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Lecture 6: Financing Based on Market Values I
Discounted Cash Flow, Section 2.3,
2.4.1-2.4.3
© 2004, Lutz Kruschwitz and Andreas Löffler
2.3 Adjusted present value
Definition 2.4 (autonomous financing): A firm is autonomously financed if its future amount of debt is certain.
Theorem 2.4 (APV): If a firm is auonomously financed, then
Proof: trivial.
tD
T
tsts
f
sfut
lt
r
DrVV
1
1
1
~~
2.3 Constant debt
Theorem 2.5 (constant debt, MoMi): If and
const, then
Proof:
TtD
.l ut t tV V D
1
1
1
1
1
1
1.
Tf sl u
t t s ts t f
Tut f t s t
s t f
ut f t
f
r DV V
r
V r Dr
V r Dr
2.3 Constant debt, againAn altered representation would be
We come back to this (or at least a similar) equation in the next lecture when talking about the Modigliani–Miller–adjustment.
0 0
0 0 0
0 0 0
0 ,0
1
1
l ut
l u lt
l u
u
lE u
V V D
V V l V
l V V
E FCFV
l k
2.3 The finite exampleWe now look at our finite example with
Hence,
This is also the value of the firm threatened by default.
.50,100,100 210 DDD
0 1 20 0 2 3
2
3
1 1 1
0.5 0.1 100 0.5 0.1 100229.75
1 0.1 1 0.1
0.5 0.1 50240.30.
1 0.1
f f fl u
f f f
r D r D r DV V
r r r
50%.
2.3 Future firm values
For later use we evaluate future firm value ,lV1
~
1 21 1 2
1 2
1 1
0.5 0.1 100 0.5 0.1 50
1 0.1 1 0.1
199.88 if up,
164.74 if down.
f fl u
f f
u
r D r DV V
r r
V
2.3 Debt and leverage ratio
Attention: The corresponding leverage ratio
is uncertain!
Hence, a certain amount of debt implies an uncertain leverage ratio! (And vice versa. . . )
11
1
50.03 % if up,
60.03 % if down,l
Dl
V
The finite case with default
Since we can use the APV-formula,
Another way of obtaining this value is by evaluating and discounting it with the riskless rate.
In addition let us determine the cost of debt,
0 240.30.lV [ ]uQ tE FCF
,2 2 3 3 2 3
22
1(1 ( ) ( | ) ( ( ) ) ( | )
1( ) 1
1(1 0.32962) 50 0.5 (48.4 0.5 50)0.5
1 0.5 150
13.281%.
D nom u
Dk dd D P u dd FCF ddd D P u dd
k ddD
2.3 The infinite example
Here and constant debt
Then
.100tD
550
1
1
00
01
00
0100
DV
r
DrV
r
DrVV
u
tt
f
fu
tt
f
tful
50%
2.4 Financing based on market values
Definition 2.5 (financing based on market values):
Financing is based on market values if debt ratios are certain.
The amount of future debt is uncertain! The tax advantages from debt are uncertain as well! APV does not apply! Instead three different
procedures. . .
tl
tD
2.4 The general procedure
To evaluate the company.1. We start with «appropriate» cost of capital.2. We assume that these costs of capital are deterministic
and apply (as usual) a corresponding valuation formula.3. We then look at the connection of these procedures: they
aregiven by textbook formulas.
There are three «appropriate» cost of capital, hence there will be
three valuation procedures: FTE, TCF, WACC.
Notice that default is not ruled out!
2.4.1 Flow to equity (FTE)
With FTE we are looking at the stockholders and their cost of
equity. The cash flow to stockholders is given by
Definition 2.6 (cost of equity): Costs of equity are conditional
expected returns
1
1 1
free cash flows
repayment and interest .
lt
t t
FCF
R I
11 1 1
,|
: 1.
ltt t t t
E lt
t
E E FCF I Rk
E
F
2.4.1 FTE approach
Theorem 2.6 (FTE): If are deterministic, then
Proof: see our general valuation theorem (Theorem 1.1).Remarks:-FTE requires deterministic cost of equity.-The theorem does not yet require financing based on market values!-The leverage ratio does not appear in FTE.-The knowledge of expected repayment is necessary.
lE
tk,~
F, ,
1 1
|.
1 ... 1
lTs s s t
t E l E ls t t s
E FCF I RE
k k
2.4.2 Total cash flow (TCF)
Now we are looking at the stockholders and debtholders, or the
cost of equity and debt.
Definition 2.7 (weighted average cost of capital – type 1):
WACC type 1 are conditional expected returns
11 |
: 1.
lltt t
t lt
E V FCFk
V
F
TCF approach
Theorem 2.7 (TCF): If deterministic, then
Proof: see our general valuation theorem (Theorem1.1)Remarks:- TCF requires deterministic WACC type 1.- The theorem does not yet require financing based on market
values!- The leverage ratio does not appear in TCF.- The knowledge of expected debt is not necessary.
Ø
tk~
FØ Ø
1 t s-1
|.
1 1
lTs tl
ts t
E FCFV
k k
2.4.2 TCF textbook formula
What is the connection between FTE and TCF? The answer is the textbook formula.
Theorem 2.8 (TCF textbook formula): It always holds that
Ø , 1 .E l Dt t t t tk k l k l
2.4.2 ProofStart with definition cost of equity and use
1 1 1 1
,1 1 1 1 1
,1 1
,1
,
1
,
1 |
1 |
1 |
1 1 |
1
lt t t t t t
E l l lt t t t t t t t
E l l l Dt t t t t t t t
E l D l lt t t t t t t
E lt
E lt E E E FCF R I
k E E V FCF R D I
k E E V FCF D k D
k E k D E V FCF
k
k
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D l lt t t t t t t t
E l Dt tt tl l
t t
E k D k V
E k E D k D V k V
E Dk k k
V V
1 1 1
l
t t tUseV E D
2.4.2 Proof
,1 1 1
,1 1 1
,1 1
,1
1 1 1
1
1
1
1 |
1 |
1 |
1 1 |
E l lt t t t t t
E l lt t t t t t
E l l l Dt t t t t t t t
E l D l l
t t t
lt
t t t t t
t
t t
k E E FCF R I
k E E FCF R I
k E E V FCF D k D
k E k D E V FC
E D D
V D
F
F
F
F
F
, Øt
, Øt
, Øt
1 1 1
.
E l D lt t t t t
E l D l lt t t t t t t t
E l Dt tt tl l
t t
k E k D k V
E k E D k D V k V
E Dk k k
V V
2.4.2 ProofUse definition of cost of debt
,1 1 1 1
,1 1
,1 1
,1 1
1 1 1
,
1 |
1 |
1 |
1 1 |
1
E l lt t t t t t t
E l l lt t t t t
E l l lt t t t t
E l D l lt t t t t t t
E lt
t t t
Dt t t
k E E E FCF R I
k E E V FCF
k E E V FCF
k E k D E V FCF
D I
D k
k
R
D
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D l lt t t t t t t t
E l Dt tt tl l
t t
E k D k V
E k E D k D V k V
E Dk k k
V V
2.4.2 ProofUse rule 5
,1 1 1 1
,1 1 1 1 1
,1 1
,1 1
,
1 |
1 |
1 |
11 |
1
E l lt t t t t t t
E l l lt t t t t t t t
E l l lt t t t t
E l l lt t t t t
E lt
Dt t t
Dt t
k E E E FCF R I
k E E V FCF R D I
k E E V FCF
k E E V F
D k
k F
D
k
D C
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D l lt t t t t t t t
E l Dt tt tl l
t t
E k D k V
E k E D k D V k V
E Dk k k
V V
2.4.2 ProofUse definition of WACC type 1
1
,1 1 1 1
,1 1 1 1 1
,1
1
1
,
,
1 |
1 |
|
1 |
1 1
1
l lt t
E l lt t t t t t t
E l l lt t t t t t t t
E l l l Dt t t t t t t t
E l Dt t t t
l
t
Et
k E E E FCF R I
k E E V FCF R D I
k E E V FCF D k D
k E k FED C
k
V F
F
F
F
F
, Ø
t
, Øt
Øt11
.
lDt t t
E l D l lt t t t t t t t
E l Dt tt
t
t
tl lt
E k D
E k E D k D V k V
E Dk k k
V V
k V
2.4.2 ProofRearrange terms,use l
t t tV E D
,1 1 1 1
,1 1 1 1 1
,1 1
,1 1
,
1 |
1 |
1 |
1 1 |
1
E l lt t t t t t t
E l l lt t t t t t t t
E l l l Dt t t t t t t t
E l D l lt t t t t t t
E lt
k E E E FCF R I
k E E V FCF R D I
k E E V FCF D k D
k E k D E V FCF
k
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D lt t t t t
E l Dt tt tl l
t
lt t t
t
E k D k V
k E k D k V
E Dk k
V
E
V
V
k
D
2.4.2 ProofltDividebyV
,1 1 1 1
,1 1 1 1 1
,1 1
,1 1
,
1 |
1 |
1 |
1 1 |
1
E l lt t t t t t t
E l l lt t t t t t t t
E l l l Dt t t t t t t t
E l D l lt t t t t t t
E lt
k E E E FCF R I
k E E V FCF R D I
k E E V FCF D k D
k E k D E V FCF
k
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D l lt t t t t t t t
E t tl Dt tl l
t t
E k D k V
E k E D k D
E D
V
V k V
k k kV
2.4.2 Proof.tApply definitionof leverageratio I QED
,1 1 1 1
,1 1 1 1 1
,1 1
,1 1
,
1 |
1 |
1 |
1 1 |
1
E l lt t t t t t t
E l l lt t t t t t t t
E l l l Dt t t t t t t t
E l D l lt t t t t t t
E lt
k E E E FCF R I
k E E V FCF R D I
k E E V FCF D k D
k E k D E V FCF
k
F
F
F
F
Øt
, Øt
, Øt
1 1
.
D lt t t t
E l D l lt t t t t t t t
E l Dt tt tl l
t t
E k D k V
E k E D k D V k V
E Dk k k
V V
2.4.2 Remarks to TCF, I
The costs of debt are not reduced by the tax rate in the TCF textbook formula. The formula holds regardless of whether the relevant variables are deterministic or stochastic.
In particular: financing based on market values is not necessary!
But: if WACC type 1 as well as cost of equity and debt arecertain, then debt ratios must be certain as well! In this case, FTE and TCF can be used alternatively.
2.4.2 Remarks to TCF, II
On the other hand, in the case of no default: if the debt ratios are uncertain, either WACC type 1 or cost of equity has to beuncertain.
If WACC type 1 is deterministic TCF has to be used; youcannot apply FTE since costs of equity are uncertain.
If cost of equity is deterministic FTE has to be used; you cannot apply TCF since WACC type 1 is uncertain.
2.4.3 Weighted average cost of capital
We are now stockholders and debtholders again.
Definition 2.8 (weighted average cost of capital – type 2):
WACC type 2 are the conditional expected returns
Remark: These are costs of capital of a firm that is on the one hand levered
and on the other hand unlevered . Apples and oranges mixed here.
F1 1 |
: 1.l u
t t t
t lt
E V FCFWACC
V
ltV~
1u
tFCF
WACC approachTheorem 2.9 (WACC): If is deterministic, then
Proof: see our general valuation theorem (Theorem 1.1)Remarks:– WACC requires deterministic WACC type 2.– The theorem above does not yet require financing based on market values!– The leverage ratio does not appear in WACC.– The knowledge of cash flow of an unlevered firm is necessary.
1 1
|
1 .... 1
uTs tl
ts t t s
E FCF FV
WACC WACC
tWACC
WACC textbook formula
What is the connection between FTE and WACC? The answer
is another textbook formula.
Theorem 2.10 (WACC textbook formula): Always
, 1 1 .E l Dt t t t tWACC k l k l
2.4.3 ProofStart with definition cost of equity
F
F
F
F
1 1 1
,1 1 1
,1 1 1 1 1
1
,
,1
1 |
1 |
1 |
1 |
lt t t t t t
E l l lt t t t t t t
E l l ut t t t t t t t t t
E l l u Dt t t t t f
E lt
t t t
E E E FCF R I
k E E V FCF D I
k E E V FCF D I I R D
k E E V FCF D r D k D
k
.
2.4.3 Proof
F
F
F
,1 1
,1 1
,1 1 1 1 1
,1 1
1 1 1
1
1 |
1 |
1 |
1
E l lt t t t t t
E l lt t t t t t
E l l ut
t t t
t t t t t t t t t
E l l ut t t f
l
t
t
t
k E E FCF R I
k E E FCF D I
k E E V FCF D I I R D
k E E V FCF D
E
V
r
D D
D F| .D
t t tk D
1 1 1
l
t t tUseV E D
2.4.3 ProofMind tax advantage
F
F
F
F
,1 1 1
,1 1
,1
1
1 1 11
,1 1
1 |
1 |
1 |
1 |
E l lt t t t t t t
E l lt t t t t t
E l lt t t t t t
E l l u Dt t t t t f t t
lt
ut t t t
t
k E E E FCF R I
k E E V FCF D I
k E E V FCF D I
k E E V FCF D r D k D
I R D
.
1 1 1
l u
t t tIFCF FCF
2.4.3 ProofUse definition of cost of debt
F
F
F
F
,1 1 1
,1 1 1
,1 1 1
,1 1
1 1
1 |
1 |
1 |
1 |
E l lt t t t t t t
E l l lt t t t t t t
E l l ut t t t t t t
E l l ut
t t t
D Dtt t t t t
k E E E FCF R I
k E E V FCF D I
k E E V FCF D I
k E E V FC
I R D
kD D k DF .t
2.4.3 Proof
F
F
F
F
,1 1 1
,1 1 1
,1 1 1 1 1
,1 1
1 |
1 |
1 |
1 |
E l lt t t t t t t
E l l lt t t t t t t
E l l ut t t t t t t t t t
E l l u Dt t t t t f t t t
k E E E FCF R I
k E E V FCF D I
k E E V FCF D I I R D
k E E V FCF D r D k D .
2.4.3 Proof, continued
Use rule 5
F,1 1
,
,
,
1 |
1 1 1
1
1 .
1
1
1E l l ut t t t t
E l D lt t t t t
E l D l lt t t t t t t t
tE l tt f tl
t
Dt
lt
k Dk E E V FCF
k E k D WACC V
E k E D k D V WACC V
E Dk r WACC
V V
2.4.3 Proof, continued
Use definition of WACC type 2
F
,
,
,
1,
11 1 1
1 1
|
1
1
1
1
E l Dt t t
E l Dt t t
E l D l lt t t t t t t t
tE l tt f tl l
t
l ut t t
lt
t
t
k E k D
k E k D
E k E D k D V
E V FCF
WACC V
WACC V
E Dk r WACC
V V
2.4.3 Proof, continuedRearrange terms and use
F,1 1
,
,
,
1 1 1 |
1
1
1
1 1 1
E l D l ut t t t t t
E l D lt t t t t
E l D lt t t t t
tE l tt f
lt t t
tl lt t
k E k D E V FCF
k E k D WACC V
k E k D WACC V
E Dk r WACC
V V
E D V
l
t t tV E D
2.4.3 Proof, continued
Divide by
F,1 1
,
,
,
1 1 1 |
1 1 1 1
1
1
E l D l ut t t t t t
E l D lt t t t t
E l D l lt t t t t t t t
tE l tt f l
ttl
t
k E k D E V FCF
k E k D WACC V
E k E D k D V WA
V
CC V
E Dk r WACC
V
l
tV
2.4.3 Proof, continued
Apply definition of leverage ratio , q.e.d.
F,1 1
,
,
,
1 1 1 |
1 1 1 1
1
1
E l D l ut t t t t t
E l D lt t t t t
E l D l lt t t t t t t t
tE l tt f tl l
t t
k E k D E V FCF
k E k D WACC V
E k E D k D V WACC V
E Dk r WACC
V V
tI
And this was to be shown. QED
2.4.3 Remarks, I
The costs of debt are reduced by the tax rate in the WACC
textbook formula. The formula holds regardless of wether the
relevant variables are deterministic or stochastic.
In particular: financing based on market values is not necessary!
But: if WACC type 2 as well as cost of equity and debt are
certain, then the debt ratios must be certain as well! In this case,
FTE and WACC can be used alternatively.
2.4.3Remarks, II
On the other hand, in the case of no default: if debt ratios are uncertain, either WACC type 2 or cost of equity has to be uncertain.
If WACC type 2 is deterministic WACC has to be used; you cannot apply FTE since costs of equity are uncertain.
If cost of equity is deterministic FTE has to be used; you cannot apply WACC since WACC is uncertain.
Summary
