loan l e v e l s . In each case, values of Bl=129 f o r the larg e lender and B2=51 i s op t i m i z i n g . The r e s u l t i n not changed whether Sa i s set equal t o . 3 , .4 or .5. 201 CHAPTER V I I I - CONCLUSION This t h e s i s has examined the i m p l i c a t i o n s of corporate bankruptcy-through the use of s t o c h a s t i c d i f f e r e n t i a l equations and absorbing boundaries. The s t o c h a s t i c process used was d i f f e r e n t from that most commonly used i n finance l i t e r a t u r e (geometric brownian motion), i n that the magnitude of the s t o c h a s t i c term was constant, r a t h e r than p r o p o r t i o n a l to the s t a t e v a r i a b l e under c o n s i d e r a t i o n . A d i f f e r e n t i a l equation and cl o s e d form s o l u t i o n were derived f o r the discounted value of expected future revenue f o r a f i r m f a c i n g p o s s i b l e bankruptcy. The same equation and s o l u t i o n were a l s o a p p l i c a b l e i n v a l u i n g a lender's stream of expected revenue and the expected salvage value recovered by a lender on bankruptcy. The d i f f e r e n t i a l equation d e r i v e d was not homogeneous, as i s often the case i n models based on geometric brownian motion. With value f u n c t i o n s a v a i l a b l e f o r both a borrower and lender, various aspects of debtor - c r e d i t o r r e l a t i o n s were examined. In a competitive loan market, a s t r a i g h t f o r w a r d explanation f o r 202 c r e d i t r a t i o n i n g was provided, without r e s o r t i n g to market imperfections or asymmetric in f o r m a t i o n . As w e l l , loan l e v e l s were found to r i s e as l e g a l changes were simulated to increase, c e t e r i s paribus, e i t h e r a lender's or borrower's salvage value on bankruptcy. However, when salvage recovery was r e a l l o c a t e d from a borrower t o a lender, the r e s u l t was ambiguous. The borrower's increased demand was o f f s e t by the lender's decreased supply. In a monopoly lending market, the borrower was g e n e r a l l y reduced to a zero p r o f i t c o n d i t i o n . However, c r e d i t r a t i o n i n g could not be demonstrated t o occur g e n e r a l l y . Where c r e d i t r a t i o n i n g was present, a l e g a l change to increase the lender's recovery on bankruptcy would r e l a x the c r e d i t c o n s t r a i n t , i n c r e a s i n g lending l e v e l s . Absent c r e d i t r a t i o n i n g , a change i n the lender's salvage recovery d i d not a f f e c t any e q u i l i b r i u m v a r i a b l e . Conversely, an increase i n the borrower's recovery increased the e q u i l i b r i u m i n t e r e s t r a t e whether or not c r e d i t r a t i o n i n g was present. A r e a l l o c a t i o n from lender to borrower produced no unambiguous r e s u l t s when c r e d i t r a t i o n i n g was present, but unambiguously increased i n t e r e s t r a t e s (by i n c r e a s i n g the maximum i n t e r e s t r a t e the borrower could afford) when c r e d i t r a t i o n i n g was absent. 203 The f i n a l case analysed was that of a monopolist lender a l l o c a t i n g l i m i t e d funds between two borrowers, one of which was smaller and r e l a t i v e l y more r i s k y . In t h i s part of the a n a l y s i s , a d i f f e r e n t s t o c h a s t i c process and maximization problem were used. Under the r e v i s e d s t r u c t u r e , c r e d i t r a t i o n i n g was shown to e x i s t f o r a l l borrowers independently of the lender's a l l o c a t i o n problem, and even when the lender's c o n s t r a i n t on t o t a l lending was not b i n d i n g . For r e l a t i v e l y low values of the lender's c o n s t r a i n t , the r i s k i e r borrower was cut out of the market a l t o g e t h e r . As more aggregate funds became a v a i l a b l e , they were a l l o c a t e d between the two borrowers. However, no borrower ever received as much as he would p r e f e r . A change i n the lender's salvage recovery could o n l y be evaluated n u m e r i c a l l y , but seemed to have r e l a t i v e l y l i t t l e e f f e c t on loan l e v e l s except at corner s o l u t i o n s . The a n a l y s i s has been c a r r i e d out i n a p a r t i a l e q u i l i b r i u m framework, l i k e most other s t u d i e s focussing d i r e c t l y on the issue of bankruptcy. I t remains to be seen whether the closed form s o l u t i o n s derived i n t h i s t h e s i s are s u f f i c i e n t l y t r a c t a b l e to be used i n a general e q u i l i b r i u m framework. 204 REFERENCES Adler, B.E. 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