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CHAPTER 11
ESTIMATING GROWTH
The value of a firm is the present value of expected future cash flows generated by
the firm. The most critical input in valuation, especially for high growth firms, is the
growth rate to use to forecast future revenues and earnings. In this chapter, we consider
how best to estimate these growth rates for firms, including those with low revenues and
negative earnings.
There are three basic ways of estimating growth for any firm. One is to look at the
growth in a firm’s past earnings – its historical growth rate. While this can be a useful
input when valuing stable firms, there are both dangers and limitations in using this
growth rate for high growth firms. The historical growth rate can often not be estimated,
and even if it can, it cannot be relied on as an estimate of expected future growth.
The second is to trust the equity research analysts that follow the firm to come up
with the right estimate of growth for the firm and to use that growth rate in valuation.
While many firms are widely followed by analysts, the quality of growth estimates,
especially over longer periods, is poor. Relying on these growth estimates in a valuation
can lead to erroneous and inconsistent estimates of value.
The third is to estimate the growth from a firm’s fundamentals. A firm’s growth
ultimately is determined by how much is reinvested into new assets and the quality of
these investments, with investments widely defined to include acquisitions, building up
distribution channels or even expanding marketing capabilities. By estimating these
inputs, you are, in a sense, estimating a firm’s fundamental growth rate. While the
determinants of fundamental growth remain the same for all firms, estimating these inputs
for high growth firms can pose special challenges.
The Importance of Growth
A firm can be valuable because it owns assets that generate cash flows now or
because it is expected to acquire such assets in the future. The first group of assets is
categorized as assets in place and the second as growth assets. Figure 11.1 presents a
financial balance sheet for a firm:
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Assets Liabilities
Investments already
made
Debt
Equity
Borrowed money
Investments yet to Owner’s funds
be made
Existing Investments
Generate cashflows today
Expected Value that will be
created by future investments
Figure 11.1: A Financial View of a Firm
Note that an accounting balance sheet can be very different from a financial balance sheet,
since accounting for growth assets tends to be both conservative and inconsistent.
For high growth firms, accounting balance sheets do a poor job of summarizing the
values of the assets of the firm because they completely ignore the largest component of
value, which is future growth. The problems are exacerbated for firms that invest in
research because the book value will not include the most important asset at these firms –
the research asset.
Historical Growth
When estimating the expected growth for a firm, we generally begin by looking at
the firm’s history. How rapidly have the firm’s operations as measured by revenues or
earnings grown in the recent past? While past growth is not always a good indicator of
future growth, it does convey information that can be valuable while making estimates for
the future. In this section, we begin by looking at measurement issues that arise when
estimating past growth and then consider how past growth can be used in projections.
Estimating Historical Growth
Given a firm’s earnings history, estimating historical growth rates may seem like a
simple exercise but there are several measurement problems that may arise. In particular,
the average growth rates can be different, depending upon how the average is estimated,
and whether you allow for compounding in the growth over time. Estimating growth rates
can also be complicated by the presence of negative earnings in the past or in the current
period.
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Arithmetic versus Geometric Averages
The average growth rate can vary depending upon whether it is an arithmetic
average or a geometric average. The arithmetic average is the simple average of past
growth rates, while the geometric mean takes into account the compounding that occurs
from period to period.
Arithmetic Average =
gt
t=-n
t =-1
n
where gt = growth rate in year t
Geometric Average =
Earnings0
Earnings-n
(1/ n)
-1 where Earnings-n = earnings in n years ago
The two estimates can be very different, especially for firms with volatile earnings. The
geometric average is a much more accurate measure of true growth in past earnings,
especially when year-to-year growth has been erratic.
In fact, the point about arithmetic and geometric growth rates also applies to
revenues, though the difference between the two growth rates tend to be smaller for
revenues than for earnings. For firms with volatile earnings and revenues, the caveats
about using arithmetic growth carry even more weight.
Illustration 11.1: Differences between Arithmetic and Geometric Averages: Motorola
Table 11.1 reports the revenues, EBITDA, EBIT and net income for Motorola for
each year from 1994 to 1999. The arithmetic and geometric average growth rates in each
series are reported at the bottom of the table:
Table 11.1: Arithmetic and Geometric Average Growth Rates: Motorola
Year Revenues % Change EBITDA % Change EBIT % Change
Net
Income % Change
1994 $ 22,245 $ 4,151 $ 2,604 $ 1,560
1995 $ 27,037 21.54% $ 4,850 16.84% $ 2,931 12.56% $ 1,781 14.17%
1996 $ 27,973 3.46% $ 4,268 -12.00% $ 1,960 -33.13% $ 1,154 -35.20%
1997 $ 29,794 6.51% $ 4,276 0.19% $ 1,947 -0.66% $ 1,180 2.25%
1998 $ 29,398 -1.33% $ 3,019 -29.40% $ 822 -57.78% $ 212 -82.03%
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1999 $ 30,931 5.21% $ 5,398 78.80% $ 3,216 291.24% $ 817 285.38%
Arithmetic Average 7.08% 10.89% 42.45% 36.91%
Geometric Average 6.82% 5.39% 4.31% -12.13%
Standard deviation 8.61% 41.56% 141.78% 143.88%
Geometric Average = (Earnings1999/Earnings1994)1/5-1
The arithmetic average growth rate is higher than the geometric average growth rate
for all four items, but the difference is much larger with net income and operating income
(EBIT) than it is with revenues and EBITDA. This is because the net and operating
income are the most volatile of the numbers, with a standard deviation in year-to-year
changes of almost 140%. Looking at the net and operating income in 1994 and 1999, it is
also quite clear that the geometric averages are much better indicators of true growth.
Motorola’s operating income grew only marginally during the period and this is reflected
in its geometric average growth rate, which is 4.31%, but not in its arithmetic average
growth rate, which indicates much faster growth. Motorola’s net income dropped by
almost 50% during the period. This is reflected in its negative geometric average growth
rate but its arithmetic average growth rate is 36.91%.
Linear and Log-linear Regression Models
The arithmetic mean weights percentage changes in earnings in each period equally
and ignores compounding effects in earnings. The geometric mean considers compounding
but focuses on the first and the last earnings observations in the series – it ignores the
information in the intermediate observations and any trend in growth rates that may have
developed over the period. These problems are at least partially overcome by using OLS1
regressions of earnings per share (EPS) against time. The linear version of this model is:
EPSt = a + b t
where,
EPSt = Earnings per share in period t
t = Time period t
1 An ordinary least squares (OLS) regression estimates regression coefficients by
minimizing the squared
differences of predicted from actual values.
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The slope coefficient b on the time variable is a measure of earnings change per time
period. The problem, however, with the linear model is that it specifies growth in terms
of dollar EPS and is not appropriate for projecting future growth, given compounding.
The log-linear version of this model converts the coefficient into a percentage
change.
ln(EPSt) = a + b t
where,
ln(EPSt ) = Natural logarithm of Earnings per share in period t
t = Time period t
The coefficient b on the time variable becomes a measure of the percentage change in
earnings per unit time.
Illustration 11.2: Linear and Log-linear models of growth: General Electric.
The earnings per share from 1991 to 2000 is provided for GE in Table 11.2 with
the percentage changes and the natural logs of the earnings per shares computed each year.
Table 11.2: Earnings Per Share: General Electric
Year
Calendar
Year EPS
% Change in
EPS ln(EPS)
1 1991 0.42 -0.8675
2 1992 0.41 -2.38% -0.8916
3 1993 0.4 -2.44% -0.9163
4 1994 0.58 45.00% -0.5447
5 1995 0.65 12.07% -0.4308
6 1996 0.72 10.77% -0.3285
7 1997 0.82 13.89% -0.1985
8 1998 0.93 13.41% -0.0726
9 1999 1.07 15.05% 0.0677
10 2000 1.27 18.69% 0.2390
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There are a number of ways in which we can estimate the growth rate in earnings per
share at GE between 1991 and 2000. One is to compute the arithmetic and geometric
averages.
Arithmetic average growth rate in earnings per share = 13.79%
Geometric average growth rate in earnings per share = (1.27/0.42)1/9-1 = 13.08%
The second is to run a linear regression of earnings per share against a time variable
(where
the earliest year is given a value of 1, the next year a value of 2 and so on):
Linear Regression : EPS = 0.2033 + 0.0952 t R2 = 94.5%
(4.03) (11.70)
This regression would indicate that the earnings per share increased 9.52 cents a year from
1991 to 2000. We can convert it into a percentage growth in earnings per share by
dividing this change by the average earnings per share over the period.
Growth rate in earnings per share =
13.10%
0.727
0.0952
Average EPS
Coefficient on linear regression = =
Finally, you can regress ln(EPS) against the time variable.
Log-linear Regression: ln (EPS) = -1.1288 + 0.1335 t R2=96.3%
(19.53) (14.34)
The coefficient on the time variable here can be viewed as a measure of compounded
percent growth in earnings per share; GE’s earnings per share grew at 13.35% a year
based upon this regression.
