The University of NSW
School of Accounting
RESEARCH SEMINAR
SESSION 1, 2003.
Simultaneous estimations
of the implied value of
franked dividends, cost of equity
and growth rates
using a modified residual
income valuation model.
presented by
Julian Yeo
University of Melbourne
Date: Friday 11th April
Time: 3:30 to 5:00 p.m.
Where: Webster 256
Simultaneous Estimations of
the Implied Value of Franked Dividends,
Cost of Equity, and Growth Rates using a
Modified Residual Income Valuation Model*
Julian Yeo
[e-mail]
Department of Accounting and Business Information Systems, The University of
Melbourne, Victoria 3010, Australia
Even though it has been more than a decade since the imputation tax system was
introduced in 1987 in Australia, the appropriate treatments for imputation tax credits for
security valuation purposes remain contentious. By using a novel approach that builds
upon the residual income valuation model, this study addresses the following two
questions: What is the market value of franked dividends?; What is the cost of equity in
the presence of imputation tax credits? These issues are of broad interests to any country
with some form of imputation tax system in place.
An estimation procedure is developed to simultaneously estimate the cost of
equity, the value of franked dividends, and growth rates in tax-adjusted residual income
for a portfolio of firms. Using a set of firms followed by I/B/E/S between 1993 and 1999,
the results show that the market value of a dollar of franked dividend is around $1.20
from 1993 to 1997 and slightly lower (around $1.15) for 1998 and 1999. The value of
franked dividends varies across industries, with a dollar of franked dividends worth no
more than a dollar of cash dividend in the finance sector for the year 1999 to $1.37 in the
building/engineering/manufacturing sector for the year 1996. The cost of equity
estimates (discount rates) applied to payoffs that exclude imputation tax credits range
from 7.37% to 9.95% over the sample period, while the cost of equity estimates applied to
payoffs that adjusted for imputation tax credits range between 7.95% and 10.93%.
Further analyses show that firms align their tax paying patterns with their tax clienteles,
portfolio of firms with higher average effective tax rates have a higher market value of
franked dividends compared to firms with lower average effective tax rates.
Preliminary draft (February 2003), please do not cite or quote without permission.
* This is based on my PhD dissertation. I am indebted to my supervisors - Peter Easton and Nasser Spear.
This paper has also benefited from numerous discussions with Greg Clinch, Kevin Davis, and David
Robinson, and Stephen Penman. I thank workshop participants at the 2002 Capital Markets Research in
Accounting Symposium at the University of Melbourne and the 2003 Summer Research School in
Accounting at UTS.
 Double taxation is bad for our economy. Double taxation is wrong.
 It s fair to tax a company s profit. It s not fair to double-tax by taxing the shareholder
on the same profits.
George W. Bush; January 7, 20031
1. Introduction
Most countries operate with a full or partial integrated tax system for corporate
tax and personal income tax.2 A fully integrated corporate and personal tax system, in a
form of an imputation tax system, was introduced in Australia in 1987. Under the
imputation tax system, the receipt of imputation tax credits (that are attached to cash
dividends) entitles investors to a tax rebate to the extent that corporate taxes have already
been paid on dividends. Dividends that are attached with franking credits are commonly
referred to as franked dividends.
Ever since the introduction of the imputation tax system in Australia, there is no
unanimous agreement amongst academics and practitioners on the appropriate treatments
of imputation tax credits on security pricing. What is the market value of a dollar of
franked dividends? What is the appropriate cost of equity estimate to be applied for
valuation purposes in the presence of imputation tax credits? Answers to the above
questions remain divergent and inconclusive. The aim of this study is to address the two
questions using a novel approach that builds upon the residual income valuation model.
The study also aims to provide a way to conceptualize the relationship between
imputation taxes, entity-level accounting variables, and market measures of firm values.
These issues are of broad interests to countries with some forms of imputation tax
systems in place and countries that are in the process of introducing an integrated
corporate and personal tax system (i.e., in the case of the US).3
1
See George Bush s growth and job plan to strengthen America s economy on January 7, 2003.
http://www.whitehouse.gov/news/releases/2003/01/print/20030107-5.html
2
Australia, Germany (up to October 2000), Italy, and New Zealand have full imputation tax
systems where all of the corporate tax can be offset against personal tax obligations. Many other countries
such as Canada, France, Spain, and the United Kingdom have partial imputation tax systems.
3
The centerpiece of Bush s economic plan in January 2003 is the elimination of the double
taxations of dividends which will cost $US364 billion over the next 10 years.
2
Imputation tax credits are created when companies pay taxes on profits. These
credits are usually transferred to investors when companies opt to pay franked dividends.4
Depending on the tax status of investors, imputation tax credits allow certain recipients to
offset their personal tax liability against the amount of tax that has been paid on
dividends by the company.5 Investors are unlikely to value imputation tax credits that
they cannot use. Since some recipients are unable to utilize credits that are distributed,
the extent to which imputation tax credits are impounded into the company s share price
(i.e., the value of imputation tax credits) depends on the amount of imputation tax credits
that will be eventually created, distributed, and utilized.6
The knowledge of the value of imputation tax credits has direct application to
firms capital budgeting exercises, optimal capital structures, and dividend distribution
policies. Under an imputation tax system, a portion of taxes paid at the corporate level is
often thought of as a pre-collection of personal tax (Officer, 1994). Effectively, the
amount of company tax will be lowered by the amount of imputation tax credits
distributed and utilized. The tax benefits of imputation tax credits need to be accounted
for in the calculation of payoffs in capital budgeting decisions.7 It has also been
suggested that the value of the tax shield on debt depends on the amount of imputation
tax credits that shareholders can utilize (Howard and Brown, 1992; Twite, 2001).8
4
Each dollar of fully franked dividends consists of two components: a cash dividend and an
imputation credit. The value of franked dividends is the joint value of cash dividends and imputation tax
credits.
5
The receipt of imputation tax credits affects the personal tax liability of investors that qualify as
Australian residents for tax purposes. The imputation tax system aims to eliminate the double taxation of
dividends that occurs both at the corporate tax level (dividends are paid out of after tax earnings) and
personal tax level for Australian residents.
6
These mark the three milestones in the life of imputation tax credits first described in Hathaway
and Officer (2001).
7
If the adjustments for imputation tax credits were to be made in the discount rate, the firm s
weighted average cost of capital (WACC) would depend on the value attached to imputation tax credits by
marginal investors (Officer, 1994).
8
Howard and Brown (1992) argue that for shareholders who can utilize imputation tax credits,
companies should not minimize company tax but rather maximize pre-tax profits, pay the applicable tax
and pass the imputation tax credits to shareholders. Twite (2001) examines the changes in capital structure
around the introduction of the dividend imputation tax system in Australia and finds evidence supporting
firm s incentives to reduce the level of debt financing and increase the level of external equity financing in
the presence of imputation tax credits. These incentives are found to vary across firms depending on the
firms effective corporate tax rate. The benefits of a tax shield must thus be weighed against the value of
imputation tax credits that shareholders can access. Also, see Kemsley and Nissim (2001) and Kemsley
and Williams (2001) for discussion on how dividend tax capitalization is related to the value of the debt-tax
shield on an after personal tax basis.
3
Lastly, the decision to retain or distribute dividends rests on whether investors are able to
utilize the imputation tax credits. It follows that knowing the value of imputation tax
credits assists in fine-tuning dividend distribution decisions.9
Even though it has been more than a decade since the imputation tax system was
introduced in July 1987 in Australia, there is still no consensus on the market value of a
dollar of franked dividends.10 Most empirical studies so far have relied on drop-off
events (i.e., the drop-off in share price when a share goes ex-dividend) across different
time periods and tax regimes to examine the value of franked dividends and imputation
tax credits (Brown and Clarke, 1993; Bruckner, Dews, and White, 1994; Partington and
Walker, 1999; Hathaway and Officer, 2001). 11 It is widely recognized that the estimates
of franked dividends (and imputation credits) via drop-off ratios exhibit considerable
variations. In addition, the ex-dividend price change may be confounded by other
unrelated events.12 For example, short-term arbitrage trading activities around the ex-
dividend dates may cause the prices used in inferring the value of franked dividends to
differ from the equilibrium price of all future franking credits incorporated in the market
prices of equity.13
Recent studies deviates from drop-off events by exploiting trading arrangements
and certain derivative securities that are unique to the Australian retail market (Cannavan,
Finn, and Gray, 2000; Chu and Partington, 2001; Twite and Wood, 2002) to infer the
9
A good example is Wesfarmers, Inc as cited in Harris, Hubbard and Kemsley (1999). In order to
decide whether franked dividends should be distributed, Wesfarmers surveyed its shareholders ability to
access the imputation tax credits. Wesfarmers found that the weighted-average shareholder tax rate was
less than half of the corporate tax rate and created a dividend reinvestment plan to pass on the benefits of
imputation tax credits.
10
It is common in the extant literature to focus on the value of a dollar of imputation tax credits.
The inference of the value of imputation tax credits is conditional on the value of cash dividends.
11
Also, see studies in other countries that focus on ex-dividend day. For example, Germany,
McDonald (2001); France, Alphonse (1999); UK, Lasfer (1995); Italy, Michaely and Murgia (1995).
12
Drop-off calculation from non-consecutive closing price data may be influenced by extraneous
information (Hathaway and Officer, 1994). Walker and Partington (1999) control for this by focusing on a
sample of firms that are permitted to trade on a cum-dividend basis during the ex-dividend period. For the
first time, researchers are able to show that the value of fully franked dividends has a value significantly
greater than $1.00. Nevertheless, their study still doesn t overcome the noise problems associated with the
drop off event windows.
13
The presence of microstructure effects may also confound the results of these studies. Frank and
Ragannatahan (1998) show that in the Hong Kong market, the bid-ask spread leads to an ex-dividend price
drop-off that is smaller than the dividend amount even though there is no tax on dividends or capital gains
tax. Bali and Hite (1998) in the US argue that dividends are continuous while prices are constrained to
discrete tick multiples, which explains the drop-off to be less than 1. Also, see Shaw (1991) for other non-
tax factors influencing the ex-dividend day price movements.
4
value of imputation tax credits. A shortcoming of these studies is their small sample size.
None of these studies has more than 30 Australian companies in their sample due to the
fact that the special trading arrangements and derivatives are only available to a selected
number of large Australian companies.
In the absence of a satisfactory answer to the market value of a dollar of franked
dividends, this study develops an estimation procedure that avoids the confounding
effects associated with drop-off studies and bypasses the sample restriction problem in
recent studies that utilize derivatives or special trade arrangements that are unique to the
Australian retail market. The estimation procedure developed in this study estimates the
value of franked dividends (and imputation tax credits) that are impounded in share
prices by marginal investors for a portfolio of firms.
Based on the premise that investor will only value future imputation tax credits
that eventually be distributed and utilized, the estimation procedure allows the inference
of the value of franked dividends based on expected streams of dividends and imputation
tax credits (i.e., the adjustments required to redefine dividends to include imputation tax
credits). This is to be contrasted with extant drop-off studies that infer the value of
imputation tax credits from a sample of firms for which there is a dividend announcement
with an identifiable date (i.e., known distribution of imputation tax credits). The results
of these studies only indicate the market value of distributed credits (Hathaway and
Officer, 2001). The estimate obtained from the proposed procedures is not conditional on
a known distribution of dividends, rather it allows the estimation of the market of franked
dividends based on the stream of expected cash dividends and expected imputation tax
credits that can be utilized by marginal shareholders.
Apart from the value of franked dividends, the appropriate discount rates to be
applied in the presence of imputation tax credits have been the subject of much
confusion. Two common approaches have been adopted to incorporate the effect of
imputation tax credits for security valuations (Lally, 2002). The first approach treats
imputation as a process of lowering company tax rate and redefining dividends to include
imputation tax credit (Hathaway and Dodd, 1993; and Officer ,1994).14 The alternative
14
The Australian Competition and Consumer Commission (ACCC) also favours the approach as
evident from their regulatory decisions.
5
approach is to treat imputation as a process of lowering personal tax on dividends (Ball
and Bowers, 1988; Australian Department of Treasury, 1991; Monkhouse, 1993).15 In
other words, in the presence of imputation tax credits, the value of imputation tax credits
may be added to the stream of expected cash dividends to make the dividend discount
model complete. The appropriate discount rate to be applied is one that is free from
imputation tax effects (hereinafter referred to as non-imputation adjusted cost of equity
estimates). The alternative is to adjust for the effect of imputation tax credits in the
discount rate (hereinafter referred to as the imputation-adjusted cost of equity estimates).
Both combinations will yield consistent results so long as the payoffs and discount rates
are defined consistently.
Researchers have attempted to modify the Capital Asset Pricing Model (CAPM)
to incorporate imputation tax credits to derive the estimates of required returns to equity
(Officer, 1987, 1988, 1994; Monkhouse, 1993; Wood, 1997). However,  one can t help
but be more confused, rather than enlightened (Hathaway, 1995, p.95) with the
divergent and inconsistent ways of incorporating the imputation tax credits systems into
the CAPM model put forth by various researchers (Brailsford and Davis, 1995a and
1995b; Easton and Howard, 1995; Hathaway, 1995). In addition, the appropriate  risk
premium to be applied for  imputation-adjusted CAPM is still another contentious and
much unresolved issue (Mehra and Prescott, 1985; Scott, 1992; Banartzi and Thaler,
1995; Davis, 2000; Faff, Hillier, and Wood, 2001).16
15
Also see work in the UK and New Zealand that adopts this approach (Ashton, 1989, 1991; Cliffe and
Marsden, 1992; Lally, 1992, 2000). This approach is more in line with the alternative form of integration
proposed in the 1992 Treasury report in the US, in which tax relief is provided by allowing shareholders to
exclude dividends from their tax assessable incomes.
16
Following Officer (1994), Wood (1997), Faff, Hillier, and Wood (2000), the pricing implications of
dividend imputation tax credits in a CAPM framework has been commonly modeled as:
E(ri ) + = rf + [E(rm ) + - rf ]
i i m
ri
where is the return on asset i;
rm
is the return on the market portfolio excluding imputation tax credits
is the imputation tax credits distributed and utilized on asset i;
i
rm is the imputation tax credits distributed and utilized from the market portfolio
The  imputation-adjusted CAPM is identical to Brennan s (1970) classical tax-based model (except for the
sign of taxation related parameters). Given that is not directly observable, much debates revolve
m
around the appropriate risk premium ( E(rm ) + ) to be applied in the presence of imputation tax credits.
m
6
Using an alternative to the CAPM model, the estimation procedures developed in
this study allows the estimation of the cost of equity implied by share prices, pro forma
book values, and forecasts of earnings. In addition, the estimation procedures allow the
separation of the cost of equity estimates from the imputation tax effect. In other words,
the estimation procedure allows the estimates of the adjustments for the stream of
expected cash dividends in the presence of imputation tax credits (i.e., value of
imputation tax credits) and the appropriate discount rate to be applied to payoffs that have
accounted for imputation tax credits (i.e., non-imputation adjusted cost of equity
estimates).
This study develops an estimation procedure that estimates the value of franked
dividends, cost of equities, and growth rates of imputation-tax-adjusted residual income
simultaneously. A system of two equations, utilizing forecasts of current year earnings
and one-year ahead earnings, is developed to estimate the value of franked dividends,
cost of equities, and growth rates that are implied by current share prices, book values,
and forecasts of earnings directly for a portfolio of firms. The essential elements of the
estimation procedure are described as follows. Two equivalent price expressions utilizing
different periods of earnings forecasts are developed using the modified residual income
valuation model. Via simple manipulation and rearrangement of the two equations, a
system of two equations with four parameters and four unknowns (cost of equity, the
value of franked dividends, and the two growth rates based on different period earnings)
can be estimated with unique solutions using the Seemingly Unrelated Regressions
(SUR) method.
The remainder of this study is structured as follows. Section 2 introduces the
modified residual income valuation model. The estimation procedure that simultaneously
estimates the cost of equity and the value of franked dividends is developed in Section 3.
Section 4 discusses data, sample profiles, and results. Section 5 concludes with a
summary, limitations of the study, and possible future research directions.
Faff et al (2000) find that using a simple CAPM, the relationship between beta and return is more
steeply sloped in Australia subsequent to the introduction of the dividend imputation tax system. However,
no such change occurs in the US market over the sample period. Therefore, they conclude that the findings
are tax-driven.
7
2. The modified residual income valuation model
As in Rubinstein (1976), the no arbitrage assumption is all that is necessary to
derive the dividend discount model. My derivation of the modified residual income
valuation model begins by incorporating dividend imputation tax credits in the payoffs of
a one-period dividend discount model. Applying the recursive substitution technique, the
modified residual income model is then derived by recognizing that dividends are equal
to earnings minus change in book values.
Under a classical tax system, a one-period dividend discount model can be
expressed on an after corporate tax but before personal tax basis as follows: 17
Pt+1 + dt+1
Pt =
(2.1)
(1+ r' )
where Pt is the market price per share at time t,
dt is the dividend payment per share at time t, and
r' is the expected rate of return in the absence of imputation tax credits.
In the presence of imputation tax credits, the value of imputation tax credits may
be added to the stream of expected cash dividends to make the dividend discount model
complete. The alternative is to adjust for the effect of imputation tax credit in the
discount rate. Both combinations will yield consistent results so long as the payoffs and
discount rates are defined consistently.
The separation of the effect of imputation tax credits from the cost of equity
estimate (i.e., adjusting the effect in the payoffs) allows the specification of the
relationship between imputation tax credits, accounting variables, and share prices.
Imputation tax credits are tax-related matters that do not affect the double-entry system.
By making the adjustment in the payoffs, imputation tax credits can now be interpreted
with reference to accounting variables (such as earnings and book values). The
specification also permits the examination of the impact of imputation tax credits on price
level and return level regressions commonly employed in the accounting literature. The
17
The inputs into the model are defined at corporate level but before personal tax (i.e., capital
gains tax is not added to the expected terminal price and personal tax is not added to dividend). Different
definitions of payoffs and discount rates can be applied to obtain equivalent price estimates, provided that
both payoffs and discount rates are defined consistently. The proof in Appendix A shows that when the
payoffs are expressed on an after corporate tax (but before personal tax) level, the appropriate discount rate
is the one that is grossed-up from after personal tax cost of equity.
8
estimation of the value of franked dividends impounded in share prices is also made
possible by adopting this approach.
It can also be argued that for practical valuation purpose, imputation tax credits
are typically incorporated in the payoffs (i.e., numerator) rather than in the discount rates
(i.e., denominator). As suggested in Hathaway (1995), it is more practical to take the
imputation tax effects in the numerator rather than trying to  squeeze imputation tax into
the cost of equity (p.21). Hathaway (1995) cites the example of a terminal project with
different expected level of taxes over its finite horizon. The adjustment of imputation tax
credits in the numerator is much easier as imputation tax credits can be added later in the
life of the project to reflect delays in paying taxes due to large capital expenditures in the
early life of the project. If the adjustments were to be made in the cost of capital, the
adjusted cost of capital must be an average effect (given that typically, a constant cost of
capital would be used over the finite horizon of the project). The same parallel can be
drawn when valuing companies.
When imputation tax credits are adjusted in the payoffs, each dollar of franked
dividend consists of two components: a cash dividend and an imputation credit. Each
dollar of franked dividends carries an imputation tax credit, which is equivalent to the
tc
amount of taxes that has been paid on the dividends that equals to .18 The one-
(1- tc)
period dividend discount model therefore becomes:19
18
Each dollar of franked dividends carries an imputation tax credit (which is the amount of taxes
tc /(1- tc )
that has been paid on the dividends) that equal to . Upon the receipt of franked dividends,
recipients must  gross-up the cash dividend amount to include the imputation tax credits in their assessable
1+ [tc /(1- tc)] =1/(1- tc)
income, . Recipients are then taxed on the  grossed up assessable income at
their marginal tax rate, but can use the imputation tax credits to offset their tax liability. In circumstances
where investors marginal tax rate is lower than the company tax rate, investors are able to utilize the
excess imputation tax credits to offset their other tax liability. On the other hand, recipients who are tax-
exempt, overseas investors, or investors with excess surplus imputation credits (prior to 1 July 2000) are
unable to utilize the imputation credits directly.
tc
Using a corporate tax rate ( ) of 30% as an example, for each $1 of pre-tax earnings, $0.30 is
paid as tax. If the after tax earnings of $0.70 (i.e., $1(1- tc )) is all paid out as dividends, it will carry an
tc
imputation tax credit of $0.30. The denominator (1- ) grosses the after-tax dividends up to their pre-tax
levels. The pre-tax amount is then multiplied by tc to calculate the amount of corporate tax paid on the
distributed dividends.
19
The model is based on the premise that investors who can utilize the franking credits are willing
to pay a higher share price, despite the firm s dividends and imputation tax credits payout policy. The
sequence of imputation tax credits and dividends is irrelevant in security valuation even though it is the
9
tc
Pt +1 + dt+1(1+ k )
1 - tc
Pt =
(2.2)
(1+ r)
where k is the proportion of dividends that are franked, and
is the value of imputation tax credits,
tc is the corporate tax rate
r
is the expected rate of return applied to payoffs that account for imputation
tax credits.
k
The inclusion of is because not all expected dividends carry imputation tax
k
credits. is the expected proportion of future dividends that are attached with imputation
tax credits. Imputation tax credits are created only when companies pay corporate taxes.
Given that it is possible for companies to report profits without paying taxes, k depends
on future imputation tax credits that will be created (i.e, taxes that will be paid on future
earnings).20 The expected level of imputation credits also depends on undistributed
imputation tax credits that companies have accumulated (in cases where insufficient tax
credits will be created, previous credits accumulated can be utilized if the company
wishes to maintain the same level of franking).21
Also, not all investors can redeem or utilize the imputation tax credits. is the
proportion of imputation tax credits that can be redeemed or utilized by investors.
Depending on the tax status of the investors and the existence of various schemes
present value of expected dividends and imputation credits determine a firm s value. When imputation tax
credits are capitalized, the share price absorbs the amount of franking credits that will be created and the
amount that marginal investors will be able to utilize. Consistent with Miller and Modigliani s (1961)
dividend irrelevance proposition, a change in expected franked dividend payout will simply displaces
expected share price a dollar for dollar. Given that it is the after-tax dividends and imputation tax credits
faced by the marginal investors that are capitalized into share prices, it is such that:
"Pt /"dt = 1
tc tc
"Pt / "dt (1+ k ) = (1+ k )
1 - tc 1 - tc
20
Section 254T of the Corporations Law specifies that dividends must be paid out of the profits
including retained profit of a company. Dividends cannot be paid out from capital as outlined in s201 of
the Corporations Law. The second Corporate Law Simplification Bill also introduced another restriction
prohibiting companies to pay dividends if do so would make the company insolvent.
21
Over infinite horizon, the distinction becomes irrelevant. In practice, a weighted average k over
the lifetime of the company is typically employed. k is influenced more by future taxes that will be created
as the expected life of the firm approaches infinity.
10
allowing recipients of imputation tax credits to utilize the credits22, the model suggests
that investors will only price the extent of imputation tax credits that are expected to be
attached to future dividends and the proportion of the imputation credits that can be
redeemed into the share price.
Recognizing how financial statements articulate the stocks and flows and using the
clean-surplus relation, 23
Bt = Bt -1 + NIt - dt
(2.3)
where Bt is the book value of equity per share at time t,
NIt is the (comprehensive) earnings per share for fiscal period t-1 to t, and
dt is the dividend payment per share at time t.24
at time t=0, the dividend term in equation (2.2) can be re-expressed using book values
and earnings:
(1- tc(1- k )) (NI1 - rB0)(1- tc(1- k )) P1(1- tc ) - B1(1- tc (1- k ))
P0 = B0 + + (2.4)
(1- tc ) (1+ r)(1- tc) (1+ r)(1- tc)
(See Appendix B)
P, P2, P3,...,etc
Recursively substituting for , the multi-period expression of the
1
residual income model adjusted for imputation tax credits can be expressed as:
"
(1- tc (1- k )) (NIt +i - rBt )(1- tc (1- k ))
Pt = Bt + (2.5)
"
(1- tc ) (1+ r)t +i (1- tc )
i=1
tc
(NIt+i - rBt )(1+ k )
tc " 1- tc
or Pt = Bt (1+ k ) + (2.6)
"
1- tc i=1 (1+ r)t +i
(See Appendix C)
22
In addition, the recipients may be selling the shares to those who would place value on the
imputation tax credits.
23
Following Ohlson (1995), it is assumed that:
"Bt / "dt = -1
,
"NIt /"dt = 0 .
24
dt
By definition, this relationship holds. The term is also flexible enough to incorporate
investments/disinvestments from shareholders other than dividends. Given that there are differential tax
treatments on issuance of dividends and transactions that involve changes in capital structure (e.g., shares
dt
buy backs), the empirical analysis restricts the definition of to only include dividends to infer the value
of franked dividends. See section 4 for further discussion on the inputs to the empirical analyses.
11
In equation (2.6), the model states that the price is equal to the tax-adjusted book
value of equity and the present value of tax-adjusted future residual incomes. The
(tc )
amount required to adjust the two inputs depends on the corporate tax rate , the
(k)
expected proportion of franking on future dividends , and the value of imputation tax
( )
credits .
The restatement of the dividend discount model in the formulation of the residual
income model allows the shifts of valuation focus from wealth distribution (i.e.,
dividends) to wealth in place (book values) and wealth creation (earnings). 25 By
incorporating the effect of imputation tax credits in the payoffs, the expected level of
franking attached to future dividends can be interpreted with reference to the
undistributed imputation tax credits the company has accumulated so far (i.e., a
proportion of current book values of equity) and future imputation tax credits that will be
created (i.e., a proportion of future earnings).
The model recognizes that taxes and book values are two separate matters. The
expected proportions of dividends (i.e., book values and earnings) that carry imputation
credits are accounted for by factor k. Nevertheless, the specification allows imputation
tax credits to be analysed with reference to book values and earnings. The point to be
stressed here is that imputation tax credits are not part of reported book values and
earnings, but the fact that they can be analyzed with reference to book values and
earnings is the appeal of the specification.26
The aim of the imputation tax system is to remove the double taxation of
dividends. Rather than paying tax on dividends at both corporate and individual levels,
certain recipients of the imputation tax credits are able to utilize the amount of corporate
taxes paid on the dividends as part of their personal tax rebates. 27 From the perspective
of the company, part of the corporate taxes paid is just a pre-collection of personal tax.
25
Dividends are discretionary corporate policy that is often not tied to a firm s ability to generate
future payoffs. Also, by focusing on book values and earnings, confounding valuation effects of dividend
signaling are avoided.
26
See section 2.3.2 for further discussions on this.
27
In other words, a large proportion of the tax that  masquerades as company tax is personal tax
collected at company level. Using data obtained from the Australian Taxation Office (ATO), Hathaway
and Officer (1996) show that 48% of the company tax paid is just a pre-payment of personal tax.
Australian companies are  effectively paying a company tax rate that is closer to 19% rather than the
statutory rate of 36% during their sample period.
12
From a valuation standpoint, the company is  effectively paying less tax when the
analysis is done on an after-corporate tax but before personal tax basis.28
Given that current book values (and earnings) and future earnings will eventually
be paid out as dividends, to the extent that future dividends will be attached with
tc
imputation tax credits that can be utilized (i.e., by the factor of (1+ k ) ), current
1- tc
book values and future earnings are worth more than their face values if investors
impound imputation tax credits in share prices.29
To further illustrate, two extreme cases are considered. In the model, the process
of grossing-up the book values and earnings is to reflect the idea that a proportion of
corporate tax paid is really a pre-collection of personal tax. Equation (2.7) is an extreme
case where investors are able to utilize all imputation tax credits and all future dividends
carry imputation tax credits (i.e., =1,k =1).
In this extreme case, all taxes paid at the corporate level are just a pre-payment of
personal taxes as investors do not pay the amount of corporate taxes paid on the
dividends again when calculating their personal tax liability. The  effective corporate
tax in this case will be nil. Therefore, in this extreme case where all imputation tax
1/(1- tc )
credits can be utilized, earnings is grossed up to a before tax level (i.e., ), with
the flow-through effect reflected in grossed-up book values (based on the clean surplus
relation). 30
28
The effective company tax referred to here is a theoretical concept that excludes the amount of
franking credits that investors are able to utilize. Under this theoretical construct, a proportion of company
tax is a pre-collection of personal tax, with the remainder defined as the  effective company tax . This is to
be distinguished from the standard business and tax law interpretation of the term  company tax .
29
One way of explaining why both book values and earnings are influenced by the same factor k
over infinite horizon is mirrored in Hanlon, Myers, and Shevlin (2001). The residual income model is a
forward-looking model in which current book value of equity and earnings are used to predict future
dividends (i.e., dividends are from after-tax earnings, therefore also future earnings). As pointed out in
Dechow, Hutton, and Sloan (1999),  given a stream of future dividends, book values and earnings could be
picked as random numbers so long as they are updated according to the clean-surplus relationship .
Both book values and earnings are simply restatements of dividends in the residual income model. If
imputation tax credits are capitalized into share prices, both book values and earnings (as proxies for future
imputation tax credits attached to dividends) will need to be grossed up to reflect the present value of
expected imputation tax credits that is impounded in price.
30
The model is also consistent with Ohlson s (1995) earnings displacement property, that is
"Et[NIt +1]/ "dit = r . If imputation credits are capitalized into share prices, the earnings component will
13
"
(NIt +i - rBt )
1
Pt = Bt + (2.7)
"
(1 - tc ) (1 + r)t +i (1- tc )
i =1
In a classical tax environment, dividends distributed are unfranked, investors are
not entitled to any tax rebates for dividends distributed (i.e., = 0,k = 0 ). Equation (2.8)
shows that the modified residual income model collapses to the one that we see under the
classical tax system. In the absence of imputation tax credits, no grossing up is required
as the price is equal to the book value of equity plus the present value of future residual
incomes.
"
(NIt+i - rBt )
Pt = Bt + (2.8)
"
(1+ r)t+i
i=1
3. The estimation procedure
Analogous to the estimation of the internal rate of return on a bond using market
values and coupon payments, the residual income model can be inverted to estimate the
cost of equity and growth rates in residual income implied by share prices for a portfolio
of firms (Easton, Taylor, Shroff, and Sougiannis, 2002).31 In the presence of imputation
tax credits, the difficulty lies in disentangling the effect of imputation tax credits from the
cost of equity estimates. To illustrate this, equation (2.6) may be re-written in terms of
g
current year residual earnings growing at perpetuity at annual growth rate .32
tc
(NI0 - rB-1)(1+ k )(1+ g1)
tc (1- tc )
P0 = B0(1+ k ) + (3.1)
(1- tc ) (r - g1)
Let x be the adjustment for imputation tax credits at time t=0. That is,
tc
x = (1+ k ) . Rearranging equation (3.1), deflating the equation by B-1 , the
(1 - tc )
equation becomes:
be grossed up by the tax adjustment factor, "Et[NIt+1(1+ k tc (1- tc))]/"dt (1+ k tc(1- tc ) = r .
Therefore, the earnings displacement property is preserved.
31
Also, see O Hanlon and Steele (2000) for a time-series application of the inverted residual
income model.
32
The same underidentification problem to disentangle the imputation tax effect from the cost of
equity estimate exists when equation (2.6) is written in terms of future period of residual earnings growing
at perpetuity.
14
P0 - B0
ëÅ‚ öÅ‚x
ìÅ‚ ÷Å‚
x
xr(1+ g1) NI0x (1+ g1)
íÅ‚ Å‚Å‚
= - +
B-1 (r - g1) B-1 (r - g1)
P0 - B0
ëÅ‚ öÅ‚
÷Å‚
x
NI0 ìÅ‚ (r - g1)
íÅ‚ Å‚Å‚
= r +
B-1 B-1 (1+ g1)
ëÅ‚
P0
ìÅ‚ - B0 öÅ‚
÷Å‚
x
NI0
íÅ‚ Å‚Å‚
= + (3.2)
B-1 0 1 B-1
(r - g1 )
where the constant term = r , = .
0 1
(1 + g1 )
The estimation of both r and g simultaneously is essential because the g corrects
for the r estimate, accounting for the fact that r is estimated using book values, current
earnings, and prices. No reasonable estimation of r can be inferred without knowing g.33
The g estimated here is the unique growth rate of the subsequent stream of residual
incomes estimated using current earnings. The estimate g will differ when multi-period
earnings or different period s earnings are used in the estimation procedure. The
important point here is that the estimated g is one that enables the right-hand-side of
equation (3.1) to be equal to prices.
Since the analysis so far has been at the individual firm level, equation (3.2) can
be written as the following for each firm j:
Pj0
ëÅ‚
ìÅ‚ - Bj0 öÅ‚
xj ÷Å‚
NI
j 0
íÅ‚ Å‚Å‚
= + (3.3)
Bj-1 j 0 j1 Bj-1
Equation (3.3) is a classic random coefficient model. The following regression
can be estimated:
Pj0
ëÅ‚ öÅ‚
ìÅ‚ - B
xj j0 ÷Å‚
NI
j 0
íÅ‚ Å‚Å‚
= + + (3.4)
1 j0
Bj-1 j0 j1 Bj-1
33
A slight change in g can cause significant changes in value estimates.
15
The error term in equation (3.4) arises because of the firm-specific random
1 jt
component of the coefficients and , where
j 0 j1
= - + ( - )(Pj / xj0 - Bj 0) / Bj-1 .34 Given that this error is heteroskedastic
1 j0 j0 0 j1 1 0
by construct, standard errors are corrected using White (1980). The estimates of the
coefficients , are non-stochastic and may be regarded as the mean of the firm-
0 1
r g
specific coefficients. The estimates and are estimates for the portfolio of J firms.35
Various other specifications can be achieved by deflating equation (3.1) by a
scalar factor. However, in order to estimate unbiased r and g, the specification needs to
isolate r or g as a constant term or a slope coefficient. Estimations utilising the ratio of
different slope coefficients and/or constant term will cause the estimate to possess a
density function with non-existent moments (Geary, 1930). Because the estimates of the
constant term and slope coefficients in this estimation procedure are imperative in
inferring the correct cost of equity and growth rate, any specification that potentially may
suffer from multicollinearity problem should also be avoided. This includes any
specification that has any combination of price, book value, and earnings as separate
independent variables.36
From equation (3.4), knowing the tax adjustment x is vital in the estimation of a
cost of equity r that is not confounded by the imputation tax effects. As outlined in the
timeline presented in Figure 1, prior to the preliminary announcement of end of the fiscal
year earnings, proforma current earnings (forecast of earnings for the period ending fiscal
year t=0) and forecast of one-year ahead earnings (for the period ending fiscal year
t=+1) are both available at the end of the fiscal period at time t=0.
34
Stated differently, equation 3.4 can be written as:
Pj0
ëÅ‚
ìÅ‚ - Bj0 öÅ‚
xj ÷Å‚
NI
j 0
íÅ‚ Å‚Å‚
= ( + ) + ( + )
Bj-1 0 1 j0 1 1 j1 Bj-1
35
See Judge, Hill, Griffiths, Lutkepohl, and Lee (1988) pp.436-438 for further discussion on the
random coefficients model.
36
For example, when earnings and book values are separate dependent variables in the regression,
the relative weighting of earnings and book value is a function of the persistence of abnormal earnings.
This may render the intended interpretation of r and g implausible. In equation (10), (P0 / x - B0)is used
as one independent variable to avoid multicollinearity problems that may bias against the constant and
slope coefficient estimates.
16
Forecast of earnings for the period
Forecast of earnings for the ending fiscal year t=+1
period ending fiscal year t=0
Fiscal year t=-1 Fiscal year t=0 Fiscal year t=+1
Preliminary earnings announcement
Figure 1: Forecasts of earnings available at end of fiscal year t=0
Equation (3.4) employs the current earnings (for the period ending t=0) to derive
the price expression. The same price equation can be expressed using forecast of
earnings at t=1.37 Equation (3.5) derives the same price expression using the future
stream of residual income based on residual income at time t=1. A different growth rate
is employed in equations (3.5) as residual earnings are growing from a different base
period (compared to equation (3.4)).
(NI1 - rB0 )x
P0 = B0x + (3.5)
(r - g2 )
B0
Deflating equation (3.5) , via simple rearrangement, it can be rewritten as:
NI1 P0(r - g2)
= g2 +
B0 B0x
NI1 P0
= g2 + (r - g2)
(3.6)
B0 B0x
A system of two equations (equations 3.9a and 3.9b) is developed with equations
(3.4) an d (3.5). A just identified system with four unknowns and four estimates is
derived to allow the estimation of the cost of equity, imputation tax adjustments, and the
two growth rates. The use of both forecasts of current period earnings and one-year
ahead earnings in obtaining the estimates of interest is intuitive appealing. Given that
37
Here, in both equations, forecasts of earnings are used rather than historical earnings. The use
of historical earnings is counter-intuitive as knowing one-year ahead earnings at time t=0 will likely to alter
investors perspective on the cost of equity demanded at time t=0.
17
both forecasts are available in the month of the fiscal year end, investors are unlikely to
estimate their required rates of returns using one piece of information without the other.
g1 g2
The system of two equations now has four unknowns ( r, x , , and ) and four
estimates ( , , , and ). The system is just identified with unique solutions.
0 1 0 1
Pj0
ëÅ‚
ìÅ‚ - Bj0 öÅ‚
÷Å‚
NI
ëÅ‚ - g1 ÷Å‚ íÅ‚ x Å‚Å‚
öÅ‚
r
j0
ìÅ‚
= r + + (3.9a)
1 j0
Bj-1 ìÅ‚ 1+ g1 ÷Å‚ Bj-1
íÅ‚ Å‚Å‚
NI Pj0
j1
= g2 + (r - g2 ) + (3.9b)
2 j0
Bj0 Bj0x
The two error terms in the system (3.9) allows for the firm-specific random
component of the four estimates.38 The aim of the exercise is to estimate r, x, and g for
the set of I/B/E/S firms. Apart from using the two equivalent price expressions with
different period residual incomes to obtain those estimates, Seemingly Unrelated
Regressions (SUR) make use of the omitted information captured in the two error terms
to obtain more efficient estimates of the four parameters with increased sampling
precision.
The impact of omitted information (other than those captured by the explanatory
variables) is likely to have similar effect impact on equations since both equations are
based on the same equivalent price expression expressed using different period residual
income (based on forecast information). Zellner s SUR or error related regression
equations accounts for the possibility that the errors for the two equations may be
contemporaneously (in the same time period) correlated in the error covariance matrix.39
38
In the absence of error terms in both equations, the relation between the two growth rates can be
ëÅ‚ - rB0 ÷Å‚ìÅ‚ r - g1 ÷Å‚
öÅ‚ëÅ‚ öÅ‚
NI1
expressed as: g2 = r - ìÅ‚
. However, the inclusion of both error terms in the two
ìÅ‚
NI0 - rB-1 ÷Å‚ìÅ‚ 1+ g1 ÷Å‚
íÅ‚ Å‚Å‚íÅ‚ Å‚Å‚
separate equations eliminates the abovementioned restriction, and allows for four independent unknowns.
In addition, given that the two growth rates appear in separate equations, the use of Seemingly Unrelated
Regression (SUR) method would have corrected for contemporaneous correlation in both error terms to
provide efficient estimates of the coefficients and c onstant terms in the system.
39
The SUR is a special form that corrects for heteroskedasticity and autocorrelation that appears
jointly.
18
The generalized least squares estimates of the four parameters, with error
covariance matrix that recognizes that the two error terms are contemporaneously
correlated, are the best unbiased estimators for the system of two equation (that is better
than least squares applied separately to each equation). Since the estimates are non-
stochastic and may be regarded as the mean of the firm-specific coefficients, it follows
that estimates r, x , g1, and g2 are the estimates for the portfolio of J firms.
The alternative to adjusting imputation tax credits in the payoffs is to adjust it in
the cost of equity estimates. Appendix D introduces the cost of equity (r*) that treats
imputation as a process of lowering personal tax on dividends. In other words, r* is the
appropriate discount rate when payoffs (e.g., forecast of current period earnings and one
year ahead earnings) are not adjusted for imputation tax credits (see Appendix E).
When the payoffs are not redefined to include imputation tax credits (i.e., x=1),
the cost of equity estimated from equation (3.4) becomes r*. If imputation tax credits are
indeed capitalised in share prices, the independent variable, (P0 / x0 - B0) / B-1 , should
posses a higher value. The failure to adjust for imputation tax credits in the payoffs in
equation (3.4) will therefore result in an estimate of cost of equity that is downwardly
0
biased. This is consistent with the notion that when x>1, it follows that r>r*.
4. Empirical analyses
The study includes firms followed by I/B/E/S that provided analysts forecasts
between 1 January 1993 and 31 December 1999. The following data are obtained from
the I/B/E/S: monthly consensus forecast for the month that coincides with the end of the
fiscal year period; earnings forecasts of current period fiscal year end and one-year ahead
period; and dividends forecast of current period at the month of the fiscal year end. The
mean and median forecasts from the month of the fiscal year-end and one-year-ahead
forecasts are obtained from the summary tapes.40
Given that firms with different months of fiscal year-end are all included in the
sample, the implied cost of equity and the value of franked dividends are not estimated at
40
The results reported in this study are based on the mean forecasts. Although not reported, the
results using median forecasts are quantitatively similar and do not alter the conclusions reached in this
study.
19
the same point in time for each firm-year observation. The estimates are estimated at
various times throughout the year and are categorized by their calendar year for reporting
purposes.41
Financial variables, including book values of equity, sales, operating profits
before and after tax, actual dividends per share are obtained from FinData. Firms with
negative book values are deleted from the sample. Share price and dividend data
including price at fiscal year-end, number of shares outstanding, and franking
information are obtained from the AGSM Share Price and Price Relatives dataset.
Forecast of earnings and
Proforma book values at t=0 is
dividends for the period
constructed using actual book values
Actual book values
ending t=0 available
at t=-1 plus forecast of earnings for
at t=-1
the period ending t=0 and minus
forecast of dividends for the period
ending t=0
Fiscal year t=-1 Fiscal year t=0 Fiscal year t=+1
Figure 2: Construction of pro forma book values (excluding changes in capital structure)
Current book values (prior to the release of preliminary earnings and dividend
results) are required as part of the estimation procedure. Rather than using actual book
values (from FinData), year-end book value is constructed via the clean surplus relation
using actual lagged book value and forecast of current period earnings and dividends (i.e.,
book value determined by taking actual lagged book value plus forecast of earnings
minus forecast of dividends).42 Like studies that utilize earnings forecasts to infer the
implied cost of equity (Easton et al, 2001; Gebhardt et al,2001; Claus and Thomas, 2001),
possible biases in the forecast data are not adjusted. Therefore, the estimates of cost of
equity and the value of franked dividends are those implied by share prices available at
41
For robustness, when the estimation is applied to firm year observation with 30 June year-end,
similar results are obtained.
42
Given that dividends forecasts are not available from I/B/E/S until late 1994, actual dividends
are used for period where forecasts of dividends are not available.
20
the end of the fiscal year end, actual lagged book values, earnings forecasts, and book
values constructed using actual lagged book values, earnings and dividends forecasts.43
Table 1 reports the descriptive statistics of key variables for the sample of 1,581
firm year observations from 1993 to 1999. 30 out of the initial 1611 firm year
observations with P/B greater than 10 are excluded.44 The exclusion avoids the
estimation procedure being driven by extreme large values.45 The median market
capitalization is $290.73 million. The median share price is $2.30 and the median current
(lagged) book value is $1.47 ($1.43), with a median P/B ratio of 1.46. The median
earnings forecast for current period and one year ahead period are $0.15 and $0.19
respectively. The median return on common equities (ROCE) measured using current
earnings and lagged book value is 11% and 12% if measured using one-year ahead
earnings forecast and current book values.
To provide a sense of variability of these key variables across years, Table 2
reports the median of these key variables by years. The median market capitalization has
decreased over the sample period due to greater coverage of smaller firms as well by
I/B/E/S in recent years. The median earnings forecasts of current period range from
$0.14 to $0.18 while the one-year ahead forecasts range from $0.17 to $0.22. The
median return on common equities do not exhibit much variations and take on only
possible 3 (2) values across years when ROCE is measured using current earnings (one-
year ahead-earnings). The median P/B ratio ranges from 1.32 to 1.71 over the sample
period.
The sample firms are partitioned into 11 categories based on their
Sector/Industry/Group code provided by I/B/E/S. The industry classification is loosely
43
Analysts forecasts may be a noisy measure of true value-generating ability. To avoid the
classic errors in variable problem (Maddala, 1977, page 292-293), actual book value is also used so that the
independent variable contains no error, while earnings forecasts is measured with errors in the dependent
variable. This does not alter the conclusions drawn in this study.
44
The 30 firm year observations excluded have a median P/B ratio of 15.05. The median price per
share for the excluded sample is $3.55 and the median book value is $0.24. 17 firm year observations are
from the consumer services industry.
45
For robustness, two other methods of removing outliers are also employed. Results are
quantitatively similar when observations with residuals that are two standard deviations from the means are
excluded, or the removal of top and bottom 5% observations of each variables also produces similar results.
In sum, the exclusion of P/B larger than 10 is a simple criteria and results are representative to those
estimated under the other two methods of removing outliers.
21
based on S&P industry groupings. Table 3 shows there are a total of 551 sample firms
included in the estimations. The three main groupings are: 24% from basic industries
(including steel, textiles, metals, chemicals), 21% of the sample firms are from the
finance industries, and 18% from consumer services (including communications, leisure,
foods and goods retailing).
Table 4 reports the medians of key variables. Firms in public utilities have larger
market capitalization ($1.49 billion), while firms in transportation have the lowest market
value of equity ($82.64 million). Firms in technology and health care have highest P/B
ratios (1.83 and 2.11 respectively), while firms in finance and transportation have lowest
P/B ratios (1.22 and 1.20 respectively). Firms in public utilities have higher median
earnings forecasts of $0.35 and $0.39 for current period and one-year ahead period
respectively.
4.1 Analyses by calendar years
The annual sample size changes from 129 firm observations in 1993 to 294 in
1998 and 278 in 1999. Given the changes in sample composition over time, inferences
on the time trend of estimates are difficult to draw since the estimates across years may
simply reflect changes in sample composition rather than underlying market wide
variables.
Table 5 reports the annual estimates of cost of equity (r*) that are applied to
payoffs that are not adjusted for imputation tax credits. In other words, the imputation
tax adjustments have been accounted for in the cost of equity estimates (in the form of
lower required rates of returns by investors who are able to utilize the imputation tax
credits). All annual estimates are positive and significant. The estimates of r* range
from as low as 7.37% in 1997 to as high as 9.95% in 1995. The r* estimates lies within
6.63% and 10.89% in the 95% confidence interval across years. The annual estimates of
g* are within 2.70% to 5.33%. The adjusted R2s range from 43 percent to 63 percent.
Table 6 reports the annual estimates of cost of equity (r) and the value of franked
dividends (x) and the two growth rates applied to current tax adjusted residual earnings
(g1) and one-year ahead tax adjusted residual earnings (g2). The estimate of r is not
confounded by the imputation tax effect, as the impact is accounted for in the redefined
22
payoffs that include imputation tax credits (x). All estimates of r and x are positive and
significant. In all sample years, except 1994, r* is lower than r, which implies a non-zero
x. The unconfounded r estimates range from 7.82% in 1997 to 10.93% in 1995. The
estimates lie within 7.04% to 11.86% within 95% confidence interval across the sample
period 1993 to 1999. The estimates of the two growth rates range from 1.88% to 4.38%.
The lowest adjusted R2 in the system of equations is 36% in 1994.
The estimation procedure shows that all estimates of x are above $1. That is, a
dollar of franked dividends is worth more than a dollar of cash dividends. There is a
steady increase in the market value of franked dividends from 1994 to 1997, with an
estimate that is around $1.20. Within 95% confidence interval, the estimates of x lie
within $1.10 to $1.32 between 1994 and 1997. The estimates are slightly lower in 1998
and 1999 (at around $1.15, with a 95% confidence interval of $1.06 to $1.22). The lower
estimates in 1998 and 1999 is consistent with Cannavan et al (2000) who report a 0.069
reduction in the value of imputation tax credits (although not significant) after the 45-day
rule. The lower estimate in the latter period implies that the 45-day trading rules may
have been effective in preventing the transfer of imputation tax credits.
In Table 7, the estimation procedure is applied to two portfolios of firms with
market capitalization above (larger firms) and below (smaller firms) the median. It is
shown that smaller firms have higher r estimates compared to larger firm across all
sample years. With the exception of 1994, the market value of franked dividends appears
to be slightly higher for smaller firms compared to larger firms. Rather crudely, this is
consistent with the notion that smaller firms are more likely to be owned by local
investors who are able to utilize the imputation tax credits than foreign investors.
4.2 Analyses by industry sectors
Further analyses are conducted with sample firms in three main industries in
Tables 8, 9, and 10. The three industries are: (i) consumer-orientated,
(ii)building/engineering/manufacturing, and (iii) finance industries. The three industries
are selected because there are the three industries with most I/B/E/S followings in the
Australian capital market. The industry results provide a sense of what the method may
23
yield if the procedure is applied to a group of similar firms with comparable operating
risk and accounting methods.
Given the changes in sample composition across years, it is difficult to compare
the relative ranking of the cost of equity across different industries. However, the results
show that firms in the finance sector tend to have higher cost of equity estimates
compared to firms in the consumer-oriented industries. Within the 95% confidence
interval, the cost of equity estimates range from 5.44% to 11.24% for firms in consumers-
oriented sector, 5.76% to 12.42% for the building/engineering/manufacturing sector, and
7.37% to 13.28% in the finance sector.
The value of imputation tax credits varies across companies in different
industries. The range of possible values of franked dividends within 95% confidence
interval is much narrower for the finance sector ($0.99 to $1.12 excluding 1993)
compared to the building/engineering/manufacturing sector ($1.15 to $1.32). The value
of franked dividend is no more than a dollar of cash dividends for the finance sector in
1999. This consistent with Cannavan et al (2000) who claim that the market value of
imputation credits is close to zero after the implementation of the 45 day trading rules for
companies with substantial foreign holdings. However, the value of franked dividends
depends on whether investors are able to utilize these credits. Therefore, it is not always
appropriate to assume a zero value for these credits. In industries such as
building/engineering/manufacturing, where the range of possible values of franked
dividends is more diverse, the results show a possible range of 1.13 to 1.51 (excluding
1993) prior to 1998 and 1.03 to 1.34 after 1998.
4.3 Analyses of tax clienteles and firms tax strategies
Bellamy (1994) and Nicol (1992) shows that there is a clientele effect associated
with company s imputation credits distribution policies. Investors who are able to utilize
imputation credits have preference over whether companies should pay franked
dividends. Shareholders who are able to utilize would prefer to invest in firms that pay
out more franked dividends while investors who are unable to utilize, prefer firms that
retain more of their profits. However, it is difficult to avoid franking credits when buying
shares, Hathaway and Officer (2001) show that 83% of dividends paid out are franked
24
dividends (with 92% fully franked and 8% are on average 50% franked). Therefore,
there are practical limits on how far this segmentation can go. Nevertheless, using ATO
data, Hathaway and Officer is able to show that there is a steady difference of 10% in the
proportion of franked dividend income to total dividend income between taxable and
non-taxable investors since 1987.
Table 11 reports the estimates of r and x, when the estimation procedure is
applied to a set of firms that typically pay franked dividends and unfranked dividends.
Firms with four consecutive dividends payments (interim and final) that are all franked or
unfranked are included in the sample.46 The values of franked dividends are not always
higher for firms that typically pay franked dividends in all years. In 1997 and 1998, firms
that pay unfranked dividends are found to have higher market value of franked dividends
priced in the shares.
Although there is a difference between the market value of franked dividends for
a set of firms that typically pay franked dividends compared to a set of firms that
typically pay unfranked dividends, the inconsistent pattern observed in 1997 and 1998
implies that the tax clientele effect may be difficult to be observed via the firms
imputation credit distribution policies, given that most dividends distributed are franked
dividends.47
For firms that have a history of issuing unfranked dividends, the results show that
value of dividend is valued more than its face values. The estimates are based on the
assumption that all after-tax earnings and imputation tax credits created in future taxes
will also eventually be paid out. Given that if the firm has a history of paying dividends,
and dividends comes from after-tax earnings. It is unlikely that the firm will not pay any
taxes that contribute towards the creation of imputation tax credits. In addition, shares
owned by foreign investors may be sold to resident individuals and institutions that price
them. Following Officer (1988) who concludes that a positive value of franking credits
will be capitalized into the price, it is not surprising to find the value of market value for
the set of firms with unfranked dividends should be greater than 1.
46
Although not reported, for robustness, firms with 2,3, 6, and 8 consecutive franked or unfranked
dividend payments are also examined. The results are not quantitatively dissimilar to those reported in
Table 12.
47
This in turn suggests that the distribution of franked dividends may have signaling effect
surrounding the announcement of the distribution of imputation tax credits.
25
In contrast to the classical system in which companies are motivated to minimize
tax and maximize their after-tax profits, companies operating under the imputation tax
system, tax minimization may be less advantageous for firms that have shareholders who
are able to utilize imputation tax credits (Hamson and Ziegler, 1990; Bellamy, 1994).
Given that not all investors are able to utilize these credits, firms are likely to organize
their tax affairs that best suit their clienteles.
Wilkinson, Cahan, and Jones (2001) show that the imputation tax system reduces
firm s incentives to minimize corporate taxes paid. When firms are partitioned using a
measure of taxes paid, firms that pay more (less) taxes should have investors who are
able to utilize imputation tax credits more (less) and therefore value those imputation tax
credits higher (lower) as reflected in x. As a measure of firm s tax paying behaviour, two
measures of  average effective tax rates , commonly used as a measure of tax burden
across firms (Callihan, 1994), are employed. The two average effective tax rates are
measured using the ratio of current tax expense over sales and the ratio of current tax
expense over net profit before tax.
Table 12 reports the estimates of r and x when firms are partitioned into those
with higher than median average effective tax rates (when sales is the denominator)48.
The market value of franked dividends are much higher for firms that have higher than
median average effective tax rates compared to firms with lower than median average
effective tax rates. The value of x ranges from 1.21 to 1.47 for higher tax paying firms
compared to the range of 1.00 to 1.18 for lower tax paying firms. The market value of
close to 1 for 1999 and 2000 for low tax paying firms again implies that firms with
shareholders that cannot utilize the imputation tax credits are unlikely to engage in tax-
maximising activities.
4.4 Further discussion on the value of imputation credits
tc
x = 1+ k
So far our discussion in restricted to x where .  can be
1- tc
interpreted as the value of a dollar of tax credit to the holder (Officer, 1994, p.4) The
value of is influenced by whether investors are able to utilize the credits, which
48
Although not reported, the results are robust when the alternative measure is used.
26
depends much on the composition of the shareholders (i.e., local shareholders that are
able to utilize the credits to offset their taxable income vs. foreign investors). With
 increasing evidence of the ease of tax arbitrage, particularly for overseas investors
Officer (1988, p.68), even though foreign investors are unable to directly utilize the tax
credits have been able to get access to the credits via various imputation tax capturing
schemes that are in place.
The value of has been a contentious issue with the ACCC favouring a value of
0.50 as a rule of thumb to be applied for valuation purposes. In inferring the value of ,
previous studies have relied on the correct inference of the value of cash dividends. If it
is believe that investors have a preference of capital gains over personal income tax for
dividends, the value of cash dividends is believed to be less than 1. Therefore, the
appropriate adjustments need to be made from the market value of a dollar of franked
dividends to calculate the market value of a dollar of imputation tax credit.
Provided that the appropriate cost of equity is estimated, the value of x can be
used to infer . Given that the analyses here are conducted using entity-level payoffs,
the preference for capital gains or personal income tax has already been accounted for in
the weighted average tax rate that grossed up the cost of equity estimate (see Proof 1). In
theory, $1 of cash dividend is worth no more or less than $1 (after considering taxes). k
is the proportion of future dividends that are attached with imputation tax credits. Over
infinite horizon, k is based on future taxes created that are likely to be attached with
dividends. Future dividends must come from after-tax earnings in which taxes would
have been paid, it is not unreasonable to assume a value of 1 for k in an infinite horizon
model as employed in the estimation procedure.
Given that tax changes occur at the beginning of the fiscal year starting from 1st of
July, Table 13 presents the annual estimates of r, x, and using portfolio of firms
grouped using 30th of June fiscal year end (rather than calendar years as previously
reported). The value of is based on the tax rates of 39% for 1 July 1993 to 30 June
1994 and 36% for dividends paid after 1 July 1994 in the sample period.49 With the
49
For 1994/1995, depending on the tax rate that has been applied to the profits from which
dividends are paid, the amount of franking must be classified as either class A (39%) or class B (33%). For
simplicity, a flat 36% is applied for that financial year to calculate the value of ?.
27
assumption that k=1, for a portfolio of firms followed by I/B/E/S, the 95% confidence
intervals of the estimated values of imputation tax credits range from 13.21% in
1998/1999 to as high as 55.94% in 1996/1997. The value of imputation tax credits
steadily increases over the years from 27.56% in 1993/1994 to 40.40% in 1996/1997. A
drop in value after the implementation of the 45-day trading rule is also observed for
1997/1998 and 1998/1999. The results show that the 0.50 rule of thumb favoured by the
ACCC is slightly higher than the estimated range of 24.07% to 40.40% for a set of
I/B/E/S observed across the sample years.
5. Conclusion and future research directions
One of the methodological innovations in this study is the development of a
modified residual income model that incorporates the imputation tax credits. The
development of the model rests on the premise that share prices impound the after tax
returns to investors. The elimination of double taxation of dividends has direct monetary
consequences on the after tax returns for investors who are able to utilize the imputation
tax credits. However, security analyses are typically conducted using entity level
accounting variables rather than proprietary after tax returns. This study demonstrates
how imputation tax credits can be incorporated for entity level analyses, not ignoring the
fact that it is proprietary after tax returns that determine share prices.50
In the modified residual income model, an emphasis is placed on separating the
effect of imputation tax credits from the cost of equity estimates.51 By doing so, the
market value of franked dividends can be estimated. In addition, empirical analyses on
the imputation tax adjusted cost of equity estimates and non-imputation-tax adjusted cost
of equity estimates can be compared and contrasted.
The estimation procedure is applied to a set of I/B/E/S followed firms for the
years 1993 to 1999. The annual estimates of cost of equity that are applied to payoffs
that are not adjusted for imputation tax credits (r*) range from as low as 7.37% in 1997 to
as high as 9.95% in 1995. The annual estimates of cost of equity (r) that are applied to
50
In this thesis, a view is put forth to explain the gap between analysis on payoffs defined on a
proprietary level and an entity level.
51
The alternative, which is the application of discount rates that incorporates the effect of
imputation tax credits with payoffs that exclude imputation credits, is also discussed in this thesis.
28
imputation adjusted payoffs range between 7.82% in 1997 and 10.93% in 1995. The
estimation procedure shows that a dollar of franked dividends is valued at above a dollar.
There is a steady increase in the market value of franked dividends from 1994 to 1997,
with an estimate that is around $1.20. Within 95% confidence interval, the estimates of
the value of franked dividends lie within $1.10 to $1.32 between 1994 and 1997. Lower
estimates of the value of franked dividends are observed at 1998 and 1999 ($1.15 and
$1.13). The market value of franked dividends also appears to be higher for firms with
smaller market capitalization compared to firms with larger market capitalization.
Several sub-samples of the data are used to further demonstrate the application of
the estimation procedure. The results show that firms in the finance sector tend to have
higher cost of equity estimates compared to firms in the consumer-oriented industries.
The value of imputation tax credits tend to vary across companies in different industries,
with the range of possible values within 95% confidence interval to be much narrower for
firms in the finance industry ($0.99 to $1.12 excluding 1993) to the range of possible
values that are wider in the building/engineering/manufacturing industries ($1.15 to
$1.32).
Some evidence is also provided to support firms that historically pay franked
dividends have a higher market values of franked dividends compared to firms that
historically pay unfranked dividends. The results also show that firms align their tax
strategies to cater for their tax clienteles. Using the average effective tax rates as a
measure of firm s tax paying behaviour, firms with higher average effective tax rates
have higher market value of franked dividends compared to firms with lower average
effective tax rates. Lastly, under the assumption that k=1 over infinite horizon, the value
of imputation tax credits, ?, is estimated to be in the range of 24.07% to 40.40% for a set
of I/B/E/S followed firms across the sample years.
The prevalence of the imputation tax system outsides of the US calls into question
whether the residual income model without adjusting for imputation tax credits is robust
in non-classical tax environments. This study also provides a new perspective that link
imputation tax credits, entity level accounting variables, and market measures together.
The one-sided impact of imputation tax credits on market variables without affecting
accounting variables suggests that imputation tax credits may confound the association
29
between the two. If imputation tax credits is a determinant of share prices, controlling for
the tax effect will improve the information content of key entity-level accounting
variables.
The application of the procedures in non-classical tax settings is limitless. This
study provides a way to link book values, earnings, imputation tax credits, and prices
together. If imputation tax credits is a determinant of share prices, controlling for the tax
effect will improve our ability to assess the information content of key entity-level
accounting variables (such as book values and earnings). Accounting for imputation tax
effect help bridges the gap between entity level accounting measures and firm values.
Australia operates a full imputation tax system since 1987 and provides an ideal setting in
examining how taxes (specifically, imputation tax credits) affect the relationship between
entity level accounting variables and firm values.
If share prices impound the expected after-tax returns on imputation tax credits,
while imputation tax credits do not impact on accounting variables. The apparent
questions are: How does imputation tax credits affect the price-levels and returns
regressions? Does tax matter? If so, how much? The modified residual income model
allows us to answer under what conditions do we expect imputation tax credits to affect
the association between market and accounting variables; to what extent does it affect the
association between market and accounting variables.
The estimation procedure also enables the estimates of cost of equity for firms
without trading history. In the case of Initial Public Offerings (IPOs), substantial claims
are made to justify the avoidance of the use of discount rates in valuing IPOs.52 The lack
of guidance by the extant literature on the appropriate discount rates for IPO firms further
 encourages its omission. Ignorance is not bliss, at least for valuation purposes when
dollars are at stake. Firms planning a once in a corporate lifetime event (i.e., IPOs) are
unlikely to derive their value estimates without doing a  full blown analysis that
calculates the present values of future payoffs. The estimation procedure can be applied
to this setting to provide guidance on the appropriate discount rates.
52
This is evident by claim such as  in many situation, it is difficult to estimate& an appropriate
discount rate (Kim and Ritter, 1999. p.412)
30
Payment of franked dividends is not the only means by which companies can
distribute their franking credits. The sale price (sum of paid up value of the share) in an
off-market buy-back can also be treated as franked dividends. The estimation model is
flexible enough to accommodate different distribution methods for imputation tax credits.
With sufficient year observation, the estimation procedure can also used to
estimate cost of equity on a time-series basis for individual firms. The estimation can be
compared with the traditional benchmark measure of risk under the current beta
technologies. Given that price-to-book and earnings (often a good proxy for size) are
used in estimating the cost of equity in the proposed estimation process, this may provide
the theoretical link to Fama and French s three-factor-model that incorporate the price-to-
book ratio and size.
31
Table 1: Descriptive statistics for key variables for all firm year observations
Variables #obs Mean Median Std. Dev. Skewness Kurtosis Min. Max
P 1581 3.45 2.30 4.25 4.29 29.96 0.06 48.00
B 1581 2.18 1.47 2.77 6.91 79.08 0.06 47.77
B-1
1581 2.11 1.43 2.71 7.05 82.50 0.05 47.75
NI 1581 0.22 0.15 0.25 3.17 19.31 (0.80) 2.37
1581 0.26 0.19 0.27 3.31 19.59 (0.27) 2.69
NI+1
NI/B-1 1581 0.12 0.11 0.10 2.74 19.09 (0.25) 1.11
(P-B)/B-1 1581 0.87 0.48 1.52 2.92 15.03 (0.96) 12.26
NI+1/B 1581 0.14 0.12 0.10 2.38 13.16 (0.24) 0.83
P/B 1581 1.81 1.46 1.36 2.45 10.91 0.05 9.66
MKT_CAP 1581 1,302.03 290.73 3,396.89 6.00 47.99 2.98 37,311.94
Pt Bt
is the price per share of at time t, is the per share book value of equity at time t,
NIt is the forecast of earnings per share at time t, MKT_CAP is the market capitalization
measured as the product of price per share and number of shares outstanding.
Table 2: Median of key variables by years
Year Obs P B B-1 NI NI+1 NI/B-1 (P-B)/B-1 NI+1/B P/B MKT_CAP
1993 129 2.75 2.02 1.98 0.18 0.20 0.10 0.66 0.12 1.67 426.42
1994 146 2.86 1.89 1.84 0.18 0.22 0.12 0.71 0.13 1.69 428.64
1995 229 1.92 1.36 1.27 0.16 0.19 0.12 0.33 0.13 1.32 205.95
1996 248 2.23 1.57 1.50 0.15 0.19 0.11 0.49 0.13 1.48 252.85
1997 257 2.54 1.47 1.40 0.15 0.19 0.10 0.74 0.12 1.71 361.95
1998 294 1.94 1.41 1.36 0.14 0.17 0.10 0.31 0.12 1.30 251.02
1999 278 1.97 1.37 1.30 0.14 0.17 0.11 0.43 0.12 1.41 259.05
Pt is the price per share of at time t, Bt is the per share book value of equity at time t, NIt is the
forecast of earnings per share at time t, MKT_CAP is the market capitalization measured as the
product of price per share and number of shares outstanding.
Table 3: Industry Composition based on number of sample firms
Industry Primary S/I/G Code Obs. % of obvs
FINANCE 10000-19999 114 0.21
HEALTH CARE 20000-29999 20 0.04
CONSUMER NON-DURABLES 30000-39999 47 0.09
CONSUMER SERVICES 40000-49999 100 0.18
CONSUMER DURABLES 50000-59999 14 0.03
ENERGY 60000-69999 34 0.06
TRANSPORTATION 70000-79999 10 0.02
TECHNOLOGY 80000-89999 16 0.03
BASIC INDUSTRIES 90000-99999 132 0.24
CAPITAL GOODS 100000-109999 55 0.10
PUBLIC UTILITIES 110000-119999 9 0.02
All Total 551 1.00
32
Table 4: Median of key variables by industry
Industry Obs P B B-1 NI NI+1 NI/B-1 (P-B)/B-1 NI+1/B P/B MKT_CAP
FINANCE 326 2.26 1.98 1.93 0.19 0.19 0.10 0.22 0.11 1.22 393.88
HEALTH CARE 42 2.18 1.21 1.26 0.14 0.15 0.11 1.16 0.12 2.11 302.73
CONSUMER NON-DURABLES 137 2.37 1.52 1.46 0.15 0.17 0.11 0.55 0.12 1.52 193.51
CONSUMER SERVICES 290 2.35 1.28 1.21 0.16 0.19 0.12 0.78 0.14 1.74 206.13
CONSUMER DURABLES 48 2.40 1.15 1.10 0.18 0.20 0.13 0.50 0.15 1.47 116.10
ENERGY 90 2.58 1.68 1.67 0.14 0.20 0.10 0.52 0.11 1.50 594.67
TRANSPORTATION 27 2.20 1.82 1.70 0.22 0.24 0.13 0.22 0.14 1.20 82.64
TECHNOLOGY 16 1.46 0.75 0.71 0.07 0.10 0.11 0.89 0.14 1.83 193.50
BASIC INDUSTRIES 429 2.10 1.37 1.30 0.12 0.16 0.10 0.54 0.12 1.53 317.10
CAPITAL GOODS 165 2.40 1.51 1.43 0.17 0.21 0.10 0.45 0.12 1.43 190.55
PUBLIC UTILITIES 11 4.64 3.03 2.92 0.35 0.39 0.12 0.61 0.13 1.59 1,487.44
Pt is the price per share of at time t, Bt is the per share book value of equity at time t, NIt is the forecast of earnings per share at
time t, MKT_CAP is the market capitalization measured as the product of price per share and number of shares outstanding.
33
Table 5: Annual estimates of the imputation-adjusted cost of equity (r*) and growth in residual income (g*)
Year n R* G* r* g* R2 Estimates of r* within
95% Confidence Intervals
(CI)
Lower Upper
bound bound
1993 129 1.1587 1.0624 0.0764 0.0307 0.6300 6.63% 8.64%
(106.8556) (69.1885)
1994 146 1.1944 1.1094 0.0929 0.0533 0.4262 8.15% 10.42%
(96.4146) (60.2919)
1995 229 1.2089 1.08156 0.0995 0.0400 0.4726 9.00% 10.89%
(115.9128) (65.5623)
1996 248 1.1783 1.0729 0.0855 0.0358 0.5413 7.82% 9.27%
(149.6958) (87.7135)
1997 257 1.1528 1.0548 0.0737 0.0270 0.5053 6.64% 8.10%
(114.4615) (72.4104)
1998 294 1.1871 1.0769 0.0896 0.0377 0.5002 8.14% 9.77%
(133.8921) (81.8987)
1999 278 1.1766 1.0817 0.0847 0.0400 0.5437 7.45% 9.48%
(106.9440) (75.8215)
r* is the estimated imputation-tax adjusted cost of equity and g* is the growth rate in tax-adjusted residual income derived from
X Pj0
jCT
= + + , with = G * -1 and = R *-G *. Also, R* = (1+ r*)2 andG* = (1+ g*)2 , where X is the aggregate
0 1 0 1 CT
B Bj0 j0
j0
two period cum-dividend earnings, Pj0 is the price per share of firm j at time 0, Bj 0 is the per share book value of equity of firm j at
time 0, and Bj-1is the per share book value of equity of firm j at time  1. T-statistics are presented in parentheses based on White
(1980) corrected standard errors. n is the number of observations. R2 is the adjusted R2.
34
Table 6: Annual estimates of the cost of equity (r), the value of franked dividends (x) and growth rates in tax-adjusted residual
income (g1 and g2)
2 2
Year n r Estimates of r within x Estimates of x g1 g2
Ra Rb
95% CI within 95% CI
Lower Upper Lower Upper
1993 129 0.0795 7.16% 8.77% 1.1888 1.0885 1.2891 0.0341 0.0302 0.5153 0.6874
(20.0410) (23.7038) (5.2764) (4.5459)
1994 146 0.0891 7.86% 9.95% 1.1665 1.1012 1.2318 0.0327 0.0426 0.3572 0.3985
(17.0366) (35.7351) (3.8168) (4.9531)
1995 229 0.1093 10.00% 11.86% 1.1971 1.1312 1.2630 0.03438 0.0465 0.4380 0.4497
(23.5414) (36.3408) (4.1382) (4.9870)
1996 248 0.0897 8.32% 9.61% 1.2149 1.1546 1.2751 0.02549 0.0431 0.5199 0.5368
(27.7645) (40.3138) (4.1072) (0.0431)
1997 257 0.07827 7.04% 8.61% 1.2337 1.1909 1.3192 0.0188 0.0317 0.4821 0.5441
(19.9041) (28.8434) (2.5315) (4.5430)
1998 294 0.0946 8.65% 10.26% 1.1571 1.0927 1.2215 0.0314 0.0423 0.4679 0.4750
(23.4179) (35.9251) (4.6495) (6.0442)
1999 278 0.0895 7.96% 9.94% 1.1349 1.0689 1.2001 0.0329 0.0438 0.5845 0.5540
(18.1152) (34.4167) (4.7334) (6.2816)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g (g ) is the growth
1 2
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0 j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using Seemingly Unrelated
2 2
Ra Rb
Regression (SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and is the
adjusted R2 from the second system equation.
35
Table 7: Annual estimates of cost of equity (r) and imputation tax adjustments (x) for firms above and below the median of
market capitalization
Firms above the median of market capitalization Firms below the median of market capitalization
2 2 2 2
Year n r x n r x
Ra Rb Ra Rb
1993 74 0.0733 1.0923 0.3717 0.7177 55 0.0862 1.2816 0.5453 0.6407
(15.3926) (15.1780) (12.2919) (16.7705)
1994 84 0.0819 1.1712 0.3027 0.4696 62 0.0986 1.1646 0.4965 0.4399
(12.9810) (21.4228) (12.7620) (31.2098)
1995 120 0.0933 1.0750 0.4940 0.6546 108 0.1294 1.2628 0.5187 0.4446
(16.0693) (23.2446) (19.7873) (30.1561)
1996 128 0.0777 1.1916 0.5908 0.5928 120 0.1012 1.2488 0.5010 0.6218
(18.9868) (39.4819) (23.8866) (25.1887)
1997 133 0.0708 1.1991 0.4299 0.5540 123 0.0902 1.2415 0.5248 0.5850
(16.3121) (20.6168) (14.1110) (20.4115)
1998 145 0.0875 1.1372 0.2935 0.2915 148 0.1099 1.1787 0.5841 0.6483
(18.4932) (44.2986) (19.9210) (23.2957)
1999 138 0.0710 1.1140 0.5403 0.5230 140 0.1032 1.1520 0.7195 0.7232
(8.4180) (18.3842) (21.9136) (29.5382)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0
j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, with heteroskedasticity and contemporaneously correlated error terms corrected using Seemingly Unrelated Regression
2 2
Ra Rb
(SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and is the adjusted R2
from the second system equation.
36
Table 8: Annual estimates of the cost of equity (r), the value of franked dividends (x) and growth rates in tax-adjusted residual
income (g1 and g2) for firms in the consumers-oriented sector
2 2
Year n r Estimates of r within x Estimates of x within g1 g2
Ra Rb
95% CI 95% CI
Lower Upper Lower Upper
1993 33 0.0899 7.66% 10.31% 1.2025 1.1455 1.2596 0.0322 0.0463 0.5266 0.4917
(13.5965) (42.1514) (2.4782) (3.0011)
1994 44 0.0899 8.06% 9.93% 1.1657 1.0953 1.3067 0.0168 0.0282 0.7955 0.8152
(19.2539) (33.0702) (1.8662) (3.0877)
1995 71 0.1071 8.91% 12.51% 1.1724 1.1054 1.2395 0.0006 0.0090 0.6302 0.5946
(11.8951) (34.9669) (0.0457) (0.5314)
1996 78 0.0907 7.97% 10.17% 1.1919 1.1314 1.2523 0.0049 0.0253 0.7226 0.6458
(16.5055) (39.4302) (0.5114) (2.3203)
1997 74 0.0655 5.44% 7.67% 1.1486 0.9157 1.3815 -0.0059 -0.0049 0.7413 0.8293
(11.7505) (9.8644) (-0.5293) (-0.5184)
1998 88 0.0858 7.46% 9.71% 1.1118 1.0259 1.1978 0.01821 0.0173 0.8291 0.8013
(15.2410) (25.8870) (2.4501) (1.8941)
1999 86 0.09801 9.09% 11.24% 1.1489 1.0978 1.2000 0.0460 0.0576 0.6894 0.6624
(13.7038) (44.9588) (4.8281) (5.4350)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g (g ) is the growth
1 2
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0 j1
= r + + , = g2 + (r - g2) + where is the forecast of earnings per share for firm j at
NI
2 j0
jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, with heteroskedasticity and contemporaneously correlated error terms corrected using Seemingly Unrelated Regression
2 2
Ra Rb
(SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and is the adjusted R2
from the second system equation.
37
Table 9: Annual estimates of the cost of equity (r), the value of franked dividends (x) and growth rates in tax-adjusted residual
income (g1 and g2) for firms in the building/engineering/manufacturing sector
2 2
Year n r Estimates of r within x Estimates of x g1 g2
Ra Rb
95% CI within 95% CI
Lower Upper Lower Upper
1993 59 0.0689 5.61% 8.17% 1.2722 1.0411 1.5033 0.0176 0.0120 0.5889 0.7496
(10.7581) (11.0104) (1.6451) (1.0840)
1994 61 0.0748 5.76% 9.20% 1.2914 1.1284 1.4543 0.0133 0.0320 0.3024 0.3501
(8.7062) (15.8459) (0.8905) (2.1967)
1995 91 0.1103 9.65% 12.42% 1.3180 1.1732 1.4628 0.0477 0.0734 0.3515 0.3766
(15.8962) (18.2096) (3.6712) (5.0490)
1996 96 0.0829 7.24% 9.34% 1.3214 1.1798 1.4630 0.0198 0.0404 0.4922 0.5585
(15.8515) (18.6646) (1.8546) (3.8819)
1997 98 0.0817 6.76% 9.59% 1.3712 1.2310 1.5113 0.0171 0.0429 0.2521 0.2824
(11.5670) (19.5622) (1.0429) (2.6861)
1998 102 0.1083 9.14% 12.51% 1.1466 1.0295 1.2638 0.0383 0.0534 0.2114 0.2304
(12.8273) (19.5742) (2.3527) (3.3414)
1999 89 0.0913 7.48% 10.79% 1.1936 1.0455 1.3416 0.0319 0.0449 0.4269 0.3592
(11.0432) (16.1235) (2.4700) (3.3209)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0
j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using Seemingly Unrelated
2 2
Regression (SUR) method. n is the number of observations. Ra is the adjusted R2 from the first system equation and Rb is the
adjusted R2 from the second system equation.
38
Table 10: Annual estimates of the cost of equity (r), the value of franked dividends (x) and growth rates in tax-adjusted
residual income (g1 and g2) for firms in the finance sector
2 2
Year n r Estimates of r within x Estimates of x g1 g2
Ra Rb
95% CI within 95% CI
Lower Upper Lower Upper
1993 23 0.0845 7.77% 9.13% 1.2146 1.1029 1.3263 -0.0154 -0.0014 0.6552 0.8028
(24.7801) (21.7432) (-1.0641) (-0.1254)
1994 27 0.1088 8.49% 13.28% 1.1030 1.0412 1.1649 -0.0103 -0.0167 0.4695 0.3996
(9.1072) (35.6849) (-0.4456) (-0.5403)
1995 41 0.1040 9.41% 11.38% 1.0781 1.0154 1.1408 0.0024 0.0015 0.6696 0.7090
(21.0815) (34.3739) (0.1828)
1996 51 0.0950 8.47% 10.53% 1.0888 1.0114 1.1662 0.0316 0.0384 0.4664 0.5462
(18.4607) (28.1324) (2.9064) (3.7704)
1997 54 0.0918 8.35% 10.00% 1.0434 1.0080 1.0788 0.0472 0.0508 0.6186 0.6672
(22.1301) (58.9687) (6.1475) (7.0112)
1998 64 0.0913 8.27% 9.99% 1.1152 1.0336 1.1968 0.0540 0.0628 0.4151 0.5259
(21.1873) (27.3350) (6.7995) (9.0056)
1999 69 0.0925 7.37% 11.13% 0.9892 0.8905 1.0880 0.0289 0.0283 0.7740 0.7823
(9.8388) (20.0326) (2.3685) (2.5928)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0
j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using Seemingly Unrelated
2 2
Regression (SUR) method. n is the number of observations. Ra is the adjusted R2 from the first system equation and Rb is the
adjusted R2 from the second system equation.
39
Table 11: Annual estimates of cost of equity (r) and imputation tax adjustments (x) for firms with four consecutive
interim/annual dividend payments that are all fully franked or fully unfranked
Firms with fully franked dividends Firms with unfranked dividends
2 2 2 2
Year n r x n r x
Ra Rb Ra Rb
1993 52 0.0862 1.2661 0.5015 0.6714 25 0.0717 1.1370 0.7622 0.7127
(17.8029) (27.9273) (6.2572) (8.2822)
1994 64 0.0940 1.1727 0.3769 0.4611 27 0.07427 1.1377 0.4599 0.6036
(12.3486) (23.6001) (6.7661) (11.3663)
1995 88 0.1236 1.1148 0.6019 0.5559 39 0.0983 1.0651 0.1500 0.4661
(21.1741) (53.9358) (9.8462) (14.6158)
1996 101 0.0972 1.2034 0.6826 0.6318 45 0.0887 1.0931 0.5403 0.5376
(25.8947) (44.5124) (12.2412) (29.2824)
1997 96 0.0819 1.1677 0.6504 0.7651 55 0.0791 1.2760 0.3212 0.4600
(15.5951) (14.8969) (15.1178) (15.2646)
1998 114 0.0941 1.1197 0.6409 0.6556 65 0.0908 1.1650 0.3793 0.3821
(15.9642) (28.9409) (13.6054) (12.3357)
1999 110 0.0874 1.1282 0.7661 0.7667 65 0.0958 1.0930 0.3232 0.2691
(11.8951) (22.4113) (9.8209) (14.5419)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0 j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using Seemingly Unrelated
2 2
Ra Rb
Regression (SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and is the
adjusted R2 from the second system equation.
40
Table 12: Annual estimates of cost of equity (r) and imputation tax adjustments (x) for firms above and below median
 average effective tax rates
Firms with above  average effective tax rates Firms with below  average effective tax rates
2 2 2 2
Year n r x n r x
Ra Rb Ra Rb
1993 42 0.0676 1.4135 0.7186 0.8278 44 0.0957 1.0995 0.4480 0.5364
(19.5866) (22.6858) (14.1169) (29.8127)
1994 45 0.0664 1.3353 0.3811 0.6397 46 0.0916 1.0808 0.2133 0.1729
(11.9794) (14.5250) (7.1676) (23.2040)
1995 52 0.0958 1.2452 0.4730 0.3818 51 0.1230 1.0908 0.3884 0.4074
(16.6149) (29.2976) (12.4058) (32.2615)
1996 54 0.0801 1.2697 0.4369 0.4714 55 0.0924 1.1777 0.6825 0.6868
(18.3862) (28.0034) (14.2614) (34.9213)
1997 55 0.0718 1.4745 0.3339 0.3829 55 0.0745 1.2384 0.7488 0.7445
(16.8133) (15.7202) (10.7778) (27.2560)
1998 56 0.0853 1.2769 0.3676 0.3779 58 0.1069 1.0442 0.1928 0.2720
(12.7875) (11.0847) (8.5084) (21.1493)
1999 51 0.0818 1.2092 0.5730 0.3705 51 0.1160 1.0046 0.2519 0.4170
(15.7150) (15.8742) (11.5267) (19.5989)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth
rate in tax-adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These
estimates are derived from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0 j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
time t, Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. T-statistics are presented
in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using Seemingly Unrelated
2 2
Ra Rb
Regression (SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and is the
adjusted R2 from the second system equation.
41
Table 13: Annual estimates (by fiscal year) of the cost of equity (r), the value of franked dividends (x) and growth rates in tax-
adjusted residual income (g1 and g2) and the value of imputation tax credits (?)
2 2
Fiscal n r x g1 g2 ? Estimates of ? within
Ra Rb
Year 95% CI
Lower Upper
1993/1994 146 0.0872 1.1762 0.0410 0.0514 27.56% 16.33% 38.79% 0.3177 0.4186
(17.8208) (32.7622) (5.0577) (6.6536)
1994/1995 219 0.1102 1.1968 0.0358 0.0472 34.98% 23.03% 46.94% 0.4458 0.4533
(22.3207) (35.5952) (4.1984) (4.8896)
1995/1996 244 0.0931 1.2064 0.0274 0.0450 36.69% 26.41% 46.97% 0.5375 0.5539
(28.3766) (41.7269) (4.4114) (7.2334)
1996/1997 252 0.0776 1.2273 0.0196 0.0316 40.40% 24.86% 55.94% 0.4541 0.5198
(18.9802) (28.0723) (2.5510) (4.3575)
1997/1998 283 0.0940 1.1542 0.0352 0.0473 27.41% 17.15% 37.68% 0.4591 0.4809
(22.5134) (39.9951) (5.0531) (6.9866)
1998/1999 308 0.0920 1.1354 0.0440 0.0543 24.07% 13.21% 34.93% 0.4691 0.4324
(21.7291) (37.1602) (7.2176) (8.3777)
r is the estimated non-imputation-tax confounded cost of equity, x is the estimated value of franked dividends, g1 (g2 ) is the growth rate in tax-
adjusted residual income when current period (one-year ahead period) tax-adjusted residual income in employed. These estimates are derived
from the system of equations:
NI Pj0 / x - Bj0
NI Pj0
(r - g1)
j0 j1
= r + + , = g2 + (r - g2) + where NI is the forecast of earnings per share for firm j at time t,
2 j0 jt
Bj-1 (1+ g1) Bj-1 1 j0 Bj0 Bj0x
Pjt is the price per share of firm j at time t, Bjt is the per share book value of equity of firm j at time t. The value of imputation tax credits (?) is
tc
x =1+ k
calculated based on where k is the proportion of imputation tax credits attached to the cash dividends, and tc is the corporate tax
1- tc
rate. T-statistics are presented in parentheses, where heteroskedasticity and contemporaneously correlated error terms are corrected using
2 2
Ra Rb
Seemingly Unrelated Regression (SUR) method. n is the number of observations. is the adjusted R2 from the first system equation and
is the adjusted R2 from the second system equation.
42
Appendix A:
Under neo-classical valuation paradigm, the share price is determined by the
present value of expected payoffs for owning the share. Different definitions of payoffs
and discount rates can be applied to obtain equivalent price estimates, provided that both
payoffs and discount rates are defined consistently. If the payoff is defined on an after
personal tax basis that accounts for capital gains tax and personal income tax on
dividends, the discount rate should also be an after personal tax cost of equity. To avoid
the identification of the marginal investors and their tax rates, a firm s valuation is
typically done on an after corporate tax but before personal tax level, (i.e. capital gains
tax and personal income tax on dividends are not added as part of the numerators). The
remainder of this appendix shows that the discount rate applied on payoffs defined on an
after corporate tax but before personal tax cost of equity level must be one that is
grossed up from the after personal tax cost of equity.
To illustrate this, consider a general price expression that describes that after tax
return that investors receive in their hands. Under no-arbitrage assumption, share price at
time t=0 can be expressed on the after personal tax basis in a one-period context
(ignoring transaction costs and tax rate heterogeneity). A1.1 incorporates capital gains
taxes and personal income taxes on dividends in the numerator:
tc
d1 (1+ k )(1- t ) + "P1 + "(P1 - P0)(1- t )
g
1 - tc p
P0 =
(A1.1)
1 + rapt
tc
where is the corporate tax rate,
tp is the personal income tax rate,
tg is the capital gains tax rate,
"
is the probability of investors selling their share at time t=153,
rapt is the after-personal tax cost of equity.
53
This is assumed to be exogenous and not contingent on Pt +1 .
43
Here, the share price at time t=0 is determined based on the expected share price
at time t=1 weighted be the probability of not selling54, the after-tax amount of cash
dividends and imputation tax credits expected at time t=1 and the capital gain tax that
marginal investors are liable for if they decide to sell their shares at time t=1. The term
" is introduced as marginal investors are able to defer the capital gains tax by not selling
the shares.55
Here, imputation tax credits have values to investors who can access and redeem
the credits. Using a simple example, investors at the margin have a personal income tax
rate that is the same as the corporate tax rate (i.e., tc = tp ), and investors at the margin are
able to redeem the imputation tax credits fully. For each dollar of unfranked dividends
that is distributed, the marginal investors will only end up with $0.70 after tax if
tc = tp = 30% ; whereas, for each dollar of franked dividends that is distributed, they will
have an after-tax dividends of $1 in their hands. In this case, all of the corporate tax paid
on the dollar of franked dividends is just a pre-collection of personal tax.
*
Let P1 (1- t ) = "P1 + "(P1 - P0 )(1 - tg ) . t* is the effective capital gains tax rate,
g g
which is defined as the proportion of price that is subject to capital gains tax (weighted by
the probability of investors selling the shares at time t=1). The equation (A1.1) can be
reduced to:
54
Over infinite horizon, Pt +1 is determined by the present value of future after-tax dividends (and
imputation tax credits) that marginal investors will receive by owning the share. That is:
tc
dt +1+i (1+ k )(1- tp)
"
1- tc
Pt +1 =
.
"
(1+ rapt)t +1+i
i=1
55
Investors at the margin are the ones setting the price. To illustrate this, let s assume a case with
no capital gains tax. When investors are taxed at differential rates, arbitrage activities will drive an
equilibrium price to a level at which marginal investors are indifferent between buying shares at the ex-
dividend price or the cum-dividend price. For each dollar of cash dividends distributed, an investor who is
taxed at 30% will receive after-tax cash dividends of $0.70. The share price should decline by $0.70 after
the issue of this dividend if all investors are taxed at the same tax rates. However, investors who are tax-
exempt will receive a full dollar (after tax) upon the receipt of the cash dividend. If each dollar of cash
dividends is only capitalized at $0.70, tax-exempt investors are able to buy the shares during cum-dividend
period and sell it after the price drop and make a profit of $0.30. The tax-exempt shareholders are expected
to engage in arbitrage activities to raise the share price to a level to which arbitrage opportunity is no longer
possible (after transaction costs). At this equilibrium price, the investors who are indifferent between
buying shares at the ex-dividend price or the cum-dividend price are referred to as  marginal investors and
the tax rates that they face are the  marginal investors tax rates .
44
tc
*
P1(1- tg ) + d1(1+ k )(1- tp)
1- tc
P0 = (A1.2)
1+ rapt
Next, lettapt be the weighted average of the effective capital gains and personal
income tax rates:
tc
d1(1+ k )
P1 1- tc
tapt = t* + t
tc g tc p
P + d1(1+ k ) P1 + d1(1+ k )
1
1- tc 1- tc
Equation (A1.2) can be re-expressed as:
tc
[P1 + d1(1+ k )](1- tapt)
1- tc
P0 = (A1.3)
1+ rapt
P1,P2
Recursively substituting for , etc, the dividend discount model (over infinite
horizon), equation (A1.3) can be written as:
tc
[dt+i (1+ k )](1- tapt )
"
1- tc
Pt =
(A1.4)
"
(1+ rapt )t +i
i==1
Equation (A1.4) is defined on an after personal tax basis. As previously
discussed, the residual income model is typically applied to after corporate tax (but
before personal tax) payoffs in order to avoid the identification of the marginal investors
and their personal tax rates for valuation purposes. Equation (A1.4) is to be distinguished
from equation (1) used in the analysis. Equation (1) is defined on an after-corporate tax
but before personal tax basis, its multi-period price expression is stated in equation
(A1.5)
tc
P1 + d1(1+ k )
1- tc
P0 =
(rep. 2.2)
(1+ r)
45
tc
dt+i (1+ k )
"
1- tc
Pt = (A1.5)
"
(1+ r)t+i
i=1
Where r is commonly referred to as the cost of equity (after corporate tax but
before personal taxes) in the accounting, finance and economic literature. Under the
assumption of a constant stream of dividends, in order to yield equivalent price
expressions in A1.4 and A1.5, the discount rate for after-company but before personal tax
payoffs must be such that:
" "
+i +i
"(1+ r)t = (1-1 ) "(1+ rapt )t (A1.6)
tapt i=1
i=1
This shows that cost of equity used in discounting after corporate tax (but before
personal tax) payoffs is the grossed up after personal tax cost of equity. Simply put, the
appropriate discount rate for after personal tax payoffs is the after-tax cost of equity.
When pre-personal tax payoffs are used, the effect of personal tax needs to be adjusted in
the discount rates (e.g., dividing each period s discount rate by the weighted average
personal tax rates is the same as using after-tax payoffs).
The proof also shows that as long as the definitions and discount rates are applied
consistently, the analyses can be done at both before or after personal tax levels. The
following equalities hold:
tc tc tc
[dt+i (1+ k )](1- tapt ) [dt+i (1+ k )] dt+i (1+ k )
" "
1- tc 1- tc " 1- tc
Pt =
= =
" " "
(1+ rapt )t+i (1+ rapt )t +i /(1- tapt ) (1+ r)t+i
i=1 i=1 i=1
(A1.7)
Appendix B:
Begins with the one-period dividend discount model adjusted for imputation tax
credits
tc
P1 + d1(1+ k )
1- tc
P0 =
(A2.1)
(1+ r)
46
Using the clean surplus relationship, and re-express dividends in terms of book values
and earnings
B1 = B0 + NI1 - d1
Ò! d1 = B0 - B1 + NI1
(A2.2)
Substitute (P2.2) into (P2.1),
1-tc (1- k )
P1 + (B0 - B1 + NI1)( )
(1-tc )
P0 =
(1+ r)
P1(1-tc) + (B0 - B1 + NI1)(1-tc (1- k ))
P0 =
(1+ r)(1-tc)
rB (1 - t (1 - k ))
Adding and subtracting in the numerators,
0 c
(B0 - B1 + NI )(1- tc(1- k )) + rB0(1- tc(1- k ) - rB0(1-tc (1- k ) + P1(1-tc )
1
P0 =
(1+ r)(1- tc)
(1- tc (1- k )) (NI1 - rB0 )(1-tc (1- k )) P1(1-tc ) - B1(1-tc (1- k ))
P0 = B0 + + (A2.3)
(1-tc ) (1+ r)(1-tc) (1+ r)(1-tc )
Appendix C:
Begin with the one-period residual income model at t = 0
(1- tc(1- k )) (NI1 - rB0)(1- tc(1- k )) P1(1- tc ) - B1(1- tc(1- k ))
P0 = B0 + +
(A3.1)
(1- tc ) (1+ r)(1- tc) (1+ r)(1- tc )
Repeat the derivation of the one-period residual income model, and express P1 in
terms of book values and earnings:
(1- tc(1- k )) (NI2 - rB1)(1- tc(1- k )) P2(1- tc ) - B2(1- tc (1- k ))
P1 = B1 + +
(1- tc ) (1+ r)(1- tc) (1+ r)(1- tc )
P1 (1- tc) - B1(1- tc (1- k ) (NI2 - rB1)(1- tc(1- k )) P2(1- tc ) - B2 (1 - tc (1 - k ))
= +
(1- tc ) (1+ r)(1- tc) (1+ r)(1- tc )
(A3.2)
Substitute the above expression into (1), and apply the recursive substitution
technique
47
(1- tc (1- k )) (NI1 - rB0)(1- tc (1- k )) (NI - rB1)(1- tc (1- k ))
2
P0 = B0 + + + ...
(1- tc ) (1+ r)(1- tc ) (1+ r)2 (1- tc )
PT (1- tc ) - BT (1- tc (1- k ))
+
(1+ r)T (1- tc )
(A3.3)
Over infinite horizon, the model becomes:
"
(1- tc (1- k )) - rBt )(1- tc (1-
t+i
Pt = Bt +
"(NI (1+ r)t+i (1- tc ) k )) (A3.4)
(1- tc )
i=1
(1- tc(1- k )) tc
(1+ k )
Given that = , the model can be re-expressed as:
(1- tc) 1- tc
tc
(NIt+i - rBt )(1+ k )
tc " 1- tc
Pt = Bt (1+ k ) + (A3.5)
"
1- tc i=1 (1+ r)t+i
Appendix D:
Unlike Appendix A, the effect of imputation tax credits is adjusted in the discount
rate in lieu of payoffs in Appendix E.
As in Appendix A, the one period after-personal tax model can be stated as:
tc
d1(1+ k )(1- t ) + "P1 + "(P1 - P0 )(1- tg )
1- tc p
P0 = (A4.1)
1+ rapt
where tc is the corporate tax rate,
tp is the personal income tax rate,
tg is the capital gains tax rate,
" is the probability of investors selling their share at time t=1,
rapt is the after-personal tax cost of equity.
Let P1(1 - t* ) = "P1 + "(P1 - P0 )(1- tg ) . t* is the effective capital gains tax rate,
g g
which is defined as the proportion of price that is subject to capital gains tax (weighted by
48
the probability of investors selling the shares at time t=1), the equation (A5.1) can be
rewritten as:
tc
P1(1- t* ) + d1(1+ k )(1- t )
g
1- tc p
P0 = (A4.2)
1+ rapt
That forms the basis of the Dividend Discount Model in which I derive the
modified residual income model. The alternative is to adjust for the imputation tax
credits in the discount rate.
Now, we will introduce the effective personal income tax rates t* and let
p
tc
d1(1-t* ) = d1(1+ k )(1- t )
. Effectively, this term captures the amount of
p
1- tc p
dividends that shareholders receive in their hands after paying off the appropriate taxes
and receiving a tax rebate from the imputation tax credits.
Similar to Appendix A, lett* be the weighted average of the effective capital gains
apt
and personal income tax rates:
P1 * d1
t* = tg + t*
apt
P + d1 P1 + d1 p
1
Equation (A5.2) can be re-expressed as:
*
(P1 + d1)(1 - tapt )
P0 = (A4.3)
1 + rapt
Over infinite horizon, with R =1+ r
"
dt (1- t* )
apt
P0 =
(A4.4)
"
t+i
Rapt
i=1
Given that the analysis is conducted on an after corporate tax payoffs (without
adjusting for imputation tax credits in payoffs), the discount rate must absorb the effect of
imputation tax credits.
" "
dt dt
P0 = =
(A4.5)
" "
t +
Rapti /(1- t* ) R*t +i
t =i t =i
apt
"
1
t+i
R* =
where
"Rapt
*
(1- tapt )
t =i
49
This is to be distinguished with the dividend discount model where the effect of
imputation tax credits is adjusted in the payoffs.
tc tc
dt (1+ k ) dt (1+ k )
"
1- tc " 1- tc
P0 = =
(A4.6)
" "
t+
Rapti /(1- tapt ) Rt +i
t =i t=i
"
1
t+i
R =
where
"Rapt
(1- tapt )
t =i
*
Note that (1- tapt ) > (1- tapt ) Ò! R* < R Ò! (1+ r*) < (1+ r) .
Appendix E:
The alternative to incorporating the effect of imputation tax credits in the payoffs
is to adjust it in the discount rates. It is shown in Appendix E that r* treats imputation tax
credits as a process of lowering personal tax rates. The following derives a method that
is similar to Easton et al. (2002) to estimate r* and growth rates (g*) in residual income
simultaneously using forecasts of pro forma current earnings and one-year ahead
earnings.
Earnings can be summed over time, provided that earnings on dividends are also
accounted for in aggregating earnings (Easton and Harris, 1991). Let the aggregate two
X
period cum-dividend earnings be . Using forecasts of pro-forma current earnings
CT
and dividends, and one period ahead earnings:
X = NI + NI + r * d
(A5.1)
CT 1 0 0
NIt
where is the forecast of earnings at time t,
dt
is the forecast of dividends at time t,
r * is the imputation tax adjusted required rate of return
When aggregate two period cum-dividend earnings are used in the numerator
instead, the residual income valuation over infinite horizon can be stated as:56
56
When calculating r* and g*, the square root of R* and G* also contain negative and imaginary
roots. However, the negative and imaginary roots are economically meaningless.
50
XCT - (R *-1)B0
P0 = B0 +
(A5.2)
(R *-G*)
R* = (1+ r*)2
where
2
G* = (1+ g*)
B0
Rearranging equation (A6.2) and deflating the equation by in a similar fashion
in Easton et al (2002) , the equation becomes:
X P0
CT
= (G * -1) + (R *-G*)
B0 B0
X P0
CT
= + (A5.3)
B0 0 1 B0
= G * -1 = R * -G *
where the constant term and .
0 1
Writing A6.3 as the following linear relation for each firm j and adding the error
57
term to account for firm-specific random component of the coefficients and , the
j0 j1
regression A6.4 can be stated as:
X Pj0
jCT
= + + (A5.4)
Bj 0 0 1 Bj 0 j0
Following Easton et al (2002), the estimates of the coefficients are non-stochastic
and may be regarded as the mean of the firm0specific coefficients. In the estimation, R*
and G* are estimates for the portfolio of J firms.58
57
= - + ( - )Pj / B . Given that the error term is heteroskedastic by
j0 j0 0 j1 1 0 j0
construct, White s (1980) is used to correct for standard errors.
58
An iteration process similar to Easton et al (2002) is also applied to overcome the apparent
circularity problem in the calculation of the dependent variable .
X
jCT
51
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