includes the price-dividend balance integral equation whose solution yields the time ... significant stochastic extension of the Miller-Modigliani theory of financial ...
SIAM J. MATH. ANAL.
(C)
Vol. 15, No. 6, November 1984
1984 Society for Industrial and Applied Mathematics 0O5
GENERAL SOLUTION OF THE STOCHASTIC PRICE-DIVIDEND INTEGRAL EQUATION: A THEORY OF FINANCIAL VALUATION* S. P. SETHI
,
N. A. DERZKO :
AND
J. LEHOCZKY
Abstract. This paper deals with the problem of the financial valuation of a firm and its shares of stock with given financing policies in a general stochastic environment. A model of the firm is described which includes the price-dividend balance integral equation whose solution yields the time path of the share price, the number of outstanding shares and the value of the firm. These are shown to be the unique conditional expectations of certain stochastic processes. A broad class of firms for which the solution formula yields finite valued solutions is characterized. This paper represents a rigorous mathematical treatment, as well as a significant stochastic extension of the Miller-Modigliani theory of financial valuation. It is also shown that the cash-flow approach and the dividend approach to valuation of a firm are not equivalent in general. A precise condition, which makes them equivalent, is also obtained.
Introduction. In this paper, we generalize the earlier work [1] of the first two authors in order to study the valuation of a firm in a general stochastic environment. The financing policies of the firm are defined by a pair of real-valued stochastic processes denoting the rates of total dividends paid out and the external equity raised at each time t_>0. The total dividend process is assumed to be nonnegative. A positive value of the external equity at time implies that the firm is issuing new stock at that time, while a negative value means that the firm is buying back its own stock. All transaction costs are assumed to be zero in the model. The rate of dividend per share at time is given by the rate of total dividends divided by the number of outstanding shares at that time. The rate of change in the number of outstanding shares at time is given by the total rate of external equity divided by the price of a share at that time. With the initial number of outstanding shares being given, the above procedure defines the stochastic process denoting the number of outstanding shares over time, provided that the price per share over time is known. The crucial piece of information for the valuation problem is, therefore, the price per share over time. This requires some assumptions about the economy in which the firm operates. We shall assume that there exists, in the economy, a spot rate of interest at which money can be borrowed or lent. The interest rate process will be assumed to be a stochastic process. If the agents in this economy are risk-neutral, then the price of a share at time can be defined as the expected total discounted value of future dividends per share payments given the information available through time t. In the absence of the risk-neutrality assumption, the reformulation of the problem is accomplished by taking the expectation with respect to an appropriate probability measure that is absolutely continuous with respect to the given underlying probability measure. This idea will be discussed in detail in 5. For now, it suffices to state that the
*Received by the editors June 30, 1982, and in revised form May 10, 1983. of Management Studies, University of Toronto, Toronto, Ontario, Canada M5S 1V4. The research of this author was supported in part by the Natural Sciences and Engineering Research Council under grant A-4619. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S IV4. Department of Statistics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. The research of this author was supported in part by the National Science Foundation under grant ECS-8101576.
tFaculty
il00
10 |
STOCHASTIC PRICE-DIVIDEND INTEGRAL EQUATION
entire analysis in this paper remains valid for any probability measure that is absolutely continuous with respect to the given measure. Within this context, the integral equations governing the price per share and the number of outstanding shares are developed and solved under fairly minimal assumptions. Moreover, we assume that some market processes, and other information processes, have influence on the dividend, external equity and discount rate processes. These could be economic indicators, technological forecasts, the announcement of the firm’s future plans, etc. These are also taken into account implicitly in valuing the firm and its shares. This paper represents an important advance over the seminal work of Miller and Modigliani [4] (MM hereafter). It provides a rigorous mathematical foundation for the MM theory in a general stochastic environment. MM claimed that the cash flow approach and the dividend approach to valuation are equivalent. This is not true in general. In fact, we show that the cash flow approach can provide valuation for a larger class of firms than can the dividend approach. We also provide the precise additional restriction under which the two approaches are equivalent. In the next section, we specify the notation. The model is developed in 2. The solution of the model and the main results of the paper are obtained in 3. The financial interpretations and discussion of results are provided in 4. In 5, we discuss how the model can be extended to more general economies. 6 concludes the paper.
-
,
,
1. Notation. Let (f, ,r} denote the underlying probability space and let T= [0, oo). Let E l(f, or) denote the space of integrable random variables over (f, or). Let the nondecreasing family {6"3t, T} of sub-a-algebras be given and be right-continuous. Assume that 0 consists of f and all the r-null sets, and 2"o --=o{ t_JtT ) T }, (E(t), T ) and (p(t), T } be real-valued right-continu,,ous Let { D(t), adapted stochastic processes defined on the probability space; these represent, respec-
tively, the rate of total dividends issued by the firm at t, the rate of total external equity raised by the firm at and the interest rate at t. Furthermore,
(1.1)
0_0, we conclude that M(t) is a martingale. This completes the proof. We have now shown that the cash flow system and V-arbitrage system are equivalent. In Theorem 3, we derive a formula for N(t) in terms of E(’) and V(’), 0_0, i.e., (B2) holds. To obtain (B1), however, the price formula (3.9) is not suitable. So we derive an alternate formula for P(t). For this we need to define processes F(t) and R(t), which will be interpreted economically in {}4. We let F(t)=--exp
(3.27)
fo D(s)
where we note that D(s)/V(s) represents the dioide#d yieM at time s. Since F(t) is of bounded variation and M(t) in (2.4) is a martingale, it follows [9, Thm. 1.2.8, p. 26] that
(3.28)
g(t)-M(O)t(o)+fotF(s)dM(s)-M(t)t(t)-foM(s)df(s )
is a martingale.
Substituting for M(t) and F(t) and simplifying gives
(3.29)
R(t)=P(t)F(t)--P(t)
exp[fotD(S)v(s) ds]
is amartingale.
Moreover, it is obvious that R(t) is a positive E -martingale, and so it converges. Thus
R(m)= ,--,lim P(t) exp
(3.30)
V(s) which, it should be noted, is uniquely defined and ,-measurable. Moreover, NR(m) =R(t) and R()-R(O)-P(O). From (3.29), therefore,
It follows that
(3.32)
lira/;(t)--0**f0
t-.oo
D(s) ds= lnF(oo) = oo V(s)
i.e.,
(B1) holds.
This completes the proof.
4. Financial interpretations and discussion of results. In this section, first we provide the financial interpretation of formula (3.7) for the value of the firm and the price formula (3.31). Then, we discuss the significance of Theorem 6 and condition (B 1). The interpretation of (3.7) is that the value of a firm at time is the conditional expectation of the total discounted future net cash outflow from the firm to the society, given the information available by time t. Formula (3.7) can also be interpreted as the expected present value of the total future dividends accruing to the stockholders of The integral of the first term in the integrand record at time conditioned on represents the expected total present value of dividends issued by the firm in the interval [ t, oo) given A portion of the total future dividends is obviously going to stock issued in the interval (t, oo). In the absence of arbitrage possibilities, the expected must equal ft E(,)d,. Clearly, the residual value of this portion, conditional on represented by the fight-hand side of (3.7), which belongs to the stockholders of record t, can now be interpreted as the present value of the firm at time t given
. . ,
t
.
||
l0
S.P. SETHI, N. A. DERZKO AND
.
LEHOCZKY
Moreover, for s
