### wacc - Leeds School of Business

```VALUATION & LEVERAGE
Capital budgeting considering risk and leverage
Introduction


Today we will discuss three approaches to valuing a risky
asset for which both debt and equity financing are used.
Initial Simplifying Assumptions:

The project has average (for the firm) risk.


The firm’s debt-equity ratio is held constant.


For simplicity the betas or costs of capital used will be for the existing
firm rather than being project specific. Often iffy but correctable.
This simplifies the application in that we don’t need to worry about
changing costs of capital over time and identifies the proper the
adjustment of our risk measure for leverage. It is a realistic and
common policy (at least in expectation).
Corporate taxes are the only relevant imperfection.

No agency, bankruptcy or issuance costs to quantify. Clearly not!
The Weighted Average Cost
of Capital Method
rw acc 
E
E  D
rE 
D
E  D
rD (1   c )
 Because
the WACC incorporates the tax savings from
debt, we can compute the levered value (V for enterprise
value, L for leverage, 0 for current or time 0) of an
investment, by discounting its future expected free cash
flow using the WACC.
V
L
0

F C F1
1  rw acc

F C F2
(1  rw acc )
2

F C F3
(1  rw acc )
3

Valuing a Project using the WACC

Ralph Inc. is considering introducing a new type of
chew toy for dogs.
 Ralph
expects the toys to become obsolete after five
years when it will be confirmed that chew toys actually
encourage dogs to eat shoes. However, the marketing
group expects annual sales of \$40 million for the first
year, increasing by \$10 million per year for the
following four years.
 Manufacturing
costs and operating expenses (excluding
depreciation) are expected to be 40% of sales and \$7
million, respectively, each year.
Valuing a Project using the WACC
 Developing
the product will require upfront R&D and
marketing expenses of \$8 million. The fixed assets
necessary to produce the product will require an
 The
equipment will be obsolete once production ceases and
(for simplicity) will be depreciated via the straight-line
method over the five year period for tax purposes.
 Ralph
expects \$1 million in new net working capital
requirements for the years the project operates.
 Ralph
has a target of 60% equity financing, currently
has \$50 million in excess cash, and pays a 35%
corporate tax rate.
Expected Future Free Cash Flow
"Income Statement:" Year
Sales
COGS
Gross Profit
Operating Expenses
Depreciation Exp
EBIT
Tax (35%)
Unlevered NI
Free Cash Flow:
Unlevered NI
Plus Deprecition Exp
Less Net Cap Ex
Less Changes in NWC
Free Cash Flow
0
-8.00
-2.80
-5.20
1
40.00
16.00
24.00
7.00
4.00
13.00
4.55
8.45
2
50.00
20.00
30.00
7.00
4.00
19.00
6.65
12.35
3
60.00
24.00
36.00
7.00
4.00
25.00
8.75
16.25
4
70.00
28.00
42.00
7.00
4.00
31.00
10.85
20.15
5
80.00
32.00
48.00
7.00
4.00
37.00
12.95
24.05
-5.20
0.00
20.00
1.00
-26.20
8.45
4.00
0.00
0.00
12.45
12.35
4.00
0.00
0.00
16.35
16.25
4.00
0.00
0.00
20.25
20.15
4.00
0.00
0.00
24.15
24.05
4.00
0.00
-1.00
29.05
8.00
“Market Value” Balance Sheet

Before the project (\$millions):
Assets
Excess Cash
\$ 50.00
Existing Assets \$ 850.00
Total Assets

\$ 900.00
Liabilities
Debt
\$ 390.00
Equity
\$ 510.00
Total Liabilities
and Equity
\$ 900.00
Cost of Capital
Debt
5%
Equity
12%
Risk Free
4%
The firm is currently at its target leverage:
 Equity
to Net Debt plus Equity (enterprise value) ratio:
\$510.00/(\$510.00 + \$390.00 - \$50.00) = 60.0%
Valuing a Project using the WACC

Ralph intends to maintain a similar net debt-equity
ratio for the foreseeable future, including any
financing related to the project. Thus, Ralph’s WACC
is:
rw acc 

E
E  D
510
rE 
(12% ) 
850
 8 .5%
D
E  D
340
850
rD (1   c )
(5% )(1  0.35)
Valuing a Project using the WACC

The value of the project, including the tax shield
from debt, is calculated as the present value of its
future free cash flows discounted at the WACC.
V
L
0

12.45
1.0 85

16.35
1.0 85
2

20.25
1.0 85
3

24.15
1.0 85
4
29.05
+
1.085
5
 \$77.96 m illion
 The
NPV (value added) of the project is \$51.76 million

\$77.96 million – \$26.20 million = \$51.76 million

It is important to remember the difference between value and
Summary of the WACC Method
1.
2.
3.
Determine the incremental free cash flow of the
investment project.
Compute the weighted average cost of capital.
Compute the value of the investment, including the tax
benefit of leverage, by discounting the free cash flow of
the investment using the WACC.
a.
4.
Note that only the tax benefit of debt is explicitly valued via this
method.
The WACC can be used throughout the firm as the
companywide cost of capital for new investments that are
of comparable risk to the firm itself and that will adopt the
firm’s debt-equity ratio.
Implementing a Constant Debt-Equity Ratio

By undertaking the project, Ralph adds new assets
to the firm with an initial market value of \$77.96
million.
 Therefore,
to maintain the target debt-to-value ratio,
Ralph must initially add \$31.19 million in new net debt.
 40%
× \$77.96 = \$31.19 (\$31.185, rounding)
 60%
× \$77.96 = \$46.78 (compare to \$51.76)
Implementing a Constant Debt-Equity Ratio

Ralph can add (net) debt in this amount either by
reducing cash and/or by borrowing and increasing
actual debt.

Suppose Ralph decides to spend \$26.20 million (the negative
FCF in year 0) of its excess cash to initiate the project.

This increases net debt by \$26.20 million
Assets
Excess Cash
\$ 23.80
Existing Assets \$ 850.00
New Project
\$ 77.96
Total Assets
\$ 951.76
Liabilities
Debt
\$ 390.00
Equity
\$ 561.76
Total Liabilities
and Equity
\$ 951.76
% of Total Value
Net Debt
39.5%
Equity
60.5%
New Market Value Balance Sheet


We need an initial increase in net debt of \$31.19
and equity of \$46.78. So…
Spend \$26.20 million on the project and pay a
\$4.99 million dividend so \$31.19 million in cash
goes out (the dividend further increases net debt
and reduces equity to achieve the desired ratio).
Assets
Excess Cash
\$ 18.81
Existing Assets \$ 850.00
New Project
\$ 77.96
Total Assets
\$ 946.78
Liabilities
Debt
\$ 390.00
Equity
\$ 556.78
Total Liabilities
and Equity
\$ 946.78
% of Total Value
Debt
40.0%
Equity
60.0%
Implementing a Constant Debt-Equity Ratio

The market value of Ralph’s equity increases by \$46.78
million.


\$556.78 − \$510.00 = \$46.78 (60% of \$77.96)
Adding the dividend of \$4.99 million into the mix, the
shareholders’ total gain is \$51.76 million.
\$46.78 + 4.99 = \$51.76 (rounding)
 Which is exactly the NPV calculated for the project
 The first try: without the dividend the equity value increased
by the project’s NPV of \$51.76 = \$561.76 - \$510.00. This
was too large an increase in equity (with an increase in net
debt of \$26.20) since Ralph seeks to maintain 60% equity
financing.

Implementing a Constant Debt-Equity Ratio

Debt Capacity
 The
amount of debt at a particular date that is
required to maintain the firm’s target debt-to-value
ratio
 The
debt capacity at date t is calculated as:
D t  d  Vt
L
d is the firm’s target debt-to-value ratio and VLt is
the project’s levered continuation value on date t (i.e. the
present value of all future FCF from time t).
 Where
Debt Capacity


In order to maintain the target financing, the amount
of new debt must fall over the life of the project.
This is true because the value of the project
depends upon the future cash flow at each point in
time. Since the project ends, value decreases. Since
value decreases, debt must also decrease.
year
Free Cash Flow
Levered Value
Debt Capacity d = 40%
0
1
2
3
4
5
\$ (26.20) \$ 12.45 \$16.35 \$20.25 \$24.15 \$ 29.05
\$ 77.96 \$ 72.14 \$61.92 \$46.93 \$26.77 \$ \$ 31.19 \$ 28.86 \$24.77 \$18.77 \$10.71 \$ -

A
valuation method to determine the levered value
of an investment by first calculating its unlevered
value and then adding the value of the interest tax
shield and deducting any costs that arise from other
market imperfections
L
V0
 APV
 V0
U
 P V (Interest T ax S hield)
 P V (Financial D istress, A gency, and Issuance C osts)
The Unlevered Value of the Project

The first step in the APV method is to calculate the
value of the free cash flows using the project’s cost
of capital if it were financed without leverage.
The Unlevered Value of the Project

Unlevered Cost of Capital
 The
cost of capital of a firm, were it unlevered:
If the firm maintains a target leverage ratio, rU
can be estimated (recall the picture) as the weighted
average cost of capital computed without taking into
account taxes (pre-tax WACC).
rU

 This
E
E  D
rE 
D
E  D
rD  P retax W A C C
is, strictly speaking, only true for firms that adjust their
debt to maintain a target leverage ratio, a common but not
universal policy.
The Unlevered Value of the Project

For Ralph, the unlevered cost of capital is:
 0.60  12.0%  0.40  5.0%
rU
 9.2%
 rw acc 
 D E  D   c rD
 8.5% 

 340 850   0.35  5%
 9.2%
The project’s value without leverage is:
V
U

12.45
1.0 92

16.35
1.0 92
 \$76.35 m illion
2

20.25
1.0 92
3

24.15
1.0 92
4
+
29.05
1.092
5
Valuing the Interest Tax Shield

The \$76.35 million is the value of the unlevered
project and does not include the value of the tax
shield provided by the interest payments on any
incremental debt associated with the project.
Interest paid in year t  rD  D t

 1
The interest tax shield is equal to the interest paid
multiplied by the corporate tax rate.
Interest tax shield for year t   c  rD  D t 1
Interest Tax Shield

From the debt capacity calculation we can find the
interest associated with the project if the financing is
kept on target.
year
Free Cash Flow
Levered Value (WACC)
Debt Capacity d = 40%
Interest
Interest Tax Shield
0
\$ (26.20)
\$ 77.96
\$ 31.19
\$ \$ -
1
\$ 12.45
\$ 72.14
\$ 28.86
\$ 1.56
\$ 0.55
2
\$16.35
\$61.92
\$24.77
\$ 1.44
\$ 0.50
3
\$20.25
\$46.93
\$18.77
\$ 1.24
\$ 0.43
4
\$24.15
\$26.77
\$10.71
\$ 0.94
\$ 0.33
5
\$ 29.05
\$ \$ \$ 0.54
\$ 0.19
Valuing the Interest Tax Shield

The next step is to find the present value of the
annual interest tax shields created by the borrowing
associated with the project.
 When
the firm maintains a target leverage ratio, its
future interest tax shields have similar risk to the project’s
cash flows, therefore they should be discounted at the
project’s unlevered cost of capital.
P V (interest tax shield) 
0. 55
1.0 92

0.5 0
1.0 92
 \$1.61 m illion
2

0. 43
1.0 92
3

0. 33
1.0 92
4
+
0.19
1.092
5
Valuing the Project with Leverage

The total value of the project with leverage is the
sum of the value of the interest tax shield and the
value of the unlevered project.
V
L
 V
U
 P V (interest tax shield)
 76.35  1.61  \$77.96 m illion
 The
NPV of the project is \$51.76 million
 \$77.96

million – \$26.20 million = \$51.76 million
These are exactly the same values found using the WACC
approach.
Summary of the APV Method
1.
2.
Determine the investment’s value
without leverage.
Determine the present value of the interest
tax shield.
a.
b.
3.
4.
Determine the expected interest tax shield.
Discount the interest tax shield.
Add the unlevered value to the present value of the
interest tax shield to determine the value of the
investment with leverage.
Could subtract the costs associated with debt as well if
there is a reasonable way to quantify them.
Summary of the APV Method

The APV method has some advantages.
 It
can be easier to apply than the WACC method when
the firm does not maintain a constant debt-equity ratio.
 The
APV approach also explicitly values market
imperfections and therefore allows managers to
measure their contribution to value.
 Note
that but for the tax shield the WACC method may also
do this.
The Flow-to-Equity Method

Flow-to-Equity
A
valuation method that calculates the free cash flow
available to equity holders taking into account all
payments to and from debt holders.
 Free
Cash Flow to Equity (FCFE), the free cash flow that
remains after providing for interest payments, debt issuance
and debt repayments
 The
cash flows to equity holders are then discounted
using the cost of (levered) equity capital.
Free Cash Flow to Equity

Recall, this is the actual cash flow to levered equity.
Free Cash Flow to Equity
Year
Unlevered NI
Less After Tax Interest
Plus Depr
Less Net Cap Ex
Less Change in NWC
Plus Net Borrowing
Free Cash Flow to Equity
\$
\$
\$
\$
\$
\$
\$
0
(5.20)
20.00
1.00
31.19
4.99
\$
\$
\$
\$
\$
\$
\$
1
8.45
1.01
4.00
(2.33)
9.11
\$
\$
\$
\$
\$
\$
\$
2
12.35
0.94
4.00
(4.09)
11.32
\$
\$
\$
\$
\$
\$
\$
3
16.25
0.80
4.00
(5.99)
13.45
\$
\$
\$
\$
\$
\$
\$
4
20.15
0.61
4.00
(8.06)
15.48
5
\$ 24.05
\$ 0.35
\$ 4.00
\$ \$ (1.00)
\$ (10.71)
\$ 17.99
Valuing the Equity Cash Flows

Because the FCFE represents expected payments to equity
holders, they should be discounted at the project’s cost of
equity capital.

Given that the risk and leverage of the project are the same
as for Ralph Inc. overall, we can use the firm’s cost of
levered equity capital of 12.0% to discount the project’s
FCFE.
N P V ( F C F E )  4.99 
9.11
1.1 2

11.32
1.1 2
2

13.45
1.1 2
3

15.48
1.1 2
4
+
17.99
1.12
5
 \$51.76 m illion

The value of the project’s FCFE represents the gain to shareholders
from the project and it is identical to the NPV computed using the
WACC and APV methods. (Assumes the debt is sold at a fair price.)
Project-Based Costs of Capital



Any specific project may have different systematic
risk than the average project for the firm.
In addition, different projects will may also vary in
the amount of leverage they will support.
In other words, let’s relax those initial simplifying
assumptions.
Estimating the Unlevered Cost of Capital

Suppose the project Ralph launches faces different
market risks than its main business.
 The
unlevered cost of capital for the new project can
be estimated by looking at publicly traded, pure play
firms that have similar business risk.
Estimating the Unlevered Cost of Capital

Assume two firms are comparable to the chew toy
project in terms of basic business risk and have the
following observable characteristics:
Firm
Equity Beta
Debt Beta
Net Debt-toEnterprise Value
Ratio
Firm A
1.7
0.05
40%
Firm B
1.9
0.10
50%
Estimating the Unlevered Cost
of Capital using Betas

We now find their unlevered or asset betas:




A
U
B
U
E

E
A
D
E

E
B
A
A

A
E
D

E
A
B
D
B

B
E
E
B
D 
A
D
D

A
A
B
D
B

B
D

0.6
0.6  0.4
0.5
0.5  0.5
1.7 
1.9 
0.4
0.6  0.4
0.5
0.5  0.5
0.05  1.04
0.1  1.0
An average of these unlevered betas is 1.02.
Note, an unlevered beta estimate for the project of
1.02 gives an unlevered cost of equity capital of:
rU  r f   U ( R P )  4%  1.02(6% )  10.12%  9.2%
Project Leverage
and the Equity Cost of Capital


Assume that Ralph plans to maintain a 20% net debt to
enterprise value ratio for its chew toy project, and it
expects its borrowing cost for the project to be 4%.
We now “relever” the unlevered beta estimate of 1.02
and using the SML we find the cost of levered equity:
 E  U 
D
E
(  U   D )  1 .0 2 
0 .2
(1 .0 2  0 .0 )  1 .2 7 5
0 .8
rE  r f   E ( R P )  4 %  1 .2 7 5(6 % )  1 1 .6 5 %

A cost of debt capital of 4% is consistent with the low
leverage chosen and a debt beta of 0.
Project Leverage and the
Weighted Average Cost of Capital

With a 20% debt to value ratio, a cost of equity
capital of 11.65%, and a cost of debt capital of
4% we can now estimate the WACC for the project.
rW A C C 
0.8
0.8  0.2
11.65% 
0.2
0.8  0.2
4% (1  0.35)  9.84
An Alternate Approach


From the observable (or measurable) data we can
get estimates of the cost of equity capital and the
cost of debt capital:
Firm A:
rE  4%  1.7  6%  14.2%
rD  4%  0.05  6%  4.3%

Firm B:
rE  4%  1.9  6%  15.4%
rD  4%  0.1  6%  4.6%
An Alternate Approach

Recall the relation between the levered cost of equity
capital and the unlevered cost of equity capital:
rE

( rU  rD )
E
Rearranging this we find:
rU 

 rU 
D
E
ED
rE 
D
ED
rD  pre-tax W A C C
In other words, (as we saw before) the unlevered cost
of equity capital equals the pre-tax WACC
Estimating the Unlevered Cost of Capital

If both firms are maintaining a target leverage
ratio, the unlevered cost of capital for each
competitor can be estimated by calculating their
pretax WACC.
Firm A: rU
 0.60  14.2%  0.40  4.3%  10.24 %
Firm B: rU
 0.50  15.4%  0.50  4.6%  10.0%

Based on these comparable firms, we estimate an
unlevered cost of capital for the project that is
approximately 10.12%.
Project Leverage
and the Equity Cost of Capital

Because Ralph plans to maintain a 20% debt to
value ratio for its chew toy project, and it expects
its borrowing cost to be 4%.
 Given
the unlevered cost of capital estimate of
10.12%, the chew toy division’s equity cost of capital
can be estimated to be:
rE
 10.12% 
0.20
0.80
 11.65%
(10.12%  4% )
Project Leverage and the
Weighted Average Cost of Capital

The division’s WACC can now be estimated to be:
rW A C C
 0.80  11.65%  0.20  4.0%  (1  0.35)
 9.84%

An alternate method for calculating the chew toy
division’s WACC is:
rW A C C
 rU  d  c rD
 10.12%  0.20  0.35  4%
 9.84%
```