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Non-cash ROE = (Net Income вЂ“ After-tax Interest income on cash)2003/ (BV of Equity вЂ“
Cash)2002

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Aracruz had net income of \$148.09 million in 2003, interest income before taxes of
\$43.04 million and faced a tax rate of 34%. The book value of equity at the end of 2002
was \$1760.58 million, of which cash represented \$273.93 million.
Non-cash ROEAracruz = (148.09 вЂ“ 43.04(1-.34))/ (1760.58-273.93) = .0805 or 8.05%
The expected growth in net income can be computed as the product of the non-cash ROE
and the equity reinvestment rate.
Expected Growth in Net Income = Equity Reinvestment Rate * Non-cash ROE
= 65.97% * 8.05% = 5.31%
Based upon fundamentals, we would expect AracruzвЂ™s net income to grow 5.31% a year.
In Practice: Paths to a Higher ROE
The expected growth rate in earnings per share and net income are dependent
upon the return on equity that a firm makes on its new investments. The higher the return
one equity, the higher the expected growth rate in earnings. But how do firms generate
higher returns on equity? Algebraically, the return on equity can be decomposed into a
return on capital and a leverage effect:
Return on Equity = Return on capital + D/E (Return on capital вЂ“ Cost of debt (1-tax rate))
where,
Return on capital = EBIT (1-tax rate)/ (Book value of debt + Book value of equity)
D/E = Book value of debt/ Book value of equity
The second term in the equation reflects the influence of debt. To the extent that a firm
can earn a return on capital that exceeds the after-tax cost of debt, its return on equity will
increase as it uses more debt. A firm with a return on capital of 12%, a debt to equity
ratio of 0.5 and an after-tax cost of debt of 4% will have a return on equity of 16%. Lest
firms view this as a free lunch, we should hasten to point out that using more debt will
also increase the firmвЂ™s beta and cost of equity and the value of equity may very well
decrease with higher borrowing, even though the return on equity and expected growth
rate may be higher.

b. Estimating Terminal Value
As with the dividend discount model, the terminal value in the FCFE model is
determined by the stable growth rate and the cost of equity. The difference between this

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model and the dividend discount model lies primarily in the cash flow used to calculate
the terminal price: the latter uses expected dividends in the period after the high growth
period, whereas the former uses the free cash flow to equity in that period:

FCFE n +1
Terminal value of equityn=
r" g n
In estimating that cash flow, the net capital expenditures and working capital needs
should be consistent with the definition of stability. The simplest way to ensure this is to
!
estimate an equity reinvestment rate from the stable period return on equity:
Equity Reinvestment rate in stable growth = Stable growth rate/ Stable period ROE
This is exactly the same equation we used to compute the retention ratio in stable growth
in the dividend discount model.
Many analysts assume that stable growth firms have capital expenditures that
offset depreciation and no working capital requirements. This will yield a equity
reinvestment rate of zero which is consistent only with a stable growth rate of zero. Using
a stable growth rate of 3 or 4% while allowing for no reinvestment essentially allows
your firm to grow without paying for the growth and will yield too high a value for the
firm.

Reconciling FCFE and Dividend Discount Model Valuations
The FCFE discounted cash flow model can be viewed as an alternative to the
dividend discount model. Since the two approaches sometimes provide different
estimates of value, however, it is worth a comparison.
There are two conditions under which the value obtained from using the FCFE in
discounted cash flow valuation will be the same as the value obtained from using the
dividend discount model. The first is obvious: when the dividends are equal to the FCFE,
the value will be the same. The second is more subtle: when the FCFE is greater than
dividends, but the excess cash (FCFE - Dividends) is invested in projects with a net

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present value of zero, the values will also be similar. For instance, investing in financial
assets that are fairly priced should yield a net present value of zero.13
More often, the two models will provide different estimates of value. First, when
the FCFE is greater than the dividend and the excess cash either earns below-market
returns or is invested in negative net present value projects, the value from the FCFE
model will be greater than the value from the dividend discount model. This is not
uncommon. There are numerous case studies of firms that having accumulated large cash
balances by paying out low dividends relative to FCFE, have chosen to use this cash to
finance unwise takeovers (the price paid is greater than the value received). Second, the
payment of smaller dividends than the firm can afford lowers debt-equity ratios;
accordingly, the firm may become underleveraged, reducing its value.
In those cases where dividends are greater than FCFE, the firm will have to issue
new shares or borrow money to pay these dividends leading to at least three negative
consequences for value are possible. One is the flotation cost on these security issues,
which can be substantial for equity issues. Second, if the firm borrows the money to pay
the dividends, the firm may become overleveraged (relative to the optimal), leading to a
loss in value. Finally, paying too much in dividends can lead to capital rationing
constraints, whereby good projects are rejected, resulting in a loss of wealth.
When the two models yield different values, two questions remain: (1) What does
the difference between the two models tell us? (2) Which of the two models is the
appropriate one to use in evaluating the market price? In most cases, the value from the
FCFE model will exceed the value from the dividend discount model. The difference
between the value obtained from the FCFE model and the value obtained from the
dividend discount model can be considered one component of the value of controlling a
firm вЂ“ that is, it measures the value of controlling dividend policy. In a hostile takeover,
the bidder can expect to control the firm and change the dividend policy (to reflect
FCFE), thus capturing the higher FCFE value. In the more infrequent case вЂ“вЂ“the value
from the dividend discount model exceeds the value from the FCFE вЂ“вЂ“ the difference has

13 Mechanically, this will work out only if you keep track of the cash build up in the dividend discount
model and add it to the terminal value. If you do not do this, you will under value your firm with the
dividend discount model.

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less economic meaning but can be considered a warning on the sustainability of expected
dividends.
As for which of the two values is more appropriate for evaluating the market
price, the answer lies in the openness of the market for corporate control. If there is a
significant probability that a firm can be taken over or its management changed, the
market price will reflect that likelihood; in that case, the value from the FCFE model
would be a more appropriate benchmark. As changes in corporate control become more
difficult, either because of a firm's size and/or legal or market restrictions on takeovers,
the value from the dividend discount model will provide a more appropriate benchmark
for comparison.

12.7. в˜ћ: FCFE and DDM Value
Most firms can be valued using FCFE and DDM valuation models. Which of the
following statements would you most agree with on the relationship between these two
values?
a. The FCFE value will always be higher than the DDM value
b. The FCFE value will usually be higher than the DDM value
c. The DDM value will usually be higher than the FCFE value
d. The DDM value will generally be equal to the FCFE value

Illustration 12.5: FCFE Valuation: Aracruz
To value Aracruz, using the FCFE model, we will use the expected growth in net
income that we estimated in illustration 12.4 and value the equity in operating assets first
and then add on the value of cash and other non-operating assets. We will also value the
company in U.S. dollars, rather than Brazilian real, because the firm generates so much of
its cashflows in dollars. Summarizing the basic information that we will be using:
- The net income for the firm in 2003 was \$148.09 million but \$28.41 million of this
income represented income from financial assets.14 The net income from non-
operating assets is \$119.68 million.

14 The pre-tax income from financial assets was \$43.04 million. We used a 34% tax rate to arrive at

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- In 2003, capital expenditures amounted to \$ 228.82 million, depreciation was \$191.51
million and non-cash working capital increased by \$10.89 million. The net cashflow
from debt was \$531.20 million, resulting in a large negative equity reinvestment in
that year.
Equity Reinvestment Rate2003 = (228.82 вЂ“ 191.51 + 10.89 -531.20)/ 119.68 = -
403.58%
We will use the average equity reinvestment rate of 65.97%, based upon the average
values from 199-2003, that we computed in illustration 12.4 as the equity
reinvestment rate for the next 5 years. In conjunction, with the non-cash return on
equity of 8.05% that we computed in that illustration, we estimate an expected growth
rate of 5.31% a year for the next 5 years.
- In illustration 4.7, we estimated a beta for equity of 0.7576 for the paper business that
Aracruz.15 With a nominal U.S. dollar riskfree rate of 4% and an equity risk premium
of 12.49% for Brazil (also estimated in chapter 4), we arrive at a dollar cost of equity
of 13.46%
Cost of equity = 4% + 0.7576 (12.49%) = 13.46%
After year 5, we will assume that the beta will remain at 0.7576 and that the equity
risk premium will decline to 8.66%.16 The resulting cost of equity is 10.56%.
Cost of equity in stable growth = 4% + 0.7576 (8.66%) = 10.56%
- After year 5, we will assume that the growth in net income will drop to the inflation
rate (in U.S. dollar terms) of 2% and that the return on equity will rise to 10.56%
(which is also the cost of equity). The equity reinvestment rate in stable growth can
then be estimated as follows:
Equity Reinvestment RateStable Growth = Expected Growth Rate/ Return on Equity
= 2%/10.56% = 18.94%
To value the equity in Aracruz, we begin by estimating the free cashflows to
equity from operations in table 12.3:

15 We used the equity beta of just the operating asssets in this valuation. If we had chosen to include the
cash from financial holdings as part of net income, we would have used AracruzвЂ™s consolidated equity beta
of 0.7040.
16 We halved the country risk premium from 7.67% to 3.84%. We are assuming that as Brazil grows, it will
become a less risky country to invest in.

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Table 12.3: Expected FCFE at Aracruz вЂ“ Years 1-5

1 2 3 4 5

Net Income (non-cash) \$126.04 \$132.74 \$139.79 \$147.21 \$155.03

Equity Reinvestment Rate 65.97% 65.97% 65.97% 65.97% 65.97%

FCFE \$42.89 \$45.17 \$47.57 \$50.09 \$52.75

Present Value at 13.46% \$37.80 \$35.09 \$32.56 \$30.23 \$28.05
FCFE = Net Income (1- Reinvestment Rate)
To estimate the terminal value of equity, we first estimate the free cashflow to equity in
year 6:
FCFE in year 6 = Net Income in year 6 (1- Equity Reinvestment RateStable Growth)
= 155.03 (1.02) (1- .1894) = \$128.18 million
The terminal value is then computed using the stable period cost of equity of 10.56%:
Terminal value of equity = 128.18/(.1056-.02) = \$1497.98 million
The current value of equity is the sum of the present values of the expected cashflows in
table 12.3, the present value of the terminal value of equity and the value of cash and
non-operating assets today:
Present Value of FCFEs in high growth phase = \$163.73
+ Present Value of Terminal Equity Value = 1497.98/1.13465 = \$796.55
Value of equity in operating assets = \$960.28
+ Value of Cash and Marketable Securities = \$352.28
Value of equity in firm = \$1,312.56
Dividing by the 859.59 million shares outstanding yields a value per share of \$1.53.
Converted into Brazilian Real at the exchange rate of 3.15 BR/\$ prevailing at the time of
this valuation, we get a value per share of 4.81 BR per share, well below the market price
of 7.50 BR/share.

In Practice: Reconciling your value with the market price
When you value a company and arrive at a number very different from the market
price, as we have with both Aracruz and Deutsche Bank, there are three possible
explanations. The first is that we are mistaken in our assumptions and that our valuations
are wrong while the market is right. Without resorting to the dogma of efficient markets,

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this is a reasonable place to start since this is the most likely scenario. The second is that
the market is wrong and that we are right, in which case we have to decide whether we
have enough confidence in our valuations to act on them. If we find a company to be
under valued, this would require buying and holding the stock. If the stock is over valued,
we would have to sell short. The problem, though, is that there is no guarantee that
markets, even if they are wrong, will correct their mistakes in the near future. In other
words, a stock that is over valued can become even more over valued and a stock that is
under valued may stay undervalued for years, wreaking havoc on our portfolio. This also
makes selling short a much riskier strategy since we generally can do so only for a few
months.
One way to measure market expectations is to solve for a growth rate that will
yield the market price. In the Aracruz valuation, for instance, we would need an expected
growth rate of 19.50% in earnings over the next 5 years to justify the current market
price. This is called an implied growth rate and can be compared to the estimate of
growth we used in the valuation of 5.31%.

III. Free Cashflow to the Firm Models
The dividend discount and FCFE models are models for valuing the equity in a
firm directly. The alternative is to value the entire business and then to use this value to
arrive at a value for the equity. That is precisely what we try and do in firm valuation
models where we focus on the operating assets of the firm and the cashflows that they
generate.

Setting up the Model
The cash flow to the firm can be measured in two ways. One is to add up the cash
flows to all of the different claim holders in the firm. Thus, the cash flows to equity
investors (which take the form of dividends or stock buybacks) are added to the cash
flows to debt holders (interest and net debt payments) to arrive at the cash flow to the
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