UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1967_A4_5 G5.pdf [ 6.01MB ]
Metadata
JSON: 831-1.0102385.json
JSON-LD: 831-1.0102385-ld.json
RDF/XML (Pretty): 831-1.0102385-rdf.xml
RDF/JSON: 831-1.0102385-rdf.json
Turtle: 831-1.0102385-turtle.txt
N-Triples: 831-1.0102385-rdf-ntriples.txt
Original Record: 831-1.0102385-source.json
Full Text
831-1.0102385-fulltext.txt
Citation
831-1.0102385.ris

Full Text

INVESTMENT DECISIONS UNDER RISK AND THE MODIGLIANI AND MILLER HYPOTHESIS by DONALD ROBIN GILLEY B.A.Sc, University of Toronto, 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Faculty of Commerce and Business Administration We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of • B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study* I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that,-copying or publi-cation of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission* Donald R. G i l l e y Department of Commerce and B u s i n eS 3 A d m i n i s t r a t i o n The University of B r i t i s h Columbia, Vancouver 8 ? Canada Date March 31. 1967 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 decisions consider the element of r i s k only q u a l i t a t i v e l y , i f at a l l . The believed r i s k should be an e x p l i c i t and quantitative part of the norm-a l excess present value or excess rate of return method of investment a n a l y s i s . These r i s k s are described by the subjective probabi-l i t y d i s t r i b u t i o n of possible investment outcomes and the co 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 . At 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 well as the project earnings r i s k and the c o e f f i c i e n t of association between these streams. Risk bears on the investment valuation through the investor'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 psychological reaction to the r i s k phenomenon. - i i -- i i i -This f e l t r i s k can be quantified through the invest-or's trade-off between income and r i s k , or h i s u t i l i t y of money function. This i s then used to modify the uncertain 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 investors or small groups of co-investors. Por corporate investment decisions i t i s preferable to relate the r i s k to a variable rate of required return or market discount. This rate then enables the uncertain ex-pected income to be evaluated d i r e c t l y In the usual manner. This method i s applicable on any e n t i t y basis including the i n d i v i d u a l project which i s the unit of investment decision. Here the venture has a unique r i s k with an appropriate c a p i t a l structure and cost of c a p i t a l funds. In f a c t , t h i s method of evaluation depends upon the existence of a valuation function 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 Modigliani and M i l l e r statement of the valuation of earnings under r i s k i s highly s i g n i f i c a n t . Here earnings r i s k i s c l a s s i f i e d on the basis - i v -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 displays cost advantages under tax. However, leverage i s restrained by an interest rate function which i s related to f i n a n c i a l r i s k and the uncertainty of creditor payments. I m p l i c i t i n the formulation of t h i s hypothesis i s an investor loss aversion attitude which might be broadened into a r i s k aversion basis of valuation. The comprehensive hypothesis, with a point of minimum cost of c a p i t a l , pro-vides, 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 v a l i d a t e . The valuation of a f t e r - t a x earnings under variable r i s k can be inferred from the Modigliani and M i l l e r hypo-t h e s i s . 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 special case, usually assumed, where the investment i n -come i s of equivalent r i s k and perfectly correlated to the e x i s t i n g corporate income. The method may be used to eva-luate a l t e r n a t i v e financing arrangements and mutually exclu-sive projects as w e l l as insurance proposals. This variable 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 analysis of investment opportunities. 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 organizational acceptance of t h i s approach, i t i s perhaps a basis f o r improved corporate investment de-cisio n s under r i s k . TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v i i i Chapter I . CAPITAL INVESTMENT DECISIONS UNDER RISK . . . 1 A. Risk i n Ca p i t a l Investment Investor Expectations D e f i n i t i o n of Risk Measurement of Risk B. Risk and Investment Decisions Attitude Towards Risk Certainty Equivalents Variable Rate of Discount or Return Correlation of Outcomes C. Summary I I . THE MODIGLIANI AND MILLER VALUATION HYPOTHESIS 55 A. Valuation Under Risk Introduction Basis of Valuation Risk Equivalency B. Cost of Cap i t a l Hypothesis Leverage and Risk Basic Propositions C. P r i n c i p l e Modifications Income Tax Interest Rates Expected Growth - v i -- v i i -Chapter Page D. Conclusion Investor Attitudes Summary I I I . INVESTMENT VALUATION AND DECISIONS UNDER RISK 100 A. Earnings Interrelationships and Valuation Interclass Relationships Tax and Leverage Effects C a p i t a l Sources Effects Earnings Correlation B. Investment Decisions under Risk The Decision Process Examples Further Considerations C• Conclusion BIBLIOGRAPHY . . . . . 132 APPENDIX I . SUMMARY OF NOTATION 136 APPENDIX I I . EQUILIBRATING MARKET MECHANISM 138 LIST OP FIGURES Figure Page 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 . . 5 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 Central Outcomes 5 „ 3. The Uncertainty - Risk - Certainty Continuum . . 10 4. Different P r o b a b i l i t y D i s t r i b u t i o n s with Same Expected Value 13 5. Different P r o b a b i l i t y D i s t r i b u t i o n s with Same Extreme Values 13 6. Indifference Maps Representing Investor Attitudes Towards Risk 25 7. The U t i l i t y of Money Function of an Investor . . 30 8. Convex Segment i n Investor's U t i l i t y Function . 30 9. Common Addition of Two Income Stream Di s t r i b u t i o n s 41 10. Addition of Two Typical Income Stream Dis t r i b u t i o n s 41 11. Average Annual Earnings from Terminal Expectation 60 12. Average Annual Earnings from Growth Expectation 60 13. 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 Earnings of Equivalent Risk 63 - v i i i -- ix -Figure Page 14. F i n a n c i a l Risk 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. Interest Rate as a Function of Leverage . . . . 83 18. C a p i t a l i z a t i o n Rates under Rising Interest 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 Indifference 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. Interclass Earnings Risk Relationship 102 2k.>• Schedule of Valuation of A f t e r Tax Expected Earnings under Risk . . . I l l 25. Hypothetical General Scheme of Valuation under Risk 115 CHAPTER I CAPITAL INVESTMENT DECISIONS UNDER RISK A. Risk i n Capital Investments Investor Expectations The c a p i t a l investment decision i s t y p i c a l l y based upon a careful estimate of the future operating receipts and payments as w e l l as the c a p i t a l funds required f o r a project. The figures derived f o r net income or net cash flow over the l i f e of the project are i m p l i c i t averages, perhaps the most probable or the median fig u r e s . Although such single valued predictions may represent a consensus of opinion or 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 forecasting tech-niques as recognition of many chance and l i t t l e understood variables which w i l l e f f e c t the investment outcome. Here we must treat these as stochastic variables. The use of a single 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 with the future and make decisions which are based on the future. Investment decisions made i n a state of i m p l i -c i t certainty i n a world of considerable lack of certainty could very w e l l be poor decisions. Of course, investors 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 forecast. Thus t h e i r expectations are not r e a l -l y single valued although they use such figures i n making t h e i r decisions. Perhaps t h i s fact i s somehow refl e c t e d i n the estimated figures or perhaps i t i s allowed f o r l a t e r at the point of decision i n some way. Cap i t a l investment i s an act of f a i t h , but rather than re l y e s s e n t i a l l y upon the human desires f o r action and security we can take the calculated r i s k . 2 The many complex variables and uncertainties i n most investment decisions 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 basis f o r r a t i o n a l and consistent decisions towards our defined goal. A l l the factors which are i m p l i c i t i n most decisions can be e x p l i c i t l y included i n formal, systematic analyses and decision c r i t e r i a . 1 * I f investors* expectations are multi-valued i t would - 3 -be preferable to use multi-valued forecasts i n the apprai-s a l and decision process. The development of a multi-valued estimate of operating cash flows or earnings i s based upon a consideration of a l l possible consequences of the invest-ment. This can be derived from cost and revenue estimates associated with various forecasts of general business con-d i t i o n s and s p e c i f i c market and competitive forces^ ( i . e . various states of "nature"). In addition to each one of these unique or mutually exclusive possible outcomes, a subjective estimate of the p r o b a b i l i t y of occurrence must be assigned to each event, with a l l p r o b a b i l i t i e s summing to unity. However, the development of multi-valued estimates may 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 simulation can be use-f u l i n developing a multi-valued estimate of a l l possible outcomes and t h e i r p r o b a b i l i t i e s . For short term forecast horizons involving more controllable 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 invest-ment required, there i s normally a high degree of c e r t a i n -t y . ThU3 we s h a l l consider the estimates of i n i t i a l c a p i t a l outlay, residual value and replacement cost as c e r t a i n t i e s and thus single-valued. - 4 -However, fo r long term forecasts of less c o n t r o l l a -ble and less predictable v a r i a b l e s , such as net income ( i n -cluding costs, revenue and economic l i f e ) a multi-valued forecast may add a great deal to the description of future p o s s i b i l i t i e s . This added sophistication may be of p a r t i -cular value i n cases of larger investment projects or 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 pr 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 possible earnings. Such a structure i s not so much a prediction as a personal b e l i e f held by the decision maker.7 i n the case of less precise b e l i e f s , at least some v i r t u a l l y worst and best outcomes and some average outcome are usually 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 Figures 1 and 2 indicate graphically 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 -milar to the single-valued estimate which i s t y p i c a l l y used. However the development of a subjective p r o b a b i l i t y d i s t r i -bution does not lend any higher degree of certainty to these - 5 -Possible States of Nature and Outcomes n ro -Q O weak dem. over sup. mod. dem. over sup. weakdem. under sup. strong dm. over sup. mod. dem. under sup. strong dm. under sup. \ \ \ Earnings F i g . 1.--Development of Pro 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 Pr 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 Central Outcomes - 6 -values. Thi3 i s because there i s only one possible outcome from one investment i n time rather than a set of random out-comes from many repetitions of the act. A single 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 possible outcomes des-cribes the gamble which the investor believes he faces i n the project. Of course the f u l l e r and more accurate i s our knowledge, the closer 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 quantitative ex-pression of b e l i e f s that makes the multi-valued estimate so useful f o r r a t i o n a l analysis and consistent behaviour. The use of multi-valued estimates r e f l e c t the unique uncertainties involved i n the performance of an investment and summarize the unique r i s k s associated with the venture. This i s perhaps p a r t i c u l a r l y useful i n t h i s era of business d i v e r s i f i c a t i o n into less related 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 appraisal and decision process. - 7 -D e f i n i t i o n of Risk In the case of believed absolute certainty of the future outcome of a present decision or act, the p r o b a b i l i -ty d i s t r i b u t i o n would collapse into one single outcome with a p r o b a b i l i t y of unity. 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 uncertainty as to what the possible out-comes might be and complete uncertainty as to any d i f f e r e n -t i a l p r o b a b i l i t y of one outcome over another. 1 0 This i a the complete ignorance case where one has no knowledge or basis f o r b e l i e f that any outcome i s any more possible or probable than any o t h e r . 1 1 Here the p r o b a b i l i t y d i s t r i b u -t i o n collapses into 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 lack of knowledge or b e l i e f rarely e x i s t s , even i n the cos-mos. Much attention has been given to the case of com-plete uncertainty but we s h a l l not consider i t here. A l -though decision theory might dictate use of the p r i n c i p l e 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 . In the case of a c o n f l i c t of interest and a w i l f u l opponent (e.g. state of market o l l -golopoly 1^) game theory might indicate the use of the maxi-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 eff e c t s of competitors' possible reaction strategies are believed known and are b u i l t into our mul t i -valued estimates. Between the two extremes of complete certainty and complete ignorance l i e s the area i n which we have some basis f o r b e l i e f i n some f i n i t e range of multiple mutually exclusive possible outcomes with 3 o m e more or less vague subjective 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 . This i s de-l s fined 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 or b e l i e f . Although there may be, i n r e a l i t y , some difference between knowledge and b e l i e f , due to the human psyche, we s h a l l treat 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 certainty 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 igno-rance or uncertainty i s a case of i n f i n i t e r i s k . Although we treat matters of r i s k as i f they were cases of random outcomes, usually we must also treat matters - 9 -of uncertainty as random events due to our lack of know-ledge of the possible deterministic forces at work. 1^ Much discussion has been generated over the concept of certainty as the degree of confidence we hold i n our IT "1 ft b e l i e f s . '* This would seem to confuse the concepts of subjective 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 objective 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 observation. 1^ The range and p r o b a b i l i t i e s of possible outcomes which we define express our b e l i e f s re-garding the certainty or uncertainty of the outcome. The f u l l e r i s our p r i o r knowledge, both information and under-standing, the less 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 uncertainty, through r i s k to complete certainty. Risk i s thus associated with the degree of believed v a r i a -b i l i t y of the possible outcome of the investment. Measurement of Risk Although the subjective 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 future outcomes describes the believed v a r i a -b i l i t y of the outcome of the investment ( i . e . the believed Absolute Certainty F i g . 3.—The U n c e r t a i n t y - 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 description of t h i s quantitative r i s k f o r ease of compre-hension and manipulation. The use of the mathematical expectation ( i . e . the weighted mean, or the sum across the product of each out-come and i t s p r o b a b i l i t y ) i s almost univer s a l l y rejected as i n d i c a t i n g anything of the r i s k of a project. I t has been advanced as an adequate description of a r i s k y project on the basis that i t considers both the magnitude of each and every possible outcome as w e l l as i t s associated pro-b a b i l i t y . But too much i s l o s t i n the aggregation and projects 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 value, i n terms of monetary out-comes ( i . e . expected monetary value) i s not related to the r i s k of a project 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-tains could be the worst extreme and have disastrous finan-c i a l s i g n i f i c a n c e . 2 1 Use of t h i s s t a t i s t i c f o r decision would lead one to chose the highest expected value, re-gardless of the degree of r i s k involved, which i s a reck-less pattern of behaviour. 2 2 Here a highly r i s k y investment - 1 2 -with 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 gain would be preferred to the status quo. A purely mathematical objection 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 occasionally been proposed as a suit a b l e , com-prehensive description of a r i s k y income. 2^ This i s equi-valent to re j e c t i n g the notion 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 defining a tolerable 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 believed associated outcomes. 25 i n ei t h e r case igno-rance i s implied as to the p r o b a b i l i t i e s of intermediate outcomes and the matter of the central or average tendency of r i s k y income i s disregarded as shown i n Figure 5 . . Thus we have no suitable measure which adequately describes both the central tendency and the v a r i a b i l i t y or r i s k of project income and we must treat r i s k independently. This i s a problem i n descriptive 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 selection of a suitable measure of dispersion. Of the various measures of central 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 P r o b a b i l i t y D i s t r i b u t i o n s with Same Extreme Values - 14 -exclusive f i n a n c i a l outcomes, the median i s the most s i -g n i f i c a n t . 2 ^ "Where the closeness of t h i s average to the actual outcome i s important, as i t i s here, the median i s the best estimate. I t i s the least error value and i s the value of the central event (50$ of the events have greater values and 50$ have lesser values.) The mean has no independent sig n i f i c a n c e , here, neither does the sum of a l l possible mutually exclusive outcomes upon which i t i a based. However, with the fact 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 of, the median and i t i s much easier to manipulate i n computations. The almost universal use of the mean, however, i s only s i g n i f i c a n t insofar 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 with a measure of central tendency, the extreme values or the range of possible outcomes could c e r t a i n l y be useful 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 proba-b 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 symme-t r i c a l , and the extreme values, being two fi g u r e s , somewhat complicate manipulation. - 15 -The most widely used measure of r i s k i s the stand-ard„.deviation (or the variance) about the mean of the pro-2Q SO b a b i l i t y d i s t r i b u t i o n . This follows from the almost universal use of the mean as an average, ( i . e . the mean i s the least squared.error value, while the variance i s the mean squared err o r . ) Of course the squared error has no independent significance and the standard deviation i s thus an a r b i t r a r y , r e l a t i v e measure of r i s k . Even under the assumption of symmetry, where the mean becomes s i g n i f i c a n t as an estimate of the median, the standard deviation, based upon the t o t a l squared error, remains e s s e n t i a l l y a r b i t r a r y . However, under the further, and often less d r a s t i c , assumption of approximate normality the standard deviation i s highly 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 r i s k . ) 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 are assumed, the use of the mean and standard deviation define an i m p l i c i t l y equivalent nor-mal d i s t r i b u t i o n . Where a pr 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 prohibit 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 deviation, as percentage skewness i n the d i r e c t i o n indicated by the sign^ 2) which simply adds a t h i r d dimension to our description of the d i s t r i b u t i o n and complicates the analysis, or ( i i ) trans-form the d i s t r i b u t i o n to symmetrical form, using reciprocals and/or powers and/or logarithms, which i s only p r a c t i c a l on an approximate basis, or ( 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 raises an impossible problem of defining the dispersion, as the standard devia-t i o n about the median or the mean absolute deviation about the median are not comparable to the U 3 u a l s t a t i s t i c . Suffice to say that the problems of skewness are of such magnitude that there i s great incentive to assume symmetry, otherwise the best a l t e r n a t i v e may be to 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 risk,33 which reduces the problems of skewness. Here a l l po s i t i v e deviations are assigned a value of zero i n the usual standard devia-t i o n computation. Thus f o r a symmetrical d i s t r i b u t i o n , twice the semi-deviation gives the standard deviation. This s t a t i s t i c summarizes the unfavourable ( i . e . - 1 7 -negative) deviations which are often considered to he the ess e n t i a l q u a l i t y of r i s k . In neglecting the dispersion of the more favourable outcomes, except as they a f f e c t the average value, the p o s s i b i l i t i e s of gain cannot perverse-l 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 deviation. There, extremely large outcomes are considered aa bad as extremely small outcomes. The main disadvantage of the semi-deviation, other than i t s obscurity, i s the added computational work i t involves. The standard deviation about the mean of the pro-b a b i l i t y d i s t r i b u t i o n of a l l possible outcomes, as a mea-sure of 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 believed v a r i a b i l i t y among outcomes i n the same basic units as the outcomes s), a l b e i t an a r b i t r a r y one, but one which i s frequently accepted as adequate. The difference i n r i s k between two projects, i n these absolute quantitative terms, could be compared but unless the reference project was an established c r i t e r i o n , or otherwise q u a l i t a t i v e l y defined, we would not be much enlightened by t h i s comparison, as to the qua l i t y of the project i n question. Although t h i s absolute s t a t i s t i c gives us some information as to the r i s k of a project, i t - 18 -i s usually necessary to relate i t i n 3ome way to the mean i n order to assess i t s significance f o r the investment de-c i s i o n . Risk i s often associated with the prospect of l o s s , although i t i s , i n f a c t , just as related to the prospect of gain. I f we assume normality the two s t a t i s t i c s , mean and standard deviation, define the pr o b a b i l i t y of loss or gain and the mean or expected 1 0 3 s or gain. Then an i s o -expected loss versus i s o - p r o b a b i l i t y of loss 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 or risk.34 This 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 pr o b a b i l i t y of loss or gain, assuming normality, but not the expected loss or gain 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. How-ever, i f we are working with perfectly d i v i s i b l e invest-ments, as may be approached i n the case of common stocks or j o i n t ventures, t h i s deficiency disappears a s the absolute size of the investment may be varied. Thus absolute v a r i a -b i l i t y would not enter into the appraisal of r i s k q u a l i t y , - 19 -although i t may enter into the decision 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 perfectly 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 project. B. Risk and Investment Decisions Attitude Towards Risk The normal c a p i t a l investment analysis e s s e n t i a l l y compares the future net income or cash flow to the present investment outlay. This comparison i s best accomplished by e i t h e r the excess present value method or the excess rate of return 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 benefit-cost) approach, the future income i s d i s -counted to i t s present value at 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 to the investor, and t h i s figure i s then compared to the present outlay required. The excess rate of return ( i . e . i n t e r n a l rate of - 2 0 -return or discounted return on investment) approach i n -volves the derivation of the rate of return on the project which equates the present outlay to the future income and t h i s figure i s then compared to 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 to the investor. These methods are normally used to reduce future c e r t a i n income, where certainty i s implied i n the single values, to present c e r t a i n income which i s related to present ce r t a i n outlay, where t h i s certainty i s assumed. Now we wish to introduce the element of r i s k into the decision pro-cess i n some way, i n order that i t may influence our evalua-t i o n of the proposed investment. Here we w i l l he concerned with reducing future r i s k y income to present c e r t a i n income fo r comparison to present c e r t a i n outlay i n the same manner. This immediately raises the questions of how to i n -corporate the measure of r i s k into the analysis and how i t a f f e c t s the investment decision. 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 quantitative measure of disper-sion or r i s k of a project, the investment decision i s i n f l u -enced through the investor's a t t i t u d e towards t h i s r i s k . - 21 -This attitude 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 future wealth of the investor which could include a l l r e a l i z a b l e benefit producing assets whether f i n a n c i a l , r e a l or 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 pr o b a b i l i t y of bankruptcy or, at the extreme, l i f e l o n g poverty. On the other hand there may be a certain f i n i t e desire f o r greater wealth and beyond, a region of l i t t l e valued gain, assuming some current l e v e l of as 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 large and very r i s k y project, i n r e l a t i o n to the investor's wealth may resolve into a case of s t r i c t loss aversion while a small project of low r i s k may not involve wealth considerations at a l l , beyond a simple pre-ference f o r probable gain. In the second place, the investor's attitude towards r i s k w i l l be psychologically conditioned through the mental stress he fe e l s as a result of the lack of cer t a i n t y . This may be a closely r e l a t e d , but nevertheless an added dimen-sion to the economic considerations above. Some people more - 22 -e a s i l y bear greater r i s k than others, even where the conse-quences may be the same, and some may even derive s a t i s f a c -t i o n from chance unknowns, where others f e e l insecure. Thus the investor w i l l have a very subjective attit 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 deviation a s t r i c t aversion to outcomes which deviate from the mean i s implied, whether these be favourable or unfavourable devia-t i o n s . On the other hand the use of the semi-deviation suggests an aversion to unfavourable deviations i n outcome but a s t r i c t indifference to favourable v a r i a t i o n s . Thus the very selection of the measure of r i s k carries implica-tions of investor attitudes which should bear on the choice. We are then faced with the problem of defining a s p e c i f i c investor's unique at 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 attitude must then be q u a n t i t a t i v e l y related to investment value. Certainty Equivalents Many approaches to the appraisal of r i s k y investments take the form of a d i r e c t modification of the average or - 23 -expected income figure as defined by the Investor's reaction to the risk. 1*' 2 Perhaps the simplest method of accomplishing t h i s i s f o r the investor to subjectively interpret 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 unassailable there are some p r a c t i c a l objections. I t ne-cessitates each i n d i v i d u a l investor having a thorough under-standing and f a m i l i a r i t y with 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 -gnificance of the p a r t i c u l a r degree of r i s k involved. This method also presumes that the investor's psyche i s not sub-ject to temporary deviations from some norm. These are perhaps unreasonably strong assumptions about the p r a c t i c i n g decision maker. To avoid some of these problems, attempts have been made to define, more rigorously and generally, the investor's attitu d e towards r i s k i n order to provide a framework within which any s p e c i f i c r i s k may be judged. The objective i s to provide a r a t i o n a l basis of incorporating r i s k into the i n -vestment analysis f o r consistent investor behavbur towards h i s goals.44,45 Obviously, the problems of attempting to - 24 -quantify and give functional expression to the Investor's attitudes are enormous. One method of accomplishing t h i s i s with r i s k indifference functions or maps generated from the investor's evaluation of many d i s t r i b u t i o n s of possible outcomes.^ Typical r i s k indifference maps are shown i n Pigure 6. The r i s k indifference map or function can be used to resolve any r i s k y set of possible outcomes into an equi-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 unit f o r any i n -vestment decision. 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 value)-probability ( i . e . inverse of dispersion) indifference method. Another method of quantifying the investor's a t t i -tude towards r i s k would be to have him weigh a l l possible outcomes according to h i s aversion to them,^ as shown i n the following example. - 26 -(1) (2) (3) (4)«(l)x(3) Possible P r o b a b i l i t y Aversion Weighted Outcomes of Outcome Schedule Outcomes (5)«(2 $5000 .05 .30 1500 75 4000 .10 • 35 1400 140 3000 .15 .40 1200 180 2000 .20 .55 1100 220 +$1000 .30 • 75 + 750 +225 0 .15 1.00 0 0 -$1000 .05 1.40 -1400 - 70 M i l 1.00 Weighted Monetary Value - 770 Expected Monetary Value - $1750. 0 The weighted monetary value of 770 i s equivalent to a single c e r t a i n monetary value of about $1050 which i s the certainty equivalent of t h i s project, as compared to the expected monetary value of $1750. The rapidly f a l l i n g preference f o r money schedule with increasing outcomes indicates a strong aversion to r i s k . The method of determining the investor's schedule of money preference i s not defined. 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 indifference. Here the expect-ed monetary value would be a s i g n i f i c a n t and adequate des-c r i p t i o n of a r i s k y project as there i s no need to d i f f e r e n -t i a t e r i s k s f o r a r i s k i n d i f f e r e n t 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 function concept. J For any r i s k y set of ex ante possible monetary outcomes, of which only one w i l l obtain 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 subjective monetary preferences. Such a useful cardinal measure of u t i l i t y i s made possible through the method of generating the preferences. The investor i s offered choices among pairs of hypothetical gambles ( i . e . acts with f i n i t e numbers of consequences con-fined to an aggregate p r o b a b i l i t y of unity) which are related 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, preferably representing extremes of s a t i s -f a c t i o n (e.g. success m G) and unpleasantness (e.g. f a i l u r e » F ) . This defines the u t i l i t y datum and scale, although any l i n e a r function of such a u t i l i t y i s also 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 investor 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 . t h ere i s a l i n e a r correspondence between P and U(X) where U(X) - P.U(G) + (l-P).U(F) ). I t i s assumed t h a t p r e f e r e n c e s are 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 do e x i s t . A f u r t h e r assumption i s of the c o n t i n u i t y , o r u n i - d i m e n s i o n a l c h a r a c t e r , of p r e f e r e n c e s . P r e f e r e n c e s are a l s o c o n s i d e r e d t o be i n d e -pendent from the context i n which the p r o b a b i l i t i e s a re o f f e r e d and f o r m u l t i - s t a g e d r i s k s i t i s assumed t h a t com-pound p r o b a b i l i t i e s a p p l y . A b a s i c assumption i s t h a t o f a b s o l u t e p r e f e r e n c e f o r h i g h e r p r o b a b i l i t i e s o f success. 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 over a l t e r n a t e gambles the i n v e s t o r ' s u t i l i t y o f any monetary outcome can be found and h i s u t i l i t y f u n c t i o n defined,5° such as shown i n F i g u r e 7 • Although not necessary, the u s u a l i n v e s t o r u t i l i t y f u n c t i o n i s of the f o l l o w i n g g e n e r a l c h a r a c t e r i s t i c s ; ( i ) p o s i t i v e s l o p e , i m p l y i n g a p o s i t i v e marginal u t i l i t y o f money o r a p r e f e r e n c e 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 u t i l i t y o f money or a p r e f e r e n c e f o r c e r t a i n t y ( i . e . an a v e r s i o n to r i s k ) . - 2.9 • -A generally observed phenomenon of investor beha-viour suggests a convex i n f l e c t i o n about the status quo i n the otherwise generally concave u t i l i t y curve,51*52,53 as shown i n Pigure 8. This describes a strong preference fo r r i s k where the stakes are small. Thus where the f i n a n -c i a l consequences become less s i g n i f i c a n t the psychological t h r i l l and s o c i a l excitement of gambling may become pro-nounced. To f a c i l i t a t e computation attempts have been made to f i t mathematical functions to the observed data, with the logarithmic form providing a very good f i t i n some cases (e.g. U = a + b.log(X+c) ) while 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 possible monetary outcomes can be transformed into a set of possible u t i l i t y outcomes having the same subjective 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 believed 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 significance ^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 s-F i g . 7.—The U t i l i t y of Money Function of an Investor Increment of Wealth >-F i g . 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 function, the expected u t i l i t y value w i l l he less than the u t i l i t y of the expected monetary value to the extent that the project i s r i s k y . The expected u t i l i t y value could he equated to some cer t a i n monetary value and thus we have, i n f a c t , de-termined an equivalent certai n monetary value to the r i s k y set of monetary outcomes. In other words, we have simply modified the monetary mean to a certainty equivalent value. The difference between the expected monetary value and i t s certainty equivalent monetary value represents a margin of security against error, both i n the forecast of pro b a b i l i t y outcomes and i n any single outcome i n a random process.57 The maximization of expected u t i l i t y i s consi-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 de-c i s i o n s . Although i t i s d i f f i c u l t to generate successfully a p r a c t i c a l u t i l i t y function, t h i s approach i s generally accepted as being fundamentally useful. 5 8 * 5 9 In these approaches the Introduction of r i s k involves the conversion of r i s k y income to equivalent certain income through the investor's trade-off between income and r i s k or hi s u t i l i t y of money function. In other words r i s k i s applied to modify the mean ris k y income value to some equivalent - 32 -r i s k l e s s mean income. As t h i s trade-off or u t i l i t y function represents the investor's current attitude towards current income, i t should be used to convert present risky income to equivalent present certain income. Thus future risky income should f i r s t be discounted to present r i s k y income before i t i s converted to equivalent present certain income The following example i l l u s t r a t e s t h i s procedure: (1) (2) (3) Possible P r o b a b i l i t y from (1) U t i l i t y (5 Future of Present Index f o r Annual Future Value at 8$ Investor Income Annual of Future of Present -(2)xi f o r 5 y r s . Income Income Value $5000 .05 $20,000 87 4.3 4000 .10 16,000 85 8.5 3000 .15 12,000 82 12.3 2000 .20 8,000 78 15.6 +$1000 .30 +$ 4,000 72 21.6 0 .15 0 64 9.6 -$1000 .05 -$ 4,000 53 2.7 1.00 Expected Present U t i l i t y Value - 74 .6 Equivalent Present Certain Monetary Value - $5700 Expected Present Monetary Value - $7000 Here the expected monetary value ( i . e . the mean of present risky income) of $7000 has been modified by the r i s k , and the investor's attitu 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 $ 5 7 0 0 value would then be compared to the required outlay f o r a decision, assuming that 8 $ i s the cost of c a p i t a l or re-quired rate of return. In the excess rate of return method, assuming that the required present, c e r t a i n investment out-lay i s $ 5 7 0 0 , the 8$ i n t e r n a l rate of return f o r the project i s then compared to the cost of c a p i t a l or required rate of return f o r a decision. However there i s a further problem. Most r e a l ca-p i t a l investment decisions, not to mention a great many f i -nancial investment decisions, are made by groups of people i n the form of partnerships or corporations rather than by sole proprietors. Where we have developed the concept and technique of a u t i l i t y function f o r the i n d i v i d u a l investor we now require the u t i l i t y function f o r a group of co-inv e s t o r s . ^ 1 S t r i c t l y speaking we cannot make interperson-a l comparisons or additions 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 resolu-t i o n i n the case of partnerships by developing a u t i l i t y function f o r the partners as a group based upon t h e i r j o i n t preferences and decisions. However, i n the process i n d i v i d u a l - 34 -attitudes would be more or less submerged depending upon the personality interactions among the partners. The re-sul t may be d e s c r i p t i v e l y accurate but normatively erroneous. Although f o r a widely held corporation i t would be f u t i l e to develop a u t i l i t y function f o r each current share-holder, we might assume that the u t i l i t y function f o r the fi r m , as defined by the preferences of the chief 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, basis of decision. Such a u t i l i t y function could provide an e f f e c t i v e means of delegating decision making i n the large f i r m . 6 4 "Variable Rates of Discount or Return We may also introduce r i s k into the investment ana-l y s i s i n the form of a modification to the rate of discount, f o r the excess present value method, or the required rate of return, f o r the excess rate of return method, rather than a modification to the mean value. 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 , or the opportunity cost of funds f o r invest-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 outlay. In the excess rate of return method the i n t e r n a l rate of ri s k y return f o r the project 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 r i s k . This method, of course, suffers 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 project are reinvested i n projects of l i k e r i s k and thus, l i k e return. A number of approaches, of varying so p h i s t i c a t i o n , have been suggested whereby r i s k may be introduced into the investment analysis i n t h i s fashion. Perhaps the most e l e -mentary approach i s to c l a s s i f y projects according to some predetermined r i s k groupings.^6 This might be on the basis of the usual degree of certainty 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 return, f or projects 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 projects i n less 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, i n d u s t r i e s , e t c . ) . - 36 -One could make a further d i f f e r e n t i a t i o n , i n the former case, between projects involving the perhaps more ce r t a i n savings i n replacement projects and the possibly more ri3ky increased earnings of expansion projects. The grouping of r i s k s could also be made on the b a s i 3 of the asset f u n c t i o n s H e r e the fi r m i s conceived as being a consolidated or integrated group of operating investments, each with i t s own appropriate c a p i t a l structure and cost of c a p i t a l . This grouping could be simply on the basis of operating function, say production, marketing and research, as i n the following example. (1) Operating Function Production Marketing Research (2) Appropriate Capital Structure of Function 50$ debt 50$ equity 30$ debt 70$ equity 10$ debt 90$ equity (3) from (2) Functional Cost of Ca p i t a l * .5x3$-1.5$ .5x9$-4.5 6.0$ .3x3$-0.9$ .7x9$-6 .3$ 7.2$ .1x3$-0.3$ .9x9$ »8.1 0 $ Proportion of Investment 50$ 30$ 20$ Consolidated Corporate Cost of Cap i t a l * assuming corporate debt at 3$, and equity at 9$. (5) •(3)x(4) 3.0$ 2.1$ 1.7$ 6.8$ - 37 -Of course the r i s k s could be further 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 within any function, such as between funda-mental and developmental projects within the research func-t i o n or between product or process groups within the market-ing or manufacturing departments. In addition the r i s k s could be c l a s s i f i e d on a geographic basis (e.g. by regions or nations). In the f i n a l a n a l y s i s , each project has i t s own unique r i s k s and associated cost of funds which i s deter-mined, not only by the earnings r i s k as we have described, but also by the c o l l a t e r a l or asset r i s k which i s related to the proportion of assured or re 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 project can be considered to have some uniquely appropriate c a p i t a l structure of i t s own. The project i s then the basis of new borrowing power, as opposed to latent general borrow-ing power which derives from the current equity i n e x i s t i n g assets and t h e i r earning power. The project can be assign-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 pr o b a b i l i t y of default and cre d i t o r c a p i t a l loss i n the case of default. The resultant earnings to the equity ( i . e . project earnings less debt charges) can then be compared to - 38 -the required equity investment ( i . e . t o t a l investment less borrowing quota), given the cost of equity funds. This whole approach hinges upon the determination of the o p t i -mum borrowing quota of a project, with a l l the complexities which that involves. Here the approach to r i s k evaluation has been de-veloped beyond the demand side of the analysis into consi-deration of the cost of funds on the supply side of the pro-blem. This raises the question of how one i s to determine the appropriate c a p i t a l structure and/or cost of funds f o r an investment project or class of given r i s k . Although we may approximate a firm's composite cost of c a p i t a l , at which i t acquires funds from various sources f o r a l l investments, t h i s does not indicate the appropriate cost of these funds which are invested i n any p a r t i c u l a r project 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 external investments of l i k e r i s k . ^ 0 The borrow-ing quota, discussed above, i s suggested as being based upon sound lending p r i n c i p l e s , however those may be defined, but we must also define the cost of equity f u n d s . ^ Thus the variable rate of discount or return approach-es to r i s k evaluation require a knowledge of the cost of r i s k c a p i t a l . This knowledge i s often i m p l i c i t l y assumed - 39 -perhaps because of the complexities and tenuous state of understanding which are involved,? 2 Suffice to say that these methods must ultimately 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 attention i n the chapters which follow. Correlation of Outcomes A l l of the foregoing approaches introduce r i s k into the c a p i t a l investment analysis as a measure of dispersion of possible project outcomes which i s applied to modify the expected value of possible outcomes or the rate r e f l e c t i n g the cost of funds to the project. However, there i s another aspect of r i s k which has been ignored thus f a r . This i s the effe c t of c o r r e l a t i o n between future income on presently held assets and that on the proposed investment. New income may be compensatory i n behaviour r e l a t i v e to established income, rather than cumu-l 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 au-thors q u a l i f y t h e i r r i s k analysis concepts due to 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 usually made over various possible states of - 40 -nature (e.g. general business conditions, competitor s t r a -tegies) the association between the l e v e l of income and the states of nature i s dropped i n further analysis. F i -gure 9 indicates the usual addition of two r i s k y income streams where the worst and best possible 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 per-f e c t l y p o s i t i v e l y correlated and thus i d e n t i c a l l y affected 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 highly p o s i t i v e l y correlated, the assumption of perfect such c o r r e l a t i o n would be an inaccurate generalization. I f i n fact the two risks-streams i n Figure 9 were perfectly negatively correlated, being affected i n exactly opposite fashion by d i f f e r e n t sta-tes of nature, t h e i r combination would y i e l d the same ex-pected value as shown but with zero dispersion, and thus zero r i s k or c e r t a i n t y . We might also encounter a s i t u a t i o n which f a l l s between these two extremes, such as perfectly uncorrelated streams as shown i n Figure 10. Thus, on a functional r i s k basis, an integrated company may be less r i s k y than simply - 4 1 -0 Income >-F i g . 9.--Common Addition of Two Income Stream Distri b u t i o n s 0 Income >-F i g . 1 0 . — A d d i t i o n of Two Typical Income Stream Dis t r i b u t i o n s - 42 -the sum of i t s functional parts.7 4 D i v e r s i f i c a t i o n into new or competitive products or industries which behave d i f f e r e n t l y under various states of business conditions may reduce the dispersion 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 state-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 , rather 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 possible states of nature, the standard deviation (S) of the combined streams i s given by, Sx+y " V Sx^ + 2 R S x S y + 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 •» +1.0 which gives, Sx+y " S x + Sy. The c o e f f i c i e n t of c o r r e l a t i o n 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 of association (A), where A m 1 - V 1 - R 2, which i s a measure of the degree of error 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 correspon-dence . - 43 -The following example i l l u s t r a t e s the significance and interpretation of c o r r e l a t i o n between project earnings and the earnings of the e x i s t i n g enterprise. Annual earnings of enterprise : x» $10,000 , S x - $4,000 thus Sx/x - 0.4 Annual earnings of project: y - $1,000 Sy- $1,000 thus Sy/y " 1.0 , which indicates that the project earnings are of much greater r e l a t i v e r i s k than the enter-prise 1s earnings. Combined annual earnings of enterprise and project; 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 project to increase capacity where the add i t i o n a l demand i s not assured, then; S x + y - 10 6V4 2+ 2 ( 4 ) ( l ) ( + 0 . 9 5 ) + l 2 " $4950 which gives Sx+y/x + y m 0 .45 . Thus both the absolute and the r e l a t i v e r i s k of the firm's earnings have increased. In 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 y i - $950, with Sy«/y'« 0 .95 . - 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 project i n a quite unrelated economic or market area which i s strongly subject to d i f f e r e n t influences, then, Sx+y - $4400 and Sx+y/x + y - 0.40. Thus, although the absolute r i s k of the firm's earn-ings would increase t h e i r r e l a t i v e r i s k i s unchanged. An equivalent perfectly p o s i t i v e l y correlated project has y ' « $1,000 and S y t - $400 with S y i / y 1 " 0.4, which suggests that the appropriate cost of funds f o r t h i s project would be i d e n t i c a l to the firm'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 strongly subject to d i f f e r e n t influences but oppositely affected by some of the same influences, then, Sx+y - $4025 and Sx+y/x+y-0.37, Thus, although the r e l a t i v e r i s k of the firm's earn-ings has decreased, the absolute r i s k has remained almost constant which suggest that the project contributes 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 correlated project would have y , s a $1,000 and Syi 0 0 which indicates that these are c e r t a i n earnings, that the project could have a borrowing quota with debt charges equal to the f u l l $1,000, and that the appropriate cost of funds f o r the project 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 project which i s i n an unrelated area and which i s oppositely affected by many of the same influences, then, S x+y D $3820 and S x + y/x + y - 0.35. Thus both the absolute as well as the r e l a t i v e r i s k of the firm's earnings have decreased and t h i s project i s more valuable than $1000 per year with c e r t a i n t y , and would have an appropriate cost of funds less than the debt rate. In other words we could afford to pay, to have t h i s project, 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. In 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 insu-rance) 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 associated with the pro-ject, and the appropriate cost of funds f o r the project. This emphasizes the fact that the project w i l l not be carr 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 investor f i r m . More s p e c i f i c a l l y , we are interested i n the project's incremental r i s k contribution to the t o t a l earnings of the firm . An investment which involves greatly increased r i s k f o r one - 46 -f i r m , may possibly be considered of lesser r i s k by a d i f f e r -ent f i r m . 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. Summary We have seen how r i s k arises from a lack of c e r t a i n -ty as exemplified by a subjective 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 constituted f i r m and the prospective invest-ment project. This r i s k i s defined by the dispersion 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 usually measured by the standard deviation of an assumed or transformed symme-t r i c a l d i s t r i b u t i o n . The j o i n t ri3k i s , i n addition, de-pendent upon the association between these earnings which i s usually 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 . The relevant r i s k of the project i s the marginal or incremental r i s k which i t contributes to the t o t a l future earnings of the f i r m . Given t h i s measure of net project r i s k , which i a baaed upon investor b e l i e f s , i t may be incorporated into the analysis i n one of two ways, both of which are baaed upon the investor's a t t i t u d e towarda ri3k. F i r s t l y , t h i a - 47 -may "be accomplished through a modification of the expected earnings to a certainty equivalent value based upon an es-tablished u t i l i t y of money function f o r the investor. A l -though the theory behind the u t i l i t y function 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 applicable to the i n d i v i d u a l investor with some doubt of i t s s t r i c t v a l i -d i t y even i n the case of small partnerships. Considering the prevalence of the corporate mode of enterprise and the magnitude of corporate investment t h i s l i m i t a t i o n i s highly s i g n i f i c a n t . Fortunately there may be an alt e r n a t i v e approach. The r i s k of a project may be incorporated into the analysis through a modification of the required earnings rate or cost of funds f o r the project based upon an established cost of c a p i t a l under r i s k function. More s p e c i f i c a l l y , an e x p l i c i t function defining investor valuation of corporate shares i s the sine qua non of t h i s approach. The development of a sati s f a c t o r y theory of share valuation involves the problem of hypothesizing a valuation model which i s capable of conclusive empirical v e r i f i c a t i o n . Thus the valuation model must be s u f f i c i e n t l y comprehensive - 48 -to include a l l of the s i g n i f i c a n t variables and suitably structured to 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 significance to the v a l i d a t i o n t e s t s . At the same time the valuation 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 solution and suitably founded on object-i v e l y measurable variables i n order to make s t a t i s t i c a l t e s t i n g f e a s i b l e . Unfortunately these two requirements tend to be contradictory and require a compromise i n valuation models ei t h e r i n favour of the f e a s i b i l i t y of empirical t e s t i n g at a loss i n the s t a t i s t i c a l significance of such t e s t s , or In favour of t h e o r e t i c a l rigour at a loss i n the f e a s i b i l i t y of empirical v a l i d a t i o n . The lack of success i n es t a b l i s h -ing such a validated theory of share valuation has been the prime obstacle to the use of the variable cost of c a p i t a l approach to investment decision making under risk.®0 The Modigliani and M i l l e r hypothesis of valuation under risk,®1 i f not wholly d e f i n i t i v e and conclusively validated, represents an in 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 defining how Investors c a p i t a l i z e risky future earnings. - 49 -We have established the significance of such a theory f o r c a p i t a l investment decisions under r i s k . Thus i t would be useful to examine t h i s hypothesis and i t s possible c o n t r i -bution towards a conceptually sound and u n i f i e d method of analyzing 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 valuation model i s developed i t would v i t a l i z e the r i s k premium rate method of analysis and provide a useful a l t e r n a t i v e to the certainty equiva-lent or u t i l i t y function approach. The value of such a scheme f o r corporate investment decisions would be enormous. ^-Frederick Lutz and Vera Lutz, The Theory of Invest-ment of the Firm (Princeton: Princeton University Press, 1951) , Ch. XV. 2 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 , Inc., 1962) , Foreword, Ch. 5 . 3 C . J . Grayson J r . , Decisions under Uncertainty (Cambridge: Harvard University Press, i 9 6 0 ) , Ch. 1 . 4D.T. Nowalki, " P r o b a b i l i t i e s and Expected Values Applied 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, 1959) . 5 L u t z and Lutz, Ch. XV. ^Robert N. Anthony (ed.),Papers on Return on Invest-ment (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 . Savage, 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 & Sons, I n c . , 1 9 5 4 ) , Ch. 4 . ®Grayson, Ch. 9 . ^ H a r o l d Bierman J r . and Seymour Smidt, 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, I960) , pp. 1 2 0 - 1 3 2 . 1 0 S a v a g e , Ch. 4 . i : L M a s s e , Ch. 5 . 1 2 S a v a g e , Gh. 4 , 9 . !3R. Duncan Luce and Howard R a i f f a , Games and  D e c i s i o n s (New Y o r k : J o h n W i l e y & Sons, I n c . , 1 9 5 7 ) , Ch. 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 and F r o f i t ( B o s t o n : Houghton - M i f f l i n , 1921"]"^ •-, Masse, Gh. 5 . ^ L u t z and L u t z , Ch. XV. •^Anthony, I . •^Savage, Ch. 4 . 2 0 M a s s e , Ch. 5 . 2 1 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 : J o h n W i l e y & Sons, I n c . , 1 9 5 9 ) , Ch. 1 0 . 2 3 L u c e and R a i f f a , Ch. 2 . 2 i*Masse, Ch. 5 . - 51 -^Grayson, 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, Inc., 1962), pp. 186-192. 27Robert W. Johnson, F i n a n c i a l Management (2nd ed., Boston: A l l y n and Bacon, Inc., 1964), Ch. 7. 2^Lutz and Lutz, Ch. XV. 2^Markowitz, Ch. 4. 3°Lutz and Lutz, Ch. XV. 3 1Ekeblad, pp. 202-203. 3 2Ekeblad, p. 251 i 3 3Markowitz, Oh. 9. 3 4Ekeblad, p. 249. 35R 0bert Lindsay and Arnold W. Sametz, Finaneial  Management (Homewood: Richard D. Irwin, Inc., 1963), Ch. 3-3 6johnson, Ch. 7. 37pearson Hunt, Charles M. Williams and Gordon Donaldson, Basic Business Finance (Homewood: Richard D. Irwin, Inc., 1961), pp. 627-629. 3 ^ E z r a Solomon, "The Arithmetic of Capital Budgeting Decisions", Journal of Business, ( A p r i l , 1956). •^Bierman and Smidt, pp. 120-132. 40 Savage, Ch. 5. 41 Grayson, Ch. 6, 9. - 52 -4 2 L u t z and Lutz, Ch. XV. -•Johnson, Ch. 7. 4 4Markowitz, Ch. 10. 45Grayson, Ch. 1. 46 Lutz and Lutz, Ch. XV. ^Bierman and Smidt, pp. 120-132. ^^Llndsay and Sametz, Ch. 3. ^John von Neumann and Oskar Morgenstern, Theory of  Games and Economic Behaviour (2nd ed., Princeton: Princeton University Press, 1947). 5°Grayson, Oh. 10. ^Savage, Gh. 5. ^ 2Markowitz, o i l. 1 0 > -^Harold Bierman J r . , Lawrence E. Fouraker and Robert K. Jaedicke, Quantitative Analysis f o r Business Decisions (Homewood: Richard D. Irwin, Inc., 1961), pp.142-153. ^Gordon M. Kaufman, S t a t i s t i c a l Decision and Related  Techniques i n O i l and Gas Exploration (Englewood C l i f f s : P r e n t i c e - H a l l , Inc., 1963), Ch. 7--^Bierman, Fouraker and Jaedicke, pp. 142-153. 5 6Luce and R a i f f a , Ch. 2. sse, Ch. 5• •^Grayson, Ch. 10. - 53 -5 9 s a v a g e , Ch. 5 6 G L u t z and Lutz, Ch. XV. 6 lBierman, Fouraker and Jaedicke, pp. 142-153. 62 u*Luce and R a i f f a , Ch.. 2. 6 3Grayson, Ch. 1G. 64 Grayson, Ch. 1 0 . ^Myron j . Gordon, The Investment, Financing and  Valuation of the Corporation (Homewood; Richard D. Irwin, Inc., 1962), Ch. 2. 66 J o e l Dean, Cap i t a l Budgeting (New York: Columbia University Press, 1951 )> Ch. V. ^7R. A . Golde and G.E. Grisard, "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 § ) . 6 ^ E z r a Solomon, "Measuring a Company's Cost of C a p i t a l , " Journal of Business, (October, 1 9 5 5 ) . 6Q . ^Lutz and Lutz, Ch. XIV. V. Roberts, "Current Problems i n the Economics of Ca p i t a l Budgeting," Journal of Business, (January, 1 9 5 7 ) . 7 1Ezra Solomon, Journal of Business, (October, 1 9 5 5 ) . 7 2Gordon, Ch. 3 -7 3Ekeblad, p p > 5 0 5 - 5 1 9 . 74 1 Golde and Grisard, Papers on Return on Investment, ed. R. N. Anthony. - 5 4 -7 5 Bierman and Smidt, pp. 1 2 0 - 1 3 2 . 76 1 Lutz and Lutz,. Ch. XVT-. ^ H a r r y M. Markowitz, P o r t f o l i o Selection: 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). 7 8 Ekeblad, pp. 5 1 1 - 5 1 9 . ^Grayson, c h . 1 0 . 8°Gordon, Ch. 3 , 14. . On Franco Modigliani 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 ) , pp. 2 6 1 - 2 9 7 . CHAPTER I I THE M O D I G L I A H I AND MILLER VALUATION HYPOTHESIS A. Valuation under Risk Introduction Although the cost of c a p i t a l funds i s of c r u c i a l significance f o r investment and financing decisions, 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 valuation and the effect of capi-t a l structure was highly descriptive but hardly precise. S c i e n t i f i c approaches to share valuation, however, have been based on the elusive present value of expected 2 1 future dividends or expected future earnings^ as are a 4 5 number of c o r o l l a r y growth models. , J 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 descriptive than d e f i n i t i v e . - 55 -- 56 -None of 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 or uncertainty i n t h e i r formulation although i t s existence i s recognized. However, Modigliani and Miller® have rigorously enunciated the net operating earnings hypothesis of j o i n t c a p i t a l funds valuation w i t h i n an e x p l i c i t scheme of r i s k . The introduction of r i s k adds a q u a l i t a t i v e dimen-sion to the quantitative p r o f i t maximization objective which can be resolved by means of a value maximization c r i t e r i o n such as the market value of the common shares of the fi r m . Here t h i s i s treated as the t o t a l common shares of the fi r m rather than the a l t e r n a t i v e , but equivalent, per share basis. Thus Modigliani and M i l l e r i n essence have formula-ted a hypothesis of the valuation of the fi r m under r i s k . Basis of Valuation The hypothesis i s developed In terms of p a r t i a l equilibrium of the i n d i v i d u a l f i r m or class of firms, by means of s t a t i c analysis which i s time related through h i s -t o r i c a l "givens" and future "expectations". In spite of the c l a s s i c debate over the role of dividends i n share v a l u a t i o n , 9 A 0 and the cleavage between - 57 -empirical 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,14>^-5> 16 expected future earnings are used as the basis of valuation. Suffice to say that Modigliani and M i l l e r are among the leading proponents of the earnings approach. 1 7 i t i s assumed here that Investors are r a t i o n a l ( i . e . the hypothesis i s pres c r i p t i v e rather than descrip-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 retained earnings are equi-valent to a f u l l y subscribed, pre-emptive issue of common stock. I t i s assumed that a l l physical assets are corpora-t e l y owned, thus are associated with shareholder ownership, and that they generate a flow of earnings i n d e f i n i t e l y into the future. 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 re-cognized to maintain the productive c a p i t a l of a l l corpora-tions i n t a c t . Such an earnings stream, which derives d i r e c t l y from a firm's assets, 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 i n t e r e s t . This i s as opposed to the gross or cash flow - 58 -basis of valuation which i s normally used f o r investment and bond valuation, where there i s a f i n i t e terminal point. The net income or earnings basis i s more usual f o r the va-l u a t i o n of firms and common stock. The difference between the two arises from the t r e a t -ment of consumed c a p i t a l assets. The earnings approach assumes reinvestment of such l i q u i d a t e d assets where rea l c a p i t a l and equity c a p i t a l are considered permanent i n natu-re . 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 decisions to reinvest earned depreciation funds. The flow of future earnings i s defined, f o r valua-t i o n purposes, by investor b e l i e f s . This believed future earnings stream can be defined by a subjective 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 variable x(n), possible 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 of 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 . Modigliani and M i l l e r have s i m p l i f i e d t h i svaluation process by postulating 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 - 5 9 -e x p e c t e d v a l u e X, o f i d e n t i c a l d i s c o u n t e d v a l u e such t h a t ; O (1+p) o (1+p) P The p r o h a h i l i t y d i s t r i b u t i o n o f p o s s i b l e a verage a n n u a l e a r n i n g s X 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 assumed t o be unanimous, r e g a r d i n g t h e magnitude and r i s k o f the l o n g r u n e a r n i n g s o f t h e f i r m . The e x p e c t e d v a l u e X i s , i n e f f e c t , t h e v a l u e e q u i v a l e n t o f the assumed i n f i n i t e f l o w o f f u t u r e e a r n i n g s f r o m a f i r m ' s a s s e t s . A l t h o u g h we speak o f i n f i n i t y as the l o n g r u n , i n p r a c t i c e t h i s i s t h e e q u i v a l e n t o f f r om a g e n e r a t i o n t o a c e n t u r y due t o the h o r i z o n e f f e c t o f the d i s c o u n t i n g p r o c e s s . "Where i n v e s t o r s b e l i e v e t h a t x ( n ) has a f i n i t e t e r m i n a l p o i n t , t h i s w i l l s i m p l y be r e f l e c t e d i n a l o w e r e q u i v a l e n t a verage a n n u a l e x p e c t e d e a r n i n g s X, as shown i n F i g u r e 11. Where f u t u r e e a r n i n g s a r e b e l i e v e d s u b j e c t t o some compound r a t e o f g r o w t h , t h i s w i l l s i m p l y r a i s e the v a l u e o f X, as shown i n F i g u r e 12. F o r e x c e s s i v e growth r a t e s i t i s n e c e s s a r y t o assume -i Q a f u t u r e r e d u c t i o n i n r a t e s , o r a l i f e c y c l e w i t h f u t u r e 1Q d e c l i n e . * I n i t i a l l y i t i s assumed t h a t f i r m s o f e q u i v a l e n t r i s k have e q u a l e x p e c t e d r a t e s o f growth o f a s s e t s and e a r n i n g s . F u r t h e r i t i s assumed t h a t new a s s e t s g e n e r a t e - 60 -A x(0) I/) rj) \ ^ \ X X Earn v \ x \ \ Annual \ \ \ \ ^ 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 present-l y held assets. Risk 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 possible values of average annual future operating earnings X has an expected value X = Z?(X).X, which we s h a l l c a l l the ex-pected earnings. The dispersion of possible earnings X about ex-pected earnings X can be described by the standard devia-t i o n of possible earnings Sx« The r e l a t i v e dispersion, or c o e f f i c i e n t of v a r i a t i o n Sx/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. equivalent earnings r i s k classes where a l l firms i n each class 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, which i s the expected earnings r i s k f o r the c l a s s , where S X J_ - standard deviation of earnings of fi 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 t follows that a l l firms can be grouped into s x 0 s x l X 0 X]_ - 62 -i f f o r f i r m f and f i r m g, X g « a.Xf, and they are of the same class of earnings r i s k , then S Xg * a.S xf as shown i n Figure 13. In addition, firms of equivalent earnings r i s k must have pe r f e c t l y correlated future earnings. This i s often misinterpreted as a requirement of t h e i r periodic earnings x ( ( n ) . 2 0 m fact i t i s only necessary, f o r homo-geneity of earnings r i s k , that long run possible earnings X he helieved perfectly correlated ( i . e . proportional to t h e i r expected values) 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 0 X i X i where X* i s the helieved value of possible earnings given the long run state of nature m. This, of course, presumes a f u l l y deterministic b e l i e f , with only the states of nature occurring randomly. The states of nature relate 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 production, market and management variables. Thus as various states of nature obtain over time and as the believed p r o b a b i l i t i e s of future states of nature change over time, firms of equi-valent earnings r i s k remain i n the same valuation c l a s s . A^ j A^ ] A 2 A 2 A 3 A 3 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 o f E a r n i n g s o f E q u i v a l e n t R i s k - 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 possible earnings X may take any shape, the r e l a t i v e dispersion and perfect c o r r e l a t i o n requirements dictate that a l l firms of equivalent earnings r i s k have d i s t r i b u t i o n s of exactly iden-t i c a l r e l a t i v e shape. Given a l l these class membership re-quirements we might suspect that there would be very few firms i n any one r i s k c l a s s , but rather a continuum of r i s k classes. This i s the basis 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 class with the 2 1 firms i n one industry. B. Cost of Capital Hypothesis Leverage and Risk For f i r m 0 which has no debt i n i t s c a p i t a l structure 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, CQ m VQ = XQ where, CQ •> the market value of the common share of f i r m 0 i n class j . VQ m the market value of f i r m 0 i n class j . - 65 -XQ m the expected earnings - the expected net earn-ings of f i r m 0 i n class 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 class j (from S X i / X i and Rj.). I t i s assumed that shares are traded i n perfect markets where there are no issue or trading costs f o r securi-t i e s and where investors have f u l l knowledge 2 2 and are not r e s t r i c t e d i n any way. Here shares which are perfect subs-t i t u t e s must s e l l at the same p r i c e . Thus f o r any other debt free f i r m i n class j , as i t s expected earnings and expected net earnings are of equi-valent r i s k to those of f i r m 0, they w i l l also be c a p i t a l -ized at the same rate P j . In other words, as the two ex-pected earnings flows are perfectly proportional substitutes they must be proportionately valued. With the introduction of debt financing i t i s assumed that a l l debt obligations, whether corporate or personal, produce a constant and ce r t a i n flow of interest payments which i s believed by a l l creditors f o r a l l borrow-ers. Thus a l l debt obligations are perfect substitutes i n q u a l i t y . I t i s further assumed that a l l debt obligations are traded i n perfect markets and thus they y i e l d the same - 6 6 -rate of return which i s the c a p i t a l i z a t i o n rate f o r ce r t a i n streams. For a f i r m i with expected earnings X± which has some debt i n i t s c a p i t a l structure the expected net earn-ings 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 structure of company i . = the constant, universal rate of interest or the c a p i t a l i z a t i o n rate f o r ce r t a i n streams. Because the interest payments rB^ are c e r t a i n , they have no dispersion and thu3 do not a l t e r the absolute d i s -persion of expected net earnings from that of expected earnings, S x i. Thus the r e l a t i v e deviation or r i s k of ex-Sxi pected net earnings = w i l l be greater than that of x i " r B i expected earnings S x i . The introduction of debt into the * i c a p i t a l structure 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 equity 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 14. This concept of f i n a n c i a l r i s k , wholly related 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 consistent 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 x / X-rB i s s u f f i c i e n t to define ?(X-rB < 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 interest payments, so that rB represents the re a l economic burden of debt. We have also 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 obligations 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 firm'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 p r o f i t leverage, dx/dQ, and the output leverage, do/dm). Basic Propositions The basic Modigliani 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 structure and i s given by c a p i t a l i z i n g i t s expected earn-ings at the rate appropriate to i t s r i s k class membership. Thus, - 69 -X i „ 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 class j Pj » the average c a p i t a l i z a t i o n rate f o r expected earnings of firms i n class 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 senior s e c u r i t i e s / obligations of f i r m i . Here f i n a n c i a l r i s k or leverage i s irr e l e v a n t f o r r i s k class 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 structure 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 r i s k c l a s s . This hypothesis follows d i r e c t l y from the expected earnings r i s k class equivalency d e f i n i t i o n and the perfect market assumption. Here, commodities of equivalent qua-l i t y must be proportionally valued. ( i . e . the market value of a l l claims against a firm's assets must equal the market value of those assets based on t h e i r earning, or - 70 -l i q u i d a t i o n value.) The hypothesis re i t e r a t e s the view 25 that there i s a t o t a l i t y of r i s k to the firm and i t s expected earnings which cannot be changed but simply redistributed among classes of se c u r i t i e s or claims. This concept has been 2 6 challenged f o r widely held firms where each Investor i s se 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 valuation anomalies which i n aggregate produce a change i n the market value of the fi r m . This c r i t i c i s m involves the r e a l i t y of the assumptions regarding the nature and valuation 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 equilibrated i s based upon the a b i l i t y of investors to 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 ) to that produced by corporations. This equivalent "homemade" investor leverage may be increased by borrowing funds (e.g. buying shares on margin) or by reducing port-f o l i o holdings of bonds and i t may be decreased by holding a d d i t i o n a l 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 ffects derive from the assump-tions of perfect markets and perfect equivalency of corpo-rate 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 essential 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. There are trading costs and knowledge i s not free. The matter of investors' r i s k preferences 2 7 and t h e i r market distribution2® i s involved. For only i f these exactly 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 perfect. However there seems to be no basis f o r not believing that investor preferences are w e l l d i s t r i b u t e d and issued secu-r i t i e s are w e l l matched to fund flows. Much of the discussion has concerned the equivalen-cy of corporate and personal debt. 2 0 . 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 unlimited l i a b i l i t y . Thus the shares of leveraged firms 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 public r e s t r i c t i o n s to margin buying. We might conclude that there are basic market forces - 72 -operating to achieve equilibrium i n share valuation but that there are i n f a c t , many obstacles to i t s Immediate and complete r e a l i z a t i o n . These however, do not material-l y detract from the rigorous conceptual proposition of Modigliani and M i l l e r . The veLghted average cost of c a p i t a l i s , V i V i where, k i = the expected y i e l d of the common stock of f i r m i i n class j . The significance of t h i s expression now l i e s i n the fact that we have shown pj to 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 to changes i n c a p i t a l structure ( i . e . a s p e c i f i c pattern of investor share valuation). I t follows that the expected y i e l d can be expressed as, k ± . p j + ( p r r ^ i Thus the expected common stock y i e l d derives from the c a p i t a l i z a t i o n rate f o r a pure equity earnings stream of the* same class plus a premium related to 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 variable rate depending upon r i s k , as follows, c i - X i - 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 follows that the value of any f i r m i i n a class j can be stated as, V i = Ci+Bi . X j - r B j + rBj. « * i k i r' pj This share valuation corollary suggests that invest-ors c a p i t a l i z e expected net earnings at a higher rate due to the Increased r i s k induced by leverage and that t h i s discounting exactly offsets the advantages to be gained by the use of apparently "low cost" 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 Modigliani 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 firm) i t follows that the required rate of return on an investment opportu-n 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 , regardless of the method of financing the investment. This derives from the market - 74 -ro or c O ro Pj N "ro CL k B / C </) (U +-> ro cr c O ro N 1 CL ro n Pj / k 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 objective where, ™1 ± 1 d l ± where, 1^ m the amount of c a p i t a l required by an investment, Thus, dVi m d(Xj/pj) ^ d l i d l i and thus, dX-i — . ± -w — r>i d ^ J where, w = the expected rate of return on an investment opportunity. This c r i t e r i o n would apply equally i n the case of financing out of earned depreciation or retained earnings and would prevent decreased returns to shareholders. C. P r i n c i p l e Modifications Income Tax Perhaps the most s i g n i f i c a n t elaboration of the basic hypothesis was i t s restatement i n a world of t a x e s 3 1 where interest payments are allowed as a deductable ex-pense. Here i t i s assumed that there i s a f i x e d average rate of tax on corporate income ( i . e . constant marginal rate) and that t h i s rate w i l l not change over time ( i . e . i t i s c e r t a i n ) . - 76 -The expected earnings ( i . e . before interest ex-pense) a f t e r tax are now, ( X i - r B i ) ( l - t ) + r B i - X i ( l - t ) + r B j t where, t » the average and marginal corporate Income tax rate. Thus the market value of the f i r m a f t e r tax w i l l be, X±{l-t) r B j t X i U - t ) + r B j t V ±* where, V^* „, the market value of the f i r m under tax. 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 tax. Thus i t follows that, * j - P j - ( p j - r ^ | f * 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 as, V j L* » C i * + B i - ( X j - r B j ) ( l - t ) + r B i h i r where, C i * - the market value of the common stock of the f i r m under tax. h i = 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 tax. - 7 7 -I t can be shown that, 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 accept-ance of investment opportunities of l i k e r i s k , are derived as follows; dV j * _ d X j ( l - t ) / P j ^ d r B j t / r ^ ^ d l d l d l Thus, contrary to the proposal of Modigliani and M i l l e r , the minimum acceptable rate of return a f t e r tax i s , d ( X j ( l - t ) + r B j t ) ^ y _ p . dpjBjt + drBjt d l J " d l d l and thus, y - PJ - (pj-r)t__± Por e n t i r e l y debt financing where dBi/dl = 1 , the marginal cost of debt c a p i t a l a f t e r tax i s p j ( l - t ) + r t . For e n t i r e l y equity financing where dBi/dl - 0 the marginal cost of new equity c a p i t a l i s p j . This would include capi-t a l raised through the issue of preferred stock. For financing out of retained 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 dividend 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 various sources, based upon the "target" c a p i t a l structure of the fi r m , i s the relevant marginal cost of c a p i t a l f o r investment decisions. A l t e r n a t i v e l y the relationships may he stated on a before tax basis as follows, -Bi P X i . q J - r tTT* . Pi (1 - t5i ) J - V T * i - t — r=r ' V i 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 tax. Also, k ±* = X j - r B j . h i _ j ? j +(pj,-ri)B_i C i * 1-t 1-t C i * 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 tax. The relationships between the average c a p i t a l i z a t i o n rates under tax are shown i n Figure 16. The significance - 79 -B / V * > Q. r ro B/C * >-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 tax as leverage increases. This effect i s due to the amount and certainty of tax savings a r i s i n g from the treatment of interest charges. However i t i s sol e l y due to t h i s f act that any permanent advantage ac-crues from the use of debt. Future s h i f t s i n the degree of leverage w i l l not, ignoring tax, 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 exactly offs 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 tax, as the interest payment tax shield accrues to shareholders, i t would be s t r i c t l y correct to consider ex-pected future leverage i n share valuation. This concerns the believed unexploited f i n a n c i a l opportunities i n c a p i t a l structure and the p r o b a b i l i t i e s as to i f and when manage-ment might seize these opportunities. In practice however, current leverage should be appropriate to operating earn-ings r i s k and cre d i t o r constraints, the tax shield saving i s r e l a t i v e l y small, and t h i s future value must s t i l l be discounted f o r present valuation. Thus t h i s sophistication i s perhaps a r e a l but complex and t r i v i a l modification to any valuation scheme - 81 -I n t e r e s t R a t e s Perhaps the most p e r s i s t e n t c r i t i c i s m ^ 2 0 f t h e 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 has been t h a t , c o n s i d e r -i n g t a x e s , an o p t i m a l c a p i t a l s t r u c t u r e o f v i r t u a l l y 100$ debt i s i n d i c a t e d . Of c o u r s e t h i s s e n s e l e s s c o n c l u s i o n , 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 o f t h e b a s i c h y p o t h e s i s w h i l e s t i l l assuming a c o n s t a n t I n t e r e s t r a t e , r e g a r d l e s s o f l e v e r a g e ( i . e . i n t e r e s t pay-ments assumed c e r t a i n . ) I t i s a g e n e r a l l y a c c e p t e d f a c t t h a t as l e v e r a g e i n c r e a s e s I n t e r e s t r a t e s may 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 -v e s t o r s .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 a t t e m p t e d t o r e s t a t e t h e i r t h e o r y i n t h e s e terms 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 con-s t r a i n t s upon t h e e x t e n t o f l e v e r a g e . They a l l o w t h a t t h e r e a r e a p l u r a l i t y o f i n t e r e s t r a t e s based upon the p r o v i s i o n s o f t h e l o a n ( e . g . term t o m a t u r i t y , c o l l a t e r a l s e c u r i t y ) 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 borrower.35 The q u e s t i o n t h e n c o n c e r n s the s p e c i f i c n a t u r e o f the f u n c t i o n f o r t h e average c a p i t a l i z a t i o n r a t e f o r c o r p o -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 function of the debt/equity r a t i o ( B / C ) 3 ^ which i s a measure of r i s k to c r e d i t o r s , given X. However a l i n e a r function, as shown i n Figure 17, would be a simpler and, perhaps adequate appro-ximation (e.g. produces a c u r v i l i n e a r function of B/v.) In the case of extreme leverage ( 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 large ( i . e . r — > - o o ) . Of course the interest rate must be some function of the l e v e l of t o t a l contractual payments (rB) i n r e l a t i o n to the quantity (X) and q u a l i t y ( S X ) of operating earnings. This measure of r e l a t i v e commitment (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 closely related to future 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 default ( i . e . P(X<rB)) i s usually i n s i g n i f i c a n t . 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 valuation purposes, i s the expected future interest payments, as per the usual security analysis procedure, rather than the current interest payments.3'3 However we have assumed that the c a p i t a l i z a t i o n rates f o r debt streams r are constant over time ( i . e . norm-a l i z e d rates with no long term secular trend). - 83 -•+-> P Pj 05 N $ Q. r OJ rO CJ B/C B/V > F i g . 17.--Interest Rate as a F u n c t i o n o f Leverage - 84 -Modigliani 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 personal, then investors can equi-v a l e n t l y make or undo le v e r a g e . ^ Thus the basic hypo-thesis w i l l not be affected and the cost of c a p i t a l w i l l not change with leverage, except f o r tax saving e f f e c t s . The necessary implication of t h i s i s that invest-ors must c a p i t a l i z e more highly leveraged net earnings streams at decreasingly greater rates. 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 investors w i l l value more high-40 l y a more r i s k y flow of net earnings. Modigliani 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 explain t h i s as a demand created by r i s k lovers or that in t e r e s t rates r i s e only moderately while non-price controls are the primary constraint on leverage. 1* 1 To the extent that shareholders only consider monetary payments i n assess-ing the cost of creditor c a p i t a l t h i s may be an accurate des c r i p t i o n of the interest rate function. However, the fact of an ultimate non-price contract-ual or i n s t i t u t i o n a l l i m i t to borrowing implies a cost of debt function which r i s e s to i n f i n i t y . The increasing - 85 -constraints upon management freedom, as leverage increases, must "be considered as having an imputed economic cost. Thus a correct valuation function f o r contractually promised streams w i l l probably be c u r v i l i n e a r or u n i - l i n e a r . We thus conclude that the equity c a p i t a l i z a t i o n rate continues to r i s e as the e f f e c t i v e interest rate r i s e s . 4 Modigliani and M i l l e r agree that net earnings w i l l be more heavily discounted as stockholders face a s i g n i f i c a n t l y ho greater r i s k of bankruptcy due to leverage. J The inevitable conclusion 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 valuation hypo-thesis 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 default rather than operating f a i l u r e . ) Expected Growth Consider also the case, which was previously con-strained, of d i f f e r i n g growth rates of assets and future - 86 -cn +-> CO Pj cr H c o +-> I a co U s \ \ \ k v B/C F i g . 1 8 . - - C a p i t a l i z a t i o n Rates under Rising Interest 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 firms whose earnings are of equivalent r i s k . ^ This raises the earnings retention-payout issue once more but we s h a l l simply accept the authors' earn-ings approach. The market value of a uniquely growing f i r m i , whose expected earnings from currently held assets, X i ( 0 ) , are of r i s k class j , i s then, „ , M "^°Xi(0) + Ij.(n) (wj(n) -pj) /pj ( 1 + P j ) n - X l ( Q ) + ^ l i ( n ) (wj(n) - p j ) /pj L /.. *n Pj n=o (1 + Pj ) where, Vj,(n) „ the market value of f i r m i at time n. X^(n) m the expected earnings of firm i from assets held at time n. Ij^ n ) . the expected increase i n assets of f i r m i during the time period from n - 1 to 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 perpetuity from assets 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 non-growing, but perhaps expanding, expected - 88 -earnings stream of r i s k class 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 assets, I^(n) .Wj_(n), are of equivalent r i s k to those from e x i s t i n g assets, X^(0). This expression f o r the market value of the fi r m i s simply the present value of the sum of expected earnings on presently held assets, X^O), and expected excess earnings on future investments, T 1(n).(w i(n) - P j ) . This l a t t e r growth term represents expected investment opportunities to he exploited at a rate of return greater than P j . This approach to valuation under growth i s s i m i l a r 46 46 to the authors 1 f u l l e r treatment and other expositions. As b e l i e f s about future growth may be imprecise, we might simply express the essential magnitude of these expectations This suggests a simpler valuation expression such as, v l ( o ) . ^ ( £ ) ^ ( ? I / P J ) - i i where, Yj_ m 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 modifications 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 future earnings and do not impair the v a l i d i t y of the basic valuation hypothesis. D. Conclusion Investor Attitudes The valuation scheme and basis of r i s k equivalency carry an i m p l i c i t assumption regarding investors' general attitud e towards r i s k . Together they define an earnings equivalency or indifference function as shown i n Figure 20. Although t h i s suggests investor aversion to r i s k as we would expect, there i s no absolute r i s k constraint nor any realm of v i r t u a l r i s k indifference. This I s , of course, due to the presumed d i v i s i b i l i t y of investment where the r e l a t i v e r i s k of earnings i s the ultimate constraint. Thus the s i g n i f i c a n t aspect of t h i s earnings indifference or i s o -value map i s constant, rather than the usual diminishing, marginal r e l a t i v e r i s k ( i . e . constant, not increasing, r i s k aversion.) In terms of the u t i l i t y approach t h i s defines cons-tant r i s k aversion within 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 re s u l t i s a - 90 -f a m i l y of b i - l i n e a r u t i l i t y f u n c t i o n s of the form shown i n F i g u r e 21. T h i s type o f u t i l i t y f u n c t i o n i s s i m i l a r to t h a t i m p l i c i t i n the use of expected l o s s as a measure 47 of r i s k . T h i s suggests t h a t i n v e s t o r s behave on the b a s i s o f l o s s a v e r s i o n r a t h e r than the broader r i s k a v e r -s i o n motive. I f one p o s t u l a t e s t h a t i n v e s t o r s as a whole have an i n c r e a s i n g a v e r s i o n to g r e a t e r a b s o l u t e r i s k r e l a t i v e to t h e i r a b s o l u t e wealth and t h a t they i n d i v i d u a l l y r e f r a i n from h o l d i n g e x c e s s i v e amounts of any one s e c u r i t y Issue, a r e f o r m u l a t i o n of the M o d i g l i a n i and M i l l e r b a s i s o f r i s k e q u i v a l e n c y would be i n d i c a t e d . F o r i n s t a n c e , the 2 ._ r e l a t i v e v a r i a n c e 3 X /X c o u l d be used as the measure of r i s k . T h i s would imply t h a t the v e r y s i z e o f a f i r m ' s a s s e t s 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 d i s a p p e a r when these a s s e t s a r e d i v i d e d among s e v e r a l f i r m s . A f u r t h e r c o r o l l a r y i s t h a t the more wid e l y h e l d i s a f i r m , the h i g h -e r w i l l be i t s market v a l u e . T h i s o b v i o u s l y n e g l e c t s the motive of c o n t r o l and the s o c i a l and i n s t i t u t i o n a l concen-t r a t i o n of the supply of funds. Complex as such a scheme may be, i t would produce a more t r a d i t i o n a l i m p l i c i t e a r n i n g s i n d i f f e r e n c e f u n c t i o n w i t h a d i m i n i s h i n g marginal c o e f f i c i e n t of v a r i a n c e , 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 function with diminishing marginal u t i l i t y of c e r t a i n money. This would suggest that investors act i n terms of r i s k aversion. Summary Although the basic independence hypothesis and the extreme leverage case have proven most newsworthy an i n c l u -sive summation of the hypothesis i s more relevant. Figure 22 portrays the combined r e s u l t of the basic hypothesis with corporate taxes, a r i s i n g interest rate function and bankruptcy r i s k . Interestingly t h i s Inclusive model defines a convex cost of c a p i t a l function which exhibits 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 consideration 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 Modigliani and M i l l e r propositions 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 forces i s a powerful contribution. But there also remains a wide quantitative gap between the two concepts. - 93 -- 94 -Contrary to the t r a d i t i o n a l view, Modigliani 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 rest on fundamentally d i f f e r e n t concepts of investor behaviour and market functioning. Modigliani and M i l l e r define a r i g i d investor-market response to leverage i n the valuation of shares. There remains the problem of attempting to reconcile these t h e o r e t i c a l abstractions with the observed world of 48 4Q SO r e a l i t y . Unfortunately empirical t e s t i n g *^»-/ 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 with a broad scatter of leverage, c u r v i l i n e a r regression 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 begin-ning and should be capable of further development. This may be i n terms of a dynamic form of analysis where a time dimension i s introduced e x p l i c i t l y into the firm's invest-ment and financing. This hypothesis should also be set i n a general equilibrium context, p a r t i c u l a r l y with respect to the c a p i t a l market. This would introduce a r i s i n g supply - 95 -of c a p i t a l funds function. > J O Some modification of the perfect market assumption, towards greater realism, might be possible i n spite of the d i f f i c u l t i e s of defining imperfections. This would concern any systematic deviations (e.g. i n s t i t u t i o n a l oligopoly, corporate access to markets, brokerage fees, personal income tax) which might create lags 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 equilibrium within reasonable periods of time. However, the concepts developed here provide a necessary foundation of a theory of the valuation of firms and shares under r i s k . This i s the necessary element re-quired to apply the variable rate of discount or return me-thod of analysis to c a p i t a l investments under r i s k . •'•Benjamin Graham, David L. Dodd, Sidney Cottle and Charles Tatham, Security Analysis, (4th ed.; New York: McGraw H i l l Book Company, Inc., 1 9 6 2 ) , Part IV. 2John Burr Williams, The Theory of Investment Value, (Cambridge : Harvard University Press, 1 9 3 8 ) , p. 5 5 -^Ezra Solomon, "Measuring a Company's Cost of C a p i t a l , " Journal 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 , " Journal of Finance, (March, 1 9 5 6 ) , pp. 29-41. - 96 -^M.J. Gordon and E. Shapiro, "Capital Equipment An 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 of Debt and Equity Funds f o r Business," Conference on Research i n Business Finance, (New York: National Bureau of Economic Research, 1 9 5 2 ) . ^Solomon, Journal of Business, (October, 1 9 5 5 ) . ^Franco Modigliani 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 Invest-ment," American Economic Review, (June, 1 9 5 8 ) > pp.2 6 1 - 2 9 7 . %)avid Durand, "The Cost of C a p i t a l , Corporation Finan-ce, and the Theory of Investment: Comment," American Economic  Review, (September, 1 9 5 9 ) , pp. 6 3 9 - 6 4 4 . 1 0Franco Modigliani 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 In-vestment: Reply," American Economic Review, (September, 1 9 5 9 ) , PP. 6 5 5 - 6 6 9 . •^Graham, Dodd, Cottle and Tatham, Ch. 3 5 . I . Friend and M. Puckett, "Dividends and Stock Pr i c e s , " American Economic Review, (September, 1 9 6 4 ) . IS ^Myron J . Gordon, The Investment, Financing and  Valuation of the Corporation, (Homewood: Richard D. Irwin, Inc., 1 9 6 2 ) . 14 Solomon, Journal of Business, (October, 1 9 5 5 ) . rdon and Shapiro, Management Science, (October, 1 9 5 6 ) . l6¥alter, Journal of Finance, (March, 1 9 5 6 ) . 1?H. H. M i l l e r and F. Modigliani, "Dividend P o l i c y , Growth and the Valuation of Shares," Journal of Business, (October, 1961), pp. 411-433-- 97 --i o N. Molodovsky, "Stock Values and Stock Pr i c e s , " F i n a n c i a l Analysts Journal, (May-June, i960). •'•^ E.S. Mead and J u l i u s Grodinsky, The Ebb and Flow  of Investment Values, (New York: Appleton-Century-Crofts, Inc., 1939). Durand, American Economic Review, (September, 1959)• 2 1 J . Fred Weston, "A Test of Cost of Cap i t a l Propo-s i t i o n s , " Southern Economics Journal, (October, 1963), PP. 105-112. 22 1 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. ^Alexander Barges, The Effect of Capital Structure  on the CQ3t of C a p i t a l , (Englewood C l i f f s : Prentice-Hall,Inc., 1962) , Ch. 2. 2 i*Ezra Solomon, The Theory of Fi n a n c i a l Management, (New York: Columbia University Press, 1963), Ch. V I I I . 25Durand, Conference on Research i n Business Finance. 26 Barges, Ch. 6. 27 Barges, Ch. 6. pO M i l l e r and Modigliani, Journal 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. J Franco Modigliani and Merton H. M i l l e r , "Taxes and the Cost of C a p i t a l , " American Economic Review, (June, 1963) , PP. 433-443. - 98 -3 2 E z r a Solomon, "Leverage and the Cost of C a p i t a l , " Journal of Finance, (May, 1963), pp. 273-279. 33prederick Lutz and Vera Lutz, The Theory of Invest-ment of the Firm, (Princeton: Princeton University Press, 1951), Ch. XVI. S4 J Solomon, The Theory of Fi n a n c i a l Management, Ch. V I I I . 35]y[odigliani and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 3 6Modigliani and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 3^¥.B. Hickman, Corporate Bond Quality and Investor  Experience, (New York: National Bureau of Economic Research, 1 9 5 B T 38sarges, Ch. 6. •^Modigliani and M i l l e r , American Economic Review, (June, 1958), pp. 261-297. 40 Solomon, The Theory of Fin 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 Modigliani and M i l l e r , American Economic Review, (September, 1959), PP. 655-669. - 99 -45Miller and Modigliani, Journal of Business, (October, 1961), pp. 411-433. Walter, Journal of Finance, (March, 1956), pp. 29-41. ^H a r r y M. Markowitz, P o r t f o l i o Selection, (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 Journal, (October, 1963), pp. 105-112. ^°Barges, The Effect of Capital Structure 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. IX. 52Solomon, Journal of Business, (October, 1955). -^Robert Lindsay and Arnold W. Sametz, Fin a n c i a l  Management, (Homewood: Richard D. Irwin, Inc., 1963), p. 181. CHAPTER I I I INVESTMENT VALUATION AND DECISIONS UNDER RISK A. Earnings Interrelationships and Valuation Interclas3 Relationships I t has been emphasized that the variable rate of discount or return approach to investment decisions c r u c i a l -l y depends upon some scheme of valuation under r i s k . The Modigliani and M i l l e r hypothesis 1 of the valuation of firms and shares under r i s k , which has been examined, may be applied to t h i s purpose. This hypothesis defines the market valuation of a fi r m of given operating r i s k ( i . e . r e l a t i v e deviation of possible earnings) under variable f i n a n c i a l r i s k ( i . e . l e -verage). However, we can also deduce the market valuation of unleveraged firms under variable operating r i s k ( i . e . among d i f f e r e n t r i s k c l asses). Consider a f i r m i n r i s k class j with expected earn-ings Xj of standard deviation Sj and with debt i n i t 3 capi-t a l structure of market value B j . Thus, assuming there are •- :ioo- -- 101 -no taxes, the expected net earnings of t h i s f i r m w i l l he Xj-rBj of r i s k S j / X j - r B j . Consider also another f i r m i n r i s k class f with expected earnings Xf of standard de-v i a t i o n Sf, which i s unleveraged, where, ^ f - _ s j X f Xj - rBj Thu3 the expected earnings of fi r m f and the ex-pected net earnings of f i r m j are of equivalent r i s k and w i l l he valued p e r f e c t l y proportionally, assuming that they are also p e r f e c t l y correlated. These earnings and r i s k re-la t i o n s h i p s are shown i n Figure 2 3 . We can express t h i s relationship as, p f - k f - kj - pj + ( P j - r ) B/C - Pj ( 1 + (1-r/pj) (rB/r) } ( X j / p j ) - ( r B / r ) - Pj ( 1 + (1-ypj) V B / I i } ) J J r/pj - rB/3Cj - p. ( ( f/Pj) X j ) } J ( r / p j ) - (rB/ Xj) - p . ( ( f/Pj) U j - r B ) / X j ) 3 ( r / p j ) - 1 + ( X j - r B ) / Xj - 102 --- 103- -(_) J _ "3 PJ 3j/(Xj-rB) Pj S j / ( X j " r B ) ) Therefore, Pf = Pj ( EL.) a ^ y . Pj 3 f / X f ) 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 class 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 classes and the interest rate, assuming that earnings of a l l r i s k s are per f e c t l y correlated and that there are no taxe s. Tax and Leverage Ef 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 fact that the tax shi e l d on int e r e s t payments, and i t s c e r t a i n t y , enhance the quantity and q u a l i t y of a f t e r tax expected earnings. This results i n a decreasing cost of c a p i t a l q and an increasing market value V* as leverage increases with the result 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 effects 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 class c a p i t a l i z a t i o n rate relationships independent of i n t r a class leverage e f f e c t s , we w i l l assume that interest pay-ments 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 and, 1 • P + (P-r) B/C* where 1 - the c a p i t a l i z a t i o n rate f o r expected net earn-ings a f t e r tax, where interest i s not a tax deductible expense. Thus, further 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 ex-pected earnings stream a f t e r tax, X ( l - t ) . Then i t follows that, ( r _ ) S j / X j ( l - t ) Pf " Pi ( Pj Sf/ Xf(1-t) } •(£-)•-' i ; 3 J / ^ ( 1 - T } Pj S f / : x f ( i - 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 class i s 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 a f t e r tax earnings r i s k "between the two classes and the interest rate, assuming a l l earnings are perfectly correlated. Although empirical 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 valuation expression would he desirable i t w i l l not be attempted here. The use of assumed values w i l l serve to i l l u s t r a t e i t s a p p l i c a t i o n . Let pj - 7 . 5 $ where S j / X W l - t ) - 0 . 3 and r - 5 . 5 $ , thus r/p-j - 0 . 7 3 3 and, S f / X f ( l - t ) S j / X j ( l - t ) s f / x f ( i - t ) Pf/Pj Pf 0 . 0 oo 0 . 7 3 3 5 . 5 0 $ 0 . 1 3 . 0 0 0 . 8 0 4 6.04 0 . 2 1 . 5 0 0 . 8 9 2 6 . 6 9 0 . 3 1 . 0 0 1 . 0 0 0 7 . 5 0 0 . 4 0 . 7 5 1 . 1 3 9 8 . 5 4 0 . 5 0 . 6 0 1 . 3 2 1 9 . 9 1 0 . 6 0 . 5 0 1 . 5 7 3 1 1 . 8 1 0 . 7 0 . 4 3 1 . 9 3 1 14 . 50 0 . 8 0 . 3 8 2 . 5 4 5 1 9 . 1 0 0 . 9 0 . 3 3 3 . 8 8 6 2 9 . 1 6 1 . 0 0 . 3 0 6 . 5 9 7 4 9 . 5 0 - 1"06 -Further to 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 in 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 interest payments, where, <lf = Pf - ( P f - r ) t .Bf/v*f We make the assumption here that the l e v e l of ex-pected net earnings r i s k S f / ( X f - r B f ) ( l - t ) » 0.75 f o r any fi r m i n any r i s k class f, i s an I n s t i t u t i o n a l l i m i t to the a v a i l a b i l i t y of debt c a p i t a l ( i . e . at t h i s point r - oo) . This constraint then places a l i m i t on c a p i t a l leverage which, f o r the Modigliani 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 structure of any fi r m , namely, V*f ( r / p f ) (1-t) B f " x S f / x f ( l - t ) + " S f / ( X f - r B f ) ( l - t ) Here the expected earnings r i s k i s , Sf r Sf 1-t X f ( l - t ) + rBft " X f ( l - t ) C x _ S f / X f ( l - t ) > t ' S f / ( X f - r B f ) ( l - t ) Thus the minimum a f t e r tax, average cost of c a p i t a l f o r firms 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, - 1 0 7 -S f / X f ( l - t ) p f Bf/V* f S f / ( X f ( l - t ) + r B f t ) q f Capital 0 . 0 5 . 5 0 $ 1 . 0 0 0 . 0 0 5 . 5 0 $ 0 . 1 6.04 0 . 9 8 0 . 0 5 5 . 7 8 0 . 2 6 . 6 9 0 . 9 4 0 . 1 2 6.13 0 . 3 7.50 0 . 9 0 0 . 1 9 6 . 6 0 0 . 4 8 . 5 4 0.84 0 . 2 7 7 . 2 6 0 . 5 9.91 0 . 7 5 0 . 3 8 8.25 0 . 6 1 1 . 8 1 0 . 6 0 0 . 5 0 9 . 9 2 0 . 7 14.50 0 . 3 0 0 . 6 6 I3:.i5 0 . 8 1 9 . 1 0 0 0 . 8 0 1 9 . 1 0 0 . 9 2 9 . I 6 0 0 . 9 0 2 9 . 1 6 1 . 0 4 9 . 5 0 0 1 . 0 0 4 9 . 5 0 Sources Ef f e c t s The marginal cost of external equity funds ( i . e . common and preferred stock and income bonds) i s Pf and of cred i t o r funds ( i . e . tax deductible contractual payments) i s P f ( l - t ) + r t . Let the value of preferred stock and i n -come bonds be rF/r - F. Then the proportion of common and preferred stock and income bonds w i l l be (C* + F)/v* and the proportion of payment deductible senior issues w i l l be (B-F )/v*. - 108"-Thua the weighted average coat of external equity and debt funds w i l l be, ( f ^ ) ( P f ( l - t ) + r t ) + (^i£)pf - P f - (Pf-r)t(B_£) The marginal cost of retained earnlng3, which i3 a major aource of funds, i a P f > ( l - t d ) / ( l - t g ) . I f D i s the expected average annual dividend then (X-rB)(l-t)-D i a the expected average annual retained earninga. Then the market value of the common aharea w i l l be, c* - ( X - r B ) ( l - t ) . ( X ; r B ) ( l - t ) - D + D_ h h * ( l - t d ) / ( l - t g ) 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 are equal (I.e. dividend payout ahould be ir r e l e v a n t to inveatora). and, h* - h(i±BL) ( l - / D % ' 1 - t d ( x - r B ) ( l - t ) 1 - t g where D/§C-rB)(l-t) i a the expected dividend payout r a t i o . The proportion of retained 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 external and in t e r n a l equity and debt funda q* w i l l now be, q»- - 0 l ± ( P f ( l - t ) + r t ) + G*-D/h^ ( l - t d } +D/h*+F,pf ) 1 A V* V* 1-tg V* , . B-F . , . . C*-D/h* , t d - t 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) + rt(B-F) - S/E. Assume that there i s no c a p i t a l gains tax ( i . e . tg _ 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 preferred stock issues due to the optimal c a p i t a l structure assumption. Thus the comprehensive cost of c a p i t a l f o r a fi r m of r i s k class f w i l l be, / , / <£> j. ,C*-0.6c*h/h*. n n , q* f . p f ( 1 - ( l - r / p f )^-.t - ( ^ ) 0.2 ) - 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* - i i o - -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 valuation under r i s k f o r optimally financed firms i s shown i n Pigure 24. I t 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 correlated expected earnings streams of any given r i s k . Earnings Correlation We have previously noted the importance of the c o r r e l a t i o n , or 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 valuation J I I I I I 1 I 1 1 0 0.1 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 Valuation of A f t e r Tax Expected Earnings under Risk - 10.2- -In our valuation model i t i 3 the co r r e l a t i o n between possible earnings X which i s relevant. This relates to ave-rage annual earnings In perpetuity over a l l possible states of nature, not annual earnings x ( t ) over time. Thus the focus i s on long run states of nature m, and the believed associated values of X m. Long run states of nature could include the following elements, market demand industry supply company operation 03 o •H -P •H rH O ft O •H a o d o o cd o •H Cl Xi o <D •P ai •H O O M i n t e r n t l . r e l a t i o n s business regulation private vs. public private vs. public international trade in t e r n a t i o n a l trade national income employment related products foreign markets market prices s c i e n t i f i c needs population growth taste changes le i s u r e time resource development resource reserves market structure transportation foreign production factor costs process technology automation product design 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 applied technology research a b i l i t y labour force s k i l l s labour rela t i o n s work attitudes public re l a t i o n s A condition of equivalent earnings r i s k i s perfect m c o r r e l a t i o n over a l l states 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- -perfect substitutes. Thus earnings streams of equivalent r i s k , as defined by S/E, when combined w i l l remain of iden-t i c a l r i s k as they are perfectly correlated, The earnings of most firms w i l l respond s i m i l a r l y to various states of general business conditions although there w i l l be notable exceptions as w e l l as more s p e c i f i c 3tates of nature which w i l l cause d i f f e r e n t e ffects i n d i f f e r e n t industries and firms. However, we might postulate that our schedule of valuation concerns standard or r e f e r -ence r i s k classes which are a l l perfectly p o s i t i v e l y corre-l a t e d among themselves as well as with 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 with equivalent earnings r i s k s/E but whose earnings are le s s p o s i t i v e l y or even negatively correlated with those of the reference class w i l l tend to reduce the r i s k of combined earnings andd thus enhance t h e i r valuation. 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 relationships between them, w i l l be an element i n the valuation of these earnings. We might expect that the greater market demand of - 1 1 4 — investors f o r less highly p o s i t i v e l y correlated 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 valuation would depend upon the r e l a t i v e proportion of corporate earnings of various degrees of c o r r e l a t i o n represented i n the market. This aspect of valuation i s then part of a general e q u i l i -brium valuation theory which we w i l l not attempt to develop here. We may conclude that 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 valuation. A hypothetical scheme i s i l l u s -t rated i n Figure 2 5 . However within the context of p a r t i a l equilibrium 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 with 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 expectations. This i m p l i c i t l y assumes that long run c o r r e l a t i o n with 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 valuation or that a l l earnings streams are very highly p o s i t i v e l y correlated. Risk, S f / E f  F i g . 25.—Hypothetical General Scheme of Valuation under Risk - 1 1 6 -B. Investment Decisions under Risk The Decision 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 , contributes to the market va-lue of the prospective investor f i r m . This can be deter-mined from our valuation schedule. 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 y where q* i s a function of S„ /E^+E and where f+y f+y f y /—2 : \ Sf+y - V S f + 2 R f y s f s y + s y Here the marginal cost of c a p i t a l f o r the project with the f i r m , or the required rate of return f o r t h i s i n -vestment project w i l l be q* ' 4 f+y 1 + (E f/E y) ( l - q * f + y / q * f ) This c r i t e r i o n carries with i t the i m p l i c i t long run objectives 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 - 1 1 7 -market value of the firm . Although t h i s valuation i s stated i n terms of ave-rage annual earnings i n perpetuity, where the assets remain i n t a c t ( i . e . earnings less asset amortization) i t i s usually preferable 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 analysis can be carried 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 project earnings are assumed to be of equivalent r i s k Sf/If, and perfectly correlated R|.y _ +X-0, to the earnings of the f i r m then, Sy/Ey = Sf/Ef s f + y - V s / + 2 S f S y + s y * . S f + S y and, 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 single valued estimate of the earnings of the pro-jec t are discounted at 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 decision. This i s the t y p i c a l simple evaluation case where r i s k i s quantita-- 118 -t i v e l y ignored, carrying with i t the i m p l i c i t assumptions stated above. Examples The process of investment evaluation and decision can be i l l u s t r a t e d by means of a few examples. If . $15,000,000 and S f . $5,500,000 Is considering the esta-blishment of a cement plant i n South America. The proposed plant would require an investment of $50,000,000. future construction outlook i n the South American country, the sole domestic supplier status with t a r i f f protection as we l l as currency exchange problems and possible govern-ment take-over. Example 1: A Canadian cement manufacturer with The possible earnings of the project r e f l e c t the 2 Ey 2 (By) By 3* (Ey) ?.(Ey).(Ey-Ey) $-1,000,000 +1,000,000 3,000,000 5,000,000 7,000,000 9,000,000 .05 .20 .30 .25 .15 .05 - 50,000 +200,000 900,000 1,250,000 1,050,000 450,000 1.152 x 10 12 1.568 0.192 0.360 1.536 1.352 6.160 x 10 $2,465,000 - 119 -I t i s believed that the c o e f f i c i e n t of associa-t i o n between the earnings of the e x i s t i n g f i r m and those of the proposed plant w i l l be approximately 0.3 due to the independence of the Canadian and South American economies, beyond general world conditions (e.g. int e r n a t i o n a l finan-cing of construction, l e v e l of exports), and the random l o c a l p o l i t i c a l p o s s i b i l i t i e s . This dictates that the c o e f f i c i e n t of c o r r e l a t i o n Bfy - +0.70. Sf 5,500,000 Ef 15,000,000 0.367, q*f - 8,12$ 5 f + y . 106V5.52+2(5.5)(2.465)0.7+2.4652 - 10 V55.41 - $7,450,000 S f ^ - 7,450,ooo „ 0 < 3 9 6 * m Q A 1 % f f + E y 18,800,000 ly - $50,000,000 * l 8 ; ^ Q Q 0 . i^ooo^ooo - 223,500,000 - 184,700,000 - $38,800,000. Thus t h i s proposal must be rejected due to i t s low return and high r i s k . However there i s an a l t e r n a t i v e whereby with a minimum of 25$ l o c a l private c a p i t a l par-t i c i p a t i o n i t i s believed that the r i s k of government take-over could be greatly reduced. - 120 -P - ( E Z ) E Z . ? , ( E Z ) B ( E Z ) . ( E Z - E Z ) $2,000,000 .05 100 x 103 0.512 x 1012 3,000,000 .10 300 0 . 4 8 4 4,000,000 .20 800 0.288 5,000,000 .25 1,250 0.010 6,000,000 .20 1,200 • 0.128 7,000,000 .10 700 0.324 8,000,000 .05 4 0 0 0.392 9,000,000 .05 450 0.722 E Z =$5,200,000 2.860 x 10 S z - $1,690,000 0.75E"Z - $3,900,000 0.75SZ = $1,270,000 Here the association between the two earnings, streams i s s i m i l a r to what i t was before excluding the random p o l i t i c a l e f f e c t s , or about 0 .4 which results i n a believed c o e f f i c i e n t of c o r r e l a t i o n , R f z - +0.80. Sf+z - 1 0 6 V 5 . 5 2 + 2 ( 5 . 5 ) ( 1 . 2 7 ) 0 . 8 + 1 . 2 7 2 . 1 0 6 V 4 3 . 0 3 .. $6,550,000 s - 6 ,550,000 _ . * _ - - 0 .347 , q*f+z - 7 .92$ E f +0.75E Z 18 ,900,000 Iz " 0-75 x 5 0 , 0 0 0 , 0 0 0 - $37,500,000 ^ 1 8 , 9 0 0 , 0 0 0 - 1 5 , 0 0 0 , 0 0 0 .0792 ToHi2 - 238,600,000 - 184,700,000 - $53,900,000 Thus t h i s alternate proposal can be accepted. This - 121 -project might he further evaluated as an optimization pro-blem i n international financing. Example 2: An o i l r e f i n i n g company with Ef "$10,000,000 and Sf=$4,000,000 are evaluating a d d i t i o n a l cracking capa-c i t y with the alternatives 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 better suit seasonal f l u c -tuations i n gasoline and f u e l o i l demand as well as diver-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 unit i s required with an investment outlay of $7,500,000. The average annual incremental possible earn-ings a f t e r taxes, but before i n t e r e s t , are as follows; B y 3(Ey) Byg(By) P ( E y ) . ( E y - E y ) 2 $ 400,000 .05 20,000 0.011 x 1012 600,000 .20 120,000 0.013 800,000 .35 280,000 0.001 1,000,000 .25 250,000 0.005 1,200,000 .10 120,000 0.012 1,400,000 .05 70,000 0.015 12 Ey -$860,000 0.057 x 10 S y - $240,000 - 122 -As these incremental earnings l a r g e l y represent savings over e x i s t i n g thermal cracking operations as well as more e f f i c i e n t refinery operations the c o e f f i c i e n t of association i s believed to be approximately 0.75 which re-sul t s i n Rfy - +0.95. S f + y . 103-\/40002+2(4000)(240)0.95+2402 - 103Vl7,883,000 - $4,230,000 2&a - 4 ' 2 t Q ' Q Q Q - - 0.39, q * f + T - 8.37$ Ef+Ey 10,860,000 ' 1 + y ^ 5f 4,000,000 , n , f£. = ^ L i i r i i r i r » 0.40, q*f - 8.48$ Ef 10,000,000 Iv - $7,500,000 ^ 10,860,000 _ 10,000,000 y V I 9 J .0848 - 129,600,000 - 117,900,000 = $11,700,000 Thus t h i s proposal, with a benefit/cost r a t i o of 11,700,000/7,500,000 - 1.56, i s acceptable. (b) Hydrocracker - a 25,000 barrel per day capacity unit i s required representing an investment outlay of $9,000,000, with possible earnings as follows; -123 -E z . p ( E z ) p ( E z ) . ( E 2 - E z ) 2 $ 600,000 800,000 .05 .10 .20 .25 .20 .10 .05 .05 30,000 80,000 200,000 300,000 280,000 160,000 90,000 100,000 0.019 0.012 0.001 0.005 0.013 0.016 0.029 0.021 x 10 1,000,000 1,200,000 1,400,000 1,600,000 1,800,000 2,000,000 E z - $1,240,000 0.116 x 10 12 S z - $340,000 The c o e f f i c i e n t of association between the earn-ings streams i s believed to 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 cracking operations as well as storage e f f i c i e n c i e s and input-output v a r i a t i o n e f f i c i e n c i e s . Thus the c o e f f i c i e n t of co r r e l a -t i o n , R f z - +0.85. S f + Z = 103V40002+2(4000)(340)0.85+3402 . 10 VI-8,426,000 . $4,290,000. - 136,000,000 - 117,900,00 - $18,100,000 Thus the hydrocracker proposal, with a benefit/ cost r a t i o of 18,100,000/9,000,000 - 2.01, i s also acceptable. - 124 -I t remains then to determine which of these two i n d i v i d u a l l y acceptable, but mutually exclusive proposals i s the more desirable. Certainly the hydrocracker has the higher o v e r a l l benefit/cost r a t i o but an analysis of the incremental'investment i s indicated. I z - y - $1,500,000 ^ V * f + Z - V * f + y - 136,000,000 - 129,600,000 -$6,400,000 Thus the incremental investment f o r the hydro-cracker a l t e r n a t i v e , with a benefit/cost r a t i o of 6.4/1.5 = 4.27, i s acceptable, even highly desirable. Differences i n the time p r o f i l e of cash flows f o r the proposals would dictate closer analysis using the mar-g i n a l cost of c a p i t a l derived as follows; q*y 8 6 0 >000 = 7.35$ - 11,700,000 q » z - . 1,240,000 „ 6.85$ z 18,100,000 J P q* z-v - 380,000 ^  5 . 9 ¥ 6,400,000 Example 3 : A P a c i f i c coast sawmill with Ef=$250,000 and Sf-$150,000 i s considering the purchase of f i r e and disaster insurance. The annual premium of $50,000 i s be-l i e v e d to be certain and i s considered as a f i n a n c i a l charge - 125 -rather than an operating expense. The possible benefits 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 certain f i x e d expenditure l i a b i l i t i e s due to loss of production. They also 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 P(Ey) Ey.P(Ey) ?(Ey).(Ey-Eyf $ 0 .10 0 203 x 10( 20,000 .25 5,000 156 40,000 .30 12,000 8 60,000 .15 9,000 34 80,000 .10 8,000 123 100,000 .05 5,000 154 120,000 .05 6,000 282 % - $45,000 96O x 106 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 negatively correlated with business earnings under "disaster" states of nature (except f o r de-ductable amounts and maximum l i m i t s ) and quite uncorrelated under "business conditions" states of nature. Thus an appro-ximate c o e f f i c i e n t of association believed 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 - loVl502+2(l50(31)(-0.85)+312 - 103Vl5,550 - $124,500 S f + Y 124,500 - L « - 0.422, q* f ,v - 8.81? Ef+Ey 295,000 I + y Sf 150,000 ==- - ^—- = 0.60 , q* f = 11.79$ Ef 250,000 / ',000 .055 i y = 5 Q'°°? . $910,000 ^ 295,000 _ 250,000 .0881 .1179 = 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 qua-l i t a t i v e r i s k aspect of earnings i n the evaluation of ca-p i t a l investment proposals. This has added another dimen-sion to the quantitative analysis that was previously less e x p l i c i t l y a part of the evaluation. Here the use of a corporate cost of c a p i t a l function has validated the r i s k premium rate of discount/return method of analysis and de-c i s i o n . - 127 -Further Considerations Rather than evaluate i n d i v i d u a l investment propo-sals i t would he desirable to develop a periodic c a p i t a l budget or set of acceptable projects from among a l l the opportunities which are presented. On the funds supply side our valuation function has assumed perfect e l a s t i c i t y of external sources. This leaves the net incremental va-lue of i n d i v i d u a l projects as the only constraint to the firm's budget of acceptable projects. The problem resolves into one, s i m i l a r to that of p o r t f o l i o construction,^ 0 f selecting from a set of i n -vestment proposals, each with given expected earnings, re-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 projects which w i l l maximize the value of the f i r m . A complete and rigorous solution to t h i s problem requires extensive compu-t a t i o n . A simpler approach i s to 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. Ey/ly 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 project decisions w i l l have l i t t l e e f f e c t on p r i o r decisions. Here i t would be necessary to re-eva-luate rejected proposals where projects accepted In the Interim have s i g n i f i c a n t l y changed the character of en t i t y earnings. - 128 -However t h i s may be a rather abstract problem i n view of the d i f f i c u l t y of generating s u f f i c i e n t a t t r a c t i v e investment proposals w i t h i n a business organization. In addition there are operational bottlenecks to project accept-ance (e.g. management approval and implementation, project 6 s t a f f i n g and technical s k i l l ) as well as investments which 7 are d i f f i c u l t to quantify (e.g. research and ad v e r t i z i n g ) . The period c a p i t a l budget question i s related to 8 Q the longer run sequential investment strategy problem. Here, current decisions are influenced by b e l i e f s concern-ing future investment opportunities, t h e i r occurrence, size and q u a l i t y . Certainly these c a p i t a l a l l o c a t i o n considerations are areas where the investment decision concepts outlined here could be further developed. C. Conclusion I t must be emphasized that the valuation model which underlies t h i s analysis i s imperfect, i n spite of the r i g i d relationships i t defines. A suitable interest rate function should form part of the scheme along with - 129 -cumulative bankruptcy 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 rates and cost of c a p i t a l . These would provide constraints to c a p i t a l leverage and define areas of optimal c a p i t a l structure. In addition t h i s valuation of the firm should be set i n the context of general c a p i t a l market flows. Then there i s a need to reconcile t h i s with empirical market behaviour i n terms of longer term tendencies. This would involve the earnings-dividend issue as to the basis of va-lu a t i o n and the fact of market imperfections. I t can only be concluded that the Modigliani and M i l l e r valuation hypothesis by no means provides a complete or accurate solution to 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 wi t h i n which the problem can be approached more e f f e c t i v e -l y . But considerable understanding and s k i l l e d i n t e r -pretation i s necessary i n order to apply i t , i n i t s present state, to p r a c t i c a l problems. Another consideration, i n making use of t h i s approach, i s the major task involved i n developing appropriate proba-b i l i t y estimates of the long run earnings of the f i r m and each project as well as t h e i r c o r r e l a t i o n . I t may also - 130 -take a number of years to develop organizational accept-ance of such an approach. Nevertheless, the Modigliani 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 r i s k . At the same time t h i s has established at l e a s t a tentative basis f o r the variable rate of discount/return approach to the analysis of c a p i t a l investment opportunities under r i s k . The great value of such a synthesis f o r the eva-l u a t i o n of a wide variety of prospective investment s i t u a -tions has been i l l u s t r a t e d . The variable rate of discount/ return method of analysis Is more di r e c t and i n t u i t i v e l y meaningful than the u t i l i t y approach, and. i n the case of corporations i t may be conceptually preferable. In addition i t provides a feasible administrative means of d e c e n t r a l i -zing corporate c a p i t a l investment decision making. Certainly much remains to be done to achieve a va-l 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 decision process f o r in c l u s i v e and possi-ble future projects. I t i s hoped that t h i s represents only a beginning to what w i l l eventually become a f u l l y developed - 131 -and generally accepted approach to corporate investment decisions under r i s k . •^-Franco Modigliani 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 Invest-ment," American Economic Review, (June, 1958), pp. 261-297. 2¥.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 nalysis," Management Science, (January, 1963), pp.277-293• 3Harry M. Markowitz, P o r t f o l i o Selection (New York: John Wiley & Sons, Inc., 1959~T ^Pearson Hunt, Charles M. Williams and Gordon Donaldson, Basic Business Finance (Homewood: Richard D. Irwin, Inc., 1961), p. 612. -*R. A. Golde and G.E. Grisard, "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, Williams 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, Inc., 1964), Ch. 7. ^Gordon M. Kaufman, S t a t i s t i c a l Decision and Related  Techniques i n O i l and Gas Exploration (Englewood C l i f f s : P r e n t i c e - H a l l , Inc., 1963), Ch. 8,9. ^Pierre Masse, Optimal Investment Decisions (Englewood C l i f f s : P r e n t i c e - H a l l , Inc., 1962), Ch. 6. BIBLIOGRAPHY Books Barges, Alexander. The E f f e c t of Ca 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 tice-Hall, Inc., 1962. Bierman, Harold, J r . , Fouraker, Lawrence E., and Jaedicke, Robert K. Quantitative Analysis f o r Business Decisions. Homewood: Richard D. Irwin, Inc., 1961. Bierman, Harold, J r . , and Smidt, Seymour. The Capital Budgeting Decision. New York: Macmillan Company, I960. Dean, J o e l . Capital Budgeting. New York: Columbia University Press, 1951. Ekeblad, Frederick A. The S t a t i s t i c a l Method i n Business. New York: John Wiley & Sons, Inc., 1962. Gordon, Myron J . The Investment, Financing and Valuation  of the Corporation. Homewood: Richard D. Irwin, Inc., 1962. Graham, Benjamin, Dodd, David L., C o t t l e , Sidney, and Tatham, Charles. Security Analysis. 4th ed. New York: McGraw H i l l Book Company, Inc., 1962. Grayson, C.J., J r . Decisions Under Uncertainty. Cambridge: Harvard University Press, i960. Johnson, Robert W. F i n a n c i a l Management. 2nd ed. Boston: A l l y n and Bacon, Inc., 1964. - 132 -- 133 -Kaufman, Gordon M. S t a t i s t i c a l Decision and Related Tech-niques i n O i l and Gas Exploration. Englewood C l i f f s : Prentice-Hall, Inc., 1963. Knight, P.H. Ri3k, Uncertainty and P r o f i t . Boston: Houghton-Mifflin Co., 1921. Lindsay, Robert and Sametz, Arnold W. Fi n a n c i a l Management. Homewood: Richard D. Irwin, Inc., 1963• Luce, R. Duncan, and R a i f f a , Howard. Games and Decisions. New York: John Wiley & Sons, Inc., 1957. Lutz, Frederick, and Lutz, Vera. The Theory of Investment of the Firm. Princeton: Princeton University Press, 1951 • Markowitz, Harry M. 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. New York: John Wiley & Sons, Inc., 1959. Masse, P i e r r e . Optimal investment Decisions. Englewood C l i f f s : Prentice-Hall, Inc., 1962. Savage, Leonard J . The Foundations of S t a t i s t i c s . New York: John Wiley & Sons, Inc., 1954. Solomon, Ezra. The Theory of Fi n a n c i a l Management. New York: Columbia University Press, 1963. Von Neumann,John, and Morgenstern, Oskar. Theory of Games and Economic Behaviour. 2nd ed. Princeton: Princeton University Press, 1947. Williams, John Burr. The Theory of Investment Value. Cambridge: Harvard University Press, 1938. Report Anthony, Robert N. (ed.) Papers on Return on Investment. Boston: Harvard Business School, D i v i s i o n of Research, 1959. - 134 -A r t i c l e s Durand, David. "The Cost of C a p i t a l , Corporation Finance, and the Theory of Investment: Comment," American  Economic Review, (September, 1959), PP. 639-44. Durand, David. "The Cost of Debt and Equity Funds f o r Business," Conference on Research i n Business  Finance. New York: National Bureau of Economic Research, 1952. Friend, I . , and Puckett, M. "Dividends and Stock P r i c e s , " American Economic Review, (September, 1964). Gordon, M. J ., and Shapiro, E. "Capital 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 Modigliani, F. "Dividend P o l i c y , Growth and the Valuation of Shares," Journal of Business, (October, 196l), pp. 411-33. Modigliani, 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. Modigliani, Franco, and M i l l e r , Merton H. "The Cost of Capi t a l Corporation Finance, and the Theory of Investment: Reply," American Economic Review, (September, 1959), pp. 655-69. Modigliani, 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 Pric e s , " F i n a n c i a l  Analysis Journal, (May-June, i960). Roberts, H..V. "Current Problems i n the Economics of Capital Budgeting," Journal 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, Ezra. "Leverage and the Cost of C a p i t a l , " Journal of Finance, (May, 1963), pp. 273-79. Solomon Ezra. "Measuring a Company's Cost of C a p i t a l , " Journal of Business, (October, 1955). Solomon, Ezra. "The Arithmetic of Ca p i t a l Budgeting Decisions," Journal of Business, ( A p r i l , 1956). Walter, J.F. "Dividend P o l i c i e s and Common Stock P r i c e s , " Journal of Finance, (March, 1956), pp.29-41. Weston, J . Fred. "A Test of Cost of Capital Propositions," Southern Economics Journal, (October, 1963), pp. 105-112. APPENDIX I SUMMARY OF NOTATION x - a possible earnings of a f i r m i n a future year. x = the mean or expected earnings of a f i r m i n a future year. X = a possible average annual future earnings of a f i r m i n perpetuity. X • the expected average annual future earnings of a firm i n perpetuity = expected earnings. S » the standard deviation 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 f i r m j = a class of average annual earnings r i s k , m = a long run state of nature. n - a point i n time (year-end) or a time period (year). A •» the c o e f f i c i e n t of association. 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 of a f i r m . C - the t o t a l market value of the common shares of a firm . B «• the t o t a l market value of the contractual debt of a fi 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 i n a c l a s s . r • the average c a p i t a l i z a t i o n rate f o r contractual streams. 136 -- 137 -k m the average c a p i t a l i z a t i o n rate f o r expected net earn-ings . V* - the t o t a l market value of a f i r m under tax. C* a the t o t a l market value of the common shares of a fir m under tax. 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 tax. k* • the c a p i t a l i z a t i o n rate f o r expected net earnings be-fore 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 f i r m a f t e r 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 tax. t - the average/marginal rate of corporate income tax. t d = the marginal rate of income tax on dividends f o r a l l shareholders. 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 quantity. U • the u t i l i t y index of a possible earnings f o r investors. w m the expected annual rate of earnings i n perpetuity from an investment. I •* the expected investment of a fir 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 firms i n the same r i s k clask 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. Firm 0 has no debt i n i t s c a p i t a l structure while fir m 1 has a leveraged c a p i t a l structure. Let Z be the expected net earnings to a p o r t f o l i o and Sz be the standard devia-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 investor holds a f r a c t i o n "a" of the t o t a l share value of fi r m 0 and thus he holds a.Co In h i s p o r t f o l i o where, Z 0 • a.X and S Z Q a.Sx S x Zo~ " a.X X~ The investor could s e l l t h i s holding a.Co a n d-purchase an amount a.Co (Cl/V"i) of shares of fi r m 1 , as we l l as an amount a.Co (Bl/V"i) of bonds, where, Cx X-rB! B x Z i • — - . a . C o - — 7 ; — + .7—.a.Co.r v l G l v l - 138 -- 139 -a(VQ/Vl) (Sx-O) + a.Go(Bi/Vi).0 = Sx a . (Vo/Vi) .X X~ I f V1 < V 0 then Z1 > Z 0 while S ^ / z ^ - S z o / z 0 and investors would prefer to hold the shares of f i r m 1 plus bonds. The switching from the shares of firm 0 to the shares of firm 1 would tend to depress CQ and thus VQ, and raise C i and thus V i u n t i l i n equilibrium V i - V Q . associated with the shares of firm 1 and create equivalent earnings r i s k . Thus 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 investor holds a f r a c t i o n "a", of the t o t a l share value of fi 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 a(Sx-O) S x Z~i a(X-rBi) X - r B i The investor could s e l l t h i s p o r t f o l i o holding, borrow an addi t i o n a l amount a.B^ and purchase an amount Here investors are able to "undo" the leverage - 140 -a(C]_+Bx) s n a r e s °f fi r m 0, where, Zo - a ^ C l + B l \ x - r.a.Bi - a(Xi.X-rBi) Co Vo and S z o a(Vi /Vo)Sx - a.B]_.0 S x v 0 v l I f V i > V 0 then Z 0 > Zi while S z o/Zo < S z l/z"i and investors would prefer the higher net earnings of lower r i s k associated with the shares of fi r m 0. The switching from shares of fi r m 1 to shares of fi r m 0 would tend to depress V]_ and raise V Q u n t i l i n equilibrium Vi = VQ. Here investors are able to "make" leverage equiva-lent to that i n the shares of firm 1 by buying shares of fir m 0 on margin. They could accomplish the same result by reducing, by an amount a.B]_, the holdings 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 . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0102385/manifest

Comment

Related Items