Discounted Cash Flow
Valuation
Literature Review and Direction for Research
Composed by Ngo Manh Duy
 Acronyms
 DCF Valuation: definition and core theories
 DCF Valuation: Main Objective and Basic Steps
TABLE OF
CONTENTS
 Theories and gaps in each step:




Methods
Cash Flows
Discount Rates
Implementation
 Research objective and research questions
 Method and research value
 Reference
Acronyms
DCF: Discounted Cash Flow
Vbv: Book value of the firm
Dbv: Book value of debt
Ebv: Book value of equity
APV: Adjusted Present Value method
CCF: Capital Cash Flow method
FCFF: Free Cash Flow to Firm method
FCFE: Free Cash Flow to Equity method
EVA: Economic Valued Added method
RI: Residual Income method
BR-adj FCF: Business risk-adjusted Free Cash
Flow method
BR-adj ECF: Business risk-adjusted Equity Cash
Flow method
BR-adj CCF: Business risk-adjusted Capital Cash
Flow method
RF-adj FCF: Risk free-adjusted Free Cash Flow
method
RF-adj ECF: Risk free-adjusted Equity Cash
Flow method
RF-adj CCF: Risk free-adjusted Capital Cash
Flow method
V: Value of the firm
D: Value of debt
E: Value of equity
Vu: Value of unlevered equity
VTS: Value of interest tax shield
V[EVA]: Value of economic values added
V[RI]: Value of residual incomes
FCF: Free cash flow
ECF: Equity cash flow
CCF: Capital cash flow
FCF//Ku: Business risk-adjusted Free cash flow
ECF//Ku: Business risk-adjusted Equity cash flow
CCF//Ku: Business risk-adjusted Capital cash flow
FCF//Rf: Risk free-adjusted Free cash flow
ECF// Rf: Risk free-adjusted Equity cash flow
CCF// Rf: Risk free-adjusted Capital cash flow
CFd: Cash flows to debtholders
CFTS: Cash flow of interest tax shield
EVA: Economic value added
RI: Residual income
I: Interest paid on book value of debt
r: Actual interest rate on book value of debt
Rf: Risk-free rate
Ku: cost of unlevered equity
Kd: cost of debt
Ke: cost of equity
K: weighted average cost of capital (WACC)
Kbt: before-tax weighted average cost of capital
(WACCbt)
T: corporate tax rate
Discounted Cash Flow (DCF) valuation is:
 a method of evaluating an investment opportunity
Definition of
DCF valuation
&
Application
 by discounting predicted future cash flows generated by the
investment at certain discount rates
 to find out the present value of the investment in monetary
term
Application:
Mostly used in valuating securities (bonds or shares), companies and
business projects.
Valuation of bonds using DCF is simple and straightforward while
DCF valuation of shares, companies and business projects is quite
complex leading several issues for debates exploration.
1. DCF Model (Theory of Interest) by Fisher (1930):
Two core
theories:
1. DCF Model
=
Where:
∏
( +
)
PV0 : Present value of cash flows (at time t = 0)
CFt : Cash flow at time t
ki : Discount rate or required rate of return for the period i
n : Number of periods generating cash flows
2. Value Additivity principle
Two core
theories:
The summation of present values of cash flows divided from the same
original cash flow will always equal the present value of the original.
This principle is first demonstrated in MM Proposition I with tax of
Modigliani and Miller (1958) :
Vt = Dt + Et = Vut + VTSt
2. Value
Additivity
principle
(1)
Where:
Vt : value of the firm at time t
Dt : value of debt at time t
Et : value of equity at time t
Vut : value of unlevered equity at time t (value of the firm when there is no
leverage, i.e. 100% equity)
VTSt : value of interest tax shield at time t
Two core
theories:
2. Value
Additivity
principle
CCF
=
ECF
+
CFd
=
FCF
+
CFTS
V
=
E
+
D
=
Vu
+
VTS
Where:
CCF: Capital cash flow (all cash flows available to capital providers )
CFTS: generated from tax deduction on interest expenses)
ECF: Equity cash flow (free cash flows to shareholders)
CFd: Debt cash flow (cash flows to debtholders)
FCF: Free cash flow (cash flows generated from business operation)
CFTS: Cash flow from interest tax shield
Objective of DCF valuation: Calculate shareholders’ (or investors’)
equity value of the investmen.
Basic steps of DCF valuation:
Main objective
& basis steps
1.
Choose a method (METHODS)
2.
Calculate cash flows (CASH FLOWS)
3.
Calculate discount rates (DISCOUNT RATES)
4.
Implement discounting (IMPLEMENTATION)
12. RFadj CCF
Theories and
gaps:
METHODS
2. CCF
11. RFadj ECF
3. FCFF
12
methods
10. RFadj FCF
4. FCFE
9. BRadj CCF
5. EVA
8. BRadj ECF
Acronyms
1. APV
7. BRadj FCF
6. RI
No.
Theories and
gaps:
METHODS
1 to 6
Acronyms
Method
Cash flows
Discount rate
Implementation
1
Adjusted
Present FCF, CFd, Ku, Kd
Value (APV)
CFTS
Vu: discount FCF at Ku
VTS: discount CFTS (at Ku or Kd or both depending
the chosen theory)
V = Vu + VTS
D: discount CFd at Kd
E =V – D
2
Capital Cash Flow CCF, CFd
(CCF)
Kbt, Kd
V: discount CCF at Kbt
D: discount CFd at Kd
E =V – D
3
Free Cash Flow to FCF, CFd
Firm (FCFF)
K, Kd
V: discount FCF at K
D: discount CFd at Kd
E =V – D
4
Free Cash Flow to ECF
Equity (FCFE)
Ke
E: discount ECF at Ke
5
Economic Value
Added (EVA)
K, Kd
V[EVA]: discount EVA at K
V = Vbv + V[EVA]
D: discount CFd at Kd
E =V – D
6
Residual Income (RI) RI
Ke
V[RI]: discount RI at Ke
E = Ebv + V[RI]
EVA, CFd
No.
Method
7 Business
Cash flows
risk- FCF//Ku,
adjusted Free Cash CFd
Flow (BR-adj FCF)
Discount rate
K, Ku, Kd
Implementation
FCF//Ku: obtained by adjusting FCF using K and Ku
so that discounting FCF//Ku at Ku will return V.
V: discount FCF//Ku at Ku
D: discount CFd at Kd
Theories and
gaps:
METHODS
7 to 9
E =V – D
8
9
Business
risk- ECF/Ku
adjusted
Equity
Cash Flow (BR-adj
ECF)
Ke, Ku
ECF//Ku: obtained by adjusting ECF with Ke and Ku
so that discounting ECF//Ku at Ku will return E.
E: discount ECF//Ku at Ku
Business
risk- CCF/Ku, CFd Kbt, Ku, Kd CCF//Ku: obtained by adjusting CCF using Kbt and Ku
adjusted
Capital
so that discounting CCF//Ku at Ku will return V.
Cash Flow (BR-adj
V: discount CCF//Ku at Ku
CCF)
D: discount CFd at Kd
E =V – D
Acronyms
No.
7 Risk
Method
Cash flows
free-adjusted FCF/Rf, CFd
Free Cash Flow (RFadj FCF)
Discount rate
K, Rf, Kd
Implementation
FCF//Rf: obtained by adjusting FCF using K and Rf so
that discounting FCF//Rf at Rf will return V.
V: discount FCF//Rf at Rf
D: discount CFd at Kd
Theories and
gaps:
METHODS
10 to 12
E =V – D
8
9
Risk free-adjusted ECF/Rf
Equity Cash Flow
(RF-adj ECF)
Ke, Rf
Risk free-adjusted
Capital Cash Flow
(RF-adj CCF)
Kbt, Rf, Kd
CCF/Rf, CFd
ECF//Rf: obtained by adjusting ECF with Ke and Rf so
that discounting ECF//Rf at Rf will return E.
E: discount ECF//Rf at Rf
CCF//Rf: obtained by adjusting CCF using Kbt and Rf
so that discounting CCF//Rf at Rf will return V.
V: discount CCF//Rf at Rf
D: discount CFd at Kd
E =V – D
Acronyms
 APV and CCF were created by Myers (1974) and Arditti and Levy
(1977) respectively why the rest were found by practitioners.
Theories and
gaps:
METHODS
Summary
 The most popular method is FCFF which is sometimes referred as
the “textbook” approach or the WACC approach.
 In the first 4 methods (APV, CCF, FCFF and FCFE) , cash flows can
be calculated independently of discount rates.
 The last 8 methods requires that discount rates and cash flows
must be calculated at the same time.
 As long as Value Additivity principle is satisfied, there are no
gaps in this literature regarding methods because all methods
follow the same core theories and share the same inputs.
 Hence, if inconsistent results across methods in practice, it
suggests that there are gaps in the last 3 steps.
1. FCF
13. RFadj CCF
Theories and
gaps:
CASH FLOWS
2. CFTS
12. RFadj ECF
3. CCF
13
Cash
Flows
11. RFadj FCF
4. CFd
10. BRadj CCF
5. ECF
9. BRadj ECF
6. EVA
8. BRadj FCF
7. RI
1. FCFt = EBITt(1 – Tt) – ∆Vbvt
2. CFTSt = TtIt = TtrtDbvt-1
3. CCFt = FCFt + CFTSt = EBITt(1 – Tt) – ∆Vbvt + TtIt
Theories and
gaps:
CASH FLOWS
Formulas
4. CFdt = It – ∆Dbvt
5. ECFt = CCFt – CFdt = EBITt(1 – Tt) – ∆Vbvt – (1 – Tt)It + ∆Dbvt
6. EVAt = EBITt(1 – Tt) – KtVbvt-1
7. RIt = EBITt(1 – Tt) – (1 – Tt)It – KetEbvt-1
8. FCF//Kut = FCFt + Vt-1(Kut – Kt)
9. CCF//Kut = CCFt + Vt-1(Kut – Kbtt)
10. ECF//Kut = ECFt + Et-1(Kut – Ket)
11. FCF//Rft = FCFt + Vt-1(Rft – Kt)
12. CCF//Rft = CCFt + Vt-1(Rft – Kbtt)
13. ECF//Rft = ECFt + Et-1(Rft – Ket)
Theories and
gaps:
There are 3 different approaches which will lead to different cash
flow results:
1.
Constant debt and no growth (Modigliani and Miller 1958)
CASH FLOWS
2.
3 approaches
(assumptions)
Constant debt ratio and perpetual growth (all other
researchers including big names such as Hamada (1972), Myers
(1974), Miles and Ezzell (1980), Fernández (2004), Damodaran
(2008))
3.
Pro-forma financial statements (practitioners)
Theories and
gaps:
The first approach (constant debt, no growth) was too unrealistic
to be applied in practice
The second approach (constant debt ratio, perpetual growth):
 allows having discount rate unchanged but only takes advantage
of the first year forecasted financial statements and
CASH FLOWS
 forces all financial statements to grow at the same rate.
3 approaches
(assumptions)
The last approach (Pro-forma financial statement):
 Hence, it’s still very unrealistic since it almost never happens in
real business.
 Applies constant debt ratio and perpetual growth in stable period
 Uses budgeted financial statements and releases all assumptions
in dynamic period.
Theories and
gaps:
CASH FLOWS
Summary
The third approach (used by practitioners) filled the gaps in the first
2 approaches.
However, inconsistent results still occur due to:
1. Incorrect cash flow formulas
2. Incorrect discount rate formulas (tackled in DISCOUNT RATE step)
3. Improper implementation (tackled in IMPLEMENTATION step)
The gap of this literature regarding cash flow is to reformulate
cash flow formulas so that they are general enough to address
almost all scenarios happening due to the dynamics of financial
statement in the third approach.
1. Rf
Theories and
gaps:
DISCOUNT
RATES
6. Ke
2. Kd
6
DISCOUNT
RATES
3. Ku
5. Kbt
4. K
Beta approach: CAPM module of Sharpe (1964)
Theories and
gaps:
DISCOUNT
RATES
Rf, Kd, Ku
Acronyms
Ku = Rf + BetaU MRP
 Top-down methods: (1) Regress market returns and stock return
to obtain historical equity beta; (2) Unlever historical beta to
acquire unlevered equity beta (BetaU)
 Bottom-up method: (1) Break down firm’s overall operation to
specific operations; (2) Find comparable BetaU for each operation;
(3) Calculate weighted average BetaU of the firm
Fama and French approach: Multi-factor model of Fama and
French (1993)
Theories and
gaps:
DISCOUNT
RATES
K, Kbt, Ke
Acronyms
These discount rates must be calculated internally using the
previous three discount rates and cash flow information.
Those discount rate formulas are affected by:
1. Assumption of capital structure
2. Theory on which discount rate is chosen to discount cash flow of
tax shield
Fixed debt
Theory
Theories and
gaps:
DISCOUNT
RATES
K, Kbt, Ke
No growth
Constant
growth
Author
Modigliani
and
Miller (1958)
Fixed debt
ratio
Discount CFTS
Discount CFTS
at Kd
at Ku
st
In
1 In
all From 2nd In
all
period
period
period
period
√
√
Luehrman (1997)
√
Myers (1974)
√
√
Harris and Pringle
(1985)
√
√
Kaplan
and
Ruback (1995)
√
√
Miles and Ezzell
(1985)
√
√
√
Lewellen
and
Emery (1986)
√
√
√
√
Theories and
gaps:
Capital Structure assumption:
 As stated in CASH FLOW section, dynamic capital structure is the
proper assumption for dynamic period while fixed capital structure
is suitable for stable period.
Theory on discount rate of CFTS:
DISCOUNT
RATES
K, Kbt, Ke
 Discount CFTS at Kd (MM 1958): CFTS is one part of cash flow
received by debtholders, hence, it should be discounted at cost of
debt Kd.
Summary of
theories
 Discount CFTS at Kd in period 1 and at Ku from period 2 onward
(Miles and Ezzell 1985): obtained through mathematic approach
under fixed debt assumption.
 Discount CFTS at Ku (Myers 1974): Fixed debt ratio assumption
leads to proportional adjustment of debt to firm value, hence,
CFTS which arises from debt should have the same risk as the firm
Ku.
 Author’s view: MM’s theory is straightforward and independent of
capital structure. Theories of Myers’ and Miles and Ezzell’s will lose
their veracity when fixed debt ratio assumption is released.
Theories and
gaps:
DISCOUNT
RATES
Ke and K
Formulas and
gaps
Cost of equity formula calculated through Ku and Kd
Ke = Ku +
(Ku − Kd ) by MM (1958)
Use fixed debt assumption under MM‘s theory
Gap: Reformulate Ke under dynamic debt level and dynamic growth in all 3
theories.
Weighted Average Cost of Capital formula calculated through Ke and Kd
(must be used with a correct Ke formula)
K =
Ke E
+ 1 − T Kd D
E +D
Popular “textbook” formula with no mathematic proof
The gap was filled by Fernández (2003) with the following formula:
K =
Ke E
+ Kd D
− T r Dbv
E +D
Theories and
gaps:
DISCOUNT
RATES
K
Formulas and
gaps
Weighted Average Cost of Capital formula calculated through Ku
and Kd (can be used alone)
Ku (E +D − D T ) + Kd D
K =
E +D
T − T r Dbv
Found by Fernández (2003, 2004) but was proven incorrect by
Fieten, Kruschwitz et al. (2005) and Cooper and Nyborg (2006)
Gap: Find correct K formula calculated through Ku and Kd in all 3
theories of CFTS discount rate.
Theories and
gaps:
DISCOUNT
RATES
Kbt
Formulas and
gaps
Before-tax Weighted Average Cost of Capital formula calculated
through Ke and Kd (must be used with a correct Ke formula)
Kbt =
Ke E
E
+ Kd D
+D
Found by Arditti and Levy (1977) with no mathematic proof but
correct reasoning.
Before-tax Weighted Average Cost of Capital formula calculated
through Ku and Kd (can be used alone)
Kbt = Ku with BetaU =
BetaD +
BetaE
Proved by Ruback (2002) through beta formula under Myers’ theory
Gap: Find correct Kbt formula calculated through Ku and Kd in
MM’s theory and Miles and Ezzell’s theory.
Theories and
gaps:
IMPLEMENTA
TION
Inconsistent results across DCF methods were experienced in
common practice along with violation of Value Additivity principle.
Apart from reasons due existing gaps in step 2 and step 3 which
were discussed before, improper implementation is one of the key
reasons.
In fact, considering all methods share the same input (evaluating the
same asset) and the same core theories, the method should arrive
at the same result.
Researcher
Taggart Jr (1989)
Findings
A consistent result in 3 methods:
APV, FCFF, FCFE
Limitations
Fixed leverage
Fixed discount rates Ku, Kd
Use textbook formula K
Theories and
gaps:
No complex example testing
Shrieves and Wachowicz
Jr (2001)
A consistent result in 3 methods:
FCFF, EVA, CCF
Only try to prove the consistency of
methods
No discount rate formulas shown
No testing
IMPLEMENTA
TION
Attempts and
their limitation
Fernández (2003)
Oded and Michel (2007)
A consistent result in 10 methods:
APV, FCFF, FCFE, BR-adj FCF, BRadj ECF, RF-adj FCF, RF-adj ECF,
CCF, EVA, RI
Perpetual growth
A consistent result in 4 methods:
APV, CCF, FCFE, FCFF
Fixed leverage and rebalancing
assumptions
Proven errors in formula
Use constant growth assumption in
example
Constant growth
Constant discount rates
No complex example testing
Massari, Roncaglio, and
Zanetti (2008)
Inconsistent results between APV
and FCFF under perpetual growth
assumptions
Fixed leverage
Perpetual growth
Use textbook formula K
Theories and
gaps:
IMPLEMENTA
TION
Summary
 Fernández (2003) was able to filled most of the gaps in this
literature by showing consistent results in 10 methods with
“Backward iteration” method which was also applied by Miles and
Ezzell (1985)
 However, he used incorrect formula, perpetual growth assumption
and applied only his incorrectly-proven theory.
Gap: Use correct cash flow formulas and discount rate formulas
to demonstrate consistent results in 12 methods in 3 theories
with dynamic debt and growth assumptions.
Research
objectives
Research
questions
Research objectives
Research questions
Under dynamic assumption of
capital structure and growth:
Under dynamic assumption of
capital structure and growth:
1.
Find generalised formulas
for cash flows
1.
What are the generalized
formulas for cash flows?
2.
Find generalized formulas
for Ke, K and Kbt
calculated through Ku and
Kd under all 3 theories
2.
What are the generalized
formulas for Ke, K and Kbt
calculated through Ku and
Kd under all 3 theories?
3.
Demonstrate consistent
results in 12 methods
under all 3 theories
3.
How can one demonstrate
consistent results in 12
methods under all 3
theories?
 Method: Qualitative method with mathematic approach.
 Research value:
Method and
Research value
 Academic: Enhance the current literature of DCF valuation with
more logical understanding, more general formulas and more
suitable implementation.
 Practice:
Allowing
practitioners
(investors,
analyst,
consultants…etc) to make better investment decisions through
making the popular DCF valuation much more reliable, logical and
understandable/
References
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