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UBC Theses and Dissertations

Investment decisions under risk and the Modigliani and Miller Hypothesis Gilley, Donald Robin 1967

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INVESTMENT DECISIONS UNDER RISK AND THE MODIGLIANI AND MILLER HYPOTHESIS by DONALD ROBIN GILLEY B.A.Sc, U n i v e r s i t y of Toronto,  1953  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the F a c u l t y of Commerce and Business A d m i n i s t r a t i o n  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1967  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  the requirements for 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 for reference  and  study*  I f u r t h e r agree that per-  mission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the Head of my Department or  his representatives.  I t i s understood that,-copying or p u b l i -  c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not without my  written  Commerce a n d  BusineS3  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 Canada ?  Date  March 31.  be  allowed  permission*  D o n a l d R. Department of  by  1967  Gilley  Administration  ABSTRACT  Although we l i v e i n a world of considerable  uncertain-  ty and chance, most c a p i t a l investment d e c i s i o n s consider the element of r i s k only q u a l i t a t i v e l y , i f a t a l l .  The b e l i e v e d  r i s k should be an e x p l i c i t and q u a n t i t a t i v e part of the norma l excess present value or excess rate o f r e t u r n method of investment a n a l y s i s . These r i s k s are described by the s u b j e c t i v e probabil i t y d i s t r i b u t i o n of p o s s i b l e investment outcomes and the c o e f f i c i e n t of v a r i a t i o n of t h i s d i s t r i b u t i o n i s a measure of the r e l a t i v e r i s k .  A t the same time, only  incremental  r i s k i s relevant which depends upon the e x i s t i n g earnings r i s k as w e l l as the p r o j e c t earnings r i s k and the c o e f f i c i e n t of a s s o c i a t i o n between these streams. Risk bears on the investment v a l u a t i o n through the i n v e s t o r ' s a t t i t u d e which i s conditioned by h i s sense of economic wealth and h i s p s y c h o l o g i c a l r e a c t i o n to the r i s k phenomenon.  - i i -  - i i i-  This f e l t r i s k can be q u a n t i f i e d through the i n v e s t or's t r a d e - o f f between income and r i s k , or h i s u t i l i t y of money f u n c t i o n .  This i s then used to modify the u n c e r t a i n  expected income to an equivalent c e r t a i n income which i s then evaluated i n the normal way.  However, t h i s i s only  f e a s i b l e f o r i n d i v i d u a l i n v e s t o r s or small groups of coinvestors. Por corporate investment d e c i s i o n s i t i s p r e f e r a b l e to r e l a t e the r i s k to a v a r i a b l e rate of required r e t u r n or market discount.  This rate then enables the u n c e r t a i n ex-  pected income to be evaluated d i r e c t l y In the usual manner. This method i s a p p l i c a b l e on any e n t i t y b a s i s i n c l u d i n g the i n d i v i d u a l p r o j e c t which i s the u n i t of investment d e c i s i o n . Here the venture has a unique r i s k w i t h an appropriate s t r u c t u r e and cost of c a p i t a l funds.  capital  In f a c t , t h i s method  of e v a l u a t i o n depends upon the existence of a v a l u a t i o n f u n c t i o n expressing the cost of corporate c a p i t a l under r i s k . The cost of c a p i t a l has been a d i f f i c u l t concept to define and measure while the aspect of r i s k has  received  l i t t l e a t t e n t i o n . Thus the rigorous M o d i g l i a n i and M i l l e r statement of the v a l u a t i o n of earnings under r i s k i s h i g h l y significant.  Here earnings r i s k i s c l a s s i f i e d on the b a s i s  - iv -  of equal c o e f f i c i e n t of v a r i a t i o n and perfect c o r r e l a t i o n . The use of debt c a p i t a l creates f i n a n c i a l r i s k but cost advantages under tax.  displays  However, leverage i s r e s t r a i n e d  by an i n t e r e s t rate f u n c t i o n which i s r e l a t e d to f i n a n c i a l r i s k and the uncertainty  of c r e d i t o r payments.  I m p l i c i t i n the formulation  of t h i s hypothesis i s  an i n v e s t o r l o s s a v e r s i o n a t t i t u d e which might be broadened i n t o a r i s k aversion basis of v a l u a t i o n .  The  comprehensive  hypothesis, w i t h a point of minimum cost of c a p i t a l , prov i d e s , a strong t h e o r e t i c a l p o s i t i o n but i s d i f f i c u l t to empirically validate. The v a l u a t i o n of a f t e r - t a x earnings under v a r i a b l e r i s k can be i n f e r r e d from the M o d i g l i a n i and M i l l e r hypothesis.  From t h i s can be derived a general expression f o r  the marginal value of an investment under r i s k .  This  includes  the s p e c i a l case, u s u a l l y assumed, where the investment i n come i s of equivalent r i s k and p e r f e c t l y c o r r e l a t e d to the e x i s t i n g corporate income.  The method may  be used to eva-  l u a t e a l t e r n a t i v e f i n a n c i n g arrangements and mutually e x c l u s i v e p r o j e c t s as w e l l as insurance proposals. This v a r i a b l e rate of discount or return concept provides a d i r e c t and i n t u i t i v e l y appealing means of adding  -  V  another dimension to the a n a l y s i s of investment o p p o r t u n i t i e s . Although there i s need f o r t h e o r e t i c a l development,  empiri-  c a l v e r i f i c a t i o n and o r g a n i z a t i o n a l acceptance of t h i s approach, i t i s perhaps a basis f o r improved corporate investment dec i s i o n s under r i s k .  TABLE OF CONTENTS Page ABSTRACT  i i  LIST OF FIGURES  viii  Chapter I . CAPITAL INVESTMENT DECISIONS UNDER RISK . . .  1  A. Risk i n C a p i t a l Investment Investor Expectations D e f i n i t i o n of Risk Measurement of R i s k B. Risk and Investment Decisions A t t i t u d e Towards Risk C e r t a i n t y Equivalents V a r i a b l e Rate of Discount or Return C o r r e l a t i o n of Outcomes C. Summary II.  THE MODIGLIANI AND MILLER VALUATION HYPOTHESIS A. V a l u a t i o n Under R i s k Introduction B a s i s of V a l u a t i o n Risk Equivalency B. Cost of C a p i t a l Hypothesis Leverage and R i s k Basic P r o p o s i t i o n s C. P r i n c i p l e M o d i f i c a t i o n s Income Tax I n t e r e s t Rates Expected Growth  - vi-  55  - vii-  Chapter  Page D. Conclusion Investor A t t i t u d e s Summary  III.  INVESTMENT VALUATION AND DECISIONS UNDER RISK  100  A. Earnings I n t e r r e l a t i o n s h i p s and V a l u a t i o n I n t e r c l a s s Relationships Tax and Leverage E f f e c t s C a p i t a l Sources E f f e c t s Earnings C o r r e l a t i o n B. Investment Decisions under R i s k The Decision Process Examples Further Considerations C• Conclusion BIBLIOGRAPHY  .....  132  APPENDIX I . SUMMARY OF NOTATION  136  APPENDIX I I .  138  EQUILIBRATING MARKET MECHANISM  LIST OP FIGURES Figure 1. 2.  „ 3.  Page Development o f P r o b a b i l i t y D i s t r i b u t i o n of Earnings from Forecasted States of Nature . .  5  Development o f P r o b a b i l i t y D i s t r i b u t i o n o f Earnings from Estimated Extreme and C e n t r a l Outcomes  5  The Uncertainty - Risk - C e r t a i n t y Continuum  . .  4. D i f f e r e 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 s w i t h Same Expected Value 5. 6.  13  D i f f e r e 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 s w i t h Same Extreme Values I n d i f f e r e n c e Maps Representing  10  13  Investor  A t t i t u d e s Towards R i s k  25  7.  The U t i l i t y o f Money Function of an Investor . .  30  8.  Convex Segment i n Investor's U t i l i t y Function  30  9.  Common A d d i t i o n of Two Income Stream Distributions A d d i t i o n of Two T y p i c a l Income Stream Distributions  10. 11. 12. 13.  .  41 41  Average Annual Earnings from Terminal Expectation  60  Average Annual Earnings from Growth Expectation  60  P r o b a b i l i t y D i s t r i b u t i o n s o f Earnings o f Equivalent Risk  63  - viii -  - ix -  Figure 14.  Page F i n a n c i a l R i s k to Net Earnings due to Leverage  67  15.  Earnings C a p i t a l i z a t i o n Rates under Leverage .  Jk  16.  Earnings C a p i t a l i z a t i o n Rates under Tax . . . .  79  17.  I n t e r e s t Rate as a Function of Leverage . . . .  83  18.  C a p i t a l i z a t i o n Rates under R i s i n g I n t e r e s t Rate  =  86  19.  Modified C a p i t a l i z a t i o n Rates  86  20.  I m p l i c i t Earnings I n d i f f e r e n c e Map  91  21.  I m p l i c i t Earnings U t i l i t y Function . . . . . . .  91  22.  Comprehensive Earnings C a p i t a l i z a t i o n Rates . .  93  23.  I n t e r c l a s s Earnings R i s k R e l a t i o n s h i p  2k.>• Schedule of V a l u a t i o n of A f t e r Tax Expected Earnings under R i s k . . . 25. H y p o t h e t i c a l General Scheme of V a l u a t i o n under R i s k  102 I l l 115  CHAPTER I CAPITAL INVESTMENT DECISIONS UNDER RISK A.  Risk i n C a p i t a l Investments  Investor Expectations The  c a p i t a l investment d e c i s i o n i s t y p i c a l l y  based upon a c a r e f u l estimate of the future  operating  r e c e i p t s and payments as w e l l as the c a p i t a l funds required for a project.  The f i g u r e s derived f o r net income o r net  cash flow over the l i f e of the p r o j e c t are i m p l i c i t averages, perhaps the most probable or the median f i g u r e s .  Although  such s i n g l e valued p r e d i c t i o n s may represent a consensus of opinion o r a best estimate of performance, t h e i r only sure q u a l i t y i s that they w i l l be wrong. This i s not so much a problem of f o r e c a s t i n g  tech-  niques as r e c o g n i t i o n of many chance and l i t t l e understood v a r i a b l e s which w i l l e f f e c t the investment outcome.  Here  we must t r e a t these as s t o c h a s t i c v a r i a b l e s . The  use of a s i n g l e valued estimate implies complete  confidence i n the forecasted f i g u r e , which suggests a state - 1 -  - 2 -  of c e r t a i n t y . O n l y the past i s c e r t a i n , whereas we must work w i t h the future and make d e c i s i o n s which are based on the f u t u r e .  Investment d e c i s i o n s made i n a state of i m p l i -  c i t c e r t a i n t y i n a world of considerable l a c k of c e r t a i n t y could very w e l l be poor d e c i s i o n s .  Of course, i n v e s t o r s  u n i v e r s a l l y recognize the p o s s i b i l i t y of outcomes d i f f e r e n t from those f o r e c a s t .  Thus t h e i r expectations are not r e a l -  l y s i n g l e valued although they use such f i g u r e s i n making t h e i r decisions.  Perhaps t h i s f a c t i s somehow r e f l e c t e d  i n the estimated f i g u r e s or perhaps i t i s allowed f o r l a t e r at the point of d e c i s i o n i n some way. C a p i t a l investment i s an act of f a i t h , but r a t h e r than r e l y e s s e n t i a l l y upon the human d e s i r e s f o r a c t i o n and s e c u r i t y we can take the c a l c u l a t e d r i s k .  2  The many complex  v a r i a b l e s and u n c e r t a i n t i e s i n most investment d e c i s i o n s are d i f f i c u l t to encompass by means of an informal and i n t u i t i v e mental judgement process.  We need a sounder b a s i s  f o r r a t i o n a l and c o n s i s t e n t d e c i s i o n s towards our defined goal.  A l l the f a c t o r s which are i m p l i c i t i n most d e c i s i o n s  can be e x p l i c i t l y included i n formal, systematic  analyses  and d e c i s i o n c r i t e r i a . * 1  I f i n v e s t o r s * expectations are multi-valued i t would  - 3 -  be p r e f e r a b l e to use multi-valued f o r e c a s t s i n the a p p r a i s a l and d e c i s i o n process.  The development of a multi-valued  estimate of operating cash flows or earnings i s based upon a c o n s i d e r a t i o n of a l l p o s s i b l e consequences of the i n v e s t ment.  This can be derived from cost and revenue estimates  a s s o c i a t e d w i t h various f o r e c a s t s of general business cond i t i o n s and s p e c i f i c market and competitive f o r c e s ^ ( i . e . v a r i o u s s t a t e s of "nature").  In a d d i t i o n to each one  of  these unique or mutually e x c l u s i v e p o s s i b l e outcomes, a s u b j e c t i v e estimate of the p r o b a b i l i t y of occurrence must be assigned to each event, w i t h a l l p r o b a b i l i t i e s summing to u n i t y . However, the development of multi-valued may  estimates  pose d i f f i c u l t i e s f o r many business p r a c t i t i o n e r s , 6 even  though they simply express i m p l i c i t b e l i e f s .  Monte Carlo  methods and the technique of computer s i m u l a t i o n can be usef u l i n developing a multi-valued estimate of a l l p o s s i b l e outcomes and t h e i r p r o b a b i l i t i e s . For short term f o r e c a s t horizons i n v o l v i n g more c o n t r o l l a b l e v a r i a b l e s , such as the i n i t i a l c a p i t a l i n v e s t ment r e q u i r e d , there i s normally a high degree of c e r t a i n ty.  ThU3  we s h a l l consider the estimates of i n i t i a l  capital  o u t l a y , r e s i d u a l value and replacement cost as c e r t a i n t i e s and thus s i n g l e - v a l u e d .  - 4 -  However, f o r long term f o r e c a s t s of l e s s c o n t r o l l a b l e and l e s s p r e d i c t a b l e  v a r i a b l e s , such as net income ( i n -  c l u d i n g c o s t s , revenue and economic l i f e ) a multi-valued f o r e c a s t may add a great deal to the d e s c r i p t i o n of future possibilities.  This added s o p h i s t i c a t i o n may be of p a r t i -  c u l a r value i n cases of l a r g e r investment p r o j e c t s o r groups of s i m i l a r investments. In the most e x p l i c i t case the estimate would be i n the form of a p r o b a b i l i t y d i s t r i b u t i o n over a l l p o s s i b l e earnings.  Such a structure i s not so much a p r e d i c t i o n as  a personal b e l i e f h e l d by the d e c i s i o n maker.7  i n the case  of l e s s p r e c i s e b e l i e f s , a t l e a s t some v i r t u a l l y worst and best outcomes and some average outcome are u s u a l l y formed i n the mind.^  Here some appropriate p r o b a b i l i t y d i s t r i b u -  t i o n must be erected on these three p r o b a b i l i t y points.9 F i g u r e s 1 and 2 i n d i c a t e g r a p h i c a l l y the form i n which these estimates are developed. The  most probable or the median value of earnings  i n a p r o b a b i l i t y d i s t r i b u t i o n of possible earnings are s i m i l a r to the single-valued  estimate which i s t y p i c a l l y used.  However the development of a subjective  probability d i s t r i -  b u t i o n does not lend any h i g h e r degree o f c e r t a i n t y to these  - 5 -  Possible  States of Nature and Outcomes  weak dem. mod. dem. weakdem. strong dm. mod. dem. strong dm. over sup. over sup. under sup. over sup. under sup. under sup.  \  n  ro  \  -Q O  \  Earnings F i g . 1.--Development of P r o b a b i l i t y D i s t r i b u t i o n of Earnings from Forecasted States of Nature  Possible Range of  Outcomes  Earnings F i g . 2.--Development of P r o b a b i l i t y D i s t r i b u t i o n of Earnings from Estimated Extreme and C e n t r a l Outcomes  - 6 -  values.  Thi3 i s because there i s only one p o s s i b l e outcome  from one investment i n time r a t h e r than a set of random outcomes from many r e p e t i t i o n s of the a c t . A s i n g l e investment i s thus not subject to the law of large numbers such as a group of Independent investments would be. Thus the statement of expectations i n the form of a p r o b a b i l i t y d i s t r i b u t i o n over a l l p o s s i b l e outcomes desc r i b e s the gamble which the i n v e s t o r b e l i e v e s he faces i n the p r o j e c t .  Of course the f u l l e r and more accurate i s our  knowledge, the c l o s e r our b e l i e f s w i l l be to the true s i t u a t i o n i n which we l i v e .  I t i s t h i s e x p l i c i t q u a n t i t a t i v e ex-  p r e s s i o n of b e l i e f s that makes the multi-valued estimate so u s e f u l f o r r a t i o n a l a n a l y s i s and c o n s i s t e n t behaviour. The use of multi-valued estimates r e f l e c t the unique u n c e r t a i n t i e s involved i n the performance of an investment and summarize the unique r i s k s a s s o c i a t e d w i t h the venture. T h i s i s perhaps p a r t i c u l a r l y u s e f u l i n t h i s era of business d i v e r s i f i c a t i o n i n t o l e s s r e l a t e d operations or areas. The purpose of t h i s paper i s to explore some concepts and methods whereby t h i s element of r i s k can be e x p l i c i t l y i n cluded i n the c a p i t a l investment a p p r a i s a l and d e c i s i o n process.  - 7  -  D e f i n i t i o n of R i s k In the case of b e l i e v e d absolute c e r t a i n t y of  the  f u t u r e outcome of a present d e c i s i o n or a c t , the p r o b a b i l i t y d i s t r i b u t i o n would collapse a p r o b a b i l i t y of u n i t y .  i n t o one  s i n g l e outcome w i t h  That t h i s s i t u a t i o n r a r e l y , i f  ever, e x i s t s i s u n i v e r s a l l y accepted, although i t may  be  approached. At the other extreme we could t r y to imagine the case of complete u n c e r t a i n t y as to what the possible comes might be and complete u n c e r t a i n t y as to any t i a l p r o b a b i l i t y of one  outcome over a n o t h e r .  10  out-  differenThis i a  the complete ignorance case where one has no knowledge or b a s i s f o r b e l i e f that any probable than any t i o n collapses  other.  outcome i s any more p o s s i b l e 1 1  or  Here the p r o b a b i l i t y d i s t r i b u -  i n t o an equal p r o b a b i l i t y of v i r t u a l l y n i l  f o r any and a l l conceivable outcomes.  Such an absolute  l a c k of knowledge or b e l i e f r a r e l y e x i s t s , even i n the cosmos. Much a t t e n t i o n has been given to the case of comp l e t e u n c e r t a i n t y but we  s h a l l not consider i t here.  though d e c i s i o n theory might d i c t a t e use of the  Al-  principle  of i n s u f f i c i e n t reason (equi-probable states of n a t u r e )  1 2  we w i l l presume s u f f i c i e n t knowledge to permit of ex ante  - 8 -  subjective p r o b a b i l i t i e s .  I n the case of a c o n f l i c t of  i n t e r e s t and a w i l f u l opponent (e.g. state of market  oll-  g o l o p o l y ^ ) game theory might i n d i c a t e the use of the maxi1  min c r i t e r i o n . ^  However we s h a l l consider that the pro-  b a b i l i t i e s and e f f e c t s of competitors' p o s s i b l e r e a c t i o n s t r a t e g i e s are b e l i e v e d known and are b u i l t i n t o our m u l t i valued e s t i m a t e s . Between the two extremes o f complete c e r t a i n t y and complete ignorance l i e s the area i n which we have some b a s i s f o r b e l i e f i n some f i n i t e range of m u l t i p l e mutually e x c l u s i v e p o s s i b l e outcomes with  3ome  more or l e s s vague  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 over i t .  T h i s i s de-  ls  f i n e d as the case of r i s k  v  which could be termed the case  of s i g n i f i c a n t knowledge o r b e l i e f .  Although there may be,  i n r e a l i t y , some d i f f e r e n c e between knowledge and b e l i e f , due to the human psyche, we s h a l l t r e a t them as synonymous here ( i . e . assume b e l i e f i s r a t i o n a l l y based upon some true state of knowledge.) Thus c e r t a i n t y i s r e a l l y a degenerate case of r i s k ( i . e . case of no r i s k ) while complete ignorance or u n c e r t a i n t y i s a case of i n f i n i t e r i s k . Although we t r e a t matters of r i s k as i f they were cases of random outcomes, u s u a l l y we must a l s o t r e a t matters  - 9 of u n c e r t a i n t y as random events due to our l a c k of knowledge of the p o s s i b l e d e t e r m i n i s t i c forces a t work. ^ 1  Much d i s c u s s i o n has been generated over the concept of c e r t a i n t y as the degree of confidence we h o l d i n our IT  b e l i e f s . '*  "1 ft  This would seem to confuse the concepts of  s u b j e c t i v e or personal p r o b a b i l i t y , based upon ex ante i n formation and b e l i e f , and that of o b j e c t i v e p r o b a b i l i t y or r e l a t i v e frequency d i s t r i b u t i o n s , based upon ex post i n formation or o b s e r v a t i o n . ^ 1  The range and p r o b a b i l i t i e s  of p o s s i b l e outcomes which we define express our b e l i e f s r e garding the c e r t a i n t y or u n c e r t a i n t y of the outcome.  The  f u l l e r i s our p r i o r knowledge, both information and understanding, the l e s s d i s t r i b u t e d our b e l i e f might become. Figure 3 i l l u s t r a t e s the continuum of cases from complete u n c e r t a i n t y , through r i s k t o complete  certainty.  Risk i s thus associated w i t h the degree of b e l i e v e d v a r i a b i l i t y of the p o s s i b l e outcome of the investment. Measurement of R i s k Although 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 of p o s s i b l e future outcomes describes the b e l i e v e d v a r i a b i l i t y of the outcome of the investment ( i . e . the b e l i e v e d  Absolute Certainty  F i g . 3.—The Uncertainty  - R i s k - C e r t a i n t y Continuum  - 11  -  r i s k inherent i n the venture) we require a more compact d e s c r i p t i o n of t h i s q u a n t i t a t i v e r i s k f o r ease of comprehension and manipulation. The use of the mathematical  expectation ( i . e . the  weighted mean, or the sum across the product of each outcome and i t s p r o b a b i l i t y ) i s almost u n i v e r s a l l y r e j e c t e d as i n d i c a t i n g anything of the r i s k of a p r o j e c t .  I t has  been advanced as an adequate d e s c r i p t i o n of a r i s k y p r o j e c t on the b a s i s that i t considers both the magnitude of each and every p o s s i b l e outcome as w e l l as i t s a s s o c i a t e d probability.  But too much i s l o s t i n the aggregation  and  p r o j e c t s of equivalent expected value may have t o t a l l y d i f f e r e n t degrees and type of v a r i a b i l i t y as shown i n F i gure 4. Thus the expected v a l u e , i n terms of monetary outcomes ( i . e . expected monetary value) i s not r e l a t e d to the r i s k of a p r o j e c t because there w i l l be only one eventual outcome ( i . e . one random t r i a l ) . ^ 2  The outcome which ob-  t a i n s could be the worst extreme and have d i s a s t r o u s f i n a n cial significance.  2 1  Use of t h i s s t a t i s t i c f o r d e c i s i o n  would l e a d one to chose the highest expected v a l u e ,  re-  gardless of the degree of r i s k i n v o l v e d , which i s a reckl e s s p a t t e r n of b e h a v i o u r .  22  Here a h i g h l y r i s k y  investment  -  12  -  w i t h an expectation of a very s l i g h t and i n s i g n i f i c a n t g a i n would be p r e f e r r e d to the status quo.  A purely  mathematical  o b j e c t i o n i s that any unbounded p r o b a b i l i t y d i s t r i b u t i o n has an expected value of i n f i n i t y ,  regardless of any other  c h a r a c t e r i s t i c s , and would have to be valued as  such.23  A l t e r n a t i v e l y the use of the values of the extreme outcomes has o c c a s i o n a l l y been proposed as a s u i t a b l e , comprehensive d e s c r i p t i o n of a r i s k y income. ^ 2  This i s equi-  valent to r e j e c t i n g the n o t i o n of p r o b a b i l i t i e s of outcomes and focusing upon extreme f i n i t e outcomes only or to d e f i n i n g a t o l e r a b l e l e v e l of i n s i g n i f i c a n t p r o b a b i l i t y and focusing on the b e l i e v e d a s s o c i a t e d outcomes. 5 2  i n e i t h e r case igno-  rance i s i m p l i e d as to the p r o b a b i l i t i e s of intermediate outcomes and the matter of the c e n t r a l or average tendency of r i s k y income i s disregarded as shown i n Figure 5 . . Thus we have no s u i t a b l e measure which adequately describes both the c e n t r a l tendency and the v a r i a b i l i t y or r i s k of p r o j e c t income and we must t r e a t r i s k  independently.  This i s a problem i n d e s c r i p t i v e s t a t i s t i c s and i n p a r t i c u l a r , one of s e l e c t i o n of a s u i t a b l e measure of d i s p e r s i o n . Of the various measures of c e n t r a l tendency which might be used to describe such d i s t r i b u t i o n s of mutually  - 13 -  F i g . 5.--Different Probability D i s t r i b u t i o n s w i t h Same Extreme Values  - 14  -  e x c l u s i v e f i n a n c i a l outcomes, the median i s the most s i gnificant. ^ 2  "Where the closeness of t h i s average to the  a c t u a l outcome i s important, as i t i s here, the median i s the best estimate.  I t i s the l e a s t e r r o r value and i s  the value of the c e n t r a l event (50$ of the events have greater values and 50$ have l e s s e r values.)  The mean has  no independent s i g n i f i c a n c e , here, n e i t h e r does the sum of a l l p o s s i b l e mutually e x c l u s i v e outcomes upon which i t i a based.  However, w i t h the f a c t or assumption of symmetry  i n the d i s t r i b u t i o n , the mean becomes i d e n t i c a l t o , or an estimate o f , the median and i t i s much e a s i e r to manipulate i n computations.  The almost u n i v e r s a l use of the mean,  however, i s only s i g n i f i c a n t i n s o f a r as i t i s a good estimate of the median (e.g. under the c r u c i a l assumption of symmetry). In conjunction w i t h a measure of c e n t r a l tendency, the extreme values or the range of p o s s i b l e outcomes could c e r t a i n l y be u s e f u l measures of r i s k . ? ' ^ 2  2  i t may be that  such values were used i n the f i r s t place to define a probab i l i t y d i s t r i b u t i o n (see Figure 2 ) .  The only c r i t i c i s m of  such s t a t i s t i c s would be that the range does not define any absolute extreme values unless the d i s t r i b u t i o n i a symmet r i c a l , and the extreme v a l u e s , being two f i g u r e s , somewhat complicate manipulation.  - 15 -  The most widely used measure of r i s k i s the standard„.deviation ( o r the v a r i a n c e ) about the mean of the pro2Q SO  bability distribution.  This f o l l o w s from the almost  u n i v e r s a l use of the mean as an average, ( i . e . the mean i s the l e a s t squared.error value, while the variance i s the mean squared e r r o r .  ) Of course the squared e r r o r has no  independent s i g n i f i c a n c e  and the standard d e v i a t i o n i s thus  an a r b i t r a r y , r e l a t i v e measure o f r i s k .  Even under the  assumption o f symmetry, where the mean becomes s i g n i f i c a n t as an estimate of the median, the standard d e v i a t i o n , based upon the t o t a l squared e r r o r , remains e s s e n t i a l l y However, under the f u r t h e r , and o f t e n l e s s  arbitrary. drastic,  assumption o f approximate normality the standard d e v i a t i o n i s h i g h l y s i g n i f i c a n t i n that i t v i r t u a l l y defines the d i s persion ( i . e . i t i s v i r t u a l l y a s u f f i c i e n t s t a t i s t i c f o r risk.)  Where a d i s t r i b u t i o n i s s l i g h t l y assymmetrical, but  symmetry and normality a r e assumed, the use of the mean and  standard d e v i a t i o n define an i m p l i c i t l y equivalent nor-  mal  distribution. Where a 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 s u f f i c i e n t l y  skewed to p r o h i b i t the assumption of symmetry we are forced to e i t h e r ( i ) introduce a measure of the degree and d i r e c t i o n  -  16  -  of skewnes3 (e.g. 100 (mean-median)-^ standard d e v i a t i o n , as percentage skewness i n the d i r e c t i o n i n d i c a t e d by the s i g n ^ ) 2  which simply adds a t h i r d dimension to our d e s c r i p t i o n o f the d i s t r i b u t i o n and complicates the a n a l y s i s , o r ( i i ) t r a n s form the d i s t r i b u t i o n t o symmetrical form, using r e c i p r o c a l s and/or powers and/or logarithms, which i s only p r a c t i c a l on an approximate b a s i s , o r ( i i i ) use the median d i r e c t l y which may be d i f f i c u l t f o r many items and r a i s e s an impossible problem of d e f i n i n g the d i s p e r s i o n , as the standard deviat i o n about the median o r the mean absolute d e v i a t i o n about the median are not comparable to the U 3 u a l  statistic.  S u f f i c e t o say t h a t the problems of skewness are of such magnitude that there i s great i n c e n t i v e to assume symmetry, otherwise the best a l t e r n a t i v e may be t o hope f o r a simple and adequate transformation. Another p o s s i b i l i t y i s the use of the semi-deviation (or semi-variance) as a measure of r i s k , 3 3 which reduces the problems o f skewness.  Here a l l p o s i t i v e d e v i a t i o n s  are assigned a value o f zero i n the usual standard d e v i a t i o n computation.  Thus f o r a symmetrical  distribution,  twice the semi-deviation gives the standard d e v i a t i o n . This s t a t i s t i c summarizes the unfavourable ( i . e .  -  17  -  negative) d e v i a t i o n s which are o f t e n considered to he the e s s e n t i a l q u a l i t y of r i s k .  I n n e g l e c t i n g the d i s p e r s i o n  of the more favourable outcomes, except as they a f f e c t the average v a l u e , the p o s s i b i l i t i e s of g a i n cannot perversel y a f f e c t the r i s k s t a t i s t i c to the extent they do i n using the standard d e v i a t i o n . There, extremely l a r g e outcomes are considered aa bad as extremely main disadvantage  small outcomes. The  of the semi-deviation, other than i t s  o b s c u r i t y , i s the added computational work i t i n v o l v e s . The standard d e v i a t i o n about the mean of the prob a b i l i t y d i s t r i b u t i o n of a l l p o s s i b l e outcomes, as a measure o f r i s k , i s a measure of absolute v a r i a b i l i t y ( i . e . i t describes the b e l i e v e d v a r i a b i l i t y among outcomes i n the same b a s i c u n i t s as the outcomes ), a l b e i t an a r b i t r a r y s  one, but one which i s f r e q u e n t l y accepted as adequate. The d i f f e r e n c e i n r i s k between two p r o j e c t s , i n these absolute q u a n t i t a t i v e terms, could be compared but unless the reference p r o j e c t was an e s t a b l i s h e d c r i t e r i o n , or otherwise q u a l i t a t i v e l y d e f i n e d , we would not be much enlightened by t h i s comparison, as to the q u a l i t y of the project i n question.  Although t h i s absolute  statistic  gives us some information as t o the r i s k of a p r o j e c t , i t  - 18 -  i s u s u a l l y necessary t o r e l a t e i t i n 3ome way t o the mean i n order t o assess i t s s i g n i f i c a n c e f o r the investment decision. R i s k i s o f t e n a s s o c i a t e d w i t h the prospect o f l o s s , although i t i s , i n f a c t , j u s t as r e l a t e d t o the prospect of g a i n . and  I f we assume normality the two s t a t i s t i c s , mean  standard d e v i a t i o n , define the p r o b a b i l i t y of l o s s o r  g a i n and the mean o r expected 1 0 3 s o r g a i n .  Then an i s o -  expected l o s s versus i s o - p r o b a b i l i t y of l o s s map can be developed from these two measures. A l t e r n a t i v e l y , the quotient of these two measures could be used, such as the c o e f f i c i e n t of v a r i a t i o n ( i . e . standard d e v i a t i o n m e a n ) , a s a measure of r e l a t i v e v a r i a b i l i t y o r risk.34  T h i s compact s t a t i s t i c i s s u f f i c i e n t to  define the p r o b a b i l i t y o f l o s s o r g a i n , assuming n o r m a l i t y , but not the expected l o s s o r g a i n value and of course the absolute v a r i a t i o n s t a t i s t i c i a l o s t i n the process. ever, i f we are working with p e r f e c t l y d i v i s i b l e  How-  invest-  ments, as may be approached i n the case of common stocks o r j o i n t ventures, t h i s d e f i c i e n c y disappears a s the absolute s i z e of the investment may be v a r i e d .  Thus absolute v a r i a -  b i l i t y would not enter i n t o the a p p r a i s a l of r i s k q u a l i t y ,  - 19 -  although i t may enter i n t o the d e c i s i o n as to the extent of p a r t i c i p a t i o n i n the investment.35 In the case of p e r f e c t l y i n d i v i s i b l e investments, as may be approached i n the case of many r e a l investments, the absolute v a r i a b i l i t y of the outcomes i s a s i g n i f i c a n t q u a l i t y of the p r o j e c t .  B.  R i s k and Investment Decisions  A t t i t u d e Towards R i s k The normal c a p i t a l investment a n a l y s i s e s s e n t i a l l y compares the f u t u r e net income o r cash flow t o the present investment o u t l a y . This comparison  i s best accomplished  by e i t h e r the excess present value method or the excess rate of r e t u r n method.^»37>38 In the excess present value ( i . e . p r o f i t a b i l i t y index or b e n e f i t - c o s t ) approach, the f u t u r e income i s d i s counted to i t s present value a t a rate r e f l e c t i n g the over a l l cost of c a p i t a l to the f i r m , or the opportunity cost of funds t o the i n v e s t o r , and t h i s f i g u r e i s then compared to the present o u t l a y r e q u i r e d . The excess r a t e of r e t u r n ( i . e . i n t e r n a l r a t e of  -  r e t u r n o r discounted  20  -  r e t u r n on investment) approach i n -  volves the d e r i v a t i o n of the rate of r e t u r n on the p r o j e c t which equates the present outlay to the future income and t h i s f i g u r e i s then compared to a rate r e f l e c t i n g the overa l l cost of c a p i t a l t o the f i r m , or the opportunity  cost  of funds t o the i n v e s t o r . These methods are normally used to reduce future c e r t a i n income, where c e r t a i n t y i s i m p l i e d i n the s i n g l e v a l u e s , to present c e r t a i n income which i s r e l a t e d to present c e r t a i n o u t l a y , where t h i s c e r t a i n t y i s assumed.  Now we  wish to introduce the element of r i s k i n t o the d e c i s i o n process i n some way, i n order that i t may i n f l u e n c e our evaluat i o n of the proposed investment.  Here we w i l l he concerned  w i t h reducing future r i s k y income to present c e r t a i n income f o r comparison t o present c e r t a i n outlay i n the same manner. This immediately r a i s e s the questions of how to i n corporate the measure of r i s k i n t o the a n a l y s i s and how i t a f f e c t s the investment d e c i s i o n .  The answer to the l a t t e r  question w i l l perhaps suggest an approach to the former one. Given some compact q u a n t i t a t i v e measure of d i s p e r s i o n or r i s k of a p r o j e c t , the investment d e c i s i o n i s i n f l u enced through the i n v e s t o r ' s a t t i t u d e towards t h i s r i s k .  - 21  -  This a t t i t u d e w i l l be, f i r s t l y , conditioned by economic considerations and i n p a r t i c u l a r , the present and expected f u t u r e wealth of the i n v e s t o r which could include a l l r e a l i z a b l e b e n e f i t producing assets whether f i n a n c i a l , r e a l o r human.39*40,41  0  n the one hand there may be a cons-  t r a i n t against excessive l o s s , assuming a preference f o r solvency, i n terms of some small maximum p r o b a b i l i t y of bankruptcy o r , a t the extreme, l i f e l o n g poverty.  On the  other hand there may be a c e r t a i n f i n i t e d e s i r e f o r greater wealth and beyond, a region of l i t t l e valued g a i n , assuming some current l e v e l o f a s p i r a t i o n f o r wealth.  Here we have  the c l a s s i c f i n a n c i a l dichotomy pf p r o f i t versus  solvency.  Thus a l a r g e and very r i s k y p r o j e c t , i n r e l a t i o n to the i n v e s t o r ' s wealth may resolve i n t o a case of s t r i c t l o s s a v e r s i o n while a small p r o j e c t of low r i s k may not i n v o l v e wealth considerations a t a l l , beyond a simple preference f o r probable  gain.  In the second p l a c e , the i n v e s t o r ' s a t t i t u d e towards r i s k w i l l be p s y c h o l o g i c a l l y conditioned through the mental s t r e s s he f e e l s as a r e s u l t of the l a c k of c e r t a i n t y . This may be a c l o s e l y r e l a t e d , but nevertheless an added dimens i o n t o the economic c o n s i d e r a t i o n s above.  Some people more  - 22  -  e a s i l y bear g r e a t e r r i s k than others, even where the consequences may be the same, and some may even d e r i v e s a t i s f a c t i o n from chance unknowns, where others f e e l insecure.  Thus  the i n v e s t o r w i l l have a very subjective a t t i t u d e towards v a r i a b i l i t y per se. I f r i s k i s measured, by the standard d e v i a t i o n a s t r i c t a v e r s i o n to outcomes which deviate from the mean i s i m p l i e d , whether these be favourable or unfavourable deviations.  On the other hand the use of the semi-deviation  suggests an a v e r s i o n to unfavourable d e v i a t i o n s i n outcome but a s t r i c t i n d i f f e r e n c e to favourable v a r i a t i o n s .  Thus  the very s e l e c t i o n of the measure of r i s k c a r r i e s i m p l i c a t i o n s of i n v e s t o r a t t i t u d e s which should bear on the choice. We are then faced w i t h the problem of d e f i n i n g a s p e c i f i c i n v e s t o r ' s unique a t t i t u d e towards a measurable r i s k , at a s p e c i f i c point i n time, based upon the state of h i s wealth and psyche.  Furthermore t h i s a t t i t u d e must then  be q u a n t i t a t i v e l y r e l a t e d to investment v a l u e .  C e r t a i n t y Equivalents Many approaches to the a p p r a i s a l of r i s k y investments take the form of a d i r e c t m o d i f i c a t i o n of the average or  - 23  -  expected income f i g u r e as defined by the Investor's r e a c t i o n to the risk. *' 1  2  Perhaps the simplest method of accomplishing  this  i s f o r the i n v e s t o r to s u b j e c t i v e l y i n t e r p r e t the s i g n i f i cance of the measured r i s k and adjust the expected income as he sees f i t . ^ 3  Although t h i s approach may be t h e o r e t i c a l -  l y u n a s s a i l a b l e there a r e some p r a c t i c a l o b j e c t i o n s .  I t ne-  c e s s i t a t e s each i n d i v i d u a l i n v e s t o r having a thorough understanding and f a m i l i a r i t y w i t h the p a r t i c u l a r s t a t i s t i c used to measure r i s k so that he may i n t e l l i g e n t l y assess the s i g n i f i c a n c e of the p a r t i c u l a r degree o f r i s k i n v o l v e d .  This  method a l s o presumes that the i n v e s t o r ' s psyche i s not subj e c t t o temporary d e v i a t i o n s from some norm.  These are  perhaps unreasonably strong assumptions about the p r a c t i c i n g d e c i s i o n maker. To avoid some o f these problems, attempts have been made t o d e f i n e , more r i g o r o u s l y and g e n e r a l l y , the i n v e s t o r ' s a t t i t u d e towards r i s k i n order to provide a framework w i t h i n which any s p e c i f i c r i s k may be judged.  The o b j e c t i v e i s t o  provide a r a t i o n a l b a s i s o f i n c o r p o r a t i n g r i s k i n t o the i n vestment a n a l y s i s f o r consistent i n v e s t o r behavbur towards his  goals.44,45  Obviously,  the problems of attempting to  - 24  -  q u a n t i f y and give f u n c t i o n a l expression t o the Investor's a t t i t u d e s are enormous. One method of accomplishing  t h i s i s with r i s k  i n d i f f e r e n c e f u n c t i o n s o r maps generated from the i n v e s t o r ' s e v a l u a t i o n o f many d i s t r i b u t i o n s o f p o s s i b l e outcomes.^ T y p i c a l r i s k i n d i f f e r e n c e maps a r e shown i n Pigure 6. The r i s k i n d i f f e r e n c e map o r f u n c t i o n can be used to resolve any r i s k y set o f p o s s i b l e outcomes i n t o an e q u i valent c e r t a i n ( i . e . r i s k l e s s ) expectation.  These c e r t a i n -  ty equivalents are then used as the common u n i t f o r any i n vestment d e c i s i o n .  This i s a more e x p l i c i t a p p l i c a t i o n of  the p r o f i t a b i l i t y ( i . e . expected v a l u e ) - p r o b a b i l i t y ( i . e . inverse of d i s p e r s i o n ) i n d i f f e r e n c e method. Another method o f q u a n t i f y i n g the i n v e s t o r ' s a t t i tude towards r i s k would be t o have him weigh a l l p o s s i b l e outcomes according t o h i s aversion t o them,^ i n the f o l l o w i n g example.  as shown  - 26  (1) Possible Outcomes  (2) Probability of Outcome  $5000  .05  4000 3000 2000 +$1000  0  -$1000  (3) Aversion Schedule  (4)«(l)x(3) Weighted Outcomes  .30 • 35 .40  .10 .15 .20 .30 .15 .05 1.00  -  .55  • 75 1.00 1.40  (5)«(2M i l  1500 1400 1200 1100 + 750 0 -1400  75 140 180 220  +225 0 - 70  Weighted Monetary Value -  Expected Monetary Value -  770  $1750. 0  The weighted monetary value of 770 i s equivalent to a s i n g l e c e r t a i n monetary value of about  $1050 which i s  the c e r t a i n t y equivalent o f t h i s p r o j e c t , as compared t o the expected monetary value of  $1750.  The r a p i d l y f a l l i n g  preference f o r money schedule w i t h i n c r e a s i n g outcomes i n d i c a t e s a strong aversion t o r i s k . The method of determining the i n v e s t o r ' s schedule of money preference i s not d e f i n e d .  I f t h i s schedule i s  l i n e a r , however, i t would imply a uniform preference f o r money and thus complete r i s k i n d i f f e r e n c e .  Here the expect-  ed monetary value would be a s i g n i f i c a n t and adequate desc r i p t i o n of a r i s k y p r o j e c t as there i s no need to d i f f e r e n tiate risks for a risk indifferent investor.  - 27  -  A more rigorous method of accomplishing t h i s same purpose i s to introduce the von Neumann and Morgenstern u t i l i t y f u n c t i o n concept.  J  For any r i s k y set of ex ante  p o s s i b l e monetary outcomes, of which only one w i l l o b t a i n ex post, a worth or u t i l i t y i s attached to each consequence. The u t i l i t y value of a consequence i s based upon s u b j e c t i v e monetary preferences. Such a u s e f u l c a r d i n a l measure of u t i l i t y i s made p o s s i b l e through the method of generating the preferences. The i n v e s t o r i s o f f e r e d choices among p a i r s of h y p o t h e t i c a l gambles ( i . e . a c t s w i t h f i n i t e numbers of consequences conf i n e d to an aggregate p r o b a b i l i t y of u n i t y ) which are r e l a t e d to a standard opportunity. A r b i t r a r y u t i l i t y values are assigned to two defined monetary events, p r e f e r a b l y representing extremes of s a t i s f a c t i o n (e.g. success » F).  m  G) and unpleasantness  (e.g. f a i l u r e  This defines the u t i l i t y datum and s c a l e , although  any l i n e a r f u n c t i o n of such a u t i l i t y i s a l s o a u t i l i t y ( i . e . aU + b = U ) . 1  I t i s shown that f o r any  intermediate  monetary event (e.g. X) there i s a p r o b a b i l i t y mixture of the two extreme events to which the i n v e s t o r w i l l be i n d i f f e rent and t h i s defines the u t i l i t y of the intermediate event  - 28 -  (i.e.  there  i s a linear  correspondence  b e t w e e n P a n d U(X)  where U(X) - P.U(G) + (l-P).U(F) ) .  It  i s assumed t h a t p r e f e r e n c e s a r e t r a n s i t i v e , a l -  though i n c o n s i s t e n c i e s i n behaviour assumption of  do e x i s t .  i s o f the c o n t i n u i t y , o r uni-dimensional  preferences.  Preferences  pendent from the context  pound p r o b a b i l i t i e s a p p l y . absolute preference  a r e a l s o c o n s i d e r e d t o be i n d e -  risks  i t i s assumed t h a t com-  A b a s i c assumption i s that o f  f o r higher probabilities  Through the e x p r e s s i o n o f p r e f e r e n c e s gambles t h e i n v e s t o r ' s u t i l i t y be  found  and h i s u t i l i t y  in  Figure 7 •  Although  (i)  the usual  investor  marginal  utility  of  f o r monetary g a i n , and ( i i ) n o n - l i n e a r ,  concave form i m p l y i n g a d i m i n i s h i n g marginal  risk).  utility  characteristics;  slope, implying a p o s i t i v e  money o r a p r e f e r e n c e  over a l t e r n a t e  f u n c t i o n d e f i n e d , 5 ° s u c h a s shown  not necessary,  money o r a p r e f e r e n c e  of success.  o f any m o n e t a r y outcome c a n  i s of the f o l l o w i n g general  positive  character,  i n which the p r o b a b i l i t i e s a r e  o f f e r e d and f o r m u l t i - s t a g e d  function  A further  f o r certainty  u t i l i t y of  ( i . e . an a v e r s i o n to  - 2.9 • A g e n e r a l l y observed phenomenon of i n v e s t o r behav i o u r suggests a convex i n f l e c t i o n about the status quo i n the otherwise g e n e r a l l y concave u t i l i t y as shown i n Pigure 8. for  curve,51*52,53  T h i s describes a strong preference  r i s k where the stakes are s m a l l .  Thus where the f i n a n -  c i a l consequences become l e s s s i g n i f i c a n t the p s y c h o l o g i c a l t h r i l l and s o c i a l excitement of gambling may become pronounced. To f a c i l i t a t e computation attempts have been made to f i t mathematical f u n c t i o n s to the observed data, with the  l o g a r i t h m i c form p r o v i d i n g a very good f i t i n some  cases (e.g. U = a + b.log(X+c) ) w h i l e the cubic or quadratic  2 54 55  gave good f i t s i n other cases (e.g. U = a + b.X + c.X ).  '  Thus any r i s k y set of p o s s i b l e monetary outcomes can be transformed i n t o a set of p o s s i b l e u t i l i t y outcomes having the same s u b j e c t i v e p r o b a b i l i t i e s .  The mean u t i l i t y  value or the expected u t i l i t y of t h i s set then summarizes i n one s t a t i s t i c the f e l t value of the b e l i e v e d average outcome and r i s k of the proposed investment f o r the s p e c i f i c investor. The variance of u t i l i t y outcomes has no s i g n i f i c a n c e ^6 which i s not imparted to the expected u t i l i t y value.-'  In  - 30 -  Total LossO  Aspired <3tain Increment of Wealth  Fig.  7.—The U t i l i t y of Money Function of an Investor  Increment of Fig.  s-  Wealth  >-  8.--Convex Segment i n Investor's U t i l i t y Function  - 31 -  the case of the t y p i c a l l y concave u t i l i t y f u n c t i o n , the expected u t i l i t y value w i l l he l e s s than the u t i l i t y o f the expected monetary value t o the extent that the p r o j e c t is risky.  The expected u t i l i t y value could he equated to  some c e r t a i n monetary value and thus we have, i n f a c t , determined an equivalent c e r t a i n monetary value to the r i s k y set  of monetary outcomes.  I n other words, we have simply  modified the monetary mean to a c e r t a i n t y equivalent v a l u e . The d i f f e r e n c e between the expected monetary value and i t s c e r t a i n t y equivalent monetary value represents a margin of s e c u r i t y against e r r o r , both i n the f o r e c a s t o f p r o b a b i l i t y outcomes and i n any s i n g l e outcome i n a random process.57  The maximization o f expected u t i l i t y i s c o n s i -  dered the t h e o r e t i c a l l y optimal c r i t e r i a f o r investment decisions.  Although i t i s d i f f i c u l t t o generate  successfully  a p r a c t i c a l u t i l i t y f u n c t i o n , t h i s approach i s g e n e r a l l y accepted as being fundamentally u s e f u l . 5 8 * 5 9 In these approaches the I n t r o d u c t i o n of r i s k i n v o l v e s the conversion of r i s k y income to equivalent c e r t a i n income through the i n v e s t o r ' s t r a d e - o f f between income and r i s k o r his  u t i l i t y of money f u n c t i o n . I n other words r i s k i s a p p l i e d  to modify the mean r i s k y income value to some equivalent  - 32 -  r i s k l e s s mean income. As t h i s t r a d e - o f f or u t i l i t y  function  represents the i n v e s t o r ' s current a t t i t u d e towards current income, i t should be used to convert present r i s k y income to e q u i v a l e n t present c e r t a i n income. Thus f u t u r e r i s k y income should f i r s t be discounted t o present r i s k y income before i t i sconverted to equivalent present c e r t a i n income The f o l l o w i n g example i l l u s t r a t e s t h i s procedure: (1) Possible Future Annual Income for 5 y r s . $5000 4000 3000 2000 +$1000 0 -$1000  (2) Probability of Future Annual Income  (3) from (1) Present Value a t 8$ of Future Income  .05  $20,000 16,000 12,000 8,000 +$ 4,000 0 -$ 4,000  .10 .15 .20 .30 .15 .05 1.00  Utility Index f o r Investor of Present Value 87 85 82  78 72  64 53  Expected Present U t i l i t y  (5 -(2)xi 4.3 8.5 12.3 15.6 21.6 9.6 2.7  Value - 74.6  Equivalent Present C e r t a i n Monetary Value - $5700 Expected Present Monetary Value - $7000 Here the expected monetary value ( i . e . the mean of present r i s k y income) of $7000 has been modified by the r i s k , and the i n v e s t o r ' s a t t i t u d e towards i t , to an equivalent present c e r t a i n income of $5700.  - 33 -  Using the excess present value method, the  $5700  value would then be compared t o the required outlay f o r a d e c i s i o n , assuming that 8 $ i s the cost of c a p i t a l or r e quired rate of r e t u r n .  I n the excess rate of r e t u r n method,  assuming that the required present, c e r t a i n investment outlay i s $ 5 7 0 0 , the  8$  i n t e r n a l rate o f r e t u r n f o r the p r o j e c t  i s then compared t o the cost o f c a p i t a l or required rate of r e t u r n f o r a d e c i s i o n . However there i s a f u r t h e r problem.  Most r e a l c a -  p i t a l investment d e c i s i o n s , not t o mention a great many f i n a n c i a l investment d e c i s i o n s , are made by groups of people i n the form of partnerships o r corporations rather than by sole p r o p r i e t o r s . Where we have developed the concept and technique of a u t i l i t y f u n c t i o n f o r the i n d i v i d u a l i n v e s t o r we now require the u t i l i t y f u n c t i o n f o r a group of coinvestors.^  1  S t r i c t l y speaking we cannot make interperson-  a l comparisons o r a d d i t i o n s of u t i l i t y as there i s no common or absolute standard of u t i l i t y . ^  2  This problem might be capable of pragmatic r e s o l u t i o n i n the case of partnerships by developing a u t i l i t y f u n c t i o n f o r the partners as a group based upon t h e i r j o i n t preferences  and d e c i s i o n s .  However, i n the process i n d i v i d u a l  - 34 a t t i t u d e s would be more or l e s s submerged depending upon the p e r s o n a l i t y i n t e r a c t i o n s among the p a r t n e r s . s u l t may be d e s c r i p t i v e l y accurate but normatively  The r e erroneous.  Although f o r a widely h e l d corporation i t would be f u t i l e t o develop a u t i l i t y f u n c t i o n f o r each current shareh o l d e r , we might assume that the u t i l i t y f u n c t i o n f o r the f i r m , as defined by the preferences  of the c h i e f executive,  approximates that of the aggregate stockholders.^3  Here  again i t would define the a c t u a l , i f not the appropriate, b a s i s of d e c i s i o n .  Such a u t i l i t y f u n c t i o n could  provide  an e f f e c t i v e means of delegating d e c i s i o n making i n the large firm.  6 4  "Variable Rates o f Discount o r Return We may a l s o introduce r i s k i n t o the investment anal y s i s i n the form of a m o d i f i c a t i o n to the rate of discount, f o r the excess present value method, or the required rate of r e t u r n , f o r the excess rate of r e t u r n method, r a t h e r than a m o d i f i c a t i o n to the mean v a l u e . ^ 6  This approach i s  derived from the premise that r i s k a f f e c t s the cost of funds to the f i r m , o r the opportunity cost of funds f o r i n v e s t ments of l i k e r i s k .  - 35 -  In the case of the excess present value method we reduce future r i s k y income d i r e c t l y to equivalent present c e r t a i n income using a discount rate which r e f l e c t s the cost of r i s k l e s s funds plus a premium f o r r i s k , and then, as before, compare t h i s to the present c e r t a i n o u t l a y .  In  the excess rate of r e t u r n method the i n t e r n a l rate of r i s k y r e t u r n f o r the p r o j e c t i s found and then compared to a rate which r e f l e c t s the cost of r i s k l e s s funds plus a premium f o r risk.  This method, of course, s u f f e r s from the well-known  reinvestment rate assumption problem which here i m p l i c i t l y assumes that a l l earnings from the p r o j e c t are reinvested i n p r o j e c t s of l i k e r i s k and thus, l i k e r e t u r n . A number of approaches, of varying s o p h i s t i c a t i o n , have been suggested whereby r i s k may be introduced i n t o the investment a n a l y s i s i n t h i s f a s h i o n .  Perhaps the most e l e -  mentary approach i s to c l a s s i f y p r o j e c t s according to some predetermined r i s k groupings.^6  This might be on the b a s i s  of the usual degree of c e r t a i n t y i n the estimates.  Here  one could use the present cost of c a p i t a l f o r the rate of discount, or required rate of r e t u r n , f o r p r o j e c t s In the normal sphere of business a c t i v i t y while t h i s rate could be increased or decreased f o r p r o j e c t s i n l e s s f a m i l i a r or d i f f e r e n t areas of a c t i v i t y (e.g. products,  industries,etc.).  - 36 -  One could make a f u r t h e r d i f f e r e n t i a t i o n , i n the former case, between p r o j e c t s i n v o l v i n g the perhaps more c e r t a i n savings i n replacement p r o j e c t s and the p o s s i b l y more r i 3 k y increased earnings o f expansion p r o j e c t s . The grouping o f r i s k s could a l s o be made on the basi3  o f the asset f u n c t i o n s H e r e  the f i r m i s conceived  as being a consolidated or i n t e g r a t e d group of operating investments, each w i t h i t s own appropriate c a p i t a l s t r u c t u r e and cost o f c a p i t a l .  This grouping could be simply on the  b a s i s of operating f u n c t i o n , say production, marketing and research, as i n the f o l l o w i n g example.  (1) Operating Function Production  Marketing  Research  (2) Appropriate Capital Structure of Function  (3) from (2) Functional Cost o f Capital*  Proportion of Investment  •(3)x(4)  50$ debt 50$ equity  .5x3$-1.5$ .5x9$-4.5 6.0$  50$  3.0$  30$ debt 70$ equity  .3x3$-0.9$ .7x9$-6 .3$ 7.2$  30$  2.1$  10$ debt 90$ equity  .1x3$-0.3$ .9x9$ »8.1  20$  1.7$  0$  Consolidated Corporate Cost of C a p i t a l * assuming corporate debt a t 3$, and equity a t 9$.  (5)  6.8$  - 37 -  Of course the r i s k s could be f u r t h e r d i f f e r e n t i a t e d and c l a s s i f i e d w i t h i n any f u n c t i o n , such as between fundamental and developmental p r o j e c t s w i t h i n the research funct i o n o r between product o r process groups w i t h i n the marketing or manufacturing departments.  I n a d d i t i o n the r i s k s  could be c l a s s i f i e d on a geographic b a s i s (e.g. by regions or n a t i o n s ) . In the f i n a l a n a l y s i s , each p r o j e c t has i t s own unique r i s k s and a s s o c i a t e d cost of funds which i s determined, not only by the earnings r i s k as we have described, but a l s o by the c o l l a t e r a l o r asset r i s k which i s r e l a t e d to the p r o p o r t i o n of assured o r r e a l i z a b l e value i n the assetswhich form the investment.^®  Thus the i n d i v i d u a l  p r o j e c t can be considered to have some uniquely appropriate c a p i t a l s t r u c t u r e of i t s own. The p r o j e c t i s then the b a s i s of new borrowing power, as opposed t o l a t e n t general borrowi n g power which d e r i v e s from the current equity i n e x i s t i n g assets and t h e i r earning power.  The p r o j e c t can be a s s i g n -  ed a borrowing quota based upon i t s l e v e l of s u f f i c i e n t l y c e r t a i n earnings and asset value such that there i s a low p r o b a b i l i t y of d e f a u l t and c r e d i t o r c a p i t a l l o s s i n the case of d e f a u l t .  The r e s u l t a n t earnings to the equity ( i . e .  p r o j e c t earnings l e s s debt charges) can then be compared t o  - 38 -  the r e q u i r e d e q u i t y investment  ( i . e . t o t a l investment  borrowing quota), given the cost of equity funds.  less  This  whole approach hinges upon the determination of the o p t i mum  borrowing quota of a p r o j e c t , w i t h a l l the c o m p l e x i t i e s  which that i n v o l v e s . Here the approach to r i s k e v a l u a t i o n has been developed beyond the demand side of the a n a l y s i s i n t o c o n s i d e r a t i o n of the cost of funds on the supply side of the problem.  This r a i s e s the question of how  one i s to determine  the appropriate c a p i t a l s t r u c t u r e and/or cost of funds f o r an investment  p r o j e c t or c l a s s of given r i s k .  Although  we  may approximate a f i r m ' s composite cost of c a p i t a l , at which i t a c q u i r e s funds from various sources f o r a l l  investments,  t h i s does not i n d i c a t e the appropriate cost of these funds which are invested i n any p a r t i c u l a r p r o j e c t of s p e c i f i c g i ven r i s k . ^ 9  one suggested approach i s to use the opportunity  cost of e x t e r n a l investments  of l i k e r i s k . ^  0  The borrow-  i n g quota, discussed above, i s suggested as being based upon sound l e n d i n g p r i n c i p l e s , however those may be d e f i n e d , but we must a l s o define the cost of equity f u n d s . ^ Thus the v a r i a b l e rate of discount or r e t u r n approaches to r i s k e v a l u a t i o n require a knowledge of the cost of risk capital.  This knowledge i s o f t e n i m p l i c i t l y assumed  - 39 -  perhaps because of the complexities and tenuous state of understanding which are i n v o l v e d , ?  2  S u f f i c e t o say that  these methods must u l t i m a t e l y be founded upon some concepts of the cost of c a p i t a l under r i s k .  This problem w i l l be  the focus of our a t t e n t i o n i n the chapters which f o l l o w .  C o r r e l a t i o n of Outcomes A l l of the foregoing approaches introduce r i s k i n t o the c a p i t a l investment a n a l y s i s as a measure of d i s p e r s i o n of p o s s i b l e p r o j e c t outcomes which i s a p p l i e d t o modify the expected value of p o s s i b l e outcomes o r the rate r e f l e c t i n g the cost of funds to the p r o j e c t . However, there i s another aspect of r i s k which has been ignored thus f a r .  This i s the e f f e c t of c o r r e l a t i o n  between future income on presently h e l d assets and that on the proposed investment.  New income may be compensatory i n  behaviour r e l a t i v e t o e s t a b l i s h e d income, rather than cumul a t i v e , which produces a d i v e r s i f i c a t i o n e f f e c t . Many authors q u a l i f y t h e i r r i s k a n a l y s i s concepts due t o t h i s e f f e c t but few have e x p l i c i t l y attacked the problem. Although the i n i t i a l p r o b a b i l i t y estimate of future income i s u s u a l l y made over various p o s s i b l e s t a t e s of  - 40 -  nature (e.g. general business c o n d i t i o n s , competitor s t r a t e g i e s ) the a s s o c i a t i o n between the l e v e l of income and the states of nature i s dropped i n f u r t h e r a n a l y s i s .  Fi-  gure 9 i n d i c a t e s the usual a d d i t i o n of two r i s k y income streams where the worst and best p o s s i b l e outcomes are each simply summed to form the combined income extremes. Of course, the i m p l i c i t assumption behind t h i s aggregation i s that the two r i s k y income streams are perf e c t l y p o s i t i v e l y c o r r e l a t e d and thus i d e n t i c a l l y a f f e c t e d by d i f f e r e n t states of nature.73  Although there i s a ten-  dency f o r many economic phenomena to be h i g h l y p o s i t i v e l y c o r r e l a t e d , the assumption of p e r f e c t such c o r r e l a t i o n would be an inaccurate g e n e r a l i z a t i o n .  I f i n f a c t the two  streams i n Figure 9 were p e r f e c t l y negatively  risks-  correlated,  being a f f e c t e d i n e x a c t l y opposite fashion by d i f f e r e n t s t a tes of nature, t h e i r combination would y i e l d the same expected value as shown but with zero d i s p e r s i o n , and thus zero r i s k or c e r t a i n t y . We might a l s o encounter a s i t u a t i o n which f a l l s between these two extremes, such as p e r f e c t l y streams as shown i n Figure 10. b a s i s , an i n t e g r a t e d company may  uncorrelated  Thus, on a f u n c t i o n a l r i s k be l e s s r i s k y than simply  -  41  -  0  Income  >-  F i g . 9.--Common A d d i t i o n of Two Income Stream D i s t r i b u t i o n s  0  Income Fig. 10.—Addition  >-  of Two T y p i c a l Income Stream D i s t r i b u t i o n s  - 42  the sum new  -  of i t s f u n c t i o n a l parts.7  D i v e r s i f i c a t i o n into  4  or competitive products or i n d u s t r i e s which behave  d i f f e r e n t l y under various states of business conditions may  reduce the d i s p e r s i o n of t o t a l income and  thus income  risk.75,76 s i m i l a r l y , geographical d i v e r s i f i c a t i o n may have the same e f f e c t . Markowitz77 has developed the most e x p l i c i t s t a t e ment and a p p l i c a t i o n of the concept of d i v e r s i f i c a t i o n and c o r r e l a t i o n of outcomes although h i s work was  confined to  the area of f i n a n c i a l , r a t h e r than r e a l , investment. In adding together the 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  two earnings streams, say X and Y, across a l l p o s s i b l e s t a t e s of nature, the standard d e v i a t i o n  (S) of the combined streams  i s given by, Sx+y "  V  Sx^ +  2RSxSy  +  Sy  2  Here R i s the c o e f f i c i e n t of c o r r e l a t i o n between x and y.  The  common i m p l i c i t assumption i s that R •»  which g i v e s , Sx+y " S  x  + Sy.  The  c o e f f i c i e n t of  +1.0  correlation  i s only s i g n i f i c a n t i n that i t defines the c o e f f i c i e n t association  (A), where A  m  1 - V 1 - R, 2  of  which i s a measure  of the degree of e r r o r free l i n e a r correspondence between two variables.78  This i s as opposed to R  2  which i s a mea-  sure of the degree of squared-error free l i n e a r correspondence .  - 43 -  The f o l l o w i n g example i l l u s t r a t e s the s i g n i f i c a n c e and i n t e r p r e t a t i o n o f c o r r e l a t i o n between p r o j e c t  earnings  and the earnings of the e x i s t i n g e n t e r p r i s e . Annual earnings of e n t e r p r i s e : x»  $10,000 ,  S  x  - $4,000  thus Sx/x - 0.4 Annual earnings of p r o j e c t : y - $1,000  Sy- $1,000  thus Sy/y " 1.0 , which i n d i c a t e s that the p r o j e c t earnings are o f much greater r e l a t i v e r i s k than the enterp r i s e s earnings. 1  Combined annual earnings of e n t e r p r i s e and p r o j e c t ; x + y - $11,000  Case 1 :  i f R f o r x and y i s +0.95 ( i . e . A - 0.070) such  as f o r an expansion p r o j e c t t o increase capacity where the a d d i t i o n a l demand i s not assured, then; S  - 10 V4 + 2 ( 4 ) ( l ) ( + 0 . 9 5 ) + l 6  x + y  2  which gives Sx+y/x + y  m  2  " $4950  0.45.  Thus both the absolute and the r e l a t i v e r i s k o f the f i r m ' s earnings have increased.  I n terms of an equivalent  perfectly p o s i t i v e l y correlated ( i . e . R - +1.0) project t h i s would be, y« - $1,000 and S i - $950, w i t h Sy«/y'« 0 . 9 5 . y  - 44 -  Case 2 :  i f R f o r x and y i s +0.30 ( i . e . A - 0.05)  such  as f o r a p r o j e c t i n a q u i t e unrelated economic or market area which i s strongly subject to d i f f e r e n t i n f l u e n c e s , then, Sx+y - $4400 and S +y/x + y x  0.40.  Thus, although the absolute r i s k of the f i r m ' s earnings would increase t h e i r r e l a t i v e r i s k i s unchanged.  An  equivalent p e r f e c t l y p o s i t i v e l y c o r r e l a t e d p r o j e c t has y ' « $1,000 and S t y  - $400 w i t h S y i / y " 0.4, which suggests 1  that the appropriate cost of funds f o r t h i s p r o j e c t would be i d e n t i c a l to the f i r m ' s present cost of c a p i t a l . Case 3 :  i f R f o r x and y i s -0.10  ( i . e . A - 0.006) such  as f o r an investment i n an unrelated area which i s s t r o n g l y subject to d i f f e r e n t i n f l u e n c e s but o p p o s i t e l y a f f e c t e d by some of the same i n f l u e n c e s , then, Sx+y - $4025 and S +y/x+y-0.37, x  Thus, although the r e l a t i v e r i s k of the f i r m ' s earnings has decreased, the absolute r i s k has remained almost constant which suggest that the p r o j e c t c o n t r i b u t e s v i r t u a l l y no r i s k to the earnings of the f i r m .  An equivalent per-  f e c t l y p o s i t i v e l y c o r r e l a t e d p r o j e c t would have y and Syi  0  ,sa  $1,000  0 which i n d i c a t e s that these are c e r t a i n earnings,  that the p r o j e c t could have a borrowing quota w i t h debt charges equal to the f u l l $1,000, and that the  appropriate  cost of funds f o r the p r o j e c t i n t h i s case would be the debt  - 45 -  rate. Case 4 :  i f R f o r x and y i s -0.30 ( i . e . A - 0.05) such  as f o r a p r o j e c t which i s i n an unrelated area and which i s o p p o s i t e l y a f f e c t e d by many of the same i n f l u e n c e s , then, S +y x  D  $3820 and S  x + y  / x + y - 0.35.  Thus both the absolute as w e l l as the r e l a t i v e r i s k of the f i r m ' s earnings have decreased and t h i s p r o j e c t i s more valuable than $1000 per year w i t h c e r t a i n t y , and would have an appropriate cost of funds l e s s than the debt r a t e . In other words we could a f f o r d t o pay, to have t h i s p r o j e c t , i n r e l a t i o n to the amount of r i s k i t absorbs, i n the same way we purchase insurance.  I n terms of an equivalent per-  f e c t l y negatively correlated project ( i . e . e f f i c i e n t insurance) t h i s would be y "  - $1,000 and Sy"» $180.  Thus the c o r r e l a t i o n of these earnings streams w i l l d i r e c t l y a f f e c t the degree of r i s k a s s o c i a t e d w i t h the project, and the appropriate cost of funds f o r the p r o j e c t . This emphasizes the f a c t that the p r o j e c t w i l l not be c a r r i e d out i n i s o l a t i o n , but must be viewed i n r e l a t i o n to the operations of the p o t e n t i a l i n v e s t o r f i r m . s p e c i f i c a l l y , we are i n t e r e s t e d i n the p r o j e c t ' s  More incremental  r i s k c o n t r i b u t i o n to the t o t a l earnings of the f i r m .  An  investment which i n v o l v e s g r e a t l y increased r i s k f o r one  - 46 -  f i r m , may  p o s s i b l y be considered  ent f i r m .  of l e s s e r r i s k by a d i f f e r -  This concept adds another ddmension to our ana-  l y s i s of r i s k i n c a p i t a l investment.  C. We have seen how  Summary r i s k a r i s e s from a l a c k of c e r t a i n -  ty as exemplified by a 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 of future earnings over various states of nature, both f o r the presently c o n s t i t u t e d f i r m and the prospective ment p r o j e c t .  invest-  This r i s k i s defined by the d i s p e r s i o n of  each p r o b a b i l i t y d i s t r i b u t i o n which i s u s u a l l y measured by the standard d e v i a t i o n of an assumed or transformed symmet r i c a l distribution.  The  j o i n t r i 3 k i s , i n a d d i t i o n , de-  pendent upon the a s s o c i a t i o n between these earnings which i s u s u a l l y measured by the c o e f f i c i e n t of c o r r e l a t i o n . relevant r i s k of the p r o j e c t i s the marginal or  The  incremental  r i s k which i t c o n t r i b u t e s to the t o t a l future earnings of the f i r m . Given t h i s measure of net p r o j e c t r i s k , which i a baaed upon i n v e s t o r b e l i e f s , i t may  be incorporated  into  the a n a l y s i s i n one of two ways, both of which are baaed upon the i n v e s t o r ' s a t t i t u d e towarda r i 3 k .  Firstly, thia  - 47 may "be accomplished through a m o d i f i c a t i o n of the expected earnings t o a c e r t a i n t y equivalent value based upon an est a b l i s h e d u t i l i t y of money f u n c t i o n f o r the i n v e s t o r . A l though the theory behind the u t i l i t y f u n c t i o n concept has been w e l l developed, and some success has been achieved with  79 t h i s approach i n p r a c t i c e , i t i s e s s e n t i a l l y a p p l i c a b l e to the i n d i v i d u a l i n v e s t o r with some doubt of i t s s t r i c t d i t y even i n the case of small p a r t n e r s h i p s .  vali-  Considering  the prevalence of the corporate mode of e n t e r p r i s e and the magnitude of corporate investment t h i s l i m i t a t i o n i s h i g h l y significant. Fortunately there may be an a l t e r n a t i v e approach. The  r i s k of a p r o j e c t may be incorporated  i n t o the a n a l y s i s  through a m o d i f i c a t i o n of the required earnings rate o r cost of funds f o r the p r o j e c t based upon an e s t a b l i s h e d cost of c a p i t a l under r i s k f u n c t i o n .  More s p e c i f i c a l l y , an  e x p l i c i t f u n c t i o n d e f i n i n g i n v e s t o r v a l u a t i o n of corporate shares i s the sine qua non of t h i s approach. The development of a s a t i s f a c t o r y theory of share v a l u a t i o n involves the problem of hypothesizing a v a l u a t i o n model which i s capable of conclusive e m p i r i c a l v e r i f i c a t i o n . Thus the v a l u a t i o n model must be s u f f i c i e n t l y comprehensive  - 48 -  to include a l l o f the s i g n i f i c a n t v a r i a b l e s and s u i t a b l y s t r u c t u r e d t o express the correct i n t e r r e l a t i o n s h i p s i n order to y i e l d a high degree of s t a t i s t i c a l s i g n i f i c a n c e to the v a l i d a t i o n t e s t s . A t the same time the v a l u a t i o n model must be s u f f i c i e n t l y simple to be capable of p r a c t i c a l manipulation and s o l u t i o n and s u i t a b l y founded on objecti v e l y measurable v a r i a b l e s i n order to make s t a t i s t i c a l testing feasible. Unfortunately  these two requirements tend to be  c o n t r a d i c t o r y and require a compromise i n v a l u a t i o n models e i t h e r i n favour of the f e a s i b i l i t y of e m p i r i c a l t e s t i n g at a l o s s i n the s t a t i s t i c a l s i g n i f i c a n c e of such t e s t s , o r In favour of t h e o r e t i c a l r i g o u r a t a l o s s i n the f e a s i b i l i t y of e m p i r i c a l v a l i d a t i o n . The l a c k of success i n e s t a b l i s h ing such a v a l i d a t e d theory of share v a l u a t i o n has been the prime obstacle t o the use of the v a r i a b l e cost of c a p i t a l approach to investment d e c i s i o n making under risk.®  0  The M o d i g l i a n i and M i l l e r hypothesis of v a l u a t i o n under risk,® i f not wholly d e f i n i t i v e and c o n c l u s i v e l y 1  v a l i d a t e d , represents an i n t e r e s t i n g e f f o r t i n t h i s d i r e c t ion.  This represents a t h e o r e t i c a l approach to the problem  of d e f i n i n g how Investors c a p i t a l i z e r i s k y f u t u r e  earnings.  - 49  -  We have e s t a b l i s h e d the s i g n i f i c a n c e of such a theory f o r c a p i t a l investment d e c i s i o n s under r i s k .  Thus i t would be  u s e f u l to examine t h i s hypothesis and i t s p o s s i b l e c o n t r i b u t i o n towards a conceptually sound and u n i f i e d method of a n a l y z i n g r i s k i n c a p i t a l investment proposals. This i s the  purpose of the remainder of t h i s paper. Once a workable r i s k v a l u a t i o n model i s developed  i t would v i t a l i z e the r i s k premium rate method of a n a l y s i s and provide a u s e f u l a l t e r n a t i v e to the c e r t a i n t y equival e n t or u t i l i t y f u n c t i o n approach.  The value of such a  scheme f o r corporate investment d e c i s i o n s would be enormous. ^-Frederick L u t z and Vera L u t z , The Theory of Investment of the Firm ( P r i n c e t o n : P r i n c e t o n U n i v e r s i t y Press, 1 9 5 1 ) , Ch.  XV.  P i e r r e Masse, Optimal Investment D e c i s i o n s (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1 9 6 2 ) , Foreword, Ch. 5 . 2  3 . J . Grayson J r . , D e c i s i o n s under U n c e r t a i n t y (Cambridge: Harvard U n i v e r s i t y P r e s s , i 9 6 0 ) , Ch. 1. C  D.T. Nowalki, " P r o b a b i l i t i e s and Expected Values A p p l i e d to Return on Investment," Papers on Return on Investment, ed. R. N. Anthony (Boston: Harvard Business School, D i v i s i o n of Research, 1 9 5 9 ) . 4  5 L u t z and L u t z , Ch.  XV.  ^Robert N. Anthony (ed.),Papers on Return on Investment (Boston: Harvard Business School, D i v i s i o n of Research, 1959), Preface.  - 50 -  ^ L e o n a r d J . S a v a g e , The F o u n d a t i o n s o f S t a t i s t i c s (New Y o r k : J o h n W i l e y & S o n s , I n c . , 1 9 5 4 ) , C h . 4 . ®Grayson, C h . 9 . ^ H a r o l d B i e r m a n J r . a n d Seymour S m i d t , The C a p i t a l B u d g e t i n g D e c i s i o n (New Y o r k : M a c m i l l a n Company, I 9 6 0 ) , p p . 120-132. 1 0  S a v a g e , Ch. 4 .  i : L  1 2  Masse, Ch. 5 .  S a v a g e , Gh. 4 ,  9.  !3R. D u n c a n L u c e a n d H o w a r d R a i f f a , Games a n d D e c i s i o n s (New Y o r k : J o h n W i l e y & S o n s , I n c . , 1 9 5 7 ) , C h . 1 3 . 14 Masse, Ch. 5 . •^F.H. K n i g h t , R i s k , U n c e r t a i n t y a n d F r o f i t H o u g h t o n - M i f f l i n , 1921"]"^  (Boston:  M a s s e , Gh. 5 .  •-,  ^ L u t z a n d L u t z , C h . XV. •^Anthony, I . •^Savage, Ch. 4 . 2 0  2 1  M a s s e , Ch. 5 . G r a y s o n , Ch. 9 .  Op  H a r r y M. M a r k o w i t z , P o r t f o l i o S e l e c t i o n (New Y o r k : John W i l e y & Sons, I n c . , 1959), Ch. 1 0 . 2  3 L u c e and R a i f f a , Ch. 2 .  2i  *Masse, Ch. 5 .  - 51 ^ G r a y s o n , Ch. 9. ^ F r e d e r i c k A. Ekeblad, The S t a t i s t i c a l Method i n Business (Hew York: John Wiley & Sons, I n c . , 1962), pp.  186-192.  7Robert W. Johnson, F i n a n c i a l Management (2nd ed., Boston: A l l y n and Bacon, I n c . , 1964), Ch. 7. 2  2  ^ L u t z and L u t z , Ch. XV.  2  ^Markowitz, Ch. 4.  3°Lutz and L u t z , Ch. XV. 31  E k e b l a d , pp. 202-203.  32  E k e b l a d , p. 251 i  33  M a r k o w i t z , Oh. 9.  34  E k e b l a d , p. 249.  35R bert Lindsay and A r n o l d W. Sametz, F i n a n e i a l Management (Homewood: Richard D. I r w i n , Inc., 1963), Ch. 30  3 johnson, Ch. 7. 6  37p arson Hunt, Charles M. W i l l i a m s and Gordon Donaldson, Basic Business Finance (Homewood: Richard D. I r w i n , I n c . , e  1961), pp. 627-629.  ^ E z r a Solomon, "The A r i t h m e t i c of C a p i t a l Budgeting D e c i s i o n s " , J o u r n a l of Business, ( A p r i l , 1956). 3  •^Bierman and Smidt, pp. 120-132.  40 41  Savage, Ch. 5. Grayson, Ch. 6, 9.  - 52 -  4 2  L u t z and L u t z , Ch. XV. -•Johnson, Ch. 7.  44  M a r k o w i t z , Ch. 10.  45Grayson, Ch. 1. 46 L u t z and L u t z , Ch. XV. ^ B i e r m a n and Smidt, pp. 120-132. ^^Llndsay and Sametz, Ch. 3. ^John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behaviour (2nd ed., P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , 1947). 5°Grayson, Oh. 10. ^ S a v a g e , Gh. 5. ^ Markowitz, 2  o i l  .  1 0 >  -^Harold Bierman J r . , Lawrence E. Fouraker and Robert K. J a e d i c k e , Q u a n t i t a t i v e A n a l y s i s f o r Business Decisions (Homewood: Richard D. I r w i n , I n c . , 1961), pp.142-153. ^Gordon M. Kaufman, S t a t i s t i c a l D e c i s i o n and Related Techniques i n O i l and Gas E x p l o r a t i o n (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1963), Ch. 7-^Bierman, Fouraker and J a e d i c k e , pp. 142-153. 5 Luce and R a i f f a , Ch. 2. 6  sse, Ch. 5• •^Grayson, Ch. 10.  - 53  5 9 s a v a g e , Ch.  5  L u t z and L u t z , Ch. XV.  6 G  6l  -  B i e r m a n , Fouraker and J a e d i c k e , pp. 1 4 2 - 1 5 3 .  62 u  *Luce  63  and R a i f f a , Ch.. 2.  G r a y s o n , Ch. 1G.  64 Grayson, Ch. 1 0 . ^ M y r o n j . Gordon, The Investment, F i n a n c i n g and V a l u a t i o n of the C o r p o r a t i o n (Homewood; Richard D. I r w i n , I n c . , 1962), Ch. 2. 66  J o e l Dean, C a p i t a l Budgeting (New York: Columbia U n i v e r s i t y P r e s s , 1951 )> Ch. V. ^7R. A . Golde and G.E. G r i s a r d , "Some Considerations i n Determining the Required Earnings Rate," Papers on Return on Investment, ed. R. N. Anthony (Boston: Harvard Business School, D i v i s i o n of Research, 1 9 5 § ) . ^ E z r a Solomon, "Measuring a Company's Cost of C a p i t a l , " J o u r n a l of Business, (October, 1 9 5 5 ) . 6  6Q . ^Lutz and L u t z , Ch. XIV. V. Roberts, "Current Problems i n the Economics of C a p i t a l Budgeting," J o u r n a l of Business, (January, 1 9 5 7 ) . 7 E z r a Solomon, J o u r n a l of Business, (October, 1 9 5 5 ) . 1  72  G o r d o n , Ch. 3 -  7 Ekeblad, 3  p p >  505-519.  74 Golde and G r i s a r d , Papers on Return on Investment, ed. R. N. Anthony. 1  -  75  76 1  54  -  Bierman and Smidt, pp.  120-132.  L u t z and Lutz,. Ch. XVT-.  ^ H a r r y M. Markowitz, P o r t f o l i o S e l e c t i o n : E f f i c i e n t D i v e r s i f i c a t i o n of Investments (Hew York: John Wiley & Sons, Inc., 1959). 78  Ekeblad, pp.  511-519.  ^Grayson, c h . 1 0 . °Gordon, Ch. 3 , 14. . 8  On  Franco M o d i g l i a n i and Merton H. M i l l e r , "The Cost o f C a p i t a l , Corporation Finance and the Theory of Investment," American Economic Review, (June, 1 9 5 8 ) , pp. 261-297.  II  CHAPTER  THE  MODIGLIAHI  AND MILLER VALUATION HYPOTHESIS  A. V a l u a t i o n under R i s k  Introduction Although the cost of c a p i t a l funds i s of c r u c i a l s i g n i f i c a n c e f o r investment and f i n a n c i n g d e c i s i o n s , i t has proven to "be a d i f f i c u l t concept to define and measure rigorously.  The t r a d i t i o n a l l i t e r a r y approach  1  to the  problems of common stock v a l u a t i o n and the e f f e c t of c a p i t a l s t r u c t u r e was h i g h l y d e s c r i p t i v e but h a r d l y p r e c i s e . S c i e n t i f i c approaches to share v a l u a t i o n , however, have been based on the e l u s i v e present value of expected 1 f u t u r e dividends2 or expected f u t u r e earnings^ as are a 4  number of c o r o l l a r y growth models.  , J  5  A n a l y t i c a l attacks  on the j o i n t funds supply problem have produced the net 6  operating earnings concept  and the marginal supply sche-  dule? which have been more d e s c r i p t i v e than d e f i n i t i v e . -  55  -  - 56 -  None o f these e f f o r t s have e x p l i c i t l y included the element of r i s k o r u n c e r t a i n t y i n t h e i r f o r m u l a t i o n although i t s existence i s recognized.  However, M o d i g l i a n i  and Miller® have r i g o r o u s l y enunciated the net operating earnings hypothesis of j o i n t c a p i t a l funds v a l u a t i o n w i t h i n an e x p l i c i t scheme of r i s k . The i n t r o d u c t i o n of r i s k adds a q u a l i t a t i v e dimens i o n t o the q u a n t i t a t i v e p r o f i t maximization  o b j e c t i v e which  can be resolved by means of a value maximization  criterion  such as the market value o f the common shares of the f i r m . Here t h i s i s t r e a t e d as the t o t a l common shares of the f i r m r a t h e r than the a l t e r n a t i v e , but e q u i v a l e n t , per share basis.  Thus M o d i g l i a n i and M i l l e r i n essence have formula-  ted a hypothesis of the v a l u a t i o n of the f i r m under r i s k . B a s i s of V a l u a t i o n The hypothesis i s developed In terms o f p a r t i a l e q u i l i b r i u m of the i n d i v i d u a l f i r m o r c l a s s of f i r m s , by means of s t a t i c a n a l y s i s which i s time r e l a t e d through h i s t o r i c a l "givens" and f u t u r e "expectations". I n s p i t e of the c l a s s i c debate over the r o l e of dividends i n share v a l u a t i o n , 9 A  0  and the cleavage between  - 57 -  e m p i r i c a l market b e h a v i o u r * ' 1 1  1 2  1 3  and t h e o r e t i c a l con-  cept s, >^-5> 16 expected f u t u r e earnings are used as the 14  b a s i s of v a l u a t i o n . S u f f i c e t o say that M o d i g l i a n i and M i l l e r are among the l e a d i n g proponents of the earnings approach. 7  i t i s assumed here that Investors are r a t i o n a l  1  ( i . e . the hypothesis i s p r e s c r i p t i v e r a t h e r than d e s c r i p t i v e .) I n i t i a l l y i t i s assumed that there are no income or c a p i t a l gains taxes.  Thus r e t a i n e d earnings are e q u i -  v a l e n t t o a f u l l y subscribed, pre-emptive issue of common stock. I t i s assumed that a l l p h y s i c a l assets are corporat e l y owned, thus are a s s o c i a t e d w i t h shareholder ownership, and that they generate a flow of earnings i n d e f i n i t e l y i n t o the f u t u r e .  Thus assets e i t h e r have i n f i n i t e economic l i f e  or s u f f i c i e n t c a p i t a l consumption expense i s earned and r e cognized t o maintain the productive c a p i t a l of a l l corporations intact. Such an earnings stream, which d e r i v e s d i r e c t l y from a f i r m ' s a s s e t s , i s the same as "net operating income" or t o t a l net income before any f i n a n c i a l charges such as interest.  This i s as opposed t o the gross o r cash flow  - 58 b a s i s of v a l u a t i o n which i s normally used f o r investment and bond v a l u a t i o n , where there i s a f i n i t e terminal p o i n t . The net income o r earnings b a s i s i s more usual f o r the val u a t i o n of firms and common stock. The d i f f e r e n c e between the two a r i s e s from the t r e a t ment of consumed c a p i t a l a s s e t s .  The earnings approach  assumes reinvestment of such l i q u i d a t e d assets where r e a l c a p i t a l and equity c a p i t a l are considered permanent i n nature .  This involves an i m p l i c i t b e l i e f about the q u a l i t y of  management d e c i s i o n s t o r e i n v e s t earned d e p r e c i a t i o n funds. The flow of future earnings i s defined, f o r valuat i o n purposes, by i n v e s t o r b e l i e f s .  This believed future  earnings stream can be defined by a 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 of the random v a r i a b l e x ( n ) , p o s s i b l e earnings In each period n.  The c a p i t a l i z e d value V of such a stream  i s the sum o f the present values of the expected earnings 3?(n) f o r each future time period n using a discount rate p which i s determined by r i s k . M o d i g l i a n i and M i l l e r have s i m p l i f i e d t h i s v a l u a t i o n process by p o s t u l a t i n g one p r o b a b i l i t y d i s t r i b u t i o n of equivalent perpetual annual earnings X, having a f i n i t e  -  expected value  X,  The  of i d e n t i c a l discounted  (1+p)  O  o  (1+p)  value  such  that;  P  p r o h a h i l i t y d i s t r i b u t i o n of p o s s i b l e average annual  earnings X t o be long  summarizes i n v e s t o r s ' b e l i e f s , w h i c h a r e  unanimous, r e g a r d i n g run e a r n i n g s of the  i n e f f e c t , the value flow we  the magnitude and  firm.  The  equivalent  equivalent  horizon  as  expected value  o f the assumed  e f f e c t of the  be  reflected  this will  believed  simply  For  discounting a finite  as  future  reduction  process.  i n rates,  due  o f X,  the  "Where i n v e s t o r s will  average annual  11.  as  is  to  Where  future  t o some compound r a t e o f  growth r a t e s -i  Although  terminal point, this  r a i s e the v a l u e  excessive  X is,  infinite  to a century  shown i n F i g u r e  subject  the  in practice this  i n a lower equivalent  e x p e c t e d e a r n i n g s X, earnings are  run,  of from a generation  b e l i e v e t h a t x ( n ) has simply  the l o n g  assumed  r i s k of  of future earnings from a firm's a s s e t s .  speak of i n f i n i t y  the  a  -  59  growth,  shown i n F i g u r e  i t i s n e c e s s a r y t o assume  Q or a l i f e  cycle with  future  1Q  decline.  * Initially  i t i s assumed t h a t f i r m s o f  equivalent  r i s k have equal expected r a t e s of growth of a s s e t s earnings.  Further  12.  i t i s a s s u m e d t h a t new  assets  and  generate  - 60 -  A x(0)  \  I/)  Annual Earn  rj)  X  ^  \  v  X  \ x  \  \  \ \  \  \  ^ A-  ^ _  p.v. X  pv. x U ) ^ J Time  >•  F i g . 11.--Average Annual Earnings from Terminal Expectation  Time F i g . 12.--Average Annual Earnings from Growth Expectation  - 61 expected earnings of equivalent r i s k to those from presentl y held assets.  R i s k Equivalency 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 random p o s s i b l e values of average annual f u t u r e operating earnings X has an expected value X = Z ? ( X ) . X , which we s h a l l c a l l the expected earnings. The d i s p e r s i o n of p o s s i b l e earnings X about expected earnings X can be described by the standard d e v i a t i o n of p o s s i b l e earnings Sx«  The r e l a t i v e d i s p e r s i o n ,  or c o e f f i c i e n t of v a r i a t i o n S /X, x  i s a measure of the be-  l i e v e d degree of operating or business r i s k of the expected earnings  stream. I t f o l l o w s that a l l f i r m s can be grouped i n t o  equivalent earnings r i s k c l a s s e s where a l l f i r m s i n each c l a s s have the same c o e f f i c i e n t of v a r i a t i o n of earnings. Thus, s  x0  X  0  s  xl X]_  which i s the expected earnings r i s k f o r the c l a s s , where SXJ_ - standard d e v i a t i o n of earnings of f i r m i i n the c l a s s , and X~i ss expected earnings of f i r m i i n the c l a s s . Thus,  -  i f f o r f i r m f and f i r m g, X  62  g  -  « a.Xf, and they are of the  same c l a s s of earnings r i s k , then S g X  Figure 13.  * a . S f as shown i n x  In a d d i t i o n , f i r m s of equivalent earnings r i s k  must have p e r f e c t l y c o r r e l a t e d f u t u r e earnings.  This i s  o f t e n m i s i n t e r p r e t e d as a requirement of t h e i r p e r i o d i c earnings x ( n ) .  m  2 0  (  f a c t i t i s only necessary, f o r homo-  geneity of earnings r i s k , that long run p o s s i b l e earnings X he h e l i e v e d p e r f e c t l y c o r r e l a t e d ( i . e . p r o p o r t i o n a l to t h e i r expected v a l u e s ) over every state of nature i n the long run.  Thus f o r a l l firm3 i n a r i s k c l a s s , m m m *6 « ^1 X Xi Xi =  =  0  where X* i s the h e l i e v e d value of p o s s i b l e earnings given the long run state of nature m.  T h i s , of course, presumes  a f u l l y d e t e r m i n i s t i c b e l i e f , w i t h only the states of nature o c c u r r i n g randomly. The  states of nature r e l a t e to general economic,  p o l i t i c a l and s o c i a l trends as w e l l as s p e c i f i c market and management v a r i a b l e s .  production,  Thus as various states  of nature o b t a i n over time and as the b e l i e v e d p r o b a b i l i t i e s of f u t u r e states of nature change over time, f i r m s of equiv a l e n t earnings r i s k remain i n the same v a l u a t i o n c l a s s .  A^j A^]  A2  A2  A3  Possible Xj F i g . 1 3 . - - P r o b a b i l i t y D i s t r i b u t i o n of Earnings  A3  >• of Equivalent  Risk  - 64 -  Although the p r o b a b i l i t y d i s t r i b u t i o n of p o s s i b l e earnings X may take any shape, the r e l a t i v e d i s p e r s i o n and p e r f e c t c o r r e l a t i o n requirements d i c t a t e that a l l f i r m s of equivalent earnings r i s k have d i s t r i b u t i o n s of e x a c t l y ident i c a l r e l a t i v e shape.  Given a l l these c l a s s membership r e -  quirements we might suspect that there would be very few f i r m s i n any one r i s k c l a s s , but r a t h e r a continuum of r i s k classes.  This i s the b a s i s of controversy generated  over the casual i d e n t i f i c a t i o n of a r i s k c l a s s w i t h the 21  f i r m s i n one i n d u s t r y .  B. Cost of C a p i t a l Hypothesis Leverage and R i s k For f i r m 0 which has no debt i n i t s c a p i t a l s t r u c t u r e the expected earnings a l l accrue to the shareholders i n which case we s h a l l c a l l them the expected net earnings. Thus, m VQ  CQ  =  XQ  where, CQ •> the market value of the common share of f i r m 0 in class j . VQ  m  the market value of f i r m 0 i n c l a s s j .  - 65 XQ  m  the expected earnings - the expected net earnings of f i r m 0 i n c l a s s j .  Pj  m  the c a p i t a l i z a t i o n rate f o r expected earnings of r i s k c l a s s j  (from S i / X i and Rj.). X  I t i s assumed that shares are traded i n p e r f e c t markets where there are no issue o r t r a d i n g costs f o r s e c u r i t i e s and where i n v e s t o r s have f u l l knowledge  22  and are not  r e s t r i c t e d i n any way. Here shares which are p e r f e c t subst i t u t e s must s e l l a t the same p r i c e . Thus f o r any other debt f r e e f i r m i n c l a s s j , as i t s expected earnings and expected net earnings are of equivalent r i s k to those of f i r m 0, they w i l l a l s o be c a p i t a l i z e d a t the same rate P j .  I n other words, as the two ex-  pected earnings flows are p e r f e c t l y p r o p o r t i o n a l they must be p r o p o r t i o n a t e l y  substitutes  valued.  With the i n t r o d u c t i o n of debt f i n a n c i n g i t i s assumed that a l l debt o b l i g a t i o n s , whether corporate o r p e r s o n a l , produce a constant and c e r t a i n flow of i n t e r e s t payments which i s b e l i e v e d by a l l c r e d i t o r s f o r a l l borrowers.  Thus a l l debt o b l i g a t i o n s are p e r f e c t s u b s t i t u t e s i n  quality.  I t i s f u r t h e r assumed that a l l debt o b l i g a t i o n s  are traded i n p e r f e c t markets and thus they y i e l d the same  -  66  -  rate of r e t u r n which i s the c a p i t a l i z a t i o n rate f o r c e r t a i n streams. For a f i r m i w i t h expected earnings X± which has some debt i n i t s c a p i t a l s t r u c t u r e the expected net earnings w i l l be, X i - rBj_, where, B i - the market value of debt i n the c a p i t a l s t r u c t u r e of company i . = the constant, u n i v e r s a l rate of i n t e r e s t or the c a p i t a l i z a t i o n rate f o r c e r t a i n  streams.  Because the i n t e r e s t payments rB^ are c e r t a i n , they have no d i s p e r s i o n and thu3 do not a l t e r the absolute d i s p e r s i o n of expected net earnings from that of expected earnings, S .  Thus the r e l a t i v e d e v i a t i o n or r i s k of exSxi pected net earnings = w i l l be greater than that of i " i xi  x  expected earnings  S x i  r  B  . The i n t r o d u c t i o n of debt i n t o the  *i c a p i t a l s t r u c t u r e thus increases the r i s k of the expected net earnings stream.  This f i n a n c i a l r i s k to expected net  earnings ( i . e . the r i s k induced by f i n a n c i a l leverage of e q u i t y c a p i t a l and expected net earnings) i s i l l u s t r a t e d i n Figure 1 4 . This concept of f i n a n c i a l r i s k , wholly r e l a t e d to  F i g . 14.--Financial Risk to Net Earnings due to Leverage  - 68 -  f i n a n c i a l leverage, has been c r i t i c i z e d .  J  However under  any c o n s i s t e n t assumption about the shape of 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 X, the s t a t i s t i c S to define ?(X-rB  <  x  / X-rB i s s u f f i c i e n t  0 ) . Here the cost of senior c a p i t a l  r should include the imputed cost of r e s t r i c t i v e  pledges,  as w e l l as i n t e r e s t payments, so that rB represents the r e a l economic burden of debt.  We have a l s o i m p l i c i t l y assumed  r i s k l e s s retirement or refunding of a l l debt o b l i g a t i o n s so that rB embodies a l l r i s k created by debt. We thus conclude that S j / X-rB  completely  re-  presents the r i s k to shareholders and that increments due to f i n a n c i a l leverage f u l l y define the f i n a n c i a l r i s k . S i m i l a r l y , the f i r m ' s operating or business r i s k S^ / X i s l a r g e l y determined by leverage (e.g. the marginal  profit  leverage, dx/dQ, and the output leverage, do/dm).  Basic P r o p o s i t i o n s The b a s i c M o d i g l i a n i and M i l l e r hypothesis i s that the market value of any f i r m i s independent of i t s c a p i t a l s t r u c t u r e and i s given by c a p i t a l i z i n g i t s expected earnings a t the rate appropriate to i t s r i s k c l a s s membership. Thus,  - 69 Xi „ V i  =  Ci+Bi  where, X i = the expected earnings of f i r m i whose r i s k defines i t s membership i n c l a s s j Pj » the average c a p i t a l i z a t i o n r a t e f o r expected earnings of f i r m s i n c l a s s j V  i  =  the market value of f i r m i  C i = the market value of a l l common shares of f i r m i Bj_ „ the market value of a l l s e n i o r s e c u r i t i e s / o b l i g a t i o n s of f i r m  i.  Here f i n a n c i a l r i s k or leverage i s i r r e l e v a n t f o r r i s k c l a s s membership.  The c o r o l l a r y i s that the average  cost of c a p i t a l of any f i r m i s independent 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 average c a p i t a l i z a t i o n rate f o r an unlevered expected net earnings flow of the same risk class. This hypothesis f o l l o w s d i r e c t l y from the expected earnings r i s k c l a s s equivalency d e f i n i t i o n and the p e r f e c t market assumption.  Here, commodities of equivalent qua-  l i t y must be p r o p o r t i o n a l l y valued.  ( i . e . the market  value of a l l claims against a f i r m ' s assets must equal the market value of those assets based on t h e i r earning, or  -  liquidation  70  -  value.)  The hypothesis r e i t e r a t e s the v i e w 5 that there i s 2  a t o t a l i t y of r i s k to the f i r m and i t s expected earnings which cannot be changed but simply r e d i s t r i b u t e d c l a s s e s of s e c u r i t i e s or c l a i m s .  among  This concept has been  26  challenged  f o r widely h e l d f i r m s where each Investor i s  s e l f - i n t e r e s t e d and has s p e c i f i c r i s k preferences.  Here a  r e d i s t r i b u t i o n of r i s k may cause v a l u a t i o n anomalies which i n aggregate produce a change i n the market value of the firm.  This c r i t i c i s m i n v o l v e s the r e a l i t y of the assumptions  regarding the nature and v a l u a t i o n of debt which w i l l be discussed l a t e r . The suggested market mechanism whereby the c a p i t a l i z a t i o n rates are e q u i l i b r a t e d  i s based upon the a b i l i t y  of i n v e s t o r s t o adjust t h e i r p o r t f o l i o s to create equivalent c a p i t a l and earnings leverage ( i . e . equivalent f i n a n c i a l r i s k ) t o that produced by c o r p o r a t i o n s .  This equivalent  "homemade" i n v e s t o r leverage may be increased by borrowing funds (e.g. buying shares on margin) o r by reducing p o r t f o l i o holdings of bonds and i t may be decreased by h o l d i n g additional  bonds i n the p o r t f o l i o or by reducing the amount  of personal debt (see Appendix I I ) .  - 71  -  These e q u i l i b r a t i n g e f f e c t s d e r i v e from the assumpt i o n s of p e r f e c t markets and p e r f e c t equivalency of corporate and personal debt.  The controversy over t h i s mecha-  nism i s r e a l l y then a debate over the e s s e n t i a l v a l i d i t y of these assumptions i n the r e a l world. Of course there are systematic market imperfections which can dampen t h i s mechanism. and knowledge i s not f r e e . preferences  27  There are t r a d i n g costs  The matter of i n v e s t o r s ' r i s k  and t h e i r market distribution ® i s i n v o l v e d . 2  For only i f these e x a c t l y match the market d i s t r i b u t i o n of s e c u r i t i e s r i s k would market p r i c i n g be t h e o r e t i c a l l y p e r f e c t . However there seems to be no b a s i s f o r not b e l i e v i n g that i n v e s t o r preferences are w e l l d i s t r i b u t e d and issued secur i t i e s are w e l l matched to fund f l o w s . Much of the d i s c u s s i o n has concerned the equivalency of corporate and personal d e b t .  20.  As corporate debt  i s of l i m i t e d l i a b i l i t y i t may not be adequately replaced by personal debt of u n l i m i t e d l i a b i l i t y .  Thus the shares  of leveraged f i r m s would tend to command a premium i n the ^0  market.  Furthermore there are c e r t a i n i n s t i t u t i o n a l as  w e l l as p u b l i c r e s t r i c t i o n s to margin buying. We might conclude that there are b a s i c market forces  - 72 -  operating t o achieve e q u i l i b r i u m i n share v a l u a t i o n but that there are i n f a c t , many obstacles t o i t s Immediate and complete r e a l i z a t i o n .  These however, do not m a t e r i a l -  l y d e t r a c t from the rigorous conceptual p r o p o s i t i o n of M o d i g l i a n i and M i l l e r . The veLghted average cost o f c a p i t a l i s , Vi  Vi  where, k i = the expected y i e l d o f the common stock o f f i r m i i n class j . The s i g n i f i c a n c e o f t h i s expression now l i e s i n the f a c t that we have shown pj t o be a constant and assumed r to be a constant which imply a s p e c i f i c behaviour of kj_ i n response t o changes i n c a p i t a l s t r u c t u r e ( i . e . a s p e c i f i c p a t t e r n of i n v e s t o r share v a l u a t i o n ) .  I t f o l l o w s that the  expected y i e l d can be expressed a s , k  ±  . p  j  +  ( p  r  r ^ i  Thus the expected common stock y i e l d d e r i v e s from the c a p i t a l i z a t i o n rate f o r a pure e q u i t y earnings stream of the* same c l a s s plus a premium r e l a t e d t o r i s k . Therefore the market value of the common stock of a f i r m i s given by c a p i t a l i z i n g i t s expected net earnings  - 73  -  at a v a r i a b l e rate depending upon r i s k , as f o l l o w s , c  i - Xi -  r  B  i  where, k i » the average c a p i t a l i z a t i o n rate f o r expected net  earnings.  Thus i t f o l l o w s that the value of any f i r m i i n a c l a s s j can be stated as, V i = Ci+Bi . X j - r B j + rBj. « * i ki r' pj This share v a l u a t i o n c o r o l l a r y suggests that i n v e s t ors c a p i t a l i z e expected net earnings a t a higher rate due to the Increased  r i s k induced by leverage  and that t h i s  d i s c o u n t i n g e x a c t l y o f f s e t s the advantages to be gained by the use of apparently  "low c o s t " debt.  These r e l a t i o n -  ships between c a p i t a l i z a t i o n rates are shown i n Figure  15.  Given the M o d i g l i a n i and M i l l e r hypothesis and  the  assumption of equivalent r i s k investments ( i . e . marginal r i s k of investments equals average r i s k of f i r m ) i t f o l l o w s that the required rate of r e t u r n on an investment opportun i t y , namely the marginal cost of c a p i t a l , w i l l be equal to the average cost of c a p i t a l p j , of f i n a n c i n g the investment.  regardless of the method  This derives from the market  - 74 -  k  ro or c O  ro N  Pj  "ro CL  B/C  /k </) (U +-> ro cr c O  ro  Pj  N  1 CL  ro  n  B/V  >•  F i g . 15.--Earnings C a p i t a l i z a t i o n Rates under Leverage  - 75 value maximization o b j e c t i v e where,  ™1 ± dl  1  ±  where, 1^ Thus,  the amount of c a p i t a l required by an investment,  m  dVi dli  and thus,  m  d(Xj/pj) ^ dli dX-i — . ± -w — d^  r>i J  where, w = the expected rate of r e t u r n on an investment opportunity. This c r i t e r i o n would apply e q u a l l y i n the case of f i n a n c i n g out of earned d e p r e c i a t i o n or r e t a i n e d earnings and would prevent decreased returns to shareholders.  C.  Principle Modifications  Income Tax Perhaps the most s i g n i f i c a n t e l a b o r a t i o n of the b a s i c hypothesis was i t s restatement i n a world of t a x e s where i n t e r e s t payments are allowed as a deductable expense.  Here i t i s assumed that there i s a f i x e d average  rate of t a x on corporate income ( i . e . constant  marginal  r a t e ) and that t h i s rate w i l l not change over time ( i . e . i t i s certain).  3 1  - 76 The expected earnings ( i . e . before i n t e r e s t expense) a f t e r t a x are now, ( X i - r B i ) ( l - t ) + r B i - X i ( l - t ) + rBjt where, t » the average and marginal corporate Income t a x rate. Thus the market value of the f i r m a f t e r t a x w i l l be, V *  X {l-t)  rBjt  ±  X i U - t ) + rBjt  ±  where, V^* „, the market value of the f i r m under t a x . Qj «  the average c a p i t a l i z a t i o n rate f o r expected earnings a f t e r t a x .  Thus i t f o l l o w s  that,  *j - Pj - ( j - ^| f * p  r  The expected net earnings a f t e r tax are now (Xi-rBj_) ( 1 - t ) and the value of the f i r m can be stated a s , VjL  * » Ci* + B i - (Xj- rBj)(l-t) hi  +  rBi r  where, C i * - the market value of the common stock of the f i r m under t a x . hi  = the average c a p i t a l i z a t i o n rate f o r expected net earnings a f t e r t a x .  -  -  77  I t can be shown t h a t , h i - pj + ( p j - r ) ( l - t £ i  The marginal cost of c a p i t a l c r i t e r i a , f o r acceptance of investment o p p o r t u n i t i e s of l i k e r i s k , are derived as f o l l o w s ; dV j * _ d X j ( l - t ) / j dl dl P  ^ drBjt/r  ^  ^  dl  Thus, contrary t o the proposal of M o d i g l i a n i and M i l l e r , the minimum acceptable d(Xj(l-t) + rBjt) ^ dl and  y  rate of r e t u r n a f t e r t a x i s , _ . dpjBjt " dl p  J  +  drBjt dl  thus, y - PJ - (pj-r)t__± Por e n t i r e l y debt f i n a n c i n g where d B i / d l = 1 , the  marginal cost of debt c a p i t a l a f t e r t a x i s p j ( l - t ) + r t . For e n t i r e l y equity f i n a n c i n g where d B i / d l - 0 the marginal cost of new equity c a p i t a l i s p j .  This would include c a p i -  t a l r a i s e d through the issue of p r e f e r r e d stock. For f i n a n c i n g out of r e t a i n e d earnings the marginal cost w i l l be, P 1  U-td)  - 78 where, t d - management's estimate of the average marginal tax  rate on d i v i d e n d income f o r a l l sharehold-  ers . t,g - management's estimate of the average marginal tax  rate on c a p i t a l gains f o r a l l shareholders  The weighted average of these marginal costs of c a p i t a l from v a r i o u s sources, based upon the " t a r g e t " c a p i t a l s t r u c t u r e of the f i r m , i s the r e l e v a n t marginal cost of c a p i t a l f o r investment d e c i s i o n s . A l t e r n a t i v e l y the r e l a t i o n s h i p s may he s t a t e d on a before t a x b a s i s as f o l l o w s ,  P  q  J  -  -Bi  X i . J - T T * . P i (1 - t 5 i ) VT* i - t — r=r ' Vi rt  where, P j * „ the average c a p i t a l i z a t i o n rate f o r expected earnings before t a x . Also, k * = X j - r B j . h i _ j ? j (pj,-ri)B_i Ci* 1-t 1-t Ci* ±  +  where, k^*. the average c a p i t a l i z a t i o n rate f o r expected net earnings before t a x . The r e l a t i o n s h i p s between the average c a p i t a l i z a t i o n r a t e s under t a x are shown i n Figure 16. The s i g n i f i c a n c e  - 79 -  Q. ro  B/V*  >  B/C *  >-  r  F i g . 16.--Earnings C a p i t a l i z a t i o n Rates under Tax  - 8p  -  of these l i e s i n the defined decreasing average cost of c a p i t a l under t a x as leverage i n c r e a s e s . This e f f e c t i s due to the amount and c e r t a i n t y of t a x savings a r i s i n g from the treatment of i n t e r e s t charges.  However i t i s  s o l e l y due t o t h i s f a c t that any permanent advantage accrues from the use of debt. Future s h i f t s i n the degree of leverage w i l l n o t , i g n o r i n g t a x , a l t e r the per share value of the common stock ( i . e . increased net earnings per share w i l l be e x a c t l y o f f s e t by a higher equity c a p i t a l i z a t i o n r a t e ) .  However  under t a x , as the i n t e r e s t payment t a x s h i e l d accrues to shareholders, i t would be s t r i c t l y  c o r r e c t t o consider ex-  pected f u t u r e leverage i n share v a l u a t i o n . This  concerns  the b e l i e v e d unexploited f i n a n c i a l o p p o r t u n i t i e s i n c a p i t a l s t r u c t u r e and the p r o b a b i l i t i e s as to i f and when management might s e i z e these o p p o r t u n i t i e s . I n p r a c t i c e however, current leverage should be appropriate to operating earnings r i s k and c r e d i t o r c o n s t r a i n t s , the t a x s h i e l d saving i s r e l a t i v e l y s m a l l , and t h i s f u t u r e value must s t i l l be discounted f o r present v a l u a t i o n . Thus t h i s s o p h i s t i c a t i o n i s perhaps a r e a l but complex and t r i v i a l m o d i f i c a t i o n to any v a l u a t i o n scheme  - 81 -  Interest  Rates P e r h a p s t h e most p e r s i s t e n t c r i t i c i s m ^  M o d i g l i a n i and M i l l e r hypothesis ing taxes, an optimal debt i s i n d i c a t e d .  capital  2  has been t h a t ,  0  f the consider-  structure of v i r t u a l l y  Of c o u r s e t h i s  senseless  100$  conclusion,  where r i s k i s s i g n i f i c a n t , f o l l o w s f r o m t h e extreme i n t e r p r e t a t i o n of the basic hypothesis Interest  rate, regardless  ments assumed  It increases  while  still  of leverage  assuming a  constant  ( i . e . i n t e r e s t pay-  certain.)  i s a g e n e r a l l y accepted f a c t that as I n t e r e s t r a t e s may  leverage  r i s e and e v e n t u a l l y c r e d i t  may become u n a v a i l a b l e due t o t h e a t t i t u d e s o f c r e d i t o r i n vestors  .33,34  A l t h o u g h M o d i g l i a n i and M i l l e r have not  e x p l i c i t l y attempted t o restate t h e i r theory  i n these  t h e y do q u a l i f y i t by r e c o g n i z i n g t h a t t h e r e w i l l be s t r a i n t s upon t h e e x t e n t are a p l u r a l i t y of the loan  of leverage.  con-  They a l l o w t h a t  there  o f i n t e r e s t r a t e s b a s e d upon t h e p r o v i s i o n s  ( e . g . term t o maturity,  collateral  as w e l l as the f i n a n c i a l c o n d i t i o n o f t h e  The q u e s t i o n the  terms  security)  borrower.35  then concerns the s p e c i f i c nature  of  f u n c t i o n f o r the average c a p i t a l i z a t i o n rate f o r corpo-  r a t e debt streams i n terms o f l e v e r a g e .  M o d i g l i a n i and  - 82 M i l l e r suggest that i t w i l l be a n©n-linear f u n c t i o n of the debt/equity  r a t i o ( B / C ) ^ which i s a measure of r i s k  to c r e d i t o r s , given X.  3  However a l i n e a r f u n c t i o n , as shown  i n Figure 1 7 , would be a simpler and, perhaps adequate approximation (e.g. produces a c u r v i l i n e a r f u n c t i o n of B/v.) the case of extreme leverage  In  ( i . e . B / v — ^ 1 . 0 ) the promised  y i e l d would become i n f i n i t e l y l a r g e ( i . e . r — > - o o ) . Of course the i n t e r e s t rate must be some f u n c t i o n of the l e v e l of t o t a l c o n t r a c t u a l payments (rB) i n r e l a t i o n to the q u a n t i t y (X) and q u a l i t y ( S ) of operating X  This measure of r e l a t i v e commitment  earnings.  (rB/x) i s the c l a s s i c  "times debt charges earned" measure of bond q u a l i t y which has proven to be c l o s e l y r e l a t e d to f u t u r e d e f a u l t . ? 3  Where  the "times charges earned" are s u f f i c i e n t l y great (e.g. over 2 or 3 ) , the p r o b a b i l i t y of d e f a u l t ( i . e . P ( X < r B ) ) i s usually insignificant. I t has been suggested that the appropriate measure of earnings  committed to c r e d i t o r s , f o r v a l u a t i o n purposes,  i s the expected future i n t e r e s t payments, as per the usual s e c u r i t y a n a l y s i s procedure, r a t h e r than the current i n t e r e s t payments. ' 3  3  However we have assumed that the c a p i t a l i z a t i o n  r a t e s f o r debt streams r are constant over time ( i . e . norma l i z e d rates w i t h no long term s e c u l a r t r e n d ) .  - 83 -  •+->  P  Pj  05 N  $  Q.  OJ CJ  r  O  r  B/C  B/V F i g . 17.--Interest  >  Rate as a F u n c t i o n of Leverage  - 84 -  M o d i g l i a n i and M i l l e r suggest that as long as the r i s i n g supply curve f o r loanable funds i s the same f o r a l l borrowers, corporate or p e r s o n a l , then i n v e s t o r s can e q u i v a l e n t l y make o r undo l e v e r a g e . ^  Thus the basic hypo-  t h e s i s w i l l not be a f f e c t e d and the cost of c a p i t a l w i l l not change w i t h leverage, except f o r t a x saving e f f e c t s . The necessary  i m p l i c a t i o n of t h i s i s that i n v e s t -  ors must c a p i t a l i z e more h i g h l y leveraged net earnings streams a t decreasingly greater r a t e s . Beyond some point the absolute equity c a p i t a l i z a t i o n rate w i l l a c t u a l l y f a l l as shown i n Figure 18. Thus i n v e s t o r s w i l l value more h i g h 40  l y a more r i s k y flow of net earnings. M o d i g l i a n i and M i l l e r attempt, u n s a t i s f a c t o r i l y , to e x p l a i n t h i s as a demand created by r i s k l o v e r s or that i n t e r e s t r a t e s r i s e only moderately while non-price c o n t r o l s are the primary c o n s t r a i n t on leverage. * 1  1  To the extent  that shareholders only consider monetary payments i n assessing  the cost of c r e d i t o r c a p i t a l t h i s may be an accurate  d e s c r i p t i o n of the i n t e r e s t rate f u n c t i o n . However, the f a c t of an ultimate non-price c o n t r a c t ual  o r i n s t i t u t i o n a l l i m i t to borrowing i m p l i e s a cost of  debt f u n c t i o n which r i s e s to i n f i n i t y .  The i n c r e a s i n g  - 85 c o n s t r a i n t s upon management freedom, as leverage i n c r e a s e s , must "be considered as having an imputed economic c o s t .  Thus  a c o r r e c t v a l u a t i o n f u n c t i o n f o r c o n t r a c t u a l l y promised streams w i l l probably be c u r v i l i n e a r or We thus conclude that the equity rate continues to r i s e as the e f f e c t i v e  uni-linear. capitalization interest  rate r i s e s .  M o d i g l i a n i and M i l l e r agree that net earnings w i l l be more h e a v i l y discounted as stockholders face a s i g n i f i c a n t l y ho  g r e a t e r r i s k of bankruptcy  due to leverage.  J  The  inevitable  c o n c l u s i o n i s that beyond some point the cost of c a p i t a l w i l l r i s e due to leverage, as shown i n Pigure 19•  Thus  there i s a l i m i t to the a p p l i c a t i o n of the v a l u a t i o n hypot h e s i s i n i t s o r i g i n a l form. This then represents a challenge to the concept of the t o t a l i t y of r i s k inherent i n operating earnings.  We  must now consider purely f i n a n c i a l r i s k which i s a d d i t i o n a l to operating r i s k ( i . e . the r i s k of f i n a n c i a l d e f a u l t r a t h e r than operating f a i l u r e . ) Expected Growth Consider a l s o the case, which was p r e v i o u s l y cons t r a i n e d , of d i f f e r i n g growth rates of assets and f u t u r e  4  - 86  -  cn +->  CO  Pj  s  H  cr c  o  +->  I  \ \ \  k  a co U  v  B/C F i g . 1 8 . - - C a p i t a l i z a t i o n Rates under R i s i n g I n t e r e s t Rate  to  (U  +->  CO  c  o  H->  co N  CO  R  CO  U  B/C F i g . 1 9 . — M o d i f i e d C a p i t a l i z a t i o n Rates  - 87 earnings among f i r m s whose earnings are of equivalent risk.^  This r a i s e s the earnings retention-payout issue  once more but we s h a l l simply accept the authors' earnings approach. The market value of a uniquely growing f i r m  i,  whose expected earnings from c u r r e n t l y h e l d a s s e t s , X i ( 0 ) , are of r i s k c l a s s j , i s then, „  , M  "^°Xi(0) + Ij.(n) (wj(n) - p j ) / p j (1+Pj)  -  X  l  (  Q  ) +  Pj  n  ^ l i ( n ) (wj(n) - p j ) / p j L /.. *n n=o (1 + Pj )  where, Vj,(n) „ the market value of f i r m i a t time n. X^(n)  m  the expected earnings of f i r m i from a s s e t s h e l d a t time n.  I j ^ n ) . the expected increase i n assets of f i r m i during the time p e r i o d from n - 1 t o n ( i . e . net investment which here equals gross i n vestment .) Wi(n) m the expected average annual rate of earnings ( i . e . y i e l d ) i n p e r p e t u i t y from a s s e t s I i ( n ) , Pj «  the average c a p i t a l i z a t i o n rate f o r a nongrowing, but perhaps expanding, expected  - 88 earnings stream of r i s k c l a s s i ( i . e . Ij_(n) = ) f o r a l l n and/or Wi(n)=0 f o r a l l n.) I t i s s t i l l assumed that the expected earnings from new a s s e t s , I^(n) .Wj_(n), are of e q u i v a l e n t r i s k to those from e x i s t i n g a s s e t s , X^(0).  This expression f o r  the market value of the f i r m i s simply the present value of the sum of expected earnings on p r e s e n t l y h e l d a s s e t s , X^O),  and expected excess earnings on future investments,  T (n).(w (n) - P j ) . 1  i  This l a t t e r growth term represents  expected investment o p p o r t u n i t i e s to he e x p l o i t e d at a rate of r e t u r n g r e a t e r than P j . This approach to v a l u a t i o n under growth i s s i m i l a r to the a u t h o r s f u l l e r treatment 46 and other e x p o s i t i o n s . 46 1  As b e l i e f s about f u t u r e growth may be imprecise, we might simply express the e s s e n t i a l magnitude of these expectations This suggests a simpler v a l u a t i o n expression such as, v o) l (  where, Yj_  m  . ^(£)  ^  (?I/PJ ) - i i  equivalent perpetual average annual expected earnings from average annual expected new i n vestment  ( i . e . from I i and wi.)  I i ox equivalent perpetual average annual expected new  investment.  - 89 -  These m o d i f i c a t i o n s e s s e n t i a l l y concern the s t r u c t ure of b e l i e f s regarding f u t u r e earnings and do not impair the v a l i d i t y of the b a s i c v a l u a t i o n hypothesis.  D. Conclusion  Investor A t t i t u d e s The v a l u a t i o n scheme and b a s i s of r i s k equivalency c a r r y an i m p l i c i t assumption regarding i n v e s t o r s ' general a t t i t u d e towards r i s k .  Together they define an earnings  equivalency or i n d i f f e r e n c e f u n c t i o n as shown i n Figure 20. Although t h i s suggests i n v e s t o r a v e r s i o n to r i s k as we would expect, there i s no absolute r i s k c o n s t r a i n t nor any realm of v i r t u a l r i s k i n d i f f e r e n c e . due to the presumed d i v i s i b i l i t y  This I s , of course,  of investment where the  r e l a t i v e r i s k of earnings i s the u l t i m a t e c o n s t r a i n t .  Thus  the s i g n i f i c a n t aspect of t h i s earnings i n d i f f e r e n c e or i s o value map  i s constant, r a t h e r than the usual d i m i n i s h i n g ,  marginal r e l a t i v e r i s k ( i . e . constant, not i n c r e a s i n g , r i s k aversion.) In terms of the u t i l i t y approach t h i s defines constant r i s k a v e r s i o n w i t h i n the context of a monotonic u t i l i t y c e r t a i n monetary equivalent r e l a t i o n s h i p .  The r e s u l t i s a  - 90 -  family  of b i - l i n e a r u t i l i t y  in  Figure  to  that  21.  This  implicit  type  i n the  f u n c t i o n s o f the  of u t i l i t y use  form  function i s  shown  similar  of expected l o s s as a  measure  47 of  risk.  This  suggests that  basis of l o s s aversion sion  i n v e s t o r s b e h a v e on  r a t h e r than the  broader r i s k  aver-  motive. If  one  postulates  that  i n v e s t o r s as a whole have  an  i n c r e a s i n g aversion to greater absolute  to  t h e i r absolute  w e a l t h and  from h o l d i n g excessive a  the  that  they  amounts o f any  r i s k equivalency  w o u l d be  relative  individually one  r e f o r m u l a t i o n o f t h e M o d i g l i a n i and  risk  security  Issue,  M i l l e r basis  indicated.  refrain  For  of  instance,  the  2 ._ relative  variance  3  X  /X  c o u l d be  used as  risk.  T h i s would imply  assets  i n v o l v e s r i s k s which i n v e s t o r s f e e l  these  assets are  corollary er w i l l motive tration  be  i s t h a t the  more w i d e l y  i t s market v a l u e .  o f c o n t r o l and of the  size  measure  of a  supply  a more t r a d i t i o n a l  the of  such a  held  is a  This obviously  s o c i a l and  of  firm's  disappear  d i v i d e d among s e v e r a l f i r m s .  Complex a s  with  t h a t the very  the  A  when  further  f i r m , the neglects  institutional  highthe  concen-  funds. scheme may  implicit  a diminishing marginal  earnings  be,  i t would  produce  indifference function  coefficient  of variance,  and  P i g . 2 1 . - - I m p l i c i t Earnings U t i l i t y Function  - 92 earnings u t i l i t y f u n c t i o n w i t h d i m i n i s h i n g marginal u t i l i t y of c e r t a i n money. This would suggest that i n v e s t o r s act i n terms of r i s k a v e r s i o n . Summary Although the basic independence hypothesis and  the  extreme leverage case have proven most newsworthy an i n c l u s i v e summation of the hypothesis  i s more r e l e v a n t .  22 portrays the combined r e s u l t of the basic  Figure  hypothesis  w i t h corporate taxes, a r i s i n g i n t e r e s t rate f u n c t i o n and bankruptcy r i s k . I n t e r e s t i n g l y t h i s I n c l u s i v e model defines a convex cost of c a p i t a l f u n c t i o n which e x h i b i t s a point of minimum cost of capital-maximum market value f o r a f i r m of any given r i s k c l a s s . The a d d i t i o n a l c o n s i d e r a t i o n of growth prospects  simply a l t e r s the magnitude of the future streams  which are to be c a p i t a l i z e d . We may  then wonder what i s unique about the  M o d i g l i a n i and M i l l e r p r o p o s i t i o n s compared to t r a d i t i o n a l views.  Of course, t h e i r more rigorous statement of the  causal f o r c e s i s a powerful c o n t r i b u t i o n . But there a l s o remains a wide q u a n t i t a t i v e gap between the two concepts.  - 93 -  - 94 -  Contrary to the t r a d i t i o n a l view, M o d i g l i a n i and M i l l e r suggest that debt i s not cheap and that the tax advantages of deht are the only permanent advantages.  Both views r e s t  on fundamentally d i f f e r e n t concepts of i n v e s t o r behaviour and market f u n c t i o n i n g . M o d i g l i a n i and M i l l e r define a r i g i d investor-market response to leverage i n the v a l u a t i o n of shares. There remains the problem of attempting to r e c o n c i l e these t h e o r e t i c a l a b s t r a c t i o n s w i t h the observed world of 48 4Q reality.  Unfortunately e m p i r i c a l t e s t i n g  *^»-  SO /  has not  given a conclusive answer nor s i g n i f i c a n t evidence.  Such  a v a l i d a t i o n i s fraught with d i f f i c u l t i e s (e.g. expected earnings data, sample homogeneity of operating r i s k w i t h a broad s c a t t e r of leverage, c u r v i l i n e a r r e g r e s s i o n form) and i s u n l i k e l y to s e t t l e the matter. The hypothesis described represents simply a beginning and should be capable of f u r t h e r development.  This  may be i n terms of a dynamic form of a n a l y s i s where a time dimension i s introduced e x p l i c i t l y i n t o the f i r m ' s i n v e s t ment and f i n a n c i n g . This hypothesis should a l s o be set i n a general e q u i l i b r i u m context, p a r t i c u l a r l y w i t h respect to the c a p i t a l market.  This would introduce a r i s i n g supply  -  of c a p i t a l funds f u n c t i o n .  95  -  > J O  Some m o d i f i c a t i o n of the p e r f e c t market  assumption,  towards g r e a t e r r e a l i s m , might be p o s s i b l e i n s p i t e of the d i f f i c u l t i e s of d e f i n i n g i m p e r f e c t i o n s . This would concern any systematic d e v i a t i o n s (e.g. i n s t i t u t i o n a l o l i g o p o l y , corporate access to markets, brokerage f e e s , personal income t a x ) which might create l a g s and f r i c t i o n s i n the market mechanisms.  These could f r u s t r a t e attainment of e q u i l i b r i u m  w i t h i n reasonable periods of time. However, the concepts developed here provide a necessary foundation of a theory of the v a l u a t i o n of f i r m s and shares under r i s k .  This i s the necessary element r e -  q u i r e d to apply the v a r i a b l e r a t e of discount or r e t u r n method of a n a l y s i s to c a p i t a l investments under r i s k . •'•Benjamin Graham, David L. Dodd, Sidney C o t t l e and Charles Tatham, S e c u r i t y A n a l y s i s , (4th ed.; New York: McGraw H i l l Book Company, Inc., 1 9 6 2 ) , Part IV. J o h n Burr W i l l i a m s , The Theory of Investment Value, (Cambridge : Harvard U n i v e r s i t y P r e s s , 1 9 3 8 ) , p. 5 5 2  ^Ezra Solomon, "Measuring a Company's Cost of C a p i t a l , " J o u r n a l of Business, (October, 1 9 5 5 ) . ^J.F. Walter, "Dividend P o l i c i e s and Common Stock P r i c e s , " J o u r n a l of Finance, (March, 1 9 5 6 ) , pp. 29-41.  -  96  -  ^M.J. Gordon and E. Shapiro, " C a p i t a l Equipment A n a l y s i s : The Required Rate, of ?ro<fit," Management Science, (October, 1 9 5 6 ) , pp. 1 0 2 - 1 1 0 . ^David Durand, "The Cost o f Debt and E q u i t y Funds f o r Business," Conference on Research i n Business Finance, (New York: N a t i o n a l Bureau of Economic Research, 1 9 5 2 ) . ^Solomon, J o u r n a l of Business, (October,  1955).  ^Franco M o d i g l i a n i and Merton H. M i l l e r , "The Cost of C a p i t a l , Corporation Finance and the Theory of Investment," American Economic Review, (June, 1 9 5 8 ) > p p . 2 6 1 - 2 9 7 . % ) a v i d Durand, "The Cost of C a p i t a l , Corporation Finance, and the Theory of Investment: Comment," American Economic Review, (September, 1 9 5 9 ) , pp. 6 3 9 - 6 4 4 . F r a n c o M o d i g l i a n i and Merton H. M i l l e r , "The Cost o f C a p i t a l , Corporation Finance, and the Theory of I n vestment: Reply," American Economic Review, (September, 10  1959),  655-669.  PP.  •^Graham, Dodd, C o t t l e and Tatham, Ch. 3 5 . I . F r i e n d and M. Puckett, "Dividends and Stock P r i c e s , " American Economic Review, (September, 1 9 6 4 ) . IS  ^Myron J . Gordon, The Investment, Financing and V a l u a t i o n of the Corporation, (Homewood: Richard D. I r w i n , Inc.,  1962).  14 Solomon, J o u r n a l of Business, (October,  1955).  rdon and Shapiro, Management Science, (October, 1956). l6  ¥alter, J o u r n a l of Finance, (March,  1956).  H. H. M i l l e r and F. M o d i g l i a n i , "Dividend P o l i c y , Growth and the V a l u a t i o n of Shares," J o u r n a l of Business, (October, 1961), pp. 411-4331?  - 97 -  -i o  N. Molodovsky, "Stock Values and Stock P r i c e s , " F i n a n c i a l A n a l y s t s J o u r n a l , (May-June, i960). •'•^E.S. Mead and J u l i u s Grodinsky, The Ebb and Flow of Investment V a l u e s , (New York: Appleton-Century-Crofts, Inc., 1939). Durand, American Economic Review, (September, 1959)• J . Fred Weston, "A Test of Cost of C a p i t a l Propos i t i o n s , " Southern Economics J o u r n a l , (October, 1963), PP. 105-112. 2 1  22 George J . S t i g l e r , The Theory of P r i c e , (New York: The Macmillan Company, 1952), p. 56. ^ A l e x a n d e r Barges, The E f f e c t o f C a p i t a l Structure on the CQ3t of C a p i t a l , (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1962) , Ch. 2. 1  *Ezra Solomon, The Theory o f F i n a n c i a l Management, (New York: Columbia U n i v e r s i t y P r e s s , 1963), Ch. V I I I . 2i  2  5Durand, Conference on Research i n Business Finance.  26 27 pO  Barges, Ch. 6. Barges, Ch. 6.  M i l l e r and M o d i g l i a n i , J o u r n a l of Business, (October, 1961), pp. 431-433. 29 Durand, American Economic Review, (September, 1959),  pp. 639-644.  3°Modigliani and M i l l e r , American Economic Review, (September, 1959), PP. 655-669.  Franco M o d i g l i a n i and Merton H. M i l l e r , "Taxes and the Cost o f C a p i t a l , " American Economic Review, (June, 1963), PP. 433-443. J  - 98 -  E z r a Solomon, "Leverage and the Cost of C a p i t a l , " J o u r n a l of Finance, (May, 1963), pp. 273-279. 3 2  33prederick Lutz and Vera L u t z , The Theory of Investment o f the Firm, ( P r i n c e t o n : P r i n c e t o n U n i v e r s i t y Press, 1951), Ch. XVI. S4  Solomon, The Theory of F i n a n c i a l Management,  J  Ch. V I I I .  35]y[odigliani and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 3 M o d i g l i a n i and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 6  ^¥.B. Hickman, Corporate Bond Q u a l i t y and Investor Experience, (New York: N a t i o n a l Bureau of Economic Research, 3  195BT  38sarges, Ch. 6. • ^ M o d i g l i a n i and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 40  Solomon, The Theory of F i n a n c i a l Management, Ch. V I I I . 41  ^•"-Modigliani and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 42  Solomon, The Theory of F i n a n c i a l Management, Ch. V I I I . ^ M o d i g l i a n i and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 44  M o d i g l i a n i and M i l l e r , American Economic Review, (September, 1959), PP. 655-669.  - 99 -  4 5 M i l l e r and M o d i g l i a n i , J o u r n a l of Business, (October, 1961), pp. 411-433. Walter, J o u r n a l of Finance, (March, 1956), pp. 29-41. ^ H a r r y M. Markowitz, P o r t f o l i o S e l e c t i o n , (New York: John Wiley & Sons, Inc., 1959), Ch. 13. ^ M o d i g l i a n i and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. % e ston, Southern Economics J o u r n a l , (October, 1963), pp. 105-112. ^°Barges, 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 Cost of C a p i t a l . -^"Solomon, The Theory of F i n a n c i a l Management, Ch. I X . 5 Solomon, J o u r n a l o f Business, (October, 1955). 2  -^Robert Lindsay and A r n o l d W. Sametz, F i n a n c i a l Management, (Homewood: Richard D. I r w i n , Inc., 1963), p. 181.  CHAPTER I I I INVESTMENT VALUATION AND DECISIONS UNDER RISK A. Earnings I n t e r r e l a t i o n s h i p s and V a l u a t i o n  Interclas3 Relationships  I t has been emphasized that the v a r i a b l e rate of discount o r r e t u r n approach to investment d e c i s i o n s l y depends upon some scheme of v a l u a t i o n under r i s k . 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  1  crucialThe  of the v a l u a t i o n of f i r m s  and shares under r i s k , which has been examined, may be a p p l i e d to t h i s purpose. This hypothesis defines the market v a l u a t i o n of a f i r m of given operating r i s k ( i . e . r e l a t i v e d e v i a t i o n of p o s s i b l e earnings) under v a r i a b l e f i n a n c i a l r i s k ( i . e . l e verage).  However, we can a l s o deduce the market v a l u a t i o n  of unleveraged f i r m s under v a r i a b l e operating r i s k ( i . e . among d i f f e r e n t r i s k c l a s s e s ) . Consider a f i r m i n r i s k c l a s s j with expected earnings X j of standard d e v i a t i o n Sj and with debt i n i t 3 c a p i t a l s t r u c t u r e of market value B j . Thus, assuming there are  •- :ioo- -  - 101 -  no t a x e s , the expected net earnings of t h i s f i r m w i l l he X j - r B j of r i s k S j / X j - r B j .  Consider a l s o another f i r m  i n r i s k c l a s s f w i t h expected earnings Xf of standard dev i a t i o n S f , which i s unleveraged, where, ^f - _ j X Xj - r B j s  f  Thu3 the expected earnings of f i r m f and the expected net earnings of f i r m j are of e q u i v a l e n t r i s k and w i l l he valued p e r f e c t l y p r o p o r t i o n a l l y , assuming that they are a l s o p e r f e c t l y c o r r e l a t e d .  These earnings and r i s k r e -  l a t i o n s h i p s are shown i n F i g u r e 2 3 . We can express t h i s r e l a t i o n s h i p a s , p  f  - k  f  - k j - pj + ( P j - r ) B/C - Pj ( 1  +  - Pj ( 1  +  (1-r/pj)  (rB/r) (Xj/pj)-(rB/r)  (1-ypj) i } ) r / p j - rB/3Cj V B / I  J  J  - p. ( ( / P j ) Xj) (r/pj) - (rB/ Xj) f  }  J  - .  ( ( /Pj) U j - r B ) / X j ( r / p j ) - 1 + ( X j - r B ) / Xj f  p  3  )  }  - 102 -  -- 103- -  (_) PJ  J _ "3 3j/(Xj-rB)  Pj  S  j/( j" X  r B  ) )  Therefore, EL.)  Pf  Pj (  =  Pj  a^y. 3  f/Xf  )  Thus the average c a p i t a l i z a t i o n rate f o r expected earnings of any r i s k c l a s s 13 defined by the rate f o r any other r i s k c l a s s , the r e l a t i v e earnings r i s k between the two c l a s s e s and the i n t e r e s t r a t e , assuming that earnings of a l l r i s k s are p e r f e c t l y c o r r e l a t e d and that there are no taxe s.  Tax and Leverage E f f e c t s Under corporate income tax the i n t e r c l a s s r e l a t i o n ships are complicated by the f a c t that the tax s h i e l d on i n t e r e s t payments, and i t s c e r t a i n t y , enhance the q u a n t i t y and q u a l i t y of a f t e r t a x expected earnings.  This r e s u l t s  i n a decreasing cost of c a p i t a l q and an i n c r e a s i n g market  value V* as leverage increases w i t h the r e s u l t that the equity c a p i t a l i z a t i o n rate h increases more slowly than i t would otherwise ( i . e . h r e f l e c t s the m u l t i p l i c a t i o n l e s s the tax saving e f f e c t s of leverage on net earnings r i s k ) . I n i t i a l l y , i n order to e s t a b l i s h the a f t e r tax i n t e r c l a s s c a p i t a l i z a t i o n rate r e l a t i o n s h i p s  independent of i n t r a  c l a s s leverage e f f e c t s , we w i l l assume that i n t e r e s t payments are not a tax deductible business expense.  Here the  a f t e r tax c a p i t a l i z a t i o n rates f o r expected earnings remain constant under leverage 1 • P + (P-r) where  and, B/C*  1 - the c a p i t a l i z a t i o n rate f o r expected net earnings a f t e r tax, where i n t e r e s t i s not a  tax  deductible expense. Thus, f u r t h e r to the previous  proposition,  Pf - I f - l j • Pj + (Pj - r ) Bj/C*j where,  p - the c a p i t a l i z a t i o n rate f o r an unlevered expected earnings stream a f t e r tax, X ( l - t ) .  Then i t f o l l o w s  that,  (  Pf " P i (  r_ Pj  )  Sj/ Xj(l-t) Sf/ Xf(1-t)  •(£-)•-' i ; Pj  3  }  J/ ^ Sf/:xf(i-t) ( 1  T }  105 -  Thus the average c a p i t a l i z a t i o n rate f o r unlevered a f t e r tax expected earnings of any r i s k c l a s s i s d e f i n e d "by the rate f o r any other r i s k c l a s s , the r e l a t i v e a f t e r tax earnings r i s k "between the two c l a s s e s and the i n t e r e s t r a t e , assuming a l l earnings are p e r f e c t l y  correlated.  Although e m p i r i c a l v e r i f i c a t i o n of t h i s i n t e r c l a s s earnings v a l u a t i o n expression would he d e s i r a b l e not be  attempted here.  The use o f assumed values w i l l  serve t o i l l u s t r a t e i t s a p p l i c a t i o n . Sj/ XWl-t) -  0.3  and r -  5.5$,  L e t pj - 7 . 5 $ where  thus r/p-j -  0.733  Sj/Xj(l-t) Sf/Xf(l-t)  s /x (i-t)  Pf/Pj  Pf  0.0  oo  0.733  5.50$  0.1  3.00  0.804  6.04  0.2  1.50  0.892  6.69  0.3  1.00  1.000  7.50  0.4  0.75  1.139  8.54  0.5  0.60  1.321  9.91  0.6  0.50  1.573  11.81  0.7  0.43  1.931  14.50  0.8  0.38  2.545  19.10  0.9  0.33  3.886  29.16  1.0  0.30  6.597  49.50  f  i t will  f  and,  -  1"06 -  F u r t h e r t o t h i s i n t e r c l a s s schedule of c a p i t a l i z a t i o n rates f o r unlevered earnings streams, there i s the i n t r a c l a s s e f f e c t of leverage on the cost of c a p i t a l , due to the tax d e d u c t i b i l i t y of i n t e r e s t payments, where, <lf  =  P f - ( P f - r ) t .B /v* f  f  We make the assumption here that the l e v e l o f expected net earnings r i s k S f / ( X f - r B f ) ( l - t ) » 0.75 f o r any f i r m i n any r i s k c l a s s f , i s an I n s t i t u t i o n a l l i m i t t o the a v a i l a b i l i t y of debt c a p i t a l ( i . e . a t t h i s point r - o o ) . This c o n s t r a i n t then places a l i m i t on c a p i t a l leverage which, f o r the M o d i g l i a n i and M i l l e r a f t e r tax model, w i l l define the optimal c a p i t a l s t r u c t u r e of any f i r m , namely, V*f B  f  ( r / p ) (1-t) f  "  S /x (l-t) " Sf/(Xf-rBf)(l-t)  +  x  f  f  Here the expected earnings r i s k i s , Sf Sf X f ( l - t ) + rBft " X f ( l - t ) r  C x  _  1-t Sf/Xf(l-t) Sf/(Xf-rB )(l-t)  > t  '  f  Thus the minimum a f t e r t a x , average cost o f c a p i t a l f o r f i r m s i n any r i s k c l a s s , where r - 5.5$ and t - 0.5, w i l l be,  -  -107  S /Xf(l-t) f  p  f  Bf/V*  f  S /(Xf(l-t)+rB t) f  f  q  f  0.0  5.50$  1.00  0.00  5.50$  0.1  6.04  0.98  0.05  5.78  0.2  6.69  0.94  0.12  6.13  0.3  7.50  0.90  0.19  6.60  0.4  8.54  0.84  0.27  7.26  0.5  9.91  0.75  0.38  8.25  0.6  11.81  0.60  0.50  9.92  0.7  14.50  0.30  0.66  I3:.i5  0.8  19.10  0  0.80  19.10  0.9  2 9 . I 6  0  0.90  29.16  1.0  49.50  0  1.00  49.50  C a p i t a l Sources E f f e c t s The marginal cost of e x t e r n a l equity funds ( i . e . common and p r e f e r r e d stock and income bonds) i s P f and of c r e d i t o r funds ( i . e . t a x deductible c o n t r a c t u a l payments) i s P f ( l - t ) + r t . L e t the value of p r e f e r r e d stock and i n come bonds be r F / r - F. Then the proportion of common and p r e f e r r e d stock and income bonds w i l l be (C* + F)/v* and the p r o p o r t i o n of payment deductible senior issues w i l l be (B-F )/v*.  - 108"-  Thua the weighted average coat o f e x t e r n a l e q u i t y and debt funds w i l l be, (f^) -Pf  (Pf(l-t) rt) +  (^i£)p  +  f  - (Pf-r)t(B_£)  The marginal cost of r e t a i n e d earnlng3, which i 3 a major aource o f funds, i a P f > ( l - t ) / ( l - t ) . I f D i s the d  g  expected average annual d i v i d e n d then ( X - r B ) ( l - t ) - D i a the expected average annual r e t a i n e d earninga.  Then the market  value of the common aharea w i l l be, c  * - (X-rB)(l-t) . (X rB)(l-t)-D h h*(l-td)/(l-t ) ;  +  g  D_ h*  where h*- the average c a p i t a l i z a t i o n rate f o r expected net earnings a f t e r corporate tax where personal income and c a p i t a l gains taxes a r e equal ( I . e . d i v i d e n d payout ahould be i r r e l e v a n t t o i n v e a t o r a ) . and, h* - h(i±BL) ( l '1-td (x-rB)(l-t) 1-tg D  /  %  where D/§C-rB)(l-t) i a the expected d i v i d e n d payout r a t i o . The p r o p o r t i o n o f r e t a i n e d earnings value i n the t o t a l value of the f i r m w i l l be (C*-D/h*)/v*.  Thus the  combined coat of e x t e r n a l and i n t e r n a l e q u i t y and debt funda q* w i l l now be,  - 0 l±(Pf(l-t)+rt) V*  q»1  G*-D/h^ V*  +  A  , .  B-F .  ,  . .  (  l-t D/h*+F, 1-tg V* d  }  +  pf  )  C*-D/h* , t - t . . d  g  The a f t e r tax expected earnings are now X ( l - t ) + rt(B-F) - E  and thus t h e i r r i s k w i l l he  S / x ( l - t ) + r t ( B - F ) - S/E. Assume that there i s no c a p i t a l gains t a x ( i . e . t g _ 0 ) , shareholders*  e f f e c t i v e marginal dividend income  tax rate t d - 20$ and the expected dividend payout r a t i o i s 6 0 $ . Note that there w i l l be no p r e f e r r e d stock issues due t o the optimal c a p i t a l s t r u c t u r e assumption.  Thus the  comprehensive cost of c a p i t a l f o r a f i r m of r i s k c l a s s f w i l l be, q*  f  . p  / , / <£> j. ,C*-0.6c*h/h*. , ( 1 - ( l - r / p )^-.t - ( ^ ) 0.2 ) n  f  f  - Pf " ( P f - r ) t | - - 0 . 2 3 f J Thus where r - 5 . 5 $ and t - 0 . 5 then, q*f  - P  f  - 0.23 - ( 0 . 5 P f - 2.98)'B/V*  n  - iio--  S f / X f P f  B/V*  Sf/Ef  q*f  O.G  5.50$  1.00  0.00 5.50$  0.1  6.04  0.98  0.05  5.77  0.2  6.69  0.94  0.12  6.11  0.3  7.50  0.90  0.19  6.58  0.4  8.54  0.84  0.27  7.22  0.5  9.91  0.75  0.38  8.20  0.6  11.81  0.60  0.50  9.82  0.7  14.50  0.30  0.66 12.99  0.8  19.10  0  0.80 18.87  0.9  29.16  0  0.90 28.93  1.0  49.50  0  1.00 49.27;  This schedule of earnings v a l u a t i o n under r i s k f o r o p t i m a l l y financed f i r m s i s shown i n Pigure 24.  It  provides an e x p l i c i t rate of c a p i t a l i z a t i o n f o r p e r f e c t l y c o r r e l a t e d expected earnings streams of any given r i s k .  Earnings C o r r e l a t i o n We have p r e v i o u s l y noted the importance of the c o r r e l a t i o n , o r strength of l i n e a r r e l a t i o n s h i p , between two earnings streams i n determining t h e i r combined r i s k and thus t h e i r combined v a l u a t i o n  J  0  0.1  I  I  I  I  I  1  I  1  1  0.2  0.3  0.4  0.5  0.6  0.7  08  0.9  1.0  Risk, Sf/Ef  ^  F i g . 24.—Schedule of V a l u a t i o n of A f t e r Tax Expected Earnings under R i s k  - 10.2- -  In our v a l u a t i o n model i t i 3 the c o r r e l a t i o n between p o s s i b l e earnings X which i s r e l e v a n t . This r e l a t e s t o average annual earnings I n p e r p e t u i t y over a l l p o s s i b l e s t a t e s of nature, not annual earnings x ( t ) over time.  Thus the  focus i s on long run s t a t e s of nature m, and the b e l i e v e d a s s o c i a t e d values of X . m  Long run s t a t e s of nature could  include the f o l l o w i n g elements, market demand 03  i n d u s t r y supply  company o p e r a t i o n  o •H -P •H rH O ft  i n t e r n t l . r e l a t i o n s business r e g u l a t i o n private vs. public private vs. public i n t e r n a t i o n a l trade i n t e r n a t i o n a l trade  O •H  n a t i o n a l income employment r e l a t e d products f o r e i g n markets market p r i c e s  resource development market s t r u c t u r e transportation f o r e i g n production f a c t o r costs  resource reserves management s k i l l product l i n e s manufrg. e f f i c i e n c y marketing a b i l i t y  s c i e n t i f i c needs  process technology automation product design  a p p l i e d technology research a b i l i t y  population growth t a s t e changes l e i s u r e time  labour f o r c e s k i l l s work a t t i t u d e s  labour r e l a t i o n s public relations  a o d o o  cd o  •H  Cl  Xi  o  <D  •P  ai  •H O O M  A c o n d i t i o n of equivalent earnings r i s k i s p e r f e c t m c o r r e l a t i o n over a l l s t a t e s of nature ( i . e . X-^ /X^ i d e n t i c a l f o r each i f o r a l l m).  This i s necessary i n order that shares-  representing such earnings streams be homogeneous and thus  - 113-  -  p e r f e c t s u b s t i t u t e s . Thus earnings streams of equivalent r i s k , as defined by S/E, when combined w i l l remain of ident i c a l r i s k as they are p e r f e c t l y c o r r e l a t e d , The earnings of most f i r m s w i l l respond s i m i l a r l y to v a r i o u s s t a t e s of general business c o n d i t i o n s although there w i l l be notable exceptions as w e l l as more s p e c i f i c 3 t a t e s of nature which w i l l  cause d i f f e r e n t e f f e c t s i n  d i f f e r e n t i n d u s t r i e s and f i r m s . However, we might postulate that our schedule of v a l u a t i o n concerns standard or r e f e r ence r i s k c l a s s e s which are a l l p e r f e c t l y p o s i t i v e l y c o r r e l a t e d among themselves as w e l l as w i t h some index of long run general domestic business a c t i v i t y M, such as I n d u s t r i a l Production or Corporation P r o f i t s . Firms w i t h equivalent earnings r i s k s/E but whose earnings are l e s s p o s i t i v e l y or even n e g a t i v e l y c o r r e l a t e d w i t h those of the reference c l a s s w i l l tend to reduce the r i s k of combined earnings andd thus enhance t h e i r v a l u a t i o n . Thus the c o e f f i c i e n t of c o r r e l a t i o n of earnings with some general business index R^, as defined by the strength of p  l i n e a r r e l a t i o n s h i p s between them,  w i l l be an element i n  the v a l u a t i o n of these earnings. We might expect that the greater market demand of  -  114  —  i n v e s t o r s f o r l e s s h i g h l y p o s i t i v e l y c o r r e l a t e d earnings streams f o r p o r t f o l i o d i v e r s i f i c a t i o n purposes, would exert an upward pressure on the p r i c i n g of such shares.  The  normative degree of such premium v a l u a t i o n would depend upon the r e l a t i v e p r o p o r t i o n of corporate earnings of v a r i o u s degrees of c o r r e l a t i o n represented i n the market. This aspect of v a l u a t i o n i s then part of a general e q u i l i brium v a l u a t i o n theory which we w i l l not attempt to develop here.  We may conclude t h a t both the c o e f f i c i e n t of v a r i a -  t i o n S|./E£ and the c o e f f i c i e n t of c o r r e l a t i o n  Rfyi  are i n -  herent q u a l i t i e s of earnings streams which define t h e i r r i s k and thus t h e i r v a l u a t i o n .  A h y p o t h e t i c a l scheme i s i l l u s -  t r a t e d i n Figure 2 5 . However w i t h i n the context of p a r t i a l e q u i l i b r i u m of the i n d i v i d u a l f i r m we s h a l l be concerned only w i t h the c o r r e l a t i o n between s p e c i f i c earnings streams which are to be combined, as defined by b e l i e f s and e x p e c t a t i o n s .  This  i m p l i c i t l y assumes t h a t long run c o r r e l a t i o n w i t h general business a c t i v i t y per se, i s not a s i g n i f i c a n t aspect of earnings v a l u a t i o n or that a l l earnings streams are very highly positively correlated.  Risk,  S /E f  f  F i g . 2 5 . — H y p o t h e t i c a l General Scheme of V a l u a t i o n under Risk  -116 B. Investment Decisions under R i s k The D e c i s i o n Process An investment proposal must he appraised i n terms of the marginal value which i t s earnings stream, of given v a r i a b i l i t y and c o r r e l a t i o n , c o n t r i b u t e s to the market value of the prospective i n v e s t o r f i r m . mined from our v a l u a t i o n schedule.  This can be deter-  Thus the c r i t e r i o n  f o r acceptance of an investment proposal i s , Ef + By _ | f _ _ ^ . ^*f+y ^*f ]  where q*  f+y  /—2 Sf+y - V S f  y  i s a f u n c t i o n of S„ /E^+E and where f+y f y +  2 R  fy f y s  s  : + s  y  \  Here the marginal cost of c a p i t a l f o r the p r o j e c t w i t h the f i r m , or the required rate of r e t u r n f o r t h i s i n vestment p r o j e c t w i l l be q* '  f+y 1 + (E /E ) (l-q* y/q* ) 4  f  y  f +  f  This c r i t e r i o n c a r r i e s w i t h i t the i m p l i c i t long run o b j e c t i v e s of maximization  of a f t e r tax expected earn-  ings Ef+Ey and minimization of earnings r i s k Sf y/Ef-+Ey +  which together define the goal of the maximization of the  -  117  -  market value of the f i r m . Although t h i s v a l u a t i o n i s stated i n terms of average annual earnings i n p e r p e t u i t y , where the assets remain i n t a c t ( i . e . earnings l e s s asset a m o r t i z a t i o n ) i t i s u s u a l l y p r e f e r a b l e to evaluate investment proposals on a cash flow basis with a s p e c i f i c terminal l i f e .  This f i n e r a n a l y s i s  can be c a r r i e d out using the derived marginal c a p i t a l i z a t i o n rate f o r E . y  In the case where the p r o j e c t earnings are assumed to be of equivalent r i s k Sf/If, and p e r f e c t l y c o r r e l a t e d R|.y _ +X-0, to the earnings of the f i r m then, Sy/Ey = Sf/Ef s  f y - Vs/+2SfSy+sy* . S +  and,  f +  S  y  Sf+y -3f+Sy - Sf Ef4Ey Ef+Ey Ef  Thus q*f+y « q*f -the present cost of c a p i t a l of the f i r m , and the investment c r i t e r i o n becomes, % x i q*f  y  Here the s i n g l e valued estimate of the earnings of the proj e c t are discounted a t the cost of c a p i t a l f o r the f i r m and compared to the investment outlay f o r a d e c i s i o n . This i s the t y p i c a l simple e v a l u a t i o n case where r i s k i s q u a n t i t a -  - 118 t i v e l y ignored, c a r r y i n g w i t h i t the i m p l i c i t  assumptions  stated above.  Examples The process of investment e v a l u a t i o n and d e c i s i o n can be i l l u s t r a t e d by means of a few examples. Example 1:  A Canadian cement manufacturer w i t h  If . $15,000,000 and  S  f  .  $5,500,000 I s considering  blishment of a cement plant i n South America. p l a n t would require an investment of  the e s t a -  The proposed  $50,000,000.  The p o s s i b l e earnings of the p r o j e c t r e f l e c t the f u t u r e c o n s t r u c t i o n outlook i n the South American country, the sole domestic s u p p l i e r status w i t h t a r i f f p r o t e c t i o n as w e l l as currency exchange problems and p o s s i b l e government take-over. Ey  $-1,000,000 +1,000,000 3,000,000 5,000,000 7,000,000 9,000,000  2 (By)  .05 .20 .30 .25 .15 .05  By 3* (Ey)  - 50,000 +200,000 900,000 1,250,000 1,050,000 450,000  ?.(Ey).(Ey-Ey)  2  12 1.152 x 10 1.568 0.192 0.360 1.536 1.352 6.160 x 10 $2,465,000  - 119 I t i s b e l i e v e d that the c o e f f i c i e n t of a s s o c i a t i o n between the earnings of the e x i s t i n g f i r m and those of the proposed plant w i l l be approximately 0.3 due to the independence of the Canadian and South American economies, beyond general world conditions (e.g. i n t e r n a t i o n a l f i n a n c i n g of c o n s t r u c t i o n , l e v e l of e x p o r t s ) , and the random local political possibilities.  This d i c t a t e s that the  c o e f f i c i e n t of c o r r e l a t i o n Bfy -  Sf Ef  5,500,000 15,000,000  5  .  f + y  -  f  f +  E  ly -  y  0.367, q*f - 8,12$  10 V5.5 +2(5.5)(2.465)0.7+2.465 6  2  - 10 V55.41 S f ^  +0.70.  2  -  $7,450,000  7,450,ooo „  *  0 < 3 9 6  m  Q  A  1  %  18,800,000  $50,000,000 *  l  8  ; ^  Q  Q  0  . i^ooo^ooo  - 223,500,000 - 184,700,000 - $38,800,000. Thus t h i s proposal must be r e j e c t e d due to i t s low r e t u r n and high r i s k .  However there i s an a l t e r n a t i v e  whereby w i t h a minimum of 25$ l o c a l p r i v a t e c a p i t a l part i c i p a t i o n i t i s b e l i e v e d that the r i s k of government takeover could be g r e a t l y reduced.  - 120 -  Z  Z  $2,000,000 3,000,000 4,000,000 5,000,000 6,000,000 7,000,000 8,000,000 9,000,000  B(E ).(E -E )  E .?,(E )  P-(E )  Z  Z  100 x 10 .05 .10 300 .20 800 .25 1,250 .20 1,200 • .10 700 400 .05 .05 450 E =$5,200,000  Z  Z  0.512 x 10  3  12  0.484  0.288 0.010 0.128 0.324 0.392 0.722 2.860 x 10  Z  S - $1,690,000 z  0.75E" - $3,900,000 Z  0.75S = Z  $1,270,000  Here the a s s o c i a t i o n between the two earnings, streams i s s i m i l a r t o what i t was before excluding the random p o l i t i c a l e f f e c t s , o r about 0 . 4 which r e s u l t s i n a b e l i e v e d c o e f f i c i e n t of c o r r e l a t i o n , R Sf+z -  - +0.80.  10 V5.5 +2(5.5)(1.27)0.8+1.27 6  2  . 10 V43.03 6  s _ . * _ E +0.75E f  f z  2  .. $6,550,000  6,550,000 Z  18,900,000  - 0 . 3 4 7 , q*f+z - 7.92$  Iz " 0-75 x 5 0 , 0 0 0 , 0 0 0 - $37,500,000 ^ 18,900,000 - 15,000,000 .0792 ToHi2  - 238,600,000 - 184,700,000 - $53,900,000 Thus t h i s a l t e r n a t e proposal can be accepted.  This  - 121 p r o j e c t might he f u r t h e r evaluated as an o p t i m i z a t i o n problem i n i n t e r n a t i o n a l f i n a n c i n g . Example and  2:  An o i l r e f i n i n g company w i t h E f  "$10,000,000  Sf=$4,000,000 are e v a l u a t i n g a d d i t i o n a l cracking capa-  c i t y w i t h the a l t e r n a t i v e s of i n s t a l l i n g a new hydrocracker or a normal c a t a l y t i c cracking u n i t . Hydrocracking i s a higher cost but more f l e x i b l e and e f f i c i e n t operation which may b e t t e r s u i t seasonal  fluc-  t u a t i o n s i n gasoline and f u e l o i l demand as w e l l as d i v e r gent market trends and crude o i l sources. (a) C a t a l y t i c Cracker - A  30,000 b a r r e l  per day  capacity u n i t i s required with an investment outlay of  $7,500,000.  The average annual incremental p o s s i b l e earn-  ings a f t e r taxes, but before i n t e r e s t , are as f o l l o w s ; B  $  y  400,000  600,000 800,000 1,000,000 1,200,000 1,400,000  3(Ey) .05  .20 .35 .25 .10 .05  Byg(By)  20,000 120,000 280,000 250,000 120,000 70,000  Ey -$860,000  P(E ).(E -E ) 0.011 x 10 12 y  0.013 0.001 0.005 0.012 0.015  y  12  0.057 x 10  S - $240,000 y  y  2  - 122 -  As these incremental earnings l a r g e l y represent savings over e x i s t i n g thermal c r a c k i n g operations as w e l l as more e f f i c i e n t r e f i n e r y operations the c o e f f i c i e n t of a s s o c i a t i o n i s b e l i e v e d t o be approximately  0.75 which r e -  s u l t s i n Rfy - +0.95.  S  . 103-\/4000 +2(4000)(240)0.95+240 2  f+y  2  - 10 Vl7,883,000 - $4,230,000 3  -  2&a  4  Ef+Ey 5f  ' t ' 2  Q  Q  Q  Q  - - 0.39, q *  10,860,000 4,000,000  f£. = ^ L i i r i i r i r  »  ' ,  0.40,  q*f -  f + T  - 8.37$  1 + y  ^  ,  8.48$  n  E f 10,000,000 Iv - $7,500,000 ^ 10,860,000 _ 10,000,000 y .0848 V  I  9  J  - 129,600,000 - 117,900,000 = $11,700,000  Thus t h i s proposal, w i t h a b e n e f i t / c o s t r a t i o of 11,700,000/7,500,000 - 1.56, i s acceptable. (b) Hydrocracker  - a 25,000 b a r r e l per day c a p a c i t y  u n i t i s r e q u i r e d representing an investment w i t h p o s s i b l e earnings as f o l l o w s ;  outlay o f $9,000,000,  -123 E .p(E ) z  $ 600,000 800,000 1,000,000 1,200,000  z  p(E ).(E -E )  30,000 .05 80,000 .10 200,000 .20 300,000 .25 280,000 .20 160,000 .10 90,000 .05 100,000 .05 E - $1,240,000  1,400,000  1,600,000 1,800,000 2,000,000  z  z  2  2  z  0.021 x 10 0.019 0.012 0.001 0.005 0.013 0.016 0.029 0.116  x  12  10  S - $340,000 z  The c o e f f i c i e n t of a s s o c i a t i o n between the earnings streams i s b e l i e v e d t o be about 0.5 as the increment l a r g e l y represents savings over e x i s t i n g thermal c r a c k i n g operations as w e l l as storage e f f i c i e n c i e s and input-output variation efficiencies. tion, R  f z  S  Thus the c o e f f i c i e n t of c o r r e l a -  - +0.85.  = 10 V4000 +2(4000)(340)0.85+340 3  f + Z  2  2  . 10 VI- ,426,000 . $4,290,000. 8  - 136,000,000 - 117,900,00 - $18,100,000 Thus the hydrocracker proposal, w i t h a b e n e f i t / cost r a t i o of 18,100,000/9,000,000  - 2.01,  i s a l s o acceptable.  - 124 -  I t remains then t o determine which of these two i n d i v i d u a l l y acceptable, but mutually e x c l u s i v e proposals i s the more d e s i r a b l e . C e r t a i n l y the hydrocracker has the higher o v e r a l l b e n e f i t / c o s t r a t i o but an a n a l y s i s of the incremental'investment i s i n d i c a t e d . I -y z  $1,500,000 ^ V *  f + Z  - V*  f + y  - 136,000,000 - 129,600,000 -$6,400,000 Thus the incremental investment f o r the hydrocracker a l t e r n a t i v e , w i t h a b e n e f i t / c o s t r a t i o of = 4.27,  6.4/1.5  i s acceptable, even h i g h l y d e s i r a b l e . D i f f e r e n c e s i n the time p r o f i l e of cash flows f o r  the proposals would d i c t a t e c l o s e r a n a l y s i s using the marg i n a l cost of c a p i t a l derived as f o l l o w s ; q*y  8 6 0  >  000  = 7.35$  - 11,700,000 q » - . 1,240,000 „ 6.85$ 18,100,000 q* -v 380,000 ^ . 6,400,000 zz  J P  5  z  Example and  Sf-$150,000  3  9 ¥  : A P a c i f i c coast sawmill w i t h  Ef=$250,000  i s c o n s i d e r i n g the purchase of f i r e and  d i s a s t e r insurance.  The annual premium of  $50,000 i s be-  l i e v e d t o be c e r t a i n and i s considered as a f i n a n c i a l charge  - 125 r a t h e r than an operating expense. The p o s s i b l e b e n e f i t s from claims are based on coverage, with deductible amounts, f o r damage to the m i l l as w e l l as f o r c e r t a i n f i x e d expenditure to l o s s of production.  l i a b i l i t i e s due  They a l s o r e f l e c t the p r o b a b i l i t i e s  of minor, major and t o t a l losses due to earthquake, f i r e or f l o o d . Ey  $ 0 20,000 40,000 60,000 80,000 100,000 120,000  P(Ey)  Ey.P(Ey)  .10 0 .25 5,000 .30 12,000 .15 9,000 .10 8,000 .05 5,000 .05 6,000 % - $45,000  ?(Ey).(Ey-Ey f  203 x 10 156 8 34 123 154 282 96O x 10 (  6  Sy - 31,000 The income from insurance claims i s , of course, v i r t u a l l y p e r f e c t l y n e g a t i v e l y c o r r e l a t e d with  business  earnings under " d i s a s t e r " states of nature (except f o r deductable amounts and maximum l i m i t s ) and q u i t e uncorrelated under "business c o n d i t i o n s " states of nature.  Thus an appro-  ximate c o e f f i c i e n t of a s s o c i a t i o n b e l i e v e d to be 0.5 would define the c o e f f i c i e n t of c o r r e l a t i o n , Rfy - -O.85  - 126 -  5  f  +  -  y  loVl50 +2(l50(31)(-0.85)+31 2  2  - 103Vl5,550 Sf  -  L  «  + Y  124,500  - $124,500  - 0.422,  q * , - 8.81? f  v  295,000 Sf 150,000 ==- - ^—= 0.60 , q * = 11.79$ Ef i = 250,000 '°°? . $910,000 / ',000 .055^ 295,000 _ 250,000 .0881 .1179 I + y  Ef+Ey  f  5Q  y  = 3,350,000 - 2,120,000 - $1,230,000 Therefore t h i s insurance contract w i l l he of net value to the f i r m and should he accepted. These examples i l l u s t r a t e the i n c l u s i o n of the qual i t a t i v e r i s k aspect of earnings i n the e v a l u a t i o n of cap i t a l investment proposals.  This has added another dimen-  s i o n t o the q u a n t i t a t i v e a n a l y s i s that was p r e v i o u s l y l e s s e x p l i c i t l y a part of the e v a l u a t i o n .  Here the use of a  corporate cost of c a p i t a l f u n c t i o n has v a l i d a t e d the r i s k premium rate of d i s c o u n t / r e t u r n method of a n a l y s i s and decision.  - 127 -  F u r t h e r Considerations Rather than evaluate i n d i v i d u a l investment propos a l s i t would he d e s i r a b l e t o develop a p e r i o d i c c a p i t a l budget o r set of acceptable p r o j e c t s from among a l l the o p p o r t u n i t i e s which are presented.  On the funds supply  side our v a l u a t i o n f u n c t i o n has assumed p e r f e c t e l a s t i c i t y of e x t e r n a l sources. lue  This leaves the net incremental va-  of i n d i v i d u a l p r o j e c t s as the only c o n s t r a i n t to the  f i r m ' s budget of acceptable p r o j e c t s . The problem resolves i n t o one, s i m i l a r to that of p o r t f o l i o c o n s t r u c t i o n , ^ f s e l e c t i n g from a set of i n 0  vestment proposals, each w i t h given expected earnings, r e l a t i v e earnings r i s k and c o r r e l a t i o n , a subset of p r o j e c t s which w i l l maximize the value of the f i r m .  A complete and  rigorous s o l u t i o n to t h i s problem r e q u i r e s extensive computation.  A simpler approach i s t o attempt to rank the pro-  posals i n some order of probable absolute a c c e p t a b i l i t y (e.g. E y / l y and/or Sy/Ey and Rfy)  such that once evaluated  i n d i v i d u a l l y l a t e r p r o j e c t d e c i s i o n s w i l l have l i t t l e e f f e c t on p r i o r d e c i s i o n s . Here i t would be necessary to re-eval u a t e r e j e c t e d proposals where p r o j e c t s accepted I n the I n t e r i m have s i g n i f i c a n t l y changed the character of e n t i t y earnings.  - 128 -  However t h i s may be a rather a b s t r a c t problem i n view of the d i f f i c u l t y of generating  sufficient attractive  investment proposals w i t h i n a business o r g a n i z a t i o n .  In  a d d i t i o n there are o p e r a t i o n a l bottlenecks to p r o j e c t  accept-  ance (e.g. management approval and implementation,  project  6  s t a f f i n g and t e c h n i c a l s k i l l ) as w e l l as investments which 7  are d i f f i c u l t t o q u a n t i f y (e.g. research and a d v e r t i z i n g ) . The period c a p i t a l budget question i s r e l a t e d to 8 Q  the longer run sequential investment strategy problem. Here, current d e c i s i o n s are influenced by b e l i e f s concerning future investment o p p o r t u n i t i e s , t h e i r occurrence, s i z e and q u a l i t y . C e r t a i n l y these c a p i t a l a l l o c a t i o n considerations are areas where the investment d e c i s i o n concepts o u t l i n e d here could be f u r t h e r developed.  C.  Conclusion  I t must be emphasized that the v a l u a t i o n model which u n d e r l i e s t h i s a n a l y s i s i s imperfect, i n s p i t e of the r i g i d r e l a t i o n s h i p s i t d e f i n e s .  A suitable interest  rate f u n c t i o n should form part of the scheme along w i t h  -  cumulative bankruptcy  129  -  r i s k , r i s i n g equity c a p i t a l i z a t i o n  r a t e s and cost of c a p i t a l .  These would provide c o n s t r a i n t s  to c a p i t a l leverage and define areas of optimal c a p i t a l structure. In a d d i t i o n t h i s v a l u a t i o n of the f i r m should be set i n the context of general c a p i t a l market f l o w s .  Then  there i s a need to r e c o n c i l e t h i s w i t h e m p i r i c a l market behaviour i n terms of longer term tendencies.  This would  i n v o l v e the earnings-dividend issue as to the b a s i s of v a l u a t i o n and the f a c t of market i m p e r f e c t i o n s . I t can only be concluded that the M o d i g l i a n i and M i l l e r v a l u a t i o n hypothesis by no means provides a complete or accurate s o l u t i o n t o the determination of the cost of c a p i t a l under r i s k .  I t does however describe a framework  w i t h i n which the problem can be approached more e f f e c t i v e ly.  But considerable understanding and s k i l l e d i n t e r -  p r e t a t i o n i s necessary i n order to apply i t , i n i t s present s t a t e , to p r a c t i c a l problems. Another c o n s i d e r a t i o n , i n making use of t h i s approach, i s the major task i n v o l v e d i n developing appropriate probab i l i t y estimates of the long run earnings of the f i r m and each p r o j e c t as w e l l as t h e i r c o r r e l a t i o n .  I t may a l s o  - 130  -  take a number of years to develop o r g a n i z a t i o n a l acceptance of such an approach. Nevertheless,  the M o d i g l i a n i and M i l l e r  hypothesis  represents a s i g n i f i c a n t development i n the more a r t i c u l a t e d e f i n i t i o n and measurement of the cost of c a p i t a l under risk.  At the same time t h i s has e s t a b l i s h e d at l e a s t a  t e n t a t i v e basis f o r the v a r i a b l e rate of discount/return approach to the a n a l y s i s of c a p i t a l investment o p p o r t u n i t i e s under r i s k . The great value of such a synthesis f o r the eval u a t i o n of a wide v a r i e t y of prospective investment s i t u a t i o n s has been i l l u s t r a t e d .  The v a r i a b l e rate of discount/  r e t u r n method of a n a l y s i s Is more d i r e c t and  intuitively  meaningful than the u t i l i t y approach, and. i n the case of corporations i t may  be conceptually p r e f e r a b l e .  In a d d i t i o n  i t provides a f e a s i b l e a d m i n i s t r a t i v e means of d e c e n t r a l i z i n g corporate c a p i t a l investment d e c i s i o n making. C e r t a i n l y much remains to be done to achieve a val i d a t e d general theory of the cost of c a p i t a l under r i s k and a comprehensive d e c i s i o n process f o r i n c l u s i v e and p o s s i b l e future p r o j e c t s . a beginning  I t i s hoped that t h i s represents  only  to what w i l l e v e n t u a l l y become a f u l l y developed  - 131 and g e n e r a l l y accepted approach to corporate investment d e c i s i o n s under r i s k . •^-Franco M o d i g l i a n i and Merton H. M i l l e r , "The Cost of C a p i t a l , Corporation Finance and the Theory of Investment," American Economic Review, (June, 1958), pp. 261-297. ¥.F. Sharpe, "A S i m p l i f i e d Model f o r P o r t f o l i o A n a l y s i s , " Management Science, (January, 1963), pp.277-293• 2  H a r r y M. Markowitz, P o r t f o l i o S e l e c t i o n (New York: John Wiley & Sons, Inc., 1959~T 3  ^Pearson Hunt, Charles M. W i l l i a m s and Gordon Donaldson, B a s i c Business Finance (Homewood: Richard D. I r w i n , I n c . , 1961), p. 612. -*R. A. Golde and G.E. G r i s a r d , "Some Considerations i n Determining the Required Earnings Rate," Papers on Return on Investment, ed. R. N. Anthony (Boston: Harvard Business School, D i v i s i o n of Research, 1959). 6;Hunt, W i l l i a m s and Donaldson, p.  613.  ^Robert W. Johnson, F i n a n c i a l Management (2nd ed., Boston: A l l y n and Bacon, I n c . , 1964), Ch. 7. ^Gordon M. Kaufman, S t a t i s t i c a l D e c i s i o n and Related Techniques i n O i l and Gas E x p l o r a t i o n (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1963), Ch. 8,9. ^ P i e r r e Masse, Optimal Investment Decisions (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1962), Ch. 6.  BIBLIOGRAPHY  Books  Barges, Alexander. The E f f e c t of C a p i t a l Structure on the Cost of C a p i t a l . Englewood C l i f f s : P r e n t i c e - H a l l , Inc., 1962. Bierman, H a r o l d , J r . , Fouraker, Lawrence E., and J a e d i c k e , Robert K. Q u a n t i t a t i v e A n a l y s i s f o r Business D e c i s i o n s . Homewood: Richard D. I r w i n , I n c . , 1961. Bierman, H a r o l d , J r . , and Smidt, Seymour. The C a p i t a l Budgeting D e c i s i o n . New York: Macmillan Company, I960.  Dean, J o e l . C a p i t a l Budgeting. New York: Columbia U n i v e r s i t y P r e s s , 1951. Ekeblad, F r e d e r i c k A. The S t a t i s t i c a l Method i n Business. New York: John Wiley & Sons, I n c . , 1962. Gordon, Myron J . The Investment, F i n a n c i n g and V a l u a t i o n of the C o r p o r a t i o n . Homewood: Richard D. I r w i n , Inc., 1962. Graham, Benjamin, Dodd, David L., C o t t l e , Sidney, and Tatham, C h a r l e s . S e c u r i t y A n a l y s i s . 4th ed. New York: McGraw H i l l Book Company, I n c . , 1962. Grayson, C.J., J r . D e c i s i o n s Under U n c e r t a i n t y . Harvard U n i v e r s i t y P r e s s , i960. Johnson, Robert W. F i n a n c i a l Management. 2nd A l l y n and Bacon, Inc., 1964.  - 132 -  Cambridge:  ed. Boston:  - 133 Kaufman, Gordon M. S t a t i s t i c a l D e c i s i o n and R e l a t e d Techniques i n O i l and Gas E x p l o r a t i o n . Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1963. K n i g h t , P.H. Ri3k, U n c e r t a i n t y and P r o f i t . B o s t o n : H o u g h t o n - M i f f l i n Co., 1921. L i n d s a y , Robert and Sametz, A r n o l d W. F i n a n c i a l Management. Homewood: R i c h a r d D. I r w i n , I n c . ,  1963•  Luce, R. Duncan, and R a i f f a , Howard. Games and D e c i s i o n s . New Y o r k : John W i l e y & Sons, I n c . , 1957. L u t z , F r e d e r i c k , and L u t z , V e r a . The Theory o f Investment of the F i r m . P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s ,  1951 •  Markowitz, H a r r y M. P o r t f o l i o S e l e c t i o n : Efficient D i v e r s i f i c a t i o n o f Investments. New Y o r k : John W i l e y & Sons, I n c . , 1959. Masse, P i e r r e . Optimal investment D e c i s i o n s . C l i f f s : P r e n t i c e - H a l l , I n c . , 1962.  Englewood  Savage, Leonard J . The Foundations o f S t a t i s t i c s . New York: John W i l e y & Sons, I n c . , 1954. Solomon, E z r a . The Theory o f F i n a n c i a l Management. Columbia U n i v e r s i t y P r e s s , 1963.  New Y o r k :  Von Neumann,John, and Morgenstern, Oskar. Theory o f Games and Economic B e h a v i o u r . 2nd ed. P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , 1947. W i l l i a m s , John B u r r . The Theory o f Investment V a l u e . Harvard U n i v e r s i t y P r e s s , 1938.  Cambridge:  Report Anthony, Robert N. (ed.) Papers on Return on Investment. B o s t o n : Harvard B u s i n e s s S c h o o l , D i v i s i o n of Research,  1959.  - 134 Articles Durand, David. "The Cost of C a p i t a l , C o r p o r a t i o n Finance, and the Theory of Investment: Comment," American Economic Review, (September, 1959), PP. 639-44. Durand, David. "The Cost of Debt and E q u i t y Funds f o r Business," Conference on Research i n Business Finance. New York: N a t i o n a l Bureau of Economic Research, 1952. F r i e n d , I . , and P u c k e t t , M. "Dividends and Stock P r i c e s , " American Economic Review, (September, 1964). Gordon, M. J ., and Shapiro, E. " C a p i t a l Equipment A n a l y s i s : The Required Rate of P r o f i t , " Management Science, (October, 1956), pp.102-10. M i l l e r , M.H., and M o d i g l i a n i , F. "Dividend P o l i c y , Growth and the V a l u a t i o n of Shares," J o u r n a l of Business, (October, 196l), pp. 411-33. M o d i g l i a n i , Franco, and M i l l e r , Merton H. "The Cost of C a p i t a l , Corporation Finance, and the.Theory of Investment," American Economic Review, (June,  1958), pp.26lS-97.  M o d i g l i a n i , Franco, and M i l l e r , Merton H. "The Cost of C a p i t a l Corporation Finance, and the Theory of Investment: Reply," American Economic Review, (September, 1959), pp. 655-69. M o d i g l i a n i , Franco, and M i l l e r , Merton H. "Taxes and the Cost of C a p i t a l , " American Economic Review, (June,  1963), PP. 433-43.  Molodovsky, N. "Stock Values and Stock P r i c e s , " F i n a n c i a l A n a l y s i s J o u r n a l , (May-June, i960). Roberts, H..V. "Current Problems i n the Economics of C a p i t a l Budgeting," J o u r n a l of Business, (January, 1957). Sharpe, W.F. "A S i m p l i f i e d Model f o r P o r t f o l i o Analysis," Management Science, (January, 1963), pp. 277-93.  - 135 Solomon, E z r a . "Leverage and the Cost of C a p i t a l , " J o u r n a l of Finance, (May, 1963), pp. 273-79. Solomon E z r a . "Measuring a Company's Cost of C a p i t a l , " J o u r n a l of Business, (October, 1955). Solomon, E z r a . "The A r i t h m e t i c of C a p i t a l Budgeting D e c i s i o n s , " J o u r n a l of Business, ( A p r i l , 1956). W a l t e r , J.F. "Dividend P o l i c i e s and Common Stock P r i c e s , " J o u r n a l of Finance, (March, 1956), pp.29-41. Weston, J . F r e d . "A Test of Cost of C a p i t a l P r o p o s i t i o n s , " Southern Economics J o u r n a l , (October, 1963),  pp. 105-112.  APPENDIX I SUMMARY OF NOTATION x - a p o s s i b l e earnings of a f i r m i n a f u t u r e year. x = the mean o r expected earnings o f a f i r m i n a future year. X = a p o s s i b l e average annual future earnings of a f i r m i n perpetuity. X • the expected average annual future earnings of a f i r m i n p e r p e t u i t y = expected earnings. S » the standard d e v i a t i o n of a p r o b a b i l i t y d i s t r i b u t i o n . i ™ a firm j = a c l a s s of average annual earnings r i s k , m = a long run state o f nature. n - a point i n time (year-end) o r a time p e r i o d ( y e a r ) . A •» the c o e f f i c i e n t o f a s s o c i a t i o n . R - the c o e f f i c i e n t of c o r r e l a t i o n . V - the t o t a l market value o f a f i r m . C - the t o t a l market value o f the common shares of a f i r m . B «• the t o t a l market value of the c o n t r a c t u a l debt of a f i r m . p = the average c a p i t a l i z a t i o n rate f o r expected earnings in a class. r • the average c a p i t a l i z a t i o n rate f o r c o n t r a c t u a l streams.  136 -  - 137 -  k  the average c a p i t a l i z a t i o n rate f o r expected net earnings .  m  V* - the t o t a l market value of a f i r m under t a x . C*  a  the t o t a l market value of the common shares of a f i r m under t a x .  p* = the c a p i t a l i z a t i o n rate f o r expected earnings of a f i r m before t a x . k* • the c a p i t a l i z a t i o n rate f o r expected net earnings before tax. q - the c a p i t a l i z a t i o n rate f o r expected earnings of a firm after tax. h - the c a p i t a l i z a t i o n rate f o r expected net earnings to owners a f t e r t a x . t - the average/marginal rate of corporate income t a x . t  = the marginal rate of income tax on dividends f o r a l l shareholders.  d  tg °= the marginal rate of tax on c a p i t a l gains f o r a l l shareholders. d » mathematical d i f f e r e n t i a l or marginal q u a n t i t y . U • the u t i l i t y index of a p o s s i b l e earnings f o r i n v e s t o r s . w  m  the expected annual rate of earnings i n p e r p e t u i t y from an investment.  I •* the expected investment of a f i r m i n new assets i n a future year. Y - the expected average annual earnings of a f i r m from future investments. Z • the expected net earnings to a p o r t f o l i o of s e c u r i t i e s .  APPENDIX I I EQUILIBRATING MARKET MECHANISM  Consider two f i r m s i n the same r i s k c l a s k and, f o r s i m p l i c i t y , having the same l e v e l of expected earnings X. F i r m 0 has no debt i n i t s c a p i t a l s t r u c t u r e while f i r m 1 has a leveraged c a p i t a l s t r u c t u r e .  L e t Z be the expected  net earnings to a p o r t f o l i o and Sz be the standard d e v i a t i o n of these net earnings.  The r e l a t i v e r i s k of these  net earnings i s then S /Z. z  Case I : The i n v e s t o r holds a f r a c t i o n "a" of the t o t a l share value of f i r m 0 and thus he holds a.Co In h i s p o r t f o l i o where, Z and  a.Sx Zo~ " a.X S  Z  Q  • a.X  0  S X~ x  The i n v e s t o r could s e l l t h i s h o l d i n g a.Co  and  -  purchase an amount a.Co (Cl/V"i) of shares of f i r m 1 , as w e l l as an amount a.Co (Bl/V"i) of bonds, where, Zi  Cx  X-rB!  • —-.a.Co-—7;— v  l  G  l  v  B  x  + .7—.a.Co.r  l  - 138  -  - 139 -  a(VQ/Vl) (Sx-O) + a.Go(Bi/Vi).0 a . (Vo/Vi) .X If V  1  < V  0  then Z > Z 1  0  =  Sx X~  while S ^ / z ^ - S  z o  /z  0  and i n v e s t o r s would p r e f e r to h o l d the shares of f i r m 1 plus bonds.  The switching from the shares of f i r m 0 t o  the shares of f i r m 1 would tend to depress CQ and thus VQ, and r a i s e C i and thus V i u n t i l i n e q u i l i b r i u m V i - V Q . Here i n v e s t o r s are able t o "undo" the leverage a s s o c i a t e d w i t h the shares of f i r m 1 and create equivalent earnings r i s k .  T h u s unleveraged firms could not command a  premium over leveraged firms i n the same r i s k c l a s s .  Case I I : The i n v e s t o r holds a f r a c t i o n "a", of the t o t a l share value of f i r m 1 and thus holds a.Ci i n h i s p o r t f o l i o where, Z]_ = aCx-rBx) and  S  z l  Z~i  a(Sx-O) a(X-rBi)  S  x  X -rBi  The i n v e s t o r could s e l l t h i s p o r t f o l i o h o l d i n g , borrow an a d d i t i o n a l amount a.B^ and purchase an amount  - 140 -  a(C]_+Bx)  °f f i r m 0 , where,  s n a r e s  Zo -  a  ^  C  l  +  B  l \ x  Co and  S  z o  - r . a . B i - a(Xi.X-rBi) Vo  a(Vi/Vo)Sx - a.B]_.0 v  If V i> V  0  0  Sx v  l  then Z > Z i while S /Zo < S / z " i 0  zo  zl  and i n v e s t o r s would p r e f e r the h i g h e r net earnings of lower r i s k a s s o c i a t e d w i t h the shares of f i r m 0 . The switching from shares of f i r m 1 t o shares of f i r m 0 would tend to depress V]_ and r a i s e V Q u n t i l i n e q u i l i b r i u m Vi  = VQ.  Here i n v e s t o r s are able to "make" leverage equival e n t to that i n the shares of f i r m 1 by buying shares of f i r m 0 on margin.  They could accomplish the same r e s u l t by  reducing, by an amount a.B]_, the h o l d i n g s of bonds from t h e i r wider p r o t f o l i o s .  Thus leveraged firms cannot command a pre-  mium over unleveraged firms i n the same r i s k c l a s s .  

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