Consistent methods of valuing companies by DCF:
Methods and assumptions
Ian Cooper*
Kjell Nyborg**
*London Business School, Sussex Place, Regent’s Park, London NW1 4SA, UK
**Norwegian School of Economics and Business Administration (NHH), Hellevein
30, 5045 Bergen, Norway, and CEPR
August 2006
In this note we discuss four common methods of valuing firms:
1.
2.
3.
4.
Discounting operating free cash flow at the weighted average cost of capital.
Discounting equity free cash flow at the cost of equity.
Valuing the firm using adjusted present value.
Discounting the capital cash flow at the unlevered cost of capital.
We examine four alternative assumptions about leverage policy and how they affect three
things: discount rates, the present value of tax savings, and how to use the above methods.
We describe alternative ways of implementing the valuation methods consistently, and
how to choose between them. Finally, we show how inconsistent application can lead to
errors that are subtle but large. The use of incorrect formulas can result in an estimate of
the present value of the tax saving that is double its correct value.
JEL classification: G12; G31; G32; M21
Keywords: discounted cash flow, APV, WACC, leverage policy, value of tax shields,
equity cash flow
1.
Introduction
Several alternative approaches can be used to incorporate the effect of debt when valuing
the equity of a firm. For instance, Fernandez (2004a) discusses ten methods. Of these,
four are relatively common:
1. Discount operating free cash flow at the weighted average cost of capital
(WACC). Then subtract the value of debt.
2. Discount equity free cash flow at the cost of equity.
3. Value the firm using adjusted present value (APV). Then subtract the value of
debt.
4. Discount the capital cash flow at the unlevered cost of capital. Then subtract the
value of debt.
All the methods are derived from the same general idea: the value of a levered firm equals
the unlevered value of the firm plus the present value of the tax shields arising from debt
financing (PVTS). However, none of the methods is fully defined without specifying the
discount rate used to perform the valuation. These discount rates depend on the
assumption regarding the leverage policy of the firm. The assumption about leverage
policy determines how to estimate the discount rate, which of the above valuation
methods may be used, and also the size of PVTS. All consistent valuation should start
from a clear assumption about leverage policy.
In this note we discuss alternative assumptions about leverage policy and how they affect
three things: discount rates, PVTS, how to use the above methods. We describe
alternative ways of implementing the valuation methods consistently, and how to choose
between them. Finally, using as examples Booth (2002) and Fernandez (2004a and b), we
show how inconsistent application can lead to errors that are subtle but large. The use of
incorrect formulas can result in an estimate of PVTS that is double its correct value.
2. Basic theory
The issue addressed by all these methods is the valuation of a company including the
present value of the tax saving from debt. In a complete capital market, value-additivity
holds and the relationship underlying all methods is:1
E+D = VL = VU + PVTS
(1)
Where:
E is the market value of equity
D is the market value of debt
VL is the value of the levered firm
VU is the value of the same firm without any leverage (the ‘unlevered firm’)
PVTS is the present value of the tax saving from debt
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All standard valuation methods are derived from equation (1). They differ only in the
assumption that they make about the future debt policy and tax status of the firm and,
consequently, the level and risk of the future tax saving from debt.2
3. Cash flows and discount rates
All the valuation methods involve forecasting a set of future cash flows and discounting
them at an appropriate discount rate. The relevant cash flows are:
Operating free cash flow (FCF): Free cash flow after tax, assuming that the firm is
financed entirely with equity.
Equity free cash flow (ECF): Free cash flow available for equity holders.
Capital cash flow (CCF): Free cash flow available to the combination of debt and
equity holders.
The relationships between them are:
ECF = FCF – Interest + Tax saved from interest – Net debt repayments
(2)
CCF = FCF + Tax saved from interest
(3)
Valuation also requires discount rates. Usually these are derived using the capital asset
pricing model from estimates of the riskless interest rate, equity beta, debt beta, and
market risk premium: 3
KE = RF + βE*Π
(4)
KD = RF + βD*Π
(5)
Where:
RF is the riskless interest rate
Π is the market risk premium
KE is the cost of equity
KD is the cost of debt
βE is the equity beta
βD is the debt beta
It is also common to derive the weighted average cost of capital:
WACC = (E/VL)*KE + (D/ VL)*KD*(1-T)
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where:
WACC is the weighted average cost of capital
E is the market value of equity
D is the market value of debt
T is the corporate tax rate.
Other related discount rates and betas, such as the unlevered cost of capital, KU, and the
unlevered beta, βU, are also used in valuation. However, the correct way to derive these
depends on the specific assumption about leverage. They are discussed in the following
sections under each alternative assumption about debt policy.
In summary, the four methods are:
Method 1: Discount FCF at WACC to give VL. Subtract D to give E.
Method 2: Discount ECF at KE to give E.
Method 3: Discount FCF at KU to give VU. Add PVTS to give VL. Subtract D to
give E.
Method 4: Discount CCF at KU to give VL. Subtract D to give E.
In each case, we assume that the method is implemented with a constant discount rate.
This is the standard approach. There are more complex implementations that involve
time-varying discount rates and we also mention these, where appropriate.
4. Four alternative assumptions about debt policy
Four alternative assumptions about debt policy are sometimes used:
1. Operating cash flow is a risky flat perpetuity combined with a constant amount of
debt.
2. Constant proportional market-value leverage.
3. Any arbitrary non-constant leverage policy, with tax savings from debt that have
the same risk as the operating free cash flows.
4. Any arbitrary non-constant leverage policy, with tax savings from debt that have
the same risk as the debt.
Not all of these assumptions about debt policy are consistent with all valuation methods.
Furthermore, they imply different discount rates and different levels of PVTS.
This section discusses each assumption, and gives the formulas for discount rates that are
consistent with it. Section 5 shows how to calculate PVTS, and Section 6 explains which
methods of valuation are consistent with each assumption, and gives criteria for choosing
between the assumptions and methods.
4.1 Operating cash flow a risky flat perpetuity combined with a constant amount of
leverage (Modigliani-Miller)
The first rigorous analysis of the impact of leverage on value was given in Modigliani and
Miller (MM) (1963). Their leverage assumption is a constant amount of debt combined
with the assumption that the operating free cash flow is a risky perpetuity. This
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assumption is highly restrictive and will not, in general, correspond to the actual situation
of a firm that is being valued. Despite this, the MM equations for discount rates and
PVTS are commonly used. For instance, they are used in a leading corporate finance
textbook, Grinblatt and Titman (2002).
The formulas that are consistent with the assumption of a constant amount of debt and
cash flows that are a risky perpetuity are:
Starting from equity and debt betas, the unlevered beta is derived by:
βU = (E/(VL-T*D))* βE + (E/(VL-T*D))*βD*(1-T)
(7)
The unlevered cost of capital is related to the costs of debt and equity by:
KU = (E/(VL-T*D))*KE + (E/(VL-T*D)*KD*(1-T)
(8)
The relationship between the WACC and KU is given by:
WACC = KU*(1 – T*(D/VL))
(9)
Alternatively, starting from the unlevered beta and unlevered cost of capital, the equity
beta is:
βE = βU + (βU - βD)*(D*(1-T)/E)
(10)
The cost of equity is:
KE = KU + (KU - KD)*(D*(1-T)/E)
(11)
With a zero debt beta:
βE = βU*(E/(VL-T*D))
(12)
The MM formulas assume that the debt level is constant and the expected operating cash
flow is a flat perpetuity. The formulas do not hold if either of these assumptions is not
true. In particular, they do not hold if the operating free cash flow is expected to grow.
4.2 Constant proportional market-value leverage (Miles-Ezzell)
One of the attractions of the MM assumptions is that they lead to simple formulas for
discount rates. Another assumption that also does this is constant proportional marketvalue leverage. This is the only assumption that is generally consistent with the standard
use of the WACC, regardless of the profile of future cash flows. Its implications were
first shown in Miles and Ezzell (1980), so it is sometimes known as the Miles-Ezzell
(ME) approach. It underlies most of the formulas given in Brealey and Myers (2006).4
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The formulas for discount rates that are consistent with constant proportional market
value leverage are:
Starting from equity and debt betas, the unlevered beta is derived by:5
βU = (E/VL)* βE + (D/VL)*βD
(13)
The unlevered cost of capital is related to the costs of debt and equity by:
KU = (E/VL)* KE + (D/VL)*KD
(14)
The relationship between the WACC and KU is given by:
WACC = KU –T*KD*(D/VL)
(15)
Alternatively, starting from the unlevered beta and unlevered cost of capital, the equity
beta is derived by:
βE = βU + (βU - βD)*(D/E)
(16)
The cost of equity is derived by:
KE = KU + (KU - KD)*(D/E)
(17)
Some people make the further assumption that the debt beta is zero:
βE = βU*(E/VL)
(18)
These ME formulas look similar to the MM formulas. The similarity arises because both
assume a constant proportion of leverage. ME simply assumes that future debt levels will
be set to maintain constant leverage. In contrast, the MM case has constant leverage
because of a very special combination of assumptions: a constant level of debt combined
with a constant expected operating cash flow. In the MM case, although the cash flow
each period is risky, once each period’s cash flow has been received the value of the firm
is always the same. Thus there are two highly restrictive features of the MM setup. One is
that it assumes a static amount of debt. The other is that it assumes a static ex-cash-flow
value of the firm. The combination gives a constant leverage ratio.
The crucial difference between the ME and MM cases lies in the riskiness of the tax
savings from interest. In the MM case the debt level is fixed, so tax savings from interest
have risk the same as debt. In the ME case, the future amount of debt and interest is tied
to future operating cash flows. So the tax savings from interest have risk equal to the
operating cash flows. This is what causes the formulas for discount rates to differ in the
two cases, even though the leverage ratio is constant in both. Failure to recognise this
distinction between ME and MM can result in significant confusion in the valuation of tax
shields, as we discuss below.
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4.3 An arbitrary leverage policy, with tax savings that have the same risk as the
operating free cash flow (extended ME)
An alternative to both MM and ME is to assume directly that the tax savings from interest
have the same risk as the operating free cash flow. This might arise if there is uncertainty
about whether the firm will be tax-paying in the future, leading to the risk of the tax
savings reflecting the riskiness of the operating cash flows.6
In that case, the correct valuation procedure is simple. It consists of valuing capital cash
flows by discounting at the unlevered cost of capital. Since the tax savings from interest
have the same risk as the operating free cash flow, they may be pooled with the operating
free cash flow. The total, which is the capital cash flow, may then be discounted at the
unlevered cost of capital. The procedure is undoubtedly an approximation, but it is
advocated for the valuation of tax savings in some highly leveraged transactions (Ruback
(2002)). It can accommodate any pattern of future cash flows and future leverage, but it
does require a forecast of the expected debt level in order to calculate the expected capital
cash flow.
In the extended ME case, the tax savings from interest have the same risk as in the ME
case, but the reason for this is different. In the ME case, a constant proportional leverage
policy forces the tax savings from interest to have the same risk as the operating cash
flows. It is assumed that the firm will always be paying taxes, but the amount of interest
will vary in line with the operating cash flows. In contrast, with the capital cash flow
method the assumption is that the firm will or will not be paying taxes in the future in a
way that interacts with the level of operating cash flows to generate risk in the tax savings
that is the same as the risk in the operating free cash flow.
Since the extended ME case makes no particular assumption about the leverage ratio, the
amount of leverage may change over time and, therefore, the beta and cost of equity will
also change over time. There is, therefore, no simple valuation procedure that discounts
equity free cash flow at a constant cost of equity.7
4.4 An arbitrary leverage policy, with tax savings that have the same risk as the debt
(extended MM)
An extension of the MM approach is to assume directly that the risk of the interest tax
saving is the same as the risk of the debt, but not to assume that the debt is a constant
amount or that the operating free cash flow is a constant perpetuity. In that case, the
amount of debt in each future period is known, and PVTS is equal to the present value of
the known amounts of tax saving from interest in each future period, discounted at the
cost of debt.
As in the extended ME case, the extended MM approach makes no particular assumption
about the leverage ratio. As a consequence, there is no simple valuation procedure that
discounts equity free cash flow at a constant cost of equity.
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5. The value of the tax saving from debt
The four assumptions about debt policy have different implications for the level and risk
of tax savings from interest, and therefore about PVTS. This section explains how to
calculate PVTS in each case.
5.1 PVTS under MM
The MM assumption gives a particularly simple formula for PVTS:
PVTS = D*T
(19)
This formula is often used for PVTS when the adjusted present value approach is
employed. However, the restrictive assumptions that underlie the formula are sometimes
forgotten. It arises from valuing a constant perpetual stream of tax savings from interest,
D*KD*T, at the discount rate appropriate to debt, KD. Note that the cash flow is
discounted at the rate appropriate to debt because, with the MM assumptions, the interest
tax savings have this level of risk. Contrast this with the discounting of the tax savings
under the ME assumption, where they have a risk equivalent to the discount rate KU.
5.2 PVTS under ME
Without further assumptions, the ME assumption does not give an explicit value for
PVTS. The amount of debt is a constant fraction of the future value of the firm, which can
have any pattern. Consequently, future tax savings from interest can also have any
pattern. Under ME, however, PVTS may be inferred from the value of the levered firm
minus the value of the unlevered firm. The levered value may be calculated by
discounting the operating free cash flow at WACC, and the unlevered value by
discounting the operating free cash flow at KU. This implied value of PVTS is the same as
one gets if one values the PVTS directly using a rather laborious procedure.
To illustrate how this is done we examine an important special case, that of a constant
growth rate. Under ME, for a firm growing at a constant growth rate, g, the value of the
unlevered firm is:
VU = FCF1/(KU – g)
(20)
where FCF1 is next period’s operating free cash flow. The value of the levered firm is:
VL = FCF1/(WACC – g)
(21)
The interpretation of (21) is that the value of the levered firm is given by discounting the
operating free cash flow at WACC. The WACC is a rate that incorporates the effect of
PVTS into the discount rate used to value the levered firm. So this procedure gives the
value of the levered firm directly.
A simple piece of algebra shows that:8
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Page 7
PVTS = VL – VU = D*KD*T/(KU – g)
(22)
This formula for PVTS can be interpreted as the present value of a set of cash flows that
starts at D*KD*T, the tax saved by interest in the first period. This grows at the rate g, as
does the entire firm. Future debt levels and future tax savings are assumed to be
proportional to the value of the operations of the firm. So the risk of the future tax saving
from interest is equivalent to the rate KU. Hence PVTS is given by the value of a growing
perpetuity starting at D*KD*T, growing at g, discounted at KU.
Given that the value of PVTS can be calculated in this way, VL may be calculated by
adding PVTS to the value of the unlevered firm calculated by equation (20). However,
with the ME assumption it is not worth going to this trouble, as one can calculate the
value of the levered firm directly from equation (21). It is important to realise, however,
that this direct procedure is entirely consistent with the use of the adjusted present value
method.
5.3 PVTS under extended MM and extended ME
The extended MM and ME cases make direct assumptions about the risk of the tax
savings from debt. So PVTS is simply given by the expected tax saving discounted at the
appropriate rate. In the extended MM case, the tax savings have the same risk as the debt
and may, therefore, be discounted at KD. In the extended ME case, they have risk
equivalent to KU.
Under the extended ME assumption with the special case of a constant growth rate, the
value of PVTS is given by (22). This is the same as the ME case because the extended
ME assumption makes the tax saving from debt have a level of risk the same as the
unlevered firm, just as in the ME case.
With the extended MM assumption and constant growth, PVTS is given by:
PVTS = D*KD*T/(KD – g)
(23)
The only difference from (22) is that the future tax savings are discounted at KD, rather
than KU.
The value given by (23) is considerably higher than (22). So it might seem that the
extended MM policy is the most attractive, in that it gives the highest value of PVTS.
However, the assumptions that underlie (23) are intrinsically implausible. It combines a
relatively riskless tax saving from interest with an expected level of debt that is assumed
to grow in line with the cash flows of the firm. It is unlikely that this combination is
realistic, as discussed in Cooper and Nyborg (2006). This illustrates the danger of simply
assuming an arbitrary leverage policy without examining its realism.
6. Which method may be used with which assumption?
Not all methods listed in Section 1 may be used with every assumption regarding debt
policy.
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With the ME assumption, all four valuation procedures listed in Section 1 are correct.
This holds regardless of the time-profile of future cash flows of the firm, as long as the
discount rates used to implement the methods are derived using the formulas that are
consistent with the ME assumption. The simplest procedure is to discount the operating
free cash flow at the WACC, the discount rate that includes the tax benefits of borrowing.
Another simple procedure is to value the equity free cash flow at the cost of equity, which
is constant because of the constant leverage assumption.9 The capital cash flow approach
is also simple to apply. The most difficult approach to apply in the ME case is APV. The
difficulty arises because the ME approach does not require explicit forecasts of future
debt levels. To apply APV, expected future debt levels must be used. However, given the
ease of applying any of the other three approaches, there is no need to use APV with the
ME assumption.
The MM approach applies only if the expected operating free cash flow is a flat
perpetuity. Given this assumption, three of the four approaches may be used. The capital
cash flow method is inconsistent with the MM assumption because it assumes that the
risk of the tax saving from interest is the same as the risk of the operating free cash
flows.10 In contrast, the MM approach assumes the risk of the tax savings is the same as
the debt. Although other methods may be used, the assumptions that underlie the MM
approach are so restrictive that it is probably best to use the APV method as it makes the
restrictive assumption about future debt policy most clear.
The extended ME case involves making a specific forecast of future debt levels, and
deriving from these the capital cash flow in each future period. Given the assumption that
the tax savings from interest have the same level of risk as the operating cash flows, they
may be discounted at the unlevered discount rate to give an APV calculation.
Alternatively, the capital cash flows may be discounted at the unlevered discount rate. In
this case, the standard equity free cash flow with a constant equity discount rate may not
be used, because the leverage ratio varies over time.11 Finally, the WACC may not be
used, since it also assumes that the leverage ratio is constant.12
The extended MM approach involves making a specific forecast of future debt levels and
deriving from these the tax saving from interest in each future period. Given the
assumption that these tax savings have the same level of risk as the debt, these may be
discounted at the debt rate to give an APV calculation. The equity free cash flow method
may not be used with a constant discount rate.13 Neither the WACC nor the enterprise
free cash flow method is consistent with the extended MM assumption, because they
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Page 9
assume that the tax savings from debt have the same risk as the operating cash flows,
whereas the extended MM assumption assumes that the tax savings have the same risk as
the debt. Thus, with the extended MM assumption, only the APV method is commonly
applied.
Table 1: Which methods are consistent with which assumptions
Method
Assumption
WACC
Equity method
APV
ME
√
√
√
MM
√*
√*
√
Extended ME
**
***
√
Extended MM
**
***
√
Capital CF
√
****
√
****
*The WACC and equity methods may be used, but recall that MM requires a flat perpetual cash flows in
these cases and that some of the formulas differ from the ME case.
**The WACC is not constant.
***The equity discount rate is not constant.
****The correct discount rate is not equal to Ku.
Table 1 summarises which methods are consistent with which assumptions. In all cases
the APV method may be applied, if used correctly. Each of the other methods has
shortcomings under some of the assumptions, unless great care is taken.
7. Which leverage assumption should you use?
The different leverage assumptions give different discount rates and values. The actual
discount rates and values that one should use are those that reflect the actual leverage
policy that a firm is expected to pursue. Usually, this will be the ME, extended MM, or
the extended ME policy. Rarely will it be the MM assumption.
8. Where to get betas and discount rates
Implementation of the valuation methods requires discount rates. Usually there are two
possible starting points for estimating these. One is to observe the equity beta, debt
required return, and leverage of the firm being valued. The other is to use information
from other companies in the industry.
When the firm’s own characteristics are used, the WACC may be calculated directly and
used as long as the leverage policy will be the ME policy and future leverage will be like
the past. Alternatively, the equity required return may be calculated from the observed
equity beta and used in the equity free cash flow method. Note, however, that the use of
these methods implicitly assumes that the observed equity beta is consistent with an ME
leverage policy. In other words, it assumes that a constant market value leverage policy
was being pursued over the period that the equity beta was measured. Otherwise, the
equity beta would not have been constant during the estimation period and, strictly
speaking, standard beta estimation methods should not be used.
Page 10
The other methods, APV and capital cash flow, require the unlevered cost of capital. To
get this from the observed equity beta, one must use either equation (7) or equation (13)
to unlever the equity beta. The choice of which to use depends on the leverage policy that
was pursued during the period of observation of the equity beta. In general, this is likely
to have been closer to the ME assumption than to the MM assumption.
An alternative to using the observed equity beta of the company being valued is to use an
industry asset beta estimated from the equity betas of other companies in the industry.
This approach must be used if the entity being valued is untraded, such as a private
company or a division of a traded company. The industry asset beta can be observed only
by starting from observed equity betas of traded companies in the industry, unleveraging
their betas, and then averaging the unlevered betas. Here again, the method used to
unleverage should be consistent with the leverage policy pursued by each company over
the period that its betas is estimated. Different firms may well have different leverage
policies and these may differ yet again from the policy relevant to the firm or project that
is being valued.
If an unlevered industry beta is used, but the equity free cash flow or WACC method are
to be employed, then the unlevered beta must be leveraged up using either equation (10)
if the assumed leverage policy for the company being valued is the MM policy or
equation (16) if the company being valued satisfies the MM assumptions.
9. The danger of inconsistency
Consistency in applying these valuation methods means two things. First, one should
choose the underlying assumption about leverage policy that most closely approximates
the actual leverage policy that will happen in the valuation being addressed. The danger
of not reflecting the actual leverage policy in valuations was first pointed out by Myers
(1974). Second, consistency means not mixing formulas and procedures based on one
assumption with those based on another. We give two examples of this second danger.
9.1 Example 1: An incorrect formula for PVTS
An illustration of what can happen if one mixes formulas based on different assumptions
is a formula for PVTS derived by Booth (2002). Booth analyses the constant growth case
discussed in section 5.2 above. His derivation of PVTS involves four apparently
innocuous steps that give an appealing and new formula for PVTS. The formula is wrong,
because it is based on a combination of formulas that are true only under different
assumptions. The steps in the derivation are as follows.
First, PVTS is defined by the difference between the levered and unlevered values:
PVTS = VL - VU
(23)
The value of the unlevered firm is given by the standard constant growth formula (Booth
(2002) equation (4)):
VU = FCF/(KU – g)
(24)
Page 11
The value of the levered firm is given by the constant growth formula applied using the
WACC as the discount rate (Booth (2002) equation before equation (4)):
VL = FCF/(WACC – g)
(25)
Finally, the relationship between the WACC and KU is given by:
WACC = KU*(1 – T*(D/VL))
(26)
A simple piece of algebra then demonstrates, given these equations, that:
PVTS = D*KU*T/(KU – g)
(27)
This formula looks appealing, but there is no assumption about leverage policy for which
it is correct.
The problem arises because, hidden in the apparently standard and innocuous equations
are inconsistent assumptions about capital structure. Equation (26) is true only in the MM
case where the amount of debt is constant and the expected operating free cash flow is a
level perpetuity. But the cash flow being valued here is a growing perpetuity, so that
equation should not be used. Equation (25) assumes the ME policy, which has a constant
market-value proportion of debt. This is inconsistent with equation (26). Booth calculates
PVTS as the difference between the levered and unlevered values of the firm. The levered
value is calculated using a combination of (25) and (26), which mix inconsistent
assumptions about leverage policy. When the difference between the two values is
calculated and used as an estimate of PVTS, the error is potentially large.14
To see the size of the error, note that Booth’s estimate of PVTS differs from the value
calculated under the ME policy in section 5.2 above by the ratio KU/KD. This ratio could
be as large as two. In other words, an apparently flawless procedure misvalues PVTS by a
factor of up to two times because it does not use a consistent assumption about leverage
policy.
9.2 Example 2: Incorrect valuation procedures
Another example of inconsistency, which is related to the Booth result, is given in
Fernandez (2004a, 2004b). Fernandez constructs a series of valuation procedures that
appear to be internally consistent. Importantly, as Fernandez says, the procedures are not
consistent with the standard literature in this area. As such, they represent a serious
challenge to accepted wisdom (Fernandez (2004b).
The proposed methods are laid out most clearly in Fernandez (2004a), which implements
of his proposed valuation methods to value the equity of a particular company. The
primary methods discussed are:
1. Operating free cash flow discounted at the WACC, minus the value of debt.
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Page 12
2. Equity free cash flow discounted at the cost of equity.
3. Capital cash flow discounted at the ‘WACC before tax’, minus the value of debt.
4. Adjusted present value (APV) minus the value of debt.
These are the four methods discussed in this note.
Although the methods are standard, as noted above none is fully defined without
specifying the formulas that generate the discount rates used to perform the valuations.
The formulas used by Fernandez are:
WACC = (E/VL)*KE + (D/VL)*KD*(1-T)
(28)
WACCBT = (E/VL)*KE + (D/VL)*KD
(29)
βE = βU + (βU - βD)*(D*(1-T)/E)
(30)
KE = RF + βE*Π
(31)
KD = RF + βD*Π
(32)
Equation (29) involves the introduction of the new notion of a ‘WACC before tax’ that
differs from the unlevered cost of capital.
The implementation methods proposed by Fernandez to value the equity are:
Method 1:
PV(WACC; FCF) – D
(33)
Method 2:
PV(KE; ECF)
(34)
Method 3:
PV(WACCBT; CCF) – D
(35)
Method 4:
PV(KU; FCF) + PVTS –D
(36)
PVTS = D*KU*T/(KU – g)
(37)
Where PV(X; K) means the present value of the cash flow stream X at the rate K.
Apart from the introduction of the notion of WACCBT, and the formula (37), all of these
discount rate formulas and valuation methods appear standard. Furthermore, the
calculations given by Fernandez, shown in Table 2 below, all give the same value for the
Page 13
equity. So there is consistency between the methods, as Fernandez applies them.15 Thus it
might appear from the use of common formulas and internal consistency of the valuation
procedures that the methods must be correct. However, they are not, for a reason similar
to the Booth example discussed above.
The formulas used are inconsistent with each other, in the sense that they are based on
different assumptions concerning debt policy. In particular, equation (30) is based on the
MM assumption, whereas the use of the WACC for a growing firm, as in (33), assumes
the ME leverage policy. Equation (29) also implicitly assumes the ME policy, as the
unlevered discount rate it gives implies the ME assumption.
The problem can be seen clearly in method 4. This uses the same formula for PVTS as
proposed by Booth, for exactly the same reason. The PVTS is the implicit value
calculated as the difference between levered and unlevered values using different
assumptions about leverage policy. In other words, to make the APV method work in
conjunction with inconsistent assumptions, Fernandez introduces the same incorrect
calculation of PVTS as Booth.
This inconsistency gives rise to another problem. In Fernandez’ example, the assumed
value of KU is 10%. However, the calculated value of WACCBT is 9.832%. Fernandez
uses WACCBT rather than KU to value the capital cash flow. The introduction of a new
discount rate ‘WACCBT’ that differs from KU is an innovation that Fernandez uses to
allow for the fact that KU is levered up using (30) and then unlevered using (29), which
involve inconsistent assumptions. The originally assumed value of KU does not reappear
at the end of this process. In fact, there should be no separate notion of ‘WACCBT’. The
unlevered discount rate should be the one assumed at the start of the valuation.
9.3 Consistent valuation applied to the Fernandez example
The correct analysis of the Fernandez example is given in Table 3. Since all four methods
are correct only under the ME assumption, the ME formulas should be used throughout.
The correct formulas are those used by Fernandez, with the exception that equation (30)
should be replaced by:
βE = βU + (βU - βD)*(D/E)
(38)
and equation (37) should be replaced by:
PVTS = D*KD*T/(KU – g)
(39)
Given these adjustments to the Fernandez approach, all four valuation methods still give
the same result as each other. They also correct the two anomalies in the Fernandez
valuation. The calculation of PVTS is now given by equation (22), the value for PVTS
with a constant growth rate and the ME assumption. Also, the unlevered discount rate
calculated by leveraging up KU and then unleveraging it is equal to KU, as it should be. So
the calculations are entirely consistent with standard theory.
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Page 14
As a result of correcting these errors, the equity value changes from 4,860 to 4,726, and
PVTS changes from 669 to 534. The ratio of the two values of PVTS is equal 1.25. This
is the ratio of KU and KD. In this example, the error caused by inconsistent application of
the valuation methods is one quarter of PVTS. In many practical situations the error
would be much larger.
10. Does it matter?
If one were to accept the Fernandez and Booth analysis, it would imply that many
standard results in valuation and capital structure would be invalid. Therefore, an
understanding that some surprising results in this area can be generated by inconsistent
application of assumptions is important. Booth and Fernandez imply that conventional
valuation methods are wrong, but this challenge drops away once one realises that their
results come from inconsistency in assumptions and not from a defect in conventional
practices.
A second issue is how much difference it makes to valuations if one adopts inconsistent
methods. Inconsistency can clearly have a significant effect on certain components of
valuation, as illustrated by the significant misvaluation of PVTS arising from internally
inconsistent assumptions. It is difficult to predict how these errors will arise, and it is
relatively simple to use internally consistent methods. So it is worth picking one of the
four assumptions about capital structure discussed above and sticking to it in any
individual valuation. This means not, for instance, levering up betas using an MM
formula and unlevering them using an ME formula.
Finally, there is the issue of how much difference it makes which of the four assumptions
about capital structure one chooses. Since Myers (1974) it has been widely appreciated
that one can make significant valuation errors by assuming a leverage policy that is
incorrect. So one should pick the leverage assumption that most closely approximates
reality. Frequently the best approximation will be the ME policy.
As a final observation, which is more of a curiosity than a recommendation, it is
interesting to note a possible reason that the MM approach has survived for so long,
despite the fact that it is very unlikely to be a realistic assumption about capital structure.
The MM approach has two features that make it very similar to the ME approach in some
common valuation situations. First, both allow the use of the WACC method. Second,
both can give very similar estimates of PVTS, despite the fact that they make such very
different assumptions. To see this, recall that with the ME assumption and constant
growth PVTS is given by:
PVTSME = D*KD*T/(KU – g)
(40)
With the MM assumption, it is given by:
PVTSMM = DT
(41)
These two expressions give surprisingly similar values in many practical cases. Their
ratio is equal to:
Page 15
PVTSME/PVTSMM = KD/(KU – g)
(42)
Consider a situation where the interest rate is 4%, the risk premium on the firm’s assets is
4%, the debt risk premium is 1%, and the expected growth rate is 5%. These are
parameter values that would not be unreasonable for an average firm. Then the ratio of
the two values of PVTS is:
PVTSME/PVTSMM = (0.04+0.01)/(0.04 + 0.05 – 0.04) = 1
(43)
So, in this case, the two values for PVTS are the same. The reason is that two opposite
effects balance. On the one hand, the expected future tax saving with the ME policy is
higher than that for the MM case because the amount of debt is expected to grow. On the
other hand, the ME tax saving is more risky and is, therefore, discounted at a higher rate.
With these particular parameter values the two effects exactly offset each other to give the
same value of PVTS. This result may partly explain why the extremely crude formula
(41), derived under highly unrealistic assumptions, has survived so long.
This shows that some results derived under the MM assumptions can, in certain
circumstances, be good approximations to results derived under the ME assumptions.
However, the examples we have given in Section 9 above show that this is not generally
true. One cannot freely mix and match formulas based on different assumptions regarding
capital structure. Consistency can be important and inconsistency can cause large errors
in the estimation of some value components.
Page 16
Table 2: Calculation of constant growth value by Fernandez
Cash flows in year 1:
FCF = 457.623, ECF = 408.663, CCF = 500.463.
Assumptions:
KU = 10%, RF = 6%, βU = 1, Π = 4%, βD = 0.5, KD = 8%.
D =1530, g = 2%, T = 0.35.
Intermediate calculations:
Given these assumptions, E = 4,859.66. Methods 1, 2, 3 below require this as an
input. Method 4 does not.
βE = βU +(D/E)*(1-T)*( βU – βD) = 1.102
KE = RF + βE*Π = 10.409%
WACC = (E/(D+E))*KE + (D/(D+E))*KD*(1-T) = 9.162%
WACCBT = (E/(D+E))*KE + (D/(D+E))*KD = 9.832%
Calculation of equity value using four methods:
Method 1:
PV(WACC; FCF) – D = 457.623/(0.09162-0.02) - 1,530 = 4,859.66
Method 2:
PV(KE; ECF) = 408.663/(0.10409 - 0.02) = 4,859.66
Method 3:
PV(WACCBT; CCF) – D = 500.463/(0.09832 – 0.02) – 1,530 = 4,859.66
Method 4:
PV(KU; FCF) + PVTS –D = 457.623/(0.10 – 0.02) + 669.375 – 1,530 = 4,859.66
PVTS = D*KU*T/(KU – g) = 1,530*0.10*0.35/(0.10 - 0.02) = 669.375
Page 17
Table 3: Correct calculation of constant growth value in Fernandez example
Cash flows in year 1:
FCF = 457.623, ECF = 408.663, CCF = 500.463.
Assumptions:
KU = 10%, RF = 6%, βU = 1, Π = 4%, βD = 0.5, KD = 8%.
D =1530, g = 2%, T = 0.35.
Intermediate calculations:
Given these assumptions, E = 4,725.788. Methods 1, 2, 3 below require this as an
input. Method 4 does not.
βE = βU +(D/E)*( βU – βD) = 1.162
KE = RF + βE*Π = 10.647%
WACC = (E/(D+E))*KE + (D/(D+E))*KD*(1-T) = 9.315%
WACCBT = (E/(D+E))*KE + (D/(D+E))*KD = 10.0%
Calculation of equity value using four methods:
Method 1:
PV(WACC; FCF) – D = 457.623/(0.09315-0.02) - 1,530 = 4,725.79
Method 2:
PV(KE; ECF) = 408.663/(0.10647 - 0.02) = 4,725.79
Method 3:
PV(KU; CCF) – D = 500.463/(0.10 – 0.02) – 1,530 = 4,725.79
Method 4:
PV(KU; FCF) + PVTS –D = 457.623/(0.10 – 0.02) + 535.5 – 1,530 = 4,725.79
PVTS = D*KD*T/(KU – g) = 1,530*0.08*0.35/(0.10 - 0.02) = 535.5
Page 18
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