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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.

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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:

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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)

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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:

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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):

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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.

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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.

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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.

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