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Lecture 5: Levered Firms

Discounted Cash Flow, Section 2.2

© 2004, Lutz Kruschwitz and Andreas Löffler

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Equity and debt

Let us have a closer look at levered firms: their value equals debt plus equity:

These are market values, not book values!. Debt is granted at time t, after one period the debtor has to pay redemption and interest. Without default

(Default will be discussed later.)

ltV

l lt t tV D E

1t f tI r D

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2.2.1 Debt ratio and leverage ratio

Debt ratio and leverage ratio are two important numbers.

Both can be uncertain:

debt ratio

implies .1-

leverage ratio

t

tt l

tt

ttt

t

Dl

V lL

lDL

E

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2.2.1 Book values We now introduce the book value of a company. These are

those values, with which the owners’ or creditors’ claims are

to be found in the balance sheets.

The book value equals book value of debt plus book

value of equity

Again debt ratio and leverage ratio can be defined

tD

tE

ltV

.lt t tV D E

and .t tt tl

tt

D Dl L

EV

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2.2.2 Earnings and taxes

Only the red items differ for levered and unlevered firms.

Assumption 2.2 (identical gross cash flows): There is no difference between levered and unlevered firms with respect to gross cash flows.

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2.2.2 Corporate income tax

Earnings before taxes are the tax base. Hence we have for the tax payments of the corporation

Then, the tax payments of the levered and the unlevered firm differ by

This difference is called the tax shield from debt.

. (2.7)Tax EBT

.u lt t tTax Tax l

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2.2.2 Tax shield and DCF

In case of no default interest is determined by the riskless rate and debt . The tax shield is

From our assumptions only debt can be uncertain in this model!

The level of tax shield is determined by the financing policy.

The aim of DCF is the valuation of these (uncertain) tax advantages. Not less, but also not more.

fr 1tD

1 .f tr D

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2.2.3 Financing policies

1. Autonomous financing: future amount of debt is fixed.

2. Financing based on market values: evaluator sets the future debt ratios based on market values.

3. Financing based on book values: the future debt ratios to book values are fixed.

4. Financing based on cash flows: amount of debt is based on the firms´s cash flows.

5. Financing based on dividends: debt managed so that previously determined dividend distributed.

6. Financing based on dynamical leverage ratio: evaluator sets the future cash flow-debt ratios.

tD~

tl~

tl

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Given debt policy

We do not want to answer the question as to which of these financing policies is particularly close to reality. Further, we will not discuss the question ofwhich of the mentioned financing policies maximizes the value of the levered company (later we will see: extended leverage increases company value).

Assumption 2.3 (given debt policy): The debt policy of the firm (although probably uncertain) is already prescribed.

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2.2.3 Value of tax shield I

Our aim is a general valuation equation for the tax shield. From the fundamental theorem (Theorem 1.2) for levered and unlevered firms we get

Using the definition of tax base (2.7)

it follows that

F

1 1

, ,

, .1

t t

l u l uQ tl u

tf

E FCF VV

r

2.8ut t tFCF FCF I

F

1 11.

1t t

l uQ t tl u

t tf

E I V VV V

r

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2.2.3 Value of tax shield IIThen

Now, look at the debt holder,

This gives finally

This is the basic equation for valuing the tax shield. It clearly

shows a dependence of the tax shield on the financing policy

(future debt levels).

F F1

... .1 1

Q t t Q T tl ut t T t

f t

E I E IV V

r r

1 (2.9)t f tI r D

F F1| |

... (2.10)1 1

Q f t t Q f T tl ut t T t

f f

E r D E r DV V

r r

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2.2.4 DefaultDefault trigger can be, for example:– lack of liquidity,– debt greater than assets,– expected lack of liquidity in the near future.At the moment we will not specify the default trigger.

If bankruptcy occurs, there are three possibilities:– restructuring the company,– liquidation of the company, or– sell-out of the remaining assets.We will only assume that all conceivable developments were taken into consideration when determining the company’s cash flows.

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2.2.4 What do we assume?

Owners have no personal liability.

Creditors and shareholders have identical information about the company, its estate, the firm’s cost of capital and its financing policy.

Assumption 2.4 (gross cash flows and default): The gross cash flows as well as the investment and accruals policy of the unlevered firm do not differ from those of the firm in danger of default.

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2.2.4 Prioritization rules

Who gets what in case of default?

Assumption 2.5 (prioritization of debt): The tax office’s claims range before those of other creditors. The cash flows are always sufficient to at least pay off the tax debts in full.

Now a new notation is necessary:

amount of credit outstanding in tamount that is paid back in t + 1interest paid in t + 1

tD

1tR

1tI

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2.2.4 Taxes and default

The tax office allows interest to be deducted from the tax base. On the other hand, the cancellation of debt

adds to the tax base («recapitalization gain»). Hence, the tax due for a firm in danger of default is

1tI

1 1t t t

was paid backshould be paid back

D D R

1 1 1 1 1 1

interest, gain,deducted added

.lt t t t t t tTax GCF Accr I D D R

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2.2.4 Value of the tax shield and default

Since gross cash flows as well as investments are similar to the unlevered firm, we have

and the main valuation equation now reads

This does not look like (2.10)?! But let us see. . .

11 1 1

11 1 1 1 1

1 1 1 1

l ltt t t

utt t t t t t

ut t t t t

FCF GCF Inv Tax

GCF Inv Tax I R D D

FCF I R D D

F1

1

|. 2.12

1

TQ s s s s tl u

t t s ts t f

E I R D DV V

r

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2.2.4 Fundamental theorem for the creditors

The debtholders behave rational. Therefore, at time s

and from rule 2 and rule 5 for all t s

And the rhs is the nominator from the main valuation equation (2.12) above!

FF 11

1

|payments in |

1 1

Q s s s sQ ss

f f

E I R DE sD

r r

F F1 1| | .f Q s t Q s s s s tr E D E I R D D

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Even in the case of default the valuation equation (2.10)

holds.

Or: default does not make the DCF theory fail. The difficulties of taking default into consideration lie much more in the fact that the relevant financing policies must be formulated with care.

2.2.4 Fundamental theorem for the creditors

1| |

...1 1

Q f t t Q f T tl ut t T t

f f

E r D E r DV V

r r

F F

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2.2.4 Cost of debt

Someone who invests today is entitled to payments

less remission of debts . Hence,

Definition 2.3 (cost of debt): The cost of debt of a levered firm is

If there is no default

We do not require the cost of debt to be deterministic today. Cost of debt will not be used itself to determine the value of firms.

F1 1 1 |1.

t t t tDt

t

E D I Rk

D

.Dt fk r

tD

1t tD I 1 1t t tD D R

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2.2.4 Finite example

Provisional leverage policy may be

Bankruptcy enters in if in state ,

Here: bankruptcy is equal to lack of liquidity.

0 1 2100, 100, 50D D D

1 for a levered firm,

0 for an unlevered firm.

lt t t t

ut

FCF I D D

FCF

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2.2.4 Does bankruptcy occur?

1Shareholder's claims 1 1 areut f t tFCF r D D

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2.2.4 The state ddIn all states except dd (and following) the creditors

demand

Only in state = dd bankruptcy can occur. Hence,

We now want to determine

dd , .D nomt fk r

dd , .D nomt fk r

dd ,D nomtk

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2.2.4 Determining

Let us look at the states that follow dd. At state ddd the claims cannot be paid off in full. The cash flow is necessary to pay interest and pay back the loan. Therefore,

On the other hand, since the tax office claims have priority,

which finally gives

dd ,Dnomkt

ddd ddd ddd

3 3 2 3

0

.lFCF D R I

ddd ddd ddd ddd

3 3 3 2 3 2

0

l uFCF FCF Z R D D

ddd ddd

3 3 2

146.8.

1l uFCF FCF D

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2.2.4 Determining

At state ddu the creditors get

Using the fundamental theorem we must have

which finally gives

and

dd ,Dnomkt

dd ddu ddu 2 2 3

,1 .D nomk D R It

2 3 2

2

|

1

Q

f

E R ID

r

F

dd , 32.962%2D nomk

ddu 3 153.44.lFCF

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2.2.4 Cash flows without default riskFCFl

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2.2.4 Cash flows with default risk FCFl

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Summary

We consider levered and unlevered firms with identical gross cash flows.

The tax advantage for the levered firm (no default) is

The financing policy of the firm determines the value of the tax shield.

DCF remains valid even in the case of default. But then therelevant financing policy must be handled with care.

1.f tr D