Content uploaded by Andreas Schueler
Author content
All content in this area was uploaded by Andreas Schueler on Apr 26, 2020
Content may be subject to copyright.
Content uploaded by Andreas Schueler
Author content
All content in this area was uploaded by Andreas Schueler on Mar 28, 2018
Content may be subject to copyright.
Content uploaded by Andreas Schueler
Author content
All content in this area was uploaded by Andreas Schueler on Mar 01, 2018
Content may be subject to copyright.
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Journal of Business Valuation and Economic Loss Analysis. 2017; 20160012
Andreas Schueler1
A Tool Kit for Discounted Cash Flow Valuation:
Consistent and Inconsistent Ways to Value Risky
Cash Flows
1Universität der Bundeswehr München, Fakultät für Wirtscha昀ts- und Organisationswissenscha昀ten, Neubiberg, Germany,
E-mail: andreas.schueler@unibw.de
Abstract:
The DCF method or multiples are used to value companies in practice. Starting with the value additivity prin-
ciple, the paper presents a general framework for DCF valuation. This framework allows defining stepwise and
aggregated approaches to value risky cash flows and identifying inconsistent approaches. The framework helps
to integrate sales, contribution margin, operating leverage, and financial leverage into valuation approaches and
shows the assumptions implied when multiples are used.
Keywords: valuation, DCF, multiples, value additivity
DOI: 10.1515/jbvela-2016-0012
1 Introduction
This article aims to provide a general framework for composite cash flow valuation and to use this framework
to separate consistent from inconsistent ways to discount risky cash flows with risk-adjusted rates of return.1
Thus, the paper may be of interest to managers, investors, financial analysts, and researchers working on em-
pirical and normative valuation issues. The framework is a valuation tool kit based on the value additivity
principle. Although this principle, as discussed by Schall (1972), for example, is well-known in the literature,
its application to the development of a general framework for identifying consistent valuation approaches and
for disaggregating company value is novel. The framework is introduced in Section 2. To summarize, one can
use either an adjusted total cost of capital to value the total cash flow or an adjusted total or partial discount rate
to value a component of the composite cash flow. The adjustments depend on whether the unlevered or levered
company value (entity value) or the value of equity are to be determined. We will present four applications of
the framework:
First, we use the framework to value cash flows ranging from unlevered free cash flows (FCF) to levered
FCF. Thus, Section 3 addresses the cash and risk impact of debt financing. The standard discounted cash flow
(DCF) approaches are derived in Sections 3.1 and 3.2: adjusted present value (APV), weighted average cost of
capital (WACC), flow to equity (FTE) and capital cash flow (CCF). The formulae developed by Modigliani and
Miller (1958, 1963) , Harris and Pringle (1985), Inselbag and Kaufold (1997), and Miles and Ezzell (1980), which
constitute the core of the valuation procedures discussed in text books such as those by Brealey, Myers, and
Allen (2014), Holthausen and Zmijewski (2014), and Koller, Goedhart, and Wessels (2015) are replicated.
Second, the framework enables the identification of approaches beyond these standard DCF approaches.
Thus, the paper does not only focus on presenting known DCF methods, as in Oded and Michel (2007), for
instance. Rather, we analyze these new approaches according to consistency and efficiency in Section 3.3. Ap-
proaches that do not fit into the framework are inconsistent, and approaches that depend upon the valuation
results to generate the valuation results (circular reference) are inefficient.
Third, the framework is sufficiently general to enable the disaggregation and aggregation of other cash flow
components. As unlevered FCF consist of cash inflows and cash outflows with different levels of risk, we use
the framework in Section 4 to value cash flows ranging from sales to unlevered FCF. As presented in the first
half of Section 4, this approach leads to a list of viable DCF approaches. Inter alia, it will become apparent how
the premium for sales risk and for financial and operating leverage can be accounted for consistently. To do so,
we develop a step-by-step valuation process.
Fourth, the paper adds to the literature on valuation with multiples, as in Holthausen and Zmijewski (2012)
or Chullen, Kaltenbrunner, and Schwetzler (2015), since we refer to the framework in the second half of Section
4 to reveal which steps of the valuation process are skipped by a valuation with multiples.
Andreas Schueler is the corresponding author.
© 2017 Walterde Gruyter GmbH, Berlin/Boston.
1
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
2 Valuation of a Stream of Composite Cash Flows
2.1 Valuation of Total Cash Flows
If the well-known value additivity principle applies, the value of the total cash flow (VG), consisting of a set of
jstreams of cash flows, equals the sum of the value of its components () ( Schall 1972; Haley and Schall 1979,
166, 202):
1(1)
A perfect capital market is assumed. The principle explains the irrelevance of the capital structure for company
value, given no taxes. Below, we introduce taxes on corporate income, and we briefly review how capital struc-
ture influences company value. The value additivity principle still holds ( Haley and Schall 1979, 205–206). In
many cases, the principle is no longer applicable with information asymmetry or transaction costs.
The value additivity principle implies that total market value is not dependent on how total cash flow is
split between its components. In this paper, we refer to this principle to divide total cash flows into separate
streams of cash flows and to identify consistent and efficient valuation approaches.
A and B are cash flows, which sum to cash flow C.2We assume that the risks of A and B are different and
that the risk equivalent discount rates (d) of A and B therefore differ. The value of the total cash flow (VC),
assuming a perpetuity with zero growth, is as follows:
(2)
It is possible to evaluate cash flow components A and B either separately or with the discount rate for cash flow
C. After rearranging (2), we obtain the definition of this rate:
(3)
The discount rates of cash flow components A and B are weighted according to their contribution to the total
value. Doing so creates a circular reference because if the weights of value A and B are unknown, this discount
rate depends on the valuation results. This interdependence problem does not occur for the purist application
of the value additivity principle, as shown in (1).
We can define the rate for discounting the total cash flow G for = 1,.., n components generally by:
1
(4)
2.2 Valuation of a Component of the Total Cash Flow
In addition to discounting C by the composite rate dC, valuation approaches can be designed to value only
parts of the total cash flow C while still resulting in the correct values. If total cash flow C again comprises two
components (A+B) and if A is to be discounted to arrive at the value of C, the discount rate of the total cash
flow C needs to be adjusted by rearranging (2):
(5)
Alternatively, one could value cash flow A by a modified discount rate A to yield the value of C3:
2Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
1
(6)
These approaches can be generalized for discounting any cash flow component CFjby either the adjusted rate
for the total cash flow (dG)
1
(7)
or the adjusted rate for a cash flow component CFj:
(8)
Obviously, a circular reference occurs in both approaches, as one needs to know the valuation result (VG) at
the start. This result is not surprising, as the approach aims to derive the total value from a single component.
Again, adding the value of the components, as shown by (1), is circular free.4The usefulness of the approach
based on dGas in (7) versus the approach based on djas in (8) depends on the information given. If, for instance,
we know the ratio of Vjto VG, one should use (8).
While Section 2.1 shows that discounting total cash flow requires the use of the discount rate fitting the
total cash flow, Section 2.2 shows that discounting a component of the total cash flow requires the adjustment
of either the rate for the total cash flow or the rate for the component.
3 From Unlevered Free Cash to the Value of Equity of a Levered Company
3.1 Propositions of Modigliani and Miller
In a first application of the framework developed in Section 2, we will illustrate it by deriving the standard DCF
approaches. To prepare this application to the valuation of unlevered versus levered companies, the section
starts with a brief review of the seminal contribution of Modigliani and Miller addressing the effects of debt
financing on company value. Modigliani and Miller (1958, 1963) show that capital structure has no influence
on company value in a perfect capital market. Assumptions required for a perfect capital market include no
taxes and no transaction costs. In addition, we assume a perpetuity-case at zero growth, a constant risk-free
rate of return, constant risk premia and no risk of default. Modigliani and Miller formulate three propositions,
which lead to three variants of the DCF valuation: the APV approach, the WACC approach and FTE approach.
Without taxes, there are no tax shields on debt financing, and there is no difference between the APV and
WACC approaches. Because the WACC approach equals the unlevered cost of equity, the unlevered FCF is
discounted with the same rate in both approaches. The value of equity is found after subtracting debt. For
the FTE approach, levered FCF (unlevered FCF after interest and plus the change of debt) is discounted by the
levered cost of equity to obtain the value of equity. Introducing a single corporate tax rate, the three propositions
are 5
Proposition 1:
(9)
Proposition 2:
1
(10)
3
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
Proposition 3:
1
1
(11)
With Inselbag and Kaufold (1997), we recommend the APV approach if debt employed is planned without
a link to company value (deterministic debt financing). The use of the WACC approach is suggested if future
debt levels are planned as a percentage of company value. In the last case, the levered cost of equity is defined
differently, which will be shown later. The textbook formula for the definition of WACC, or the right side of eq.
(11), remains valid.
A fourth approach, the total cash flow (TCF) or capital cash flow (CCF) approach, is not in use often, in
theory or in practice (see Ruback 2002). It requires that the levered FCF plus tax shields are discounted with
the following rate:
(12)
3.2 Reconciling Standard DCF Approaches with the General Valuation Framework
The propositions of Modigliani and Miller serve as a first illustration for an application of the general framework
developed in Section 2. Proposition 1 and the APV approach are straightforward applications of the value
additivity principle in accordance with (1). Proposition 2 follows the definition of the total cash flow, which is
the levered FCF here, using eq. (4) for the valuation of more than two cash flow components. The unlevered
FCF (A) minus the interest payment (B) plus the change in debt, which is zero in a perpetuity setting, plus the
periodic tax shield on interest expenses (C) leads to the levered FCF (D). This results in the definition of the
levered cost of equity as in (10):
Thus far, we have assumed a deterministic definition of debt employed. The general definition of a discount
rate in (4) can also be used to define the levered cost of equity for a target debt ratio as a percentage of total
company value. Following Miles and Ezzell (1980) (ME) or Harris and Pringle (1985) (HP), we arrive at
11
11
1
1
(13)
(14)
For the definition of the cost of equity, according to Miles and Ezzell (1980), the present value of the tax shield
is divided into a risk-free component (CI), the present value of the risk-free tax shield of the first year of the
planning horizon and a risky component (CII), comprising the risky tax shields of the following years. The
following formula is applied:
4Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
WACC and WACC* (proposition 3) can also be derived within the general framework. The definition of WACC*
in (12) also follows (4), with the total cash flow (C) now interpreted as the sum of unlevered FCF (A) and tax
shields (B):
The relationship between WACC* and WACC follows from (5) or the general eq. (7):
(15)
Total levered company value (VL) is the present value of the unlevered FCF (A) discounted by WACC. Using
(6) or the general eq. (8), we arrive at (11):
1
The WACC approach can be viewed as an approach that delegates the valuation of the tax shields (B) to the
discount rate and values the unlevered FCF (A), which is a component of total cash flow C (unlevered FCF plus
tax shield).
The standard DCF approaches are applications of the general framework derived in Section 2 to discount
either the total cash flow or one or more components with risk equivalent cost of capital. Table 1 summarizes
the standard approaches by aligning the surplus to be valued with the valuation result.
Table 1: Standard approaches to DCF valuation.
The approaches on the diagonal are synchronized because the surplus can be immediately associated with
the valuation result. Using the APV approach, unlevered FCF are discounted by the unlevered cost of equity to
arrive at the unlevered company value. Using the CCF approach to arrive at the levered company value, total
cash flows are discounted with WACC*, the weighted average of the cost of capital for equity investors (levered
cost of equity) and debt investors (cost of debt). Finally, using the FTE approach, levered FCF, i. e., the cash
flows paid out to the owners, are discounted with the cost of capital for equity investors (levered cost of equity)
to calculate the value of their shares (value of equity). As mentioned above, the WACC approach is a modified
approach because it shifts part of the total cash flow, the tax shield, to the discount rate (WACC). Thus, the CCF
approach can be considered better than its reputation from a conceptual point of view.
3.3 Modified Approaches
The general framework presented in Section 2 helps to widen the view to other approaches for valuing a levered
company. This is our second application of the framework.
5
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
It is possible to identify more approaches in addition to the standard DCF variants, as Table 2 illustrates.
Additional methods become apparent if one fills out the empty cells of Table 1 by finding the discount rate
that matches the surplus to be valued with the valuation result. In accordance with Modigliani and Miller, the
presentation of the DCF methods uses the term ‘cost of capital’. We use the more general term ‘discount rate’
when approaches other than the standard DCF methods are derived and analyzed. These discounts rates are
labelled dIto dV.
Table 2: Standard approaches and additional methods based on adjusted discount rates.
The table shows the discount rates necessary depending on the combination of surplus to be valued and valuation result. The standard
DCF methods are highlighted.
At first sight, approaches I and II seem to accelerate the valuation routine, since they get to the value of
equity not by discounting levered FCF but only by unlevered FCF (I) or unlevered FCF plus tax shield (II). For
doing so, we have to shift components of the levered FCF from the surplus (numerator) to the discount rate
(denominator). Approaches I and II are hybrid methods in that regard.
The discount rates III –V have a different function: The respective surplus already contains components,
such as the tax shield and/or the interest expenses, which are not part of the valuation result, such as the
unlevered or the levered company value. The discount rates should backpedal in that they value a surplus that
considers more components than the valuation result. Approach IV, for instance, discounts the levered FCF, the
unlevered FCF after debt related payments, to the unlevered company value, the value before considering debt
financing. Although we could adjust discount rates accordingly, we do not consider these approaches efficient
because of the ‘backpedaling’idea, which also decreases the ease of understanding. While approaches I and II
attempt to abbreviate the valuation process by focusing on the final valuation result, the value of equity, from
the beginning, approaches III to V extend the process. Therefore, we will not further discuss these approaches
and do not recommend them.6
Before we discuss the definition of discount rates I and II and the efficiency of the hybrid approaches I and
II, we complete the list of potential methods by methods based on adjusted surpluses. Hence, we could look
for the definition of the surplus to be discounted, while matching the valuation result and the discount rate.
Table 3 consists of all possible approaches.
Table 3: Standard approaches and additional methods based on adjusted discount rates and adjusted surpluses.
6Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
The table shows standard approaches and additional methods based on adjusted discount rates dIto dVas in Table 2. In addition,
approaches are listed that link the standard cost of capital figures (unlevered cost of equity rU, WACC, WACC* and levered cost of equity
rL) to other than the standard valuation results. The surplus to be valued has to be defined accordingly. For example, instead of
calculating unlevered company value VU, approach VI uses rUto get to the levered company value VLand approach VII uses it to get to
the value of equity E. The surplus has to be defined accordingly (first column).
The main difference between approaches I to V and approaches VI to X is that the last group of approaches
adjust the surplus by a term that corrects for the lack of fit between the cost of capital used (rU, WACC, WACC*,
rL) and the valuation result (VU, VL, E). A valuation can be based on consistently defined surpluses using the
general framework developed in Section 2. There is no inherent advantage to these methods, which try to link
a standard cost of capital to an unfitting valuation result. This lack of fit requires the use of an artificial surplus.
The resulting surpluses mix elements of cash flows with cost of capital. In addition, managers do not report
or plan them. Therefore, the approaches to efficiently addressing debt financing are still the standard DCF
methods and the hybrid approaches I and II.
For approach I, the market value of equity equals the present value of the unlevered FCF discounted with
the rate dI. Unlike the APV and WACC approach, approach I is able to generate the value of equity from the
unlevered FCF in one step. This might raise the hope for a sped-up valuation process at first. Any cash flow
component caused by debt financing, such as tax shields, interest payments and the change in debt, are part of
the definition of the discount rate. Using (7), we arrive at
1
1
(16)
The discount factor dIequals WACC multiplied by the ratio of the levered company value (entity value) to the
market value of equity because the valuation result is the value of equity and not the levered company value:
1
(17)
With eq. (6), dIis defined equivalently as follows:
(18)
This discount factor has to meet the same requirements for implementation as the WACC and has circular
references since its definition depends on the ratios VL/E(or VU/E), as WACC depends on the ratios E/VL
and D/VL. Thus, there is no disadvantage for dIcompared with WACC in terms of methodological efficiency.
There is a little advantage, because different from the WACC approach debt has not to be subtracted from the
present value VL, but the value of equity is derived without an additional step. Both approaches are consistent.
However, one could argue that the intuition behind WACC, as the average of the cost of equity and cost of debt,
is more appealing for practical implementation than dI, as dIis an adjusted version of the cost of equity or WACC.
Thus, in terms of ease of understanding, approach I might be considered inferior to the WACC approach.
Approach II is another hybrid approach using an adjusted discount rate. As with approach I, it aims at
deriving the value of equity in one step. The surplus to be valued is the total cash flow, the sum of unlevered
FCF and the periodic tax shields. Using (7) and (8), as we did for approach I, results in the discount factor dII:
(19)
Approach II and the CCF approach have the same conceptual basis. Both are consistent. Both approaches en-
counter the problem of being dependent on the valuation results or target ratios. In that regard, there is no
difference in the level of efficiency. As with dI, debt has not be subtracted in order to derive the value of equity.
Regarding the ease of understanding, WACC* is preferable to dII, as was WACC compared with dI.
7
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
We introduce a numerical example to illustrate our preliminary results. Table 4 contains the example data,
starting with volume and prices for a company that uses debt financing (200). The corporate tax rate is 40 %.
We assume a perpetuity case. We refrain from showing inconsistent and inefficient approaches. The levered
cost of equity (rL), WACC, and WACC* are calculated using (10), (11), and (12), respectively. Discount factors
dIand dII are derived from eqs (18) and (19). The APV approach is the only method that does not encounter
circular references. The other approaches need the valuation results for the determination of the cost of capital
and discount factors, respectively. The FTE approach, for example, leads to the value of equity (680) by dividing
levered FCF (91.2) by the levered cost of equity (0.1341), which depends upon the value of equity.
Table 4: Example.
Summing up Section 3, the general framework confirms the fundamental and advanced DCF equations
and helps to reconcile them in terms of consistency and efficiency. Furthermore, it lays the groundwork for
identifying consistent alternatives. Most of them are considered inefficient due to circular references, due to
mixing cost of capital with cash flows, or due to a ‘backpedaling’problem. Still, the framework helps to draw
these conclusions besides identifying these alternative methods. Approaches I and II are comparable to the
WACC and the CCF approach, respectively.
They are also an intermediate step towards the valuation with multiples, which we will discuss in Section
4. The discount factors dIand dII can be considered a reciprocal of a ‘Value of equity to unlevered FCF’(dI) or
a‘Value of equity to total cash flow’multiple (dII). Thus far, we have analyzed how payments related to debt
financing can be accounted for using various approaches. In Section 4, we will disaggregate the unlevered FCF
into its main operative components. We will first demonstrate how to value cash flow components in consistent
steps and then show which steps are omitted when using multiples.
4 From Sales to the Value of Equity
4.1 Stepwise Approaches
Usually, the literature sets up valuation methods by beginning with the unlevered FCF. It will now be demon-
strated how the general valuation tool kit developed in Section 2 can be applied to the components of unlevered
FCF. This represents our third application of the framework.
For this purpose, we divide unlevered cash flow into its components, beginning with sales. The compo-
nents of unlevered FCF can be connected in an additive or multiplicative way. For example, the gross margin
(m), sales after variable costs, is linked with sales, volume (n) times price (p), in a multiplicative manner. The
components can be statistically dependent or independent from each other. To simplify, we will assume that the
components are independent. Otherwise, we would have to account for covariance terms. As the implementa-
tion of the general valuation framework will show, this assumption is not overly restrictive; rather, it simplifies
the presentation.
8Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
Unlevered FCF is divided into the contribution margin (CM), fixed costs (Cfix) and investments (I) after taxes:
1
1
1(20)
OCF denotes the operative cash flow. The contribution margin and fixed costs are subject to taxation. The tax
payments on the contribution margin are as risky as the contribution margin. If fixed costs are to be regarded
fixed in a strict manner, the tax relief on the fixed costs is risk free. In a perpetuity setting with zero growth,
investments (capital expenditures) equal depreciations and reduce taxable income.
Analogous to the treatment of debt financing used by the APV approach, which involves separating the val-
uation of unlevered FCF (VU), tax shields (VTS) and interest payments and changes in debt (D), the components
of the unlevered FCF can be valued separately. In accordance with the value additivity principle, the sum of
the components after taxes yields the unlevered company value:
(21)
S denotes sales, VSdenotes the present value of expected sales, VCM denotes the present value of expected
contribution margins (in currency units), VCfix denotes the present value of fixed costs, VOCF denotes the present
values of expected operating cash flows, and VIdenotes the present value of expected investments.
Analogous to the FTE approach, it is possible to discount unlevered FCF by the unlevered cost of equity to
the unlevered company value in one step. However, the unlevered cost of equity depends on the risks inherent
in the elements of unlevered FCF. Using (4), the unlevered cost of equity can be derived and rearranged with
(21):
Salesrisk
premium
Riskpremium
operatingleverage
Riskpremium
investmentrisk
(22)
Before discussing the specific discount rates that define rU, it should be noted that to discount unlevered FCF
with rUwe also need to overcome a circular reference because the valuation results need to be known before rU
can be determined. We encountered a similar problem caused by debt financing while discussing the WACC
approach and the FTE approach. Back then, it could be solved by using iterative calculations or by using a target
capital structure. While iterations can be a solution for the problem at hand, fixed ratios cannot. For instance,
it is not plausible to assume that the value of capital expenditures can be planned as a percent of unlevered
company value. However, one could stick with the APV-like summing up the values of the components as
suggested by (21). Note that even if a valuation starts with unlevered FCF as opposed to levered FCF, circular
references do exist, as (22) reveals. Nevertheless, neither literature nor practice usually addresses them.
The starting point for deriving the unlevered cost of equity is the rate for discounting sales (rS), which is
also applied to the contribution margin if a constant gross margin is assumed. It equals the sum of the risk-free
rate and the premium for the business risk (zS), which is derived from the risks due to volumes and prices
that are state and time contingent. A possible approach to arrive at the risk premium would be to apply the
regression-based market model following Sharpe (1963). One could estimate sales betas by regressing historic
sales figures with market returns assuming that historical data enables the forecast of future risk premia.7
The second part in the second line of eq. (22) addresses the impact of fixed costs. Since they are not state
contingent by definition, they are risk free. Thus, with a positive sales risk premium, the discount rate rUin-
creases from rSdue to this risk-free cash outflow. The ratio of the present value of fixed costs to the unlevered
company value can serve as an indicator of operating leverage (OL).8The higher the operating leverage, the
higher the risk, and the higher the cost of capital. Finally, the rate rUdepends on the difference between rSand
the risk equivalent rates for discounting capital expenditures. The empirical derivation of disaggregated risk
premia requires more research, which is beyond of the scope of this paper.9
There are intermediate valuation approaches in-between APV-like disaggregation (21) and FTE-like aggre-
gation using (22). For example, one could define the valuation approach by discounting operating cash flows
9
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
to their present value (VOCF). After subtracting the present value of the cash outflows for investments (VI), we
get to the unlevered company value (VU). The rate for discounting operating cash flows following (3) is defined
as follows:
(23)
The discount rate contains the sales risk premium and a premium for the risk caused by operating leverage,
defined here by the ratio of the present value of fixed costs to the value of the operating cash flows. This OCF-
based approach also has an interdependency problem because the value components have to be known before
the respective streams of cash flows can be valued. The approaches can be completed with (6) or (8), leading to
the unlevered company value:
(24)
The denominator of the right-hand side of eq. (24) reveals that we do not have to rely on the discount rate rOCF
but that we can instead work with the unlevered cost of equity. This might be an advantage, if a reliable estima-
tor of the unlevered beta value is available. Then, we can apply the CAPM-based calculation of the unlevered
cost of equity: rU=i+βU· market risk premium. However, we again encounter a problem because we have to
adjust rUby the ratio of capital expenditure to unlevered company value.
Inserting the complex definition of the unlevered cost of equity in (22) into the definition of the levered cost
of equity (10) and rearranging the formula,10 we arrive at:
Sales risk
premium
Risk premium
operating leverage
Risk premium
investment risk
Risk premium
financial leverage
(25)
Table 5 continues the example. We use eq. (22) to derive the unlevered cost of equity, eq. (23) to derive the rate
to discount OCF, and eq. (25) for the calculation of the levered cost of equity step by step.
Table 5: Example continued.
10 Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
Figure 1 transfers the results of a stepwise valuation of the components of the FCF as shown in Table 5 to a
full version of the APV approach.
Figure 1: APV valuation of the example.
Table 6 illustrates the decomposition of the unlevered and levered cost of equity for our example.
Table 6: Decomposition of the cost of equity.
11
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
Figure 2 shows how the levered cost of equity and its components depend upon the financial leverage.
Figure 2: Cost of capital depending on financial leverage.
Finally, Figure 3 illustrates how both operating and financial leverage influence the levered cost of equity.
12 Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
Figure 3: Levered cost of equity depending on financial and operating leverage.
4.2 Leapfrogging Approaches
For our fourth application of the framework, we will apply it to the valuation using multiples. With (6) and
(8), one could arrive at an approach that extends the idea of finding company value by discounting a single
cash flow component to the maximum, which is performed by discounting sales to find the value of equity (or
company value) in a single step. Such an approach would shift most of the valuation process to the definition
of the discount rate. Although this idea may seem odd at first, it is exactly what the sales multiple does.
A common definition of a sales multiple (MS), in the case of a perpetuity with zero growth, which uses the
unlevered company value (unlevered enterprise value), is11
11
(26)
Variable dep denotes the depreciation ratio as a percent of sales. Following Holthausen and Zmijewski (2012),
we use unlevered company value, not levered company value, as a point of reference to avoid distortions caused
by differences between the capital structure of the peer companies and the company to be valued. In practice,
sales multiples usually refer to the levered enterprise value or the value of equity (price). In either case, sales
multiples attempt to value the company in one immense step. Table 4 shows these approaches in the upper
right corner. Rearranging (26) by dividing sales by the reciprocal value of the multiple, we get:
11
1
(27)
The implied discount rate dSequals the unlevered cost of equity divided by the components of the transition
from sales to unlevered FCF, which are assumed to be connected in a multiplicative manner. Although the
multiplicative link between the components is often assumed in practice, it masks the problem that the present
values and the risk associated with the overleaped components of FCF are not considered. One implication is
that there are no risk-free cash outflows. The more general case also allows for additive links. By referring to the
disaggregation of the unlevered FCF in Section 4.1, we can write the valuation by the sales multiple by using
(7):
13
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
1
1
(28)
With (8) we arrive at
(29)
These equations illustrate that the application of the sales multiple shifts the bulk of the valuation work into the
multiple (discount rate) without even specifying its elements. This approach may be used when the elements
in the denominator of (28) and (29) of the peer group and the company to be valued are indeed comparable.
To be able to assess the level of comparability, one has to have at least an idea of the parameter values of the
comparable companies and the company to be valued. Equation (29) could simplify the valuation if we know
rSand the ratio of the present value of sales to the unlevered company value. However, the reasoning is circular
because rStimes VSequals S, and we return to the starting definition of a sales multiple.
A valuation by multiples not only shifts the bulk of the valuation work into the discount rate and tries to
derive company value by valuing only one element of the unlevered FCF but also implicitly assumes that the
appropriate discount rate for all cash flow components is identical. Multiples, such as the sales multiple, violate
the value additivity principle. The analysis also shows how much information a multiple is supposed to contain.
To confirm that a multiple is appropriate for the company to be valued, we must dissect the surplus and the
discount rate. However, as this is what the stepwise DCF approach from the previous section accomplishes, a
valuation by multiples is not self-contained and needs the guidance of a DCF valuation.
In this paper, we do not discuss additional multiples. Table 7 lists some multiples and summarizes the
previous approaches. The approaches placed on the diagonal process surpluses and cost of capital that are
directly linked to the valuation result. As shown above, approaches I and II, as well as the WACC approach,
which lie above the diagonal, deviate from these approaches to a minor extent. We dismiss approaches below
the diagonal because of a lack of efficiency; they take a step backward because they link a surplus to the present
value of a surplus that comprises fewer components. The greater the distance from the diagonal, the lower the
efficiency of the approach. The approaches in the upper right area, far away from the diagonal, are multiples
that leapfrog to the valuation result.
Table 7: Summary of valuation approaches.
The table contains possible approaches for different valuation results based on standard surpluses and cost of capital. In addition,
approaches I to V and a few multiples are shown.
5 Implications for Valuation Practitioners
In the previous sections, we developed a conceptual framework to (1) reconcile existing methods, (2) develop
additional methods, (3) disaggregate cash flows and cost of capital consistently, and (4) to scrutinize multiples.
14 Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
The first application, the reconciliation of existing methods, demonstrated the applicability of the framework
to deal with the valuation of levered companies. Practitioners familiar with DCF valuation may be reassured
of their understanding of the approaches to DCF.
The second application, the development of additional methods, helps practitioners to separate consistent
from inconsistent methods. In a M&A transaction, for instance, it might be valuable to recognize if cost of capital
do not match the surplus to be valued. Furthermore, practitioners who are using the WACC approach might
consider applying the dIapproach instead, because it offers a small shortcut to the value of equity, since the
value of debt need not to be subtracted from the valuation result.
Decomposing cash flows and cost of capital consistently (3rd application) supports both company outsiders
(investors, financial analysts and others) and insiders (managers) in better understanding what drives the risk
equivalent cost of equity. This increases the transparency of the valuation process. Usually, the risk premium is
calculated according to the widely used CAPM as beta times market risk premium. A closer look reveals that
this risk premium is composed of several risk premia like the premia on sales risk (price risk & volume risk),
operating leverage, investment risk and financial risk.
The weight of a risk component is industry and company specific. Investment risk and operating leverage
are not as relevant for the retail industry that is less capital-intensive than the automotive industry. Cyclical
industries such as airlines or energy exploration will have a higher sales risk premium than non-cyclical indus-
tries. Depending upon the industry, one could also expect offsetting effects between different risk premia. One
example might be an insurance company that seeks to match its cash inflows from asset management with its
cash outflows for damage claims with respect to volume and risk. On the other hand, a company with a high
operating leverage (high fixed costs) has a higher cost of capital than a company with a low operating leverage
has even if it is otherwise comparable. This is due to a reinforcing effect between the risky cash inflows from
sales and the risk-free cash outflows for fixed costs.
Investors and their advisors searching for risk premia in order to derive the cost of capital for a company
valuation often employ the average risk premium (beta) of comparable companies (peer group). Our analysis
clarifies that the usual adjustment of this average risk premium (beta) to the capital structure of the company to
be valued incorporates the difference in the financial risk between the company and its peer group. However,
comparability does not end with financial risk but rather needs to include sales risk, operating leverage, and
investment risk as well.
Fourth, our analysis emphasizes the pitfalls of using multiples, because the requirements regarding com-
parability also apply to using the multiples of a peer group. We demonstrated this for a sales multiple that not
only neglects other components of cash flows than sales but also the components of the risk-equivalent cost of
capital.
In summary, a knowledge of components of cash flows and cost of capital, their weight, sign, and interre-
lation with other components is helpful for managers and other valuation practitioners. It provides additional
opportunity for empirical research.
6 Conclusions
Based on the value additivity principle, we develop a general framework for valuing risky cash flows with
risk-adjusted discount rates (a DCF tool kit). The first application of the general framework is the reconciliation
of the three standard DCF approaches (APV, FTE and WACC approach) and the CCF method. Although the
WACC approach is a hybrid approach because it shifts part of the total cash flow in the definition of the discount
rate (WACC), this deviation from a purist interpretation of a cost of capital does not stop it from being used on
a regular basis. Nevertheless, the CCF approach can be considered better than its reputation.
A second application of the framework reveals other consistent approaches to DCF not yet discussed in the
literature. Approaches that are not part of this list are inconsistent. The range of consistent approaches covers
those that value components of the total cash flow separately. If one decides against the valuation of compo-
nents, either the surplus to be discounted or the discount rate has to be adjusted. We advise against the use of
adjusted surpluses because they mix cash flows with cost of capital. Furthermore, managers or financial ana-
lysts do not work with such surpluses. Thus, approaches based on adjusted discount rates should be considered
instead.
Thirdly, our paper shows how sales, contribution margin, operating leverage, and financial leverage can be
built into a stepwise valuation method. This includes the decomposition of the cost of equity into different risk
premia. The weight of the risk premia and their interrelation depend upon industry and company specifics.
This implies that the requirements for the use of risk premia of comparable companies are higher than is often
assumed.
15
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
Schueler DE GRUYTER
Our fourth application of the framework reveals not only that multiples, such as the sales multiple, delegate
most of the work to the discount rate without questioning the plausibility of the implied components, but which
components of cash flows and cost of capital are neglected in the process.
Finally, the tool kit to value companies by its components can prove beneficial because it can cope with
challenges not covered by the paper, such as the integration of financial leasing or different layers of debt in the
valuation.
Notes
1Thus, this paper does not consider risk neutral valuation as in Cox, Ross, and Rubinstein (1979) or the valuation of certainty equivalents
as, e. g., in Fama (1977) .
2For ease of presentation, future cash flows are to be interpreted, but they are not explicitly written, as expected values. Time indices are
not shown. All present values are attributed to period t = 0.
3Using x for the definition of the adjusted discount rate, we get (6) by rearranging .
4In addition to the additive link between A and B assumed above, cash flow C might be defined as cash flow A multiplied by a factor b. In
this special case, the valuation based on A is as follows: .
5FCFU: unlevered FCF; FCFL: levered FCF; VL: levered company value; VU: unlevered company value; VTS: value of tax shields; E: market
value of equity; D: value of debt; rL: levered cost of equity; rU: unlevered cost of equity; WACC: weighted average cost of capital; i: risk-free
rate. τC: corporate tax rate. For the more general definition of the levered cost of equity on the right-hand side of (10), see Inselbag and
Kaufold (1997) .
6This also applies to additional approaches not shown in Table 2, which aim at either the unlevered company value, the levered company
value or the value of equity by discounting the unlevered FCF minus interest. For a different approach to come up with a list of possible
valuation approaches, which is not complete and mixes methods based on risk-adjusted discount rates with methods based on certainty
equivalents, see Fernandez (2008) .
7Brealey, Myers, and Allen (2014), 227–228, and Berk and DeMarzo (2011), 396, discuss the disaggregation of FCF and its use for the
adjustment of beta.
8For a textbook discussion of operating leverage, see Brealey, Myers, and Allen (2014), 253.
9The papers of Rosenberg (1974), Rosenberg and Guy (1976), and Hill and Stone (1980) could be useful in this context.
10We fist get:
. Several risk premia are added to the risk-free rate of return.
All premia are weighted in percent of the value of equity. Comparing this definition with the usual definition (10) shows that the premium
for the financing risk depends on the entry point (last term on the right-hand side). In this formula, this is the rate for valuing sales, and in
(10), it is the unlevered cost of equity. The value weights of the other components are different from (22) in (25), too. One might argue that
these effects distort a clear view on the impact of debt financing. Therefore, we recommend using (25). Both equations deliver the same
cost of equity.
11For a discussion of multiples, see Damodaran (2006), chapters 7–9, for instance.
References
Berk, J., and P. DeMarzo. 2011. CorporateFinance., 2nd. Harlow: Pearson Education Limited.
Brealey, R. A., S. C. Myers, and F. Allen. 2014. Principles of Corporate Finance., 11th. New York: McGraw-Hill.
Chullen, A., H. Kaltenbrunner, and B. Schwetzler. 2015. “Does Consistency Improve Accuracy in Multiple-Based Valuation?”Journal of Business
Economics 85: 635–662.
Cox, J. C., S. A. Ross, and M. Rubinstein. 1979. “Option Pricing: A Simplified Approach.”Journal of Financial Economics 7: 229–263.
Damodaran, A 2006. Damodaran on Valuation:Security Analysis for Investment and Corporate Finance., 2nd. New York: John Wiley & Sons.
Fama, E. F 1977. “Risk-Adjusted Discount Rates and Capital Budgeting under Uncertainty.”Journal of Financial Economics 5: 3–24.
Fernandez, P. (2008).Equivalence of Ten Di昀ferent Methods for Valuing Companies by Cash Flow Discounting University of Navarra Working Paper.
Haley, C. W., and L. D. Schall. 1979. TheTheory of Financial Decisions., 2nd. New York: McGraw-Hill.
Harris, R. S., and J. J. Pringle. 1985. “Risk-Adjusted Discount Rates –Extensions from the Average-Risk Case.”Journal ofFinancial Research 8:
237–244.
Hill, N. C., and B. K. Stone. 1980. “Accounting Betas, Systematic Operating Risk, and Financial Leverage: A Risk-Composition Approach to the
Determinants of Systematic Risk.”Journal of Financial and Quantitative Analysis 15: 595–637.
Holthausen, R. W., and M. E. Zmijewski. 2012. “Valuation with Market Multiples: How to Avoid Pitfalls When Identifying and Using Compa-
rable Companies.”Journalof Applied Corporate Finance 24 (3): 26–38.
Holthausen, R. W., and M. E. Zmijewski. 2014. Corporate Valuation: Theory, Evidence & Practice. Cambridge: Cambridge Business Publishers.
Inselbag, I., and H. Kaufold. 1997. “Two DCF Approaches for Valuing Companies under Alternative Financing Strategies.”Journal ofApplied
Corporate Finance 10: 114–120.
Koller, T., M. Goedhart, and D. Wessels. 2015. Valuation: Measuringand Managing the Value of Companies., 6th. Hoboken: John Wiley & Sons.
Miles, J. A., and J. R. Ezzell. 1980. “The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification.”Journalof
Financial and Quantitative Analysis 15: 719–730.
Modigliani, F., and M. Miller. 1958. “The Cost of Capital, Corporation Finance and the Theory of Investment.”The American Economic Review 48:
261–297.
Modigliani, F., and M. Miller. 1963. “CorporateIncome Taxes and the Cost of Capital: A Correction.”The American Economic Review 53: 433–443.
16 Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
Automatically generated rough PDF by ProofCheck from River Valley Technologies Ltd
DE GRUYTER Schueler
Oded, J., and A. Michel. 2007. “Reconciling DCF Valuation Methodologies.”Journal of Applied Finance 17: 21–32.
Rosenberg, B 1974. “Extra-Market Components of Covariance in Security Markets.”Journalof Financial and Quantitative Analysis 9: 263–274.
Rosenberg, B 1976. “Prediction of Beta from Investment Fundamentals.”FinancialAnalysts Journal 32: 60–72.
Rosenberg, B.,and J. Guy. 1976. “Prediction of Beta from Investment Fundamentals.”FinancialAnalysts Journal 32 (3): 60–72. and (4):62–70.
Ruback, R. S 2002. “Capital Cash Flows: A Simple Approach toValuing Risky Cash Flows.”Financial Management 31: 85–103.
Schall, L. D 1972. “Asset Valuation,Firm Investment, and Firm Diversification.”The Journal of Business 45: 11–28.
Sharpe, W. F 1963. “A Simplified Model for Portfolio Analysis.”Management Science 9: 277–293.
17
Angemeldet | andreas.schueler@unibw.de Autorenexemplar
Heruntergeladen am | 10.12.17 07:55
