Schmalenbach Business Review ◆ Vol. 54 ◆ April 2002 ◆ pp. 136 – 147
Frank Richter*
SIMPLIFIED DISCOUNTING RULES, VARIABLE GROWTH,
AND LEVERAGE**
ABSTRACT
It was shown earlier that unconditional expected cash flows can be discounted at a constant risk-adjusted discount rate if these cash flows follow a multiplicative binomial
process with a constant growth rate (“simplified discounting rule”). This paper extends this
analysis to the case of variable growth rates of expected cash flows. The earlier analysis
was also based on the assumption of all-equity-financing. The impact of a financing strategy based on deterministic leverage ratios is included in this paper as well. It analyses the
conditions to apply the weighted average cost of capital as a discount rate. The paper
reflects the results of Modigliani/Miller (1963), Miles/Ezzell (1980), and Löffler (1998) visà-vis the background of this theory and discusses issues of practical application.
JEL-Classification: G32.
1 INTRODUCTION
In a recent paper various simplified discounting rules were discussed that value
uncertain cash flow streams based on their unconditional expectation with a constant risk-adjusted discount rate 1. However, the model presented there is restricted
to constant growth rates of unconditional expected cash flows and a financing
strategy that includes only equity. This current paper aims to extend the earlier
analysis in both dimensions: It presents a model with time-varying growth rates
that includes the earlier model as a special case. In addition, debt financing is
introduced on the basis of not necessarily constant, but deterministic, leverage
ratios, defined as the market value of debt divided by the market value of debt
and equity. Sufficient conditions for the application of the weighted average cost
of capital (WACC) as a means to cover the tax implications of the suggested
financing strategy will be analysed.
* Prof. Dr. Frank Richter, Institute for Mergers & Acquisitions University Witten/Herdecke, e-mail:
[email protected].
** I would like to thank the referee for helpful comments.
1 This approach is called a simplified discounting rule because it takes into account the stochastic
process of the uncertain cash flow stream only on the basis of its unconditional expectations.
There is no need, e.g., to enumerate the binomial model by valuing unconditional expectations at
all nodes of the tree if the conditions for the simplified discounting rule are fulfilled. See Richter
(2001).
136
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Discounting Rules
The paper begins with assumptions on the structure of future cash flows and the
fundamental theorem used to value this stream. Then the analysis examines a
single cash flow in a time setting with just one period. Next, I extend the analysis
to value a cash flow stream in a multiperiod setting, which takes into account the
history of the cash flow stream. Thus, the derived model is able to include conditional expectations of future cash flows. Issues of practical application are
addressed in the last paragraph.
2 BASIC
ASSUMPTIONS AND DEFINITIONS
(A) BINOMIAL MODEL
The binomial model is characterized by the following process for future cash
flows:
{
C̃ t ∈ C̃ t −1u t;C̃ t −1d t
}
with 2 : t = 1,2,KT.
(1)
I assume ut ≥ 1 and 0 < dt ≤ ut. Thus, future cash flows are certain only if
ut = dt = 1 for all t. P and Q are two probability measures that can be applied to
the cash flow process. This cash flow process is called a martingale under P if
gt = put + (1 – p)dt = 1, for all t. A submartingale (respectively, supermartingale)
describes a process with gt > 1 (or gt < 1, respectively) 3. These processes are characterized by deterministic growth rates and conditional expectations, i.e.
~ |F ] = C
~ g.
EP[C
t
t −1
t−1 t
I use the binomial model here because it can easily be generalized, e.g., by transforming it into diffusion models in continuous time provided appropriate growth
factors. Furthermore, it is probably the simplest model for analysing the valuation
of uncertain cash flow streams, although it includes important features such as the
change in expectations based on the arrival of new information (embedded in
realized cash flows). Finally, it can be applied in practice without any further analytical knowledge than is needed for a discounted cash flow analysis.
(B) FUNDAMENTAL
VALUATION THEOREM
The measure Q is defined as an equivalent probability measure to P. Doing so
does not mean that the probabilities p and q are identical, but rather suggests that
an event, which occurs with some probability under P, also occurs under Q. The
equivalent probability measure Q is defined such that it transforms the conditional
2 t = 0 symbolises the present and C0 ≥ 0 is the known non-negative cash flow of the period that has
just ended.
3 See e.g. Irle (1998), p. 44; Karatzas/Shreve (1991), p. 11; Duffie (1988), p. 135.
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137
F. Richter
expected value of an uncertain cash flow in T (given its history or filtration in
T − 1) 4 into its economic value in T 5:
VT [C̃T ] = E Q [C̃T | FT−1 ].
(2)
The expectation under Q is called the certainty equivalent of the uncertain cash
flow. Investors value an uncertain cash flow with the certainty equivalent of some
amount equal to a certain cash flow of the same amount. The probability q that is
used to quantify the conditional expectation under Q shall be called “certainty
equivalent probability” 6.
The risk-free rate is used to discount certainty equivalents obtained by applying
the certainty equivalent probability in quantifying the expectation:
(3)
Vt [C̃T ] = E Q [C̃T | Ft ](1 + rf ) −(T− t) .
Equation (3) serves as the fundamental valuation theorem for the following analysis.
3 VALUATION
(A) ALL
OF SINGLE CASH FLOWS
EQUITY FINANCING
The index “e” indicates unlevered values (in the sense of “equity financed only”,
“without any leverage”), unlevered cash flows and unlevered cost of capital. The
fundamental valuation theorem of course applies to unlevered cash flow streams.
The first step is to use the theorem in a single period setting in order to determine
unlevered cost of equity 7:
!
Vte−1 = E Q [C̃et | Ft −1 ](1 + rf ) −1 = E P [C̃ et ](1 + rte ) −1 .
(4)
Inserting the structure of the cash flow process yields:
(1 + rte ) = x t (1 + rf )
with: xt =
pu t + (1 − p)d t
.
qu t + (1 − q)d t
(5)
4 See Duffie (1988), pp. 130 – 137, for an introduction to conditional expectations and the concept of
filtration.
5 This valuation theorem is also the basis for the so-called “real options” approach. See, e.g.,
Brennan/Schwartz (1985).
6 I suggest distinguishing between the measure Q as used here and the so-called equivalent martingale measure used in the context of option pricing theory. The probability q to value a cash flow
stream can be different from the probability used to value options.
7 The expectation under P without reference to a filtration indicates unconditional expectations.
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The certainty equivalent and the expectation under P differ only by a non-stochastic factor, xt, which can be used to “gross up” the risk-free rate to obtain the
appropriate risk-adjusted discount rate. I call this property of the model the “deterministic growth relation” that determines xt as the risk adjustment.
(B) DETERMINISTIC
LEVERAGE RATIO
The levered value, as indicated by the index “l”, consists of the unlevered value
plus the value of tax savings due to interest expense under a simple tax regime
that prefers debt financing rather than equity financing:
Vtl = Vte + Vtτ .
(6)
Under the assumption of a fixed leverage ratio the value of the tax savings is:
Vtτ−1 = E Q [τrf l t −1 Vtl−1 | Ft −1 ](1 + rf ) −1 .
(7)
Combining expressions shows that the unlevered value equals the levered value
times a non-stochastic factor, yt:
Vtl−1 = Vte−1y −1
t
with: yt = 1 −
τrf l t −1
.
1 + rf
(8)
I call this property the “deterministic leverage relation” that determines yt as the
financing adjustment. Using this property in combination with the deterministic
growth relation leads to the following expression for the levered value:
Vtl−1 =
E P [ C̃ t ]
.
(1 + rf )x t y t
(9)
The interesting question is now whether the multiplicative adjustment of the riskfree rate still obtains in a multiperiod setting. But before I come to that, I analyse
the discount rate, which reflects both the risk adjustment and the financing adjustment.
(C) WEIGHTED
AVERAGE COST OF CAPITAL
Inserting the expressions for x and y leads to the discount rate:
1 + rte 
τrf l t −1 

1 −
1 + rf 
1 + rf 
= (1 + rte ) (1 − τ * l t −1 )
with: τ * = τrf / (1 + rf )
≡ 1 + wacc t .
(1 + rf )x t y t = (1 + rf )
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(10)
139
F. Richter
This formula can be rewritten to obtain the so-called “textbook formula” presented
by Modigliani/Miller (1963):
wacc t = rtl (1 − l t −1 ) + rf (1 − τ)l t −1 .
(11)
The levered cost of capital is calculated under the assumptions made as follows:
rtl = rte + (rte − rf )(l − τ * )
l t −1
.
l − l t −l
(12)
Notice that this is the formula for the cost of equity that follows from the analysis
of Miles/Ezzell (1980) and Löffler (1998). These formulas require the deterministic
financing and the deterministic growth relations to hold.
(D) IMPLICATION
FOR THE VALUATION OF THE TAX SHIELD
The value of the tax shield is the difference between the values of the levered firm
and the unlevered firm. Nevertheless, I apply the fundamental valuation theorem
to provide further insight into the implicit assumptions of (10) and (11) – (12):
Vtτ = E Q [τrf D t | Ft ](1 + rf ) −1.
(13)
Given the filtration Ft, all variables with the index t are known at that point in
time. The interest payment and the tax shield in period t + 1 depend on the debt
value Dt, which is known in t. Thus, the value of the tax shield in t is the tax rate
applied to the interest payment, discounted at the risk-free rate:
Vtτ = τrf D t (1 + rf ) −1 = τ * D t .
(14)
As can be derived from (8), τ *Dt is indeed the difference between the levered and
the unlevered firm value, because Dt = ltV tl.
4 VALUATION
(A) ALL
OF CASH FLOW STREAMS
EQUITY FINANCING
In a multiperiod setting, the unlevered cost of equity is defined as:
!
V0e = ∑ Tt =1E Q [ C̃et | F0 ](1 + rf ) −t = ∑ Tt =1E P [ C̃et ](1 + R et ) −t .
(15)
To isolate the unlevered cost of equity, start at the end of the planning horizon at
t = T and assume VTe = VTl = VTτ = 0. These values exclude the final cash flow paid
out in that period. For the unlevered value in the previous period one gets:
e
ṼT−1
= E Q [ C̃eT | FT−1 ](1 + rf ) −1.
140
(16)
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Discounting Rules
Moving backwards in time yields the value at T − 2 8:
e
e
ṼT−2
= E Q [ C̃ eT−1 + ṼT−1
| FT−2 ](1 + rf ) −1
(17)
= E Q [ C̃ eT−1 + C̃ eT (1 + rf ) −1 | FT−2 ](1 + rf ) −1
= E Q [ C̃ eT−1 | FT−2 ](1 + rf ) −1 + E Q [ C̃eT | FT−2 ](1 + rf ) −2 .
Continuing the procedure of backward iteration leads to the present value of the
unlevered cash flow stream:
V0e = ∑ Tt =1E Q [ C̃et | F0 ](1 + rf ) −t .
(18)
Notice that the present value at t = 0 is not a random variable. It is, as can be seen
by the filtration F0, the unconditional expectation of the future cash flow stream
under the equivalent probability measure Q, discount at the risk-free rate.
Next, use the deterministic growth relation and again the law of iterated expectations:
E P [ C̃eT ] = C̃ eT−1g T .
(19)
Taking expectations on both sides and expanding the new expression yields:
E P [ C̃eT ] = E P [ C̃ eT−1 ]g T = C̃ eT−2 g T g T−1 .
(20)
Continuing this procedure until the present leads to:
E P [ C̃eT ] = Ce0 ∏ Tt =1g t .
(21)
Note that the unconditional expectation of the future cash flow stream is not a
random variable. With that, define the auxiliary function Xt as follows:
Xt =
E P [ C̃et ]
E Q [ C̃et
| F0 ]
=
t
∏ i=1(pu i + (1 − p)d i )
t
= ∏ i=1
xi .
t
∏ i=1(qu i + (1 − q)d i )
(22)
Given that neither the unconditional expectation of future cash flows under P is
random, nor is the conditional expectation under Q with respect to the filtration
F0, the multiperiod growth relation is deterministic as well. Thus, I can rewrite the
valuation function as:
V0e = ∑ Tt =1E Q [ C̃et | F0 ](1 + rf ) −t
(23)
−t
= ∑ Tt =1E P [ C̃et ]X −1
t (1 + rf ) .
8 We use the law of iterated conditional expectations; see, e.g., Duffie (1988), p. 84.
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141
F. Richter
Based on that the multiperiod unlevered cost of equity is determined:
1 + R et =
X t (1 + rf ) .
t
(24)
This is the extension of the simplified discounting rule no. 3 in Richter (2001),
which is based on constant growth. g = constant implies Xt = xt. Using this expression in (24) shows that if the growth rate is constant, so too is the unlevered cost
of equity.
(B) DETERMINISTIC
NON-CONSTANT LEVERAGE RATIO
Lets proceed as in the case of pure equity financing by using the recursive
approach, which begins with the value at the end of the planning period.
However, here both the unlevered cash flow and the tax savings due to debt
financing have to be taken into account. The levered firm value in T is zero, the
value in T − 1 is:
l
VT−1
= E Q [ C̃eT + τrf D T−1 | FT−1 ](1 + rf ) −1
(25)
= E Q [ C̃eT | FT−1 ](1 + rf ) −1 + τrf D T−1 (1 + rf ) −1
l
= E Q [ C̃eT | FT−1 ](1 + rf ) −1 + τrf l T−1VT−1
(1 + rf ) −1
l
⇒ VT−1
= E Q [ C̃eT | FT−1 ](1 + rf ) −1 y −1
T .
In period T − 2 we have:
l
l
VT−2
= E Q [ C̃ eT−1 + VT−1
+ τrf D T−2 | FT−2 ](1 + rf ) −1
(26)
−1
l
= E Q [ C̃ eT−1 + C̃ eT (1 + rf ) −1 y −1
+ τrf l T−2 VT−2
(1 + rf ) −1
T | FT−2 ](1 + rf )
−2 −1
l
e
−1
⇒ VT−2
= E Q [ C̃ eT−1 | FT−2 ](1 + rf ) −1 y −1
T−1 + E Q [ C̃ T | FT−2 ](1 + rf ) y T−1 y T .
Continuing this procedure until the present yields the following valuation formula 9:
V0l = ∑ Tt =1E Q [ C̃et | F0 ]
1
(1 + rf ) t Yt
t
t
with : Yt = ∏ i=1
y i = ∏ i=1
(1 − τ * l i−1 ).
(27)
Using the deterministic growth relation implies immediately:
V0l = ∑ Tt =1
E P [ C̃et ]
(1 + rf ) t X t Yt
.
(28)
9 This adjustment was suggested earlier in Richter (1998), p. 383, formula (11). However, it did not
have the foundation we have now.
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The auxiliary function X transforms the F0-expectation under Q into the unconditional expectation under P. Once this is done, I have unconditional expectations
for the cash flow that must be discounted at the cost of capital. The tax shield can
be taken into account using the auxiliary function Y.
In a final step, the auxiliary function Z is used in combination with the risk and
the financing adjustments to establish the main result of this paper:
V0l
= C e0 ∑ Tt =1
Zt
(1 + rf ) −t
X t Yt
(29)
with :
pu i + (1 − p)d i
qu i + (1 − q)d i
Xt
t
= ∏ i=1
Yt
t
= ∏ i=1
(1 − τ * l i−1 )
“the financing adjustment”
Zt
t
= ∏ i=1
(pu i + (1 − p)d i )
“the growth adjustment”.
“the risk adjustment”
By comparing X and Z it becomes clear that the growth expectation under P is not
needed, because terms cancel out. Therefore the valuation formula simplifies:
V0l
= C e0 ∑ Tt =1
Z ′t
(1 + rf ) −t
Yt
with :
t
Z ′t
= ∏ i=1
(qu i + (1 − q)d i )
(30)
“the combined growth and risk adjustment”.
Formula (30) uses a risk-adjusted growth rate to generate certainty equivalents,
which in turn must be discounted at the risk-free rate.
(C) WEIGHTED
AVERAGE COST OF CAPITAL
The risk-free rate can also be combined with the risk and financing adjustments to
obtain the weighted average cost of capital similar to the single-period case:
V0l = C e0 ∑ Tt =1
Zt
(1 + WACC t ) t
(31)
with:
1 + WACC t =
t
X t Yt (1 + rf )
“the combined risk and financing adjustment”.
The discussion around the application of WACC has become quite sophisticated.
On the one hand, it is impressive to see that the structure of the Modigliani/Miller
formula still applies, if adjusted for time-dependent leverage ratios and for τ*
instead of τ. This conclusion becomes evident if (24) and (31) are rewritten as
follows:
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143
F. Richter
(1 + R et ) t = X t (1 + rf ) t = ∏ ti=1 x i (1 + rf ) = ∏ ti=1(1 + rie )
(32)
(1 + WACC t ) t = X t Yt (1 + rf ) t = ∏ ti=1(1 + rie )(1 − τ * l i−1 ) = ∏ ti=1(1 + wacci ).
On the other hand, implicit and restrictive assumptions become transparent, such
as the deterministic growth and leverage relations.
(D) IMPLICATION
FOR THE VALUATION OF THE TAX SHIELD
Next, I clarify the implication of the multiperiod WACC approach for the value of
the tax savings. In period T, the value of the tax shield is zero, because no interest
payments will follow afterwards. In the previous period I have:
τ
VT−1
= E Q [τrf DT−1 | FT−1 ](1 + rf ) −1 = τrf DT−1 (1 + rf ) −1 .
(33)
This result is consistent with the result obtained in the single period case. Now, in
the previous period the tax shield of period T − 1, V τT − 1, must be taken into
account as well as:
−1
τ
τ
VT−
2 = E Q [τrf DT− 2 + VT−1 | FT− 2 ](1 + rf )
(34)
τ
= τrf DT− 2 (1 + rf ) −1 + E Q [VT−1
| FT− 2 ](1 + rf ) −1 .
To transform the remaining expectation under Q into an expectation under P, I
make use of the fact that the value of the tax shield equals the value of the
levered value minus the value of the unlevered value:
τ
l
e
E Q [VT−1
| FT− 2 ](1 + rf ) −1 = E Q [VT−1
− VT−1
| FT− 2 ](1 + rf ) −1
(35)
e
−1
= (y −1
T−1 − 1)E Q [VT−1 | FT− 2 ](1 + rf )
−1
e
= (y −1
| FT− 2 ](1 + rf ) −1 .
T−1 − 1)E Q [C̃ T (1 + rf )
As before, the deterministic growth relation is used to obtain the expectation
under P. Start on the right-hand side before imposing the same manipulation to
the left-hand side:
−1
−2 −1
e
e
(y −1
| FT− 2 ](1 + rf ) −1 = (y −1
T−1 − 1)E Q [C̃ T (1 + rf )
T−1 − 1)E P [C̃ T ](1 + rf ) x T−1 .
(36)
The factor xT−1− 1 transforms the right-hand side of the equation into an expectation
under P. So it does the same with the left-hand side, because the uncertain variables differ only by a non-stochastic factor, which is (yT−1− 1−1)(1 + rf )−1. Thus, the
value of the tax shield can be transformed as follows:
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sbr 54 (2/2002)
Discounting Rules
−1
τ
τ
τ
e
E Q [VT−1
| FT− 2 ](1 + rf ) −1 = E P [VT−1
](1 + rf ) −1 x −1
T−1 = E P [VT−1 ](1 + rT−1 ) .
(37)
Collecting results yields:
τ
e
VΤτ − 2 = τrf DT− 2 (1 + rf ) −1 + E P [VT−1
](1 + rT−1
) −1
(38)
e
= τrf DT− 2 (1 + rf ) −1 + τrf E P [DT−1 ](1 + rf ) −1 (1 + rT−1
) −1 .
Continuing this procedure until t = 0 gives the present value of the tax shield:
V0τ = ∑ Tt =1 τrf E P [Dt ](1 + rf ) −1 (1 + R te−1 ) −(t −1) .
(39)
This result contains those of Miles/Ezzell (1980) and Löffler (1998) for the case of
constant growth rates: The expected tax savings in t are discounted once at the
risk-free rate and (t − 1) times at the unlevered cost of equity.
5 PRACTICAL
APPLICATION
(A) SPECIFYING
KEY PARAMETERS OF THE MODEL
To apply the valuation model derived here, investors need to be able to articulate
their expectations in terms of the probability p, the growth factors u (“maximum
growth”), and d (“minimum growth”) – p, u, d determine the expected growth
rate g. Of course, a proxy for risk-free rate rf must exist.
In addition, one needs to know how p translates into q. In a first step it is helpful
to assume that there is a constant factor γ, such that q = γ p with 0 < γ ≤ 1. I assume
that p and g are constant through time. This is not a restriction, but assures that
identical information on up and down movements is being transformed systematically in the same manner into growth rates of expected cash flows and certainty
equivalents. Notice that for γ → 0 we have the case of maximum risk aversion,
which in turn yields the minimum value of the cash flow. This is the minimum
value because only down movements are taken into account, provided that q is the
certainty equivalent probability for the up movement. Investors are risk-neutral for
the case of γ = 1, implying p = q and a valuation based on the subjective expectation without any further consideration of risk. Risk-averse investors assign a value
to the uncertain cash flow stream that is higher than the minimum value and lower
than the maximum value obtained by risk-neutral investors.
I want to point out that without any further specification of γ, the range of possible values between the minimum V[C, γ = 0] and the maximum V[C, γ = 1] is determined. All that is needed is the specification of the stochastic process for the
unlevered cash flow stream, the risk-free rate, the corporate tax rate, τ, and the
leverage ratio l 10.
10 There are ways to determine a point-estimate within the range of possible values, e.g., by applying
an equilibrium capital asset pricing model. These, however, are not the focus of this paper.
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F. Richter
(B) TWO-STAGE
MODEL
Given that the assumptions on the growth profile of future cash flows and the
financing policy cannot be forecast for many years, practitioners usually differentiate the planning period into two sub-periods: In the first period from t = 0 until
t = t* the growth rate and the leverage ratio can be time dependent. In the second
sub-period, i.e. for t > t*, simplifying assumptions are used (see next paragraph):
*
V0l = C e0 ∑ tt =1
Z′
Z ′t
(1 + rf ) −t + C e0 ∑ Tt =t * +1 t (1 + rf ) −t
Yt
Yt
(40)
⇔ Ce0 ∑ t*t =1
* Z ′* Z ′
*
Z ′t
t
(1 + rf ) −t (1 + rf ) −t
(1 + rf ) −t + C e0 ∑ T−t
t =1
Yt * Yt
Yt
⇔ Ce0 ∑ t*t =1
Z ′*
*
* Z′
Z ′t
t
(1 + rf ) −t + C e0 t (1 + rf ) −t ∑ T−t
(1 + rf ) −t .
t =1
Yt
Yt *
Yt
Finally, re-substitute T for t* and expand the forecast horizon to infinity to complete the separation for the two-stage model.
Z′
Z′
Z′
V0l = C e0 ∑ Tt =1 t (1 + rf ) − t + Ce0 T (1 + rf ) − T ∑ ∞t = T+1 t (1 + rf ) −(t − T) .
Yt
YT
Yt
144
42444
3
14444
24444
3
:= m T
(41)
:= m ∞
The second component of this valuation equation is called “terminal value” or
“continuing value”. Notice, that this expression here represents the certainty equivalent of this value that must be discounted at the risk-free rate. For convenience
define the valuation multiples m.
(C) CONVERGENCE
ASSUMPTIONS FOR THE TERMINAL VALUE
If certain factors such as the growth rate and the leverage ratio are constant, then
the valuation multiples simplify. I use the result from the previous section to illustrate this:
m T = ∑ Tt =1
Z ′t
(1 + rf ) −t
Yt
t
with: Z ′t = ∏ i=1
(qu i + (1 − q)d i ) .
(42)
First, consider the case in which the growth rate of the unconditional expectation
is constant, i.e., ut = u and dt = d for all t, and define (1 + a) = qu + (1 − q) d as the
risk-adjusted growth factor. Second, assume the leverage ratio to be constant for
all t. With this, one gets:
m T = ∑ Tt =1(1 + a) t (1 + rf ) −t y −t
=∑
146
T
t =1(1 +
(43)
a) ((1 + rf )(1 − τ l))
t
*
−t
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Discounting Rules
= ∑ Tt =1(1 + a) t (1 + rf (1 − τl)) −t
T
 

(1 + a) 
1+ a
=
1 − 
 .
rf (1 − τl) − a 
 1 + rf (l − τL)  
Finally, if T approaches infinity one gets a version of the constant growth model
if, of course rf (1 − τl) > a:
m∞ =
1+a
.
rf (1 − τl) − a
(44)
To determine the weighted average cost of capital under the present scenario,
solve the following equation for (1 + wacc):
1+g
1+a
=
rf (1 − τl) − a wacc − g
1+g
⇔ 1 + wacc =
(rf (1 − τl) − a) + 1 + g
1+a
r (1 − τl) − a 

 1 + rf (1 − τl) 
= (1 + g)  1 + f
 = (1 + g) 





1+a
1+a
*
= x(1 + rf (1 − τl)) = x(1 + rf )(1 − τ l)
= (1 + re )(1 − τ * l).
(45)
Comparison with (10) shows that this is equals one plus the weighted average
cost of capital.
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