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An investigation of Gordon's common stock valuation model Smith, Patrick John 1968

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AN INVESTIGATION OF GORDON'S COMMON STOCK VALUATION MODEL  by  PATRICK JOHN SMITH B.A., U n i v e r s i t y of B r i t i s h Columbia, 1958  A t h e s i s submitted i n p a r t i a l f u l f i l l m e n t o the requirements f o r the Degree of Master o f B u s i n e s s A d m i n i s t r a t i o n  i n t h e Department of Commerce and B u s i n e s s A d m i n i s t r a t i o n  We a c c e p t t h i s t h e s i s as conforming required standard  THE UNIVERSITY OF BRITISH September, 1968  to the  COLUMBIA  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l f u l f i l m e n t o f the  requirements  f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. thesis  I f u r t h e r agree t h a t p e r m i s s i o n  f o r e x t e n s i v e copying of  this  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my  Department o r by h i s  representatives.  It  is understood that copying  o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  Department o f  COMMERCE  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada SEPTEMBER 13,  1968  ABSTRACT  The  purpose of t h i s study was t o determine whether a model  based upon the d i v i d e n d  formulation  o f Myron J . Gordon  provides  an adequate e x p l a n a t i o n  o f t h e v a r i a t i o n i n common s t o c k p r i c e s .  In p a r t i c u l a r , the hypothesis tested i s that i n v e s t o r s mainly consider  dividends,  t h e r a t e of growth i n d i v i d e n d s  c h a r a c t e r i s t i c s o f the f i r m s The  and t h e r i s k  i n v a l u i n g shares o f common  model was t e s t e d by u s i n g m u l t i p l e r e g r e s s i o n  on a c r o s s - s e c t i o n o f U.S. companies i n . t h e machinery  stock. analysis  industry  i n each o f the y e a r s 1956 t o 1965. A l t e r n a t i v e measures o r proxies  f o r "normalized" earnings,  growth, b u s i n e s s r i s k and  f i n a n c i a l r i s k were used i n t e s t i n g t h e model. E m p i r i c a l r e s u l t s s u p p o r t e d the p r o p o s i t i o n t h a t Gordon's model p r o v i d e s prices.  an adequate e x p l a n a t i o n  o f the v a r i a t i o n i n s t o c k  On average, between 76% and 81% o f the t o t a l v a r i a t i o n was  explained  by the model over the t e n - y e a r p e r i o d .  each y e a r ranged between .68 and .90. dividend,  The R^ v a l u e s f o r  The c o e f f i c i e n t s o f the  growth and s i z e v a r i a b l e s were s i g n i f i c a n t a t the 5%  l e v e l o r b e t t e r i n almost a l l t h e y e a r s t e s t e d and except f o r growth in  two y e a r s ,  these c o e f f i c i e n t s had the s i g n i n d i c a t e d by the theory.  B u s i n e s s r i s k and f i n a n c i a l r i s k performed p o o r l y  as  explanatory  variables.  The c o e f f i c i e n t s f o r both v a r i a b l e s were n o t s t a t i s t i c a l l y  significant  a t the 5% l e v e l i n most y e a r s and the b u s i n e s s  coefficient  f r e q u e n t l y d i d n o t have t h e expected  ii  sign.  risk  TABLE OF CONTENTS PAGE  ABSTRACT  i i  LIST OF TABLES  v  i  CHAPTER I.  INTRODUCTION TO THEORETICAL COMMON STOCK VALUATION MODELS  1  A.  Purpose o f the Study  1  B.  Importance of the Study  2  C.  Methodology  2  D.  O u t l i n e of P r e s e n t a t i o n  2  E.  P r e s e n t V a l u e Approach t o V a l u a t i o n  3  F.  Growth Models of V a l u a t i o n  4  1.  "No Growth"  4  2.  Simple Growth  4  3.  Constant Growth  6  G.  H. II.  A l t e r n a t i v e F o r m u l a t i o n s o f Constant Growth.  9  1.  D i v i d e n d Approach  9  2.  Investment O p p o r t u n i t i e s Approach  10  3.  Stream of E a r n i n g s Approach  12  4.  D i s c o u n t e d cash f l o w approach  13  Reasons  f o r A d o p t i n g Gordon's Model  14  GORDON'S STOCK VALUATION MODEL  16  A.  16  Assumptions  iii  CHAPTER  PAGE B.  III.  Interpretation  C. Assumptions under which k is an Increasing function of br  23  REVIEW OF GORDON'S EMPIRICAL INVESTIGATIONS ...  25  A.  25  Linear Models  B. Refinement  32  C. Nonlinear Model  34  D. Nolev Model  37  E.  Sim lev Model  43  F.  Adlev Model  48  G.  Eko Model  48  H. Conclusions IV.  18  from Empirical Findings .......  54  FORMULATION AND TESTING OF THE MODEL  55  A.  Selection of Variables  55  1.  Basic Model  55  2.  Risk Variables  55  a.  Business Risk  56  b.  Financial Risk  56  c.  Size  56  3.  Complete Model  57  B. Data  ...  C. Measurement of the Variables  57 58  1.  Dependent Variable - Price (P) ........  58  2.  Independent Variables  59  iv  CHAPTER  PAGE  D.  V.  a.  Dividends  (D)  b.  Growth Rate ( b r )  59  c.  B u s i n e s s R i s k (u)  63  d.  F i n a n c i a l R i s k (h)  65  e.  Size  65  (S)  59  66  Methodology  RESULTS AND CONCLUSIONS  67  A.  Regression Results  68  B.  Conclusions  79  82  BIBLIOGRAPHY  v  LIST OF TABLES  TABLE 1-1  PAGE Investment O p p o r t u n i t i e s Approach t o  Constant  Growth  11  3-1  R e g r e s s i o n of P r i c e on D i v i d e n d and  Income ..  3-2  R e g r e s s i o n of P r i c e on D i v i d e n d and Earnings  Retained  3-3  3-4  3-5  26  30  R e g r e s s i o n of P r i c e on D i v i d e n d , R e t a i n e d E a r n i n g s , Change i n D i v i d e n d , Change i n R e t a i n e d E a r n i n g s , A l l D e f l a t e d by Book Value  33  R e g r e s s i o n of P r i c e on D i v i d e n d , Growth Rate, and U n c e r t a i n t y V a r i a b l e s .  36  R e g r e s s i o n of P r i c e on E x p o n e n t i a l l y Averaged D i v i d e n d and Growth Rate and  on R i s k V a r i a b l e s *  40  3-6  Regression S t a t i s t i c s  f o r the Simlev Model ..  45  3-7  Regression S t a t i s t i c s  f o r the Adlev Model ...  49  3-8  Regression S t a t i s t i c s  f o r the Eko Model  52  5-1  R e g r e s s i o n S t a t i s t i c s f o r Ln P = lncc + In D + a In (1 + b r ) + 0 3 l n (1 + u) + 0:4 l n (1+h) + InS Q  0:5 2  5-2  R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a + cq. l n D + a In (1+br) + a l n (1+u) + 0:4 l n (1+h) + 0:5 InS A f t e r Change i n Growth V a r i a b l e from G  2  "i b  5-3  r  3  . Y_ -. D . _Y .- D  , 71 7  t o  R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a + ct]_ In D + a In (1+br) + a l n (1+u) + 0 4 In (1+h) + 0:5 InS A f t e r Change i n B u s i n e s s R i s k V a r i a b l e °Y °Y from u = .-*» to ~ Q  2  5-4  69  3  -7 0  /->  R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a + a]_ In D + a In (1+br) + a In (1+u) + 0:4 In (1+h) + a5 InS A f t e r Change t o O p e r a t i n g Income w i t h D  2  u =  22  W  3  and br =  75 A  vi  PAGE  TABLE 5-5  R e g r e s s i o n S t a t i s t i c s f o r Ln P = I n a + a-^ I n D + a i n (1+br) + 0:3 i n (1+u) + a4 I n (1+h) + 0:5 InS A f t e r Change t o O p e r a t i n g Income w i t h Q  2  0  5-6  Y  77  R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a + a-^ I11 D + a I n (1+br) + 0:3 In (1+u) + 0:4 I n (1+h) + InS A f t e r Change to O p e r a t i n g Income w i t h D  2  °Y u = — W  -,. U+P.S.+C.L. and h = — E  vii  CHAPTER I  INTRODUCTION TO THEORETICAL COMMON STOCK VALUATION MODELS  A.  The  Purpose o f t h e Study  purpose o f t h i s study  i s to determine whether a model  based upon t h e d i v i d e n d f o r m u l a t i o n o f Myron J . Gordon p r o v i d e s an adequate e x p l a n a t i o n The  f o r t h e v a r i a t i o n i n common s t o c k p r i c e s .  Gordon model, which i s commonly l a b e l e d t h e "Gordon and S h a p i r o  model,"1 equates t h e v a l u e o f a share value of a l l i t s future  o f common s t o c k t o t h e p r e s e n t  dividends.  I t w i l l be shown t h a t under c e r t a i n assumptions the p r i c e o f a share  equals  the c u r r e n t d i v i d e n d d i v i d e d by t h e d i f f e r e n c e  between t h e r a t e o f r e t u r n r e q u i r e d by i n v e s t o r s and t h e r a t e o f growth i n t h e d i v i d e n d .  The i n v e s t o r s ' r e q u i r e d r a t e o f r e t u r n  a l s o i s the r a t e a t which the d i v i d e n d e x p e c t a t i o n  i s discounted.  T h i s r a t e o f p r o f i t depends upon t h e growth r a t e o f t h e d i v i d e n d and the r i s k c h a r a c t e r i s t i c s of t h e f i r m . then,  The h y p o t h e s i s  t o be t e s t e d ,  i s t h a t i n v e s t o r s , i n v a l u i n g a share o f common s t o c k , m a i n l y  consider  the d i v i d e n d , the growth r a t e o f the d i v i d e n d and t h e  r i s k c h a r a c t e r i s t i c s of t h e company.  ^The i n i t i a l a r t i c l e on t h i s model appeared i n Myron J . Gordon and E l i S h a p i r o , " C a p i t a l Equipment A n a l y s i s : The Required Rate of P r o f i t , " Management S c i e n c e , I I I (October, 1956), 102-110.  1  2 2.  By  Importance of the Study  t e s t i n g a s t o c k v a l u a t i o n model i n c o r p o r a t i n g a  about i n v e s t o r b e h a v i o r we determinants  of share  these determinants  by  hope to g a i n i n f o r m a t i o n on  p r i c e s and  the  the r e l a t i v e weights a t t a c h e d  the s t o c k market.  u s e f u l not o n l y to i n v e s t o r s s e e k i n g a l s o to c o r p o r a t e managers s e e k i n g of  hypothesis  to  T h i s i n f o r m a t i o n would  to maximize t h e i r wealth  to maximize the market  be  but  price  the company's s t o c k .  C.  The model was  Methodology  t e s t e d by u s i n g m u l t i p l e r e g r e s s i o n a n a l y s i s  on a c r o s s - s e c t i o n of f i r m s i n the machinery i n d u s t r y i n the U n i t e d S t a t e s i n each of the y e a r s  1956  l e a s t - s q u a r e s was  the parameters of the r e g r e s s i o n  equations.  used to e s t i m a t e  H i s t o r i c a l d a t a was  to 1965.  The method of  p r o v i d e d by Standard  and  Poor's  Compustat s e r v i c e .  D.  The concept  O u t l i u e of P r e s e n t a t i o n  r e s t of t h i s chapter w i l l  i n t r o d u c e the p r e s e n t  of v a l u a t i o n , d i s c u s s s e v e r a l growth models of  value  stock  v a l u a t i o n , d e s c r i b e the a l t e r n a t i v e f o r m u l a t i o n s of c o n s t a n t and  conclude  for  t e s t i n g purposes.  of  by s t a t i n g  the reasons  f o r adopting  Gordon's model  Chapter I I d e s c r i b e s the t h e o r e t i c a l b a s i s  the Gordon model, i n c l u d i n g i t s assumptions, components  interpretation.  growth  Chapter I I I reviews  Gordon's i n v e s t i g a t i o n s .  the s t a t i s t i c a l  and  r e s u l t s of  Chapter IV d e s c r i b e s the f o r m u l a t i o n  t e s t i n g of the model used i n t h i s study,  and  i n c l u d i n g the measurement  3  problems encountered. c o n c l u s i o n s of the  E.  Although  Chapter  V p r e s e n t s the r e s u l t s  and  study.  P r e s e n t Value Approach to V a l u a t i o n  t h e r e i s evidence  t h a t p r e s e n t v a l u e approaches t o  s t o c k v a l u a t i o n date back at l e a s t  to 1869,  2  J.B. W i l l i a m s i s  g e n e r a l l y c r e d i t e d as the o r i g i n a t o r of s t o c k v a l u a t i o n models. In a book p u b l i s h e d i n 1938,  W i l l i a m s d e f i n e d the investment  a s t o c k as the p r e s e n t v a l u e of i t s f u t u r e i n f i n i t e stream t  Vo =  t  I  =  iTj-vt  3  +  TT-JV  Tr v  2  2  +  H3V  3  +  ...  (1-1)  value at s t a r t  = d i v i d e n d i n year t 1  -, by 1 + i i  of d i v i d e n d s :  = o o  where Vo = investment ir  v a l u e of  definition  = i n t e r e s t r a t e sought  by the  investor.  By r e p l a c i n g Vo w i t h Po, v w i t h — i — E q u a t i o n (1) can be w r i t t e n 1 + i i n a more f a m i l i a r form as  Po =  E t = 1  D  t  E  (1 + i ) t  I f the d i v i d e n d stream  =  Di _  1 + i  +  D?  —  (1 + i ) 2  +  D3 (1 + i ) 3  +  ...  (1-2)  remains c o n s t a n t , as i s the case w i t h p r e f e r r e d  s t o c k , the e q u a t i o n reduces  to  Po =  (1-3)  S e e Robert M. S o l d o f s k y , "A Note on the H i s t o r y of Bond T a b l e s and Stock V a l u a t i o n Models," J o u r n a l of F i n a n c e , XXI (March, 1966), 103-111. 2  J o h n Burr W i l l i a m s , The Theory of Investment Value N o r t h - H o l l a n d P u b l i s h i n g Company, 1938), p. 56. 3  (Amsterdam:  4 F. Growth Models of Valuation An investor who purchases a share of common stock for the future dividends r e a l i s t i c a l l y w i l l expect a growing stream of dividends.  Although the growth rate w i l l vary over time, shareholders  can be considered as forming a subjective probability distribution of future growth rates at any particular period of time.  The mean  of this distribution can be considered as the constant normal growth rate. This section describes and illustrates various types of growth encountered in stock valuation models. 1. "No Growth"  ^  A static company can be defined as one which expects to receive a constant annual net income, Y , from its existing assets and has t  no future investment opportunities that w i l l provide a rate of return greater than the return expected by shareholders.  Hence, there is  no growth possible for earnings or dividends and the price of the company's share is the sum of an annuity discounted at the shareholder's required rate of return, k, P  =  I  t=l and  .  (1 + k)t  =  Y k  k = -XP  (1-5)  being the reciprocal of the price-earnings ratio. 2.  Simple Growth To illustrate simple growth i t is useful to adopt the model of  Ezra Solomon, which is based upon the investment opportunities approach  A  to v a l u a t i o n . of  The p r i c e o f a common share i s equated  t o t h e sum  t h e p r e s e n t worth o f t h e c o n s t a n t annual n e t e a r n i n g s  e x i s t i n g a s s e t s and t h e premium r e s u l t i n g from ability  from  the c o r p o r a t i o n ' s  to i n v e s t a c o n s t a n t amount a t r a t e s o f r e t u r n g r e a t e r than  the s h a r e h o l d e r s r e q u i r e d r a t e o f r e t u r n . Let  P  =  market p r i c e o f a l l - e q u i t y  company  Y  =  c o n s t a n t annual n e t e a r n i n g s from  existing  assets b  =  f r a c t i o n of earnings r e t a i n e d f o r  investment  r  =  c o n s t a n t r a t e o f r e t u r n on f u t u r e  investment  opportunities k  =  shareholder's required rate of r e t u r n  r  >  k  Then an investment  o f bY i n each y e a r s t a r t i n g i n p e r i o d 0 ( t h e  present) w i l l generate  a d d i t i o n a l p e r p e t u a l income streams of  ( b Y ) r , each w i t h a p r e s e n t v a l u e a t t h e time t h e investment  i s made  equal to t  T =l  (bY)r (1 + k ) t  bYr ~k~  which i s t h e v a l u e of the f i r s t payment. of  t h i s stream  (1-6)  The p r e s e n t worth  today  of a d d i t i o n a l payments bYr/k i s ~ ^  bYr/k (1 + k ) t  _  bYr k^  To produce t h i s a d d i t i o n a l income, i t was n e c e s s a r y each year to  i n v e s t bY which has a p r e s e n t v a l u e o f bY/k.  The n e t p r e s e n t  E z r a Solomon, The Theory o f F i n a n c i a l Management (New York: Columbia U n i v e r s i t y P r e s s , 1963), pp. 59-62.  6 v a l u e of the a d d i t i o n a l income from investment  opportunities,  therefore, i s bYr  bY —  k  k  l  , (1-8)  which s i m p l i f i e s t o M .  k  x  (1-9)  k  1  Then the p r i c e of a common share i s  The  second  term i s the investment  o p p o r t u n i t i e s growth  I t i n c r e a s e s p r o p o r t i o n a t e l y w i t h the p r o f i t a b i l i t y being r-k.  factor.  o f investment,  I f r=k, the growth f a c t o r e q u a l s z e r o , w h i l e i f r<k, t h e  p r i c e i s decreased.  3.  Constant Growth While  t h e s i m p l e growth model assumed a c o n s t a n t amount of  investment  each y e a r , t h e dynamic growth model assumes i n c r e a s i n g  investment  at a constant r a t e .  Myron J . Gordon has developed f r e q u e n t l y used i s based (a)  5  a d i v i d e n d model which i s  f>  f o r growth models o f s t o c k v a l u a t i o n .  upon the f o l l o w i n g  The model  assumptions:  t h e c o r p o r a t i o n engages i n no o u t s i d e e q u i t y  financing; (b)  i t does n o t use debt  financing;  (c)  i t w i l l e a r n a r e t u r n , r , on investment i n  every f u t u r e p e r i o d ;  "*For example, see Solomon, Theory o f F i n a n c i a l Management, pp. 62-67. % y r o n J . Gordon, The Investment, F i n a n c i n g and V a l u a t i o n of the C o r p o r a t i o n , Homewood, 111., R i c h a r d D. I r w i n , I n c . , 1962, c h s . A and 5.  7  (d)  i t w i l l retain the fraction, b, of its income  in every future period. Since there is no outside financing, the expected dividend is a function of the corporation's current income, the investment or retention rate (which are the same) and its rate of return on investment. The dividend in any future period is certain D  t  =  (1 - b) Y  (1-11)  t  If the net income when t = o was Yo then Y  1  = Yo + rbYo  = Yo (1 + rb)  (1-12)  since during period t = o, retained earnings of bYo have been invested at rate r . Yo and rbYo.  Income in t=l therefore equals the sum of  Equation (1 - 12) is a compound interest expression  which, i f Y grows continuously at the rate g = br, can be written. t  Y  = Yo (1 + rb)  t  t  = Yo erbt  (1-13)  The product rb is also necessarily the rate at which earnings and reinvestment as well as dividends w i l l grow. If k is the stockholder's required return on the dividend D , t -*» ' the share value at the end of t = o is t  Po  = / * D e-ktdt  (1-14)  t  which can be rewritten using equations (1-11) and (1-13) as Po  = Io (1 - b) Y erbt -ktdt  (1-15)  Po  =  (1-16)  0  e  (1 ~ b)Yo, (k > rb) k - rb  8 Equation  (1-16) can be f u r t h e r s i m p l i f i e d ^ t o  Po = _2£_,  _  (k>g)  (1  k - g  To i l l u s t r a t e :  17)  A c o r p o r a t i o n pays a d i v i d e n d of $1.00 and has a  d i v i d e n d growth of 5% per annum w h i l e  i n v e s t o r s r e q u i r e a r e t u r n of  10% p e r annum.  A s t o c k bought a t t h i s p r i c e would p r o v i d e d i v i d e n d y i e l d of  .$3-rPP = 5% p l u s a c a p i t a l g a i n of 5% per annum $20.00  on the i n c r e a s e i n s t o c k p r i c e . seen by s o l v i n g E q u a t i o n k  =  2° + Po  ( 1 - 3 ) , g = o.  T h i s r e l a t i o n s h i p r e a d i l y can be  (1-17) f o r k g  In the case o f a c o n s t a n t equation  the i n v e s t o r w i t h a  (1-18)  d i v i d e n d stream p r e v i o u s l y r e f e r r e d t o i n  The p a r t i c u l a r m e r i t o f t h i s model f o r t e s t i n g  purposes i s t h a t s t o c k p r i c e i s g i v e n i n terms o f c u r r e n t  values  i n s t e a d o f expected v a l u e s o f b, Y and r .  ^The same r e s u l t can be o b t a i n e d  using  algebra:  Di  D?  D3  (1 + k)  (1 + k)2  ( i + it) 3  Do (1 + g) (1 + k)  £ t=l  (Do (1 + g)2 (1 + k)2  Do (1 + ?-,)*• (1 + k ) t  +  =  Do k - g  Do (1 + g)3 (1 + k)3  +  9  G.  A l t e r n a t i v e F o r m u l a t i o n s o f Constant  S e v e r a l hypotheses,  Growth:  c o n s i s t e n t w i t h c o n s t a n t r a t e s of growth,  have been advanced t o e x p l a i n common s t o c k v a l u e s . of t h e s e adopt  t h e d i v i d e n d s , investment  e a r n i n g s or d i s c o u n t e d cash f l o w  1.  The most  important  o p p o r t u n i t i e s , stream of  approach.  D i v i d e n d Approach I t has been p o p u l a r s i n c e t h e time o f J.B. W i l l i a m s t o equate  the p r i c e o f a common share p e r p e t u a l stream  of dividends.  above i l l u s t r a t e s The  to t h e p r e s e n t v a l u e o f i t s f u t u r e  tendency  The Gordon-Shapiro model d i s c u s s e d  the d i v i d e n d approach t o v a l u a t i o n . of w r i t e r s o f t h e " D i v i d e n d " s c h o o l i s t o c o n s i d e r  the d i v i s i o n o f e a r n i n g s between d i v i d e n d s and r e t e n t i o n s c r i t i c a l to the v a r i a t i o n i n p r i c e o f a common s h a r e . 0.  Harkavy concluded  I n h i s 1953 s t u d y ,  t h a t w h i l e d i v i d e n d s a f f e c t p r i c e more i n the  s h o r t - r u n than i n t h e l o n g - r u n , " . . . g i v e n two s t o c k s s i m i l a r i n a l l r e s p e c t s b u t d i v i d e n d payout,  a h i g h e r p r i c e w i l l be p a i d f o r t h e  s t o c k o f t h e company d i s t r i b u t i n g a g r e a t e r p r o p o r t i o n o f i t s earnings i n dividends."^  See Graham, Dodd and C o t t l e , S e c u r i t y A n a l y s i s : P r i n c i p l e s and Technique, 4 t h ed. (New York: McGraw-Hill Book Company, I n c . , 1962); 0. Harkavy, "The R e l a t i o n Between Retained E a r n i n g s and Common Stock P r i c e s f o r L a r g e , L i s t e d C o r p o r a t i o n , " J o u r n a l of F i n a n c e , V I I I (September, 1953), 283-297; J . E . W a l t e r , " D i v i d e n d P o l i c i e s and Common Stock P r i c e s , " J o u r n a l o f F i n a n c e , XI (March, 1956), 29-41; M.J. Gordon, " D i v i d e n d s , E a r n i n g s , and Stock P r i c e s , " Review o f Economics and S t a t i s t i c s , XLI (May, 1959), 99-105; D. Durand, "The Cost of C a p i t a l , C o r p o r a t i o n F i n a n c e and the Theory of Investment: Comment," American Economic Review, XLIX (September, 1959), 639-655; G.R. F i s h e r , "Some F a c t o r s I n f l u e n c i n g Share P r i c e s , " Economic J o u r n a l , LXXI (March, 1961), 121-141; J . L i n t n e r , " D i v i d e n d s , E a r n i n g s , Leverage, Stock P r i c e s and the Supply of C a p i t a l to C o r p o r a t i o n s , " Review o f Economics and S t a t i s t i c s , XLIV (August, 1962), 243-269. 0.  Harkavy, I b i d . , 297.  10 2.  Investment O p p o r t u n i t i e s To b e s t  i s equivalent  illustrate  t h a t the investment o p p o r t u n i t i e s  to the d i v i d e n d  M i l l e r w i l l be d e s c r i b e d . ^ opportunities  Approach approach  approach, t h e model of M o d i g l i a n i and While Solomon adopted an investment  approach i n h i s "simple growth" model, he used the  Gordon-Shapiro model as the b a s i s f o r h i s "dynamic growth" model which assumes a c o n s t a n t  r a t e of r e i n v e s t m e n t r a t h e r  than a  constant  amount of r e i n v e s t m e n t . ^ The  investment o p p o r t u n i t i e s  approach s t a t e s t h a t the p r i c e  of a common share i s determined by the e a r n i n g e x i s t i n g assets  power o f the c o r p o r a t i o n ' s  and the premium a r i s i n g from the a b i l i t y  to make investments y i e l d i n g a r a t e o f r e t u r n g r e a t e r r e q u i r e d by the s h a r e h o l d e r s .  To i l l u s t r a t e  this  o f the f i r m  than t h a t  approach,  let Y  ~=  b  =  f r a c t i o n o f income r e t a i n e d and r e i n v e s t e d ;  r  =  perpetual  k  =  shareholders'  constant  annual income from e x i s t i n g  assets;  r a t e of r e t u r n on new investment required  rate of return;  r > k At  the end o f y e a r 1, the c o r p o r a t i o n  income a t the end o f y e a r 2 o f ( b Y ) r .  i n v e s t s bY, which r e t u r n s an At t h i s time, a second  i s made c o n s i s t i n g o f bY + b(bYr) o r bY (1 + b r ) .  The f i r m  investment continues  to i n v e s t as shown i n T a b l e 1-1 so t h a t both income and r e i n v e s t m e n t i n c r e a s e over time a t the r a t e b r .  M.H. M i l l e r and F. M o d i g l i a n i , " D i v i d e n d P o l i c y , Growth, and the V a l u a t i o n of Shares," J o u r n a l of B u s i n e s s , XXXIV (October, 1961), 411-433. Solomon, Theory of F i n a n c i a l Management, pp. 62-67.  11 TABLE Investment  1-1  O p p o r t u n i t i e s Approach t o Constant Growth  End of  Constant  Year  Income  Total A  Income  Income  Investment  1  Y  -  Y  2  Y  (bY)r  Y ( l + br)  3  Y  bY(l + br)r  Y(l + br)  bY(l + b r )  The annual investment  I  t  2  1 1 - 1  bY(l + b r )  i s made e q u a l to Itr  k  (1 + k ) t  The n e t b e n e f i t i n y e a r t , t h e r e f o r e , e q u a l s  i  S i n c e the investment  t  A  ~r^*  x  t  i n year t e q u a l s bY  (1 + b r ) £""1, the p r e s e n t  v a l u e of the t o t a l net b e n e f i t s e q u a l I  bY(l + br)t-l  t=l  .  (r - k)/k  (1 + k ) t  _r_zJi.  {  }  I  b Y  k  k  x  - br  {1 + b r ) ^  t=l  ;  1  k  (1 + k ) t  5  The p r i c e of a share under the investment e q u a l s the sum  opportunities  2  bY(l + b r ) ^  t  Itr  t=l  Y(l + br)  b Y ( l + br)  g e n e r a t e s an annual income I r w i t h a p r e s e n t  v a l u e a t the time the investment eo  r  t _ 2  bY  approach  of the p r e s e n t v a l u e s of the c o n s t a n t income and  the  1  12  income from investment  P  =  I t=l  =  1 k  o p p o r t u n i t i e s , that i s ,  2_ (1 + k ) t  +  {  +  ^L_ k - br  second term e q u a l s zero and  r ^ _ k k  }  (  1  _ i  9  )  k t h a t r > k, f o r i f r = k,  i f r < k, P w i l l  (1-19) can be s i m p l i f i e d  common f a c t o r  {  }  For growth to o c c u r , i t i s n e c e s s a r y  Equation  { _ _ k - br  the  decline.  f u r t h e r by t a k i n g out  the  Y/k,  =  Y k  (  1 + b(r - k), k - br  .ijizja  .  (1  20)  k - br Since  D  fc  =  (1 - b ) Y  and br = g, e q u a t i o n  3.  (1-11)  t  (1-20) i s e q u i v a l e n t t o  (1-17).  Stream of E a r n i n g s Approach The v a l u e of a share a l s o can be d e f i n e d as the sum  of  p r e s e n t v a l u e of f u t u r e e a r n i n g s l e s s the p r e s e n t v a l u e of investments  P F  I f we  the  the  i n the r e s p e c t i v e year of the e a r n i n g s ; t h a t i s ,  =  i t=l  l t " t> (1 + k ) t Y  I  .  12  the investment  _2i) U  assume t h a t the c o r p o r a t i o n f i n a n c e s i t s investments  through r e t a i n e d e a r n i n g s ,  ( 1  solely  i n p e r i o d s 1, 2, 3,...,  •*- In the M&M f o r m u l a t i o n , o u t s i d e f i n a n c i n g can be taken account w i t h o u t a l t e r i n g the r e s u l t . 2  into  13  00  i s bYo,  b Y o ( l + br) , b Y o ( l + b r )  bYo(l + b r )  2  p r e s e n t v a l u e of the t o t a l investment I  bYo(l +  t=l  t _ 1  .  The  is  br)t-l  (1 + k ) t  w h i l e the p r e s e n t worth of the t o t a l net income i s 00  Z t=l  Yo(l +  br)*"  1  (1 + k ) t  Hence, Po  =  I t=l  Yo(l  + b r ) t ~ l - bYo(l + (1 + k ) t  Y o ( l - b)  I  br)t-i  t=l  (1 + b r ) ^  1  (1 + k ) t  (1-22) which i s e q u i v a l e n t . t o (1-20).  4.  Discounted  cash f l o w approach  T h i s approach equates  the p r i c e of a common share to the p r e s e n t  v a l u e of a l l f u t u r e net cash flows between the c o r p o r a t i o n and i t s shareholders. income Y  t  and  stock, 2 .  t o the c o r p o r a t i o n are the net  i n a d e b t - f r e e c o r p o r a t i o n , from  E q u a t i n g the s o u r c e s and uses of  or In  s o u r c e s of funds  the s a l e of  The uses of funds are f o r i n v e s t m e n t s ,  t  Dj..  The  Y  t  + g  D  t  = Y  t  t  = + 3  I  +  t  D  additional  I , and d i v i d e n d s , t  funds  t  ~ t  (1-23)  1  t  o t h e r words, the s h a r e h o l d e r s expect  to r e c e i v e a d i v i d e n d f l o w  e q u a l to the t o t a l cash flow of the f i r m l e s s the investment  costs;  14  hence p r i c e CO  p  (Yt - i t  I  t=l  (1 + k ) t  Assuming, as i n the p r e v i o u s =  P  + et) (1-24)  formulations  that &  - o  - IQ. (1 + k ) t  Z  t=l  which i s the same as the stream of e a r n i n g s  H.  t  Reasons f o r Adopting  approach  (1-21).  Gordon's Model  I t might w e l l be asked s i n c e the a l t e r n a t i v e f o r m u l a t i o n s e q u i v a l e n t , what a r e the advantages i n a d o p t i n g formulation.  First,  w i l l be d i s c u s s e d  simple  of r , b, and Y which a r e more  r e a d i l y a s c e r t a i n e d than expected v a l u e s . expectations  Gordon's d i v i d e n d  i t i s u s e f u l i n t e s t i n g because i n a  form i t i n c o r p o r a t e s c u r r e n t v a l u e s  are  Hypotheses d e a l i n g w i t h  i n Chapter I I I which expands  the  simple model i n t o an e m p i r i c a l model. Secondly, we an i d e a l world p o i n t e d out  have assumed i m p l i c i t l y  of c e r t a i n t y .  M o d i g l i a n i and M i l l e r  section  correctly  t h a t once the investment p l a n i s g i v e n , d i v i d e n d  becomes i r r e l e v a n t . - ' retained earnings affecting  i n the p r e v i o u s  3  The. f i n a n c i n g of investment can be  or by s a l e a d d i t i o n a l common shares  the p r i c e of s h a r e s .  of g r e a t e r d i v i d e n d s of t h e i r shares  c o u l d accomplish  or by p l e d g i n g  I f market i m p e r f e c t i o n s to postpone p r e s e n t are r i s k averse,  through  without  the cash  the same by s e l l i n g  equivalent a portion  them as c o l l a t e r a l f o r a l o a n . e x i s t , however, i n v e s t o r s may  consumption of d i v i d e n d s .  any  13, ' M i l l e r and  Investors wishing  policy  not  Assuming t h a t i n v e s t o r s  s h i f t i n g of the time p a t t e r n of d i v i d e n d s  M o d i g l i a n i , "Dividend  wish  P o l i c y , " 411-433.  into  15 the future, may cause them to attach greater risk to future dividends.  Accordingly, the required rate of return k w i l l be  increased.  The effect of dividend policy on the required rate of  return of shareholders w i l l be discussed more fully in the next chapter which presents Gordon's empirical model.  CHAPTER I I  GORDON'S STOCK VALUATION MODEL  Gordon's s t o c k v a l u a t i o n model s t a t e s t h a t t h e p r i c e of a share i s e q u a l t o t h e c u r r e n t d i v i d e n d between the s h a r e h o l d e r s ' growth of t h e d i v i d e n d ;  o  r  Po  =  po  =  required  d i v i d e d by t h e d i f f e r e n c e  r a t e o f r e t u r n and t h e r a t e o f  that i s ,  (2-1)  k - rb LLJZJL&O  (2-2)  k - rb  A.  Assumptions  The model makes t h e f o l l o w i n g assumptions r e g a r d i n g t h e corporation: (1)  t h a t i t i s s u e s no new  shares;  (2)  that i t maintains a constant  (3)  t h a t i t w i l l e a r n a r a t e of r e t u r n , r , on i t s  debt-equity  ratio;  inve s tmen t and (4)  that i t w i l l r e t a i n a constant income i n every f u t u r e  Accordingly,  f r a c t i o n , b, o f i t s  period.  the d i v i d e n d w i l l grow a t t h e r a t e b r .  There i s ample e v i d e n c e t h a t c o r p o r a t i o n s ,  "particularly  engaged i n m a n u f a c t u r i n g , undertake r e l a t i v e l y l i t t l e  16  outside  those equity  17  financing."-'-  Furthermore, the work of Gordon g i v e s e v i d e n c e t h a t  corporations  attempt to m a i n t a i n  Accordingly,  "...  a corporation's  a constant  i t i s q u i t e reasonable  future dividends  debt-equity  ratio.  2  to assume t h a t i n e s t i m a t i n g  i n v e s t o r s do not  consider  the  p o s s i b l e f u t u r e s t o c k s a l e s by a c o r p o r a t i o n as b e i n g m a t e r i a l t h a t they expect the c o r p o r a t i o n equity  to m a i n t a i n  i t s existing  and  debt  ratio." The  r e m a i n i n g two  assumptions perhaps are more o b j e c t i o n a b l e .  While a c o r p o r a t i o n i s u n l i k e l y  to r e t a i n a f r a c t i o n b of i t s income  Iti every f u t u r e p e r i o d , t h i s i s not  important.  i s what i n v e s t o r s expect the c o r p o r a t i o n i n v e s t o r w i l l estimate Lintner provide  b and  The  r for future periods.  rational The  f i n d i n g s of  e v i d e n c e t h a t f i r m s f o l l o w a p o l i c y of paying  s t a b l e f r a c t i o n of t h e i r normal e a r n i n g s G i v e n b,  to do.  What i s r e l e v a n t  the r e t e n t i o n and  as  a  dividends.^  hence the investment r a t e , a  c o r p o r a t i o n i s expected to earn a r e t u r n of r on i t s investment i n every f u t u r e p e r i o d . that r w i l l vary  T h i s assumption does not e x c l u d e the  i f b varies.  I t does e x c l u d e the p o s s i b i l i t y  t a k i n g on d i f f e r e n t v a l u e s when b i s c o n s t a n t . hypothesis p o l l e d on  possibility  The  of r  behavioral  behind t h i s assumption i s t h a t " . . . i f i n v e s t o r s were the change they expect i n the r a t e of p r o f i t a  corporation  Hl.J. Gordon, "The S a v i n g s Investment and V a l u a t i o n of a C o r p o r a t i o n , " Review of Economics and S t a t i s t i c s , XLIV (February, 1962), 38. The p r e s e n t a t i o n i n t h i s c h a p t e r r e l i e s l a r g e l y on the above a r t i c l e . See M.J. Gordon, " S e c u r i t y and a F i n a n c i a l Theory of Investment," Q u a r t e r l y J o u r n a l of Economics, LXXIV (August, 1960), 472-492, esp. 476-479. o  Gordon, Review of Economics and  Statistics,  (February,  1962),  ^ J . L i n t n e r , " D i s t r i b u t i o n of Incomes of C o r p o r a t i o n s among Retained E a r n i n g s , and Taxes," American Economic Review, XLVI (May, 1956), 97-113.  39.  Dividends,  18  w i l l e a r n , t h e t y p i c a l r e s u l t w i l l be a frequency mean z e r o and a s m a l l s t a n d a r d d e v i a t i o n . empirical results w i l l  d i s t r i b u t i o n with  The q u a l i t y of the  t u r n i n l a r g e measure on t h e a c c u r a c y o f t h i s  speculation."-'  B.  Interpretation  The q u e s t i o n whether the p r i c e o f a s h a r e i s independent d i v i d e n d p o l i c y w i l l depend upon t h e assumption  of  made w i t h r e s p e c t t o  the b e h a v i o r o f r and k as b i s v a r i e d . Case 1:  I f r and k a r e independent  „  9 P / 3 b  The  ( r  _  k )  ( k  3°  rb)  o f b, then  y  (2-3)  c o n d i t i o n when the share's p r i c e i s maximized can be  found by s e t t i n g  the p a r t i a l d e r i v a t i v e of P w i t h r e s p e c t to b  e q u a l t o z e r o , i n which case r equals k. p r i c e i s independent  At t h i s p o i n t , a share's  o f the r e t e n t i o n r a t e b.  w i t h b and c o n v e r s e l y i f r < k, P w i l l  I f r > k, P w i l l  f a l l w i t h b.  rise  The f a c t t h a t P  w i l l r i s e or f a l l w i t h b when r £ k i s n o t due t o f i n a n c i n g the investment If  through  r e t e n t i o n b u t t o the p r o f i t a b i l i t y o f investment.  funds were r a i s e d  p r i c e would s t i l l Case 2;  through  s a l e w h i l e b was kept  change.  I f r and k a r e n o t independent  or  fall  is  not a constant.  c o n s t a n t , the  o f b, P w i l l n o t r i s e  i n d e f i n i t e l y w i t h b, however, because t h e r a t e of r e t u r n r , Provided  there a r e no i n d i v i s i b i l i t i e s i n  5Gordon, Review o f Economics and S t a t i s t i c s , ( F e b r u a r y , 1962), 39  19  the f i r m ' s investment o p p o r t u n i t i e s r w i l l f a l l as b i n c r e a s e s . In t h i s  event,  = f r - k - b (1 - b)  3P/3b  3r/3bT ^  By  — (k - r b ) 2  (2-4)  s e t t i n g 3P/3b = 0, we can a l s o s e t the e x p r e s s i o n  {r - k - b (1 - b) 3r/3b} when r = k.  =  0.  When b = 0, p r i c e i s maximized  I f r > k a t b = o, P i s maximized a t a p o s i t i v e  of b s i n c e as b i n c r e a s e s , r f a l l s negative  because 3r/3b i s n e g a t i v e ,  and b (1 - b) 3r/3b which i s increases i n absolute  At b = 1, P = 0 r e g a r d l e s s o f t h e v a l u e o f r and k. the p r o f i t a b i l i t y zero value  value  amount. Despite  o f i n v e s t m e n t , a c o r p o r a t i o n ' s shares w i l l have  i f i t i s expected never to pay a d i v i d e n d .  Gordon admits,  however, t h a t h i s model i s n o t the b e s t means of d e a l i n g w i t h the s i t u a t i o n of a non-dividend In o r d e r  paying  t o use E q u a t i o n  corporation,^  (2-2) to s o l v e f o r the optimum p r i c e  we must be a b l e t o observe the v a r i a b l e s . obtained  from h i s t o r i c a l d a t a ,  must be d e r i v e d .  Equation  & PoZ so t h a t the l e f t  While Yo, r and b can be  the s h a r e h o l d e r s '  required return  (2-2) can be r e w r i t t e n as  =  side i s equal  based on i t s c u r r e n t d i v i d e n d .  k - br  (2-5)  t o d, a c o r p o r a t i o n ' s d i v i d e n d Then a sample o f s i m i l a r  of e q u i v a l e n t r i s k can be taken and used t o e s t i m a t e d = a where a  6  Q  i s an e s t i m a t e  I b i d . , 40 f n .  G  - a^br of k and a^ should  yield  corporations  the parameters of (2-6)  equal  1.  20 Gordon, however, found t h a t a^ was s i g n i f i c a n t l y l e s s than 1.^  But i f k i s a c o n s t a n t ,  negative  f o r a s u f f i c i e n t l y h i g h growth r a t e and t h e share's p r i c e  would pass through i n f i n i t y . greater  independent o f b r , d c o u l d become  I t i s u n l i k e l y , however, t h a t b r i s  than k as r would have to be g r e a t e r  r > l k w i t h 0 < b < 1, would not l i k e l y future period.  Yet according  than g.k or a t minimum  t o our assumption,  investors  expect t h i s r a t e o f r e t u r n on investment i n every Moreover, a c o r p o r a t i o n whose r e t u r n on investment  exceeded i t s c o s t o f c a p i t a l would d o u b t l e s s f i n a n c i n g u n t i l r = k.  Nevertheless,  are expected to earn a h i g h  there  engage i n o u t s i d e a r e some f i r m s which  r a t e o f r e t u r n on investment f o r a long  time and which do n o t engage i n o u t s i d e  f i n a n c i n g whose s h a r e s  still  s e l l at f i n i t e p r i c e s . An but  a l t e r n a t i v e explanation  i s t h a t k i s n o t independent o f b r  i s an i n c r e a s i n g f u n c t i o n of b r .  Gordon suggests t h a t  this  requirement would be s a t i s f i e d by the e x p r e s s i o n d  =  Then as b r i n c r e a s e s ,  a  (1 + b r ) ~ l  (2-7)  a  Q  the r e q u i r e d  d i v i d e n d y i e l d , d, would  though not as r a p i d l y and would a s y m p t o t i c a l l y These a l t e r n a t i v e f u n c t i o n s  approach  f o r d are i l l u s t r a t e d  fall  zero.  below:  Dividend Chart 2-1 V a r i a t i o n i n Dividend Y i e l d with Expected Rate o f Growth i n the D i v i d e n d  M.J. Gordon, " D i v i d e n d s , E a r n i n g s , and Stock P r i c e s , " of Economics and S t a t i s t i c s , XLI (may, 1959), 99-105.  Review  21  S i n c e d = k - b r , the e x p r e s s i o n the o r i g i n a l model Po  Po  =  Yo  =  i n (2-7)  Xft Or -~bJk - br  (1 - b) -1 o  can be s u b s t i t u t e d i n  to g i v e  (1 + b r ) l  (2-8)  a  a  When b = o, P = Yo  (1 - b) — A s o  b r i s e s , the m u l t i p l i e r  increases  a  to —1 *o  (1 + b r ) l .  Taking  a  the c u r r e n t d i v i d e n d Y o ( l - b) as  given,  the l a r g e r the expected growth of the d i v i d e n d b r , the h i g h e r p r i c e i n v e s t o r s w i l l pay (2-8)  f o r the s h a r e , p r o v i d e d  are  positive.  For  t h i s f o r m u l a t i o n to be  c o s t of c a p i t a l k must be  theoretically  the  the parameters of  sound, the  firm's  assumed to be an i n c r e a s i n g f u n c t i o n of  b r , the expected r a t e of growth i n i t s d i v i d e n d . k - br implies  =  a  (1 + b r ) ~ l  that k = a (l + br)  _ a  Q  ak/Sbr  and  =  -  =  1  a  l + br  product  (2-9)  a ( l + br)"^  ±  1  +  when b = o, a  (1 + b r ) " " l f a l l s a  to pay  1.  Assuming a a ^ Q  As b and hence br r i s e ,  >  1,  3k/8br  positive.  I t f o l l o w s t h a t a-^ can be willing  1  l  0  In (2-9) Q  a  a a± should e q u a l a p p r o x i m a t e l y  w i l l become permanently  +  (2-10)  k w i l l f a l l as b i n c r e a s e s from 0.  rises a  +*l)  1  Q  (1 + b r ) The  (2-7)  a  D  Q  i s a f i r m ' s c o s t of c a p i t a l .  As  br  thus moderating the i n c r e a s e of k w i t h considered  f o r growth i n the  "as  br.  the p r i c e i n v e s t o r s are  dividend."^  'Gordon, Review of Economics and  Statistics,  (February,  1962), 41.  22 The greater the value of a-^> the larger the price investors are willing to pay for dividend growth. Formerly, under the assumption that k was a constant, price was independent of b when r = k = a . Q  the dividend and investment rates.  In this case, P depends on  When br = o, k = a „ . * o  P w i l l change with b because k changes.  If r = a . o'  Not only does the value of  a share change due to the return on investment but also because of the fraction of earnings retained or paid out in dividends. If the assumption that 3k/3br > o is correct, i t follows that there is an optimum price associated with a finite dividend rate when the rate of return is a constant.  The optimum dividend or  retention rate can be found by differentiating (2-8) with respect to b, 9  p/  8 b  = M l + brlfl  [Taj (1 - b) - 7 ]  When 3P/3b = o, the expression |—  (  •= o.  2  _  n  )  Solving  for b, b  = — ajL + 1  - 7 - — ~ r(a^ + 1)  (2-12)  Then Po is maximized by this function of b i f r is independent of b. Moreover, the optimum retention rate is an increasing function of the rate of return, r , and of a-^, the price investors are willing to pay .for growth in the dividend, Gordon notes that the proposition that k is an increasing function of br does not imply that Po is a decreasing function of br. From (2-8) Po  = Yo (1 - b) ~ ao  (1 + b r ) l a  23 i t can be seen that for a given b, share price w i l l always increase through a rise in r and for a given r and a^, Po w i l l rise and then f a l l with an increase in b. C. Assumptions under which k is an Increasing function of br: The basic premise of Gordon's model is that the value of a share is equal to the present value of the future stream of dividends arising from the share.  The present value may be arrived at by  either of two methods:  by discounting the expected value of the  dividends at an interest rate which reflects risk or by discounting the certainty equivalent of the future dividends at the pure rate of interest. The second assumption is that the uncertainty of dividend increases with its time in the future.  In terms of the standard  deviation, a, this means that at the end of year n, an > on - 1. Similarly, on + 1 > on. The effect of increasing uncertainty on k is indeterminate; that i s , k may be less than, equal to, or greater than k + 1. R  If however, k  n  is an increasing function of t then a corporation's  t  cost of capital k being a weighted average of the k ' s is an t  increasing function of the rate of growth in the dividend.^  A mathematical proof of this proposition is provided by Ramesh Gangolli in Gordon, I b i d . , 49-50. Also see H. Chen, "Valuation under Uncertainty," Journal of Financial and Quantitative Analysis, II (September, 1967), 313-325. Chen points out that while the risk of a future dividend increases with time i t is more important that we know the rate at which this risk increases. He shows that i f risk (1) increases at a constant rate over time k = k + \ for a l l t=l, 2, 3, . . . ; (2) increases at a decreasing rate over time, kpk >k3>.. .>k >k .(. ±; or (3) increases at an increasing rate over y  t  2  t  t  t  t  Deductive argument cannot solve the problem of how the k ' s behave t  and i t should be regarded as a question of fact.  Substantiation  for this proposition must be based on a model incorporating this proposition in explaining common share prices.  time, k^<lc2<k^<.. .<k <k . Since Gordon's proposition that k is an increasing function of b is based on the assumption that the k ' s increase with t, either i t must be assumed that the risk of future dividends must increase at an increasing rate or i t must be shown that k increases with b regardless of whether the k ' s increase with t, to substantiate the theory. t  1  +  t  t  CHAPTER I I I  REVIEW OF GORDON'S EMPIRICAL INVESTIGATIONS  A.  L i n e a r Models  Gordon's e a r l i e s t e m p i r i c a l work was r e p o r t e d i n h i s paper " D i v i d e n d s , E a r n i n g s and Stock P r i c e s . "  1  I t attempted  to e s t a b l i s h  whether i n v e s t o r s used e a r n i n g s , d i v i d e n d s or a combination o f the two i n a r r i v i n g a t the v a l u a t i o n o f a s h a r e .  The f o l l o w i n g  linear  f u n c t i o n was used:  where  =  a  P  =  year-end  D  =  dividends paid during year  Y  =  e a r n i n g s p a i d d u r i n g year  D  E i g h t samples  +  a  P  1  D + a  2  Y  (3-1)  price  c o n s i s t i n g o f f o u r i n d u s t r i e s f o r the y e a r s 1951  and 1954 were used t o e s t i m a t e the parameters  of (3-1).  i n d u s t r i e s groups used f o r t e s t i n g were Chemicals Steel  The  (32) , Foods ( 5 2 ) ,  (34) and Machine T o o l s ( 4 6 ) . I n o r d e r t o o b t a i n samples o f  s u f f i c i e n t s i z e some f r i n g e c l a s s i f i c a t i o n s were i n c l u d e d such as p h a r m a c e u t i c a l m a n u f a c t u r e r s under Chemicals and f o r g i n g  manufacturers  and o t h e r s t e e l f a b r i c a t o r s w i t h the b a s i c p r o d u c e r s under  ^•Review of Economics 2  and S t a t i s t i c s , XLI (May, 1959),  I b i d . , 100.  25  Steel.  2  99-105.  26  In a d d i t i o n , c o n s i d e r a b l e v a r i a t i o n e x i s t e d among the f i r m s w i t h r e s p e c t to such a t t r i b u t e s as s i z e , p r o f i t a b i l i t y , market s t r u c t u r e s for  s u p p l y and demand and  presents  type of s h a r e h o l d e r .  T a b l e 3-1  the l e a s t - s q u a r e s e s t i m a t e s of the parameters,  below  their  s t a n d a r d e r r o r s (the standard d e v i a t i o n of the sample o b s e r v a t i o n s about the r e g r e s s i o n o r e s t i m a t i n g l i n e ) , coefficients  (R) and  the m u l t i p l e c o r r e l a t i o n  the c o e f f i c i e n t s of d e t e r m i n a t i o n  TABLE  (R ) •  3-1  R e g r e s s i o n of P r i c e on D i v i d e n d and Income  Sample 1951 Chemicals  Constant Term  Coefficient and s t a n d a r d e r r o r of D Y  Multiple Correlat i o n  Coefficient of determination R z  -.8 (5.2)  16.7 (3.1)  .93  .865  .1  7.0 (1.5)  5.5 (.9)  .90  .810  5.5  6.6 (1.8)  2.0 (.6)  .86  .740  Machine T o o l s 2.4  12.0 (1.2)  .8 (.5)  .90 .  .810  -3.0  25.7 (5.2)  .3 (3.3)  .92  .846  Foods  -.4  10.4 (2.2)  5.6 (1.0)  .91  .828  Steels  8.7  8.4 (1.7)  2.0 (.8)  .94  .884  Machine T o o l s 6.3  5.5 (1.4)  4.1 (.6)  .89  .792  Foods  Steels  1954 Chemicals  -7.0  27 Gordon expected that i f the dividend coefficient is considered an estimate of the rate of the profit the regression statistics could be evaluated as follows: a.  Since the dividend yields on good preferred stocks  during these years was 4-5% and companies acquired through mergers had been selling at five times earnings before income tax, i t would be reasonable to expect the dividend coefficient being the reciprocal of the dividend yield, to range between 10 and 25; b.  the rate of profit (and hence the dividend coefficient)  should be ranked according to industry characteristics in the following order: (1)  Chemicals were characterized by size,  growth and stability; (2)  foods represented a stable industry;  (3)  steels included large companies and was  vulnerable to cyclical trends; and (4)  machine tools consisted of comparatively  small corporations which were cyclically vulnerable; and c.  ,Since 1951 and 1954 differed in terms of business expectations,  the coefficients between years should reflect the difference. The results for the model incorporating both earnings and dividends were hardly encouraging.  The fact that the dividend coefficient  is large and statistically significant in seven out of eight samples would appear to discount the earnings hypothesis.  Also, the income  coefficients except for Chemicals in 1951 were very low as a measure  28  of for  the p r i c e i n v e s t o r s p l a c e d on e a r n i n g s . Chemicals  v a r i e s widely  from -.8  The  i n 1951  dividend  to 25.7  in  coefficient 1954.  Because t h e r e i s h i g h i n t e r c o r r e l a t i o n between the independent v a r i a b l e s making the c o e f f i c i e n t s h i g h l y u n s t a b l e t h e r e i s l i t t l e p r e d i c t i v e v a l u e i n t h i s model. If  i n v e s t o r s purchase a share of s t o c k f o r i t s e x p e c t e d . f u t u r e  d i v i d e n d s t h i s h y p o t h e s i s might be r e p r e s e n t e d by  the f o l l o w i n g  equation: P  =  a  Q  + aiD + a  where r e t a i n e d e a r n i n g s  2  (Y - D)  the growth i n the d i v i d e n d . per share of common then , b  _ "  r  (Y - D) (Y)  Gordon p r e f e r r e d to use  (Y - D)  (3-2)  i s used as a proxy measure of b r ,  I f B r e p r e s e n t s the book or a s s e t  00  W  _  ~  Y - D  value  /o o\ (3_3)  the a b s o l u t e measure of growth (Y - D) r a t h e r  than to d e f l a t e r e t a i n e d e a r n i n g s by book v a l u e because p r i c e d i v i d e n d are a b s o l u t e q u a n t i t i e s i n the m o d e l . approach however i s t h a t two  companies may  3  A weakness of  f i n d a lower  this  have the same r e t a i n e d  e a r n i n g s but have v a s t l y d i f f e r e n t r a t e s of r e t u r n on I n v e s t o r s , i t i s assumed, w i l l  and  investment.  r a t e of growth s t a r t i n g  from a h i g h book v a l u e p r e f e r a b l e to a h i g h e r r a t e of r e t u r n based on a low  a s s e t v a l u e which w i l l be d i s c o u n t e d more h e a v i l y i n the  distant future. The  d i v i d e n d c o e f f i c i e n t s proved more r e a s s u r i n g than under  the f i r s t model. too l a r g e .  3  The  Ibid.,  Only  the 1951  c o e f f i c i e n t f o r machine t o o l s  range of v a r i a t i o n was  101.  appeared  much s m a l l e r than p r e v i o u s l y  29  tested although s t i l l 1954  quite large.  S t e e l s - 1951 and machine t o o l s -  were low and c h e m i c a l s - 1954 was h i g h .  was much  The s t a n d a r d  error  reduced.  S i n c e the r e t a i n e d e a r n i n g s c o e f f i c i e n t r e p r e s e n t e d the p r i c e i n v e s t o r s were w i l l i n g  to pay f o r d i v i d e n d growth, i t was p r e d i c t a b l e  t h a t the c o e f f i c i e n t s s h o u l d be p o s i t i v e . g e n e r a l l y poor, p a r t i c u l a r l y The  Otherwise,  t h e r e s u l t s were  as t h e c o e f f i c i e n t s a r e low.  second model d i d p r o v i d e a s u p e r i o r means o f e s t i m a t i n g  p r i c e v a r i a t i o n w i t h r e s p e c t t o the d i v i d e n d g i v e n the f i r m ' s r e t a i n e d e a r n i n g s as compared to e a r n i n g s . I t s h o u l d be noted is  t h a t the d i v i d e n d c o e f f i c i e n t i n T a b l e 3-2  the sum o f the d i v i d e n d and e a r n i n g s c o e f f i c i e n t s i n T a b l e 3-1.  The  i n c r e a s e i n d i v i d e n d i n (3-1) causes  an e q u i v a l e n t r e d u c t i o n i n  r e t a i n e d e a r n i n g s w h i l e under (3-2) t h e r e t a i n e d e a r n i n g s a r e h e l d constant.  I n a d d i t i o n , the m u l t i p l e c o r r e l a t i o n and R  2  statistics  a r e the same f o r each year i n the two t a b l e s , s i n c e p r i c e i s a l i n e a r f u n c t i o n o f the same v a r i a b l e s .  Finally,  the e a r n i n g s and  r e t a i n e d e a r n i n g s c o e f f i c i e n t s and t h e i r s t a n d a r d e r r o r s a r e the same s i n c e a change i n e a r n i n g s i s the same as a change i n r e t a i n e d earnings with the dividend held constant.  Not o n l y a r e the c o e f f i c i e n t s  low but i n t h e case o f machine t o o l s - 1951 and c h e m i c a l s - 1954, the c o e f f i c i e n t s a r e n o t s t a t i s t i c a l l y cent  s i g n i f i c a n t a t the f i v e per  level. The  r e t a i n e d e a r n i n g s c o e f f i c i e n t s a r e more encouraging  viewed as the p r i c e i n v e s t o r s a r e w i l l i n g d i v i d e n d than as e a r n i n g s c o e f f i c i e n t s . is  when  to pay f o r growth i n the I f the growth  coefficient  low, an i n c r e a s e i n d i v i d e n d s w i t h a c o r r e s p o n d i n g d e c r e a s e i n  30  TABLE 3-2 R e g r e s s i o n o f P r i c e on D i v i d e n d and R e t a i n e d E a r n i n g s  Constant Term  Sample 1951 Chemicals  Coefficient and s t a n d a r d error of D Y - D  -7.0  15.9 (2.7)  .1  12.5 (1.1)  Steels  5.5  Machine T o o l s  Multiple Correlation  R  2  .93  .865  5.5 (.9)  .90  .810  8.6 (1.5)  2.0 (.6)  .86  .740  2.4  12.8 (1.0)  .8 (.5)  .90  .810  -3.0  30.0 (2.6)  .3 (3.3)  .92  .846  Foods  -.4  15.9 (1.5)  5.6 (1.0)  .91  .828  Steels  8.7  10.4 (1.4)  2.0 (.8)  .94  .884  Machine T o o l s  6.3  9.6 (1.2)  4.1 (.6)  .89  .792  Foods  1954 Chemicals  retained earnings w i l l  16.7 (3.1)  •  i n c r e a s e the p r i c e o f a share more than when  the growth c o e f f i c i e n t i s h i g h e r . The d i v i d e n d h y p o t h e s i s i n Model 2 i s s u p e r i o r i n p r e d i c t i n g how p r i c e w i l l  change w i t h t h e d i v i d e n d g i v e n the r e t a i n e d e a r n i n g s  than when g i v e n the f i r m ' s e a r n i n g s .  The d i v i d e n d c o e f f i c i e n t s  are g r e a t e r than f o r Model 1 and the s t a n d a r d e r r o r s a r e l e s s . There a r e c e r t a i n l i m i t a t i o n s , however, i n t h i s k i n d o f model. A s c a l e f a c t o r , p a r t of which i s r e f l e c t e d by the presence of both  31  high-priced and low-priced stocks in the samples help cause correlation between variables and variation in the coefficients among industries. The influence of scale on the coefficients might be reduced by deflating the variables, by book value for instance, or by the use of logs, or by a combination of the two. Secondly, the current values of dividends and earnings in any one year may vary from the normal values resulting from the average over some prior period.  Hence a combitiation of current values and  past averages for dividends and retained earnings might provide a better model. The absence of other variables such as corporation size, - debt-equity ratio and variability of earnings which the investor may take into account in valuing a share, would bias the retained earnings and dividend coefficients. Lastly, a model using the actual growth rate rather than an index of growth represented by retained earnings might be improvement.  Empirically, there are problems, however.  an The use  of book value as measure of return on investment is questionable because of variation in accounting practice.  The use of past  values to predict future earnings might be questioned especially i f other variables are used by investors to predict future earnings. Retained earnings are quite unstable and each industry may show a different pattern of variation over time.  32 B.  Refinement  The refined model next tested by Gordon was: P a  W  where  o  +  a  l w  +  + a g + a (g-g)  3 2  3  (3-4)  4  P/W =  year-end price divided by book value in current year,  D/W =  average dividend for the prior five years divided by the book value,  D/W =  current year's dividend divided by book value  g  =  average retained earnings for the prior five years divided by book value,  g  =  current year's retained earnings divided by book value.  The assumption is made that in arriving at the current dividend and its growth, investors look at the average over the prior five years and the amount by which the current years values depart from these averages.  Deflation by book value was undertaken to eliminate  the scale effect although its use is debatable.^ In interpreting the coefficients,  i f a^ = a  2  or a  3  = a4 implies  that investors ignore the five-year average dividend or growth in dividend in favor of the current values; i f a the current dividend is ignored; i f a^ > a  2  2  = o this implies that  this "implies that  investors adjust to a change in the dividend with a log, i . e . , elasticity of expectations is less than one."^ the reverse is true.  4  I b i d . , 104 fn.  5  I b i d . , 105.  the  When a^ < a , of course, 2  /  33  TABLE 3-3 R e g r e s s i o n of P r i c e on D i v i d e n d , Retained E a r n i n g s , Change i n D i v i d e n d , Change i n Retained E a r n i n g s , A l l D e f l a t e d by Book V a l u e .  Constant Term  Sample 1951 Chemicals  C o e f f i c i e n t & s t a n d a r d <e r r o r of D/W D/W-D/W S-R  Multiple Correlation  R  2  -.23  12.42 (2.63)  9.79 (5.98)  18.74 (5.96)  14.36 (5.60)  .80  .640  Foods  .04  14.04 (1.04)  8.06 (2.49)  3.16 (1.39)  4.57 (1.58)  .90  .810  Steels  .15  9.88 (1.05)  6.38 (1.87)  1.45 (1.09)  .41 (1.06)  .88  .874  Machine T o o l s  .12  12.62 (1.17)  5.93 (2.75)  .12 (.99)  1.11 (.80)  .91  .828  .54  17.38 (2.92)  12.71 (8.93)  .12 (6.39)  3.44 (4.78)  .79  .624  -.03  15.51 (1.04)  8.74 (2.82)  5.15 (1.66)  5.96 (1.67)  .92  .846  Steels  .18  9.69 (.99)  3.85 (1.13)  2.02 (.68)  2.85 (.67)  .91  .828  Machine T o o l s  .05  11.65 (1.16)  6.06 (1.74)  3.70 (1.12)  1.92 (1.04)  .87  .757  1954 Chemicals  Foods  In comparison to T a b l e the d i v i d e n d  3-2 t h e r e i s a  c o e f f i c i e n t s whose range of f l u c t u a t i o n among samples  has been reduced.  Moreover, a l lbut the chemicals  s i g n i f i c a n t a t the f i v e per cent l e v e l . a^ > 2 a  s l i g h t improvement i n  i t can  be i n f e r r e d  c o e f f i e n t s are  S i n c e i n every  sample,  t h a t i n v e s t o r s w i l l not take an i n c r e a s e  i n d i v i d e n d s i n t o account u n t i l the average of d i v i d e n d s has i n c r e a s e d .  34  The growth  c o e f f i c i e n t s are d i s a p p o i n t i n g i n that  the  c o e f f i c i e n t s f o r g are p o o r e r than f o r Y - D i n T a b l e 3-2 of t h e s e are not s t a t i s t i c a l l y samples, growth  the change i n growth  showing  significant.  Furthermore, i n f i v e  c o e f f i c i e n t s exceeded  those f o r average  t h a t i n most c a s e s , i n v e s t o r s p r e f e r r e d  change i n r e t a i n e d e a r n i n g s over p a s t performance. used do not y i e l d  and t h r e e  The  a current averages  r e l i a b l e e s t i m a t e s of what i n v e s t o r s are w i l l i n g  to pay f o r r e t a i n e d e a r n i n g s or  growth.  While t h i s model i s a g r e a t improvement over the f i r s t t h e r e were s e v e r a l ways, Gordon f e l t f a c t o r might be t r e a t e d d i f f e r e n t l y the v a r i a b l e s . due  F i v e of the  a l s o be  The  scale  through another r e l a t i o n s h i p among  v a l u e s are lower than i n T a b l e  to the d e f l a t i o n by book v a l u e .  might  i t c o u l d be improved.  model  The r e p r e s e n t a t i o n of  3-2  growth  improved.  C.  N o n l i n e a r Model  The r e s u l t s of t e s t i n g a n o n l i n e a r model were r e p o r t e d i n Gordon's 1960  paper "The Optimum D i v i d e n d Rate."^  estimating equation including r i s k variables ln(P/B)  t  =  lna  Q  + g t l n a i + a2ln + a ln S 3  where  t  (D/B)  The  new  was  t  + a4ln U  (3-5)  P  t  =  average of h i g h and low p r i c e s f o r the months September, October and November of y e a r t ,  B  t  =  book v a l u e per share at end of y e a r t ,  S l . J . Gordon, "The Optimum D i v i d e n d Rate," Management S c i e n c e s : Models and T e c h n i q u e s , e d i t e d by C. West Churchman and M i c h e l V e r h u l s t (New York: Pergamon P r e s s , 1960).  35 g  t  =  p r o d u c t of c u r r e n t r e t e n t i o n r a t e b = ( Y - D t ) / Y t and the c u r r e n t rate, r = Y /B , t  t  t  t  earnings  t  D  t  =  d i v i d e n d p a i d d u r i n g year  S  t  =  s i z e of the c o r p o r a t i o n measured by t o t a l book v a l u e of the common e q u i t y at the end of year t and  -  i n s t a b i l i t y of p a s t e a r n i n g s measured by the s t a n d a r d d e v i a t i o n of the r e t u r n , r , on common e q u i t y over the p e r i o d 1934-1954.  U  t,  An a d d i t i o n a l r i s k v a r i a b l e c o n s i s t i n g of the r a t i o of net worth p l u s long-term debt to net worth was was  included o r i g i n a l l y  but  d e l e t e d because the c o e f f i c i e n t v a l u e s were u n s a t i s f a c t o r y . ^  D e f l a t i o n by book v a l u e was p r i c e and  dividends  i n c l u d e d i n the  due  undertaken to a v o i d c o r r e l a t i o n between  to l o w - p r i c e d  and  h i g h - p r i c e d shares  being  samples.  While the e i g h t samples c o n s i s t e d of the same i n d u s t r i e s and years  as used p r e v i o u s l y , the c r i t e r i a used f o r a f i r m ' s  d i f f e r e d somewhat, r e d u c i n g  the sample listed  inclusion  sizes:  (1)  the f i r m must be  on a s e c u r i t y exchange;  (2)  d a t a on income and net worth must be c o n t i n u o u s l y s i n c e 1934; and  (3)  the d i v i d e n d i n the sample year must be l e a s t 2% of the book v a l u e per s h a r e .  published  at  ' I b i d . , 101. A l s o see M.J. Gordon, The Investment, F i n a n c i n g and V a l u a t i o n of the C o r p o r a t i o n (Homewood, I l l i n o i s : R i c h a r d D. I r w i n , I n c . , 1962), 146-147. The e i g h t samples were s p l i t e v e n l y between p o s i t i v e and n e g a t i v e c o r r e l a t i o n , o n l y t h r e e were s t a t i s t i c a l l y s i g n i f i c a n t a t the f i v e per cent l e v e l and two had the wrong s i g n .  36  TABLE 3-4 . R e g r e s s i o n o f P r i c e on D i v i d e n d , Growth Rate, and U n c e r t a i n t y V a r i a b l e s . C o e f f i c i e n t & Standard Sample  E r r o r of  Dividend InD/B  Growth rate g  Size InS  .825 (.132)  4.56 (2.29)  .077 (.038)  -.081 (.104)  .757  Foods (49)  1.023 (.063)  5.51 (1.07)  .024 (.026)  -.283 (.064)  .865  S t e e l s (34)  .733 (.095)  2.10 (.90)  .002 (.029)  -.002 (.132)  .740  Machine T o o l s (46)  .874 (.078)  1.54 (.82)  .041 (.042)  -.085 (.082)  .757  .822 (.130)  -.11 (1.92)  .088 (.039)  -.018 (.100)  .792  Foods (45)  .953 (.055)  3.87 (1.08)  .084 (.024)  -.165 (.057)  .903  S t e e l s (29)  .S35 (.091)  3.71 (.85)  .022 (.021)  .136 (.098)  .846  Machine T o o l s (43)  .694 (.081)  3.24 (.87)  .028 (.044)  -.131 .089  .774  1951 Chemicals (32)  1954 Chemicals (31)  Uncertainty InU  R2  The a d d i t i o n of s i z e and u n c e r t a i n t y v a r i a b l e s were an improvement All  the s i z e c o e f f i c i e n t s had the c o r r e c t s i g n and t h r e e were  statistically had  significant.  the wrong s i g n w h i l e  regarded  F o r the u n c e r t a i n t y c o e f f i c i e n t , o n l y one  two were s t a t i s t i c a l l y  the d a t a as s u p p o r t i n g  Gordon  the hypotheses t h a t p r i c e i s d i r e c t l y  r e l a t e d t o s i z e and i n v e r s e l y r e l a t e d  Gordon, "Optimum D i v i d e n d  significant.  t o the i n s t a b i l i t y  Rate," 102.  of past  earning  37  While the d i v i d e n d c o e f f i c i e n t s a  2  a l l but f o o d s - 1951 a r e l e s s than one. a  2  = 1.  1954,  are a l l h i g h l y  significant,  The theory p r e d i c t s t h a t  The growth c o e f f i c i e n t s w i t h one e x c e p t i o n ,  have the r i g h t s i g n and a r e s i g n i f i c a n t .  chemicals  -  The d i v i d e n d and growth  c o e f f i c i e n t s a r e an improvement over the r e s u l t s o f the l i n e a r models but  c o u l d be improved f u r t h e r .  D.  The  Nolev Model  next e m p i r i c a l model employed by Gordon was r e p o r t e d i n  h i s paper "The S a v i n g s ,  Investment, and V a l u a t i o n o f a C o r p o r a t i o n ,  the b a s i s o f which a r e used i n d e v e l o p i n g  Po  =  Yo (1-b) —  was expanded to take of  (1 +  Chapter I I .  Equation  (2-8),  br) l a  i n t o account c o r p o r a t i o n s i z e and i n s t a b i l i t y  earnings:  where  Po  =  a  = l/a  s  Yo(l-b)  a  Yo (1-b) ( l + b r ) l a  s  (l+u) 2 a  S 3 a  (3-6)  Q  =  Do  r  =  Yo/Bo  b  =  1 - Do/Yo  r b  =  YcL^JDo B  S  =  index o f s i z e = sum of net p l a n t account and working c a p i t a l  u  =  index  of i n s t a b i l i t y o f e a r n i n g s .  To a r r i v e a t Do, the n o r m a l i z e d  current dividend, i t i s hypothesized  t h a t i n v e s t o r s use an e x p o n e n t i a l l y weighted average o f a c o r p o r a t i o n ' s  ^Review of Economics and S t a t i s t i c s , XLIV ( F e b r u a r y ,  1962), 37-51.  38 past dividends.  S i m i l a r l y , i n v e s t o r s take an e x p o n e n t i a l l y weighted  average of p a s t e a r n i n g s e a r n i n g s Yo. normalized  The  to a r r i v e at the n o r m a l i z e d  r e t u r n on investment  r i s assumed to be  r e t u r n on e x i s t i n g a s s e t s , where Bo  regard  index, u, i s d e f i n e d as the a b s o l u t e  to s i g n ) average of the y e a r - t o - y e a r  Y / B , over t  the p e r i o d 1947  t  Sample d a t a was  of 1958.  The  (i.e.,  The  without  to the sample y e a r .  1954  to 1957.  from the i n d u s t r i a l groups of the Value the end  of t = o.  change i n r a t e of r e t u r n ,  d e r i v e d from 48 food and  c o r p o r a t i o n s f o r the y e a r s  the  i s the a c t u a l net  worth or book v a l u e per share of s t o c k at the end instability  current  The  48 machinery f i r m s were s e l e c t e d  L i n e Investment Survey at  c r i t e r i a employed excluded  f i r m s under  the  following conditions: (1)  i f d a t a on e a r n i n g s available since  (2)  and  d i v i d e n d s were not  1947;  i f abnormal market i n t e r e s t i n the  shares  o c c u r r e d d u r i n g the sample y e a r s ; (3)  i f the f i r m ' s d i v i d e n d f e l l below 2% of i t s book v a l u e i n two p e r i o d 1951  The  normalized  to  or more y e a r s d u r i n g  the  1958.  earnings  and  d i v i d e n d s v a r i a b l e s are d e f i n e d  as f o l l o w s :  If  D{.  =  3D  t  +  (1 - B ) D t _ i ; and  Y£  =  XY  t  +  (1 ~ X)Y*_!.  the smoothing c o n s t a n t s  implied  3 and  X e q u a l one  then the i n v e s t o r i s  to c o n s i d e r o n l y the most r e c e n t v a l u e s of d i v i d e n d s  and  39 earnings, whereas i f 8 and X equal zero, only the normalized earnings and dividends in the prior period is considered.  The following  combinations of 8 and X were tested on the sample data: (1) 8 =  1.0;  (2) 3 = (3) 3 = Values for  X = 1.0  (4) 3  =  .8;  X=  .3  .8;  X=  .7  (5) 3  =  .6;  X=  .5  .8;  X=  .5  (6) 3  =  .6;  X=  .3  and rb were calculated.  The set 3 =  .6 and X = .5  was chosen because in every sample i t provided the highest multiple correlation. ^ 1  I n i t i a l values of .05 and .08 times the year-end  book value were assigned arbitrarily to 1957.  and Y^ respectively for  Each subsequent year's values were determined by the equations  for Dt and Yt except that D£ was always taken as the greater of the actual dividend or two per cent of B . T  The log form of (3-6) was used to yield a linear expression, ln P =  lna5 + aAln D + ailn (1 + br) + 1  + a l n (1 + u) + a lnS 2  The dividend coefficient a  3  A  (3-7)  is included in order that this parameter  can be estimated also and is not forced to equal one. The results are presented in Table 3-5. The dividend coefficient is significantly less than one, ranging from .82 to .93 with the standard error in each sample being very small.  The significance is that heretofore i t has been  assumed that the coefficient equals one, meaning that a doubling of the dividend, other things being equal, would double the price. Gordon suggests several possible explanations for this. 1 0  I b i d . , Appendix B., 50-51.  Investors  TABLE 3-5 R e g r e s s i o n o f P r i c e on E x p o n e n t i a l l y Averaged D i v i d e n d and Growth Rate and on R i s k V a r i a b l e s *  ln P Mean  lna5  l n D* Coeff. Mean  l n (1 + b r ) Mean Coeff.  InS  In (1 + u) Mean Coeff.  Mean  Coeff.  R2  FOOD SAMPLE 1954  3.70 (.57)  2.55 (.15)  .671 (.494)  .83 (.06)  .039 (.021)  11.80 (1.49)  .031 (.024)  -5.49 (1.31)  4.36 (.85)  . .071 (.036)  .904  1955  3.69 (.59)  2.56 (.13)  .672 (.503)  .93 (.05)  .040 (.021)  9.87 (1.29)  .029 (.022)  -4.28 (1.23)  4.41 (.84)  .055 (.032)  .931  1956  3.59 (.54)  2.62 (.14)  .654 (.500)  .92 (.05)  .042 (.020)  8.76 (1.34)  .027 (.020)  -4.44 (1.35)  4.47 (.84  .027 (.032)  .912  1957  3.54 (.52)  2.49 (.12)  .600 (.488)  .83 (.05)  .042 (.018)  9.87 (1.25)  .026 (.010)  -6.21 (1.20)  4.51 (.85)  .065 (.026) .  .931  MACHINERY SAMPLE 1954  3.53 (.47)  2.42 (.22)  .609 • (.412)  1955  3.62 (.51)  2.57 (.19)  1956  3.67 (.55)  1957  3.36 (.52)  .88 (.09)  .047 (.025)  4.16 (1.32)  .035 (.023)  -1.97 (1.61)  3.68 (.73)  .122 (.045)  .799  .619 (.465)  .83 (.07  .045 (.022)  6.07 (1.39)  .035 (.020)  -2.32 (1.48)  3.76 (.75)  .093 (.039)  .869  2.41 (.17)  .553 (.518)  .82 (.05)  .054 (.024)  7.68 (1.12)  .035 (.019)  -1.77 (1.40)  3.87 (.77)  .116 (.034)  .912  2.23 (.17)  .557 (.518)  .85 (.05)  .055 (.021)  3.91 (1.25)  .033 (.017)  -4.52 (1.60)  3.98 (.81)  .149 (.033)  .893  * The s t a n d a r d d e v i a t i o n s and s t a n d a r d e r r o r s o f the means and c o e f f i c i e n t s r e s p e c t i v e l y a r e shown i n p a r e n t h e s e s below t h e mean and c o e f f i c i e n t .  41  may place a lower price relative to dividend on high priced shares than on low priced shares.  Alternatively, there may be some normal  level for the dividend and investors expect low dividends w i l l more likely rise than w i l l high dividends.  Lastly, errors in  measurement may bias the coefficient downwards.H The results do show an improvement in s t a t i s t i c a l significance and in the narrower range of fluctuation for the dividend and growth coefficients.  The fact that the machinery results are poorer than  those for the food sample could be attributed to investors being less confident that the expected rate of growth w i l l be achieved and accordingly placing a lower price on the stock.  Alternatively,  an inadequate measurement for the expected rate of growth for the 12 machinery sample may have been used. The instability of earnings coefficient has a negative sign and exceeds its standard error for every sample, supporting the hypothesis that price varies inversely with earnings instability. Similarly, the size coefficient has the correct sign in every sample and is significant at the five per cent level in six of eight years. To further validate the results, Gordon examined the food sample and determined that there was no significant bias imparted to the coefficient estimates through the existence of correlation between the squares of the residuals and the growth rate and dividend 13 variables.  Although the dependent variable, price, was not  1 1  I b i d . , 46.  1 2  Ibid.  •^Ibid., 47. Heteroscedasticity is discussed in P.G. lloel, Introduction to Mathematical Statistics (New York: John Wiley & Sons, Inc., 1947), p. 106.  42  deflated  t h i s problem of h e t e r o s c e d a s t i c i t y  through the use  a p p a r e n t l y was  avoided  of a l i n e a r e q u a t i o n i n l o g s .  A second s t a t i s t i c a l problem i s t h a t of " f i r m e f f e c t s " which a r i s e s when the same sample i s used i n s u c c e s s i v e over time.  I f the average of  the r e s i d u a l s  s i g n i f i c a n t l y from zero i t i s not the e x i s t e n c e  of one  the model and  peculiar  the v a r i a n c e bias The  of  the  to each f i r m .  to e l i m i n a t e  e s t i m a t e of  to be  excluded  the  to  from  increase  the dependent v a r i a b l e and  possibly  independent v a r i a b l e s . ^ * 1  t e s t on  f i r m e f f e c t s were h i g h l y  the problem and  the a c t u a l and possibly  the  to random d i s t u r b a n c e s but  These f i r m e f f e c t s  of Kuh's a n a l y s i s of v a r i a n c e  determined t h a t  for a firm departs  or more independent v a r i a b l e s  the parameter e s t i m a t e s of use  due  cross-sections  the  food  samples  significant.  In  order  to s e c u r e a c l o s e r correspondence between  e s t i m a t e d v a l u e of  the dependent v a r i a b l e and  also  r e v i s e d parameter e s t i m a t e s a d d i t i o n a l v a r i a b l e s might have  included  i n the model or d i f f e r e n t measurement r u l e s f o r  the  independent v a r i a b l e s used."*""* T h i s model has price variation.  shown g r e a t  The  promise as an attempt to e x p l a i n  c o e f f i c i e n t s f o r growth and  d i v i d e n d s are  highly  s i g n i f i c a n t i n terms of  t h e i r s i z e i n r e l a t i o n to t h e i r s t a n d a r d  and  range of v a r i a t i o n among samples.  show a f a i r l y  small  v a r i a b l e s used have improved the f o r improvement i n the  r e s u l t s but  treatment of r i s k and  there i s s t i l l  The  stock  errors  risk  room  the i n c l u s i o n of  leverage  S e e Edwin Kuh, "The V a l i d i t y of C r o s s - S e c t i o n a l l y E s t i m a t e d B e h a v i o r E q u a t i o n s i n Time S e r i e s A p p l i c a t i o n s , " E c o n o m e t r i c a , XXVII ( A p r i l , 1959), 202. 1 A  1 5  1962),  M . J . Gordon, Review of Economics and 47.  Statistics  (February,  43  c o u l d be u n d e r t a k e n .  Finally,  more a c c u r a t e measurement of  the  variables i s possible. Using  the same sample d a t a , Gordon t e s t e d two a d d i t i o n a l  models, which he  l a b e l e d the "Simlev"  and  " A d l e v " models.  r e s u l t s of these models were r e p o r t e d t o g e t h e r w i t h the model of e q u a t i o n V a l u a t i o n of the  (3-6)  i n h i s book, "The  "Nolev"  Investment, F i n a n c i n g ,  and  Corporation.-^  E.  The  The  Simlev  Model  l o g a r i t h m i c form used to i n c o r p o r a t e f u r t h e r r i s k v a r i a b l e s  was ln P  where  =  ln a  +  aA l n (1+h)  P  =  P r i c e of a common share  D  =  Current  br  =  I — D Growth r a t e of d i v i d e n d = { — ~ ™ }  Y  =  Earnings normalized  =  Earnings  (o/W)  t  + a i In D + a  Q  +  =  Inn + ^ a  l n (1 + o/W)  3  l n u + ay InS  (3-8)  instability  Y {-}  by e x p o n e n t i a l index  W  =  Y — D —  W  smoothing  (leverage free) =  W  t  t  W  + Lt  net worth + net debt per share = o p e r a t i n g per  assets  share,  h  =  Debt-equity  TT  =  Operating  ratio  =  L/W  . asset l i q u i d i t y  index -  7 INV + 5 00A + 3 PE _ - _ _ _ . o  "Myron J . Gordon, The Investment, F i n a n c i n g , and V a l u a t i o n of the C o r p o r a t i o n (Homewood, I l l i n o i s : R i c h a r d D. I r w i n , I n c . , 1962). 1  +  dividend  t  t  5  In (1+br) + a  Y  0/w) where W  + a  2  44  where INV  = normal i n v e n t o r y , 00A  =  other  o p e r a t i n g a s s e t s such as d e f e r r e d charges investments i n o t h e r companies, and and  and  PE = p l a n t  equipment net of d e p r e c i a t i o n a l l of which  a r e a r b i t r a r i l y weighted a c c o r d i n g degree of liquidity."^  y  =  Debt m a t u r i t y  index =  1  +  L  /  1 + where  L  =  (current l i a b i l i t i e s ) (long-term pension  liabilities)  (cash and and  L'  =  the sum  debt) +  +  W  L'/W  (intermediate-term  (liability +  r e s e r v e s such as  ( p r e f e r r e d stock)  government bonds) -  (accounts  of the items c o m p r i s i n g  according  to t h e i r m a t u r i t y  debt) +  receivable)  L, weighted  so t h a t u  18 increases with S  =  Size  The Simlev  -  assets  -  maturity.  current  dividend coefficients  liabilities.  were o n l y s l i g h t l y lower f o r the  model compared to the Nolev model.  dividend versus  The  use  of the  an e x p o n e n t i a l l y smoothed average of the  current dividend  would seem p r e f e r a b l e because of the ease of computation.  The  coefficients,  again  s i m i l a r l y , are h i g h l y s i g n i f i c a n t  b e t t e r f o r the food sample.  performing  The measurement of growth appears  s l i g h t l y b e t t e r than f o r the Nolev model.  The  problems of measuring  r e t u r n on investment i n the machinery i n d u s t r y remain  Ibid.,  74-75,  Ibid.,  80,  162.  163.  growth  unsolved.  TABLE 3-6 Regression  1954  Statistics  1955  f o r the S i m l e v  1956  Model  1957  1958  .  Average  FOOD SAMPLE 2.488 .170  2.458 .146  2.584 .169  2.553 .134  2.668 .158  2.550 .155  .784 .066  .859 .058  .885 .064  .789 .049  .711 .056  .804 .059  10.972 1.870  10.854 1.508  8.946 1.789  10.114 1.849  10.199 1.930  10.217 1.789  -5.619  -4.442  -3.885  -8.562  -3.661  -5.234  2.127  1.812  2.432  2.201  2.440  2.202  -.522 .159  -.416 .151  -.255 .163  -.343 .120  -.175 .47  -.342 .148  A s s e t l i q . , 35  .322 .197  .343 .181  .175 .202  .285 .156  .164 .186  .258 .184  Debt mat., a§  .008 .462  -.019 .412  -.286 .425  -.247 .374  -.572 .478  -.223 .430  S i z e , a-j  .114 .038  .094 .034  .051 .037  .087 .029  .106 .035  .090 .035  .913  .939  .904  .933  .901  .918  Const, term, l n a  D  D i v i d e n d , a^  Growth, a.2  Earn,  inst.,  Leverage, a  a  3  A  Coeff. of det., R  2  TABLE 3-6 CONTINUED  1954  1955 MACHINE  1956  1957  1958  Average  SAMPLE  2.516 .182  2.533 .176  2.205 .163  2.337 .174  3.019 .176  2.522 .174  .845 .079  .813 .067  .858 .052  .919 .058  .824 .056  .852 .062  7.468 1.415  7.619 1.742  8.819 1.495  4.370 1.618  2.197 2.157  6.095 1.685  -3.112 1.711  -3.317 1.576  -.854 1.638  -3.880 1.862  -.625 1.887  -2.358 -„ 1.735  .119 .149  -.202 .169  .026 .153  .255 .173  .162 .172  .072 .163  Asset l i q . , 3 5  -.400 .370  -.337 .352  -.016 .300  -.094 .321  .602 .312  -.049 .331  Debt mat., a^  -.088 .458  .161 .663  -.628 .495  -1.029 .574  .456 .627  -.226 .563  .082 .047  .101 .047  .158 .040  .117 .045  .048 .047  .101 .045  .851  .880  .913  .890  .881  .883  Const, term,  lna  Q  D i v i d e n d , a^  Growth, a„  Earn, i n s t . , a 3  L e v e r a g e , a^  S i z e , aj  Coeff. of det., R  2  47  The  r i s k v a r i a b l e s produced  instability  c o e f f i c i e n t s while having  statistically samples.  significant  I n two  f o r two  the r i g h t  The  earnings  s i g n are not  food samples and  t h r e e machinery  of the machinery samples the c o e f f i c i e n t i s  exceeded by i t s s t a n d a r d e r r o r . fluctuate widely.  The  In a d d i t i o n , the  coefficients  c o e f f i c i e n t i s h i g h e r but l e s s  than f o r the Nolev Model. without  mixed r e s u l t s .  significant  N e v e r t h e l e s s , Gordon b e l i e v e d t h a t  the use of a leverage-free>  e a r n i n g s i n s t a b i l i t y measurement  v e r s u s a l e v e r e d measurement the r e s u l t s would have been p o o r e r . The  l e v e r a g e c o e f f i c i e n t performed  f a i r l y w e l l f o r the food  sample but p o o r l y f o r the machinery sample. samples had attributed  the wrong s i g n .  The  Four of f i v e machinery  l a c k of i n v e r s e c o r r e l a t i o n may  to the c o r r e l a t i o n of l e v e r a g e w i t h e a r n i n g s  d i r e c t l y , and i n d i r e c t l y levered earnings.  through  and  be  instability  e a r n i n g s i n s t a b i l i t y based  More p l a u s i b l y perhaps,  measure p r o f i t a b i l i t y  19  on  the l e v e r a g e v a r i a b l e  may  s e c u r i t y r a t h e r than l e v e r a g e s i n c e secure  20 p r o f i t a b l e f i r m s are more l i k e l y  to use debt  financing.  A better  measurement of r e t u r n than r e t u r n on book v a l u e of e q u i t y may  be  necessary. The o p e r a t i n g a s s e t l i q u i d i t y and debt m a t u r i t y v a r i a b l e s performed  p o o r l y and Gordon concludes  or n o t h i n g to the model. Only All  the s i z e v a r i a b l e performed  the samples had 1 9  Ibid.,  165.  2 0  Ib_id.,  166.  2 1  Ibid.,  the r i g h t s i g n and  166-167.  t h a t they c o n t r i b u t e l i t t l e  21 w e l l as a r i s k  variable.  e i g h t of t e n samples were  48 statistically significant.  This  V7as  an improvement on the Nolev model  for the food sample while the size coefficients  for the machinery  sample were the same. F.  ln P  Q  =  Adlev Model  In a + a ln D + a Q  ±  2  In (1+br) + a^ ln (l+67w) +  + a4 ln (1+h - ih/k) + a$ lnir + ay ln S  (3-9)  A new leverage variable (1+h - ih/k) was substituted for (1+h) where i = interest on debt and k = rate of return required by investors on a share i f a corporation's leverage and retention rates are both equal to zero.  Also, the risk variable u was omitted  because of the poor results in the previous model. The leverage coefficient in the food samples performed exceptionally well.  Not only did a^ increase in size over the five  year period but also the range of variation was reduced while the ratio of size to standard error increased from approximately two in the Simlev model to from three to five.  The machinery sample  values, however, failed to show improvement. G.  In P  =  Eko Model  ln a + a^ InD + a Q  + a a  5  3  In (1+br+vq)  2  ln (1+o/W) + a  4  ln (1+h - ih/k) +  ln IT + ay ln S + aQ ln (1 + q)  (3-10)  The Eko model is an expansion of the Adlev model to include outside equity financing.  The rate of growth in the dividend, br,  TABLE 3-7 Regression 1954  Statistics 1955  f o r t h e A d l e v Model 1956  1957  1958  Average  FOOD SAMPLE 2.454 .158  2.435 .142  2.545 .168  2.541 .126  2.632 .150  2.521 .149  .790 .058  .855 .055  .874 .063  .787 .044  .708 .051  .803 .054  12.401 1.640  11.609 1.449  8.959 1.725  10.442 1.535  10.038 1.685  10.690 1.607  Earn. i n s t . , a3  -5.427 1.872  -3.977 1.678  -3.312 2.258  -8.675 1.988  -4.000 2.251  -5.078 2.009  Leverage, a  -1.062 .195  -.854 .200  -.617 .223  -.834 .171  -.852 .222  -.844 .202  -.272 .161  .342 .155  .222 .182  .283 .131  .245 .154  .273 .157  .107 .034  .091 .032  .056 .036  .085 .026  .113 .033  .090 .032  .925  .942  .904  .941  .911  .925  Const, term, l n a  Q  Dividend,  Growth, a  2  A  A s s e t l i q . , a^  S i z e , aj  C o e f f . of d e t . , R  2  1  TABLE 3-7 CONTINUED  1  1954  1955  1956  Average  1957  1958  2.238 .166  2.330 .181  2.975 .178  2.524 .176  .851 .052  .875 .059  .832 .056  .843 .062  MACHINERY SAMPLE Const, term,  lna  2.556 .179  c  .844 .077  D i v i d e n d , a-^  Grov;th, &2  2.523 .178 .815 .066  7.207 1.375  7.442 1.705  8.846 1.517  3.758 1.638  2.176 2.135  5.886 1.674  -3.163 1.542  -3.351 1.542  -.274 1.650  -4.317 1.992  -.788 1.995  -2.379 1.744  Leverage, a A  -.337 .325  -.353 .276  -.096 .274  -.118 .295  .428 .379  -.095 .310  Asset l i q . ,  -.337 .325  -.250 .314  .061 .283  .052 .331  .666 .334  .038 .317  .076 .045  .106 .044  .142 .039  .131 .046  .064 .046  .104 .044  .857  .878  .909  .882  .879  .881  Earn, i n s t . ,  a  3  o  S i z e , ay  Coeff. of det., R  2  51 is increased by the equity accretion rate vq and the stock financing variable q is added where v  =  fraction of funds invested by new shareholders during a period that accrues to the equity of existing shareholders at the start of the period  q  =  funds raised through outside equity financing as a fraction of the net worth per share.  The outside financing rate was measured by N  where  t-1  W  t-1  N  =  number of shares outstanding  W  = book value or net worth per share.  For the food sample, the outside equity financing rate coefficient ag has the right sign in every year but is not statistically significant in any year and fluctuates widely. of growth variable coefficient a  2  Moreover, the accuracy  has been reduced by the addition  of vq resulting from correlation between vq and q.  This correlation,  in turn, has created correlation between the outside financing rate 22 and share price. The'machinery sample again proved to be even worse.  The  outside equity financing coefficient has the wrong sign in two years and did not exceed its standard error.  The growth coefficient  was reduced in every year and the ratio to its standard error was reduced in some years. 2 2  I b i d . , 172.  TABLE 3-8 Regression S t a t i s t i c s 1954  1955 ,  f o r the E k o M o d e l t  1956  1957  1958  Average  FOOD SAMPLE Const, term, l n a  c  Dividend, a^  Growth, a  2  Earn, i n s t . , a3  L e v e r a g e , a^  2.476 .158  2.403 .136  2.663 .185  2.611 .130  2.755 .144  2.582 .151  .803 .059  .883 .053  .892 .070  .764 .046  .672 .050  .803 .056  11.876 1.615  11.515 1.362  7.143 1.875  8.741 1.404  9.248 1.479  9.705 -1.547  -5.164 1.897  -2.953 1.671  -3.117 2.579  -7.850 2.031  -4.051 2.188  -4.627 2.070  -.998 .195  -.588 .217  -.466 .256  -.759 .178  -.781 .216  -.712 .212  A s s e t l i q . , a$  .297 .164  .279 , .156  .204 .206  .306 .138  .300 .149  .277 .163  S i z e , a-j  .103 .037  .086 .031  .040 .040  .083 .027  .095 .033  .081 .034  -3.316 1.967  -5.221 1.239  -2.218 1.816  -1.246 1.384  -.649 1.153  -2.530 1.512  .924  .947  .886  .938  .919  .923  S t o c k i s s u e , ag  Coeff. of det., R  2  TABLE 3-8 1954  1955  CONTINUED 1956  1957  1958  Average'  MACHINERY SAMPLE  2.660 .195  2.543 .182  2.312 .156  2.401 .173  3.012 .178  2.586 .177  .876 .084  .853 .067  .850 .053  .860 .058  .836 .062  .855 .065  4.917 1.227  5.617 1.412  7.744 1.212  3.738 1.216  .742 1.542  4.552 1.322  -2.352 1.685  -2.858 1.600  -.186 1.576  -4.566 1.941  -.588 2.016  -2.110 1.764  .277 .281  -.274 .297  -.038 .292  -.142 .318  .322 .408  .029 .319  -.218 .358  -.169 .324  .069 .274  .029 .322  .646 .341  .051 .324  S i z e , ay  .064 .050  .111 .046  .135 .037  .116 .046  .062 .048  .098 .045  S t o c k i s s u e , ag  .382 .9S3  -.709 .768  -1.539 .716  -.814 .937  .891 1.103  -.358 .901  .830  .872  .920  .892  .879  .879  Const, term, l n a  D  D i v i d e n d , a-^  Growth, &2  Earn. i n s t . , a  Leverage, a  Asset l i q . ,  3  4  a5  Coeff. of det., R  2  54 H.  Conclusions  from E m p i r i c a l F i n d i n g s  Of the t h r e e models t e s t e d , the A d l e v model performed b e s t particularly  f o r the food sample.  While t h e o r e t i c a l l y  outside  e q u i t y f i n a n c i n g s h o u l d be r e c o g n i z e d , the Eko model d i d not  perform  as w e l l p o s s i b l y because of the d i f f i c u l t y i s measuring t h a t v a r i a b l e . The  f o o d sample r e s u l t s  d i v i d e n d , growth and  f o r the Adlev model show t h a t the  l e v e r a g e v a r i a b l e s are h i g h l y s i g n i f i c a n t  have a s m a l l range of f l u c t u a t i o n . the r i g h t  s i g n and  The  remaining  exceed t h e i r standard  v a r i a b l e s have  e r r o r i n every  year.  Moreover, i n terms of s t a t i s t i c a l s i g n i f i c a n c e a t the f i v e cent l e v e l , the s i z e earnings  coefficient  i s significant  i n s t a b i l i t y i n t h r e e y e a r s and  i n two y e a r s . coefficient  operating asset l i q u i d i t y  With b e t t e r measurement, the e a r n i n g s i n s t a b i l i t y  c o u l d be expected  modestly w e l l .  Only  i n every year and  it  to be  stronger.  the d i v i d e n d c o e f f i c i e n t  s t a b l e over  i s not s t a t i s t i c a l l y s i g n i f i c a n t size  i n two y e a r s .  i n the l a s t y e a r .  coefficients  lb±d.  ,  175.  the  sign i n  i n only three years.  have the wrong s i g n  t h a t the c o e f f i c i e n t s  r e l i a b l e e s t i m a t e s of the parameters."23  23  growth r a t e  While  have the r i g h t  asset l i q u i d i t y c o e f f i c i e n t s  Gordon concludes  The  t h r e e y e a r s , then drops u n t i l  every year they each are s t a t i s t i c a l l y s i g n i f i c a n t l e v e r a g e and  only  i s highly significant  the f i v e year p e r i o d .  performs w e l l i n the f i r s t  e a r n i n g s i n s t a b i l i t y and  The  per  i n four years,  The machinery sample under the A d l e v model performed  coefficient  and  "do not  provide  CHAPTER IV FORMULATION AND TESTING OF THE MODEL  T h i s chapter present- the model used f o r t e s t i n g and s p e c i f i e s the r e g r e s s i o n v a r i a b l e s and the a l t e r n a t i v e measures used of the v a r i a b l e s .  The d a t a and methodology a l s o w i l l be d e s c r i b e d .  A.  1.  f o r some  S e l e c t i o n of V a r i a b l e s  B a s i c Model The b a s i c e m p i r i c a l model o f e q u a t i o n  (2-8)expresses  the dependent  v a r i a b l e , p r i c e , i n the f o l l o w i n g f u n c t i o n a l r e l a t i o n s h i p t o the independent  P  v a r i a b l e s , d i v i d e n d s and the growth r a t e :  o  =  D  o ~  (1 + b r ) l a  By r e p l a c i n g l / a w i t h ao and a^ w i t h a Q  coefficient  which i s allowed  2  and by i n c l u d i n g a d i v i d e n d  to vary i n s t e a d of being constrained  to e q u a l one, the above e q u a t i o n can be m o d i f i e d t o p  o  =  %  D  o  a  For t e s t i n g purposes, InP =  2.  In a  i  C  1 +  b r  )°  t " )  2  4  1  the l o g a r i t h m i c form i s used: 0  a  + ai ln D +  2  l n (1 + br)  (4-2)  Risk Variables R i s k i s d e f i n e d i n t h i s study as the v a r i a b i l i t y o f the c o r p o r a t i o n ' s  expected  r e t u r n s from investment,  due t o u n f o r e s e e n  circumstances.  These f u t u r e r e t u r n s a r e random v a r i a b l e s w i t h a j o i n t  55  probability  56 distribution.  The  i n v e s t o r i s assumed to be a b l e to determine  s u b j e c t i v e l y 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 v a r i a n c e or d i s p e r s i o n about i t s expected If  value.''"  the s t a n d a r d d e v i a t i o n of expected  index of r i s k ,  the p r i c e of a share over  returns i s considered  an  time other t h i n g s b e i n g  e q u a l , w i l l be a d e c r e a s i n g f u n c t i o n of the p r o b a b i l i t y  distribution's  dispersion. a.  Business  variability of  r i s k can be  of a f i r m ' s o p e r a t i n g income.  c o n s i d e r e d as  I t a r i s e s from the  i t s f u t u r e investments.  p r o j e c t s w i l l have d i f f e r e n t p r o b a b i l i t y  return.  The  the a b i l i t y and  Business  the f i r m ' s e x i s t i n g a s s e t s and  investment of  Risk:  interest  nature  Different  distribtuions  r i s k of a d e c l i n e i n o p e r a t i n g income w i l l  of the c o r p o r a t i o n to pay  the  charges  endanger  on i t s debt  d i v i d e n d s to s h a r e h o l d e r s , b.  F i n a n c i a l Risk:  F i n a n c i a l r i s k a r i s e s from the method of  f i n a n c i n g which causes v o l a t i l i t y or a d d i t i o n a l f l u c t u a t i o n s i n the s h a r e h o l d e r ' s e a r n i n g s .  The  use o f debt  f i n a n c i n g may  increase  the r a t e of r e t u r n to s h a r e h o l d e r s but i t w i l l a l s o i n c r e a s e the variability earnings to its  of the expected  to the s h a r e h o l d e r s  return. i t has,  The v a r i a b i l i t y  of  expected  and w i l l be argued, i s r e l a t e d  the f i n a n c i a l s t r u c t u r e of the c o r p o r a t i o n b e s i d e s the n a t u r e  of  investments. c.  Size:  An  s i z e of a f i r m and  inverse relationship the s t a b i l i t y  s i z e should be r e f l e c t e d  G. Illinois:  i s p o s t u l a t e d between the  of i t s o p e r a t i n g income.  While  i n the measurement of b u s i n e s s r i s k  by  David Q u i r i n , The C a p i t a l E x p e n d i t u r e D e c i s i o n (Homewood, R i c h a r d D. I r w i n , I n c . , 1967), Ch. X.  57 reducing the variability of expected earnings, a size variable is included nevertheless on the proposition that investors w i l l consider, independently, the variability of income and size in arriving at their measure of risk.  Two companies with historically  identical standard deviations of operating income, for example, could, i t is proposed, be in different risk classes i f they differed in size, other things remaining the same.  It is conceivable that the  shares of large corporations may command a premium because of greater investor interest and greater marketability of the shares. 3.  Complete Model If we let u  =  index of business risk  h  =  index of financial risk  S  =  index of size  then our complete empirical model is 2  InP  =  ln a + 0  ctAln  InD + a In (1 + br) + a-jln (1 + u) + 2  (1 + h) + a lnS 5  B.  (4-3)  Data  The historical data used in this study is provided by Standard and Poor's Compustat service.  The Compustat tape which was  used here contains annual fiiiancial information for approximately nine hundred industrial companies for the period 1946 to 1965. I t should be noted that since ln 1 = o, the use of ln (1+u) and ln (1+h) allows the risk variables to remain positive provided u and h are positive. 2  58 T h i s i n f o r m a t i o n was  g a t h e r e d from company r e c o r d s , from  S e c u r i t i e s Exchange Commission and from o u t s i d e s o u r c e s . the  For the study a t hand, d a t a were taken  s e v e n t y - s e v e n companies i n the machinery  United States.  More w i l l be s a i d on t h i s  C.  1.  Some of  d a t a has been a d j u s t e d by Standard and Poor's f o r g r e a t e r  c o m p a r a b i l i t y among f i r m s . for  the  i n d u s t r y i n the  later.  Measurement of the V a r i a b l e s  Dependent V a r i a b l e - P r i c e  (P)  The Compustat tape p r o v i d e s annual a b s o l u t e h i g h , low and p r i c e s f o r the companies l i s t e d bid  on n a t i o n a l s t o c k exchanges and  p r i c e s f o r over-the-counter i s s u e s .  close the  These p r i c e s a r e g i v e n on  a c a l e n d a r - y e a r b a s i s , r e g a r d l e s s of the f i s c a l year-end and have been a d j u s t e d by Standard and Poor's f o r s t o c k s p l i t s  and s t o c k  d i v i d e n d s except when these occur between a f i s c a l year-end the  and  c a l e n d a r year-end. For  the purpose of t h i s s t u d y , the year-end c l o s i n g p r i c e  been chosen.  has  Arguments can be and have been o f f e r e d f o r o t h e r measures.  Y e a r l y averages o f d a i l y average p r i c e s p r o b a b l y would r e c e i v e a l o t of  support but these are not r e a d i l y a v a i l a b l e .  There i s a l s o  3 support f o r u s i n g the mean of the h i g h and low p r i c e s d u r i n g the y e a r . T h i s i s r e a d i l y a v a i l a b l e on the Compustat tape. attempt  Our model i s an  to d e s c r i b e those phenomena which determined  individual  share  p r i c e s a t a s p e c i f i c p o i n t i n time r a t h e r than average p r i c e s over  -^Charles E. Edwards and James G. H i l t o n , "A Note on the H i g h Low P r i c e Average as an E s t i m a t o r of Annual Average Stock P r i c e s , " J o u r n a l of F i n a n c e , XXI (March, 1966), 112-115.  59 the  year.  2.  Independent V a r i a b l e s a.  Dividends  (D).  The  current per-share  y e a r of the c r o s s - s e c t i o n i s used. splits  and  The  data i s adjusted  s t o c k d i v i d e n d s on the Compustat tape and  p r e f e r r e d s t o c k payments i n l i e u of cash and The  d i v i d e n d i n the f o r stock  excludes  a l s o excludes  spin-offs.  c u r r e n t d i v i d e n d i s chosen as the b e s t measure of normal  d i v i d e n d s on the ground t h a t w h i l e i t i s t r u e t h a t f l u c t u a t i o n s i n dividends occur,  t h e r e i s evidence  degree of s t a b i l i t y earnings.  4  i n dividends  Thus, " n o r m a l i z e d "  d i v i d e n d s ) would not be b.  t h a t c o r p o r a t i o n s seek a  i n the f a c e of r i s i n g or  dividends  falling  (some average of p a s t  appropriate.  Growth Rate ( b r ) .  Growth r a t e s can be c a l c u l a t e d u s i n g  v a r i o u s measures i n c l u d i n g d o l l a r s a l e s , net p r o f i t invested, per-share  fair  on t o t a l  capital  earnings, t o t a l assets, fixed assets, t o t a l  income or r e t a i n e d e a r n i n g s . r a t i o of r e t a i n e d e a r n i n g s "growth" v a r i a b l e s . - *  The  Increase  to p e r - s h a r e  i n p r i c e or changes i n the earnings  are a l s o p o s s i b l e  time p e r i o d from which r e s u l t s might  taken f o r the purposes of p r e d i c t i o n might be an average of s e v e r a l p r e v i o u s  net  be  the p r e v i o u s year  or  years.  I n s o f a r as Gordon's model i s concerned w i t h I t s a t t e n t i o n to the growth i n d i v i d e n d s , we  w i l l d e f i n e growth as the product  of  A  Cf. Chapter I . A l s o see J . Fred Weston and Eugene F. Brigham, M a n a g e r i a l F i n a n c e , 2nd ed. (New York: H o l t , R i n e h a r t and Winston, 1966), pp. 460-464. J . Fred Weston, The Scope and Methodology of Finance (Englewood C l i f f s , N.J: P r e n t i c e - H a l l ^ Toe.",T9oTTJ','" pp. v x f r = T x y Benjamin Graham, David L. Dodd and Sidney C o t t l e , S e c u r i t y A n a l y s i s : P r i n c i p l e s and Technique, 4th ed. (New York: McGraw-Hill Book Company, I n c . , 1962), p. 235. 5  60 the r a t e o f e a r n i n g s r e t e n t i o n , b and t h e p r o f i t a b i l i t y o f investment, r;  t h a t i s , growth,  g The Y,  =  br  r e t e n t i o n r a t e i s measured by t h e f r a c t i o n o f "normal r e t a i n e d i n each  b  =  earnings,"  period:  M  Y  E q u i v a l e n t l y , t h e r e t e n t i o n r a t e , b = 1 - r where - = the payout rate. The  p r o f i t a b i l i t y o f investment, r , can be measured i n such  ways as (1)  r e t u r n on common e q u i t y , E :  (2)  r e t u r n on t o t a l r  =  Y  =  shareholders'  Y  E + PS  W  where PS = p r e f e r r e d (3)  r e t u r n on t o t a l Y E+PS+D  r  funds, W:  =  stock;  capital,  C:  Y C '  where D = n o n c u r r e n t debt; and (A)  r e t u r n on t o t a l r  The  =  a s s e t s , A,  Y ~ . A  r e t u r n on common e q u i t y o r n e t worth measures have a s t r o n g  a p p e a l from the v i e w p o i n t o f the s h a r e h o l d e r s , interested  who as owners a r e  i n the r e t u r n on the money they have i n v e s t e d  i n the p a s t .  61  It  i s a measure t h a t can be used to compare d i f f e r e n t  investments.  A l t e r n a t i v e l y , the r e t u r n on t o t a l c a p i t a l or t o t a l measures r e c e i v e s t r o n g support The  a f t e r - t a x earnings  expense, i n o r d e r i s minimized.  should  from investment a n a l y s t s . ^  be a d j u s t e d  Hence these measures p r o v i d e  power of f i r m s w i t h d i f f e r e n t  incremental  to i n c l u d e i n t e r e s t  t h a t the e f f e c t of changes i n c a p i t a l s t r u c t u r e  measure of c o r p o r a t i o n performance and  Superior  to any  a c o n s i s t e n t long-term  a comparison of the  average r a t e s of r e t u r n i s the  r a t e of r e t u r n .  Without access  r e c o r d s , however, i t would be extremely d i f f i c u l t Our  tacit  earning  capital structures.  of these  or m a r g i n a l  t h i s measure.  assets  to c o r p o r a t i o n  to even approximate  assumption i s t h a t i n v e s t o r s use  average r a t e of r e t u r n based upon the h i s t o r i c  an  c o s t or book v a l u e  of a s s e t s . For  t e s t i n g purposes both the r e t u r n on t o t a l a s s e t s and  worth were chosen. income and dividends.  The  f i r s t was  the second i n c o n j u n c t i o n w i t h net  A  Y _  br  =  Since per-share  a first  regressed  Cf.  income b e f o r e  respectively.  earnings  approximation,  A  n  - — — W  s u b s t a n t i a l f l u c t u a t i o n s i t was As  to o p e r a t i n g  Hence, the measurements of growth a r e :  Y and  used i n c o n j u n c t i o n  net  are s u b j e c t t o , i n some i n s t a n c e s , felt  earnings  per  on time f o r the p e r i o d 1956  Graham, Dodd and  necessary share to 1965  to n o r m a l i z e  earnings.  f o r each c o r p o r a t i o n were using  the  linear  C o t t l e , S e c u r i t y A n a l y s i s , pp.  462-463.  62 function,  Y  =  t  a +  bT  The n o r m a l i z e d e a r n i n g s , Y, each y e a r o b t a i n e d from The  second  assumption  the t r e n d  s t e p was  and  line.  to e l i m i n a t e any b i a s a r i s i n g  of a linear, t r e n d .  a l s o were f i t t e d  The  V7as d e f i n e d as the p r e d i c t e d v a l u e i n  In a d d i t i o n , two  from  l o g a r i t h m i c equations  f o r each company:  ln  Y  Y  = a + b (ln T).  t  t  - a +  bT  computed t r e n d v a l u e f o r each f i r m ' s n o r m a l i z e d e a r n i n g s  taken from whichever e q u a t i o n of the t h r e e p r o v i d e d the b e s t chosen on the b a s i s of m i n i m i z i n g Z(Y - Y ) ; 2  the squared  t h a t i s , the sum  d e v i a t i o n s between a c t u a l e a r n i n g s and  estimated  B e s i d e s e l i m i n a t i n g a l i n e a r b i a s , the use of these l o g a r i t h m i c f u n c t i o n s i s supported The  was fit, of earnings.^  two  by the n a t u r e of e a r n i n g s growth.  a r i t h m e t i c s t r a i g h t l i n e r e p r e s e n t s a c o n s t a n t amount of growth  per y e a r , whereas the l o g a r i t h m i c s t r a i g h t l i n e s r e p r e s e n t percentage The Our  the  r a t e of growth. s t a r t i n g p o i n t of the e a r n i n g s t r e n d i s a l s o  i n i t i a l assumption  determine  constant  was  f u t u r e expected  important.  t h a t i n v e s t o r s use h i s t o r i c a l d a t a to v a l u e s of d i v i d e n d s , growth and  A c c o r d i n g l y , the t r e n d e q u a t i o n was  risk.  changed to cover a p e r i o d up  ^ P r o p e r l y , goodness of f i t i s based upon the t r e n d l i n e w i t h the lowest s t a n d a r d e r r o r of e s t i m a t e . Since _ „ JZ (y - Yj 2 v n - k (n - k) was the same f o r each t r e n d l i n e the use of z(Y - Y) i s equivalent. T h i s t o p i c i s t r e a t e d at l e n g t h i n Mordecai E z e k i e l and K a r l A. Fox, Methods of C o r r e l a t i o n and R e g r e s s i o n A n a l y s i s . 3rd ed. (New York: John Wiley" & Sons, I n c . , 1959), Ch. V I I .  63 to and  i n c l u d i n g the c r o s s - s e c t i o n y e a r . The  actual period  covered should  the average b u s i n e s s c y c l e but f i n a n c i a l information deplete trend  the sample.  equation;  s e c t i o n year.  on  be  s h o r t enough so  that l a c k of  the Compustat tape vzould not  A ten y e a r p e r i o d was  The  seriously  chosen f o r each y e a r ' s  t h a t i s , from t - 9 to t where t was  the  cross-  machinery sample, o r i g i n a l l y c o n s i s t i n g of  s e v e n t y - s e v e n f i r m s , had  been reduced to f i t y - s i x  the 1956-1965 t r e n d e q u a t i o n s were used. ten y e a r s ,  long enough to encompass  By  companies when  taking  the p e r i o d back  an a d d i t i o n a l t h i r t e e n companies were d e l e t e d  leaving  a sample of f o r t y - t h r e e f i r m s . Both o p e r a t i n g and  net  income (income b e f o r e  income or e a r n i n g s b e f o r e  were t e s t e d  taxes and  p r e f e r r e d and  for t h e i r explanatory  f i x e d charges)  common  dividends  powers.  c.  Business Risk  (u):  The  s e l e c t i o n of a sample c o n s i s t i n g of one  i n d u s t r y or more  a c c u r a t e l y , a group of c l o s e l y r e l a t e d i n d u s t r i e s i s an attempt c o n t r o l f o r the v a r i a t i o n i n share p r i c e due  to b u s i n e s s  to  risk.  A pure "homogeneous r i s k c l a s s " i s however, u n l i k e l y to e x i s t where the number of companies i n any comprising with l i k e  an i n d u s t r y  c l a s s exceeds one.  sample, w h i l e m a n u f a c t u r i n g s i m i l a r p r o d u c t s  technology r e t a i n d i f f e r e n c e s as a r e s u l t among o t h e r  of s e r v i n g d i f f e r e n t g e o g r a p h i c a l varying  Companies  areas or market segments,  things,  maintaining  degrees of p r o d u c t d i v e r s i f i c a t i o n or q u a l i t y , or having  different abilities  of management.  B u s i n e s s or o p e r a t i n g operating  r i s k r e l a t e s to the v a r i a b i l i t y  income or e a r n i n g s b e f o r e  i n t e r e s t and  taxes.  of  A useful  64 measure o f t h i s r i s k , u, i s t h e s t a n d a r d d e v i a t i o n " of income, Y; that i s ,  u  =  f (0y)  As u i n c r e a s e s , t h e d e s i r a b i l i t y  t o most i n v e s t o r s of t h e share o f  common s t o c k t o which u a t t a c h e s d e c l i n e s and t h e p r i c e , t h i n g s r e m a i n i n g unchanged, Two measures  and  (1)  U  (2)  U  where  Y W  w i l l be r e d u c e d .  of b u s i n e s s r i s k were t e s t e d :  Y w average income over a t e n - y e a r p e r i o d net  worth.  The f i r s t measure, c a l l e d used i n t e s t s by Barges  the " c o e f f i c i e n t of v a r i a t i o n , " has been and B e n i s h a y  x u  and f r e q u e n t l y i s recommended  as a measure o f r e l a t i v e r i s k by f i n a n c i a l w r i t e r s . " ^ measure was used by Gordon P  other  (see Chapter I I I ) .  The second  Both a r e d e s i g n e d t o f  A l t e r n a t i v e measures of d i s p e r s i o n such as skewness might a l s o have been used but a r e beyond the scope o f t h i s s t u d y . C f . Fred D. A r d i t t i , " R i s k and t h e Required Return on E q u i t y , " J o u r n a l o f F i n a n c e , XXII (March, 1967), 19-36.  9 of  A l e x a n d e r Barges, The E f f e c t o f C a p i t a l S t r u c t u r e on t h e Cost C a p i t a l (Englewood C l i f f s , New York: P r e n t i c e - H a l l , I n c . , 1963).  H a s k e l B e n i s h a y , " V a r i a b i l i t y i n E a r n i n g s - P r i c e R a t i o s of C o r p o r a t e E q u i t i e s , " American Economic Review, L I (March, 1961), 81-94. 1 0  Hc.F. Stephen H. A r c h e r and C h a r l e s A. D'Ambrosio, B u s i n e s s F i n a n c e : Theory and Management (New York: The M a c M i l l a n Company; London: C o l l i e r - M a c M i l l a n L i m i t e d ; 1966), and James C. Van Home, F i n a n c i a l Management and P o l i c y (Englewood C l i f f s , New J e r s e y : P r e n t i c e - H a l l I n c . , 1968).  65  account f o r t h e i n f l u e n c e o f f i r m s i z e on t h e measurement o f business r i s k . the  I n t h e homogeneous r i s k c l a s s r e f e r r e d  t o above,  b u s i n e s s r i s k o f each f i r m would be t h e same. d.  F i n a n c i a l R i s k : (h)  As a proxy f o r f i n a n c i a l r i s k , we chose two measures r e f l e c t i n g the to  and  long-term debt p a y i n g a b i l i t y of a f i r m ; h o l d debt i n i t s c a p i t a l  t h a t i s , the c a p a c i t y  structure:  (1)  times - f i x e d charges - earned = o p e r a t i n g income f i x e a charges  (2)  debt-equity r a t i o .  Unfortunately  the d a t a on f i x e d charges was u n a v a i l a b l e on t h e  Compustat tape f o r most f i r m s i n the machinery the  sample so g r e a t l y  sample and d e p l e t e d  t h a t the f i r s t measurement was abandoned.  With r e s p e c t t o t h e d e b t - e q u i t y r a t i o , long-term debt was chosen i n t h e numerator.  P r e v i o u s t e s t s by M o d i g l i a n i and M i l l e r  i n c l u d e d p r e f e r r e d s t o c k s i n c e both p r e f e r r e d s h a r e h o l d e r s and bondholders r e c e i v e payment b e f o r e r e s i d u a l e a r n i n g s may be distributed  to common s h a r e h o l d e r s .  to  include current l i a b i l i t i e s ,  in  the numerator e.  long-term debt and p r e f e r r e d  Size (S): to measure s i z e , such as t o t a l book v a l u e  common e q u i t y , t o t a l o f n e t p l a n t account and working  assets l e s s current l i a b i l i t i e s , market v a l u e of e q u i t y . of  dollars.  calculate.  shares  of the l e v e r a g e v a r i a b l e .  Numerous ways e x i s t of  A s e p a r a t e t e s t was conducted  capital,  book v a l u e of i n v e s t e d c a p i t a l and  T h i s study used  t o t a l assets i n m i l l i o n s  The c h o i c e was based on expediency - i t i s simple to There a r e no apparent  one measure over a n o t h e r .  t h e o r e t i c a l grounds  to favor  66 D. Methodology  The machinery i n d u s t r y f o r the p e r i o d 1956 s e l e c t e d f o r t e s t i n g purposes. work which covered the p e r i o d utility  i n d u s t r y and  others,  1954  to 1958.  classified  a v a i l a b l e on  general I t was  Poor's as f o l l o w s :  been  chosen  There was  a  s e v e n t y - s e v e n , whose The  firms  are  machine t o o l s ( 8 ) , handling  (8), i n d u s t r i a l  ( 6 ) , s p e c i a l t y ( 1 6 ) , steam g e n e r a t i n g  (11), (4)  (5).  n e c e s s a r y to s e t c e r t a i n c r i t e r i a f o r a f i r m ' s i n c l u s i o n  i n the sample. information  would r e s u l t .  ( 6 ) , c o n s t r u c t i o n and m a t e r i a l s  industrial  electrical  A ten year p e r i o d was  the Compustat tape.  m e t a l f a b r i c a t i n g (13), o i l w e l l and  the  a t t e n t i o n has  l a r g e number of companies, t o t a l l i n g  by Standard and  agricultural  Unlike  relatively l i t t l e  i n the hope t h a t a d d i t i o n a l i n f o r m a t i o n  f i n a n c i a l d a t a was  was  Thus t h i s study extends Gordon's  devoted to the machinery i n d u s t r y .  sufficiently  to 1965  To a f f o r d c o m p a r a b i l i t y  on a company had  y e a r p e r i o d o t h e r w i s e the  to be  f i r m was  between y e a r s ,  financial  a v a i l a b l e i n each y e a r of the deleted  from the sample.  ten  Moreover,  s i n c e l o g a r i t h i m s were used, the v a r i a b l e s f o r each f i r m were r e q u i r e d to have v a l u e s The  greater  than  zero.  m u l t i p l e r e g r e s s i o n a n a l y s i s was  performed by  the U.B.C.  TRIP program at the Computing C e n t r e .  A separate routine  required  of the e a r n i n g s t r e n d  f o r the l e a s t - s q u a r e s  fitting  was regression.  CHAPTER V RESULTS AND  In the i n i t i a l select fit  t e s t s , the most important o b j e c t i v e s were t o  the a p p r o p r i a t e  trend  period  f o r normalized earnings.  defined  CONCLUSIONS  and t o choose t h e c u r v e o f b e s t  For t h i s purpose, the v a r i a b l e s were  as f o l l o w s : P  =  year-end p r i c e i n c r o s s - s e c t i o n  D  =  current  br  =  growth r a t e of d i v i d e n d  =  Y - D A  Y  =  normalized earnings i n c r o s s - s e c t i o n  A  =  t o t a l assets  u  =  index of b u s i n e s s  dividend  year  i n cross-section  year  year  a t book v a l u e risk  = C y / ? ( c o e f f i c i e n t of v a r i a t i o n ) Cy Y  s t a n d a r d d e v i a t i o n o f n e t income about trend  =  =  average o f n e t income from t-9 to t where t i s cross-section  h  line  year  =  index of f i n a n c i a l  =  L/E  L  =  long-term debt  E  =  book v a l u e o f common  S  =  s i z e o f the f i r m i n m i l l i o n s of d o l l a r s o f t o t a l  67  risk  equity asset  68 A. Regression Results The sample estimates of the regression statistics are shown in Table 5-1.  The coefficient values in each year represent the  influence on price that investors attribute to each variable. The dividend coefficient ranges in magnitude from .5671 in 1965 to .8463 in 1956 and averages ,7088.  This means that for a cross-section  of firms in 1956, for example, a 100% change in the dividend of a stock is associated with an 85% change in the price of that stock. The coefficients of the other variables can be similarly interpreted although the particular logarithmic form of the growth, business risk and financial risk variables complicates the analysis. The dividend, growth and size coefficients performed well.  statistically  The dividend coefficient was "highly significant,"  that i s , statistically significant at the 1% level and had the expected sign indicated by the theory in every year. The growth coefficient has the expected sign in every year and is highly significant in eight of the ten years while the size coefficient is significant at the 5% level in eight of ten years including six years where i t is significant at the 1% level and has the correct sign in a l l years. at the 10% level.  In one year, size is significant  The growth coefficient fluctuates considerably,  however, from a high of 10.2215 in 1956 to a low of 1.9181 in 1965. The drastic decline after 1962 both in magnitude and ratio of coefficient to its standard error may indicate a disenchantment on the part of investors with "growth" in the stock market.  TABLE 5-1 R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a 0:3 l n (1 +  1956  1957  1958  1959  u) +  0 4  1960  Q  + ct^ l n D + a  l n (1+h) + a  1961  5  2  I n (1 + b r ) +  InS  1962  1963  1964  1965  2.4222 .2078  2.4886 .1006  2.5068 .1296  2.6916 .1970  Average  MACHINERY SAMPLE, n=43 C o n s t a n t , In a  D  D i v i d e n d , a-^  Growth, a2  1.8561 .1601  2.3755 .1803  2.2224 .1880  2.3696 .1820  .8051 .0577  a  .7065 .0773  10.2215 2.2990  3  7.7367 1.8480  a  7.3805 1.6293  a  8.2272 2.0462  Financial Risk, 0 4 S.E  .7635 .3106  S i z e , ct^  .0590 .0428  .1165 .0362  R2  .8106  .8613  b  .2648 .2509  .5364 .2338 a  .4788 .1732  .1163 .0350  a  -.0206 .2921  b  .1098 .03S0  a  .8082  .8957 a b c  .6706 .0915  a  .8240 .0583  a  a  7.2825 2.1055  a  8.5411 2.0838  a  3.3269 1.3872  a  a  9.7356 2.1529  a  .3525 .2599  a  a  •8137 .0736  .1159 .1406  .6235 .0877  .5831 .0788  a  .0786 .1508  a  a  .8463 .0867  Business Risk, 0 3  2.6198 .1962  2.0085 .1946  .350?c .1791 -.0360 .2920  a  .1525 .0396 .7788  a  .1660 .1456  .0420 .2757  .4536 .1149  .1176 .3063  -.3426 .2845  -.2275 .1678  .0768° .0417  .0976 .0384  .7283  .7536  s i g n i f i c a n t a t t h e .01 l e v e l s i g n i f i c a n t a t the .05 l e v e l s i g n i f i c a n t a t t h e .10 l e v e l  b  .1090 .0213 .8983  .6480 .0752  a  2.5519 2.1234 .3053 .1470  b  -.3103 .1909 b  .1381 .0278 .7995  a  ,5671 .0934  2.3561 .1736 a  .7088 .0780  1.9181 2.3181  5.9222 1.9993  .0346 .1861  .2378 .1773  -.2554 . .2441  .0490 .2573  ,1562 .0352 .7027  a  .1132 .0356 .8037  70 The business and financial risk coefficients performed poorly. Neither coefficient was significant at the 5% level and both, at times, carried unexpected signs.  The coefficient for business  risk was positive in every year while the coefficient for financial risk was negative in six years. The coefficient of determination, R , which is the ratio of explained variance to total variance of the dependent variable is considered high for cross-sectional regression.  It averages .8037  over the ten-year period, with a range of .7027 to .8957. In the previous test net income before preferred and common dividends was used to normalize earnings and normal earnings in turn was used in the measurement of growth.  Since net income should  be associated with the rate of return on net worth, W, rather than on total assets, the growth variable was changed to Y where  W  W  W = book value of common equity and preferred stock.  Table 5-2 presents the regression statistics. There is l i t t l e change in the dividend, growth and size coefficients.  They have the expected sign in every year and the  dividend coefficient is highly significant in every year. The growth coefficient is highly significant in seven years out of ten years and significant at the 5% level in another, while assets are highly significant in eight years and significant at the 10% level in another year.  This is a slight improvement in significance.  One notable change is the decrease in the fluctuation of the growth coefficient which ranges from 1.5636 to 7.7929.  TABLE 5-2 R e g r e s s i o n S t a t i s t i c s f o r Ln P = l n a  Q  +  In D + a  2  In (1+br) + a  ln(l+u) +  3  Y — D  Y — D  a A l n ? l + h ) + a r InS A f t e r Change i n Growth V a r i a b l e from b r = H  n = 43  Constant, l n a  Q  D i v i d e n d , a-^  Growth, ct  Business  Risk, a  1957  1958  1959  1960  1961  1962  1963  1964  1965  2.4290 .1679  1.8820 .1521  2.2079 .1817  2.3943 .1725  2.0530 .1798  2.6452 .1862  2.4094 .1991  2.5101 .1002  2.5284 .1262  2.7096 .1963  6.9011 1.3302  a  .8180 .0709  a  .8073 .0557  a  .7165 .0742  3  5.6S06 1.2334  a  5.8653 1.1892  a  6.1985 1.3944  .0650 .1417  .1110 .1349  .4156 .2519  F i n a n c i a l R i s k , 0:4  .4586 .2770  -.0127 .2346  .2603 .2251  Size, 0 5  .0556 .0403  .1157 .0350  R2  .8318  .8701  3  .  1956  .8538 .0818  2  to — - — ,w  A  J  a  .1179 .0339  a  a  ,.4569 .1681  a  -.2904 .2822 .1085 .0368  a  .9022  .8204 a b c  .5946 ' .0741  a  .6222 .0853  7.7929 1.4942  a  5.9456 1.5479  .3287° .1697 -.3473 .2721  a  .1441 .0375 .8021  a  a  .6757 .0882  a  .8211 .0593  a  7.0063 1.5279  a  2.2474 1.0485  ..1159 .1434  .0601 .2656  -.1090 .3034  -.6319 .2805  b  .0739° .0404  .0988 .0369  a  .7429  .7716  s i g n i f i c a n t a t t h e .01 l e v e l s i g n i f i c a n t a t t h e .05 l e v e l s i g n i f i c a n t a t the .10 l e v e l  a  b  .6463 .0756  a  1.5636 1.4808  .5686 .0926  Average  2.3769 .1662 a  .7124 .0758  1.6957 1.5898  5.0897 1.3836  a  .3032 .1491  b  .0157 .1864  .2326 .1729  -.3230° .1738  -.3678 .1926  c  -.2938 .2322  -.1657 .2474  .1362 .0280  a  .4485 .1179  .1078 .0216 .8954  a  .7978  .1517 .0357 .7062  a  .1110 .0346 .8140  72 The business risk coefficients overall, are slightly lower in magnitude with the same level of significance. in sign.  There is no improvement  Financial risk, however, has the sign predicted by the theory  in eight out of ten years but is significant at the 5% level in only one of those years and significant at the 10% level in two others. The coefficient of determination showed slight improvement to .8140 over ten years. An alternative measure of business risk next was tested by changing u to  u  W  The results are presented in Table 5-3. There was no significant change in the dividend, growth, financial risk and size coefficients.  The R values also were very 2  close to previous results, averaging .8129 over the testing period. Business risk as an explanatory variable appeared to have less influence than the l i t t l e i t had before.  Although one year had the  expected sign, the fluctuation of the coefficient over the years studied increased considerably, with a range of -.2715 to 2.7266. The use of net income in deriving normalized earnings also affects the measurement of business risk.  The theory indicates that  the variability of earnings applies to net operating income; that i s , net income before income taxes, fixed charges and depreciation. Accordingly, in the next test, operating income was used in conjunction with the risk and growth proxies u and  br  -  W Y - D A  TABLE 5-3 Regression Statistics for Ln P = lna  Q  + cx-^ In D + ct ln (1+br) + ct In (1+u) + 2  3  0v  0"Y  otA In (1+h) + 015 InS After Change in Business Risk Variable from u = - ~ to —* -> Y w 0  n=43  Constant, lna  Q  Dividend, Growth, or-2  1956  1957  1958  1959  2.4376 .1635  1.8946 .1528  2.2912 .1707  2.4574 .1717  .7878 .8535 . .0696 .0794 a  6.0950 1.2752  2.0992 .1725  a  ,8010 .056S  a  .6974 .0749  a  5.6627 1.2162  a  5.7381 1.4690  a  6.8462 1.3545  . 1960  2.5972 .0977  2.5644 .1227  2.7423 .1829  2.4169 .1599  .9120 1.0582  .0672 1.8386  -.1108 .3031  -,6363 .2818  .0270 .2364  .2653 ..2318  -.3538 «.3171 .2908 , .2738  Size, 0 5  .0552 .0402  .1174 .0355  R2  .8319  .8677  a b c  2.4329 .1811  c  .4584° .2757  .8112  2.6524 .1835  2.1142 1.1565  b  Financial Risk,0:4  .8985  Average  5.7857 1.5867  2.6683 1.1667  a  1965  a  2.0937 1.8569  .1102 .0378  1964  7.3587 1.5564  a  .0210 .7918  a  1963  .6190 .0838  .5823 .0730  .3869 .7751  ,1115 .0342  1962  a  a  Business Risk, 0:3  a  1961  a  .6700 . .0851  a  6.9556 1.5213  .1440 .0378•  .0747 .0405  .8001  .7436  a  c  significant at the .01 level significant at the .05 level significant at the .10 level  a  .7960 .0574  b  b  a  1.5863 1.1220  -.3163 .1753  a  .5615 .0889  a  .7000 .0740  1.8969 1.6654  4.8571 1.4369  a  2.5226 1.0575  b  -.2715. 1.3186  1.3241 1.1755  c  -.3595 .1888  c  -.3012 .2315  -.1644 .2489  .1036 .0218 .8940  .6320 .0712 .6457 1.6025  2.7266 .7349  .0973 .0368 .7713  a  a  .1379 .0273 .8045  a  .1479 .0360 .7065  a  .1100 .0348 .8129  74  The growth proxy  i n c l u d e s the r a t e of r e t u r n on t o t a l a s s e t s w h i l e  the o t h e r v a r i a b l e s were l e f t unchanged. are p r e s e n t e d  i n Table  The  regression s t a t i s t i c s  5-4.  Over the ten-year p e r i o d , the d i v i d e n d c o e f f i c i e n t s l i g h t l y J b w e r , averaging every year.  g e n e r a l l y lower and  1965,  the magnitude of the growth c o e f f i c i e n t .  a b s o l u t e l e v e l s were evidenced p a r t i c u l a r l y when the c o e f f i c i e n t  the c o e f f i c i e n t was and  .6706, but remained h i g h l y s i g n i f i c a n t  significant  significant  became n e g a t i v e .  a t the 5% i n o n l y two  other years.  The  a t the 5% l e v e l i n e i g h t out of the ten y e a r s .  at  s l i g h t l y h i g h e r i n most y e a r s but  size  significant  tests,  remained  r i s k v a r i a b l e s d i d not show improvement e i t h e r . f o r f i n a n c i a l r i s k was  having  the expected  coefficient  s i g n i n s i x y e a r s but not b e i n g B u s i n e s s r i s k had  significant  f o r b u s i n e s s r i s k was  The  about the same as i n p r e v i o u s  the 5% l e v e l i n any of the y e a r s .  s i g n i n f i v e y e a r s but was  in  a t the 1% l e v e l i n o n l y f i v e y e a r s  ranged  coefficient  The  In a d d i t i o n ,  coefficient  The  in  The use of o p e r a t i n g income i n the growth v a r i a b l e  c o n s i d e r a b l y reduced  1964  ranged  the  statistically expected  i n none of t h e s e y e a r s .  significant  The  a t the 5% l e v e l i n o n l y  three years. To determine  whether the poor r e s u l t s f o r r i s k were caused  the use of u = O y / W , the same model was coefficient  of v a r i a t i o n was  that i s ,  Y  used  t e s t e d except  as the proxy  t h a t the  f o r business  risk,  by  TABLE 5-4 Regression Statistics for Ln P = l n a + a-^ 'In D + a 0  cu ln (1+h) +  ln (1+br) + 0 3 ln (1+u) +  2  InS After Change to Operating Income with u = -~ and br = W  n=46  Constant, lna  Q  Dividend, Growth, a  2  Business Risk, 0 3  1956  1957  1958  1959  1960  1961  1962  1963  1964  1965  2.2045 .2529  1.7033 .2401  2.4379 .2469  2.6406 .2613  1.7678 .2591  2.6422 .2381  2.4450 .2081  2.4567 .1292  2.3029 .1606  2.7316 .2005  .7335 .0879  a  .7417 .0702  3.9339 1.0171  a  2.8276 .7298  a  .7843 .0643  a  2.0340 .7891  a  .6071 .0781  a  ,6140 .0834  b  2.5643 .8423  a  3.0750 .8299  -.7484 1.0038  -.2792 1.1210  -.9063 .9548  -1.2969 1.0312  3.0859 1.2724  .6074 .3693  .2479 .2677  .3134 .2581  -.0549 .3132  -.3036 .3170  Size, 0:5  .0679 .0512  .1312 .0428  R  .7306  .8217  Financial Risk, a  2  4  " A  a  ,1069 .0400  .0912 .0447  a  .8619  .7552 a b c  b  a  .6477 .0862  a  b  .1702 .0450 .7438  a  a  .6317 .0844  1.9302 .7660  b  2.4841 .7183  1.2616 1.2260  -.3089 1.0681  1.8876 .9124  4.1315 1.5138  .1356 .3218  -.3515 .2859  -.2389 .1829  .0601 .0440  .0865 .0379  .6994  .7386  significant at the .01 level significant at the .05 level significant at the .10 level  a  ,7364 .0606  a  •9154C  a  .5400 b  b  .1202 .0239 .8585  a  .6472 .0729  a  .5626 .0840  Average  2.3332 .2197 a  .6706 .0772  -.1659 .7497  1.9424 .7690  a  .7804 1.5512  .7607 1.1655  -.3493C .1836  -.3044 .2325  .0298 .2732  .1728 .0273  .1583 .0324  -.1747 .7080  .7806  a  .6881  a  .1165 .0389 .7678  76 As Table 5-5 shews, the coefficient for business risk performed poorly.  Only three years had the sign indicated by the  theory compared to five years in Table 5-4 and only one year was significant compared to three years previously.  In neither case  did the business risk coefficient have the correct sign in the year in which i t was significant.  There was, however, a considerable  reduction in the range of fluctuation over the years. More importantly, the use of the coefficient allowed the growth coefficient  to "improve."  of variation  In a l l ten years,  compared to eight years previously, growth has the expected sign. In addition, eight of the years are significant, being one more than before.  The range of fluctuation is slightly reduced as well.  Comparison of Table 5-2 with 5-3 and Table 5-4 with 5-5 indicates that there is a slight advantage in using ay/* instead of ay/W as a business risk proxy. Using the same model reported i n Table 5-4, the financial risk proxy was changed from h = L/E to h where  =  L + P.S. + C.L. ———————-—  E  P.S.  = book value of preferred stock  C.L.  =  current l i a b i l i t i e s .  The results tabulated in Table 5-6 show no appreciable improvement in the performance of this risk coefficient nor in the effects on any other variable.  compared to previous tests Although one more year has  the expected sign, none of the years are significant at the 5% level The range, however, was slightly reduced.  TABLE 5-5 Regression Statistics for Ln P = lna  Q  + aj_ In D + a  ln (1+br) + a  2  3  ln (1+u) +  °Y  0:4 ln (1+h) + 0 5 InS After Change to Operating Income with u = -= n=46  Constant, lna  D  Dividend, a. Growth, a  1957  1958  1959  1960  1961  1962  1963  2.2257 .2677  1.6426 .2569  2.3029 .2752  2.6515 .2769  1.9180 .2843  2.7063 .2438  2.4623 .2328  2.3921 .1541  .7355 .0907  a  3.6805 .9375  2  Business Risk, a  1956  3  .7465 .0709  a  .7802 .0649  2.8001 .7151  a  2.0977 .8283  a  .6001 .0798  a  b  2.36l4 .8717  a  .5986 .0887  a  .6458 .0891  a  3.2535a .8924  1.9654b .7808  .6296 .0852  2.4280a .7538  -.1658 .3040  .0677 .3129  .0527 .3117  -.3609 .3146  .4853 .3861  .1676 .3333  -.1013 .3125  Financial Risk, 0 4 .5662 .3619  .2602 .2700  .3529 .2642  -.1199 .3159  -.2300 .3310  .1546 .3245  -.3540 .2861  Size, 0.5  .0700 .0512  .1365 .0424  .0523 .0438  .0856 .0383  R  .7289  .8216  .6934  .7387  2  3  a  .1170 .0406  .0977 .0434  a  .7536  .8589 a b c  b  .1478 .0461 .7173  a  a  significant at the .01 level significant at the .05 level significant at the .10 level  1964  .7380 .0623  a  .6464 .0771  1.3444b .5326  .7779 .6331  .4263 .2442.  .8725 .3909  -.2240 .1868 .1239 .0247 .8545  Average  2.1986 " 2.6664 .2380 .2049  c  b  1965  a  b  -.2800 .1923 a  .1752 .0287 .7685  a  .5767 .0882  2.3166 .2435 a  .6697 .0797  .0412 .7547  2.0750 .7700  .2843 .3990  .1728 .3309  -.2712 .2362  .0145 .2769  .1597 .0322 .6901  a  .1166 .0391 .7626  - TABLE 5-6 Regression Statistics for Ln P ctA ln (1+h) + a n=46  Constant, lna  1956  o  Dividend,  2.1212 .2560  3  l  n  a  + 0  a  l  n  D  +  1  a  2 ^  n  (  1 + b r  )  +  a  3  l  n  (  1+u  )  +  OY  5  InS After Change to Operating Income with u = — and h = W  TH-P  +r T  S  E  ,1957  1958  1959  1960  1961  1962  1963  1964  1965  Average  1.6556 .2368  2.3926 .2486  2.6473 .2546  1.8020 .2509  2.6173 .2304  2.5080 .2040  2.4893 .1284  2.3366 .1645  2.7169 .2055  2.3287 .2180  .7128 .0875  a  .7386 .0672  a  Growth, (*2  3.5783 .9857  a  2.8206 .7217  3  Business Risk, 0:3  -.5817 1.0059  -.2004 1.1155  -1.0405 .9805  -1.3003 1.0304  3.0540 1.2681  Financing Risk, 0 4  .3133 .2543  .2427 .1953  .0120 .1987  -.0349 .2042  -.2062 .2115  Size, ct5  .0824 .0502  .1267 .0419  R2  .7229  .8247  a  .7657 .0646 l„9799 .8034  b  .1300 .0403 .8568  a  a  .6072 .0781  a  .6129 .0836  a  .6390 .0875  2.5719 .8416  a  3.0981 .8288  a  b  .0909 .0442 .7552  b  .1705 .0449 .7440  a  a  .6546 .0883  a  .7522 .0600  1.9304 .7681  b  2.4412 .7277  a  .8889 .5472  1.2737 1.2296  -.3718 1.1010  1.9040 .9273  4.1710 1.5557  .0464 .2536  -.1480 .2532  -.1223 .1525  -.2015 .1765  a  a  .1623 .7337 b  .0656 .0462  .0761° .0405  .1164 .0255  .6983  .7310  .8548  a significant at the .01 level b significant at the .05 level c significant at the .10 level  .6662 .0746  a  a  .1709 .0309 .7682  a  .6206 .0850  a  .6770 .0776  .3053 .7819  1.9452 .7740  .7976 1.5767  .7706 1.1791  .1303 .2022  .0032 .2102  .1256 .0363 .6781  a  .1155 .0401 .7634  79  B.  Conclusions  The purpose of this thesis was to determine whether a stock valuation model based upon the dividend formulation of Myron J . Gordon provides an adequate explanation of the variation in stock prices over an extended period of time.  It can be concluded that the  empirical results support the theory that investors mainly consider the current dividend, its expected rate of growth and its risk in determining the price of a share.  Tests conducted on a sample taken  from the machinery industry, showed that on average over a ten-year period, between 76% and 81% of the total variance in share price was explained by the model.  The lowest R value in any year was .6781, 2  while the highest value was .9022. The coefficients  of dividends, growth and size were statistically  significant at the 5% level or better in most of the years tested. The dividend coefficient, every year.  in particular, was highly significant in  Furthermore, except for the growth coefficient  years, these coefficients  in two  had the expected sign indicated by the  theory.  In terms of stability, only the growth coefficient  fluctuated  widely.  The change in absolute level and stability of growth can  be partly explained by the measurements used for corporate earnings or return on investment.  The most important reason, however, may  be changing investor valuation of growth in the stock market. Except for a short period in early 1962, the stock market has been an expansionary phase since 1960.  The tests show that after  1962, however, there was a drastic decline in the growth coefficient which corresponds to the market's reassessment of "growth stocks."  80  That this decline continued until 1965 probably relates to the industry's prospects rather than to growth expectations for the overall market. The good performance of the size variable is a mixed blessing. The fact that size appeared to be highly correlated with both financial and business risk may have contributed to the failure of the other risk variables.  More importantly, the lack of statistical  significance  in most years would indicate that the measurement of risk was at fault. In particular, measures of business risk other than the standard deviation of income might be more successful.  There is also a strong possibility  that in assessing risk, investors consider such factors as quality of management, which is not derived from the balance sheet or income statement of the corporation.  Lastly, i t is possible that the machinery  industry, i t s e l f , is not broadly representative of the stock market. Although the model proved generally successful in predicting share price variation, its usefulness to investors for security selection is limited.  Cross-section averages for an industry may  ignore conditions peculiar to certain companies within the industry. Omission of other important determinants of stock prices common to an industry also is possible i f less likely.  A more common problem  is that of accurate measurement of the variables.  Assuming the  problem of measurement can be overcome i t is probable that the nature of the relationship between variables w i l l change from year to year and this study indicates this is frequently the case. The weight attached to growth in dividends and earnings was found to be particularly vulnerable to change over time.  Lastly,  81  the mechanistic use of past or current earnings - even i f normalized to the extent that its values deviate from investors expectations limit the model's usefulness. The main value of Gordon's model is in specifying the variables important to stock valuation and in clarifying the nature of the relationships.  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