N
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FO
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The Stock as a Portfolio
of Durations: Solving Black’s
Dividend Puzzle Using Black’s
Criteria
A
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122
tion of the perfectly negative correlation
between the value of a stock’s dividends and
terminal value as dividend policy changes.
It also shows Baskin’s [1989] inverse relation
between dividends and duration as a function of the less-than-perfectly-negative correlation between the durations of the stock’s
dividends and terminal value. The latter’s
imperfect correlation fundamentally gives
relevance to dividend policy, even in perfect
markets.
Gordon [1963] and Lintner’s [1962]
“bird in the hand” theory argued for higher
value for stocks paying dividends, as current dividends are less risky compared
with expected future dividends. Miller and
Modigliani [1961] argue that this is fallacious,
as a firm’s riskiness depends on the riskiness of its earnings, rather than its dividend
stream. Fama [1974] also finds support for the
view that the dividend decision is separate
from the investment decision. Thus, many
investors have resorted to market imperfections to address the puzzle of why firms pay
cash dividends.
These include signaling effects owing to
information asymmetry. Fama et al. [1969],
Pettit [1972], Watts [1973], Ross [1977],
Kwan [1981], Aharony and Swary [1980],
Asquith and Mullins [1983], Brickley [1983],
Miller and Rock [1985], and Richardson
et al. [1986] generally find positive signaling
R
TI
n “The Dividend Puzzle,” Black [1976]
claims that the harder one looks at the
questions “Why do corporations pay
dividends?” and “Why do investors
pay attention to dividends?,” the more the
answers “… seem like a puzzle, with pieces
that just don’t fit together.” This puzzle has
endured in the corporate finance literature.
Yet in Exploring General Equilibrium, Black
[1995] questioned the existence of puzzles,
suggesting that “when people claim to find
‘puzzles’ in the data, they usually mean that
the data seem to conf lict with their models.”
In questioning the existence of puzzles in
1995, Black suggests that the problem with
his 1976 dividend puzzle is with the models,
not the data. This article proposes to resolve
this Black versus Black debate.
First, it views a stock as a two-asset
portfolio with dividends and terminal value
payoffs.1 Second, it differentiates this portfolio in terms of how these two assets affect
the stock’s value versus duration. Third, it
shows that the correlation between the stock’s
duration as related to dividends and terminal
value is not perfectly negative, unlike the
stock’s values. Fourth, it uses this differentiation to argue that dividends are not puzzling,
even in perfect markets, as duration is not
invariant to dividend policy.
This article shows Miller and Modigliani’s [1961] dividend irrelevance as a func-
IS
holds the Charles R. and
Dorothy S. Carter Chair in
Business Administration at
the University of Texas at
El Paso, in El Paso, TX.
[email protected]
C
I
OSCAR VARELA
LE
IN
A
OSCAR VARELA
THE STOCK AS A PORTFOLIO OF DURATIONS : SOLVING BLACK’S DIVIDEND P UZZLE USING BLACK’S CRITERIA
SUMMER 2015
Copyright © 2015
JPM-VARELA.indd 122
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effects associated with dividends. Fuller and Blau [2010]
also find that dividends signal future earnings for lowquality firms, and eliminate the free-cash-f low problem
for higher quality firms, reducing agency costs. Rozeff
[1982] finds that dividends serve to bond managers
and owners, reducing agency costs, and DeAngelo and
DeAngelo [2007] tie agency costs to financial f lexibility,
as dividends reduce agency costs by limiting free cash
f low and facilitate access to equity capital when necessary, increasing f lexibility.
Dividends have also been related to clientele effects
by Miller and Modigliani [1961], Elton and Gruber
[1970], Black and Scholes [1974], Allen et al. [2000],
and Bell and Jenkinson [2002]. Baker and Wurgler
[2004], however, proposed a catering theory of dividends, where managers cater to investor demand for
increases or decreases in dividends. Dividend policy
has also been related to taxes, generally favoring lower
dividends, as in Farrar and Selwyn [1967] and Brennan
[1970], although Miller and Scholes [1978] argue for
indifference and Masulis and Trueman [1988] for cash
dividends. Myers and Majluf ’s [1984] pecking-order
theory argues that differential internal versus external
costs in raising capital rule against dividend payments
unless a residual is available.
In contrast to this literature, the present article’s
motive for (cash) dividends does not rely on market
imperfections. The timing of cash dividends, as Baskin
[1989] showed, affects the sensitivity of the stock’s value
to changes in its cost of equity (its duration), with such
sensitivities less (more) the higher (lower) the dividend
payments. The present article shows that the main
reason for this result is that, as a portfolio of two assets,
the contributions of dividends and terminal value to the
stock’s duration are not perfectly negatively correlated.
In perfect markets, dividend policy affects the stock’s
duration—but not its value. Firms that pay no dividends
create a higher-risk environment for their owners, with
the effect more significant for longer-lived firms.2
THEORETICAL FRAMEWORK
For the purposes of argument, assume a firm with a
finite, five-year life, where D 1 to D 5 represents the fiveyear dividend stream, P 5 represents the terminal value
of all earnings accumulated (via retention) through year
five, and ke is the cost of equity.
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JPM-VARELA.indd 123
The intrinsic value of this firm’s stock is
P0 =
D1
D2
D3
D4
+
+
+
(1 + ke )1 ((1 ke )2 (1 + ke )3 ((1 ke )4
+
D5
P5
+
5
(1 + ke ) ((1 ke )5
(1)
This stock’s value is derived from an underlying
earnings stream over its five-year life (t = 1 to 5), where
E1 to E 5 represents the five-year earnings stream, rr is
the retention rate of earnings, (1 − rr) is the dividend
payout rate, roe is the return on equity, and (rr) (roe) is
the growth rate of earnings and dividends. Let E 0 be
time zero earnings, such that the earnings stream over
the firm’s five-year life is the compounded value of E 0 at
growth rate (rr) (roe); that is, E 0 [1 + (rr) (roe)] t for t = 1
to 5. These earnings are divided between dividend payments E 0 [1 + (rr) (roe)] t (1 − rr) and retained earnings E 0
[1 + (rr) (roe)] t (rr) for t = 1 to 5. Under these conditions,
the stock’s intrinsic value is
P0 =
E0 [1 + ( )( )]1 (1
(1 + ke )1
)
+
E0 [1 + ( )( )]2 (1
((1 + ke )2
+
E0 [1 + ( )( )]3 (1
(1 + ke )3
)
+
+
E0 [1 + ( )( )]5 (1
(1 + ke )5
)
+
+
[
0
+
[
0
[1 (rr )(roe )]2 ( )] [
+
(1 + ke )2
[1 (rr )(roe )]4 ( )] [
+
(1 + ke )4
)
E0 [1 + ( )( )]4 (1
((1 + ke )4
[
0
0
0
)
[1 (rr )(roe )]1 ( )]
((1 + ke )1
[1 (rr )(roe )]3 ( )]
((1 + ke )3
[1 (rr )(roe )]5 ( )]
((1 + ke )5
(2)
where the first (second) five terms represent the present
value of the five-year dividend (retained earnings)
stream. As a convenience, we show retained earnings
as a stream, rather than as its fifth-year terminal value
P 5, although the values are the same when adjusted for
time value, since roe, the rate at which retained earnings
is compounded to its terminal value, is equal to ke, the
rate at which the terminal value is discounted.
If the retention rate rr changed, shown as ∂rr (by
0.01, for example), then the change in the dollar value
of the five dividend stream terms is equal but opposite in
THE JOURNAL OF PORTFOLIO M ANAGEMENT
123
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sign to the change in the dollar value of the five earnings
retention stream terms (terminal value), such that
⎡ E0 [1 + ( ∂ )( )]1 ( − ∂rr ) E0 [1 + ( ∂ )( )]2 ( − ∂rr )
+
⎢
(1+ke )1
((1 + ke )2
⎣
Substituting Equation (4) for dividends and the
terminal value, obtain
Duration =
− E0 [1 + ( )( )]1 (1 − )(1)
(1 + ke )1 (P0 )
+
E0 [1 ((∂ )( )]3 ( − ∂rr ) E0 [1 ((∂ )( )]4 ( − ∂rr )
+
(1 + ke )3
((1 + ke )4
+
+
E0 [1 ((∂ )( )]5 ( − ∂rr )] ⎤
⎥
(1 + ke )5
⎦
− E0 [1 + ( )( )]2 (1 − )(2)
(1 + ke )2 (P0 )
+
− E0 [1 + ( )( )]3 (1 − )(3)
(1 + ke )3 (P0 )
+
− E0 [1 + ( )( )]4 (1 − )(4)
(1 + ke )4 (P0 )
⎡[
= −⎢
⎣
+
+
0
[
[
[1 ((∂∂rr )(roe )]1 ( ∂ )] [
+
(1 + ke )1
0
[1 ((∂∂rr )(roe )]3 ( ∂ )] [
+
(1 + ke )3
0
[1 ((∂∂rr )(roe )]2 ( ∂ )]
((1 + ke )2
0
[1 ((∂∂rr )(roe )]4 ( ∂ )]
((1 + ke )4
[1 ((∂∂rr )(roe )] ( ∂ )] ⎤
⎥
(1 + ke )5
⎦
5
0
(3)
− E0 [1 + ( )( )]5 (1 − )(5)
+
(1 + ke )5 (P0 )
+
where (∂rr) (roe) is the change in the growth rate from
the ∂rr change in the retention rate, E 0 [1 + (∂rr) (roe)] t
(−∂rr) is the change in dividends, and E 0 [1 + (∂rr) (roe)] t
(∂rr) is the change in retained earnings at time t from
the ∂rr change in the retention rate for t = 1 to 5. Price
is invariant to dividend policy, as the stock is a perfectly
hedged portfolio of cash dividends and retained earnings
with values that are are perfectly negatively correlated.
Consistent with Miller and Modigliani [1961], the stock
is risk-free with respect to its cash dividend policy.
In contrast, although the stock’s value behaves like
a perfectly hedged portfolio of dividends and terminal
values, the stock’s duration does not. The stock’s duration—the sensitivity of its value to a change in the cost
of equity—explicitly depends on the timing of its dividends and terminal value, where we use our five-year
life example to calculate duration as follows:3
Duration =
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JPM-VARELA.indd 124
∂ 0 × (1
( + ke )
− D1[1]
=
∂( + ke ) × (P0 ) ((1 + ke )1 × (P0 )
+
− D2 [2]
− D3 [3]
+
2
(1 + ke ) × (P0 ) ((1 + ke )3 × (P0 )
+
− D4 [4]
− D5 [5]
+
4
(1 + ke ) × (P0 ) ((1 + ke )5 × (P0 )
+
− P5 [5]
(1 + ke )5 × (P0 )
(4)
+
+
+
+
−[
0
−[
0
−[
0
−[
0
−[
0
[1 + (rr )(roe )]1 ( ) ](1 +
(1 + ke )1 (P0 )(
)(1 + ke )4
)4 (5)
[1 + (rr )(roe )]2 ( ) ](1 +
(1 + ke )2 (P0 )(
)(1 + ke )3
)3 (5)
[1 + (rr )(roe )]3 ( ) ](1 +
(1 + ke )3 (P0 )(
)(1 + ke )2
)2 (5)
[1 + (rr )(roe )]4 ( ) ](1 +
(1 + ke )4 (P0 )(
)(1 + ke )1
)1 (5)
[1 + (rr )(roe )]5 ( ) ](1 +
(1 + ke )5 (P0 )(
)(1 + ke )0
)0 (5)
(5)
where the numerator of the last five terms in Equation (5) represents the terminal value, given each year’s
earnings, earnings growth, retention of earnings, and
reinvestment rates. The calculation of the stock’s duration depends not just on the value of the terminal cash
f low, but also on its timing, unlike the case for the
stock’s value. We apply and weight all cash f lows related
to the terminal value into the terminal value year (year
5), with any change in rr having differential effects on
duration from dividends versus terminal values. Duration is not invariant to dividend policy, as identified by
changes in the retention rate rr.
THE STOCK AS A PORTFOLIO OF DURATIONS : SOLVING BLACK’S DIVIDEND P UZZLE USING BLACK’S CRITERIA
SUMMER 2015
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If the retention rate rr changes, as shown as ∂rr (by
0.01, for example), then the change in duration related to
the first five dividend stream terms is not equal (although
it is opposite in sign) to the change in duration related
to the second five earnings retention stream terms (terminal value). The durations of the dividend stream and
terminal value are not perfectly negatively correlated.
This result can be shown by rearranging Equation
(5) as follows, after modifying it for a change in rr as
specified by ∂rr, such that
−[[
+
+
+
+
0
−[[
[1 ( ∂rr
r )(roe )]1 ( − ∂ )(1)
(1 + ke )1 (P0 )
[1 ( ∂rr
r )(roe )] ( − ∂ )(2)
(1 + ke )2 (P0 )
2
0
−[[
0
−[[
0
−[[
0
+
+
+
[1 ( ∂rr
r )(roe )] ( − ∂ )(3)
(1 + ke )3 (P0 )
JPM-VARELA.indd 125
−E
E0 [1 ( ∂ )( )]2 ( − ∂rr )(2)
(1 + ke )2 (P0 )
+
−E
E0 [1 ( ∂ )( )]3 ( − ∂rr )(3)
(1 + ke )3 (P0 )
+
−E
E0 [1 ( ∂ )( )]4 ( − ∂rr )(4)
(1 + ke )4 (P0 )
+
−E
E0 [1 ( ∂ )( )]5 ( − ∂rr )(5)
(1 + ke )5 (P0 )
[1 ( ∂rr
r )(roe )]4 ( − ∂ )(4)
(1 + ke )4 (P0 )
+
[1 ( ∂rr
r )(roe )]5 ( − ∂ )(5)
(1 + ke )5 (P0 )
+
0
−[[
0
−[[
0
−[[
0
−[[
0
[1 ( ∂rr
r )(roe )]1 ( )](1
(1 + ke )1 (P0 )(1
)( ke )4
)4 (5)
+
[1 ( ∂rr
r )(roe )]2 ( )](1
(1 + ke )2 (P0 )(1
)( ke )3
)3 (5)
+
[1 ( ∂rr
r )(roe )]3 ( )](1
(1 + ke )3 (P0 )(1
)( ke )2
)2 (5)
[1 ( ∂rr
r )(roe )]4 ( )](1
(1 + ke )4 (P0 )(1
)( ke )1
)1 (5)
[1 ( ∂rr
r )(roe )]5 ( )](1
(1 + ke )5 (P0 )(1
)( ke )0
)0 (5) ⎤
⎥
⎦
(6)
where the terms on the left relate to dividends and those
on the right relate to the terminal value.
Simplifying the right-hand side of Equation (6)
further, given that roe = ke,
SUMMER 2015
+
⎡ −[[
≠ −⎢
⎣
3
⎡ −[[
≠ −⎢
⎣
+
−E
E0 [1 ( ∂ )( )]1 ( − ∂rr )(1)
(1 + ke )1 (P0 )
0
−[[
0
−[[
0
−[[
0
−[[
0
[1 ( ∂rr
r )(roe )]1 ( ∂rr )](5)
(1 + ke )1 (P0 )
[1 ( ∂rr
r )(roe )]2 ( ∂rr )](5)
(1 + ke )2 (P0 )
[1 ( ∂rr
r )(roe )]3 ( ∂rr )](5)
(1 + ke )3 (P0 )
[1 ( ∂rr
r )(roe )]4 ( ∂rr )](5)
(1 + ke )4 (P0 )
[1 ( ∂rr
r )(roe )]5 ( ∂rr )](5) ⎤
⎥
(1 + ke )5 (P0 )
⎦
(7)
it follows that if ∂rr ≠ 0, that the sum of the terms on the
right-hand side of Equation (7) are greater (in absolute
value) than the sum of the terms on the left-hand side of
Equation (7), as all terms on the right side are otherwise
similar, except that they are multiplied by five. In addition, if ∂rr > 0 such that cash dividends are decreased,
duration unambiguously rises, and if ∂rr < 0 such that
cash dividends are increased, duration unambiguously
falls. Duration is not independent of dividend policy,
because the components of duration are not perfectly
negatively correlated.
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SIMULATION FRAMEWORK, PRICES
AND DURATIONS
Assume firms with lifespans of 100 years (lifespan
1) and 1000 years (lifespan 2), roe (return on equity)
of 12%, ke (cost of equity) of 12%, and initial earnings of $0.12 per share. Assume that rr (retention rate of
earnings)—that is 1 − rr (dividend payout rate), varies
from 100% (0%) to 0% (100%) in 25% increments, producing five dividend policies and cash f low simulations
for each lifespan. Exhibit 1, panel A, shows this initial
framework.
The five cash f lows for each lifespan are affected by
different growth rates, as the growth rate equals (rr) (roe).
In Exhibit 1, Panel B (Panel C) shows the five lifespan 1
(2) simulation scenarios, labeled simulations L1.1 to L1.5
(L2.1 to L2.5) for the 100 (1,000)-year life firms.
Exhibit 2 shows the price and duration of the firm’s
stock for the initial case, where ke equals roe. In Exhibit 2,
panel A (panel B) shows the stock price and duration
for the simulation scenarios pertaining to lifespan 1 (2),
given the initial conditions previously described.
The results show that the value of the firm’s stock
at time 0 is $1.00, consistent with Modigliani–Miller,
regardless of the life of the firm or its dividend policy.
Under equilibrium conditions, in a steady state such that
the firm’s roe equals ke, the firm that pays (does not pay)
dividends simultaneously reduces (increases) its growth rate
EXHIBIT 1
Simulation Scenarios
Notes: The prefix “L1” and “L2” indicates that the simulations involve the 100- and 1,000-year firm lifespans, respectively. Each of the five simulations
within each category is related to the percentage of earnings retained by the firm. Simulations L1.1 and L2.1 indicate 100% retention, L1.2 and L2.2 refer
to 75% retention, L1.3 and L2.3 to 50% retention, L1.4 and L2.4 to 25% retention, and L1.5 and L2.5 to 0% retention.
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EXHIBIT 2
Stock Price and Duration for Simulation Scenarios
Notes: The prefixes “L1” and “L2” indicates that the simulations involve the 100-and
1,000-year firm lifespans, respectively. Each of the five simulations within each category
is related to the percentage of earnings retained by the firm. Simulations L1.1 and L2.1
indicate 100% retention, L1.2 and L2.2 refer to 75% retention, L1.3 and L2.3 to
50%, L1.4 and L2.4 to 25%, and L1.5 and L2.5 to 0% retention. The initial situations are shown in Exhibit 1, panel A.
and future terminal value, at terms of trade that
keep the present value of the firm unchanged.
Even as the stock’s value is invariant to its
dividend policy, its duration is not. The duration
of the stock for lifespan 1 (2) simulations vary
from 100 (1000) for a 100% retention policy to
18.59 (18.66) for a 50% retention policy to 9.33
(9.33) for a 0% retention policy.4 By causing
changes in duration, dividend policy and changes
in dividend policy can create low-or high-risk
environments for stockholders. This result is
more deeply explored through an examination
of the effects of hypothetical changes in ke on
the stock’s price in the next section.
THE STOCK AS A PORTFOLIO OF
CASH FLOWS AND DURATIONS
The stock may be viewed as a security
that consists of a portfolio of two cash f lows:
EXHIBIT 3
Components of the Stock’s Price and Duration for Lifespan 1 and Lifespan 2 Simulations when Return on
Equity (12%) Equals Cost of Equity (12%)
Notes: The prefixes “L1” and “L2” indicate that the simulations involve the 100- and 1,000-year firm lifespans, respectively. Each of the five simulations within each category is related to the percentage of earnings retained by the firm. Simulations L1.1 and L2.1 indicate 100% retention, L1.2 and
L2.2 refer to 75% retention, L1.3 and L2.3 refer to 50% retention, L1.4 and L2.4 refer to 25% retention, and L1.5 and L2.5 refer to 0% retention. An
indication that a value ≈ zero means that the value is marginally greater than zero when values are accounted for by more than four decimal places.
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dividends and terminal values. When related to the value
of the stock, Miller and Modigliani’s [1961] invariance
propositions suggest that when ke equals roe, the stock’s
payoffs from its two cash f low components are perfectly
negatively correlated. Increasing cash dividends increases
the stock’s value related to its dividends but reduces its
terminal value by an equal amount related to its retention policy.
Exhibit 3, panels A.1 and B.1, shows the components of the stock’s price for all simulation scenarios
when ke equals roe. The results for lifespan 1 in panel A.1
show that as the firm retains less of its earnings, a greater
amount of its invariant $1.00 price can be attributed to
dividends and a lower amount to its terminal value. All
scenarios in lifespan 1 where the firm pays dividends
(starting with simulation L1.2 with 25% dividend payouts) show that the largest component of a firm’s stock
price is derived from the dividend stream. The change
in the value of the dividend stream completely offsets
the change in the value of the terminal retained earn-
ings as dividend payouts change, such that the stock’s
price is unaffected by the dividend policy. The results for
lifespan 2 in panel B.1 are generically the same, except
that for such a long-lived firm (1,000 years), virtually all
of the stock’s value is derived from the dividend stream,
starting with simulation L2.2 with 25% dividend payouts, even though some earnings are retained.
The stock as a portfolio of durations has durations
that, even when ke equals roe, are not invariant to dividend policy. Increases in cash dividends can increase or
decrease the duration component related to cash dividends (depending on whether the initial payout is zero
or positive), as more of the stock’s anticipated payoffs
occur earlier in time. The associated decreases in terminal value decrease the component of duration related
to terminal value. The change in the stock’s duration
from an increase in cash dividends and decrease in terminal value is not perfectly negatively correlated, with
an overall decrease in duration. We obtain opposite
results for decreases in dividends.
EXHIBIT 4
Components of the Stock’s Price and Duration for Lifespan 1 and Lifespan 2 Simulations when the Cost of
Equity Rises from 12% to 13% while Return on Equity Equals 12%
Note: See notes to Exhibit 3.
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In Exhibit 3, panels A.2 and B.2 show the components of the stock’s duration for all simulation scenarios
when ke equals roe. The results for lifespan 1 in panel
A.2 show that as the firm retains less earnings, its stock’s
duration falls, with decreases in both terminal value (as
a greater percentage of earnings are paid out through
dividends) and in the duration component related to the
dividend stream. The only exception is the case where
no dividends are paid, when duration from dividends is
zero and duration from the retained earnings’ terminal
value is 100. As the firm retains more earnings, its stock’s
duration rises, with increases in both the duration component related to the dividend stream and to terminal
value, as a lesser percentage of earnings are paid out
through dividends. Again, the only exception is the case
where no dividends are paid. The results for lifespan 2 in
panel B.2 are generically the same, except that for such
a long-lived firm (1,000 years) virtually all the stock’s
duration is derived from the dividend stream, starting
with simulation L2.2 with 25% dividend payouts, even
if some earnings are retained.
FURTHER DISCUSSION WHEN THE COST
OF EQUITY CHANGES
The value of the stock is not invariant to the dividend policy when ke changes, because its duration is
not invariant to the dividend policy. Thus, even the
present value of the stock’s dividend and terminal value
cash f lows are no longer perfectly negatively correlated.
Exhibits 4 and 5 illustrate these results.
Exhibit 4 shows the effects of an increase from
12% to 13%, and Exhibit 5 the effects from a decrease
from 12% to 11% in ke on the stock price and duration,
and the components of these as related to dividends and
terminal values, for all dividend policy simulations. In
EXHIBIT 5
Components of the Stock’s Price and Duration for Lifespan 1 and Lifespan 2 Simulations when the Cost of
Equity Falls from 12% to 11% while Return on Equity Equals 12%
Note: See notes, to Exhibit 3.
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these exhibits, panels A.1 and B.1 show the effects on the
stock price and its components for company life spans
of 100 and 1,000 years. Panels A.2 and B.2 show the
effects on the duration and its components for company
life spans of 100 and 1,000 years.
Increases in ke decrease the stock’s total value
(Exhibit 4, panels A.1 and B.1), and decreases in ke
increase the stock’s total value (Exhibit 5, panels A.1
and B.1). More importantly, the extent of stock price
changes varies with dividend policy. The volatility in the
stock’s price is greatest when the firm retain all earnings
(simulations L1.1 and L2.1), compared with when they
retain no earnings (simulations L1.5 and L2.5), regardless of the length of the firm’s life, although the greatest
volatility occurs for firms with longer lives (lifespan 2
as opposed to lifespan 1). Except for the case in which
the firm retains most or all of its earnings, changes in
the stock’s price are mostly derived from changes in the
dividend stream’s value, and greater dividend payouts
best insulate the stock price from greater volatility.
Finally, changes in ke affect duration, such that as
ke rises, duration falls (Exhibit 4, panels A.2 and B.2,
compared with Exhibit 3, panels A.2 and B.2), and as
ke falls, duration rises (Exhibit 5, panels A.2 and B.2,
compared with Exhibit 3, panels A.2 and B.2). The
duration change is primarily attributed to the firm’s dividend stream, especially for reasonable dividend payouts
and longer-lived firms. The basic principle—that stocks
paying high dividends have lower durations—holds
regardless of whether ke rises or falls. Future changes in
ke affect the stock’s values the most when fewer dividends
are paid, such that as ke changes, the stock’s value is not
invariant to its dividend policy. Nevertheless, it may be
that as ke changes, the firm re-evaluates its investment
choices such that roe adjusts to equal the new ke, producing a value, but not a duration, that subsequently is
invariant to the dividend policy.
different sensitivities, depending on payouts, leading to
the relevance of dividend policy.
One may wonder why the Miller and Modigliani
[1961] argument that homemade dividends lead to dividend policy irrelevance cannot be extended to homemade durations. After all, if stockholders can create their
own dividend stream without affecting value, then could
not this invariance also be applicable to duration? It
is instructive to view the valuation model in a portfolio context, where the stock is a portfolio of one asset
with its return equal to a stream of cash dividends, and
another asset with its return equal to a terminal value.
When a stock’s values from these two assets are perfectly
negatively correlated, it composes a perfectly hedged
portfolio: the stock’s value is invariant to the payment
of cash dividends. However, the sensitivity of the stock’s
value to subsequent changes in the cost of capital is not
invariant to the dividend payment. That is, the durations
of the dividends and terminal value, when considered
separately, are not perfectly negatively correlated. As the
composition of the stock’s payoff components change,
so does its duration, particularly for longer-lived firms
that pay low dividends.
The value of the firm’s stock in perfect markets is
invariant to its dividend policy when the cost of equity
equals the return on equity. Nevertheless, stockholders
are cognizant of the risks associated with a stock investment, given changes in the cost of equity. Risks rise as
dividend payouts fall, such that any increase in the cost of
equity detrimentally affects the stock’s value by a greater
amount, the lower the dividend payments.
Dividend policy may not matter (relative to value)
when the markets are in equilibrium. But stockholders
are wise to demand dividends that can insulate a stock’s
value from losses when the cost of equity rises.
SUMMARY
Thanks to the College of Business Administration at
the University of Texas at El Paso for Summer 2010 research
support for this work. I also thank an anonymous referee
and Editor Frank J. Fabozzi for their comments, as well as
Professors Jun “Q J” Qian and Oguzhan Karakas of Boston
College.
1
This idea is analogous to stripping bonds into their
interest and principal components.
2
Bierman and Smidt [2007, pp. 102-103] applied the
same argument to capital budgeting projects. In analyzing
This article focuses on the relevance of dividend
policy with respect to the duration of the firm’s stock. It
shows that projected earnings streams for firms with different finite lives have different risks, depending on how
the earnings are paid out as dividends, because duration depends on the payout stream. Changes in stock
values that come from changes in required returns have
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ENDNOTES
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two projects, they found that both had the same value, but
one had a lower duration. As they stated, “If we think that
the rate of interest may increase, we might prefer the investment with the shorter duration, since its present value would
decline by less as a result of the interest rate increase.”
3
We use a finite life assumption because it more explicitly shows the relation of dividend policy to duration, in contrast to the standard infinite life assumption.
4
Holt [1962] examined how the duration of a growth
rate affects share prices, but did not examine how the dividend stream affects the stock’s duration.
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