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Similarly, the equity value-to-book formula can be simpli¬ed by making assumptions
about long-term ROEs and growth.


1. Relation Between Current and Future Abnormal Earnings
Several assumptions about the relation between current and future net income are pop-
ular for simplifying the abnormal earnings model. First, abnormal earnings are assumed
to follow a random walk. The random walk model for abnormal earnings implies that an
analyst™s best guess about future expected abnormal earnings are current abnormal earn-
ings. The model assumes that past shocks to abnormal earnings persist forever, but that
future shocks are random or unpredictable. The random walk model can be written as
follows:
Forecasted AE 1 = AE 0

Forecasted AE1 is the forecast of next year™s abnormal earnings and AE0 is current period
abnormal earnings. Under the model, forecasted abnormal earnings for two years ahead
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are simply abnormal earnings in year one, or once again current abnormal earnings. In
other words, the best guess of abnormal earnings in any future year is just current abnor-
mal earnings.7
How does the above assumption about future abnormal earnings simplify the dis-
counted abnormal earnings valuation model? If abnormal earnings follow a random
walk, all future forecasts of abnormal earnings are simply current abnormal earnings. It
is then possible to rewrite value as follows:
AE 0
Stock value = BVE 0 + -----------
-
re
The stock value is the book value of equity at the end of the year, plus current abnormal
earnings divided by the cost of capital.
Of course, in reality shocks to abnormal earnings are unlikely to persist forever. Firms
that have positive shocks are likely to attract competitors that will reduce opportunities
for future abnormal performance. Firms with negative abnormal earnings shocks are
likely to fail or to be acquired by other ¬rms that can manage their resources more ef-
fectively. The persistence of abnormal performance will therefore depend on strategic
factors, such as barriers to entry and switching costs, discussed in Chapter 2. To re¬‚ect
this, analysts frequently assume that current shocks to abnormal earnings decay over
time. Under this assumption, abnormal earnings are said to follow an autoregressive
model. Forecasted abnormal earnings are then:
β AE 0
Forecasted AE 1 =
β!is a parameter that captures the speed with which abnormal earnings decay over time.
If there is no decay,!β!is one and abnormal earnings follow a random walk. If β!is zero,
abnormal earnings decay completely within one year. Estimates of!β!using actual com-
pany data indicate that for a typical U.S. ¬rm,!β is approximately 0.6. However, it varies
by industry, and is smaller for ¬rms with large accruals and one-time accounting
charges.8
The autoregressive model implies that stock values can again be written as a function
of current abnormal earnings and book values9:
β AE 0
Stock value = BVE 0 + ------------------------------
-
1 + re “ β
This formulation implies that stock values are simply the sum of current book value plus
current abnormal earnings weighted by the cost of equity capital and persistence in
abnormal earnings.


2. ROE and Growth Simplifications
It is also possible to make simpli¬cations about long-term ROEs and equity growth to
reduce forecast horizons for estimating the equity value-to-book multiple. Firms™ long-
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term ROEs are affected by such factors as barriers to entry in their industries, change in
production or delivery technologies, and quality of management. As discussed in Chap-
ter 10, these factors tend to force abnormal ROEs to decay over time. One way to model
this decay is to assume that ROEs follow a mean reverting process. Forecasted ROE in
one period™s time then takes the following form:
ROE 0 + β ( ROE 0 “ ROE )
Forecasted ROE 1 =
ROE is the steady state ROE (either the ¬rm™s cost of capital or the long-term industry
ROE) and β!is a “speed of adjustment factor” that re¬‚ects how quickly it takes the!ROE
to revert to its steady state.10
Growth rates are affected by several factors. First, the size of the ¬rm is important.
Small ¬rms can sustain very high growth rates for an extended period, whereas large
¬rms ¬nd it more dif¬cult to do so. Second, ¬rms with high rates of growth are likely to
attract competitors, which reduces their growth rates. As discussed in Chapter 10, book
value growth rates for real ¬rms exhibit considerable reversion to the mean.
The long-term patterns in ROE and book equity growth rates imply that for most com-
panies there is limited value in making forecasts for valuation beyond a relatively short
horizon, three to ¬ve years. Powerful economic forces tend to lead ¬rms with superior
or inferior performance early in the forecast horizon to revert to a level that is compara-
ble to that of other ¬rms in the industry or the economy. For a ¬rm in steady state, that
is, expected to have a stable ROE and book equity growth rate (gbve), the value-to-book
multiple formula simpli¬es to the following:
ROE 0 “ re
Equity value-to-book multiple = 1 + ----------------------------
-
re “ gbve
Consistent with this simpli¬ed model, there is a strong relation between price-to-
book ratios and current ROEs. Figure 11-1 shows the relation between these variables for
¬rms in the Retail: Apparel industry as reported by Dow Jones Interactive on July 16,
1999. The correlation between the two variables is 45 percent. Two ¬rms, The Limited


Figure 11-1 Relation Between ROE and Price-to-Book Multiples

100%
80%
ROE




60%
40%
20%
0%
0% 500% 1000% 1500% 2000% 2500%
Price- to-Book Ratio
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and The Gap, have ROEs that are considerably higher than those for other ¬rms in the
industry (88 percent and 48 percent, respectively). Earnings for The Limited include a
$1.7 million special gain from the spin-off of a subsidiary, indicating that the high ROE
is unlikely to be sustained. Absent this gain, The Limited™s ROE would be approximately
14 percent, in keeping with its price-to-book value. The Gap has shown a steady increase
in earnings during the last four years. Its high price-to-book ratio suggests that investors
expect this level of performance to be sustainable.
Of course, analysts can make a variety of simplifying assumptions about a ¬rm™s ROE
and growth. For example, they can assume that they decay slowly or rapidly to the cost
of capital and the growth rate for the economy. They can assume that the rates decay to
the industry or economy average ROEs and book value growth rates. The valuation for-
mula can easily be modi¬ed to accommodate these assumptions


THE DISCOUNTED CASH FLOW MODEL
The ¬nal valuation method discussed here is the discounted cash ¬‚ow approach. This is
the valuation method taught in most ¬nance classes. Like the abnormal earnings ap-
proach, it is derived from the dividend discount model. It is based on the insight that div-
idends can be recast as free cash ¬‚ows,11 that is:
Dividends = Operating cash flow “ Capital outlays + Net cash flows from debt owners
As discussed in Chapter 9, operating cash ¬‚ows to equity holders are simply net in-
come plus depreciation less changes in working capital accruals. Capital outlays are cap-
ital expenditures less asset sales. Finally, net cash ¬‚ows from debt owners are issues of
new debt less retirements less the after-tax cost of interest. By rearranging these terms,
the free cash ¬‚ows to equity can be written as follows:
NI “ ∆ BVA + ∆ BVND
Dividends = Free cash flows to equity =
where NI is net income, ∆ BVA is the change in book value of operating net assets (in-
cluding changes in working capital plus capital expenditures less depreciation expense),
and ∆ BVND is the change in book value of net debt (interest-bearing debt less excess
cash).
The dividend discount model can therefore be written as the present value of free cash
¬‚ows to equity. Under this formulation ¬rm value is estimated as follows:
Equity value = PV of free cash flows to equity claim holders
NI 1 “ ∆ BVA 1 + ∆ BVND 1 NI 2 “ ∆ BVA 2 + ∆ BVND 2
Equity value = --------------------------------------------------------------------- + --------------------------------------------------------------------- + . . .
( 1 + re ) ( 1 + r )2 e
Alternatively, the free cash ¬‚ow formulation can be structured by estimating the value
of claims to net debt and equity, and then deducting the market value of net debt. This
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approach is more widely used in practice, because it does not require explicit forecasts
of changes in debt balances.12 The value of debt plus equity is then:
Debt plus equity value = PV of free cash flows to net debt and equity claim holders
NOPAT 1 “ ∆ BVA 1 NOPAT 2 “ ∆ BVA 2
Debt plus equity value =
------------------------------------------------- + ------------------------------------------------- + . . .
( 1 + WACC ) ( 1 + WACC ) 2
Valuation under the discounted cash ¬‚ow method therefore involves the following
steps:
Step 1: Forecast free cash ¬‚ows available to equity holders (or to debt and equity hold-
ers) over a ¬nite forecast horizon (usually 5 to 10 years).
Step 2: Forecast free cash ¬‚ows beyond the terminal year based on some simplifying as-
sumption.
Step 3: Discount free cash ¬‚ows to equity holders (debt plus equity holders) at the cost
of equity (weighted average cost of capital). The discounted amount represents
the estimated value of free cash ¬‚ows available to equity (debt and equity hold-
ers as a group).
Returning to the Down Under Company example, there is no debt, so that the free
cash ¬‚ows to owners are simply the operating pro¬ts before depreciation. Since Down
Under is an all-equity ¬rm, its WACC is the cost of equity (10 percent), and the present
value of the free cash ¬‚ows is as follows:
PV of Free
Year Free Cash Flows PV Factor Cash Flows
1 $40m 0.9091 $36.4m
2 50 0.8264 41.3
3 60 0.7513 45.1
Equity value $122.8m




COMPARING VALUATION METHODS
We have discussed three methods of valuation derived from the dividend discount mod-
el: discounted dividends, discounted abnormal earnings (or abnormal ROEs), and dis-
counted cash ¬‚ows. What are the pluses and minuses of these approaches? Since the
methods are all derived from the same underlying model, no one version can be consid-
ered superior to the others. As long as analysts make the same assumptions about ¬rm
fundamentals, value estimates under all four methods will be identical.
However, there are several important differences between the models that are worth
noting:
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• they focus the analyst™s task on different issues;
• they require different levels of structure for valuation analysis; and
• they have different implications for the estimation of terminal values.


Focus on Different Issues
The methods frame the valuation task differently and can in practice focus the analyst™s
attention on different issues. The earnings-based approaches frame the issues in terms
of accounting data such as earnings and book values. Analysts spend considerable time
analyzing historical income statements and balance sheets, and their primary forecasts
are typically for these variables.
De¬ning values in terms of ROEs has the added advantage that it focuses analysts™
attention on ROE, the same key measure of performance that is decomposed in a stan-
dard ¬nancial analysis. Further, because ROEs control for ¬rm scale it is likely to be eas-
ier for analysts to evaluate the reasonableness of their forecasts by benchmarking them
with ROEs of other ¬rms in the industry and the economy. This type of benchmarking is
more challenging for free cash ¬‚ows and abnormal earnings.


Differences in Required Structure
The methods differ in the amount of analysis and structure required for valuation. The
discounted abnormal earnings and ROE methods require analysts to construct both pro-
forma income statements and balance sheets to forecast future earnings and book values.
In contrast, the discounted cash ¬‚ow method requires analysts to forecast income state-
ments and changes in working capital and long-term assets to generate free cash ¬‚ows.
Finally, the discounted dividend method requires analysts to forecast dividends.
The discounted abnormal earnings, ROE, and free cash ¬‚ow models all require more
structure for analysis than the discounted dividend approach. They therefore help ana-
lysts to avoid structural inconsistencies in their forecasts of future dividends by speci¬-

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