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Working Papers Series
Intrinsic Bubbles: The Case of Stock
Prices A Comment
By: Lucy F. Ackert and William C. Hunter
Working Papers Series
Research Department
WP 99-26
Intrinsic Bubbles: The Case of Stock Prices
A Comment
Lucy F. Ackert* and William C. Hunter**
Forthcoming: American Economic Review (December 1999)
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Abstract
Some recent empirical evidence suggests that stock prices are not properly modelled as the
present discounted value of expected dividends and that empirical models incorporating nonlinear
bubble components better fit the data. In this paper we show that the nonlinearity in the
relationship between prices and dividends may arise from how managers choose dividend payout.
In particular, we propose a model of managed dividends which can explain observed long-term
trends in stock prices.
This model of managed dividends is shown to be observationally
equivalent to the popular intrinsic bubbles model.
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Intrinsic Bubbles: The Case of Stock Prices
A Comment
Deviations in stock prices from those predicted by the simple present value model based on
constant discount rates, ordinary cash dividends, and rational expectations appear to be substantial
and persistent over time. However, until a recent paper in this Review by Kenneth A. Froot and
Maurice Obstfeld (1991a), no other parsimonious model of stock price has found empirical
support. Froot and Obstfeld model stock price using a rational >intrinsic= bubble which depends
exclusively on economic fundamentals, i.e., aggregate dividends, and not on the extraneous or
extrinsic factors which often underlie bubble terms. Intrinsic bubbles are appealing because they
are able to generate persistent deviations from present-value prices, but the deviations are driven
exclusively by changes in fundamental value. Despite this appeal, the intrinsic bubbles model has
not ended the search for alternatives to the simple present-value model. These bubbles are
arbitrary and problematic in that their existence depends on rather stringent assumptions about
investor behavior and the dynamic inefficiency of the economy. Froot and Obstfeld assert that
"(e)ven if one is reluctant to accept the bubble interpretation, the apparent nonlinearity of the
price:dividend relation requires attention" (1991a, p. 1208).
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Our purpose is to propose another interpretation of their results. However, there is an important
theoretical distinction between their alternative hypotheses and the model of dividend regulation
we outline. The model we offer is not suggestive of short-run speculative profit opportunities nor
does it imply that Athe market is literally stuck for all time on a path along which price:dividend
ratios eventually explode@ (Froot and Obstfeld, 1991a, p. 1190). To explain the apparent
nonlinear relationship between stock prices and dividends, we appeal to observed managerial
behavior.
There is no generally accepted theory of optimal dividend policy. In fact, the pioneering
work of Merton H. Miller and Franco Modigliani (1961) shows that dividend policy is irrelevant
in the absence of taxes and transactions costs. John Lintner's (1956) classic study, which suggests
that dividends are a distributed lag on earnings, provided a foundation for our understanding of
how firms choose dividends. Empirical studies by Eugene F. Fama and Harvey Babiak (1968),
R. Richardson Pettit (1972), Ross Watts (1973), Marsh and Merton (1987), and Bong-Soo Lee
(1996) provide empirical support for Lintner=s model though models of economic behavior that
predict dividend smoothing by managers have only recently been proposed by Vincent A. Warther
(1994) and Drew Fudenberg and Jean Tirole (1995). Yet, Robert J. Shiller (1984) points out that
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that is able to explain the relationship between prices and dividends. Although our model may
not seem to appropriately describe dividend policy at the firm-specific level, it is a reasonable
representation for an aggregate index of firms. See Marsh and Merton (1987) for motivation of
studies on aggregate-dividend behavior.
In our model price is a function of fundamentals alone; however, fundamental values are
unobservable. Instead, we observe a managed dividend series. We show that the observable
effect of dividend control on the price-dividend relation is identical to the effect of intrinsic
bubbles. Thus, the nonlinear relation between prices and dividends may be attributable to how
managers choose to manage dividends which, in effect, makes Froot and Obstfeld=s intangible
bubble tangible. This result improves our understanding of the relationship between prices and
dividends.
The note is structured as follows. In section I, we posit a simple discounted present value
model of stock price. Section II provides concluding remarks.
I.
Dividend Control in a Present-Value Model
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process given in equation (1) is Markov and the probability distribution of Dt increments depends
only on its current level and the parameters.
The intrinsic value of the firm's shares under the present-value model is obtained by
infinity
P t = E[
∫D
t +s
- ks
e ds]
0
discounting the expected future dividend stream, i.e.,
where Pt is the time t stock price, E is the expectations operator, and k is the discount rate. With
a constant growth rate in dividends (µ) and in the absence of dividend regulation by management,
Pt =
Dt
k -µ
equation (2) has the familiar simple solution
where k > µ. In the case of dividend regulation or control, a problem arises when evaluating the
present value relation in equation (2) because the true dividend or fundamental {Dt} is
unobservable. Instead, we observe the managed dividend process {dt}.2
The dividend management process we envision is one in which actual cash dividends
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supported, the dividend is reduced to a level that is consistent with the permanently lower earnings
capacity. Similarly, if future earnings capacity is deemed sufficient to support a permanently
higher dividend, the dividend is increased above the current level. Earnings in excess of total
dividends paid are retained in the firm at a given reinvestment rate.
As is well known, most firms exhibit a bias against lowering cash dividends which
suggests that managers place a lower bound on the level of cash dividends. This bound represents
a barrier below which management is reluctant to reduce dividends, even when earnings capacity
is consistent with a lower payout level. The firm may resort to liquidating assets in order to
maintain the level of dividend payment. However, if the level of dividend payment cannot be
supported by earnings, management may choose to shift the lower barrier downward to
accommodate the change in fundamentals. Ordinarily, this lower barrier serves as a reflecting
barrier for the dividend process.
On the up side, evidence regarding a firm=s actual cash payments suggests that
management has the flexibility to increase dividends if earnings capacity supports the increase.
However, beyond some level, dividend increases must be justified by strong evidence of a
permanent increase in the firm=s earnings capacity. The upper bound for the dividend process can
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cash dividends has lower, l, and upper, u, reflecting barriers. In this specification, {dt} is a
regulated geometric Brownian motion process with dynamics given by equation (1) in the absence
of regulation. As a result of dividend management, when the dividend process {Dt} reaches the
upper barrier (u) or the lower barrier (l), the observed dividend process {dt} is reflected back
towards the interior of the band [l, u]. Thus, the dynamics of dividends suggested by this model
are quite different from those suggested by Lintner.
Following Samuel Bentolila and Giuseppe Bertola (1990, page 386), the managed dividend
d t = Dt U t / Lt
stream {dt} can be related to the true dividend process {Dt} as follows
where {Lt} is a lower regulator defined as the unique, nondecreasing, continuous process
which increases only when dt equals l keeping dt $ l and {Ut} is an upper regulator defined as
the unique, nondecreasing, continuous process which increases only when dt equals u keeping
dt # u (see also J. Michael Harrison, 1985, page 20). In our model, the barriers, u and l, are
exogenously specified.
The stock price can be expressed as a function of managed cash dividends following
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where A and B are constants determined by boundary conditions and ß1 and ß2 are the positive and
Q ≡ 1/2 σ 2 β ( β - 1) + (k - µ )β - k = 0,
negative roots to the quadratic equation
and the other variables are as previously defined. Our formulation rests on the simplifying
assumption that incremental retained earnings resulting from the dividend management process
earn exactly the capitalization rate, k. In this case, a stockholder is indifferent between receiving
the incremental earnings (which may be negative) and the capitalized cash flow in some future
period. On this indifference see Myron J. Gordon (1962, Chapter 5). As the reader can verify,
equation (5) is equivalent to the general solution provided by Froot and Obstfeld (1991a, footnote
8, page 1192) for their intrinsic bubbles model. Although our general solution includes two
nonlinear terms, Froot and Obstfeld exclude the second nonlinear term in their estimation because
the estimate of the second term was imprecise and its inclusion did not contribute to explaining
movements in stock price. In our empirical examination of the model we also find that estimates
vary widely, though the coefficients of both nonlinear terms are significantly different from zero
in some sample periods (Ackert and William C. Hunter, 1996).
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type of model may be more realistic, the analysis of such models is extremely complex.
II.
Concluding Remarks
Froot and Obstfeld proposed a model of stock price that includes intrinsic bubbles and
showed that the intrinsic bubbles model is superior to a simple constant growth rate model in
predicting changes in actual stock prices. Their model is better able to track changes in actual
stock prices because of the inclusion of a nonlinear bubble. However, as Froot and Obstfeld
recognize, the rational bubbles specification is not the only one that can explain stock price
movements. In this extension of their analysis, we show that the nonlinearity in the relationship
between prices and dividends may arise from how managers choose dividend payout.
In
particular, we propose a model of managed dividends which can explain observed long-term
trends in stock prices. In contrast to Froot and Obstfeld, the long-term trends implied by the
model we develop do not depend on bubbles, but instead result from observed management
behavior. As Froot and Obstfeld=s intrinsic bubbles model and the dividend control model
described herein are observationally equivalent, the same gains in predicting actual stock prices
that arise in the bubbles model can be derived from the model of dividend control.
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References
Ackert, Lucy F. and William C. Hunter. AAn Empirical Examination of the Price-Dividend
Relation.@ Working paper, Federal Reserve Bank of Atlanta, November 1996.
Ackert, Lucy F. and Brian F. Smith. "Stock Price Volatility, Ordinary Dividends, and Other
Cash Flows to Shareholders." Journal of Finance, September 1993, 48(4), pp. 1147-1160.
Bagwell, Laurie Simon and John B. Shoven. "Cash Distributions to Shareholders." Journal of
Economic Perspectives, Summer 1989, 3(3), pp. 129-140.
Bentolila, Samuel, and Giuseppe Bertola. AFiring Costs and Labour Demand: How Bad Is
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Driffill, John, and Martin Sola. AIntrinsic Bubbles and Regime Switching.@ Journal of Monetary
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Froot, Kenneth A. and Maurice Obstfeld. "Intrinsic Bubbles: The Case of Stock Prices."
American Economic Review, December 1991a, 81(5), pp. 1189-1214.
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Harrison, J. Michael. Brownian Motion and Stochastic Flow Systems. New York: John Wiley
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Krugman, Paul R. ATarget Zones and Exchange Rate Dynamics.@ Quarterly Journal of
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Svensson, Lars E.O. "An Interpretation of Recent Research on Exchange Rate Target Zones."
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Notes
1. See Froot and Obstfeld (1991a), footnote 4, in particular, where they discuss models of fads
and bubbles that are consistent with their results. John Driffill and Martin Sola (1998) show that
a stock price formulation based on a dividend switching model better explains stock prices than
a bubble model. Our model can be viewed as another interpretation and is compatible with Froot
and Obstfeld=s conceptualization.
2. The observed, ordinary cash dividend is not the only cash flow received by shareholders. The
finance literature has long recognized that firms distribute cash flows to shareholders through
other methods (Miller and Modigliani, 1961). The importance of other cash payments to
shareholders, in addition to ordinary cash dividends, is well documented (see, for example, John
B. Shoven, 1987; Laurie Simon Bagwell and Shoven, 1989; Lucy F. Ackert and Brian F. Smith,
1993). The difference between the fundamental and ordinary cash dividend may reflect other cash
distributions such as share repurchases and takeover distributions, among others.
3. The solution given in (5) requires that Paul R. Krugman=s (1991) Asmooth pasting@ conditions
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