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ing short-term and long-term debt”indicating that both short-term and long-term debt
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should be considered a part of capital when computing the WACC. Servicing of other li-
abilities, such as accounts payable or accruals, should already have been considered as
we computed free cash ¬‚ows. Thus, internal consistency requires that operating liabili-
ties not be considered a part of capital when computing the WACC.
The tricky problem we face is assigning a market value to equity. That is the very
amount we are trying to estimate in the ¬rst place! How can the analyst possibly assign
a market value to equity at this intermediate stage, given that the estimate will not be
known until all steps in the DCF analysis are completed?
One common approach to the problem is to insert “target” ratios of debt to capital
[Vd / (Vd + Ve )] and equity to capital [Ve / (Vd + Ve )] at this point. For example, one might
expect that a ¬rm will, over the long run, maintain a capital structure that is 40 percent
debt and 60 percent equity. The long-run focus is reasonable because we are discounting
cash ¬‚ows over a long horizon.
Another way around the problem is to use book value of equity as a starting point as
a weight for purposes of calculating an initial estimate of the WACC, which in turn can
be used in the discounting process to generate an initial estimate of the value of equity.
That initial estimate can then be used in place of the guess to arrive at a new WACC, and
a second estimate of the value of equity can be produced. This process can be repeated
until the value used to calculate the WACC and the ¬nal estimated value converge. In this
chapter, we use book value debt and equity to estimate a company™s WACC.

The cost of debt (rd ) should be based on current
ESTIMATING THE COST OF DEBT.
market rates of interest. For privately held debt, such rates are not quoted, but stated in-
terest rates may provide a suitable substitute if interest rates have not changed much
since the debt was issued. The cost of debt should be expressed on a net-of-tax basis,
because it is after-tax cash ¬‚ows that are being discounted. In most settings, the market
rate of interest can be converted to a net-of-tax basis by multiplying by one minus the
marginal corporate tax rate.

Estimating the cost of equity (re ) can be dif-
ESTIMATING THE COST OF EQUITY.
¬cult, and a full discussion of the topic lies beyond the scope of this chapter. At any rate,
even an extended discussion would not supply answers to all the questions that might be
raised in this area, because the ¬eld of ¬nance is in a state of ¬‚ux over what constitutes
an appropriate measure of the cost of equity.
One possibility is to use the capital asset pricing model (CAPM), which expresses the
cost of equity as the sum of a required return on riskless assets, plus a premium for sys-
tematic risk:

r f + β [ E ( rm ) “ r f ]
re =
where rf is the riskless rate;
[E(rm ) “ rf ] is the risk premium expected for the market as a whole, expressed
as the excess of the expected return on the market index over the riskless rate;
and β is the systematic risk of the equity.
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To compute re, one must estimate three parameters: the riskless rate, rf ; the market
risk premium [E(rm) “ rf ], and systematic risk, β. For rf , analysts often use the rate
on intermediate-term treasury bonds, based on the observation that it is cash ¬‚ows
beyond the short term that are being discounted.4 When rf is measured in that way, then
average common stock returns (based on the returns to the Standard and Poor™s 500
index) have exceeded that rate by 7.6 percent over the 1926“1998 period (Ibbotson
Associates [1998]).5 This excess return constitutes an estimate of the market risk pre-
mium [E(rm) “ rf ]. Finally, systematic risk (β) re¬‚ects the sensitivity of the ¬rm™s value
to economy-wide market movements.6
Although the above CAPM is often used to estimate the cost of capital, the evidence
indicates that the model is incomplete. Assuming stocks are priced competitively, stock
returns should be expected just to compensate investors for the cost of their capital.
Thus, long run average returns should be close to the cost of capital, and should (accord-
ing to the CAPM) vary across stocks according to their systematic risk. However, factors
beyond just systematic risk seem to play some role in explaining variation in long-run
average returns. The most important such factor is labeled the “size effect”: smaller
¬rms (as measured by market capitalization) tend to generate higher returns in subse-
quent periods. Why this is so is unclear; it could either indicate that smaller ¬rms are
riskier than indicated by the CAPM, or that they are underpriced at the point their market
capitalization is measured, or some combination of both. Average stock returns for U.S.
¬rms (including NYSE, AMEX, and NASDAQ ¬rms) varied across size deciles from
1926“1993 as shown in Table12-10.


Table 12-10 Stock Returns and Firm Size
Fraction of total
Market value of largest Average annual NYSE value
company in decile, stock return, represented by
Size decile in 1998 (millions of dollars) 1926“1998 decile (in 1998)
.........................................................................................................................
1-small $ 10,764.3 21.0% 0.1%
2 27,647.9 17.9 0.3
3 53,218.4 17.1 0.7
4 78,601.4 16.0 1.0
5 114,517.6 15.6 1.4
6 170,846.6 15.5 2.1
7 273,895.7 14.8 3.3
8 476,920.5 14.1 5.8
9 1,052,131.2 13.7 12.8
10-large 5,985,553.1 12.1 72.6
.........................................................................................................................
Source: Ibbotson Associates (1998).
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The table indicates that, historically, investors in ¬rms in the top two deciles of the
size distribution have realized returns of only 12.1 to 13.7 percent. Note, however, that
if we use ¬rm size as an indicator of the cost of capital, we are implicitly assuming that
large size is indicative of lower risk. Yet, ¬nance theorists have not developed a well-
accepted explanation for why that should be the case.
One method for combining the cost of capital estimates is based on the CAPM and the
“size effect.”7 The approach calls for adjustment of the CAPM-based cost of capital,
based on the difference between the average return on the market index used in the
CAPM (the Standard and Poor™s 500) and the average return on ¬rms of size comparable
to the ¬rm being evaluated. The resulting cost of capital is:
rf + β [ E ( rm ) “ rf ] + rsize
re =
In light of the continuing debate on how to measure the cost of capital, it is not sur-
prising that managers and analysts often consider a range of estimates. In particular,
there has been considerable debate in recent times about whether or not the historical
risk premium of 7.6 percent is valid today. Many analysts argue that a variety of changes
in the U.S. economy make the historical risk premium an invalid basis for forecasting
expected risk premium going forward. Some recent academic research has provided ev-
idence that suggests that the expected risk premium in the market in recent years has de-
clined substantially, to the range of 3 to 4 percent.8 Since this debate is still unresolved,
it is prudent for analysts to use a range of risk premium estimates in computing a ¬rm™s
cost of capital.
To estimate the cost of capital for Sigma, we start with the assumption that its after-
tax cost of debt is 5 percent, its cost of equity is 12 percent using the CAPM model, and
the market risk premium is 7.6 percent. The weighted average cost of capital of 9.2 is
computed using book value weights of 40 percent debt and 60 percent equity. Clearly,
this estimate is only a starting point, and the analyst can change the estimate by changing
the assumed market risk premium as well as using an iterative approach discussed above
to re¬ne the weights used in computing WACC.


COMPUTING ESTIMATED VALUES
We show below the estimated value of Sigma™s assets and equity, each using three dif-
ferent methods. Value of assets is estimated using abnormal operating ROA, abnormal
NOPAT, and free cash ¬‚ows to debt and equity. Value of equity is estimated using
operating ROE, abnormal NOPAT, and free cash ¬‚ow to equity. These values are com-
puted using the ¬nancial forecasts and the terminal value forecasts with sales growth
of 10 percent from 1999 to 2003, terminal growth rate of 3.5 percent, and abnormal
pro¬ts in terminal years persisting at the year 2003 level. Asset values are estimated
with a WACC of 9.2 percent, and equity values are estimated with a cost of equity of
12 percent.
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ABNORMAL RETURNS METHOD
= Book value of net assets at the beginning of 1999
Estimated value of assets
— (1 + PV of abnormal operating ROA for 1999“2003
+ Terminal value)
= $715 million — (1 + 0.215 + 0.269)
= $715 million — (1.484)
= $1,061.1 million
= Book value of equity at the beginning of 1999
Estimated value of equity
— (1 + PV of abnormal ROE for 1999“2003 +
Terminal value beyond 2003)
= $429 million — (1 + 0.338 + 0.265)
= $429 million — (1.603)
= $687.7 million
ABNORMAL EARNINGS METHOD
= Book value of assets at the beginning of 1999
Estimated value of assets
+ PV of abnormal NOPAT for 1999“2003
+ Terminal value beyond 2003
= $715 million + $153.7 million + $192 million
= $1,060.7 million
= Book value of equity at the beginning of 1999
Estimated value of equity
+ PV of abnormal earnings for 1999“2003
+ Terminal value beyond 2003
= $429 million + $144.8 million + $113.5 million
= $687.3 million
FREE CASH FLOW METHOD
= PV of free cash ¬‚ow to debt and equity for 1999“2003
Estimated value of assets
+ Terminal value beyond 2003
= $170.8 million + $890.3 million
= $1,061.1 million
= PV of free cash ¬‚ow to equity for 1999“2003
Estimated value of equity
+ Terminal value beyond 2003
= $205 million + $482.6 million
= $687.6 million
Value estimates presented above show that the abnormal returns method, abnormal
earnings method, and the free cash ¬‚ow method result in the same value estimates (ex-
cept for small differences due to rounding errors)”the estimated value of Sigma™s assets
is about $1061 million, and the estimated value of its equity is about $687 million.9 Note
also that Sigma™s terminal value represents a signi¬cantly larger fraction of the total
value of assets and equity under the free cash ¬‚ow method relative to the other methods.
As discussed in Chapter 11, this is due to the fact that the abnormal returns and earnings
methods rely on a company™s book value of assets and equity, so the terminal value
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estimates are estimates of incremental values over book values. In contrast, the free cash
¬‚ow approach ignores the book values, so the terminal value forecasts are estimates of
total value during this period.
The primary calculations in the above estimates treat all ¬‚ows as if they arrive at the
end of the year. Of course, they are likely to arrive throughout the year. If we assume for
the sake of simplicity that cash ¬‚ows will arrive mid-year, then we should adjust our
value estimates upward by the amount 1 + « -- , where r is the discount rate.
r
-
 2


Other Practical Issues in Valuation
The above discussion provides a blueprint for doing valuation. However, in practice, the
analyst has to deal with a number of other issues that have an important effect on the
valuation task. We discuss below three frequently encountered complications”account-
ing distortions, negative book values, and excess cash.

We know from the discussion in
DEALING WITH ACCOUNTING DISTORTIONS.
Chapter 11 that accounting methods per se should have no in¬‚uence on ¬rm value (ex-
cept as those choices in¬‚uence the analyst™s view of future real performance). Yet the
abnormal returns and earnings valuation approaches used here are based on numbers”
earnings and book value”that vary with accounting method choices. How, then, can the
valuation approach deliver correct estimates?
Because accounting choices must affect both earnings and book value, and because
of the self-correcting nature of double-entry bookkeeping (all “distortions” of account-
ing must ultimately reverse), estimated values will not be affected by accounting
choices, as long as the analyst recognizes the accounting distortions.10 As an example,
let™s assume that managers are aggressive in their accounting choices, choosing to pro-
vide for a lower allowance for uncollected receivables even though they have informa-
tion to the contrary, thus causing the current period™s abnormal earnings and the ending
book value to be higher by $100. For the time being, let™s say the accounting choice has
no in¬‚uence on the analyst™s view of the ¬rm™s real performance. That is, the analyst is
assumed to recognize that management™s current estimate of future customer defaults is
arti¬cially lower and can make accurate forecasts of future defaults.
Our accounting-based valuation approach starts with the current period™s abnormal
earnings, which are $100 higher as a result of the accounting choice. However, the

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