Dividends: Relevance, Rigidity, and Signaling∗
Sigitas Karpavičius†
August 9, 2012
Abstract
This paper uses a dynamic general equilibrium model to explain a puzzle of
dividend smoothing. In contrast to the Modigliani-Miller theory, I show that
firm value depends on payout policy. The analysis implies that firms with
more stable dividend policy are more valuable. This explains why dividends
are rigid over time. Firms use share repurchases and special dividends, in
addition to constant regular cash dividends, in order to reduce the likelihood
of dividend cuts in bad times while keeping the same historical average payout.
However, I do not find support for dividend signaling theory, rather, results
depend on a firm’s performance measure and the environment in which a firm
operates.
Key words: Payout Policy; Dividends; General Equilibrium Model
JEL classifications: G35; D21; D58
∗
I would like to thank Takeshi Yamada, Fan Yu and seminar participants at the University of
Adelaide, Flinders University, and 3rd Conference on Financial Markets and Corporate Governance
for their helpful comments and suggestions.
†
Sigitas Karpavičius is at Flinders University.
Address: Flinders Business School, Flinders University, GPO Box 2100, Adelaide SA 5001,
Australia. E-mail: sigitas.karpavicius@flinders.edu.au. Phone: +61 8 8201 2707. Fax:
+61 8 8201 2644.
1
Introduction
Approximately 90% of CFOs agree or strongly agree that they smooth dividends
from year to year and try to avoid reducing dividends (Brav, Graham, Harvey,
and Michaely, 2005). Dividend smoothing behavior was also recorded by other
surveys (Baker and Powell, 1999; Bernheim, 1991; Lintner, 1956). According to the
survey, managers believed that “the market puts a premium on stability or gradual
growth in rate” of dividends (Lintner, 1956). In addition, empirical studies document
dividend smoothing behavior. In approximately 80% of cases, firms do not change
their quarterly dividends (see Aharony and Swary, 1980; Loderer and Mauer, 1992;
Nissim and Ziv, 2001). The number reduces to 25% for annual dividend per-share
changes from 1966 to 2005 for all Compustat firms (Guttman, Kadan, and Kandel,
2010).
There were several attempts to explain the puzzle of rigid dividends. One stream
of literature argues that firms use dividends as a costly signal regarding future
earnings prospects (Bernheim, 1991; Bhattacharya, 1979; John and Williams, 1985).
Thus, firms tend not to decrease dividends as it would be a bad signal. However,
these studies fail to explain why firms use dividends but not share repurchases to
signal and why firms smooth dividends.
Allen, Bernardo, and Welch (2000) find that some firms prefer to pay dividends
rather than repurchase shares due to clientele effect. When institutional investors
are relatively less taxed than individual investors, firms paying dividends attract
more institutions. Allen, Bernardo, and Welch (2000) argue that their static model
is able to explain dividend smoothing; however, they do not show this in a dynamic
framework. The model in Kumar (1988) shows that small changes in productivity
do not lead to dividend changes. Similarly, Garrett and Priestley (2000) assume
1
managers minimize the costs of adjusting dividends toward the target dividend and
find that for an unexpected increase in permanent earnings, dividends will increase
by less than one third of the increase in permanent earnings. Thus, the findings
of Garrett and Priestley (2000) and Kumar (1988) are consistent with the concept
of rigid dividends but these studies cannot explain this phenomenon. Guttman,
Kadan, and Kandel (2010) argue that managers use a partially pooling dividend
policy according to which the same dividend is paid for a range of different earnings
realizations. However, Guttman, Kadan, and Kandel (2010) use the two-period
model and do not distinguish between dividends and share repurchases.
In the recent study, Lambrecht and Myers (2012) use a dynamic agency model
to explain dividend smoothing. They argue that managers smooth total payout
(dividends plus net repurchases) in order to smooth their rents. Lambrecht and
Myers (2012) show that the payout-to-rent ratio is constant in equilibrium. The
authors assume that manager’s utility function features habit formation process and
that managers are financially constraint meaning that they do not have any savings.
The assumptions imply that managers prefer non-volatile, smooth consumption. In
order to achieve it, managers need to have a smooth flow of rents that leads to
smooth payout to shareholders.
Leary and Michaely (2011) divide the existing theoretical models on dividend
smoothing into two groups: models based on information asymmetry and agencybased models. Leary and Michaely (2011) argue that empirical data of US industrial
firms provide more support for the models based on agency costs of free cash flows.
Leary and Michaely (2011) find that dividend smoothing is more prevalent among
financially unconstrained firms with weaker corporate governance and low levels of
asymmetric information.
2
This paper provides an alternative explanation for why dividends tend to be
constant from year to year that is unrelated either to models based on information
asymmetry or models based on agency costs. Besides dividend smoothing, I explain
several other stylized facts about dividends and payout policy. Specifically, the goal
of this paper is: a) to test whether firm value depends on payout policy; b) to explain
the puzzle of dividend smoothing; c) to analyze dividend information content.
The main methodological tool employed in the analysis is a dynamic stochastic
general equilibrium model.1 The model is derived from microeconomic principles
(i.e. utility maximization) and describes the behavior of the firm as a whole by
analyzing the interaction of several firm’s decisions. The model replicates the simplified behavior of a real profit-seeking firm. I consider an infinitely lived firm in
discrete time. The model assumes that the firm’s manager acts in the best interest
of current shareholders. A manager maximizes a certain objective function that
positively depends on equity value subject to the evolution of shareholder value and
asset composition of a firm. The model incorporates the main items of balance sheet
and income statement. Those items are endogenous variables. The model includes
also nine stochastic processes (shocks or exogenous variables), such as technology,
interest rate, or corporate income tax rate. In each period, to respond to the changes
in the environment (i.e. shocks), the manager needs to make several decisions regarding production volume and price, investment, the amount of raw materials used
in production, debt level, and share issues/repurchases. The optimal choices of the
manager are expressed by first-order conditions. Thus, the model consists of the
evolution of shareholder value, asset composition constraint, first-order conditions,
several variable definitions, and exogenous processes. The relationship among all
endogenous variables and their dynamics are jointly determined in equilibrium. The
1
Similar models are extensively used in asset pricing and macroeconomics; however, to my best
knowledge, there have been no attempts to adopt them in corporate finance.
3
solution of the model is a unique stable rational expectations equilibrium. The
dynamics of endogenous variables are expressed by the policy functions that are
linearized functions that depend on magnitude of shocks and past values of endogenous state variables that are those endogenous variables which appear at the
previous period.2 The model is calibrated assuming that the variables are measured
quarterly.
Such a model is superior to traditional methods used in the theoretical corporate finance research due to several reasons. First of all, the model reflects the
behavior of a real firm. Its manager maximizes shareholder value and in each period
makes operating, financial, and investment decisions. Thus, the dynamic of endogenous variables reflects the optimal managerial decisions that are consistent with the
shareholder wealth maximization. Secondly, the model is dynamic. In contrast,
many theoretical models used in corporate finance research are either static or twoperiod. The dynamic models allow to analyze the behavior of firms during the long
time period and draw the conclusions from simulations. More importantly, the dynamic models are superior in most cases as the impact of a shock on an endogenous
variable could be different in the short-term and in the long-term. In addition, static
or two-period models might suffer from a manager’s myopic behavior. A dynamic
setting allows to introduce long-term incentives for the firm’s manager. Thirdly, the
model includes several stochastic processes in order to make model more realistic.
Usually, dynamic models rely on a single technology shock or assume that share
price or profit follow Brownian motion. In this paper, I use simulations corresponding to a random draw of shocks to analyze the long-term performance of firms with
2
The model takes into account endogeneity issue. The dynamic of endogenous variables is
explained by the exogenous processes and past values of endogenous state variables. From the
perspective of instrumental variable approach widely used in corporate finance research, those
endogenous variables, which appear in the model at the previous period, can be viewed as instrumental variables.
4
different payout policies. Shocks impact endogenous variables differently. Let’s consider technology shock. It affects production level directly: positive shock increase
production output, and vice versa. But how does it affect leverage? The effect will
be indirect and will be determined by other endogenous variables such as net income, current debt level, effective interest rate etc. The effect on leverage might be
positive, might be negative and it is also possible that leverage might be unaffected.
If we consider shock to interest rate then the impact on leverage would be direct and
straightforward; however, it would be difficult to say how productive capital stock
would be affected. So if a model includes only one exogenous process, it is likely that
simulations using random draw of shocks might not be able to show the true impact
of parameter of interest on long-term firm performance. Further, the simulations
will be unrealistic as firms are affected not only by exogenous changes in technology but by many other factors. Since the majority of shocks impact payout policy
indirectly, the model that includes several shocks is preferred to one with single
exogenous process. Fourthly, the manager is assumed to have rational expectations
about the future; therefore, the solution of the model is a rational expectations equilibrium. Fifthly, the equilibrium relationship between any two endogenous variables
is determined by structural parameters that are policy-invariant. The dynamic of
the variables depends on the shock pattern over certain time period. If shocks are
different enough during two time periods, it possible that the correlation coefficient,
that reflects the dynamic relationship, between two endogenous variables, for example, dividends and leverage, will be positive in one period and negative in the other
period. This could help explain why the relationship between endogenous variables
changes over time. Finally, such models can be used to generate impulse response
functions, run simulations, compute theoretical moments, and variance decomposition of endogenous variables. These results help validate the model, especially if the
5
number of shocks is sufficiently large.
The model allows us to examine the impact of payout policy on firm value within
the dynamic framework. I assume that dividends per share are the weighted sum
of constant amount of cash and net income per share. This implies that dividends
consist of constant and variable parts. I assume that the weights are set by the
board of directors or shareholders and the firm’s manager takes them as given and
cannot alter them. The definition of dividends is consistent with the views of CFOs.
Brav, Graham, Harvey, and Michaely (2005) report that 40% of CFOs claim that
they target dividends per share. In constrast, only 28% of the responded CFOs
say that they target dividend payout, and 27% of the survey respondents target
growth in dividends per share. Thus, I assume that the constant part of dividends
is a certain amount of cash distributed to shareholders each period. Lintner (1956)
reports that current earnings are the most important factor of a firm’s decision to
alter existing dividend yield. Therefore, I set the variable part of dividends per
share equal to net income per share multiplied by its weight. One can interpret the
latter part of dividends as special dividends and share repurchase that is followed
by the respective stock split (in order to keep the number of shares outstanding
unchanged).
The definition of dividends and the functions defining the manager’s optimal
decisions imply that share price is equal to the discounted sum of constant parts of
dividends and is not impacted by the variable part. Thus, firm value depends on
payout policy. The result is in contrast to the Modigliani-Miller theory (Miller and
Modigliani, 1961) but is consistent with DeAngelo and DeAngelo (2006). This result
helps explain why dividend-paying firms tend to keep their dividends constant. The
board of directors chooses dividend policy that maximizes firm value. Since equity
value only depends on constant part of dividends, firms tend to smooth dividends
6
(that is to set the weight of constant part of dividends to a value close to one
keeping the dividend-to-price ratio constant) in order to maximize share price. The
result that dividend-smoothing firms are more valuable is consistent with the survey
conducted by Lintner (1956).
Despite share price is equal to the discounted sum of constant parts of dividends,
it does not mean that if a firm increases dividends, it’s stock price would increase as
well. I show that share price and so value of the firm depend on firm’s net income
which is determined by several other factors, such as productivity. Thus, to improve
firm value, the manager should maximize firm’s intertemporal earnings rather than
increase dividends.
Further, I analyze why firms use special dividends and share repurchase in addition to rigid cash dividends. If a firm chooses not to use special dividends and share
repurchases, then, according to the definition of the model, a firm commits to pay a
constant stream of cash dividends in both good times and bad times. There could be
a hypothetical long period of time with adverse environment and continuous losses.
And if a firm continues to pay dividends, it increases its default likelihood. Thus,
it is more likely that a firm will cease paying dividends or will at least cut them. I
use simulated dataset and show that greater weight of constant amount of cash (or
lower weight of variable dividend part) leads to higher default probability. It is more
realistically that a firm would cut dividends rather than default. The model implies
that dividend cuts result in lower share prices. Thus, firms use alternative payout
mechanisms besides constant regular cash dividends to avoid dividend omissions/
cuts or to reduce their bankruptcy risk while keeping the same historical average
dividend yield. Thus, firms with more stable dividends are riskier. I show that
permanent or even temporal increases in the constant part of dividends, keeping the
amount of total dividends the same, lead to higher share prices. Previously, I find
7
that firms with more stable dividend stream are more valuable. Thus, a positive
risk-return trade-off exists.
Finally, I analyze dividend information content. I compute asymptotic autocorrelation coefficients between firm performance, proxied by either net income, net
income per share or share price, and dividends up to 12th lead for different values
of autocorrelation coefficients of all shocks. The results are inconsistent as in some
cases, autocorrelation coefficients are positive; however, in other cases, they are negative. Thus, I do not find support for dividend signaling theory. The results depend
on whether share price or net income are used as a firm’s performance measure, the
environment in which a firm operates (autocorrelation coefficients of the shocks),
and the lead value of autocorrelation. This suggests that dividends cannot be used
to predict a firm’s future performance and is in contrast to a large body of literature
on dividend signaling.
One possible reason why no support for dividend signaling theory is found is
the structure of the model. It is assumed that dividends depend on the current but
not expected future net income. In the model, the manager does not purposely do
any signaling to investors and does not derive any utility from using dividends as
a signaling device. Thus, this could be a reason why we do not observe a positive
correlation between dividends and future firm performance. The results above mean
that if we do not assume that managers signal the market using dividends, the
relationship between dividends and future firm performance is unlikely to be positive
as larger dividends suppress future growth opportunities and so negatively impact
future share price and profit.
The rest of the paper is structured as follows. Section 2 develops a dynamic
stochastic general equilibrium model. Obtained results are detailed in Section 3.
8
Finally, Section 4 concludes.
2
The model
In this section, I develop a dynamic stochastic general equilibrium model for a
firm. The model tries to replicate the life and behavior of a representative firm in
a dynamic world with a changing environment. I consider an infinitely lived firm
in discrete time. It is assumed that a firm’s manager has rational expectations
about the future and acts completely in the best interest of shareholders. In each
time period, the firm’s manager has to choose how much to raise capital in the
external equity and debt markets, how much to produce, and how much to invest
in capital stock (i.e. fixed assets used in production). The decisions are made with
respect to the changes in the environment that are defined by several shocks such
as productivity or interest rate shocks. The firm’s manager does not know when
the future shocks will occur but knows their distribution. Thus, the decisions of
the manager are made knowing that the future value of innovations are random but
will have zero mean. A firm produces a single tradable final good that is sold in a
competitive market.
2.1
A firm
I assume that the firm’s manager acts in the best interest of current shareholders
and maximizes a certain objective function that depends positively on shareholder
value, Et . An intertemporal objective function of the firm’s manager is given by:
E0
∞
X
β t exp (ζt ) Ut ,
t=0
9
(1)
where β is the subjective discount factor. ζt is the preference shock and is assumed
to follow an independent first-order autoregressive (AR(1)) process:
ζt = ρζ ζt−1 + ηtζ , where ηtζ ∼ N(0, σζ2 ).
(2)
The instantaneous objective function, Ut , is
Et1−σ
,
1−σ
Ut =
(3)
where Et is shareholder value and σ is the coefficient of constant relative risk aversion
(the inverse of elasticity of substitution).
The firm’s manager maximizes the objective function subject to the evolution of
book value of equity and asset composition of the firm:
b
m
m
Ptb Nt = Pt−1
Nt−1 + Pt−1
Nt − Pt−1
Nt−1 − ΦN
− ΦD
(Dt − Dt−1 )2
(Kt − Kt−1 )2
− ΦK
,
Dt−1
Kt−1
Kt = κ exp(κt )(Dt + Ptb Nt ).
(Nt − Nt−1 )2
− dt Nt−1 + πt
Nt−1
(4)
(5)
Ptb is book value of equity per share at time t and Nt is the number of shares
outstanding. The left hand side of Equation (4) is the book value of equity. Ptm is
m
m
the market value of equity per share. Therefore, Pt−1
Nt − Pt−1
Nt−1 is the proceeds
from issuance of common stock.3 The relationship between market value and book
m
m
If Nt < Nt−1 then Pt−1
Nt − Pt−1
Nt−1 is equal to the funds spent to repurchase shares and
will not be counted as a part of a firm’s payout. This term controls for market timing activities.
For example, if the share price falls below its fair value, a firm might decide to repurchase some
shares as it would be in line with the interests of shareholders. This should not affect the inferences
of this paper.
3
10
value of equity is given by:
Ptm = Ptb q exp(qt ),
(6)
where q is market-to-book ratio in the steady state.4 q can be viewed as a “firm“
premium and shows by how much the value of separate assets increases when they
are combined together. q > 0 means that assets have a higher value when combined
together than when they are separate. qt is shock to market-to-book ratio and
follows the AR(1) process:
qt = ρq qt−1 + ηtq , where ηtq ∼ N(0, σq2 ).
(7)
Positive qt means that stock is overvalued, and vice versa. Shareholder value, Et , is
equal to book value of equity multiplied by steady-state market-to-book ratio:
Et = qPtb Nt .
(8)
Et can be seen as a proxy for fair value of equity. It is assumed that the utility
function is not impacted by the shock to market-to-book ratio in order to generate
market-timing behavior of the firm’s manager. So if qt is positive, the manager
learns that stock is overvalued in the market and this will trigger an equity issue.
Similarly, the optimal response for a firm to negative qt is share repurchase. It is
assumed that the firm’s manager maximizes fair value of equity rather than market
value of equity to make the model more realistic and consistent to the empirical
evidence. If qt > 0 (i.e. stock is overvalued) then market value of equity maximizing
manager would never issue more shares. However, market timing is one of the key
factors determining firm’s capital structure (Baker and Wurgler, 2002).
4
In this paper, market-to-book ratio is computed as market value of equity over book value of
equity rather than market value of assets over book value of assets. The term “steady state” refers
to the deterministic steady state.
11
Dividends at time t, dt , are paid to those who owned shares at time (t − 1).
Thus, investors who purchase shares at time t are not entitled to receive dividends
in this period. πt represents net income. Dt−1 is debt a firm pays back in period
t; thus, Dt is new borrowing. For simplicity, it is assumed that debt consists of
one-period securities. Kt is capital stock at time t. The quadratic number of shares
2
2
t−1 )
t−1 )
, ΦD (Dt −D
, and
outstanding, debt, and capital adjustment costs, ΦN (Nt −N
Nt−1
Dt−1
2
t−1 )
ΦK (Kt −K
, are assumed in order to reduce volatility of the number of shares
Kt−1
outstanding, Nt , debt, Dt , and capital stock, Kt . ΦN ≥ 0, ΦD ≥ 0, and ΦK ≥ 0 are
the number of shares outstanding, debt, and capital adjustment cost parameters.
Due to quadratic debt adjustment costs, short-term debt becomes equivalent to
long-term debt with variable interest rates. Equation (4) implies that it is costly
for a firm to increase or decrease its capital stock. Quadratic capital (the number
of shares outstanding) adjustment costs are assumed in order to reduce volatility in
capital stock (the number of shares outstanding).
Equation (5) implies that a firm can invest only the κ exp(κt ) fraction of its
financial assets into capital stock. κ is a constant, and κt is the AR(1) process:
κt = ρκ κt−1 + ηtκ , where ηtκ ∼ N(0, σκ2 ).
(9)
If the shock is positive (κt > 0), a firm invests a larger portion of its financial
resources into capital stock, and vice versa.
Stock of physical capital, Kt , evolves according to:
Kt = (1 − δ)Kt−1 + It ,
where δ is the capital depreciation rate. It stands for investment.
12
(10)
Firm’s net income is given by:
πt = [St − exp(ct )Ct − δKt−1 − Dt−1 rt−1 ] × [1 − τ exp(τt )],
(11)
where St is sales revenue. Ct is an amount of production input (for example, labor
and raw materials). It is assumed that the unit cost of Ct is one. ct is the shock to
the input price:
ct = ρc ct−1 + ηtc , where ηtc ∼ N(0, σc2 ).
(12)
τ is corporate income tax in steady state. τt is shock to corporate income tax
that follows the AR(1) process:
τt = ρτ τt−1 + ηtτ , where ηtτ ∼ N(0, στ2 ).
(13)
rt is the interest rate for a debt obtained in time t, Dt . The interest rate at which
a firm can borrow funds evolves according to the following equation:
rt = r
∗
exp(rt∗ )
1 + Φr
Dt
Dt + Ptb Nt
,
(14)
where r∗ is a constant and equal to the hypothetical interest rate on corporate bonds
for firms with zero leverage in the steady state. rt∗ is shock to the interest rate on
long-term corporate bonds and follows the AR(1) process:
∗
∗
∗
rt∗ = ρr∗ rt−1
+ ηtr , where ηtr ∼ N(0, σr2∗ ).
(15)
The last term in Equation (14) is the risk premium related to a firm’s financial
leverage. Φr > 0 is the parameter of risk premium. The definition of interest
rate implies that it is an increasing function of a firm’s financial leverage. In the
13
model, debt has advantages (such as tax deductibility of interest expenses and lower
costs) and disadvatages (debt adjustment costs and increased bankruptcy risk). The
trade-off implies that there should be an optimal capital structure.
Sales revenue, St , is the product of output volume, Yt , and the price per output
unit, pt :
St = Yt pt .
(16)
The price per output unit depends on demand for a firm’s products and is given by
the following equation:
pt = exp(γt )p̄
Ȳ
Yt
η
,
(17)
where Ȳ and p̄ are respectively the demand for a firm’s products and market price
in the steady state.5 γt is the demand shock that affects product price:
γt = ργ γt−1 + ηtγ , where ηtγ ∼ N(0, σγ2 ).
(18)
Parameter η is price elasticity of demand.
To produce a single tradable good, a firm uses the following Cobb-Douglas technology:
α
Yt = A exp(At )Kt−1
Ct1−α ,
(19)
where A is the total factor of productivity. At is the productivity shock that follows
the AR(1) process:
At = ρa At−1 + ηta , where ηta ∼ N(0, σa2 ).
(20)
α is capital share. Equation (19) implies that production output is the increasing
5
Throughout this paper, variables with bars denote steady-state values.
14
function of capital stock and other production inputs.
To analyze the rigidity of dividends, I assume that dividends, dt , consist of
constant and variable parts. The constant part is equal to the steady-state dividends,
¯ multiplied by ψ exp(ψt ). It is equivalent to a certain amount of cash per share
d,
distributed to shareholders at the end of each period. Lintner (1956) argues that
current earnings are the most important determinant of a firm’s decision to change
existing dividend yield. Thus, I assume that the variable part of dividends per share
is net income per share multiplied by its weight, [1 − ψ exp(ψt )]:
πt
,
dt = ψ exp(ψt )d¯ + [1 − ψ exp(ψt )]
Nt−1
(21)
where ψ is the weighting parameter and exp(ψt ) is its shock:
ψt = ρψ ψt−1 + ηtψ , where ηtψ ∼ N(0, σψ2 ).
(22)
One can interpret the constant part of dividends as normal cash dividends. The
variable part of dividends consists of special dividends and share repurchase that
is followed by the respective stock split (in order to keep the number of shares
outstanding unchanged). The definition of dividends implies that dividends per
share are the weighted sum of constant amount of cash and net income per share.
It is assumed that the magnitude of weighting parameter is chosen by the board
of directors or shareholders and the manager does not participate in choosing the
value of ψ but takes it as given. Thus, the composition of dividends is determined
exogenously from the perspective of managers.
In each period, the firm’s manager observes the weight of constant dividend part
as well as the values of other shocks and chooses strategy {Ct , Kt , Nt , Dt }t=∞
t=0 to
15
maximize her expected lifetime utility subject to constraints (Equations (4) and (5)),
initial values of debt, capital stock, share price, the number of shares outstanding
and a no-Ponzi scheme constraint of the form:
Dt+j
≤ 0.
i=1 (1 + ri )
lim Et Qj
j→∞
(23)
Maximization of objective function (Equation 1) subject to the evolution of
shareholder value and asset composition of a firm (Equations (4) and (5)) yields
the following first-order conditions:
∂
: exp(ct )Ct = (1 − α)(1 − η)St ,
∂Ct
(24)
−σ
Kt
exp(ζt ) (q)1−σ
Kt
1
∂
:
− Dt
+ λt −
− 2ΦK
−1
∂Kt
κ exp(κt )
κ exp(κt )
κ exp(κt )
Kt−1
(
"
2
1
q exp(qt ) Nt+1
Kt+1
+ βEt λt+1
− 1 + ΦK
+
κ exp(κt ) κ exp(κt )
Nt
Kt
St+1
−ΦK + ψ exp(ψt+1 ) (1 − τ exp(τt+1 )) α(1 − η)
−δ
Kt
2 ##)
Dt
∗
∗
+Φr r κ exp(rt ) exp(κt )
= 0,
(25)
Kt
Kt−1
∂
q exp(qt−1 )
Nt
: λt
− Dt−1 − 2ΦN
−1
∂Nt
Nt−1
κ exp(κt−1 )
Nt−1
Nt+1
Kt
= βEt λt+1 q exp(qt )
− Dt
(Nt )2 κ exp(κt )
#)
2
Nt+1
−ΦN
+ ΦN + ψ exp(ψt+1 )d¯ ,
Nt
16
(26)
−σ
∂
Kt
Dt
1−σ
: − exp(ζt ) (q)
− Dt
−1
+ λt 1 − 2ΦD
∂Dt
κ exp(κt )
Dt−1
(
"
"
#
2
Nt+1
Dt+1
= βEt λt+1 1 + q exp(qt )
− 1 − ΦD
−1
Nt
Dt
Dt
∗
∗
,
+ ψ exp(ψt+1 ) [1 − τ exp(τt+1 )] r exp(rt ) 1 + 2Φr κ exp(κt )
Kt
(27)
where λt is a Lagrange multiplier. Equation (24) defines the optimal level of production input, Ct , and Equations (25)-(27) are Euler conditions.
The equilibrium of the model is defined by the evolution of shareholder value,
asset composition constraint, first-order conditions, several variable definitions (in
total 15 equations) and nine shocks.6 The number of endogenous variables is equal
to the number of equations; thus, the model can be solved. First of all, I analyze
the properties of the model assuming a non-stochastic environment. It would help
to understand long-term equilibrium relationships among the model’s variables. To
do so, I solve for the non-stochastic steady state of the model by using the following
procedure: all shocks are set to zero, the time subscripts are dropped, and the steadystate values of each endogenous variable are expressed in terms of parameters.
When all shocks are set to zero and the time subscripts are dropped, the model
reduces to 13 equations: Equation (17) cancels out and the steady-state expressions
of Equations (4) and (21) are identical. To express the steady-state values of each
endogenous variable in terms of parameters and constants, the number of endogenous
variables must be equal to the number of equations. Thus, I assume that steady¯ are known.
state values of the number of shares outstanding, N̄ , and dividends, d,
6
Specifically, the equilibrium of the model is defined by Equations (4)-(6), (8), (10), (11), (14),
(16), (17), (19), (21), (24)-(27) and nine exogenous processes: Equations (2), (7), (9), (12), (13),
(15), (18), (20), and (22).
17
Table 1: Calibration of the parameters
Panel A
2.2
Panel B
Coefficient
Value
Coefficient
N̄
d¯
q
A
κ
τ
r∗
α
η
β
ψ
ΦD
ΦK
ΦN
Φr
δ
σ
1
1
1.25
1
0.2
0.3
0.015
0.25
0.2
0.97
0.75
0.25
0.25
8
1
0.025
2
ρζ
ρq
ρκ
ρc
ρτ
ρr∗
ργ
ρa
ρψ
Panel C
Value
Coefficient
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
σζ
σq
σκ
σc
στ
σr ∗
σγ
σa
σψ
Value
0.30
0.05
0.02
0.10
0.02
0.30
0.10
0.05
0.05
Calibration
The model is calibrated assuming that the variables are measured quarterly. The
calibration of the model is summarized in Panel A of Table 1. I assume that steady¯ are equal to
state values for the number of shares outstanding, N̄ , and dividends, d,
one. The steady-state market-to-book value, q, is 1.25.
Quarterly discount factor, β, is set to 0.97. It corresponds to a 12% annual
discount rate or the approximate long-term average return on equity. The coefficient
of manager’s risk aversion, σ, is two.7 I assume that ψ is 0.75. It implies that the
weight of constant dividend is 0.75. Following macroeconomic literature, quarterly
7
Murphy (1999) uses 1, 2, and 3 as a value of relative risk aversion. I repeat the analysis with
other two values and find that different values of σ do not affect the results.
18
capital depreciation rate, δ, is set to 0.025. The total factor of productivity, A, is
one. Capital share in the production function, α, is equal to 0.25. Further, I assume
that a firm can invest only 20% of its financial resources in the productive capital
(κ = 0.2).
The steady-state quarterly interest rate on corporate bonds, r∗ , is set to 0.015.
It implies that the hypothetical annual interest rate for firms without debt is 6%.
Thus, debt financing is cheaper than equity financing. The corporate income tax
rate, τ , is 0.3. I assume that price elasticity of demand, η, is 0.2. It implies that
if the production supply increases by 10%, the sale price decreases by 2%, and vice
versa. The parameter of risk premium, Φr , is set to one. It implies that if a firm’s
leverage increases by one percentage point, the interest rate will increase by 1.5 basis
points if r∗ is 0.015. To introduce some persistence in the artificially generated time
series, I calibrate debt and capital adjustment cost parameters, ΦD and ΦK , to 0.25.
In unreported simulations, I found that the number of shares outstanding tend to
be quite volatile in contrast to empirical data. Thus, ΦN is set to 8.
The calibrated parameter values imply quite reasonable firm characteristics in
the steady state. Book (market) leverage is 0.38 (0.33). Quarterly dividend yield is
0.04 implying that shareholders earn 16% per year on their investment. Net margin
(net income, π̄, over sales, S̄) is equal to 0.22. Gross margin (difference between
sales and costs, (S̄ − C̄), over sales) is equal to 0.40. The steady-state value of
tangibility (fixed assets, K̄, over total assets, (D̄ + P̄ b N̄ )) is 0.2.8
8
Table 5 in Appendix A presents the steady-state values of all variables.
19
3
Results
In this section, I analyze several stylized facts about dividends and payout policy.
First of all, I study the relevance of payout policy, in particular the constant part of
dividends, and whether firm value is impacted by dividends. Secondly, I analyze why
firms use share repurchases and special dividends besides constant cash dividends.
Finally, I analyze the dividend information content.
3.1
Why are dividends rigid?
I re-arrange steady-state expressions of Equations (5) and (26) and get that book
and market values of equity per share in the steady state are as follows:9
βψ d¯
,
q(1 − β)
βψ d¯
=
.
1−β
P̄ b =
P̄ m
(28)
(29)
β is the subjective discount factor; thus, (1 − β) is a discount rate. Equations (28)
and (29) imply that steady-state share price is the sum of discounted constant parts
of dividends. There is a coefficient β in the numerator because it is assumed that new
shareholders start receiving dividends only in the next period. If this assumption is
relaxed then market value of equity per share, P̄˜ m , is as follows:
βψ d¯
ψ d¯
P̄˜ m =
+ ψ d¯ =
.
1−β
1−β
(30)
Equation (30) implies that share price is equal to the sum of discounted future
dividends. Equation (30) is similar to zero growth dividend discount model except
9
See Appendix B for more details.
20
that the latter considers whole dividends: constant and variable parts.
Firms with a more stable dividend policy (with greater ψ) have higher share
price:
∂ P̄ b
= 1 > 0,
∂ψ
∂ P̄ m
= 1 > 0.
∂ψ
(31)
(32)
A firm that pays constant dividends each period is less risky for investors than a comparable firm which pays volatile dividends when the expected or average dividend
is the same. Thus, such stocks are preferred by investors. The result is consistent
with the survey conducted by (Lintner, 1956).
Equations (28) and (29) do not mean that if a firm increases dividends, its share
¯ Thus,
price will also increase. Firm’s net income in the steady state is π̄ = N̄ d.
d¯ =
π̄
N̄
and the equations can be rewritten as:
βψ N̄π̄
,
P̄ =
q(1 − β)
βψ N̄π̄
P̄ m =
.
1−β
b
(33)
(34)
Equations (33) and (34) show that share price and so value of the firm depend on
firm’s net income which is determined by several other factors, such as productivity.
Thus, if managers would like to improve firm value, they should focus on profit
maximization rather than just increase dividends.10
Equations (28) and (29) have several important implications. First of all, they
show that dividends are not irrelevant as argued by Miller and Modigliani (1961).
10
The impact of dividends on share price is discussed in Section 3.3.
21
Modigliani and Miller’s dividend irrelevancy proposition implies that dividend policy does not impact firm value in the absence of transaction costs and tax advantage or disadvantage associated with dividends. The model of this paper includes
transaction costs (for example, the number of shares outstanding adjustment costs);
however, Equations (28) and (29) imply that the transaction costs do not play any
role in determining equity value per share. The model assumes that dividends are
not taxed or that a firm manager makes her decisions ignoring taxes on dividends.
It is easy to show that if dividends were taxed then equity value per share would
reduced as certain part of firm’s profit would be distributed not to shareholders but
to the government. Thus, dividends matter. The result is different from the one
in Miller and Modigliani (1961) possibly due to different assumptions regarding the
dividends. Miller and Modigliani (1961) assume 100% free cash flow payout in every
period whereas in this paper, dividends consist of constant part and variable part.
Thus, in good times when net income exceeds its steady-state value, a certain part
of net income is retained. Similarly, in bad times, dividends are greater than net
income at the expense of reduced firm’s assets.
Second, firm value depends on the stability of expected dividends or how dividends are paid. This outcome is able to explain why dividend-paying firms tend
to keep their dividends constant. The board of directors or shareholders choose
dividend policy that maximizes firm value. In most cases, it implies that the value
of the coefficient of dividend rigidity, ψ, will be high. Third, firm value is not impacted by the variable part of dividends. Combining the evolution of book value of
equity (Equation (4)) and the definition of dividends (Equation (21)) results in the
22
following representation of the evolution of book value of equity:
(Nt − Nt−1 )2
¯ t−1
− ψ exp(ψt )dN
Nt−1
(Dt − Dt−1 )2
(Kt − Kt−1 )2
+ ψ exp(ψt )πt − ΦD
− ΦK
.
(35)
Dt−1
Kt−1
b
m
m
Ptb Nt = Pt−1
Nt−1 + Pt−1
Nt − Pt−1
Nt−1 − ΦN
According to Equation (35), if an investors purchase one share at time t (for Ptm ),
m
) in the next period. Assuming
the value of investment will be (ψ exp(ψt )d¯ + Pt+1
that β is the effective discount factor, the present value of the investment’s expected
m
m
future value is β(ψ d¯ + Pt+1
).11 Further, the value of Pt+1
in the next period (t + 2)
m
); thus, the present value of the investment’s expected
will be (ψ exp(ψt )d¯ + Pt+2
m
future value is βψ d¯ + β 2 (ψ d¯ + Pt+2
). If we continue this process, we will get that
P
t ¯
the present value of the investment’s expected future value is equal to ∞
t=1 β ψ d or
βψ d¯
.
1−β
This would suggest that the variable part of dividends could be considered as
a waste; however, it increases payout-to-price ratio.
The important question that arises is whether shareholders could achieve homemade constant dividends at a lower or the same cost than a firm. If yes, then
shareholders by borrowing and lending as well by purchasing and selling shares could
replicate constant dividends paid by a firm; and we could conclude that dividends
are irrelevant as in Miller and Modigliani (1961). It would be more realistically to
assume that borrowing costs are lower for dividend-paying firms than for individual
shareholders. Empirically, dividend-paying firms are large and profitable (Fama
and French, 2001). Such firms are more likely to have a better access to capital
markets than individual shareholders. This leads to the conclusion that a stability
of dividends is important and is valued by shareholders. Thus, dividend policy
affects firm value.
11
The expected value of exp(ψt ) is one as the expected value of ψt is zero.
23
The fourth is implication is that firm value depends on the time preference (β)
of the firm’s manager. However, it is not impacted by the risk aversion of the firm’s
manager. It means that equity value does not depend whether firm’s manager is
risk averse, risk neutral or risk loving. Further, the results are in contrast to the
findings of Lambrecht and Myers (2012) who find support for dividend irrelevancy
proposition. Next, I explain why not all firms choose ψ to be equal to one.
3.2
Why is the constant dividend weight not equal to one for
all firms?
If ψ is equal to one then the equity value of a firm is maximized while keeping
the same historical average dividend yield. If a firm chooses such payout, a firm
commits to pay a constant stream of dividends in both good times and bad times.
It is natural that the shares of such firm are more valuable than shares of a firm
with variable dividends. Especially, when in bad times, dividends as well as any
other financial resources are more valuable than the ones in good times. Is there
any risk and return trade-off? Yes, there is. If a firm commits to pay out future cash
flows to the owners of a firm, disregarding the economic and business environments,
there could be a hypothetical long period of time with adverse environment and
continuous losses. And if a firm continues to pay dividends, it increases its default
likelihood. Thus, it is more likely that a firm will cease paying dividends or will at
least cut them. According to Equations (28) and (29), the outcome of dividend cuts
is a lower share price. Thus, such firms are riskier, as I will show numerically by
using simulations.
Perturbation techniques based on a generalized Schur decomposition are used
to solve the first order approximated solution of the model. It is a unique stable
24
rational expectations equilibrium. The dynamic of endogenous variables is explained
by the exogenous processes and past values of endogenous state variables that are
those endogenous variables which appear at the previous period. The first order
approximation of any variable xt has the following form:
xt = x̄ + A(yt−1 − ȳ) + But , where
(36)
yt−1 is the vector of endogenous state variables, ut is a vector of shocks, A and B
are matrices.12
I assume that the AR(1) coefficients for all shocks are equal to 0.5 (see Panel B in
Table 1). It implies that the magnitude of shocks diminishes over time and two years
(eight quarters) later it is less than 1% of its initial magnitude. Thus, the shock
has almost no impact after two years. This assumption reflects the competitive and
dynamic environment in which a firm operates. Further, the standard deviations
of the shocks on variables with small steady-state values (interest rate, rt∗ , and
preference shock, ζt ) are set to 0.3 (see Panel C in Table 1). The standard deviations
of the shocks that directly affect production and financing decisions are set to either
0.05 or 0.1 depending on their impact on endogenous variables. It turned out that
the productivity shock, At , and the shock to market-to-book ratio, qt , affect the
variables significantly; thus, their standard deviations are set to 0.05. The standard
deviations of shock to the input price, ct , and the demand shock, γt , are 0.1. It
is assumed that the standard deviations of shock to corporate income tax rate, τt ,
and shock to tangibility, κt , are 0.02 as empirically, tax rates and tangibility ratios
are relatively constant. At last, I assign an arbitrary value of 0.05 to the standard
deviation of the shock to the weighting parameter of constant dividend part, ψt .
12
See Appendix C for more details about the solution of the model.
25
The calibration implies that endogenous variables are the least impacted by ψt and
τt .13
I will use simulated data and count the number of cases when firms default. I
assume that a firm defaults if its share price or the number of shares outstanding
become non-positive. In this exercise, the greater number of shocks is preferred if
one wants to generate a certain number of defaults and assume the reasonable levels
of shocks’ standard deviations. For example, the standard deviation of productivity
shock, At , is 0.05. It means that there is a probability of approximately 68% that
the unexpected change in productivity in any given period will be between –0.05 and
0.05. It is very unlikely that a model with a single shock (for example, productivity
shock with the standard deviation of 0.05) would be able to generate even a single
case of firm default. Then one would need to increase the shock’s standard deviation
to, let’s say, 0.30. However, this number is very unreasonable. Especially, if we keep
in mind that dividend payers are usually large firms operating in mature industries.
It would be quite unlikely that such a firm is subject to the unexpected change in
productivity in a quarter by +/– 30%. The solution is to use a model that features
several shocks. Such model would be able to generate a certain number of defaults.
A model with more shocks is more realistic as a firm is impacted by many exogenous
factors (for example, unexpected change in demand for its products, interest rates,
tax rates etc.). Further, one does not need to make unrealistic assumptions about
shocks’ standard deviations. Thus, the model is more credible.
To generate an artificial dataset with heterogeneous firms, I use the following
procedure. I simulate 1,000 firms for 2,100 time periods and then drop the first
100 time periods. In each simulation, shocks are random. Thus, these 1,000 time
13
The calibration of exogenous variables leads to quite reasonable dynamic characteristics of the
model (see Table 6 and Table 7 in Appendix A for correlation matrix and variance decomposition,
respectively).
26
series should be unique. Then I compute in how many cases firms go bankrupt.
There are at least two methods to compare the results with hypothetical cases when
ψ is not equal to 0.75. Firstly, one can run another 1,000 simulations using the
same shock patterns for different values of ψ; however, if ψ changes, then a firm’s
characteristics also change. For example, if ψ increases, a firm’s debt is unaffected
but its equity value increases; thus, the firm has lower leverage. Therefore, in this
case not identical firms are compared for different values of ψ. More importantly, it
implies that firms with greater ψ have lower leveraged; thus, they are less risky and
it is more difficult to prove that greater ψ increase the likelihood of default. I run
1,000 simulations again using the same shock patterns when ψ is one and compute
the number of defaults. Panel A in Table 2 presents the results. As expected, I
find that the number of defaults increases with ψ = 1. In period 2,000, there are
939 defaults when ψ = 0.75 and 941 defaults when ψ = 1. However, market (book)
leverage is 0.19 (0.22) if ψ = 1. Thus, the default risk due to high leverage is offset
by the smaller value of ψ. Thus, firms with more rigid payout are riskier and more
likely to default.
Secondly, I use the previously simulated dataset (when ψ = 0.75) and compute
hypothetical share prices under altered payout scheme. This method also has some
flaws. Hypothetical share price is not used in the model. The firm’s manager
still uses a simulated share price rather than a hypothetical one to make decisions,
also interest rate depends on a simulated share price but not on a hypothetical
one. Nevertheless, together both methods should provide us convincing results. I
compute hypothetical share prices and the number of firm defaults when ψ is equal
to one. Panel B in Table 2 shows the results. I find that the number of defaults
increases to 947 in this specification.
Despite the number of defaulted firms increases only by a few if ψ changes from
27
Table 2: Firm defaults
This table presents the number of firm defaults for alternate values of ψ. The sample
includes 1,000 simulated firms for 2,000 time periods. In Panel A, I run another 1,000
simulations using the same shock patterns for different values of ψ. In Panel B, I use
the previously simulated dataset (when ψ = 0.75) to compute hypothetical share
prices and the number of firm defaults under altered payout scheme.
Panel A
Panel B
ψ
ψ
Time period
0.75
1
1
100
200
300
400
500
600
700
800
900
1,000
1,100
1,200
1,300
1,400
1,500
1,600
1,700
1,800
1,900
2,000
72
200
300
388
469
535
612
664
712
745
779
812
838
856
872
886
905
921
932
939
72
207
310
397
478
544
621
675
720
753
790
818
842
861
877
890
908
923
934
941
78
212
310
404
489
559
629
684
728
762
792
821
848
866
881
895
914
926
939
947
0.75 to one, it does not mean that the impact of ψ is negligible. This reflects the
relationship between this coefficient and the likelihood of default. A number of unreported simulations reveal that the frequency of defaults depends on the covariance
matrix of shocks. If the standard deviations of shocks increase, the number of defaults rises, and vice versa. However, in all cases the number of defaults increases
with ψ. The results imply that firms might decide to choose ψ < 1 in order to reduce
28
the default probability while keeping the same historical average payout-to-price ratio. This explain why firms use share repurchase and special dividends rather than
only rigid cash dividends to disgorge excess cash to shareholders.
The exogenous variable ψt impacts the weights of constant and variable parts of
dividends while keeping the magnitude of the steady-state dividend unchanged. If
the shock is positive then the weight of a constant part of dividend increases, and
vice versa. Figure 1 presents the effects of the shock of one standard deviation on
a firm’s share price, Ptb . The responses are expressed as the deviations in levels
from the steady state. I compute the reaction of share price for different values of
shock’s AR(1) coefficient, ρψ .14 I find that in general, share price increases with the
constant part of dividends. However, we observe the temporal (for approximately
nine periods) negative impact. The impact is long lasting.15 For example, in case
ρψ = 0.0001, a manager and investors know that the constant part of dividends will
effectively be higher only for one period. The impact on share price is long-term
but relatively small (see Figure 1a).
To provide more detail, I analyze the benchmark case (when ρψ = 0.5). The
shock is temporal and vanishes in approximately nine periods. Keeping in mind
the dynamic nature of the model, the shock has different impact on variables in the
short term and long term.
When a firm’s manager realizes that there is a shock, she needs to make certain
decisions that will maximize shareholder value in the long run. This could mean
increasing a share price if the shock is favorable or mitigating the impact on the share
price of the unfavorable shock. In this case the constant part of dividends temporally
14
I do not show impulse responses when ρψ is zero as then the impact on share price is less
than 10−10 . Further, the model does not converge if ρψ is set to one. However, I compute impulse
responses when ρψ is equal to the values close to zero and one, 0.0001 and 0.9999, respectively.
15
For example, the positive impact is observed until 80th period.
29
5.E-07
0.05
ψ
4.E-07
0.04
3.E-07
Pb
2.E-07
0.03
1.E-07
0.02
Quarter
0.E+00
1
-1.E-07
0.01
5
9
13
17
21
25
29
33
37
-2.E-07
Quarter
0.00
1
5
9
13
17
21
25
29
33
37
-3.E-07
-4.E-07
(a) ρψ = 0.0001
0.0020
0.05
Pb
ψ
0.0015
0.04
0.0010
0.03
0.0005
Quarter
0.02
0.0000
0.01
-0.0005
1
Quarter
-0.0010
37
-0.0015
0.00
1
5
9
13
17
21
25
29
33
5
9
13
17
21
25
29
33
37
(b) ρψ = 0.25
0.010
0.05
Pb
ψ
0.008
0.04
0.006
0.03
0.004
0.002
0.02
Quarter
0.000
0.01
-0.002
Quarter
-0.004
37
-0.006
0.00
1
5
9
13
17
21
25
29
33
1
5
9
13
17
21
25
29
33
37
(c) ρψ = 0.5
Figure 1: The impact of the increase in the constant part of dividends (continued on
the next page). This figure plots the effects of the shock of one standard deviation
on a firm’s share price, Ptb . The responses are expressed as the deviations in levels
from the steady state.
30
0.05
0.05
Pb
ψ
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.00
Quarter
1
Quarter
-0.01
37
-0.02
0.00
1
5
9
13
17
21
25
29
33
5
9
13
17
21
25
29
33
37
(d) ρψ = 0.75
0.6
0.05
Pb
ψ
0.5
0.04
0.4
0.03
0.3
0.02
0.2
0.01
0.1
0.0
37
-0.1
0.00
1
5
9
13
17
21
25
29
33
Quarter
Quarter
1
5
9
13
17
21
25
29
33
37
(e) ρψ = 0.9999
Figure 1: The impact of the increase in the constant part of dividends (continued
from the previous page). This figure plots the effects of the shock of one standard
deviation on a firm’s share price, Ptb . The responses are expressed as the deviations
in levels from the steady state.
increases by one standard deviation (from 0.75 to 0.75 × exp(0.05) ≈ 0.79) due to,
for example, the decision of the board of directors (see Figure 2a). This implies that
the manager needs to take into account that the constant amount of cash that will
be distributed to the shareholders, disregarding firm performance, increases. More
rigid dividends increase the bankruptcy risk; thus, the manager’s best response is to
the number of shares outstanding by buying back some shares (see Figure 2b). In the
short term, approximately 0.1% of shares are repurchased. Two financing sources
are used to generate funds for share repurchase: the decreased investment in capital
and additional borrowing (see Figures 2c and 2d). This leads to the lower capital
31
0.012
0.05
ψ
Pb
0.010
0.04
Pm
N
0.008
0.006
0.03
0.004
0.02
0.002
Quarter
0.000
0.01
-0.002
Quarter
5
9
13
17
21
25
29
33
5
9
13
17
21
25
29
33
37
-0.004
0.00
1
1
37
-0.006
(a)
(b)
0.003
0.003
I
K
C
Lb
0.002
r
D
0.002
Quarter
0.001
0.001
Quarter
0.000
-0.001
0.000
1
5
9
13
17
21
25
29
33
37
-0.001
-0.002
-0.002
-0.003
-0.003
-0.004
-0.004
-0.005
1
5
9
13
17
21
25
29
33
37
-0.005
(c)
(d)
0.0010
0.0008
Y
p
S
d
π
EPS
0.0006
0.0005
Quarter
0.0000
1
5
9
13
17
21
25
29
33
0.0004
37
0.0002
-0.0005
Quarter
0.0000
1
-0.0010
5
9
13
17
21
25
29
33
37
-0.0002
-0.0015
-0.0004
-0.0020
-0.0006
(e)
(f)
Figure 2: The impact of the increase in the constant part of dividends. This figure
plots the impact of the shock of one standard deviation. The responses are expressed
as the deviations in levels from the steady state.
stock. As a firm uses Cobb-Douglas technology, it is optimal to reduce the amount
of production input, Ct (see Figure 2c). Since both production factors decrease,
the production volume also decreases which leads to greater price per output unit.
32
However, the impact of lower production volume offsets higher price and the sales
revenues, St , slightly decrease.
Additional borrowing and share repurchase lead to higher book leverage, Lbt
Lbt = Dt /(Dt + Ptb Nt ) , and effective interest rate, rt . Smaller sales revenues and
higher borrowing costs negatively impact net income, πt (see Figure 2f). As the
number of shares outstanding decreases more than net income, earnings per share
(EPS) (net income, πt , scaled by the number of shares outstanding, Nt ) increase
and so do dividends, dt .
After nine periods, the weight of constant part of dividends almost reaches its
initial level meaning that the payout policy does not increase firm’s bankruptcy risk.
Thus, the firm issues more shares, rebalances its capital structure, and accordingly
adjusts its production decisions. This leads to greater share price (see Figure 2a).
Further, I analyze dividend information content.
3.3
Dividend information content
Dividend signaling theory argues that managers have better information about the
firm’s expected future cash flows. Thus, managers may use dividends as a costly
signal to alter market perceptions concerning future earnings prospects. The best
known signaling models are the ones developed in Bhattacharya (1979), Miller and
Rock (1985), and John and Williams (1985). The general implication of signaling
models is that we should observe the positive relation between dividends and future
firm performance. The empirical studies provide mixed evidence about whether
dividends predict future earnings (see Allen and Michaely (2003) and Kalay and
Lemmon (2008) for a detailed literature review).
To analyze the model’s consistency with signaling theory, I compute asymptotic
33
autocorrelation coefficients between firm performance and dividends. Firm performance is proxied by earnings or the book value of equity per share, Ptb . I use two
proxies of earnings: net income, πt , and EPS. Empirical studies using stock returns
as firm performance measures generally support dividend signaling theory (Grullon,
Michaely, and Swaminathan, 2002; Michaely, Thaler, and Womack, 1995). However,
if firm performance is proxied by earnings then no support is found (see, for example, Benartzi, Michaely, and Thaler, 1997; Grullon, Michaely, Benartzi, and Thaler,
2005). The empirical studies used data from different tax regimes and business cycles. This could be a reason for mixed evidence. For robustness, the analysis is
repeated in several environments, that is for different values of AR(1) coefficients of
all shocks: 0.0001, 0.25, 0.5, 0.75, and 0.9999. I compute autocorrelation coefficients
up to 12th lead.
I find that the autocorrelation coefficients between dividends, dt , and share
prices, Ptb , do not provide a clear answer whether dividends convey a good news
or bad news about future share price. The sign of the autocorrelation coefficient
depends on lead number and the environment. The autocorrelation coefficients are
negative in the current period and for the first two leads (see Table 3). If ρ = 0.9999
then the autocorrelation coefficients between dividends and share price are always
negative except for the 12th lead when all five autocorrelation coefficients are positive. Thus, the results are mixed and do not provide a strong support for dividend
signaling theory.
The absolute values of autocorrelation coefficients are small; only a few of them
exceed 0.20. The results suggest that in general the relationship between dividends
and share price becomes stronger when shocks are more autocorrelated. The values
of autocorrelation coeficients increase with the number of lead. Since the model is
quarterly; thus, the autocorrelation coefficient for 12th lead shows the strength of
34
Table 3: Autocorrelation coefficients between dividends and lead share prices
This table presents the autocorrelation coefficients between dividends, dt , and share
prices, Ptb , for different values of AR(1) coefficients of all shocks. The lead value is
given in the parentheses.
d(0)
P b (0)
P b (1)
P b (2)
P b (3)
P b (4)
P b (5)
P b (6)
P b (7)
P b (8)
P b (9)
P b (10)
P b (11)
P b (12)
ρ = 0.0001
ρ = 0.25
ρ = 0.5
ρ = 0.75
ρ = 0.9999
–0.054
–0.035
–0.017
0.001
0.017
0.032
0.047
0.061
0.074
0.086
0.097
0.107
0.117
–0.087
–0.051
–0.025
–0.003
0.019
0.038
0.057
0.075
0.091
0.107
0.121
0.135
0.147
–0.126
–0.082
–0.046
–0.014
0.015
0.042
0.067
0.090
0.112
0.133
0.152
0.170
0.186
–0.171
–0.128
–0.086
–0.045
–0.007
0.030
0.065
0.099
0.130
0.160
0.187
0.213
0.237
–0.154
–0.138
–0.122
–0.107
–0.092
–0.078
–0.064
–0.051
–0.038
–0.026
–0.015
–0.004
0.007
relationship between today’s dividends and share price three years later.
The results presented in Table 3 have the important implications from the postdividend announcement drift perspective. The positive autocorrelation coefficients
when the lead value is greater than five and if ρ 6= 0.9999 imply that if dividends
are raised today, the share price will increase one, two, and three years later. This
implies that the post-dividend announcement drift does not mean that investors do
not understand the signal rather the drift is an artifact driven by the environment
and the simulations using a dynamic model.
Next, I compute the autocorrelation coefficients between dividends, dt , and two
measures of earnings for different values of AR(1) coefficients of all shocks. Table 4
reports the results. I find that autocorrelation coefficients are positive for dividends
35
Table 4: Autocorrelation coefficients between dividends and lead earnings
This table presents the autocorrelation coefficients between dividends, dt , and net
income, πt , (see Panel A) and between dividends, dt , and earnings per share (EPS)
(see Panel B) for different values of AR(1) coefficients of all shocks. The lead value
is given in the parentheses.
d(0)
ρ = 0.0001
ρ = 0.25
ρ = 0.5
ρ = 0.75
ρ = 0.9999
Panel A. Autocorrelations between dividends and net income
π(0)
π(1)
π(2)
π(3)
π(4)
π(5)
π(6)
π(7)
π(8)
π(9)
π(10)
π(11)
π(12)
0.854
–0.062
–0.055
–0.048
–0.041
–0.035
–0.028
–0.022
–0.017
–0.011
–0.006
–0.001
0.004
0.773
0.120
–0.035
–0.065
–0.064
–0.056
–0.047
–0.037
–0.028
–0.019
–0.011
–0.003
0.005
0.635
0.240
0.052
–0.033
–0.066
–0.075
–0.070
–0.061
–0.048
–0.035
–0.021
–0.008
0.005
0.365
0.213
0.108
0.037
–0.008
–0.034
–0.047
–0.049
–0.044
–0.034
–0.021
–0.005
0.012
–0.049
–0.047
–0.045
–0.043
–0.042
–0.040
–0.038
–0.036
–0.035
–0.033
–0.032
–0.030
–0.029
Panel B. Autocorrelations between dividends and EPS
EPS(0)
EPS(1)
EPS(2)
EPS(3)
EPS(4)
EPS(5)
EPS(6)
EPS(7)
EPS(8)
EPS(9)
EPS(10)
EPS(11)
EPS(12)
0.993
0.077
0.072
0.066
0.061
0.055
0.050
0.045
0.040
0.036
0.031
0.027
0.023
0.993
0.340
0.170
0.121
0.102
0.091
0.082
0.074
0.066
0.059
0.051
0.044
0.037
0.994
0.596
0.390
0.280
0.217
0.179
0.153
0.134
0.117
0.103
0.090
0.078
0.066
0.995
0.832
0.703
0.599
0.515
0.445
0.386
0.336
0.293
0.255
0.221
0.190
0.162
0.990
0.943
0.897
0.852
0.807
0.762
0.718
0.675
0.632
0.590
0.549
0.508
0.469
and net income if the lead value is zero or 12 and if AR(1) coefficients of shocks are
equal to 0.0001 or 0.25, or 0.5, or 0.75 (see Panel A). Otherwise, the relationship
36
between dividends and future share price is negative. The results are consistent with
Benartzi, Michaely, and Thaler (1997) who find that contemporaneous correlation
between dividends and earnings is positive; however, future earnings are unrelated
to current dividends. The results presented in Panel A of Table 4 generally suggest
that there is a negative relationship between dividends and future earnings and do
not support dividend signaling theory.
Panel B in Table 4 shows that the autocorrelation coefficients between dividends
and EPS are positive for all 12 leads and in all environments. The relationship is
stronger for smaller values of leads and greater values of AR(1) coefficients of all
shocks. According to the model, EPS is the only measure of firm performance which
future values can be predicted using current level of dividends.
Overall, the results reported in Tables 3 and 4 do not support dividend signaling theory. The results depend on a firm’s performance measure (share price vs.
earnings), the environment in which a firm operates (how autocorrelated shocks
are), and the methodology (a lead number). This suggests that the empirical evidence of dividend information content could be mixed due to at least three reasons.
First, empirical studies use different firm performance measures and get conflicting
results. Tables 3 and 4 show that even the results obtained using simulated data
are significantly impacted by the choice of firm performance measure. For example,
let’s consider the relationship between dividends and 8th lead of firm performance
(i.e. firm performance in two years) and assume that AR(1) coefficients of shocks
are smaller than 0.9999. If firm performance is proxied by share price, we would
find support for dividend signaling theory as there is a positive correlation between
dividends and future share price. However, if we use net income to measure firm
performance, we would conclude that dividends cannot be used to predict firm’s
future performance. The results would not support dividend signaling theory as
37
greater dividends lead to lower net income. Thus, the choice of firm performance
measure really matters. Second, previous studies use different sample periods to
analyze dividend information content. It is very likely that different sample periods
feature different pattern of shocks. In one period, shocks might be more autocorrelated than shocks in the subsequent period. Similarly, the magnitude of shocks
might be quite different in various time periods. Let’s consider the results provided
in Table 3. Suppose we want to analyze the relationship between dividends and 8th
lead of share price. If shocks are strongly autocorrelated (all AR(1) coefficients are
equal to 0.9999) then we would report a negative relationship between the variables
and conclude that dividends do not carry any positive news on future firm performance. However, if shocks are not strongly autocorrelated then we would find a
positive relationship between dividends and future share price and conclude that
the results are consistent with dividend signaling theory. Thus, the environment
also matters.16 Third, empirical studies might consider different lead values of firm
performance. Let’s consider the results provided in Table 3. If we choose 2nd lead of
share price then we would report that dividends cannot predict future share prices.
However, if we choose 12th lead of firm’s share price then the asymptotic autocorrelation coefficients between the variables are positive disregarding the environment;
thus, we would find support for dividend signaling theory. Therefore, the choice
of lead value is also important. All this suggests that mixed empirical evidence
could be due to different firm performance measures, different environment, and
non-identical methodology used in the analysis. For example, it is possible that
shocks have AR(1) coefficients with values close to one during 1980-1990 period and
that shocks are not autocorrelated during 1990-2000 period.17 If it is a case, then
16
The environment would also change if we altered the standard deviations of the shocks or if
we assume that shocks are correlated between each other (in this paper, I assume that shocks are
uncorrelated with each other).
17
The variances of shocks might also change over time.
38
one would find conflicting empirical evidences during different sample time periods:
the relationship between dividends and future firm performance would be positive
supporting dividend signaling theory during one sample period and the relationship
would be negative during the other sample period implying that dividends cannot
be used to predict firm’s future performance.
The intuition of the obtained results is quite simple. Larger dividends suppress
future growth opportunities and so negatively impact future share price and profit.
However, the correlation between contemporary net income and dividends is positive
(if ρ ≤ 0.75) because the part of net income is distributed as dividends (see Equation
21). If ρ ≥ 0.9999 then dividend stream depends less on the current earnings; thus,
we do not observe the positive relationship between net income and dividends.
One possible explanation for why no support for dividend signaling theory is
found is the model. It is assumed that dividends depend on the current but not
expected future net income. In the model, the manager does not purposely do any
signaling to investors and does not derive any utility from using dividends as a
signaling device. Thus, this could be a reason why we do not observe a positive
correlation between net income and dividends. The results above means that if we
do not assume that managers signal the market using dividends, the relationship
between dividends and future firm performance is unlikely to be positive as greater
dividends today lead to lower future growth and lower profits.
To show the impact of exogenous increase in dividends without the increase in
productivity, I modify the definition of dividends, dt :
dt = ψ exp(ψt )d¯exp(ωt ) + [1 − ψ exp(ψt )]
39
πt
,
Nt−1
(37)
¯
where exp(ωt ) is the shock to d:
ωt = ρω ωt−1 + ηtω , where ηtω ∼ N(0, σω2 ).
(38)
σω is set to 0.1. Figure 3 presents the impact of the dividend shock of one
standard deviation on a firm’s share price, Ptb . The responses are expressed as the
deviations in levels from the steady state. For robustness, I consider five different
values of shock’s AR(1) coefficient, ρω and find that in all cases the impact on share
price, Ptb , is negative. The impact increases with shock’s AR(1) coefficient. If shock’s
AR(1) coefficient is close to one then the manager’s long-term decisions are similar
to those if the change in dividends is permanent. For example, if AR(1)=0.9999,
ψ400 = 0.096 (ψt value in period 400) and only marginally less than its initial value
ψ1 = 0.10. Thus, the manager understands that the only way not to default is to
reduce the number of shares outstanding. Thus, a firm repurchases a significant
portion of its shares: in the short term, the number of shares outstanding is reduced
by one third; in the long term, a firm re-issues some shares until the number of
shares outstanding is approximately 10% less than in the steady state. The number of shares outstanding decreases by the same proportion as dividends increased;
thus, total dividends paid (Nt × dt ) and dividend yield are similar to their steadystate values. Thus, the manager’s decision to the increase dividends is similar to
the reverse stock split that positively impacts share price in the long term. The
negative short-term impact on share price indicates that if the board of directors
increases dividends without increasing productivity (or without respective shock to
the total factor of productivity, A) then shareholder value is destroyed. Thus, one
could expect that a firm increases dividends only if its expected future profitability
improves.
40
0.00
0.00
1
5
9
13
17
21
25
29
33
-0.02
37
Quarter
1
5
9
13
17
21
25
29
33
37
Quarter
25
29
33
37
Quarter
25
29
33
-0.02
Pb
Pb
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
-0.10
-0.10
-0.12
-0.12
(a) ρω = 0.0001
0.00
0.10
ω
1
0.08
-0.03
0.06
-0.06
5
9
13
17
21
Pb
0.04
-0.09
0.02
Quarter
-0.12
0.00
1
5
9
13
17
21
25
29
33
37
-0.15
(b) ρω = 0.25
0.00
0.10
1
ω
0.08
-0.05
0.06
-0.10
5
9
13
17
21
37
Quarter
Pb
0.04
-0.15
0.02
Quarter
-0.20
0.00
1
5
9
13
17
21
25
29
33
37
-0.25
(c) ρω = 0.5
Figure 3: The impact of dividend increase (continued on the next page). This figure
plots the effects of the shock of one standard deviation on a firm’s share price, Ptb .
The responses are expressed as the deviations in levels from the steady state.
41
0.00
0.10
1
ω
0.08
-0.10
0.06
-0.20
5
9
13
17
21
25
29
33
37
Quarter
Pb
0.04
-0.30
0.02
Quarter
-0.40
0.00
1
5
9
13
17
21
25
29
33
37
-0.50
(d) ρω = 0.75
0.00
0.10
1
ω
5
9
13
17
21
25
29
33
37
-0.25
0.08
Quarter
-0.50
Pb
0.06
-0.75
0.04
-1.00
0.02
-1.25
Quarter
-1.50
37
-1.75
0.00
1
5
9
13
17
21
25
29
33
(e) ρω = 0.9999
Figure 3: The impact of dividend increase (continued from the previous page). This
figure plots the effects of the shock of one standard deviation on a firm’s share price,
Ptb . The responses are expressed as the deviations in levels from the steady state.
4
Conclusion
This study reports that dividend policy is not irrelevant as argued in Miller and
Modigliani (1961). I show that in the shareholder wealth maximization framework,
share price depends on the smoothness of dividends. I assume that dividends per
share are the weighted sum of constant amount of cash and net income per share.
This paper shows that only the constant part of dividends matters as variable part
does not affect share price. Firms tend to keep their dividends constant in order to
maximize share price. This helps explain the dividend smoothing behavior of firms.
Further, I show that firms might choose to pay not only constant cash regular
42
dividends in order to reduce the likelihood of dividend omission or cuts that arise
from a firm’s commitment to disgorge constant amounts of cash each period. This
justifies why firms use special dividends and share repurchases besides cash dividends
that are relatively constant. At last, I investigate dividend information content and
find conflicting results. The sign of autocorrelation coefficient between dividends
and firm performance depends of firm performance measure, methodology, and how
autocorrelated shocks are. Thus, the results are inconclusive.
This paper contributes to the existing literature at least in two areas. First of all,
this paper provides an alternative explanation for why dividends are rigid over time.
I show that firm value is maximized, keeping the same dividend yield, if dividends
are constant or at least rigid. The result is consistent with the opinion of managers
that “the market puts a premium on stability or gradual growth in rate” (Lintner,
1956).
Further, this paper makes a methodological contribution. The study introduces
a new type of model that is extensively used in macroeconomics to the literature of
corporate finance. The model features nine exogenous processes that help imitate a
realistic environment in which a firm operates.
The model tries to replicate the behavior of a real firm. The model is developed
using several realistic assumptions. It assumes that a firm’s manager maximizes
shareholder value and in each period she needs to make several decisions such as
how much to produce or how much to borrow in the credit market. A manager
makes decisions to respond to the shocks that affect firm performance. The model
incorporates the main aspects of the firm’s life: operating, financing, and investment
decisions making under uncertainty. In this paper, the dynamic stochastic general
equilibrium model explains the puzzle of rigid dividends. I believe that this class
43
of models can be further used in the field of corporate finance to explain a firm’s
behavior.
44
A
Additional tables
Table 5: Steady-state values
This table presents the values of the variables in the steady state. The model is
calibrated as in Table 1.
Variable
Value
C̄
D̄
I¯
K̄
P̄ b
P̄ m
S̄
Ȳ
p̄
r̄
λ̄
π̄
2.75
11.92
0.16
6.26
19.40
24.25
4.58
3.38
1.36
0.02
0.13
1.00
45
46
Ptb
1
0.792
0.983
–0.126
0.523
–0.847
0.930
0.311
0.481
0.426
–0.406
0.565
–0.373
Variable
Ptb
Nt
Ptm
dt
πt
Dt
Kt
It
St
Ct
rt
Yt
pt
0.792
1
0.790
–0.439
0.409
–0.943
0.951
0.123
0.354
0.296
–0.412
0.457
–0.441
Nt
0.983
0.790
1
–0.127
0.515
–0.834
0.923
0.355
0.473
0.420
–0.402
0.556
–0.368
Ptm
–0.126
–0.439
–0.127
1
0.635
0.380
–0.303
0.066
0.674
0.693
0.096
0.566
0.398
dt
0.523
0.409
0.515
0.635
1
–0.421
0.489
0.075
0.992
0.963
–0.257
0.969
0.030
πt
–0.847
–0.943
–0.834
0.380
–0.421
1
–0.920
–0.106
–0.368
–0.313
0.351
–0.465
0.419
Dt
0.930
0.951
0.923
–0.303
0.489
–0.920
1
0.254
0.438
0.379
–0.448
0.538
–0.433
Kt
It
0.311
0.123
0.355
0.066
0.075
–0.106
0.254
1
0.073
0.066
–0.127
0.084
–0.049
Table 6: Correlation matrix
This table presents the correlation matrix (theoretical moments).
0.481
0.354
0.473
0.674
0.992
–0.368
0.438
0.073
1
0.974
–0.187
0.969
0.063
St
0.426
0.296
0.420
0.693
0.963
–0.313
0.379
0.066
0.974
1
–0.161
0.973
–0.056
Ct
–0.406
–0.412
–0.402
0.096
–0.257
0.351
–0.448
–0.127
–0.187
–0.161
1
–0.232
0.196
rt
0.565
0.457
0.556
0.566
0.969
–0.465
0.538
0.084
0.969
0.973
–0.232
1
–0.184
Yt
–0.373
–0.441
–0.368
0.398
0.030
0.419
–0.433
–0.049
0.063
–0.056
0.196
–0.184
1
pt
Table 7: Variance decomposition
This table shows the variance decomposition (in %), that is the independent contribution of each shock to the variance of each endogenous variable.
Variable
ηtζ
ηtq
ηtκ
ηtτ
ηtc
ηtr
ηtγ
ηtA
ηtψ
Ptb
Nt
Ptm
dt
πt
Dt
Kt
It
St
Ct
rt
Yt
pt
0.88
1.38
0.84
0.28
0.26
0.91
1.16
2.35
0.22
0.17
0.21
0.34
0.22
21.26
33.66
24.63
6.90
6.43
22.31
28.15
66.56
5.38
4.01
5.03
8.15
5.39
0.07
0.07
0.07
0.08
0.06
0.08
0.39
18.55
0.08
0.06
0.01
0.11
0.08
0.03
0.03
0.03
0.04
0.04
0.04
0.03
0.01
0.01
0.00
0.01
0.01
0.01
17.93
14.94
17.16
21.51
21.69
17.09
16.07
2.85
22.23
41.96
3.34
33.69
22.29
2.05
1.78
1.96
1.85
1.61
4.50
2.41
0.46
0.46
0.34
80.66
0.70
0.46
49.8
41.5
47.67
59.76
60.25
47.47
44.64
7.93
61.75
46.08
9.27
42.02
61.64
7.97
6.64
7.63
9.56
9.64
7.59
7.14
1.27
9.88
7.37
1.48
14.97
9.91
0.01
0.01
0.01
0.00
0.00
0.01
0.01
0.02
0.00
0.00
0.00
0.00
0.00
47
B
Solving for P̄ b and P̄ m
Equation (6) is
Ptm = Ptb q exp(qt ),
In the steady state, it can be written as follows:
P̄ m = P̄ b q.
(39)
Equation (5) is:
Kt = κ exp(κt )(Dt + Ptb Nt ).
It is expressed in the steady state as follows:
K̄ = κ(D̄ + P̄ b N̄ ).
(40)
Equation (40) implies that the steady-state value of equity, (P¯b N̄ ), has the following
form:
P̄ b N̄ =
K̄
− D̄.
κ
Equation (26) is:
q exp(qt−1 )
Kt−1
Nt
λt
− Dt−1 − 2ΦN
−1
Nt−1
κ exp(κt−1 )
Nt−1
Nt+1
Kt
= βEt λt+1 q exp(qt )
− Dt
(Nt )2 κ exp(κt )
#)
2
Nt+1
−ΦN
+ ΦN + ψ exp(ψt+1 )d¯ .
Nt
48
(41)
Its steady-state expression is as follows:
q
N̄
K̄
− D̄
κ
q
=β
N̄t
K̄
− D̄ + ψ d¯ .
κ
(42)
Equations (39), (40), and (42) imply the following relation:
P̄ m = β P̄ m + ψ d¯ .
Now we can solve for P¯m and then for P¯b :
βψ d¯
,
1−β
βψ d¯
P̄ b =
.
q(1 − β)
P̄ m =
49
(43)
C
Solution of the model
The equilibrium of the model is defined by the evolution of shareholder value, asset
composition, first-order conditions, several variable definitions (in total 15 equations)
and nine shocks. The model can be written in the following form:18
Et [f (xt+1 , xt , xt−1 , ut )] = 0, where
(44)
xt is a vector of endogenous variables:
xt = [Et , Ptb , Nt , P m , dt , πt , Dt , Kt , It , St , Ct , rt , Yt , pt , λt , qt , κt , ct , τt , rt∗ , At , ζt , γt , ψt ]0 .
ut is a vector of exogenous variables:
∗
ut = [ηtq , ηtκ , ηtc , ηtτ , ηtr , ηtA , ηtζ , ηtγ , ηtψ ]0 .
ut depends on its past values and independent and identically distributed innovations, t , with zero mean and variance-covariance matrix Σ :
ut = F (ut−1 , t ),
(45)
Et (t ) = 0,
Et (t 0t ) = Σ .
The solution of the model can be expressed by certain functions called policy functions. They relate the endogenous variables with the contemporary values of exoge18
The description of the solution of the model is mainly based on Griffoli (2011).
50
nous variables and the lagged values of endogenous state variables:
xt = g(yt−1 , ut ), where
(46)
yt−1 is the vector of endogenous state variables.19 In the steady state, x̄ = g(ȳ, 0).
The values of endogenous variables in period (t + 1) can be written as:
xt+1 = g(yt , ut+1 ),
= g(g(yt−1 , ut ), ut+1 ).
(47)
Thus, xt+1 is the following function:
xt+1 = F (yt , ut , ut+1 ), where
(48)
F (yt , ut , ut+1 ) = f (g(g(yt−1 , ut ), ut+1 ), g(yt−1 , ut ), yt−1 , ut ),
(49)
Et [F (yt−1 , ut , ut+1 )] = 0.
(50)
In the steady state, the model is expressed as follows:
f (ȳ, ȳ, ȳ, 0) = 0.
(51)
The first order Taylor approximation of the model results in:
Et [F (1) (yt−1 , ut , ut+1 )] = Et f (ȳ, ȳ, ȳ, 0) + fyt+1 [gy (gy ŷ + gu u) + gu u0 ]+
+ fyt (gy ŷ + gu u) + fyt−1 ŷ + fu u , where
ŷ = yt−1 − ȳ, u = ut , u0 = ut+1 , fyt+1 =
gu =
19
∂f
,
∂yt+1
f yt =
∂f
,
∂yt
fyt−1 =
∂f
,
∂yt−1
gy =
∂g
.
∂ut
A vector of endogenous variables, xt , also includes all endogenous state variables, yt .
51
(52)
∂g
,
∂yt−1
The expected value of future shocks is zero. Thus, Equation (52) can be rewritten
as:
Et [F (1) (yt−1 , ut , ut+1 )] = f (ȳ, ȳ, ȳ, 0) + fyt+1 [gy (gy ŷ + gu u)]+
+ fyt (gy ŷ + gu u) + fyt−1 ŷ + fu u =
= (fyt+1 gy gy + fyt gy + fyt−1 )ŷ + (fyt+1 gy gu + fyt gu + fu )u = 0.
(53)
Equation (53) has two unknowns: gy and gu . They can be solved assuming that
the contents of each parentheses must be equal to zero and using generalized Schur
decomposition. To ensure that the solution is unique and stable, the Blanchard and
Kahn condition and the rank condition must be satisfied. The first order approximated solution of the model is:
xt = x̄ + gy ŷ + gu u.
This is a unique stable rational expectations equilibrium.
52
(54)
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