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An analysis of the cost of capital hypotheses Spence, John David 1968

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AN'ANALYSIS OF THE COST OF CAPITAL HYPOTHESES by JOHN DAVID SPENCE B. Ap. Sc., Univers i ty of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION in the department of COMMERCE & BUSINESS ADMINISTRATION We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1968 In p re sen t i ng t h i s t he s i s in 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 fo r reference and Study. I f u r t h e r agree that permiss ion fo r ex tens i ve copying of t h i s t he 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 r ep re sen t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be a l lowed without my w r i t t e n permi s s ion . Department of Commerce and Business Administration The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date June, 1968 i i ABSTRACT The c o s t of c a p i t a l has r e c e i v e d much t h e o r e t i c a l and e m p i r i c a l s tudy i n r e c e n t y e a r s . Two c o n t r a d i c t o r y v iews have emerged c o n c e r n i n g the e f f e c t of c a p i t a l s t r u c t u r e on the c o s t o f c a p i t a l . W r i t e r s such as M o d i g l i a n i and M i l l e r m a i n t a i n t h a t the c o s t of c a p i t a l i s i ndependent of the r e l a -t i v e p r o p o r t i o n o f l i a b i l i t i e s to owner s ' e q u i t y and depends on l y on the r i s k a s s o c i a t e d w i t h the type o f b u s i n e s s the f i r m i s i n . The o p p o s i t e v iew i s t a ken by those w r i t e r s who s uppo r t what i s known as the t r a d i t i o n a l v i e w . These w r i t e r s m a i n t a i n t h a t j u d i c i o u s use of debt can reduce the f i r m ' s c o s t o f c a p i t a l . The purpose of t h i s paper i s to de te rm ine wh ich a p -p roach appears to be the more a c c u r a t e i n a r e a l w o r l d s i t u a -t i o n . We f i r s t i n v e s t i g a t e the many d i f f i c u l t i e s a s s o c i a t e d w i t h the e m p i r i c a l t e s t s wh ich have been a p p l i e d to e v a l u a t e the two c o n f l i c t i n g h y p o t h e s e s . The prob lems a s s o c i a t e d w i t h t he se t e s t s lead us to r e j e c t them as a means o f r e s o l -v i ng the c o s t of c a p i t a l c o n t r o v e r s y . I n s t e a d , we choose a t h e o r e t i c a l a p p r o a c h . Based on s u g g e s t i o n s made by M o d i g l i a n i and M i l l e r and by the t r a d i t i o n a l w r i t e r s , we p o s t u l a t e the way i n wh ich debt and e q u i t y c a p i t a l i z a t i o n r a t e s are e x p e c -ted to respond to i n c r e a s e s i n the amount of debt i n the c a p i -t a l s t r u c t u r e . The M o d i g l i a n i and M i l l e r h y p o t h e s i s and the t r a d i t i o n a l h y p o t h e s i s a re s t u d i e d i n d e t a i l and computer models f o r each h y p o t h e s i s are then d e v e l o p e d . The h y p o t h e -i i i t i c a l l y de te rm ined c a p i t a l i z a t i o n r a t e s a re used as i n d e p e n -dent v a r i a b l e s i n the models to deve l op r e l a t i o n s h i p s between the dependent v a r i a b l e s and the d e b t - e q u i t y r a t i o . The r e s -ponse of the dependent v a r i a b l e s to changes i n l e ve r age i s s t u d i e d to see i f i t r e p r e s e n t s r a t i o n a l i n v e s t o r b e h a v i o r . Rea l w o r l d f a c t o r s a re i n t r o d u c e d i n t o the a n a l y s i s . E f f e c t s of c o r p o r a t e income tax on both the M o d i g l i a n i and M i l l e r h y p o t h e s i s and the t r a d i t i o n a l h y p o t h e s i s are i n v e s -t i g a t e d . In a d d i t i o n , we s tudy the way i n wh i ch l e g a l r e s t -r i c t i o n s and l i m i t e d p e r s o n a l l i a b i l i t y may r e s t r i c t the Mo-d i g l i a n i and M i l l e r a r b i t r a g e p r o c e s s . The t h r e e computer models used i n the a n a l y s i s a re d e -s c r i b e d i n d e t a i l i n the a p p e n d i c e s . These models have been deve l oped so as t o be as f l e x i b l e as p o s s i b l e . A 1 i I * parame-t e r s are s p e c i f i e d by the u s e r , so the models may be adapted to a v a r i e t y of s i t u a t i o n s . Program l i s t i n g s and the o u t p u t used i n our a n a l y s i s a re a l s o i n c l u d e d . 1 V TABLE OF CONTENTS Page CHAPTER I INTRODUCTION 1 Types of Risk 3 Models of the Firm 5 The Dividend E f fec t S8 Limitat ions of Empirical Tests 13 CHAPTER II THE MODIGLIANI AND MILLER HYPOTHESIS 23 Assumptions 23 Proposit ion I 25 Proposit ion II 29 Ef fects of Corporate Tax 30 Development of a Computer Model 32 CHAPTER III THE TRADITIONAL HYPOTHESIS 38 The Net Operating Income Approach 38 The Net Income Approach 39 Two Possible Computer Models 41 Marginal Cost of Debt 45 CHAPTER IV AN ANALYSIS OF THE MODIGLIANI AND MILLER HYPOTHESIS 49 Arbitrage 50 Limited Liabi I i ty 55 V Page Behavior of Debt and Equity Holders 57 Analysis of Results from Computer Model. 60 Analysis of Subjective Probab i l i ty D i s -t r ibut ions of Earnings 71 CHAPTER V ANi.ANALYSIS OF THE TRADITIONAL HYPOTHESIS.. 84 The Tradi t iona l Hypothesis 84 The Net Income Hypothesis 89 CHAPTER VI CONCLUSIONS 92 BIBLIOGRAPHY 98 APPENDIX I LIST OF SYMBOLS 1033 APPENDIX II MODIGLIANI AND MILLER MODEL 104 Descr ipt ion of Program 104 Program L i s t i n g 106 Output 108 APPENDIX III TRADITIONAL MODEL 11 6 Descr ipt ion of Program 1 1 6 Program L i s t i n g 118 Output 120 vi Page APPENDIX IV NET INCOME MODEL 124 Descr ipt ion of Program 124 Program L i s t i n g 126 Output 128 vi i LIST OF TABLES Table Page I Value of the Corporat ion, Net Operating Income Mode I 39 II Value of the Corporation, Net Income Model 39 III E f fec t of Capital Structure on Value, Net Income Model 40 IV Use of Arbitrage 53 V Subjective Probab i l i ty D i s t r ibu t ion of Corporate Income 72 VI D i s t r ibut ion of Returns to Shareholders for Levered and Unlevered Firms, (Symmetrical D i s t r ibut ion) 72 VII D i s t r ibu t ion of Returns to Shareholders for Levered and Unlevered Firms, (Skewed D i s t r ibut ion) 74 VIII D i s t r ibu t ion of Returns to S HarehoIders for Levered and Unlevered Firms, S E f fect s of Limited L i a b i l i t y 75 IX Interest Coverage of Residual Income to Equity Holders 80 v i i i LIST OF FIGURES Figure Page I Modigliani and M i l l e r Hypothesis, Cost of Funds and Value of the Firm as a Function of the Debt-Equity Ratio (No Corporate Income Tax) 62 II Modigliani and M i l l e r Hypothesis, Cost of Funds and Value of the Firm as a Function of the Debt-Equity Rat io, {50% Corporate Tax) 65 III Modigliani and M i l l e r Hypothesis, E f fec t of Corporate Income Tax Rate on the Value of the Firm 69 IV Before-Tax Earnings of Shareholders for Various Levels of Corporate Earn ings . . . . 79 V Trad i t iona l Hypothesis, Cost of Funds and Value of the f i rm as a Function of the Debt-Equity Ra;tio (No Cor-porate Incomie Tax) 85 VI Trad i t iona l Hypothesis, Cost of Funds and Value of the Firm as a Function of the Debt-Equity Rat io, (50$ Cor-porate Tax) 88 VII Net Income Hypothesis, Cost of Funds and Value of the Firm as a Functi on of the Debt-Equity Ratio (50$ Corporate Tax) 90 1 CHAPTER I INTRODUCTION The c o s t o f c a p i t a l has r e c e i v e d a g r e a t dea l o f bo th t h e o r e t i c a l and e m p i r i c a l a t t e n t i o n i n r e c e n t y e a r s . How-e v e r , the many s t u d i e s , i n s t e a d of p r o v i d i n g a p r e c i s e and wo rkab l e d e f i n i t i o n , have r e s u l t e d i n much c o n t r o v e r s y . That t h i s c o n t r o v e r s y w a r r a n t s f u r t h e r s tudy can be seen f rom the f o l l o w i n g d e f i n i t i o n , wh i ch shows the impor tance o f the c o s t 1 . of c a p i t a l i n f i n a n c e . "The c o s t of c a p i t a l f o r a f i r m i s a d i s c o u n t r a t e w i t h the p r o p e r t y t h a t an i n ve s tmen t w i t h a r a t e of p r o f i t above (below) t h i s r a t e w i l l r a i s e ( l owe r ) the v a l ue o f the f i r m . " A l t h o u g h the c o s t o f c a p i t a l i s e a s i l y d e f i n e d , i t i s very d i f f i c u l t to d e t e r m i n e . In p a r t t h i s d i f f i c u l t y r e s u l t s f rom the f a c t t h a t the f i n a n c e f u n c t i o n i s composed o f t h r e e c l o s e l y i n t e r r e l a t e d 2. probI ems: " 1 . How l a r g e s h o u l d an e n t e r p r i s e be , and how f a s t shou Id i t grow? 2. In what f o rm s hou l d i t h o l d i t s a s s e t s ? 3. What s h o u l d be the c o m p o s i t i o n of i t s l i a b i l i t i e s ? " We ob se r ve t h a t the c o s t o f c a p i t a l wh i ch de te rm ine s the c u t o f f p o i n t f o r f u t u r e a s s e t e x p e n d i t u r e s i s i t s e l f i n f l u e n c e d by the t ype and amount o f a s s e t s p u r c h a s e d . As Gordon s t a t e s , the c o s t o f c a p i t a l i s not a c o n s t a n t but " a f u n c t i o n of the l e v e l o f the f i r m ' s i n ve s tmen t w i t h the pa ramete r s of the f u n -2 ct ion depending on the f i rm ' s dividend ra te , debt-equity r a t i o , 3. rate of return on investment, and/or other var i ab les . " A major d i f f i c u l t y in determining the cost of cap i ta l for use as an investment decis ion c r i t e r i a results from the fact that i t does not compensate for the uncertainty a t t a -ched to the possible returns from the f i rm ' s assets. Authors 4. 5. such as Cord and Paine have studied ways in which r isk may be introduced, and have found that in many cases, pro-jects with expected returns which exceed the estimated average cost of cap i ta l should be rejected due to abnormal r i s k , while other projects with low or negative rates of return should be accepted because they reduce overa l l corporate r i sk . They have determined an asset p ro t fo l i o which provides the optimum combination of return and r i s k . Many authors have also studied the companion problem of the optimum composition of l i a b i l i t i e s and owners' equity with which to finance the asset p o r t f o l i o . It is in th is area that much controversy has ar i sen. Writers such as 6. Modigliani and M i l l e r character ize a group which maintains that no matter what combination of debt and equity the f i rm has in i t s cap i ta l s t ructure, the cost that the f irm pays to acquire i t s cap i ta l is a constant. Many other t rad i t iona l 7. wr i ter s , in a group of which Solomon is representat ive, maintain that the judicious use of debt can reduce the f i rm ' s average cost of capi ta l and therefore reduce the cutoff rate for investment project acceptance. It is. the question of whether or not the average cost of capita l can be reduced 3 through the use of debt that we wish to study in th is paper. We w i l l study th i s problem in i s o l a t i o n , but before proceding, i t should be noted that an optimal cap i ta l structure for the f i rm can only be determined through a considerat ion of i t s assets. As Johnson ind icates , "there are certa in instances where a major investment w i l l change the ent i re cost of c a p i -ta l funct ion for a f i rm. A major investment in jet a i r c r a f t apparently lowered the break-even point on the a i r l i n e s , with a consequent reduction in the i r r i sk c l a s s , even though they 8. remained in the same type of business." If we are to disregard the asset side of the f i rm ' s balance sheet, our analysis must then be made by considering a l l investment to be in assets which are s imi lar in r i sk to those already held by the f i rm. Types of Risk As Schwartz has noted, the Individual f i rm faces two 9. types of r i s k . The f i r s t of these is the external or bus i -ness r i s k . This is determined from -the s t a b i l i t y of earnings of the f i rm , and the safety, l i q u i d i t y and marketabi l i ty of i t s assets. The f i rm has no control over this type of r i sk since i t is d ictated by the type of business the f i rm is in and is not subject to any control by the f i nanc i a l decis ion makers. Business r i sk can be excluded from our analysis by considering cap i ta l structures only for firms in what Modig-10. l i an i and M i l l e r have ca l led "equivalent return" classes 11 . or in what Wippern has ca l l ed equivalent r i sk c lasses . 4 There are many ways i n wh ich these e q u i v a l e n t r e t u r n or r i s k cil;asses can be d e f i n e d . A s t a n d a r d s t a t i s t i c a l method would be to i n c l u d e a l l f i r m s i n the same r i s k c l a s s i f they have s i m i l a r c o e f f i c i e n t s o f v a r i a t i o n f o r income, where the c o -e f f i c i e n t of v a r i a t i o n i s measured by the s t a n d a r d d e v i a t i o n of income d i v i d e d by the mean expec ted income. I t s h o u l d be noted t h a t t h i s method measures v a r i a t i o n r e l a t i v e to s i z e , so f i r m s of g r e a t l y d i f f e r i n g s i z e can be i n c l u d e d i n the same r i s k c l a s s . M o d i g l i a n i and M i l l e r use an a l t e r n a t i v e but s i m i l a r d e f i n i t i o n of r i s k c l a s s . They v iew the f i r m as y i e l d i n g a s t ream o f p r o f i t s over t i m e , but the e lements of t h i s s t r eam are u n c e r t a i n and ex tend i n d e f i n i t e l y i n t o the f u t u r e . However, the mean v a l ue o f the s t ream over t ime i s f i n i t e and r e p r e s e n t s a random v a r i a b l e wh i ch can be d e s c -r i b e d by a s u b j e c t i v e l y a s s i g n e d p r o b a b i l i t y d i s t r i b u t i o n . Then, the average va lue over t ime o f the s t ream of income a c c r u i n g to the f i r m i s the annual r e t u r n to the f i r m , and the ma thema t i c a l e x p e c t a t i o n of t h i s average i s the expec ted annual r e t u r n . whose e lement s are s u b j e c t to the j o i n t p r o b a b i l i t y d i s t r i -b u t i o n , The f i r m gene ra te s ansincome s t r e a m , X(1), X(2)- X ( T ) , Average annual r e t u r n to the f i r m i s : T X = Mm I Z X ( t ) T - * o o T t=1 . . 5 so the expected annual return i s : X = E(X) Then for firms in any given equivalent return c la s s , the ra t io of the annual return to the expected annual return w i l l be a constant, ( -w- = constant). A l l shares in any expec-X ted return class are equally des irable from the point of view of business r i s k . The other type of r i sk for the f i rm is the internal or f i nanc i a l r i sk which depends on i t s cap i ta l s t ructure. This r i sk depends on the proportion of f ixed commitment l i a -b i l i t i e s to equity c a p i t a l . In this paper, we w i l l deal with f irms in a given r i sk c l a s s , and see i f there is some combination of debt and equity which w i l l resu l t in a maxi-mum value for the f i rm , and, therefore, a minimum cost of capi t a l . Models of the Firm There are two types of models which can be developed to study the e f fec t of leverage on the cost of c a p i t a l . In the f i r s t model, the company subst i tutes debt for equity. 12. This is the type of model presented by Mao. In this case, the company is considered to have a f ixed group of assets which y i e l d earnings of a given r i sk c l a s s . I n i t i a l l y , the company has only equity in i t s cap i ta l s t ructure. The ow-ners issue debt and reta in the proceeds, thereby keeping the earnings and assets constant. That i s , the company issues debt and uses the proceeds to redeem outstanding stock held by the owner s . The advantage o f t h i s model i s t h a t i t keeps the a s s e t s c o n s t a n t , and so a l l o w s a d i r e c t compar i son o f the v a l ue o f the f i r m a t v a r y i n g l e v e l s of d e b t . 13. The second type o f model i s the one used by So lomon. In t h i s m o d e l , as debt i s i s s u e d , i t i s used to a c q u i r e a d -d i t i o n a l a s s e t s , so the company i s a l l o w e d to expand. Th i s model p e r m i t s the ready i d e n t i f i c a t i o n of the m a r g i n a l c o s t o f each. i n c rement of d e b t . However, to d e r i v e u s e f u l r e s u l t s , we must assume t h a t the new a s s e t s pu rchased produce e a r -n i n g s of the same y i e l d and q u a l i t y as the o r i g i n a l a s s e t s . Fo r our work, we s h a l l use a model of the f i r s t t y p e . As has been p r e v i o u s l y n o t e d , an expan s i on o f a s s e t s can e a -s i l y change the r i s k c l a s s of the f i r m , so i d e a l l y a s s e t s s h o u l d be kept c o n s t a n t . A l s o , t h i s model i s t he type used by M o d i g l i a n i and M i l l e r , whose h y p o t h e s i s we want t o i n v e s -t i g a t e . The models to be deve loped w i l l be based on the f o l -l ow ing a s s u m p t i o n s : 1. The company has a f i x e d group of a s s e t s on which i t ea rns a c o n s t a n t r a t e of r e t u r n of a g i v e n q u a l i t y . E q u i -v a l e n t ^ , the company has e xpec ted e a r n i n g s of X , wh ich be -long to a g i ven r i s k c l a s s . T h i s means t h a t we are no t c o n -s i d e r i n g b u s i n e s s r i s k i n our a n a l y s i s . We de te rm ine how changes i n l e v e r a g e , and t h e r e f o r e f i n a n c i a l r i s k , can i n f -luence the v a l ue of companies w i t h e a r n i n g s o f a g i ven b u s i -ness r i sk . 2. The company i s c o n s i d e r e d to have on l y two s ou r ce s 7 of funds: long term debt and equity. 3 . The structure of market c a p i t a l i z a t i o n rates is given and does not change over time. The debt and equity markets es tab l i sh c a p i t a l i z a t i o n rates to apply to firms in a gi ven r i sk c l a s s . In determining these c a p i t a l i z a t i o n rates , the markets w i l l consider such factors as v a r i a b i l i t y in earnings, length of time the f i rm has been in business, the nature of the bus i -ness, the i r estimations for further success, and the ease with which secur i t i e s can be traded on the market. Since a l l these factors have been considered in assigning c a p i t a l i z a t i o n rates to earnings of a given r i sk c l a s s , the c a p i t a l i z a t i o n rate within a par t i cu la r r i sk c lass should be a funct ion only of leverage. That i s , by working within a given r isk c l a s s , we have extracted a l l business rf:sk from our analysis and are dea-l ing only with f i nanc i a l r i s k . This f i nanc i a l r i sk is then the r i sk of default on debt. If the ra t io of equity to debt i s high, the interest coverage within a given r i sk c lass w i l l a l -so be high. Also, in the case of defau l t , the asset value in re l a t i on to the cliaim from senior secur i t i e s is large. Then the c a p i t a l i z a t i o n rates for debt and equity within any given r i sk c lass should be a functioneonIy of the re la t ionsh ip of debt to equity within the f i rm. In our s tudies, we shal l use c a p i t a l i z a t i o n rates which increase as the absolute amount of debt in the cap i ta l structure increases. Functions of th is type are s imi lar to those in which the c a p i t a l i z a t i o n rate is a pos i t ive funct ion of the debt-equity ra t io or the ra t io of debt to total value. In the models to be developed, debt is 8 used to replace equity so any increase in debt resu l ts in a de-crease in equity. Therefore, pos i t ive functions of the amount of debt are equivalent to negative functions of the amount of equity, the equity to debt r a t i o , or the rat io of equity to tota l value. The Dividend E f fec t It has been shown that the debt and equity c a p i t a l i z a -t ion rates should be functions only of the amount of debt in the cap i ta l structure for earnings of a given r i sk c l a s s . Then could two firms in the same r i sk class with the same ca -p i t a l j s t ruc tu re have d i f f e ren t equity c a p i t a l i z a t i o n rates due to a d i f ference in dividend pol icy? The controversy surroun-ding the dependence of the equity c a p i t a l i z a t i o n rate on d i v i -dend pol icy 1s almost as great as that surrounding the e f fect of cap i ta l structure on the cost of c a p i t a l . While i t Is not the purpose of th is p^per to resolve the dividend controversy, some ind icat ion wi l l be given at th is time as to why we think equity c a p i t a l i z a t i o n rates should be Independent of dividends, that i s , we propose that expected future earnings are the source of the value of a stock, so the cost of equity is mea-sured by an expected earnings-pr ice r a t i o . Of those writers who regard dividends as being the determining factor of stock p r i ce s , the foremost is M. J . 14. Gordon, who has provided both empirical and theoret ica l mo-dels to support his views. In his empirical f i nd ings , he uses regression analysis to show that share price is depen-dent on dividends, dividend growth ra te , earnings v a r i a b i l i t y , 9 and c o r p o r a t e s i z e . U n f o r t u n a t e l y , e m p i r i c a l s t u d i e s of t h i s type r e v e a l l i t t l e about s t o c k p r i c e dependence on d i v i d e n d s . I t i s e xpec t ed e a r n i n g s and not p r e s e n t e a r n i n g s t h a t d e t e r -mine s t o c k p r i c e . I t seems l i k e l y t h a t c o r p o r a t e d i v i d e n d p o l i c y i s more s t a b l e than p r e s e n t e a r n i n g s and a l s o r e f l e c t s management ' s e x p e c t a t i o n s of f u t u r e e a r n i n g s . D i v i d e n d po -l i c y i s s e t by i n s i d e r s who s h o u l d have the be s t p o s s i b l e knowledge o f the f i r m ' s e xpec ted f u t u r e pe r f o rmance . D i v i -dends t h e r e f o r e are a b e t t e r s u r r o g a t e f o r e xpec ted f u t u r e e a r n i n g s than are p r e s e n t e a r n i n g s , wh i ch tend to f l u c t u a t e w i d e l y . Gordon a l s o p r o v i d e s a t h e o r e t i c a l argument s u p p o r t i n g 1 5 . h i s d i v i d e n d h y p o t h e s i s . S i n c e u n c e r t a i n t y i n c r e a s e s w i t h t i m e , i n v e s t o r s p r e f e r a c e r t a i n sum today to a l a r g e r , u n -c e r t a i n sum i n the f u t u r e . Th i s leads Gordon to p o s t u l a t e t h a t the r a t e a t wh ich d i v i d e n d s are d i s c o u n t e d must i n c r e a s e w i t h t i m e . I f t h i s i s t r u e , then an i n c r e a s e i n e a r n i n g s r e -t e n t i o n w i l l l ead to an i n c r e a s e i n the e q u i t y c a p i t a l i z a -t i o n r a t e . L i n t n e r f o l l o w s an a n a l y s i s somewhat s i m i l a r to 1 6 . G o r d o n ' s . He f i n d s t h a t , s i n c e u n c e r t a i n t y i n c r e a s e s w i t h t ime and s i n c e u n c e r t a i n t y i s d i s c o u n t e d by the i n v e s t o r , p r e -sen t d i v i d e n d payout s s h o u l d be i n c r e a s e d . Th i s r e s u l t s be -cause " t h e r e l e v a n t ma r g i n a l c o s t o f c a p i t a l i s not on l y g r e a -t e r than c u r r e n t e a r n i n g s y i e l d s by amounts t h a t i n c r e a s e w i t h the s i z e of the budget , but i s n e c e s s a r i l y r i s i n g a t the optimum p o i n t . " U n f o r t u n a t e l y , both L i n t n e r and Gordon have combined 1 0 the i n ve s tmen t and f i n a n c i n g d e c i s i o n . In t h e i r mode l s , t h e i r on l y s ou rce of funds f o r f u r t h e r c o r p o r a t e e xpan s i on i s f rom r e t a i n e d e a r n i n g s . The company i s p r e ven ted f r om i s s u i n g a d -d i t i o n a l debt or e q u i t y , and so w i l l be f a c e d by a c o n t i n u a l l y chang ing d e b t - e q u i t y r a t i o . In a d d i t i o n , the p o l i c i e s deve -loped f r om t h e i r models would lead to e v e n t u a l l i q u i d a t i o n of a l l companies by e s t a b l i s h i n g too h i gh a c u t o f f r a t e f o r e x p e n d i t u r e . What these a u t h o r s have f a i l e d to do i s to c o n -s i d e r the o p p o r t u n i t y c o s t of p a i d out d i v i d e n d s . I f the i n -v e s t o r shows a p r e f e r e n c e f o r d i v i d e n d s r a t h e r than e a r n i n g s r e t e n t i o n and g rowth , we must ask what he wants these d i v i -dends f o r . I f he answers t h a t he needs the funds f o r c o n -s u m p t i o n , then he s h o u l d not have bought the s t o c k i n the f i r s t p l a c e . I f he wants d i v i d e n d s to purchase o t h e r s t o c k s and d i v e r s i f y h i s p o r t f o l i o , then he c o u l d p r o b a b l y accom-p l i s h t h i s d i v e r s i f i c a t i o n more e f f e c t i v e l y by s e l l i n g some s t o c k . I f the i n v e s t o r has an i n ve s tmen t wh ich y i e l d s him a h i g h e r u t i l i t y than the s t o c k , then i n s t e a d of p u t t i n g on l y h i s d i v i d e n d s i n t h i s i n v e s t m e n t , he s hou l d s e l l h i s s t o c k a l s o . Many o t h e r au tho r s have p r o v i d e d both e m p i r i c a l and t h e o r e t i c a l work to show t h a t any p r e f e r e n c e f o r d i v i d e n d s 1 7 . i s i r r a t i o n a l . M i l l e r and M o d i g l i a n i have shown t h a t i f we assume: 1. p e r f e c t c a p i t a l market s i n wh ich a l l t r a d e r s have equal and c o s t l e s s acce s s to i n f o r m a t i o n about p r i c e and o t h e r d a t a , and t h e r e are no b roke rage c o s t s , t r a n s f e r f e e s 11 or tax i n c e n t i v e s ; 2. r a t i o n a l b e h a v i o r i n wh ich i n v e s t o r s p r e f e r more w e a l t h to l e s s and are i n d i f f e r e n t between cash payments and c a p i t a l g a i n s ; and 3 . p e r f e c t c e r t a i n t y i n wh ich every i n v e s t o r knows the f u t u r e p r o f i t and i n ve s tment programs f o r the c o r p o r a t i o n ; then under these c i r c u m s t a n c e s , we w i l l get the e q u i v a l e n t v a l ue f o r a share by d i s c o u n t i n g ca sh f l o w , by u s i n g an i n v e s t -ment o p p o r t u n i t y app roach , by d i s c o u n t i n g s t reams of d i v i -dends , o r by d i s c o u n t i n g s t reams of e a r n i n g s . Mo reove r , they can ex tend these r e s u l t s t o the case o f u n c e r t a i n t y i f every t r a d e r p r e f e r s more w e a l t h to l e s s w e a l t h , r e g a r d l e s s o f the fo rm t h i s w e a l t h may t a k e , and b e -l i e v e s t h a t o t h e r t r a d e r s behave t h i s way a l s o . L i n d s a y and Sametz s u p p o r t the use o f e a r n i n g s i n 1 8 . share e v a l u a t i o n . They s t a t e : " E a r n i n g s are the fundamenta l determinant; o f s t o c k p r i c e s ; the s t o c k h o l d e r ' s p r e f e r e n c e f o r cash d i v i -dends ve r su s c a p i t a l a p p r e c i a t i o n depends p r i m a r i l y on h i s p e r s o n a l m a r g i n a l income tax r a t e . But s i n c e s t o c k h o l d e r s tend to ho l d those s t o c k s whose cash payout r a t i o f i t s t h e i r own d e s i r e s , s t o c k p r i c e s are i n f l u e n c e d p r i m a r i l y by s t a t i s t i c s on e a r n i n g s , no t d i v i d e n d s . " T h e i r v iews s u p p o r t our c o n t e n t i o n t h a t , i f two f i r m s w i t h the same d e b t - e q u i t y r a t i o (same f i n a n c i a l r i s k ) and i n the same r i s k c l a s s (same b u s i n e s s r i s k ) have d i f f e r e n t d i v i d e n d payout p o l i c i e s , both s t o c k s w i l l s e l l a t the same p r i c e , f o r each f i r m w i l l be ab l e t o a t t r a c t a group o f s t o c k h o l d e r s t h a t approves of i t s d i v i d e n d p o l i c y . I r w i n F r i e n d and M a r s h a l l P u c k e t t have p r e s e n t e d both 12 i n t u i t i v e and e m p i r i c a l r e s u l t s to r e f u t e the d i v i d e n d hypo-19. t h e s i s . They t h i n k t h a t , s i n c e i n v e s t o r s c o n t i n u e to buy sha res a t the p r e v a i l i n g market p r i c e , i t i s i n d i c a t e d t h a t t h i s p r i c e o f f e r s a r a t e o f r e t u r n a t l e a s t as h i g h as c o u l d be o b t a i n e d f rom o t h e r i n ve s tmen t s of a comparab le r i s k . Then, i f i n v e s t o r s are w i l l i n g to buy and ho l d these s h a r e s , they s hou l d be i n d i f f e r e n t i f the p r e s e n t va lue of the a d d i -t i o n a l f u t u r e r e t u r n s r e s u l t i n g f rom e a r n i n g s r e t e n t i o n equa l s the amount of d i v i d e n d s f o r e g o n e . A l s o , t h e r e i s a tax a d -vantage f a v o r i n g e a r n i n g s r e t e n t i o n as opposed to d i v i d e n d payou t . They c i t e t h r e e b e h a v i o r a l a s sumpt ions nece s sa r y i f r e t a i n e d e a r n i n g s are to c o n s i s t e n t l y r e c e i v e a lower market 20. v a l u a t i o n than d i v i d e n d s . These a r e : " 1 . The average h o l d e r o f common s t o c k p o s s e s s e s , a t the marg in of h i s p o r t f o l i o , a very s t r o n g p r e f e -rence f o r c u r r e n t income over f u t u r e income (a s i t u a t i o n which c o u l d h a r d l y be expec ted to p e r s i s t over t i m e . ) 2. The expec ted i n c r e a s e i n e a r n i n g s a r i s i n g f rom i n c r e a s e d per share i n ve s tmen t i s v iewed as i n v o l -v i n g a much h i g h e r degree o f r i s k than t h a t a t t a -c h i n g to e a r n i n g s on e x i s t i n g c o r p o r a t e a s s e t s . 3 . The p r o f i t a b i l i t y of i n c r e m e n t a l c o r p o r a t e i n v e s t -ment, as v iewed by s h a r e h o l d e r s , i s e x t r e m e l y low r e l a t i v e to the c o m p e t i t i v e y i e l d p r e v a i l i n g i n the s t o c k m a r k e t . " S i n c e new s t o c k , wh i ch i m p l i e s the s u b s t i t u t i o n of c u r r e n t f o r f u t u r e income, can be i s s u e d at raear market p r i c e s , t h i s s e r ve s to r e f u t e the f i r s t two a s s umpt i on s . S i n c e m a r g i n a l p r o f i t r a t e s i n most i n d u s t r i e s appear to be q u i t e h i g h , and i n growth i n d u s t r i e s , i n c r e m e n t a l i n ve s tment i s h i g h l y p r o -13 f i t a b l e , the t h i r d a s sumpt ion must a l s o be f a l s e . They note t h a t i n any s t a t i s t i c a l s t u d i e s where p r i c e i s r e g r e s s e d a g a i n s t both d i v i d e n d s and r e t a i n e d e a r n i n g s , i f the c o e f f i c i e n t s of these terms d i f f e r , then the payout p o s i -t i o n i s no t i n e q u i l i b r i u m and s t o c k p r i c e c o u l d be i n c r e a s e d by i n c r e a s i n g e i t h e r d i v i d e n d s o r e a r n i n g s r e t e n t i o n . T h e i r e m p i r i c a l r e s u l t s show t h a t , when o t h e r p e r t i n e n t v a r i a b l e s a re i n c l u d e d , the c o e f f i c i e n t s f o r p r i c e and r e t a i n e d e a r -n i ng s are n e a r l y e q u a l . In the models t h a t f o l l o w , we w i l l assume t h a t the e q u i t y c a p i t a l i z a t i o n r a t e i s i ndependent o f d i v i d e n d payou t , and depends o n l y on l e v e r a g e . L i m i t a t i o n s of E m p i r i c a l Te s t s To t e s t the e f f e c t of l e ve rage on the c o s t of c a p i t a l , many e m p i r i c a l t e s t s have been made. The f i r s t of these was made by M o d i g l i a n i and M i l l e r t o s uppo r t t h e i r c o n t e n t i o n t h a t the c o s t o f c a p i t a l was i ndependent o f l e v e r a g e . They d e f i n e d the c o s t of c a p i t a l as t o t a l e a r n i n g s a f t e r tax d i -v i d e d by the market va l ue of a l l s e c u r i t i e s , and found t h a t t h i s c o s t , when exp re s sed as a f u n c t i o n of the r a t i o of debt to t o t a l v a l u e , was i ndependent o f the debt i n the c a p i t a l s t r u c t u r e . That i s , i f X 1 = a + bk, — V where: X * i s a f t e r - t a x e a r n i n g s , V i s t o t a l v a l ue o f the f i r m , L i s the amount of debt i n the c a p i t a l s t r u c t u r e , 14 the va l ue of b i s not s t a t i s t i c a l l y s i g n i f i c a n t . One of the major problems i n any s t a t i s t i c a l work of t h i s type i s to make c e r t a i n t h a t a l l da ta i s f o r f i r m s i n the same r i s k c l a s s . In a r e c e n t pape r , Wippern a t tempted to de te rm ine i f " o b j e c t i v e d e t e r m i n a b l e r i s k c l a s s e s e x i s t ? 21 . And do these c l a s s e s c o r r e s p o n d to i n d u s t r y g roups ? " As a measure of b u s i n e s s r i s k , he used the v a r i a b i l i t y of o p e -r a t i n g e a r n i n g s per sha re f o r f i r m s i n e i g h t i n d u s t r i e s , i n -c l u d i n g o i l , e l e c t r i c u t i l i t i e s , paper and rubber compan ie s . V a r i a b i l i t y was measured by the a n t i l o g of the s t a n d a r d e r r o r around the l o g a r i t h m i c r e g r e s s i o n of annual e a r n i n g s ove r a ten yea r p e r i o d . H i s r e s u l t s showed t h a t f o r the proxy v a -r i a b l e cho sen , t h e r e was as much v a r i a t i o n w i t h i n p a r t i c u l a r i n d u s t r y groups as t h e r e was among d i f f e r e n t g roups . He c o u l d on l y c onc l ude t h a t " i n d u s t r y groups do not p r o v i d e an adequate b a s i s on wh ich to i n s u r e homogeneity of b a s i c b u s i -22. ness u n c e r t a i n t y . " The data used by M o d i g l i a n i and M i l l e r were f o r e l e c -t r i c u t i l i t i e s and o i l compan ie s . As F i s h e r has n o t e d , the e l e c t r i c u t i l i t i e s do not c o n s t i t u t e a v a l i d group on wh i ch 23. to t e s t the e f f e c t of f i n a n c i a l r i s k on the c o s t o f c a p i t a l . These u t i l i t i e s a re c o n t r o l l e d by r e g u l a t o r y bod i e s wh i ch p r e -vent them f rom m a x i m i z i n g p r o f i t . Then, i f a d e c l i n e i n e a r -n i ng s were to o c c u r , the r e g u l a t o r y bod i e s would r e l a x t h e i r r e s t r i c t i o n s to a l l o w e a r n i n g s t o r e t u r n to a " f a i r " l e v e l . A p u b l i c u t i l i t y w i t h the same appa ren t b u s i n e s s r i s k or f l u c t u a t i o n i n e a r n i n g s as a m a n u f a c t u r i n g company would be 15 much l e s s l i k e l y to d e f a u l t on i t s d e b t . A l s o , as Barges p o i -n t s o u t , f o r the u t i l i t y sample o n l y 8 of the 43 u t i l i t i e s s t u d i e d had debt to t o t a l market va l ue r a t i o s between Cffo and 24. 50%, and most o f the o b s e r v a t i o n s were between 50% and 80%. Then, i f the c o s t of c a p i t a l c u r ve was a c t u a l l y s a u c e r - s h a p e d , t h e r e would not be enough o b s e r v a t i o n s i n the d e c l i n i n g c o s t p o r t i o n . The f i n d i n g s of Wippern and Barges i n d i c a t e a f u n -damental p rob lem i n u s i n g e m p i r i c a l t e s t s to de te rm ine the e f f e c t of l e v e r a g e on the c o s t of c a p i t a l . I f t h e r e i s an o p t i m a l c a p i t a l s t r u c t u r e wh ich r e s u l t s i n a minimum c o s t of c a p i t a l , we would expec t a l l f i r m s i n the same r i s k c l a s s t o have t h i s c a p i t a l s t r u c t u r e . The f a c t t h a t t h e r e i s a wide range of c a p i t a l s t r u c t u r e s f o r the sam-p l e s f rom the o i l i n d u s t r y p r o b a b l y i n d i c a t e s t h a t t he se f i r m s are not i n the same r i s k c l a s s . Weston found t h a t the o i l companies i n the r i s k c l a s s used by M o d i g l i a n i and M i l l e r i n -25. e l u d e d the f o l l o w i n g : " f u l l y i n t e g r a t e d o i l compan ie s , o i l companies s t r o n g in r e f i n i n g , o i l companies s t r o n g i n d i s -t r i b u t i o n ; some r e g i o n a l i n t h e i r o p e r a t i o n s , some w i t h heavy i n ve s tmen t s i n t r o u b l e d i n t e r n a t i o n a l r e g i o n s ; some w i t h s t a b l e , a s s u red or r i s i n g income f rom p e t r o c h e m i c a l s or u ran ium or o t h e r m i n e r a l s . " The o i l compan ie s , t h e r e f o r e , c o u l d not be l ong to the same r i s k c l a s s . In the e q u a t i o n used by M o d i g l i a n i and M i l l e r i n t h e i r e m p i r i c a l s t u d i e s , V, wh ich may be s u b j e c t to random v a r i a -t i o n , appears i n the denominato r of both the dependent and i ndependent v a r i a b l e s . Th i s tends to improve the c o r r e l a t i o n 16 and bias the resu l t s against the t rad i t i ona l view. If the firms in the sample used by Modigliani and M i l l e r belonged to the same r i sk c l a s s , then we should only need to include some var iable descr ibing f i nanc ia l r i s k . Un-for tunate ly , since at least those firms in the o i l industry belonged to d i f f e ren t r i sk c lasses , other variables such as growth and f i rm s ize should have been included. Weston found that when growth was included, the lack of any change in the cost of cap i ta l with changes in the capi ta l structure was the resu l t of the negative cor re la t ion between leverage and 26. earn i ngs growth. F i n a l l y , to refute Modigliani and M i l l e r ' s empirical r e su l t s , we should note that th is data indicates that the " a f t e r - t ax " cost of cap i ta l is independent of leverage. In a subsequent paper, they discover that due to corporate tax, the use of debt should actual ly resu l t in a decreasing cost of c a p i t a l , something the i r resu l t s do not show. Modigliani and M i l l e r undertook a second study for the purpose of determlnlng the cost of cap i ta l in the e l e c t r i c 27. u t i l i t y industry. They used two stage least-squares reg-ression to express the value of the f i rm as a function of the tax deductabi I i ty of the interest on debt, f i rm s i ze , ear-nings and rate of growth. Their model allowed them to i n t ro -duce debt as an addit ional explanatory variable and they found that the c o e f f i c i e n t of the debt term was not s t a t i s -t i c a l ly s i gn i f i c an t for any of the three years studied. There are many reasons why the resul ts of the i r tests 17 are not useful in our ana lys i s . As noted prev ious ly, the e l e -c t r i c u t i l i t y industry is not representative of industry in general. In f a c t , Wippern does not even consider i t in his 28. empirical invest igat ion of the leverage e f f e c t . He states: "The e l e c t r i c u t i l i t y industry, one that is most f r e -quently chosen to test for cap i ta l structure e f f e c t s , was excluded from th is sample because i t is bel ieved to be an inappropriate group from which to draw con-clusions regarding shareholder responses to f i nanc ia l r i s k . Interest charges are included among the expen-ses allowed by the commissions in determining e l e c t r i c u t i l i t y rates . Further, the regulatory agencies appear to have a s i gn i f i c an t influence over the f inanc ia l structure adopted by f irms in th i s industry. It i s , therefore, doubtful whether f ixed commitment f inancing exposes the e l e c t r i c u t i l i t y shareholder to f i n a n -c i a l r i sk in the same manner and/or to the same extent as the shareholder of a non-regulated f i r m . " Modigliani and M i l l e r do not attempt to rjustify their choice of the e l e c t r i c u t i l i t y industry as a test for the i r 29. model. They note that: "corporate income taxes are deductible in computing the earnings allowed on the rate base. To the extent that tax is thus passed on, the ultimate value of the tax subsidy on interest is correspondingly reduced." However, as Gordon observes, the i r model describes industr ies in which the before tax earnings is an exogenous random va-r i ab le whereas the resu l t of u t i l i t y regulat ion is to make 30. after tax earnings the exogenous var iab le . Furthermore, as Modigliani and M i l l e r admit, the i r ob-ject ive is to estimate the cost of cap i ta l and not to test con-f l i c t i n g views about the e f fect s of leverage on va luat ion. They do not want a precise estimate of the leverage e f f e c t , but only want to make cer ta in that leverage w i l l not s i g n i f i -cantly influence their re su l t s . From the resu l t s of the mathe-1 8 matical models of Chapter V, the e f fec t of leverage on the cost of cap i ta l is not great. This is pa r t i cu l a r l y so for r e l a t i v e l y safe industr ies such as regulated u t i l i t i e s where the required debt y i e ld should increase only s l i g h t l y i f at a l l to compensate for the increased f i nanc i a l r i sk of leverage. We must re ject the resul ts of th is test made by Modig-l i an i and M i l l e r . They have considered an industry which is far from being t y p i c a l . Business and f i nanc i a l r i sk is almost non-existent due to regu lat ion. Moreover, the cap i ta l s t ruc -ture is contro l led by the regulatory agency so that the range of cap i ta l structures required to test the cost of cap i ta l hypothesis can not be found in the e l e c t r i c u t i l i t y industry. The empirical studies done by those espousing the t r ad i t i ona l view are also of l imited usefulness. Weston has analyzed Modigliani and M i l l e r ' s data, and after including terms to compensate for growth and s i z e , he found that these data actual ly supported the t rad i t i ona l view. However, as noted e a r l i e r , the e l e c t r i c u t i l i t i e s do not const i tute a va l id sample and the o i l companies are not a homogeneous r i sk c l a s s , so the data cannot be considered as conclus ive. 31 . Barges used ra i l roads , department stores and cement companies to test the e f fect s of leverage on the cost of c a -p i t a l . His data for ra i l roads indicated that the cost of ca -p i t a l was a saucer-shaped curve. Unfortunately, most of his e f f o r t was devoted towards determining how the equity c a p i -t a l i z a t i o n rate changed with leverage. Data from these tests can be of only l imited usefulness in choosing between Modig-1 9 Mani and M i l l e r ' s approach and the t rad i t i ona l approach to the e f fec t of leverage on the cost of c a p i t a l . Ronald Wippern has t r i ed to avoid the problem of r isk c lass determination by including business and f inanc ia l r i sk in a s ingle var iab le . He suggests that the subjective proba-b i l i t y d i s t r i bu t i on of future returns can best be determined from a study of past performance. He determines the l ogar i -thmic regression of income on time for a ten year period. Then, knowing the standard error for the regression l ine and the current income predicted from the regression l i ne , he determines the minimum expected income within a given c o n f i -dence i n t e r v a l . By comparing the f ixed interest charges which are a funct ion of leverage to the minimum expected income, he obtains a proxy uncertainty var iable which includes both business and f i nanc i a l r i s k . The resu l t s of his study tend to suggest that e f fec t i ve use of leverage can reduce the cost of cap i ta l beyond the advantage provided by tax deductabi I ity of the interest charges. However, future business r i sk is a funct ion of much more than past earnings f l uc tua t ions . Also, current f i xed interest charges may d i f f e r greatly from future expected interest charges. Unt i l a l l invest igators can agree on exact measures of f i nanc ia l and business r i s k s , any con-clus ions based on arb i t rary r i sk parameters w i l l remain sus-pect. We doAnot think that empirical tests can be developed which w i l l determine whether the average cost of cap i ta l is a funct ion of leverage. We have noted that i t is almost im-possible to get data for a homogeneous r i sk c lass which is 20 necessary i f we are to exclude business r i sk and study only the influence of f i nanc ia l r i s k . The y i e l d on equit ies w i l l always be influenced by market imperfect ions.. Ins t i tut iona l r e s t r i c t i on s and imperfect knowledge on the part of investors w i l l prevent un l i s ted or unknown stocks from s e l l i n g at as high a pr ice as l i s ted stocks, even though they may belong to the same r i sk c l a s s . Since re s t r i c ted l i s t s ex is t for banks and other Ins t i tu t ions , cer ta in stocks w i l l have their pr ices bid up. If the investor is i r r a t i ona l enough to be influenced by dividends, stocks in the same r isk c lass could se l l at d i f f e ren t pr ice-earnings mul t ip les . In gathering data for empirical t e s t s , measurement errors may re su l t . We have no way of determining expected earnings for a growth stock, so any c a p i t a l i z a t i o n rate ca lculated from current earnings and pr ice w i l l be low. Due to the many problems involved in empirical te s t ing , we think that the cost of capi ta l contro-versy can be resolved only from a theoret ica l analysis of the behavioral cha rac te r i s t i c s suggested by the models developed by Modigliani and M i l l e r and the t r a d i t i o n a l i s t s . In the f o l -lowing chapters we shal l develop each model and examine the type of behavior required from investors i f the models are to be cor rec t . Footnotes: 1. Myron J . Gordon, The Investment Financing and Valuation  of the Corporation. Richard D. Irwin, Homewood, III., 1963, p. 218. 21 2. Ezra Solomon, The Theory of F inancia l Management. Columbia Univers i ty Press, New York, 1963, pp. 8-9 3. M. J . Gordon, op. c i t . , pp. 2 1 8 - 2 1 9 4. Joel Cord, "A Method for A l locat ing Funds to Investment Projects When Returns are Subject to Uncerta inty" , Management Science. Vo l . 10, No. 2, January 1964, PP. 335-341. 5. Nei l .R. Paine, "Uncertainty and Capital Budgeting", The Accounting Review. Vo l . XXXIX, No. 2, Apr i l 1964, PP. 330-332. 6. Modiglaini and M i l l e r , "The Cost of C a p i t a l , Corporation Finance, and the Theory of Investment", The American  Economic Review. Vo l . XLVIII, No. 3, June 1958, pp; 261-297 7. Ezra Solomon, "Leverage and the Cost of C a p i t a l , " The Journal of Finance. Vo l . XVIII, No. 2, May 1963, pp. 273-79 8. Robert Johnson, "An Integration of Cost of Capital Theo-r i e s " , Fourth Summer Symposium of the Economy D iv i s ion of the American Society for Engineering Education, 1966, pp. 11-17 9. E l i Schwartz, "Theory ofCapita l Structure of the F i rm" , Journal of Finance. Vo l . XIV, No. 1, March I 9 6 5 , pp. 18-39 10. Modigliani and M i l l e r , op. c i t . , p. 265 11. Ronald Wippern, "A Note on the Equivalent Risk Class As-sumption", The Engineering Economist. Vo l . XI, Spring 1966, pp. 13-22 12. James C. T. Mao, Quantitat ive Analysis of F inancia l Dec i -s i ons. Macmillan Company, New York, Forthcoming. 13. Ezra Solomon, op. c i t . , p. 2 7 5 14. M. J . Gordon, op. c i t . , espec ia l ly chapters 4 & 5. 15. M. J . Gordon, "The Savings, Investment and Valuation of a Corporat ion" , Review of Economics and S t a t i s t i c s . Vo l . XLIV, Feb. 1962, pp. 37-51 1 6 . John L in tner , "Optimal Dividends arid Corporate Growth Under Uncerta inty" , The Quarterly Journal of Economics. Vo l . LXXVIII, No. 1, 1964, pp. 49-95 22 17. M i l l e r and Modig l ian i , "Dividend Po l i cy , Growth and the Valuation of Shares", Journal of Business of the Un i -vers i ty of Chicago. Vo l . XXXIV, No. 4, Oct. I96I, pp. 411-433 18. Lindsay and Sametz, F inancia l Management, an Analyt ica l Approach. Richard D. Irwin, Homewood, III., 1963, p. T45 19. Friend arid Puckett, "Dividends and Stock P r i c e s " , The American Economic Review. Vo l . LIV, Sept. 1964, pp. 656-82 20. i b i d . , p. 662 21. R. Wippern, op. c i t . , p. 15 22. i b i d . , p. 20 23. Lawrence F i sher , "Determinants of Risk Premiums on Cor-porate Bonds", Journal of P o l i t i c a l Economy. Vo l . LXVII, No. 3, June 1959, pp. 217-237 24. Alexander Barges, The E f fec t of Capital Structure on the Cost of Capi t a l . P ren t i ce -Ha l l , Englewood C l i f f s , N.J., I963, p. 22 25. J . Fred Weston, "A Test of the Cost of Capital Propos i -t i o n " , Southern Economic Journal . Vo l . XXX, No. 2, Oct. .1963, p. 109 26. i b i d . , p. 109 27. Modigliani and M i l l e r , "Some Estimates of the Cost of Ca-p i t a l to the El lectr ic U t i l i t y Industry, 1954-57", American Economic Review. Vo l . LVI, No. 3, June 1966, pp. 333-391 28. Ronald Wippern, "F inanc ia l Structure and the Value of the F i rm" , Journal of Finance. Vo l . XXI, No. 4, December 1966, p. 615-633 29. Modigliani and M i l l e r , "Some Estimates of the Cost of Capital to the E l e c t r i c U t i l i t y Industry, 1954-57" 30. Myron J . Gordon, "Some Estimates of the Cost of Capital to the E l e c t r i c U t i l i t y Industry, 1954-57 - Comment", American Economic Review. Vo l . LVII, No. 5, Dec. 1967, * p. 1267-1277. A. Barges, op. c i t . , chapter 5. •23 CHAPTER II THE MODIGLIANI AND MILLER HYPOTHESIS As sumpt ions 1 . M o d i g l i a n i and M i l l e r propose t h a t the average c o s t of c a p i t a l to the f i r m i s i ndependent of the f i r m ' s c a p i t a l s t r u c t u r e . They m a i n t a i n t h a t the r e a l c o s t of debt to the f i r m a f t e r t a k i n g i n t o account the i n c r e a s e i n the e q u i t y c a -p i t a l i z a t i o n r a t e r e s u l t i n g f rom i n c r e a s e d l e v e r a g e , i s such t h a t the m a r g i n a l c o s t o f c a p i t a l must be equal to the a v e -rage c o s t of c a p i t a l . T h i s i m p l i e s t h a t the t o t a l market v a l u e of a f i r m i s u n a f f e c t e d by the c o m p o s i t i o n of i t s c a p i -t a l s t r u c t u r e . In t h e i r i n i t i a l mode l , M o d i g l i a n i and M i l l e r make the f o l l o w i n g a s s u m p t i o n s : 1. They f i r s t assume t h a t a l l f i r m s can be grouped i n homogeneous r i s k c l a s s e s . As shown i n the i n t r o d u c t i o n , t h i s i m p l i e s t h a t i n v e s t o r s r e g a r d income f rom a l l f i r m s i n a g i v en c l a s s as e q u a l l y r i s k y . A l t h o u g h the a b s o l u t e amount o f income may d i f f e r ( the f i r m s may be of d i f f e r e n t s i z e s ) , the v a r i a t i o n i n income as compared to mean expec ted income i s equa l t h r oughou t the c l a s s , so a u n i f o r m c a p i t a l i z a t i o n 2. r a t e , Ko, may be a p p l i e d to the e a r n i n g s of a l l f i r m s i n the r i s k c l a s s . 2. A l l i n v e s t o r s have a s s i g n e d the same s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n to the r e t u r n s of a p a r t i c u l a r f i r m 24: i n a g i v e n r i s k c l a s s . T h i s means t h a t a l l p r e s e n t and f u t u r e i n v e s t o r s w i l l have i d e n t i c a l e s t i m a t e s of the e xpec ted a ve -rage income, X. 3 . A l l s t o c k s and bonds are t r a d e d i n p e r f e c t c a p i t a l 3 . m a r k e t s . As d e f i n e d by M o d i g l i a n i and M i l l e r , " i n p e r f e c t c a p i t a l m a r k e t s , n o buyer o r s e l l e r (o r i s s u e r K o f s e c u r i t i e s i s l a r g e enough f o r h i s t r a n s -a c t i o n s to have an a p p r e c i a b l e impact on the then r u l i n g p r i c e . A l l t r a d e r s have equa l and c o s t l e s s a cce s s to i n f o r m a t i o n about the r u l i n g p r i c e and about a l l o t h e r r e l e v a n t c h a r a c t e r i s t i c s of shar.es. No b roke rage f e e s , t r a n s f e r t a x e s or o t h e r t r a n s -a c t i o n s c o s t s are i n c u r r e d when s e c u r i t i e s are bought , s o l d o r i s s u e d , and t h e r e are no tax d i f f e r e n t i a l s e i t h e r between d i s t r i b u t e d and u n d i s t r i b u t e d p r o f i t s o r between d i v i d e n d s and c a p i t a l g a i n s . " S i n c e market s are p e r f e c t , i n v e s t o r s are assumed to be a b l e to borrow u n l i m i t e d amounts a t the same bo r r ow ing r a t e f a c e d by c o r p o r a t i o n s . 4 . In the f i r s t model deve loped by M o d i g l i a n i and M i l l e r , they do not c o n s i d e r c o r p o r a t e income t a x . T h i s means t h a t any tax i n c e n t i v e r e s u l t i n g f rom debt f i n a n c i n g i s n e g I e c t e d . Work ing w i t h i n t h i s f ramework , we w i l l now deve lop the arguments put f o r t h ;by M o d i g l i a n i and M i l l e r . We w i l l f i n d t h a t t h e i r model i s c o n s i s t e n t w i t h the a s sumpt ion s made, but when r e a l w o r l d c h a r a c t e r i s t i c s such as c o r p o r a t e t a x e s , l i m i t e d l i a b i l i t y , and i n v e s t o r a t t i t u d e toward r i s k a re i n -t r o d u c e d , the model w i l l l e ad to p a r a d o x i c a l r e s u l t s . Throughout our a n a l y s i s , the f o l l o w i n g symbols w i l l be u sed . S i n c e a l l models may be c o n s i d e r e d as d e s c r i b i n g d i f f e r e n t f i r m s i n the same r i s k c l a s s but w i t h a d i f f e r e n t 25 c o m p o s i t i o n of debt and e q u i t y , or as d e s c r i b i n g the same f i r m w i t h e a r n i n g s c h a r a c t e r i s t i c o f a g i v e n r i s k c l a s s but w i t h a changed d e b t - e q u i t y r a t i o , no s u b s c r i p t s w i l l be used to r e f e r to d i f f e r e n t f i r m s . X = e xpec ted annual r e t u r n on a s s e t s , where X i s de -t e r m i n e d as i n the i n t r o d u c t i o n . L = amount of debt i n the c a p i t a l s t r u c t u r e ( a t mar-k e t v a l u e . ) S = amount o f e q u i t y i n the c a p i t a l s t r u c t u r e ( a t market v a l u e . ) V = L + S = t o t a l market va lue of the company. Kj = debt c a p i t a l i z a t i o n r a t e . I t i s de te rm ined by the debt market and i s assumed to remain c o n -s t a n t over t i m e . S i n c e i t i s a va lue a s s i g n e d to a p a r t i c u l a r r i s k c l a s s , i t can on l y be a f u n c t i o n of the c a p i t a l s t r u c t u r e o f f i r m s w i t h i n o t h e r i s k c l a s s to wh ich i t i s a s s i g n e d . K e = e q u i t y c a p i t a l i z a t i o n r a t e . I t i s a s s i g n e d by the e q u i t y market to e q u i t y of f i r m s w i t h i n a p a r t i c u l a r r i s k c l a s s , so i s a f u n c t i o n on l y o f c a p i t a l s t r u c t u r e . I t i s assumed to remain c o n s t a n t over t i m e . Y = r e t u r n to i n v e s t o r f rom a g i v e n i n ve s tmen t p o r t -f o l i o . K 0 = average c o s t o f c a p i t a l to f i r m s w i t h i n a p a r -t i c u l a r r i s k c l a s s . ( I t r e p l a c e s p i n Mo-d i g l i a n i and M i l l e r ' s a n a l y s i s . ) A comp le te l i s t of symbols i s g i v en i n Append ix I. P r o p o s i t i o n I C o n s i d e r any f i r m i n a g i v e n r i s k c l a s s and hav i ng e a r n i n g s X. Then the v a l ue of t h i s f i r m i s g i ven by: V = L + S = * K 0 26 M o d i g l i a n i and M i l l e r ' s P r o p o s i t i o n I s t a t e s t h a t V, " t h e mar-ke t v a l u e of any f i r m , i s i ndependent of i t s c a p i t a l s t r u c -t u r e and i s g i v e n by c a p i t a l i z i n g i t s e xpec ted r e t u r n at the 4. r a t e , K 0 , a p p r o p r i a t e to i t s c l a s s . " A l t e r n a t i v e l y s t a t e d , (2) X _ x _ K nrs - — ~ °> so f o r f i r m s i n any g i v en r i s k c l a s s , " t h e average c o s t o f c a p i t a l to any f i r m i s c o m p l e t e l y i ndependent of i t s c a p i t a l s t r u c t u r e and i s equal to the c a p i t a l i z a t i o n r a t e of a pure 5. e q u i t y s t ream o f i t s c l a s s . " Work ing w i t h i n t h e i r a s s u m p t i o n s , they prove P r o p o s i -t i o n I by showing t h a t i f two f i r m s i n the same r i s k c l a s s , but h a v i n g d i f f e r e n t degrees of l e ve r age i n t h e i r c a p i t a l s t r u c t u r e , have d i f f e r e n t v a l u e s , then th rough what they c a l l a r b i t r a g e , s ha re s w i l l be bought and s o l d u n t i l , a t e q u i l i b -r i u m , the f i r m s w i l l have the same v a l u e . Suppose both Company 1 and Company 2 a re i n the same r i s k c l a s s , and have equa l e xpec ted e a r n i n g s , X^ = X2. Now suppose Company 2 i s l e v e r e d and has a va lue h i g h e r than u n -l e v e r e d Company 1. An i n v e s t o r hav i n g an amount of s h a r e s , S2> i n Company 2 and t h e r e f o r e owning a f r a c t i o n of the com-pany, <x = £ 2 , w i l l r e c e i v e the f o l l o w i n g r e t u r n f rom h i s 3 2 p o r t f o l i o : (3) Y 2 = « ( x - K j L 2 ) I f he w i she s t o s e l l h i s s ha re s i n Company 2, he would r e -c e i v e an amount, cx S2. B e f o r e he can i n v e s t i n the u n l e v e r e d 27. company, i f he w i she s to keep h i s f i n a n c i a l r i s k i n Company 1 the same as i t was i n Company 2, he must a c q u i r e some p e r s o -nal d e b t . Now, by p l e d g i n g h i s new h o l d i n g s i n Company 1 as c o l l a t e r a l , he w i l l borrow an amount, o< L g . T h i s means t h a t t h rough h i s "homemade l e v e r a g e " he has p r e s e r v e d the same d e b t - e q u i t y r a t i o he had i n Company 2. That i s , h i s new i n -ves tment has the same f i n a n c i a l r i s k as h i s o l d one. Wi th the p roceeds f rom h i s s a l e of s t o c k p l u s h i s p e r s o n a l b o r r o -w i n g , he can purchase an amount <X(S2 + L2) o f the sha re s of the u n l e v e r e d Company 1. He now owns a f r a c t i o n : |1 _ °<(S2 + Lg) S T - s . A f t e r he pays the i n t e r e s t on h i s p e r s ona l d e b t , h i s t o t a l e a r n i n g s f rom t h i s new p o r t f o l i o a r e : Y1 = cx(s? + L?) X - K j « L 2 S i ( 4 ) = w ! 2 x - K j O < L 2 v1 Compar ing e q u a t i o n s (3) and (4 ) , t h i s a r b i t r a g e p r oce s s w i l l remain p r o f i t a b l e as long as Y1 > Y 2 . A r b i t r a g e w i l l o n l y s t op when Yj = Y2, and a t t h i s t ime w i l l equal V2. M o d i g l i a n i and M i l l e r c o n c l u d e t h a t l e v e r e d companies cannot command a premium over u n l e v e r e d companies because i n v e s t o r s can use p e r s o n a l l e ve r age as a p e r f e c t s u b s t i t u t e f o r c o r p o -r a t e l e v e r a g e . Now c o n s i d e r the p o s s i b i l i t y t h a t the l e v e r e d Company 2 has a lower v a l ue than the u n l e v e r e d Company 1. That i s , V2 < V1. Now an i n v e s t o r w i l l f i n d i t p r o f i t a b l e to " s w i t c h " 28 the s ha re s of Company 2 f o r h i s s ha re s o f Company 1. H i s r e t u r n f rom h i s h o l d i n g s i n Company 1 i s : (5) Yj = l l X = a x . Si I f h i s f i n a n c i a l r i s k i s to remain c o n s t a n t , he must purchase an amount of e q u i t y , = $2 and an amount o f d e b t , s2  ~ 2 = 1 - L4 '< • Th i s w i l l g i ve him a c l a i m to the same share o f e a r n i n g s i n Company 2 t h a t he had i n Company 1. By a c q u i r i n g t h i s mixed p o r t f o l i o , the i n v e s t o r " undoe s " the l e ve r age of the f i r m . M o d i g l i a n i and M i l l e r s t a t e t h a t " i t i s t h i s p o s s i b i l i t y o f undo ing l e ve rage wh ich p r e v e n t s the va l ue of l e v e r e d f i r m s 6. f rom b e i n g g r e a t e r than t h a t of u n l e v e r e d f i r m s . " The r e t u r n f rom h i s new h o l d i n g s of e q u i t y and debt i n Company 2 i s now: V2 = £2 (X - K j L 2 ) + Kj l 2 S 2 = ±L (X " K |L. 2) + Kf S 1 V 2 v 2 (6) = H x = « f i x v 2 v 2 Compar ing e q u a t i o n s (5) and (6), i f V 2 i s l e s s than S-| , then i t w i l l pay the s t o c k h o l d e r to s e l l h i s s ha re s i n Company 1 and r e p l a c e them w i t h the debt and e q u i t y o f Com-pany 2. Only when the v a l ue of the two f i r m s are equa l (S^ = V2, where f i r m 1 has on l y e q u i t y i n i t s c a p i t a l s t r u c -29; t u r e ) w i l l the a r b i t r a g e s t o p . P r o p o s i t i o n II P r o p o s i t i o n II s t a t e s t h a t " t h e e xpec ted y i e l d o f a share o f s t o c k i s equal to the a p p r o p r i a t e c a p i t a l i z a t i o n r a t e , K 0 , f o r a pure e q u i t y s t ream i n the c l a s s , p l u s a p r e -mium r e l a t e d to f i n a n c i a l r i s k equal t o the d e b t - t o - e q u i t y 7. r a t i o t imes, the sp read between K 0 and K i . " That i s , (7) K e = K 0 + ( K 0 - K,-) L/S Th i s i s e s t a b l i s h e d as f o l l o w s . The e q u i t y c a p i t a l i z a t i o n r a t e can be g i v e n by: (8) K e = X - K j L S But f rom P r o p o s i t i o n I, e q u a t i o n ( 1 ) , we know t h a t X = K 0 ( S + L) Making t h i s s u b s t i t u t i o n f o r X i n e q u a t i o n (8), we get equa -t i o n (7), wh i ch i s P r o p o s i t i o n I I . P r o p o s i t i o n : 11 as d e t e r -mined i n t h i s f a s h i o n has no i ndependent l o g i c . I t i s d e r i -ved f rom an e x p r e s s i o n f o r e xpec ted y i e l d and f rom the a v e -rage c o s t o f c a p i t a l h y p o t h e s i s g i v en i n P r o p o s i t i o n I. P r o p o s i t i o n II can a l s o be deve loped f rom the d e f i n i -t i o n of the average c o s t of c a p i t a l . L e t wi = debt = L t o t a l v a l ue V w 2 = e q u i t y - 3 t o t a l v a l ue V A l s o , V = L + S. Then, ( 9 ) K 0 = w ^ i + w 2 K e 30 W2 Then s u b s t i t u t i n g f o r wj and w2» K e = K 0 - 7 Ki S V (7) = K 0 + ( K 0 - K j ) L S S i n c e P r o p o s i t i o n I has not been u s e d , the above r e l a -t i o n s h i p can be used to d e s c r i b e the b e h a v i o r of K e in the models of both M o d i g l i a n i and M i l l e r and the t r a d i t i o n a l w r i -t e r s . However, the model as used by M o d i g l i a n i and M i l l e r keeps K 0 c o n s t a n t , so K e i s a l i n e a r f u n c t i o n o f l e v e r a g e . The t r a d i t i o n a l w r i t e r s t r e a t K 0 as a v a r i a b l e wh ich depends on the c a p i t a l s t r u c t u r e and the v a l u e s of K e and K j . E f f e c t s o f C o r p o r a t e Tax M o d i g l i a n i and M i l l e r then remove a s sumpt i on 4 and c o n s i d e r c o r p o r a t e tax i n t h e i r a n a l y s i s . When c o r p o r a t e tax i s i n c l u d e d , the va l ue o f a f i r m w i t h i n a r i s k c l a s s becomes a f u n c t i o n of the tax r a t e and degree of l e ve r age as w e l l as the e xpec t ed a f t e r - t a x r e t u r n s . T h i s means t h a t t h e r e i s a tax advantage to u s i n g debt so t h a t i n c r e a s e d l e ve rage w i l l r e s u l t i n a h i g he r v a l ue f o r the f i r m and , t h e r e f o r e , a lower c o s t o f c a p i t a l . M o d i g l i a n i and M i l l e r note t h a t the v a l u a -t i o n i m p l i e d by t h e i r model now comes c l o s e r to t h a t p r e d i c -ted by the t r a d i t i o n a l mode l . However, i t i s t h e i r v iew t h a t t h i s r e d u c t i o n i n the c o s t o f c a p i t a l t h rough the use o f debt o c c u r s o n l y because of p r e s e n t tax laws and not because debt 31 is inherently cheaper. As before, the long run average earnings before in te -rest and taxes can be denoted by X for the f i rm in a pa r t i cu -lar r i sk c l a s s . Then, i f we let Z = X / X, a l l f irms in the same r i sk c lass w i l l have the same value of Z. The ave-rage earnings, X, can be expressed as X*Z. Now, X^, the ave-rage earnings af ter tax but before in teres t , is given by = (1 - X)(X - KjL) + KjL = (1 - t )X + t K j L (10) = (1 - T ) X Z + X KjL where ~t is the corporate tax ra te , but E(X^ ) 5 X^ = (1 -~ti)X +X K jL . So, replacing (1 -~t )x" in equation (10) by x"^ - t K j L gives (11) X^ = ( X ^ - " t K j L ) Z + t K j L Now the d i s t r i bu t i on of X r is seen to depend on the tax rate and the degree of leverage (where ( K j L ) / X is a measure of leverage) as well as on the value of Z for the r isk c lass to which the company belongs. From equation (10) we see that the long-run average stream of a f ter - tax earnings is composed of two parts. The' f i r s t is an uncertain stream, (1 - T ) X Z , where the uncerta in-ty associated with this stream is dependent on the r i sk class to which the f i rm . be Iongs. The second is a v i r t u a l l y cer -tain stream resu l t ing from the tax deductabi I ity of interest payments. This stream is certa in to the extent that the tax 3 2 deductabi I ity can be applied against current income or car -r ied forward or back to o f f set income of other periods. Since the two streams have d i f fe rent degrees of uncerta inty, they should be cap i t a l i zed d i f f e r e n t l y . The value of an un-levered company, V u , can be determined by c a p i t a l i z i n g a f t e r -tax earnings at a rate, Ko« Then, V u = .(1 -~K)X K o where K0 is now the appropriate cost of cap i ta l after tax. The cer ta in stream generated by the debt should be cap i t a l i zed at a rate K\. Then the value of a levered f i rm is determined by c a p i t a l i z i n g the r isky portion at a rate K0 and the cer ta in portion at a rate Kj. Then, V L = (1 - t ) X + t K j L K0 K,-( 1 2 ) = V u + t L Equation ( 1 2 ) implies that the value of a levered f i rm is equal to the value of an unlevered f irm of the same r i sk c l a s s , plus an addit ional amount which increases with both the amount of debt in the cap i ta l structure and with the le -vel of corporate taxes. We note that the addit ional term resu l t ing from the tax saving has been cap i t a l i zed at a more favorable rate than the uncertain stream. Development of a Computer Model To appraise the va l i d i t y of Modigliani and M i l l e r ' s hypothesis, we wish to study the shape of the function for Ke and the value of the f i rm after tax as the amount of debt 33 i n the c a p i t a l s t r u c t u r e changes . A computer program i s de -ve l oped based on the a s sumpt ion s and e q u a t i o n s of M o d i g l i a n i and M i l l e r to p r o v i d e the r e q u i r e d i n f o r m a t i o n . Th i s p rogram, and i t s o u t p u t are g i v en i n Appendix I I . As noted i n the i n t r o d u c t i o n , the t ype of model used by M o d i g l i a n i and M i l l e r i n t h e i r t h e o r e t i c a l deve lopment a s -sumes t h a t the f i r m has a f i x e d group of a s s e t s and r e c e i v e s c o n s t a n t e a r n i n g s f rom them, c h a r a c t e r i s t i c of the r i s k c l a s s t h a t the f i r m be long s t o . Th i s means t h a t i n our a n a l y s i s , X i s t r e a t e d as a c o n s t a n t . The f i r m i s v a l ued in the case when t h e r e i s no c o r p o r a t e t a x , by c a p i t a l i z i n g i t s e a r n i n g s a t a r a t e c h a r a c t e r i s t i c o f i t s r i s k c l a s s as g i v en i n equa -t i o n ( 1 ) . ( D V = J L K o When c o r p o r a t e tax i s i n c l u d e d i n the a n a l y s i s , the f i r m i s v a l u e d by e q u a t i o n ( 1 2 ) . (12) V, = (1 - ~ t ) X + t L K o The debt c a p i t a l i z a t i o n r a t e i s f i r s t de te rm ined by u s i n g a h y p o t h e t i c a l f u n c t i o n to d e s c r i b e the b e h a v i o r of the debt m a r k e t . We have noted the many d i f f i c u l t i e s encoun -t e r e d when e m p i r i c a l da ta i s a n a l y z e d to de te rm ine f u n c t i o n s f o r K e , K j , e t c . Then, i n s t e a d of u s i n g e m p i r i c a l d a t a , we p o s t u l a t e what we c o n s i d e r to be a r e a s o n a b l e f u n c t i o n f o r the dependence o f Kj on the l e v e l o f debt i n the c a p i t a l s t -r u c t u r e , based on c h a r a c t e r i s t i c s t h a t both M o d i g l i a n i and M i l l e r and the t r a d i t i o n a l w r i t e r s seem to agree upon. Both 34 s c h o o l s agree t h a t Kj s h o u l d be a lmos t c o n s t a n t f o r the f i r s t s m a l l i n c r emen t s of d e b t . Then, as the amount of debt i n c r e a -s e s , the c a p i t a l i z a t i o n r a t e s hou l d i n c r e a s e at an i n c r e a s i n g r a t e . The f o l l o w i n g f u n c t i o n s have been deve l oped to d e t e r -mine Kj i n the M o d i g l i a n i and M i l l e r model and a l s o i n the t r a d i t i o n a l and net income models of the f o l l o w i n g c h a p t e r . (13) Kj = a + bl_ where: a = a c o n s t a n t a r b i t r a r i l y chosen a t 5%. I t s hou l d be noted t h a t the v a l ue of ' a 1 depends both on c u r r e n t economic c o n d i t i o n s and on the r i s k c l a s s to wh i ch the f i r m b e l o n g s . b = r a t e a t wh ich the c a p i t a l i z a t i o n r a t e i n c r e a s e s f o r each s u c c e s s i v e u n i t o f debt i n the c a p i -t a l s t r u c t u r e . L = amount o f debt i n the c a p i t a l s t r u c t u r e . E q u a t i o n (13) shows t h a t the r e q u i r e d debt y i e l d i s a l i n e a r f u n c t i o n of the amount o f d e b t . Th i s l i n e a r f u n c t i o n does not r e p r e s e n t r a t i o n a l b e h a v i o r and,can be shown to g i ve p a r a d o x i c a l r e s u l t s when used i n the t r a d i t i o n a l mode l . (14) Kj = a + b L 2 Th i s f u n c t i o n shows an i n c r e a s i n g a v e r s i o n to debt as the l e v e l o f debt i n c r e a s e s . In t h i s r e s p e c t , i t r e p r e s e n t s r a t i o n a l b e h a v i o r . (15) Kj = a + b L 3 Th i s f u n c t i o n shows an even g r e a t e r i n v e s t o r a v e r s i o n to i n c r e a s e d l e v e l s of debt than d i d e q u a t i o n ( 1 4 ) . Equa -t i o n (14) may be c h a r a c t e r i s t i c o f b e h a v i o r of debt h o l d e r s i n i n d u s t r i e s wh i ch have a r e l a t i v e l y s t a b l e l e v e l o f e a r -n i n g s , such as r e g u l a t e d u t i l i t i e s , w h i l e e q u a t i o n (15) c o u l d be c h a r a c t e r i s t i c of the b e h a v i o r of debt h o l d e r s f o r an i n -35' d u s t r y where e a r n i n g s f l u c t u a t e more r a p i d l y , such as the auto i n d u s t r y . I t has been sugges ted t h a t the debt c a p i t a l i z a t i o n r a t e may rema in c o n s t a n t as sma l l amounts of debt are i n t r o d u c e d i n t o an a l l - e q u i t y c a p i t a l s t r u c t u r e . Then, when the amount of debt exceeds a c e r t a i n l e v e l , the debt c a p i t a l i z a t i o n r a t e i n c r e a s e s s h a r p l y w i t h subsequent i n c rement s of d e b t . T h i s s u gge s t s use of the f o l l o w i n g t ype s of e q u a t i o n s to d e s -c r i b e the dependence of the debt c a p i t a l i z a t i o n r a t e on the amount of debt i n the c a p i t a l s t r u c t u r e : ( 1 6 ) Kj = a + b (L - A) (17) Ki = a + b (L - A ) 2 ( 1 8 ) Kj = a + b (L - A ) 3 (19) Kj = a , f o r L < A I t s h o u l d be noted t h a t e q u a t i o n s ( 1 3 ) , ( 1 4 ) , and. (15) r e p -r e s e n t l i m i t i n g cases of the above e q u a t i o n s when A = 0. T h i s program p e r m i t s the u se r to s p e c i f y the v a l ue s f o r the pa ramete r s a , b, and A and a l s o whether he w i she s to use a f i r s t , second or t h i r d o r d e r e q u a t i o n to de te rm ine the c o s t of d e b t . The u se r s p e c i f i e s the v a l ue o f K 0 to use i n e q u a t i o n s (1) and (12). In our a n a l y s i s , we have used a va l ue o f 1% which c o r r e s p o n d s to the a f t e r - t a x e q u i t y c a p i t a l i z a t i o n r a t e used i n the t r a d i t i o n a l model when t h e r e i s no debt i n the c a p i t a l s t r u c t u r e . The M o d i g l i a n i and M i l l e r model f i r s t c a l c u l a t e s the va l ue of the f i r m assuming t h e r e i s no c o r p o r a t e t a x . S i n c e i t uses the a f t e r - t a x c a p i t a l i z a t i o n r a t e to do t h i s , the 56 r e s u l t s are not r e a l i s t i c , but do not a f f e c t the b a s i c shape of the c u r v e s , so the da ta i s s t i l l u s e f u l f o r our a n a l y s i s . A f t e r the va lue of the f i r m i s f o u n d , the va l ue o f the e q u i t y , S, i s d e t e r m i n e d . The r a t i o s of debt to t o t a l v a l ue and e q u i t y to t o t a l v a l ue are then d e t e r m i n e d . The e q u i t y c a p i t a l i z a t i o n r a t e b e f o r e tax i s then u n i q u e l y de te rm ined f rom e q u a t i o n ( 2 0 ) . (20) K e = ( K 0 - w i K t ) W2 U s i n g e q u a t i o n ( 1 2 ) , the v a l u e of the f i r m a f t e r tax i s then c a l c u l a t e d . Knowing t h e . v a l u e of the f i r m a f t e r tax and the amount of debt i n the c a p i t a l s t r u c t u r e , the va l ue of the e q u i t y i s then d e t e r m i n e d . Then the e q u i t y c a p i t a l i z a t i o n r a t e i s found by d i v i d i n g the e a r n i n g s a c c r u i n g to the e q u i t y by the a f t e r - t a x v a l ue of the e q u i t y . In the M o d i g l i a n i and M i l l e r mode l , the c o s t of e q u i t y i s dependent on the c o s t of debt and the v a l ue of the f i r m as g i v e n by e q u a t i o n (1) i n the b e f o r e - t a x c a s e , or e q u a t i o n (12) i n the a f t e r - t a x c a s e . In t h i s mode l , K e i s the dependent v a r i a b l e and K 0 and Kj are the i ndependent v a r i a b l e s . Th i s program uses an i t e r a t i v e p rocedu re and o p e r a -t e s as f o l l o w s . The program uses the debt c a p i t a l i z a t i o n f u n c t i o n , e a r n i n g s c a p i t a l i z a t i o n r a t e K 0 , e a r n i n g s x", tax r a t e X , and i nc rement to debt as s p e c i f i e d by the u s e r . I n i t i a l l y , the program assumes t h a t t h e r e i s no debt i n the c a p i t a l s t r u c t u r e . In s u c c e s s i v e i t e r a t i o n s , the program i n c r e a s e s the debt by an i nc rement s p e c i f i e d by the u s e r . i>i The o u t p u t f r om the program then g i ve s the f o l l o w i n g v a l u e s f o r v a r y i n g amounts of deb t : the va l ue of the f i r m b e f o r e and a f t e r t a x , the a f t e r - t a x c o s t of c a p i t a l , the b e f o r e - t a x and a f t e r - t a x d e b t - e q u i t y r a t i o , the va l ue o f e q u i t y b e f o r e and a f t e r t a x , the average c o s t of d e b t , the average c o s t of e q u i t y b e f o r e and a f t e r t a x . The program adds s u c c e s s i v e i nc rement s o f debt and t e r m i n a t e s when a l l the e q u i t y has been r e p l a c e d by d e b t . F o o t n o t e s 1. The a n a l y s i s i n t h i s c h a p t e r i s based p r i m a r i l y on t h e i r p a p e r s , "The C o s t of C a p i t a l , C o r p o r a t i o n F i n a n c e , and the Theory of I n v e s t m e n t " , pp. 261-97, and " C o r p o r a t e Income Taxes and the Co s t o f C a p i t a l : A C o r r e c t i o n " , Amer ican Economic Rev i ew. V o l . L111, No. 3, June , 1963, pp. 433-43. 2. In t h e i r a n a l y s i s , M o d i g l i a n i and M i l l e r use f as the c a p i t a l i z a t i o n r a t e a p p l i e d to e a r n i n g s o f a g i v en r i s k c l a s s . We w i l l use K 0 i n s t e a d of p so t h a t the symbols used i n the M o d i g l i a n i and M i l l e r a n a l y s i s w i l l be c o n s i s t e n t w i t h those used i n the net income and t r a d i t i o n a l a n a l y s i s . 3. M i l l e r and M o d i g l i a n i , op. c i t . , p. 412. 4. M o d i g l i a n i and M i l l e r , "The Co s t o f C a p i t a l , C o r p o r a t i o n F i n a n c e and the Theory of I n v e s t m e n t " , p. 268. 5. i b i d . , p. 268. 6. i b i d . , p. 272. 7. i b i d . , p. 270 58 CHAPTER I I I THE TRADITIONAL HYPOTHESIS The Net O p e r a t i n g Income Approach The M o d i g l i a n i and M i l l e r h y p o t h e s i s d e s c r i b e d i n the 1 . p r e v i o u s c h a p t e r i s what Durand has c a l l e d the net o p e r a t i n g income approach to v a l u a t i o n of the f i r m . When t h i s approach i s u s ed , the net o p e r a t i n g income i s c a p i t a l i z e d u s i n g a r a t e c h a r a c t e r i s t i c o f the r i s k c l a s s t o wh ich the f i r m b e l o n g s , t o de te rm ine the t o t a l v a l ue o f the f i r m . The va l ue o f the e q u i t y i s then found by s u b t r a c t i n g the amount o f debt f rom the t o t a l v a l u e . Then, as long as the debt c a p i t a l i z a t i o n r a t e does not change, the e q u i t y c a p i t a l i z a t i o n r a t e i s a l i n e a r f u n c t i o n o f the d e b t - e q u i t y r a t i o and i s g i ven by M o d i g l i a n i and M i l l e r ' s P r o p o s i t i o n I I . An example w i l l make t h i s c l e a r . Suppose a company be long s to a r i s k c l a s s i n wh ich t o t a l e a r n i n g s of $ 1 , 0 0 0 a re c a p i t a l i z e d a t 1 0 $ . Sup -pose t h i s company has 02 , 5 0 0 of debt at 4$ i n i t s c a p i t a l s t r u c t u r e . Then the va l ue o f the e q u i t y i s g i ven as f o l l o w s : 39 Tab le I VALUE OF THE CORPORATION NET OPERATING INCOME MODEL Net O p e r a t i n g Income $1,000 C a p i t a l i z e d a t 10$ x 10 T o t a l Va lue 10,000 Less T o t a l Debt 2.500 Va lue of E q u i t y $7,500 We s h o u l d note t h a t because of P r o p o s i t i o n I, the t o -t a l v a l ue of debt and e q u i t y must a lways be $10 ,000 , r e g a r d -l e s s o f t h e i r p r o p o r t i o n s . The Net Income Approach The a l t e r n a t i v e approach i s what Durand c a l l s the net income method. In t h i s method, i n t e r e s t cha rges are f i r s t s u b t r a c t e d f rom the net o p e r a t i n g income. The r e m a i n i n g net income i s then c a p i t a l i z e d a t a r a t e a p p l i c a b l e to t h a t r i s k c l a s s to de te rm ine the v a l ue o f the e q u i t y . I f the e q u -i t y c a p i t a l i z a t i o n r a t e i s 10$, then f o r the f i r m d e s c r i b e d above, the v a l u e o f the e q u i t y i s now de te rm ined as f o l l o w s : Tab le II VALUE OF THE CORPORATION NET INCOME MODEL Net O p e r a t i n g Income $1,000 Less I n t e r e s t - 100 Net Income 900 C a p i t a l i z e d a t 10$ x 10 $9,000 4'0 Now the va lue of the e q u i t y i s seen to be $9,000 i n s t e a d of $7 ,500 as de te rm ined by the net o p e r a t i n g income method. F u r t h e r m o r e , the v a l ue of the f i r m i s now g i v e n by the t o t a l of debt and e q u i t y , so i t i s now $11,500. We note t h a t the va l ue of the f i r m i n c r e a s e s as the amount o f debt in the c a p i t a l s t r u c t u r e i n c r e a s e s . Tab le I I I EFFECT OF CAPITAL STRUCTURE ON VALUE NET INCOME MODEL Amount o f Debt 0 1,250 2,500 Va l ue of E q u i t y 10,000 9,500 9,000 T o t a l Va l ue $10,000 $10,750 $11,500 The p roponent s o f the net income method do not sugges t t h a t the p rocedu re o f r e p l a c i n g debt w i t h e q u i t y and the reby i n c r e a s i n g the v a l ue of the f i r m can c o n t i n u e i n d e f i n i t e l y . The v iew of those s u p p o r t i n g the ne t income approach to v a l -u a t i o n i s t h a t f o r f i r m s i n a g i v e n r i s k c l a s s , the market va l ue of the f i r m w i l l f i r s t r i s e as the amount of debt i n the c a p i t a l s t r u c t u r e i s i n c r e a s e d f rom zero to some p o i n t de te rm ined by the c a p i t a l m a r k e t ' s e v a l u a t i o n o f the f i n a n -c i a l r i s k a s s o c i a t e d w i t h v a r y i n g degrees o f l e v e r a g e . Be -yond t h i s p o i n t , i n v e s t o r s r e q u i r e a h i g h e r y i e l d on e q u i t y to compensate f o r the i n c r e a s e d f i n a n c i a l r i s k r e s u l t i n g f rom l e v e r a g e . Th i s means t h a t any " s a v i n g " f r om the use o f debt i s o f f s e t by the i n c r e a s e i n " c o s t " o f e q u i t y , so the v a l ue of the f i r m remains a lmos t c o n s t a n t . At even h i g he r amounts #1 of d e b t , both debt and e q u i t y p u r c h a s e r s r e g a r d t h e i r h o l d i n g s as very r i s k y . The much h i g h e r y i e l d t h a t they demand r e s u l t s i n a r a p i d dec rea se i n the v a l ue o f the f i r m . Two P o s s i b l e Computer Models We now w i sh t o d e s c r i b e two d i f f e r e n t mode l s . We de -v e l o p a " n e t income m o d e l " and a " t r a d i t i o n a l m o d e l " . I t s h o u l d be noted t h a t t h i s t e r m i n o l o g y i s s i m i l a r to t h a t used 2. by Weston, but many w r i t e r s tend to use the two terms i n t e r -c h a n g e s ^ y . The net income model may be rega rded as d e s c r i b i n g a p o s s i b l e extreme of i n v e s t o r b e h a v i o r . I t i s used here s i n c e i t was sugges ted by M o d i g l i a n i and M i l l e r as a t h e o r e t i c a l 3. a l t e r n a t i v e to t h e i r h y p o t h e s i s . They s t a t e : " W i t h o u t do i n g v i o l e n c e to t h i s p o s i t i o n ( t r a d i t i o n a l h y p o t h e s i s ) , we can b r i n g out i t s i m p l i c a t i o n s more s h a r p l y by i g n o r i n g the q u a l i f i c a t i o n ( t h a t the e a r -n i n g s - p r i c e r a t i o or i t s r e c i p r o c a l , the t i m e s - e a r -n i n g s m u l t i p l i e r , of a f i r m ' s e q u i t y w i l l be on l y s l i g h t l y a f f e c t e d by moderate amounts o f debt i n the c a p i t a l s t r u c t u r e ) and t r e a t i n g the y i e l d as a v i r -t u a l c o n s t a n t ove r the r e l e v a n t r a n g e . " Th i s means t h a t i n what we have chosen to c a l l the net income mode l , the e q u i t y h o l d e r i s r ega rded as c o n s i d e r i n g the moderate use of debt as not i n c r e a s i n g f i n a n c i a l r i s k s u f f i c i e n t l y to w a r r a n t any i n c r e a s e i n the e q u i t y c a p i t a l i -z a t i o n r a t e . A f t e r a g i v e n l e v e l o f debt i s reached (depen -d i ng on the r i s k c l a s s to wh ich the f i r m b e l o n g s ) , the e q u i t y and debt c a p i t a l i z a t i o n r a t e s i n c r e a s e r a p i d l y . In what we c a l l t he t r a d i t i o n a l mode l , we d e s c r i b e the "42 i n v e s t o r as t h i n k i n g t h a t any use of debt r e s u l t s in i n c r e a s e d f i n a n c i a l r i s k . Even moderate use o f debt r e s u l t s i n an i n -c r e a s e i n the e q u i t y c a p i t a l i z a t i o n r a t e . The f i r s t d e r i v a -t i v e w i t h r e s p e c t to debt of the e q u i t y c a p i t a l i z a t i o n r a t e i s a p o s i t i v e i n c r e a s i n g f u n c t i o n of the amount of d e b t . The e q u i t y h o l d e r r e ga rd s i n c r e a s i n g l e v e l s o f debt as r e s u l t i n g i n a more than p r o p o r t i o n a t e i n c r e a s e i n f i n a n c i a l r i s k . The net income model i s deve l oped as f o l l o w s . The c a -p i t a l i z a t i o n r a t e s f o r both debt and e q u i t y are s e t by the m a r k e t . As i n the model deve loped to t e s t the M o d i g l i a n i and M i l l e r h y p o t h e s i s , a c h o i c e of t h r e e f u n c t i o n s i s a v a i l a b l e to de te rm ine the i n t e r e s t payments on d e b t . These f u n c t i o n s t ake the same fo rm as i n the M o d i g l i a n i and M i l l e r mode l . ( 1 6 ) K ( 1 7 ) K ( 1 8 ) K ( 1 9 ) K = a + b (L - A) ••= a + b (L - A ) 2 = a + b (L - A ) 3 = a , f o r L < A As i n the p r e v i o u s mode l , Kj = a , i n a l l ca se s where L < A. A l s o , s i n c e e q u i t y c a p i t a l i z a t i o n r a t e s are now s e t by the m a r k e t , we have a c h o i c e of t h r e e f u n c t i o n s f o r K e : (21) K e = c + d (L - A) (22) K e = c + d (L - A ) 2 (23) K e = c + d (L - A ) 3 (24) K e = c , f o r L < A By u s i n g f u n c t i o n s o f t h i s t y p e , the e q u i t y y i e l d r e -mains c o n s t a n t u n t i l the amount o f debt i n the c a p i t a l s t r u c -t u r e i s i n c r e a s e d to the l e v e l a t wh ich L = A. Th i s i s i n 43 k e e p i n g w i t h the o b j e c t i v e s o f the net income mode l . Once the amount of debt exceeds A, the r e q u i r e d y i e l d i n c r e a s e s a t a r a t e wh i ch depends on the f u n c t i o n s e l e c t e d . E q u a t i o n (21) i m p l i e s i r r a t i o n a l b e h a v i o r s i n c e a v e r s i o n to f i n a n c i a l r i s k i s e xp re s sed as a l i n e a r f u n c t i o n of the amount of d e b t . E q u a t i o n s (22) and (23) d e s c r i b e i n v e s t o r s who r e q u i r e y i e l d s wh i ch i n c r e a s e r a p i d l y when the l e v e l o f debt exceeds A. In the a n a l y s i s wh ich f o l l o w s , we assume t h a t , i n i t i a l -l y , the debt h o l d e r s r e q u i r e 5$ i n t e r e s t on t h e i r h o l d i n g s and t h a t the y i e l d on e q u i t y must be 1%. C o r p o r a t e e a r n i n g s must be s u f f i c i e n t to pay 5% to the debt h o l d e r s and a l s o r e t u r n 1% a f t e r tax t o the e q u i t y h o l d e r s . These numbers are chosen because they are thought to be r e a l i s t i c f o r i n -v e s t o r s h o l d i n g s e c u r i t i e s of a f i r m i n a moderate r i s k c l a s s . I t s h o u l d be noted t h a t the programs can be r e a d i l y adapted to o t h e r i n t e r e s t r a t e f u n c t i o n s . In a d d i t i o n , the l e v e l at wh i ch the c a p i t a l i z a t i o n r a t e s f i r s t i n c r e a s e can be v a r i e d by chang i ng A and the r a t e a t wh ich they i n c r e a s e can be changed by s p e c i f y i n g b and d. The t o t a l market va lue o f the f i r m f o r the net income and t r a d i t i o n a l models i s de te rm ined f rom an e q u a t i o n such as 4 . the one g i v en by L i n d s a y and Sametz: V = L + S (25) = L + X - K j L K e f o r the case when t h e r e are no c o r p o r a t e t a x e s , o r , as g i ven 5 . by Mao, when c o r p o r a t e t axe s a re i n c l u d e d : (26) V = L + (X - K j L ) ( l --t) Ke where a i l v a r i a b l e s have been p r e v i o u s l y d e f i n e d . In e q u a t i o n (26), the second term on the r i g h t hand s i d e e x p r e s s e s the va l ue of the e q u i t y as b e i n g the sum i n p e r p e t u i t y of the ne t a f t e r - t a x e a r n i n g s t o e q u i t y . I t s hou l d be noted t h a t X i s the e xpec ted average r e t u r n on a s s e t s as s p e c i f i e d i n M o d i g l i a n i and M i l l e r ' s method o f r i s k c l a s s de -t e r m i n a t i o n , so a l l growth and f l u c t u a t i o n i n e a r n i n g s have been c o n s i d e r e d i n d e t e r m i n i n g X. The va l ue o f the e q u i t y can then be a c c u r a t e l y de te rm ined by a p p l y i n g a c a p i t a l i z a t i o n r a t e c h a r a c t e r i s t i c of i t s r i s k c l a s s . The programs o p e r a t e as f o l l o w s . The u se r s p e c i f i e s the type of f u n c t i o n he wants f o r Ke and K i , i n c l u d i n g the l e v e l of d e b t , A, up to wh i ch K e and Ki are c o n s i d e r e d to r e -ma i n- c o n s t a n t . He s p e c i f i e s the expec ted average e a r n i n g s on a s s e t s , X, wh ich i n our a n a l y s i s i s c o n s i d e r e d to be $75. The tax r a t e and i n c rement t o debt are a l s o s p e c i f i e d . The programs then pe r f o rm a s e r i e s of c a l c u l a t i o n s to de te rm ine the va l ue o f the f i r m , v a l ue o f e q u i t y , debt and e q u i t y y i e l d s , average c o s t of c a p i t a l and o t h e r pa rameter s f o r v a r i o u s l e v e l s of d e b t . The e q u a t i o n s used are e q u a t i o n (26) and e q u a t i o n (9), wh ich e x p r e s s e s the average c o s t o f c a p i t a l as a f u n c -t i o n of the c o s t and amounts of debt and e q u i t y i n the c a p i -t a l s t r u c t u r e : (9) K 0 = w-jKj + w 2 K e The t r a d i t i o n a l model i s i d e n t i c a l to the net income 45 model i n a l l r e s p e c t s e x cep t f o r the f u n c t i o n s d e s c r i b i n g K e and K j . The t r a d i t i o n a l w r i t e r s sugges t t h a t e q u i t y h o l d e r s are a ve r se to even sma l l i n c r e a s e s i n d e b t . Then, i n s t e a d o f k eep i n g K e c o n s t a n t u n t i l a debt l e v e l of A i s r e a c h e d , the c a p i t a l i z a t i o n r a t e s hou l d i n c r e a s e s l i g h t l y as soon as any debt i s i n t r o d u c e d i n t o the c a p i t a l s t r u c t u r e . E q u a t i o n s t h a t can be used to d e s c r i b e the r e q u i r e d debt y i e l d s a re the same as e q u a t i o n s ( 1 3 ) , (14) and ( 1 5 ) : (13) K i = a + bL (14) Ki = a + b L 2 (15) K i = a + b L 3 The e q u i t y c a p i t a l i z a t i o n r a t e s can be g i v en by the f o l l o w i n g e q u a t i o n s : (27) K e = c + dL (28) K e = c + d L 2 (29) K e = c + d L 3 A program l i s t i n g , i n s t r u c t i o n s f o r u s e , and ou tpu t o f the t r a d i t i o n a l model i s g i v en i n Appendix I I I . The net income model i s g i v e n i n Appendix IV. M a r g i n a l Co s t of Debt A l s o i n c l u d e d i n the programs f o r the t r a d i t i o n a l mo-de l and the net income model a re p r o cedu re s to c a l c u l a t e the m a r g i n a l c o s t of debt f o r the type o f model deve l oped by Mo-d i g l i a n i and M i l l e r and f o r the model deve loped by the t r a d i -t i o n a l w r i t e r s . A l t h o u g h the ma r g i n a l c o s t s w i l l not be 46 a n a l y z e d i n d e p t h , t h e i r r e l a t i o n s h i p to the average c o s t cu r ve and the maximum va l ue of the f i r m w i l l be c o n s i d e r e d i n the f o l l o w i n g c h a p t e r s . In the model deve loped by M o d i g l i a n i and M i l l e r , the t o t a l v a l u e o f a s s e t s i s kept c o n s t a n t and debt i s used to r e -duce e q u i t y h o l d i n g s (dL = - d S ) . In d e t e r m i n i n g the m a r g i -na l c o s t o f d e b t , we need c o n s i d e r o n l y the c o s t of the new i nc rement o f debt p l u s the way i n wh ich the new debt i n f l u e n -ces the r e t u r n r e q u i r e d on the p r e v i o u s d e b t . The m a r g i n a l c o s t of debt can be de te rm ined f rom d_(K,-L) = L d K j + K, dL dL I f we de te rm ine the ma r g i n a l c o s t of debt f o r the Mo-d i g l i a n i and M i l l e r model f rom e q u a t i o n s ( 1 6 ) , ( 1 7 ) , and ( 1 8 ) , we g e t : (30) d_ (Ki L )16 = bL + a + b (L - A) dL (31) d_ ( K i L ) i 7 = 2 b L 2 - 2AbL + a + b (L - A ) 2 dL (32) d__ ( K j L ) 1 8 = 3 b L 3 - 6AbL 2 + 3 b L A 2 + a dL + b(L - A ) 3 I f we use e q u a t i o n s ( 1 3 ) , ( 1 4 ) , and (15) to de te rm ine the r e q u i r e d y i e l d on d e b t , the m a r g i n a l c o s t of debt i s g i ven by the f o l l o w i n g e q u a t i o n s : ( 3 3 ) d_ (K j L ) 1 3 = bL + a + bL dL (34) d_ (K j L ) 1 4 = 2 b L 2 + a + b L 2 dL (35) d_ (K j L)i5 = 3 b L 3 + a + b L 3 dL 47 In the model deve loped by the t r a d i t i o n a l w r i t e r s , KQ i s the dependent v a r i a b l e and Ke and Kj a re the i ndependent v a r i a b l e s . We noted p r e v i o u s l y t h a t i n the M o d i g l i a n i and M i l l e r mode l , K e was the dependent v a r i a b l e and K 0 and Ki were the i ndependent v a r i a b l e s . Th i s meant t h a t a un ique c r o s s -r e l a t i o n s h i p was assumed to e x i s t between the debt and e q u i t y s e c t o r s o f the m a r k e t , wh i ch kept K 0 and V c o n s t a n t . The t r a d i t i o n a l w r i t e r s do not assume t h a t t h i s r e l a t i o n s h i p e x i s t s . S i n c e K e and K i are thought by them to be de te rm ined i n d e p e n -d e n t l y , the m a r g i n a l c o s t of debt i n the t r a d i t i o n a l and net income models must i n c l u d e a term to compensate f o r the change i n the c o s t of e q u i t y r e s u l t i n g f rom i n c r e a s e d use o f d e b t . The m a r g i n a l c o s t o f debt i s now composed of two components. The f i r s t component a ccoun t s f o r the i n c r e a s e d c o s t o f d e b t . In an i t e r a t i v e mode l , i t may be g i v en by the f o l l o w i n g e x p -r e s s i o n : (36) Kj m a r g = K j ( L + A ) ( L + A ) - K j L A where A i s the amount by wh i ch the debt has been i n c r e a s e d and K i ( L + A \ , i s the i n t e r e s t c o s t a t the h i ghe r debt l e v e l . The second component, g i v i n g the e f f e c t of i n c r e a s e d debt on the v a l u e of e q u i t y , i s g i v e n f o r sma l l i n c rement s by: (37) K\ ^,-5 = (X - K j L ) (1 - T ) f~K e(L+A) - 1 A [ Ke(L) where K 6(L+A) i s t n e c a p i t a l i z a t i o n r a t e a t the h i ghe r debt l e v e l , and K ^ L ) i s the c a p i t a l i z a t i o n r a t e a t the lower l e v e l . 48 The tota l marginal cost of debt can now be determined from the sum of the marginal cost of increases in debt char-ges and the marginal cost of the change in the value of the equ i ty: K i marg = K i 'marg + K i "marg for smaII changes. Footnotes 1. David Durand, "Cost of Debt and Equity Funds for Business: Trends and Problems of Measurement", Conference on Research on Business Finance, New York, National Bu-reau of Economic Research, 1952 , pp. 215-247. 2. J . F. Weston, op. c i t . , pp. 1 0 5 - 1 1 2 3 . Modigl iani and M i l l e r , "The Cost of C a p i t a l , Corporation Finance and the Theory of Investment", p. 2 6 8 . 4. Lindsay and Sametz, op. c i t . , p. 129 5. James C. T. Mao, op. c i t . , chapter 1 1 . 49 CHAPTER IV AN ANALYSIS OF THE MODIGLIANI AND MILLER HYPOTHESIS In Chap te r I I , the M o d i g l i a n i and M i l l e r a r b i t r a g e a r -gument has been used to show t h a t i f markets are p e r f e c t , the v a l u e o f two f i r m s w i t h the same e a r n i n g s and i n the same r i s k c l a s s must be equa l even i f t h e i r c a p i t a l s t r u c t u r e s are d i f f e r e n t . We now w i sh to examine some of the i m p l i c a t i o n s of the M o d i g l i a n i and M i l l e r h y p o t h e s i s and to see how more r e a l i s t i c a s sumpt ions can i n f l u e n c e t h e i r r e s u l t s . 1 . M o d i g l i a n i and M i l l e r s t a t e ; "Our P r o p o s i t i o n s I and I I , as no ted e a r l i e r , do not depend f o r t h e i r v a l i d i t y on any a s sumpt ion about i n d i v i d u a l r i s k p r e f e r e n c e s . Nor do they i n v o l v e any a s s e r t i o n as to what i s an adequate compensa t i on to i n v e s t o r s f o r assuming a g i v e n degree o f r i s k . They r e l y mere ly on the f a c t t h a t a g i ven commodity cannot c o n s i s t e n t l y s e l l a t more than one p r i c e i n the ma rke t ; or more p r e c i s e l y , t h a t the p r i c e of a commodity r e p r e s e n t i n g a " b u n d l e " of two o t he r com-m o d i t i e s cannot be c o n s i s t e n t l y d i f f e r e n t f rom the we i gh ted average of the p r i c e of the two component s . " I t i s our c o n t e n t i o n t h a t M o d i g l i a n i and M i l l e r ' s P r o -p o s i t i o n II jjs^ a d e s c r i p t i o n of e q u i t y h o l d e r b e h a v i o r i n r e a c t i o n to i n c r e a s i n g debt i n the c a p i t a l s t r u c t u r e . Mo re -o v e r , we hope to show t h a t the b e h a v i o r d e s c r i b e d by M o d i g -l i a n i and M i l l e r i s not r e a l i s t i c i n a r e a l w o r l d c o n t e x t where c o r p o r a t e income t a x , l i m i t e d p e r s o n a l l i a b i l i t y and r i s k - a v e r s e i n v e s t o r s e x i s t . A l s o , we sugges t t h a t due to such f a c t o r s as t a x e s , l i m i t e d l i a b i l i t y and i n s t i t u t i o n a l r e -50 s t r i c t i o n s , the bundle of commod i t i e s may depend on the compo-s i t i o n of i t s p a r t s . A r b i t r a g e M o d i g l i a n i and M i l l e r ' s P r o p o s i t i o n I i s based on t h e i r ' a r b i t r a g e i d e a . We w i l l now examine some of the f a c t o r s t h a t may impede t h i s a r b i t r a g e or s w i t c h i n g . M o d i g l i a n i and M i l l e r m a i n t a i n t h a t whenever the v a l u e of two s i m i l a r companies w i t h equal e a r n i n g s d i f f e r s , i t w i l l pay i n v e s t o r s to s e l l t h e i r s ha re s i n the h i g h e r - v a l u e d company and buy sha re s i n the l o w e r - v a l u e d company. There a re s e v e r a l f a c t o r s wh i ch i m -pede t h i s p r o c e s s . F i r s t , i f the va l ue of sha re s i n the " o v e r -p r i c e d " company has i n c r e a s e d , the i n v e s t o r w i l l be s u b j e c t to c a p i t a l ga i n s t a x . Th i s may w e l l wipe out a l l ga in s f rom purchase of the unde r v a l ued s h a r e s . A l s o , the i n v e s t o r must pay b roke rage f e e s on h i s a r b i t r a g e t r a n s a c t i o n . The com-b i n a t i o n of these two c o s t s may make i t more p r o f i t a b l e f o r him to r e t a i n h i s " o v e r p r i c e d " s h a r e s . I f a r b i t r a g e i s to be e f f e c t i v e , a g r e a t amount o f mo-ney must be a v a i l a b l e f o r these t r a n s a c t i o n s . M o d i g l i a n i and M i l l e r t h i n k t h a t f unds can be r a i s e d by p u r c h a s i n g e q u i t y on marg in or by p l e d g i n g sha re s o f the u n d e r v a l u e d company as s e c u r i t y f o r pe r s ona l l o a n s . There are s e v e r a l f a c t o r s wh ich l i m i t the e f f e c t i v e n e s s o f marg in b u y i n g . When M o d i g l i a n i and M i l l e r p u b l i s h e d t h e i r f i r s t p ape r , l e g a l r e s t r i c t i o n s r e q u i r e d the i n v e s t o r t o own o u t r i g h t , 5 0 $ o f the t o t a l equ -i t y ; Th i s has now been i n c r e a s e d to 9 0 $ , so the amount of 5'V funds a v a i l a b l e on marg in i s very s m a l l . Even t h i s sma l l amount of marg in buy ing i s w i t h h e l d f rom the i n s t i t u t i o n s wh ich accoun t f o r a l a r ge f r a c t i o n of e q u i t y p u r c h a s e s . Mu-t u a l f u n d s , most p e r s o n a l t r u s t f u n d s , c l o s e d end t r u s t , f i r e , c a s u a l t y and l i f e i n s u r a n c e companies are a l l p r e v e n t e d f rom buy i ng s t o c k on m a r g i n , e i t h e r th rough d i r e c t r e s t r i c t i o n s imposed by t h e i r c h a r t e r , o r th rough the d i c t a t e s of p ruden t b e h a v i o r . When we i n t r o d u c e the p o s s i b i l i t y of pe r s ona l bank l o a n s , s e cu red by the e q u i t y p u r c h a s e s , many d i f f i c u l t i e s are e n c o u n t e r e d . I f i n v e s t o r s a re to be a b l e to undo c o r p o r a t e l e ve rage th rough p e r s o n a l l e v e r a g e , the i n t e r e s t r a t e s p a i d by the c o r p o r a t i o n and by the i n d i v i d u a l must be i d e n t i c a l , as can be seen f rom a s tudy of e q u a t i o n s (3) t h rough (6). M o d i g l i a n i and M i l l e r " c o n j e c t u r e t h a t the cu r ve f o r bond y i e l d s as a f u n c t i o n of l e ve rage w i l l t u r n out to be a non -2. l i n e a r o n e . " They agree w i t h t r a d i t i o n a l t h e o r i s t s t h a t , because of the i n c r e a s e d r i s k of d e f a u l t of i n t e r e s t p a y -ments a s s o c i a t e d w i t h e x c e s s i v e amounts of l e v e r a g e , debt y i e l d s must i n c r e a s e as the amount of debt i n the c a p i t a l s t r u c t u r e i n c r e a s e s . However, i f "homemade" and c o r p o r a t e l e ve r age are to be s u b s t i t u t e s , the i n d i v i d u a l s e c u r i n g the p e r s o n a l l oan and the c o r p o r a t i o n must f a c e i d e n t i c a l f u n c -t i o n s f o r K i . Th i s i s not p o s s i b l e when we c o n s i d e r t h a t c o r p o r a t e debt i s b a c k e d ' o n l y by the a s s e t s of the c o r p o r a -t i o n w h i l e the p e r s o n a l loan i s s e cu red by u n l i m i t e d p e r s o n a l l i a b i l i t y . At extreme l e v e l s o f c o r p o r a t e l e v e r a g e , the 52 indiv idual can obtain debt funds at a lower cost than can tfee corporat ion. Only i f the indiv idual is ident ica l in a l l re s -pects to the corporation can corporate and homemade leverage be considered as equivalent. However, Modigliani and M i l l e r see another source of 3. funds: "Under normal condi t ions, moreover, a substantial part, of the arbitrage process could be expected to take the form not of having arbitrage operators go into debt on personal account to put the required leverage into the i r p o r t f o l i o s , but simply of having them reduce the amount of corporate bonds they already hold when they acquire underpriced unlevered stock." In a real l i f e s i t ua t i on , any action such as this w i l l most l i ke ly change the overa l l f inanc ia l r i sk of the investment p o r t f o l i o . Before purchasing the underpriced, unlevered stock, the investor has in his po r t fo l i o a combination of bonds and stocks which y i e l d him earnings of a given mean and variance or of a given f i nanc i a l r i s k . Any attempt to s e l l some of the bonds in his ex i s t ing por t fo l i o to provide funds to create homemade leverage for the purchase of the undervalued security w i l l probably a l te r the f i nanc i a l r i s k . Unless the investor holds the same proportion of bonds and stock in the ove rp r i -ced secur i ty , the procedure described by Modigliani and M i l l e r w i l l re su l t in a greater amount of funds being invested in a par t i cu la r secur i ty , thereby changing the mean and variance of returns from the p o r t f o l i o . If Modigliani and M i l l e r allow f inanc ia l r i sk to change, they introduce complications into the i r analysis which are impossible to resolve without f i r s t determining u t i l i t y functions for investors. 53 Table IV shows the value of two companies with the same earnings and in the same r i sk c lass but with d i f fe rent capita l s t ructures . The equity of Company Y is cap i t a l i zed at a higher rate to compensate for the f inanc ia l r i sk from leverage. Table IV USE OF ARBITRAGE Company X Company Y Expected Earnings $1,000 $1,000 Debt - 3,000 Interest (4%;) - 120 Returns to Shareholders 1,000 880 Cap i t a l i z a t i on Rate 10% 11% Value of Equity 1:0,000 8,000 Value of Firm 10,000 11,000 D/E Ratio 0 37.5 Through the Modigliani and M i l l e r arbitrage argument, the investor, who is assumed to hold $1,000 of equity in Com-pany Y, w i l l s e l l these shares. To keep his f i nanc ia l r i sk constant, he w i l l borrow $375 and invest th is tota l of $1,375 in Company X. In his old p o r t f o l i o , he earned 11% on $1,000 or $110. In th is new one, he earns: $1,375(.1) ~ $375(.04) = $122.50. However, i f he uses the $375 he borrowed to purchase more shares in Company Y, he would have an income of $136.25, or an increase of $13.75 over what he could receive i f he inves-ted in Company X. The question of whether th is added $13.75 is s u f f i c i e n t to o f f set the increased f inanc ia l r i sk cannot be answered by any of the models deve loped to d a t e . Barges d e s c r i b e s these c i r c u m s t a n c e s wh ich l i m i t the u s e f u l n e s s of the a r b i t r a g e p roce s s i n e q u a t i n g the v a l ue o f 4. s i m i l a r f i r m s w i t h d i f f e r e n t c a p i t a l s t r u c t u r e s . C o n s i d e r f i r m s X and Y as d e s c r i b e d p r e v i o u s l y . S i n c e the i n v e s t o r can r e c e i v e a g r e a t e r r e t u r n at the same r i s k by s w i t c h i n g h i s i n ve s tmen t f rom Company Y to Company X, the i m p l i c a t i o n i s t h a t t h i s d i s p a r i t y i n market va lue w i l l be e r a s ed by a r -b i t r a g e . S a l e of s t o c k of Company Y w i l l d e f l a t e i t s p r i c e , d r i v i n g i t s y i e l d up, w h i l e the purchase of s t o c k i n Company X w i l l i n c r e a s e i t s p r i c e , d r i v i n g i t s y i e l d down. Barges p o s t u l a t e s t h a t these changes i n y i e l d s may induce the f o l l o -wing r e a c t i o n , wh ich s e r ve s to c oun te r the e f f e c t i v e n e s s o f a r b i t r a g e . S i n c e the y i e l d o f s t o c k X has d e c l i n e d , some o f the o r i g i n a l s h a r e h o l d e r s o f Company X may t h i n k t h a t t h i s new lower y i e l d i s i n s u f f i c i e n t to compensate them f o r t h e i r r i s k . Mo reove r , these i n v e s t o r s note t h a t the y i e l d of s t o c k Y has been d r i v e n up , w h i l e i t s r i s k has remained c o n s t a n t . Then some of the s h a r e h o l d e r s of Company X who do not l i k e margin- buy ing w i l l now f i n d s t o c k Y more a t t r a c t i v e . Wh i le the M o d i g l i a n i and M i l l e r a r b i t r a g e p r oce s s i s a c t i n g to cause i n v e s t o r s to s e l l s t o c k Y ( d e p r e s s i n g i t s p r i c e ) and buy s t o c k X ( i n f l a t i n g i t s p r i c e ) , an induced r e a c t i o n r e s u l -t i n g f rom the a r b i t r a g e p roce s s i s c a u s i n g ano the r group of i n v e s t o r s to s e l l s t o c k X ( d e p r e s s i n g i t s p r i c e ) and buy s t o c k Y ( i n f l a t i n g i t s p r i c e ) . The a c t i o n s of t he se two groups of i n v e s t o r s may c o u n t e r b a l a n c e , r e s u l t i n g i n unchan -55 ged s t o c k p r i c e s . We note t h a t i n v e s t o r s s e l l i n g s t o c k X and buy i ng s t o c k Y are i n c r e a s i n g t h e i r f i n a n c i a l r i s k , but they may w e l l r e g a r d the i n c r e a s e d y i e l d as adequate c o m p e n s a t i o n . We note t h a t i f the induced r e a c t i o n i s to p r o c e d e , we must have a group of i n v e s t o r s who d i s l i k e marg in b u y i n g . I f t h i s were not s o , the o r i g i n a l i n v e s t o r s i n Company X c o u l d improve t h e i r r e t u r n by t a k i n g a marg ined p o s i t i o n i n Company X. We now examine the way i n wh ich l i m i t e d l i a b i l i t y i n f -luences the r i s k s of marg in b u y i n g . L i mi ted L i abi I i t y M o d i g l i a n i and M i l l e r a s s e r t t h a t a r b i t r a g e i s p o s s i b l e s i n c e homemade l eve rage i s a s u b s t i t u t e f o r c o r p o r a t e l e v e -r a g e . However, l i m i t e d p e r s o n a l l i a b i l i t y p r o t e c t s the e q u i t y h o l d e r i n the l e v e r e d company. The i n v e s t o r u s i n g p e r s o n a l l e ve r age r e c e i v e s no such p r o t e c t i o n . Suppose an i n v e s t o r ho l d s $7,000 o f e q u i t y i n Company Y and t h a t the rema inder of the company ' s c a p i t a l s t r u c t u r e i s composed of $3,000 wo r t h of d e b t . I f Company Y has a g r e a -t e r v a l u e than Company X, M o d i g l i a n i and M i l l e r sugges t t h a t the i n v e s t o r can improve h i s r e t u r n by s e l l i n g h i s s ha re s of Company Y f o r $7,000. To keep the f i n a n c i a l r i s k of h i s new p o r t f o l i o equal to t h a t o f h i s o l d one , he must borrow $3,000 and i n v e s t the t o t a l o f $10,000 i n Company X. M o d i g l i a n i and M i l l e r m a i n t a i n t h a t r e t u r n has been i n c r e a s e d but f i n a n c i a l r i s k remains c o n s t a n t . We submit t h a t M o d i g l i a n i and M i l l e r are c o r r e c t o n l y i f l i m i t e d p e r s o n a l l i a b i l i t y f o r e q u i t y 56 h o l d e r s does not e x i s t . In our example, the s h a r e h o l d e r o f Company Y c o u l d lo se a maximum of $7,000. A f t e r he has e n -gaged i n homemade l e v e r a g e , h i s maximum p o s s i b l e l o s s has been 5-i n c r e a s e d to $10,000. As Durand n o t e s , i n p r a c t i c e the r i s k of l o s i n g the e n t i r e $10,000 i s sma l l s i n c e the bank or b r o ke r w i l l s e l l out the s t o c k h o l d e r to meet marg in requ i ' rements , but the p r o t e c t i o n of maximum l o s s g r e a t l y i n c r e a s e s the r i s k o f s m a l l e r l o s s . In t h i s c o n t e x t , we s hou l d note another d i f f e r e n c e between p e r s o n a l and c o r p o r a t e l e v e r a g e . When a c o r p o r a t i o n b o r r o w s , i n t e r e s t on debt has f i r s t c l a i m to the e a r n i n g s of the c o r p o r a t i o n . When the s t o c k h o l d e r engages i n p e r s o n a l l e v e r a g e , t h i s i s no l onger s o . Any d i v i d e n d pay -ment i s now a t the d i s c r e t i o n of management. Wh i l e the f i r m may have ample e a r n i n g s , i t may dec i de to r e t a i n them. In t h i s c a s e , the s h a r e h o l d e r who has engaged i n p e r s o n a l b o r -row ing to c r e a t e homemade l eve rage w i l l be f o r c e d to l i q u i -date some of h i s h o l d i n g s to meet pe r s ona l i n t e r e s t c h a r g e s . Th i s can r e s u l t i n l o s s e s i f the l i q u i d a t i o n o c c u r s a t an i n -oppor tune t i m e . In summary, we have noted t h a t t he r e are many r e a l w o r l d r e s t r i c t i o n s wh i ch may p r e v e n t the a r b i t r a g e p r oce s s of M o d i g l i a n i and M i l l e r f rom e q u a t i n g the va l ue of two f i r m s w i t h equa l e a r n i n g s and i n the same r i s k c l a s s but hav i ng d i f -f e r e n t c a p i t a l s t r u c t u r e s . R e s t r i c t i o n s on marg in buy i n g l i -m i t i t s u s e f u l n e s s . A l s o , we have seen t h a t homemade l e v e -rage i s more r i s k y than c o r p o r a t e l e v e r a g e . I f a group of . i n v e s t o r s e x i s t who are a ve r se to marg in b u y i n g , the a r b i t -57 rage p roce s s may induce r e s u l t s wh ich c a n c e l , out i t s e f f e c -t i v e n e s s . B e h a v i o r of Debt and E q u i t y H o l d e r s Even though we have s e r i o u s doubts about the e f f e c t i v e -ness o f the a r b i t r a g e p r o c e s s , we w i sh to examine some o f the b e h a v i o r a l a s sumpt ion s i m p l i e d by P r o p o s i t i o n I I , g i v e n t h a t P r o p o s i t i o n I i s c o r r e c t . A l s o , we want to examine the e f f e c t of c o r p o r a t e t a x e s and the t a x - d e d u c t a b i I i t y of i n t e r e s t pay -ments on the M o d i g l i a n i and M i l l e r mode l . We sugges t t h a t P r o p o s i t i o n II i s a d e s c r i p t i o n of i n -v e s t o r r e a c t i o n to the i n t r o d u c t i o n of debt i n t o the c a p i t a l s t r u c t u r e . P r o p o s i t i o n I I , deve loped i n Chap te r II and r e -peated be l ow, g i v e s the e q u i t y c a p i t a l i z a t i o n r a t e as a f u n c -t i o n o f the d e b t - e q u i t y r a t i o : (7) K e = K 0 + ( K 0 - K t ) L / S and as we no ted p r e v i o u s l y , i t i s based on the a s sumpt ion t h a t the we i gh ted average c o s t of c a p i t a l i s a c o n s t a n t depen -dent on the r i s k c l a s s to wh ich the f i r m b e l o n g s . T h e r e f o r e , Ke, wh ich i s the dependent v a r i a b l e i n the M o d i g l i a n i and M i l l e r mode l , becomes a f u n c t i o n of the i ndependent v a r i a b l e s K 0 , wh i ch i s i n d i c a t i v e o f i n v e s t o r a p p r a i s a l of the r i s k c l a s s , K K j , wh ich i s de te rm ined by the debt ma r ke t , and L, the amount of debt i n the c a p i t a l s t r u c t u r e . Both the t r a d i t i o n a l w r i t e r s and M o d i g l i a n i and M i l l e r a re agreed t h a t the debt c a p i t a l i z a t i o n r a t e s hou l d i n c r e a s e as the f i n a n c i a l r i s k , de te rm ined by the amount of debt i n the c a p i t a l s t r u c t u r e , i n -58 c r e a s e s . What they a r e . n o t agreed upon i s the way i n wh ich the e q u i t y c a p i t a l i z a t i o n r a t e s h o u l d i n c r e a s e . In o r de r to b e t t e r a n a l y z e the b e h a v i o r a l i m p l i c a t i o n s of M o d i g l i a n i and M i l l e r ' s mode l , l e t us de te rm ine some o f the reasons why the debt and e q u i t y c a p i t a l i z a t i o n r a t e s s h o u l d i n c r e a s e w i t h i n -c r e a s e d use o f d e b t . Bo th debt and e q u i t y h o l d e r s are conce rned w i t h the r i s k of d e f a u l t . The debt h o l d e r s are conce rned w i t h the f a c t t h a t they may not r e c e i v e t h e i r , i n t e r e s t payments. The e q u i t y h o l d e r s are conce rned w i t h the f a c t t h a t i f the debt h o l d e r s do not r e c e i v e t h e i r i n t e r e s t payments , the e q u i t y h o l d e r s may lo se c o n t r o l of the company. Both groups of i n -v e s t o r s are then i n t e r e s t e d i n the amount by wh ich the f i r m ' s a s s e t s can d e c l i n e i n v a l u e b e f o r e they become l e s s than i t s l i a b i l i t i e s and the f i r m becomes i n s o l v e n t . The l i k e l i h o o d of i n s o l v e n c y can be de te rm ined by the e q u i t y - d e b t r a t i o . Fo r example , i f the r a t i o i s 1 9 * 1 , the f i r m ' s a s s e t s may f a l l 95% i n v a l ue b e f o r e the debt i s no l onge r c o v e r e d . How-e v e r , i f the r a t i o i s 1:4, d e f a u l t w i l l o c c u r i f the a s s e t s lo se o n l y 20% of t h e i r v a l u e . Fo r the f i r s t f i r m , the r i s k f o d e f a u l t i s very s m a l l , f o r even i f the e a r n i n g s f l u c t u a t e g r e a t l y , the chances ' of not be ing ab l e to c o ve r the bond i n -t e r e s t i s r emote . In the second c a s e , even s l i g h t f l u c t u a -t i o n s i n e a r n i n g s may r e s u l t i n the f i r m ' s be i n g unab le to meet i t s l a r ge i n t e r e s t b i l l , and i f the f i r m d e f a u l t s , the a s s e t s would p r o b a b l y not be wor th enough to pay the bonds o f f i n f u l l . Bo th bond and s t o c k h o l d e r s of the second com-59 pany s hou l d r e q u i r e h i g h e r y i e l d s i n o rde r to compensate them f o r t h e i r r i s k . Use; of debt may i n f l u e n c e the f i r m ' s f u t u r e i n ve s tmen t s t r a t e g y . A h i g h l y l e v e r e d f i r m miay be f o r c e d to pass up i n -ves tments wh ich an u n l e v e r e d f i r m i n the same r i s k c l a s s can u n d e r t a k e . The h i g h l y l e v e r e d f i r m may have i n s u f f i c i e n t funds to meet r e q u i r e d i n t e r e s t payments and s t i l l u nde r t a ke a l l the p l anned i n ve s tmen t s wh ich have a p o s i t i v e p r e s e n t v a l u e . S i n c e these p r o f i t a b l e i n ve s tmen t s can be unde r taken by the u n l e v e r e d f i r m , i t s va l ue w i l l be i n c r e a s e d r e l a t i v e to the l e v e r e d f i r m . I f the h i g h l y l e v e r e d f i r m r e s o r t s to e x t e r n a l debt or e q u i t y m a r k e t s , i t may f i n d t h a t any b o r -rowing w i l l r e s u l t i n c r e d i t o r s ' impos ing r e s t r i c t i o n s on the f i r m ' s f i n a n c i a l p o l i c i e s . E q u i t y h o l d e r s w i l l e xp r e s s a r e -l u c t a n c e to purchase sha re s o f these unsound c o r p o r a t i o n s and can be e n t i c e d to do so on l y by o f f e r i n g s u b s t a n t i a l p r e -miums i n y i e l d . I f e x c e s s i v e use o f l e ve rage does r e s u l t i n r e s t r i c -t i o n s on the f i r m ' s i n ve s tmen t p o l i c i e s , we can expec t a change i n the expec ted average income, X*, and the f i r m may be p l a c e d i n a d i f f e r e n t r i s k c l a s s . The f a c t t h a t use of debt i n c r e a s e s the f i n a n c i a l r i s k of the b u s i n e s s and r e s u l t s i n o p e r a t i n g l i m i t a t i o n s wh ich cause debt and e q u i t y h o l d e r s to r e q u i r e h i g h e r y i e l d s tends t o s uppo r t M o d i g l i a n i and M i l -l e r ' s c o n t e n t i o n t h a t the r e a l c o s t o f debt i s the same as the average c o s t of c a p i t a l . There are s e v e r a l o t h e r f a c t o r s , however, wh ich tend 60 to make debt " c h e a p e r " than e q u i t y . S i n c e c o r p o r a t e income i s t axed and i n t e r e s t on debt i s a tax d e d u c t i o n , t h e r e i s an advantage us u s i n g debt i n both M o d i g l i a n i and M i l l e r and i n the t r a d i t i o n a l mode l . A l t h o u g h use o f debt i n c r e a s e s the e xpec ted per share r e t u r n s , i t a l s o i n c r e a s e s the v a r i a b i l i t y of these r e t u r n s . However, due t o the e x i s t e n c e of l i m i t e d l i a b i l i t y , the ga in s p o s s i b l e t h rough l e ve rage are i n f i n i t e , w h i l e the l o s s e s a re l i m i t e d , so the d i s t r i b u t i o n o f r e t u r n s tends to be f a v o r a b l y skewed. S i n c e debt c o n t r a c t s are made i n monetary terms i n s t e a d o f r e a l d o l l a r t e r m s , a bonus w i l l a c c rue to e q u i t y h o l d e r s i f debt i s c o n t r a c t e d p r i o r to an i n f l a t i o n . In a r e c e s s i o n , debt i s a d i s a d v a n t a g e to the f i r m . U s i n g our computer p rog rams, we can now examine these i m p l i c a t i o n s f o r a h y p o t h e t i c a l f i r m . A n a l y s i s o f R e s u l t s From Computer Model We use our computer model to de te rm ine the v a l ue of the f i r m as p r e d i c t e d by the M o d i g l i a n i and M i l l e r mode l . In the f o l l o w i n g c h a p t e r , we r e p e a t the c a l c u l a t i o n s u s i n g models based on the net income h y p o t h e s i s and on the t r a d i -t i o n a l h y p o t h e s i s . The net o p e r a t i n g income h y p o t h e s i s o f M o d i g l i a n i and M i l l e r beg in s by p o s i t i n g a c o n s t a n t v a l ue f o r K 0 . The ne -c e s s a r y e q u i l i b r i u m va l ue of Ke i s then de te rm ined f rom P r o -p o s i t i o n I I , u s i n g a debt y i e l d , K j , as s e t by the ma rke t . We w i s h to s tudy the M o d i g l i a n i and M i l l e r model by u s i n g a h y p o t h e t i c a l f u n c t i o n to d e s c r i b e the dependence of K i . o n L. 61 Whi le t h i s f u n c t i o n has been chosen i n t u i t i v e l y , i t s h o u l d be noted t h a t i t s f o rm i s c h a r a c t e r i s t i c of r a t i o n a l i n v e s t o r b e h a v i o r and i s s uppo r ted by many e m p i r i c a l s t u d i e s . The r e -s u l t s of our a n a l y s i s a re not dependent on the e xac t shape of the Kj f u n c t i o n , but on l y on i t s f o r m . As long as i n v e s -t o r s are r i s k - a v e r s e , ( d 2 K j i s p o s i t i v e ) , our f i n d i n g s d L * -w i l l h o l d . The f u n c t i o n f o r Kj t h a t we use i s : Ki = .05 + 5 x 10-9 (i_ _ -125)3 Th i s f u n c t i o n d e s c r i b e s a debt market i n wh i ch bond p u r c h a s e r s do not r e q u i r e any a d d i t i o n a l compensa t ion f o r r i s k u n t i l t h e r e i s $125 of debt i n the c a p i t a l s t r u c t u r e . T h i s d e s -c r i p t i o n o f the b e h a v i o r of debt h o l d e r s i s the same as t h a t g i ven by the net income mode l . F i g u r e I g i ve s the va lue o f the f i r m , average c o s t of d e b t , m a r g i n a l c o s t of d e b t , average c o s t of e q u i t y , and a v e -rage c o s t o f c a p i t a l f o r a f i r m e a r n i n g $75, and f o r wh ich t h e r e i s no c o r p o r a t e income t a x . In k eep i n g w i t h M o d i g l i a n i and M i l l e r ' s method of a n a l y s i s , our da ta i s p l o t t e d u s i n g the d e b t - e q u i t y r a t i o as a measure of l e v e r a g e . The da t a used i n c o n s t r u c t i n g t h i s graph i s g i v en i n Appendix I I . From a s tudy of the da ta and g r aph , the f o l l o -wing i m p l i c a t i o n s of M o d i g l i a n i and M i l l e r ' s model are n o t e d . As long as t h e r e i s no c o r p o r a t e t a x , the v a l ue o f the f i r m and i t s we i gh ted average c o s t o f c a p i t a l a re c o n s t a n t . As g i v en by P r o p o s i t i o n I I , as long as K i remains c o n s t a n t , K e i s a l i n e a r i n c r e a s i n g f u n c t i o n o f the d e b t - e q u i t y r a t i o . In 62 f ~ 5 A. T \ 1. u H I-w i - -V ) - f - .! =1 E-f -1 -P '-I r-i ? i r c > VT i ' 1 s-3" B-r--< i r r-di - \- si- /v 1 f at fl tf :-F 1 jt f ( tr a i \ L" D 1 :< ll a 7 11 ZD L rj El T n '" 1 r: :K t- i NIC 1 ( M. ?- -> y - i- .-i >r w < »•< i 1C i \< 1— -I _ < 1 _y.ftt.-i J E — i-< X J -_ j _ 5-0 | || V-— — U! \ u. 0 -3- .« to D ul U: h& -hi -Id —"i i s V J 1 3> i . i \1 .b l n \ ( / ( ? • c 5-O I 1 o -1 • p 3 * 5 r 1 I JE i I E LQJUJJ r v .0 {. c 63 t h i s c a s e , Ki remains c o n s t a n t u n t i l the amount of debt i n the c a p i t a l s t r u c t u r e exceeds A, where A = 1 2 5 . When L exceeds A, wh i ch i n t h i s model o c c u r s a t a d e b t - e q u i t y r a t i o of ap -p r o x i m a t e l y . 1 2 5 , Kj then beg in s to i n c r e a s e . However, i f Kj i s to i n c r e a s e and K 0 remain c o n s t a n t , Ke must now i n c r e a s e a t a d e c r e a s i n g r a t e . Mo reove r , f o l l o w i n g a p roo f g i ven by Rob i chek and Mye r s , we can show t h a t when the m a r g i n a l c o s t of debt exceeds the average c o s t o f e q u i t y , the average c o s t of e q u i t y must s t a r t to d e c l i n e w i t h f u r t h e r i n c r e a s e s i n 6. the d e b t - e q u i t y r a t i o . The t o t a l i n t e r e s t p a i d by the c o r p o r a t i o n i s K j L , so the m a r g i n a l r a t e i s g i ven by: (38) M = d_ ( K i L ) = Ki + L dKj dL d L ~ A l s o , the average c o s t o f c a p i t a l i s g i ven by: (9) K 0 = w-|Ki + w 2 K e K 0 = L Kj + S K e V V (39) K 0 V = LK i + SK e Then, w r i t i n g e q u a t i o n (39) i n d i f f e r e n t i a l f o r m : (40) VdK 0 + K 0 dV = K j dL + LdKj + K e dS + SdK e But M o d i g l i a n i and M i l l e r m a i n t a i n t h a t V and K 0 a r e ^ b o t h c o n s t a n t , so : (41) 0 = Kj dL + LdKj + K e dS + SdK e However, i n the M o d i g l i a n i and M i l l e r mode l , debt i s be i ng sub -s t i t u t e d f o r e q u i t y , s o : dL = -dS Then, making t h i s s u b s t i t u t i o n i n e q u a t i o n (41) g i v e s : 64: (42) 0 = Kj dL + LdKj + SdK e - K edL The point at which Ke s tarts to decline occurs when i t s f i r s t der ivat ive dK e = 0. Then, solving for this point from equa-tion (42) gives: (43) 0 = Kj dL + LdKj - K edL Then: (44) K e = Ki + L dKj_ dL = M From Proposit ion I, when Kj = Ko, then Ke must also equal Ko. (9) K0 = wiKi + w2Ke but i f Kj = K0 then K0 = wiKo + w2Ke (1 ~ wpKn = Ke (45) ( w 2 ) But w1 + W2 = 1, so w'2 = 1 - , so Ke = Ko. Since Ke is dec l in ing in this region, for debt-equity rat ios beyond those at which Ko = K i , a necessary condit ion is that Ke < Kj . From the above ana lys i s , we note that i f Ko is to rem-ain constant for a l l debt-:equity r a t i o s , the fol lowing beha-vior is required for Ke i f Kj is to increase with increasing use of debt. Ke f i r s t increases at a constant rate as long as Ki is constant. Then, once Ki s tarts to increase, Ke i n -creases, but at a decreasing rate. When the marginal cost 65 F i c r -( ._ | )-] -( k •i 4 I-] 4 JE )- \ M 4 4 :E f t 4 I-Y4 G "4 ft ^€ -I •£ 1 l\ IUL 1 JL It i 1 _ 1ft 1 n I )J )h F ij it* JC ) \ A ll n IF It! r r IV 1 ^£ > p | j r- c IF •j 11 I I:E ::F HI 1 1 — "E a F "t If L -C :.c .C !> i ! | f c ;r (1 Vf )\ 4 p V •V > v. >- V / 4 / / / It / >-1 « cs OS 1 H © * * fvo / © il (o II a: j | 1 LU !-r III f -44-/ II IL W \ T -7 / w b4- _ / " H P 1 1 VA L-oeL- ft 3< \ * c Q 4 4 6 6 i 1 i • •Jo .8 I. a 3. T 'I i : I :r D l E..Q O (1 FRACj TIC M 66 of debt exceeds the average cost of equity, K e must s tar t to dec l ine . Furthermore, once K\ exceeds the average cost of c a -p i t a l , the average cost of debt is greater than the average cost of equity. This behavior implies that, as debt is i n t ro -duced into the capi ta l s t ructure, thereby Increasing f i n a n -c i a l r i s k , equity holders require a l inear increase in their y i e l d . As greater amounts of debt are used, the compensation for f i nanc i a l r i sk increases but at a decreasing rate. Then a point is reached at which the compensation required dec-reases aasmore debt is employed. F i n a l l y , the senior claim is c ap i t a l i zed at a higher rate than the subordinate c la im. Modigliani and M i l l e r state that the shares of an overlevered company would be purchased by r i sk lovers, but i t is d i f f i -cu l t to conceive of an investor who is so fond of r i sk that he prefers both a lower y i e ld and more uncertainty to a higher y i e ld and greater cer ta in ty . We maintain that i f investors are averse to r i sk and are operating in perfect markMs, the interest rate on debt cannot be greater than the y ie ld on equity, for the debt holders are in a preferred pos it ion to the equity holders. 7. Robichek and Myers think that for r i sk-averse inves-tors and perfect markets, the cost of debt cannot behave in such a way as to force the equity c a p i t a l i z a t i o n rate to de-c l i n e . They state that i f the company adds an increment of debt,A L, the interest rate on th is addit ional debt cannot be greater than the expected return required by stockhol -ders as long as the debt holders are in a preferred pos i t ion . 67 to the stockholders with regards to a l l cash flows and assets of the company. However, an analysis of the i r argument reveals that they define the marginal cost of the increment of debt, A L, as being the interest paid only on that increment of debt. They maintain that addit ional increments of debt do not i n -crease the cost of ex is t ing debt. When we include a compo-nent of marginal cost to cover the increased cost of debt a l -ready held by the company, we note from Figure II that the average cost of debt is s t i l l much lower than the average cost of equity when the cost of equity s tarts to dec l ine. Also, even i f we accept Robichek and Myers' de f i n i t i on of marginal cost , we s t i l l cannot explain the behavior of equi -ty holders. From Figure II, we note that as soon as Kj i n -creases, K e increases at a decreasing rate. Robichek and Myers provide no explanation for th i s paradoxical behavior of equity holders. The i l l o g i c a l behavior required by the Modigliani and M i l l e r model tends to support the t rad i t i ona l pos i t ion which maintains that both debt and equity c a p i t a l i z a t i o n rates are establ ished independently by the markets. Figure II gives the cost of funds and the value for a f i rm with the same earnings and debt charges as used in F i -gure I, but paying a 50% corporate income tax. As in Figure I, the average cost of equity is observed to decrease once the marginal cost of debt exceeds i t . Also, the average cost 68 of debt, equity, and cap i ta l again intersect at one point. However, when corporate tax is introduced into the ana lys i s , the value of the f i rm is seen to increase as debt is i n t r o -duced into the cap i ta l s t ructure. The average cost of c a p i -tal is observed to decrease for small increments of debt, reach a minimum where the marginal cost of debt is equal to the average cost of c a p i t a l , and then increase. The data for th is f i gure are given in Appendix II, except for the marginal cost of debt which is determined from the net income model of Appendix IV. Figure III shows the influence of tax rate on the value of the f i rm for d i f fe rent debt-equity r a t i o s . The cost of debt is once again given by the funct ion: Kj = .05 + 5 x 10-9 (|_ - A ) 3 Data is p lotted for a f i rm with expected earnings of $75, and paying tax at rates of 0%, 30%, 50% and 70%. In a l l cases, the e f fec t of corporate income tax is to increase the value of the f i rm as the amount of debt in the cap i ta l structure is increased. It is obvious that i f we resort to an a l l -debt cap i ta l s t ructure, the value of the f i rm w i l l be equal to $1,071 (the value of the f i rm when there is no tax) , regardless of the preva i l ing corporate tax rate. The impl icat ion of Modigliani and M i l l e r ' s model is to suggest that the f irm use the maximum amount of debt pos-s i b l e . Also, from Figure III, we observe that the percentage 1 1 l n i • • P \ \| A M n u T rF «- 3-( D T Hf I < \ J L _ I \ I y I n [ j 1 < -— r- I &< )f \- \-Fi I-lie M •fi « -f h \-ri \ j r =r - 1 :t J1 1 )f n HI "F r r f If (i h r -O ic >e — -9 € e 3 C 8 5 (Y u-1 1 1 _> UJ T i 1 1 U . J r i | | U l < C -1^ r (J 3 OO • 1 1 D [ ^ . o 1 1 II :c D i L JVT Y 3 r FR* T IO ) 70 i n c r e a s e i n v a l u e f o r any g i v en i nc rement of debt i n c r e a s e s w i t h i n c r e a s i n g tax r a t e s . I f M o d i g l i a n i and M i l l e r ' s model p r o v i d e s a v a l i d d e s c r i p t i o n o f i n v e s t o r b e h a v i o r , we would e xpec t c o r p o r a t i o n s to i n c r e a s e t h e i r use of debt as c o r p o -r a t e income tax i n c r e a s e s . In p r a c t i c e , the pa s t 50 y e a r s have seen a g r e a t i n c r e a s e i n c o r p o r a t e tax but a reduced use of debt f i n a n c i n g . Data f o r F i g u r e I I I i s g i v en i n Appendix I I . Two o t h e r o b s e r v a t i o n s about M o d i g l i a n i and M i l l e r ' s h y p o t h e s i s can be made f rom F i g u r e s II and I I I . F i r s t , s i n c e use of debt f i n a n c i n g does i n c r e a s e the va l ue of the f i r m , the r e s u l t s of t he M o d i g l i a n i and M i l l e r h y p o t h e s i s and the t r a -d i t i o n a l h y p o t h e s i s are s i m i l a r . T h e r e f o r e , i t i s very d i f -f i c u l t to prove or d i s p r o v e e i t h e r h y p o t h e s i s based on e m p i -r i c a l e v i d e n c e . As M o d i g l i a n i and M i l l e r a d m i t ; " t h e tax advantages of debt f i n a n c i n g are somewhat g r e a t e r than we o r i g i n a l l y sugges ted and , to t h i s e x t e n t , t he q u a n t i t a t i v e d i f f e r e n c e between the v a l u a t i o n s i m p l i e d by our p o s i t i o n and '8, by the t r a d i t i o n a l view i s n a r r o w e d . " Our second o b s e r v a t i o n i s t h a t our t h e o r e t i c a l work, based on M o d i g l i a n i and M i l l e r ' s model and a s s u m p t i o n s , r e -f u t e s t h e i r own e m p i r i c a l e v i d e n c e . In t h e i r e m p i r i c a l work, i n wh i ch they r e g r e s s e d c o s t of c a p i t a I a g a i n s t the r a t i o of debt t o t o t a l v a l u e , they found t h a t the c o s t of c a p i t a l was not s i g n i f i c a n t l y dependent on the d e b t - t o - t o t a I - v a l u e r a t i o . T h e i r d e f i n i t i o n of c o s t of c a p i t a l was the r a t i o of t o t a l e a r n i ngs a f t e r t axe s t o the market va lue of a l l s e c u r i t i e s . 71 I f M o d i g l i a n i and M i l l e r d i d use a f t e r - t a x d a t a , our F i g u r e II s u gge s t s t h a t the c o s t o f c a p i t a l s hou l d dec rea se i n i t i a l -ly w i t h i n c r e a s e s i n the d e b t - t o - v a l u e r a t i o . A n a l y s i s of S u b j e c t i v e P r o b a b i l i t y D i s t r i b u t i o n s of E a r n i n g s In the p r e c e d i n g a n a l y s i s , we have c o n s i d e r e d X as the e xpec ted average va l ue o f the income to the f i r m . However, we know t h a t a l t h o u g h X i s the most l i k e l y v a l u e , many o t h e r l e v e l s of income are p o s s i b l e . We now w i s h to c o n s i d e r a p r o b a b i l i t y d i s t r i b u t i o n f o r the income of f i r m s i n a p a r t i -c u l a r r i s k c l a s s . U s i n g i dea s deve loped by B a r g e s , we can de te rm ine how f u t u r e expec ted economic c o n d i t i o n s may f a -vor the use of l e ve rage and how the e x i s t e n c e of l i m i t e d c o r -p o r a t e l i a b i l i t y can i n f l u e n c e the p r o b a b i l i t y d i s t r i b u t i o n i n such a way t h a t p e r s o n a l and c o r p o r a t e l e v e r a g e are not p e r f e c t s u b s t i t u t e s . .10. L e t us c o n s i d e r two c o r p o r a t i o n s , i d e n t i c a l i n a l l r e s p e c t s excep t t h a t Company A i s 100$ e q u i t y f i n a n c e d w h i l e Company B has $200 wor th of debt i n i t s c a p i t a l s t r u c t u r e on wh ich i t pays i n t e r e s t a t 5$ , so t h a t r e s i d u a l e a r n i n g s to e q u i t y h o l d e r s are reduced by $10. The h y p o t h e t i c a l i n v e s -t o r w i l l be c o n s i d e r e d to a r r i v e at the f o l l o w i n g s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n to d e s c r i b e income f rom f i r m s i n t h i s r i s k c l a s s . Tab I e V SUBJECTIVE PROBABILITY DISTRIBUTION OF CORPORATE INCOME Returns to Company Subjective Probabi l i ty D i s t r ibu t ion | 2 5 10$ 50 2 5 $ 15 30% 100 25% 125 10$ X =' | 7 5 In Company A, the d i s t r i bu t i on of shareholders ' re -turns are the same as the d i s t r i bu t i on of returns to the Company. In Company B, the shareholders ' returns are reduced by the interest payments of $ 1 0 . The fo l lowing table gives returns to shareholders in the two companies. TabIe VI DISTRIBUTION OF RETURNS TO SHAREHOLDERS FOR LEVERED & UNLEVERED FIRMS (SYMMETRICAL DISTRIBUTION) Qompany A Company B Subjective (unlevered) (levered) Probab i l i ty D i s t r ibut ion '0 $ 2 5 $15 1 5 0 40 2 5 $ 75 65 3 0 $ 100 9 0 2 5 $ 125 115 10$ 73 Company A - X* = $75 Company B - x" = $65 For the u n l e v e r e d and! l e v e r e d company, the a b s o l u t e range of the d i s t r i b u t i o n s are the same. However, the lower l i m i t . o f $25 f o r the u n l e v e r e d company r e p r e s e n t s a p o t e n t i a l l o s s of 67$ = 75 - 25L, and the upper l i m i t of $125 r e p r e s e n t s 75 a p o t e n t i a l g a i n of 67$ = 125 - 75 . Fo r the l e v e r e d company, 75 the p o t e n t i a l l o s s i s now 77$ = 65 - 1m5 , and the p o t e n t i a l 65 ga in i s now 77$ = 115- 65 . The p o t e n t i a l ga in s and l o s se s f o r Company B are g r e a t e r than those of Company A, making the s ha re s of Company B more " r i s k y " than those o f Company A. B e f o r e c o n s i d e r i n g how much compensa t ion e q u i t y h o l d e r s s h o u l d r e q u i r e f o r t h i s added r i s k , t h a t i s , how much h i g h e r w i l l K e be f o r the l e v e r e d f i r m than i t i s f o r the u n l e v e r e d f i r m , we s h a l l s tudy how i n v e s t o r s ' e x p e c t a t i o n s o f f u t u r e economic c o n d i t i o n s can i n f l u e n c e the d e s i r a b i l i t y of l e v e -r a g e . Suppose the i n v e s t o r t h i n k s t h a t the. p r o b a b i I i t y of i n f l a t i o n i n the f u t u r e i s g r e a t e r than the p r o b a b i l i t y of p r i c e s t a b i l i t y or d e f l a t i o n . I f i n f l a t i o n o c c u r s , debt i n -t e r e s t and p r i n c i p a l repayments remain f i x e d i n monetary t e rms . However, i t i s g e n e r a l l y assumed t h a t the money e a r -n ing s o f c o r p o r a t i o n s r i s e w i t h an i n c r e a s e i n genera l p r i c e l e v e l . Th i s i n c r e a s e w i l l p a r t i a l l y or c o m p l e t e l y o f f s e t any d e c l i n e i n the r e a l v a l ue o f the d o l l a r amount of the o r i g i n a l e a r n i n g s e x p e c t a t i o n s o f s h a r e h o l d e r s . The i n v e s t o r 74 w i l l now expec t h i g he r v a l u e s o f f u t u r e income, so 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 e a r n i n g s f o r a l l f i r m s in t h i s r i s k c l a s s w i l l become skewed, and c o u l d take the f o l l o w i n g f o r m : Tab le VII DISTRIBUTION OF RETURNS TO SHAREHOLDERS FOR LEVERED & UNLEVERED FIRMS (SKEWED DISTRIBUTION) Company A (un 1 evered) Company B ( I e v e r e d ) Sub j e c t i ve P r o b a b i l i t y D i s t r i b u t i o n $25 $15 10$ 50 40 20$ 75 65 25$ 100 90 20$ 125 115 15$ 150 140 10$ When p o s s i b l e s h a r e h o l d e r ga i n s and l o s se s are c a l c u -l a t e d f rom :the skewed d i s t r i b u t i o n f o r Company A, the p o t e n -t i a l l o s s i s 67$ = 75 - 25 , and the p o t e n t i a l ga in i s 75 100$ = 1 5 0 - 7 5 . Fo r Company B, the p o t e n t i a l l o s s i s 75 77$ = 6 5 - 1 5 and the p o t e n t i a l g a i n i s 115$ = 140 - 65 . 65 65 In both c a s e s , the p o t e n t i a l l o s s e s are the same as f o r the s ymmet r i c a l d i s t r i b u t i o n , but the p o t e n t i a l pe rcen tage g a i n f o r Company B i s g r e a t e r than t h a t f o r Company A. I f Com-pany B had employed an even h i g h e r l e v e l o f l e v e r a g e , the skewness of r e t u r n s to s h a r e h o l d e r s of Company B would be IS f u r t h e r i n c r e a s e d as would the p o t e n t i a l ga in s o f s t o c k B compared to s t o c k A. I f a s t r o n g i n f l a t i o n a r y s e n t i m e n t p r e -v a i l s i n the minds o f s h a r e h o l d e r s , they w i l l f i n d t h a t l e v e -rage w i l l i n c r e a s e t h e i r p o s s i b l e r e t u r n s . I t i s obv i ou s t h a t , i f i n v e s t o r s e xpec t f u t u r e d e f l a t i o n , sha re s i n u n l e -ve red companies w i l l be more v a l u a b l e to them. In our p r e v i o u s examples f o r the l e v e r e d company, t h e r e was no chance of d e f a u l t on i n t e r e s t payments whether the e a r -n i ng s d i s t r i b u t i o n was s y m m e t r i c a l or skewed. Now, u s i n g a s y m m e t r i c a l d i s t r i b u t i o n of e xpec ted e a r n i n g s , we can show t h a t f o r h i g h l y l e v e r e d compan ie s , the e x i s t e n c e o f l i m i t e d p e r s o n a l l i a b i l i t y f o r e q u i t y h o l d e r s can r e s u l t i n a f a v o r a -b l y skewed d i s t r i b u t i o n o f e xpec ted r e t u r n s . Suppose Company B now has $1,000 of debt i n i t s c a p i t a l s t r u c t u r e . Then the d i s t r i b u t i o n of r e t u r n s i s as f o l l o w s : Tab le V I I I DISTRIBUTION OF RETURNS TO SHAREHOLDERS FOR LEVERED & UNLEVERED FIRMS EFFECTS OF LIMITED L IABIL ITY Company A (un I eve red ) Company B ( l e v e r e d ) Sub j e c t i ve P r o b a b i l i t y D i s t r i b u t i o n $25 $-25 10$ 50 0 25$ 75 25 30$ 100 50 25$ 125 75 10$ 76 For the shareholders in Company B, the maximum possible return is now $75. If l imited l i a b i l i t y did not ex i s t , the maximum possible loss would be $-25 and the maximum possible gain and loss would be symmetrical about the mean. With the existence of l imited personal l i a b i l i t y for equity holders, the maximum possible loss is $0. Thus, the negative region of the d i s t r i bu t i on of returns is truncated. The mean of the d i s t r i bu t i on must therefore be sh i f ted upward, and the pos s i -ble gain expressed as a percentage of this new mean is grea-ter than the possible loss. The existence of l imited l i a b i -l i t y l imits the shareholder 's possible loss, but does not l imit his possible gain, resu l t ing in a d i s t r i bu t i on which is favorably skewed. This skewed d i s t r i bu t i on does not ex is t when personal leverage is used as a subst itute for corporate leverage, so the two forms of leverage cannot be regarded as being ident ica l in r i s k . We now examine how leverage influences the d i s t r i b u -t ion of y ie lds to equity holders. By doing t h i s , we hope to determine the way in which the equity holders should react to the increased f inanc ia l r i sk resu l t ing from leverage. In the preceding part of our ana lys i s , we assumed that the interest required by the debt market was given by the funct ion Kj = .05 + 5 x 10~9 (L - 125) 3 . If Proposit ion II were to hold, the equity c a p i t a l i z a t i o n rate was uniquely determined by the cost of debt and the r i sk c lass to which the f i rm belonged. In this part of our ana lys i s , we want to de-termine i f the equity c a p i t a l i z a t i o n rate derived from Mo-d i g l i a n i and M i l l e r ' s P r o p o s i t i o n II i s a r e a l i s t i c d e s c r i p -t i o n of i n v e s t o r b e h a v i o r . More s p e c i f i c a l l y , s hou l d the e q u i t y c a p i t a l i z a t i o n r a t e i n c r e a s e a t a c o n s t a n t r a t e as debt i s f i r s t i n t r o d u c e d i n t o the c a p i t a l s t r u c t u r e and then i n c r e a s e a t a d e c r e a s i n g r a t e , and f i n a l l y dec rea se f o r s u c -c e s s i v l e y l a r g e r amounts of debt ? We use the f o l l o w i n g p r o c e d u r e . To p r o v i d e a d i r e c t compar i son w i t h the M o d i g l i a n i and M i l l e r mode l , we assume t h a t the debt market s t i l l r e q u i r e s an i n t e r e s t y i e l d g i ven by the f u n c t i o n Kj = .05 + 5 x 1 0 " 9 (L - 1 2 5 ) 2 . The e q u i t y i n v e s t o r i s assumed to a r r i v e at the f o l l o w i n g c o n c l u s i o n s c o n c e r n i n g f i r m s i n t h i s r i s k c l a s s . The i n v e s t o r e x p e c t s an average annual income o f X = $75- However, now he c o n s i d e r s the p o s s i b l e d i s p e r s i o n of t h i s income. He; c o n c l u d e s t h a t t h e r e i s a 10$ chance t h a t annual income w i l l be no g r e a t e r than $25 per y e a r . He a l s o c o n c l u d e s t h a t t h e r e i s a 10$ chance t h a t annual income w i l l exceed $125. The e x a c t f o rm of the s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n i s not r e q u i r e d in our a n a l y s i s ; however, we s h a l l assume t h a t the d i s t r i -b u t i o n i s s y m m e t r i c a l . S i n c e the e q u i t y i n v e s t o r i s d e a l i n g w i t h f i r m s i n the same r i s k c l a s s , a l l i n f l u e n c e o f bu s i ne s s r i s k has been removed f rom our a n a l y s i s and we c o n c e n t r a t e on l y on f i n a n c i a l r i s k . The e q u i t y h o l d e r i s now conce rned w i t h the p o s s i b i l i t y t h a t the f i r m w i l l d e f a u l t on i t s i n -t e r e s t payments. We sugges t t h a t some i n d i c a t i o n of the chance of d e f a u l t can be de te rm ined f rom the e a r n i n g s a c c -r u i n g t o e q u i t y h o l d e r s a f t e r the i n t e r e s t on debt has been 78 p a i d . As the amount of debt i n the c a p i t a l s t r u c t u r e i n c r e a -s e s , the r e s i d u a l e a r n i n g s to e q u i t y h o l d e r s d e c r e a s e s . A l s o , the t o t a l i n t e r e s t b i l l i n c r e a s e s , so the r a t i o of r e s i d u a l e a r n i n g s to s h a r e h o l d e r s d i v i d e d by the i n t e r e s t c o s t d e c r e a -ses a t an ever i n c r e a s i n g r a t e . We can make ad ju s tmen t s to our net income model and use i t to de te rm ine the da ta r e q u i -red f o r our a n a l y s i s . As we have s een , the M o d i g l i a n i and M i l l e r model f o r c e s a p a r t i c u l a r b e h a v i o r upon the i n v e s t o r . Then, to see i f t h i s b e h a v i o r i s r a t i o n a l , we can not use the M o d i g l i a n i and M i l l e r m o d e l , but must use a model wh ich w i l l p r o v i d e us w i t h the da ta on wh ich the i n v e s t o r bases h i s b e h a v i o r . S i n c e the net income model de te rm ine s the v a l ue of e q u i t y f rom the f u n c t i o n (40) s = (X - K i L ) (1 - t ) Ke t h i s model can be used to de te rm ine the r e s i d u a l e a r n i n g s a c -c r u i n g t o the e q u i t y h o l d e r s i f we s e t Ke = 1 . Then i n the da ta p r e s e n t e d i n Appendix IV, the column headed Va lue o f E q u i t y , on page 129 o f t h i s append ix g i v e s the b e f o r e - t a x r e t u r n s to s h a r e h o l d e r s i n s t e a d o f the va lue of the e q u i t y . Data are p r o v i d e d f o r ca se s i n wh i ch e xpec ted c o r p o r a t e e a r -n i ng s a re $ 2 5 , $ 7 5 and $ 1 2 5 . The b e f o r e tax e a r n i n g s of the s h a r e h o l d e r s i s shown i n F i g u r e IV f o r v a r i o u s amounts o f d e b t . From t h i s f i g u r e , we ob se r ve t h a t the r e s i d u a l e a r n i n g s to e q u i t y f i r s t d e c -rea se a t a l i n e a r r a t e and c o n t i n u e to do so as long as Kj i s c o n s t a n t . As soon as the amount of debt exceeds A and Kj s t a r t s to i n c r e a s e , the r e s i d u a l e a r n i n g s to e q u i t y d e c -i t T i r I J D l J \ ST I- J r t K > zl i f R ~F r< )1 3 S ( n I I 0 & I I 7 J-3- / _ i . j 3. _ i >Q r A SLGi •c r —--A B>- £ J: 5 L-l_ uJ b 1 ou. 1 1 0 >-h 1 y 1 II C p< M X IS 1 1 1 i r r < i I LU (V U .o u. lit k / Of 2 \ A © 1 |\ c 2 a /I b o 1 r: 1 3 T M_OJJ_N © •L 86 rease a t an i n c r e a s i n g r a t e . Mo reove r , s i n c e the t o t a l i n -t e r e s t b i l l i s c o n t i n u a l l y i n c r e a s i n g , the r a t i o o f r e s i d u a l e a r n i n g s to i n t e r e s t payments i s d e c r e a s i n g r a p i d l y . The f o l l o w i n g t a b l e g i v e s the r a t i o of r e s i d u a l b e f o r e - t a x e a r -n i ng s to e q u i t y h o l d e r s d i v i d e d by the t o t a l i n t e r e s t b i l l f o r e xpec ted e a r n i n g s o f $25, $75 and $125 and f o r v a r i o u s l e v e l s of debt i n the c a p i t a l s t r u c t u r e . Tab le IX INTEREST COVERAGE OF RESIDUAL INCOME TO EQUITY HOLDERS Amount of R a t i o of B e f o r e - T a x Income to E q u i t y H o l d e r s Debt D i v i d e d by I n t e r e s t Charges X = 25 x = 75 x = i ; 0 oo oO oO 50 9.0 29.0 49.0 1 00 4.0 14 .0 24 .0 1 50 2.3 9.0 15-6 200 1 .4 6.2 11.0 250 .7 4.0 7.4 300 .09 2.3 4.4 350 - 1.0 2.3 400 - .2 1 .0 450 — — .3 From Tab l e IX, when expec ted e a r n i n g s are $25, w i t h zero debt i n the c a p i t a l s t r u c t u r e , the i n t e r e s t cove rage i s i n f i n i t e . A f t e r $50 of debt has been i n t r o d u c e d i n t o the c a p i t a l s t r u c t u r e , t h e r e i s s t i l l enough a d d i t i o n a l income 8* to c o ve r i n t e r e s t 9 t i m e s . The nex t $50 of debt i n t r o d u c e d , making a t o t a l of $100 of debt i n the c a p i t a l s t r u c t u r e , r e -duces the i n t e r e s t coverage to 4.0. Th i s i s s t i l l a. f a i r l y s a f e l e v e l o f coverage and the e q u i t y h o l d e r f a c e s l i t t l e r i s k of d e f a u l t . However, c o n s i d e r h i s b e h a v i o r as d e s c r i b e d by M o d i g l i a n i and M i l l e r . From F i g u r e I, we obse rve t h a t K e i s a l i n e a r l y i n c r e a s i n g f u n c t i o n of the d e b t - e q u i t y r a t i o as long as the amount of debt i s l e s s than $125. T h i s i m p l i e s t h a t the i n v e s t o r , who i s a t t e m p t i n g to compensate f o r f i n a n -c i a l r i s k by i n c r e a s i n g h i s e q u i t y c a p i t a l i z a t i o n r a t e , a s s o -c i a t e s as much r i s k w i t h the f i r s t $50 o f debt wh ich reduces i n t e r e s t coverage f r om i n f i n i t y to 9.0 as he does w i t h the next $50 wh i ch reduces i t f rom 9.0 to 4.O.- We would expec t t h a t i n v e s t o r s would f i n d l i t t l e r i s k a s s o c i a t e d w i t h an i n -t e r e s t coverage o f 9.0 and p r obab l y would not r e q u i r e any a d -d i t i o n a l compensa t i on f o r b e a r i n g t h i s r i s k . The nex t i n c -rement of debt f u r t h e r i n c r e a s e s r i s k , but a coverage of 4.0 s hou ld not r e q u i r e much compensa t i on i n the fo rm of h i g h e r e q u i t y y i e l d s . The nex t $50 o f debt added i n c r e a s e s t o t a l debt to $150 and reduces i n t e r e s t coverage to 2.3. Now t h e r e i s a f a i r amount of f i n a n c i a l r i s k a s s o c i a t e d w i t h the c a p i -t a l s t r u c t u r e . S i n c e the amount o f debt has exceeded $125, debt h o l d e r s r e q u i r e i n c r e a s e d compensa t ion f o r t h e i r r i s k , so Ki s t a r t s to i n c r e a s e . We would expec t a n o t i c a b l e i n c -rea se i n K e , but M o d i g l i a n i and M i l l e r say t h i s i s not s o . I f K e does i n c r e a s e , i t i n c r e a s e s a t a d e c r e a s i n g r a t e , imp-l y i n g t h a t e q u i t y h o l d e r s now a s s o c i a t e l e s s r i s k w i t h each 82 s u c c e s s i v e i n c rement of d e b t . As can be seen f rom F i g u r e I, at even h i g h e r l e v e l s of l e v e r a g e , when f i n a n c i a l r i s k becomes ex t reme, Ke may t u r n downward and even be l e s s than the y i e l d r e q u i r e d when t h e r e i s no debt i n the c a p i t a l s t r u c t u r e . Mo-d i g l i a n i and M i l l e r m a i n t a i n t h a t t h i s may be caused by the r i s k - s e e k i n g i n v e s t o r who gambles on the f i r m ' s r e c e i v i n g the maximum p o s s i b l e e a r n i n g s of $ 1 2 5 r a t h e r than the m i n i -mum e a r n i n g s of $ 2 5 . However, as Tab l e IX shows, even i f e a r -n i ng s are $ 1 2 5 , a t h i g h l e v e l s of l e ve rage t h e r e i s s t i l l a good chance of d e f a u l t . We m a i n t a i n t h a t M o d i g l i a n i and M i l -l e r ' s e q u i t y h o l d e r who v iews the f i r s t few f i f t y d o l l a r i n -c rements to debt as be i ng e q u a l l y r i s k y and v iews f u r t h e r i n -c rements as d e c r e a s i n g the f i n a n c i a l r i s k i s a c t i n g • i r r a t i o -n a l l y . We must agree w i t h H i r s h l e i f e r who " f e e l s very easy i n a s s e r t i n g c o n v e x i t y of the i n d i f f e r e n c e c u r v e s ( d i m i n i s -h i n g ma r g i n a l r a t e of s u b s t i t u t i o n between c e r t a i n income and '11 . expec ted u n c e r t a i n i n c o m e ) . " We sugges t an e q u i t y c a p i -t a l i z a t i o n r a t e wh ich remains c o n s t a n t or i n c r e a s e s on l y s l i g h t l y w i t h the f i r s t few i n c rement s o f d e b t . Then, as more debt i s i n t r o d u c e d , i n c r e a s i n g the f i n a n c i a l r i s k , the e q u i t y c a p i t a l i z a t i o n r a t e i n c r e a s e s a t an ever i n c r e a s i n g r a t e to compensate f o r the i n v e s t o r ' s a v e r s i o n to t h i s i n c r e a s e d r i s k . In the f o l l o w i n g c h a p t e r , we w i l l s tudy the p o s i t i o n t aken by the t r a d i t i o n a l w r i t e r s who m a i n t a i n t h a t debt and e q u i t y market s i n d e p e n d e n t l y de te rm ine the f u n c t i o n s f o r K e and K j . 83 Footnotes: 1. Modigliani and M i l l e r , "The Cost of C a p i t a l , Corporation Finance and the Theory of Investment", p. 279 2. i b i d . 3. i b i d . 4. A. Barges, loc. c i t . 5. D. Durand, "The Cost of Capi taJ, Corporation Finance and the Theory of Investment: Comment", American Economic Review. Vo l . XLIX, No. 4, Sept. 1959, pp. 639-55 6. Robichek and Myers, Optimal Financing Decis ions. Prentice-H a l l , Englewood C l i f f s , N.J. , 1965, chapter 5. 7. i b i d . , pp. 34-36 8. Modigliani and M i l l e r , "Corporate Income Taxes and the Cost of C a p i t a l " , American Economic Review, Vo l . LI I I, No. 3, June 1963, p. 434 9. A. Barges, op. c i t . , chapter 6 10. Throughout this analysis we w i l l assume that the f i rm does not pay any corporate income tax. 11. J . H i r s h l e i f e r , "Risk, the Discount Rate and Investment Dec i s ions " , American Economic Review. Vo l . LI, No. 2, May 1961, p. 115 84-: CHAPTER V AN ANALYSIS OF THE TRADITIONAL HYPOTHESIS The T r a d i t i o n a l H y p o t h e s i s In t h i s c h a p t e r we s tudy the r e s u l t s o b t a i n e d u s i n g the t r a d i t i o n a l and net income mode l s . In these mode l s , K e and Ki a re de te rm ined by the e q u i t y and debt markets r e s p e c -t i v e l y . W h i l e the M o d i g l i a n i and M i l l e r model t r e a t s K e as the dependent v a r i a b l e , i n these mode l s , K 0 i s the dependent v a r i a b l e and i s de te rm ined f rom the we i gh ted average o f the c o s t s o f debt and e q u i t y i n the c a p i t a l s t r u c t u r e . The t r a -d i t i o n a l model c o n s i d e r s s m a l l i n c r e a s e s i n debt as c a u s i n g s l i g h t i n c r e a s e s i n the c o s t of bo th debt and e q u i t y . As subsequent amounts of debt are added, the c o s t of both debt and e q u i t y i n c r e a s e s a t an i n c r e a s i n g r a t e . F u n c t i o n s wh i ch meet these r e q u i r e m e n t s a r e : Kj = .05 + 1 x 1 0 ~ 9 L 5 K e = .07 + 1 x 10-9 L 5 The above two f u n c t i o n s were chosen as be i ng r e p r e s e n -t a t i v e of a mode ra t e l y r i s k y c o r p o r a t i o n wh i ch would have a maximum va l ue a t a d e b t - e q u i t y r a t i o of about 40% or a t a d e b t - t o t a l v a l u e r a t i o of about 29%. The company i s assumed to have an e xpec ted annual income of $75. The r e s u l t s o f the model f o r ca se s i n wh ich the company pays no c o r p o r a t e i n -come tax and i n wh i ch i t pays 50% c o r p o r a t e income tax are 85 | [ - i- i- 1 - \ / 1 - V 1 [)" r P 1 VI -| 1 >( =tf _! n 13 S -J 3 1 •1 j )7 £ —i i J.! :: K )"F II -F c__ 01 D-_ V _ I _l _F U 1 J u J - J !^  f h 1 III 1-r- -( 31 T Mf f Di _-[ 3-"-• -dr/hi J-,— IN -F r c •> i V i i t i l V ( ) - -( r •t- )- r »- ~ l T( :-( )T \ U K 13 < | - n 8 n c 1 AJE l l LU 1- s i 2 \l —:<5- P, In 6 .0 I— rvl -V | 1 i 1 =_ » — i 1 m <* H b_ i • r _.' r F -H- J d- <1 0 4 0 M K1 u> (I i | -v— r — t-- I.C X1 — -c r i | 0 •-1 . c > 31 c E —-D E LQ.O i n 11© -^AGT \c • •86 g i ven i n Appendix I I I . We s h a l l f i r s t a n a l y z e the case i n wh i ch the f i r m does not pay any c o r p o r a t e t a x . F i g u r e V shows the va lue o f the f i r m , the m a r g i n a l and average c o s t o f d e b t , the average c o s t of e q u i t y and the average c o s t o f c a p i t a l , a l l p l o t t e d a g a i n s t the d e b t - e q u i t y r a t i o . The r e s u l t s are not i n t ended to be r e a l i s t i c , but are p r e s e n t e d as an a i d i n d e t e r m i n i n g the r e l a t i o n s h i p between the m a r g i n a l c o s t o f debt and the a v e -rage c o s t of c a p i t a l . We note s e v e r a l d i f f e r e n c e s between F i g u r e V and F i -gure I. In both c a s e s , Kj i s an i n c r e a s i n g f u n c t i o n of the amount of debt i n the c a p i t a l s t r u c t u r e . In the t r a d i t i o n a l mode l , K e i s a l s o an i n c r e a s i n g f u n c t i o n of the amount of d e b t . I n s t ead of the va l ue of the f i r m be i ng c o n s t a n t , i t now i n c r e a s e s as debt i s i n t r o d u c e d , reaches a maximum at a d e b t - e q u i t y r a t i o of about .08, and then s t a r t s to d e c l i n e . The we i gh ted average c o s t of c a p i t a l i s no l onger c o n s t a n t . I t i s now s a u c e r - s h a p e d . As debt i s i n t r o d u c e d , the w e i g h -ted average c o s t of c a p i t a l d e c r ea s e s s l o w l y . When t h e r e i s no c o r p o r a t e income t a x , i t r eache s a minimum at the same d e b t - e q u i t y r a t i o a t wh ich the v a l ue of the f i r m i s m a x i m i -zed . I t then i n c r e a s e s s l o w l y as more debt i s added. One o t h e r o b s e r v a t i o n s hou l d be made about the average c o s t o f c a -p i t a l c u r v e . A l t h o u g h t h e r e i s a d i s t i n c t minimum, the cu r ve i s very f l a t . The model has been run u s i n g many o t h e r t ype s o f f u n c t i o n s f o r K e and K\ and i n a l l ca se s the a ve -rage c o s t c u r ve was on l y s l i g h t l y d i s h e d . I f the f u n c t i o n s 87 we have used are a t a l l r e a l i s t i c , t h i s s h a l l o w c u r v a t u r e w i l l make i t a lmos t i m p o s s i b l e to use e m p i r i c a l da ta to t e s t the v a l i d i t y o f the M o d i g l i a n i and M i l l e r h y p o t h e s i s o r the t r a -d i t i o n a l h y p o t h e s i s . We a l s o note t h a t the minimum c o s t of c a p i t a l o c c u r s where the ma r g i n a l c o s t of debt f i r s t exceeds the average c o s t o f c a p i t a l . Up to t h i s p o i n t , any use of debt s e r v e s to reduce the we i gh ted average c o s t of c a p i t a l , but beyond t h i s p o i n t , i t w i l l r a i s e i t . We s hou l d a l s o note t h a t the m a r g i n a l c o s t used i n t h i s case i s d i f f e r e n t f rom t h a t used i n the M o d i g l i a n i and M i l l e r a n a l y s i s . The mar-g i n a l c o s t o f debt i n t h i s model i n c l u d e s a term to compen-s a te f o r any change i n the v a l ue of e q u i t y . The ma r g i na l c o s t s o f debt f o r both the M o d i g l i a n i and M i l l e r model and the t r a d i t i o n a l model are g i v e n i n the da ta of Appendix I I I , where SO i n d i c a t e s the t r a d i t i o n a l c a l c u l a t i o n method and MM i n d i c a t e s the M o d i g l i a n i and M i l l e r c a l c u l a t i o n method. U s i n g the same f u n c t i o n s f o r Ke and K j , the da t a i s p l o t t e d i n F i g u r e VI f o r a f i r m pay i ng 50$ c o r p o r a t e income t a x . The f i r m now has a maximum va l ue a t a d e b t - e q u i t y r a t i o of 40$. However, when tax i s i n t r o d u c e d i n t o the a n a l y s i s , the minimum c o s t of c a p i t a l o c c u r s a t a d e b t - e q u i t y r a t i o of 20$, long be fo re the f i r m reache s i t s maximum v a l u e . We note t h a t m a x i m i z i n g the va lue of the f i r m i s not e q u i v a l e n t to m i n i m i z i n g the we i gh ted average c o s t of c a p i t a l when c o r p o -r a t e t a x e s e x i s t . However, the minimum average c o s t o f c a p i -t a l i s s t i l l t h a t v a l ue a t wh i ch the ma r g i n a l c o s t o f debt ( i n c l u d i n g a c o s t f o r the chang ing va l ue of e q u i t y ) i s equal 88 i ~i 1 - - y -• a *^ rr M H 3 V +> F 1 1 V f 3 3 -> L/ J 1 t J 1 j r 3 J \ h \ J U )F rf F F l/l' it / A 1 M( r T 1 Tf _L 1 IT :i r 1 r ) o) -I "_TIL J "\ i F \t y • u ' t R( -v* ( =4 >< M »-: / \ V •i ji i \ 1 VALA 1 1 o <:. ) < 1 III • cr: -t; Vi 1 1 I J E V- p \ ~>">: IX ) U'fd— —LuJ— hri I i UJ i — n n (/> /i . \-i •~r 7n i. 1 Li-M / X / / >. 1 1 1 1 I ! ' K kii> f I 1 -z )--1 1 i • \ .( 4 l.'o P IE I r :i :( T 1 B J IQ. R f i J ro PACTl bi J 89 to the average c o s t of c a p i t a l . The Net Income H y p o t h e s i s In the net income mode l , the c o s t s of debt and e q u i t y remain c o n s t a n t u n t i l a c e r t a i n amount of debt i s i n t r o d u c e d i n t o the c a p i t a l s t r u c t u r e . Only when t h i s l e v e l i s e x c e e -ded i s the f i n a n c i a l r i s k g r ea t enough to w a r r a n t an i n c r e a s e i n the c a p i t a l i z a t i o n r a t e s . So t h a t t h i s a n a l y s i s w i l l be c o n s i s t e n t w i t h t h a t of Chap te r IV, we use the same f u n c t i o n f o r Kj as was used i n the M o d i g l i a n i and M i l l e r mode l . Ki = .05 + 5 x 10-9 ( L - 125)3 A l s o , K e = .07 + 5 x 10~9 ( L _ 125)3 Aga in the f i r m i s assumed to have e xpec ted annual e a r -n i ng s of |75 and pay c o r p o r a t e income tax a t a r a t e o f 50%. Data f o r the above case and f o r the case i n wh i ch the f i r m does not pay any c o r p o r a t e tax are g i v e n i n Appendix IV. From the da t a f o r the case i n wh i ch the f i r m pays no c o r p o -r a t e t a x , the minimum c o s t of c a p i t a l and maximum va l ue of the f i r m o c cu r a t the same d e b t - e q u i t y r a t i o . The minimum c o s t of c a p i t a l i s equal to the ma r g i n a l c o s t of debt as i t was i n the t r a d i t i o n a l mode l . The da t a f o r the case i n wh ich the f i r m pays 50% c o r -p o r a t e tax i s g i v en i n F i g u r e V I I . As i n the t r a d i t i o n a l mode l , the v a l ue of the f i r m i n c r e a s e s as debt i s added, reaches a maximum, and then dec r ea se s w i t h the a d d i t i o n of s t i l l g r e a t e r amounts of d e b t . In t h i s c a s e , the cu rve i s more s y m m e t r i c a l than i t was i n the t r a d i t i o n a l mode l . The F i f i r r \ -i 4 r K I L V • - M T •N € to + Y •p e -T t E rs , 1 C F C iiF F L L P IL 1 V 1 t 1 h t h J h IV /* F L -b IC J - iE IE DF -1 E •C -L -I -Y B -I n_ c - ft n Q -q F -f e -r -a -t -i 7 -8 -x -) -i /_ r b 1 r i VA.L [tie- . r *i i u \TT_! \ r -2 1 > 2: c_ i~7i l l I—j u 1 1? i UJ / / W*i UJ ! / 1© i - / «/> ±1 o i X 1 / -LLi / 1 k „ D M /  / •— j B € / / 6 U la J ••< K I i ' b 1 i t r 5 5 1 ) _ 5 T 1 \: r: ;: E G 0 ACI •j 91 c u r v e s f o r Ke and Kj are c o n s t a n t u n t i I the amount o f debt e x -ceeds $125 (a d e b t - e q u i t y r a t i o of about .23). The ma rg i na l c o s t cu rve c o i n c i d e s w i t h the average c o s t of debt c u r ve as long as the c o s t of debt remains c o n s t a n t . When the c o s t of debt s t a r t s to i n c r e a s e , the ma r g i n a l c o s t of debt i n c r e a s e s a t a much g r e a t e r r a t e than i t d i d i n the t r a d i t i o n a l mode l . The we i gh ted average c o s t of c a p i t a l i s a minimum where i t i s equal to the m a r g i n a l c o s t of d e b t . The c o s t of c a p i t a l i s not a minimum at the d e b t - e q u i t y r a t i o where the va l ue o f the f i r m i s max im i zed . In t h i s mode l , however, the d e b t - e q u i t y r a t i o s a t wh i ch the c o s t of c a p i t a l i s m i n i m i z e d and a t wh i ch the va l ue o f the f i r m i s max imized are much c l o s e r than they were i n the t r a d i t i o n a l mode l . We a l s o note t h a t the net i n -come model r e s u l t s i n a more d i s h e d average c o s t cu r ve w i t h a more pronounced minimum. 92 CHAPTER VI CONCLUSIONS In t h i s s t u d y , we have a t tempted to de te rm ine i f the c o s t of c a p i t a l o f the f i r m s hou l d be u n a f f e c t e d by i t s c a p i -t a l s t r u c t u r e , as sugges ted by M o d i g l i a n i and M i l l e r , or i f the j u d i c i o u s use o f debt can r e s u l t i n a dec rea se i n the c o s t of c a p i t a l as sugges ted by the t r a d i t i o n a l w r i t e r s . We were f o r c e d to r e j e c t e m p i r i c a l methods as p rocedu re s f o r t e s t i n g the two c o n f l i c t i n g h ypo the se s . We found t h a t i t was d i f f i -c u l t , i f not i m p o s s i b l e , to d i v i d e cpmpanies i n t o groups i n wh ich b u s i n e s s r i s k was c o n s t a n t . Mo reove r , even i f these groups c o u l d be d e t e r m i n e d , the f i r m s i n a p a r t i c u l a r r i s k c l a s s would u s u a l l y have n e a r l y i d e n t i c a l c a p i t a l s t r u c t u r e s . I n s t e a d , we s t u d i e d the a s sumpt ions on which each mo-de l was ba sed . M o d i g l i a n i and M i l l e r proposed t h a t the a ve -rage c o s t of c a p i t a l , Ko , and the va l ue of the f i r m , V, were independent of the f i n a n c i a l s t r u c t u r e . From t h i s , they de -ve l oped P r o p o s i t i o n I I , wh i ch exp re s sed K e as a f u n c t i o n of the d e b t - e q u i t y r a t i o . They v a l i d a t e d these p r o p o s i t i o n s th rough t h e i r a r b i t r a g e argument. There are s e v e r a l f a c t o r s wh ich can impede the a r b i -t r a g e p r o c e s s . Lega l r e s t r i c t i o n s p r e v e n t many l a r ge groups of i n v e s t o r s f rom buy i ng on m a r g i n . F u r t h e r m o r e , even those i n v e s t o r s who can buy on marg in can borrow on l y 10$ of the t o t a l f unds r e q u i r e d . A f u r t h e r r e s t r i c t i o n to the a r b i t -93 rage p r oce s s o c c u r s i f i n v e s t o r s view p e r s o n a l b o r r o w i n g f o r marg in pu rchase s as be i n g more r i s k y than c o r p o r a t e b o r r o -w i n g . M o d i g l i a n i and M i l l e r m a i n t a i n t h a t a r b i t r a g e w i l l o c c u r , f o r p e r s o n a l or homemade l eve rage i s a p e r f e c t s ub s -t i t u t e f o r c o r p o r a t e l e v e r a g e . We sugges t t h a t t h i s c o n t e n -t i o n i s not c o r r e c t , f o r pe r s ona l l e ve rage r e s u l t s i n u n l i m i -ted p e r s o n a l l i a b i l i t y , w h i l e c o r p o r a t e l e ve rage r e s u l t s i n l i m i t e d p e r s o n a l l i a b i l i t y . A l s o , M o d i g l i a n i and M i l l e r n e g l e c t the p o s s i b i l i t y t h a t the new y i e l d r e l a t i o n s h i p s b rough t about by the a r b i t r a g e p r oce s s may not p r o v i d e s a t i s -f a c t o r y r i s k compensa t i on to the o r i g i n a l s h a r e h o l d e r s . They do not c o n s i d e r the p o s s i b i l i t y t h a t the o r i g i n a l s h a r e h o l -der s i n the unde r v a l ued company, upon hav i n g t h e i r e q u i t y y i e l d reduced by a r b i t r a g e o p e r a t o r s , may f i n d the sha re s of the o v e r v a l u e d company more d e s i r a b l e . B e f o r e a v a l i d model can be deve loped a l ong the l i n e s pursued by M o d i g l i a n i and M i l l e r , we need to know the t r a d e o f f the i n v e s t o r r e q u i r e s between h i g h e r y i e l d and i n c r e a s e d f i n a n c i a l r i s k . The M o d i g l i a n i and M i l l e r computer model i s run u s i n g a h y p o t h e t i c a l f u n c t i o n to d e s c r i b e the y i e l d r e q u i r e d by the debt market i n re sponse to i n c r e a s e s i n l e v e r a g e . I t s hou ld be emphas ized t h a t the y i e l d f u n c t i o n i s not based on any e m p i r i c a l e v i d e n c e . However, the f u n c t i o n does d e s c r i b e a r i s k - a v e r s e i n v e s t o r . A l t h o u g h the e x a c t d e s c r i p t i o n of b e h a v i o r may not be c o r r e c t , i t s hou l d be noted t h a t any o t h e r f u n c t i o n w i t h a p o s i t i v e second d e r i v a t i v e (a r i s k - a v e -r se i n v e s t o r ) w i l l g i ve r e s u l t s s i m i l a r to what we have o b -94 t a i n e d . From t h i s a n a l y s i s , we obse rve t h a t the e q u i t y c a p i -t a l i z a t i o n r a t e de te rm ined f rom P r o p o s i t i o n II cannot r e p r e -sen t r a t i o n a l b e h a v i o r on the p a r t o f e q u i t y h o l d e r s . As debt i s i n t r o d u c e d i n t o the c a p i t a l s t r u c t u r e , the e q u i t y h o l -de r s r e q u i r e l i n e a r l y i n c r e a s i n g compensa t i on f o r r i s k as long as the c o s t o f debt i s c o n s t a n t . As soon as the c o s t o f debt s t a r t s to i n c r e a s e , the y i e l d r e q u i r e d by the e q u i t y h o l d e r s i n c r e a s e s a t a d e c r e a s i n g r a t e . When the m a r g i n a l c o s t of debt exceeds the average c o s t of e q u i t y , any f u r t h e r i n c r e a -ses i n l e ve r age r e s u l t i n a dec rea se i n the r e q u i r e d y i e l d on e q u i t y . At s t i l l h i g h e r l e v e l s of d e b t , we have a paradox i n wh ich the s e n i o r c l a i m i s c a p i t a l i z e d a t a h i g h e r r a t e than the s u b o r d i n a t e c l a i m . From a n i a n a l y s i s o f the M o d i g l i a n i and M i l l e r h y p o t h e -s i s , we note t h a t t h e r e are many f a c t o r s p r e s e n t i n the r e a l w o r l d wh ich can impede the a r b i t r a g e p r o c e s s . However, i f we admi t t h a t a r b i t r a g e i s p o s s i b l e , then the b e h a v i o r of e q u i t y h o l d e r s as de te rm ined f rom P r o p o s i t i o n II i s i r r a t i o n a l . Mo reove r , an i n t r o d u c t i o n o f c o r p o r a t e tax i n t o the M o d i g -l i a n i and M i l l e r model r e s u l t s i n a c o n t i n u a l l y i n c r e a s i n g va l ue o f the f i r m f o r i n c r e a s i n g use of d e b t , and sugges t s the o p t i m a l c a p i t a l s t r u c t u r e to be one composed 100$ of d e b t . The t r a d i t i o n a l w r i t e r s t h i n k t h a t the c o s t of debt as a f u n c t i o n o f l e ve r a ge i s de te rm ined by i n v e s t o r s i n the debt ma r ke t . The c o s t of e q u i t y i s de te rm ined i n a s i m i l a r f a s h i o n f rom the p r e f e r e n c e s o f i n v e s t o r s i n the e q u i t y ma rke t . The 95 t r a d i t i o n a l w r i t e r s do not t h i n k t h a t any c r o s s r e l a t i o n s h i p e x i s t s between the c o s t of debt and e q u i t y to keep the average c o s t of c a p i t a l and v a l ue of the f i r m c o n s t a n t . In t h e i r a p -p r o a c h , Ke and Kj are de te rm ined i n d e p e n d e n t l y by the markets and the average c o s t o f c a p i t a l i s de te rm ined by w e i g h t i n g the c o s t o f debt and o f e q u i t y by t h e i r p r o p o r t i o n s i n the c a p i t a l s t r u c t u r e . M o d i g l i a n i and M i l l e r m a i n t a i n t h a t the r i s k a s s o c i a t e d w i t h income i n any r i s k c l a s s cannot be a l -t e r e d by chang i n g the c o m p o s i t i o n o f the f i r m ' s l i a b i l i t i e s . 1 . They s t a t e t h a t : " a g i v e n commodity cannot c o n s i s t e n t l y s e l l a t more than one p r i c e i n the marke t ; or more p r e c i s e l y , t h a t the p r i c e of a commodity r e p r e s e n t i n g a " b u n d l e " of two o t h e r commod i t i e s cannot be c o n s i s t e n t l y d i f f e -r e n t f rom the we i gh ted average of the p r i c e s of the two component s . " The t r a d i t i o n a l w r i t e r s contend t h a t due to market i m p e r f e c -t i o n s , the v a l u e of the whole w i l l depend on the c o m p o s i t i o n o f i t s p a r t s . Durand has shown how i n s t i t u t i o n a l r e s t r i c -t i o n s r e s u l t i n the c r e a t i o n of a superpremium wh ich i s a t t a -2. ched to h i gh grade c o r p o r a t e bonds. L i f e i n s u r a n c e compa-n i e s , pen s i on t r u s t s , e t c . , a l l f a c e l e g a l r e s t r i c t i o n s wh ich compel them to purchase h i gh grade c o r p o r a t e bonds. The d e -mand c r e a t e d by these r e s t r i c t i o n s r e s u l t s i n lower i n t e r e s t c o s t s than would occu r i n p u r e l y c o m p e t i t i v e m a r k e t s . The d e d u c t a b i I i t y of i n t e r e s t cha rge s on c o r p o r a t e debt means t h a t t h e r e i s a tax advantage to u s i n g d e b t . L i m i t e d l i a b i -l i t y p r o t e c t s the e q u i t y h o l d e r i n the l e v e r e d company, w h i l e no such p r o t e c t i o n i s o f f e r e d the i n v e s t o r who c r e a t e s p e r -sona l l e ve r age t h rough buy i ng on m a r g i n . The t r a d i t i o n a l 96 writers think that these and other re s t r i c t i on s w i l l hinder the arbitrage process and inval idate Proposit ion I and Pro-pos i t ion II which is based on i t . As Vickers notes, "the assumption of market perfect ion should not form the s tar t ing point of analysis or provide the 3. paradigm." Unfortunately, Modigliani and M i l l e r have s tu -died the special case of perfect competit ion. The general case involves an imperfectly competitive environment where the f i rm can discriminate against i t s sources of c a p i t a l , thereby inf luencing the e l a s t i c i t y of i t s interest cost func-t i on . In f a c t , Schwartz has developed a procedure by which the f i rm discr iminates sole ly against the suppl iers of debt 4. funds to influence i t s cost of debt. We think that due to the many di f ferences that exist between real world markets and perfect markets, the t r a d i t i o -nal model is l i ke ly to give more r e a l i s t i c results than the Modigliani and M i l l e r model. In add i t ion, we suggest that the behavior of the equity market predicted by the Modigliani and M i l l e r model is i r r a t i o n a l . Our analysis does not allow us to choose between the t rad i t i ona l and net income hypothe-ses, but th is is of minor importance since both models i n d i -cate that the f i rm can influence i t s cost of cap i ta l through a judicious choice of the amount of debt and equity in i t s cap i ta l s t ructure. Footnotes 1. Modigliani and M i l l e r , "The Cost of C a p i t a l , Corporation Finance and the Theory of Investment", p. 281 97 2. D. Durand, loc. c i t . 3. Douglas V ickers , " E l a s t i c i t y of Capital Supply, Monop-son i s t i c D i scr iminat ion, and Optimum Capita l S t ructure" , Journal of Finance. 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XXI, No. 4, December, I 966 1 0 3 APPENDIX I L IST OF SYMBOLS K e - e q u i t y c a p i t a l i z a t i o n r a t e Kj - debt c a p i t a l i z a t i o n r a t e K 0 - we i gh ted average c o s t o f c a p i t a l L - amount of debt i n the c a p i t a l s t r u c t u r e ( a t market v a l ue ) S - amount of e q u i t y i n c a p i t a l s t r u c t u r e ( a t market va lue ) V - ' ' t o t a l market v a l ue of company V L - a f t e r - t a x market va l ue of a l e v e r e d company V U - a f t e r - t a x market va l ue of an u n l e v e r e d company X - average annual b e f o r e - t a x e a r n i n g s of c o r p o r a t i o n X - e xpec ted average annual b e f o r e - t a x e a r n i n g s o f c o r p o r a -t i o n X - e xpec ted average annual a f t e r - t a x e a r n i n g s of c o r p o r a -t i o n Y - r e t u r n to i n v e s t o r f rom a g i v e n p o r t f o l i o of e q u i t y and debt 1.04 APPENDIX II MODIGLIANI AND MILLER MODEL D e s c r i p t i o n of Program The program u se r i s g i ven the c h o i c e o f t h r e e debt c a -p i t a l i z a t i o n f u n c t i o n s by c h o o s i n g a v a l ue f o r the v a r i a b l e SELECT. I f SELECT = 1, the f u n c t i o n used i s K i = a + b(L - A) SELECT = 3, Ki = a + b (L - A ) 2 SELECT = 5, K{ = a + b (L - A ) 3 The user s p e c i f i e s v a l u e s f o r a and b. The f o l l l o w i n g data i s a l s o s p e c i f i e d by the u s e r : XBAR = the e a r n i n g s to be c a p i t a l i z e d ( expec ted net o p e r a t i n g income b e f o r e t a x ) . TAX = average c o r p o r a t e tax r a t e . RHO = c a p i t a l i z a t i o n r a t e f o r e a r n i n g s o f f i r m s i n t h i s r i s k c l a s s . In the b e f o r e - t a x c a s e , i t i s the va l ue of K 0 a p p l i e d to the p a r t i c u l a r r i s k c l a s s . In the a f t e r - t a x c a s e , i t i s the r a t e a t wh i ch e a r n i n g s of u n l e v e r e d f i r m s i n the p a r t i c u l a r r i s k c l a s s are c a p i t a l i z e d . DELT = amount by wh ich the l e v e l o f debt in the c a -p i t a l s t r u c t u r e s h o u l d be i n c r e a s e d f o r the nex t c a l c u l a t i o n . ABAR = A = the I eve I o f debt below wh ich K i remains c o n s t a n t . A may be s e t equal to zero to get f u n c t i o n s s i m i l a r to those d e s c r i b e d i n the t r a d i t i o n a l h y p o t h e s i s . SENT = s e n t i n e l wh ich w i l l t e r m i n a t e program i f not 105 s e t equal to 0 . The pa rameter s are read o f f one da ta c a r d w i t h the f o r -mat and o r d e r as I i s ted be I ow.. Parameter Format End ing Column SELECT F 5 . 0 5 A E15 . 0 2 0 B E 1 5 . 0 3 5 XBAR F 5 . 0 40 TAX F 5 . 0 4 5 RHO F5 . 0 5 0 DELT F 5 . 0 5 5 ABAR F 5 . 0 60 SENT F 5 . 0 6 5 A sample da ta c a r d i s g i v en below, 0 o o o o o[a 1 2 3 4 5 6 111111 222222 3 3 3 3 D 3 5 5 5 D5 6 6 6 B 6 7 7 7 7 7 7 8 8 8 8 L 8 2'2 33 5  5 699 9 9 9999999 999 1 2 3 1 5 6 OPERATION 0 0 0 0 0 0 O f 8 9 10 II 12 13 14 11111 2222222 3 3 3 3 3 3 5 5 5 5 5 5 6 6 6 6 6 6 6 777777 K B 9 10 11 12 13 14 15 IBM 1420030 33 5. E-2 D uu • VARIABLE FIELD .50 .070 fO. 125 Li' U U L) 0 COMMENTS 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o Ho o !]o Go 0 0 !J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 i Q l 1 1 Q 1 1 1 1 1 1111 1 11 1 1 SEQUENCE d x  0 0 0 0 0 0 0 73 74 75 76 77 78 79 80 1 1 1 1 1 1 1 1 |2222L2222222222222222222222222222222222222Q2222222222222222222222 L33333333333333D3333333033Q333Q3333333Q3333D33333333333333333333 444444|4444444|44444444444444444444444444444444444444444444444444444444444444444 5 05 055555555555505055555055550555555555555550555555555555555555555 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 • 7 7 7 7 7 7 7 7 7 7 ] 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 999 99999999999999999 r 9999999999999999999999999999999999999 17 18 19 20 21 22 23 24 25 25 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 9 9 9 9 9 9 9 9 73 74 75 76 77 78 79 80 PROGRAM LISTING MODIGLIANI AND MILLER HYPOTHESIS 106 £ M O D I G L I A N I A N D M I L L E R M O D E L - E Q U I T Y C A P I T A L I Z A T I O N R A T E D E T E R M I N A T I O N C C O S T O F D E B T I S G I V E N B Y T H E M A R K E T - W E S P E C I F Y S H A P E f E A R N I N G S , T A X E S C C D E F I N E R E A L V A R I A B L E S R E A L K Q A T , L , K I B T , K 6 B T , K E A T  2 R E A D ( 5 ' , L O O ) S E L E C T , A , B , X B A R , T A X , R H G , D E L T , A B A R , S E N T C S E N T I N A L W I L L T E R M I N A T E P R O G R A M I F N O N - Z E R O I F ( S E N T . N E . 0 . ) G G T O 3 V B T = X B A R / R H O W R I T E ( 6 , 2 0 0 ) X B A R , T A X , A , B , V B T , S E L E C T , R H O , D E L T , A B A R , S E N T W R I T E ( 6 , 2 0 2 )  C I N I T I A L I Z E O R I G I N A L V A L U E S L = 0 . 0 I C O N T I N U E C D E T E R M I N E . W H I C H F U N C T I O N T O U S E F O R K I B T I F ( L - A B A R ) 1 0 , 1 0 , 1 1 1 0 K I B T = A  G O T O 1.2 I I C O N T I N U E I F;( S E L E C T . E Q . 1 . ) K I B T = A + B * ( L - A B A R ) I F , { S E L E C T . E Q . 3 . ) K I B T = A + B * ( L - A B A R ) * * 2 . I F ( S E L E C T . E Q . 5 . ) K I B T = A + B * ( L - A B A R ) * * 3 . 1 2 C O N T I N U E  C T H E F O L L O W I N G S T A T E M E N T S D E T E R M I N E K E B T ( E Q U I T Y B E F O R E T A X ) S ^ V B T - L W 1 = L / V B T W 2 = S / V B T W 3 = W 1 / W 2 K E B T = ( R H 0 - W 1 * K L B T ) / W 2  C T H E F O L L O W I N G S T A T E M E N T S D E T E R M I N E K E A T ( E Q U I T Y A F T E R T A X ) V A T = ( X . B A R * . ( l . - T A X ) ) / R H O + T A X * L S A T = V A T - L C D E T E R M I N E E A R N I N G S T O E Q U I T Y { E Q E A R N ) E Q B A R N = ( l . - T A X ) * ( X B A R - K I B T * L ) C K E A T ^ E A R N L N G S T O E Q U I T Y / V A L U E O F E Q U I T Y  K E A T = E Q B A R N / S A T W 1 A T = L / V A T W 2 A T = S - A T / V A T W 3 A T = W I A T / W 2 A T K 0 A T = W 1 A T * K I B T + W 2 A T * K E A T W R I T E ( 6 , 2 0 1 ) V A T , K O A T , W 3 , W 3 A T , L , S , S A T , K I B T , K E B T , K E A T  2 0 1 F O R M A T I I X , 1 F 9 . 3 , 1 F 1 0 . 6 , 2 F 1 0 . 6 , 3 F 1 0 . 3 , 3 F 1 0 . 6 ) C I N C R E M E N T D E B T B Y A N A M O U N T D E L T _ L - L + D E L T C C H E C K T O S E E I F E N O U G H V A L U E S H A V E B E E N C O M P U T E D I F ( W 2 - . . C O 0 1 ) 2 , 2 , 1 3 S T O P  PROGRAM LISTING (cont'd) MODIGLIANI AND MILLER HYPOTHESIS 1 0 7 ":Ys 1 0 0 F O R M A T ( F 5 . 0 , 2 E 1 5 . 0 , 6 F 5 . 0 ) 2 0 0 F 0 R M A T ( l H l ' i 3 7 X , . 2 5 H T H E V A L U E O F T H E F I R M F O R / 1 3 9 X , 2 3 H V A R Y I N G A M O U N T S O F D E B T / / 2 3 7 X , 2 7 H M O D I G L I A N I A N D M I L L E R M O D E L / / 3 1 5 X , 4 0 H T H E F I R M H A S D E T E R M I N I S T I C E A R N I N G S O F $ , F 5 . 0 /  4 1 5 X , 3 5 H T H E F I R M H A S A N A V E R A G E T A X R A T E 0 F , F 6 . 3 / 5 1 5 X , 4 1 H T H E C O S T O F D E B T I S G I V E N B Y T H E F U N C T I O N , 6 F 1 6 . 9 , I H + . , F 1 3 . U , 1 H L / 8 1 5 X , 3 7 H T H E V A L U E O F T H E F I R M B E F O R E T A X I S $ , F 1 0 . 3 / 9 2 0 X , 7 . H S E L E C T = , F 5 . 0 , 1 0 X , 4 H R H 0 = , F 6 . 3 , 5 X , 7 5 H D E L T = , F 5 . 0 , 5 X , 5 H A B A R = , F 5 . 0 , 5 H S E N T = , F 5 . 0 / / / )  2 0 2 F 0 R M A T < 4 X , 5 H V A L U E , 4 X , 7 H A T C O S T , 3 X . T H B T D E B T , 3 X , 7 H A T D E B T , 1 3 X , 5 H V A L U E , 5 X , 5 H V A L U E , 5 X , 5 H V A L U E , 5 X , 7 H A V E R A G E , 3 X , 7 H A V E R A G E , . 2 2 X . 7 H A V E R A G E / 4 X , 5 H A . F T E R , 6 X , 2 H 0 F , 6 X , 6 H E Q U I T Y , 4 X , 6 H E Q U I T Y , 3 6 X , 2 H 0 F , 6 X , 5 H B T 0 F , 5 X , 5 H A T O F , 6 X . 4 H C 0 S T , 5 X , 7 H C 0 S T B T . 2 X , 4 7 H C 0 S T A T / 5 X , 3 H T A X , 5 X , 7 H C A P I T A L , 7 4 X , 5 H R A T I Q t , 5 X , 5 H R A T 1 0 , 5 X , 4 H D E B T ,  5 5 X , 6 H E 0 U I T Y , 4 X , 6 H E 0 U I T Y , 4 X , 7 H O F D E B T , 2 X , 9 H 0 F E Q U I T Y , 6 1 X , 9 H 0 F E Q U I T Y / / / ) E N D _ " S E N T R Y T H E V A L U E O F T H E F I R M F O R V A R Y I N G A M O U N T S O F D E B T M O D I G L I A N I A N D M I L L E R M O D E L T H E F I R M H A S D E T E R M I N I S T I C E A R N I N G S O F $ 7 5 . T H E F I R M . H A S A N A V E R A G E T A X R A T E O F 0 . 5 0 0 T H E C O S T € F D B B T _ I S G I V E N B Y T H E ^ F U N C T I O N 0 . 0 5 0 0 0 0 0 0 + ^ 0 . 0 _ 0 _ q 0 0 Q p O 5 . ( L - _ - i a s > _ T H E V A L U E O F T H E F I R M B E F O R E T A X I S $ 1 0 7 1 . 4 2 9 S E L E C T ^ 5 . R H O = 0 . 0 7 0 D E L T = 1 0 . A B A R - 1 2 5 . S E N T = - 0 . V A L U E A T C O S T B T D E B T A T D E B T V A L U E A F T E R O F E Q U I T Y E Q U I T Y O F T A X C A P I T A L R A T I O R A T I O D E B T V A L U E V A L U E A V E R A G E ^ . A V E R A G E _ A . V E R A G J L _ B T O F " A T O F " C O S T C O S T B T C O S T A T E Q U I T Y E Q U I T Y O F D E B T O F E Q U I T Y O F E Q U I T Y 53 5 . 714 0 . 0 7 C 0 G 0 0 . cc-ocoo 0 . 0 0 0 0 0 0 0 . 000 1071 . 429 535 .714 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 7 0 0 0 0 5 4 0 . 714 " 0 . 0 6 9 8 1 5 0 . 0 09421 0 . 0 1 8 8 4 3 10 . 0 0 0 1061 . 429 530 . 714 0 . 0 5 0 0 0 0 0 . 0 7 0 1 8 8 0 . 070188 5 4 5 . 714 0 . 0 6 9 6 3 4 0 . 019 022 0 . 0 3 8 0 4 3 20 . 0 00 1051 . 429 525 .714 0 . 0 5 0 0 0 0 0 . 0 7 0 3 8 0 0 . 0 7 0 3 8 0 55 0 . 714 0 . 0 6 9 4 5 5 0 . 02 8'807 0 . 0 5 7 6 1 3 30 . 0 0 0 1041 . 429 520 .714 0 . 0 5 0 0 0 0 0 . 0 7 0 5 7 6 0 . 0 7 0 5 7 6 5 5 5 . 714 0 . 0 6 9 2 8 0 0 . 03 8 781 0 . 0 7 7 5 6 2 40 . 000 1031 . 429 515 .714 0 . 0 50000 0 . 0 7 0 7 7 6 0 . 070776 56 0 . 714 0 . 0 6 9 1 0 8 0 . 048951 0 . 0 9 7 9 0 2 50 . 0 00 1021 . 4 2 9 510 .714 0 . 0 5 0 0 0 0 0 . 0 7 0 9 7 9 0 . 070979 56 5. 714 0 .0'6 8939 0 . 0 59322 0 . 1 1 8 6 4 4 60 . 0 00 1011 . 429 505 . 714 0. 0 50000 0 . 0 7 1 1 8 6 0 . 0 7 1 1 8 6 5 7 0 . 714 0 .068773" " 0 . 0 6 9 9 0 0 " 0 . 1 3 9 8 0 0 70 .COO 1001 . 429 500 . 714 0 . 0 5 0 0 0 0 0 .071. 398 0 . 071398 5.75. 714 0 . 0 6 8 6 1 0 0 . 0*80692 0 . 1 6 1 3 8 3 80 . 000 991 . 429 495 . 714 0 . 0 50000 0 . 0 7 1 6 1 4 0 . 071614 .5 8 0 . 714 0 . 0 6 8 4 5 0 0 . 0 91703 0 .1.83406 90 .GOO 981 . 429 490 .714 0 . 0 5 0 0 0 0 0 . 0 7 1 8 3 4 0 . 0 71834 5 8 5 . 714 0 .0 ' 68293 0 . 1 0 2 9 4 1 0 . 2 0 5 8 8 2 100 . 0 00 971 . 429 485 . 714 0 . 0 5 0 0 0 0 0 . 0 7 2 0 5 9 0 . 072059 5 9 0 . 714 0 .0 r 68138 0 . •114 413 0 . 2 2 8 8 2 6 110 . 0 0 0 961 . 429 480 . 714 0 . 0 5 0 0 0 0 0 . 0 7 2 2 8 8 0 . 072288 59 5. 714 0 . 0 6 7 9 8 6 0 . 126126 0 . 2 5 2 2 5 2 120 .GOO 951 . 429 475 .714 0 . 0 50000 0 . 0 7 2 5 2 3 ._ 0 .072 52 3 _ 6 0 0 . 714 0 . 067 83 6 d . 13 8 08 8 " 0 . 2 7 6 1 7 6 " 1 3 0 . 0 0 0 " 941 . 429 4 70 .714 0 . 050001 0 . 0 7 2 7 6 2 0 .072762 6 0 5 . 714 0 . 0 6 7 6 9 1 0 . ,150307 0 . 3 0 0 6 1 4 140 . 0 0 0 931 . 429 465 . 714 0 . 0 5 0 0 1 7 0 .0 730 04 0 . 0 73004 6 1 0 . 714 0 . 0 6 7 5 5 3 0 . 1 6 2 791 0 . 3 2 5 5 8 1 .15 0 . 000 921 . 4 2 9 460 .714 0 . 0 50078 0 . 0 7 3 2 4 3 0 . 073243 6 1 5 . 714 0 . C67429 0 . 175549 0 . 3 5 1 0 9 7 160 . 0 0 0 911 . 429 455 .714 0 . 0 50214 0 . 0 7 3 4 7 3 0 . 073473 § 6 2 0 . 714 0 . 0 6 7 3 2 4 0 . 1 8 8 5 9 0 0 . 3 7 7 1 7 9 170 . 0 0 0 901 . 429 450 .714 0 . 050456 0 . 0 7 3 6 8 6 0 . 0 73686 6 2 5 . 714 0 . 0 6 7 2 4 3 0 . 2 01923 0 . 4 0 3 8 4 6 180 . 0 0 0 891 . 429 445 . 714 0 . 0 50832 . 0 .0 73 870 0 . 073870 6 3 0 . 714 0 . 0 6 7 1 9 4 0 . 215559 0 . 4311 18 190 . 0 0 0 881 . 429 440 .714 0 . 051373 0 . 0 7 4 0 1 5 0 . 07401.5 6 3 5 . 714 0 . 0 6 7 1 8 6 0 . 229508 0 . 4 5 9 0 1 6 200 . 0 0 0 871 . 429 435 .714 0 . 052109 0 . 0 74106 0 . 074106 6 4 0 . 7 14 0 . 0 6 7 2 2 6 0 . 243781 0 . 4 8 7 5 6 2 210 . 0 0 0 861 . 429 430 .714 0 . 053071 0 . 0 7 4 1 2 7 0 . 0 /• 41 2 7 645. 714 0. C67323 0. 258389 0 .516779 220 .000 851. 429 425 .714 0.054287 o: 074060 0. 074060 6 5C. 714 0. 067488 0. 273345 0 .546689 230 .000 841. 429 420 .714 0.055788 0. 0738 85 0. 073885 655. 714 0 . 067732 0. 288660 0 .577320 240 .000 831. 429 415 .714 0.057604 0 . 073578 0. 073578 bbO. 7 14 ' 0. 0'6 8064 0. "364 348" 0 .608696 250 .000 821. 429 410 .714 0.059766 " "o". 0 7 3115 0. 073115 665. 714 0. 068497 0. 320423 0 .640845 260 .000 811. 429 405 .714 0.062302 0. 072467 0. 072467 670. 714 0. 069043 0. 336898 0 .673797 270 .000 801. 429 400 .714 0.065243 0. 071603 0. 071603 6^75. 714 0. 069714 0. 353791 0 .707 581 280 .000 791. 429 395 .714 0.068619 0. 070488 0. 070488 '6 80. 714 0. 070524 0. '371 115 0 .742230 290 .000 781 . 429 390 .714 0.072461 0. 069087 0. 069087 685. 714 0. 071487 0. .388889 0 .7 77778 300 . 000 771. 429 385 .714 .0.076797 Q67357 _ 0 . 067357 690. 714 " ~0. 072616 0. 407129 0 .814259 310 .000 761. 429 380 .714 0. 081658 0. 06 52 54 0. 065254 69 5. 714 0. 073927 0. 425856 0 .85171.1 320 .000 751 . 429 375 .714 0.087074 0. 062729 0. 062729 70 0. 714 0. 075434 0. .445087 0 .890173 330 .000 741. 429 370 .714 0.093076 0 . 059729 0. 059729 705. 714 0.077152 0. 464844 0 ,929688 340 .000 731. 429 365 .714 0.099692 Ov. 056198 0. 056198 ' 7L0. 714 0. 079099 0. 485149 0 .970297 350 .000 721. 429 360 .714 0.106953 0. 052072 0. 052072 715. 714 0. 081290 0. 506024 1 .012048 360 . 000 711. 429 355 .714 __0. 114889 047285 0. 047285 7 20.714 0. 08 3 74 1 0. 527495 ~ 1 .054990 370 ~.000~ 701. 429 " 350 .714 0. 123531 b . 04176 3 0. 041763 " 725. 714 0. 086470 0. 549587 1 .099174 380 .000 691. 429 345 .714 0.132907 0. 035427 0. 035427 7 3G. 714 0. 08 9494 0. 572327 1 . 144654 390 .000 681. 429 '340 .714 0.143048 0. 02 8193 0. 028193 735. 714 c. 092831 • 0. 595745 1 . 191489 400 .000 671. 429 3 35 .714 0.153984 0 . 019967 0. 019967 7^0. 714 JD. 096499 0.619870 1 .239741 410 .000 661. 429 330 .714 0.165746 0. 010650 0. 010650 745.714 0. 100516 0. 64 4 73 7 1 .289474 420 .000 651. 429 325 .714 0.178362 0. 000135 0. 000135 750. 714 0. 104901 0. 670379 1 .340757 430 .000 641. 429 320 .714 0.191863 - 0 . 011694 - 0 . 011694 755. 714 0. 109673 0. 696833 1 .393665 440 .000 631. 429 315 .714 0.206279 - 0 . 024964 - 0 . 024964 76 0. 714 0. 114851 0. 724138 1 .448276 450 .000 621. 429 310 .714 0.221641 - 0 . 03 9 809 - 0 . 039809 7 6 5. 714 0. 120456 0. 752336 1 .504673 460 .000 611. 429 305 .714 0.237977 - 0 . 056375 - 0 . 056375 7 70. 714 0. 126506 _.Q. 781473 1 .562945 470 .000 601. 429 300 . 714 0.255318 _T .O . 07_4 82X _-o„. 074821 7 75. 714 0. 133021 0. 811594 1 .623188 480 .000 591. 429 295 .714 0.273694 - b . 095317 - 0 . 095317 7 80. 714 0. 14G023 0. 842752 1 .685504 490 .000 581. 429 290 .714 0.293136 - 0 . 118048 - 0 . 118048 .7 85. 714 0. 147532 0. 875C00 1 .750000 500 .000 571. 429 285 .714 0.313672 - 0 . 143213 - 0 . 143213 790. 714 0. 155568 0.908397 1 .816794 510 .000 561. 429 280 .714 0.335333 - 0 . 171028 - 0 . 171028 795. 714 0. 164153 0. 943005 1 .886010 52 0 .000 551. 429 275 .714 0.3 58149 - 0 . 201726 - 0 . 201726 800. 714 0. 173 308 0. 978892 1 .957784 530 .000 541. 429 270 .714 0.382151 - o . 235562 - 0 . 2355 62_. 80 5. 7 14 0. 183054 1. 016129 2 .032258 540 .000 531. 429 265 .714 0.407367 - 0 . 272808 - 0 . 2 728 08 810. 714 0. 193413 1. 054795 2 . 109589 550 .000 521. 429 260 .714 0.43 3828 - 0 . 313764 - 0 . 313764 815. 714 0. 204407 1. 094972 2 .189944 560 .000 511. 429 255 .714 0.461564 - 0 . 358752 - 0 . 358752 8 20. 714 0. 216059 1. 13 6752 2 .273504 570 .000 501. 429 250 .714 0.490606 - 0 . 408124 - 0 . 408124 O 825. 714 0. 228390 1. 180233 2 .360465 580 .000 491. 429 245 .714 0.520982 - o . 462264 - 0 . 462264 8:30. 714 0 . 241423 1. 225519 2 .451039 590 .coo 481. 429 240 .714 0.5 52 72 3 - o . 521586_ -p . . .52_1.5.86_ 8 35. 714 0. 255180 1. 272727 2 .545455 600 .000 471. 429 2 35 , .714 0.585859 - 0 . 586548 586548 840. 714 0. 269685 1. 321981 2 .643963 610 .000 461. 429 230, .714 0.620421 - 0 . 657646 - o . 657646 84 5. 714 0. 284961 1. 373418 2 .746836 620 .000 451. 429 225, .714 0.656437 - 0 . 735423 -0 . 735423 T H E V A L U E O F T H E F I R M F O R V A R Y I N G A M O U N T S O F D E B T M O D I G L I A N I A N D M I L L E R M O D E L T H E F I R M H A S D E T E R M I N I S T I C E A R N I N G S O F $ 75. T H E F I R M H A S A N A V E R A G E T A X R A T E O F 0.500 T H E C O S T O F D E B T I S G I V E N B Y T H E F U N C T I O N 0.05000000+ 0.00000000! L . . T H E V A L U E O F T H E F I R M B E F O R E T A X I S $ 1071.429 , S E L E C T = 5. R H O = 0.070 D E L T = 10. A B A R = - O . S E N T = - 0 . V A L U E A T C O S T B T D E B T A T D E B T V A L U E A F T E R ~ O F E Q U I T Y E Q U I T Y O F T A X C A P I T A L R A T I O R A T I O D E B T V A L U E V A L U E A V E R A G E A V E R A G E A V E R A G E _ B T O F A T O F C O S T C O S T B T C O S T A T E Q U I T Y E Q U I T Y O F D E B T O F E Q U I T Y O F E Q U I T Y 535- 714 0.070000 0. -C C 0 C 0 0 0. 00 0!00 0 0 .000 1071 . 429 535 .714 _ o . 0 50000 . 0 .070000 0. 070000. . _ 540. 7T4 0 .069815 0. 009421 0. 018843 10 .000 1061 . 429 530 .714 0. 0 50001 0 .070188 0. 070188 54 5 . 714 0 .069634 0. 019022 0. 038 04 3 20 .000 1051 .429 525 .714 0. 050008 0 .070380 0. , 070380 5 50. 714 0 .069456 0. 028807 0. 057613 30 .000 1041 . 429 520 .714 0. 050027 0 .070575 0. 070575 55 5. 714 0 .069283 0. 03 8 781 0. 077562 40 .000 1031 . 429 515 .714 0. 050064 0 .070773 0. 070773 56G. 714 0 .069114 0. 048951 0. 097902 • 50 .000 1021 . 429 510 .714 0. 050125 0 .070973 0. 070973 •56 5. 714 0 .068951 0. 059322 . 0. 118644 60 .000 1011 .429 505 .714 .0. 050216 0 .071174 0. 071174 " 570. 714 0 .068795 0. 069900 0. 139800 70 .000 1001 . 429 500 .714 0. 0 50 343 0 .071374 0. 071374 57 5. 714 0 .068646 0. 0'80692 0. 161383 80 .000 991 .429 495 .714 0. 050512 0 .071573 0. 07157 3 580. 714 0 .068507 0. 091703 0. 183406 90 .000 981 . 429 490 .714 0.050729 0 .071767 0. 071767 , 585. 714 0 ,0'68378 0. 102941 0. 205882 100 .000 971 . 429 485 .714 0. 051000 0 .071956 o . 071956 59 0. 71'4 0 .0'68262 0. 114 413 0. 228826 110 .000 961 . 429 480 .714 0. 051331 0 .072136 0. 072136 .5.9 5. 714 0 .068160 0. 126126 0. 252252 120 .000 951 . 429 475 .714 0. 051728 0 .072 3 05 . 0. 072305 60 0. 714 0 ,0'6 8 074 0. 13808 8 0. 276176 130 .000 941 . 429 470 .714 0. 052197 0 .072458 0. 07245 8 605. 714 0 .068006 0. 15 0307 0. 300614 140 .000 931 . 429 465 .714 0. 052744 0 .072594 0. 072594 — 610. 714 0 .067958 0. 162791 0. 325581 150 .000 921 . 429 460 .714 0. 053375 0 .072706 0. 072706 Q 615.714 0 .067934 0. 175549 0. 351097 160 .000 911 . 429 455.714 0. 054096 0 .072792 0. 072792 620.714 0 .067934 0. 188590 0. 377179 170 . 000 901 . 429 450 .714 0. 054913 0 .072845 o . 072845 .. 625. 714 0 .067962 0. 201923 0. 403846 180 .000 891 . 429 445 .714 0. 055832 o, .072861 0. 0728 61. 630. 714 0 .063021 0. 215559 0. 431118 190 . 000 881 . 429 440 .714 0. 056859 0 .072833 0. 0728 3 3 63 5. 714 0 .068112 0. 229508 0. 459016 200 .000 871 . 429 435 .714 0. 058000 o, .072754 0. 072754 640. 714 0 .068240 0. 243781 0. 487562 210 .000 861 . 429 430 .714 0. 059261 0, .072618 0. 072618 64 5.714 0.068407 0. 258389 0 .516779 220 .000 851. 429 425 .714 0.060648 0. 072416 0. 0724 16 650.714 0 .068616 0.273345 0 .546689 230 .000 841 . 429 420 . 714 0.062167 0. 072141 0. 072141 655.714 0 .068870 0. 288660 0 .577320 240, .000 831. 429 415 .714 0.063824 0. 071783 0. 071783 660.714 0 .069172 0. 3043.48 0 .608696 250 .000 - 821. 429 410 . 714 0.065625 0. 071332 0. 071332 665.714 0 .069527 0. 32 042 3 0 .640845 260 .000 811. 429 405 .714 0.067576 0. 070777 0.0707 77 670.714 0 .069936 0. 336898 0 .673797 270 .000 801. 429 400 . 714 0.069683 0. 070107 0. 070107 675.714 0 .070404 0. 353791 0 .707581 280 .000 791. 429 395 .714 0.071952 0. 069309 0. 069309 r6 8 0.714 0 .070935 0.3711L5 0 .742230 290 .000 781. 429 390 .714 0.074389 0. 068371 0. 0683 71 685.714 . 0.071531. 0. 38 8 8 89 0 .777778 300 .000 771. 429 385 .714 0.077000 0. 067278 0. 0672 7 8 6 90.714 0 .072197 0. 40712 9 0 .814 259 31.0 .000 761. 429 380 .714 0.079791 0. 066014 0. 066014 69 5.714 0 .072936 0. 425 85 6 0 .851711 320 .000 751. 429 37 5 .714 0.082768 0. 064563 0. 064563 700.714 0 .073753 0.445087 0 .8901 73 330 .000 741. 429 370 .714 0.085937 0. 062907 o. 062907 70 5.714 0 .074650 0. 464844 0 .929688 340 .000 731. 429 365 .714 0.089304 0. 061027 0 . 061027 710.714 0 .075633 0. 485149 0 .970297 350 .000 721. 429 360 .714 0.092875 0. 058902 0. 058902 715.714 0 .076704 0. 506024 1 .012048 360 .000 711. 355 .714 0.096656 0. 056511 0. 056511 720.71:4 13.07786 8 0.527495 1 .054990 370 .000 701 . 429 350 . 714 0.100653 0. 053831 0. 053831 725.714 0.079-130 0. 549587 1 .099174 380 .000 691. 429 345 .714 0.104872 0. 050835 0. 050835 730.714 0 .0 80493 0. 572327 1 .144654 390 .000 681. 429 340 .714 0.109319 0. 047497 0. 047497 735.714 0 .081961 0. 5.9574 5 1 . 191489 400 .000 671. 429 335 .714 0.114000 0. 043787 0. 043787 .740 .714 ..-0-8 3 5 39 0. 619870 1 .239741 410 .000 661. 429 330 .714 0.118921 0. 0396 75 0. 039675 74 5.714 0 .085232 0.644737 1 .289474 42 0 .000 651. 429 325 .714 0.124088 0. 035127 0. 035127 750.714 0 .087042 0. 670379 1 .340757 430 .000 641. 429 320 .714 0.129507 0. 030108 0. 030108 755.714 0 .088976 0.696833 1 .393665 440 .000 631. 429 315 .714 0.135184 0. 024578 0. 024578 760.714 0 .091037 0. .724138 1 .448276 450 .000 621. 4 29 310 .714 0.141125 0. 018496 0. 018496 76 5.714 0 .093230 0. 752336 1 .504673 460 .000 611. 429 305 .714 0. 147336 0. 011817 0. 011817 77 0.714 0 .095559 0. 7814 73 1 .562945 470 .000 601. 429 300 .714 0. 153823 0. 004495 0. 004495 775.714 0 .098028 0. 811594 1 .623188 480 .000 59 1. 429 295 .714 0.160592 - 0 . 003524 - 0 . 0035 2 4 7 80.714 0 .100644 0. 842752 1 .685504 490 .000 581. 429 290 .714 0.16 764 9 - 0 . 012294 - 0 . 012294 785.714 0 .103409 0.875000 1 .750000 500 .000 571. 429 285 .714 0.175000 - 0 . 021875 - 0 . 021875 790.714 0 . 10 6 329 0. 908397 1 .816 794 510 .000 561. 429 280 .714 0.182651 - 0 . 032332 - 0 . 032332 795.714 0 .109409 0.943005 1 .886010 520 .000 551. 429 275 .714 0.190608 - 0 . 0437.34 - 0 . 043734 '800. 714 0 . 112652 0. 978892 1 .957784 530 .000 541. 429 270 .714 0.198877 - 0 . 056157 - 0 . 0561.5 7 80 5.714 0 . 116065 1. 0161.29 2 .032258 540 . 000 531. 429 265 .714 0.207464 - 0 . 069681 - 0 . 069681 810.7 1-4 0 .119651 1. 054795 2 .109589 550 .000 521. 429 260 .714 0.216375 - 0 . 084396 - 0 . 0843 96 815.714 0 .123416 1. 094972 2 .189944 560 .000 511. 429 255 .714 0.225616 - 0 . 100395 - 0 . 100395 820.714 0 .12736 5 1. 136752 2 .273 504 570 .000 501. 429 250 .714 0.235193 - 0 . 1 17783 - 0 . 117783 825.714 0 .131501 1. 180233 2 .360465 580 .000 491. 429 245 .714 0.245112 - 0 . 136673 - 0 . 136673 ~ •830. 714 0 .135831 1. 225519 2 .451039 590 .000 481. 429 240 .714 0.255379 - 0 . 157186 - 0 . 157186 II 8.35.714 0 .140359 1. 272727 2 .545455 600 . 000 471. 429 235 .714 0.266000 - 0 . 1 79455 - 0 . 179455 840.714 0 .145090 1. 321981 2 .643963 610 .000 461. 429 230 .714 0.276981 - 0 . 203625 -0 . 203625 845.714 0 .150029 1. 373418 2 .746836 620 .000 451 . 429 225 .714 0.288328 - 0 . 229856 - 0 . 229856 T H E V A L U E O F T H E F I R M F O R V A R Y I N G A M O U N T S O F D E B T M O D I G L I A N I A N D M I L L E R M O D E L T H E F I R M H A S D E T E R M I N I S T I C E A R N I N G S O F $ 7 5 . T H E F I R M H A S A N A V E R A G E T A X R A T E O F 0 . 3 0 0 T H E C O S T O F D E B T I S G I V E N B Y T H E F U N C T I O N _ _ 0 . 0 5 0 0 0 0 0 0 + 0 . 0 0 0_00 0 0 0 5 ( j - - I 2 . S ) 3 T H E V A L U E O F T H E F I R M B E F O R E T A X I S $ 1 0 7 1 . 4 2 9 S E L E C T = 5 . R H O = 0 . 0 7 0 D E L T = 1 0 . A B A R = 1 2 5 . S E N T = - 0 . A V E R A G E A V E R A G E A V E R A G E _ J C O S T C O S T B T C O S T A T O F D E B T O F E Q U I T Y O F E Q U I T Y Y A L U E _ _ A T C 0 S T B T D E B T A T D E B T V A L U E V A L U E V A L U E A F T E R " O F ' E Q U I T Y E Q U I T Y O F B T O F A T O F T A X C A P I T A L - R A T I O R A T I O D E B T E Q U I T Y E Q U I T Y 750 .000 0 .070000 0. CGCCOO 0 .000000 0 .000 107 1 . 429 750 .000 0. 050000 0 .070000 0. 070000 7,53 .ceo" 0 .069 920' 0. 009421 0 .013459 10 .000 1061 .429 743 . 000 0. 050000 6 .070188 0. 07 0188 756 .occ 0 .069841 0. 019022 0 .027174 20 .COO 1051 . 429 736 .000 0. 050000 0 .070380 0. 070380 7 59 .000 •o .069763 0. 028807 0 .041152 30 .000 1041 . 429 729 .000 0. 050000 0 .070576 0. 070576 762 .000 0 .069685 0. 03 8 781 0 .055402 40 .000 1031 . 429 722 .00 0 0. 050000 0 .070776 0. 070776 7 65 .CCO 0 .069608 0. 048951 0 .069930 50 .'000 1021 .429 715 . COO 0. 050000 0 .070979 0. 070979 768 .coo 0 .069531 0. 059322 0 .084746 60 .000 1011 . 429 708 . 00 0 0. 050000 0 .071 186 0. 071186 771 .000 0 .069455 0. 069900 0 .099 85 7 70 .000 1001 . 429 70.1 .000 0. 050000 0 .071398 0. 071398 ' 7 74 .CCO 0 .069380 0. 080692 0 .115274 80 .000 991 . 429 694 .000 0. 050000 0 .071614 0. 071614 77 7 • COO 0 .069305 0. 091703 0 .131004 90 . 000 981 . 429 687 . 000 0. 050000 0 .071834 0. 071834 180 .000 0 .069231 0. 102941 0 .147059 100 . 000 971 . 429 680 .000 0. 050000 0 .072059 0. 072059 783 .coo 0 .069157 0. 114 413 0 .163447 110 . 000 961 . 429 673 .000 0. 050000 0 .072288 0. 072288 786 .CCO 0 .069084 0. 126126 0 .180180 120 .000 951 . 429 666 .00 0 o. 050000 0 .072523 0. 072523 789 .000 0 .069011 d. '13 8 088 " 0 .197269 130 .000 """941 . 429 659 .000 0. 0 50001 o" .072762 0. 072762 79 2 .000 0 .0'68940 0. 150307 0 .214724 140 .000 931 . 429 652 .000 0. 050017 0 .073004 0. 073004 79 5 .coo 0 .068872 0.162791 0 .232558 150 .000 921 . 429 645 .00 0 0.05C078 0 .073243 0. 073243 -798 .CCO 0 .0'68810 0. 175549 0 .250784 160 .000 91 1 . 429 638 .000 0. 050214 0 .073473 0. 073473 -801 .000 0 .068756 0. 188590 0 .269414 170 . 000 901 . 429 631 .000 0. 050456 0 .073686 0. 073686 804 .CCO 0 .068713 0. 201923 0 .288462 180 .000 891 . 429 624 .000 0. 050832 0 .073870 0. 073870 ~~ 8 07. CCO 0 .068684 0. 215559 0 .307942 190 . 000 831 . 429 617 .000 0. 051373 0 .074015 0. 074015 810 .OCO 0 .068675 0. 229508 0 .327869 200 .000 871 . 429 610 .000 o. 052109 0 .074106 0. 074106 813 .CCO 0 .068688 0. 243781 0 .348259 210 .000 861 . 429 603 .000 0. G53071 0 .074-1? 7 0. 074-127 816.000 0 .068729 0.258389 0 .369128 220. .000 851. 429 596, .000 0.054287 0 .074060 0. •074060 8 19.CCD 0.068303 0.273345 0 .390492 2.30, .000 841.429 589, .000 0.055788 0 .073885 0. 07 38 8 5 8 2 2. ooo 0 .068914 0. 288660 0 .412371 240, .000 831. 429 582, .000 0.057604 0 .073578 0. 073578 825. 000 0 .069 070 " b . 3 0434 8 ' 0 .434783 250, .000 821. 429 575 .000 0.0 59766 0 .073115 0. 07 3115 828. ceo 0 .069275 0. 320423 0 .457746 260 . 000 811. 429 568 .000 0.062302 0 .072467 0. 072467 8 31. 000 0 .069536 0. 336898 0 .481283 270 .000 801. 429 561 .000 0.065243 0 .071603 0. 071603 834. CCO 0 .069861 0. '353791 0 . 505415 280 .000 791 . 429 5 54 .000 0.068619 0 .070488 0. 070488 8:3 7. ceo 0 .070256 0. 371115 0 .530165 290 . 000 781. 429 547 . 000 0.072461 0 .069087 0 . 069087 840. 000 0 .070728 0. 388889 0 .555556 300 . 000 771. 429 540 .000 0.076797 0 .06.7 35 7 0. -Q673 57 843. CCO 0 .0 71286 " 0. 40 7129 0 .581614 310 .000 761. 429 5 33 .00 0' 0.0 81658 0 .06 5 2 54 0. 065254 846. 000 0.071938 0. 425856 0 .608365 320 .COO 751. 429 526 .000 0.087074 0 .062729 0. 062729 849. coo 0 .072691 0. 445087 0 .635838 330 .000 741. 429 519 .000 0.093076 0 .059729 0. 059729 852. 000 0 .073555 0. 464844 0 .664063 340 .000 731. 429 512 .000 0.099692 0 .056198 0. 056198 855. GCO 0 .074538 0. 485149 0 .693069 350 .000 721. 429 505 .000 0.106953 0 .052072 0. 052072 8 58. 000 0 .075650 0. .506024 0 .722892 360 .000 711. 429 498 .000 . 0.114889 0 .047285 0. .0472 85 861. coo 0 .076901 b'. 52.7495 0 .753564 37~0 .000 701. 429 491 .000 0.123531 6 .041763 0. 041763 864. CCO 0 .078300 0. 549587 0 .785124 380 .000 691. 429 484 .000 0.132907 0 .035427 0. 03 54 2 7 867.COO 0 .079858 0. 572327 "0 .817610 390 .000 681. 429 477 .000 0.143048 0 .028193 0. 028193 8 70. GCO 0 .081584 0. 595745 0 .851064 400 .000 671. 429 470 .000 0.153984 0 .019967 0. 019967 -8-1-3. e+>e 0' .083490 0. 619870 0 .885529 410 .000 661. 429 4 63 .000 0.165746 0 .010650 0. 010650 8 76. OGO 0 .085586 0. "64473 7 0 .921053 42 0 . 000 651. 429 456 .000 0.178362 0 .000135 0. 000135 879. COO 0 .087884 0. 670379 0.957684 430 .000 641. 429 449 .000 0.191863 -0 .011694 -0. 011694 8 82. CCO 0 .090396 0.696833 0 .9954 7 5 440 .000 631. 429 442 .000 0.206279 -0 .024964 - 0 . 024964 885. GOO 0 .093132 0. 7 2 413 8 1 .034483 450 .000 621. 429 43 5 .000 0.2 21641 -0 .039809 - 0 . 039309 8 88. CCO 0 .096105 0. 752336 1 .074766 460 .000 611. 429 428 .000 0.237977 -0 .056375 - 0 . 056375 891. OCO 0 .099326 0. 781473 1 .116390 ..47 0 .000 601. 429 421 . 000 0..2 5 5318_. _T_Q .074821 - o . 074821 894. CCO 0 .102810 0.811594 1 .159420 480 .000 591. 429 414 .000 0.273694 -0 .095317 - 0 . 095317 8S7. GCO 0.106567 0. 842752 1 .203931 490 .000 581. 429 407 .000 0.293136 -0 .118048 - 0 . 118048 900. OCO 0 .110612 0. •875C00 1 .250000 500 .000 571. 429 400 .000 0.313672 -0 . 143213 - 0 . 143213 903. COO 0 . 1 14957 0. 908397 1 .297710 510 .000 561 . 429 393 .000 0.335333 -0 .171028 - 0 . 171028 ' 906. GOO 0 .119615 0. 943005 1 .347.150 520 .000 551. 429 386 .000 0.358149 -0 .201726 - o . 201726 909. 000 0 .124601 0. 978892 1 .398417 530 .000 541. 429 379 .000 0.382151 -0 .235562^ -0 . .2355 62 .9 12. OCO 0 .129927 1 . 016129 1 .451613 540 .000 531 . 429 372 .000 0.40 7367 -0 .272808 - 0 . 2 72 8 08 915. GOO 0 .135608 1. 054795 1 . 506849 550 . 000 521. 429 365 .000 0.433828 -0 .313764 -0 . 313764 918. GCO 0 .141659 1 . 094972 1 .564246 560 .000 511. 429 358 .000 0.461564 -0 .358752 - 0 . 358752 12 921. 000 0 .148093 1. 13 6752 1 .623932 570 .000 501. 429 351 .000 0.490606 -0.408124 - o . 408124 V>J' 924. GCO 0 .154925 1. 180233 1 .686047 580 . 000 491 . 429 344 .000 0.520982 -0 .462264 - o . 462264 .927. OCO 0 .162170 1 . 225519 1 . 750742 590 .000 481. 429 337 . COO 0.552723 -o .521586 .rO. 521586 9 30. OCO 0 .169844 1. 272727 1 .818182 600 .000 471. 429 330 .000 0.585859 -0 .586548 - 0 . 5 86543 933. CCO 0 .177960 1. 321981 1 .888545 610 . 000 461. 429 323 . 000 0.620421 -0 .657&46 -0. 657646 936. COG 0 .186536 1. 373418 1 .962025 620 .000 451. 429 316 .000 0.656437 -0 .735k23 -0 . 7 3542 3 T H E V A L U E O F T H E F I R M F O R V A R Y I N G A M O U N T S O F D E B T M O D I G L I A N I A N D M I L L E R M O D E L T H E F I R M . H A S D E T E R M I N I S T I C E A R N I N G S O F $ 7 5 . T H E F I R M H A S A N A V E R A G E T A X R A T E O F 0 . 7 0 0 T H I . _ c t L S T _ Q F D E B T I S G I V E N B Y T H E F U N C T I O N 0.05000000+ 0 . 0 0 0 0 0 0 0 O 5 ( L - \ Z $ f T H E V A L U E O F T H E F I R M B E F O R E T A X I S $ 1 0 7 1 . 4 2 9 S E L E C T = 5 . R H O = 0 . 0 7 0 D E L T = 1 0 . A B A R = 1 2 5 . S E N T = - 0 . V A L U E A T C O S T B T D E B T A T D E B T V A L U E _ V A L U E V A L U E A V E R A G E A V E R A G E A V E R A G E A F T E R O F E Q U I T Y E Q U I T Y O F " " B T O F A T O F C O S T " C O S T " B T C O S T A T T A X C A P I T A L R A T I O R A T I O D E B T E Q U I T Y E Q U I T Y O F D E B T O F E Q U I T Y O F E Q U I T Y 3 2 1 . 4 2 9 0 . 0 7 0 0 0 0 O . C C O C O O 0 . 0 0 0 0 0 0 0 . 0 0 0 1 0 7 1 . 4 2 9 3 2 1 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 7 0 0 0 0 3 2 8 . 4 2 9 0 . 0 6 9 5 7 4 0 . 0 0 9 4 2 1 0 . 0 3 1 4 0 4 1 0 . 0 0 0 1 0 6 1 . 4 2 9 3 1 8 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 1 8 8 0 . 0 7 0 1 8 8 3 3 5 . 4 2 9 0 . 0 6 9 1 6 5 0 . 0 1 9 0 2 2 0 . 0 6 3 4 0 6 2 0 . 0 0 0 1 0 5 1 . 4 2 9 3 1 5 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 3 8 0 0 . 0 7 0 3 8 0 3 4 2 . 4 2 9 0 . 0 6 8 7 7 3 0 . 0 2 8 8 0 7 0 . 0 9 6 0 2 2 3 0 . 0 0 0 1 0 4 1 . 4 2 9 3 1 2 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 5 7 6 0 . 0 7 0 5 7 6 3 4 9 . 4 2 9 0 . 0 6 8 3 9 7 0 . 0 3 8 7 8 1 0 . 1 1 2 9 2 7 1 4 0 . 0 0 0 1 0 3 1 . 4 2 9 3 0 9 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 7 7 6 0 . 0 7 0 7 7 6 3 5 6 . 4 2 9 0 . 0 6 8 0 3 6 0 . 0 4 8 9 5 1 0 . 1 6 3 1 7 0 5 0 . 0 0 0 1 0 2 1 . 4 2 9 3 0 6 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 0 9 7 9 0 . 0 7 0 9 7 9 3 6 3 . 4 2 9 0 . 0 6 7 6 8 9 0 . 0 5 9 3 2 2 0 . 1 9 7 7 4 0 6 0 . 0 0 0 1 0 1 1 . 4 2 9 3 0 3 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 1 1 8 6 0 . 0 7 1 1 8 6 3 7 6 . 4 2 9 0 . 0 6 7 3 5 4 0 . 0 6 9 9 0 0 0 . 2 3 3 0 0 0 7 0 . 0 0 0 1 0 0 1 . 4 2 9 3 0 0 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 1 3 9 8 0 . 0 7 1 3 9 8 3 7 7 . 4 2 9 0 . 0 6 7 0 3 3 0 . 0 8 0 6 9 2 0 . 2 6 8 9 7 2 8 0 . 0 0 0 9 9 1 . 4 2 9 2 9 7 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 1 6 1 4 0 . 0 7 1 6 1 4 3 8 4 . 4 2 9 0 . 0 6 6 7 2 2 0 . 0 9 1 7 0 3 0 . 3 0 5 6 7 7 9 0 . 0 0 0 9 8 1 . 4 2 9 2 9 4 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 1 8 3 4 0 . 0 7 1 8 3 4 3 9 1 . 4 2 9 0 . 0 6 6 4 2 3 0 . 1 0 2 9 4 1 0 . 3 4 3 1 3 7 1 0 0 . 0 0 0 9 7 1 . 4 2 9 2 9 1 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 2 0 5 9 0 . 0 7 2 0 5 9 3 9 8 . 4 2 9 0 . 0 6 6 1 3 5 0 . 1 1 4 4 1 3 0 . 3 8 1 3 7 7 1 1 0 . 0 0 0 9 6 1 . 4 2 9 2 8 8 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 2 2 8 8 0 . 0 7 2 2 8 8 4 0 5 . 4 2 9 0 . 0 6 5 8 5 6 0 . 1 2 6 1 2 6 0 . 4 2 0 4 2 0 1 2 0 . 0 0 0 9 5 1 . 4 2 9 2 8 5 . 4 2 9 0 . 0 5 0 0 0 0 0 . 0 7 2 5 2 3 0 . 0 7 2 5 2 3 4 1 2 . 4 2 9 "0 . 0 6 5 5 8 7 0 . 1 3 8 0 8 8 0 . 4 6 0 2 9 3 1 3 0 . 0 0 0 9 4 1 . 4 2 9 2 8 2 . 4 2 9 0 . 0 5 0 0 0 1 0 . 0 7 2 7 6 2 0 . 0 7 2 7 6 2 4 1 9 . 4 2 9 0 . 0 6 5 3 3 1 0 . 1 5 0 3 0 7 0 . 5 0 1 0 2 2 1 4 0 . 0 0 0 9 3 1 . 4 2 9 2 7 9 . 4 2 9 0 . 0 5 0 0 1 7 0 . 0 7 3 0 0 4 0 . 0 7 3 0 0 4 — . 4 2 6 . 4 2 9 0 . 0 6 5 0 9 5 0 . 1 6 2 7 9 1 0 . 5 4 2 6 3 6 1 5 0 . 0 0 0 9 2 1 . 4 2 9 2 7 6 . 4 2 9 0 . 0 5 0 0 7 8 0 . 0 7 3 2 4 3 0 . 0 7 3 2 4 3 X 4 3 3 . 4 2 9 0 . 0 6 4 8 8 7 0 . 1 7 5 5 4 9 0 . 5 8 5 1 6 2 1 6 0 . 0 0 0 9 1 1 . 4 2 9 2 7 3 . 4 2 9 0 . 0 5 0 2 1 4 0 . 0 7 3 4 7 3 0 . 0 7 3 4 7 3 4 4 0 . 4 2 9 0 . 0 6 4 7 1 9 0 . 1 8 8 5 9 0 0 . 6 2 8 6 3 2 1 7 0 . 0 0 0 9 0 1 . 4 2 9 2 7 0 . 4 2 9 0 . 0 5 0 4 5 6 0 . 0 7 3 6 8 6 0 . 0 7 3 6 8 6 4 4 7 . 4 2 9 0 . 0 6 4 6 0 2 0 . 2 0 1 9 2 3 0 . 6 7 3 0 7 7 1 8 0 . 0 0 0 8 9 1 . 4 2 9 2 6 7 . 4 2 9 0 . 0 5 0 8 3 2 0 . 0 7 3 8 7 0 0 . 0 7 3 8 7 0 4 5 4 . 4 2 9 0 . 0 6 4 5 4 8 "0. 2 1 5 5 5 9 0 . 7 1 8 5 3 1 1 9 0 . 0 0 0 " 8 8 1 . 4 2 9 2 6 4 . 4 2 9 0 . 0 5 1 3 7 3 0 . 0 7 4 0 1 5 0 . 0 7 4 0 1 5 4 6 1 . 4 2 9 0 . 0 6 4 5 7 2 0 . 2 2 9 5 0 8 0 . 7 6 5 0 2 7 2 0 0 . 0 0 0 8 7 1 . 4 2 9 2 6 1 . 4 2 9 0 . 0 5 2 1 0 9 0 . 0 7 4 1 0 6 0 . 0 7 4 1 0 6 4 6 8 . 4 2 9 0 . 0 6 4 6 8 7 0 . 2 4 3 7 8 1 0 . 8 1 2 6 0 4 2 1 0 . 0 0 0 8 6 1 . 4 2 9 2 5 8 . 4 2 9 0 . 0 5 3 0 7 1 0 . 0 7 4 1 2 7 0 . 0 7 4 1 2 7 475.429 0. 064910 0.258389 0 .861298 220 .000 851 .429 255.429 0.054287 0.074060 0.074060 482.429 0-. .065237 0. 273345 0 .911149 230.000 841 .429 252.429 0.055788 0.073885 0. 073885 489.429 0.065745 0. 288660 0 .962199 240 .000 831 .429 249.429 0.057604 0.073578 0.073578 4 96.429 0. 066392 0. 304348 1 .014493 250 .000 821 .429 246.429 0.059766 0.073115 0.073115 503.429 0. 067217 0. 320423 1 .068075 260.000 811 .429 243.429 0.062302 0.072467 0. 072467 510.429 0.068239 0. 336898 1 . 122995 270.000 801 .429 240.429 0.065243 0.071603 0. 071603 517.429 0. 069477 0. 353791 1 .179302 280.000 791 .429 237.429 0.068619 0.070488 0. 070488 5524.429 0. 070952 0. 371115 1 .237051 290.000 781 .429 234.429 0.072461 0.069087 0. 069087 531.429 0. 072686 0. 388889 1 .296296 300 .000 771 .429 231.429 0.076797 0.067357 0. 067357 53 8.429 0. 074699 0. 407129 1 .357098 310 .000 761 .429 228.429 0.081658 0.065254 0. 06 52 54 545.429 0. 077012 0. 425856 1 .419518 320 .000 751 . 429 225.429 0.087074 0.062729 0.062729 552.429 0. 079649 0. 445087 1 .483622 330 .000 741 . 429 222.429 0.093076 0.059729 0. 059729 559.429 0. 082632 0. 464844 1 .549479 340 .000 731 . 429 219.429 0.099692 0.056198 0. 056198 566.429 0. 085984 0. 485149 1 .617162 350 .000 721 .429 216.429 0.106953 0.052072 0. 052072 57 3.429 0. 089727 0. 506024 1 .686747 360 .000 711 .429 213.429 0.114889 0.047285 0. 047285 580.429 0. 093887 0. 527495 1 .758316 370 .000 701 . 429 210.429 0. 123531 0.041763 0. 041763 587.429 0.098486 0. 549587 1 .831956 380 .000 691 .429 207.429 0.132907 0.035427 0.035427 594.429 0. 103548 0. 572327 1 .907757 390 .000 681 .429 204.429 0.143048 0.028193 0. 028193 601.42 9 0.109100 0. 595745 1 .985816 400 .000 671 . 429 201.429 0.153984 0.019967 0. 019967 6 08.429 t>. 115164 0. 619870 2 .066235 410 .000 661 .429 198.429 0.165746 0.010650 0. 010650 615.429 0. 121766 0.644737 2 .149123 420 .000 651 .429 195.429 0.178362 0.000135 0. 000135 622.429 0. 128932 0. 670379 2 .234595 430 .000 641 .429 192.429 0.191863 -0.011694 -0 . 011694 629.4 29 0. 136686 0. 696833 2 .322775 440 .000 631 . 429 189.429 0.206279 -0.024964 - 0 . 024964 636.429 0. 145054 0. 724138 2 .413793 450 .000 621 .429 186.429 0.221641 -0.039809 -0 . 039809 643.429 0. 154063 0. 752336 2 .507788 460 .000 611 .429 183.429 0.237977 -0.056375 -0 . 056375 . _650.429 0. 163738 0. 781473 2 .604909 470 .000 601 . 429 180.429 0.255318 -0.074821 - 0 . 074321 657.429 0. 174105 0.811594 2 .705314 480 .000 591 .429 177.429 0.273694 -0.095317 -0 . 095317 664.429 0. 185190 0. 842752 2 .809173 490 .000 581 .429 174.429 0.293136 -0.118048 -0 . 118048 671.429 0.197020 0. 875000 2 .916667 500 .000 571 . 429 171.429 0.313672 -0.143213 -0 . 143213 678.429 0. 209623 0. 908397 3 .027990 510 .000 561 .429 168.429 0.335333 -0.171028 -0 . 171028 685.429 0. 223023 0. 943005 3 .143351 520 .000 551 .429 165.429 0.358149 -0.201726 -0 . 201726 692.429 0. 237249 0. 978892 3 .262973 530 .000 541 . 429 162.429 0.382151 -0.235562 - 0 . 235562 699.42 9 0. 252327 1. 016129 3 .387097 540 .000 531 .429 159.429 0.407367 -0.272808 -0 . 2 72808 ,706.429 0. 268284 1. 054795 3 .515982 550 .000 521 .429 156.429 0.433828 -0.313764 -0.313764 . 713.429 0. 285149 1. 094972 3 .649907 560 .000 511 .429 153.429 0.461564 -0.358752 -0 . 358752 720.429 0. 302947 1. 136752 3 .789174 570 .000 501 .429 150.429 0.490606 -0.408124 -0 . 408124 7 27.429 0. 321707 1. 180233 3 .934109 580.000 491 .429 147.429 0.520982 -0.462264 - 0 . 462264 734.429 0. 341455 1. 225519 4 .085064 590 .000 481 .429 144.429 0.552723 -0.521586 -0 . 521586 741.429 b. 362 221 T . 272727" 4.242424 600 .000 471 . 429 141.429 0.585859 -0.586548 -0 . 5 86548 748.429 0. 384031 1. 321981 4.406605 610 .000 461 . 429 138.429 0.620421 -0.657646 -0 . 657646 755.429 0. 406913 1. 373418 4 .578059 620 .000 451 . 429 135.429 0.656437 -G.735423 -0 . 735423 116 APPENDIX III TRADITIONAL MODEL Descr ipt ion of Program The user is given the choice of three functions for required y i e ld on debt and three functions for required y i e ld on equity by choosing values for the variables SELECT and TELECT. If SELECT 1 , the funct ion Ki = a + bL SELECT - 3, Kj = a + bL 2 SELECT = 5, Ki a + b L3 If TELECT 2, Ke = c + dL TELECT = 4, K e c + dL2 TELECT 6, K e = c + dL^ The user supplies the fo l lowing addit ional parameters: A = a in equation for debt function B = b in equation for debt funct ion C = c in equation for equity function D = d in equation for equity funct ion XBAR = the net operating income or expected average annual earnings of the f i rm before tax 117 TAX = average c o r p o r a t e tax r a t e DELT = SENT = amount by wh ich debt s hou l d be i n c r e a s e d f o r each s u c c e s s i v e c a l c u l a t i o n s e n t i n e l wh i ch w i l l t e r m i n a t e program i f . not s e t equal to z e r o . The pa ramete r s a re read o f f one da ta c a r d i n the f o r -mat and o r d e r l i s t e d be low. Parameter Format Endi ng ( SELECT F 5 . 0 5 TELECT F 5 . 0 10 A E 1 0 . 0 20 B E 1 0 . 0 30 C E 1 0 . 0 . 40 D E 1 0 . 0 50 XBAR F 5 . 0 . 55 TAX F 5 . 0 60 DELT F 5 . 0 65 SENT F 5 . 0 70 A sample da ta c a r d i s g i v e n below, 5. NAME 0 0 0 0 0 0 1 2 3 4 5 6 111111 11111 22222222222 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2  3 3 3 3 • 3 4 4 5 5 5 D5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 i 8 8 8 9 9 9 9 9 9 1 2 3 4 5 6 55 OPERATION 0 0 0 0 0 0 0 8 9 10 11 12 13 14 5 5 5 5 5 5 | J D6 6 6 6 6 6 77777777 0  8 8 8 9-9 9 9 9 9 9 9 8 9 10 II 12 13 14 U M N 2 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 o 3 0 0 0 0 0 0 0 0 5.E-2 i.E-9 7.E-2 1. E-9 50 10. VARIABLE FIELD 0 0 u I ] u COMMENTS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5B 59 60 61 62 63 64 65 66 67 68 69 70 71 72 i i i i i i i i i i " i i i i i i i i i i i i i i i i i i i ] n i i i i i i i i i i i i i i n i i i i i i i i i . 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 222[2222222222222222222022222222222222222222222222222222 0 3 3 3 3 3 3 3 3 3 C 3 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 3." 3 3 3 3 3 3 3 3 3 ^ 3 3 3 3 3 3 0 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 05 05 5 5 5 5 5 5 5 5 05 5 5 5 5 5 5 5 5 0 5 5 5 5 5 5 5 5 5 0 5 5 5 5 5 ^ 5 5.5 5 ~i 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 77777 77777777777777707777777777777777Q7777777777777777777 8 L'8 8 8 8 8 8 8 8 8 |J8 8 8 8 8 8 8 8 8 [ 8 8 8 8 8 8 8 8 8 :] 8 8 8 8 8 n i j 8 8;] 8 8 8 8 8 8!] 8 8 8 8 8 8 8 999999999999909999999999999999999H9999999999999999999999 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 4849 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 SEQUENCE    0 0    73 74 75 76 77 78 79 8 1 1 1 1 1 1 1 1 22222222 3 3 3 3 3 3 3 3 4         6 6 6 6 6 6 6 6 77777777 9 9 9 9 9 9 9 9 73 74 75 76 77 78 79 80 i 18 PROGRAM LISTING TRADITIONAL HYPOTHESIS C VALUE OF THE FTRiM - DETERMINISTIC MODEL C TRADITIONAL MODEL C MODEL ALLOWS CHOICE OF THREE TYPES OF FUNCTION FOR DEPENDENCE OF C DEBT AND EQUITY CAPITALIZATION RATES ON AMOUNT OF DEBT C WE SPECIFY INTEREST FUNCTIONStEARNINGStTAX RATE.AND INCREMENT C C DEFINE REAL VARIABLES REAL L,KESTOR»ICSTOR»KIAVG,KEAVG,KIMARW »INTCOS,KOAVG, 1KMARGL,KMARGS,KMARGR 2 READ(5,100)SELECT,TELECT,A,B,C,D,XBAR,TAX,DELT,SENT 100 FORMAT(2F5.0,4E10.0,4F5.0)  C SENTINAL WILL TERMINATE PROGRAM IF NON-ZERO IF(SENT.NE.O.)G0 TO 3 WRITE{6,200)XBAR,TAX,CD,A,B,SELECT,TELECT,DELT,SENT WRITE(6,202) C INITIALIZE ORIGINAL VALUES L=0.0  KESTOR=C ICSTOR=0.0 1 CONTINUE C DETERMINE WHICH FUNCTIONS TO USE FOR KIAVG AND KEAVG IF(SELECT.EQ.1.)KIAVG=A+B*L IFtSELECT.EQ.3.)KIAVG=A+B*L**2.  IF tSELECT.EQ.5.)KIAVG=A+B*L**3. IF(TELECT.EQ.2.)KEAVG=C+D*L IF(TELECT.EQ.4.)KEAVG=C+D*L**2. IF(TELECT.EQ.6.)KEAVG=C+D*L**3. C THE NEXT FUNCTIONS DETERMINE THE MARGINAL INTEREST RATE WHERE C KIMARW = D/DL OF (KIAVG*L )  C = KIAVG + D/DL OF KIAVG MULTIPLIED BY L IF(SELECT.EQ.1.)KIMARW=B*L+A+B*L IF(SELECT.EQ.3.)KIMARW=2.*B*L**2.+A+B*L**2. IF(SELECT.EQ.5.)KIMARW=3.*B*L**3.+A+B*L**3. C DETERMINE THE INTEREST COST OF DEBT INTCOS=KIAVG*L C DETERMINE THE AFTER TAX EARNINGS FOR EQUITY EQEARN=(XBAR-INTCOS)*(l.-TAX) C DETERMINE VALUE OF EQUITY S=EQEARN/KEAVG C DETERMINE TOTAL VALUE OF FIRM V=L + S C DETERMINE DEBT RATIO W1=L/V C DETERMINE EQUITY RATIO W2=S/V c DETERMINE THE DEBT EQUITY RATIO W3=W1/W2 PROGRAM LISTING (cont'd) TRADITIONAL HYPOTHESIS C DETERMINE AVERAGE COST OF CAPITAL KOAVG=Wl*KIAVG+W2*KEAVG C THE FOLLOWING FIVE STATEMENTS ARE USED TO DETERMINE THE MARGINAL COST C OF DEBT BY AN INCREMENTAL APPROACH. WE CONSIDER THE ADDED DEBT C CHARGES AND ALSO THE EFFECT OF ADDED DEBT ON THE COST OF EQUITY KMARGL=(INTCOS-ICSTOR)/DELT KMARGS=({EQEARN/KESTOR)*KEAVG-EQEARN)/DELT KMARGR=KMARGL+KMARGS KESTOR=KEAVG ICSTOR=INTCOS WRITE 16,201)V,L,S.KIAVG,KEAVG,W1.W3.K0AVG.KIMARW,KMARGR» 1W2 »KMARGL,KMARGS 201 F0RMAT(1X,1F9.3,2F10.3,10F10.6) C INCREMENT BY AN AMOUNT DELT L=L+DELT C CHECK TO SEE IF ENOUGH VALUES HAVE BEEN COMPUTED IF(W2-.00001)2,2,1 3 STOP 200 F0RMAT(1H1,37X,25HTHE VALUE OF THE FIRM FOR/ 1 39X,23HVARYING AMOUNTS OF DEBT// 8 42X,17HTRADITIONAL MODEL// 2 15X,40HTHE FIRM HAS DETERMINISTIC EARNINGS OF $,F5.0/ 3 15X,35HTHE FIRM HAS AN AVERAGE TAX RATE 0F.F6.3/ 4 15X,43HTHE COST OF EQUITY IS GIVEN BY THE FUNCTION,F14.9,1H+, 5 F13.11,1HL/15X,41HTHE COST OF DEBT IS GIVEN BY THE FUNCTION, 6 F16.9,1H+,F13.11,1HL/20X,7HSELECT=,F5.0,8X,7HTELECT=,F5.0,5X, 7 5HDELT=,F5.0,5X,5HSENT=,F5.0///) 202 FORMAT<4X,5HVALUE,5X,6HAM0UNT,4X,5HVALUE, 1 4X,7HAVERAGE,3X,7HAVERAGE,3X,8HFRACTION,4X,4HDEBT,4X, 2 7HAVERAGE,3X,8HMARGINAL,2X,8HMARGINAL,2X,8HFRACTION,2X, 3 8HMARGINAL,2X,8HMARGINAL/6X,2H0F,8X,2HQF,8X,2H0F,6X,4HC0ST,6X, 4 4HC0ST,8X,2H0F,6X,6HEQUITY,4X,4HC0ST,5X,7HC0ST MM,3X,7HC0ST SO, 5 6X,2H0F,4X,9HC0ST DEBT,IX,1OHCOST EQUTY/5X,4HFIRM,6X,4HDEBT, 6 5X,6HEQUITY,3X,7H0F DEBT,2X,9H0F EQUITY,4X,4HDEBT,6X,5HRAT10, 7 2X,10H0F CAPITAL,1X.7H0F DEBT,3X,7H0F DEBT,4X,6HEQUITY,3X, 8 7HP0RTI0N,3X,7HP0RTI0N///) END $ENTRY O THE VALUE OF THE FIRM FOR VARYING AMOUNTS OF DEBT TRADITIONAL MODEL THE FIRM HAS DETERMINISTIC EARNINGS OF 75.' """""' "  THE FIRM HAS AN AVERAGE TAX RATE OF 0.000 THE COST OF EQUITY IS GIVEN BY THE FUNCTION 0.07000000+ 0 . 0 0 0 0 0 0 0 01 L 3 THE COST ODDEST IS G I V E N B Y THE" F UN C T ION "'" 0". 05000000 + • 0 . GO 0000001 L 3 "SELECT= 5. TELECT= 6. DELT= 10. S E N T - - 0 . , VALUE _ AMOUNT VALUE- AVERAGE AVERAGE FRACTION DEBT AVERAGE MARGINAL MARG[NAL FRACTI ON OF OF OF ' COST "COST O F " "EQUITY" "COST COST MM" COST SO" OF'"" FIRM DEBT EQUITY OF DEBT OF EQUITY DEBT RATIO OF CAPITAL OF DEBT OF DEBT EQUITY." 1071.'429 0 .000 1071 .42 9 0 .050000 0. 070000 0 .000000 0 .000000 0.070000 0 . 050000 -0 .000000 1 .000000 • """10 74. 27 0 ' 1 C ."oco "1064 . 2 7 0 " 0 .050001' 0 .0 70001 0 .009309 0 . 009396 0. 06981 5" 0 .050004" 0 .050107 0 .990691 1077.020 20 . 000 1057 .020 .0 .050008 0.070008 . 0 .018570 0 .018921 0.069637 0 .050032. 0 .050755 0 .981430 1079.584 30 . 000 1049 . 584 0 .050027 0 .070027 0 .027788 0 .028583 0.069471 0 .050108 0 .052060 0 .972212 108 1.868 40 . GOO 1041 . 868 C .050064 0 .070064 0 .036973 0 . 038393 0.069325 0 .050256 0 .054032 0 .963027 1083.779 50 .CCO 103 3 .7 79 0 .050125 0 .070125 0 .046135 0 .048366 0.069202 0 .050500 0 . 056680 0 .95 3865 108 5.22 3 60 . COO 1025 . 22 3 0 .050216 0 .070216 0 .055288 0 .058524 0.069110 0 . 05 0 864 0 .060013 0 .944712 • """1086. 107" 7C .000 " 1016 ~. 107 " "d .050343" " 0 .0 70 34 3" 0 .064450" 0 . 068890 0.069054 0 .051372 0 . 06403 3 " 0 .93 55 50 1086.34 0 80 . o c o 1006 . 34 0 0 .050512 0 .070512 0 .073642 0 .079496 0.069039 0 .052048 0 . 06 87 43 0 . 926358 1085.83 5 90 . 000 995 .835 0 .050729 0 .070729 0 .082886 0 .090376 0.069071 0 .052916 0 .074141 0 .917114 1084.507 100 . 000 9C4 . 5C7 0 .051000 0 .0 7 1000 0 .092208 0 .1015 74 0.069156 0 . 054000 0 . 080221 0 . 907 7 92 1082.278 11C . 000 9 72 .278 0 .051331 0 .071331 0 . 101637 0 .113136 0.069298 0 . 05 5 3 24 0 .036973 0 .898 363 1079.0-77 120 . OCO 959 . 077 0 .051728 0 .071723 0 .111206 0 .125120 0.069504 0 .056912 0 .0943 8 2 0 . 88 87.94 "1074.837 130 . 000 " 94 4 .837 0 .052197" 0 .072197 " 0 . 120949" 0 . 137590" 0.069778 0 .05 3 7 88" 0 . 102428 0 .8 79051 1069.504 140 . 000 9 29 . 504 0 .052 74 4 0 .072744 0 .130902 0 . 15061 3 0.070126 0 . 060976 0 . 1.11084 0 .869098 ,106 3.0 32 150 .000 913 .032 0 . C53375 0 .073375 0 .141 106 0 .164288 0.070553 0 . 063500 0 . 120321 0 .858394 105 5.38 8 160 . CCO 895 . 388 0 .054096 0 .074096 0 .151 603 0 .178694 0.071064 0 . 066 384 0 .130103 0 .848397 1046.547 170 . 000 876 . 547 0 .0549 13 0 .074913 0 . 162439 0 . 193943 0.071664 0 .069652 0 .1403 8 9 0 . 83 7561 1036.502 180 . COO 8 56 . 502 0 .055832 0 .075832 0 . 173661. 0 .210157 0.072359 0 .073328 0 . 151133 0 .826339 102 5.2 54 190 .000" • 8 35 .254 0 .056359 0 .0 76859 0 .18 5 320 0 . 22 7476 0.073153 0 .07 74 36 0 .1.62287"" "o .814680 1012.821 200 .000 812 .821 0 .058000 0 .078000 0 .197468 0 .246057 0.074 051 0 .082000 0 .173799 0 . 802532 999.230 210 .COO 789 .230 0 . 05926 1 0 .079261 0 .210162 0 .266082 0.075058 0 .087044 0 .185612 0 .739838 o ro o V CM 0"! ro O J CO so s!- C O :n C C CO o CO ro C O 0 s P~! . - ^ LP — ro i co' m co 0 s CM CO CN 0^ CNJ CNJ 0 s p- C O —, 0 s CM C O . — i rH Is- o CP ro .—, CM ro, L O LO. cc S T o CO to CNJ P - O CP- L P CM c CM LP ai CNJ <f ro o rH CC a s O; O o CM IS- IN! 00 o —i 0- L P ,—i L P C O 0 s CO LP o uo -j •—« in uo 0 0 ; >cr r-- ro rH c- CC- -0 f—i P- <r rH C O tn CN' CP P— C O C O C O ro' C M p- p~ Is- P— Is- -•u o o O - o LP L P L P in <r -J' C O ro ro CM CM ,—1 o O. C o o o O o o o o o o o o O o o o O o O O o O CO o C ° i I O 1 o so IN! O CC CN! LP: ro o CP CP- <f ro •JO <r r- 0- PI ro rH <r o r- rH 0 s sr o sr 0s- (Nj C O <f a s vO C0 O .—1 00 o p- ro ,-o C M cl co o; CNI 0 s CN] Is- fN! so (NJ OvJ: rH p- rH CM • — i v0 0> C Q IT' 0' , — i o p- <NJ L 0 -o Is-: P - cr- CN! Is- 0 s rH •O co 0 s rH CN! ro ro r<" r>-. r>~ CM C M ,—1 0s- 00 . •-. <r Pv! o 0s- o <N] ro S T in IS- CO 0 s o .—. (>1 vf m Is- CO 0 s o rH CM CM ro •4" L O -o- Is-t\j CN! CM CNJ (Nj CNi CN! (Nj; ro C O ro ro ro ro ro ro •cr <r -cr <r --CT, •4-o o O O O o O C O o o o o o o O O o o o o o O c o,o CM C O SO O S T CN] CC o O <r CM CC -0 o sT CM1 co o -cj- C M co o *cr ,Nj' CO 0 s 0 s o o ro o LP, o v0 r- i—i o CM rH p- o CO U O CM ro o <r 0s o in o (NJ in ro Is- CO in o rH O CM LP < J <r CM o ro O r- LO ro CM ro CM CO in CN O co Is- Is- CO CP rH C O P—• o CM 0 s r- o LP vO CC o vT CP- U O 0 0 0 s 0 s O ,-H CNJ CN! ro L P o CO CP O CNI LP, o cc '—. CM x+ o 0 s rH ro 0 s c o •—i -"' -1 rH rH rH rH CM ( M CM CM CM CNi ro ro ro rO ro -cr St"! >t o o o o o o O o O o o o O O C O o o c O O O O o O o: O 0s- 0 s CNJ ro co o n in rH co CP CO CP celco ro p- •cr CP P - CO CM r\j CP IS- rH C O Is- a-- -0 rH rH p- C O o —1 ro O O rH C O ro CO CNI P- C M C O L O 0s <r ,-— (Ni CO S3 in o CO rH o ro CM CM vT co ro r-« o CM tn r—t CO CO o o r- CO O ro m Is- •0s CM sl- i O ro O 0 s ro Is- •—« LP. CP C O ro CP -CT co Is-' r- Is- co CO CO CO co CO CP CP CP O o o o rH H CM CM CM ro ro uo o o o q o o O O o o O o o i—t rH rH r~i rH rH rH rH rH . — i .rH rH ,—1 o o o o o o o o o O o o o o o o o o o o O o O o o O o o ro . — i ro. p- CNJ o fT. -cr in CO rH LP, m _ o r\j CP ro O o o uo c cr CP ro CO ro r- Is- r- ro CP LP r— 0s- CO 0 s CM Is- CM CM rH -o .—t CO Is- ro o ro sr CT ro ro <r CO in in v0 in CNJ O CM .—i <r s0 0s- CO ro CM ( M cr Is- r-i p- in -o O 0s- CN! o LP p- CP CM CP CM O CC ro o cc- CP co ro CM rH co .—i ro 0 s ro vO rH rH Is- vT ro CM -cr co I P LP, CM p- CM •CP o in O^J 0 s c\j ro ro.rO ro sr ST IP in vO •JO Is- C O CP O rH ro m CO rH vO ro ro ro CN) IP o O O o O o o o O o o O O o rH rH rH rH rH C M CM ro vT o o O rH —1 ro • CO Is- (NJ co CNJ sr -0 CNI (0s in CM CM o >c (NJ Is- CM rH ro, Is- rO L P o- o Is- r\j i in LO rH c Is- — i Is- ro Is- m Is- o CM CM co C v0 Is- 1^ co C0 CM CP O r— 00 Is- ro<r sr -n m 0 s f—t -J- Is- CNJ CP o vT ro p- o-- P - <r .—1 p- in vO CP CO rH rH cr oO ro Is- CNJ Is- ro i—i cr co o o 00 <r rH o H >T cr LP <J-IP 00 CO <r Is-r\j ro in o CO c ,— i ro in CO o CM LP CO rH -d- Is- o <t- ao CM sO rH vO rH -0 (Ml f\i CM CNJ CN! CN! ro ro ro ro <r IP uo uo Is- Is- CO CO cr- CP o O o o o o O o o O o o O o o o o o o O o o o O o o o rH CO Is- sr IO vO ro CNJ 0 s C . — i CO P - uo *o CO rvj 0 s o rH CO Is- LP sO ro' CNJ sT CNJ CN] Is- co-m co O CP o ro O Is- LP. LP Is- rH o CM CO o CO CM ro CM' CP-O •—t CO o in O 0 s ro o Is- r~ 0 s ro cd vO O co ro o cr c L P rH ro CO in o CM ro in r- 0 s , — i CP- CM in ,0s CM <S o 0s- CO o- CP LP ~H Is- ro, C CO 00 co co CO co 0 s 0 s 0 s 0 s O o o rH rH CM (NJ r\J ro ro •sT <r LP vO v0 p- CO o o o o o o o o o o rH rH rH ^ rH rH rH —' rH rH r—1 rH rH rH rH rH o o o o o o o o o o o o O o Oi O o o O o o o o O O o. o CO Is- sr L O -JO ro CNJ CP c rH CO p- <r in o CP CM CP o 1—1 CO Is- <r U O *0 oo CM sr o (N) CNI I— CC LP CO o 0 s o ro o p- LP L P P - rH o (NJ CO o C0 CM ro CM er -JO- rH C O m 0s- oo o Is- p- 0 s ro C0 vO O C0 ro o 0 s o LP — H ro co U O C O <NJ ro in r- cr- rH vr Is- cr CM LP 0s- CNI o o CP <r CO sT CP tn rH Pr ro CO o o •c o Is- r- Is- p- CO CO CO CP CP o o O H rH CM CM ro <f sr LP o o o O o o o o o o o o o O c o rH rH rH r^ rH . — i rH rH rH rH ' o o O o o O O' o o o o o O D o o O o o o O o o o o O o U P cr L O r^ <r CNJ CNJ rH m cr rH Is- vO o r- Is- ro CM ro rH <!- CP rH rH f\J in ON o Is- 0s- rO in O P - vC P - sO ro in' IS- p- O O rH Is- C ro r- LP CP, r— 0 s ro Is- vT in o o P - o r\j ro <r CP C M <r cr CO rH O ro . — i in in sr oo rH sr in s0 <£) vO IP. ro OJ o CO 0^ -T 'CNJ o (7s CO C O 0 s O rH pp LP rH so ro rH CO Hi CNI 0 s ro o Is- vT C r- CO If." rH CO in CN; o r- r-, r» r- Is- so so vO in in IP in <]- <r <f ro ro ro (NJ CM C M #-H * ^ •—^  rH 1 o o o o o o O o O o o o O o o O O O O o O O o O o o' o o o o o o o o o O o o o O o o O O O o o O o o O o o o o o o o o o o o O o o o O o ° O o o o o O o o o o o o o o o o o o O o o o o o O o o o o o o o o o o o o o: o r\j ro sr in vO Is- CO 0 s o CM ro vT LP Is- CO cr o .—1 (N! ro LO Is-' CO CM (N) CNJ (N! CNJ CNJ (Nj ro:co j ro ,-o rO ro ro ro <r <r <r -cr i LO C P LP, r- <o rH CNJ CNJ uo 0s- rH Is- >o o r- Is- ro (NJ ro rH -cr CT> , — i CP C M in cr o Is- 0 s ro in a Is- •c P - o ro LP r- O o rH r- o 0 0 P - LP uo in Is- 0 s ro Is- LP. CJ> o Is- 6 CM ro <• O , C P C M vT CP CO .—1 o ro H in <f sr co rH sT m o <o o in ro C M o CO <o >r C M rH o 0s CO CO C P o rH ro in CO CO in ro rH 0s Is- in ro rH •4- C M o co 0s-p- in C M O CO 0s- C P 0s- cr CO CO co CO co r> r- P«-, r> Is- o in in in in in in sr; i sr THE VALUE OF THE FIRM FOR VARYING AMOUNTS OF DEBT TRADITIONAL MODEL ro THE FIRM THE FIRM THE COST THE COST OF DEBT SELECr= 5 HAS DETERMINISTIC EARNINGS GF $ 75, HAS AN AVERAGE TAX RATE OF 0.5OO OF EQUITY IS GIVEN BY THE FUNCTION IS GIVEN BY THE T E L E C T = .0.07000000+ 0. FUNCTION ""0.05000000+ ' 0, 6. DELT= 10. S E M T = OOGOOOOOIL* OOCOOOOOIL3' - 0 . . VALUE '"'OF'" FIRM AMOUNT " " OF" "' DEBT VALUE "'" OF EQUITY AVERAGE COST OF DEBT AVERAGE "COST OF EQUITY FRACTION OF DEBT DEBT EQUITY RAT 10 AVERAGE COST OF CAPITAL MARGINAL COST' MH ' OF DEBT MARGINAL COST SO OF DEBT FRACTION OF" EQUITY 'i 535 .714 0 .000 535 .714 0 .050000 0 .07CC00 0 .ooocoo 0 . OOGOOO 0 .070000 0 .050000 -0 .000000 1 .000000 ) 5 4 2 . 1.3 5 ' T 0 ."bob" - - 5 3 2 .13 5 " 0 .050001 ' 0 .0 70001. 0 .018 446" "o .018792 0 .069632 " "o .050004 0 .050054 0 .981554 1 548 .5 10 20 .000 523 .510 0 .050008 0 .070008 0 . 036462 0 .037842 0 .069279 0.050032 0 .050385 0 .963538 | 5 5 4 . 792 30 .COO 524 .792 0 .05002 7 0 . 0 7002 7 0 .054074 0 .057166 0.068946 0 .050103 0 .051062 0 .9459 2 6 j 56 0 . 9.3 4 40 .000 5 20 . 934 0 .050064 0 .0 70064 0 .071310 0 .076735 0 .068638 0 .050256 0 .052103 0 . 928690 1 566 .889 50 .000 516 .889 0 .050125 0 .070125 0 .088201 0 .096732 0 .068361 0 .050500 0 .053525 0 .911799 , !i 5 72 .611 60 .000 512 .611 0 .050216 0 .070216 0 . 10478.3 0 . 1 1704 8 0 .068120 0 .05 0 8 64 0 .05534 2 0 .6952 17 1 "5 7 8 .053" 7 C ."coo " "" 508 . 0 53 " " 0 .050343 ' 0 .070343 0 .121096 0 .13778 1 ' 0 .067921"' 0 .051372 "o .057569 0 .878904 'j 583 . 170 80 . 000 503 .170 0 .050512 0 .070512 0 .137 181 0 .158992 0 .067768 0 .052048 0 . 0602 1.9 0 . 862 3 1.9 1 587 . 91 7 90 .000 497 .917 0 .05 07 29 0 .070729 0 . 1.53083 0 . 130753 0 .067667 0 . 0 5 ? 9 1.6 c .063303 o . 84 69 17 ,j 592 . 2 54 100 . 00 0 492 .254 0 .051 COO 0 .07 1000 0 .168847 0 . 203147 0 . 0 6 7 6 ?. 3 0 .0540 00 0 .066 830 0 .83115 3 i, 596 . 139 110 . 000 486 .139 0 .051331 0 .071331 0 . 184521 0 . 226273 0 .067641 0 .05 5 32.4 0 . 070807 0 . 8 1. 5479 ? 5 9 9 .538 120 . 000 4 79 .53 3 0 .05172 8 0 .071728 0 . 200 154 0 .250241 0 . 06772.5 0 .056912 0 . 07 52 39 0 .7 9 9 84 6 { '6 0 2 . 418 """ 1.3 0 . CCO 472 .413 "0 .052197 0 .072197 0 .215797 0 .275180 0 .067 3 81 ' o .058788 0 . 0801.26 0 .734203 •? 604 . 75 2 140 .COO 464 .752 0 .C52744 0 .072744 0 .231500 0 . 30 12 36 0.068114 0 . 06 09 76 0 .085469 0 .768500 ' 606 .516 150 . 000 456 .516 0 .053 375 0 .073375 0 .247314 0 . 328575 0 .068429 0 . 063 500 0 . 09 12 65 0 . 75 2 6 86 'i 607 .694 160 . oco 447 . 6 94 0 .054096 0 .074096 0 . 263290 0 .357387 0 .068830 0 . 066 3 84 0 .09 75 07 0 . 7367 10 ; 6 08 . 274 170 .coo 438 .2 74 0 .054913 0 .074913 0 .2 79 479 0 . 387085 0 .069323 0 .069652 o .]04187 0 .720521 } 608 .251 130 .oco 428 .251 0 .055832 0 .075832 0 .295931 0 .420314 0 .069913 0 .073 328 0 . 1.11294 o . 704069 ! 60 7 .627 190 .00 0 417 . 627 0 .056 8 59 0 .076859 0 .312692 0 . 454951 0 .070605. 0 .077436 0 .118 316 0 .68 7 3 08 j 606 .410 200 .000 406 .4 10 0 .058COO 0 .078000 0 . 3298 1.0 0 .492 114 0 .071404 0 .082000 0 . 1 26739 0 . 6 70 1.90 ' 604 .615 . 210 .000 39 4 .615 0 .05926 1 0 .079261 0 . 347328 0 .532164 0 .072 314 0 .037044 0 . 1 35046 0 .652672 602 . 263 220 . 00 0 3 82 .2 63 0 . 06064 8 0 .08064 8 0 . 365289 0 . 575520 0 . 073342 0 .092592 0 . 1437 2 3 0 .6347 11 599 . 379 2.30 . 000 • 369 . 379 0 .06 2167 0 .082167 0 .383730 0 .622666 0. 074492 0 .098663 0 . 1527 51 0 .6 162 70 .5.9 5 . 997 240 . coo 355 .997 0 .063824 d .083824 0 .402686 0 .674162 0. 075770 0 . 1.05 296 0 . 162.1 1 3 0 . 597314 592 . 153 " 250 . 000 34 2 . 1 53 ' 0 . 065625' 0 .085625 0 .422188 0 .730667 0 . 0 77181 0 . 1 12500 0 .17 17 9 5 0 .577312 587 . 888 260 . o co 327 . 88 8 0 .0675 76 0 .087576 0 .442261 0 . 792954 0 .073731 0 .120304 0 . 181780 0 . 557 73 9 583 . 24 6 270 . coo 313 . 246 0 .069683 0 .089683 0 .462927 0 . 861.944 0. 080424 0 .1287 32 0 . 1.92054 0 . 537073 578 .272 280 . 00 0 293 . 272 0 .071952 0 .09 1952 0 .484201 0 .93874 0 0 . 082268 0 .137 803 0 .202605 0 . 515 799 573 .016 2 90 . 000 283 .016 0 .074389 0 .094389 0 . 506094 1 .024677 0 . 0 842 67 0 .147556 0 .213424 0 .493 9 06 567 .526 300 . coo 267 . 526 0 .077000 0 .097000 0 . 528610 I .12 138 7 0 . 086428 0 . 1 5 30.0 0 0 .224502 0 .471390 " 5 6 1 .850 " 3 1 0 .000 ' "251 . 850 0 .079791 0 .09979 1 0 .551748 1 .230890 0. 008756 0 .169164 0 .235335 0 .448252 . 556 .038 ,320 .000 236 . 03 3 0 .082 76 3 0 .102768 0 .575 501 1 .355716 0 . 091258 0 .131072 0 .24 7420 0 .42 4499 550 . 135 330 . 000 220 . 135 0 . 035937 0 .105937 0 . 599353 1 .499083 0 . 093940 0 .193748 0 .259257 0 .400147 54 4 . 186 340 . oco 2 04 . 186 0 .089304 0 .109304 0 . 624787 1 .665150 0. 096 80 3 0 .2072 16 0 .271350 0 .375213 538 .234 350 . 000 188 . 234 0 .092875 0 .112875 0 .650275 1 . 359391 0 . 099869 0 .221500 0 . 283703 0 .3 497 2 5 5 32 . 318 360 . 000 172 .318 0 .096656 0 .116656 0 .676 288 2 . 089162 0. 103130 0 .236624 0 . 2 9632 7 0 .323712 5 2 6 .475 •-- ^ ? o .COO" ' 156 . 4 75 " '" "o" .1006 5 3 " 0 .1206 5 3" " 0 . 702787" 2 .364592 " 0. '106 5 97 " "o" .252612 ' 0 . 3092 3b" ' 0 .297213 ' 520 .739 . 380 . 000 1.4 0 .7 39 0 .104872 0 .124872 0 . 729733 2 . 700039 0 . 110277 0 .269488 0 .322429 0 .270267 ; 515 . 139 ' 39 0 . 000 125 . 139 0 .109319 0 .129319 0 . 757073 3 . 116545 0 . 114177 0 .2372 76 0 . 335936 o .242922 ' 509 . 701 400 . oco 109 . 701 0 . 114000 0 .134000 0 .7 84 773 3 .646258 0. 118305 0 .306000 0 . 349 7 69 0 .215227 504 .451 410 . 000 .9 4 .451 0 .118921 0 .13 3921 0 .312765 4 .3408 86 0. 122666 0 . 325684 0 . 363947 0 .187235 1 49 9 . 406 420 .000 79 .406 0 .124 08 8 0 .14 4088 0 .840993 5 .289240 o . 12 7 2 6 8 0 . 346352 0 . 373490 0 . 159002 4.94 . 586 4 30 . COO 64 .536 0 . 129507 0 .149507 0 .869415 6 .65783 1 0 . 132119 0 . 363028 0 . .393420 0 . 1305 8 5 4:9 0 .002 4 40 . COO 50 . 002 0 .135184 0 . 1. 55184 0 . 897955 8 .799632 0. 137225 o .390736 0 .4 0 8 7 5 9 0 . 102045 .4 8 5 . 667 450 .000 35 . 667 0 .141125 0 .161125 0 .926560 12 .616629 0. 142594 0 .414500 0 ..424530 0 . 0 73440 481 . 590 4 60 .000 21 . 590 0 .147336 0 . 1.6 733 6 0 . 955 170 21 .306521 0. 148233 0 .43 9344 0 . 440 75 7 0 . 04 48 30 4.7 7 . 7 76 470 . c oo 7 . 776 0 .153823 0 .173823 0 .983725 60 .444437 0. 154 148 . . .0.46529? 0 .4 5 74 64 0 . 0 16 2 7 5_ 474 . 2 30 430 . c oo - 5 . 770 0 .160592 0 .180592 1 .012 168- 83 . 183973 0. 1.60349 0 .49 2 368 0 . 4 74673 - 0 .012 16 3 i_ i ro 124 APPENDIX IV NET INCOME MODEL Descr ipt ion of Program The user is given the choice of three functions for required y i e ld on debt and three functions for required y ie ld on equity by choosing values for the var iables SELECT and TELECT. If SELECT = 1, the funct ion used is Kj = a + b(L - A) SELECT = 3 Kj = a + b(L - A ) 2 6 SELECT = 5 Kj = a + b(L - A ) 3 If TELECT = 2, , Ke = c + d(L - A) TELECT = 4, Ke = c + d(L - A ) 2 TELECT = 6, Ke = c + d(L - A ) 5 A l l the data spec i f ied by the user is ident ica l to that used in the t rad i t iona l model of Appendix III. In addi t ion, the user must specify ABAR. ABAR = A = the level of debt below which Ke- and Kj remain constant. 1 25 The parameter s are read o f f one da ta c a rd i n the f o r -mat and o r d e r l i s t e d be low. Parameter Format SELECT F5 .0 TELECT F5 . 0 A E10.0 B E10.0 C E l 0 .0 D E10.0 XB AR F 5 . 0 TAX F5 . 0 DELT F5 . 0 ABAR F5.0 SENT F 5 . 0 End ing Column 5 10 20 30 40 50 55 60 65 70 75 A sample da ta c a r d i s g i ven below, "57 0 0 0 0 0 0 1 2 3 4 5 6 2 2 2 2 2 2 2i 3 3 3 3 03 3 3 11 OPERATION 0 0 0 0 0 0 offo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o Qo o oQo 0 0 0 0 0 0 0 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 i i i i i i i i i i n i 1 i i i i i i i i i i i i i i i i i i i i i i i m i i i i i i i 1 i i i 1 1 i i a > i i i i i i   2 2 2 2 2 3 03 3 3 3  : 3 3 ; 5.E-0 t i U VARIABLE FIELD 5.E-9 U U LitJ 0 5.E-9 Gu 0 75. .50 10. 125. u •> i u COMMENTS LI SEQUENCE 6 \ 0 0 0 0   73 74 75 76 77 78 79 80 1 1 1 1 1 1 ! 2 2 2 Q 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^ 2 2 2 2 C 3 3 3 3 3 3 3 3 3 Q3 3 3 3 3 3 3 3 3 03 3 3 3 3 3 3 3 3 Q 3 3 3 3 3 3 3 J3 3 " 3 3 3 3 3 3 ^ 3 3 3 3 ]3 3 4444441444444414444444444444444444444444444444444444444444444444444444444444444 r 5 5 5 5 5 5 5 05 Q5 5 5 5 5 5 5 5 5 T5 5 5 5 5 5 5 "5,^ 5 5 5 5 5 ;]5 5 5 5 5^ 5 5 5 5 5 5 5 5 ;]5 5 5 6 6 6 6 6 6 G 6 E 6 6 6 6 6 6 6 6 G 6 6 G 6 6 G 6 6 6 6 6 6 6 6 G 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 G E 6 6 6 6 6 ; 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 " 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 0 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 [,8 8 8 8 8 8 8 8 8 T 8 8 8 8 8 8 8 8 8 T8 8 8 8 8 8 8 8 8 M 8 8 8 8 8 8 8IJ8 8!] 8 8 8 8 8 8.^  8 8 8 8 "8 8 5 C5! 9 9 9 9 9 9 9 9 9 9 9 9 9 9 OS 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 09 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 4849 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 G5 67 68 69 70 71 72 2 2 2 2 2 2 2 2 3 3 3  3 3 3 3 4 5 5 5 5 5 5 5 5         7 7 7 7 7 7 7 7 9 9 9 9 9 9 9 9 73 74 75 76 77 78 79 B0 :1-26 PROGRAM LISTING NET INCOME MODEL C VALUE OF THE FIRM - DETERMINISTIC MODEL C NET INCOME MODEL C MODEL ALLOWS CHOICE OF THREE TYPES OF FUNCTION FOR DEPENDENCE OF C DEBT AND EQUITY CAPITALIZATION RATES ON AMOUNT OF DEBT C IN THIS MODEL CAPITALIZATION RATES ARE CONSTANT UNTIL  C L IS GREATER THAN ABAR C WE SPECIFY INTEREST FUNCTIONS,EARNINGS,TAX RATE,AND INCREMENT C C DEFINE REAL VARIABLES REAL L,KEST0R,ICSTOR,KIAVG,KEAVG,KIMARW,INTCOS,KOAVG, 1KMARGL,KMARGS,KMARGR  2 READ(5,100)SELECT,TELECT,A,B,C,D,XBAR,TAX,DELT,ABAR,SENT 100 FORMAT(2F5.0,4E10.0,5F5.0) C SENT INAL WILL TERMINATE PROGRAM IF NON-ZERO IF(SENT.NE.0•)G0 TO 3 WRITE (6,200) XBAR,TAX, C D , A, B, SELECT, TELECT, DELT, ABAR, SENT WRITE(6,202)  C INITIALIZE ORIGINAL VALUES L = 0.0 KESTOR=C ICSTOR=0.0 1 CONTINUE C DETERMINE WHICH FUNCTIONS TO USE FOR KIAVG AND KEAVG  IF(L-ABAR)10,10,11 KIAVG=A KEAVG=C KIMARW=A GO TO 12 CONTINUE  IF(SELECT.EQ.1.)KIAVG=A+B*(L-ABAR) IF(SELECT.EQ.3.)KIAVG=A+B*(L-ABAR)**2. IF(SELECT.EQ.5.)KIAVG=A+B*(L-ABAR)**3. IF(TELECT.EQ.2.)KEAVG=C+D*(L-ABAR) IF(TELECT.EQ.4.)KEAVG=C+D*(L-ABAR)**2. IF(TELECT.EQ.6.)KEAVG=C+D*(L-ABAR)«*3.  C THE NEXT FUNCTIONS DETERMINE THE MARGINAL INTEREST RATE WHERE C KIMARW = D/DL OF (KIAVG*L) C = KIAVG + D/DL OF KIAVG MULTIPLIED BY L IF(SELECT.EQ.1.)KIMARW=B*L+A+B*(L-ABAR) IF(SELECT.EQ.3.>KIMARW=2.*B*L**2.-2.*ABAR*B*L+A+B*(L-ABAR)**2. IF(SELECT.EQ.5.)KIMARW=3.*B*L**3.-6.*ABAR*B*L**2.+3.*B*L*ABAR**2. 1 +A+B*(L-ABAR)**3. 12 CONTINUE C DETERMINE THE INTEREST COST OF DEBT IJ5UXaS=K I AVG*L C D€TER'iMINE THE AFTER TAX EARNINGS FOR EQUITY EQif A'R'N\= ( XBAR-1 NTCOS ) »(1 .-TAX )  10 11 PROGRAM LISTING (cont'd) NET INCOME MODEL 1 2 7 C DETERMINE VALUE OF EQUITY S=EQEARN/KEAVG C DETERMINE TOTAL VALUE OF FIRM V = L + S C DETERMINE DEBT RATIO  W1=L/V C DETERMINE EQUITY RATIO W2=S/V C DETERMINE THE DEBT EQUITY RATIO W3=Wi/W2 C DETERMINE AVERAGE COST OF CAPITAL  KOAVG=Wl*KIAVG+W2*KEAVG C THE FOLLOWING FIVE STATEMENTS ARE USED TO DETERMINE THE MARGINAL COST C OF DEBT BY AN INCREMENTAL APPROACH. WE CONSIDER THE ADDED DEBT C CHARGES AND ALSO THE EFFECT OF ADDED DEBT ON THE COST OF EQUITY KMARGL=(INTCOS-ICSTOR)/DELT KMARGS=((EQEARN/KESTOR)»KEAVG-EQEARN)/DELT  KMARGR=KMARGL+KMARGS KESTOR=KEAVG ICSTOR=INTCOS WRITE(6,201)V,L,S,KIAVG,KEAVG,W1,W3,K0AVG,KIMARW,KMARGR, 1W2,KMARGL,KMARGS 201 F0RMATtlX,lF9.3,2F10.3,10F10.6)  C INCREMENT BY AN AMOUNT DELT L=L+DELT C CHECK TO SEE IF ENOUGH VALUES HAVE BEEN COMPUTED IF(W2-.00001)2,2,1 3 STOP 200 FORMAT(1H1,37X,25HTHE VALUE OF THE FIRM FOR/  1 39X,23HVARYING AMOUNTS OF DEBT// 8 42X.16HNET INCOME MODEL// 2 15X,40HTHE FIRM HAS DETERMINISTIC EARNINGS OF $,F5.0V 3 15X.35HTHE FIRM HAS AN AVERAGE TAX RATE 0F,F6.3/ 4 15X,43HTHE COST OF EQUITY IS GIVEN BY THE FUNCTI0N,F14.9,1H+, 5 F13.ll,1HL/15X,41HTHE COST OF DEBT IS GIVEN BY THE FUNCTION, 6 F16.9,1H+,F13.11,1HL/20X,7HSELECT=,F5.0,8X,7HTELECT=,F5.0,5X, 7 5HDELT=,F5.0,5X,5HABAR=,F5.0,5HSENT=,F5.0///) 202 FORMAT<4X,5HVALUE,5X,6HAMQUNT,4X,5HVALUE, 1 4X,7HAVERAGE,3X,7HAVERAGE,3X,8HFRACTION,4X,4HDEBT,4X, 2 7HAVERAGE,3X,8HMARGINAL,2X,8HMARGINAL,2X,8HFRACTI0N,2X, 3 8HMARGINAL,2X,8HMARGINAL/6X,2H0F,8X,2H0F,8X,2H0F,6X,4HC0ST,6X, 4 4HC0ST,8X,2H0F,6X,6HEQUITY,4X,4HC0ST,5X,7HC0ST MM,3X,7HC0ST SO, 5 6X,2H0F,4X,9HC0ST DEBT,IX,10HC0ST EQUTY/5X,4HFIRM,6X,4HDEBT, 6 5X,6HEQUITY,3X,7H0F DEBT,2X,9H0F EQUITY,4X.4HDEBT,6X,5HRATI0, 7 2X,10H0F CAPITAL,IX,7H0F DEBT,3X,7H0F DEBT,4X,6HEQUITY,3X, 8^7iP©rri0N,3X,7HP0RTI0N///) mm THE VALUE OF THE FIRM FOR VARYING AMOUNTS OF DEBT NET INCOME MODEL ro co FIRM HAS DETERMINISTIC EARNINGS OF % FJ.RM HAS AN AVERAGE TAX RATE OF 0 .000 _ COST OF EQUITY IS GIVEN BY THE. FUNCTION THE COST OF, DBBT IS GIVEN BY T H E FUN C.T I ON SELECT- 5 . TELECT- ' 6 . THE THE THE 75 . 0 . 07000000+ 0 . 000000005(L-ia5r ""O..-05000000+ o . ooooooooso-.-iasf DEL T= 10. A B A R = 125 .SENT- - 0 . ..VALUE AMOUNT VALUE AVERAGE AVERAGE FRACTION DEBT AVERAGE MARGINAL MARG IN A L. FRACT1 ON OF OF OF COST COST OF EQUITY C OS T COST MM COST SO OF FIRM DBBT EQUITY OF DEBT 0 F EQUITY DEBT RATIO OF CAPITAL OF DEBT OF DEBT EQUI'T Y 1071 .429 0 .000 1071 .42 9 0 .050000 0 .070000 0 .000000 0 .000000 0 .070000 0 .050000 -o .000000 1 ,000000 10:74. 286 10 .000 1064 .286 0 .050000 0 .070000 0 .009309 0 .009396 0 .069814 0 .050000 Q . 0 500 00" 0 .990691 10-7 7. 143 20 . 000 1057 . 143 0 .050000 0 .070000 0 .018568 0 .013919 0 .069629 0 .050000 0 .050000 0 .981432 1080 .000 30 . 000 1050 . COO 0 .050000 0 .070000 0 .027778 0 .028571 0 .069444 0 .050000 0 .05 00 00 0 .97222 2 10 8 2 . 8 5 7 40 . 000 1042 . 05 7. 0 .050000 0 .070000 0 .036939 0 .033356 0 .069261 0 .050000 0 .050000 0 . 963061 1085. 714 50 .000 1035 .714 0 .050000 0 .070000 0 . 046053 0 . 0482 76 0 .069079 0 .050000 0 .050000 0 . 95 3947 108 8 .571 60 . 000 1028 . 571 0 .050000 0 .070000 0 .055118 0 .05 8 333 0 .068898 0 .050000 0 . 050000. 0 .944882 1091 .42 9 70 . 000 • 1.0 21 .42 9 0 .050000 0 .07 0000 0 .064136 0 .068531 0 .068717 0 .050000 0 .050000 0 .935364 1094 .286 80 . 00 0 1014 . 2 8 6 0 .050000 0 .0 70000 0 .073 107 0 .078873 0 .068538 0 .050000 0 .050000 0 .926393 10.97. 143 90 . 00.0 1007 . 1 A3 0 .050000 0 .070000 0 .002031 0 .089362 0 .068359 0 .050000 0 .050000 0 .917969 1 1 0 0 . 0 00 100 .000 10C0 . COO 0 .050000 0 .070000 0 . 090909 0 . 100000 0 .068182 0 .0 50 000 o .050000 0 . 90 9 09 1 1 102 .857 110 . 000 992 .857 0 .050000 0 . 070000 . 0 .099741 0 .110791 0 .068005 0 .050000 0 .050000 0 .900259 1105 .714 120 . coo 985 .714 0 .050000 0 .070000 0 . 108 527 0 . 12 1739 0 .067829 0 .050000 0 . 050000 0 .89147 3 . "l 108 . 562 130 .000 ' " 978 . 562 0 .050001 0 .070001 0 . 1 17269 0 . 132848 0 .067655 0 .050049 "o .050069 0 . 332731 1111 .161 140 . 000 971 .161 0 .050017 0 .070017 0 . 125994 0 . 144157 0 .067497 0 .05 0489 0 .0518 07 0 .874006 1 1 1 3.04 3 150 .000 963 . 04 3 0 .050078 0 .070078 0 . 134 766 0 . 155756 0 . 06 73 03 0 .051484 0 . 0 5 6 3 3 9 0 . 36 52 3 4 1113 .7 32 160 . 000. 953 .7 32 0 .050214 0 .070214 0 .143661 0 . 167762 0 .067341 0 .053154 0 .0652 78 0 . 356339 1 I 12 .75 7 170 .000 942 .757 0 .050456 0 .0 7045 6 0 . 152774 0 .180322 0 .067400 0 .055619 0 .077138 0 . 8472 26 .1109.670 180 . OOCL 9 29 . 670 0 .0508 32 0 .070832 0 .162210 0 . 193617. 0 .067588 0 .058999 0 .092 394 0 . 837790 1104 .057 190 . COO 914 .057 0 .051373 0 .071373 •0 .172093 0 . 207864 0 .067931 0 .063^14 0 . 1.10967 0 . 82 7907 . ' 1095 .550 200 .000 895 .55 8 0 . 0521.09 0 .0 721.09 0 . 182 555 0 . 223324 0 .068458 0 . 06 8 98 4 0 .132714 0 .817445 10 8 3 .883 210 . 00 0 s 873 .83 3 0 .053071 0 .073071 0 . 193748 0 .240307 0 .069196 0 .075829 0 .157417 0 . 806 2.5 2__ ST CM SO oo rH ro cr cr ro r- sr o ^t C D — i r- in s0 sj o ro CO 00 -f 0 • cr o ,—* iM ST o IT. iM c\> L0 LO, Co r-. o 00 r- rH r— ,—I rH ^  CO o r- uo rH CM m, LO, O LO. o o Is-o t —< Oj o ro ro CM CO ,—i ,—i 0s' C M CM CO cr m SJ Cx| — J sr rH 00 CC m rO _H cr rH Is- St o in O in r-- ro so CO o CO r— r- Is- r- r- 0 - -0 o •o LO m st S t ro 00 t\. o o O o o o O o O o c o o o O O a O O o o c o 1 L D rH ro •o in CM <r CM •O CM cr o SO o cr- LO LO c CO m at 0 s r- 3 s ro C M o~. OJ O J ro, ro L O r- sr ro, — i 00 o o- ST 0 s ' CO so r- O C M o rH ro LO Is- Cr' C M 3D CO -j- CO 00 co st sr in CO CM c •O Is- s2 ro cc f—i CM CM C L0 o ro SJ I S-co s t Is- r-. CO •4-r- o LO co C C M ,~o, LO CO cr o rH CM C M C M ro rr. ro <r <t' uo in in in S3 o SO SO sO o -0 Is-1 ° C O O o O O O: ! o o o o o O o O o o o o o 0 s ST si- 0 s Cr 0 s 0 s St O'- \ - t St 0 s s t CT' cr- sr so CM rH LO r- 0 s C M CQ o O cr er 0", co O o sr ro r- r-H r~ o CO CM lO ro rO m JO o CO m o St I S- rH *o in CO r- oo 0J C0 sr r i in CO ro, O o o vT o 0s rH uo C M rr. o ro ro Is- LO SO ,—* 00 cr o —1 rO in -0 o ,—1 sf vO O' ro Is- .—1 m O in. o SJ 00 Cr o O rH ,—1 rH --: rH CM C M C M ro ro ro sr sr in in s0 sO r- Is-O o c o o c o o O O o o c o O c c o c o O O c CO so o SO CO C M <r Is- •C -o o Oj so sr H CO cr in cr ro cr Is- o ro CO SO rH C M C M cr in ro sr rH r— St ro co rH ro CO 00 ,—. sr 0 V I S- cr in <r co O CO o 0 s r- .—1 o sO cc CM SI' ro o o i—< C M ' sr SO 0 s O J m, 0 s ro CO ro cr 0D ro O co r- Is- Is- CO o I S- r— r- Is- Is- CO CO CO cr cr O O —« 0 J r 0 o~, st m -0 r- cr o o o o o o o o O o o o o o o o o o Q o o o o O O O O O O o a o O o o in ro, o 0 s O J ro ro sj- rH cr r- .—< 00 st ro co o r- in CO CO in ro i-H r- (T-r- «£>' i—I O ro O ro cr so r- ^ - I-— •51- CO rH 00 sr O 0 s o O CO CM cr so O co in CO o cr H cr cr cr o st C M ro , — i <r O CO sT so OvJ o C M ro 00 00 cr -—4 o sr in CO o ro SO o Cr in ro CM St cr co St rH I S- o rH ->f oo C M ro ro ro sr <r <r in s 0 r- 00 cr r-H sr '00 m ro r-H CO ,—i o O o O o O o o o,o o O o _H _ i rH C M ro L O O C M rH 1 ro OO Is- CO sr sr 0s ,—r r- st rH rH Is- ro SO St o a- ro in SO ro CO rr- v0 .—i o m in o o rO cr c- r- in cr sr CO lo- ro sr C M CO cr ro CM CO C M CO co in rH cr C M sr co r- <• cr st ro cr r-in CO ro -0 vO Is- i-H CO co o Is- r- ,—* o S t ro Is- CO in CO cr O •—i ro sr o co o ro m, CO C M in cr St cr sr C s0 ro ,—t cr CO r\l C M OJ C M C M C M CO ro ro S t sr St in in s0 r- Is- CO cr 0s- o O £2 O C o o o O o O o •o O o o C o O o o o rH CO st O C M ro 0 s rH r~ CO st ON; ro 0s- Is- CO sr sO 00 ro co o:; o o O St r-H cr LO Is- r- •cr in co ro o st CO sO so oo Is- SO Is- ro OJ <r r- SO O Q cr CO in cr o 0s lo- ro cc sr LO Is- Cr C M in aj OvJ O rH Is- ro. cr vO sr ro 00 ro, ro in CO ,—i r- I S- r- r- CO CO CO cr- r^ O O — i rH C M ro st in sO Is- CO 0 s -H o O o o O c o o O C M o O o o O o o o o O O c o o o O c O O o o O r- CO sr oo C M ro Cr I—1 Is- CO sr sO C M m cr r- CO sr 00 oo ro co CO o -0 O <*- i—i vO 0 s m I S- Is- Cr in cc ro c sr CO sr SO so oo Is- SO IO - ro 00 vO I— vO O o so 0s co m 0s o cr r- ro, CO sr LO . rr C O C M in CO o>; -0 ,—i Is- ro, Cr sO St ro, C M 0 s ro LO CO rH in m in IT, -JO so o Is- Is- CO C0 cr cr o r-J C M rO sr in SO Is- cr o o o o O o o O o o o o O ,—1 i—1 rH rH rH rH rH rH rH o o o o O o o o o o o o O o o o O o O o o o cr -o ro CO rH L0 CO o <r vO Oj CO ro sr cc sO sr st, sr SO CM CT CT r 0 oo CO o CO o in. —1 St 00 o CO cr 0 s C M in C M -t o CO C M CM 0 s sr —1 IT, CO CD C M O s t 0 s ro, O ,—1 CO o cr sr sr CO O CO oo sr ro cr ro vO CO o rH ro L0, cr sr o Is- r- r» o in sr 00 CQ in r| Is- rsj CO ro CO S t cr St cr sr o o. rH r- ro ro CO CO Is- r- Is- SO in in sr sr ro ro C M C M (Mi rH rH 1 o o C O o CD o O o O o O O o o o o o O o o c o a o o O o o o O o O O o o o o o o O o o o o o o o o o o O c O o o •o; o o a o o o o o o o o o o o o o o o o O o o o o o o o o o CM ro st in SO I S- CO cr o rH C M ro sr in SO Is- cc cr o rH 00 oo C M C M <N OJ OJ 00 C M O M ro ro ro T> ro ro rO ro ro ro sr st st sr 0 s o m CO in co <r cr rH 00 CO co! sr o SO sr sr sr sr C M cr cr ro -X) 90 o co o in |H sr CM o co! cr 0s C M in 00 sr cr 00 C\i C M cr ST .—i in CO 00 r- C M SO sr 0s- ro o rH CO o 0 s in CO o ro CM sr ro o- ro -o 00 o rH ro in cr •st o Is- r- r~ o sr 00 in C M O r- ST o r- ro cr o fM co sr o r- sr o sr CM cr O o O O 0s cr cr 00 CO r«-. r- >o s0 in m in sr sr sr ro <-i r-l t THE VALUE OF THE F I R M FOR VARYING AMOUNTS OF D E F5 T NET INCOME MODEL o .THE F I R M HAS D E T E R M I N I S T I C EARNINGS OF i 75 THE F I R M HAS AN AVERAGE TAX RATE OF 0 . 5 0 0 OF EQUITY.. IS G I V E N BY THE F U N C T I O N OF DEBT I S G I V E N BY THE F U N C T I O N THE THE COST COST S E L E C T -0 . 0 7 0 0 0 0 0 0 + , 0 . 0 5 0 0 0 0 0 0 + 5 . T E L E C T = 6 . D E L T = 1 0 . 0 . 0 0 0 0 0 0 0 0 5 0 — i£5) 0 . 00 0 00 00O5( L- l a s ) 3 ' A B A R - 12 5 . S E N T - - 0 . •VALUE AMOUNT V A L U E AVE RAG E _ AVERAGE _ F R A C T I O N DEBT AVERAGE MARGINAL MARGINAL F R A C T I O N O F OF • O F 'COST " C O S T ~ O F " " " ~ " E Q U I T Y " C O S T COST MM C O S T SO OF F I R M DEBT EQUITY O F DEBT O F EQUITY DEBT RATIO OF C A P I T A L OF DEBT OF DEBT EQUITY 53 5 . 714 _ 0 . 0 0 0 535 . 7 1 4 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 5 0 0 0 0 - 0 . 0 0 0 0 00 _ 1 . 00 0 00.0 5 A 2 . 1 4 3 ~ ~10 / b o o " " 532 . 1 4 3 6" . 0 5 0 0 0 0 " 0 . 0 7 C O 0 O 0 . 0 1 8 4 4 5 ' 0 . 0 1 8 7 9 2 ' 0 . 0 6 9 6 3 1 ' 0 . 0 5 0 0 0 0 " "o . 050000"* "o . 9 3 1 5 5 5 5 4 8 . 5 7 1 20 . 0 0 0 528 . 571 0 . 0 5 0 0 0 0 0 . 0 7 C 0 O 0 0 . 0 3 6 4 5 8 0 . 0 3 7838 0 . 0 6 9 2 7 1 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 96 3 542 5 55 . 0 0 0 30 . 0 0 0 525 . COO 0 . 0 5 0 C 0 0 0 . 0 7 0 0 0 0 0 . 0 5 4 0 5 4 0 . 0 5 7 1 4 3 0 . 0 6 8 9 1 9 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 94 5 94 5 5 6 1 . 4 2 9 40 . 0 0 0 521 . 429 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 7 I 247 0 . 0 7 6 7 1 2 0 . 0 6 8 5 7 5 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 9 2 8 7 5 3 567 . 8,5 7 50 . 000 517 . 857 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 0 8 8 0 5 0 0 . 0 9 6 5 5 2 0 . C 6 8 2 3 9 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 91.1 9 5 0 5 7 4 . 2 8 6 60 . OCO 514 . 2 36 0 . 0 5 0 G 0 0 0 . 0 7 C 0 0 0 0 . 1 0 4 4 7 8 0 . 1 1 6 6 6 7 0 . 0 6 7 9 1 0 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 8 9 5 5 2 2 5 8 0 . 7 1 4 70 . 0 0 0 510 . 7 1 4 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 1 2 0 5 4 1 0 . 1 3 7 0 6 3 0 . 0 6 7 5 8 9 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 b . 3 794 59 ' 5 8 7 . 143 80 . 0 0 0 507 . 1.43 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 1 3 6 2 5 3 0 . 1 5 7 7 4 6 0 . 0 6 7 2 75 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 863 74 7. 59 3 . 571 9C . COO 503 . 5 71 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 1 5 1 6 2 5 0 . 1 7 8 7 2 3 0 . 0 6 6 9 6 8 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 8 4 8 375 6 0 0 . CCO 100 . 000 500 . COO 0 . 0 5 0 0 0 0 0 . 0 7 C 0 0 0 0 . 1 6 6 6 6 7 0 . 2 0 0 0 0 0 0 . 0 6 6 6 6 7 0 . 0 5 0 0 0 0 0 . 0 5 00 00 0 . 8 3 3 3 3 3 6 0 6 . 4 2 9 110 . 0 0 0 49 6 . 42 9 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 1 3 1 3 9 0 •0 . 2 2 1 5 8 3 0 . 0 6 6 37 2 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 8 1 8 6 1 0 612 . 85 7 120 . 0 0 0 492 . 8 5 7 0 . 0 5 0 0 0 0 0 . 0 7 0 0 0 0 0 . 1 9 5 8 0 4 0 . 2 4 3 4 7 8 0 . 0 6 6 0 6 4 0 . 0 5 0 0 0 0 0 . 0 5 0 0 0 0 0 . 3 0 4 1 9 6 6 1 9 . 2 8 1 130 . o o o - 4 0 9 . 2 3 1 ' 0 . 0 5 0 0 0 1 0 . 0 7 0 0 0 1 0 . 2 0 9 9 2 1 0 . 2 6 5 6 9 6 0 . 0 6 5 8 0 2 'o . 0 5 0 0 4 9 ' 0 . 0 5 0 0 3 9 0 . 7 9 0 0 7 9 ' 6 2 5 . 5 8 0 140 . 0 0 0 4 85 . 530 0 . 0 5 0 0 1 7 0 . 0 7 0 0 1 7 0 . 2 2 3 7 9 2 0 . 2 8 8 3 1 5 0 . 0 6 5 5 4 1 0 . 0 5 0 4 8 9 0 .051 .017 0 . 7 7 6 2 0 8 6 3 1 . 5 2 2 150 . GOO 4 8 I . 522 0 . 0 5 0 0 7 8 0 . 0 7 0 0 7 8 0 . 2 3 7 5 2 2 0 . 3 1 1 5 1 2 0 . 0 6 5 3 2 8 0 . 0 5 1 4 84 0 . 0 5 3 8 8 7 0 . 7 6 2 4 7 3 6 3 6 . 8 6 6 160 . CCO 4 76 . 3 6 6 0 . 0 5 0 2 1 4 0 . 0 7 0 2 1 4 0 . 2 5 1 2 3 0 0 . 3 3 5 5 2 4 0 . 0 6 5 1 9 0 0 . 0 5 3 1 5 4 0 . 0 5 3 7 6 6 0 . 7 4 8 770 641 . 3 7 9 1 70 . 000 471 . 379 0 . 0 5 0 4 5 6 0 . 0 7 0 4 5 6 0 . 2 6 5 0 5 4 0 . 3 6 0 6 4 4 0 . 0 6 5 155 0 . 0 5 5 6 1 9 0 . 0 6 5 7 2 7 0 . 73494 6 6 4 4 . 8 3 5 130 . 000 _ 464 . 335 0. . 0 5 0 8 3 2 0 . 0 7 0 8 3 2 0 . 2 7 9 1 4 1 0 . 3 8 7 2 3 4 0 . 0 6 5 2 4 9 0 . 0 5 8 9 9 9 0 . 0 7 4 3 1 1. 0 . 72 085 9 64 7 . 0 2 9 190 . 0 0 0 457 . 0 2 9 0 . 0 5 1 3 7 3 0 . 0 7 1 3 7 3 0 . 2 9 3 6 5 0 0 . 4 1 5 7 2 9 0 . 0 6 5 5 0 0 0 .06.34 14 0 . 0 0 6 0 4 1 0 . 7 0 6 3 5 0 6 4 7 . 7 7 9 2 0 0 . 0 0 0 447 . 7 7 9 0 . 0 5 2 1 0 9 0 . 0 7 2 1 0 9 0 . 3 0 8 7 4 7 0 . 4 4 6 6 4 9 0 . 0 6 5 9 3 4 0 . 0 6 8 9 8 4 0 . 0 9 9 4 0 6 0 . 69 1 25 3 6 4 6 . 94 1 210 . 0 0 0 436 . 9 4 1 0 . 0 5 30 7 1 0 . 0 7 3 0 7 1 0 . 3 2 4 6 0 4 0 . 4 8 0 6 1 4 0 . 0 6 6 5 7 9 0 . 0 7 5 629 0 . 1 1 4 0 5 6 0 . 67 5 396 6 4 4 . 6 4 0 . 6 3 4 . 6 2 6 . 6 1 7 . 6 0 6 . 4 15 143 14 6 4 6 9 231 592 2 2 0 . 230 . 24 0 . 2 5 0 . 2 6 0 . 270 . OCO 00 0 coo 0 0 0 000 0 0 0 4 24 . 4 1 0 . 3 94 . 3 7 6 . 3 5 7. 3 3 6 . 415 14 3 146 46 9 231 592 0 . 0 5 4 2 3 7 0 . 0 5 5 7 8 8 0 . 0 5 7 6 0 4 0 . 0 5 9 7 6 6 0 . 062 302 0 . 0 6 5 2 4 3 0 7 4 2 8 1 . 0 7 5 7 8 8 0 7 7 6 0 4 . 0 7 9 7 6 6 0 3 2 3 0 2 0 8 5 2 4 3 0 . 0 0, 0 0, 0 , 3 4 1 3 9 5 3 5 9 2 9 2 3 7 8 4 6 2 .399062 4 2 1 2 36 4 4 5 110 5.1 8 3 6 1 5 6 0 7 7 3 6 0 8 9 1 1 6 6 4 0 6 5 7 2 7 8 2 1 302 157 0 . 0 6 7 4 5 9 0 . 0 6 8 60 2 0 . 0 7 0 0 3 5 0 . 0 7 1 7 8 4 0 . 0 7 3 8 7 7 0 . 0 7 6 3 4 1 0 8 4 0 6 9 09 3 8 2 4 10 5 2 14 1 183 59 1 3 3 3 7 9 1 5 0 3 9 4 0 . 1. 3 2 3 0 7 0 . 1 5 1 6 3 3 0 . 1726 80 0 . 1 9 5 2 6 6 0 . 2 1 9 1 9 2 0 . 2 4 4 2 5 4 0 . 6 5 8 6 0 5 0 . 6 4 0 7 0 8 0 . 6 2 1 5 3 8 0 . 6 0 0 9 3 8 0 . 5 7 8 7 6 4 0 . 5 5 4 8 9 0 5 9 4 . 5 8 1 . _ 5 6 8 . 5 5 4 . 5 4 0 . i 5 2 5 . 754 9 4 3 4 0 2 378 110 8 2 0 2 8 0 . 2 9 0 . 30 0 . 3 1 0 . 3 2 0 . 330 . 000 0 0 0 COO 00 0 00 0 0 0 0 3 1 4 . 291 . . 2 6 8 . 2 44 . 2 2 0 . 1 9 5 . 754 9'+3 402 37 8 110 82 0 0, 0, 0, 0 , 0, 0 . 0 6 8 6 1 9 07 2 4 6 1 0 7 6 7 9 7 0 3 1 6 5 8 0 3 7 0 7 4 0 9 3 0 7 6 0 . 0 8 8 6 1 9 0 . 0 9 2 4 6 1 0 . 0 9 6 7 9 7 0 . 1 0 1 6 5 8 0 . 1 0 7 0 7 4 0 . 1 1 3 0 7 6 0. 0 0 0, 0 0. 4 7 0 7 8 3 4 9 8 3 3 1 5 2 7 7 9 6 559 185 5 92 47 2 627 591. 0 . 8 8 9 5 8 4 0 . 9 9 3 3 4 5 1. 1 177.27 1 . 2 6 8 5 2 8 1 . 4 5 3 8 2 1 1 . 6 8 5 2 1 7 0 . 0 7 9 2 0 4 0 . 0 0 2 4 9 4 0 . 0 8 6 2 4 1 0 . 0 9 0 4 7 4 0 . 0 9 5 2 2 5 0 . 1 0 0 5 2 4 0 . 1 6 9 5 2 4 0 . 1 9 0 3 8 9 0 . 2 1.4609 0 . 2 4 0 8 0 4 0 . 2 69 5 94 0 . 3 0 1 0 9 9 0. 0, 0, 0, 0, 0 , 2 7 0 2 5 6 2 9 7 0 1 9 3 2 4 3 9 2 352 2 6 0 ' 38054.7 4 0 9 2 19 0 . 5 2 9 2 1 7 0 . 5 0 1 6 6 9 0 . 4 72 2 04. 0 . 4 4 0 8 1 5 0 . 4 0 7 5 2 8 0 . 3 7 2 409 5 1 1 . 4597. 4 8 4 . 4 .72. 4 6 0 . 4 4 8 . 7 L 1 9 54 6 9 4 04 7 0 9 9 913 34 0 . 3 5 0 . 3 6 0 . 3 7 0 . 380 . 39 0 . COO 0 0 0 0 0 0 boo" 0 0 0 OCO 4 3 8 . 4 2 8 . 4 2 0 . 4 1 2 . 52 7 9 6 2 222 297 4 0 0 . 4 1 0 . 4 2 0 . 4 3 0 . COO 0 0 0 CCO 0 0 0 171 . 1 4 7 . 1 24 . 1 0 2 . 8 0 . 5 3 . 711 954 694 04 7' 099 913 •38 . 1 8 . 0 . - 1 7 . 52 7 962 22.2 70.3 0, 0, .0. 0, 0 . 0, 0 9 9 6 9 2 1 0 6 9 5 3 1 1 4 3 0 9 1 2 3 5 3 1 1 3 2 9 0 7 1 4 3 0 4 3 0 . 1 19692 0 . 1 2 6 9 5 3 0 . 1 3 4 8 8 9 0 . 1 4 3 5 3 1 0 . 1 5 2 9 0 7 0 . 1 6 3048 0, 0 0, 0, 0 , 0, 6 6 4 4 3 8 7 0 2 8 7 6 7 4 2.736 7 0 3 8 21 8 2 5 9 0 9 8 6 8 7 6 6 1 . 2 , 2, 3 , 4, 6, 9 8 0 0 7 4 3 6 5 6 0 2 8 3 7 0 6 4 6 2 5 7 8 8 7 4 4 1 2 5 6 1 9 9 5 6 0. 0, 0. 0 . 0 . 1 3 6 3 8 9 0 . 1 4 5 6 7 3 1 0 6 4 0 3 1 1 2 8 9 6 1 2 0 0 3 5 12 78 54 0 . 3 3 5 4 3 9 0 . 372.734 0 . 4 1 3 1 0 4 0 . 4 5 6 6 6 9 0 . 5 0 3 5 4 9 0 . 5 5 3 3 6 4 0, 0, 0, 0 . 0 . 0 , 4 3 8 2 8 3 4 677 86 4 9 7 8 0 5 . 5 2 8 4 4 6 5 5 9 3 3 7 5 9 2 1 2 3 0 . 3 3 5562 0 . 2 9 7 1 2 4 0 . 2 5 7 264 0 . 2 16179 0 . 1 7 4 091 0 . 1 3 1 2 3 4 0 . 1 5 3 9 3 4 0 . 165 746 0 . 1 7 3 3 6 2 0. 1 9 1 8 6 3 0 . 1 7 3 9 3 4 0 . 1 3 5 7 4 6 0 . 1 9 3 362 0 . 21 1.36.3 0 . 9 1 2 14 4 0 . 9 5 5 79 5 0 . 9 9 9 4 7 2 ' : 1 . 0 4 2 9 3 7 -1 0 . 3 3 2 2 8 0 2 1 . 6 2 1955 & a * >;< "V 1* 2 4 . 2 8 9 9 6 9 0 . 1 55 74 1 0 . 1 6 6 6 3 0 0 . 1 7 8 3 7 2 0 . 1 9 1 0 0 4 0 . 6 0 7 7 3 4 0 . 6 6 5 2 7 9 0 . 7 2 6 6 1 9 0 . 7 9 1 8 7 4 0 . 6 2 5458 0 . 6 6 0 0 0 6 0 . 6 9 5926 0 . 7 3 3 3 8 7 0 . 0 3 7 0 5 6 0 . 0 4 4 205 0 . 0 0 0 5 2 8 • 0 . 0 4 2 9 3 7 o j THE VALUE OF THE FIRM FOR VARYING AMOUNTS OF DEBT j ' ' ' NET INCOME MODEL - -[ ; ; • v*i I THE FIRM HAS DETERMINISTIC EARNINGS OF $ 25. W I THE FIRM HAS AN AVERAGE TAX RATE OF 0.000 j _. .._ THE COST OF EQUITY IS GIVEN BY THE FUNCTION 1.0OOO0000+ -O.OOOOOOOOL j THE COST OF DEBT IS GIVEN BY THE FUNCTION 0.05000000+ O.OOOOOOOCfL-las) 3 ~ I SELECT= 5. TELECT= 6. DELT= 10. ABAR= 125.SENT= - 0 . VALUE AMOUNT VALUE AVERAGE AVERAGE FRACTION DEBT AVERAGE MARGINAL MARGINAL FRACTION OF OF OF COST COST OF EQUITY COST COST MM COST SO OF FIRM DEBT EQUITY OF DEBT OF EQUITY DEBT RATIO OF CAPITAL OF DEBT OF DEBT EQUITY 25oCC0 0.000 25.COO 0.050000 1.000000 0.000000 0 .000000 1.000000 0.050000 0.000000 1 .000000 34.500 10.000 24.500 0.050000 1.000000 0.289855 0 .408163 0.724638 0.050000 0.050000 0 .710145 44.000 20.000 24.000 0.050000 1.000000 0.454545 0 .833333 0.568182 0.050000 0.050000 0 .545455 53.500 30.000 23.500 0.050000 1.000000 0.560748 1 .276596 0.467290 0.050000 0.050000 0 .439252 63.000 40.000 23.COO 0.050000 1.000000 0.634921 1.739130 0.396825 0.050000 0.050000 0 .365079 72.500 5 0 o 0 0 0 22.500 0.050000 1.000000 0.689655 2 .222222 0.344828 0.050000 0.050000 0 .310345 82.000 60.000 22.COO 0.050000 1.000000 0.731707 2 .727273 0.304878 0.050000 0.050000 0 .268293 91.5C0 70.000 21.500 0.050000 1.000000 0.765027 3 .255814 0.273224 0.050000 0.050000 0.234973 101.000 80.000 21.COO 0.050000 1.000000 0.792079 3 .809524 0.247525 0.050000 0.050000 0 .207921 110.500 90.000 20.500 0.050000 1.000000 0.814480 4 .390244 0.226244 0.050000 0.050000 0 . 185520 ' 120.OCO 100.000 20.000 0. 050000 1.000000 0. 833333 5. 000000 0.208333 0.050000 0.050000 0.166667 129.500 110.000 19.500 0. 050000 1.000000 0. 849421 5. 641026 0. 193050 0.050000 0.050000 0.150579 139.000 120.000 19.000 0. 050000 1.000000 0. 863309 6. 315789 0.179856 0.050000 0.050000 0.136691 148.5C0 130.000 18.500 0.050001 1.000000 0.875421 7.027058 0.168350 0.050049 0.050008 0.124579 157.998 140.000 17.998 0.050017 1.000000 0. 886089 7. 778799 0. 158230 0.050489 0.050228 0.113911 167.488 150.000 17.488 0. 050078 1.000000 0. 895585 8. 577172 0.149264 0.051484 0.050936 0.104415 176.966 160.000 16.966 0. 050214 1.000000 0. 904130 9. 430793 0.141270 0.053154 0.052258 0.095870 186.423 170.000 16.423 0. 050456 1.000000 0. 911907 10. 351624 0.134104 0.055619 0.054316 0.088093 . 195.850 180.000 15.850 0. 050832 1.000000 0. 919069 11. 356279 0.127649 0.058999 0.057228 0.080931 205.239 190.000 15.239 0. 051373 1.000000 0. 925750 12.467923 0.121809 0.063414 0.061116 0.074251 214.578 200.000 14.578 0. 052109 1.000000 0.932061 13. 719185 0.116508 0.068984 0.066098 0.067939 223.855 210.000 13.855 0.053071 1 .000000 0. 938 107 15. 156798 0.1 11679 0.075829 0.072296 0.061893 C) f 233.057 220.000 13.057 0.054287. 1.000000 0.943976 16.849344 0.107270 0.084069 0.079828 0.056024 j 242.169 230.000 12.169 0.055788 1.000000 0.949751 18.900902 0.103234 0.093824 0.088816 0.050249 ! 251.175 240.000 11.175 0.057604 1.000000 0.955509 21.476605 0.099532 0.105214 0.099378 0.044491 1 260.059 250.000 10.059 0.059766 1.000000 0.961322 24.854368 0.096132 0.118359 0.111636 0.038678 | 268.802 260.000 8.802 0.062302 1.000000 0.967256 29.540376 0.093005 0.133379 0.125708 0.032744 277.384 270.000 7.384 0.065243 1.000000 0.973379 36.563780 0.090128 0.150394 0.141716 0.026621 285.787 280.000 5.787 0.068619 1.000000 0.979752 48.387862 0.087478 0.169524 0.159778 0.020248 293.986 290.000 3.986 0.072461 1.000000 0.986440 72.746994 0.085038 0.190889 0.180016 0.013560 301.961 300.000 1.961 0.076797 1.000000 0.993506********** 0.082792 0.214609 0.202548 0.006494 309.686 310.000 -0.314 0.081658 1.000000 1.001014********** 0.080727 0.240804 0.227496 -0.001014 THE VALUE OF THE FIRM FOR VARYING AMOUNTS OF DEBT NET INCOME MODEL V>! 4> THE FIRM HAS DETERMINISTIC EARNINGS OF $ 75. THE FIRM HAS AN AVERAGE TAX RATE OF 0.000 THE COST OF EQUITY IS GIVEN BY THE FUNCTION 1.00000000+ -0.00000000L THE COST OF DEBT. IS GIVEN BY THE FUNCTION 0.05000000+ 0. 0O0O000G5 ( L - i £ 5 ) 3 SELECT= 5. TELECT= 6. DELT= 10. ABAR= 125.SENT= - 0 . VALUE AMOUNT VALUE AVERAGE AVERAGE FRACTION DEBT AVERAGE MARGINAL MARGINAL FRACTION OF OF OF COST COST OF EQUITY COST COST MM COST SO OF FIRM DEBT EQUITY OF DEBT OF EQUITY DEBT RATIO OF CAPITAL OF DEBT OF DEBT EQUITY 75.CCO 0.000 75 .COO 0. 050000 1 .000000 0.000000 0 .000000 1.000000 0.050000 0.000000 1.000000 j 84.500 10.000 74.500 0.050000 1 .000000 0.118343 0 .134228 0.887574 0.050000 0.050000 0.881657 | 94.000 20.000 74 .COO 0. 050000 1 .000000 0.212766 0 .270270 0.797872 0.050000 0.050000 0.787234 i 103.5C0 30.000 73 .500 0. 050000 1 .000000 0.289855 0 .408163 0.724638 0.050000 0.050000 0.710145 j 113.000 40.000 73 .COO 0. 050000 1 .000000 0.353982 0 .547945 0.663717 0.050000 0.050000 0.646018 j 122.500 50.000 72 .500 0. 050000 1 .000000 0.408163 0 .689655 0.612245 0.050000 0.050000 0.591837 ! 132.CC0 60.000 72 .000 0. 050000 1 .000000 0.454545 0 .833333 0.568182 0.050000 0.050000 0.545455 141.5C0 70.000 71 .500 0. 050000 1 .000000 0.494700 0.979021 0.530035 0.050000 0.050000 0,505300 151.000 80.000 71 .COO 0. 050000 1 .000000 0.529801 1 .126761 0.496689 0.050000 0.050000 0.470199 160.500 90.000 70 .500 0. 050000 1 .000000 0.560748 1 .276596 0.467290 0.050000 0.050000 0.439252 17C.CC0 100.000 70 .COO 0.050000 1 .000000 0.588235 1.428571 0.441176 0.050000 0.050000 0.411765 179.500 110.000 69 .500 0. 050000 1 .000000 0.612813 1 .582734 0.417827 0.050000 0.050000 0.387187 189.000 120.000 69 .000 0. 050000 1 .000000 0.634921 1 .739130 0.396825 0.050000 0.050000 0.365079 198.5C0 130.000 68 .500 0. 050001 1 .000000 0.654912 1 .897812 0.377834 0.050049 0.050008 0.345088 207.998 140.000 67 .998 0.050017 1 .000000 0.673085 2 .058895 0.360581 0.050489 0.050228 0.326915 217.488 150.000 67 .488 0. 050078 1 .000000 0.689692 2 .222608 0.344846 0.051484 0.050936 0.310308 226.966 160.000 66 .966 0. 050214 1 .000000 0.704952 2 .389283 0.330446 0.053154 0.052258 0.295048 236.423 170.000 66.423 0. 050456 1 .000000 0.719052 2 .559372 0.317229 0.055619 0.054316 0.280948 245.850 180.000 65 .850 0. 050832 1 .000000 0.732 153 2 .733474 0.305064 0.058999 0.057228 0.267847 255.239 190.000 65 .239 0. 051373 1 .000000 0.744400 2 .912364 0.293842 0.063414 0.061116 0.255600 264.578 200.000 64 .578 0. 052109 1 .000000 0.755920 3 .097024 0.283470 0.068984 0.066093 0.244080 273.855 210.000 63 .855 0. 053071 1 .000000 0.766029 3 .288692 0.273867 0.075829 0.072296 0.233171 O 283. 057 220.000 63.057 0. 054287 1 .000000 0 .777229 3.488913 0.264964 0. 084069 0. 079828 0. 222771 292. 169 230.000 62.169 0. 055788 1 .000000 0 .787216 3.699609 0.256701 0.093824 0. 088816 0. 212784 301. 175 240.000 61.175 0. 057604 1 .000000 0 .796879 3.923174 0.249025 0. 105214 0. 099378 0. 203121 310. 059 250.000 60.059 0.059766 1 .000000 0 .806299 4. 162602 0.241890 0. 118359 0. 111636 0. 193701 318. 802 260.000 58.802 0.062302 1 .000000 0 .815554 4.421655 0.235256 0. 133379 0. 125708 0. 1844.46 327. 384 270.000 57.384 0. 065243 1 .000000 0 .824719 4.705115 0.229089 0. 150394 0. 141716 0. 175281 335. 787 280.000 55.787 0. 068619 1 .000000 0 .833863 5.019129 0.223356 0. 169524 0. 159778 0. 166137 343. 986 290.000 53.986 0. 072461 1 .000000 0 .843057 5.371721 0.218032 0. 190889 0. 180016 0. 156943 351. 961 300.000 51.961 0. 076797 1.000000 0 .852367 5.773568 0.213092 0.214609 0. 202548 0.147633 359. 686 310.000 49.686 0. 081658 1 .000000 0.861863 6.239184 0.208515 0. 240804 0. 227496 0. 138137 367.136 320.000 47.136 0. 087074 1 .000000 0 .871611 6.788837 0.204284 0. 269594 0. 254978 0. 128389 374.285 330.000 44.285 0. 093076 1 .000000 0 .88168 1 7.451726 0.200382 0.301099 0.285116 0. 118319 381. 105 340.000 41 .105 0. 099692 1 .000000 0 .892143 8.271547 0.196796 0. 335439 0.318028 0. 107857 387. 566 350.000 37.566 0.106953 1 .000000 0 .903071 9.316834 0.193515 0. 372734 0. 353836 0. 096929 393. 640 360.000 33.640 Oo 114889 1 .000000 0 .914542 10.701599 0.190529 0. 413104 0.392658 0.085458 399. 294 370.000 29.294 0. 123531 1 .000000 0 .926636 12.630713 0.187832 0.456669 0.434616 0. 073364 404. 495 380.000 24.495 0. 132907 1 .000000 0 .939442 15.513122 0.185416 0.503549 0. 479828 0.060558 409. 211 390.000 19.211 0. 143048 1 .000000 0 .953053 20.300620 0.183279 0. 553864 0. 528416 0. 046947 413.406 400.000 13.406 0. 153984 1 .000000 0 .967571 29.836818 0.181420 0. 607734 0.580498 0. 032429 417.044 410.000 7.044 0. 165746 1 .000000 0 .983109 58.203133 0.179837 0. 665279 0. 636196 0c 016891 420. 088 420.000 0.088 0. 178362 1 .000000 0 . 9 9 9 7 9 0 * * * * * * * * * * 0.178534 0. 726619 0. 695627 0. 000210 422. 499 430.000 -7.501 0. 191863 1 .000000 1 .017754- 57.324680 0.177515 0. 791874 0. 758915 -0. 017754 THE VALUE OF THE-FIRM FOR VARYING AMOUNTS OF DEBT NET INCOME MODEL THE FIRM HAS DETERMINISTIC EARNINGS OF $ 125. THE FIRM HAS AN AVERAGE TAX RATE OF 0.000 THE COST OF EQUITY IS GIVEN BY THE FUNCTION 1.00000000+ THE COST OF DEBT IS GIVEN BY THE FUNCTION 0.05000000+ SELECT= TELECT= DELT= 10. -O.OOOOOOOOL 0.0000000QS(L'ia5)3 ABAR= 125.SENT= - 0 . VALUE . AMOUNT VALUE . . AVERAGE AVERAGE FRACTION DEBT AVERAGE MARGINAL MARGINAL FRACTION OF OF OF COST COST OF EQUITY COST COST MM COST SO OF FIRM DEBT EQUITY OF DEBT OF EQUITY DEBT RATIO OF CAPITAL -OF DEBT OF DEBT EQUITY 125. CCO 0.000 125.COO 0.050000 1 .000000 o.ooocoo 0.000000 1.000000 0.050000 0.000000 1. 000000 134. 500 10.000 124.500 0.050000 1 .000000 0.074349 0.080321 0.929368 0.050000 0.050000 0. 925651 144. COO 20.000 124.000 0. 050000 1 .000000 0.138889 0. 161290 0.868056 0.050000 0.050000 0. 861111 153.5C0 30.000 123. 500 0. 050000 1 .000000 0.195440 0.242915 0.814332 0.050000 0.050000 0. 804560 163. 000 40.000 123. COO 0. 050000 1 .000000 0.245399 0.325203 0.766871 0.050000 0.050000 0. 754601 172. 500 50.000 122. 500 0. 050000 1 .000000 0.289855 0.408163 0.724638 0.050000 0.050000 0. 710145 182.OCO 60.000 122. 000 0. 050000 1 .000000 0.329670 0.491803 0.686813 0.050000 0.050000 0. 670330 191. 500 70.000 121. 5C0 0. 050000 1 .000000 0.365535 0.576132 0.652742 0.050000 0.050000 0. 634465 201. COO 80.000 121. 000 0. 050000 1 .000000 0.398010 0.661157 0.621891 0.050000 0.050000 0. 601990 210. 500 90.000 120. 500 0. 050000 1 .000000 0.427 5 53 0.746888 0.593824 0.050000 0.050000 0. 572447 220. CCO 100.000 120. 000 0. 050000 1 .000000 0.454545 0.833333 0.568182 0.050000 0.050000 0. 545455 229. 500 110.000 119. 500 0. .050000 1 .000000 0.479303 0.920502 0.544662 0.050000 0.050000 0. 520697 239. 000 120.000 119. 000 0. 050000 1 .000000 0.502092 1.008403 0.523013 0.050000 0.050000 0. 497908 248.5C0 130.000 118. 500 0. 050001 1 .000000 0.523139 1.097047 0.503018 0.050049 0.050008 0. 476861 257. 998 140.000 117. 998 0. 050017 1 .000000 0.542641 1. 186464 0.484501 0.050489 0.050228 0.457359 267. 488 150.000 117. 488 0. 050078 1 .000000 0.560772 1.276723 0.467310 0.051484 0.050936 0. 439228 276. 966 .160.000 116. 966 0. 050214 1 .000000 0.577689 1.367922 0.451319 0.053154 0.052258 0.422311 286.423 170.000 116.423 0. 050456 1 .000000 0.593529 1.460198 0.436418 0.055619 0.054316 0.406471 295. 850 180.000 115. 850 0. 050832 1 .000000 0.608416 1.553730 0.422511 0.058999 0.057228 0. 391584 305. 239 190.000 115. 239 0. 051373 1 .000000 0.622463 1.648746 0.409515 0.063414 0.061116 0.377537 314. 578 200.000 114. 578 0. 052109 1 .000000 0.635772 1.745534 0.397358 0.068984 0.066098 0. 364228 323.855 210.000 . 113. 855 0. 053071 1 .000000 0.648438 1.844449 0.385975 0.075829 0.072296 0. 351562 333.057 220.000 113.057 0. 054287 1.000000 0. 660548 1. 945923 0.375311 0. 084069 0.079828 0.339452 342.169 230.000 112.169 0. 055788 1.000000 0. 672183 2. 050482 0.365317 0.093824 0.088816 0.327817 351.175 240.000 111.175 0. 057604 1.000000 0. 683420 2. 158760 0.355948 0. 105214 0.099378 0.316580 360.059 250.000 110.059 0. 059766 1.000000 0. 694331 2. 271517 0.347166 0. 118359 0.111636 0.305669 368.802 260.000 108.802 0. 062302 1.000000 0. 704986 2.389673 0.338936 0.133379 0.125708 0.295014 377.384 270.000 107.384 0. 065243 1.000000 0. 715451 2. 514333 0.331227 0. 150394 0.141716 0.284549 385.787 280.000 105.787 0. 068619 1.000000 0.725790 2. 646839 0.324013 0. 169524 0.159778 0.274210 393.986 290.000 103.986 0. 072461 1.000000 0. 736066 2. 788826 0.317270 0. 190889 0.180016 0.263934 401.961 300.000 101.961 0.076797 1.000000 0. 746341 2. 942303 0.310975 0. 214609 0.202548 0.253659 409.686 310.000 99.686 0. 081658 1.000000 0. 756677 3. 109765 0.305112 0. 240804 0.227496 0.243323 417.136 320.000 97.136 0. 087074 1.000000 0. 767136 3. 294343 0.299662 0. 269594 0.254978 0.232864 424.285 330.000 94.285 0. 093076 1.000000 0.777779 3. 500025 0.294613 0. 301099 0.285116 0.222221 431.105 340.000 91.105 0. 099692 1.000000 0. 788671 3. 731967 0.289953 0. 335439 0.318028 0.211329 437.566 350.000 87.566 0. 106953 1.000000 0. 799879 3. 996966 0.285671 0. 372734 0.353836 0.200121 443.640 360.000 83.640 0. 114889 1.000000 .0.811469 4. 304169 0.281760 .0.413104 0.392658 0.188531 "* 449.294 370.000 79.294 0.123531 1.000000 0. 823515 4. 666198 0.278214 0. 456669 0.434616 0. 176485 454.495 380.000 74.495 0. 132907 1.000000 0. 836092 5. 100987 0.275030 0. 503549 0.479828 0. 163908 459.211 390.000 69.211 0.143048 1.000000 0. 849282 5. 634923 0.272206 0. 553864 0.528416 0. 150718 463.406 400.000 63.406 0. 153984 1.000000 0.863173 6. 308526 0.269742 0. 607734 0.580498 0. 136826 467.044 410.000 57.044 0. 165746 1.000000 0. 877861 7. 187397 0.267641 0. 665279 0.636196 0.122139 470.088 420.000 50.088 0. 178362 1.000000 0. 893450 8. 385239 0.265908 0. 726619 0.695627 0.106550 472.499 430.000 42.499 0. 191863 1.000000 0. 910055 10.117916 0.264551 0.791874 0.758915 0.089945 474.237 440.000 34.237 0.206279 1.000000 0. 927806 12. 851563 0.263581 0.861164 0.826179 0.072194 475.262 450.000 25.262 0.221641 1.000000 0.946847 17.813510 0.263013 0. 934609 0.897536 0.053153 475.531 460.000 15.531 0. 237977 1.000000 0. 967340 29. 618880 0.262864 1. 012329 0.973109 0.032660 475.CCO 470.000 5.COO 0.255318 1.000000 0.989473 93.990874 0.263158 1.094444 1.053015. 0.010527 473.627 480.000 -6.373 0.273694 1.000000 1.013456-75.314352 0.263921 1.181074 1.137377 -0.013456 

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