Lecture 8: Financing Based on
Book Values
Discounted Cash Flow, Section 2.5
© 2004, Lutz Kruschwitz and
Andreas Löffler
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2.5.1 The problem
WACC is one of the most popular methods of evaluating a firm.
But: it requires a deterministic leverage to market values, or
 if share price goes up  debt has to go up as well and
 if share price goes down  debt goes down as well and
 if share price goes up/down  debt goes up/down as well.
This assumption seems very unrealistic. But what happens
with the valuation formula if we replace the market values by
book values?
That is the aim of this section!
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2.5.1 The assumptions
Let us talk about book values. We assume for the
book value of debt
~ ~
Dt  Dt
and no default or I t 1  rf Dt .
But that does not make sense with equity:
usually
E t  Et !
What drives the book value of equity?
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2.5.1 Book value of equity
~
What influences book value of equity E t ? Three elements
are possible:
1. Increase of subscribed capital (which we assume to
be zero).
2. Retained earnings after tax given by
EBIT
3.
t 1

 It 1 1    .
Paid dividends given by cash flows to shareholders,
or
l


FCF t 1  It 1  Dt  Dt 1 .
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2.5.1 Clean surplus relation
Hence, the book value of equity has to satisfy


E t 1  E t  EBITt 1  It 1 1   
l
l


 FCFt u1   It 1  Dt  It 1  Dt 1

which can be simplified to



E t 1  E t  EBITt 1 1     FCFtu1  Dt  Dt 1 .
l
l
Assumption 2.17 (clean surplus relation)
E t 1  Dt 1  E t  Dt  EBITt 1 1     FCFtu1.
l
l
This leads to our next theorem.
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2.5.1 Operating assets relation
Theorem 2.13 (operating assets relation): The book
value of the firm follows
V t 1  V t  EBITt 1 1     FCFt u1
l
levered firm
l
fromunlevered firm
This relation invokes elements from the levered as well
as from the unlevered firm.
Since cash flows and EBIT are random, the book value
will be a random variable as well.
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2.5.1 Financing based on book values
Definition 2.9 (financing based on book
~ on book values
values): A company is financed based
lt
if debt ratios to book values are deterministic.
To get an idea of the company‘s value more information
necessary: we need to know about investment and
accruals since they drive the book value.
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2.5.1 Investment and accruals
Hence,
FCFt u1   EBITt 1  Inv t 1  GCFt 1  EBITt 1  Accrt 1.
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2.5.1 Investment and accruals
Hence,
EBITt 1 1     FCFt u1  Inv t 1  Accrt 1 .
 2.21
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2.5.1 Investment policies
The rhs of (2.21) depends on the investment policy and accruals
EBITt 1 1     FCFt u1  Inv t 1  Accrt 1 .
 2.21
Now three investment policies are differentiated:
1. Investment due to full distribution;
2. replacement investments;
3. investment based on cash flows.
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2.5.2 Full distribution policy
The profit after interest and taxes is
EBIT
t 1
Owners receive

 It 1 1    .


FCFt u1   It 1  Dt  It 1  Dt 1 .
In case of a full distribution policy both amounts are identical.
Hence,
Definition 2.10 (full distribution policy): A levered firm carries out a
full distribution policy if
FCFtu1  EBITt 1 1     Dt  Dt 1 .
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2.5.2 Remarks
It is doubtful whether full distribution is a realistic behaviour, but it
has a strong tradition in Germany.
It is equivalent to investments completely financed by debt:
Inv t 1  Accrt 1  D t  D t 1
And it implies that book value of equity stays constant:
Vt l1  Vt l  EBITt 1 1     FCFt u1 .
Vt l1  V tl  D t  D t 1
E t 1  E t .
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2.5.2 Valuation
The last equation implies: the book value of equity is
deterministic. But if the leverage ratio is deterministic as well,
debt is deterministic:
D t 1  L t 1 E .
l
0
Then the adjusted present value approach is appropriate:
Theorem 2.14 (full distrubution): If the firm is financed based on
book values and carries out full distribution, then
Vt l  Vt u 
T

s t 1
 rf L s 1 E l0
1  r 
s 1t
.
f
Proof: immediate.
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2.5.3 Replacement investment
Definition 2.11 (replacement investment): A firm takes on
replacement investments if for all t
Inv t 1  Accrt 1.
Then
EBITt 1     FCFtu  Inv t  Accrt  0.
Hence, there is no difference between profits and cash flows of the
unlevered firm. Hence, using the operating liabilities relation
l
l
V t 1  V t .
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2.5.3 Valuation
The last equation implies: the book value of the firm is deterministic.
But if the leverage ratio is deterministic as well, debt is deterministic
Dt 1  l t 1V 0
l
Then the adjusted present value approach is appropriate.
Theorem 2.15 (replacement investment): If the firm is financed
based on book values and carries out replacement investment, then
Vt l  Vt u 
T

s t 1
 rf L s 1V l0
1  r 
s 1t
.
f
Proof: immediate.
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2.5.4 Investment based on cash flows
Now let us turn to a more realistiv case - investments are tied to
cash flows:
- increasing cash flows raise investments,
- decreasing cash flows lower investments.
Investments are aligned with the unlevered firm (investments are
independent of leverage!), or
Inv t   t FCFt u
 t  0.
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2.5.4 Gross cash flows and investment
To motivate our assumption let us look at gross cash flows. They are
paid to and for
1. tax authority,
2. investments, and
3. shareholder and creditors.
If investments get a fixed portion, it should depend on gross cash
flow after taxes

Inv t  ßt GCFt   EBITt

0  ßt  1.
But this implies our assumption since (see (2.22))
1  ßt
Inv t 
FCFt u .
ßt
: t
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2.5.4 Accruals
Assumption 2.9 (no discretionary accruals): Accruals
follow for all t
Accrt 
1
Inv t 1  ...  Inv t n  .
n
It is necessary that past investments Inv-1 to Inv-n+1 must be
known. Our assumption is certainly satisfied if depreciation is
the only element of accruals and if we apply straight-line
depreciation.
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2.5.4 Valuation
Given all assumptions a valuation theorem can be verified (see
Theorem 2.17). Unfortunately, this formula is very muddled...
Important drivers for value are:
1. tax rate, riskless interest rate (, i.e. market value increases with
them), cost of capital k E ,u (, i.e. value decreases with them);
2. today´s book value (), past investments ();
3. leverage ratio l to book values ();
4. investment ratio a ();
5. depreciation period n (~) - requires explanation.
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2.5.4 Market value and depreciation
If accruals (depreciation) increase




(given the investment!) book values increase
(given the leverage ratio) debt increases
tax advantages increase
market values increase.
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2.5.4 Long-term constant debt ratios
A simpler formula can be obtained for infinite horizon. We
assume
-an infinite lifetime
-a constant debt ratio and
l a constant investment parameter α, and
-disregard past investments.
Then (Theorem 2.18)
 n 1

V0l  V0u 1 
rf   l    D0 .
2


This formula looks very like Modigliani-Miller, except for the term in
brackets!
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2.5.5 Infinite example
We use
n  2,
  50%,
l  50%
No investments before t = 0. Then
n 1
rf   l V0u
2
2 1
 500  0.5  100 
0.1  0.5  0.5  0.5  500
2
 559.375
V0l  V0u   D0 
(Remark: Here we use the approximation formula)
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Summary
Financing based on book values requires knowledge of the
investment policy.
The operating liabilities relation reveals movement of book values.
Three investment policies can be considered:
- full distrubition (or debt financed investments)  APV;
- replacement investments  APV;
- investments based on cash flows  MoMi-like formula.
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