When does the share price equal the present value of future dividends?

Economic Theory 8, 307-319 (1996)

Econom/c Theory

�9 Springer-Verlag 1996

When does the share price equal the present value of future dividends?

A modified dividend approach*

Suresh P. Sethi Faculty of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, M5S 3E6 CANADA

Received: April 26, 1995; revised version August 3, 1995

Summary. This paper discusses an explicit necessary and sufficient condition on the dividend stream of a publicly traded company, under which the price of the company's share is equal to the present value of the future dividends that will accrue to it. When it is not, the share price equals the present value of the future per share dividend plus the limiting per share value of the company "at infinity". It uses a well-accepted generalization of the Miller-Modigliani framework, and assumes that the firm is an infinite horizon firm which may engage in repurchasing its own shares. It develops a proper dividend approach that can value such a firm for any dividend stream. The paper concludes by clarifying some remarks in the Miller- Modigliani paper.

1 Introduction

Many corporations repurchase their own shares. The purpose of this paper is to discuss a precise condition on a firm's financing policies including repurchase of its own shares, under which the price of a share of its stock is the present value of future dividends accruing to it as well as to develop a modified dividend stream approach of valuation suitable for firms that may issue negative stock. Moreover, the paper clarifies certain remarks or claims made by Miller and Modigliani (MM hereafter) in their classic paper [2].

In [2], MM provide a valuation formula (see their equation (9)) for an infinite horizon firm given its investment decisions I(t) and stream of net profits X(t) over time, under the assumption of perfect capital markets, rational behavior, and perfect certainty. Simply stated, the MM valuation formula equates the value of the firm to the present value of the net distributions to stockholders over the infinite horizon. This can be seen by using their identity (4) in their formula (9).

* For helpful discussions and comments, I thank Laurence Booth. Mike Gordon, Robert Jarrow, Raymond Kan, Rajnish Mehra, David Quirin, and Rishin Roy. Support from Social Sciences and Humanities Research Council of Canada is greatly appreciated.

308 S.P. Sethi

While MM do not explicitly provide for the repurchase of the firm's own shares, it is widely believed that their valuation formula holds even if negative stock issues are allowed. It is not generally known or appreciated, however, that when the firm repurchases its own shares, its share price may or may not equal, depending on its financing policy, the present value of future per share dividends. This is because while the present value of the firm vanishes at t -~ oo as required by what is known as the transversality condition, there may still be a residual per share value of the firm in the limit, since the number of outstanding shares may also go to zero as t ---4 o0.

In Section II of their paper, MM show that four more or less distinct valuation approaches in the literature are equivalent. While their equivalence proof is incomplete, they conclude that each of these approaches results in the same valuation of the firm as the MM formula. We shall focus only on the so-called "stream of dividends approach", which they indicate to be "by far the most popular one in the literature of valuation." It is known from Sethi et al. [3] that if share repurchases are allowed, the stream of dividends approach as used in MM has meaning only for a class of financing decisions, while it is meaningless, i.e., it provides no consistent valuation, for the remaining class. In other words, they delineate a necessary and sufficient condition on a firm's financing policies under which the price of a share of its stock is the present value of the future dividends which will accrue to it. Absence of this condition implies that there exists no share price which is the present value of future per share dividends and which, at the same time, yields the value of the firm when multiplied by the number of outstanding shares. However, in this case, the share price equals the present value of future per share dividends plus the limiting present value of the firm per share as time goes to infinity.

The purpose of this paper is to provide an economic interpretation of the condition. Briefly, the absence of the condition means that the present value of firm per share does not vanish "at infinity". It also means that a share in the dividend reinvestment plan would grow to only a finite number of shares as t ~ oo. The insight obtained in the process leads to a subtle modification of the MM dividend stream approach that can value a firm no matter what the dividend stream. The refor- mulated or "properly formulated" dividend stream approach of valuation restores its equivalence to the MM valuation formula and establishes the "irrelevancy of dividends" for firms that may repurchase their own shares.

To keep things simple, we shall assume a discrete time, deterministic framework, although the analysis can be repeated with the help of higher mathematics and the results obtained in [4, 5], when there are uncertainties present and/or when the time is assumed continuous. Also as in MM, we shall without loss of generality assume no borrowing, as far as the purpose of this paper is concerned.

Before providing a plan of this paper we should emphasize that the MM paper has been around for a long time. A great deal has been written on it. However with the exception of the papers [3, 4, 5], no one to our knowledge has examined in depth the impact of allowing share repurchases on the value of the firm. If MM did not allow repurchases, then this paper represents an extension of their paper. If they implicitly allowed repurchases, then their paper is not completely correct and this

A modified dividend approach 309

paper completes their paper along with some clarification. In either case, given that the MM paper is a required reading in many introductory as well as advanced finance courses, it seems worthwhile to devote a few pages to adding some precision to it and provide the needed clarification. Moreover, a bit of additional insight is obtained in the process.

The plan of the paper is as follows. In the next section we introduce our notation which is more or less consistent with MM. In Section 3 we define a general deterministic infinite horizon firm, obtain its value using the fundamental principle of valuation as given in MM, and develop expressions for the share price and the number of outstanding shares over time. In Section 4, we specify the necessary and sufficient condition for the share price to equal the present value of future per share dividends. We illustrate the condition with five illustrative examples. Furthermore, we provide a proper formulation of the so-called dividend approach to valuation. In Section 5 we provide economic interpretation of the results. In Section 6 we point out some mathematical errors in the MM paper and provide clarification of some of the issues involved. Moreover, we provide a nontrivial proof that a constant payout firm alluded in MM will always have the share price equal present value of future per share dividends no matter what its dividend and external equity streams are.

2 Notation

We introduce a notation similar but not identical to that in MM. Let

Z = {0, 1, 2 . . . . }, the set of discrete time periods; ' tEZ' and 't _> 0' will be used interchangeably,

p(t) = price (ex any dividend in period t - 1) of a share in the firm at the start of period t,

d(t) = dividend per share paid by the firm during period t, m(t + 1) = the number of new shares sold during t at the ex dividend price p(t + 1);

a negative value for m(t + 1) means share repurchase, n(t) = the number of shares outstanding at the start of t; let n(0) = no, 0 < no < c~, v(t) = n(t)p(t) = the total value of the firm at the start of period t, ~(t) = n(t)d(t) = the total dividends paid during period t to stockholders of record at

the start of the period, g(t) = m(t + 1)p(t + 1) = the amount of equity capital raised during period t at the

ex-dividend closing price p(t + 1); a negative value for g(t) means equity capital retired,

p = the market rate of discount or the stockholder's required rate of return per period.

Also by convention

• f ( i ) = O i f t < _ s - - 1, s , t~Z , i = s

I f ( i ) = 1 i f t < s - - 1, s , t~Z . i = s

310 s.P. Sethi

It is convenient to perform the entire analysis in the present-value terms. For this we define

D(t) = (1 + p)-t~(t) = (1 + p)-'n(t)d(t),

E(t) = (1 § p)- 'g(t) = (1 § p)- 'm(t § 1)p(t § 1),

1"(o) = p(o), P( t + 1) = (1 + p ) - ' p ( t + 1),

v(0) = v(0) = vop(O),

V(t + 1) = (1 + p)-'v(t + 1) = n(t + 1)P(t + 1),

so that the relevant present-valued variables are denoted as capital letters. Hence- forth, the value terms such as dividends, price, etc. will mean their present values, unless otherwise specified.

3 The framework

A firm can be denoted by the pair {D(t), E(t), t ~ Z} satisfying the following assumptions:

(A1) D(t) >_ O, t >_ 0, (A2) E iZ o {D(i) + IE(i) l } < oo (A3) Z,~t [D(i) - E(i)] > D(t), t _> 0.

Note that we do not deal with the question of how one determines D(t) and E(t), t e Z . One possible way to obtain them is by solving an optimization problem that maximizes the value of the firm. However, our purpose here is to treat every possible firm {D(t),E(t), t ~ Z } that satisfies Assumptions (A1)-(A3), regardless of how its specification is obtained.

It is easily shown that under Assumptions (A1)-(A3), the fundamental principle of valuation (see [1]) expressed as

P(t) = P(t § 1) § - -

along with the usual transversality condition

D(t) n(t)' (1)

lim V(t) - lira n(t)P(t) = 0, (2) t-+CO t-*CO

where the number of outstanding shares n(t), assuming no stock splits, is defined as

t - 1 E ( i )

n(t)= n o+ ~ P(i + l)' i = 0

establishes the value of the firm at time t to be

V(t) = ~ [D( i ) - E(i)], t > 0. (3) i= t

Subsequently, the number of outstanding shares can be expressed as

' - ' v(i + 1) n(t) = n o l-[ V(i + 1) - E(i)' t = 0, (4)

i=O


and finally the share price can be obtained as

V(t) _ 1 [D(i) - E(i)] l-[ 1 t > 0. (5) P(t) = n(t) no i=t j=o V(j + I) ' -

Moreover, 0 < V(t) < 0% 0 < n(t) < 0% and 0 < P(t) < oo for all t > 0. In particular, Assumption (A2) ensures that the present value of the firm is well defined and finite in any period. Note that this assumption implies that 52i~= o D(i) is finite and

i 52~=o E( ) is well defined and finite. By a sum to be well-defined, we mean that it does not depend on the order of summation. Assumption (A3) requires that the firm in any period may not issue a dividend which is equal or in excess of its value in that period. This ensures that the value of the firm stays positive and the number of outstanding shares remains positive and finite. Note that one could relax " > " in (A3) to an " > " at the risk of having a firm liquidate itself within a finite horizon. Indeed, if s denotes the first time when (A3) is an equality, then it is easy to see that the dividend D(s) is the liquidating dividend, with the consequence that V(s + 1)= 0, n(s + 1)= ~ , and P(s + 1)= 0. But then it is no longer an infinite horizon firm under consideration in this paper. The transversality condition (2) is a standard boundary condition on the value of firm at "infinity" in present value terms. It imposes a uniqueness on the solution of the difference equation (1), and rules out so-called "bubbles" and "Ponzi schemes." It should be noted that such transversality conditions are quite common in the economic literature such as that of infinite horizon economic growth models; see, e.g., Malliaris and Brock (1982, p. 249).

The question that will be answered precisely in the next section is: When and when not is the share price present value of future per share dividends? In other words, given D(t), n(t) as in (4), and P(t) as in (5), we would like to derive a necessary and sufficient condition under which

P(t) = ~ D(i) d(i) i=t ~ - i=, ~ (1 + p)V (6)

4 Necessary and sufficient condition for dividend capitalization - a modified dividend approach

First we write the general solution of the difference equation (1) as

D(i) P(tl:c~+/~,=,n-- ~ , t_>0, (7)

for some constant c~. Thus, (5) can be written as (7) with

c~ = lim P(t)= lim --1 [D( i ) - E(i)]- 1- I 1 . (8) ~-.oo ~-*~ no ~:, i=o V(j + 1)

Comparing (6) and (7) reveals that (6) holds if and only if ~ = 0. Before proceeding further, let us give some examples. Assume n o = 1 in all these examples without loss of generality.

312 S.P. Sethi

Example 1. The firm never pays any dividend. It indulges in repurchasing its own shares, which is responsible for its positive valuation. More specifically,

D(t) = O, E(O = - (�89 t ~ Z .

For this firm, we have

V( t )=( i ) t - i , n(t)=(�89 V( t )=2, t e Z ,

lim n(t) = 0, c~ = lim P(t) = 2. t - ~ O0 t - -~ oO

Note that Example I is well known in the finance literature. But its resolution by MM is not quite appropriate as we shall indicate in Section 6. The next example is not well known (and surprising to some) in the field of finance, even though it is not much different from Example 1 from a mathematical viewpoint.

Example 2. The firm pays a positive but declining dividend. It also repurchases its shares. More specifically,

D(t) i it it = ( 9 ] , t ~ z . - ~ [ ( ~ ) - (~)(~), e ( t ) - ~ ~ '


t-I I t-i

V( t ) = i t H t2, A, t e Z , (~), n(t)=,=o= 2 - ( i ) * ' P(t)=~=oI-I [1-c-l~*+lq

l imn(t)=O, c~= lira P( t )= f l [1 - i v + i ] 2, ~ 0.2887881. t ~ t ~ ~ = 0

Example 3. The firm is similar to that in the example just above. Only the specifics are different. These are

D(t)=(�89 t, E(t)=--( �89 t+l, t e Z .

Moreover, the specifics are different in an important way since e = 0 in this case as shown below, whereas it is positive in Example 2. For this firm, we have

V(t) = (�89 n(t) = (2)t, P(t) = (�89 t e Z ,

lira n(t) = 0, ~ = lim P(t) = O. t --~ oo t - + o o

Example 4. The firm pays the same amount of dividends as in Example 2, but issues no positive or negative stock. Thus

D(t) = (�89 E(t) = O, t~Z .


V(t)=(-~)(�88 n ( t ) = l , P(t)=(-~)(�88 t e Z ,

lim n(t) = 1, ~ = lim P(t) = 0. t ---~ c~ t ~ o o

Example 5. The firm pays dividends as well as issues new stock. The specifics are

D(t)=(-~)(-~) t, E(t)=(�89189 t+l, t e Z .



V(t) = (�89 n(t) = 2 t, P(t) = (~)t, t eZ ,

lira n(t) = CO, a = lira P(t) = O.

These examples clearly show that in some cases such as Examples 3-5, P(t) -40 as t ~ CO, which implies that the price is present value of future per share dividends, while in others such as Examples 1 and 2, P(t) -4 a > 0 as t -4 CO, which implies the opposite. We should also note that in cases of the first type, we can have new share issues or share repurchases and we can have the number of outstanding shares go to zero, go to infinity, or stay the same, as t -4 CO. In cases of the second type, we can have an identical zero dividend or positive dividend in every period. Furthermore, it is possible to prove that in these cases, we must have share repurchases as t -* oo and the number of outstanding shares will approach zero as t ~ Co.

Remark 1. The observation with respect to the number of outstanding shares apply only if there are no stock splits. For instance, if the stock in Example 1 is split 2 for 1 in each period t ~ Z before each repurchase, then the number of outstanding shares after stock splits will approach 1 as t-4 CO. Moreover, the definition of the share price can be appropriately modified in the presence of stock splits. We do not explicitly consider stock splits for simplicity in exposition.

In what follows, we explicitly characterize these two classes of firms. Intuitive interpretation of the characterization is provided in Section 5.

Theorem 1. The share price equals the present value of the dividends accruing to it, i.e., relation (6) holds, if and only if the sum of the dividend yields is infinite, i.e.,

D(i) i= o V(i---))= CO' (9)

which in turn holds if only if

fi[1 i=0 - v ( i ) J = ~ . (10)

Proof. The first part is proved in [3]. The second part follows immediately from the first.

It follows from Theorem 1 that when the dividend yields sum to something finite, i.e.,

{ V(i) i ='~aO ~ ( CO, (11)

or, equivalently, when

I D(i) l -~ 1 - ~ j < Co, (12) i=0

then dividend capitalization alone does not yield the share price and the stream of dividends approach, as specified in M M (see their equation (13)), does not provide

314 S.P. Sethi

any positive value, let alone the correct valuation provided by the MM formula or by formula (3) in this paper. Indeed it is proved in Sethi, Derzko and Lehoczky [5J that the system of equations defining the dividend stream approach to valuation, i.e.,

D(i) t~l E(i) V(t) = n(t)P(t), (13) P(t) = -n~' n(t) = n o + P(i + 1)'

i = t i=0

has no solution satisfying 0 < V(t) < 0% 0 < P(t) < oo and 0 < n(t) < oo, t > O, if, and only if, inequality (11) holds. In other words, when (11) holds, the stream of dividends approach (13) is simply meaningless. Moreover, since (13) can be obtained from the fundamental valuation principle (1) and condition lim~. oo P(t) = O, the latter condition cannot in general be a sensible transverality condition. Why this is so will become transparent later in this section.

This does not at all mean that a firm with financing policies satisfying (11) is economically "absurd". What it means instead is that the dividend stream approach (13) is not properly formulated when repurchases are allowed. To see why, consider a finite horizon case with horizon length T for a moment. In this case, a proper formulation of the first relation in (13) would be

P(t) = ~ i D(i) V(T) ~ t - ~ ~ n(T) ' (14)

where V(T) is some "properly arrived" liquidation value of the firm at the terminal time T. One of the main reasons why the finite horizon version is not popular in theoretical analyses of the problem is the difficulty of arriving at the terminal liquidation value. Note that the mathematics would go through for any arbitrary nonnegative terminal liquidation value, but the specification of the terminal value would have implications for the share price. In fact, the only proper terminal value at time T is in terms of its dividend and external equity streams from time T to infinity. But, then the firm is not different from an infinite horizon firm.

The difficulty of arriving at a terminal value of the firm at T can be overcome by reformulating (14) in the infinite horizon case as

T~I n(i) . V(T) P(t)= mt - ~ + n--(-~' V T > 0 a n d lirno~ V(T)=0. (15)

Note that the transversality condition is the same as (2), and is a proper replacement of an otherwise arbitrary terminal liquidation value in the finite horizon formulation. Of course, the first part of (15) implies that

P(t)= lim ~r~i~ + V ( T ) ] (16) r-o~ L i=, (') n(T)'J

Let us now replace the dividend stream approach (13) by

V ( T ) _ n(t) = n o + P(t) = lira - ~ -t- n(T) J' P(i + 1)' T ~ O O L i = t i=0

V(t) = n(t)P(t), lim V(T) = O, (17) T ~ o o


which we shall prove shortly to be the "properly formulated" approach, in contrast to MM's equation (13). The reason we can no longer set l imT_.~P(T)=

V(T) limr-~ ~ n(T) = 0 is because while V(T) ~ 0 as T ~ ~ , n(T) may approach 0 faster as

T goes to infinity. This is precisely what happens in Examples 1 and 2. Note however that in Example 3, V(T) /n (T)~ 0 even though n(T) also goes to zero, as T ~ ~ .

Theorem 2. The dividend stream approach (17) is properly formulated in the sense that it is equivalent to the valuation approach defined by (1) and (2).

Proof. We shall first show that (17) implies (1) and (2). It is sufficient to show that the first relation in (17) implies (1). This is easily seen as follows:

T-1

T~o~LI=t n(i) i=,+1 n(i) J n(t)"

To prove that (1) and (2) imply (17), we show that (5) satisfies the first relation in (17). By comparing (5) and (7), we see that

O(i) P(t) = a + n(i)

i= t

= lira V(T) D(i)

r ~ n--~ + -- n(i) i = t

= lira [ ~ 1 D(i) , V ( T) 7 (18)

The remainder of the proof is straightforward. []

In the next section, we provide intuitive explanation of our results.

5 Discussion of results

In view of Remark 1, let us note that the following discussion assumes there are no stock splits. Of course, the discussion could be modified appropriately if there were. Needless to say, the basic conclusions will not change because of stock splits.

Let us first explain the meaning of (16). It says that given a neighborhood v(t)

[a, ~ + 6), fi > 0, of ~ = limt~ ~ n(t--~' the per share value of the firm will be in this

neighborhood for sufficiently large T. If the limit ~ > 0, then this per share value at large T is not zero and cannot therefore be ignored as in (13).

But a > 0 if and only if (11) holds. Therefore, (11) could be interpreted to be the case of a firm which is not giving sufficient total dividends in relation to its share price, and the difference is passed on to its stockholders by the firm's repurchase activities. More precisely, whenever a > 0, one can show that (11) implies n(t) ~ 0 as t ~ o% i.e., the firm retires all of its stock as t ~ ~ . Thus in view of(7), all stockholders in the limit are properly compensated for their investment in the firm's shares.

One should be careful, however, not to jump to the erroneous conclusion that n(t) ~ 0 is a sufficient condition for the share price not to equal the present value of

316 S.P. Sethi

future per share dividends. Example 3 is the counter-example constructed to avoid just such a hasty conclusion.

While we have now provided some intuition behind the failure of dividend capitalization to represent the share price in terms of the insufficiency of dividend stream, it is not intuitively obvious why condition (11) translates into insufficiency of dividends. We now attempt to provide an explanation.

First we use n(t) = V(t) /P(t) to transform (1) into

P(t) = P(t + 1)g(t), (19)

where

O(t)= 1-t n(t)P(t + l) - ~ J

It is clear that g(t) - 1 is the share growth rate if all the dividends are reinvested to buy more shares. The interpretation of (18) is obvious. The left hand side is the value of one share at time t and the right hand side is the value of a share at time (t + 1) times one plus the number of shares that would be obtained by reinvesting the dividend received on the one share during period t. Clearly the LHS and RHS must be equal.

Next, we solve the difference equation backward in time as

'hi= o ' I~[ = o D(T)I- i V(~) J (21, P(O) = P(t) g(z) = P(t) 1 -- .

Clearly, the right hand side of (20) represents the value of a portfolio of shares at time t that would result from having a share at time zero under a dividend reinvestment plan. Since P(0) > 0, one can see from (20) that if a share in the dividend reinvestment plan grows to only a finite number of shares, then we must have limt_, oo P(t) > 0. This is the case when the share price is not equal to the present value of future per share dividends. Clearly, then a dividend stream is insufficient if and only if a share in the dividend reinvestment plan grows to a finite number of shares as t ~ oo. On the other hand, if the number of shares in a portfolio would approach infinity under the dividend reinvestment plan as t ~ 0% then dividends are sufficient and the share price can be obtained by simply capitalizing the future per share dividends.

We now wish to clarify some aspects of the MM paper [2].

6 Clarification of some issues in the M M paper

We begin with the examination of MM's derivation of the equivalence of their valuation formula (9) and the dividend stream approach (13). If share repurchases are allowed, then (13) is obviously not "properly formulated" in view of (16) and the attendant explanation in Section 4.

On the other hand, if share repurchases are not allowed, then the equivalence is a correct result, but MM's derivation of the result is incomplete. Their derivation begins with relation (6) (their relation (13)), which presupposes limt_, o~ P(t) = 0. Then they obtain their equation (17), which according to them implies the MM formula. But this implication holds only if limt~ ~ V(t) = 0. However, one does not have the


luxury of assuming limt_~ oo V(t) = 0, if one wants to derive the value of the firm beginning with (6). It must be proved, which they do not.

In addition, one has also to begin with the MM formula (9) or our formula (3) and show that P(t) satisfies (6), to complete the proof of equivalence.

Next we discuss their footnote 12, which is quite confusing at least to us. For convenience of the reader, we reproduce the footnote below.

"The statement that equations (9), (12), and (14) are equivalent must be qualified to allow for certain pathological extreme cases, fortunately of no real economic significance. An obvious example of such a case is the legendary company that is expected never to pay a dividend. If this were literally true then the value of the firm by (14) would be zero; by (9) it would be zero (or possibly negative since zero dividends rule out X(t) > I(t) but not X(t) < I(t)); while by (12) the value might still be positive. What is involved here, of course, is nothing more than a discontinuity at zero since the value under (14) and (9) would be positive and the equivalence of both with (12) would hold if that value were also positive as long as there was some period T, however far in the future, beyond which the firm would pay out e > 0 per cent of its earnings, however small the value of ~."

Recall that I(t) and X(t) denote the investment decision and the net profit in period t, respectively. Note further that N ( t ) - E ( t ) - X ( t ) - I(t). We should also mention that in what follows we shall not comment on MM formulas (12) and (14).

First we note that their parenthetical remark imputing a possible negative valuation of the firm by their formula (9) is simply a result of not specifying something like our Assumption (A3) in advance. Indeed, a careful formulation of the firm under consideration in the MM paper requires Assumptions (A1)-(A3) stated in Section 3 of this paper.

Furthermore, if they do not allow repurchases, which appears to be the case in the footnote when they say "zero dividends rule out X(t) > I(t)", then equivalence holds in all cases and no qualification in addition to Assumptions (A1)-(A3) is needed; note that zero dividend does not rule out X(t) > I(t) since profit in excess of investment can be used to repurchase shares if allowed. But the MM do not rule out X(t) < I(t), which means that E(t) > 0 even though D(t) =_ O. Thus, MM's legendary company does not satisfy Assumption (A3). While MM have labeled such a firm "pathological", we have excluded them altogether by requiring the firms to satisfy Assumption (A3). On the other hand, if D(t) = 0 and E(t) =- 0 for the legendary firm (i.e., X(t) - I(t)), then MM equation (9) and their equation (13) both assign the same value zero to the firm. Even though equivalence holds for this firm, we have excluded it also by our Assumption (A3) because the firm is uninteresting or of "no real economic significance"; see the paragraph following equation (5), where we discuss such a firm.

If repurchases are allowed, then the cases for which equivalence does not hold are precisely those given by our condition (11). A particular case is Example 1, which could be termed a legendary zero-dividend firm engaging in share repurchase. But there is nothing "pathological" or "extreme" about this case. It is not "pathological" since it satisfies Assumption (A3), and a properly formulated stream of dividends

318 S.P. Sethi

approach, (17), is an equivalent valuation approach. It is not "extreme" since the valuation formula (3) is not discontinuous at zero dividends. The value of the firm is linear and, therefore, continuous in total dividends. Indeed there are cases such as Example 2 with strictly positive dividends that are not different from Example 1 in principle. We note parenthetically that while Example 1 seems to be a well-known example for which the dividend stream approach (13) does not work, Example 2 was formulated for the first time in [3]. In this example, there is a positive dividend payout in every period and still the dividend stream approach (13) does not work.

In view of the explanation provided in the last section, all these cases are just as sensible as the others. Their valuation is still provided by (2), signifying the "irrelevance of dividend policy, given investment policy." Values still depend on "the earning power of the firm's assets and its investment policy - and not by how the fruits of the earning power are "packaged" for distribution."

Finally, MM's perception of discontinuity has lead them to modify the "legendary" company by having it "payout e > 0 percent of its earnings, however small the value of e." Thus modified, the constant payout firm is no longer the legendary company mentioned at the beginning of the footnote. Furthermore, it is shown below that the modification transforms the firm into one that pays "sufficient" dividends in the sense of (9), with or without repurchases.

In view of Theorem 1, it is sufficient to show that the constant payout firm in question satisfies Assumption (A1)-(A3) and condition (9) for any e > 0.

Let I(t) > 0 and X(t) denote the investment decision and net profit in period t, respectively. For the constant payout firm, X(t)> O, since dividends must be nonnegative. We have

(1 + p)-t[X(t) - I(t)] ~ D(t) - E(t) = (1 + p)-teX(t) - E(t).

1 Moreover, let Zi=o( +P)-t[X(t) + l(t)] < c~, so that we can use (3) to obtain

V(t) = - I(i)(1 + p)- ' . i = t

It is easy to verify that Assumptions (A1) and (A2) hold. Assumption (A3) also holds if we restrict the firm from liquidating itself within a finite horizon; see Section 3. It remains only to verify condition (9), namely,

D(i) ~, D(i)

i=oV(i) = ~ ID(J) 1 = ~ i=~ ~ j= i -- I(j)(1 +p)-J e

Since I(j) > O, it is sufficient to show that

D(i) OOD. : ~ . i=o Zj=i (J)

But this follows from Theorem 1. To see this, consider another firm with the same dividend stream D(i) and E(i)- 0. It is easy to verify that this firm satisfies Assumptions (A1)-(A3). Since n(i)=n o for this firm, we can conclude


1 = - - l i m , ~ oo V(t) = 0, whe re V(t) = ~,i~176 D(i) is the va lue of th is f irm. In v iew of (7),

1,1 o the s h a r e p r i ce equa l s the p r e sen t va lue of the d i v i d e n d s a c c r u i n g to it. By T h e o r e m 1, therefore , this f i rm sat isf ies c o n d i t i o n (9), name ly ,

, : o V ( i ) = i=o Zj~iD(J) ~ '

w h i c h is w h a t we w a n t e d to es tab l i sh .

References

1. Malliaris, A. G., Brock, W. A.: Stochastic methods in economics and finance. New York: North- Holland, 1982

2. Miller, M. H, Modigliani, F.: Dividend policy, growth and the valuation of shares. J. Business 34, 411-433 (1961)

3. Sethi, S. P., Derzko, N., Lehoczky, J.: Mathematical analysis of the Miller-Modigliani theory. Oper. Res. Lett. 1, 148 152 (1982)

4. Sethi, S. P., Derzko, N., Lehoczky, J.: General solution of the stochastic price-dividend integral equation:a theory of financial valuation. SIAM J. Math. Anal. 15, 1100-1113 (1984)

5. Sethi, S. P., Derzko, N., Lehoczky, J.: A stochastic extension of the Miller-Modigliani framework. Mathematical Finance 1, 57-76 (1992)

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