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warrants.xls: This spreadsheet allows you to value the options outstanding in a firm,

allowing for the dilution effect.

Illustration 12.10: Value of Equity Options

Disney has granted considerable numbers of options to its managers. At the end of

2003, there were 219 million options outstanding, with a weighted average exercise price

of $26.44 and weighted average life of 6 years. Using the current stock price of $26.91,

an estimated standard deviation30 of 40, a dividend yield of 1.21%. a riskfree rate of 4%

and an option pricing model, we estimate the value of these equity options to $2.129

billion.31 The value we have estimated for the options above are probably too high, since

we assume that all the options are exercisable. In fact, a significant proportion of these

options (about 50%) are not vested32 yet, and this fact will reduce their estimated value.

We will also assume that these options, when exercised, will generate a tax benefit to the

firm equal to 37.3% of their value:

After-tax value of equity options = 2129 (1-.373) = $1334.67 million

29 We assume that all options will be exercised, and compute the number of shares that will be outstanding

in that event.

30 We used the historical standard deviation in BoeingвЂ™s stock price to estimate this number.

31 The option pricing model used is the Black-Scholes model. It is described in more detail in the appendix.

32 When options are not vested, they cannot be exercised. Firms, when providing options to their

employees, firms often require that they continue as employees for a set period before they can exercise

these options.

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Reconciling Equity and Firm Valuations

This model, unlike the dividend discount model or the FCFE model, values the

firm rather than equity. The value of equity, however, can be extracted from the value of

the firm by subtracting out the market value of outstanding debt. Since this model can be

viewed as an alternative way of valuing equity, two questions arise - Why value the firm

rather than equity? Will the values for equity obtained from the firm valuation approach

be consistent with the values obtained from the equity valuation approaches described in

the previous chapter?

The advantage of using the firm valuation approach is that cashflows relating to

debt do not have to be considered explicitly, since the FCFF is a pre-debt cashflow, while

they have to be taken into account in estimating FCFE. In cases where the leverage is

expected to change significantly over time, this is a significant saving, since estimating

new debt issues and debt repayments when leverage is changing can become increasingly

messy the further into the future you go. The firm valuation approach does, however,

require information about debt ratios and interest rates to estimate the weighted average

cost of capital.

The value for equity obtained from the firm valuation and equity valuation

approaches will be the same if you make consistent assumptions about financial leverage.

Getting them to converge in practice is much more difficult. Let us begin with the

simplest case вЂ“ a no-growth, perpetual firm. Assume that the firm has $166.67 million in

earnings before interest and taxes and a tax rate of 40%. Assume that the firm has equity

with a market value of $600 million, with a cost of equity of 13.87%, and debt of $400

million, with a pre-tax cost of debt of 7%. The firmвЂ™s cost of capital can be estimated:

& 600 # & 400 #

Cost of capital = (

13.87% )$ ! + (7% )( - 0.4)$

1 ! = 10%

1000 " 1000 "

% %

EBIT( - t ) 166.67( - 0.4 )

1 1

= $1,000

Value of the firm = =

Cost of capital 0.10

Note that the firm has no reinvestment and no growth. We can value equity in this firm

by subtracting out the value of debt.

Value of equity = Value of firm вЂ“ Value of debt = $ 1,000 - $400 = $ 600 million

Now let us value the equity directly by estimating the net income:

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Net Income = (EBIT вЂ“ Pre-tax cost of debt * Debt) (1-t) = (166.67 - 0.07*400) (1-0.4) =

83.202 million

The value of equity can be obtained by discounting this net income at the cost of equity:

Net Income 83.202

= $ 600 million

Value of equity = =

Cost of equity 0.1387

Even this simple example works because of the following assumptions that we made

implicitly or explicitly during the valuation.

1. The values for debt and equity used to compute the cost of capital were equal to

the values that we obtained in the valuation. Notwithstanding the circularity in

reasoning вЂ“ you need the cost of capital to obtain the values in the first place вЂ“ it

indicates that a cost of capital based upon market value weights will not yield the

same value for equity as an equity valuation model, if the firm is not fairly priced

in the first place.

2. There are no extraordinary or non-operating items that affect net income but not

operating income. Thus, to get from operating to net income, all we do is subtract

out interest expenses and taxes.

3. The interest expenses are equal to the pre-tax cost of debt multiplied by the

market value of debt. If a firm has old debt on its books, with interest expenses

that are different from this value, the two approaches will diverge.

If there is expected growth, the potential for inconsistency multiplies. You have to ensure

that you borrow enough money to fund new investments to keep your debt ratio at a level

consistent with what you are assuming when you compute the cost of capital.

fcffvsfcfe.xls: This spreadsheet allows you to compare the equity values obtained

using FCFF and FCFE models.

Illustration 12.11: FCFF Valuation: Disney

To value Disney, we will consider all of the numbers that we have estimated

already in this section. Recapping those estimates:

- The operating income in 2003, before taxes and adjusted for operating leases, is

$2,805 million. While this represents a significant come back from the doldrums of

2002, it is still lower than the operating income in the 1990s and results in an after-tax

return on capital of only 4.42% (assuming a tax rate of 37.30%).

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- For years 1 through 5, we will assume that Disney will be able to raise its return on

capital on new investments to 12% and that the reinvestment rate will be 60%. (See

illustration 12.6). This will result in an expected growth rate of 7.20% a year.

- For years 1 through 5, we will assume that Disney will maintain its existing debt ratio

of 21% and its current cost of capital of 8.59% (see illustration 12.7).

- The assumptions for stable growth (after year 10) and for the transition period are

listed in illustration 12.8.

In table 12.4, we estimate the after-tax operating income, reinvestment and free cashflow

to the firm each year for the next 10 years:

Table 12.4: Estimated Free Cashflows to the Firm - Disney

Expected EBIT (1- Reinvestment

Year Growth EBIT t) Rate Reinvestment FCFF

Current $2,805

1 6.38 % $2,984 $1,871 53.18 % $994.92 $876.06

2 6.38 % $3,174 $1,990 53.18 % $1,058.41 $931.96

3 6.38 % $3,377 $2,117 53.18 % $1,125.94 $991.43

4 6.38 % $3,592 $2,252 53.18 % $1,197.79 $1,054.70

5 6.38 % $3,822 $2,396 53.18 % $1,274.23 $1,122.00

6 5.90 % $4,047 $2,538 50.54 % $1,282.59 $1,255.13

7 5.43 % $4,267 $2,675 47.91 % $1,281.71 $1,393.77

8 4.95 % $4,478 $2,808 45.27 % $1,271.19 $1,536.80

9 4.48 % $4,679 $2,934 42.64 % $1,250.78 $1,682.90

10 4.00 % $4,866 $3,051 40.00 % $1,220.41 $1,830.62

In table 12.5, we estimate the present value of the free cashflows to the firm using the

cost of capital Since the beta and debt ratio change each year from year 6 to 10, the cost

of capital also changes each year.

Table 12.5: Present Value of Free Cashflows to Firm вЂ“ Disney

Year Cost of capital FCFF PV of FCFF

1 8.59 % $876.06 $806.74

2 8.59 % $931.96 $790.31

3 8.59 % $991.43 $774.21

4 8.59 % $1,054.70 $758.45

5 8.59 % $1,122.00 $743.00

6 8.31 % $1,255.13 $767.42

7 8.02 % $1,393.77 $788.91

8 7.73 % $1,536.80 $807.42

9 7.45 % $1,682.90 $822.90

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10 7.16 % $1,830.62 $835.31

PV of cashflows during high growth = $7,894.66

To compute the present value of the cashflows in years 6 through 10, we have to use the

compounded cost of capital over the previous years. To illustrate, the present value of

$1536.80 million in cashflows in year 8 is:

1536.80

Present value of cashflow in year 8 =

(1.0859) 5 (1.0831)(1.0802)(1.0773)

The final piece of the valuation is the terminal value. To estimate the terminal value, at

the end of year 10, we estimate the free cashflow to the firm in year 11:

!

FCFF11 = EBIT11 (1-t) (1- Reinvestment RateStable Growth)/

= 4866 (1.04) (1-.40) = $1,903.84 million

Terminal Value = FCFF11/ (Cost of capitalStable Growth вЂ“ g)

= 1903.84/ (.0716 - .04) = $60,219.11 million

The value of the firm is the sum of the present values of the cashflows during the high

growth period, the present value of the terminal value and the value of the non-operating

assets that we estimated in illustration 12.9.

PV of cashflows during the high growth phase =$ 7,894.66

60,219.11

PV of terminal value= =$ 27,477.81

(1.0859) 5 (1.0831)(1.0802)(1.0773)(1.0745)(1.0716)

+ Cash and Marketable Securities =$ 1,583.00

+ Non-operating Assets (Holdings in other companies) =$ 1,849.00

!

Value of the firm =$ 38,804.48

Subtracting out the market value of debt (including operating leases) of $14,668.22

million and the value of the equity options (estimated to be worth $1,334.67 million in

illustration 12.10) yields the value of the common stock:

Value of equity in common stock = Value of firm вЂ“ Debt вЂ“ Equity Options

= $38,804.48 - $14,668.22 - $1334.67 = $ 22,801.59

Dividing by the number of shares outstanding (2047.60 million), we arrive at a value per

share o $11.14, well below the market price of $ 26.91 at the time of this valuation.

12.10. в˜ћ: Net Capital Expenditures and Value

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In the valuation above, we assumed that the reinvestment rate would be 40% in

perpetuity to sustain the 4% stable growth rate. What would the terminal value have been

if, instead, we had assumed that the reinvestment rate was zero, while continuing to use a

stable growth rate of 4%?

In Practice: Adjusted Present Value (APV)

In chapter 8, we presented the adjusted present value approach to estimate the

optimal debt ratio for a firm. In that approach, we begin with the value of the firm

without debt. As we add debt to the firm, we consider the net effect on value by

considering both the benefits and the costs of borrowing. To do this, we assume that the

primary benefit of borrowing is a tax benefit and that the most significant cost of

borrowing is the added risk of bankruptcy.

The first step in this approach is the estimation of the value of the unlevered firm.

This can be accomplished by valuing the firm as if it had no debt, i.e., by discounting the

expected free cash flow to the firm at the unlevered cost of equity. In the special case

where cash flows grow at a constant rate in perpetuity, the value of the firm is easily

computed.

FCFFo ( + g )

1

Value of Unlevered Firm =

!u - g

where FCFF0 is the current after-tax operating cash flow to the firm, ПЃu is the unlevered

cost of equity and g is the expected growth rate. In the more general case, you can value

the firm using any set of growth assumptions you believe are reasonable for the firm.

The second step in this approach is the calculation of the expected tax benefit

from a given level of debt. This tax benefit is a function of the tax rate of the firm and is

discounted at the cost of debt to reflect the riskiness of this cash flow. If the tax savings

are viewed as a perpetuity,

(Tax Rate )(Cost of Debt )(Debt )

=

Cost of Debt

Value of Tax Benefits = (Tax Rate )(Debt )

= tc D

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The tax rate used here is the firmвЂ™s marginal tax rate and it is assumed to stay constant

over time. If we anticipate the tax rate changing over time, we can still compute the

present value of tax benefits over time, but we cannot use the perpetual growth equation

cited above.

The third step is to evaluate the effect of the given level of debt on the default risk

of the firm and on expected bankruptcy costs. In theory, at least, this requires the

estimation of the probability of default with the additional debt and the direct and indirect

cost of bankruptcy. If ПЂa is the probability of default after the additional debt and BC is

the present value of the bankruptcy cost, the present value of expected bankruptcy cost

can be estimated.

= (Probability of Bankruptcy)(PV of Bankruptcy Cost )

PV of Expected Bankruptcy cost

= ! a BC

This step of the adjusted present value approach poses the most significant estimation

problem, since neither the probability of bankruptcy nor the bankruptcy cost can be

estimated directly.

In theory, the APV approach and the cost of capital approach will yield the same

values for a firm if consistent assumptions are made about financial leverage. The

difficulties associated with estimating the expected bankruptcy cost, though, often lead

many to use an abbreviated version of the APV model, where the tax benefits are added

to the unlevered firm value and bankruptcy costs are ignored. This approach will over

value firms.

Valuing Private Businesses

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