1. 1
Valuation, Capital Budgeting and the Cost of Capital Equations
OR
What is Wrong with Capital Budgeting and Just About Everything Else
Mike Dempsey* and Graham Partington**
*Monash University **University of Sydney
ABSTRACT
Note: Although this paper is set in the context of an Australian problem, the important point,
is that the application is universal. In my presentation it will become evident that the models
presented herein have the dual advantage of being simple and having wide generality in
application. The approach suggested has advantages for analysis, pedagogy and practice.
What is wrong with the current approach will become clear in my presentation.
Since the introduction of the Australian imputation tax system there have been problems both
in the measurement of the market value of franking (imputation tax) credits and in their
application to estimating cash flows and the cost of capital. In the present paper, we provide a
convenient and robust resolution to the above problems in the context of an internally
consistent set of equations for the cost of capital, asset valuation, and the CAPM. The
equations apply under both classical and imputation tax systems and under differential
taxation of dividends, capital gains, and interest. The simple form of the CAPM presented
here is shown to encompass more complex versions of the CAPM which attempt to
accommodate the effect of personal taxes. The valuation equations require an estimate of the
market value of $1 of the firm’s dividends, within which is embedded the market value of the
imputation tax credits. This provides the key to solving several difficult problems.
JEL classification G12, G31, G38
Keywords: WACC, valuation, taxes, imputation tax system, regulation, CAPM.
Acknowledgements
We are particularly grateful to Stephen Gray for his suggestions. We acknowledge also the helpful comments of
participants at the 2004 conference of the Accounting and Finance Association of Australia and New Zealand, in
particular Bruce Grundy; and also helpful comments of participants at Seminars at UNITEC, and at the
Universities of Deakin, Melbourne and Macquarie. We thank Graham Bornholt, John Pierre Fenech, Alastair
Marsden, Giang Truong, and Scott Walker for their help; and we thank the CMCRC for its support.
1

2. 1
1.
Introduction
The cost of capital is important in valuation, capital budgeting, and in setting prices for
regulated utilities. Under the Australian and New Zealand imputation tax systems, two related
questions that arise are: How do imputation tax credits affect the measurement of a project’s
(or firm’s) cash flows? and, How do imputation tax credits affect the cost of capital used to
discount such cash flows? Notwithstanding a well-developed literature that addresses the
impact of imputation (for example, Cliffe and Marsden, 1992; Howard and Brown, 1992;
Monkhouse, 1993, 1996; Officer, 1994; Lally, 1992, 2002, 2004; Lally and van Zijl, 2003;
and Faff, Hillier and Wood, 2000), the above questions remain open.
Officer’s (1994) version of the CAPM with associated weighted average cost of
capital (WACC) valuation model has been the framework preferred by Australian regulators.
The primary tax parameter that arises in Officer’s formulation of imputation tax credits is ,
which represents - as an outcome of the firm’s franking (imputation tax) credits - that portion
of corporate tax that is “not really company tax but rather is a collection of personal tax at the
company level.” Thus Hathaway and Officer (1995) model γ as the product of two elements:
the ratio of imputation tax credits distributed to corporate tax paid (IC/TAX) and their
utilisation ratio (U).
Empirical estimates of the firm’s utilisation ratio (U) are derived from an assessment
of the market value of the firm’s distributed imputation tax credits. Such assessments,
however, are problematical. The estimates are typically derived from a separation of the
market value of the firm’s imputation tax credits from the market value of their associated
dividends. Unfortunately, the required disentanglement is confounded by a high degree of
colliniarity between the tax credits and dividends as well as the problem of allocating costs
between the credits and the dividends. The outcome has been a general degree of confusion
1

3. 2
and inconsistency in regard to the estimation of gamma and the cost of capital and valuation
equations.
The significance of the present paper is the provision of a WACC valuation
framework that accounts for imputation tax while avoiding the above problems. The
framework is achieved by formulating the cost of capital and valuation equations so as to
require only the value of the combined package of imputation tax credits and their associated
cash dividends (q). No separate estimates of the above problematical variables - imputation
tax credits distributed in proportion to company tax paid, utilisation ratios, market valuation
of tax credits, or even gamma - are called for.
Our proposed model nevertheless retains generality and applies for differential taxes
on dividends and capital gains, for differences in tax rates on debt and equity, and for uneven
cash flows under both classical and imputation tax systems. For example, the specification for
the CAPM encompasses the Officer (1994) CAPM, the Brennan-Lally CAPM (see Lally,
2000), and Lally and van Zijl (2003) CAPM. The equations allow additionally for the impact
of taxation on valuation depending on whether the project is funded externally or by retaining
earnings.
The paper proceeds as follows. Section 2 discusses problems in the measurement of
the value of imputation credits and in the use of gamma. Section 3 presents the set of
equations for valuation which avoid the need to estimate either gamma or the utilisation ratio.
The valuation model and an application in price regulation are discussed. It is also shown how
extant versions of the CAPM adjusted for personal taxes can be encompassed within one
simple CAPM. Empirical estimates of the package value of dividends and credits are
discussed in section 4. In Section 5, there is a comparison of valuations based on the approach
traditionally used, Officer’s (1994) approach and the approach advocated here. Section 6
concludes the paper.
2

4. 3
2.
Problems with the valuation of franking credits
The attempt to value imputation tax credits based on the market value of the firm’s dividends
requires a delineation of the market value of $1 of the firm’s cash dividends (q) in terms of
the value of the dividend’s cash and imputation tax credit components. The relationship may
be represented as: q = qcash + F.φ, where qcash is the market value of $1 of the cash component
of dividends (which is to say, cash as income), F is the face value of the franking (imputation
tax) credits attached to one dollar of dividends, and φ is the market value of one dollar of
distributed imputation tax credits.
A typical approach in valuing franking credits has been to regress the ex-dividend
price change on the face value of the cash dividends with a separate variable for the face
value of the franking credits. The level of colliniarity between dividends and franking credits
however is invariably high and poses a serious threat to reliable estimation of the market
value of franking credits. Attempts to finesse estimates of the variables are therefore reliant
on variations of franking levels and changes in corporate tax rates over time. Bellamy and
Gray (2005) and Cannavan, Finn and Gray (2004) conclude that the colliniarity between
distributed dividends and their compensating tax credits is so severe that an estimate of the
market value of the cash dividend is a pre-requisite to estimating the value of franking credits.
Ultimately, the package value of dividends and franking credits is determined not only
by their magnitude, but also by such variables as the corporate tax rate, personal tax rates for
dividends and capital gains, transactions costs and discounting for time lags in cash flows. So
we have here an additional problem, which is one of allocation. How should the tax effects,
transactions costs and discounting effects be allocated in separating the package value into
dividend and franking components? In Walker and Partington (1999) the personal tax effects,
discounting effects and transactions costs are implicitly allocated to the valuation of the cash
component of dividends. In contrast, Cannavan et al. (2004) implicitly address the allocation
3

5. 4
by assuming that the cash component of the dividend is fully valued. Differing again,
Hathaway and Officer (2004) implicitly address the allocation by assuming that one dollar of
the cash component of a franked dividend and one dollar of an unfranked dividend have the
same market value.
Inconsistencies also arise in Officer’s (1994) CAPM where it is assumed that the cash
dividend is fully-valued, but that the associated imputation tax credits are not (Hathaway and
Officer (1995), for example, find that both the dividend and the franking credits are worth
less than their face value). Nevertheless models with such inconsistencies continue to
underpin Australian regulatory determinations. For example, Gray and Hall (2006) criticise
applications of the Officer approach in regulatory decisions and show that the implied market
risk premium and the implied dividend yield are inconsistent with historical data. They argue
that the inconsistency may be eliminated by allocating franking credits a value of zero.
Exemplifying the “allocation” problem, however, Truong and Partington (2007) contend that
such inconsistency can also be resolved by attributing a positive value to the franking credits
retained within the firm.
3.
The Cost of Capital and Valuation Equations
3.1. The fundamental valuation model
Assuming that the firm’s dividends DIVt occur at the end of each period, t, we commence
with a simple definition of the cum-dividend expected return R. Ex-ante, this is equal to the
market’s expectation of the firm’s share price growth over the period immediately following
the dividend payment at period t until immediately prior to the dividend expected at t+1. This
expected growth is defined as:
R
[E(Pcumt+1) - Pext] / Pext
4
(1)

6. 5
where E(Pcumt+1) represents the market’s expected cum-dividend share price (Pcumt+1) at
time t+1, and Pext represents the ex-dividend share price at time t. Consistent with the prior
literature we assume that R is constant for a given stock.
The relation between stock price and dividends is expressed via a capitalisation qfactor which as previously defined, is the ratio of the capitalised value of the dividend to the
face value of the dividend. The expected cum-dividend price at t + 1 can therefore be defined
as:
E(Pcumt+1)
E(Pext+1) + q.E(DIVt+1)
(2)
We assume that for a given company the value of q is constant. Ultimately, q is determined by
the market at a given time in respect of a given company’s dividends. Current practice in
respect to gamma ( is to assume a constant market-wide value on the basis of equalisation
across stocks by arbitrage. A similar practice could be applied in respect to q, but it would
need to be conditioned by the level of franking. If franking credits have value, the value of q
will differ across companies depending upon the extent to which the dividend is franked.1
Rearranging expression 1 yields:
Pex t = E(Pcumt+1) / (1 + R)
(3)
Substituting from expression 2 for E(Pcumt+1) in equation 3 and iterating forwards, yields the
ex-dividend market share price (Pex0 ) as a function of the firm’s expected dividends, E(DIVt):
N
P ex 0
t 1
E ( P ex N )
(1 R) N
E ( DIVt ). q
(1 R) t
(4)
as in Dempsey (1996). Equation 4 computes the firm’s current equity value (Pex0) by
discounting the expected growth in equity market value E(DIVt).q, which arises from future
dividends. In contrast with the traditional discounted dividend model, expected dividend
1
In a majority of cases the dividend is fully franked or unfranked, requiring knowledge of either q for fully
franked dividends or qcash for unfranked dividends. For partially franked dividends we would need to interpolate
between q franked and qcash.
5

7. 6
payments are not simply expressed at their face value, but are instead converted to their
market value immediately prior to payment. This simple transformation has the important
result that the equation maintains “dimensional consistency”. That is, prices and capitalised
dividends are both measured in market values.
When evaluating a project, however, common practice is not to discount a stream of
expected dividends - since dividend payments occur at the level of the firm, not at the level of
the project - but rather to discount the project’s expected after corporate tax cash flow as
though the project were financed only by equity. This cash flow is often referred to as the
unlevered cash flow. Most projects, however, are levered. We therefore require a formulation
which shows how the unlevered cash flow might meaningfully be discounted from the
perspective of a levered project. In developing such a formulation, authors such as Miles and
Ezzell (1980), Clubb and Doran (1992), Sick (1990), Taggart (1991), Officer (1994),
Monkhouse (1996) and Dempsey (1998, 2001) have typically relied on the following explicit,
or implicit assumptions:
i.
the market value of the project’s debt financing is maintained as a constant proportion
(L) of the market value of the project (Vt) at the end of each time period t,
ii.
the firm’s bonds are expected to pay interest at a coupon rate (rb) per period and if
redeemed such redemption is at their par value,
iii.
the firm’s cash distributions to shareholders are either in the form of dividends and/or
the repurchase of the shares at their issue price only. There are no share repurchases at
above the issue price of the shares.
iv.
the firm’s share holders may expect both a capital gain and a dividend yield,
v.
the rate of corporate tax, Tc, is constant for the life of the project.
vi.
required rates of return on debt and equity are constant for the life of the project
6

8. 7
With the above assumptions, as shown in Appendix A, equations 1 and 2 lead to the
current market valuation, V0, of a project as:
N
V0
t 1
[Yt .(1 Tc ) Qt .Tc ]. q
(1 WACC ) t
VN
(1 + WACC ) N
(5)
with:
WACC = (1 - L).R + L.rb.(1 - Tc ).q
(6)
as in Dempsey (2001), where Yt is the net cash flow before interest and tax, Qt represents the
non-cash deductions from Yt required to yield taxable income for the unlevered firm at the
end of period t, and VN is the terminal value of the project. In equation 5, the cash flow
[Yt .(1 Tc ) Qt .Tc ] is the usual textbook definition of the after-tax cash flow.
Equation 6 is similar to the standard textbook definition of WACC except that the cost
of equity R is measured as the cum-dividend return, or, equivalently, the expected growth rate
in equity value prior to the next ex-date, and the cost of debt rb is multiplied by q. One way to
interpret the weighting of q on the cost of debt is to recognise that higher values of q reduce
the tax effectiveness of debt relative to equity.
Equations 5 and 6 provide a generally applicable basis for project valuation. No
further adjustments for imputation or for other sources of differential taxation of dividends
and capital gains are required.
3.2 The valuation model and regulation
The standard approach of a utility regulator trying to set prices is that investors should receive
the cost of capital on the value of the funds that they have invested. In implementing this
approach, prices can be set such that the resulting expected income stream (correctly
discounted) equates with the book value of assets B (as assessed by the regulator in the range
for example between depreciated actual cost (DAC) and depreciated optimised replacement
7

9. 8
cost (DORC) as a
proxy for investment). This is the approach currently adopted by
Australian regulators.
Although considerable regulatory attention has been devoted to the issue of how the
allowable return (discount rate) is affected by the value of imputation credits, much less
attention seems to have been paid to the role of taxes (including imputation credits) in
determining the opportunity cost of the funds required to finance the investment. To see what
is at issue here, consider the case where the investment is financed from retained earnings that
could otherwise be paid out as franked dividends. If imputation credits have value, the market
value of the funds invested is greater than their face value and the investment base for price
regulation should be adjusted accordingly. For example, if retained earnings equal to the book
value B0 were distributed as dividends, the market value V0 of that cash would be given by V0
= B0.q. Assuming that the project cash flows have the same values for q, and that the project
has a terminal value of zero, then substituting for V0 in equation 5, q cancels out giving:
N
B0
t 1
[Yt .(1 Tc ) Qt .Tc ]
(1 WACC ) t
(7)
with the WACC defined according to equation 6.
As indicated in section 3.1, with increasing q, the tax advantage to debt is effectively
reduced. Thus for a given book value B0 and cost of equity (R) a higher determined q value
leads to a higher required Yt and hence higher prices. This is an interesting result; in most
regulatory hearings, the utilities argue for a zero value for imputation credits (equivalent to a
lower q) in order to justify higher prices. Under the proposed system, it is possible that such
argument would actually be reversed.
A key implication of this analysis is that if the investment comes from funds that if
distributed would have a q greater than one2 the investment basis should be increased to the
2
Harris, Hubbard and Kemsley (2001) present empirical evidence that the tax benefits of imputation are
capitalised into retained earnings in Australia, thereby increasing the opportunity cost of using retained earnings
8

10. 9
market value foregone (decreased if q is less than one). The current regulatory setting is based
on the regulator’s assessment of what is the most efficient way of structuring the business.
For example, the firm’s actual financing structure is substituted with what the regulator
considers to be the most efficient financing structure. It may be that the same principle would
need to be applied in regard to the firm’s structure of retained earnings and external equity
financing.
3.3. The cost of equity and the CAPM
From equations 1 and 2, the value of R can be measured as either: R = (Pcumt+1 - Pext)/Pext or
as R = (Pext+1 + q.DIVt+1 - Pext)/Pext . For capital budgeting purposes, however, we require
the equilibrium expected return and in Australia this return is usually estimated using the
CAPM.
Officer (1994) adapts the traditional CAPM for imputation by redefining the returns
on individual shares and the market risk premium to include imputation tax credits. Thus
Officer defines the return on equity as Rofficer = (Pext+1+ DIVt+1 + U.ICt+1 - Pext)/Pext, where IC
is the face value of imputation tax credits accompanying the dividend.3 Assuming equality of
taxes on dividends and capital gains, Officer simply substitutes his definition of equity returns
into the traditional CAPM.
We follow a similar approach and substitute into the traditional CAPM according to
the definition of return R in equation 1 (with the corresponding measure of the return on the
market Rm) 4 together with the equity beta ( ) defined with respect to the covariance between
However, the method used by Harris et al. has been the subject of substantial criticism, see for example,
Dhaliwal, Erickson, Frank and Banyi (2003).
3
R from the q method can be written in a form similar to ROfficer as R = (Pext+1 + qcash.DIVt+1 + φ ICt+1 - Pext)/Pex,
where qcash is the market value to book value for the cash component of the dividend. The parameter qcash
captures differences (other than imputation) between the taxation of dividends and capital gains that are assumed
away in Officer’s (1994) model where qcash is constrained to equal one.
4
In measuring the cum-dividend market return, Rm, it is convenient to use the equation for return measurement
as R = (Pext+1+ q.DIVt+1 - Pext)/Pext. Hence we require an additional parameter, the value of qm for the dividend on
the market portfolio.
9

11. 10
these two returns. We also adjust the risk-free rate by the factor qB which allows that the
market value of interest payments on risk-free debt may vary from their face value due to tax
effects.5 The resulting equation for the CAPM is then expressed:
R = rf.. qB
+
.[ Rm - rf.. qB ]
(8)
where rf.is the interest rate on risk-free bonds that retain their par value. Combining equations
1 and 2 the cum-dividend market return R may be expressed:
R = (Pext+1 + q.DIVt+1 - Pext)/Pext.
If we assume that the variation in q is due only to imputation, the above expression gives the
same definition of return as Officer (1994), since q.DIVt+1 is then equal to DIVt+1 + U.ICt+1.
Given the foregoing assumption, and setting qB equal to one, results in the same CAPM as in
Officer (1994). Setting both q and qB equal to one gives the traditional CAPM. Additionally,
both the Lally and van Zilj (2003) CAPM under imputation with differential taxation of
dividends and capital gains and Lally’s simplified version of the Brennan-Lally CAPM are
equivalent to equation 8. These latter equivalences are demonstrated in Appendix B.
4.
The value of q
Since Modigliani and Miller (1961) there has been an extensive debate regarding the value of
q. The Australian ex-dividend evidence under the classical tax system is that q is less than one
(Brown and Walter, 1986). Following the introduction of the imputation system in Australia,
however, Brown and Clarke (1993) found that the value of q initially declined, and that
consistent with a difference in q for franked and unfranked stocks, franking credits appeared
to have value. Subsequent ex-dividend work also finds a difference in q between franked and
unfranked stocks (Bellamy, 1994; Hathaway and Officer, 1995, 2004). However, Bellamy
5
Alternatively the risk-free rate might be scaled by q and the resulting term rf .q would then be interpreted as the
return on zero beta equity.
10

12. 11
and Gray (2005) argue that q is equal to one and that franking credits have no value to the
price-setting investor.
In summary the ex-dividend evidence tends to suggest that q differs for franked and
unfranked dividends. However, due to the one day gap between observation of cum-dividend
and ex-dividend prices, measurements of q are extremely noisy (Walker and Partington,
1999). The interpretation of ex-dividend studies is also problematic. There is, for example,
the issue of whether prices about the ex-dividend date are set by short-term dividend
arbitrageurs or long-term investors.
Several more recent studies have sought to avoid the above measurement problems for
Australian markets. Walker and Partington (1999) study cum-dividend trading in the exdividend period and obtain measures of q that are consistently greater that one for fully
franked dividends.
By-passing the ex-dividend approach completely, Twite and Wood (1997) infer the
value of dividends by comparing the price of individual share futures with the underlying
shares. A similar approach is used by Cannavan, Finn, and Gray (2004). Both studies suggest
a q value greater than one, although Cannavan et al. argue that the value of q became one
after the introduction of the 45 day rule which restricts trading in imputation credits.
Another approach which seeks to avoid traditional measurement problems, takes
advantage of rights issues where the newly issued shares do not qualify for the forthcoming
dividend. Such studies have been conducted for Australia (Chu and Partington, 2001), and for
the UK (Armitage, Hodgkinson and Partington, 2006), and show that dividends which benefit
from imputation tend to have a q greater than one.
These alternatives to traditional ex-dividend studies offer more precise estimates of q,
which are generally greater than one for franked dividends. For fully franked dividends,
Twite and Wood (1997) provide an estimate of 1.13, Cannavan et al. (2004) provide an
11

13. 12
estimate of 1.1, changing to 1.0 post the introduction of the 45 day rule designed to restrict
ex-dividend arbitrage. Walker and Partington (1999) report an average of 1.23 by trade and an
average of 1.15 by event.
Valuation methods are compared in Section 4 below for values of q of 1.02 and 1.15,
which we take to be representative of the currently competing values for q. These particular
values are based on Hathaway and Officer (2004) and Walker and Partington (1999)
respectively.
5.
A numerical comparison
In this section we consider the possible impact on investment decisions of adopting the q
method in place of existing methods. To this end, we compare NPVs from both (1) the
traditional valuation approach, ignoring franking credits, and (2) application of Officer’s
(1994) valuation approach incorporating franking credits, with (3) the q method. We compute
the firm’s cost of equity capital using the CAPM appropriate to each approach. We assume
that it is possible to consider a single average stock with a beta of one across all 3 approaches;
that the risk-free rate of interest is 5 percent, and that the market risk premium is 7 percent.
And we assume that the franked dividend yield on the market is 4 percent and that the stock
has a dividend yield of 4 percent.
The project considered involves a $30 million initial investment with an expected
unlevered cash flow after corporate tax of $5 million a year for ten years. We consider both
the case that the project is financed externally and the case that it is financed internally. The
firm’s market leverage ratio is set at 40 percent for the life of the project with a pre-tax cost of
company debt (rb) set at 7.1 percent as in Lally (2004). The corporate tax rate is 30 percent.
We consider both the case that the firm pays sufficient dividends that distributed imputation
tax credits equate with the firm’s corporate tax liability (IC/TAX = 1) so that gamma (γ)
equates directly with the utilisation rate (U) of such credits; and the opposing case where the
12

14. 13
cash flows lead to dividends that do not give rise to franking credits (γ = 0), as would be the
case, for example, if the project gave rise to earnings taxed overseas. For the sake of our
example, we shall take it that φ and U are interchangeable (in other words, if, say, one half of
the firm’s credits are utilised (U = ½), then the market value of such credits is half their face
value (φ = ½)).
In the case that the utilisation of credits (U) = 0 and the capitalisation factors q (= qf
for franked, qu = qcash for unfranked) the three valuation methods are identical, giving an NPV
for the project = $1.82 million. We also calculate the NPV of the project across a range of
input values for the utilisation rate of tax credits (U) and the capitalisation factor (q). Since
the firm is an average firm we apply the estimates for average q and U suggested in prior
studies.
The first estimates that we use for the market value of $1 of distributed imputation tax
credits (φ) and of cash dividends (qcash) are derived from Hathaway and Officer (2004) who,
in an ex-dividend analysis estimate that franking credits are worth 51 percent of their face
value (consistent with this, they estimate, from a study of taxation statistics, that about 50
percent of franking credits distributed are redeemed). The grossed-up value of a fully-franked
$1 dividend is $1/(1-Tc) = $1/(1 - 0.3) = $1.43, and assuming that the credit is distributed
implies a face value of the associated franking credit as $1.43(0.3) = $0.43, and hence a
market value of the franking credit as $0.43(0.51) = $0.22. Hathaway and Officer also
estimate that the cash component of the dividend is valued at 80 percent of its face value,
which implies a q for fully-franked dividends as (0.80 + 0.22) = 1.02 (using: q = qcash + F.φ).
The second set of estimates for q are derived from Walker and Partington (1999) who
calculate q equal to 1.15 and that franking credits have a market value that is 88 percent of
their face value. The implied q for an unfranked dividend is then equal to 1.15 – 0.43(0.88) =
0.77. In this case, the higher valuation assigned to franking credits is a consequence of
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15. 14
Walker and Partington’s allocation of all personal tax effects and transaction costs to the cash
dividend when splitting up the package value of cash dividends and franking credits.
We now consider three ways in which a project analyst might choose to define the
firm’s cost of capital: (1) the traditional approach, (2) an application of Officer’s (1994)
model, and (3) the q model approach, and follow through the implications for valuation.
(1)
The traditional method
In this case the cost of equity is given by substitution of the values for the risk-free rate, beta
and the market risk premium into the traditional CAPM.
Rtraditional = rf. +
.[E(Rm) - rf ]
= 0.05 + 1(0.07) = 0.12
(2)
Application of Officer’s (1994) model
Officer (1994) proposes that a share’s imputation tax credits are viewed by the investor as a
constituent of return but do not affect the investor’s overall required return. In applying the
Officer WACC, we therefore retain the traditional CAPM:
Rofficer = rf. +
.[E(Rm) - rf ] = 0.12
However, the value of imputation tax credits implies an effective reduction in the cost of
equity from the firm’s perspective, which leads to the representation of the cost of equity as
Rofficer.(1 - Tc)/[1 - Tc (1-γ)] in the Officer WACC (as clarified in Appendix C).
(3)
The q method
The q-model implies a conceptually different approach to the CAPM. Two adjustments are
required. First, the traditional measurement of the market return must be converted to a cumdividend return, and second, the risk-free rate must be scaled by qB. This gives:
R = rf..qB
+
.[E(Rm) + (q-1)DYm - rf..qB ]
14

16. 15
where, DYm is the dividend yield on the market. Based on analysis of ex-interest price
changes on the ASX we estimate qB to be 0.7.6 Using the 1.02 estimate for q, based on
Hathaway and Officer (2004), gives:
R (H/O) = (0.05)(0.70)
+ 1[0.12 + (0.02)(0.04) – (0.05)(0.70 )] = 0.121
while using Walker and Partington’s (1999) 1.15 value (above) for q gives:
R (W/P) = (0.05)(0.70)
+ 1[ 0.12 + (0.15)(0.04) – (0.05)(0.70 )] = 0.126
The weighted average cost of capital can now be calculated for the above approaches and
details of these calculations are given in Appendix C.
In Table 1, the cash flow in the traditional method and Officer’s method is $5 million.
This implies that in Officer’s method the adjustment for the value of imputation credits is
made in the WACC. Consequently, the WACC in Officer’s method falls with increasing
gamma ( . Under the q method, the cash flows are multiplied by q and so increase with q.
However, under the q method, the WACC also increases with q. There are two reasons for
this. First, in computing the cum-dividend value for E(Rm), we add an adjustment that is an
increasing function of q (as above). Second, the tax shield benefit of debt is less effective as q
increases (as equation 6).
Application of the WACCs to their corresponding definitions of cash flow leads to the
NPV values in Table 1. Panels A-C summarise the outcome of applying: (1) the traditional
approach (where cash flows of $5 million a year for ten years unadjusted for any imputation
tax credits are discounted by the traditional WACC as per Appendix B), (2) an application of
Officer’s model (where cash flows of $5 million unadjusted for any imputation tax credits are
discounted by the Officer WACC as per Appendix C), and (3) the q value approach (where
6
We acknowledge the contribution of Scott Walker in deriving this estimate. The estimate is based on the ratio
of the ex-interest price change to the interest payment for corporate interest bearing securities traded on the ASX
between 1999 and 2004.
15

17. 16
cash flows of $5 million multiplied by the above calculated q-factors (eg. with q = 1.15, cash
flow is $5.75 million) are discounted by the q-adjusted WACC as per Appendix C).
The project NPV values are calculated as the project discounted cash flows minus the
$30 million initial investment funds. The first two columns represent “base cases.” The first
column serves to illustrate that with a utilisation ratio (U) = 0, the traditional method and
application of Officer’s model are in agreement (NPV = $1.82m); whereas the q method leads
to a negative NPV = -$3.53m (where the effect of q on cash flows (q = 0.8) has more than
offset the effect of a lower cost of capital). The second column illustrates that with q = 1 (for
franked and unfranked dividends) the traditional method and the q method are in agreement
(NPV = $1.82m); whereas application of Officer’s model leads to a substantially more
positive NPV = $4.77m in the case of fully-franked credits. This is because the Officer model
adjusts the cost of equity downwards in the WACC to reflect the value of the tax credits to
investors (as clarified in Appendix C).
Under the q method, NPVs are an increasing function of q, while under Officer’s
method the NPVs are an increasing function of gamma. The NPV calculations also depend
upon whether we assume fully-franked dividends (Panel B) or unfranked dividends (Panel C).
For unfranked dividends, q = qcash, which lowers the capitalised value of the project cash
flows under the q method (on account of qcash is estimated to be less than 1). Hence the q
method here leads to a negative NPV for unfranked dividends, as compared with a positive
value under the other two methods.
Except for issues costs, the distinction between finance for the project raised
externally, or alternatively obtained from retained earnings, is commonly ignored in capital
budgeting. The q method accommodates this important distinction. Thus in the q method of
Panels B and C, we have assumed that the external project funding of $30 million has an
opportunity cost to investors of $30 million. In which case, a positive NPV implies that the
16

18. 17
project is justified when financed externally. In Panels D and E, we have assumed that the
project is financed by the firm’s retained earnings which if distributed would merit tax
credits at the same rate as the project’s distributable earnings stream. For this reason, the $30
million initial investment has been scaled by q to represent its opportunity cost to investors.
In this case, a positive NPV implies that the project is justified when financed by the firm’s
retained earnings. Thus in column one under the q method, where q is less than one, the NPV
switches from negative (Panel B) to positive (Panels D and E) as the outcome of the cost of
internal financing having been reduced below $30 million. In contrast, in columns three and
four of Panel D, the NPVs are lowered substantially because the cost of the investment has
been scaled up (q>1). The implication here is that franking credits have been accumulated by
the firm. In contrast, the negative NPVs switch from negative (Panel C) to positive (Panel E)
under the q method. This is because the reduction in value of the cash flow for unfranked
dividends is offset in Panel E by the reduced opportunity cost of the investment (q<1). The
foregoing analysis highlights an issue for regulation, namely the issue of whether prices
should be regulated in accordance with the firm’s ability to attract new equity, or with its
ability to justify reinvestment of funds within the firm to its shareholders.
TABLE 1 about here
6.
Conclusion
The introduction of imputation tax credits has raised two related questions, namely how do
imputation tax credits contribute to a firm’s (or project’s) cash flow, and how do imputation
tax credits affect the cost of capital by which such cash flows are to be discounted. We have
observed that inconsistencies and implementation errors have characterised the prevailing
response in the literature. The present paper offers a convenient and robust resolution in the
context of an internally consistent set of equations for the CAPM, weighted average cost of
capital, and valuation. An advantage of the proposed approach is that it retains both simplicity
17

19. 18
and generality and is applicable under both imputation and classical tax systems. This is
demonstrated with respect to the incorporation of personal tax effects into the CAPM. The
paper shows that a simple CAPM encompasses several versions of the CAPM including the
traditional CAPM, Officer’s (1994) CAPM, the Brennan-Lally CAPM (see Lally, 2000), and
Lally and van Zijl’s (2003) CAPM.
By defining returns as cum-dividend returns and converting dividends to capitalised
values through multiplication by the q-factor, the need to calculate Officer’s gamma is
removed from the cash flow estimates. Additionally, adjustments for taxes on dividends, as
well as for tax differentials between dividends and capital gains, are removed from the equity
discount rate. The outcome, we believe, is that the q approach holds potential for both more
robust and accurate valuations.
The approach offers additional benefits. For example, the q-factor provides a
convenient way to handle differences in the firm’s opportunity costs of external finance and
retained earnings. The analysis raises the question of whether prices for regulated entities
should be set in accordance with a firm’s ability to attract new equity, or with its ability to
justify retention of its earnings for reinvestment purposes.
We have presented a numerical example to demonstrate that quite different valuations
can arise using the q method relative to either the traditional method or the Officer (1994)
method. Replacement of the requirement for explicit estimates of the value of imputation
credits and the tax liabilities of investors with the requirement for empirical estimates of q,
implies the need for reliable values of q itself, which remains an uncompleted task. Promising
work has however been achieved in this regard, and we hope that the paper will motivate
further effort in researching such values.
18

20. 19
Appendix A
Demonstration of the valuation equations (Equations 5 – 6)
Provided we can demonstrate that at the end of each time period t:
Vt
[Yt 1.(1- Tc ) Qt 1.Tc ].q
1 WACC
Vt
1
(A1)
with WACC identified as equation 6:
WACC = (1 - L).R + L.rb.(1 - Tc ).q
(6)
then equations 5 and 6 follow directly by forward iteration of equation A1.
To this end, we adapt an approach from Clubb (1992). Thus we observe that the
expected payoff to equity and debt capital at the end of time t+1 can be expressed from the
definitions as:
[(Yt+1 - L.Vt .rb).(1- Tc) + Qt+1.Tc].q
+ L.Vt .rb. qB + Vt+1
(A2)
where the three sets of terms represent, respectively, (i) the market equity value of the
expected after interest and tax earning steam, (ii) the market bond value of the expected
interest stream, which is achieved by multiplying the interest payment (L.Vt .rb) by the market
bond value of $1 of interest payment (defined to be qB), and (iii) the expected market value of
the firm to equity and bondholders at the end of the period t+1. Such payoff may alternatively
be represented from the definitions as:
(1-L).Vt.(1+R) + L.Vt .(1+ rb.qB)
(A3)
where the two sets of terms represent the expected market returns to equity and bonds,
respectively. Combining expressions A2 and A3, we derive:
[(Yt+1).(1- Tc) + Qt+1.Tc].q + Vt+1
= (1-L).Vt.(1+R) + L.Vt .(1+ r b.qB) - L.Vt .r b.[qB – (1-Tc).q]
= Vt.[1 + (1 - L).R + L.rb.(1 - Tc ).q]
19

21. 20
The left-hand side of the above equation is the numerator in equation A1. Substituting
accordingly in this equation, and solving for WACC equation 6 results.
Appendix B
Equivalence of Lally and van Zilj’s (2003) CAPM and the Brennan-Lally CAPM
with Equation 8.
The Lally–van Zijl (2003) CAPM is expressed (their equations 9 and 10) as:
r j = rf . +
j.(rm
- rf ) +
j
(B1)
and
j=
T1 [- rf..(1 – j ) + (d j – d m. j)] + T2 [dj . U.(ICj /DIVj) - d m .U.(ICm /DIVm). j ]
(B2)
where r j represents the expected return on company j inclusive of imputation tax credits (rm
similarly for the market portfolio);
j
represents the traditional covariance of asset j’s return rj
with the market return rm; dj represents the company’s dividend yield (dm similarly for the
market portfolio); U represents the market-wide utilization rate for the imputation credits
(ICj) attached to the cash dividend DIVj for company j (ICm , DIVm similarly for the market
portfolio); and T1, T2 both represent weighted averages over [(ti - tg) /(1 - tg)], where ti, tg,
represent shareholders’ marginal tax rate liabilities on interest and capital gains. Rearranging
the Lally–van Zijl equations B1 and B2 as:
r j = rf..(1 – T1) + d j .T1 + d j .U.(ICj /DIVj).T2
+
.[rm - rf..(1 – T1) - dm .T1 - d m .U.(ICm /DIVm).T2]
and setting T1 and T2 as (ti - tg)/(1 - tg) = T (consistent with Lally and van Zijl, 2003), and
rearranging, we have:
r j - d j [1+ U.(ICj /DIVj)] + d j [(1+ U.(ICj /DIVj)].(1- T ) - rf..(1 – T)
=
.{rm - dm [1+ U.(IC m /DIVm)] + dm [(1+ U.(ICm /DIVm)].(1- T) - rf..(1 – T) }
20

22. 21
In the above, the term (1 - T), which may be expressed = 1 - (ti - tg)/(1 - tg) = (1- ti)/(1- tg), is
the same as qB. To see this, we follow Elton and Gruber (1970) and Auerbach (1979) who
identify the capitalisation factor q of the firm’s dividends as:
q = (1 - td) /(1 - tg)
(B3)
where td, tg, represent investors’ effective marginal rate of tax on, respectively, dividends and
capital gains (as effective rates, tg includes the effect of, for example, offsets against losses
and delay in the payment of capital gains, and td includes, for example, the effect of
imputation tax credits). By analogy, the market value of $1 of the firm’s interest payments is
determined as:
qB = (1 - ti) /(1 - tg)
(B4)
and hence we may express (1 - T) = qB. Thus we note that the Lally–van Zijl equation as
written above has simply subtracted the dividend yield plus associated utilised imputation tax
credits – the dj [1+ U.(IC j /DIV j)] term - from rj (and rm) (which leaves the capital gain
component of the return) and then added back the self-same dividend yield plus associated
imputation tax credits multiplied by qB = (1 - T). Hence the cum-dividend return Rj (for the
general company j) equates with rj - dj [1 + U.(ICj /DIVj)] + dj [(1 + U.(ICj /DIVj)].(1- T ). And
hence the Lally–van Zijl equations B1 and B2 may be expressed equivalently:
Rj - rf..qB
=
j.(
Rm - rf..qB )
which is equation 8. Similarly Lally’s simplified version of the Brennan-Lally CAPM:
rj - rf..(1- T ) =
j.[
rm - rf..(1- T )]
(B5)
follows as an exact equation on equating qB = (1-T ) as above and on identifying the expected
return r with R (and rm with Rm).
21

23. 22
Appendix C
Computation of WACCs for the numerical example in Section 5
For both the traditional and application of Officer’s model approaches, the cash flow to be
discounted is the $ 5 million. In the case of the q method, the $ 5 million is multiplied by q (qf
for franked, qu = qcash for unfranked dividends).
(1)
The traditional method
With no adjustment for imputation the WACC is given as:
WACCtraditionl = (1 - L).Rtraditional + L.rb.(1 - Tc )
= 0.6 (0.12) + 0.4 (0.071)(1 - 0.3) = 0.092
(2) Application of Officer’s (1994) model
Consistent with Officer (1994) the specification of the WACC depends upon which definition
of cash flow is used. The cash flows in the example are defined as unlevered net cash flows
after corporate tax and the matching specification for the WACC is:
WACCofficer
(1 L).Rofficer .
(1 Tc )
L.rd .(1 Tc )
[1 Tc (1 )]
Using the estimate of the utilisation rate (U) for franking credits from Hathaway and Officer
(2004) (above = 0.51, with IC/TAX assumed = 1 here) gives:
WACC officer, f H / O
(1 0.4)0.12
(1 0.3)
[1 0.3(1 0.51)]
(0.40 )0.071 (1 0.3)
0.079
By contrast, if a project of the firm were expected to generate only unfranked dividends, the
project WACCofficer,u (on maintaining the firm’s cost of equity but with gamma equal to zero in
the WACC equation) would be:
WACCofficer, u (H/O) = 0.092
22

24. 23
while using Walker and Partington’s (1999) estimate of the utilisation rate (U) for franking
credits (above = 0.88), the values of WACC, for franked and unfranked dividends
respectively, are:
WACCofficer, f (W/P) = 0.072
WACCofficer, u (W/P) = 0.092
(3) The q method
Applying equation 6 the WACC is given by:
WACC = (1 - L).R + L.rb.(1 - Tc ).q
Using Hathaway and Officer’s (2004) estimates of q (= qf (1.02) for franked dividends, and =
qu = qcash (0.80) for unfranked dividends) the values of WACC to be applied to franked and
unfranked dividends respectively, are:
WACCf (H/O) = (1 – 0.4)0.121 + (0.4)0.071(1 – 0.3)1.02 = 0.093
WACCu (H/O) = (1 – 0.4)0.121
+ (0.4)0.071(1 – 0.3)0.80 = 0.088
while using Walker and Partington’s (1999) estimates of q (= qf (1.15) for franked dividends,
and = qu = qcash (0.77) for unfranked dividends) the values of WACC, for franked and
unfranked dividends respectively, are:
WACCf (W/P) = (1 – 0.4)0.126 + (0.4)0.071(1 – 0.3)1.15 = 0.098
WACCu (W/P) = (1 – 0.4)0.126 + (0.4)0.071(1 – 0.3)0.77 = 0.091
23

25. 24
REFERENCES
Armitage, S., Hodgkinson. L. and G. Partington, 2006, The market value of UK dividends
from shares with differing entitlements, Journal of Business Finance and Accounting, 33,
220-244.
Auerbach, A. J., 1979, Share valuation and corporate equity policy, Journal of Public
Economics, 11, 291-305.
Bellamy, D. E., 1994, Evidence of imputation clienteles in the Australian equity market, Asia
Pacific Journal of Management, 11, 275-287.
Bellamy, D. and S. Gray, 2005, Using stock price changes to estimate the value of dividend
franking credits, Paper presented at the Accounting and Finance Association of Australia and
New Zealand Annual Conference, Melbourne.
Brown, P. and T. Walter, 1986, Ex-dividend day behaviour of Australian share prices,
Australian Journal of Management, 11, 139 – 152
Brown, P. and A. Clarke, 1993, The ex-dividend day behaviour of Australian share prices
before and after dividend imputation, Australian Journal of Management, 18, 1-40.
Cannavan, D., Finn, F. and S. Gray, 2004, The value of dividend imputation tax credits in
Australia, Journal of Financial Economics, 73, 167-197.
Chu H. and G. Partington, 2001, The value of dividends: evidence from a new method, Paper
presented at the Accounting Association of Australia and New Zealand Annual Conference,
Wellington.
Cliffe, C. and A. Marsden, 1992, The effect of dividend imputation on company financing
decisions and the cost of capital in New Zealand, Pacific Accounting Review, 4, 1-30.
Clubb, C. D. B., 1992, A pedagogic note on the weighted average cost of capital and project
evaluation, Accounting and Business Research, 19, 465-467.
Clubb, C. D. B. and P. Duran, 1992, On the weighted average cost of capital with personal
taxes, Accounting and Business Research, 23, 44-48.
Dempsey. M., 1996, The cost of equity capital at the corporate and investor levels allowing a
rational expectations model with personal taxations, Journal of Business Finance and
Accounting, 23, 1319-1331.
24

26. 25
Dempsey. M., 1998, The impact of personal taxes on the firms weighted average cost of
capital and investment behaviour: A simplified approach using the Dempsey discounted
dividends model, Journal of Business Finance and Accounting, 25, 747-763.
Dempsey. M., 2001, Valuation and cost of capital formulae with corporate and personal
taxes: A synthesis using the Dempsey discounted dividends model, Journal of Business
Finance and Accounting, 28, 357-378.
Elton, E. J. and M. J. Gruber, 1970, Marginal stockholder tax rates and the clientele effect,
The Review of Economics and Statistics, 52, 68-74.
Faff, R., Hillier, D. and J. Wood, 2000, Beta and Return: Implications of Australia’s Dividend
Imputation Tax system, Australian Journal of Management, 25, 245-260.
Gray S. and J. Hall, 2006, The relation between franking credits and the market risk premium,
Accounting and Finance, 46, 405-428.
Harris, T., Hubbard, R. and D. Kemsley, 2001, The share price effects of dividend taxes and
tax imputation credits, Journal of Public Economics 79, 569-596.
Hathaway, N. and R. Officer, 1995, updated 2004, The Value of imputation tax credits,
Working Paper, University of Melbourne.
Howard P. and R. Brown, 1992, Dividend policy and capital structure under the imputation
tax system: Some clarifying comments, Accounting and Finance, 32, 51-61.
Lally, M., 1992, The CAPM under dividend imputation, Pacific Accounting Review, 4, 31-44.
Lally, M., 2000, Valuation of companies and projects under differential personal taxation,
Pacific-Basin Finance Journal, 8, 115-133
Lally, M., 2002, The cost of capital under dividend imputation, Report prepared for the
Australian Competition and Consumer Commission.
Lally, M and T. van Zijl, 2003, Capital gains tax and the capital asset pricing model,
Accounting and Finance, 43, 187-210.
Lally, M., 2004, The cost of capital for regulated industries, Report prepared for the
Queensland Competition Authority.
Miles, J. A. and J. R. Ezzell, 1980, The weighted average cost of capital, perfect capital
markets, and project life: A clarification, Journal of Financial and Quantitative Analysis, 15,
719-730.
Modigliani, F. and M.H. Miller, 1961, Dividend policy, growth and the valuation of
Shares, Journal of Business, 34, 411-433.
25

27. 26
Monkhouse R. H. L., 1993, The Cost of equity under the Australian dividend imputation tax
System, Accounting and Finance, 33, 1-18.
Monkhouse, R. H. L., 1996, The valuation of projects under the dividend imputation tax
system, Accounting and Finance, 36, 185-212.
Officer, R. R., 1994, The cost of capital of a company under an imputation tax system,
Accounting and Finance, 31, 1-17.
Sick, G. A., 1990, Tax-adjusted discount rates, Management Science, 36, 1432-50.
Taggart, R. A. Jr., 1991, Consistent valuation and cost of capital expressions with
corporate and personal taxes, Financial Management, 20, 8-20.
Truong, G. and G. Partington, 2007, The relation between franking credits and the market risk
premium: A comment, Accounting and Finance, Forthcoming.
Twite, G. and J. Wood, 1997, The pricing of Australian imputation tax credits: Evidence from
individual share futures contracts, Working Paper Series: Australian Graduate School of
Management, 97-002.
Walker, S. and G. Partington, 1999, The value of dividends: Evidence from cum-dividend
trading in the ex-dividend period, Accounting and Finance, 39, 275-296.
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28. 27
Table 1
$NPV
Description
q f = q u = qcash
= 0.80
γ=U=0
( discounted cash flows )
[ firm cost of equity, WACC ]
q f = q u = qcash
q f = 1.02
qu = qcash = 0.80
= 1.0
γ = U = 0.88
γ = U = 0.51
q f = 1.15
qu = qcash = 0.77
γ = U = 0.88
PANEL A – no adjustment for dividend tax effects
The traditional method
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
PANEL B – the project distributes fully-franked dividends
Officer’s (1994) method
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$4.77
( -$30.0, 10 x $5.0 )
[8.7%, 7.2%]
$3.71
( -$30.0, 10 x $5.0 )
[9.8%, 7.9%]
$4.77
( -$30.0, 10 x $5.0 )
[8.7%, 7.2%]
The q method
-$3.53
( -$30.0, 10 x $4.0 )
[11.2%, 8.3%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$2.30
( -$30.0, 10 x $5.1 )
[12.1%, 9.3%]
$5.50
( -$30.0, 10 x $5.75)
[12.6%, 9.8%]
PANEL C - the project distributes unfranked dividends
Officer’s (1994) method
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%, 9.2%]
The q method
-$3.53
( -$30.0, 10 x $4.0 )
[11.2%, 8.3%]
$1.82
( -$30.0, 10 x $5.0 )
[12.0%. 9.2%]
-$4.14
( -$30.0, 10 x $4.0 )
[12.1%. 8.8%]
-$5.38
( -$30.0, 10 x $3.85)
[12.6%. 9.1%]
PANEL D – the project distributes fully-franked dividends (as PANEL B) and is financed internally with cash that could
otherwise be distributed as fully-franked dividends
$2.47
$1.82
$1.70
$1.00
The q method
( -$24.0, 10 x $4.0 ) ( -$30.0, 10 x $5.0 ) ( -$30.6, 10 x $5.1 ) ( -$34.5, 10 x $5.75)
[11.2%, 8.3%]
[12.0%. 9.2%]
[12.1%. 9.3%]
[12.6%. 9.8%]
PANEL E – the project distributes unfranked dividends (as PANEL C) and is financed internally with cash that could
otherwise be distributed as unfranked dividends
$2.47
$1.82
$1.86
$1.52
The q method
( -$24.0, 10 x $4.0 ) ( -$30.0, 10 x $5.0 ) ( -$24.0, 10 x $4.0 ) ( -$23.1, 10 x $3.85)
[11.2%, 8.3%]
[12.0%. 9.2%]
[12.1%. 8.8%]
[12.6%. 9.1%]
The table compares the NPVs as well as the computed cost of equity and WACC for a project that requires an
investment of $30 million and is expected to generate unlevered cash flows of $5 million a year for ten years.
The project is valued using the traditional method of ignoring imputation, adjusting for imputation using
Officer’s (1994) model, and encapsulating the market value of dividends using q (the ratio of the market value
of dividends to their face value (qf for franked, qu = qcash for unfranked dividends)). Discounting is performed by
the weighted average cost of capital which differs according to the valuation method used (detailed WACC
calculations are given in Appendix B). In the case of the traditional method and application of Officer’s model,
the $5million is the discounted cash flow; in the q method, the $5 million is multiplied by q. In the case of
methods that allow for the value of imputation credits the analysis is extended to consider the valuation for both
unfranked and franked dividends. The q method of analysis is further extended to consider the difference in
valuations that arises when financing the project externally as opposed to financing with retained earnings, in
which latter case the opportunity cost of the $30 initial investment is taken to be such cost as a distribution to
shareholders. The first two columns represent “base cases.” The first column shows that application of Officer’s
model reduces to the traditional method when U = 0. The second column shows that the q method reduces to the
traditional method when q = 1. Alternative values for q and U in columns 2 and 3 are based respectively on
Officer and Hathaway (2004) and Walker and Partington (1999).
27