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
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 )
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:
Net Income = (EBIT вЂ“ Pre-tax cost of debt * Debt) (1-t) = (166.67 - 0.07*400) (1-0.4) =
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%).
- 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
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
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:
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
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
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
FCFFo ( + g )
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
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
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
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
Valuing Private Businesses