Valuing companies by discounted cash flows 10 methods and 1 solution 2

INTERNATIONAL JOURNAL OF MANAGEMENT (IJM)
International Journal of Management (IJM), ISSN 0976 – 6502(Print), ISSN 0976 –
6510(Online), Volume 4, Issue 2, March- April (2013)
ISSN 0976-6502 (Print)
ISSN 0976-6510 (Online)
Volume 4, Issue 2, March- April (2013), pp. 63-77
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VALUING COMPANIES BY DISCOUNTED CASH FLOWS: 10
METHODS AND 1 SOLUTION

Nisarg A Joshi1, Jay Desai2, Arti Trivedi3
1
(M.B.A, Shri Chimanbhai Institute of Management & Research, Ahmedabad, Gujarat, India
2
3

ABSTRACT

This paper shows 10 valuation methods based on equity cash flow; free cash flow;
capital cash flow; APV (Adjusted Present Value); business's risk-adjusted free cash flow and
equity cash flow; risk-free rate-adjusted free cash flow and equity cash flow; economic profit;
and EVA.
All 10 methods always give the same value. This result is logical, as all the methods
analyze the same reality under the same hypotheses; they differ only in the cash flows or
parameters taken as the starting point for the valuation.
We present all ten methods, allowing the required return to debt to be different from the cost
of debt. Seven methods require an iterative process. Only the APV and business risk-adjusted
cash flows methods do not require iteration.

Keywords: valuation, DCF, discounted cash flow, WACC, free cash flow

1. INTRODUCTION

To value a company by cash flow discounting, we may use different cash flows,
which will always have different risks and will require different discount rates. As in each
case we are valuing the same company, we should get the same valuation, no matter which
expected cash flows we use. In this paper we present ten different approaches to valuing
companies by discounting cash flows. Although some authors argue that different methods
may produce different valuations, we will show that all methods provide the same value
under the same assumptions.

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The most common method for valuing companies is the Free Cash Flow method. In
that method, interest tax shields are excluded from the cash flows and the tax deductibility of
the interest is treated as a decrease in the weighted average cost of capital (WACC). As the
WACC depends on the capital structure, this method requires an iterative process: to
calculate the WACC, we need to know the value of the company, and to calculate the value
of the company, we need to know the WACC. Seven out of the ten methods presented require
an iterative process, while only three (APV and the two business risk-adjusted cash flows
methods) do not. The prime advantage of these three methods is their simplicity. Whenever
the debt is forecasted in levels, instead as a percentage of firm value, the APV is much easier
to use because the value of the interest tax shields is quite easy to calculate. The two business
risk-adjusted cash flows methods are also easy to use because the interest tax shields are
included in the cash flows. The discount rate of these three methods is the required return to
assets and, therefore, it does not change when capital structure changes.

Section 1 shows the 10 most commonly used methods for valuing companies by discounted
cash flows:

1. equity cash flows discounted at the required return to equity;
2. free cash flow discounted at the WACC;
3. capital cash flows discounted at the WACC before tax;
4. APV (Adjusted Present Value);
5. the business's risk-adjusted free cash flows discounted at the required return to assets;
6. the business's risk-adjusted equity cash flows discounted at the required return to
assets;
7. economic profit discounted at the required return to equity;
8. EVA discounted at the WACC;
9. the risk-free rate-adjusted free cash flows discounted at the risk-free rate; and
10. The risk-free rate-adjusted equity cash flows discounted at the required return to
assets.

All ten methods always give the same value. This result is logical, since all the
methods analyze the same reality under the same hypotheses; they differ only in the cash
flows taken as the starting point for the valuation.

In section 2 the 10 methods and a theory are applied to an example. The theory is:

1) Fernandez (2007)[6] assumes that the company will have a constant debt-to-equity ratio in
book value terms. In this scenario, the risk of the increases of debt is equal to the risk of the
free cash flow.

2. LITERATURE REVIEW

There is a considerable body of literature on the discounted cash flow valuation of
firms. The main difference between all of these papers and the approach proposed in sections
I and II is that most previous papers calculate the value of tax shields as the present value of
the tax savings due to the payment of interest.

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Myers (1974)[18] introduced the APV (adjusted present value) method, but proposed
calculating the VTS by discounting the tax savings (N T r) at the required return to debt (Kd).
The argument is that the risk of the tax saving arising from the use of debt is the same as the
risk of the debt. This approach has also been recommended in later papers in the literature,
two being Taggart (1991)[22] and Luehrman (1997)[12]. One problem with the Myers (1974)[18]
approach is that it does not always give a higher cost of equity than cost of assets and this
hardly makes any economic sense.
Harris and Pringle (1985)[9] propose that the present value of the tax saving due to the
payment of interest should be calculated by discounting the interest tax savings (N T r) at the
required return to unlevered equity (Ku), i.e. VTS = PV [Ku; N T r]. Their argument is that
the interest tax shields have the same systematic risk as the firm’s underlying cash flows and,
therefore, should be discounted at the required return to assets (Ku).
Ruback (1995 and 2002)[19][20], Kaplan and Ruback (1995)[11], Brealey and Myers
(2000, page 555)[23], and Tham and Vélez-Pareja (2001)[21], also claim that the appropriate
discount rate for tax shields is Ku, the required return to unlevered equity. Ruback (1995 and
2002)[19][20] presents the Capital Cash Flow (CCF) method and claims that WACCBT = Ku.
Based on this assumption, Ruback (1995 and 2002) [19][20] gets the same valuation as Harris
and Pringle (1985)[9].
A large part of the literature argues that the value of tax shields should be calculated
in a different manner depending on the debt strategy of the firm. Hence, a firm that wishes to
keep a constant D/E ratio must be valued in a different manner from a firm that has a preset
level of debt. Miles and Ezzell (1980)[13] indicate that for a firm with a fixed debt target (i.e. a
constant [D/(D+E)] ratio), the correct rate for discounting the interest tax shields is Kd for the
first year and Ku for the tax saving in later years. Inselbag and Kaufold (1997)[10] and Ruback
(1995 and 2002) [19][20] argue that when the amount of debt is fixed, interest tax shields should
be discounted at the required return to debt. However, if the firm targets a constant debt/value
ratio, the value of tax shields should be calculated according to Miles and Ezzell (1980)[13].
Finally, Taggart (1991)[22] suggests using Miles & Ezzell (1985)[14] if the company adjusts to
its target debt ratio once a year and Harris & Pringle (1985) [9] if the company adjusts to its
target debt ratio continuously.
Damodaran (1994, page 31)[3] argues that if all the business risk is borne by the
equity, then the formula relating the levered beta (βL) to the asset beta (βu) is βL = βu +
(D/E) βu (1– T). This formula is exactly formula (29) assuming that βd = 0. One
interpretation of this assumption is (see page 31 of Damodaran, 1994)[3] that “all of the firm’s
risk is borne by the stockholders (i.e., the beta of the debt is zero)”. In some cases, it may be
reasonable to assume that the debt has a zero beta, but then the required return to debt (Kd)
should also be the risk-free rate. However, in several examples in his books Damodaran
(1984 and 2002)[25] considers the required return to debt to be equal to the cost of debt, both
of which are higher than the risk-free rate.
Fernandez (2007)[6] shows that for a company with a fixed book-value leverage ratio,
the increase of debt is proportional to the increases of net assets and the risk of the increases
of debt is equal to the risk of the increases of assets. In that situation, when the debt value is
equal to its book value, VTS0 = PV0 [Kut; Dt-1 Kut Tt]. This expression does not mean that
the appropriate discount for tax shields is the unlevered cost of equity, since the amount being
discounted is higher than the tax shields (it is multiplied by the unlevered cost of equity and
not the cost of debt). This result arises as the difference of two present values. In the case of
no growth perpetuities, equation (12) is VTS = DT. The value of tax shields being DT for no
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growth perpetuities is quite a standard result. It may be found, for example, in Bodie and
Merton (2000)[24], Modigliani and Miller (1958 and 1963)[16,17], Myers (1974)[18], Damodaran
(2002)[25] and Brealey and Myers (2000)[23]. In contrast, the Miller (1977)[15] view that T∗= 0
corresponds to a CAPM where the intercept is RF(1 − TC). A similar eﬀect can be seen in the
formula for the required return on assets: It contains the most striking results of the valuation
performed on the company according to Fernández (2004)[4], Damodaran (1994)[3], Ruback
(2002)[20] and Myers (1974)[18]. It may be seen that:
1) Equity value (E) grows with residual growth (g), except according to Damodaran (1994)[3].
2) Required return to equity (Ke) decreases with growth (g), except according to Damodaran
(1994)[3] and Ruback (2002)[20].
3) Required return to equity (Ke) is higher than the required return to assets (Ku), except for
Myers (1974)[18] when g > 5%.
Please note that these exceptions are counterintuitive.

3. RESEARCH METHODOLOGY

Ten discounted cash flow valuation methods

Method 1: Using the expected equity cash flow (ECF) and the required return to equity
(Ke).

Equation (1) indicates that the value of the equity (E) is the present value of the expected
equity cash flows (ECF) discounted at the required return to equity (Ke).

E0 = PV0 [Ket; ECFt] (1)

The expected equity cash flow is the sum of all expected cash payments to shareholders,
mainly dividends and share repurchases.

Equation (2) indicates that the value of the debt (D) is the present value of the expected debt
cash flows (CFd) discounted at the required return to debt (Kd). Dt is the increase in debt,
and It is the interest paid by the company. CFdt = It - Dt

D0 = PV0 [Kdt; CFdt] (2)

Definition of FCF: the free cash flow is the hypothetical equity cash flow when the company
has no debt. The expression that relates the FCF with the ECF is:

FCFt = ECFt - Dt + It (1 - T) (3)

The expected debt cash flow in a given period is given by equation (4)

CFd t = Nt-1 rt - (Nt - Nt-1) (4)

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Where N is the book value of the financial debt and r is the cost of debt. Nt-1 rt is the interest
paid by the company in period t. (Nt - Nt-1) is the increase in the book value of debt in period
t. When the required return to debt (Kd) is different than the cost of debt (r), then the value of
debt (D) is different than its book value (N). Note that if, for all t, rt = Kdt, then N0 = D0. But
there are situations in which rt > Kdt (i.e. if the company has old fixed-rate debt and interest
rates have declined, or if the company can only get expensive bank debt), and situations in
which rt < Kdt (i.e. if the company has old fixed-rate debt and interest rates have increased).

Method 2: Using the free cash flow and the WACC (weighted average cost of capital).

Equation (5) indicates that the value of the debt (D) plus that of the shareholders' equity (E) is
the present value of the expected free cash flows (FCF) discounted at the weighted average
cost of capital (WACC):

E0 + D0 = PV0 [WACCt; FCFt] (5)

The free cash flow is the hypothetical equity cash flow when the company has no debt. The
expression that relates the FCF with the ECF is:

ECFt = FCFt + (Nt - Nt-1) - Nt-1 rt (1 - Tt ) (6)

Definition of WACC: the WACC is the rate at which the FCF must be discounted so that
equation (5) gives the same result as that given by the sum of (1) and (2). By doing so, the
WACC is (7):

WACCt = [Et-1 Ket + Dt-1 Kdt (1-T)] / [Et-1 + Dt-1] (7)

T is the tax rate used in equation (3). Et-1 + Dt-1 are neither market values nor book values:
in actual fact, Et-1 is the value obtained when the valuation is performed using formulae (1)
or (4). Consequently, the valuation is an iterative process: the free cash flows are discounted at
the WACC to calculate the company's value (D+E) but, in order to obtain the WACC; we need
to know the company's value (D+E).

Some authors, one being Luehrman (1997)[12], argue that equation (4) does not give us the
same result as is given by the sum of (1) and (2). That usually happens as a result of an error
in calculating equation (6), which requires using the values of equity and debt (Et-1 and Dt-1)
obtained in the valuation. The most common error when calculating the WACC is to use
book values of equity and debt, as in Luehrman (1997)[12] and in Arditti and Levy (1977)[1].
Other common errors when calculating WACCt are:

- Using Et and Dt instead of Et-1 and Dt-1.
- Using market values of equity and debt, instead of the values of equity and debt (Et-1 and
Dt-1) obtained in the valuation.

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Method 3: Using the capital cash flow (CCF) and the WACCBT (weighted average cost of
capital, before tax).

The capital cash flows are the cash flows available for all holders of the company's
securities, whether these be debt or shares, and are equivalent to the equity cash flow
(ECF) plus the cash flow corresponding to the debt holders (CFd).

Equation (8) indicates that the value of the debt today (D) plus that of the shareholders'
equity (E) is equal to the capital cash flow (CCF) discounted at the weighted average cost of
capital before tax (WACCBT).

E0 + D0 = PV[WACCBTt; CCFt] (8)

Ruback (2002)[20] also proves in a different way that the CCF methos is equivalent to FCF
method. Therefore the expression WACCBT shows the rate at which the CCF must be
discounted so that equation gives the same result as that given by the sum of (1) and (2).

WACCBT t = [Et-1 Ket + Dt-1 Kdt] / [Et-1 + Dt-1] (9)

The expression that relates the CCF with the ECF and the FCF is (9):

CCFt = ECFt + CFdt = FCFt + It T. Dt = Dt - Dt-1 ; It = Dt-1 Kdt (10)

Method 4: Adjusted present value (APV)

Equation (11) indicates that the value of the debt (D) plus that of the shareholders' equity (E) is
equal to the value of the unlevered company's shareholders' equity, Vu, plus the value of the tax
shield (VTS):

E0 + D0 = Vu0 + VTS0 (11)

Ku is the required return to equity in the debt-free company. Vu is given by (12). Ku is the
required return to equity in the debt-free company (also called the required return to assets).

Vu0 = PV0 [Ku t ; FCFt ] (12)

Most descriptions of the APV suggest calculating the VTS by discounting the interest tax
shields using some discount rate. Myers (1974)[18], Taggart (1991)[22] and Luehrman
(1997)[12] propose using the cost of debt (based on the theory that tax shields are about as
uncertain as principal and interest payments). However, Harris and Pringle (1985) [9], Kaplan
and Ruback (1995)[11], Brealey and Myers (2000)[23] and Ruback (2002)[20] propose using the
required return to the unlevered equity as the discount rate.4 Copeland, Koller and Murrin
(2000)[2] assert that “the finance literature does not provide a clear answer about which
discount rate for the tax benefit of interest is theoretically correct.” They further conclude
“we leave it to the reader’s judgment to decide which approach best fits his or her situation.”

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Fernández (2004)[4] shows that the value of tax shields is the difference between the present
values of two different cash flows, each with its own risk: the present value of taxes for the
unlevered company, and the present value of taxes for the levered company. Following
Fernández (2007)[6], the value of tax shields without cost of leverage is:

Method 5: Using the business risk-adjusted free cash flow and Ku (required return to assets).

Equation (13) indicates that the value of the debt (D) plus equity (E) is the present value of
the expected business risk-adjusted free cash flows (FCFKu), discounted at the required
return to assets (Ku):

E0 + D0 = PV0 [Kut ; FCFtKu] (13)

The definition of the business risk-adjusted free cash flows (FCFKu) is (14):

FCFtKu = FCFt - (Et-1 + Dt-1) [WACCt - Kut ] (14)

Method 6: Using the business risk-adjusted equity cash flow and Ku (required return to
assets).

Equation (15) indicates that the value of the equity (E) is the present value of the expected
business risk-adjusted equity cash flows (ECFKu) discounted at the required return to assets
(Ku):

E0 = PV0 [Kut; ECFt Ku] (15)

The definition of the business risk-adjusted equity cash flows (ECFKu) is (16):

ECFtKu = ECFt - Et-1 [Ket - Kut ] (16)

Method 7: Using the economic profit and Ke (required return to equity).

Equation (17) indicates that the value of the equity (E) is the equity's book value (Ebv) plus
the present value of the expected economic profit (EP) discounted at the required return to
equity (Ke).

E0 = Ebv0 + PV0 [Ket; EPt] (17)

The term economic profit (EP) is used to define the accounting net income or profit after tax
(PAT) less the equity's book value (Ebvt-1) multiplied by the required return to equity.

EPt = PATt - Ke Ebvt-1 (18)

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Method 8: Using the EVA (economic value added) and the WACC (weighted average cost of
capital).

Equation (19) indicates that the value of the debt (D) plus equity (E) is the book value of equity
and the debt (Ebv0+ N0) plus the present value of the expected EVA, discounted at the WACC:

E0 + D0 = (Ebv0+ N0) + PV0 [WACCt; EVAt] (19)

The EVA (economic value added) is the NOPAT (Net Operating Profit After Tax) less the
company's book value (Dt-1 + Ebvt-1) multiplied by the weighted average cost of capital
(WACC). The NOPAT (Net Operating Profit After Taxes) is the profit of the unlevered company
(debt-free).

EVAt = NOPATt - (Dt-1 + Ebvt-1) WACC t (20)

Method 9: Using the risk-free-adjusted free cash flows discounted at the risk-free rate

Equation (21) indicates that the value of the debt (D) plus equity (E) is the present value of the
expected risk-free-adjusted free cash flows (FCF RF), discounted at the risk-free rate (RF):

E0 + D0 = PV0 [RF t ; FCFtRF] (21)

The definition of the risk-free-adjusted free cash flows (FCFRF) is (22):

FCFtRF = FCFt - (Et-1 + Dt-1) [WACCt - RF t ] (22)

Method 10: Using the risk-free-adjusted equity cash flows discounted at the risk-free rate

Equation (23) indicates that the value of the equity (E) is the present value of the expected risk-
free-adjusted equity cash flows (ECFRF) discounted at the risk-free rate (RF):

E0 = PV0 [RF t; ECFt RF] (23)

The definition of the risk-free-adjusted equity cash flows (ECFRF) is (24)

ECFtRF = ECFt - Et-1 [Ket - RF t ] (24)

We could also talk of an eleventh method; using the business risk-adjusted capital cash flow and
Ku (required return to assets), but the business risk-adjusted capital cash flow is identical to the
business risk-adjusted free cash flow (CCFKu = FCFKu). Therefore, this method would be
identical to Method 5.

We could also talk of a twelfth method; using the risk-free-adjusted capital cash flow and RF
(risk-free rate), but the risk-free-adjusted capital cash flow is identical to the risk-free-adjusted
free cash flow (CCFRF = FCFRF). Therefore, this method would be identical to Method 9.

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4. SAMPLE DATA & ANALYSIS

The fictitious company NJ Inc. has the balance sheet and income statement forecasts
for the next few years shown in Table II. After year 3, the balance sheet and the income
statement are expected to grow at an annual rate of 3%. Although the statutory tax rate is
40%, the effective tax rate will be zero in year 1 because the company is forecasting losses,
and 36.36% in year 2 because the company will offset the previous year’s losses. The cost of
debt (the interest rate that the bank will charge) is 9%. Using the balance sheet and income
statement forecasts in Table II, we can readily obtain the cash flows given at the bottom of
Table II.
Table III contains the valuation of the company NJ Inc. using the ten methods
described in Section I. The unlevered beta (βu) is 1. The risk-free rate is 6%. The cost of debt
(r) is 9%, but the company feels that it is too high. The company thinks that the appropriate
required return to debt (Kd) is 8%. The market risk premium is 4%. Consequently, using the
CAPM, the required return to assets is 10%. As the cost of debt (r) is higher than the required
return to debt (Kd), the value of debt is higher than its nominal value. The value of debt also
fulfils equation. The first method used is the APV because it does not require an iterative
process and, therefore, is easier to implement. To calculate the value of the unlevered equity
and the value of tax shields, we only need to compute two present values using Ku (10%) is
the enterprise value and the equity value. Next is the calculation of the equity value as the
present value of the expected equity cash flows. Please note that they are calculated through
an iterative process because for equation (15) we need to know the result of equation (1), and
for equation (1) we need to know the result of equation (15). The equity value in lines 8 and 6
is exactly the same. Next is the WACC according to equations (8), (14) and (19). Then there
is the calculation of the equity value as the present value of the expected free cash flows
minus the debt value. They are also calculated through an iterative process because for
equations (8) and (19) we need to know the result of equation (7), and for equation (7) we
need to know the result of equation (8) or (19). Please note that in year 1 WACC = Ku = 10%
because the effective tax rate is zero. The equity value is exactly the same. Next is the
WACCBT according to equation (9). Thereafter is the calculation of the equity value as the
present value of the expected capital cash flows minus the debt value. They are also
calculated through an iterative process because for calculating the WACCBT we need to
know the equity value and vice-versa. Note that in year 1 WACCBT = WACC = Ku = 10%
because the effective tax rate is zero.
Next Line is the expected residual income according to equation. Next line is the
calculation of the equity value as the present value of the expected residual income plus the
book value of equity. They are calculated through an iterative process because for calculating
the residual income we need to know the required return to equity, and for this, we need the
equity value.
Then the expected business’s risk-adjusted equity cash flows will be calculated. It is
the calculation of the equity value as the present value of the expected business’s risk-
adjusted equity cash flows. As the present value is calculated using Ku (10%), there is no
need for an iterative process.
Next is the expected business’s risk-adjusted free cash flows according. It is the
calculation of the equity value as the present value of the expected business’s risk-adjusted
free cash flows minus the value of debt. As the present value is calculated using Ku (10%),
there is no need for an iterative process.
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Next line is the expected risk-free-adjusted equity cash flows according to equation. It
is the calculation of the equity value as the present value of the risk-free rate adjusted equity
cash flows. They are calculated through an iterative process because for calculating the risk-
free-adjusted equity cash flows we need to know the required return to equity, and for that,
we need the equity value.
Finally it is the expected risk-free-adjusted free cash flows according to equation.
Next is the calculation of the equity value as the present value of the risk free rate-adjusted
free cash flows minus the debt value. They are calculated through an iterative process
because for calculating the risk-free-adjusted free cash flows we need to know the WACC,
and for that, we need the equity value.

5. RESULTS

The method for calculating the company using cash flow is given in Table. 1. The
figures of forecasted profit and loss account of a fictitious company NJ Inc. is given in Table
2. Finally, the valuation of NJ Inc. using 10 methods is given in Table 3.

Table I: Ten Valuation Methods

Method Cash Flow Discount Rate Value
1 ECF (Equity Cash Flow) Ke (Cost of Levered Equity) E (Equity)
CFd (Debt Cash Flow) Kd( Required Return to Debt) D (Debt)
2 FCF (Free Cash Flow) WACC E+D
3 CCF (Capital Cash Flows) WACCBT (Weighted Average E+D
Cost of Capital Before Taxes)
4 FCF (Free Cash Flow) Ku (Cost of unlevered equity) Vu (Unlevered
Equity)
D T Ku Ku (Cost of unlevered equity) VTS (Value of
Tax Shields)
5 RI (Residual Income) Ke E – Ebv
6 EVA (Economic Value WACC E + D – Ebv – N
Added)
7 FCF//ku (Business risk- Ku E+D
adjusted free cash flow)
8 ECF//ku (Business risk- Ku E
adjusted equity cash flow)
9 FCF//Rf (Risk free-adjusted Rf E+D
free cash flow)
10 ECF//Rf (Risk free-adjusted Rf E
equity cash flow)

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Table II: Balance Sheet and Income Statement Forecasts of NJ Inc.

Balance Sheet 0 1 2 3 4 5
Working Capital
Requirements 800 890 1,000 1,100 1,133 1,166.99
Gross Fixed Assets 1,200 1,300 1,450 1,660 1,895.10 2,134.90
-
Accumulated Depriciation -200 -405 -615 -818.2 1,026.20
Net Fixed Assets 1,200 1,100 1,045 1,045 1,076.90 1,108.70

TOTAL ASSETS 2,000 1,990 2,045 2,145 2,209.90 2,275.69

Debt 1,500 1,500 1,500 1,550 1,597 1,644.40
Equity (Book Value) 500 490 545 595 612.9 631.29
TOTAL LIABILITIES 2,000 1,990 2,045 2,145 2,209.90 2,275.69

Income Statement 1 2 3 4 5
EBIDTA 325 450 500 515 530.45
Depriciation 200 205 210 216.3 222.79
Interest Payments 135 135 135 139.5 142.29
PBT -10 110 155 159.2 165.37
Taxes 0 40 62 63.68 66.15
PAT -10 70 93 95.52 99.22

Taxes 0% 36.36% 40% 40% 40%

Cash Flows 1 2 3 4 5
PAT -10 70 93 95.52 99.22
Depreciation 200 205 210 216.3 222.79
Increase of Debt 0 0 50 46.5 47.9
Increase in Working Capital -90 -110 -100 -33 -33.99
Investment in Fixed Assets -100 -150 -210 -235.1 -239.8

ECF (Equity Cash Flow) 0 15 43 90.22 96.12
FCF (Free Cash Flow) 135 100.91 74 127.42 133.59
CFd (Debt Cash Flow) 135 135 85 93 94.39
CCF (Capital Cash Flow) 135 150 128 183.22 190.51

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Table III: Valuation of NJ Inc.

Formula 0 1 2 3 4 5
Ku 10.00% 10.00% 10.00% 10.00% 10.00%
D = PV(Kd; CFd) 1,743.73 1,748.23 1,753.09 1,808.33 1,844.50 1,881.39

Vu = PV (Ku; FCF) 1,525.62 1,543.18 1,596.59 1,682.25 1,715.90 1,750.21
VTS = PV[Ku; D T Ku + T
(Nr - DKd)] 762.09 838.3 860.33 878.33 895.9 913.82
E + D = VTS + Vu 2,287.71 2,381.48 2,456.92 2,560.58 2,611.80 2,664.03
E = VTS + Vu – D 543.98 633.25 703.83 752.25 767.29 782.64

Ke 16.41% 13.51% 12.99% 12.88% 12.88%
E = PV(Ke; ECF) 543.98 633.25 703.83 752.25 767.29 782.64

WACC 10% 7.41% 7.23% 7.26% 7.26%
E = PV(WACC; FCF) – D 543.98 633.25 703.83 752.25 767.29 782.64

WACCBT 10% 9.47% 9.43% 9.44% 9.44%
E = PV(WACCBT; CCF) -
D 543.98 633.25 703.83 752.25 767.29 782.64

RI -92.05 3.78 22.21 17.12 17.46
E = PV(Ke; RI) + Ebv 543.98 633.25 703.83 752.25 767.29 782.64

EVA -75 8.55 26.12 21.84 22.28
E = Ebv - (D-N) +
PV(WACC; EVA) 543.98 633.25 703.83 752.25 767.29 782.64

ECF//Ku -34.87 -7.25 21.95 60.18 61.38
E = PV(Ku; ECF//Ku) 543.98 633.25 703.83 752.25 767.29 782.64

FCF//Ku 135 162.71 142.02 204.85 208.94
E = PV(Ku; FCF//Ku) – D 543.98 633.25 703.83 752.25 767.29 782.64

ECF//RF -56.63 -32.58 -6.19 30.09 30.69
E = PV(RF; ECF//RF) 543.98 633.25 703.83 752.25 767.29 782.64

FCF//RF 43.49 67.46 43.75 102.42 104.47
E = PV(RF; FCF//RF) – D 543.98 633.25 703.83 752.25 767.29 782.64

74


6. CONCLUSION

The paper shows that ten commonly used discounted cash flow valuation methods
always give the same value. This result is logical, since all the methods analyze the same
reality under the same hypotheses; they differ only in the cash flows taken as the starting
point for the valuation.
We present all ten methods, allowing the required return to debt to be different from
the cost of debt. Seven methods require an iterative process. Only APV and the business risk
adjusted cash flows methods do not require iteration; that makes them the easiest methods to
use.
The relevant tax rate is not the statutory tax rate, but the effective tax rate applied to
earnings in the levered company on each year.
The value of tax shields is not the present value of tax shields using either the cost of
equity or debt to discount the tax write-offs. It is the difference between the present values of
two different cash flows, each with its own risk: the present value of taxes for the unlevered
company and the present value of taxes for the levered company.

Abbreviations

βd = Beta of debt
βL = Beta of levered equity
βu = Beta of unlevered equity = beta of assets
D = Value of debt
E = Value of equity
Ebv = Book value of equity
ECF = Equity cash flow
ECF//Ku = business risk-adjusted equity cash flows
ECF//RF = expected risk-free-adjusted equity cash flows
RI = Residual income
EVA = Economic value added
FCF = Free cash flow
FCF//Ku = business risk-adjusted free cash flows
FCF//RF = risk-free-adjusted free cash flows
g = Growth rate of the constant growth case
I = Interest paid
Ku = Cost of unlevered equity (required return to unlevered equity)
Ke = Cost of levered equity (required return to levered equity)
Kd = Required return to debt
N = Book value of the debt
NOPAT = Net Operating Profit After Tax = profit after tax of the unlevered company
PAT = Profit after tax
PBT = Profit before tax
PM = Market premium = E (RM - RF)
PV = Present value
r = Cost of debt
RF = Risk-free rate
RM = Market return
75


ROA = Return on Assets = NOPATt / (Nt-1+Ebvt-1)
ROE = Return on Equity = PATt / Ebvt-1
T = Corporate tax rate
VTS = Value of the tax shields
Vu = Value of shares in the unlevered company
WACC = Weighted average cost of capital
WACCBT = Weighted average cost of capital before taxes
WCR = Working capital requirements = net current assets

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