ISSN (Print) 0473-453X

Discussion Paper No. 1070

ISSN (Online) 2435-0982

THE LOCK-IN EFFECTAND
THE CORPORATE PAYOUT PUZZLE
Chris Mitchell

Revised August 2021
December 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Lock-In Effect and the Corporate Payout Puzzle
Chris Mitchell∗
August 15, 2021

Abstract
Taxes on capital gains are deferred until realization, whereas dividends are taxed upon
accrual. This often makes dividends tax-disadvantaged relative to share repurchases,
which leads to the payout puzzle: why do firms pay dividends? This paper demonstrates
that tax deferment can also provide a solution to the payout puzzle: if shareholders
demand repurchase premiums when selling equity back to a firm - as compensation
for accelerated realizations - then dividends can become tax-efficient. This mechanism
is appealing because it explains dividend payments without appealing to asymmetric
information, incomplete contracts, repurchase constraints, or shareholder irrationality.

Keywords: Payout Policy, Capital Taxation.
JEL classification: G35, H24.

∗

Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe, Hyogo, 657-8501,
Japan. Phone: +81-7-8803-6885. E-mail: mitchell@econ.kobe-u.ac.jp. I would like to thank Chris Bennett,
Kul Bhatia, James Davies, Mike McCausland, Miguel Molico, Peter Streufert, Tamon Takamura, Yaz Terajima, seminar participants at Osaka University, Sophia University, the University of Ottawa, and Western
University for many helpful comments on earlier drafts of this paper. I retain full responsibility for any possible errors. I would also like to acknowledge financial support from JSPS KAKENHI Grant-in-Aid Number
15H05728.

1

1Introduction

Shareholders derive income from dividends and capital gains. Both sources of income haveidentical value in perfect capital markets without taxation, but their values can diverge whencapital income is taxed. In the United States, for instance, dividends and long-term capitalgains have a top federal tax rate of 20%, and when you factor-in the ability to postpone
taxes on accrued capital gains until realization, the latter’s effective tax rate is lower (likely
in the range of 11% - 16% for many shareholders, as discussed below). This tax differential
makes dividend payments somewhat of a puzzle, since firms can repurchase equity and generate tax-favored capital gains for their shareholders; a concept enshrined in Fisher Black’s
1976 paper “The Dividend Puzzle.” The current paper develops a model of corporate payout
policy to demonstrate that dividends may be paid (by some firms) for the very same reason
that dividends are currently tax disadvantaged in the US: shareholders often postpone therealization of capital gains for tax purposes. This incentive to delay equity sales - known as
the lock-in effect - may require some firms to “over pay” when repurchasing equity, which
can make tax-disadvantaged dividends optimal. The argument below is built on evidence
that both retail and institutional investors often behave in accordance with the lock-in effect,
and that equity prices typically appreciate during repurchase programs.
Payout policy - the choice between dividends and share repurchases - is shown by Miller
and Modigliani (1961) to be independent of firm value when capital markets are perfect,
investment policy is fixed, investors are rational, and taxes are nil. This follows from theequilibrium condition that wealth-maximizing shareholders are indifferent between receiving$1 in cash (a dividend payment) and a stream of cash flows with present value $1 (via higher
ownership concentration following a share repurchase). This irrelevance result fails to hold if
one form of payout is tax disadvantaged, however, which is often thought to be the case with
US dividends owing to: higher statutory tax rates prior to 2003; and the postponement of
capital-gains taxes until realization, which lowers their effective tax rate. The second factor
produces the lock-in effect mentioned above, which is manifest in the trading behavior ofboth retail and institutional investors.
Feldstein et al. (1980) appears to be the first study to provide empirical evidence that2retail investors time their capital-gains realizations to lower tax burdens. This effect isalso identified in Auten and Clotfelter (1982), which shows that realizations are particularly
responsive to the transitory components of capital-gains taxation; a result confirmed by Burman and Randolph (1994) and Auerbach and Siegel (2000). This evidence is supplemented
by Brown and Ryngaert (1992) and Landsman and Shackelford (1995), which show thatshareholders typically demand higher prices when selling equity with larger accrued capitalgains. Finally, the evidence in Barber and Odean (2003) and Ivkovic et al. (2005), that investors harvest capital losses in December, is also consistent with tax-motivated realizations.
These empirical regularities are consistent with Chey et al.’s (2006) estimate that 90% of all
directly-held stock is placed within taxable accounts.1
Retail investors have gradually substituted away from direct equity ownership and towards institutional ownership. This places greater emphasis on the latter’s trading behavior
when considering lock-in effects. Rather than being a homogeneous group, institutional investors differ greatly regarding their tax exposure: ranging from tax-exempt pension funds
and non-profit organizations; to partially-taxed insurance companies (with tax relief for policyholder income); to fully taxable corporations, hedge funds, and mutual funds. Chetty
and Saez (2005) estimates that only 15% of institutional investors are fully nontaxable on adollar-weighted basis, and since mutual-fund inflows are positively related with a fund’s tax
efficiency (Bergstresser and Poterba, 2002 and Dimmock et al., 2018), and fund-manager
compensation is often tied to asset size, these managers have an incentive to be tax efficient.
This seems to be the case; Huddart and Narayanan (2002), Jin (2006), Sialm and Starks(2012), and Dimmock et al. (2018) all provide evidence that investment funds with taxsensitive clients behave in accordance with the lock-in effect, and Sialm and Starks (2012)
and Sialm and Zhang (2020) provide evidence that pre-tax returns are typically not sacrificed
by tax-efficient funds.2
1

However, some investors consistently make investment mistakes by selling appreciated assets too soon
while failing to realize capital losses in a timely manner; Shefrin and Statman (1985) dub this phenomenon
the disposition effect, and both Barber and Odean (2003) and Ivkovic et al. (2005) document its occurrence.
The empirical evidence suggests that wealthier households are less disposed to making investment errors (seeIvkovic et al., 2005 and Campbell, 2006).
2
The down-side of tax efficiency is greater capital-gains overhang - higher levels of accrued capitalgains within a fund, and thus, a higher potential for capital-gains taxes in the future - which is shownby Bergstresser and Poterba (2002) to reduce future inflows. However, since future realizations are timedby the fund itself, the actual impact on shareholder taxes is relatively small according to Sialm and Zhang

3With this backdrop of shareholder lock-in effects, the model’s primary mechanism isstraightforward to describe. Firms in the model use profits to pay dividends, repurchaseshares, and make capital investments. Managers act in the interest of shareholders by choosing investment/payout policy to maximize firm value. Once investment is determined, management decides what fraction of aggregate payout to distribute as dividends and whatfraction to use repurchasing shares. Both forms of payout are subject to capital-income taxation, and dividends are taxed more heavily. Shareholders are heterogeneous regarding their
desired holding periods and level of accrued capital gains on firm equity, and shareholderwealth maximization is shown to create a wedge between equity’s intrinsic value and the ask
price of shareholders with an accrued capital gain and non-zero investment horizon - compensation for the lock-in effect. This differential - referred to as the “lock-in premium” herein- is weakly positive and increasing in both capital gains and investment horizon. Firms may
therefore be required to pay lock-in premiums when repurchasing equity, which is acceptableprovided these premiums remain small, since the alternative is paying a tax-disadvantaged
dividend. However, since firms typically repurchase large quantities of equity, especially overmultiple years,3 the marginal shareholder’s lock-in premium can become sufficiently largethat paying a tax-disadvantaged dividend becomes optimal. At this point, firms pay dividends. This mechanism is consistent with Brav et al.’s (2005) finding that 86% of surveyed
executives cite their company’s stock price - relative to its true/intrinsic value - as an important, or very important, determinant of repurchase activity.
This mechanism is appealing because it explains dividend payments without appealing toasymmetric information, repurchase constraints, incomplete contracts and/or irrationality:4(2020).
3
The announcement size of a typical repurchase program is reported to be 6.6% in Ikenberry et al. (1995),
6.8% in Gaspar et al. (2012), 7% in Stephens and Weisbach (1998), 7.5% in Chan et al. (2010), and 8% in
Jagannathan et al. (2000). In addition, Stephens and Weisback (1998) reports that many firms repurchasemore equity than they announce, and Skinner (2008) reports that repurchasing firms do so every 2 out of
3 years on average (since 1990). Taken together, share repurchases can add significant demand-side pricepressure.
4
As discussed in Section 2, these factors are highlighted in the following studies. 1) Repurchase constraints: King (1977), Auerbach (1979), Bradford (1981), and Grullon and Michaely (2002). 2) Asymmetric
information: Bhattacharya (1979), John and Williams (1985), Ofer and Thakor (1987), Brennan and Thakor(1990), Bernheim (1991), and Allen et al. (2000). 3) Incomplete contracting: Easterbrook (1984), Jensen(1986), Allen et al. (2000), and Morck and Yeung (2005). And 4) Irrationality: Shefrin and Statman (1984),

4the only assumption from Miller and Modigliani (1961) to be relaxed is zero taxes.5 This
is not meant to question the validity of these restrictions, only to point out that skeptics oftheir applicability - for a particular firm at a particular time - may find the current model’spaucity of qualifying assumptions appealing. In any case, manages hold heterogeneous viewson which factors drive payout policy - as documented in Brav et al. (2005) - so any modelof payout policy is unlikely to explain the behavior of all firms at all times, and many of the
explanations offered (including this one) are not mutually exclusive.
The remainder of this paper is organized as follows. Section 2 discusses the previous
literature and how this paper fits within it. Section 3 provides an empirical context forthe model by illustrating that dividends are tax disadvantaged in the United States. Section 4 contains the main payout model. Section 5 extends this model by endogenizing theshareholder distribution. Section 6 concludes.

2

Previous Literature

The theoretical literature on why firms pay tax-disadvantaged dividends can be segmented
by the four restrictions mentioned above, i.e., constraints on share repurchases, constraints
on information sets, the incompleteness of contracts, and irrationality.6 Starting with thefirst, King (1977), Auerbach (1979), and Bradford (1981) assume that firms avoid equity
buybacks, in part, due to Section 302 of the Internal Revenue Code (implicit in King, 1977
and Bradford, 1981), whereby share repurchases that resemble dividend payments (proportional repurchases) are taxed as such.7 In addition, Grullon and Michaely (2002) argues thatsome firms were concerned that repurchasing equity prior to the adoption of rule 10b-18 in
Bagwell and Shoven (1989), and Baker and Wurgler (2004).
5
Some papers (e.g., Bagwell and Shoven 1989) note that firms may tailor payout policy to certain tax
clienteles: firms that cater to shareholders with a dividend-tax advantage (e.g., certain non-pass-throughcorporations) will pay dividends, while those catering to shareholders with a dividend-tax disadvantage (e.g.,hedge funds and mutual funds) will repurchase shares. However, since most of the market’s capitalization isconcentrated in firms that pay dividends and repurchase stock (Skinner 2008), shareholders that gravitate
towards either dividend-only firms or repurchase-only firms will be grossly under diversified.
6
Dividends may also be paid due to the transactions costs of buying and selling equity, where the latter
affects shareholders that consume out of realized capital gains. For a survey of the payout literature, see
Allen and Michaely (2003).
7
See Brennan and Thakor (1990) for a description of Section 302’s guidelines.

51982 would have triggered antimanipulative provisions of the Securities and Exchange Act.8
The current paper places no restrictions on payout policy beyond the standard non-negative
dividend constraint.
Regarding the second restriction, Bhattacharya (1979) and Ofer and Thakor (1987) argue
that dividend payments signal firm value by exposing firms to costly external finance: dividend taxes are necessary for Bhattacharya’s (1979) separating equilibria, while they affectthe payout mix in Ofer and Thakor (1987). John and Williams (1985) and Bernheim (1991)
also require dividend taxes to generate separating equilibira: dividend payments reduce equity dilution in John and Williams (1985), while they fine-tune payout taxes in Bernheim(1991). In addition to addressing information problems between managers and shareholders,Brennan and Thakor (1990) and Allen et al. (2000) show that dividends can address information problems among shareholders: dividend payments forestall the transfer of wealth from
uninformed to informed shareholders during repurchase programs in Brennan and Thakor(1990), while they attract informed shareholders in Allen et al. (2000). The current paper
assumes that all market participants have the same information.
Regarding the third restriction, Jensen (1986) seems to be the quintessential argument
in favor of using dividend payments to extract free cash flow, although it does champion
interest payments for this purpose. Morck and Yeung (2005) argues that bankruptcy concerns make dividend payments the superior option. Allen et al. (2000) and Morck and Yeung(2005) argue that dividends attract large institutional investors that reduce collective-action
problems (Morck and Yeung 2005) and increase monitoring effort. Easterbrook (1984) also
highlights monitoring, but focuses on the external monitoring brought about because highdividend payments lead to external finance more often (as in Bhattacharya, 1979 and Ofer
and Thakor, 1987). Easterbrook (1984) also notes that dividends can offset managerial riskaversion by increasing leverage, and therefore, equity value. The current paper assumes thatmanagers act in the interest of shareholders and only invest in positive net-present-value
projects.
Finally, there are two sets of explanations pertaining to irrationality: irrational managers
8

Rule 10b-18 provides guidelines for repurchasing equity; see Jagannathan et al. (2000) for details.

6and irrational shareholders. Regarding the first, Bagwell and Shoven (1989) propose thatmanagers have gradually learned to appreciate the benefits of share repurchases over time,
and that Nixon’s price and wage ceilings of the 1970s - which led to “voluntary” dividend
controls according to Ofer and Thakor (1987) - catalyzed their learning-by-doing process.
When this argument is coupled with Lintner’s (1956) dividend-stickiness model, the paymentof current dividends is understandable.9 With regard to shareholder irrationality, Shefrin
and Statman (1984) argues that firms pay dividends because shareholders suffer from regret aversion and/or self-control issues; and since preferences are guided by prospect theory(rather than expected-utility theory), mental accounting creates a value for dividends. In
addition, Baker and Wurgler (2004) argues that shareholders have time-varying preferences
for dividend-paying firms (or conversely, growth-oriented firms), and that firms cater to these
inexplicable preferences by initiating, continuing, or omitting dividend payments. The current paper assumes that both managers and shareholders are perfectly rational, and thatshareholders are expected-wealth maximizes.
The current study is also related to theoretical work on share repurchases and equitysupply curves. Two relevant papers are Stulz (1988) and Bagwell (1991), which study
upward-sloping equity-supply curves in the context of corporate takeovers, and note thatlock-in effects may contribute to their positive gradients. Stulz (1988) argues that share
repurchases are useful for increasing takeover-bid premiums but reduce success probability,
whereas Bagwell (1991) highlights their usefulness as a takeover defense, since repurchasing
equity increases acquisition cost. Evidence for upward-sloping equity-supply curves, in thecontext of Dutch repurchase auctions, is provided by Bagwell (1992). The current paper
differs from Stulz (1988) and Bagwell (1991) in a number of important ways. First, it deals
with dividend policy (a regularly-occurring consideration for most firms) and not takeover
strategy (a comparatively less-frequent consideration). Second, payout policy in the current
model involves a trade off between taxes on dividends and capital gains, whereas in Stulz(1988) and Bagwell (1991) it involves wealth transfers between target-firm shareholders andacquiring-firm shareholders. Third, a greater emphasis is placed on characterizing the relationship between shareholder characteristics and lock-in premiums in the current study,
9

Brav et al. (2005) also documents that some managers pay dividends because of inertia; wishing they
had never started.

7

which is important for understanding its static and dynamic properties. And furthermore,
the current model provides a more holistic analysis of the firm’s investment/payout policy
and how dividends and share repurchases evolve over time.
In addition to Stulz (1988) and Bagwell (1991), Barclay and Smith (1988), Chowdhry
and Nanda (1994), and Huang and Thakor (2013) all highlight the positive relationshipbetween share repurchases and stock prices. This relationship is attributable to higher bidask spreads in Barclay and Smith (1988), the information content of share repurchases in
Chowdhry and Nanda (1994), and the reduction in shareholder-manager disagreement in
Huang and Thakor (2013). The current paper differs from these studies because it considers
a complete-information framework with perfectly-rational agents.

3

Empirical Payout

This section provides an empirical context for the payout model by illustrating the US payout puzzle over the period 1972-2017. This is done by establishing both the prevalence ofdividend payments over this period and their tax disadvantage relative to share repurchases.
Since capital gains are taxed upon realization, an investor’s effective tax rate on accrued
capital gains is weakly lower than their statutory tax rate on realized capital gains (denotedby τg ). To see this, suppose that an asset appreciates by $1 today and is liquidated in T
periods. The tax liability thus created - due in T periods - can be satisfied by investingτg /(1 + rf )T in a risk-free security with after-tax interest rate rf . As such, one measure of
the effective tax rate is τg /(1 + rf )T ≤ τg , which is decreasing in rf and T .10 This raises
the question: what is the “marginal investor’s” effective tax rate - denoted by τe - i.e., theinvestor who sets equity prices? This question is partially addressed by Protopapadakis(1983) and Chay et al. (2006), which suggest that US equities have effective tax rates ofapproximately 55%-80% of the statutory rate on average.11
10This explanation is similar to that in Constantinides (1983) and is well articulated by King (1977, p. 59):
“Deferral is equivalent to an interest-free loan from the revenue authorities to the taxpayer of an amountequal to the tax liability on the accrued gain, and hence is also equivalent to a reduction in the effective rateof tax.” Section 5 below provides a detailed discussion of effective tax rates on accrued capital gains.
11
Poterba (1987) suggests 25%, which is partially supported by Ivkovic et al. (2005).

8

Since share repurchases generate accrued capital gains for most investors, it is standard
to apply the marginal investor’s effective tax rate τe when valuing this form of income.12
Alternatively, since dividends are taxed upon accrual, no adjustment is needed for the dividend tax rate (denoted by τd ). If we characterize the “dividend tax preference parameter”
from Poterba (2004) as:θt =(1 − τd,t ),(1 − τe,t )(1)where τd,t is the top federal tax rate on dividends in year t, and τe,t is 80% of the top federal
tax rate on long-term capital gains in year t (top rates are used because capital income mostly
accrues to high-income individuals), then Table 1 illustrates that dividends were likely to
be tax-disadvantaged in every year between 1972-2017, since the after-tax income from a $1
gross dividend payment was approximately θt < 1 of the after-tax income from a $1 share
repurchase in year t.13
From Table 1 it would appear that US corporations should have avoided dividend payments altogether between 1972-2017. This, of course, was not the case. Despite being tax
disadvantaged, dividends were a significant component of corporate payout in every year.
This is illustrated by Figures 1 and 2, which plot aggregate dividend payments and share
repurchases (in 2017 dollars), and the percentage of firms paying each, respectively, over theperiod 1972-2017 by Compustat-listed firms headquartered in the US.14 As illustrated by Figure 1, dividends were the largest component of corporate payout until 1997, and althoughshare repurchases assumed this position afterwards - except in 2009 - aggregate dividends
remained sizable and grew in most years (average growth rate of 4.8% per year from 1997 to2017). Furthermore, as illustrated by Figure 2, more firms paid dividends than repurchased
12

Share repurchases also generate realized capital gains for selling shareholders; an important feature of
the analysis below.
13
Appendix A describes each tax-rate series and discusses θt . Also, see Jacob and Jacob (2013) for evidence
that dividends were tax-disadvantaged relative to accrued capital gains in the following G-10 countries overthe period 1990-2008: France, Germany, Japan, the Netherlands, Sweden (with the possible exception of
1994), Switzerland, and the United States.
14
In keeping with the previous literature, both utilities and financial institutions - which typically pay highdividends - are excluded from the sample due to their unique regulatory environments (Standard Industrial
Classification codes 4900-4949 and 6000-6999, respectively). See Appendix A for a description of each payout
variable.

9

Table 1. Top Federal Tax Rate on Dividends and Long-Term Capital Gains (US: 1972-2017)

Year
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994

Div.
τd,t %
(1)
70.0
70.0
70.0
70.0
70.0
70.0
70.0
70.0
70.0
70.0
50.0
50.0
50.0
50.0
50.0
38.5
28.0
28.0
28.0
31.0
31.0
39.6
39.6

Cap. Gainsτg,t %
τe,t %
(2)
(3)
35.0
28.0
35.0
28.0
35.0
28.0
35.0
28.0
35.0
28.0
35.0
28.0
35.0
28.0
28.0
22.4
28.0
22.4
28.0
22.4
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0
28.0
22.4
28.0
22.4
28.0
22.4
28.0
22.4
28.0
22.4
28.0
22.4
28.0
22.4
28.0
22.4

θt
(4)
0.42
0.42
0.42
0.42
0.42
0.42
0.42
0.39
0.39
0.39
0.60
0.60
0.60
0.60
0.60
0.79
0.93
0.93
0.93
0.89
0.89
0.78
0.78

Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017

Div.
τd,t %
(5)
39.6
39.6
39.6
39.6
39.6
39.6
39.1
38.6
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
20.0
20.0
20.0
20.0
20.0

Cap. Gainsτg,t %
τe,t %
(6)
(7)
28.0
22.4
28.0
22.4
28.0
22.4
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
15.0
12.0
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0
20.0
16.0

θt
(8)
0.78
0.78
0.78
0.72
0.72
0.72
0.73
0.73
0.97
0.97
0.97
0.97
0.97
0.97
0.97
0.97
0.97
0.97
0.95
0.95
0.95
0.95
0.95

Columns 1 and 5 (2 and 6) report the top federal tax rate on dividends (long-term capital gains) in the United States from1972 to 2017 (in years when the maximum tax rate changed mid-year, the highest rate is used), while columns 3 and 7 report
estimates of the effective tax rate on accrued capital gains using an effective-to-statutory ratio of 0.8. Finally, columns 4 and8 report the dividend tax preference parameter (Equation 1) for each year; it is below unity in all 46 years, which implies that
dividends were tax-disadvantaged relative to share repurchases over the entire period.

shares during most of the period 1972-1994. However, the percentage of dividend-paying
firms has declined substantially over time: from a maximum of 69.1% in 1977 to a minimumof 15.6% in 2001 (Fama and French, 2001 attributes much of this decline to firm-composition
effects). The percentage of firms paying dividends has picked up recently, however, reaching
29% by 2017.
10

Figure 1. Aggregate Dividends and Share Repurchases (US: 1972-2017)
700

Billions ($2017)

600

Dividends
Repurchases

500
400
300
200
100
0
1972

1975

1978

1981

1984

1987

1990

1993

1996

1999

2002

2005

2008

2011

2014

2017

Year
This figure plots the aggregate level of real dividends (solid line) and share repurchases (dashed line) by Compustat-listed firms
headquartered in the United States (excluding utilities and financial institutions) over the period 1972-2017 (in 2017 dollars).

Figure 2. Fraction of Firms Paying Dividends and Repurchasing Equity (US: 1972-2017)80% of Firms

60

40

20
Dividends
Repurchases
0
1972

1975

1978

1981

1984

1987

1990

1993

1996

1999

2002

2005

2008

2011

2014

2017

Year
This figure plots the fraction of Compustat-listed firms headquartered in the United States (excluding utilities and financial
institutions) that paid dividends (solid line) and repurchased equity (dashed line) in each year between 1972-2017.

Although share repurchases have become more prominent in recent years, dividends havenot disappeared and remain a significant component of corporate payout. The model developed below provides an explanation for this.

11

4

Model

This section develops the payout model. It begins by characterizing firms and shareholders,
and then it derives the model’s steady-state equilibria.

4.1

Firms

Firms in the model are all-equity financed; this allows us to abstract from issues related tocapital structure and to focus on payout policy. Shareholders discount income generated by
the firm, net of all applicable taxes, at the constant rate 1/(1 + ρ) per-period, where ρ > 0.
Consistent with investor rationality, shareholders value after-tax income from dividends andcapital gains identically. There are two levels of taxation: firms pay corporate-profits tax at
the rate τc , while shareholders pay income tax at the rate τd on dividends and τg on realized
capital gains. These rates are constant across firms, shareholders, and time. Furthermore,
there is no risk in the model, all information is symmetric, and firms live forever.
Firms begin each period t with a given level of non-depreciating capital Kt carried overfrom the previous period. This is used to generate end-of-period profits according to thefunction π(K), where π 0 (K) > 0 and π 00 (K) < 0. When gross profits are generated, firms
simultaneously pay corporate-profits tax, issue dividends, repurchase equity, and make capital investments, in that order. Firms can also issue equity (negative share repurchases) anddivest capital (negative investment), but they cannot pay negative dividends. Managers act
in the interest of shareholders by maximizing firm value (V ). This is accomplished by selecting a feasible sequence of dividend payments, share repurchases, and capital investments
subject to the firm’s beginning-of-period capital stock.
Given these assumptions, an end-of-period dividend payment Dt is worth (1−τd )Dt /(1+ρ)
at the beginning of period t, whereas an end-of-period capital gain ∆Vt is typically worth
more than (1 − τg )∆Vt /(1 + ρ), since, as discussed in Section 3, the effective tax rate on∆Vt is typically lower than τg .15 If we continue to denote the marginal investor’s effective
tax rate by τe , then these gains are worth (1 − τe )∆Vt /(1 + ρ) at the beginning of period t.
15Note that Vt is the firm’s beginning-of-period market value, whereas Dt and ∆Vt are the end -of-period
dividend payment and capital gain (ex-dividend), respectively.

12

Taken together, the firm’s intrinsic/market value at the beginning of period t is:16
Vt =(1 − τd )Dt (1 − τe )∆Vt
Vt++.
(1 + ρ)(1 + ρ)(1 + ρ)(2)

Capital gains can be generated in two ways within the model: higher continuation values, i.e., Vt+1 > Vt ; and higher ownership concentration following a share repurchase, i.e.,
Vt+1 /(1 − δt ) > Vt+1 , where δt > 0 is the fraction of equity repurchased in period t, whichequals:
Rt
δt =
,
(3)
Rt + Vt+1
where Rt is the intrinsic value of repurchased equity.17 Therefore, capital gains at the endof period t are:

Rt
∆Vt = Vt+1 / 1 −Vt+1 + Rt

− Vt ,

which simplifies to:∆Vt = Vt+1 + Rt − Vt .

(4)

Substituting Equation 4 into Equation 2 produces the following firm-value identity:
Vt =Vt(1 − τd )Dt (1 − τe )(Vt+1 + Rt − Vt )++,(1 + ρ)(1 + ρ)(1 + ρ)and solving for Vt produces:
Vt = 1 +ρ(1 − τe )−1 (1 − τd )Dt + Rt + Vt+1 .
(1 − τe )(5)This is a common firm-value equation but deserves a brief explanation. The end-of-period
value of a gross dividend payment Dt is equal to the after-tax cash receipt (1 − τd )Dt plus
16

The terms “intrinsic value” and “market value” are used interchangeably. Section 5 refines the latter
concept by introducing market transactions that involve outside investors.
17
This equation can be derived as follows. Denote by Nt the number of perfectly-divisible outstanding
shares at the beginning of period t, then Rt = (Nt δt ) (Vt+1 /Nt (1 − δt )). I.e., the firm repurchases Nt δt of
its shares during the repurchase program, and each of these is worth Vt+1 /Nt (1 − δt ), where this valuation
follows from the firm’s continuation value of Vt+1 being split equally among the remaining Nt (1 − δt ) shares.
Canceling the Nt terms, and rearranging this expression, produces Equation 3.

13

the subsequent capital-loss offset that is created once the stock becomes ex-dividend. Whenboth of these are fully capitalized into the stock’s price, the value of a marginal dividend just prior to its payment - is equal to (1 − τd )/(1 − τe ). Conversely, share repurchases - justprior to the repurchase program - have no tax implications for marginal investors, since theprogram’s full value is already capitalized into the stock’s price. This implies that both Rt
and Vt+1 have a unit coefficient at the end of period t. However, the story is different whencapital gains are expected to materialize in the future. Here, marginal investors rightly anticipate the future tax liability on these gains, and the discount factor is adjusted (reduced)
accordingly. This can be seen from the square-bracketed term of Equation 5, which has ρ
inflated by the factor 1/(1 − τe ).18 This modified discount factor is applied to end-of-period
dividends and share repurchases equally, as both constitute a capital gain when viewed fromthe beginning of a period (i.e., both increase equity’s market value).
Moving on, the firm’s intrinsic value from Equation 5 can be transformed into the following
infinite sequence of payout:
Vt =

∞ 
X
s=t

ρ
1+
(1 − τe )−(s−t+1) (1 − τd )Ds + Rs ,(1 − τe )(6)

by applying the transversality condition:lim 1 +T →∞

ρ(1 − τe )−TVT = 0.

Equation 6 is useful for describing the corporate payout puzzle. Recall that firms use after-tax
profits to pay dividends, repurchase shares, and/or make capital investments. It is usuallyassumed that firms can repurchase all equity at its intrinsic value when capital markets are18
For the interested reader, the discount factor [1 + ρ/(1 − τe )]−1 can be explained as follows. An increasein either dividends or share repurchases translates into a higher end-of-period cum-dividend share price, i.e.,a capital gain. The value of a marginal capital gain is (1 − τe ) at the end of a period, which translates
into a beginning-of-period value of (1 − τe )/(1 + ρ). However, when this amount is capitalized into thefirm’s beginning-of-period market value, it not only increases the stock price, it also reduces the marginal
investor’s future tax liability, since the stock now appreciates by (1 − τe )/(1 + ρ) less. This creates a future
tax savings of τe (1 − τe )/(1 + ρ) with a present value of τe (1 − τe )/(1 + ρ)2 . This, in turn, increases thefirm’s stock price once again and further reduces the marginal investor’s future tax liability. Continuing
this process indefinitely produces an infinite sequence of progressively smaller tax savings with present value[τe (1 − τe )]/[(1 + ρ)(1 + ρ − τe )]. When this sum is added to the discounted value of the original capital gain(i.e., (1 − τe )/(1 + ρ)), we arrive at the discount factor in Equation 5.

14

perfect (i.e., complete, competitive, frictionless, and free of informational asymmetries). This
assumption is inconsistent with capital-gains taxation, however - with important implications
below - but for the sake of describing the payout puzzle, assume that it holds for the moment,i.e., that Rs - the intrinsic value of repurchased equity in period s - equals the amount spent
repurchasing shares in that period (denoted by As henceforth). In this case, a firm’s period-s
budget constraint is:(1 − τc )π(Ks ) = Ds + Rs + Is .
(7)
It becomes immediately obvious from Equations 6 and 7 that setting Ds = 0 ∀ s is optimal
whenever τd > τe , since a marginal substitution of dividends with share repurchases in period s ≥ t raises the firm’s period-t value by [1 + ρ/(1 − τe )]−(s−t+1) (τd − τe )/(1 − τe ) > 0.
Given that τd is generally greater than τe (see Table 1 of Section 3) the payout puzzle follows
immediately.19
Since Ds = 0 ∀ s is empirically violated (see Figures 1 and 2 of Section 3) the perfectinformation literature has typically appealed to either: exogenous factors that place lower
bounds on dividends (either explicit or implicit), such as the “intrinsic value of dividends”
argument in Shefrin and Statman (1984); or exogenous factors that place upper bounds onshare repurchases, such as appeals to Section 302 of the Internal Revenue Code. The former
explanation is commonly referred to as the Traditional view or Conventional view, while thelatter is commonly referred to as the New view or Tax Capitalization view.
In contrast to these explanations, the one pursued here does not require exogenous bounds
on dividends or share repurchases. The crucial assumption is that capital gains are taxed
upon realization rather than accrual. In this way, the model provides a self-contained rational
for dividend payments. This is accomplished by deriving the firm’s “repurchase function”(the mapping from At into Rt ), which is based on shareholder-wealth maximization withina realization-based capital-gains tax system. This is done next.
19

Conversely, if τd < τe , firm-value maximization entails unlimited equity issuance and unlimited dividend
payments. Only in the special case when τd = τe could there be an interior solution.

15

4.2

Shareholders and the Mapping from At into Rt

Shareholders are identical in all respects except for their investment horizons and level ofaccrued capital gains on firm equity. The mechanisms that produce each source of heterogeneity are not modeled in this section (they are modeled in Section 5 when the shareholder
distribution is endogenized); in practice, the former depends on factors such as retirement
planning, portfolio management, and the timing of consumption expenditures, while thelatter depends on the timing of past share purchases. Consistent with the notion of an“investment horizon,” shareholders in the model reinvest all dividend income in the same(or identical) stock and fully liquidate their equity positions upon reaching that horizon.20
In this way, shareholders seek to maximize the amount of after-tax wealth they have uponreaching their investment horizon. Pursuing this objective within a realization-based capitalgains tax system produces the lock-in effect described above: i.e., the incentive to postpone
the sale of equity with an accrued capital gain and non-zero investment horizon.
To illustrate this point, consider two scenarios regarding a period-t equity position withunit market value and proportional tax basis β (the position’s acquisition cost dividend by
its current market value): 1) the position is held for H periods (where H is the investment
horizon); and 2) the same position is sold at the beginning of period t, and the after-tax
proceeds are reinvested for H periods (in the same equity). Note that β < 1 indicates acapital gain, whereas β > 1 indicates a capital loss. If we denote the rate-of-return from
dividends (capital gains) by rd (rg ), then under Scenario 1, the position’s after-tax value atthe beginning of period t + H will be:
#
"
H
X


(1 + rg + (1 − τd )rd )h−1 ,(1 − τg ) (1 + rg + (1 − τd )rd )H + τg β + (1 − τd )rd

(8)

h=1

where the first part of this expression is the position’s market value multiplied by (1 − τg ),while the second is the position’s tax basis - the original tax basis (β) plus the reinvested
dividend income - multiplied by τg ; i.e., since we applied (1 − τg ) to the position’s entire
value (part one), we must add back its tax basis scaled by τg , since this part of the position
20

The full-reinvestment assumption is not necessary for the main results below, however, it seems to be
the most natural. Furthermore, reinvestment in debt securities is also permissable - as in Section 5.

16

escapes taxation. See Appendix B for a derivation of Expression 8. With regard to thesecond scenario, the period-t equity sale produces after-tax wealth of:(1 − τg ) + τg β,(9)i.e., the position’s period-t market value (1) multiplied by (1 − τg ) plus the period-t tax basis(β) scaled by τg . When this amount is reinvested in the same equity, the new position’safter-tax value becomes:(1 − τg ) ((1 − τg ) + τg β)(1 + rg + (1 − τd )rd )H"+ τg ((1 − τg ) + τg β) 1 + (1 − τd )rdHX(1 + rg + (1 − τd )rd )h−1

h=1

!#

(10)

at the beginning of period t + H. This expression is similar to 8 above, except that theposition’s market value and tax basis are both (1 − τg ) + τg β at the beginning of period t(instead of 1 and β, respectively) because of the period-t tax payment (refund) when β < 1(β > 1). See Appendix B for a derivation of Expression 10.
The difference in after-tax wealth across these two scenarios (Expression 8 minus Expression 10) is characterized by the following function:Ω(H, β) =







τg (1 − β)[(1 − τg )rg + (1 −Ph−1τd )rd ] H
h=1 (1 + rg + (1 − τd )rd )0

if H > 0,(11)

if H = 0,

which has the following five relevant properties:
1. Ω(H, β) ≥ 0 if β < 1: shareholders have a disincentive to sell equity with an accrued
capital gain,
2. Ω(H, β) ≤ 0 if β > 1: shareholders have an incentive to sell equity with an accrued
capital loss,21
21

The tax refund from selling equity with a capital loss requires offsetting income (e.g., a capital gain onother assets).

173. Ω(H, β) = 0 if β = 1: shareholders are indifferent between selling/retaining equity inthe absence of a capital gain or loss,
4. Ω0β (H, β) ≤ 0: the wealth differential is decreasing in the investor’s tax basis (increasingin the capital gain), where Ω0β (·) is the derivative of Ω(H, β) with respect to β,
5. Ω(H + 1, β) − Ω(H, β) ≥ 0: the wealth differential is increasing in the investor’s investment horizon.
See Appendix B for a derivation of Equation 11. The first property of Ω(H, β) is due to theforegone capital income that would have accrued on the shareholder’s gross wealth used to
pay the capital-gains tax in period t, whereas the second property is due to the additional
income that is generated on the period-t tax refund.
Given the properties of Ω(H, β), shareholders with a capital loss are happy to liquidate
their equity positions immediately, whereas those with a capital gain are not indifferent
between: 1) liquidating their position at the firm’s intrinsic value, and 2) retaining it for adesired number of periods. In order to create such indifference, a shareholder must be offereda “lock-in premium” in addition to equity’s intrinsic value. This premium is shareholderspecific owing to differences in tax basis and investment horizon (properties 4 and 5, respectively) and is characterized by the following function (as a fraction of equity’s intrinsic
value):
hiΩ(H,β)
(1 − β) τg
if β < 1,1−τg Ω(H,β)+τg (1−β)
L(H, β) =(12)0if β ≥ 1,

where Ω(H, β) is from Equation 11. See Appendix C for a derivation of this result. As
with Ω(H, β), the lock-in premium L(H, β) is increasing in H and decreasing in β. Whenshareholders are offered this premium (in addition to equity’s intrinsic value) they become
indifferent between liquidating their equity position and holding it for a desired numberof periods. This indifference makes L(H, β) a pure cost from the perspective of firms andshareholders during a repurchase program. The only beneficiary is the tax authority via
accelerated realizations.
It will be assumed throughout that each investor is paid their specific lock-in premium
18when selling equity back to a firm, which is the minimum compensation that a wealthmaximizing shareholder is willing to accept.22 This minimizes the cost of repurchasing
equity while maintaining shareholder rationality, which in turn, reduces the likelihood that
dividends are paid in equilibrium. As such, the current pricing assumption is taken to be a“worst-case scenario” for dividend payments.
Recall that shareholders are heterogeneous with respect to H and β; this gives rise toa diverse set of shareholder types. Denote the density of this set by f (H, β), which isassumed to be distributed discretely over H (in one-period intervals) and continuously overβ. Furthermore, assume that f (H, β) has full support over the domain {H, β | 0 ≤ H ≤H, 0 ≤ β ≤ β} for some H > 0 and β > 0, and that f (H, β) is constant across time.
The last two assumptions are relaxed in Section 5 when the shareholder distribution is
endogenized. But for the time-being, it is convenient to assume that f (H, β) has reached along-run steady-state23 and has full support.24

4.3

The Repurchase Function

Now that we have the lock-in premium function L(H, β), the shareholder density f (H, β),and a pricing assumption, we can derive the mapping from At into Rt . To make this derivation straight-forward, first decompose the shareholder density f (H, β) into H + 1 sections
corresponding to each possible investment horizon from 0 to H, and denote these by f H (β).
Second, designate a β value for each investment horizon, and denote these by β(H). Finally, note that firm-value maximization entails repurchasing equity at the lowest cost, i.e.,
22This assumption is consistent with Ikenberry et al. (1995), which documents the gradual increase inshare prices during a typical repurchase program: specifically, Ikenberry et al. (1995) shows that a stock’s
cumulative abnormal return (CAR) gradually increases over the three-year period following an open-market
repurchase announcement, a time during which most repurchase activity takes place according to Stephensand Weisbach (1998). Chan et al. (2010) also documents this pattern for CARs during most of the two-year
period following a repurchase announcement.
23
As shown in Section 5, share repurchases necessarily alter the shareholder distribution by removing
investors with relatively low lock-in premia (i.e., those with short investment horizons and/or few capitalgains). However, a constant shareholder distribution is still possible given countervailing dynamics: a shareholder’s investment horizon becomes progressively shorter as time passes, and shareholder turnover between
successive repurchase programs creates a new set of shareholders with relatively low capital gains.
24
This assumption merely ensures a continuously-differentiable mapping from At into Rt .

19

from shareholders with the lowest lock-in premia.25 Therefore, a firm’s period-t repurchasefunction is derived by selecting β(H), H = 0, 1, .., H to maximize:
Rt = max {β(H)}

subject to:

At = 

H Z β
X

H=0

H Z β
X

H=0

β(H)

β(H)



f H (β)dβ  (Rt + Vt+1 ),

(13)



[1 + L(H, β)]f H (β)dβ  (Rt + Vt+1 ),(14)

where Vt+1 is the firm’s continuation value. These two equations state that firms maximize
the total mass of repurchased equity subject to spending At on the repurchase program. In
the absence of lock-in premia (i.e., L(H, β) = 0 ∀ H and ∀ β) - which is typically assumed the repurchase function collapses to R(At ) = At . However, when lock-in premia are characterized by Equation 12, firms pay investor-specific premiums to repurchase equity, and thus,
R(At ) ≤ At . This property of the repurchase function is stated in Proposition 1 along withthree others.
Proposition 1 The repurchase function has the following 4 properties for any shareholder
density with full support:
1. R(At ) ≤ At ,2. R0 (At ) > 0,

3. R00 (At ) ≤ 0,

and

4. R(At ) ∈ C 1 .

See Appendix D for a proof of these results.
Proposition 1 states that R(At ) is increasing in the amount spent and below (possibly weakly)
the 45◦ line. As more equity is repurchased, the marginal shareholder’s lock-in premium increases; it follows that R(At ) is globally concave in the amount spent. Finally, the repurchasefunction is continuously differentiable.
With the repurchase function now derived, we return to the firm’s maximization problem.
25

Dittmar and Field (2015) provides evidence that firms typically repurchase equity in a cost-effective
manner.

20

4.4

The Firm’s Problem

Recall that firms seek to maximize their value by selecting a sequence of dividend payments,
share repurchases, and capital investments from the feasible set. Formally, firms seek to:max{Ds ,As ,Is }∞
s=t

Vt =

∞ 
X
s=t

ρ
1+
(1 − τe )−(s−t+1) (1 − τd )Ds + Rs ,(1 − τe )subject to:• the repurchase function: Rs = R(As ),• the per-period budget constraint: (1 − τc )π(Ks ) = Ds + As + Is ,• the law of motion for capital: Ks+1 = Ks + Is ,• and the non-negative dividend constraint: Ds ≥ 0.
The Karush-Kuhn-Tucker necessary conditions for optimality are as follows (for any periods ≥ t):
−(s−t+1)ρ(1 − τd )
D
1+
− λB
(15)
s − λs = 0,
(1 − τe )(1 − τe )1+ρ
(1 − τe )−(s−t+1)

R0 (As ) − λBs = 0,

(16)

0
K
K
λB
s (1 − τc )π (Ks ) − λs + λs−1 = 0,

(17)

K
− λB
s − λs = 0,(18)

λis ≥ 0 for i = B, K, D,

(19)

λD
s Ds = 0,

(20)

K
D
where λBs , λs , and λs are the period-s Lagrange multipliers on the budget constraint, lawof motion for capital, and non-negative dividend constraint, respectively.

21

4.5

Steady-State Payout

To derive the steady-state level of payout, and to show that dividends are consistent with
firm-value maximization, use Equations 15 and 16 to get:
R0 (A∗s ) =(1 − τd )+ λDs ,(1 − τe )where A∗s = A∗ ≥ 0 is the equilibrium amount spent repurchasing shares in period s. Thisequation states that equilibrium share repurchases have a marginal value equal to that ofdividends plus the shadow value of relaxing the non-negative dividend constraint λD
s . FromD
Equations 19 and 20, λs can take one of two qualitatively-distinct values in period s:∗
1. λD
s > 0 (when Ds = 0),∗
2. λD
s = 0 (when Ds > 0),

where Ds∗ is the equilibrium level of dividends in period s.
The first of these (λDs > 0) prevails whenever the marginal value of a share repurchaseexceeds that of a dividend payment for the firm’s entire payout. This is usually the case in
perfect-information models since R0 (As ) = 1 > (1 − τd )/(1 − τe ) ∀ As . However, as argued in
Section 4.2, the repurchase function’s first derivative is not equal to unity for all As . Rather,
it is strictly less than unity whenever a marginal shareholder has strictly-positive capital gains
and a non-zero investment horizon. Furthermore, since marginal lock-in premia increase as
more equity is repurchased, the marginal value of a share repurchase declines monotonicallyas more equity is sought (property 3 of Proposition 1). Therefore, it becomes entirely possiblethat R0 (As ) falls below (1 − τd )/(1 − τe ) as As increases. However, management prevents thisfrom happening in equilibrium by switching from share repurchases to dividend payments
when R0 (As ) = (1 − τd )/(1 − τe ). This is captured by the second case above (λDs = 0). In
0
∗
∗
∗
∗
this situation, R (As ) = (1 − τd )/(1 − τe ) ∀ s, and Ds = (1 − τc )π(K ) − As > 0 ∀ s, whereK ∗ is the steady-state level of capital; thus providing the perfect-information explanation
for dividends.

22

5

Endogenous Shareholder Distribution

This section endogenizes the shareholder distribution using numerical methods. This serves
three purposes. First, it illustrates that dividends are more of a certainty in the current
framework than a mere possibility for long-lived firms. Second, it establishes the relationship between equity demand curves, effective tax rates, and the firm’s market value (byintroducing market transactions that involve outside investors). And third, it illustrates
that dividends are optimal for various types of marginal investor.
To begin with, assume that two investor types arrive at the beginning of each period:“long-term” investors with initial investment horizons of HL ; and “short-term” investors
with initial investment horizons of HS ≤ HL .26 Long-term (short-term) investors are endowed with ωL (ωS ) of wealth at the beginning of each period, and both have access to thesame two investment opportunities: a risk-free security with after-tax interest rate ρ andthe firm’s equity. To perpetuate the shareholder distribution over time, every shareholder
liquidates a fraction λ of their equity position at the beginning of a period (rebalancing theirportfolios), which is sold to outside investors. The proceeds from these equity sales (and any
dividend income received) are reinvested in the risk-free security until the shareholder’s investment horizon is reached. Furthermore, in keeping with the model’s dynamic nature, theinvestment horizon of every shareholder is reduced by 1 at the end of each period. Management’s task is to select a sequence of dividend payments and share repurchases to maximize
firm value subject to a constant level of net income (normalized to 1 in every period).27
As before, capital gains and losses are generated through changes in a firm’s market value
and through share repurchases. Unlike before, the repurchase function is now time-specific(due to the shareholder distribution’s endogeneity). If we denote the period-s repurchasefunction by Rs (·), and recall that δs is the fraction of equity repurchased in period s (i.e.,δs = Rs (As )/(Rs (As ) + Vs+1 )), then Equation 21 characterizes the period-s rate-of-return26

Two investor types are used here to simplify the analysis and to improve transparency, more can be
added without changing the qualitative results below.
27
Net income is exogenous in the current exercise to simplify the analysis - there is no need to specify thefirm’s profit function or initial capital stock. One can assume that firms have reached their efficient scale ofproduction at a net income of 1.

23

from capital gains:rg,s =

Vs+1− 1.
(1 − δs )Vs(21)

The foundation for Equation 21 is developed on page 13 and is elaborated upon below (it isalso explained in Appendix E). Since the period-s rate-of-return form dividends is:rd,s =

Ds
,
Vs

(22)

a period-t equity investment made by an investor with initial holding period H - an H-type
investor for brevity - has the following after-tax value at the beginning of period t + H:"

(1 − λ)H−1 τg + (1 − τg )

+λ

H−1
X

Vt+H

Vt

QH−1

w=0 (1 − δt+w )#"(1 + ρ)H−h (1 − λ)h−1 τg + (1 − τg )h=1

+ (1 − τd )

H−1
X

Vt+h

Vt

Qh−1

w=0 (1 − δt+w )(1 + ρ)H−h−1 (1 − λ)h

h=0

#

Dt+h
, (23)Qh−1
Vt w=0 (1 − δt+w )where this expression accounts for: 1) the after-tax value of equity held for H periods (thefirst part), 2) the after-tax value of equity sold at the beginning of each period (the fractionλ) and reinvested in the risk-free security until period t + H (the second part), and 3) theafter-tax value of reinvested dividend income (the third part). See Appendix E for a derivation of Expression 23.
Going forward, Subsection 5.1 discusses the relationship between effective tax rates (nowendogenous) and the firm’s market value, Subsection 5.2 explains the dynamics of the shareholder distribution, while Subsection 5.3 presents the main result: dividends are optimal forvarious marginal-investor types.

24

5.1

Effective Tax Rates and Firm Value

This section explains the relationship between a firm’s market value and the marginal investor’s effective tax rate on accrued capital gains. It also explains the evolution of a firm’s
market value within a period.
To begin with, consider a steady-state in which Dt = D and Rt (At ) = R for all t (therepurchase function is endogenized below).28 Since there are two investment opportunities the risk-free asset and the firm’s equity (which is also risk-free) - the value of equity to an
H-type investor - denoted by V H accordingly - solves the following equation:"(1 + ρ)H = (1 − λ)H−1 τg + (1 − τg )

+λ

H−1
X

H−h

(1 + ρ)



R+VH
R
h−1

(1 − λ)h=1+ (1 − τd )

"

H−1
X

H #

τg + (1 − τg )H−h−1

(1 + ρ)

h=0



R+VH
R

(1 − λ)h #h Dt+h

VH



R+VH
R

h

, (24)

where the left-hand-side of Equation 24 is the risk-free asset’s total return after H periods,
while the right-hand-side is the after-tax return on an equity investment - conditional on V H- over the same period (Expression 23 with the following substitutions: Dt = D, Rt (At ) = R,
and Vt = V H ∀ t). This equation states that an H-type investor is willing to pay V H for
the firm’s equity, and no more, since their outside option is generating an after-tax returnof (1 + ρ)H via the risk-free asset. To provide a concrete example, suppose that D = 0.25,
R = 0.75, τd = τg = 0.2, λ = 0.1, and ρ = 0.1. Under these parameters, Panel 1 of Figure 3
plots the mapping from H into V H and illustrates their positive relationship, which arises
because longer investment horizons imply longer tax deferment.
This result is associated with an investor’s “effective tax rate” on accrued capital gains:
the nominal tax rate - applied to capital gains as they accrue - that equates an investor’s
28
A steady-state is used here reduce the number of variables under consideration; the results hold for any∞∞
sequences {Ds }∞t , {Rs }t , and {Vs }t+t that generate capital gains.

25

Figure 3. Firm Valuation and Effective Tax Rates by Investment Horizon
Effective Tax Rate (τe )Firm Valuation (V H )Panel 1: Firm Valuation by H
8.6
8.5
8.4
8.3
8.2
8.1
8
1

5

10

15

20

Panel 2: Effective Tax Rate by H
.2
.18
.16
.14
.12
1

5

Investment Horizon (H )

10

15

20

Investment Horizon (H )Firm Valuation (V H )Panel 3: Firm Valuation by Effective Tax Rate
8.6
8.5
8.4
8.3
8.2
8.1
8
.12

.14

.16

.18

.2

Effective Tax Rate (τe )Panel 1 plots the value of equity to an H-type investor when D = 0.25, R = 0.75, τd = τg = 0.2, λ = 0.1,
and ρ = 0.1. Panel 2 plots an H-type investor’s effective tax rate using the same parameters. Panel 3 plots
the relationship between an H-type investors’s effective tax rate and their valuation of firm equity.

after-tax return with those under a realization-based tax.29 To derive an H-type investor’s
effective tax rate - denoted by τe,H accordingly - you equate Expression 23 (their after-tax
return under a realization-based tax) with Expression 25 below (their after-tax return underan accrual-based tax - which is derived in Appendix F) and solve for τe,H .
29

There is some disagreement in the public-finance literature over what constitutes the effective tax rate.
Is it the rate - applied to capital gains as they accrue - that equalizes the present value of tax revenues asin Boadway et al. (1984)? Or is it the rate that equalizes an investor’s after-tax returns as in Glenday andDavis (1990)? Since the current analysis deals with investor behavior (not public finances), the latter rate
is appropriate.

26

(1 − λ)

H−1

H−1
Y
w=0

Vt+w+1τe,H + (1 − τe,H )(1 − δt+w )Vt+w

H−1
X

+λ

H−h

(1 + ρ)(1 − λ)

h−1

w=0

h=1

+ (1 − τd )

H−1
X

h−1
Y

H−h−1

(1 + ρ)(1 − λ)h=0h Dt+h

Vt+w+1τe,H + (1 − τe,H )(1 − δt+w )Vt+w

h−1Y

Vt+h w=0



Vt+w+1τe,H + (1 − τe,H )(1 − δt+w )Vt+w. (25)

An investor’s effective tax rate is monotonically decreasing in their investment horizon due
to the associated tax deferment. This is illustrated by Panel 2 of Figure 3, which plots the
mapping from H into τe,H . In the special case of H = 1, τe,1 = τg since no taxes are deferred,but τe,H becomes strictly less than τg for all H > 1. The positive relationship between H andV H (Panel 1 of Figure 3), and the negative relationship between H and τe,H (Panel 2 of Figure 3), implies the negative relationship between τe,H and V H (plotted in Panel 3 of Figure 3).
Next, to establish the firm’s market value at the beginning of a period, we require theequity demand curve and the quantity of equity sold. To determine the former, we require theinvestment horizon of potential investors and their aggregate wealth. Suppose that potential
investors have initial investment horizons of HL = 20 and HS = 2, and aggregate wealthof ωL = 0.5 and ωS = ∞. Based on these values (and V H from Equation 24), the inverse
demand curve for equity - the mapping from the fraction of equity purchased (Q) into theunit price of equity (P ) - is characterized by the following function:

P =



V 20



ωL

Q



V

2

h
i
if Q ∈ 0, VωL
20
h
i
ωL ωL
if Q ∈ V , V
h 20 i2
if Q ∈ VωL , 1

 0.5 

8.585 if Q ∈ 0, 8.585

 0.5 0.5 
P = 0.5
if Q ∈ 8.585, 8.065Q8.065 if Q ∈ 0.5 , 1 ,

=⇒

8.065

2

and plotted in Figure 4. To establish the firm’s market value, we also need the quantity ofequity traded; three examples are depicted in Figure 4. The first (right-most dashed line
at Q = 1) is indicative of an “initial public offering” in which all equity is sold to outside
investors. The firm’s market value is 8.065 in this case, reflecting the marginal investor’s
effective tax rate (τe,2 = 0.191) and their corresponding valuation. The second example (the27dashed line at Q = 0.1) is indicative of a medium-sized equity sale, and here too, the market
value of equity is 8.065. The third example (the dashed line at Q = 0.03) is indicative of a
relatively-small equity sale in which long-term investors become marginal; as a result, thefirm’s market value increases to 8.585, reflecting the marginal investor’s lower effective tax
rate (τe,20 = 0.122) and higher valuation.
Figure 4. The Equity Demand Curve and the Firm’s Market Value
Price of Equity (P)

8.6
8.5
8.4
8.3
8.2
8.1
8
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fraction of Equity Purchased (Q)This figure plots the equity-demand curve when potential investors have initial investment horizons of HL= 20 and HS = 2, aggregate wealth of ωL = 0.5 and ωS = ∞, and when Dt = 0.25 ∀ t, Rt (At ) = 0.75 ∀t, τd = τg = 0.2, λ = 0.1, and ρ = 0.1. The firm’s equilibrium market price is determined by the marginal
investor’s valuation.

The preceding discussion warrants a quick note on terminology. The intrinsic/market
value of equity is what outside investors are willing to pay for it (marginal investors) andnot what firms are willing to pay during a repurchase program. If it were the latter, then firms
could make themselves arbitrarily “valuable” by paying exorbitant prices during a repurchaseprogram for small amounts of equity. This would damage the firm’s financial position andreduce its true intrinsic/market value. However, since value-maximizing firms repurchaseequity economically (from shareholders with the lowest lock-in premia), the distinction between maximizing Vt and the price paid during a repurchase program is immaterial, as both
involve maximizing Vt .
Finally, to help with the discussion below, Figure 5 illustrates the evolution of a firm’s
market value within a period (assume a unit measure of equity).30 The firm’s market value at
30

Strictly speaking, equity is only traded at the beginning of a period (to outside investors) and during

28

the beginning of period t is Vt - consistent with the marginal investor’s effective tax rate (τe,H )∞∞and the equilibrium sequences {Ds }∞t , {Rs }t , and {Vs }t+1 . The firm’s cum-dividend market
value at the end of period t is the sum of: the market value of dividends (1 − τd )Dt /(1 − τe,H )- recall that dividends create an immediate tax liability of τd Dt and reduce the capital-gains
tax liability; the market value of repurchased equity Rt ; and the firm’s continuation value
Vt+1 - this may be smaller than, larger than, or equal to Vt depending on who the period∞∞t + 1 marginal investor is and the equilibrium sequences {Ds }∞t+1 , {Rs }t+1 , and {Vs }t+2 (Vt+1
is drawn lower than Vt for convenience). As time progresses from the beginning of periodt until the end (cum-dividend), the firm’s market value steadily increases as the discountperiod shortens (i.e., the discount factor [1 − ρ/(1 − τe,H )]−1 increases with time). As equity
becomes ex-dividend, the firm’s market value falls by (1 − τd )Dt /(1 − τe,H ) - reflecting the
reduction in cash balances. Since a fraction δt of equity is repurchased in period t, the measure of outstanding shares drops to (1 − δt ) after the repurchase program; each of these has amarket value of Vt+1 /(1−δt ) - reflecting the firm’s continuation value (Vt+1 ) and the measureof outstanding shares post-repurchase. Since the market value of all equity is identical (eachshare is fully fungible: having equivalent cash-flow and control rights), repurchased equity
has a market value of δt Vt+1 /(1 − δt ) (i.e., the measure of equity repurchased multiplied by
the unit value of equity), which equals Rt by definition of the repurchase function, and thus,
the firm’s market value is Rt + Vt+1 just before the repurchase program.31 Upon completionof the repurchase program, the firm’s market value falls once more to Vt+1 - reflecting thefurther reduction in cash balances. This dynamic occurs in every period.

5.2

Shareholder Distribution and Lock-In Premia

This section endogenizes the shareholder distribution to explain its evolution over time; italso illustrates why firms are more likely to pay dividends as they age. The exogenous
variables are now: initial investment horizon (HL and HS ), investor wealth (ωL and ωS ),statutory tax rates (τd and τg ), the risk-free interest rate (ρ), the fraction of equity solda repurchase program (to the firm). A firm’s within-period market value is taken to be the amount that aperiod-t marginal investor is willing to pay at each point in time.
31
Another way to see this market-value identity is as follows. The market value of repurchased equity isδt Vt+1 /(1 − δt ) = Rt from above, while the market value of non-repurchased equity is (1 − δt )Vt+1 /(1 − δt ) =Vt+1 (i.e., the measure of non-repurchased equity multiplied by the per-unit value of equity). Their sum is
Rt + Vt+1 .

29

Figure 5. The Evolution of a Firm’s Market Value Within a Period(1−τd )
D + Rt + Vt+1
(1−τe,H ) t(1−τd )
D(1−τe,H ) t

Rt + Vt+1[

1 + (1−τρe,H )]−1 ((1−τd )
D + Rt + Vt+1
(1−τe,H ) t

)





Vt+1
δt (1−δ= Rt 
t)

= Vt



Vt+1

t+1

t

This figure illustrates the evolution of a firm’s market value within a period. The firm’s market value is Vt atthe beginning of period t. It steadily increases until equity is just about to become ex-dividend, at this point,
it equals [(1 − τd )/(1 − τe,H )]Dt + Rt + Vt+1 (where τe,H is the marginal investor’s effective tax rate). Whenequity becomes ex-dividend, the firm’s market value drops to Rt + Vt+1 , and after the repurchase program iscomplete, it drops once more to Vt+1 . Note that all end-of-period activities take place simultaneously withthe beginning of period t + 1.

at the beginning of each period (λ), and the payout sequences controlled by management
32∞({Ds }∞
t=0 and {As }t=0 ). The model is solved under the parameters reported in Table 2.
Table 3 reports the firm’s equilibrium market value (Vt ) and the fraction of equity repurchased (δt ) in periods 1-4 and in the steady-state (these values are helpful for the discussion
below). Appendix G provides a detailed discussion of the model’s solution algorithm.

Table 2. Parameter Values
HL
20

HS
2

ωL
0.5

ωS
∞

τd
0.2

τg
0.2

ρ
0.1

λ
0.1

Dt
0.25 ∀ t

At
0.75 ∀ t

This table reports the parameter values used in the numerical exercise below (Section 5.2).

At the beginning of period 1, all equity is sold to outside investors (a unit measureof equity). Long-term (short-term) investors have an effective tax rate of τe,20 = 0.118(τe,2 = 0.191) and value equity at 8.44 (8.045). Since short-term investors are marginal, thefirm’s market value is V1 = 8.045 at the beginning of period 1. Long-term investors purchase
6.2% of the firm’s equity (ωL /V1 = 0.0622), and short-term investors purchase the remaining
32

The wealth of each investor type is chosen to ensure that both participate in every beginning-of-period
equity sale. This is maintained throughout.

30

Table 3. Equilibrium Values of: Vt and δt

Vt
δt

1
8.04480.08522
8.04310.0853

Period3
8.04040.0853

48.03780.0854

Steady-State
7.94260.0847This table reports the firm’s beginning-of-period market value (Vt ) and the fraction of equity repurchased(δt ) at the end of periods 1-4 and in the steady-state.

93.8%. Since all equity is purchased at the firm’s market price, no capital gains are generated
at the beginning of period 1 (i.e., β = 1 for all shareholders). Panel 1 of Figure 6 depicts
the beginning-of-period shareholder density.
The shareholder distribution evolves in two ways during a period: investment horizons are
reduced by 1, and every shareholder’s β is reduced (increased) when capital gains (losses)
are generated. Since a fraction δ1 = 0.0852 of equity is repurchased in period 1 (Table 3),
the firm’s market value increases to V2 /(1 − δ1 ) = 8.792 just before the repurchase program,which reduces every shareholder’s proportional tax basis to β = 0.915. This fall in β, andthe reduction in every shareholder’s investment horizon, is reported in Panel 2 of Figure 6,
which depicts the end-of-period shareholder density (just before equity is repurchased). Every shareholder requires a lock-in premium at this point given their non-zero investment
horizon and accrued capital gains: short-term shareholders require a 0.1% premium, while
long-term shareholders require a 6.3% premium; these values are reported in Panel 3 of Figure 6.
Since A1 = 0.75 from Table 2, all equity can be repurchased from short-term shareholders at the unit price of (1.001)V2 /(1 − δ1 ) = 8.8 (these shareholders require a 0.1% lock-in
premium from above), which enables the firm to repurchase a fraction δ1 = 0.0852 of its
equity. This reduces the measure of outstanding shares to (1 − δ1 ) = 0.915 and produces
the aforementioned capital gain. Short-term shareholders are left with 93.2% of the firm’s
equity after the repurchase program - their initial measure of equity (0.938) minus the measure repurchased (0.085) divided by the measure outstanding after the repurchase (0.915)- which has a total market value of (0.932)V2 = 7.496, whereas long-term investors are left
with 6.8% of the firm’s equity with a total market value of (0.068)V2 = 0.547. Finally, it
31Tax Basis (β)

Tax Basis (β)(.68, .76](.76, .84](.84, .92](.92, 1](.68, .76](.76, .84](.84, .92](.92, 1](.68, .76](.76, .84](.84, .92](.92, 1]1 2 3 ...16 17 18 19 20
Investment Horizon (H)

6.2

1 2 3 ...16 17 18 19 20
Investment Horizon (H)

6.1

6.2

0

6.0

6.1

6.2

1 2 3 ...16 17 18 19 20
Investment Horizon (H)

3.7

77.9

Panel 7 (Ownership % in Cell)

0

83.9

3.8

Panel 4 (Ownership % in Cell)

0

93.8

Panel 1 (Ownership % in Cell)

Beginning-of-Period Density(.68, .76]

6.1

6.2

1 2 3 ...16 17 18 19 20
Investment Horizon (H)

3.8

0

6.0

6.1

6.2

1 2 3 ...16 17 18 19 20
Investment Horizon (H)

0.1
77.9

Panel 8 (Ownership % in Cell)

0(.76, .84] 3.7(.84, .92](.92, 1](.68, .76]

6.2

1 2 3 ...16 17 18 19 20
Investment Horizon (H)93.8

Panel 5 (Ownership % in Cell)

0(.76, .84] 83.9(.84, .92](.92, 1](.68, .76](.76, .84](.84, .92](.92, 1]Panel 2 (Ownership % in Cell)

End-of-Period Density

0

2

4

8
6

0

2

4

6

0

2

4

6

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Fraction of Equity

Panel 90 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Fraction of Equity

Panel 30 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Fraction of Equity

Panel 3

Marginal Lock-In Premium (%)

Figure 6. Shareholder Densities and Marginal Lock-In Premia

Panels 1, 4, and 7 depict the beginning-of-period shareholder density (just after equity is purchased by outside investors) in periods 1,
2, and 3, respectively. The numbers contained in each cell indicate the percentage of equity owned by each shareholder type. Panels
2, 5, and 8 depict the end-of-period shareholder density (just before equity is repurchased) in periods 1, 2, and 3, respectively. Panels
3, 6, and 9 plot the marginal shareholder’s lock-in premium (as a percentage of equity’s contemporaneous market value) according
to the fraction of equity repurchased, in periods 1, 2, and 3, respectively.

Tax Basis (β)

Tax Basis (β)
Tax Basis (β)
Tax Basis (β)

Lock-In Prem.
Lock-In Prem.
Lock-In Prem.

32

is convenient to assume that firms split their stock at the end of each period to maintain aunit measure of equity (this has no bearing on real values).
All shareholders liquidate a fraction λ = 0.1 of their equity at the beginning of period
2, which reduces the investment horizon on these shares to H = 0; short-term (long-term)shareholders retain 83.9% (6.1%) of the firm’s equity (these values are reported in Panel 4 of
Figure 6). Similar to period 1, long-term (short-term) investors that arrive in period 2 haveeffective tax rates of τe,20 = 0.118 (τe,2 = 0.191) and value equity at 8.438 (8.043). Since
short-term investors are marginal, the firm’s market value is V2 = 8.043 at the beginningof period 2. Long-term investors purchase 6.2% of the firm’s equity, whereas short-term
investors purchase the remaining 3.8% (these values are reported in Panel 4 of Figure 6).
Note that only 10% of the firm’s equity changes hands at the beginning of period 2, since
its value to a marginal investor is V2 = 8.043, whereas its “locked-in” value to an existing
shareholder is either: (0.001)V2 = 8.051, for shareholders with H = 1 and β = 0.915; or(0.063)V2 = 8.55, for shareholders with H = 19 and β = 0.915.33
The dynamics during period 2 are similar to those of period 1: investment horizons are
reduced by 1, and every shareholder’s β is reduced by 8.5% due to the capital gains thatare generated because a fraction δ2 = 0.0853 of equity is repurchased. Both adjustments arereflected in Panel 5 of Figure 6. Four shareholder types emerge at the end of period 2: thosewith H = 0 and β = 0.837 (due to the compound capital gains from periods 1 and 2), theseshareholders require no lock-in premium because H = 0; those with H = 1 and β = 0.915,
these shareholders require a 0.1% lock-in premium; those with H = 19, β = 0.915, and arequired lock-in premium of 6.3%; and those with H = 18, β = 0.837, and a required lock-in
premium of 6.9% (these premia are reported in Panel 6 of Figure 6). Since 83.9% of thefirm is owned by shareholders with H = 0 and β = 0.837, all equity can be repurchased at
the market price of V3 /(1 − δ2 ) = 8.79; this enables a fraction δ2 = 0.0853 of equity to be
repurchased when A2 = 0.75.
Equity sales at the beginning of period 3 are composed of two types: a fraction λ = 0.1 of33

In principle, long-term investors are willing to pay (0.001)V2 = 8.051 at the beginning of period 2, butnever will, since their demand is fully satiated at the lower market price of V2 = 8.043.

33

every shareholder’s portfolio is liquidated, and all equity owned by shareholders with H = 0(the remaining short-term shareholders from period 1) is also liquidated. The first constitutes 1.8% of the firm’s total equity, whereas the latter constitutes 82.4%. Similar to periods
1 and 2, long-term investors (short-term investors) that arrive in period 3 have effective tax
rates of τe,20 = 0.118 (τe,2 = 0.191) and value equity at 8.435 (8.04), which results in a market
value of V3 = 8.04. Long-term investors purchase 6.2% of the firm’s equity, and short-term
investors purchase the remaining 77.9%. These values are reported in Panel 7 of Figure 6.
By the end of period 3 there are five shareholder types according to H and β (reported
in Panel 8 of Figure 6). Their required lock-in premia are: 0%, 0.1%, 6.3%, 6.9%, and 7.4%
(these values are reported in Panel 9 of Figure 6). The firm can repurchase 3.7% of its equity
at the market price of V4 /(1 − δ3 ) = 8.787 from shareholders with H = 0 and β = 0.837,
and can repurchase the remaining 4.8% (note that δ3 = 0.0853 from Table 3) at a 0.1% premium from shareholders with H = 1 and β = 0.915. Taken together, the firm repurchases afraction δ3 = 0.0853 of its equity at the end of period 3.
The model evolves according to the themes described above from period 4 onwards andreaches a stead-state by period 30. Panel 1 (Panel 2) of Figure 7 depicts the beginningof-period (end-of-period) steady-state shareholder density, while Panel 3 plots the required
lock-in premium of each shareholder type. Firms repurchase a fraction δt = 0.0847 of theirequity in each steady-state period and pay marginal lock-in premia of 4.2% (marginal shareholders are type H = 3 and β = 0.222).
These results illustrate why long-lived firms are more likely to pay dividends (Grullon andMichaely, 2002). Share repurchases are desirable when a large number of shareholders haverelatively low lock-in premia (i.e., limited capital gains and/or short investment horizons)
as in the first three periods of the model described above. However, since share repurchases
target these inexpensive investors - removing them form the shareholder distribution - theposterior shareholder distribution becomes less favorable for subsequent buybacks, save forthe gradual reduction in every shareholder’s investment horizon and the addition of shortterm investors. Furthermore, the act of repurchasing equity generates capital gains, which
increases every shareholder’s lock-in premium by reducing their β; this is seen by comparing

34

Figure 7. Shareholder Densities and Marginal Lock-In Premia: Steady State
Panel 1: Beginning-of-Period Density (Ownership % in Cell)

Tax Basis (β)(.85, 1]

6.2 6.3

3.7(.7, .85]

5.9 6.0 6.1(.55, .7]

5.7 5.8
5.3 5.4 5.5 5.6(.4, .55]
4.9 5.0 5.1 5.1 5.2(.25, .4]
2.5 4.8(.1, .25]
0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 16 17 18 19 20
Investment Horizon (H)Panel 2: End-of-Period Density (Ownership % in Cell)

Tax Basis (β)(.85, 1]

3.7

6.3(.7, .85]

6.0 6.1 6.2
5.8 5.9(.55, .7](.4, .55]

5.4 5.5 5.6 5.7(.25, .4]

5.0 5.1 5.1 5.2 5.3
2.5 4.8 4.9(.1, .25]
0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 16 17 18 19 20
Investment Horizon (H)Marginal Lock-In Premium

Panel 3: Marginal Lock-In Premium (%)
8
6
4
2
0
0

.1

.2

.3

.4
.5
.6
Fraction of Equity

.7

.8

.9

1

Panel 1 depicts the beginning-of-period shareholder density (just after equity is purchased by outside investors) in the steady-state; the numbers contained within each cell indicate the percentage of equity owned
by each investor type. Panel 2 depicts the end-of-period shareholder density (just before equity is repurchased) in the steady-state. Panel 3 plots the marginal shareholder’s lock-in premium (as a percentage ofequity’s contemporaneous market value) according to the fraction of equity repurchased, in each steady-state
period.

35

the steady-state shareholder density (in which β < 0.55 for over 60% of the firm’s equity)with the shareholder density in the first three periods of the model (in which β > 0.68 for all
equity). Taken together, repurchasing equity is desirable in the early stages of payout (Bravet al. 2005), whereas dividends become more desirable as time passes, ceteris paribus.

5.3

Optimal Dividends

This section presents the main result: dividends are optimal for various marginal-investor
types. The parameters used here (reported in Table 4) are similar to the ones used above(Table 2), except that HS is now variable, as are the payout sequences controlled by management.

Table 4. Parameter Values
HL
20

ωL
0.5

ωS
∞

τd
0.2

τg
0.2

ρ
0.1

λ
0.1

This table reports the parameter values used in the numerical exercises below (Section 5.3).

The simplest case to consider is when HS = 1 since the marginal investor’s effective tax
rate equals the statutory rate in every period (i.e., τe,1 = τg = 0.2), and dividends have aunit marginal value regardless of the payout mix (i.e., (1 − τd )/(1 − τe,1 ) = 1). Since themarginal value of a share repurchase is weakly less than 1 - equal to (less than) 1 in theabsence (presence) of lock-in premia - excluding dividends is sub-optimal. This is illustrated
by Figure 8, which plots the firm’s market value under the following two policies: an all∞∞dividend policy ({Dt }∞t=1 = 1, {At }t=1 = 0), and an all-repurchase policy ({Dt }t=1 = 0,
{At }∞
t=1 = 1). The firm’s market value under the first policy is higher in every period
since the second involves strictly-positive lock-in premia from period 13 onwards (marginal
shareholders require a 9.5% premium for ∀ t ≥ 13, and they are type H = 7 and β = 0.216).
Furthermore, since investors are forward looking, the payout inefficiency in later periods is
capitalized into the firm’s earlier-period market values: i.e., even though lock-in premia arenil for t < 13, the firm’s market value under the all-repurchase policy is strictly lower inevery period. These results suggest that dividends are optimal when marginal investors have36

no dividend-tax disadvantage.34

Figure 8. Firm Value when HS = 1: Two Payout Policies
8.1

Firm Value (Vt )8
7.9

Dt = 1, At = 0 ∀ t
Dt = 0, At = 1 ∀ t

7.8
7.7
7.6
7.5
1

5

10

15

20

25

30

35

40

45

50

Period (t)This figure plots the firm’s market value at the beginning of periods 1-50 when HS = 1 under the parameter∞values in Table 4. The solid line corresponds to the all-dividend policy ({Dt }∞t=1 = 1, {At }t=1 = 0), while∞∞
the dashed line corresponds to the all-repurchase policy ({Dt }t=1 = 0, {At }t=1 = 1).

The next case considers HS = 2, and as such, the marginal investor’s effective tax rateis strictly less than τg - resulting in a dividend-tax disadvantage - whenever capital gains
are generated. Since the marginal value of dividends is strictly less than 1 in this case (i.e.,(1 − τd )/(1 − τe,2 ) < 1), the payment of lock-in premia is now acceptable. However, as
before, omitting dividends is sub-optimal. This is illustrated by Figure 9, which plots thefirm’s market value under the two policies described above; and once again, the all-dividend
policy is superior. However, both policies are sub-optimal and dominated by the following:∞{Dt }∞
t=1 = 0.35, {At }t=1 = 0.65. As Figure 9 illustrates, this intermediate policy results inhigher market values in every period despite the payment of strictly-positive lock-in premia:
which are 1.5% in every steady-state period (marginal shareholders are type H = 1 and
β = 0.229). Taken together, these results suggest that dividends can remain optimal whenmarginal investors have modest dividend-tax disadvantages, but that an all-dividend policy
is unlikely to be optimal.35
34

The all-dividend policy is not uniquely-optimal: policies that include share repurchases are also optimal
if no lock-in premia are paid (all such policies result in: {Vt }∞t=1 = 8). However, every optimal policy includes
dividends, as evidenced by Figure 8.
35
A reasonable conjecture is that firms should forego dividend payments altogether and only repurchaseequity from shareholders when their investment horizons reach H = 0 - i.e, no lock-in premia - by storing

37

Figure 9. Firm Value when HS = 2: Three Payout Policies

Firm Value (Vt )8.1
8

Dt = 1, At = 0 ∀ t
Dt = 0, At = 1 ∀ t
Dt = 0.35, At = 0.65 ∀ t

7.9
7.8
7.7
7.6
1

5

10

15

20

25

30

35

40

45

50

Period (t)This figure plots the firm’s market value at the beginning of periods 1-50 when HS = 2 under the parameter∞values in Table 4. The solid line corresponds to the all-dividend policy ({Dt }∞t=1 = 1, {At }t=1 = 0), the∞∞dashed line corresponds to the all-repurchase policy ({Dt }t=1 = 0, {At }t=1 = 1), while the dotted-and∞dashed line corresponds to the following policy: {Dt }∞t=1 = 0.35, {At }t=1 = 0.65.

The final case considers HS = 3, and like the previous two, excluding dividends is suboptimal. This is illustrated by Figure 10, which plots the firm’s market value when dividends
are excluded and when share repurchases are excluded. Unlike the previous two cases, thefirm’s market value under the all-repurchase policy is initially higher - due to the marginal
investor’s relatively-low effective tax rate (τg,3 = 0.184) and the favorable shareholder distribution in early periods - and then falls below that of the all-dividend policy - as theshareholder distribution becomes less favorable for buybacks. As with the previous case∞(HS = 2), the policy {Dt }∞t=1 = 0.35 and {At }t=1 = 0.65 dominates the two extreme policies(as Figure 10 illustrates). The firm’s market value under the intermediate policy is higher
when HS = 3 compared to when HS = 2 (in every period) since capital gains become morevaluable as the marginal investor’s effective tax rate decreases. Furthermore, since market
values are higher when HS = 3, less equity is repurchased for a given amount spent (At ),ceteris paribus, which raises every shareholder’s β (compared to when HS = 2) and makes
repurchasing equity more desirable. Marginal lock-in premia are 0.2% in every steady-state
period when HS = 3, and marginal shareholders are type H = 2 and β = 0.9258.
excess profits in the risk-free security. However, the firm’s cash holdings would grow without bound under
this strategy due to the limited measure of equity with H = 0 in each period; this, in turn, would violate
the firm’s transversality condition (limT →∞ [1 + ρ/(1 − τe )]−T VT = 0).

38

Figure 10. Firm Value when HS = 3: Three Payout Policies
8.2

Firm Value (Vt )8.1
8

Dt = 1, At = 0 ∀ t
Dt = 0, At = 1 ∀ t
Dt = 0.35, At = 0.65 ∀ t

7.9
7.8
7.7
1

5

10

15

20

25

30

35

40

45

50

Period (t)This figure plots the firm’s market value at the beginning of periods 1-50 when HS = 3 under the parameter∞values in Table 4. The solid line corresponds to the all-dividend policy ({Dt }∞t=1 = 1, {At }t=1 = 0), the∞∞dashed line corresponds to the all-repurchase policy ({Dt }t=1 = 0, {At }t=1 = 1), while the dotted-and∞dashed line corresponds to the following policy: {Dt }∞t=1 = 0.35, {At }t=1 = 0.65.

Dividends remain optimal ∀ HS ≤ 7 and become sub-optimal thereafter due to themarginal investor’s relatively-low effective tax rate: τe,8 = 0.155 under the all-repurchasepolicy. However, if HL and/or ωL were to increase, then dividends could become optimal
again, as both adjustments put upward pressure on lock-in premia.
These results suggest that dividends are optimal when marginal investors have no dividendtax disadvantage (HS ≤ 1) and can remain optimal when they do (HS > 1). However, if
dividends become significantly tax disadvantaged, then excluding dividends can become optimal.

6

Conclusion

This paper developed a model of corporate payout policy to explain one aspect of the payout puzzle. Shareholders in the model have heterogeneous investment horizons and heterogeneous accrued capital gains on firm equity. It was shown that shareholder-wealth maximization - within a realization-based capital-gains tax system - can create a wedge between
equity’s intrinsic value and the ask price of many shareholders. This wedge - called the“lock-in premium” herein - was shown to be an increasing function of investment horizon
39

and accrued capital gains. Paying lock-in premia is acceptable provided they remain small,but when firms repurchase large quantities of equity - especially in multiple successive years- marginal lock-in premia can become sufficiently expensive that paying tax-disadvantaged
dividends becomes optimal.
Unlike most payout models that offer a solution to the payout puzzle, the current one
does not rely on informational asymmetries, repurchase constraints, incomplete contracting,or irrationality. The key assumption is that capital gains are taxed upon realization and notaccrual, thus providing a self-contained explanation for dividend payments within a perfectinformation framework.

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A

Tax Rates and Payout Variables

Dividends were taxed as ordinary income from 1972-2002 (subject to a small exclusion notexceeding $400 from 1972-1986) according to Congressional (2014). The top federal tax rateon ordinary income was: 70% from 1972-1981, 50% from 1982-1986, 38.5% in 1987, 28%from 1988-1990, 31% from 1991-1992, 39.6% from 1993-2000, 39.1% in 2001, and 38.6% in
2002. The Jobs and Growth Tax Relief Reconciliation Act of 2003 reduced the maximum
tax rate on dividends to 15%. This rate prevailed until the 2012 American Taxpayer Relief
Act increased the maximum rate to 20% in 2013; this is the current rate.
Between 1972-1978 (October 31), the tax rate on capital gains (in excess of $50,000)
was 50% of the rate on ordinary income (U.S. Treasury, 1985); the maximum tax rate on
ordinary income was 70% in each of these years. Between 1978 (November 1) and 1986, capital gains were taxed at 40% of the rate on ordinary income (Auten and Cordes, 1991); the45

maximum rate on ordinary income was 70% (50%) between 1979-1981 (1982-1986), however,realizations after June 9, 1981 were subject to a maximum ordinary-income-tax rate of 50%(instead of the 70% that prevailed in 1981) according to U.S. Treasury (1985). The Tax
Reform Act of 1986 increased the maximum rate on long-term capital gains to 28% in 1987.
This lasted until May 6, 1997, when the Taxpayer Relief Act of 1997 reduced the maximum
rate on long-term capital gains to 20% (Auten, 1999).36 This rate prevailed until the Jobs
and Growth Tax Relief Reconciliation Act of 2003 reduced the maximum rate to 15% in
2003, which lasted until the 2012 American Taxpayer Relief Act increased the maximum
rate to 20% in 2013; this is the current rate.
Note that all tax rates described above (and used in Table 1) do not incorporate: phaseout provisions, the minimum tax, alternative tax rates, income tax surcharges, the Medicare
contribution, and interactions with other tax provisions. They are only meant to capture
the top statutory tax rate on dividends and realized capital gains.
When calculating θt from Equation 1, it is assumed that marginal investors face the top
statutory tax rate on dividends and long-term capital gains. An alternative characterizationPof θt would be sj=1 wj,t (1 − τd,j,t )/(1 − τe,j,t ), where τd,j,t is the marginal tax rate on dividends in year t for investors in income class j ∈ [1, s], τe,j,t is the effective tax rate on accrued
capital gains in year t for investors in income class j (a proportion of the statutory rate), andthe wj,t ’s are equity-ownership weights across income class and time; this approach is used in
Poterba (1987) and Bernheim and Wantz (1995) and measures the average θt among shareholders . It is not clear which of the two measures is a better approximation of the marginal
investor’s θt . Conversely, Allen and Michaely (2003) point out that marginal investors may
have τd = τe (for instance, non-taxed institutional investors), and therefore, may not havea dividend-tax disadvantage. As illustrated in Section 5, dividends are still optimal whenmarginal investors have τd = τe provided that some equity is repurchased from shareholders
with a lock-in effect. Finally, it should be noted that Table 1 does not account for dynamic
tax-trading strategies: i.e., when equity is traded around the ex-dividend day (by investors
with different tax situations) to lower the overall tax burden; see Allen and Michaely (2003)
36
A provision in the 2001 Tax Act reduced the maximum rate to 18% for assets purchased after 2001
and held for at least 5 years. This provision is ignored in Table 1 due to the unusually-long holding-period
restriction.

46

for a discussion.
Table 5 describes the payout variables used in Figures 1 and 2 and reports their Compustat
data codes in parentheses. These variables are similar to the ones used in Grullon andMichaely (2002) and Moser (2007), among others.
Table 5. Payout Variable Descriptions
Variable

Data Description (Compustat Variable Code)Dividends

Ordinary dividends on common equity (DVC).

Share Repurchases

Purchases of common and preferred stock (PRSTKC)
minus preferred stock redemption value (PSTKRV).

Description of each payout variable used in Figures 1 and 2 of Section 3.

B

Derivation of Expression 8 and Ω(H, β)

To derive the difference in after-tax wealth under the following two scenarios: 1) holding anequity position for H > 0 periods; and 2) selling it immediately, paying the capital-gains
tax (or receiving the capital-loss offset), and reinvesting the net proceeds for H > 0 periods,
we will first calculate the marginal after-tax wealth under each scenario and then take thedifference.
Under the first scenario, the after-tax value of a period-t equity position (of size 1) held
until the beginning of period t + 1 (with a period-t tax basis of β) is:(1 − τg )[(1 + rg ) − β] + β + (1 − τd )rd .

(26)

This is derived as follows. The position’s market value at the beginning of period t + 1 is
(1 + rg ), and therefore, the position’s total capital gain/loss is [(1 + rg ) − β]. This is subject
to capital-gains tax (or a capital-loss offset) at the rate τg , which leaves (1 − τg )[(1 + rg ) − β]once the tax is paid (or the refund is received). Since the tax basis (β) is not subject tocapital-gains taxation, you add this back to the investor’s after-tax wealth. Finally, add
47

the after-tax dividend payment from period t (i.e., (1 − τd )rd ) to get Expression 26. This
expression can be rewritten as:(1 − τg ) [(1 + rg + (1 − τd )rd )] + τg [β + (1 − τd )rd ] .
which is also equal to:"(1 − τg )[(1 + rg + (1 − τd )rd )1 ] + τg β + (1 − τd )rd

1
X
h=1

#

(1 + rg + (1 − τd )rd )h−1 .

(27)Next, suppose that the position is held until the beginning of period t + 2 (instead of
period t + 1). Its after-tax value at the beginning of period t + 2 is:(1 − τg )[(1 + rg )(1 + rg + (1 − τd )rd ) − β − (1 − τd )rd ]+ β + (1 − τd )rd + (1 − τd )rd (1 + rg + (1 − τd )rd ) (28)This is derived as follows. First, the position’s market value at the beginning of period t + 1
is composed of two parts: 1) the period-t return from capital gains - i.e., (1 + rg ) - plus
the period-t dividend reinvestment - i.e., (1 − τd )rd . Taken together, the position’s market
value at the beginning of period t + 1 is (1 + rg + (1 − τd )rd ). Second, since this investment
generates a return from capital gains in period t + 1 as well (the return is (1 + rg ), as before),the position’s market value at the beginning of period t + 2 is (1 + rg )(1 + rg + (1 − τd )rd ).
Third, calculate the investor’s tax basis at the beginning of period t + 2; this is composed of
two parts: 1) the tax basis at the beginning of period t (i.e., β), and 2) the period-t dividend
reinvestment (i.e., (1 − τd )rd ). Taken together, the investor’s tax basis at the beginning of
period t + 2 is: β + (1 − τd )rd . Fourth, given the position’s market value at the beginning of
period t + 2 (from step two) and the investor’s tax basis (from step three), the total capitalgain/loss at the beginning of period t + 2 is: (1 + rg )(1 + rg + (1 − τd )rd ) − β − (1 − τd )rd .
This is subject to capital-gains taxation (or a capital-loss offset), which leaves: (1 − τg )[(1 +rg )(1 + rg + (1 − τd )rd ) − β − (1 − τd )rd ] once the tax is paid (or the refund is received). Fifth,
add the investor’s tax basis to the after-tax capital gain/loss that we just calculated (thispart of the position is untaxed). And sixth, add the after-tax dividend payment received
in period t + 1, which is equal to the position’s size at the beginning of period t + 1 (i.e.,48

(1 + rg + (1 − τd )rd ) from step one) multiplied by the after-tax dividend yield (1 − τd )rd note that dividends in period t + 1 are larger than in period t because the investment size islarger - this equals: (1 − τd )rd (1 + rg + (1 − τd )rd ). Finally, after summing the amounts fromsteps four, five, and six, we get Expression 28. Note that this expression can be rewritten as
follows:"#
2X(1 − τg )[(1 + rg + (1 − τd )rd )2 ] + τg β + (1 − τd )rd(29)(1 + rg + (1 − τd )rd )h−1 .
h=1

By examining Expressions 27 and 29, we can see that continuing this process indefinitely
results in the following general expression for the position’s after-tax value at the beginningof period t + H:
"
#
H
X


H
h−1
(1 − τg ) (1 + rg + (1 − τd )rd ) + τg β + (1 − τd )rd(1 + rg + (1 − τd )rd ),h=1

which is equivalent to Expression 8 from Section 4.2.
To facilitate the derivation of Ω(H, β), it is helpful to note that:H(1 + rg + (1 − τd )rd ) = (rg + (1 − τd )rd )HX(1 + rg + (1 − τd )rd )h−1 + 1,

(30)

h=1

which allows us to rewrite Expression 8 as:[(1 − τg )rg + (1 − τd )rd ]HX(1 + rg + (1 − τd )rd )h−1 + (1 − τg ) + τg β.

(31)

h=1

Next, to derive the position’s after-tax value under the second scenario, note that the net
proceeds from the period-t equity sale are 1 − τg (1 − β). After reinvesting these proceeds (inthe same equity) for H periods, the investor’s after-tax wealth at the beginning of periodt + H becomes:"(1 − τg (1 − β)) (1 − τg ) (1 + rg + (1 − τd )rd )H49"+ τg 1 + (1 − τd )rdHX(1 + rg + (1 − τd )rd )h−1

h=1

##

. (32)

This expression is similar to Expression 8 above, except that equity’s initial size is 1−τg (1−β)
instead of 1, and the initial tax basis is 1 − τg (1 − β) instead of β. If we apply the sametransformation as before (i.e., Equation 30), then Expression 32 can be rewritten as:"#
HX(1 − τg (1 − β)) [(1 − τg )rg + (1 − τd )rd ](1 + rg + (1 − τd )rd )h−1 + 1 .

(33)

h=1

Finally, subtracting Expression 33 from Expression 31 produces:H
X(1 + rg + (1 − τd )rd )h−1 ∀ H > 0.
Ω(H, β) = τg (1 − β) [(1 − τg )rg + (1 − τd )rd ]

(34)

h=1

Furthermore, since investors with H = 0 desire to sell their equity immediately, it follows
that Ω(0, β) = 0 ∀ β. Combining this result with Equation 34 produces Equation 11 from
Section 4.2:
τ (1 − β)[(1 − τ )r + (1 − τ )r ] PH (1 + r + (1 − τ )r )h−1 if H > 0,
g
d d
g
g g
d d
h=1
Ω(H, β) =0if H = 0.

C

Derivation of L(H, β)The lock-in premium is defined as the remuneration above equity’s intrinsic value that creates
indifference between: 1) liquidating an equity position at its intrinsic value plus this premium,and 2) retaining the position for a desired number of periods. If we denote the lock-in
premium by L(H, β) (as a percentage of equity’s intrinsic value), then liquidating equity
under scenario 1 provides marginal after-tax proceeds of:
1 + L(H, β) − τg (1 + L(H, β) − β).
When these proceeds are promptly reinvested in the same stock (or identical stock), and50

held for H > 0 periods, the investor’s marginal after-tax wealth becomes:h(1 + L(H, β) − τg (1 + L(H, β) − β)) [(1 − τg )rg+ (1 − τd )rd ]HX(1 + rg + (1 − τd )rd )

h−1

h=1

i
+ 1 . (35)See Appendix B for an explanation of how this expression is derived. Alternatively, theequity position’s marginal after-tax value under scenario 2 is:[(1 − τg )rg + (1 − τd )rd ]HX(1 + rg + (1 − τd )rd )h−1 + (1 − τg ) + τg β,

(36)

h=1

after H > 0 periods. Finally, equating Expressions 35 and 36, and solving for L(H, β),produces:Ω(H, β)τg.
(1 − β)1 − τg Ω(H, β) + τg (1 − β)Combining this expression with the fact that only shareholders with capital gains require alock-in premium (i.e., β < 1), we have Equation 12 from Section 4.2:

L(H, β) =Dh(1 − β) τg1−τgΩ(H,β)Ω(H,β)+τg (1−β)

0iif β < 1,
if β ≥ 1.

Properties of the Repurchase Function:
Proposition 1

The firm’s repurchase function is characterized by Equations 13 and 14:


Rt = max {β(H)}

H Z β
X

H=0

β(H)



f H (β)dβ  (Rt + Vt+1 ),51

subject to:At = 

H Z β
X

H=0

β(H)



[1 + L(H, β)]f H (β)dβ  (Rt + Vt+1 ).

These equations state that for every H, firms choose a corresponding β(H) to maximize themass of repurchased equity subject to spending At on the repurchase program. The cost ofeach repurchased share is equal to the firm’s intrinsic value during the repurchase program(Rt +Vt+1 ) plus the shareholder-specific lock-in premium, where the firm’s continuation value(Vt+1 ) is independent of the period-t payout mix: it depends on the capital stock at the beginning of period t + 1 and the payout sequences from period t + 1 onwards.
Recall that δt is the fraction of equity repurchased in period t, and therefore:
H Z β
X

H=0

f H (β)dβ = δt .

(37)

β(H)

Furthermore, if we assume that there is a unit measure of equity at the beginning of periodt (for simplicity), then the per-unit value of equity during the period-t repurchase programis:
Vt+1
Rt + Vt+1 =.
(38)
1 − δtGiven Equations 37 and 38, the period-t repurchase function can be rewritten as:
Rt = max δt{β(H)}

subject to:
A t = δt



H Z β
X

Vt+1
+
(1 − δt )
H=0

β(H)

Vt+1
,
(1 − δt )

L(H, β)f H (β)dβ Vt+1.
(1 − δt )It is trivial to show that R(At ) is weakly below the 45◦ line (due to L(H, β) ≥ 0 ∀ H and∀ β) and increasing in the amount spent (given that L(H, β) is finite for any finite H). The
non-trivial task is to show that R(At ) is concave. If we take the inverse of R(At ) we have:52At = min δt{β(H)}

subject to:



H Z β
X

Vt+1
+
(1 − δt )
H=0

β(H)

Rt = δt

L(H, β)f H (β)dβ Vt+1,(1 − δt )Vt+1.
(1 − δt )The marginal change in Rt given a change in β(H), for a particular H ∈ [0, H], is:∂Rt
Vt+1
Vt+1 f H (β(H))
Vt+1 f H (β(H))= −f H (β(H))
− δt
=
−
,
∂β(H)
1 − δt(1 − δt )2(1 − δt )2
As Rt increases, at least one of the β(H) > 0 must decrease. Without loss of generality,b where Hb ∈ [0, H], then:suppose that a marginal increase in Rt is made through β(H),b∂β(H)(1 − δt )2
.
=−
b
∂Rt
Vt+1 f H (β(H))

b is:Furthermore, the change in At from a marginal change in β(H)bb β(H))fb H (β(H))V
b t+1
∂At
Vt+1 f H (β(H))
L(H,
=−
−
b
(1 − δt )2(1 − δt )
∂β(H)


−

H Z β
X

H=0

β(H)



L(H, β)f H (β)dβ bVt+1 f H (β(H)).
(1 − δt )2Therefore, we have:
H Z β
X
∂At
b β(H))(1b= 1 + L(H,− δt ) +L(H, β)f H (β)dβ,∂Rt
H=0 β(H)which implies that At is an increasing function of Rt (when the increase is made throughbβ(H)),
since δt < 1, and L(H, β) ≥ 0 ∀ H & β. The derivative of ∂At /∂Rt with respect53

b is:β(H)∂ 2 Atb∂Rt ∂β(H)b β(H))(1b= L0β (H,− δt ),b β(H))b is the first derivative of L(H, β) with respect to β evaluated at Hb andwhere L0β (H,b This, in turn, equals:β(H).
b β(H))
b = − τg
L0β (H,1 − τg"b β(H))b − Ω0 (H,b β(H))(1
b
Ω(H,
− β)2
β
b β(H))b + τg (1 − β)]2
[Ω(H,

#

< 0,

b β(H))
b < 0 is the first derivative of Equation 11 with respect to β evaluated atwhere Ω0β (H,b and β(H).
b Therefore:Hb β(H))(1b− δt )3L0β (H,∂ 2 At
=−
> 0,
b
∂Rt2
Vt+1 f Hb β(H)which implies that A(Rt ) is a convex function using any β(H) > 0 to expand the set of
repurchased equity, and thus, the inverse of this function (R(At )) is concave. Furthermore,
the continuity of R(At ) and R0 (At ) follows from the continuity of L(H, β) and δt with respect
to β.

E

Deriving the Return on Equity

The after-tax return on equity for an H-type investor has three components: 1) the returnfrom equity held for H periods, 2) the return from equity sold at the beginning of each period
and reinvested in the risk-free asset, and 3) the return from dividends that are reinvested in
the risk-free asset.
As mentioned above, capital gains/losses are generated in two ways: through changes in
the firm’s market value, and through share repurchases. If we assume a unit measure ofequity at the beginning of period t (for simplicity), then when a fraction δt of equity isrepurchased, the measure of outstanding shares becomes (1 − δt ) post-repurchase. Since themarket value of equity at the beginning of period t + 1 is Vt+1 by definition, the per-unit
value of non-repurchased equity is Vt+1 /(1 − δt ). Furthermore, since the market value of54

repurchased equity and non-repurchased equity must be the same, the firm’s total market
value just before to the repurchase program is the sum of: δt Vt+1 /(1 − δt ) - the measure of
repurchased equity multiplied by the unit value of equity - plus (1 − δt )Vt+1 /(1 − δt ) - themeasure of non-repurchased equity multiplied by the unit value of equity. Therefore, thefirm’s total market value just before the repurchase program is Vt+1 /(1 − δt ). If we divide
this by Vt (the firm’s market value at the beginning of period t), we get the return from
capital gains in period t (Equation 21 rearranged):
1 + rg,t =Vt+1.
Vt (1 − δt )Furthermore, recall that the rate-of-return from dividends in period t is equal to (Equation 22):Dt.
rd,t =Vt
To derive an H-type investor’s total return from equity, we will derive their after-tax
return from each component above (Components 1 - 3) in turn. With regard to the firstcomponent, when the investment horizon is H = 1, the after-tax return from capital gains
on a marginal investment (of size 1) from period t to period t + 1 is:τg + (1 − τg )Vt+1.
Vt (1 − δt )(39)

This is derived as follows. The rate-of-return from capital gains is rg,t = Vt+1 /Vt (1 − δt ) − 1
from Equation 21, which results in an after-tax rate-of-return equal to (1 − τg )[Vt+1 /Vt (1 −δt ) − 1] and an after-tax return of 1 + (1 − τg )[Vt+1 /Vt (1 − δt ) − 1] (by adding 1 - the tax
basis); this equals Expression 39 above.
Next, derive Component 1 when the investment horizon is H = 2. The compound returnfrom capital gains in periods t and t + 1 is:

Vt+1
Vt (1 − δt )



Vt+2
Vt+1 (1 − δt+1 )

55



=

Vt

Vt+2
,
w=0 (1 − δt+w )Q1

which results in an after-tax return at the beginning of period t + 2 equal to:
"

#
Vt+2
(1 − τg )− 1 + 1,Q
Vt 1w=0 (1 − δt+w )which can be rearranged as follows:τg + (1 − τg )

Vt

Vt+2
.
w=0 (1 − δt+w )

(40)

Q1

Furthermore, since the investor sells a fraction λ of her equity position at the beginningof period t + 1 (since her investment horizon is reached at the beginning of period t + 2),
Expression 40 is applied to a fraction (1 − λ) of the original investment, resulting in:
"

#
Vt+2
(1 − λ) τg + (1 − τg ) Q1.
Vt w=0 (1 − δt+w )Using a similar process to that above, the after-tax return from a marginal investment
made in period t and held until period t + 3 (i.e., H = 3) is:τg + (1 − τg )

Vt

Vt+3
,
w=0 (1 − δt+w )Q2

which is applied to a fraction (1 − λ)2 of the marginal investment, since a fraction λ of theposition is sold at the beginning of period t + 1 and period t + 2, resulting in the following
expression for Component 1 when H = 3:
(1 − λ)2"#
Vt+3
τg + (1 − τg ) Q2.
Vt w=0 (1 − δt+w )Continuing this process indefinitely results in the following general expression for an Htype investor’s Component 1:"(1 − λ)H−1 τg + (1 − τg )

Vt+H

Vt

QH−1

w=0 (1 − δt+w )

#

.

(41)

Moving on to Component 2. When the investment horizon is H = 1, this component is56

equal to zero, since the entire equity position is liquidated at the beginning of period t + 1.
When the investment horizon is H = 2, this component equals:



Vt+1
(1 + ρ)λ τg + (1 − τg ).
Vt (1 − δt )(42)

This is derived as follows. At the beginning of period t + 1 the marginal investment hasappreciated in value to Vt+1 /Vt (1 − δt ). When a fraction λ of the position is sold, the aftertax proceeds are (1 − τg )[λVt+1 /Vt (1 − δt ) − λ] + λ, since the tax basis on this portion ofequity is λ. This can be rewritten as:
λτg + (1 − τg )λVt+1.
Vt (1 − δt )(43)

Furthermore, since the investment horizon is reached at the beginning of period t + 2, andthis sale happens at the beginning of period t + 1, the after-tax proceeds are reinvested for
one period (in the risk-free asset). Therefore, Expression 43 is multiplied by (1 + ρ) to arriveat Expression 42.
When the investment horizon is H = 3, the equity sale at the beginning of period t + 1 isreinvested for two periods (from the beginning of period t + 1 until the beginning of periodt + 3), and therefore, you multiply Expression 43 by (1 + ρ)2 to arrive at the period t + 3
value of the period t + 1 equity sale, which is:Vt+1(1 + ρ) λ τg + (1 − τg ).
Vt (1 − δt )
2



(44)

In addition to the period t + 1 equity sale, a fraction λ of the remaining position is also soldat the beginning of period t + 2 (the investment horizon is reached at the beginning of periodt + 3). Since the remaining equity at the beginning of period t + 2 is a fraction (1 − λ) of
the original position, due to the fraction λ sold at the beginning of period t + 1, λ(1 − λ) of
the original position is sold at the beginning of period t + 2. The after-tax proceeds of thissale are:
#
"
Vt+2
,
(45)
λ(1 − λ) τg + (1 − τg ) Q1
Vt w=0 (1 − δt+w )57

since, as before, the compound return from capital gains in periods t and t + 1 is:

Vt

Vt+2
.
w=0 (1 − δt+w )Q1

Since the after-tax proceeds from this sale are reinvested for one period (until the beginningof period t + 3), you multiply Expression 45 by (1 + ρ) to arrive at the period t + 3 value ofthe period t + 2 equity sale, i.e.:
"

#
Vt+2
.
(1 + ρ)λ(1 − λ) τg + (1 − τg ) Q1
Vt w=0 (1 − δt+w )(46)

Finally, Component 2 for this investor is the period t + 3 value of both beginning-of-period
equity sales (Expression 44 plus Expression 46), which equals:
"
2
X
3−S
S−1
λ
(1 + ρ) (1 − λ)τg + (1 − τg )
S=1

Vt+S

Vt

QS−1

w=0 (1 − δt+w )#.

Continuing this process indefinitely results in the following general expression for an Htype investor’s Component 2:
λ

H−1
X
S=1

"

(1 + ρ)H−S (1 − λ)S−1 τg + (1 − τg )

Vt+S

Vt

QS−1

w=0 (1 − δt+w )

#

.

(47)

Moving on to Component 3. A marginal investment at the beginning of period t results
in a fractional ownership of 1/Vt of the firm’s equity. This entitles the owner to Dt /Vt of theperiod-t dividend, which has an after-tax value of:(1 − τd )

Dt
,
Vt

(48)

and constitutes Component 3 when the investment horizons is H = 1 (since all equity is soldat the beginning of period t + 1).
When the investment horizon is H = 2, a marginal investment also receives the aftertax dividend from Expression 48. However, since the investment horizon is reached at the58

beginning of period t + 2, these dividends are reinvested in the risk-free asset for one period,
producing the following value at the beginning of period t + 2:
(1 + ρ)(1 − τd )

Dt
.
Vt

(49)

In addition to receiving period-t dividends, the investor also receives a dividend payment
in period t + 1. Since the firm repurchases a fraction (1 − δt ) of its equity in period t, amarginal investment made at the beginning of period t confers ownership rights to a fraction
1/Vt (1 − δt ) of the firm’s equity at the beginning of period t + 1. However, since a fraction
λ of this position is sold (by the investor) at the beginning of period t + 1, the investor’sownership fraction declines to (1 − λ)/Vt (1 − δt ) before dividends are paid in period t + 1.
Taken together, the owner is entitled to (1 − λ)Dt+1 /Vt (1 − δt ) of the period t + 1 dividend,
which has an after-tax value of:(1 − τd )(1 − λ)

Dt+1.
Vt (1 − δt )(50)

Therefore, when the investment horizon is H = 2, Component 3 is the sum of Expression 49
and 50, which equals:(1 − τd )1X(1 + ρ)1−S (1 − λ)S

S=0

where

Q−1

Dt+S
,
QS−1
Vt w=0 (1 − δt+w )w=0 (1 − δt+w ) = 1 (for the case of period-t dividends).

When the investment horizon is H = 3, the after-tax dividends received in period t are
reinvested for two periods, so Expression 48 is multiplied by (1 + ρ)2 , while the after-tax
dividends received in period t+1 are reinvested for one period, so Expression 50 is multipliedby (1 + ρ). Finally, the after-tax dividend received in period t + 2 is:(1 − τd )(1 − λ)2

Vt

Dt+2
,
w=0 (1 − δt+w )Q1

since a fraction λ of the position sold at the beginning of period t + 1 and t + 2, andfurthermore, the firm repurchases a fraction δt (δt+1 ) of its equity in period t (t + 1). Taken
59together, when the investment horizon is H = 3, Component 3 is equal to:(1 − τd )2X(1 + ρ)2−S (1 − λ)S

S=0

Dt+S
.
QS−1
Vt w=0 (1 − δt+w )Continuing this process indefinitely results in the following general expression for an Htype investor’s Component 3:(1 − τd )H−1
X(1 + ρ)H−S−1 (1 − λ)S

S=0

Dt+S
.
QS−1
Vt w=0 (1 − δt+w )(51)

Now that Components 1, 2, and 3 are derived for an H-type investor (Expressions 41, 47,
and 51, respectively), we can take their sum to arrive at Expression 23 - the total after-tax
return on a marginal equity investment made at the beginning of period t.

FDerivation of Expression 25

Recall that capital gains have a period-t rate-of-return equal to:
rg,t =

Vt+1− 1,(1 − δt )Vt(52)

from Equation 21. When you apply an effective tax rate to these gains (levied upon accrual),an H-type investor’s after-tax rate-of-return is:(1 − τe,H )rg,t = (1 − τe,H )




Vt+1
−1 ,
(1 − δt )Vt

where τe,H is the investor’s effective tax rate. The total return is therefore:
1 + (1 − τe,H )rg,t = 1 + (1 − τe,H )




Vt+1
Vt+1
− 1 = τe,H + (1 − τe,H ),(1 − δt )Vt(1 − δt )Vt

60

(53)

by adding 1. Given Equation 53, the compound after-tax return for an H-type investor fromperiod t to period t + S, based on an accrual-based tax, is therefore:
S−1
Y

w=0

Vt+w+1τe,H + (1 − τe,H )(1 − δt+w )Vt+w



.

(54)

Recall from Expression 23 that the after-tax compound return from capital gains under arealization-based tax (from period t to period t + S) is:τg + (1 − τg )

Vt+S
.
QS−1
Vt w=0 (1 − δt+w )(55)

Therefore, substitute Expression 54 for Expression 55 in Expression 23 to convert the aftertax return from a realization-based tax to an accrual-based tax (one of these substitutions
is for t + H (the first line of Expression 23) and the other is for t + S (the second line of
Expression 23)).
Next, to convert the return from dividends based on a realization-based tax to that of an
accrual-based tax, note that the pre-tax compound return from capital gains from period t
to period t + S under a realization-based tax is:
Vt+S
,
QS−1
Vt w=0 (1 − δt+w )(56)

from Equation 52. Expression 56, in conjunction with the pre-tax rate-of-return from dividends in period t (i.e., Equation 22):rd,t =

Dt,
Vt

produces the following component of Expression 23:
Dt+S
Vt+S
Dt+S
= QS−1.
QS−1
Vt+S Vt w=0 (1 − δt+w )Vt w=0 (1 − δt+w )(57)

To convert this component of Expression 23 into one based on an accrual-based tax, you61

substitute Expression 54 for Expression 56 in Equation 57 to arrive at:
S−1 
Dt+S Y
Vt+w+1τe,H + (1 − τe,H ).
Vt+S w=0(1 − δt+w )Vt+w(58)

Finally, substitute Expression 58 for Equation 57 in Expression 23 to arrive at Expression 25 (along with the previous two substitutions, i.e., Expression 54 for Expression 55 in
Expression 23 - twice).

G

Solution Algorithm

This section describes the model’s solution algorithm.
Step 1) Select the parameter values, which are:• Statutory tax rates on dividends and realized capital gains: τd and τg , respectively.
• The after-tax rate-of-return on the risk-free asset: ρ.
• The fraction of equity sold by shareholders at the beginning of each period: λ.
• The initial investment horizon of new investors: HL and HS .
• The wealth of new investors in each period: ωL and ωS .
• The number of periods to analyze: N .
Step 2) Select the payout sequences controlled by management:• Dividends: {Dt }N
t=1 .
• The amount spent repurchasing equity: {At }N
t=1 .
Step 3) Make an initial guess regarding:• The sequence of firm values: {Vt }N
t=1 .
• The fraction of equity repurchased in each period: {δt }N
t=1 .
62

N
N
Step 4) With the sequences {Vt }N
t=1 , {Dt }t=1 , and {δt }t=1 , and the parameters τd , τg , λ, andρ, determine each shareholder’s required lock-in premium in each period. This is done inthree sub-steps:

1. Calculate the buy-and-hold after-tax return of each shareholder with an initial investment horizon H that purchases equity at the beginning of period t (Expression 23):"

(1 − λ)H−1 τg + (1 − τg )

+λ

Vt+H
QH−1
Vt w=0 (1 − δt+w )
"

H−1
X

#

(1 + ρ)H−S (1 − λ)S−1 τg + (1 − τg )

S=1

+ (1 − τd )

Vt+S

Vt

H−1
X

QS−1

w=0 (1 − δt+w )(1 + ρ)H−S−1 (1 − λ)S

S=0

#

Dt+S
, (59)
QS−1
Vt w=0 (1 − δt+w )where the first part of this expression is the future after-tax return from equity retainedin the firm, the second is the future after-tax return from all beginning-of-period equity
sales (the fraction λ of the position sold at the beginning of each period) that are
reinvested in the risk-free asset, and the third is the future after-tax return from all
dividend income reinvested in the risk-free asset.37
2. Calculate the same investor’s after-tax return when their position is sold at the end of
period t+b (where b ∈ {B ∈ N|0 ≤ B ≤ H −1}) for the amount (1+ L̃)Vt+b+1 /(1−δt+b )(where L̃ is the multiplicative repurchase premium that we are solving for), and theproceeds of this equity sale are reinvested in the risk-free asset for the remaining H −b−1
periods. This equals:((1 + ρ)H−b−1 (1 − λ)b37"(1 + L̃)Vt+b+1
τg + (1 − τg ) Qb
Vt w=0 (1 − δt+w )#To calculate lock-in premia we require both contemporary and future values of: Dt , Vt and δt . Specifically,
to calculate premia for periods 1 through N (and for all shareholders) we require Vt (Dt and δt ) over theperiods 1 to N + HL (1 to N + HL − 1) - see Expressions 59 and 60. As such, for all t > N , Vt is set to VN ,Dt is set to DN , and δt is set to δN .

63

"
b
X
+λ
(1 + ρ)b+1−S (1 − λ)S−1 τg + (1 − τg )
S=1

+ (1 − τd )

b
X

b−S

(1 + ρ)

S=0

Vt+S

QS−1

Vt

w=0 (1 − δt+w )S(1 − λ)

Vt

#

Dt+S

QS−1

w=0 (1 − δt+w )), (60)

where the part of this expression inside of the large curly brackets is the investor’s aftertax return at the end of period t + b (after selling their equity back to the firm), whichis reinvested in the risk-free asset for H − b − 1 periods (the risk-free asset’s return is(1 + ρ)).
For example, suppose that the investor’s initial holding period is H = 3, and that b = 1.
In this case, the equity position is held for 2 periods (from the beginning of period t
until the end of period t + 1), the relevant market value is Vt+2 (i.e., the firm’s market
value at the very end of period t + 1 is the same as the market value at the beginningof period t + 2 - both occur simultaneously). Finally, since the investor’s initial holding
period is H = 3, the proceeds from this equity sale (and the amount already invested
in the risk-free asset) are reinvested for 1 period at the rate-of-return ρ.
3. Finally, solve for the value of L̃ that equates Expression 59 with Expression 60. This is
the shareholder’s required lock-in premium (as a fraction of equity’s market value).
This procedure is carried out for each shareholder in each cohort (periods 1 to N ) in eachpotential period of ownership. For example, shareholders with an initial investment horizonof H who purchase equity at the beginning of period t may own equity at the end of periods
t through t + H − 1; their investment horizon becomes H = 0 afterwards (at the beginningof period t + H), and thus, no equity is retained afterwards.
Step 5) Determine the evolution of the shareholder distribution starting with the first period.
This is done in four sub-steps:
1. Determine the fraction of equity sold at the beginning of a period, which is composedof two types:• Equity sold by shareholders with an investment horizon of H = 0.
64• A fraction λ of every shareholder’s equity position.
Remove this equity from the shareholder distribution (i.e., remove a fraction λ of equityfrom all shareholders, and remove all equity from shareholders with H = 0). Next, given
the firm’s beginning-of-period market value (from the sequence {Vt }N
t=1 ) and the wealthof long-term investors (i.e., ωL ), determine how much equity they can afford - thisconstitutes their initial ownership share. Next, allocate the remaining equity (sold at
the beginning of a period) to the group of short-term investors - this constitutes theirinitial ownership share. Since all equity is purchased at the firm’s market price, every
shareholder’s proportional tax basis is initially β = 1.
2. Right before the repurchase program commences, reduce every shareholder’s investment
horizon by one period. Also, reduce every shareholder’s β by an amount commensuratewith the capital gains generated in that period (recall that period-t capital gains are
Vt+1 /(1 − δt ) − Vt ).
3. Sort all shareholders by their required lock-in premium (determined in Step 4) fromlowest to highest, and remove equity from shareholders in that order until the total
fraction of equity removed equals the fraction of equity repurchased - according to thesequence {δt }N
t=1 .
4. Re-normalize the total mass of equity to 1 (after the repurchase) by dividend the massof every shareholder’s remaining equity position by (1 − δt ). This constitutes the shareholder distribution at the beginning of the subsequent period (just before new investors
arrive).
Note, the four sub-steps above are repeated for all periods (from 1 to N ) to determine theshareholder-distribution’s evolution.
Step 6) Now that we know which shareholders have their equity repurchased in each period(from Step 5) and the cost of that equity (from Step 4), we can calculate the cost of repurchasing a fraction δt of the firm in period t. When the cost of this repurchased equity isgreater than (less than) At , decrease (increase) the value of δt to arrive at a new sequence{δt }N
t=1 .

65

N
Step 7) With the new sequence {δt }N
t=1 (from Step 6) and the original sequence {Dt }t=1 ,calculate the firm’s market value in each period (based on the marginal shareholder’s valuation). This is done using backward induction starting with the last period, where Dt , Vtand δt are held constant from period N onwards.

N
Step 8) With the updated sequences {Vt }N
t=1 (from Step 7) and {δt }t=1 (from Step 6), andNN
the original sequence {Dt }N
t=1 , repeat Steps 4-7 until {Vt }t=1 and {δt }t=1 converge. This
produces the model’s dynamic equilibrium - all other variables/distributions/functions canbe obtained afterwards.

66